■^pfvi. 1 KA.GM'.niant IKOl y ^ ^NEW AMERICAN PRACTICAL NAVIGATOR: EPITOME OF NAVIGATION; CONTAINING ALL THE TABLES NECESSARY TO BE USED WITH THE NAUTICAL ALMANAC IN DETERMINING THE LATITUDE, AND THE LONGITUDE BY LUNAR OBSERVATIONS, AND KEEPING A COMPLETE RECKONING AT SEA; ILLUSTRATED BY PROPER RULES AND EXAMPLES: THE WHOLE EXEMPLIFIED IN A JOURNAL, KEPT FROM BOSTON TO MADEIRA, IN WHICH ALL THE RULES QF NAVIGATION ARE INTRODUCED i ALSO, THE DEMONSTRATION OF THE USUAL RULES OF TRIGONOMETRY ; TROBLiftlS IB MENSURATION, SURVEYING, AND GAUGING , DICTIONARY OF SEA TERMS ; AND THE MANNER OF PERFORMING THE MOST USEFUL EVOLUTIONS AT SEA : WITH AN APPENDIX, CONTAINING METHODS OF CALCULATING ECLIPSES OF THE SUN AND MOON, AND OCCULTATIONS OF THE FIXED STARS ; RULES FOR FINDING THE LONGITUDE OF A PLACE BY OBSERVATIONS OF ECLIPSES, OCCULTATIONS, AND TRANSITS OF THE MOON'S LIMB OVER THE MERIDIAN ; ALSO A NEW METHOD FOR FINDING THE LATITUDE BY TWO ALTITUDES. BY NATHANIEL BOWDITCH, LL. D. Fellow of the Royal Societies of London, Edinburgh, and Dublin; of the Astronomical Society in London: "j (At American Philosophical Society, h eld at Philadelphia ; of the American Academy of Arts and Sciences; of the Connecticut Academy of Arts and Sciences ; of the Literary and Philosophical Society of JWm York ; Corresponding Member of the Royal Societies of Berlin, Palermo, &-c., — and, since his decease, continued by his son, J. INGERSOLL BOWDITCH. THIRTIETH NEW STEREOTYPE EDITION. NEW-YORK: PUBLISHED BY E. & G. W. BLUNT, PROPRIETORS, No. 179 WATER-STREET, CORNER OF BITRLING SLIP. STEREOTYPED AT THE BOSTON TYPE AND STEREOTYPE FOUNDRY. 1861. X . «NEW AMERICAN PRACTICAI. NAVIGATOR: BEING AN EPITOME OF NAVIGATION; CONTAINING ALL THE TABLES NECESSARY TO BE USED WITH THE NAUTICAL ALMANAC IN DETERMINING THE LATITUDE, AND THE LONGITUDE BY LUNAR OBSERVATIONS, AND KEEPING A COMPLETE RECKONING AT SEA; ILLUSTRATED BY PROPER RULES AND EXAMPLES: THE WHOLE EXEMPLIFIED IN A JOURNAL, KEPT FROM BOSTON TO MADEIRA, IN WHICH ALL THE RULES OF NAVIGATION ARE INTRODUCED i ALSO, THE DEMONSTRATION OF THE USUAL RULES OF TRIGONOMETRT ; FROBLiMS IS MENSURATION, SURVEYING, AND GAUGING , DICTIONARY OF SEA TERMS *. AND THE MANNER OF PERFORMING THE MOST USEFUL EVOLUTIONS AT SEA : WITH AN APPENDIX, CONTAINING METHODS OF CALCULATING ECLIPSES OF THE SUN AND MOON, AND OCCULTATIONS OF THE FIXED stars; RULES FOR FINDING THE LONGITUDE OF A PLACE BY OBSERVATIONS OF ECLIPSES, OCCULTATIONS, AND TRANSITS OF THE MOON'S LIMB OVER THE MERIDIAN; ALSO A NEW METHOD FOR FINDING THE LATITUDE BY TWO ALTITUDES. BY NATHANIEL BOWDITCH, LL. D. Fellow »f the Royal Societies of London, Edinburgh, and Dublin; of the Astronomical Society in London; of the American Philosophical Society, held at Philadelphia ; of the Jimerican Academy of Arts and Sciences; of the Connecticut Academy of Arts and Sciences; of the Literary and Philosophical Society of J^Tew York; Corresponding Member of the Royal Societies of Berlin, Palermo, &-c., — and, since his decease, continued by his son, J. INGERSOLL BOWDITCH. THIRTIETH NEW STEREOTYPE EDITION. NEW-YORK: PUBLISHED BY E. & G. W. BLUNT, PROPRIETORS, No. 170 WATER-STREET, CORNER OF BURLING SLIP. STEREOTYPED AT THE BOSTON TYPE AND STEREOTYPE FOUNDRY. 1861. NOTICE TO THE 30th EDITION. Some corre'ctrbn's in tlie Latitude and Longitude of points on the coast of Cuba iia^^e Ijee.n made. ' TLe Pole 'Star 'table, o: page 206, has been altered to correspond nearly to the year 1860. Table LV. has been corrected from the " Tide Tables for the English and Irish Ports for 1860." From Mr. Portales I have received valuable aid. The article on " Tides," on pages 120, &c., was prepared by Dr. Bache. Captain Josiah Snow, of the ship Asterion, informs me, that the doubtful shoal in Macassar Straits, laid down in 4° 50' S. and 116° 50' E., is about three miles broad in the shoalest part, and bears N. W. by W. from the North Seras seven miles ; in some places there is not over six feet of water. By good observations he places it in 116° 58' E. On Sahul Bank, off Timor, in lat. 10° 50' S. and 127° 40' E., he passed a shoal or reef a quarter of a mile long East and West ; some of the rocks were even with the water's edge. Captain Snow, in the lat. 10° 54^ S. and 127° 05' E., passed a line of breakers two miles long, running East and West, and from appearances thought there must be many shoal spots in the ' neighborhood. He thinks nearly all the islands East of Mindoro are laid down erroneous, and should not be depended upon — especially Semerara, the largest, which should be 12 miles further north. From Mr. Daniel P. Upton, I learn that, in 1859, the Dutch bark "Hoop von Capello" struck on a sharp coral rock, about 20 feet long, in the Straits of Augier, having 16 feet of water on it. The compass bearings are — The Cap N. W. by W. 4° W. Point Tanjong Lemuing S. -j W. 4 Point Lio-ht S. W. The " Caimsmore" Rock is a small, dangerous, and very abrupt rock, 30 or 40 feet in diameter, on which the ship Caimsmore was totally lost 26th June, 1858. Lat. 30° 42' 10" N. Long. 122° 34' 40" E. The geographical position of some of the Shoals, Rocks, &c., in the " Coral Sea," have been corrected from the Surveys of 1859, and new positions added on p. 451. I860. J. INGERSOLL BOWDITOH. Entered according to Act of Congress, la the year of our Lord ISSY, by E. & G. W. Blunt, in the Clerk's Office of the District Court for the Southern District of K"ew York. For new Nautical Publications, &c., of E. &. G. ^Y. Blunt, see adv'ertibiment at the end. Printed by Joseph Russell, 79 John bx. IN MEMORrAM ■\ ^lf^^~iyti^7rp^. ^^r^^J^^^A. ir 1^. '^^u^/yAyy^^/^z/.'^f^ . ) //iyyy^>trAy^'^^4'y^^'y^/^^ — - Z'^// ,y/f-j/-^,'/y'j ^^'-^^ y^(^/y^'^^r^ /yf ^^ /^ iS^ '\ i y y y ^ y -^ /- y — r y ^<^^ z^^?^/ lyy.yf, m^^J^//Z^r/7^y^,,,.^^^ .^,, / Jr '/^^^7 , /^i M^Vy^/Z^y^/^y^^: ^ <'^^^'^^ y^U/ //-/ /7ji.^ /^ Ar/ cf'/i 7^ /tl /-// i/rM^ m>j/A^^ ^ / / ^ ".^^^^ ^/^?//t^^J_ ^/V.-'yTyQ/^y/l/Wify/j,,. ^iV >j 'J7C ^ ^mrJA^^j,^ ^^''^^'^y ^^'/^y^y/^7, A^6. ^c f/. tPage 1. CORRECTIONS AND ADDITIONS TO GEOGRAPHICAL POSITIONS IN TABLE LIV. OF BOWDITCH S NAVIGATOR, EDITION OF 1851. Kindly furnished by Dr. Bache, Superintendent of the U. S. Coast Survey, by authority of the Treasury Department, J^AME OF PLACES. Cape Elizabeth West Light Cape Elizabeth East Light , Wood Island Light Agamenticus Hill Plum Island East Light Plum Island West Light Beverley Spire Ipswich East Light Ipswich West Light Squara Light Straits mouth Island Light Thatcher's Island South Light Thatcher's Island North Light Ten pound Island Light Eastern Point Light Eaker's Island Light Salem, Tall Spire Marblehead, Black Top Church Nahant Hotel BOSTON, State-House Cambridge Observatory Dome Bunker Hill Monument Scituate Light Boston Light Long Island Light Plymouth Light Race Point Light Cape Cod Light Long Point Light WeUfleet Light Billingsgate Point Light Nausett Centre Light.. Nausett South Liglit Chatham South Light ilouomoy Light , New Bedford Light Cape Pogue Light Great Point Light Brant Point Beacon Saukaty Head Light Nantucket Harbor Light Nantucket Old South Shoal .... Nantucket Old South Shoal .... Davis' New South Shoal Fishing Rip, 5^ fath Barnstable Light Point Gammon Light Edgartown Light 43 33.8 43 33.9 43 27.4 43 i3.4 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 48.4 48.4 43.0 4i.i 4i.i 39.7 39.7 38.2 38.3 36.1 34.8 32.2 3l .2 3o.4 25. I 21 .5 22.9 22 .6 12.3 19.6 19.8 00.2 o3.7 02.4 02 .0 55.8 5i.6 5i.6 5i.i 4o.2 33.5 35.5 25.2 23.4 17.4 17.0 16.4 o5.5 o4.2 58. o o3 07.5 43.3 36.5 23.4 Longitude, D. M. 70 1 1 . 8 70 II .7 70 19.4 70 41.2 70 48.7 / 70 48.8 ] 70 52.4 70 45.6 70 45.8 70 40.6 70 35.0 70 34.2 ) 70 34.2 J 70 39.6 70 39.5 70 46.8 70 53.6 70 5o.5 70 54 -O 71 o3.5 71 07-4 71 o3.3 70 43.6 70 53. I 70 57.1. 70 35.7 70 i4.3 70 o3.3 70 09.8 70 01 .7 70 03.9 69 56.7 69 56.6 69 56.6 69 59.3 70 53.7 70 26.7 70 02.4 70 o5.2 i, 69 57.6 70 o4.4 69 5o.o 69 5i .4 69 5 1 . 5 69 26 69 29 70 16.5 70 1 5. 6 70 29.8 REMARKS. Trig. Point of C. S. Newbury port Lights. Spire with Tm-rets. Cape Ann, Boston Bay. Gurnet South Light. Highland Light. Clark's Point Light. Nantucket. Eastern Spot. Western Spot. Nantucket. Q95^RQi Pa«e 2.] TABLE LIY. Corrections and Additions. ^r^ME OF PLACES. West Chop Light Nobska Light Bird Island Light Tarpauhn Cove Light Ned's Point Light Pahner's Ishmd Light New Bedford, Baptist Spire Round Hill Light Cuttyhunk Light Gay Head Light No Man's Land Newport Spire Goat Island Light Nayat Light Warwick Light Wickford Light Providence, Baptist Church Dutch Island Light Beavertail Light Point Judith Light Watchill Light Block Island Lie;ht Stonington Light Mystic Light Say brook Light Little Gull Island Light. New London Light Falkner's Island Liglit . . New Haven Light Stratford Point Light.... Black Rock Light , Sheffield Island Light .., Captain's Island Light.. Latitude. D. 4i 4i 4i 4i 4i 4i 4i 4i 4i 4i 4i M. 28.9 30.9 4o.i 28.1 39.0 37.6 38.2 32.3 24.8 20.9 l5.2 Plum Island Light Montauk Light Cedar Island Light , Oldfield Point Light Eaton's Point Light Sands Point Light NEW YORK, City Hall.. Robin's Reef Light Navy Yard, Flagstaff Castle Garden, Flagstaff Fire Island Light Prince's Bay Light 4i 4i 4i 4o 4o 4o 4o 4o 40 40 4o 40 19.6 19.0 16.2 12.3 19.0 12.7 14.9 09.1 08.5 02.9 58.9 10.4 04.2 02.4 58.6 57.2 5i .9 42.7 39.4 42.0 42.0 37.9 3o.4 Longitude. M. 35.8 39.0 42.7 45.1 47-4 54.2 55.3 55.0 56.7 49-8 48.5 4i 29.2 71 18.5 4i 29.6 71 19.3 4i 43.5 71 20.0 4i 4o.o 71 22.4 4i 34.2 71 26.0 4i 49.6 71 24.2 4i 29.8 71 23.9 4i 26.9 71 23.6 4i 21.6 71 28.6 4i 18.2 71 5i .2 4i i3.4 71 34.2 71 54.0 71 59.0 72 20.3 72 06.1 72 o5.i 72 38.9 72 53.9 73 05.9 73 12.7 73 24.8 73 37.1 72 12.4 71 5i.i 72 i5.3 73 06.8 73 23.4 73 43.5 74 00. 1 74 o3.6 73 58.5 74 00.5 73 12.8 74 12.5 REMARKS. Trig. Point of C. S. Rhode Island Light Brooklyn. New York. TABLE LIV. Corrections and Additions. [Page 3. J^^ME OF PLACES. Longitude, REMARKS. Sandy Hook Light Navesink Light Ocean House Flagstaff Barnegat Light Tucker's Island Light.. Cohansey Light Egg Island Light CajDe May Light PHILADELPHIA, State-House "Wilmington Light Bombay Hook Light . Mispilion Light Breakwater Light Cape Henlopen Light Susquehanna Light Turkey Point Light Baltimore, Washington Monument Poole's Island Light North Point Lower Light North Point Upper Light Bodkin Light Annapolis, State-House Sharp's Island Light Clay Island Light Point Lookout Light National Observatory WASHINGTON City, Dome of Cap. Fog Point Light Assateague Light Smith's Point Light I Watts' Island Light ! New Point Comfort Light, Old Point Comfort Light . . Smith's Island Light ,. Cape Charles 1 Cape Henry Light D. M. 4o 27.7 4o 23.7 4o 22.8 39 46.0 39 3o.3 39 20.3 39 10.5 38 55.8 39 56.9 38 53.6 38 53.3 D. M. 73 59.8 73 58.8 73 58.2 74 06.0 74 16.8 75 21 .3 75 08.0 74 57.3 75 08.7 39 43.3 75 30.9 39 21.8 75 3o.3 38 56.6 75 18.5 38 47-9 75 06.1 38 46.6 75 04.7 39 32.4 76 04.8 39 26,9 76 00.2 39 17.8 76 36.6 39 17.4 76 i5.7 39 II .6 76 26.2 39 1 1. 8 76 27.3 39 08.0 76 25.1 38 58.7 76 29.1 38 37.7 76 22.6 38 i3.9 75 58.1 38 02.3 76 19.0 77 02.8 77 00.2 38 02.1 76 02.2 37 54.6 75 21 . I 37 53.2 76 i4.o 37 46.9 75 53.3 37 18.0 76 16.4 37 00.0 76 18. I 37 07.8 75 52.2 37 07.3 75 57.9 36 55.5 76 00.2 Southern Light. Little Egg Harbor Light. Washingtoa. Mouth of Potomaa Trig. Point of C. S. *■)*. Page 4.] TABLE LIV. Corrections and Additions, J^AME OF PLACES. KOI Devil Hill Bodies' Island Light Ne-R' Inlet, South Point... Cape Hatteras Light Ocracoke Light Fort Pinkney Charleston Light SAVAKNAH, Exchange. Cape Florida Light Key West Light Sand Key Light , MobUe, Barton's Academy Choctaw Point Light Grant's Light , Mobile Point Light , Sand Island Light Biloxi Light Pass Christian Light , Round Island Light , Cat Island Light Ship Island West , Chandeleur Light Galveston, Entrance Galveston, Cathedral Point Lobos South Farallon Point Pinos Point Conception Point Loma Latitude. D. M. 36 01 . 1 35 47-3 35 4i.i 35 i5.2 35 o6.5 32 46.4 32 4i -9 32 04.9 3o 23.8 3o 18.9 3o 17.5 3o 13.9 3o 12.9 3o o3.4 29 20.5 29 18.3 3? 47.0 37 4i.6 36 38. o 34 26.9 32 4o.2 Longitude. 25 39.9 80 o5.o 24 33.0 81 47.3 24 27.2 81 51.9 D. M. 75 39.7 75 3i.6 75 28.5 75 30.9 75 58.9 79 ^^-^ 79 52.5 81 o5.2 3o 4i 4 88 01 9 do 4o 2 88 01 I 3o 17 6 88 07 5 3o i3 8 88 00 5 3o II 3 88 02 88 53.1 89 14.0 88 34.1 89 08.7 88 57.0 5i.8 94 45.0 94 47.0 122 32.0 122 59.2 * 120 25.7 REMARKS. Trig. Point of C. S. Trig. Point of C. S. Charleston. Trig. Point of C. S. San Francisco Bay. Monterey. San Diego Bay. [Page 5. Stations on tJie Pacific Coast ; determined astronomically hy the U. S. Coast Survey. STATIONS. Latitude. Longitude. San Diego San Nicolas San Catalina San Pedro Prisoner's Harbor Santa Barbara Point Conception San Luis Obispo San Simeon Point Pinos Santa Crnz Presidio Hill Piinta de los Reyes Bodega Bay Havens' Anchorage Mendocino City Shelter Cove Bucksport Trinidad Bay Crescent City Telegraph Hill Cuyler's Harbor San Clemcnte Ewing Harbor Uraquah River Cape Hancock Point Hudson False Dungeness Scarborough Harbor Lunimie Island Astor Point Heard's Islands, a new discovery, . -j 32 33 33 32 2i 34 M 35 35 36 36 37 3? 38 38 39 4o 4o 4i 4i 3? 4i Sy. l4 12- 26 34- 43 19. 01 lO^ 24 24' 26 56. 10 37. 38 24' 37 59. 57 26. 47 36. 59 34. 18 20. 47 57. 18 06. 01 i3. 46 37. o3 20. AA 44' 48 06. 96 N. 71 84 59 20 71 3o 48 43 86 93 i5 20 37 87 16 67 09 o4 10 43 44 21-73 4i 45-3i 16 34-85 07 03-02 07 52 -03 21 48-78 44 01-74 II 27-61 17 i3 19 25 18 28 18 16 19 40 19 4o 20 25 20 43 21 10 21 54 22 00 22 26 22 57 23 02 23 34 23 47 24 o3 24 10 24 08 24 II 22 23 20 20 18 34 24 28 24 09 24 02 22 44 23 27 24 37 22 4o 23 49 25-00 TV 00-00 45-00 o3-oo 00-00 18-00 39-00 3i -00 22-00 25-00 10-00 i5-oo 4o-io 28-80 00-70 25-65 02-85 43 -80 07.95 13-95 19.42 27-00 00 -DO 47 -40 57-00 00-81 33-00 21.00 12-00 37.35 31.65 53 o3 53 00 73 3o E. 72 3o Positions 0/ points in the North Pacific Ocean., prepared ly Lieut. Bent., dy direction of Commodore Perry., commander of the late expedition to Japan. NAMES OF PLACES. Latitude. Longitude. Formosa. — S. E. point Islands, 's true lat, — 960" X versed sine par. in lat. If we suppose the sum of these three parts to be S seconds, and the moon's horizontal semi-diameter to be D minutes. Part IV. corresponding to S and D, will be S X ^ *= 256 Table XLV. The arguments at the side being B and 12 — B hours, and the second difference at the top A, the correction of this table will be 288 Table XLVI. gives the variation of the altitude of any heavenly body, arising from a change of 100 seconds in the declination. Table XLVII. contains the proportional logarithms as in Table XXII., increasing the argument at the bottom of the table by 5°, and inverting the order of the numbers. Table XL VIII. contains the third correction of a lunar obseivation in Lyons's improved method. These numbers may be easily computed from Shephard's tables, using the moon's parallax 57' 30", which is nearly its mean value. X PREFACE. Table XLIX. For computing the parallax in altitude of a planet, supposing its horizontal parallax to be 35". Table L. Proportional parts, to reduce the numbers of Table XLIX. to the values corresponding to the actual horizontal parallax of a planet. Table LI. To change mean solar time into sideral time. Table LII. To change sideral time into mean solar time. Table LIII. Variation of the compass in different parts of the world, deduced from Barlow's chart. Table LIV. contains the latitudes and longitudes of the most remarkable ports, harbors, &.c. in the world, from the latest and best authorities. Table LV. contains the times of high water on the full and change of the moon, with the vertical rise of the tide, at many ports, harbors, &c. in the world. This table, (like the preceding,) depending wholly on observations, is therefore liable to be erroneous, though great pains have been taken to make it as correct as possible, using for this purpose the observations collected by Dr. Whewell. Table LVI. Extracts from the Nautical Almanac for the year 1836, cor responding to the examples which are given in this work. The tables have all been newly cast from a clear and beautiful type, and above ninety pages have been added to the collection. Various improvements have been made in the body of the work, which is now for the first time completely stereotyped. Among the additions made to the work, may be mentioned the description of a portable transit instrument, with its uses in regulating a chronometer, and in finding the longitude by observations of the moon's transits over the meridian of the place of observation ; methodi for making allowance for any observed change in the rate of a chronometer ; new methods and improvements in the computation of lunar observations, &c. In preparing this edition, I have been very much assisted by my son, J. Ingersoll Bowditch, who computed most of the new tables, and care- fully examined those which were taken from other works. By associating him with me, many improvements have been made which otherwise would not have been introduced. N. BOWDITCEI Boston, October 1, 1837. CONTENTS. Page Signs and abbreviations used in this work xvi Decimal arithmetic 1 Geometry 4 Demonstration of the most useful propositions of geometry 7 Demonstration of tlieorems in plane trigonometry 13 Geometrical problems 15 Construction of the plane scale 18 Description of Gunter's scale 20 Description and use of the sliding rule 23 Description and use of the sector 25 To find the logarithm of any number, and the contrary 28 Multiplication by logarithms 30 Division by logarithms 31 Involution by logarithms 31 Evolution by logarithms 32 The rule of three by logarithms 32 To calculate compound interest by logarithms 32 To find the log. sine, tangent, »&c. corresponding to any number of degrees and minutes 33 To find the degrees, minutes, and seconds, corresponding to any log. sine, cosine, «Sbc. 35 To find the arithmetical complement of any logarithm 35 Plane trigonometry 36 Table of solutions of the various cases of trigonometry 37 Right-angled plane trigonometry , 38 Questions to exercise the learner in right-angled plane trigonometry 41 Oblique trigonometry 41 A short introduction to astronomy and geography 45 Explanations of the terms used in astronomy and geography 47 Examples in geography 51 Plane sailing 52 A table of the angles which every point of the compass makes with the meridian 53 A table of solutions of the several cases of plane sailing 53 Questions to exercise tlie learner in plane sailing 58 Traverse sailing - 59 Parallel sailing 63 Theorems for solving the several cases of parallel sailing 63 A table showing how many miles of meridian distance correspond to a degree of longi- tude at every degree of latitude 64 Questions to exercise the learner in parallel sailing 65 Middle latitude sailing 66 Theorems in middle latitude sailing 67 Table of solutions of the several cases of middle latitude sailing 68 Table to correct the middle latitude 76 Questions to exercise the learner in middle latitude sailing 77 Mercator's sailing 78 To find the meridional parts corresponding to any degree and minute 78 Xii CONTENTS. Page Table of solutions of the various cases of Mercator's sailing 79 '"■o work a compound course by middle latitude or Mercator's sailing 86 Construction and use of Mercator's chart 87 Problems useful in navigation and surveying 89 To find the difference between the true and apparent directions of the wind 97 To determine the height of a mountain by barometers 97 Mensuration 99 " Gauging ■ 103 Surveying 106 To find the contents of a field by the table of difference of latitude and departure 107—- To survey a coast in sailing along shore 109 To survey a harbor by observations on shore Ill Methods of surveying a small bank or shoal "where great accuracy is required 112 To reduce soundings, taken at any time of the tide, to low water 115-1- To reduce a drauglit to a smaller scale 115 Of winds 117 Directions for sailing from America to India 118 Tides 120 To find the time of high water by a Nautical Almanac 121 To find the time of high water by the tables C and D 122 Tables for calculating tlie time of high water 123 Currents 124 Gulf stream 124 Of the log-line and half-minute glass 126 Description and use of a quadrant of reflection 1 28 To adjust a quadrant 129 To take an altitude by a fore observation 130 To take the sun's altitude by a back observation 131 Advice to seamen in the choice of a quadrant 1 31 Description and use of a sextant of reflection 133 To adjust a sextant 134 To measure the angular distance of the sun from the moon 135 To measure the angular distance of the moon from a sVai , 136 Verification of the mirrors and colored glasses 136 Description and uses of the circle of reflection 137 Adjustments of a circle of reflection 133 To observe the meridian altitude of an object by a circle 140 To measure the angular distance of the sun from the moon by a circle 141 To measure the angular distance of the moon from a star by a circle . 142 Verification of the mirrors and colored glasses 143 Description and use of a portable transit instrument 145 Adjustments of a transit instrument 146 To observe the transit of any heavenly body over the meridian 150 Tables for correcting the adjustments of a transit instrument 151 On parallax, refraction, and dip of the horizon • 153 To find the distance of the land in order to calculate the dip 155 To find the sun's declination 156 Variation of the compass 158 To observe an amplitude or azimuth by a compass 158 To calculate the true amplitude 159 To calculate the true azimuth 160 Questions to exercise the learner in calculating an azimuth 160 Having tlie true and magnetic amplitude or azimuth, to find the variation 161 To calculate the variation by azimuths, observed at equal altitudes, before and after passing the meridian 161 Variation observed 163 On the dip of the magnetic needle 164 To find the latitude by a meridian altitude of the sun or a fixed star 166 To find the time of the moon's passing the meridian 170 To find the moon's declination 170 CONTENTS. xiii Page To find the latitude by the moon's meridian altitude 171 To find the latitude by the meridian altitude of a planet 174 ( of the sun 176 of a star 176 of a planet 177 To find the latitude by double altitudes \ of the moon 177 of two different objects, taken within a few minutes of each other, by one observer .... 178 ^ of two different objects, taken at different times 178 To estimate the effects of small errors in the observations 179 First method of calculating double altitudes 180 Second method 185 Third method 189 Questions to exercise the learner in working double altitudes 193 Fourth method, when the declinations are different 193 Fifth method, to fifld the latitude from altitudes and distances used in taking a lunar observation 197 To find the latitude by one altitude of the sun, having your watch previously regulated 200 To find the latitude by the mean of several altitudes of the sun, taken near noon by a sextant or circle 202 To find the latitude on shore by means of an artificial horizon 204 ' To find the latitude by the polar star 206 To find the time at sea, and regulate a watch 208 Examples to exercise the learner in finding the mean time 210 Second method of finding the mean time at sea 210 Third method of finding the mean time at sea 211 To find the time at sea by the moon's altitude 213 To find the time at sea by a planet's altitude 215 To find the apparent time by an altitude of a fixed star 217 To regulate a chronometer by equal altitudes of the sun 219 To regulate a chronometer by means of a transit instrument 221 To find the longitude at sea by lunar observations 225 Method of finding the stars used in lunar observations 226 General remarks on the taking of a lunar observation 228 To work a lunar observation 229 Examples of lunar observations 232 Second method of working a lunar observation 239 Third method of working a lunar observation 242 Fourth method, or Witchell's improved method of finding the true distance 243 Table of corrections for second differences 245 Method of taking a lunar observation when you have only one observer 246 To calculate the sun's altitude at any time 247 To calculate the moon's altitude 248 To calculate a planet's altitude 249 To calculate a star's altitude 250 Method of combining several lunar observations, and determining the error of the chro- nometer 251 To find the longitude by the eclipses of Jupiter's satellites 252 To find the longitude by an eclipse of the moon 253 To find the longitude by a time-keeper or chronometer 253 do do do do 289 To allow for the change of rate in a chronometer 257 Precautions in using a chronometer 259 On a variation chart 259 Method of keeping a reckoning at sea 260 To find the lee-way, and allow for it 261 To correct the dead reckoning 263 Rules for working a day's work 264 Examples for working a day's work .• 266 Journal from Boston to Madeira 270 ZIT CONTENTS. ARRANGEMENT OF THE TABLES. Table. Page I Difference of latitude and departure for points 1 II. Difference for degrees 17 III. Meridional parts 62 IV. Sun's declination 63 IV. A. Equation of time 68 V. For reducing the sun's declination 72 VI. Sun's right ascension 77 VI. A. Correction for the daily variation of the equation of time 77 VII. Amplitudes 78 VIII. Right ascensions and declinations of the fixed stars * 80 IX. Sun's rising and setting 84 X. For finding the distance of terrestrial objects at sea 8t X. A. Parallax in altitude of a planet 86 XI. Proportional parts , . . , . 87 XII. Refraction of the heavenly bodies 88 XIII. Dip of the horizon 88 XIV. Sun's parallax in altitude 88 XV. Augmentation of the moon's semi-diameter 88 XVI. Dip for different heights and distances 88 XVII. To find the correction and logarithm of a lunar observation when a star or either of the planets Venus, Mars, Jupiter or Saturn is observed 89 XVIII. To find the correction and logarithm of a lunar observation when the sun is used 97 XIX. To find the correction and logarithm of a lunar observation depending on the moon's altitude 98 XX. For finding the third correction of a lunar observation 130 — — • XXI. For turning degrees and minutes into time, and the contrary 131 XXII. Proportional logarithms 132 XXIII. For finding the latitude by two altitudes of the sun 148 XXIV. Natural sines and cosines 160 XXV."f Log. sines, tangents, &c. to points and quarter points 169 — f" . XXVI. Logarithms of numbers . /P.,.'. .•■. . .-. .' ^.^^4^ 169 flt^^JU XXVII. Logarithmic sines, tangents, and secants 185 ' XXVIII. To find the time of the moon's passing the meridian 230 XXIX Correction of the moon's altitude for parallax and refraction 230 XXX. To find the variation ot the moon's decimation, &c 231 XXXI. To find the sun's right ascension 23? XXXII. Variation of the sun's altitude in one minute from noon 23S XXXIII. To reduce the numbers of Table XXXII. to other given intervals from noon 243 XXXIV. Errors arising from a deviation of one minute in the parallelism of the surfaces of the central mirror 244 XXXV. Errors arising from a deviation of the telescope from a plane parallel to the plane of the instrument 2^14 XXXVI. Corref .ion of the mean refraction for various heights of the thermometer and barometer 244 XXXVII. Longitudes and latitudes of the fixed stars 245 XXXVIII. Reductions of latitude and horizontal parallax 246 XXXIX. Aberration of tlie planets in longitude 246 XL. Equation of the equinoxes in longitude 246 XLI. Aberration of the fixed stars in latitude and longitude 246 XLII. Aberration of the fixed stars in right ascension and declination 247 XLIII. Nutation in right ascension and declination 248 XLIV. Augmentation of the moon's semi-diameter, found by the nonagesimal . . . 249 XLV Equation of second differences » 250 -r CONTFaNTS. XV Table. Page. XLVI. Table, showing the variation of the altitude of an object arising from a change of 100 seconds in its declination 251 XLVIl. Logarithms in Lyons's improved method 253 XLVIII. Third correction in Lyons's improved method , 275 XLIX. Correction for a planet whose horizontal parallax is 35" 326 L. Reduction to any other parallax 328 LL To change solar time into sideral time 329 LIL To change sideral time into solar time 329 LIII. Variation of the compass, by Barlow 330 LI V. Latitudes and longitudes 332 LV. Tide table 379 LVI. Extracts from the Nautical Almanac 383 Catalogue of the Tables, with examples of the uses of those not explained in other parts of the work 385 APPENDIX. Addition and subtraction, using the signs as in algebra 395 Problem L To find the longitude, latitude, &c. of the moon 395 Problem 11. To find the horary motion of the moon 398 Problem IIL To find the ecliptic conjunction or opposition of the moon and sun, or a star 400 Problem IV. To find the altitude and longitude of the nonagesimal 402 Table to facilitate the calculation 403 Abridged rule for calculating the altitude and longitude of the nonagesimal 403 Problem V. To calculate the moon's parallax in latitude and longitude 404 Problem VI. To calculate the longitude of a place from the observed beginning and end of a solar eclipse 407 Problem VII. To calculate the longitude of a place from the observed beginning and end of an occultation 410 Problem VIII. To find the longitude of a place from the beginning or end of a solar eclipse . . . . , 413 Problem IX. To find the longitude of a place from the beginning or end of an occultation 414 Problem X. To project an eclipse of the moon 415 Problem XI. To calculate an eclipse of the sun 417 Problem XII. To project an occultation of a fixed star 421 Problem XIII. To calculate the beginning or end of an eclipse or occultation 425"' Problem XIV. To find the apparent time at Greenwich from the moon's longi- tude 426 Problem XV. To find the longitude of a place by measuring the distance of the moon from a fixed star not marked in the Nautical Almanac 427 Problem XVI. To find the longitude of a place by the moon's passage over the meridian 429 Pnoblem XVII. Given the latitude of the moon, and longitude of the moon and sun, to find their angular distance 433 Problem XVIII. Given the longitudes and latitudes of the moon and a star, to find their angular distance 434 Problem XIX. Given the right ascension and declination, to find the longitude and latitude 435 Problem XX. Given the longitude and latitude, to find the right ascension and declination 436 Spheric trigonometry 436 Improvement of Napier's rules for the circular parts 436 Theorems in spherics 439 Redfield's theory of storms, &c 440 Problem XXI. To find the longitude of a place from the beginning or end of a solar eclipse 443 Problem XXII. To find the longitude of a place from the beginning or end of an occul- tation 446 INDEX TO TABLE LIV. LATITUDES AND LONGITUDES. PAGE ADRIATIC, coast of 350 AFRICA, North coast 351 West coast 354 South coast 856 East coast 356 Red Sea 356 ALBANIA, coast of 350 ALGIERS, coast of 352 AMERICA — Eastern coast. Greenland 343 • Hudson's and Davis's Bay and Straits 343 Labrador 342 • Newfoundland 342 Gulf of St. Lawrence 341 Canada 342 Nova Scotia 841 ■ United States 332 Mexico 836 Honduras 337 Mosquitoes 83*7 Panama 83'7 Darien 337 Cartagena 337 . Maracaybo 337 — Caracas 337 Cumana 337 Surinam 838 . Maranham 838 Brazil 338 River Plata 339 . Patagonia 839 Terra del Fuego 839 AMERICA— Western coast. • Patagonia 340 Chili 340 Peru 840 Quito 840 ■ Panama 840 Mexico 841 California 341 Oregon 341 British and Russian possessions.... 341 ANAMBAS ISLANDS 365 ANDAMAN « 362 APO BANK 369 ARABIA, coast of 357 ASCENSION— Island 355 PA. 01 ASIA: Red Sea 356 Arabian coast 357 Gulf of Persia..: 857 Malabar 357 Ceylon 358 Coromandel 358 Bengal 858 Pegu 359 Malay 359 Siam 359 Cochin China 359 Hainan — Island 860 Cliina, coast of, to Canton 860 " from Canton to Kamtskatka, with adjacent islands 370 AZOF, sea of 851 BANCA— Island 364 BANDA SEA 370 BASHEE ISLANDS 369 BENGAL, coast of 358 BENIN, coast of 355 BIAFRA 855 BLACK SEA 351 BORNEO— Island 368 BOTHNIA 348 BRAZIL, coast of 338 CALIFORNIA, coast of 341,451 CANARY ISLANDS 354 CANDIA 353 CAPE BRETON— Island 841 CAPE VERDE ISLANDS 354 CARACAS, coast of 337 CARIBBEAN SEA 333 CAROLINE ISLANDS 374 CARTAGENA, coast of '. 837 CELEBES— Island 368 CERAM— Island 870 CEYLON— Island 853 CHAGOS ARCHIPELAGO 361 CHILI, coast of 340 CHINA, Southern coast 360 Eastern coast 370 CHINA SEA, islands in 365 Southeast part 366 COCHIN CHINA, coast of 359 INDEX TO TABLE LIV. xvu COMORO ISLAJSnOS 361 CONGO, coast of 355 COROMANDEL COAST 358 CORSICA— Island 352 CRIMEA, coast of 351 CUMANA, coast of 337 CYPRUS— Island 353 DARIEN, coast of 337 DENMARK, coast of 347 Islands 348 EGYPT, Mediterranean 351 Red Sea 356 ENGLAND, South coast 343 East coast 344 West coast 346 EOLIAN ISLANDS 352 FALKLAND ISLANDS 355 FEEJEE GROUP 451 FERRO ISLANDS 345 FRANCE, coast of North coast .T 344,347 West coast 349 South coast 350 FRIENDLY ISLANDS 377 GALAPAGOS ISLANDS 375 GERMANY, Baltic 348 North Sea 347 GEORGIA— Island 355 GOOD HOPE, Cape of 355 GREECE, Western coast 350 Southern coast 351 Eastern coast 351 GRECIAN ARCHIPELAGO 353 GREENLAND, coast of 343 GUINEA, coast of 355 GULF OF VENICE, coast 350 • " Islands 353 GULF OF PERSIA 357 GULF OF BOTHNIA 348 Finland 348 Guinea 355 Mexico 336 Persia 357 Slam 359 St. Lawrence 841 Tartaiy 371 Tonquin 860 Venice 350 HAINAN— Island 360 HOLLAND, coast 347 HONDURAS, coa^t 337 ICELAND 343 lONLAN ISLANDS 353 IRELAND, East coast 346 IRELAND, North coast 346 South coast 347 West coast 345 ISLANDS, Anambas 355 Andaman 352 Apo Bank ; 359 Ascension 355 Asia, eastern coast from Canton ... 370 Baltic 348 Banca 364 Banda Sea 370 Bashee 369 between Batavia and New Guinea, south of the Celebes 367 between Cape Verde, Cape of Good Hope, and Cape Horn 355 Borneo, Celebes, Luconia, with those adjacent as far east as New Guinea 368 Borneo 368 Canary 354 Caudia 353 Cape Breton 341 Cape Verde 354 Caroline 874 Celebes 368 Ceram 370 Ceylon , 358 Chagos Arcliipelago 361 China, Eastern coast 370 China Seas 365 " and adjacent 862 " S. E. part 366 Comoro 361 Corsica 35?. Cyprus 853 Dangerous Archipelago 378 Denmark, coast of 348 Eolian 352 Falkland 355 Feejee 451 Ferro 345 Friendly S77 • Galapagos 375 Georgia 355 Grecian Archipelago 353 Gulf of St. Lawrence 341 Gulf of Venice 353 Hainan 360 Indian Ocean, between Caj)e of Good Hope and Sumatra, iLc... S60 Iceland 363 Isle of Wight 344 Isle of Man 347 Ivica 352 Ionian 353 Italian 352 Japan 871 Java 863 Java Sea 367 INDEX TO TABLE LIV. ISLAISTDS, Kirigsmill Group Laccadive Archipelago Luconia Madagascar Madeira Magdalen Mahe Bank Majorca Maldivia Archipelago Marquesas • Mediterranean , Mindanao Mindoro Minorca ly I'lecas ]Natnnas ■ Navigator New Caledonia Newfoundland New Hebrides . New Holland, adjacent to New Pliillipines New South Shetland New South Wales ' New Zealand Nicobar Orkney Pacific Ocean, north 313. " " south Panay Paracels Paternosters Paumotu Group Fratas Shoals ■ Prince Edward Radack PtaUck Sandwich Land Sandwich Islands 315 Sardinia Shetland — Sicily -= " Islands adjacent Society Solomons Sooloo Sea 369, St. Helena Spitzbcrgen Straits of Banca " BiUington " Gaspar Straits east of Java Straits of Malacca " Macassar • " Sincapore " Sunda " Torres Staten ... Sumatra west of.. PAGE 450 362 369 361 354 342 361 352 361 318 362 369 369 852 810 365 450 316 342 316 312 314 318 312 311 362 345 ,450 315 369 865 861 450 866 342 314 314 856 ,450 352 345 852 853 811 316 451 356 343 864 864 364 361 365 868 364 364 315 839 362 363 ISLANDS, Tambelan 365 Terra del Fuego 339 Timor 368 Van Dieman's Land 312 Wasliington 318 Western 354 West India 333 XuUa Besseys 369 Zealand 341 ISLE OF MAN 341 ITALIAN ISLANDS 352 ITALY 350 IVICA 352 JAPAN ISLANDS 311 JAVA 363 Sea, islands in 361 Straits east of Java 361 JUTLAND, coast of. 341 KINGSMILL GROUP 450 LABRADOR, coa# of. 342 LACCADIVE ARCHIPELAGO 362 LAPLAND, coast of 349 LOANGO, coast of 355 LUCONIA— Island 369 MACASSAR STRAITS 868 MADEIRA ISLANDS 854 MADAGASCAR—Island 361 MAGDALEN ISLANDS 342 MAHE BANK 361 MAJORCA— Island 352 MALABAR, coast of 851 MALAY, coast of 359 MALDIVIA ARCHIPELAGO 361 MARMORA, sea of 351 MARANHAM, coast of 388 MARACAYBO, coast of 331 MARQUESAS ISLANDS 318 MEDITERRANEAN SEA : East coast 351 South coast 351 North coast 849 Islands in 352 MEXICO, South and East coast 836 West coast 341 MINORCA— Island 852 MINDANO— Island 369 MINDORO— Island 369 MOLUCCAS— Islands 3l0 MOREA, coast of 851 MOROCCO, North coast 852 West coast 354 MOSQUITOES, coast of 831 NAPLES, coast of 350 NATUNAS ISLANDS 365 NAVIGATOR ISLANDS 460 INDEX TO TABLE LIV. NEW CALEDONIA— Island 376 NEW" HEBRIDES— Island 376 NEW HOLLAND 372 NEW PHILLIPINE ISLANDS 374 NEW SOUTH SHETLAND— Island 378 NEW SOUTH WALES, coast of. 372 NEW ZEALAND— Island 377 NICOBAR ISLANDS 362 NORTHWEST COAST OF AMERICA ... 341 NORWAY, Cattegat.... 347 West coast 349 North coast 349 NOVA SCOTIA, coast of 341 OREGON, coast of 341 ORKNEY ISLANDS 345 PACIFIC OCEAN, N., Islands in 373, 450 PACIFIC OCEAN, S., Islands in S75, 450 PANAY— Island 369 PANAMA, East coast 337 AYest coast 340 PARACELS ISLANDS 365 PAUMOTU GROUP 450 PATAGONIA, E. coast of 339 W. coastof 340 PATERNOSTERS— Islands 367 PEGU, coastof. 359 PERU, coastof 340 PICO ISLANDS 354 PORTUGAL, coastof 349 PRATAS SHOALS 366 PRINCE EDWARD'S ISLAND 342 QUITO, coastof 340 RADACK ISLANDS 374 RALICK ISLANDS 374 RED SEA, coastof 356 RIVER PLATA, coastof 339 RUSSIA, Black Sea 351 Baltic Sea and Islands 348 Gulfs of Finland and Bothnia 348 N. Eastern coast 371 Sea of Azof. 351 SACHALIN— Island 371 SANDWICH LAND., 356 SANDWICH ISLANDS 375, 450 SARDINIA— Island 352 SCOTLAND, East coast 345 N. and West coasts 345 SENEGAMBIA, coastof 354 SHETLAND ISLANDS 345 SIAM, coastof. 359 PAOB SICILY— Island 352 SICILIAN ISLANDS 353 SOCIETY ISLANDS 377 SOLOMONS ISLANDS 376 SOOLOO SEA 369, 451 SPAIN, North coast 349 South coast 349 SPITZBERGEN— Island 343 ST. ANTHONY— Island 354 ST. HELENA— Island 355 STATEN ISLAND 339 SUMATRA ^62 Islands west of 363 SURINAM, coastof 338 SWEDEN, Cattegat and Sound 347 Baltic 348 TAMBELAN ISLANDS 365 TARTARY, coastof 371 TENERIFFE— Island 354 TERRA DEL FUEGO, coastof 339 TIMOR— Island 368 TORRES STRAITS, Islands in 375 TRIPOLI, coastof 351 TUNIS, coastof 351 TURKEY 351 Adriatic or West coast 350 UNITED STATES OF AMERICA : Eastern coast 332 Western coast 341 VAN DIEMAN'S LAND 372 WASHINGTON ISLANDS .-... 378 WEST INDIES : Bahama Bank, Great 335 " " Little 336 Bermuda 336 Caycos Islands 335 Cuba, North side 335 Cuba, South side 334 Jamaica 334 Passage Islands 335 Porto Rico 334 Salt Key 336 St. Domingo 334 Vu-gm Islands 334 Windward Islands 333 WESTERN ISLANDS 354 WHITE SEA 349 XULLA BESSEYS— Island 369 ZEALAND— Island , 341 SIGNS AND ABBREVIATIONS USED IN THIS WORK. |- IS the sign of addition, and denotes that whatever number or quantity follows the sigu, must be added to those that go before it; thus, 9-|-8 signifies that 8 is to be added to 9 ; or A + B implies that the quantities represented by A and B are to be added to- gether. The sign -j- is called the positive sign. - the sign of subtraction, and denotes that the number following it must be subtracted from those going before it; thus, 7 — 5 signifies that 5 must be subtracted from 7. The sign — is called the negative sign. X is the sign of multiplication, and shows that the numbers placed before and after it are to be multiplied together ; thus, 7x9 signifies 7 multiplied by 9, which makes 63; and 7x8x2 signifies the continued product of 7 by 8 and, by 2, which makes 112. Multiplication is also denoted by placing a point between the quantities to be multi- plied together ; thus, A . B signifies that A is to be multiplied by B. -f- is the sign of division, and signifies that the number that stands before it is to be divided by the number following it ; as, 72 -r- 12 shows that 72 is to be divided by 12 Division may also be denoted by placing two points between the numbers; thus, 72 72 : 12 represents 72 divided by 12 ; or by placing the numbers thus, — > which signifies 72 divided by 12. ( ) or -. Either of these marks is used for connecting numbers together ; thus, 3 -(- 4 X 6, or (3 -|- 4) X ^j signifies that the sum of 3 and 4 is to be multiplied by 6 t= is the sign of equality, and shows that the numbers or quantities placed before it are equal to those following it; thus, 8 X 12 = 96 ; or, 8 multiplied by 12 are equal to 96; and 7 4- 2 X 4 = 36. •: : are the signs of proportion, and are used thus ; 7 : 14 : : 10 : 20, that is, as 7 is to 14, so is 10 to 20 ; or, A : B : : C : D, that is, as A is to B, so is C to D. ** signifies degrees ; thus, 45° represents 45 degrees. signifies minutes ; thus, 24', or 24 minutes. " signifies seconds; thus, 44", or 44 seconds. '" signifies thirds, or sixtieth parts of seconds ; thus, 44'", or 44 thirds. In noting any time, d is the mark for days, h for hours, m for minutes, &c S. signifies sine. N. S. signifies natural sine. Sec. signifies secant. Tan. signifies tangent. Cosine, Cotangent, or Cosecant of an arc, signifies the sine, tangent, or secant of tlia complement of that arc respectively. <^ signifies angle. /\ signifies triangle. /\'s, triangles, □ signifies a square. or @, the sun. Q) or "^ , the moon. * a star. L. L. lower limb. U. L. upper limb. N. L. nearest limb. S. D. semi- diameter. P. L. proportional logarithm, N. A. Kautical Almanac. Z. D. zenith distance. D. R. dead reckoning. DIRECTIONS FOR THE BINDER. Plate I. to front the title-page. II. to front page 17. III. to front page 48. IV. to front page 60. V. to front page 52. VI. to front page 64. VII. to front page 112. Plate VIII. to front page 116. IX. to front page 136. X. to front page 144. XI. to front page 150. XII. to front page 156. XIII. to front page 426, Appendix DECIMAL ARITHMETIC. Mant persons wlio have ac(iuircJ coiisiderublp skill in common arithmetic, are unacquauited with the method of calcuhitiiiji by decimals, which is of great use in Navigation ; for which reason it was thought proper to prefix the foUowuig brief explanation. Fractions, or Vulgar Fractions, are expressions for any assignable part of a unit ; they are usually denoted by two numbers, placed the one above the other, with a line between them ; thus | denotes the fraction one fourth, or one part out of four of some whole quantity, considered as divisible into four equal ])arts. The lower number, 4, is called the denominator of the fraction, showing into how many parts the whole or integer is divided ; and the upper numbei', 1, is called the numerator, and shows how many of those equal parts are contained in the fraction. And it is evident that if the numerator and denominator bo varied in the same ratio, the value of the fraction will remain unaltered ; thus, if the numerator and denominator of the fraction, ^, be multiplied by 2, 3, or 4, &c., the fractions arising will be -|, -^^, 3^, &c., which are evidently equal to ^. A Decimal Fraction is a fraction whose denominator is always a unit with some number of ciphers annexed, and the numerator any number whatever ; as, ^2^, tB^ttj T^aTT) &c. And as the denominator of a decimal is always one of the numbers 10, 100, 1000, &c., the inconvenience of writing the denominator may be avoided, hf placing a point between the integral and the fractional part of the number ; thus, -fjy is \vi-itten .3 ; and xV\ is written .14 ; the viired number 3^^^^ consisting of a whole number and a fractional one, is written 3.14. In setting down a decimal fraction, the numerator must consist of as many places as tliere are ciphers in the denominator ; and if it lias not so many figures, the defect must be supplied by placing ciphers before it ; thus, ^'^^j =z .16, li^'iJ = '016, fTjxyiyu =^ .0016, &c. And as ciphers on the right hand side of integers increase their value in a tenfold proportion, as, 2, 20, 200, &c., so, when set on the left hand of decimal fractions, they decrease their value in a tenfold jiroportion, as, .2, .02, .002, &c.; but ciphers set on the right hand of these fractions make no alteration in their value, neither of increase or decrease ; thus, .2 is the same as .20 or .200. The common arithmetical operations are performed the same way in decimals as they are in integers ; regard being had only to the particular notation, to distinguish the uitegral from the fractional part of a sum. ADDITION OF DECIMALS. Addition of decimals is performed exactly like that of whole numbers, placing the numbers of the same denomination under each other, in which case the decimal separatmg points will range straight in one column. EXAMPLEfc Miles. Feet. Inches. 26.7 1.26 272.3267 32.15 2.31 .0134 143.206 1.785 2.1576 .003 2.0 31.4 Sura 202.059 7.355 305.8077 1 2 DECIMAL ARITHMETIC. SUBTRACTION OF DECIMALS. Subtraction of decimals is performed in the same manner as in whole numbere, by observing to set the figures of the same denomination and the separating points dii-ectly under each ct*ier. EXAMPLES. From 3L267 36.75 L254 1364.2 Take 2.63 .026 .316 25.163 Difftrence 28.637 36.724 .938 1339.037 MULTIPLICATION OF DECIMALS. Multiply the numbers together the same as if they were whole numbers, and ponit ofi'as many decimals from the right hand as tliere are decimals in both factors together; and when it happens that there are not so many figures in the product as there must be decimals, then prefix as many ciphers to the lefl; hand as will supply the defect EXAMPLE IV. Multiply .17 by .06. .17 EXAMPLE I. Multiply 3.25 by 4.5. 3.25 4.5 1.625 13.00 Answer 14.625 In one of the factors is one decimal, and In the other two ; their sum, 3, is the uuml)er of decimals of the product. EXAMPLE II. Multiply 0.5 by 0.7. 0.5 0.7 Answer 0.35 EXAMPLE III. Multiply 3.25 by .05. 3.25 ■05 Answer .1625 .06 Answer .0102 In each of the factors are two decimals the product ought therefore to contain 4 and, there being only three figures in the product, a cipher must be prefixed. EXAMPLE V. Multiply .18 by 24. .18 _24 72 36 Answer 4.32 EXAMPLE VI. Multiply 36.1 by 2.5. 36.1 2.5 18.05 72.2 Answer 90.25 DIVISION OF DECIMALS. Division of decimals is performed in the same manner as in whole numbei-s ; only observing that the number of decimals in the quotient must be equal to the excess of the number of decimals of the dividend above those of the divisor. When the divisor contains more decimals than the dividend, ciphers must be affixed to the right hand of the latter to make the number equal or exceed that of the divisor. EXAMPLE II. Divide 3.1 by .0062. Previous to the division, I affix a number of ciphers to the right hand of 3.1, which does not alter its value. .0062)3.100000(500.00 310 EXAMPLE I. Divide 14.625 by 3.25. 3.25 ) 14.625 { 4.5 1300 1625 1625 In this example, there are two decimals in the divisor, and three in the dividend ; hence there is one decimal in the quotient. 00000 Therefore the answer is 500 00 or 500. DECIMAL ARITHMETIC. EXAMPLE in. Divide 0.35 by 0.7. .7 ) .35 ( .5 .35 EXAMPLE IV. Divide 9.6 by .06. .06 ) 9.60 160 Answer. Here, by affixing a cipher to 9.6, it becomes 9.60, and has then two decimals in it, which is the same number as is in the divisor; therefore the quotient is an integral number. EXAMPLE V. Divide 17.256 by 1.16. L13) 17.25600 (14.875 116_ 565 464 1016 928 880 812 680 580 100 REDUCTION OF DECIMALS. If you wish to reduce a vulgar fraction to a decimal, you may add any number of ciphers to tlie numerator, and divide it by the denominator ; the quotient will be the decimal fraction ; the decimal point must be so placed that there may be as many figures to the right hand of it as you added ciphers to the numerator ; if tiiere ai"e not as many figures m the quotient, you must place ciphers to the left hand to make up the number EXAMPLE I. Reduce i^ to a decimal. 5 )1.0 .2 Answer. EXAMPLE II. Reduce f to a decimal. 8 ) 3.000 .375 Answer. EXAMPLE III. Reduce 3 inches to the decimal of a foot. Since 12 inches =z 1 foot, this fraction is ^% 12 ) 3.00 .25 Answer. EXAMPLE IV. Reduce 3^ mches to the decimal of a foot 3J- = J ; this divided by 12 is /j. 24 ) 7.000 ( .291 Answer, nearly. 40 24 16 EXAMPLE V. Reduce 1 foot and 6 inches to the decimal of a yard. Here 1 foot 6 inches ■=z 18 inches. And 1 yard rr: 36 inches; therefore this fraction is ^|. 36 ) 18.0 ( .5 Answer. 180 If you have any decimal fraction, it is easy to find its value in the lower denominations of the same quantity ; thus, if the fraction was the decimal of a yard, by multiplying it by 3 we have its value in feet and parts ; if we multiply this by 12, the product is its value in inches and pai-ts ; and in the same manner the valuea may be obtained in other cases. EXAMPLE VII. Answer, 3 yards, feet, 9 inches. EXAMPLE VI. Required the value of 7.231 days ired the value of 3.25 yards. 3.25 7.231 24 3 .75 924 4G2 12 5.544 60 32.640 60 38.400 Answer, 7 days, 5 hours, 32 minute% 38 seconds, and 4 tenths of a second. GEOMICTRY. Geometry is the science which treats of the description, properties, and relation3 of magnitudes in general, of which there are three kinds or species ; viz. a Ihie, which has only length without either breadtli or thickness ; a suptrficies, comprehended by length and breadth ; and a solid^ wliich has length, breadth, and thickness. I. A roiJv'T, considered mathematically, has no leng*Ji, breadth, or thickness. II. A STRAIGHT LINE, Or RIGHT LINE, is tlic shortcst distance between the two pointa wiiich limit its length, as AC. 4 c III. A PLANE SUPERFICIES is that ui which any two pomts being taken, the straight line between them lies wholly in that surface. IV. Parallel lines are such as are m the same plane, and which, A '■ — 3 extended infinitely, do never meet, as AB, DC. j) c V. A CIRCLE is a plane figure, bounded by a uniform cune line ; it is commonly described with a pair of compasses ; one point of which is fixed, whilst the other 13 turned round to the place where the motion first began ; the fixed point is called tlie CENTRE, and the line described by the other point is called the circumference. VI. The radius of a circle, or semi-diameter, is a right line drawn fi-om the centre to the circumference, as AC ; or it is that line which is taken between the points of the compasses to describe the cii cle. A diameter of a circle is a right line drawn through the centre, and terminated at both ends by the chcumference, as ACB ; and is the double of the radius, AC. A diameter divides the circle and its circumference into two equal parts. VII. An ARC of a circle is any part or portion of the circumference, as DFE. VIII. The CHORD of an arc is a straight line joining the ends of the arc; it divides the circle into two unequal parts, called segments, and is a chord to them both ; as DE is the chord of the arcs DFE and DGE. IX. A semicircle, or half circle, is a figure contained under a diameter and the arc terminated by that diameter, as AGB or AFB. Any iwirtof a circle contained between two radii and an arc, is called a sector. X. A QUADRANT is half a semichcle, or one fourth part of a whole circle, as the Dguve CAG. Note. All circles, whether gi-eat or small, are supposed to have their circumference divided into 3G0 equal parts, called degrees; and each degree into 60 equal parts, called minutes ; mid each minute into GO equal parts, called seconds ; and so on into tliirds, GEOMETRY. fouiths,* &c. ; and an arc is said to be of as many degrees as it contains parts of the 3G0, into Yvliich tlie circumference is divided. XI. An ANGLE is the inchuation of two luies which meet, but not in the same direction. An angle is usually expressed by the letter placed at the angular point, as the angle A. But when two or more angles are at the same point, it is tlien necessary to express each by three letters, and the letter at the angular point is placed between the otlier two. Thus the angle formed by the lines AB, AC, is called the angle BAG, or CAB ; and that formed by AB, AD, is called the angle BAD, or DAB. An angle is measured by the arc of a circle comprehended between the two le^s that form the angle ; the centre of the circle being the angular point, and the whole circumference considered as equal to 360°. Thus the angle A is measured by the arc BC described round the point A as a centre, and the angle is said to be of as many degrees as the arc is ; that is, if the arc BC is 30°, then the angle BAC is said to be an angle of 30 degi-ees. XIT. If a right line, AB, fall upon another, DC, so as to incline neither to the one side nor the other, but makes the angles ABC, ABD, equal to each other, then the line AB is said to be perpendicular to the line DC, and each of these angles is called a right angle, being each equal to a quadrant, or 90°; because the sum of the two angles, ABC, ABD, is measured by the semicircle DAC, described on the diameter DBC, and centre B. XIII. An ACUTE ANGLE is Icss than a right angle, as ABC. A B I> XIV. An OBTUSE ANGLE is greater than a right angle, as GEH. The least number of right lines that can include a space are three, which form a figure called a triangle, consisting of six parts, viz. three sides and three angles ; it is distinguished into three sorts, viz. a nght-angled triangle, an ohtuse-angled tnangle, and an acute-angled triangle. XV. A RIGHT-ANGLED TRIANGLE lias ouc of its anglcs right ; the side opposite the right angle is called the hypotenuse ; and the other two sides are called legs; that which stands upright is called the perpendicular, and the other the base; thus BC is the hypotenuse, AC the perpendicular, and AB the base ; the angles opposite the two lesrs are both acute. A H M A XVI. An ACUTE-ANGLED TRIANGLE Iias cacli of its angles acute, as DEG. XVII. An OBTUSE-ANGLED TRIANGLE has ouc of its auglcs obtusc, or greater than a right angle, as BAF ; the other two angles are acute. N'ote. All triangles that are not right-angled, whether they arc acute or obtuse, are in general called oblique-angled triangles, without any other distuiction. * A new division of the circumference of tiie circle has lately been adopted by several eminent French mathematicians, in which the quadrant is divided into 100°, each degree into 100', each minute into 100", &c., and tables of logarithms have been published conformable thereto. The general adoption of this division would tend greatly to facilitate most of the calculations of navigation and astronomy. 6 GEOMETRY. H XVIII. A QUADRILATERAL figure is one bounded by four sides, as ACDB. If the opposite sides are parallel, they ai-e called paral- lelograms. Thus, if AC be parallel to BD, and AB pai-allel to CD, the figure ACDB is a parallelogram. A parallelogram having all its sides equal, and its angles right, is called a square, as B. When the angles are right, and the opposite sides only equal, it is «alled a rectangle, as A XIX. The sine of an arc is a line drawn from one end of the arc pei-pendicular to a diameter drawn through the other end of the same arc ; thus RS is the sine of the arc AS, RS being a line drawn from one end, S, of that arc, pei-pendicular to DA, which is the diameter passing through the other end, A, of the arc. XX. The COSINE of an arc is the sine of the complement of that arc, or of what that arc wants of a quadrant ; thus, AH being a quadrant, the arc SH is the complement of the arc AS ; SZ is the sine of the arc SH, or the cosme of the arc AS. XXI. The VERSED SINE of an arc is that part of the diameter contained between the sine and the arc ; thus RA is the versed sine of the arc AS and DCR is the versed sine of tlie arc DHS. XXII. The tangent of an arc is a right line drawn perpendicular to the diameter, passing through one end of the arc, and terminated by a line di-awn from the centi'e through the other end of the arc ; thus AT is the tangent of the arc AS. XXIII. The cotangent of an arc is the tangent of the complement of that arc to a quadrant ; tlius HG is the tangent of the arc HS, or the cotangent of the ai'c AS. XXIV. The SECANT of an arc is a right line drawn from the centre through one end of the arc to meet tlie tangent drawn from the other end ; thus CT is the secant of the arc AS. XXV. The COSECANT of an arc is the secant of the complement of that arc to a quadrant ; thus CG is the secant of the arc SH, or cosecant of the arc AS. XXVI. What an ra-c wants of a semicircle is called the supplement of the arc ; thus the arc DHS is the supplement of the arc AS. Tlie sine, tangent, or secant of an arc, is the same as the sine, tangent, or secant of its supplement ; thus the sine of 80° =z sine of 100°, and the sine of 70° = sine of 110°, &c. XXVII. If one line, Alj,fall any way upon another, CD, the sum of the two angles, ABD, ABC, is always equal to two right angles. For, on the point B as a centre, describe the circular arc CAD, cutting the line CD in C and D ; then (by Art. G), this arc is equal to a seinichcle, but it is also equal to the sum of the arcs CA and AD, the measures of the two angles ABC, ABD ; therefore the sum of the two angles is equal to a semicircle, or two right angles. Hence it is evident that all tlie angles which can be made from a point in any line, towards on« Bide of the line, are equal to two right angles, and that all the angles which can b' made about a point, are equal U four right angles. J) GEOMETRY. 7 XXVIII. If a line, AC, cross another, BD, in the point E, the opposite angles will he equal, viz. BEA =r CED, and BEC = AED. Upon the point E as a centre, describe the circle ABCD ; then it is evident that ABC is a semicircle, as also BCD (by AH. G) ; therefore the arc ABC =: arc BCD ; taking from both the common arc BC, there remains arc AB = arc CD ; that is, the angle BEA is equal to the angle CED. After the same manner we may prove that the angle BEC is equal to the angle AED. XXIX. If a line, GH, cross two parallel lines, AB, CD, it makes the external opposite angles equal to each other ; viz. GEB = CFH, and AEG = HFD. For since AB and CD are parallel to each other, they may be considered as one hroad line, and GH crossing it ; then the vertical or opposite angles, GEB, CFH, are equal (by Art. 28), as also AEG = HFD. XXX. —j^-^ If a line, Gil, cross tivo parallel lines, AB, CD (see the figure), ^ -^ the alternate angles, A^F and EFD, or CFE and FEB, are equal. For GEB =: AEF [Art. 28), as also CFH =: EFD (by the same Art), but GEB = CFH by the last ; therefore AEF is equal to EFD; in the same way may we prove FEB=: CFE. XXXI. If a line, GH, cross two parallel lines, AB, CD (see the preceding figure), the external angle, GEB, is equal to the internal opposite one, EFD, or AEG equal to CFE. For the angle AEF is equal to the angle EFD by the last, and AEF = GEB (by Art. 28) ; therefore GEB = EFD ; in the same way we may prove AEG rr CFE. XXXII. If a line, GH, cross tivo parallel lines, AB, CD (see the preceduig figure), the sum of the two internal angles, BEF and DFE, or AEF and CFE, is equal to two right angles. For since the angle GEB is equal to the angle EFD (by Ad. 31), to both add the angle BEF, and we have GEB + BEF = BEF + EFD ; but GEB -\- BEF = two right angles [Art. 27). Hence, BEF -|- EFD =: two righl angles ; and in the same manner we may prove AEF -f- CFE = two right angles. XXXIII. In any triangle, ABC, one of its legs, as BC, being produced towards D, the externa, angle, ACD, is equal to the sum of the internal and opposite angles, ABC, BAC. To prove this, through C draw CE parallel to AB ; then, since CE is parallel to AB, and the lines AC, BD cross them, the angle ECD:r=ABC (by ^7-/. 31), and ACE = BAC (by ^r^ 30); adding these together we have ECD 4- ACE = ABC + BAC ; but B ECD + ACE z= ACD ; therefore ACD = ABC + BAC. XXXIV. Hence it may be proved that if any two lines, AB and CD, be crossed by a third line, EF, arid the alternate angles, AEF and EFD, be equal, the lines AB aiia CD will be parallel. For, if they are not parallel, they must meet each other on one side of the line EF (su])pose at G),and so form the triangle EGF, / one of whose sides, GE, being produced to A, the exterior angle, r/ n AEF, must (by the preceding article) be equal to the sum of the two angles EFG and EGF ; but by sui)position it is equal to the angle EFG alone; therefore the angle AEF must be equal to the sum of the two angles EFG and EGF, and at the same time equal to EFG alone, which is absurd; therefore tlie ImesAB,CD, camiot meet, and must be parallel. GEOMETRY. XXXV. In any right-lined triangle, ABC, the sum of the three angles is equal to ticu right angles. To prove this, you must produce BC (in the fig. ^4/-i. 83) towards D; then (by^rf.33), the external angle ACD = ABC -|- BAG ; to both add the angle ACB, and we have ACD -[- ACB = ABC + B AC + ACB ; but ACD -[- ACB = two right angles (by Jlrt. 27). Hence, ABC-f-BAC-f- ACB = two right angles; therefore the sum of the tlu'ce angles of any plaui triangle, ACB, is equal to two right angles. XXXVI. Hence in any plain triangle, if one of its angles he known, the sum of the other two will he also knoivn. For by the last article the sum of all three angles is equal to two right angles, or 180°; hence, by subtractmg the given angle from 180°, the remainder will be the sum of the other two. In any right-angled triangle, the two acute angles taken together are just equal to a right angle ; for, all three angles being equal to two right angles, and one angle bemg right by supposition, the sum of the other two must be equal to a right angle ; consequently, any one of the acute angles being given, the other one may be foimd by subtracting the given one from 90 degrees. XXXVII. If in any two tnangles, ABC, DEF, two legs of the one, AB, AC, he equal to two legs of the other, DE, DF, each to each respectively, that is, AB = DE, and AC z=z DF, and the angles BAC, EDF, included between the equal legs he equal ; then the remaining leg of the one will he equal to the remaining leg of the other, and the angles opposite to the equal legs will be equcd ; that is, BC rz: EF, ABC = DEF, and ACB = DFE. For if the triangle ABC be supposed to be lifted up and Eut upon the triangle DEF, with the point A on the point -A ^ ►, and the hue AB upon DE, it is plain, since AB=z:DE, that the pouit B will fall upon E ; and since the angles BAC, EDF are equal, the Ime AC will fall upon DF ; and these __ lines being of equal length, the pomt C will fall upon F ; consequently the line BC will fall exactly upon the line EF, and the triangle ABC will in all respects be exactly equal to the triangle DEF, and the angle ABC wiU be equal to the angle DEF, also the angle ACB will be equal to the angle DFE. XXXVIII. After the same manner it may be proved that if in any two triangles, ABC, DEF (see the preceding figure), tiuo angles, ABC and ACB, of the one he equal to two angles, DEF, DFE, of the other, and the included side, BC, be equal to EF, the remaining sides and included angles tvill also be equal to each other respectively; that is, ABnrDE, AC = DF, and the angle BAC = the angle EDF. For if the triangle ABC be supposed to be lifted up and laid upon the triangle DEF, the point B being upon the point E, and the line BC upon the line EF, then, since BCinEF, the point C will fall upon the point F ; and, as the angle ACB = the angle DFE, the line CA will fall upon the line FD ; by the same way of reasoning, the line BA will fall upon the line ED; therefore the pouU of intersection. A, of the two lines, BA, CA, will fall upon D, the point of "intersection of the lines ED, FD ; consequently AB = DE, AC = DF, and the angle BAC = the angle EDF. XXXIX. If two sides of n triangle are equal, the angles opposite these sides tvill also be equal; that is, if AB=z AC, the angles ABC, ACB, tvill also be eqtial. For, draw the line AD, bisecting the angle BAC, and meeting the line BC in D, dividing the triangle BAC into two triangles, ABD, ACD, in which the side AB = AC, the side AD is common to both triangles, and the angle BAD =: the angle DAC ; consequently (by Jlrt. 37), tlie angle ABD must be equal to tlie angle ACD. The converse of this proposition is also true; that is, if two angles of a triangle are equal, the opposite sides are also equal. This is demonstrated nearly in tlie same manner, by means of Art. 38. GEOMETRY. XL. Any angle at the circumference of a cirde is equal to half the angle at the centre, standing upon the same arc. Thus the angle CAD is half the angle BCD, standing upon the same arc, BD, of the cu'cle BEDA whose centi-e is C. To demoustfate this, draAV through A and the centre C, the right line ACE; then (by Art. 33) the angle CAD + angle CDAr= angle ECD ; but AC := CD (being two radii of the same circle) ; therefore (by ^/-^ 39), the angle CAD = tlie angle CD A, and the sum of these two angles is the double of either of them ; tliat is, CAD -\- CDA — tivice CAD ; therefore ECD = twice CAD ; in the same manner it maybe proved that BCE = twice BAC. and by adding these together, we have ECD 4- BCE =: twice CAD + twice BA C ;' that is, BCD = twice BAD, or Bx'VD equal to half of BCD. The demonstration is sbnilar Avhen B, D, fall on the same side of E. XLI. An angle at the circumference is measured by half the arc it subtends. For an angle at the centre, standing on the same arc, is measm-ed by the whole arc (by Art. 11); but since an angle at the centre is double that at the circumference [Art. 40), it is evident that an angle at the cuxumfei-ence must be measured by half the arc it stands upon. Hence all angles, ACB, ADB, AEB, &c., at the circumference of a circle standing on the same chord, AB, are equal to each other ; for they ai-e all measm-ed by the same arc, viz. half the arc AB. XLII. An angle in a segment greater than a semicircle is less than a light angle. Thus, if ABC be a segment gi-eater than a semicii-cle, the arc AC on which it stands must be less than a semicircle, and the half of it less than a quadrant or a right angle ; but the angle ABC in the segment is measured by the half of the ai*c AC ; therefore it is less tlian a right angle. An angle in a semicircle is a right angle. For since DEE is a semicircle, the arc DKF must also be a eemicircle ; but the angle DEF is measured by half the arc DKF, that is, by half a semicu'cle or by a quadi'ant ; thcrefoi'e the angle DEF is a right one. An angle in a segment less than a semicircle is greater than a rigid angle. Thus, if GHI be a segment less than a semi-circle, the arc GKI on which it stands must be gi-eater than a semicircle, and its half greater than a quadrant or right angle ; tlierefore the angle GHI, which is measured by half the arc GKI is gi-eater than a riglit angle. XLIII. If from the centre, C, of the circle ABE there be let fall the fcrpcndicidar CD on tht chord AB, it ivill bisect the chord in the point D. Draw the radii CA, CB; ihen (by Art. 39) the angle CBA = the angle CAB, and as the angles at D are right, the angle ACD must be equal to the angle BCD (by Art. 3G). Hence in the triangles ACD, BCD, we have the angle ACD equal to the angle BCD, CA = CB. and CD common to both triangles, consequently (by Art. 37) AD = DB ; that is, AB is bisected at D. XLIV. If from the centre, C, of the circle ABE thei-e be draivn a perpendicular, CD, to thf chord AB, and it be continued to meet the circle in F, it loill bisect the arc AFB in F (See the preceding figure.) For in the last article it was proved that the angle ACD =r the angle BCD ; hence (by Art. 11) the arc AFz=the arc FB. 2 10 GEOMETRY. XLV. Any line bisecting a chord at right angles is a diameter. For since (by JlrtA'3) a line dl•a^vn from the centre pei-pendicular to a chord, bisects that chord at riglit angles, therefore conversely a line bisecting a chord at right aiiglea ipust pass through the centre, and consequently be a diameter. XLVI. TTic sine of any arc is equal to half the chord of twice that arc. For (in the last scheme) AD is the sine of the arc AF, and AF is equal to half the aj'C AFB, and AD half the chord AB ; hence the proposition is manifest. XLVII. If two equal and parallel lines, AB, CD, be joined by two others, AC, BD, these ivill 6t also equal ami parallel. To demonstrate this, joui the two oj)posJte angles A and D with the line AD ; then it is evident, that the line AD divides the quadrilateral ACDB uito two triangles, ABD, ACD, in which AB is equal to CD, by sup])osition, and AD is common to both triangles ; and since AB is i)arallel to CD, the angle BAD is equal to the angle ADC (by Art. 30) ; therefore, in the two triangles, the sides AB, AD, and the angle BAD, are equal respectively to the sides CD, AD, and the angle ADC ; hence (by Art. 37) BD is equal to AC, and the angle DAC equal to the angle ADB ; therefore (by Art. 34) the lines BD, AC, must be parallel. Co7\ Hence it follows, that the quadrilateral ABDC is a parallelogram, since the opposite sides are parallel. It is also evident that, in any parallelogram, the line joining the opposite angles (called the diagorud), as AD, divides the figure into two equal parts, since it has been proved that the triangles x\BD, ACD, are equal to each other. XLVIII. It follows also from the preceding article, that a triangle, ACD (see the preceding figure), on the same base, and betiveenthe same parallels loith a parallelogram, ABDC, is the half of that parallelogram. XLIX. From the same article it also follows, that the opposite sides of a parallelogram are equal ; for it has been proved, tiiat, ABDC being a parallelogi-am, AB is equal to CD, and AC equal to BD. L. All parallelograms on the same or equal bases, and between the same parallels, are equal to each other ; that is, if BD and GH be equal, and the lines BH, AF, be parallel, the parallelograms ABDC, BDFE, and EFHG, ivill be equal to each other. For AC is equal to EF, each being equal to BD (by Art. 49) ; to both add CE, and we liave AE, equal to CF ; therefore in the two triangles ABE, CDF, AB is equal to CD, AE is equal to CF, and the angle BAE is equal to DCF {hy Art. 31); therefore the two trianglesABE, CDF, are equal (liy Art. 37), and taking the triangle CKE from both, the figure ABKC is equal to the figure KDFE, to both which add tlie triangle KBD, and we have the parallelogi-am ABDC, equal to the parallelogram BDFE. In the same way it may be proved that the parallelogram EFHG is equal to the parallelogram BDFE ; therefore the three parallelograms ABDC, BDFE, and EFHG, arc equal to each other. Cor. Hence it follows, that triangles on the same base, and between the same parallels, are eqiud, since they are the half of the })arallelograms on the same base and between the same parallels (by Art. 48). LI. In any right-angled triangle, the square of the hypotenuse is equal to the su77i of the squares of the two sides. Thus, if BAG be a right-angled tiiangle, the square of the hypotenuse BC, viz. BCMH, is equal to the sum of the squares made on the two sides, AB and AC, viz. to ABDE and ACGF. To demonstrate this, through the point A draw AKL per|)cndicular to the hypotenuse BC. Join AH, AM, DC, and BG ; then it is evident, that DB is equal to BA (by Art. 18' GEOMETRY. II and BH equal to BC ; therefore iii the triangles DBC, ABH, the two legs, DB, BC, of the one, are equal to the two legs, AB, BH, of the other; and the moluded angles, DBC and ABH, m-e also equal ; (for DBA is equ;d to CBH, being both right ; to each add ABC, and we have DBC, equal to ABH); therefore the triangles DBC, ABH, are equal (by Jlrt.S7) ; but the trian- gle DBC is half of the square ABDE (by ArtA8), and the ti'iangle ABH is half the parallelogram BKLH (by the same article) ; consequently the square ABDE is equal to the parallelogi-ani BKLH. In the same way it may be proved that the square ACGF is equal to the parallelogram KCML. Therefore the sum of the squares ABDE and ACGF is equal to the sum of the parallelograms BKLH and KCML; but the sum of these parallelograms is equal to the square BCJ^HI ; therefore the sum of the squares on AB and AC is equal to the squai-e on BC. Coj: Hence, in any right-angled triangle, if we have the hypotenuse and one of the legs, we may easily find the other leg, by taking the square of tlie given leg from the square of the hypotenuse; the square root of the remainder will be the sought leg. Thus, if the hypotenuse was 13, and one leg was 5, the other leg would be 12, for the square of 5 is 25, and the square of 13 is 169; subtracting 25 from IG'J leaves 144, the square root of which is 12. If both legs are given, the liy])otenuse may also be found by extracting the square root of die sum of the squares of tlie legs; thus, if one leg was 6, and the other 8, the square of the first is 36, the stiuare of the second is 04 ; adding 36 and 64 together gives 100, whose square root is 10, which is the sought hyi)Otenuse. LIL Four quantities are said to be proportional, when the magnitude of the Jirst compared tinth the second is the same as the magnitudt of the third compared with the fourth. Thus 4, 8, 12, and 24, are proportional, because 4 is half of 8, and 12 is half of 24 ; and if we take equi-multi))les, A X «> •'i X b, of the quantities a and 6, and other equi-multiples, B y, a, B Xb, of the same quantities a and b, the four quantities, A X. a, A S<, b, B y^ a, B X b, will be proportional ; for Ay, a com])ared with Ayb'is of the same magnitude as a compared with 6, and By a compared with B yb is also of the same magnitude as a compared with b. LIIL In any triangle, AGg, if a line, Ee, be draivn parallel to either of the sides, as Gg, the side AG ivill be to AE 05 Ag to Ae, or as Gg to Ee. To demonstrate this, upon the line AG take the line AB so that a certain multii)le of it may be equal to AE, and another multiple of it may be equal to AG; this may be always done accurately when AE and AG are connuensura- ble; if they are not accurately commensurable, the quantity AB may be taken so small that certain multiples of it may differ from AE and AG respectively by quantities less dian any assignable. On the line AG, take BC, CD, DE, EF, FG, &c., each equal to AB; and through these points draw the lines Bb, Cc, &c., parallel to Gg, cutting the line Ag in the points b, c, d, e, &c. ; draw also the lines BM, CL, DK, &.C., parallel to Ag, cutting the former parallels in the points N, O, P, &c., and the line Gg in the points M, L, K, &c. Thon the triangles ABb, BCN, CDO, &c., are similar and equal to each otlier ; for the lint* Bb, CN, are parallel; dierefore the angle ABb — BCN (by Art. 31), and by the san-e article the angle BAb is equal to CBN (becaiLse BN is parallel to Ab), and by i;onstruction ABz=BC; therefore (by Art. 38) die triangles ABb and BCN are eqivi. to each other; and in the same manner we may prove tliat the others, CDO, DEP, EFQ, &c., are equal to ABb. Therefore Al)=: BN=:CO = DP, &,c., and Bb ^ CN = DO = EP, &c. ; but (by Art. 49) BN == be, CO = cd, DP = oO 40 50 ••» B Hniic Scale Yi<;;S. Rinini 1 -1 j; y . i^ : ■ ^-jq ? , 1 biittfj m^ ^^^ ^tt P ''\V />' 12 -. iSUTft 7 ■ ' ''> '"'',7- EA.GWBUJN'r. 1B61 GEOMETRICAL PROBLEMS. 17 another ai-c cutting the former in tlie points G, II, through which ch-iw tlio right Hne GHC, cutting the former right line EFC in the point C ; upon the ])oiiit C as a centre with an extent equal to CA, CB, or CD, as radius, describe the sought circle. PROBLEM X. 7'o divide a circle into 2, 4, 8, IC, or 32 equal parts. Draw a diaineter through the centre, dividuig the circle into two equal parts ; bisect this diameter by another, drawn perpendicular tliereto, and the circle will be div'ided into four eijual i)arts or quadrants; bisect each of these quadrants again by right lines di'awn through the centre, and the cu'cle will be divided into eight equal j)arts ; and so you may continue the bisections any number of times. This problem is useful in constructing the mariner's compass. PROBLEM XL To divide a given line into any nwnber of equal parts Let it be required to divide the line AB into five ecjual })arts. From the pohit A draw any line, AD, making an angle with the lin« AB ; then through the pohit B draw a line", BC, parallel to AD ; and from A, with any small ojjening in your compasses, set off a number of equal parts on the line AD, less by one than the proposed number (which number of equal parts in this examjtle is 4); then from B, set off the same number of the same parts on the line BC; then join 4 and 1, 3 and 2, 2 and -3. I and 4, and these lines will cut the given line as requii-ed. J , f- 18 CONSTRUCTION OF THE PLANE SCALE Isi. With the nuliu5 you intend for your scale, describe a semicircle, ADB (Plate II fig. 1), and from the centre, C, draw CD perpendicular to AB, which will divide the semicu-cle into two quadrants, AD, BD ; continue CD towards S, draw BT perpen- dicular to CB, and join BD and AD. 2dly. Divide the quadrant BD into 9 equal parts ; then will each of these be 10 degrees; subdivide each of these parts into single degrees, and, if your radius will admit of it, into minutes or some aliquot parts of a degree greater than mmutes. 3dly. Set one foot of the comi)asscs in B, and transfer each of the divisions of the quadrant BD to the right line BD, tlum will BD be a line of chords. 4thly. From the points 10,20, 30, &c., in the quadrant BD, draw right lines parallel to CD, to cut the radius CB, and they will divide that luie into a line of sines which must be ninnbered from C towards B. 5thly. If the same line of sines be numbered from B towards C, it will become a line of versed sines, which may be continued to 180°, if the same divisions be transferred on the same line on the other side of the centi'e C. Gthly. From the centre C, through the several divisions of the quadrant BD, draw right lines till they cut the tangent BT ; so will the line BT become a line of tangents. 7thly. Setting one foot of the compasses in C, extend the other to the several divisions, 10, 20, 30, &c., in the tangent line, BT, and transfer these extents severally to the right line, CS ; then will that hue be a line of secants. 8thly. Right lines drawn from A to the several divisions, 10, 20, 30, &c., in the quadrant BD, will divide the radius CD into a line of semi-tangents. Dthly. Divide the quadrant AD into eight e(]ual parts, and from A, as a centre, tj-ansfer these divisions severally into the line AD ; then will AD be a line of rhumbs, each division answering to 11° 15' upon the line of chords. The use of this line is for protracting and measuring angles, according to the connnon division of the marmer's compass. If the radius AC be divided into 100 or 1000, &c., equal parts, and tlie lengths of the several sines, tangents, and secanis, corresj)onding to the several arcs of the quadrant, be measured thereby, and these numbers be set down in a table,* each in its proper column, you will by these means have a collection of numbers by which the several cases in trigonometry may be solved. Right lines, graduated as above, beuig f)laced severally upon a ruler, form the instrument called the Plane Scale 'see Plate II. fig. 2), by wliich the lines and angles of all triangles may be measured. All right lines, as the sides of plane triangles, &c., wlicn they are considered sunply as such, without having any relation to a circle, are measured by scales of equal parts, one of which is subdivided e(iually into 10, and this serves as a common division to all the rest. In most scales, an inch is tak(Mi for a common measure, and what an inch is di\ ided into is generally set at the end of the scale. By any conunon scale of equal parts, divided in this manner, any number less than 100 may be readily taken ; but if the number should consist of three places of figures, the value of the third figure cannot be exactly ascertained, and in this case it is better to use a diagonal scale, by which any number consisting of three places of figures, may be exactly found. 'The figiu-e of this scale is given in Plate 11. fig. 3;. its construction is as follows: — Having jjrepared a ruler of convenient breadth (or your scale, draw near the edges thereof two right lines, af, eg, parallel to each other ; divide one of these lines, as of into equal parts, according to the size of your scale;} and, through each of these divisions draw right lines perpendicular to q/", to meet eg*; then divide the breadth into 10 equal ])arts, and through each of these divisions draw right lines parallel to af and eg; divide the lines ab, cd, into 10 etjual parts, and from the pomt a to the first division * In Tabic XXIV. are given the sine and cosine to every iniinite of the quadraJit, to five places of decimals. t The length of one of these equal parts at the end of the scale to which this dcscriiJtion refers is ah or cd ; the length of one of liie equal parts of the scale of the other end being ihe half of cib. COiNSTRUCTION OF THE PLANE SCALE. 19 in tlie line cd, draw a diagonal line ; then, parallel to that line, draw diagonal linos through all the other divisions, and the scale is complete. Then, if any number, consisting of three places of figures, as 256, be required from the larger scale, gd, you must place one foot of the compasses on the figure 2 on the line gd, then the extent from 2 to the point d will represent 200. The second figure being 5, count five of the smaller divisions from d towards c, and the extent from 2 to that point will be 250. Move both points of the compasses downwards till they are on the sixth parallel line below gd, and open them a little till the one pohit rests on the vertical line dra^vn througli 2, and the other on the diagonal line drawn through 5 ; the extent then in the compasses will represent 256. In the same way the quantities 25.6, 2.56, 0.256, &c., are measured. Besides the lines already mentioned, there is another on the Plane Scale, marked ML, which is joined to a line of chords, and shows how many miles of easting or westing correspond to a degree of longitude in every latitude.* These several lines are generally put on one side of a ruler two feet long ; and on the other side is laid do\vn a scale of the logai-ithms of the sines, tangents, and numbers, which is commonly called Gunter's Scale ; and, as it is of general use, it requires a particular descri[)tion. * As it would confuse the adjoined figure to describe on it the line of longitudes, it is neglected, but tlie crnslruction is as follows ; divide the line CB into GO equal parts (if it can be done), and through each point draw lines parallel to CD, to intersect the arc BD ; about 15, as a centre, transfer the severiil points of intersection to the line of chords, BD, and then number it from D towards B, from to GO, aiid it will be the line of longitudes, corresponding to the degrees on the line of cliords. 20 GL'NTER'S SCALE. On Gunter's Scale are eight lines, viz. 1st. Sine rhumbs, marked (SR), correspontling to the logarithms* of the natural smes of every point of the maruier's compass, numbered from the left hand towards the right, with 1, 2, 3, 4, 5, 6, 7, to 8, where is a brass pm. This line is also divided, where it can be done, into halves and quarters. 2dly. Tangent rhumbs, marked (TR), correspond to the logarithms of the tangents of eveiy point of the compass, and are numbered 1, 2, 3, to 4, at the right hand, where there is a })in, and thence towards the left hand with 5, 0, 7 ; it is also divided, where it can be done, mto lialvcs and quarters. 3d]y. The line of numbers, marked (Num.), corresponds to the logarithms of numbers, and is marked thus: near tlie left hand it begins at 1, and towards the right hand are 2, 3, 4, 5, G, 7, 8, 9 ; and 1 in the middle, at which is a brass pin ; then 2, 3, 4, 5, 6, 7, 8, 9, and 10, at the end, where there is another pin. The values of these numbers and their intermediate divisions depend on the estimated values of the extreme numbers 1 and 10 ; and as this line is of great imy)ortance, a particular description of it Avill be given. The first 1 may be counted for 1, 10, 100, or 1000, &c., and then the next 2 will be 2, 20, 200, or 2000, &c., respectively. Again, tlie first 1 may be reckoned 1 tenth, 1 hundredth, or 1 thousandth part, «fcc. ; then the next will be 2 tenth, or 2 hundredth, or 2 thousandth parts, &c. ; so that if the first 1 be esteemed 1, the middle 1 will be 10 ; 2 to its right, 20 ; 3, 30 ; 4, 40 ; and 10 at the end, 100. Again, if the first 1 is 10, the next 2 is 20, 3 is 30, and so on, making the middle 1, 100; the next 2 is 200, 3 is 300, 4 is 400, and 10 at the end is 1000. In like manner, if the fii-st I be esteemed 1 tenth part, the next 2 will be 2 tenth parts, and the niiddle 1 will be 1 ; the next 2, 2 ; and 10 at the end will be 10. Again, if the first 1 be counted 1 hundredth pait ; the next, 2 hundredth parts ; the middle 1 will be 10 hundredth parts, or 1 tenth part; and the next 2, 2 tenth parts; and 10 at the end will be but one whole number or imeger. As the figures are increased or diminished in their value, so in like manner must all the intermediate strokes or subdivisions be increased or diminished; that is, if the first I at the left hand be counted 1, then 2 (next followmg it) will be 2, and each subdivision between them will be 1 tenth part; and so all the way to the middle 1, Avhicli -will be 10 ; the next 2, 20 ; and the longer strokes between 1 and 2 are to be counted from 1 thus, 11, 12 (where is a brass pm); then 13, 14, 15, sometimes a longer stroke than the rest ; then IG, 17, 18, 19, 20, at the figure 2 ; and hi the same manner the short strokes between the figures 2 and 3, 3 and 4, 4 and 5, &c., are to be reckoned as units, Agam, if 1 at the left hand be 10, the figures between it and the middle 1 will be conunon tens, and the subdivisions between each figure will be units ; from the middle 1 to 10 ut the end, each figure will be so many hunch-eds ; and between these figures each longer division will be 10. From this description it will be easy to find the divisions representing any given lunnber, thus: Supj)ose the j)oint representing the number 12 were required; take the division at tlie figure 1 in tlie middle, for the firet figure of 12 ; then for the second figure count two tenths, or longer strokes to the right hand, and this will be the point representing 12, where the brass pin is. Again, suj)pose the number 22 were required; the first figure 2 is to be found on the scale, and for the second figure 2, count 2 tenths onwards, and that is the point representuig 22. Again, sup])ose 1728 were required; for the first figure 1, 1 talce the middle 1, for tlie second figure 7, count onwards as before, and that will be 1700. And, as the remaming figures are 28, or nearly 30, I note the jjouit which is nearly fV of the distance between the marks 7 and 8, and this will be the point representing 1728. * The description and use of logarilhms arc given in page 23, et seq. The log. sines, tangents, &c., are marked on these scales by means of a line of equal parts, corresponding to the size of the scale. UKrfCRlPTIOiN AND USE OF GUTTERS SCALE. 21 If the point representing 435 was i-eqinred, from the 4 in tlie second inten-al count towards 5 on the right, three of the larger divisions and one of the smaller (this smaller division being midway between the marks 3 and 4), and that will be the division expressing 435. In a similar manner other numbers may be found. All fractions found in this line nuist be decimals; and if they are not, they must be reduced into decimals, which is easily done by extending the compasses from the denominator to the numerator ; that extent laid the same way, from 1 m the middle or right hand, will reach to the decimal required. Example. Requu-ed the decunal fraction equal to ^. Extend from 4 to 3 ; that extent will reach from 1 on the middle to .75 towards the left hand. The like may be observed of any other vidg-ar fraction. Multii)lication is performed on this Ihie by extendhig from 1 to the multiplier; that extent will reach from the multiplicand to the product. Suppose, for example, it were requu-ed to find the product of 16 multiplied by 4 ; extend from 1 to 4 ; that extent will reach from 10 to G4, the product required. Division being the reverse of multiplication, therefore extend from the divisor to unity ; that extent will reach from the dividend to the quotient. Sa))pose C4 to bo divided by 4 ; extend from 4 to 1 ; that extent will reach from 64 to IG, the quotient. Questions in the Rule of Three are solved by this luie as follows : Extend from the first terir to the second ; that extent will reach fi'om the thu'd term * to the fourth. And it ought to be particularly noted, that if you extend to the left, from the first number to the second, you nuist also extend to the left, from the third number to the fourth ; and the contrary. ExASiPLE. If the diameter of a circle be 7 inches, and the circumference 22, what is the circumference of another circle, tJie diameter of which is 14 inches .'' Extend ti"om 7 to 22 ; that extent -will reach from 14 to 44, the same way. The superficial content of any parallelogram is foimd by extending from 1 to the breadth ; that extent will reach from the length to the superficial content. Example. Siqipose a plank or board to be 15 uiches broad and 27 feet long, the content of which is required. Extend from 1 to 1 foot 3 inclies (or 1.25) ; that extent will reach from 27 feet to 33.75 feet, the superficial content. Or extend from 12 inches to 1.5, &c. The solid content of any l)ale, box, chest, &c., is found by extending from 1 to the breadth ; that extent will reach from the depth to a fourth number, and the extent from 1 to that fourth number will reach from the length to the solid content. Example I. What is the content of a square j)illar, whose lenglh is 21 feet 9 inches, and breadth 1 foot 3 inches ? The extent from 1 to 1.25 will reach from 1.25 to 1.56, the content of one foot in length ; again, the extent from 1 to 1.56, will reach from the length 21.75 to 33.9, or 34, the solid content m feet. Example II. Suppose a squai-e piece of timber, 1.25 feet broad, .56 deep, and 36 long, be given to find the content. Extend from 1 to 1.25; that extent will reach from .56 to .7 ; then extend from 1 to .7 ; that extent will reach from 36 to 25.2, the solid content. In like manner may the contents of bales, &-c., be found, which, being divided by 40, will give the number of tons. 4thly. Tiie line of sines, marked (Sin.), coiTcsponduig to the log. sines of the degi-ees of the quadrant, liegins at the lefi; hand, and is numbered to the right, 1, 2, 3, 4, 5, &c., to 10 ; then 20, 30, 40, &:c., ending at 90 degrees, where is a brass centre-pin, as there is at the right end of all the luies. 5thly. The line of versed shies, marked (V. S.), corresponding to the log. versed sines of the degrees of the quadrant, begins at the right hand against 90° on tlie sines, and from thence is niuiibei-ed towards the left hand, 10, 20, 30, 40, &c., ending at the lefi hand at about 169° ; each of the subdivisions, from 10 to 30, is in general two degrees ; from thence to 90 is suigle degrees ; from thence to the end, each degree is divided mto 15 minutes. Gthly. The line of tangents, marked (Tang.), corresponding to the log. tangents of the degrees of the quadrant, begins at the left hand, and is numbered towards the right, 1, 2, 3, &c. to 10, and so on, 20, 30, 40, and 45, where is a brass pin under 90° on the shies ; from thence it is numbered backwards, 50, 60, 70, 80, &c. to 89, ending at the lefi; hand where it began at 1 degree. The subdivisions arc nearly similar to those of the sines. When you have any extent in your comjiasses, to be set off" from any number less than 4.5° on the line of tangents, towards the right, and it is found to reach * Or ynii may cxteiul from the first to tlie third ; for lliat extent will roach from the second to the fourth. This inelliod must be adopted vviicn usiiisj the lines of sines, tangents, &c., if the first and third terms are of the same name, and different from the second and fourth. 22 DESCRIPTION AND USE OF GUNTERS SCALh. beyond the mark of 45°, you must see how far it extends beyond that mark, and set it off from 45° towards the left, and see what degi-ee it falls upon, which will be the number sought, which must exceed 45° ; if, on the contrary, you are to set off such a distance to the right from a number greater than 45°, you must proceed as before, only remembering, that the answer must be less than 45°, and you must always consider the degi'ees above 45°, as if they were marked on the continuation of tlie line to the right hand of 45°. 7thly. The line of the meridional parts, marked (Mer.), begins at the right hand, and is numbered, 10, 20, 30, &c., to the left hand, where it ends at 87 degrees. This line, with the line of equal parts, marked (E. P.), under it, are used together, and only in Mercator's Sailing. The upper line contains the degrees of the meridian, or latitude in a Mercator's chart, corresponding to the degrees of longitude on the lower line. The use of this Scale in solving the usual problems of Trigonometry, Plane Sailing, Middle Latitude Sailing, and Meixator's Sailing, will be given in the course of this work ; but it will be unnecessary to enter uito an explanation of its use in calculating the common pi-oblems of Nautical Asti'onomy as it is much more accui'ate to perform those calculations by logarithms. 23 ON THE SLIDING RULE. The Sliding Rule consists o^ di fixed part and a slider, and is of the same dimensiona. and has the same Imes marked on it as on a common Gunter's Scale or Plane Scale, which may be used, with a pair of compasses, in the same manner as tliose scales ; and as a description of those lines has already been given, it will be uimecessary to repeat it here, it being sufficient to observe, that there are two lines of numbers, a line of log. sines, and a line of log. tangents, on the slider, and that it may be shifled so as to fix any face of it on cither side of the fixed part of the scale, accordmg to the nature . of the question to be solved. In solving aiay problem in Ai'ithmetic, Trigonometry, Plane Sailing, &c., let the proposition be so stated that the first and third terms may be alike, and of course the second and fourth terms alike ; then biing the first term of the analogy on the fixed part, against the second term on the slider, and against the third term on the fixed part ivill be found the fourth term 07i the slider ;* or, if necessary, the first and third terms may be found on the slider, and the second and fourth on the fixed part. Multiplication and division are performed by this rule, in considering unity as one of the terms of the analogy. Thus, to perform multiplication ; set 1 on the line of numbere of the fixed pai-t, against one of the factors on the line of numbers of the slider ; then agamst the other factor, on the fixed part, will be found the product on the slider. Example, To find the product of 4 by 12 ; draw out the slider till 1 on the fixed part comcides with 4 on the slider ; then opposite 12 on the fixed part will be found 48 on the slider. To perform division ; set the divisor on the line of numbers of the fixed part against 1 on the slider ; then against the dividend on the fixed pait will be found the quotient on the slider. Example. To divide 48 by 4 ; set 4 on the fixed part against 1 on the slider ; then against 48 on the fixed pait will be found 12 on the slider. EXAMPL*ES IN THE RULE OF THREE. If a ship sail 25 miles m 4 houre, how many miles will she saU in 12 houre at the same rate ? Bring 4 on the line of numbei-s of the fixed part against 25 on the line of numbers of the slider ; then against 12 on the fixed part will be found 75 on the slider, which is the answer required. Example. If 3 pounds of sugar cost 21 cents, Avhat will 27 jiounds cost? Bring 3 on the line of numbers of the fixed part, against 21 on tlie line of numbers of the slider ; then against 27 on the fixed i)art will be found 189 on the slider. EXAMPLE IN TRIGONOMETRY. In the oblique-an -^, or^, or any other even parts of a minute, the like parts may be taken of the difference of the logarithms, and added or subtracted as above, which may be frequently done by inspection. These proportional parts may also be found very nearly by means of the three columns of differences for seconds, given, for the first time, in the nintli edition of this Avork. The first column of J'fferences, which is to be used with the two columns marked A, A, is placed between Tlien, as 60" : 38" : : 29 : 18, which, beuig added to the number correspondmg to 24° 16', gives 9.61400, the log. sine of 24° 16' 38". 34 LOGARITHMS. these columns. The second column of differences, which is to be used with the two cohuiais B, B, is placed between these two columns. In like manner, the third column of differences, between the columns C, C, is to be used with them. The correction of the tabular logarithms in any of tlie columns A, B, C, for any number of seconds, is found by entering the left-hand column of the table, marked S' at the top, and finding the number of seconds ; opposite to this, in the column of differences, will be found the corresponding correction. Thus, in tlie table, page 215, which contains the log. sines, tangents, &c., for 30°, the corrections corresponding to 25", are 9 for the columns A, A, 12 for the columns B, B, 3 for the columns C, C ; so that, if it were required to find the sine, tangent, or secant of 30° 12' 25", we must add these corrections respectively to the numbers corresponding to 30° 12' ; thus. Col. a. Col. B. Col. C. Logs, for 30° 12' .... Sine 9.70159 Tangent .... 9.76493 Secant .... 10.0C335 Corrections for 25" in S' + 9 -f 12 -f 3 Logs, for 30° 12' 25" 9.70168 9.76505 10.06338 these corrections being all added, because the logarithms increase in proceeding from 30° 12' to 30° 13'. Instead of taking out the logarithms for 30° 12', and adding the correction for 25", we may take out the logarithms for 30° 13', and subtract the correction for 60" — 25", or 35", found in the margin S' ; thus, Logs, for 30° 13' ... . Sine 9.70180 Tangent .... 9.76522 Secant . . . . 10.06342 Corr. for 35" in col. S', ? ,o 17 a or 25" in col. G' .... $ ~ ^'^ ~^^ ~^ Logs, for 30° 12' 25" .... 9.70167 9.76505 10.06338 The corrections are in this case subtracted, because the logaiithms decrease in proceeding backward 35" from 30° 13', to attain 30° 12' 25". The tangents and secants, in this example, are the same by both methods ; the sines differ by one unit, in the last decimal place, and this will frequently happen, because the difference of the logarithms for 1', sometimes differ one or two units from the mean values which are used in the three columns of differences. The error arising from this cause is generally diminished by using the smallest angle * S', when the seconds of the pro- posed angle are smaller than 30" ; or the greatest angle G', when the number of seconds are greater than 30". Thus, in the above example, where the angle S' = 30° 12', and the angle G' r= 30° 13', it is best to use the angle S' when the gifen angle is less tlian 30° 12' 30", but the angle G' when it exceeds 30° 12' 30". thus, if it be required to find the sine of 30° 12' 51", it is best to use the angle G'=:30° 13', and find the correction by entering the margin marked S', with the difference 60" — 51" =^9", opposite to which, in the column of diflerences, is 3, to be subtracted from log. sine of 30° 13' = 9.70180, to get the log. sine of 30° 12' 51" = 9.70177. To save the trouble of subtracting the seconds from 60", wc may use the right-hand margin, marked G', and the correction may then be found by the following rules: — Rule 1. When the smallest angle S' is used, find the seconds in the column S', and take out the corresponding correction, which is to be applied to the logarithm corresponding to S' ; by adding, if the log. of G' be greater than the log. of S'; otherwise, by subtracting. Rule 2. When the greater angle G' is used, find the seconds in the column G', and take out the corresponding correction, which is to be applied to the logarithm corresponding to G'; by adding, if the log. of S' be greater than the log. of G'j- otherwise, by subtracting; so that, in all cases, the required logarithm may tall be- tween the two logarithms corresponding to the angles S' and G'. The correctness of these rules Avill evidently appear by comparing them with the preceding exam])les ; and by the inverse process we may find the angle correspond- ing to a given logarithm, as in the next article. We have given at the bottom of the page, in this table, a small table for finding the proportional jjarts for the odd seconds of time, corresponding to the column of Hours A. M. or P. M. ; to facilitate the process of finding the log. sine, cosine, ifcc- correspomling to the nearest second of time in the column of hours, or, on the con- trary, to find the nearest second of time corresponding to any given log. sine, cosine. &c. Thus, in the preceding examples, where the angle 8' = 30° 12', and the * If wc neglect the seconds in any proposed angle whose sine, &c. is required, we get llie angle denoted above by S', and this angle increased by 1', is represented by G'' ; so that the proposed angle falls between S'antI G' ; S' being a smaller, and G' a crreater angle than that whose log. sine, <^c., is required ; the letters S'and G', accented for minutes, being used because they are easily rememtored as tlse viiitials of smalhr and greater 1 B. C. Tangent 9.7G493 Secant 10.06335 + 11 + 3 LOGARITHMS 35 angle G'r=30° 13'; the times corresponding in the column of Hours P.M., are S,_4h jm 36s. G' = i^ 1-" 44'; and if we wish to find the log. sine, cosine, &c., corresponding to any intermediate time, as, for example, 4'' 1"" 39% which differs 3* from the angle S', we must find the tabular logarithm corresponding to S', and apply the correction for 3% given by the table at the bottom of the page, as in the following examples : — A. Logs, for S' r= 4" 1" 30^ Sine 9.70159 Correction for -}- 3' -|- 8 Logs, for 4h 1"^ 39^ Sine 9.70167 Tangent 9.76504 Secant 10.06338 Nearly tlie same results are obtained by using the angle G', in the manner we have before explained : — Logs, for G' = 4" 1-" 44' Sine 9.70180 Tangent 9.76522 Secant 10.06342 Correction for — 5' — 13 — 18 5 Logs, for 4 " 1-" 39' Sine 9.70167 Tangent 9.765 04 Secant 10.06337 These corrections must be applied by addition or subtraction, according to the directions given above, so as to make the required logarithm fall between those which correspond to the times S' and G'. The inverse process will give the time corresponding to any logarithm. Thus, if the log. sine 9.70167 be given, the difference between this and 9.70159, corre- sponding to S' = 4'' 1" 36', is 8 ; seeking this in the cohnnn A, in the second line of the table at the bottom of the page, it is found to correspond to 3' ; adding this to the time S' =:4'' 1" 36', we get 4'' l"" 39' for the required time. We may proceed in the same manner with the logarithms in the columns 13, C ; using the numbers coiTesponding, marked B, C, respectively, in the table at the bottom of the page. To find the degrees, minutes, and stconds, corresponding to any given logarithm sine, cosine, t^'c. hij Table XXVII. Find the two nearest numbers to the given log. sine, cosine, &c., in the column marked sine, cosine, &c., respectively, one being greater, and the other less, and take their difference, D ; take also the difference, d, between the given logarithm and the logarithm corrcsj)onding to the smallest number of degrees and minutes ; then say, As the first found difference is to the second found difference, so is 60" to a number of seconds to be annexed to the smallest number of degrees and minutes befoi-e found. The three columns of differences may also be used, by an inverse operation to that which we have explained in the preceding article. EXAMPLE V. Find the degrees, minutes, and seconds (less than 90°), corresponding to the log. sme 9.61400. Next less log. S' ==24° 16' 9.61382 Log. of smallest angle S' = 24° 16' is 9.61382 Greater G' = 24 17 9.61411 Given log 9.61400 D=:29 d=l8 Then say. As 29 : 18 :: 60": 38", nearly ; which, annexed to 24° 16', give 24° 16' 38", answering to losr. sine 9.61400. Subtracting 24° 16' 38" from 180°, there remain 155° 43' 22", the log. sine of which is also 9.61400. The quantity 38" may also be found by inspection in the side column S' of the page opposite d=zl8, in the column of differences between the two columns, A, A. If we use the angle G', we shall have (/' equal to 11, the difference of the logarithms 9.61411 and 9.61400, and the corresjionding number of seconds in column G', is 37", making 24° 16' 37". To find the arithmetical complement of any logarithm. The arithmetical complement of any logarithm is what it wants of 10.00000, and is used to avoid subtraction. For, when working any proportion by logarithms, you may add the arithmetical complement of the logarithm of the first term, instead of suljtracting the logarithm itself, observing to neglect 10 in the index of the sum of the logarithms. The arithmetical complement of any logarithm is thus found : — Begin at tlie index, and icnite down ichat each figure ivants of 9, except the last significant figure, which take from 10.* Thus, the arithmetical complement of 9.62595 is 0.37405 ; tha t of 1.86567 is 8.13433; and that of 10.33133 is 89.66867, or 9.66867. * When llic index of (he given log-arithm is greater than 10, as in some of the numbers of Table XXV II., tlie left-hand figure of it must be neglected ; and when there are any ciphers to the right hand of the last significant figure, j'ou may place the same number of ciohers to the right hand of the other ^iRiuf^s of the arithmetical complement 36 PLANE TRIGONOMETRY. Plane Trigonometry is the science which shows how to find the measures of the sides and angles of plane triangles, some of them being akeady known. It is divided into two parts, right-angled and oliique-angled ; in the former case, one of the angles is a right angle, or 90° ; in the latter, they are all obhque. In every plane triangle there are six parts, viz. three sides and three angles ; any three of which beuig given (except the three angles), the other three may be found by various methods, viz. by Gunter's scale, by the slidmg rule, by the sector, by geometrical construction, or by arithmetical calculation. We shall explain each of these methods ; * but the latter is by far the most accurate ; it is perlbrmed by the help of a few theorems, and a ti-igonometrical canon, exliibituig the natural or the logarithmic smes, tangents, and secants, to every degree and minute of the quadrant.t The theorems alluded to are the followuig : — THEOREM 1. In any riglit-angled triangle, if the hypoteniisc ie made radius, one side ivill be tlie sine of the opposite angle, and the other its cosine ; but if either of the legs be made radius, the other leg will be the tangent of the opposite art^le, and the hypotenuse will be the secant of the same angle. A HiidiusC ATanffeni •D 1st. If, in the right-angled plane triangle ACB (fig. 1), we make the hypotenuse AB radius, and upon the centre. A, describe the arc BE, to meet AC produced in E, then it is evident that BC is the sine of the arc BE (or the sine of the angle BAC), and that AC is the cosine of the same angle ^ and if the arc AD be described about the centre B (fig. 2), AC will be the sine of the angle ABC, and BC its cosine. 2dly. If the leg AC (fig. 3) be made radius, and the arc CD bo described about the centre A, CB will be the tangent of that arc, or the tangent of the angle CAB ; and AB will be its secant. 3dly. If the leg BC (fig. 4) be made radius, and the arc CD be described about the centre B, CA will be the tangent of that arc, or the tangent of the angle B, and AB will be its secant. Now, it has been already demonstrated (in Art. 55, Geometiy) that the sme, tangent, secant, &c. of any arc in one circle is to the sine, tangent, secant, &c. of a similar arc in another cu'cle as the radius of the former cu'cle to the radius of the latter. And since in any right-angled triangle there are given either two sides, or the angles and one side, to find the rest, we may, if we wish to find a side, make any side radius ; then say. As the tabularnumbcr of the same name as the given side is to the given side of the triangle, so is the tabidar number of the same name as the requii-ed side, to the requh-ed side of the triangle. If we wish to find an angle, one of the given sides must be made radius ; then say, As the side of the triangle made radius is to the tabidar * It will not be necessary to add any furlher description of the uses of the sector or sliding rule; for what we have already given will be suHicicnt for any one tolerably well versed in the use of Gunter's scale. t See Tables XXIV. and XXVII. PLANE TRIGONOMETRY. 37 radius, so is the other given side to the tabular sine, tangent, secant, &c. by it repre- sented ; which, bemg sought for m the table of sines, &c., will coirespond to the degi'ees and minutes of the required angle. THEOREM II. In all plane triangles, the sides are in direct proportion to the sines of their opposite angles (by Art. 58, Geometiy). Hence, to find a side, Ave must say. As the sme of an angle is to its opposite side, so is the sine of either of the other angles to tlie side opposite thereto. But if we v/ish to find an angle, we must say. As any given side is to the sine of its ojiposite angle, so is either of llie other sides to the suie of its opposite angle. THEOREM HI. In every plane triangle, it will be, as the sum of any two sides is to their difference, so is the tangent of half the sum of the two opposite angles to the tangent of half their difference (by Art. 59, Geometry). THEOREM IV. As the base of any plane triangle is to the sum of the two sides, so is the difference of the two sides to twice the distance of a perpendicular {let fall upon the base from the opposite angle) from the middle of the base (by Aii. GO, Geometiy). THEOREM V. In any plane triangle, as the rectangle contained by any tioo sides including a sought angle, is to the rectangle contained by the half sum of the three sides and the same half sum decreased by the other side, so is the square of radius to the square of the cosine of half the sought angle (by Ad. 61, Geometiy). In addition to these theorems, it will not be amiss for the learner to recall to mind the following aiticles : — 1. In eveiy triangle, the greatest side is opposite to the greatest angle, and the greatest angle opposite to the gi-eatest side. 2. In eveiy tiiangle equal sides subtend equal angles. {Aii, 39, Geometrj\) 3. The three angles of any plane triangle are equal to 180°. {Art. 35, Geometiy.) 4. If one angle of a triangle be obtuse, the rest are acute ; and if one angle be right, the other two together make a right angle, or 90° ; therefore, if one of the acute angles of a right-angled triangle be known, the other is found by subtractmg the known angle from 90°. If one angle of any triangle be knoAvn, the sum of the other two Is found by subtracting the given angle from 180° ; and if two of the angles be knoAvn,the third is found by subtracting their sums from 180°. 5. The complement of an angle is lohat it wants of 90°, and the supplement of an angle is what it luants of 180°. In the two following tables we have collected all the lailes necessaiy for solving the vai'ious cases of Right-angled and Oblique-angled Trigonometry. FORMULAS L\ RIGHT-ANGLED TRIGONOMETRY. Case. Given. Sought. Solutions. 1 Hyp. AC. Angles. Leg BC. Leg AB. Bad. : hyp. AC : : sine A : leg BC. Rad. : hyp. AC : : sine C : leg AB. 2&3 Leg BC. Angles. Leg AB. Hyp. AC. Bad. : leg BC : : tang. C : leg AB. ( Rad. : leg BC : : sec. C : hyp. AC. ( Or, sine A : leg BC : : rad. : hyp. AC. 4 &5 Hvp. AC. Leg AB. Angles. Leg BC. Hvp. AC : rad. : : leg AB : sine C, whose corap. is A. Rad. : hyp. AC : : sine A : leg BC. 6 Both legs. AB & BC. Angles. Hyp. AC. Leg BC : rad. : : leg AB : tang. C, whose conip. is A. ( Sine C : leg AB : : rad. : hyp. AC. } Or, rad. : leg BC : : sec. C : hvp. AC. 88 RIGHT-ANGLED TRIGONOMETRY. A a C D -^ D (r <^ FORMULAS LN OBLIQUE-ANGLED TRIGONOMETRY Case. Given. Sought. Solutions. 1 The angles and side AB. Side BC. Side AC. Sine C : side A6 : : sine A : side BC. Sine C : side AB : : sine B : side AC. 2&3 Two sides, AB, BC, and angle opposite to one of tiiem. Angle A. Angle B. Side AC. Side AB : sine C : : side BC : sine A, which added to C, and the sum subtracted from 180°, gives B. Sine C : side AB : : sine B : side AC. 4&5 Two sides, AC, AB, and the in- cluded angle A. Angles C and B. Side BC. Subtract half the given angle. A, from 90° ; the remainder is half the sum of the other angles. Then say. As the sum of the sides, AC, AB, is to their ditference, so is the tangent of the half sum of the other angles to the tangent of half their difference; which added to and subtracted from the half sum, will give the two angles B and C ; the greatest angle being opposite to the greatest side. Sine B : side AC : : sine A : side BC. 6 All three sides. All the angles. Let fallaperpendicular,BU, opposite to the required angle; then, as AC : sum of AB, BC : : their difference : twice UG, the distance of the perpendicular from the middle of the base ; hence, AD, CD, are known, and the triangle ABC is divided into two right-angled triangles, BCD, BAD; then, by Cases IV. and V. of Right-angled Trigonometry, we may tind the angle A or C. Either angle, as A. Either of the angles, as A, may also be found by the follow- ing rule. From half the sum of the three sides subtract the side BC opposite to the sought angle ; take the logarithms of the half sum and remainder, to which add the arithmetical complements of the logarithms of the sides AB, AC (including the sought angle) ; half the sum of these four logarithms will be the logarithmic cosine of half the sought angle. In calculating by logarithms by any of the pi-eceding i-ules, you must remember, that Git logarithm of the first term of the analogy is to be subtracted from the sum of the logarithms of the second and third terms ; the reviainder will he the logaiithm of the sought fourth term. When the first temi is radius (whose logarithm is 10.00000), you need only reject a unit in the second left-hand figure of the mdex of the sum of the second and thu'd terms. But when the i-adius occurs m the second or third term, you must suppose a unit to be added to the second left-hand figui-e of the mdex of the other term, and subtract thei'efrom the logarithm of the fii'st term. RIGHT-ANGLED TRIGONOMETRY. Solution of the six cases in Right-angled Trigonometry. CASE 1. The angles and hypotenuse given, to find the legs. Given the hypotenuse AC 250 leagues, and the angle C, opposite to the side AB, z= 35° SO', to find the base CB, and perpendiculai* AB. li\ PROJECTION. Draw tlie base CB of any length ; with an extent equal to the chord of 60°, and on C as a centre, describe the arc DE ; from E to D lay off" 35° 30' taken from the line of chords ; * through C and D iJ J? * In all projections of this kind, the angles are measured from the line of chords ; the radius used for describing arcs by which the angles are to '^e measured, being equal to the chord of C0°, the sides of i RIGHT-ANGLED TRIGONOMETRY. 39 draw the line AC, which make equal to 250 ; from A let fall the perpendicular AB to cut CB in B, and it is done ; for CB will be 203.5, and AB equal to 145.2. BY LOGARITHMS. By making the hypotenuse CA radius, it will be, To find the base BC. As radius 10.00000 is to the hypotenuse AC 250. . 2.3!)794 So is the sine angle A 54° 30' . . 9.910G9 To the base BC 203.5 2.30863 To find the perpendicular AB. As radius 10.00000 Is to the hypotenuse AC 250 . . 2.39794 So is the sine angle C 35° 30' . . 9.76395 To the pei-pendicular AB 145.2 2.16189 BY GUNTER'S SCALE. In all proportions which are calculated by Gunter's scale, when the first and second terms are of the same kind, the extent from the first tenn to the second will reach from the third to the fourth. Or, when the first and third tenns are of the same kind. The extent from the first term to the thu'd will reach from the second to the fourth ; that is, we must set one point of the compasses on the division expressing the first term, and extend the other point to the division expressing the third term ; then, without altering the openuig of the compasses, we must set one point on the division representuig the second term, and the other point will fall on the division showing the fourth term or answer. In the present example the Avork is as follows : — Extend from radius, or 90°, to 54° 30' on the line of sines ; that extent will reach from 250, the hypotenuse, to 203.5, the base on the line of numbers ; and th e extent from radius or 90°, to 35° 30' on the line of sines, will reach from 250 to 145.2 on tlie line of numbers. Observe the same method in all the following examples, except in those proportions where the word secant is mentioned, which cases must be virought by consideruig the hypotenuse radius,* there being no line of secants on the common Gunter's scale, although it can easily be marked on the line of sines. JVote. The radius, according to the nature of the proportion, may be either of the following quantities : — 8 points on the line of rhumbs. 1 90° on the Ime of smes. 4 points on the Ime of tangent rhumbs. | 45° on the line of tangents. CASES II. A>D III. Tlie angles and one leg given, to find the hypotenuse and other leg. The angle ACB 33° 15', the leg BC 163 miles, given, to find the hypotenuse and the other leg. BY PROJECTION. Draw the line BC, which make equal to 163 miles ; on B erect the perpendicular BA ; on C, as a centre, with the chord of 60°, sweep the arc BD, which make equal to 33° 15' ; draw CD, and continue 't to cut AB in A, and it is done ; for AB being measured on the same scale that BC was, will be 106.9, and AC 194.9 miles. BY LOGARITHMS. By making the base BC radius, it will be, To find the perpendicular AB. As radius 45° 10.00000 Is to the base BC 163 2.21219 So is tangent angle C 33° 15' . . 9.81666 To tlie perpendicular AB 106.9 2.02885 To find the hypotenuse AC. As radius 90° 10.00000 Is to the base EC 163 2.21219 So is secant angle C 33° 15' . . . 10.07765 To the hypotenuse AC 194.9 . . 2.28984 the triangles are measured b}- scales of equal parts, as was before observed. Instead of using the line of chords, it is much more convenient to set oft' the angles by means of a protractor, or circular arc, ou which the degrees are marked. Its construction is so simple that it needs no explanation. * Or by usuig in the analogy, radius : cosine angle, instead of secant angle : radius ; and radius : sine angle, instead of cosecant angle : radius. 40 RIGHT-ANGLED TRlGONOMETRTf . BY GUNTER. Extend from 45° to 33° 15' on the line of tangents ; that extent will reach from the base 1G3 to the pei-pendicular 106.9, on the Ime of numbers. 2dly. Extend from 56° 45' to radius on the Ime of smes ; that extent wiU reach from the base 163 to the hypotenuse 194.9, on the Ime of numbers. CASES IV. AND V. The hypotenuse and one leg given, to find the angles and other leg. Given the leg AB 91, and the hypotenuse AC 170, being to find the angle ACB BAC, and the leg BC. BY PROJECTION. Draw BC at pleasure ; on B erect the pei-pendicular BA, which make equal to 91 ; take 170 in your compasses, and, with one foot on A, describe an arc to cut BC in C ; join A and C, and it is done; for the angle C is 32° 22', the angle A 57° 38', and BC 143.6. BY LOGARITHMS. By making the hypotenuse radius, we shall have, To find the angle C. As the hypotenuse 170 2.23045 Is to radius 10.00000 So is the perpendicular 91 1.95904 To sine ande C 32° 22' 9.72859 To find the base BC* As radius 10.00000 Is to the hypotenuse 170 2.23045 So is the sme angle A 57° 38' . . 9.92667 To the base BC 143.6 2.15712 BY GUNTER. Extend from the hypotenuse 170 to the pei-pendicular 91, on the line of numbers ; that extent will reach from radius to the angle C, or thecomplementof angle Ar:32°22' on the line of sines. 2dly. Extend from radius to the angle A 57° 38', on the Ime of sines ; that extent will reach from the hypotenuse 170 to the base 143.6, on tlie line of numbers. CASE VI. The legs given, to find the angles and hypotenuse. Given the legs AB 178, and BC 141, to find the angle BAC or ACB, and the hypotenuse AC. BY PROJECTION. Make BC equal to 141, and on B ei-ect the perpendicular BA, which make equal to 178 ; join AC, and it is done ; for the angle C is 51° 37' ; consequently tlie angle A 38° 23', and the hypotenuse 227.1. BY LOGARITHMS. By makmg the base radius, we shall have, To find the angle C. As the base 141 2.14922 Is to radius 10.00000 So is the pei-pendicular 178 . . . 2.25042 To tangent amrlc C 51° 37'. . . . 10.10120 To find the hypotenuse AC.f As radius 10.00000 Is to the base 141 2.14922 So is the secant angle C 51° 37' 10.20696 To the hji)otenuse AC 227.1 . . 2.35618 BY GUNTER. The extent from 141 to 178 on the line of numbers will reach from radius, or 45 degi-ees, to the angle C 51° 37', on the line of tangents. 2dly. The extent from the angle C 51° 37' to radius, or 90°, on the line of sines, will reach from the perpendicular 178, to the hypotenuse 227.1, on the Ime of numbere. * When you take the log. sines, or tangents, to the nearest minute only, it is best to use tliis canon for finding BC, wliich is more correct than the one found by making the perpendicular radius, because the variatirin of the log. sine of an arc is less than the corresponding variation of the log. tangent. t VVIion finding AC, it is best to make the greatest side radius, for the reason mentioned in the last note; 60 that in the present example it would be rather preferable lo use the perpendicular 178 for the radi js OBLIQUE TRIGONOMETRY. 41 QUESTIONS To exercise the learner in Right-angled Plane Trigonometry. Qiiestion 1. The hypotenuse 496 miles, and the angle opposite to the base 56° 15', given, to find the base and perpendiculai*. Answer. Base 412.4, and the perpendiculai* 275.6 miles. Quest. 2. The perpendicular 275 leagues, and the angle opposite to the base 56° 15', given, to find the hypotenuse and base. Ans. The hypotenuse 495, and base 411.6 leagues. Qiiest. 3. The base 33 yards, and the angle opposite to the peipendicular 53° 26', given, to find the hypotenuse and perpendicular. Ans. Hypotenuse 55.39, and the perpendicular 44.49 yards. Quest. 4. The hypotenuse 575, and peipendiculai* 50 miles, given, to find the base Ans. Base 572.8 miles. Qiiest. 5. The hypotenuse 59, and the base 33 miles, given, to find the per- pentl'iculai'. Ans. Pei-pendicular 48.9 miles. Quest. 6. The base 33, and pei-pendicular 52 leagues, given, to find the hypatenuse Ans. Hypotenuse 61.59 leagues. OBLIQUE TRIGONOMETRY. CASE 1. Two angles and one side given, to find either of the legs. Given the angle BAG =z 100°, the angle AGB = 54°, and the leg AB =220, to find the sides. BY PROJECTION. Subti-act the sum of the angles A and C fi'om 180°; the remamder will be the angle B = 26°. Draw the indefinite line BE, also the line BH, making the angle EBH =r 26° ; oit BH set off" BA 220. On A make the angle BAG 100° ; tlien AC will intersect the line BE in the pomt C, which completes the triangle, and BC will measure (on the same scale fi'om which BA was laid down) 268 nearly, and AC 119. BY LOGARITHMS, bv Theorem II. To find BC. As the sme of the angle G 54°. . 9.90796 Is to the side AB 220 2.34242 So is the sine of the angle A 100° 9.99335 12.33577 9.90796 To the nde BC 267.8 2.42781 To find AG. As sine angle C 54° 9.90796 Is to the side AB 220 2.34242 So is the sine angle B 26° 9.64184 11.98426 9.90796 To the side AC 119.2 2.07630 BY GUNTER. The extent from the angle C =: 54° to the angle A, or its supplement 80°, on tlie smes, will reach from AB =: 220 to BC := 268, on the line of numoers. 2dly. The extent from the angle C=:54° to the angle Brr:26°, on the sines, will reach from AB = 220 to AC := 119, on the luie of numbei-s. 6 42 OBLIQUE TRIGONOMETRY. CASES II. AND III. Ttoo sides, mid an angle opposite to one of them, being given, to find the other angles, and the third side. JVote. It may be proper to observe, that if the given angle be obtuse, the angle sought will be acute ; but when the given angle is acute, and opposite to a shorter given side, then it is doubtful whether the required angle be acute or obtuse ; it ouglit therefore to be given by the conditions of the problem. EXAMPLE. Let there be given the side BC 137, the side AB 213, and the angle A 23h°, to find the otlier side AC, and the angles ABC, BCA. BY I'ROJECTION. Draw the indefinite line FE ; make the angle DAE = 23^° ; on AD set off AB =213 ; then on B, with 137 in your compasses, taken from the same scale, describe an arr cutting FE in the points C and G; join B(/, BG, and it is done ; for the triangle may be either ACB or AGB, according as the angle C or G is acute or obtuse ; if that angle be acute, the triangle will be ABC ; the side AC will measure 303, the angle ACB will measure 38J°, and the angle ABC will measure 118° nearly ; but if the angle at the base be obtuse, the triangle will be AGB ; the side AG will measure 88, the angle AGB will measure 141 §°, and the angle ABG 15°, nearly. If die side BC had been given greater than AB, there could have been only one answer to this problem; for m that case, the point G would have fallen on the continuation of the line CA towards F, in which case the angle A of the triangle would become equal to FAB, instead of being equal to its supplement, as is required by the conditions of the problem. BY LOGARITHMS, by Theorem II. To find the angle C or G, As the side BC 137 2.13672 Is to Uie sine of angle A 23.^° 9.G0070 So is the side AB 213.....". 2.32838 Subtract 61 49 From 180 Angle ABC... 118 11 11.9S?908 2.13672 or G 141 41 23 30 9.79236 or 165 11 180 ABG 14 49 To find AC. As sine angle C 38° 19' 9.79210 Is to AB 213 2.32838 So is sineangle ABC 118° 11' 9.94319 12.27357 9.79240 To the side AC 302.8 2.48117 To find AG. As sine angle G 141° 41' 9.79240 IstoAB 213 2.32838 So is sine angle ABG 14° 49' 9.40778 11.73616 9.79240 To the side AG 87.9 1.94376 BY GUNTER. 1st. The extent from BCzz:137 to AB = 213, onthe line of numbers, will reach from A =: 23.^° to 38° 19', on tlie line of sines, which is equal to tlie angle C ; its supplement, 141° 41', being equal to the angle G. 2dly. The extent from the angle C = 38° 19' to 01° 49' (the sup])lement of the angle ABC, 118° 11') on the sines, will reacli from x\Br=213 to 303, nearly, on the line of numbers ; therefore the side AC = 303. Or, the extent from 38° 19' (the supplement of the angle G) to the angle ABG:= 14° 49', on the sines, will reach from AB :=: 213 to 88, on the line of numbers ; hence \G = 88. OBLIQUE TRIGONOMETRY. 43 CASES IV. AND V. Two sides and their contained angle being given, to find either of the other angles and the third side. Given the side AB HO miles, AC 80 miles, and angle BAG 96° 0', to find the angles BCA and CBA and the side BC. BY PROJECTION. Draw the indefinite right line AD, on which set ofFAB = 110; make the angle EABr=96°; and on AE set oft' AC rr 80 ; join BC, and it is done ; fiar BC will measure on the fi^rmer scale 143, and the angles B and C will measme 33° SS' and 50° 5', respectively, yi'^ ^ 1 „^ ' °^ jj on the line of chords. To find the angles B and C, by Theorem III. As sam of sides AC and AB 190 2.27875 Is (o their ditTercnce 30 1.47712 So is tang:. A BY LOGARITHMS. To find the side BC, by Theorem II. sum opp. angles / .^o or complement of ^ angle A ) 9.93444 11.43156 2.27875 To tangent of half difference.. . 8° 5' = 9.15281 Sum is angle C 50 5 Difference is angle B 3.'5 55 As sine angle B 33° 55' 9.74662 Is to AC 80 1.90309 So is sine angle A 96° 0' ^ „ ornn or its supplement 84 J J.JJioi 1.90070 9.74662 TosideBC 142.6 2.15408 BY GUNTER. 1st. The extent firom the sum of the sides, 190, to their difference, 30, on the line of numbers, will reach from the half sum of the angles B and C, 42°, to their half difference, 8° 5', on the line of tangents. The sum of this half sum and half difference gives the angle C 50° 5', and their diffei'ence the angle B 33° 55' ; the greatest angle being ojiposite to the greatest side. 2dly. The extent from the angle B 33° 55', to the angle A 96° (or its supplement, 84°) on the line of sines, will reach from the side AC 80, to the side BC 142.6, on the line of numbers. CASE VI The three sides of a plane triangle given, to find the angles. The sides AB 85, BC 57, AC 108, given, to find the angles ABC, BAC, BCA. BY PROJECTION. Draw the line AC, and make it equal to 108 ; take 85 iu your compasses, and, with one foot or the point A, describe an arc ; then take the disiaMce 57 in your compasses, and, with one foot on C, describe another arc intersecting the former arc in the point B ; join AB, CB, and it is done ; for the angle A being measured will be found = 3U°, B — 97°, and the angle C — 5Ii°, neai-ly. ^\ q A id BY LOGARITHMS, by Theorem IV. Suppose BD to be ch-awn perpendicular to AC, and that AG : GC. Side AB = 85 Side BC = 57 Sum ot the sides 142 Difference of the sides 28 HalfbaseAC 54 DG 18.4 Sum is greatest segment AD 72.4 Difference is least segment DC 35.6 As the base AC 108 Log. 2.03342 Is to the sum of the sides AB and BC 142 Log. 2.15229 So is the difference of the sides AB and BC 28 Log. 1.44716 3.59943 2.03342 To twice DG 36.8 Log. 1.56603 Its half is DG 18.4 44 OBLIQUE TRIGONOMETRY Having divided the triangle into two right-angled triangles, the hypotenuses and bases of which ai-e given, we may find the angles by Theorem I. To find the angle BAD. As the hypotenuse AB 85 Log. Is to radius 90° Log-. So is the greatest seg. AD 72.4 . . .Log. Tb cosine BAD = 31° 36' Log. 1.92942 10.00000 1.83974 9.93032 To find the angle BCD. As the hypotenuse BC 57 Log. 1.75587 Is to radius 90° Log. 10.00000 So is the least segment DC 35.6. . .Log. 1.55145 To cosine of BCD = 51° 21' Log. 9.79558 BAD = 31 36 ' Sum 82 57 Subtract from 180 00 Remains angle ABC 97 03 BY GUNTER'S SCALE. 1st. The extent fi-om the base AC z=: 108, to the sum of the sides 142, on the line of numbers, wiU reach fi-om the difference of the sides 28, to twice DG 36.8, on the same line of numbers. 2dly. The extent from the hypotenuse AB = 85, to the gi-eater segment AD 72.4, on the fine of numbers, avUI reach, on the smes, fi-om the radius 90°, to 58° 24', which is the complement of the angle BAD. 3dly. The extent from the hypotenuse BC 57, to the least segment, DC 35.6, or the Ime of numbers, will reach on the sines from the radius 90°, to 38° 39', which is the complement of the angle BCA. This case may be solved without dividing the ti-iangle into two right-angled triangles, by Theorem V. Having the angle A, we may find the angle C by Theorem II. AsB<;67 Log. 1.75587 Is to sme angle A 31° 36' Log. 9.71932 So is AB 83 Log . 1.92942 11.64374 1.75587 To find the angle A. BC= 57 AB = 85 Arith. Comp. Log. 8.07053 AC = 108 Arith. Comp. Log. 7.96658 Sum "250 Half sum 125 Log. 2.09691 Half sum less BC 68 Log . 1.83251 Sum ) 19.96658 Half sum.... 15° 48' Cosine Log. 9.98329 f >oul led is . . 31 36 = angle A. To the sine of angle C 51° 23' ....Log. 9.89287 45 ASTRONOMY AND GEOGRAPHY. Astronomy is the science which treats of the motions and distances of the heavenly bodies, and of the appeai-aiices thence arising. Geography is the science wliich treats of the situations and distances of the various pails of the surface of tlie earth. The common opinion of astronomers of the present day is, that the universe is composed of an mfinite number of systems or worlds ; that in every system there are certain bodies moving in free space, and revolving, at diiferent distances, round a sun, placed hi or near the centre of the system ; and that these suns, and other bodies, ai'e the stars which are seen in the heavens. The Solar System, so called, is that in which our earth is placed, and in which the sun is supposed to be fixed near the centre, with several bodies, similar to our earth, revolving round at different distances. This hypothesis, which is fully confii-med by obsei-vation, is called the Copernican System, from Nicholas Copernicus, a Polish philosopher, who revived it about tlie year* 1500, after it had been buried in oblivion many ages. Stai"s are distinguished into two kinds, fixed and tvandering. Tlie fonrier are supposed to be suns m the centres of their systems, shming Avith their oavu light, and preservuig nearly the same situation with respect to each other. They are usually distinguished by thek brightness, the largest being called of the first magnitude, and the smallest visible to the naked eye being of the sixth or seventh magnitude. A Constellation is a number of stars which appear near to each other on the concave surface of the heavens, and astronomers, for the sake of remembering them with gi'eater ease, suppose them to be circumscribed by the outlmes of some anmial or other figure. Wandermg stars are tliose bodies within our systerh, or celestial sphere, which revolve round the sun ; they appear lummous by reflectmg the light of the sun, and are of three kinds, namely, primary planets, secondary planets, and comets. The Primary Planets are bodies which revolve round the sun as the centre of their com^ses, the motions beuag regularly performed m tracks or paths, called orbits, that ai"e nearly cu'cular and concentrical with each other. A Secondai-y Planet, Satellite, or Moon, is a body which, AvhUe it is carried round the sun, reitolves also round a primaiy planet. Comets are bodies which move round the sun in veiy excentrical orbits, with vast atmospheres about them, and tails derived from the same. There are seventeen primary planets, which, reckoned in order from tho sun, are as follows : — Mercury, Venus, the Earth, Mars, Vesta, Juno, Pallas, Ceres, Astrea, Hebe, Iris, Flora, Metis, Jupiter, Saturn, Uranus, and Neptune. Mercury and Venus are called inferior planets, because their orbits are within the earth's ; the others are called superior planets, as theu- orbits mclude that of the earth. The Sun, the first and greatest object of astronomical knowledge, is placed near the centre of the orbits of all the planets, and turns round its axis m 25;^ days. Its diameter is 88.3,000 English miles, and its mean distance from the earth 95 millions of miles. IMercury is the least of all the planets knoAvn before the discovery of Vesta, Juno, Pallas, and Ceres, and is the nearest to the sun, his mean distance from that luminary being 37 millions of miles. His periodic revolution in liis orbit round the sun is performed hi 87 days 23 hours, and his diameter is about 3200 miles. Vexus is the brightest of all the planets. Her diameter is 7687 miles ; her mean distance from the sun, C9 millions of miles ; and her periodic revolution is performed m 224 days 17 hours. When this planet is m that part of her orbit which is west of the sun, she rises before him in the morning, and is called the morning star ; v»'hen she is in the eastern part of her orbit, she shmes m the evening, after he sets, and is called the evening star. The next planet is the Earth, the diameter of which is 7914 miles, the distance from the sun 95 millions of miles, and the time of revolution round the sun, one year. The earth turns round its axis from west to cast in 23 hours Tij ninutes, which occasions the apparent diurnal motion of the sun and all the J- - n?c l>odies romul il 4G ASTRONOMY AND GEOGRAPHY. from east to west in the same time, and is, of course, the cause of then- rising and setting, of day and night. Tlie axis of the earth is inchned about 23° 28' to the plane of its orbit,* and keeps nearly in a dkection parallel to itself, throughout its annual course, which causes the retiu-n of spring and summer, autuimi and winter. Tlius the diumal motion gives us the grateful vicissitude of night and day, and the annual motion the regular succession of the seasons. The earth is attended by a satellite called the x^IooN, whose diameter is 2161 miles. Her distance from the centre of the earth is 240,000 miles. She goes round her orbit in 27 days 8 hours ; but, reckoning from change to change, m 29^ days. Her orbit is inclmed to the ecliptic in an angle of 5° 9', cutting it in two points diametrically opposite to each othei-, called her nodes. As the moon shmes only by the reflected light of the sim, she must appear different when in different situations with respect to that luminary. When she is m conjunction with the sun, her dark side is turned towards the earth, which renders her invisible ; this is called new moon: when she is m opposition, her light side is wholly visible from tlie earth ; this is called full moon. If at the time of new moon she is near to either of her nodes, she may intercept a part oft... iun's light, and thus cause an eclipse of the sun; and if she is near either of her nodes at the time of full moon, she may pass mto the shadow of the earth, and cause an eclipse of the moon. In a sunUar manner, when the moon passes between an observer on the earth and a star, it is called an occuUation of the star. The instant when the moon's limb fii-st covers the star is called the immersion, and the moment of its reappearance is called the emersion. When IMercury or Venus passes between the sun and an observer, and appears to pass over the sun's disk, it is called a transit of Mercury or Venus. Eclipses, occultations, and transits, ai-e of gi-eat importance in ascertahiing the longitudes of places on the earth. Eclipses of the moon furnish a convincing proof of the rotundity of the earth, since the shadow of the earth, seen upon the moon when eclipsed, is always circular. This is further confirmed by the appearance of objects at sea; for when a ship is makmg towards the land, the mariners first descry the tops of steeples, trees, &c., pomting above the water ; the lower parto being hid, by reason of the curvature of the earth. The earth is not a perfect globe or sphere, but is a little flattened at the poles, beuig nearly of the figure of an oblate spheroid, the equatorial diameter being about 2(3 miles longer than the polar ; but since this difference bears but a small comparison to the whole diameter, we may, for all the practical purposes of navigation, consider the earth as a ])erfect sphere, as will be done in the I'est of this work. The natural divisions of the earth will be given hereafter. Mars is the next planet to tlie earth. His diameter is 4189 miles. His distance from the sun is 144 millions of miles, and his periodic revolution is performed in about 687 days. He revolves round his axis in 24 hours 40 minutes, appearmg of a dusky- reddish hue, and is supposed to be encompassed with a very gi-eat atmosphere. Between Mars and Jupiter are situated eleven planets,named asteroids, viz. Vesta, Juno, Pallas, Ceres, Astreft,f Hebe, Iris, Flora, Metis, Hygeia and Parthenope. Vesta was discovered by Dr. Olbers, of Bremen, on the 29th of 3Iarch, 1807. Its mean distance from the sun is about 224 millions of mUes. Its periodic revolution is j»ei"forn>8d in 1325 days. Juno was discovered by Mr. Harding, of Lilienthal (near Bremen), on the first of September, 1804. It appeai-s like a star of the eighth magnitude. Its distance from the sun is about 254 millions of miles. Its periodic revolution is performed in 1.593 days. The incluiation of its orbit to the ecliptic is 13° 4', and the excentricity of the orbit t 0.25. Pallas was also discovered by Dr. Olbers, Mai'ch 28, 1802. Its diameter, according to Dr. Hcrschel, is only 110 miles. It appears like a star of the eighth magnitude. Its mean distance Irom the sun is about 263 millions of miles. Its periodic revolution is performed in 1686 days. The incluiation of its orbit to the ecliptic is 34° 35', and the excentricity of the orbit 0.242. Ceres was discovered by Mr. Piazzi, of Palerino, on the first of Januaiy, 1801. Its diameter, according to Dr. Herschel, is only 160 miles. It appears like a star of the seventh or eighth magnitude. Its distance from the sun is about 263 millions of miles, and its periodic revolution is performed in 1685 daj-s, being at nearly the same distance from the sini as Pallas. The inclination of the orbit of Ceres to the ecliptic is 10° 37', * The inclination decreases at present about 50" in lOOyenrs, by reason of the attraction of the planet* im the earth. It is also affected by the nutation given in Table XLIII., which sometimes amounts to 9" t Aslrea was discovered by Mr. Hencke, of Dresden, Dec. 8, 1845. Hebe do. do. do. do. July 4, 1847. Iris do. do. Mr. Hind, London, Aug. 13, 1847. Flora do. do. do. do. Oct. 18, 1847. Metis do. do. Mr. Graham, Sligo, May, 1848. Hygcia do. do. M. Gasparis, Naples, April, 1849. Parthenope do. do. do. do. May 11, 1850. t In estimating the excenlricities of the planets, their mean distance from the sun is put equal to unltj. ASTRONOMY AND GEOGRAPHY. 47 and the excentricity 0.077. The situations of the nodes of the two planets, Ceres and Pallas, and the incluiations of then- orbits, are very different from each other, so that when those planets are in the same plane, they are at a gi-eat distance from each other, notwithstanding their mean distances from the sun are nearly equal. It has been supposed by some, that these small bodies are fragments of a fonner planet. Jupiter is situated still higher in the system, and is the largest of all the planets, bemg easily distmguished from them by his peculiar mamitudc and light. His diameter is 89,170 miles ; his distance from the sun 494 millions of miles ; and the time of his periodic revolution is 4332J days. Though Jupiter is the largest of all the planets, yet his diurnal revolution is die swiftest, being only 9 hours and 5G minutes. Jupiter is attended by four satellites, invisible to the naked eye; but through a telescope they make a beautiful appearance. In speaking of them, we distinguish them according to their places, into the first, second, and so on ; by the first we mean that which is nearest to the planet. The appearance of these satellites is marked in the XlXtli page of the Nautical Almanac for some particular hour of the night ; tlie times wlien they are ecUpsed, by passhig into the shadow of Jujiiter, are also given in the Nautical Almanac ; these eclipses are of some use in determining the longitudes of i)laces on the earth. Before the discovery of the planet Uranus, Saturn was reckoned the most remote planet of our system. He shines with but a pale and feeble light. His diameter is 79,042 miles; his distance from the sun 907 millions of miles; and his j)enodic revolution in his orbit is performed in about 29 years 1G7 days. This i)lanct is surrounded with a broad, fiat ring, has a diurnal revolution round its axis, and is attended by seven satellites. By some observations made by Dr. Herschel, it appeared that the largest diameter of Saturn corresponds to die latitude of 45° ; but from later obsei-vations he has been induced to believe, that this uTegularity is owing to an optical deception, arising from the refraction of the light in passing through the atmosphere of the ring. Uranus, Herschel, or Georgium Sidus, was discovered in the year 1781, by Dr. Her- schel, though it had been seen several times, but had been considered as a fixed star. Its diameter is 35,109 miles; its distance from the sun is 1823 millions of miles; and its periodic revolution in its orbit is performed in 83 ^ years. Dr. Herschel has also dis- covered six satellites attending this planet. Neptune, the most remote planet of our system, was seen by Dr. Galle, of Berlin, Oct. 23, 1846. Its mean distance from the sun is 2867 millions of miles — its diameter is 34,750 miles, and its period of revolution is 165| years. Mr. Lascelles has discovered one satellite. The astronomy of comets is yet in its infancy. The return of one of them in the year 1758 was foretold by Dr. Ilalley, and it happened as he predicted ; and it apjieared again in 18.35. He also foretold tlie return of another in the year 1790, but it never appeared. This was owing to the inaccuracy of the obsei-vations of the comet at its former appearance ; for Mr. Mechain, having collected all the observations, and calculated the orbit again, found it to differ essentially from that determined by Dr. Halley. Olber's comet, which appeared in 1815, has a revolution of 72 years; and Encke's comet, which lias been observed in several successive approaches to the pprihehon, com})letes its revolution in the short period of 1204 days. Biela's comet has also been observed several times, with a periodical revolution of about G^ years. Comets move romid the sun in all directions ; but the planets and satellites, except one of the satellites of Uranus, move from west to east when seen from the sun ; but if viewed from any other of the planets, as the earth, they would appear to revolve round it as a centre ; but the sun would be the only one that moves uniformly the same ^vay, for the other jilaiiets would sometimes appear to move from west to east, and then to stand still ; tiien they would seem to move from east to west ; and, after -Standing some time, diey would again move from west to east ; and so on, continually. The motion of a jjlanet from west to east is called the direct motion, or according to the order of the signs. The contraiy motion, from east to west, is called retrograde. When the planet appears to stand still, it is said to be stationary. To illustrate what has already been said relative to the motions and distances of the planets and satellites, we have given die adjoining Plates III. and IV., which require no exjilaiiation. In noting die situations of the stars and planets, astronomers have been under the necessity of imagining various lines and circles on the sphere ; and geographers have done the same for fixing the situation of [)laces on the earth. The most remarkable of these are the following: — A great circle is diat whose plane passes through the centre of the sphere ; and a mnall circle is that whose plane does not pass through that centre. A diameter of a sphere, perpendicular to any great cuxle, is called the axis of that circle; and the extremities of a diameter are called its poles. Hence the pole of a great circle is 90° from every point of it u[)on the surface of the spliere ; but as tlio 48 ASTRONOMY AND GEOGRAPHY. axis is perpendicular to the cu'cle when it is perpendicular to any two radii, a point on the surface of a sphere 90° distant from any two points of a great cuxle, will be the pole. All angular distances on the surface of a sphere, to an eye at the centre, are measured by arcs of great cu-cles. Hence all triangles formed upon the surface of a sphere, for the solution of spherical problems, must be formed by the arcs of great cuxles. Secondaries to a gi'eat cu'cle are great cu-cles which pass through its poles, and consequently must be pei-pendicular to theu* gi-eat cu'cles. The axis of the earth is that diameter about which it pei-forms its diurnal motion and the extremities of this diameter are called the poles. The terrestrial equator is a great cuxle of the earth perpendicular to its axis. Hence the axis and poles of the earth are the axis and poles of its equator. That half of the earth which lies on the side of the equator in which Europe and the United States of America are situated, is called the northern hemisphere, and the other the southern ; and the poles are respectively called the north and south poles. The latitude of a place upon the earth's siu-face is its angular distance from the equator, measured upon a secondaiy to it. These secondaries to the equator are called vieridians. The longitude of a place on the earth's surface is an arc of the equator intercepted between the meridian passing through the place, and another, called ihefirsi meridian, passing through that place from which you begin tomeasui-e ; or it is the angle formed at the pole by these two meridians. The Americans and English generally place the fii'st meridian at Greenwich ; the Fi-ench place it at Paris, the Spaniards at Cadiz ; some geographers place it at Teneriffe, and others at other places. Thi'oughout this work, Greenwich wiU be reckoned as the first meridian. The longitude is counted from the first meridian, both eastward and westward, till it meets at the same meridian on the opposite point; therefore the longitude (and also the difference of longitude between any two places) can never exceed 180°. If the plane of the terrestrial equator be pi'oduced to the sphere of the fixed stars, it tnai'ks out a cu'cle called the celestial equator ; and if the axis of the earth be produced in like manner, the pomts of the heavens, to which it is produced, are called poles, being the poles of the celestial equator. The star neai-est to each pole is called the pole star. Secondaries to the celestial equator are called circles of declination ; of these 24, which divide the equator into equal parts, each containing 15°, are called hour circles. Small ckcles pai-allel to the celestial equator are called parallels of decUnation. The sensible horizon is that cucle m the heavens whose plane touches the earth at the spectator. The rational horizon is a great circle in the heavens, passing through the earth's centre, parallel to the sensible horizon. If the I'adius dra^vn from the centre of the earth to the place where the spectator stands be produced both ways to the heavens, the point vertical to him is called the zenith, and the jx)mt opposite, the nadir. Hence the zenith and nadir are the poles of the rational horizon. Secondaries to the horizon are called vertical circles, because they are pei-pendicular to the horizon. On these cu-cles, therefore, the altitude of a heavenly body is measured. The secondai-y common to the celestial equator, and the horizon of any place, is the celestial meridian of that place. This meridian corresponds with the terrestrial meridian of the same place, which passes through the poles of tiie earth, the zenith and nadir crossing the equator at right angles, and cutting the horizon in the north and south pomts ; that poiiit being called north which passes through the north pol(>, and the opposite direction is called south. The vertical circle which cuts the meridian of any place at right angles is called the prime vertical ; the points where it cuts the horizon are called the east and ivest points, and to an observer, v/ith his face directed towards the south, the east point will be to his left hand, and the tvest to his right hand. Hence the east and west points are 90° distant from the north and south. These four are called the cardinal points. The meridian of any place divides the heavens into two hemispheres, lying to the east and west ; that lyuig to the east is called the eastern hemisphere, and the other the ivestern hemisphere. When the sun is at its greatest altitude on the meridian of any place, it is noon, or mid-daj'. The azimuth of a heavenly "body is its distance on the horizon, when refen-ed to it by a secondary, from the north or south i)oints. The amplitude is its distance from the east or west points, at the time of rising or setting. The ecliptic is that great circle in the heavens which the sun appears to describe in the course of a year. The cchptic and equator, being gi-eat cucles^ must bisect each other, and tlieir angle of inclination is called the oUiquity of the ecliptic ; and the points 'vhcre they uiterscct are called the equinoctial poiiits. The times when the sun comes Thi/^m THE g^OJLAK ^T^^TBM. e .?..■•■■■< .•■''' « /..? « 5' '6 / / / 1 \ \ \ » \ 3 \ \ ■•■-., ■■ 9 /•rfr* o/^y/'f ?»?» Aipiarius r.-, PiscfS X- The order of these is according to the a])parcnt motion ot'the sun. The first point of Arie* coincides with one of the equinoctial points, and the tirst point of Libra with the otlur. The first sly signs are called noiihern, lying on the north side of the equator ; and the last six are called southern lying on the south side. The zodiac is a space extending eight degrees on each side the ecliptic, within which the motion of all the planets is contained, except the newly-discovered planets. The right ascension of a body is an arc of the equator intercepted between the first point of Aries, and a circle of declination passing through the body, measured according to the order of the signs. Right ascension of the meridian, or mid-heaven, is the distance of the meridian from the fij-.st point of Aries, and is found by addmg the apparent time past noon to the sun's right ascension. The ascensional difference of any oljject is the difference betN-v-wen the right ascension of the object and that point of the equator which rises or sets with it. The declination of a star or any celestial object is its angular distance from the equator, measured upon a secondary to it passing through the object. The longitude of a star or any celestial olyect is an arc of the ecliptic intercepted Ijetwccn the first point of Aries, and a secondary to the ecliptic passing through the body, measured according to the order of the ^gns. If the observer be on the earth, the longitude is called the geocentric longitude ; but if seen from the sun, it is called the heliocentric longitude ; the body in each case being referred perpendicularly to the ecliptic in a plane passing through the eye. jYoimgesimal degree of the ecliptic is its highest point at any given tune, and is 90° from the |)oints where the ecliptic iut^-sects the horizon. The latitude of a star or any celestial object is its angular distance from the ecliptic, measured upon a secondary to it drawn through the body. If the body be observed frojn the earth, its angular distance from the ecliptic is called tlie geocentric latitude ; but if ol)served from the sun, it is called the heliocentric latitude. The secondary cu-cle drawn peri)endicular to the ecliptic is called a circle of latitude. The tropics are two parallels of declination touching the ecliptic. One, toucliing it at the beginning of Cancer, is called the tropic of Cancer ; and the other touching it at the beginning of Capricorn, is called the tropic of Capricorn. The two points where the tropics touch the ecliptic are called the solstitial points. Colures are two secondaries to the celestial equator, one passing through the equinoctial ])oints, called the cquinocticd colure ; and the other passing through the solstitial points, called the solstitial colure. The times when the sun comes to the solstitial j)oints are called the solstices. Aberration of a star, or any heavenly body, is a small aj)parent motion, occasioned bv the progressive velocity of light. This is calculated by^neans of Tables XXXIX., XLL, or XLII. JVutation is a small apparent motion of the heavenly bodies, occasioned by a red motion of the earth's axis, arising from the attractions of the sun and moon on the spheroidal form of the earth. The effect of this on the right ascension and declination is given in Ta!)le XLIIL, and on the longitude in Table "XL. ; the correction in this last tal)le being generally called the equation of the equinoxes in longitude. Precession of the equinoctial points is a small motion of about 50i" per year, occasioned by the same cause as the nutation. By this motion the equinoctial j)oi|^ are carried backward from east to west ; consequently, tiie heavenly l)odies appear to move forward the same quantity front west to east. The annual variations ol" the places of the stars from precession, and the secular equations arising from the change of tiie earth's orbit by the attraction of the planets, are given in Tables VIII. and XXXVII. The arctic and antarctic circles are two jiarallels of declination, the former about the north, and the latter about the south pole, the distance of which, from the two [wies, is equal to the distance of the tropics from the equator, which is about 2;F 28'.^ These arc also called polar circles. Tlie two tropics and two polar circles, when referred to the earth, divide it into five parts, called zones; the two i)arts within t!ie polar circles are called the /ri^i^ zones; the two parts between the polar circles and tropics are called the temperate zones; and the part between the tropics is called the torrid zone. iiesides the imagi)iary divisions of the earth, there are various natural divisions of its surface, such as continents, oceans, islands, seas, rivers, «S:-c. A continent is a large tract of land, wherein are several p.nii)ires, kingdoms, and countries conjoined ; as Europe, Asia, Africa, and America. 1 50 ASTRONOMY AND GEOGRAPHY. An island is a jtart of tlie earth tliut is envu'oned or euconipasscd roimd l)y tlie sea as Long Island, Block Island, &c. A peninsula is a portion of land almost suiTounded with water, save one narrow neck Avhich joins it to the continent; as the Morea. An isthmus is a narrow neck of land joining a peninsida to tlie adjacent land, by whicli the people may pass from one to the other ; as the isthmus of Darien. A promontory is a high part of land stretching itself into the sea, the extremity of which is called a cape or htadland. A mountain is a rising part of dry land, overtopping the adjacent coimtry. An ocean is a vast collection of water, separating continents from one another, and washing theii* borders or shores ; as the Atlantic and Pacific Oceans. A sea is part of the ocean, to whicli we must sail through some strait , as the Mediterranean and Baltic Seas. This term is sometimes used for the whole body of salt water on the globe. A strait is a narrow jiart of the ocean lying between two shores, and opening a way into some sea ; as the Straits of Gibraltar, that lead into the Mediterranean Sea. A creek is a small narrow part of the sea or river, that goes up but a little way mto the land. A bay is a gi-eat uilet of the land ; as the Bay of Biscay, and the Bay of Mexico ; otherwise a bay is a station or road for ships to anchor in. A river is a considerable stream of water issuing out of one or various s}irings, and continually gliding along in one or more channels, till it discharges itself uito the ocean : the smaller streams are called rivulets. A lake is a large collection of waters in an inland place ; as the Lakes Superior and Huron in America. A g""//is a part of the ocean or sea, neai-ly suj^-ounded by the land, except where it communicates with the«ea; as the Gulf of Venice. Thus we have given the most useful definitions of Astronomy and Geogi-aphy, and to assist the learner there is also given Plate V., in which those terms are exj)lained at one view. We may further observe, that, as the latitude of any place upon the earth is counted from the equator upon an arc of the meridian, the difference of latitude between two places, both north or both south, is found by subtracting the less latitude from the greater; hut if one latitude he north, and the other south, the difference is found oy adding both latitudes together. 1. Consequently, if a ship in north latitude sails northerly, or in south latitude southerly,she increases her latitude ; but in north latitude sailing southerly, or in south latitude sailing northerly, she decreases her latitude, because she sails nearer to the equator, from whence the latitude is reckoned. 2. Wherefore, in north latitude sailing northerly, or in south latitude sailing southerly, the difference of latitude, added to the latitude left, gives the latitude in. 3. In north latitude sailing southerly, or in south latitude sailing northerly, the difference of latitude, subtracted from tne latitude left, gives the latitude in. 4. JFhcn the latitude decreases, and the difference of latitude is greater than the latitude sailed from, subtract the latitude left from the difference of latitude, and the remainder ivill he the latitude in, but of a different iiame, for it is evident, in this case, that the skip has crossed the equator. 5. The difference of longitude between two i)laces, being both east or west, is found by subtracting the less longitude from the grecdcr ; but if one be in east longitude and the oti£r in west, their sum is the difference of longitude, tvhen it does not exceed 180^, bid if iPcxceeds 180°, that sum must be subtracted from 360°, and the remainder ivill be the difference of longitude. 0. Therefore" in east longitude sailing easterly, or in west longitude sailhig westerly, the difference of longitude, added to the longitude left, gives the longitude in, when that sum does not exceed 180°; but if it exceeds 180°, the simi, subtracted * from 8fi0°, leaves the longitude in, but of a different name from that-lcfV. 7. In east longitude sailing westerly, or in west longitude sailing easterly, the difference of longitude, subtracted from the longitude left, gives the longitude in ; bid when tlie difference of longitude is greatest, the longitude left must he subtracted from that difference, and the rcviavnder will he the longitude in, but of a different name from the longitude left. * In this Rile it is supposed, that the sum of the longitude left, and tlie difference of loniptude, is less than 360-^, whicli is always the case when the ditVereiice of longitude is less than 180°, which we have gr«nerally supposed to be the case in these rules. Flate IT. .,^t^' ^^fl7^^^^^^^ „.A„4 With tht Times of their I'mcMc RKokitam. «>'■ c ■ ifi^sr as. 3?i^ 13.42. r.V-* SAT ntPT !Liia ,< Teket:crs, will reach from radius or 90° to 33° 52', the complement of the course on the line of sines. And the extent from radius, to 56° 08' on the line of sines, will reach from the distance 244, to the depaiture 202.6, on the line of numbers. BY INSPECTION. Seek in the tables till against the distance, taken ui its column, is found the given difference of latitude in one of the following colunuis; adjouiing to it will stand the i'LANb: SAILING. 57 departure ; wliich if less than the difference of latitmle, tlie course is to be found at the top ;* bdt if greater, the conrse is to be found at tiie bottom. Thus the distance 244, and tlie difference of latitude I'.iG, are found to correspond to a course of 5 points, or S. E. by E., and to the departure 203.9, nearly. CASE V. Distance and departure given, to Jind the course and difference of latitude. Suppose a ship sails 244 miles between the north and west, from the latitude of 32° 25' north, until her departure is 203 miles; what eourse has siie steered, and what latitude is siie in .'' BY PROJECTION. Draw the line AB equal to the departure 203; and, perpondioular thereto, the line BC, to represent die meridian ; then take the distance 244 in your compasses, and, fixing one foot in A, as a centre, describe an arc, cutting BC in C ; join AC, and it is done ; for the angle ACB will be the course, and BC the difference of latitude DY LOGARITHMS. I)C77arfiire 203 To find the course. As the distance 244 2.38739 is to raflius 10.00000 So is the departure 203 2.30750 To tlie sine of course 56° 18' . . . 9.9201 1 To find the difference of latitude. As radius 10.00000 Is to the distance 244 T 2.-38739 So is cosme course 5G° 18' .... 9.74417 To the difference of lat. 135.4. . 2.13156 Hence the course is N. 50° 18' W., or N. W. by W. nearly. To tlie latitude sailed from 32° 25' add the difference of latitude 135 or 2° 15' ; the sum 34° 40' is the latitude the ship is in. BY GUNTER. Extend from the distance 244, to the departiu-e 203, on the line of numbers ; that extent v/ill reach troni radius to tlie course 5G° 18' on the line of sines. 2dly. Extend from radius to the complement of the course 33° 42', on the line of sines ; that extent will reach from the distiuice 244, to the difterence of latitude 135.4, on the luie of numbers. BY INSPECTION. Seek in the tables till against the distance taken in its column is found the given departure in one of the following columns; adjoining to it will stand the difference of latitude ; and if it be greater than the deiiarture, tlie course is to be found at the top ; but if less, the course is to be found at the bottom. Thus the distance 244, and the departure 203, agree to a course of 5 points, or N. W. by W., and a difference of latitude 135.G miles, nearly. CASE VI. Difference of latitude and departure given, to find the course and distance. A ship sails between the north and west till her difference of latitude is 136 miles, and her dej>aiture is 203 miles ; requu'ed her course and distance. BY PROJECTION. Draw AB=:136, and perpendicular to it BCz=203; join C and A ; then will the angle CAB be the course 56° 11', and AC the distance 244.4 miles. Dcpartu-re -zos B VI "^ ^^ <-> """"^^ ^^ N. c» * It may also be known whether the course he marked at the top or bottom of the table, by observing whether the difference of latitude and dcparinre correspond wiih the marks ai tlie top or bottom. Thus I le distance 2J4, and difference ol latitude 13G, correspond to the course 5 points, because the column iu /'hich 136 is found, is marked latitude at the bottom ; the same may be observed in the foUowiDg case*. B 5S PLANE SAILING. BY LOGARITHMS. To find the distance. As radius 1000000 Is to the difference of lat. 130. . 2.1.3354 So is secant of course 56° U' . . 10.25451 To the distance 244.4 2.38805 To find the couree. As the difference of latitude 136 2.13354 Is to radius 10.00000 So is the departure 203 2.30750 To tangent of course 50° 11'. . . 10.17396 Hence her course is N. 50° 11' \V., or N. W. by W., and the distance sailed is 244.4 miles. BY GUNTER. Extend from the difference of latitude 130, to the departure 203, on the line of numbers; that extent will reach from radius to 50° 11', the course on the line of tangents. 2(lly. For the distance we must consider it as nidius (unless there is a line of secants on the scale), and extend from the course 56° 11', to the radius, or 90°, on the line of sines ; that extent will reach from the depai'ture 203, to the distance 244.4, on the luie of numbers. BY INSPECTION. Seek m tlie tables till the given dilfei-ence of latitude and departure are found together m their respective columns ; then against them will be the distance in its column, and tlie course will be found at the top of that table if the departure be less than the difference of latitude, otherwise at the bottom. Thus with tlie difference of latitude 136, and the departure 203, enter the tables, and these luunbers will be found to coiTcspond nearly to 5 points, or N. W. liy W. course, and a distance equal to 244 miles. QUESTIONS To exercise the learner in tlie foregoing rules. Question I. A ship in 2° 10' south latitude, sails N. by E. 89 leagues ; what latitude is she ill, and what is her departure ? Answer. Latitude in 2° 12' N., and departure 17.30 leagues. Quest. II. A sliij) sails S. S. W. from a jjort in 41° 30' north latitude, and then, by observation, is in 30° 57' north latitude; required the distance run, and dejjarture. Ans. Distance run 98.5 leagues, departure 37.7 leagues. Quest. III. A ship sails S. S. W. h W. from a port in 2° 30' south latitude, until hei departure be 59 leagues ; requhed the distance run, and latitude in. Jlns. Distance run 125.2 leagues, latitude in 8° 1' south. Quest. IV. If a ship sails .300 miles south-westward from 21° 59' south latitude, until by observation she be in 24° 49' south latitude, what is her course and de])arture ? Ans. The course is S. W. by W. k W., or S. 01° 49' W., and her depaiture from the meridian is 317.3 miles. Quest. V. Suppose a ship sails 354 miles noith-eastward from 2° 9' south latitude, until her departiu-e be 150 miles, what is her course and latitude in ? Ans. Her course is N. 25° 4' E., or N. N. E. i E. nearly, and she is in lat. 3° 12' N. Quest. VI. Sailing between the north and the west, from a port in 1° 59' south latitude, and then arriving at another port in 4° 8' north latitude, which is 209 miles to the westward of the first port, requu-ed the course and distance from the fu-st port to the second. Ans. The course is N. 29° 40' W., or N. N. W. | W. nearly, and the distance of the ports is 422.4 miles, or 140.8 leagues. Quest. VII. Four days ago we were in latitude 3° 25' S., and have since that tune sailed in a direct course N. W. by N. at the rate of 8 miles an hour; reauh-ed our present latitude and departure. Ans. Latitude in 7° 14' N., departure 420.7 miles. Quest. VIII. A ship in the latitude of 3° 52' south, is bound to a \)on bearing N. W. by W. h W. in the latitude of 4° 30' north ; how far does that port lie to tlie westward, and what is the ship's distance from it? Ans. The port lies 939.2 miles to the westward, and the db-ect distance is 1065 miles. Quest. IX. A sliip from the latitude of 48° 17' N., sails S. W. by S. until she has depressed the north pole 2 degrees; what direct distance has she sailed, and how many miles has slic sailed to the westward ? Ans. Diiitance run 144.3 miles, and has sailed to the westward 80.2 miles. Ki TRAVERSE SAILING. A TRAVERSE is ail irregular track which a sliip makes l)y sailing on several different courses ; these are reduced to a single course by means of two or more cases of Plane Sailing, either by geometrical construction, or by arithmetical calculation.* The geometrical construction is performed as follows : — Describe a circle with the chord of G0°, to represent the compass, and lay off on its circumference the various courses sailed. From the centre, upon the first course, set off the first distance, and mark its extremity ; through this extremity, and j)arallel to the second course, di-aw the second distance of its proper length ; througli the extremity of the second distance, and j)arailel to the third course, draw tlie third distance of its proper length ; and thus proceed till all the distances are drawn. A line, drawn from the extremity of the last distance to the centre of the circle, will rej)resent the distance made good ; a Ime, drawn from the same point, perpendicular to the meridian, will represent the departure , and the part of the meridian intercepted between this and the centre, will represent the difference of latitude. The arithmetical calculation to work a traverse is as follows: — Make a traverse table consisting of six columns ; title' them. Course, Distance, N., S., E., W. ; begin at the left side, and write the given coiu'ses and distances in their respective colunuis. Find the difference of latitude and departure for each of these courses, by Gimter's scale, or by Tables I. or 11. (as in Case I. Plane Sailing), and ^vl•ite them in their proper columns ; that is, when the course is southerly, the difi'erence of latitude must be set in the column S. ; when northerly, in the column N. : the departure, when westerly, m the column W. ; and when easterly, in the column E. Add up the columns of northhig southing, easting, and westing; take the difference between the northing and southing, and also between the easting and westing ; the former difference will be the difTerence of latitude, which will be of the same name as the greater ; and the latter will be the departure, which will be also of the same name as the gi-eater. With this diffei'ence of latitude and departure the course and distance made good are to be found as in Case VI. Plane Sailing. EXAMPLE I. Su]ipose a ship takes her departure from Block Island, in the latitude of 41° 10' K, the middle of if bearing N. N. W., distance by estimation 5 leagues, and sails S. E. 34, VV. by S. 1(), W. N. W. 39, and S. by E. 40 miles; required the latitude she is in, and her bearing and distance from Block Island. BY PROJECTION. Let L represent the nfiddle of Block Island ; draw the meridian LM, and on L, as a centre, with a chord of GO^, describe a circle to represent the com])ass, on whicl) mark the various courses sailed, and the bearing of the land at the tune of taking the departure; 0])po- site to this bearing draw the S. S. E. line LA, which make equal to 15 miles, the estimated «listance of the land ; then will A represent the place of the ship at the time of taking the departure : through A draw AB equal 34 miles, ])arallel to the S. E. line ; then will B be the place of die ship after sailing her first course: in like manner draw BC equal to 16 miles, jiarallel to the AV. by S. line ; CD equal to 39 miles, parallel to * This mclhod of reducing compound courses to a single one is perfectly accurate in sailing on a plane, and is nearly so in sailing a sliori distance on the splicrica! surface of the earth ; and though in this case it is liable to a small error in high latitudes, yet in general the rule is sufficiently accurate for educnig the several courses -and distances sailed in one day to a single course and distance. (JO TRAVERSE SAILING. the W. N. W. liiie, and DE equal to 40 miles, parallel to the S. by E. line ; then wiL E represent the place of the ship after sailing her several courses. Join EL, and draw EM perj)endicular to LM ; then will LE be the distance of Block Island, (30.8 miles; and the angle ELlMm 12° 16', will be the course made good ; LM the difference of latitude, and EM the departure. TO FliND THE SAME BY LOGARITHMS. For the first course S. S. E. 15 miles. To find the difference of latitude. As radius 90° 10.00000 Is to cosine course 2 pouits .... 9.9()5(32 So is distance 15 1.17G09 To difference of latitude 13.9 . . 1.14171 For depailiu-e. As radius 90° '. 10.00000 Is to shie course 2 points 9.58284 So is distance 15 1.17609 To departure 5.7 0.75893 Second course S. E. 34 miles. For difference of latitude. As radius 90° 10.00000 Is to cosine course 45° 9.84949 So is distance 34 1.53148 To difference of latitude 24 ... . 1.38097 For departure. As radius 90° 10.00000 Is to sine course 45° 9.84949 So is distance 34 1.53148 Third course W For difference of latitude. As radius 90° 10.00000 Is to cosine course 78° 45' 9.29024 So is distance 16 1.20412 To difference of latitude 3.1 .. . 0.49436 Fomth course W For diffei'ence of latitude. As radius 90° 10.00000 Is to cosine course 67° 30' 9.58284 So is distance 39 1.59106 To difference of latitude 14.9 . . 1.17390 To departure 24 1.3S097 by S. 16 miles. For departure. As radius 90° 10.00000 Is to sine course 78° 45' 9.99157 So is distance 16 _L20412 To departure 15.7 1.19569 N. W. 39 miles. For depaiture. As radius 90° 10.00000 Is to sine course 67° 30' 9.9()5G2 So is distance 39 1.59106 To departure 36 1.55668 Fifth course S. by E. 40 miles. For difference of latitude. As radius 90° 10.00000 Is to cosine course 11° 15' 9.99157 So is distance 40 1.60206 To difference of latitude 39.2 . . 1.59363 For departure. As radius 90° 10.00000 Is to sine course 11° 15'. . « 9.29024 So is distance 40 1.60206 To departure 7.8 . 0.89230 Though this uiethod of finding the difference of latitude and departure by logarithms is accurate, yet the calculations may be more easily made by the tables of diflerence of latitude and departure, as in Case I. Plane Sailing. Place all these courees, distances, &c., in the traverse table ; then add up all tlie westings, eastings, northings, and soutlungs, separately, and set dov^Ti their respective sums at the bottom of each column; and as tlie westing is gi-eater than tlie easting, subtract the casting therefrom ; the difference, 14.2, shows that the ship's departm-e is so much west of her first meridian. Again, the southing being greater than the northing, subtract the north- ing from it, and the remainder, 6.5.3, shows how far the ship is to the southward of her fii'st place. TRAVERSE TABLE. Courses. Dist. Diff. of Lat. Departure. N. S. E. W. S. S^E. S. E. W. by S. W. N. W. S. by E 15 34 IG 31) 40 14.9 13.9 24.0 3.1 39.2 5.7 24.0 7.8 15.7 36.0 From sum Rpmaiuder take . . . 14.9 80.2 14.9 37.5 51.7 37.5 C5.3 14.2 TRAVERSE SAlLliNG. t)l To fiud the direct coui-se or bearing of Block Island from the ship. As the difference of latitude 65.3 1.81491 Is to radius 45^ 10.00000 So is the departure 14.2 1.15221) To tangent course 12° IC D.33738 Which, because the difference of latitude is soutlierlv, and tlie departure westerly, is S. 12° 16' W. Whence Block Island beai-s from the ship N. 12° W E-, or N. by E. 1° 1' E. To find the distance of the islai^l. As sine of course 12° IG" 9.32728 Is to the departure 14.2 1.15229 So is radius 90° 10.00000 To die distance 66.8 1.82501 BY INSPECTION. Find the course and distance by Case VI of Plane Sailinff. EXAMPLE II. A ship from Mount-Desert rock, in the latitude of 43° 50' N., sails for Cape Cod, in the latitude of 42' 3' N., its departure from the meridian of Mount-Desert rock being supposed to be 84 miles west ; but by reason of contrary winds, she is obliged to sail on the following courses, viz. south 10 miles, W. S. W. 25 miles, S. VV. 30 miles, and W. 20 miles. Reipiired the bearing and distance of the two places, the course aiid distance sailed by the ship, and the bearing and distance of her intended port. BY PROJECTION. Latitude of Mount-Desert rock 43° 50' N. Latitude of Cape Cod 42 3 N. Difference of latitude 1 47 =r 107 jiiiles. Let C represent Mount-Desert rock ; draw the meridian CF, which make ernial to 107 miles, the difference of latitude between the two places, and peri>endicular thereto the line FE, equal to the departure, 84 miles ; then is E the place of Cape Cod. WitK the chord of 60° sweep about the centre, C, a circle, S. W., to represent the compass, and upon it note the various courses sailed. The first course being south, the distance 10 miles, is set off from C towards F upon the meridian, and this point represents th« place of the ship after sailing her first course; continue setting off the various course; and distances as in the last example, viz. W. S. W. 25 miles, S. W. 30 miles, and west 20 milps. to tlie point A ; then will A represent the jilaco of the ship after m TRAVERSE SAILING. sailing these courses. Join CE, AC, AE ; draw AB perpendicular to the meridian CF, and AD parallel thereto; then will AC = 76.2 miles be the distance made good; AEr=G9.1 miles, the distance of Cape Cod from the ship; CE the distance of the two places =1 13G miles ; ACB = 57° 36', the course made good ; EAD = 16° 34', the course to Cape Cod ; and ECF the course from Mount-Desert rock to Cape Codr:38° 8', &c BY LOGARITHMS. To find the bearuig and distance of the two places by Case VI. Plane Sailmg. To find the bearing. As difference of latitude 107. . . 2.02938 Is to radius 45° 10.00000 So is departure 84 1.92428 To tangent course 38^ 8' 9.89490 To find the distance. As radius 90° 10.00000 Is to difterence of latitude 107. 2.029:38 So is secant course 38° 8' 10.10426 To the distance 136 2.13364 Whence the course from Mount-Desert rock to Cape Cod is S. 38° 8' W., distance 136 i.'.iles. The same may be found by the scale, or by inspection. Tiie difference of latitude and TRAVERSE TABLE, departure for the several courses being calculated, by Case I. Phme Sailing, and arranged in the traverse table, it appears that the difference of latitude made good by the ship is 40.8 miles, and the departure 64.3 miles ; then, by Case VI. Plane Sail- ing, these numbers are found to cor- respond to a course of S. 57° 30' W. and distance 76.2 miles. Subtract the difference of latitude made good by the ship, 40.8 miles, from the whole difference of latitude, 107 miles, and there remain 66.2 miles, which is the difference jf latitude between the ship and Cape Cod. In the same manner, by subtracting the ship's departure, 64.3 miles, from the whole departure, 84 miles, tliere remain 19.7nule-! for the dej)arture between the ship and Cape Cod. With this diftljrence of latitude 66.2, and departure, 19.7, the bearing of Cape Cod is found, by Case VI. Plane Sailing S. 16° 34' W., and its distance, 69.1 miles. All the pr(!ceding calculations may be made by logarithms, by the scale, or by inspection. But v/e shall leave them to exercise the learner, anil for the same pm-pose shall add the foUowmg example. Courses. Dist. Diff. of Lat. Dcjjarture. N. S. E. W. South. w. s. w. s. w. w. 10 25 30 20 10.0 9.C 21.2 23.1 21.2 20.0 DifF. of lat, 40.8 Depart. G4.3 EXAMPLE III. A ship in the latitude of 37° 10' N., is bound to a i)or which lies 180 miles west of the meridian of the ship; but she sails the following courses, viz. S. W. l)y W. 27 mil W. by S. 25 miles, VV. by N. 18 miles, S. S. E. 32 m S. by E. 25 miles, S. 31 miles, and S. S. E. 39 miles. Re is in, and her departm-c from the meridian, with the course and distance to her intended port. The difference of latitude and departure made on each course, are given in the adjoined traverse table; hence it ai)pears that the difierence of latitLide made good is 169.4 miles ; the dc{)arture, 47.4 miles ; and by Ciise VI. Plane Sailing, the course S. 15° 38' W., and tlistance, 175.9 miles; and the course to the intended port, S. 58° 42' W., distance 155.2 miles ; the latitude being in 34° 21' N. t in the latitude of 33° 0' N., by reason of contrary wiiids, es, W. S. W. h W. 30 miles, lifes, S. S. E. i E. 27 miles, ([uired the latitude the ship TRAVERSE TABLE. Courses. Dist. Diff. of Lat. Departure. N. S. E. W S. W. by W. w. s. w. h w. W. by S. W. by N. S. S. E.* S. S. E. 1 E. S. by E. South. S. S. E.* 27 30 25 18 32 27 25 31 3'J 3.5 15.0 8.7 4.9 20.G 23.2 24.5 31.0 3G.0 12.2 13'9 4.9 14.9 22.4 287 24.5 17.7 3.5 172.9 3.5 45.9 93.3 45.9 Diff. of lat. 169.4 Depart. 47.4 | * Iiisicad of piitliiif^ the course S. S. E. 32 miles, and S. S. E. 39 mile-s you might make me enm only, cailiiiij it .S. S. E. 'I miles. 63 PARALLEL SAILING. In Plane Sailing, the earth is considered as an extended plane ; but tins supposition Ls very erroneous, because the em'th is nearly of a spherical figure, in which the meridians all meet at '^the poles ; consequently the distance of any two meridians measured on a parallel of latitude (wliicii distance is called the meridian distance) decreases in proceedhig from the equator to the j)o1gs. To illustrate this, let Pli represent die semi-axis of the earth, IJ the centre, V the pole, PCA a quadrant of the meridian, AB the radius of the equator, and CD (parallel thereto) the radius of a parallel of latitude. Then it is evident that CD will be the cosine of AC, or the cosine of the latitude of the point C, to the radius AB ; now, if the quadrantal arc PCA be supposed to revolve round die axis PB, the point A will describe the circumference of the equator, and C the circumference of a parallel of latitude ; and the former circumference will be to the latter as AB to CD (as may easily be deduced from Art. 55, Geometry), that is, as radius to the cosine of the latitude, or the point C ; hence it follows, that the length of any arc of the equator intercej)ted between two meridians, is to die length of a corresponding avc of any parallel intercepted between the same meridians, as radius is to the cosine of the latitude of diat parallel. Hence we obtain the following theorems. THEOREM I. ♦ Tfie circumference of the equator is to the circumference of any other parallel of latitude, ns radius is to the cosine of that latitude. ' TIIE0RE3I II. j As the length of a degree of the equator is to the mendian distance corresponding ta a degree on any other parallel of latitude, so is radius to the cosine of that parallel of latitiuk. TlIEOREiM III. As radius is to the cosine of any latitude, so are the miles of diference of longitude between two mcridinns [or their distance in miles upon the equator) to the distance of these two meridians on that parallel oflatiliule in miles. THEOREM IV. As the cosine of any latitude is to rculius, so is the length of any arc on that parallel of latitude [intercepted between two meridians) in miles to the length of a similar arc on the equator, or viiles of difference of longitude. THEOREM V. As the cosine of any latitude is to the cosine oj any other latitude, so is the length of any arc on the fust parallel of latitude in miles, to the length of tlie same arc on the other in miles. By means of Theorem IH. the following table was calculated, wliich shoAVs the meridian distance corrcsjionding to a degree of longitude in every latitude ; and may be made to answer for any degree or minute by taking |)roportional parts. fil parallp:l sailing. The following Table shows for every degree of latitude how many miles distant the ttoo meridians are, ichose difference of longitude is one degree. Lat. Miles. L.\T. Miles. Lat. Miles. Lat. Miles. Lat. Miles. 1- 59.99 19° 56.73 37° 47.92 55° 34.41 73° 17.54 2 59.90 20 56.38 38 47.28 56 33.55 74 16.54 •3 59.92 21 56.01 3') 46.63 57 32.68 4 a 15.53 4 59.85 22 55.63 40 45.96 58 31.80 76 14.52 5 59.77 23 55.23 41 45.28 59 30.90 77 13.50 6 59.C7 24 54.81 42 44.59 60 30.00 78 12.47 7 59.55 25 54.38 43 43.88 61 29.09 79 11.45 8 59.42 26 53.93 44 43.16 62 28.17 80 10.42 y 59.2G 27 53.46 45 42.43 63 27.24 81 9.39 10 59.09 28 52.98 Aij 41.68 64 2(1.30 20.36 82 8.35 11 58.90 29 52.48 47 40.92 65 83 7.31 12 58.G9 30 51.96 48 40.15 66 24.40 84 6.27 13 58.46 31 51.43 49 39.36 67 23.44 85 5.23 14 58.22 32 50.88 50 38.57 63 22.48 86 4.19 15 57.96 33 50.32 51 37.76 69 21.50 87 3.14 16 57.G8 34 49.74 52 36.94 70 20.52 88 2.09 17 57.38 35 49.15 53 36.11 71 19.53 89 1.05 18 57.06 36 48.54 54 35.27 72 18..54 90 0.00 When a sliip sails east or west on the surface of the earth sup])osefl to be snjherical, she describes a parallel of latitude, and this is called Parallel Sailing. In tliis case, the distance sailed (or dej)artiire) is equal to the distance between the meridians sailed from and arrived at in that ])arallel ; and it is easy, by Theorem IV. (preceding) to find the difference of longitude from the distance, or the distance from the difference of longitude, as will ajjpear plain by the following examples. » CASE I. The differtnce of longitude between two places in the same parallel of latitude being given, to find the distance between them. Suppose a ship in the latitude of 49° 30', north or south, sails directly east or west, until her difference of longitude be 3° 30' ; required the distance sailed. •> BY PROJECTION. Take the sine of 90° from the plane scale, and, with one foot of the compasses on (fig. 1) as a centre, describe the arc EQ,* with the difference of longitude, 210 miles, in the compasses, and one foot in E, J as a centre, describe an arc cutting Ea in Q; jom PE, PQ. Take the sine of the com{)lement of the latitude 40° 30' in your compasses, and with one foot in P, as a centre, describe the arc FG, cutting PE, PQ, in F, G ; then the length of tlie chord FG being measured on the same scale of equal parts, will be the departure 13().4 miles. Or this projection may be made in the following manner. Draw AD (!ig. 2) of an iiid(;finite length ; make tlic angle DAC eijual to tlie latitude 49° 30', and AC c(|Hal to the diflerence Fig. 2. of longitude 210 miles ; draw CD perpendicular to AD; then will the line AD be the distance or departure reipiired. BY LOGARITHMS. To find the departiu'c or distance. As radius 90° 10.00000 Is to the ditrereiK-e of longitude 210 2.32222 So is cosine latitude 49° 30' 9.81254 To the (li.sariihin, by tlie preceding; operation, was found cijual to 2.25521, diilering a little from the logarithm of 180.0. 10 74 MIDDLE LATITUDE SAILINO. BY GUNTER. 1st. The extent from the distance 300, to tlie difference of latitude 240, on the line of numbers, will reach from radius 90°, to the complement of the course, equal to 53° 8' on the line of sines. 2tlly. Tlie extent from radius 90°, to the course 36° 52', on the line of sines, will reach from the distance 300, to the departure 180, on the line of numbers. 3dly. The extent from the complement of the middle latitude 51°, to the radius 90° on the line of sines, will reach from the departure 180, to the difference of longitude 231.6, on the Ime of numbers. BY INSPECTION. Find the course (as in Case IV. Plane Sailing) by seeking in Table II. till against the distance taken in its colimin is found the difference of latitude in one of the tbllowhig cohunns; adjoining to it will stand the departure ; which if less than the difference of latitude, the course is to be found at the top of the table, but if greater, at the bottom ; then take the middle latitude as a course, and find the departure in the column of diflerence of latitude, against which, in the distance column, will stand the difference of longitude. Thus the distance 300, and the difference of latitude 240, are found to correspond nearly to a course of 37°, and a departure of 180.5 ; tlien, taking the middle latitude 39° as a course, I seek the dejjarture 180.5, in the latitude column, corresponding to which, in the distance column, is tlie difference of longitude 232. CASE VI. One latitude, course, and departure given, to find the difference of latitude, distance, and difference of longitude. A ship in the latitude of 50° 10' S., and longitude of 30° 00' E., sails E. S. E. until her de})arture is 160 miles; required her distance sailed, aiid latitude and longitude hi. BY PROJECTION. Draw the meridian ACD, and parallel thereto, at a distance equal to the departure 160 miles, draw the line EB ; make the angle CAB equal to the coiu'se 6 points, and draw AB meeting EB in B ; from B let fall upon AD die perpendicular BC ; then is AC the difference of latitude 66.3 miles, and AB the distance sailed 173.2 miles ; having thus obtained the middle latitude 50° 43', make the angle CBD equal diereto, arid draw BD meeting ACD hi D ; then will BD be the difference of longitude 252.7 miles. BY LOGARITHMS. A ^^^«^. E S Departure ^^^ B 6 A D To find the difference of latitude. As radius 4 points 10.00000 Is to the departure 160 2.204 12 So is cotangent coui*se 6 points. 9.61722 To the difference of lat. 66.3. . . 1.8213 4 Latitude left 50° 10' S. Difference of latitude 66 1 06 S. Latitude in 51 1 6 S. Sum of latitudes lOI 26 Middle latitude 50 43 * To find the distance. As sine course 6 points 9.96562 Is to the departure 160 2.20412 So is radius 8 points 10.00000 To the distance 173.2 2.23850 To find the difference of longitude. As cosine middle latitude 50° 43' 9.80151 Is to the departure 160 2.20412 So is radius 90° 10.00000 To the difference of long. 252.7 2.40261 Longitude left 30° 00' E. Difference of longitude 253 4 13 E. Longitude in 34 13 E. The correclion of ihis lalilucle in llif lable at tlie end of Case VII. is insensible. MIDDLE LATITUDE SAILING. 75 BY GUNTER. 1st The extent from the course G points, to the radius 4 points, on the line marked TR, will reach from the departure 100, to tlie difference of latitude Gb.3, on the Imo of numbei-s. ^ . .i v i i cjr» 2div The extent from 6 points, to the radius, or 8 points, on the line marked bK, will reach from the departure 160, to the distance 173.2, on the line of numbers. 3dlv. The extent from the complement of the middle latitude 39° 1/', to the radiua 90°, on the sines, will reach from the departure lGO,to the dilFerence of longitude 2o2.7, pi.iu n' REFRA^CTIOy G5 - FUf.2 \ •■•3> % :ji-- :::::;- 5^- .V. PAHALL AX So is cosine coui-se 50° 58 .... y./!Ji)l» To the difference of lat. 135.4. . 2.131 62 To find the difference of longitude. As cosine middle lat. 48° 23' . . . 9.82226 Is to the departure 167 2.22272 So is radius 10.00000 To the difference of long. 251.5 2.40046 N. 'able I. or Table II. (as in Case III. •esponding to which, in the columns and the distance and difference of le as a course, seek the departure iu the distance column, will stand the nts, and seek for the dejiarture 160, numbers give the distance 173, and 13', or (51° nearly) as a course, and n, opposite to which, in the distance 254 miles, nearl3\ he vimdian given, to find the course^ ence of longitude. ide of 2.5° 0' W., sails south-easterly n be 167 miles ; required the course in. 167 miles, and ABC ; take an es, and with one in A ; join AD ; miles, and BAD latitude in, and 3 middle latitude, je the difference ,iMS. tudelefl 49°30'N. erence of latitude 135 . . . 2 15 S. itude in 47 15 N. n of the latitudes 96 45 Idle latitude 48 23* Longitude lefl 25° 00' W Difference of longitude 252. . 4 12 E. Longitude m '-^0 48 W. * The correction of this lalilude m the table is V, nuiking tlie corrected inidille latitude 48° 2V 74 MIDDLE LATITUDE SAILING. BY GUNTER. 1st. The extent from the distance 300, to tlie difference of latitude 240, on the line of numbers, will reach from radius 90°, to the complement of the course, equal to 53° 8' on the ILue of sines. 2dly. The extent from radius 90°, to the course 36° 52', on the line of sines, will reach from the distance 300, to the departure 180, on the line of numbers. 3dly. The extent from the complement of the middle latitude 51°, to the radius 90° on the line of sines, will reach from the departure 180, to the diffei'ence of longitude 231.6, on the line of numbers. BY ; Find the course (as in Case IV. Plar distance taken in its column is found t columns ; adjoining to it will stand the latitude, the course is to be found at th then take the middle latitude as a coi difference of latitude, against which, ii of longitude. Thus the distance 300, and the difl .• •.■ • nearly to a course of 37°, and a departu . ■ as a course, I seek the departure 180.5 . . •*.',, ^ ' . -, -. in the distance column, is the differenc '; ,- ■- One latitude, course, and departure givi differei A ship in the latitude of 50° 10' S., her de})arture is 160 miles; required latitude and longitude in. BY PROJECTI Draw the meridian ACD, and paral equal to the dei)arture 160 miles, dra the angle CAB equal to the coin-se i meeting EB in B ; from B let fall upo BC ; then is AC the difference of latit the distance sailed 173.2 miles ; ha^ middle latitude 50° 43', make the angle draw BD meeting ACD m D ; then wL of longitude 252.7 miles. BY L( To find the difference of latitude. As radius 4 points 10.000( Is to the departure 160 2.204. So is cotangent course 6 points. 9.617' To the difference of lat. 66.3. . . 1.82U Latitude left 50° 10' t Difference of latitude Qij 1 06 i Latitude in 51 16 S. Sum of latitudes 101 26 Middle latitude 50 43 * lis, x^woiiiv^ lumuic lailiuue OU" 4^3' I'.BULtI Is to the departure 160 2.20412 So is radius 90'-' 10.00000 To the difference of lonu:. 252.7 2.40261 Longitude left 30° OO' E. Difference of longitude 253 4 13 E. Longitude in 34 13 E. * The correclion of this latitude in thf table at the end of Case VII. is insciisiWe. MIDDLE LATITUDE SAILING. 75 BY GUNTER. 1st. The extent from the course 6 points, to the radius 4 points, on the line marked TR, will reach from the departure 160, to the diffci-ence of latitude 66.3, on the line of niimbei-s. 2diy. The extent from 6 points, to the radius, or 8 points, on the line marked SR, will reach from the departure 100, to the distance 173.2, on the line of numbers. 3dly. The extent from the complement of tlie middle latitude 39° 17', to the radius 90°, on the sines, will reach from the depai'tui-e 100, to the difference of longitude 252.7, on the Ime of numbere. BY INSPECTION. Find the coui-se among the points or degi-ees. Table I. or Table II. (as in Case III. Plane Sailing), and the departure in its column, corresponding to which, in the colunms of distance and difference of latitude, will be found the distance and difference of latitude respectively ; then with the middle latitude as a course, seek the departure in the column of latitude, coiTCsponding to which, in the distance column, will stand the difference of longitude. Thus, I enter Table I., above E. S. E., or points, and seek for the departure 100, the nearest to which is 159.8; the corresponding numbcre give the distance 173, and the difference of latitude 06.2 miles. Enter Table II. with the middle latitude 50° 43', or (51° nearly) as a course, and seek for the dejiarture 100, in the latitude column, opposite to wliich, in the distance column, will be found the difference of longitude 254 miles, nearl3\ CASE VII. One latitude, distance sailed, and departure from the meridian given, to Jind the course^ difference of latitude, and difference of longitude. A ship in the latitude of 49° 30' N,, and longitude of 25° 0' W., sails south-easterly 215 miles, until her departure from the meridian be 167 miles ; required the course Bteered, and the latitude and longitude the ship is in. BY PROJECTION. Draw the line BD equal to the departure 107 miles, and peiiJendicidar thereto draw the meridian line ABC ; take an extent equal to the distance 215, in your compasses, and with one foot in D, as a centre, describe an arc cutting AB in A ; join AD ; then will AB be the difference of latitude 135.4 miles, and BAD the course, S. 50° 58' E. Hence we have the latitude in, and middle latitude; make the angle BDC equal to the middle latitude, and draw DC cutting ABC in C ; then DC will be the difference of longitude 251.5 miles. BY LOGARITHMS. To find the course. As the distance 215 2.-33244 Is to the radius 90° 10.00000 So is the departure 107 2.22272 To sine course 50° 58' 9.89028 To find the difference of latitude. As radius 10.00000 Is to the distance 215 2.33244 So is cosine coui-se 50° 58' 9.79918 To the difference of lat. 135.4. . "2.131(32 To find the difference of longitude. As cosine middle lat. 48° 23' . . . 9.82220 Is to the departure 107 2.22272 So is radius 10.00000 Latitude left 49° 30' N. Difference of latitude 135 .. . 2 15 S. Latitude in 47 15 N. Sum of the latitudes 96 45 Middle latitude 48 23* To the difference of long. 251.5 2.40040 Longitude left 25° 00' W Difference of longitude 252. . 4 12 E. Longitude in . 20 48 W * The correctiou of this latitude in the table is 1', making the corrected middle latitude 48° £J/ 76 MIDDLE LATITUDE SAILIiNG. BY GUNTER. 1st. The extent from the distance 2L5, to the departure 167, on the line of niunbei-s, will reach from the radius 90°, to the course 50° 58' on the line of sines. Sdly. The extent from radius 90°, to the com})lement of the course 39° 02', on the line of sines, will reach from the distance 215, to the difference of latitude 135.4, on the line of numbers. 3dly. The extent from the complement of the middle latitude 41° 37', to the radius 90°, on the Une of sines, will reach from the depaitm-e 107, to the difference of longitude 251.5, on the line of numbers. BY INSPECTION. As in Case V. Plane Sailing, find the course by seeking iii Table II. till against the distance, in its column, is found the given departure in one of the following colunnis, adjoinmg to which, in the other column, will be the difference of latitude, which if gi-eater than the departure, the course will be at the top, but if less the course will be found at the bottom. Then take the middle latitude as a course, and find the departure in the column of difference of latitude, against which, m the distance column, will be found the difference of longitude. Thus the distance 215, and the depaiture 167, are found nearly to con-es])ond to a course of 51 degrees, and a difference of latitude of 135.3 ; then with the iiiiddle latitude 48°, as a course, I enter the table, and seek for the departure 167, in the latitude column; the distance con-espouduig 250 is the difference of longitude nearly. In all the preceding examples, we have used the middle latitude, without any coiTection, in computing the difference of longitude ; but when absolute accin-acy is required, this latitude must be corrected. We have given in the following table the value of this correction in the most common cases. It requires no particular explana- tion : one examj)le will serve to show its use. Suppose, therefore, tae two latitudes to be 40° and C0°. Here the middle latitude is 50°, and the difference of latitude 20° ; tlie tabular correction corresponding to these numbers is 57' ; adding this to 50°, we get tlie cori-ected middle latitude 50° 57', which is to be used instead of 50°, when great accuracy is required. We have inserted in the notes at the bottom of the pages, in the preceding examples, the values of this correction, but have not introduced it into the calculations, because it is generally unnecessary on account of its smallness TABLE. Tliis Table contains the correction, in minutes, to be added to the Middle Latitude to obtain the corrected Middle Latitude. Mid. Lat. Difference of Latitude. Mid. Lat.. o 15 18 21 24 30 35 40 45 50 1° 1 .0 2° 1 1 3° 2 1 2 2 2 2 2 2 2 3 4° / 3 3 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 5° / 5 4 4 3 3 3 3 3 4 4 4 4 5 5 5 6 6 7 6° / 7 6 5 5 5 4 5 5 5 6 6 6 7 7 8 8 9 10 7° 1 9 8 7 7 6 6 6 6 7 8 8 9 9 10 11 12 13 14 8° / 12 10 9 9 8 8 8 8 9 10 11 11 12 13 14 15 16 18 9° / 15 13 12 11 10 10 10 11 11 13 14 14 15 16 18 19 21 23 10° / 18 16 15 14 13 12 13 13 14 16 17 18 19 20 22 24 26 29 12° / 26 23 21 20 18 18 18 19 20 22 24 26 27 29 32 34 38 42 14° 1 36 32 29 27 25 24 25 26 28 31 33 35 37 40 43 47 52 58 16° / 47 41 37 35 32 32 32 34 36 40 43 46 49 52 57 62 68 76 18° / 59 52 47 44 41 40 41 43 46 51 55 58 62 67 72 79 88 98 20° / 72 64 58 54 50 49 50 53 57 63 68 72 77 83 90 99 110 124 o 15 18 21 24 30 35 40 45 50 55 53 GO 55 58 60 62 64 66 68 70 72 62 64 66 68 70 72 This Table is to be entered at the top with the difference of the two latitudes, and at the side with the middle latitude ; under the former, and opposite to the latter, is the correction, in minutes, to be added to the middle latitude, to obtain the corrected middle latitude. MIDDLE LATITUDE SAlLl^5.3. Find in this table one half the meridional difference of latitude 101.5, opposite to which, in the departm-e column, stands 125.1 ; this doubled gives 250.2, for the difference of longitude, nearly. Having explained the method of calculatmg suigle courses by Middle Latitude and Mercator's Sailmg, it now remains to explain the method of calculatmg compound courses. To do this, you must consti-uct a traverse table, and find the difference of latitude and departure for each com-se and distance, as in Traverse Sailing, and from thence the whole difference of latitude, departure, and latitude in ; with the departure and latitudes, find the difference of longitude and longitude in, as in Case II. of Middle Latitude or Mercator's Sailing. This method is exact enough for working any single day's work at sea, except in high latitudes, where it will be a little erroneous ; in this case the difference of longitude and longitude in, may be calculated for every single com-se and short distance ; but in general this nicety in calculation may be neglected. To Ulustrate the method of working compound courses, we shall here work an example by Middle Latitude and Mercator's Sailing. EXAMPLE. A ship from Cape Henlopen, in the latitude of 38° 47' N., longitude 75° 5' W., sails the following true courses, viz. E. by S. 20 miles, E. N. E. 15 miles, S. E. 26 miles, south 16 miles, W. S. W. 6 mUes, N. W. 10 miles, and east 30 miles; requu-ed her latitude and longitude. By constructing the ti'averse table with these courses and distances, it appears that the ship has made 27.8 miles of southing, and 69.3 miles of easting ; and by subtracting the southuig from the latitude of Cape Henlopen, there remams the latitude in 38° 19' N. Cape Henlopen's latitude 38° 47' N. Latitude in 38 19 N. Sum of latitudes 77 6 Middle latitude 38 33 TRAVERSE TABLE. Dist Diff. of Lat. Departure. N. S. E. W. E. by S. 20 3.9 19.6 E. N. E. 15 5.7 13.9 S. E. 26 18.4 18.4 South. 16 16.0 W. S. W. 6 2.3 5.5 N. W. 10 7.1 7.1 East. 30 30.0 12.8 40.6 81.9 12.6 12.8 12.6 DifF. of lat. . .27.8 69.3 Dep. Meridional parts 2528 Meridional pai-ta 2492 36 By inspection of Table II. it appears that the difference of latitude 27.8, and departure 69.3, con-espond to a course of 68° nearly, and a distance of 75 miles ; and in the same page of the table, opposite to the meridional difierence of latitude, found in the column of latitude, stands the difference of longitude 89 miles in the departure column ; this being subtracted from the longitude of Cape Henlopen, 75° 5' W., leaves the longitude ui 73° 36' W., by Mercator's Sailing. Or, with the middle latitude 38° 33' to 39°, as a course, find the departure 69.3, m the latitude column, opposite to which is 89 in • the distance column, which is the difference of longitude by ftliddle Latitude Sailmg ; consequently the longitude in is 73° 36' W., as above. Thus we see that such examples are performed as m Traverse Sailing, and Case II. of Mercator's or Middle Latitude Sailing, cither by inspection, as above, or by the scale of logaiithms. MERCATOR'S SAILING. 87 QUESTIONS FOR EXERCISE. Question l. A ship in tiie latitude of 49° 57' N., and longitude of 15° IC W., sails south-westerly until her depai'ture is 789 miles, and then, by observation, is in tlae latitude of 39° 20' N. ; requu-ed her coui-se, distance, and longitude in. Answer. Course S. 51° 05' W., distance 1014 miles, longitude in 33° 50' W. Quest. II. A ship in the latitude of 42° 30' N., and longitude of 58° 51' W., sails S. W. by S. 591 miles ; tlie latitude, and longitude in, ai'e required. Ans. Latitude in 34° 19' N., longitude in 65° 51' W. Quest. III. A ship from the latitude of 49° 57' N., and longitude of 30° 00' W., sails S. 39° W. till she arrives in the latitude of 45° 31' N. ; requijred the distance run, and longitude in. Ans. Distance 342.3, longitude in 35° 21' W. Quest. IV. A ship from the latitude of 50° 10' S., and longitude of 30° 00' E., sails E. S. E. until her departure is 957 miles ; required the distance sailed, and the latitude and longitude in. Jlns. Distance 1036 miles, latitude in 56° 46' S., longitude in 56° 50' E. Quest. V. A ship in the latitude of 49° 30' N., and the longitude of 25° 00' W., sails south-easterly 645 miles, makuig 500 miles depaiture ; requned the course steered, and the latitude and longitude m. Ans. Com-se S. 50° 49' E., latitude in 42° 42' N., longitude m 12° 57' W. Having gone through the necessaiy problems in Mercator's Sailing, we shall now show how Alercator's Chart may be constructed by means of the Table of Meridional Parts. To construct a 3Iercator' s Chart to commence at the equator. Suppose it was required to construct the Chart in the Plate prefixed to this work, which begins at the equator, and reaches to the parallel of 50 degi-ees, and contains 95 degi'ees of longitude west from the meridian of Greenwich. Draw the line AD representing the equator ; then take from any scale of equal ])arts the number of mmutes contained in 95 degrees, viz. 5700, which set off from A to D ; subdivide this line into 95 equal parts, representing degi'ees of longitude. Through A and D draw the lines AB, DC, perpendicular to AD, and make each of tliem equal to 3474, which are the meridional parts, correspondi)ig to 50 degrees. Join BC, which must be subdivided in the same manner as the line AD ; and through the correspond- ing points of the lines AD, BC, must be drawn (at the distance of 10° or 20°) the lines parallel to AB, representing meridians of the earth ; these lines must be numbered 0, 10, 20, &c., beginning at the luie AB, which represents the meridian of Greenwich. Set off in like manner upon the meridians AB, DC (beginning from the equator AD), the meridional parts corresponding to each degree of latitude from 0° to 50° ; and tlirough the corresponding points (at the distance of 10° or 20°) draw lines parallel to the equator AD, to represent the parallels of latitude. Then the upper part of the chart will represent the nortli, the lower the south, the right hand the east, and the left hand the west (which is generally supposed in charts, unless the contrary is expressly mentioned). If the chart does not commence at the equator, but is to serve for a certain portion of the globe contained between two parallels of latitude on the same side of the equa- tor, you must draw the meridians as directed in the last example ; then subtract the meridional parts of the least latitude of the chart from the meridional parts of the other latitudes, and set off these differences on the extreme meridians ; draw lines through the corresponding pomts, and they will be the parallels of latitude on the chart. If the chart is to be bounded by parallels of latitude on different sides of the equator, you must draw a line representing the equator, and perpendicular to it draw the lines to represent the meridians, continuing tliem on both sides of the equator ; then set off the pfiTallels of latitude on both sides of the equator, in the same manner as in the first example. Take from the Table of Latitudes and Longitudes of places the latitude and longitude of each particular place contained within the bounds of the chart, and lay a rule over its latitude, and another crossing that over its longitude ; the point Avhere these meet will represent the proposed place upon the chart. The most remarkable {)omt of a sea- coast being thus laid down, lines may be drawn from point to point, which will form the outlines of the sea-coast, islands, &c. ; to which may be annexed the depths of water expressed in common Arabian numbers, the time of high water on the full and change days expressed m Roman numbei-s, the setting of the tide expressed b} an arrow, and whatever else may be thought convenient for the chart to coutaui. 88 MERCATOR'S SAILING. This chart is not to be considered as a just representation of the earth's surface, for the figures of islands and countries are distorted towards the poles, as is evident from the construction ; but the degi'ees of latitude and longitude are increased m the riame proportion, so that the bearmgs between places will be the same on the chart as on the globe ; and as the meridians are right lines, it follows, that the rhumbs, which forai equal angles with the meridians, will be straight lines, which render this projection of the eaith's surface much more easy and proper for the mai'mer's use than any other. Having tlie latitude and longitude of a ship or place, to find the coi'responding point on the chart. Rule. Lay a ruler across the chart in the given parallel of latitude ; take in your compasses the nearest distance between the given longitude and the nearest meridian drawn acros's the chart ; put one foot of the compasses m the point of intersection of the ruler and meridian, and extend the other along the edge of the ruler on the same side of the meridian as the place lies, and that point will represent the pkce of the ship. If the longitude on the chart be counted from a different meridian from that you reckon from, you must i-educe the given longitude to the longitude of the chart, by adding or subtracting the diffei'ence of longitude of those meridians, and then mark off the ship's place, as before du-ected. Or you may draw a meridian line through the place you reckon your longitude from ; then measure off the ship's longitude on the equator, and apply it to the edge of the ruler from this meridian, and you will obtain the ship's place. To find the hearing of any place from the ship. Rule. Lay a ruler across the given place and the place of the ship ; set one foot of the compasses in the centime of some compass near the ruler, and take the neai*est distance to the edge of the ruler ; slide one foot of the compasses along that edge, keeping the other extended to the greatest distance from the rulei*, and observe what point of the compass it comes nearest to, for that will be the bearmg required. To find the distance of any place from the ship. Rule. Take the distance between the ship and the given place in your compasses, and apply it to the side of the chart or gi-aduated meridian, setting one foot as much above one place as the other is below the other place ; the number of degrees betweea the points of the compasses will be the distance nearly. When the places bear north and south of each other, this rule is accurate ; but when they bear nearly east and west, and the distance is lai-ge, it will err considerably ; but in general it is exact enough for common purposes ; if gi-eater accuracy is required, it is best to find the distance by calculation. If any one wishes to estimate tlie distance accurately by the chart, he must proceed in the following manner : — 1. If the place be in the same longitude that the ship is hi, then the preceding rule is accurate. 2. If the place be in the same latitude as the ship, or bear east or west, the distance cannot be obtained without calculating it by Case I. of Parallel Sailing. 3. If the place be neither m the same latitude, nor in the same longitude as the ship, the distance must be found in the followmg manner : — Lay a ruler over both places, and draw through one of them a parallel to the equator ; take the difference of latitude between both places in your compasses f^om the equator ; slide one foot on that par- allel, keeping the other extended so that both points shall be on the same meridian, and note the point of the ruler which is touched by the other foot of the compasses ; take the distance from this point to the given place through which the parallel was drawn, and apply it to the equator, and you will have the sought distance. The bearing rn;! distance of any two places from each other may be found in the same manner as ilic bearbig and distance of any place from the ship. EXAMPLE. Required the bearing and distance between the east end of Long Island and the north part of Bermudas. A ruler being laid over both places, as du'ected in the precedmg rule, it will be found to lie parallel to the N. W. by N. and S. E. by S. line ; and the distance between the two places being taken in the compasses, and applied to the graduated meridian, will measure about 10 degrees or 600 miles ; therefore these places bear from each other N. W. by N. and S. E. by S., and their distance is GOO miles, nearly. 89 PROBLEMS USEFUL IN NAVIGATION AND SURVEYING. PROBLEM I. Coasting along shore, I saio a cape of land bearing jY. JV*. E., and after sailing W. JV. W yO miles, it bore JV. E. by E. ; required the distance of the ship from the cape at both staiioiis. . C BY PROJECTION. Describe the compass ESW, and let its centre A represent the place of the ship at the first station ; draw the W. N. W. line AB equal to 20 miles, and B will represent the second station. Draw the N. N. E. line AC, of an indefinite length, and the line BC parallel to the N. E. by E. Ime of the compass ; the point of intei'section C will represent the place of the cape ; and the distance BC, being measm-ed, will be found 36 miles ; and AC 30 miles. BY LOGARITHMS, (by Case L of OBLiquE Trigonometry.) The difference between N. N. E. and W. N. W. is 8 points or 90°, therefore BAG is a right angle ; also the difference between the N. E. by E. and N. N. E. is 3 points, equal to the angle ACB ; and the difference between the N. E. by E. pomt and the point opposite to W. N. W. is 5 pomts, equal to the angle ABC. To find the distance BC. As sine angle ACB 3 pts. Ar. Co. 0.25526 Is to the distance AB 20 1.30103 So is sme angle BAC 8 points. . 10.00000 To the distance BC 36.0 1.5.5629 To find the distance AC. As sine ACB 3 points . . .Ar. Co. 0.25526 Is to the distance AB 20 So is sme angle ABC 5 points . To the distance AC 29.93 1.47614 1.30103 9.91985 The above solutions are by Case I. Oblique Trigonometiy, though they might have been done, in this example, by Case II. of Right-angled Trigonometry, because the angle BAC is a right angle. If the bearings of the middle pomt C of an island (or any remarkable peak) be dctennined in this manner, we may, at tlie same time, find the limit of the dimensions of the island, by measuring with a quadrant or sextant, held in a horizontal position, the angular distances between that middle point and the extremes of the island. For by drawuig the lines ADE, AGF, making the angles Dx\C, GAG, with AC, equal to the angular distances observed at A, and in the same manner by drawing the lines BDG, BEF, making angles with BC equal to the angular distances obsei-ved at B, you would obtain the quadrilateral figure DEFG, within which the island is to be placed. If similar observations could be procured at H, they woukl in general take off the comers at D and F ; and observations at I would generally take off the comers at E and G ; and by observing the pi'ojecting pomts and coves in the island, while sailuig round it, and drawing a figure conformable thereto, within the limiting ejiace thus found, the form and dimensions of the island may be obtoinerl to a consid- 'irable degi-ee of accuracy. 12 90 PRUBLEMS USEFUL IN NAVIGATION AND SURVEYING. PROBLEM IL bting at sea, ive saia two headlands, ivhose bearing from one another by the chart tvas W. by JV., and E. by S., and the distance 15 miles ; the westernmost bore from us S. S. W., and the easternmost S. E. by E. : required our distance from each of them. line ■---— D ...K BY PROJECTION. Draw the compass NESW, and through the centre A, di-aw the E AR, the S. S. W. line AB, and the S. E. by E. line AC, and contmue the two latter indefi- nitely ; upon the fonner, AJR, take AD equal to 15 miles ; througli D draw DC parallel to AB, to meet AC in C, and draw CB parallel to AD. Then A will be the place where the headlands B and C were obsei-ved ; and the distance AB of the westernmost headland, being measured, is found to be 5.8 miles, and the distance AC of the easternmost headland 15 miles. BY LOGARITHMS, Between the S. S. W. line AB, and the S. E. by E. line AC, are 7 points r= angle BAC ; and between the S. E, by.E. line AC, and the E. by S. Ime AD, are 2 points =: angle CAD mangle ACB (because AD, BC, are parallel); therefore ACB4-BAC=:9 points ; and smce aU three angles ACB, BAC, ABC, are equal to 16 pomts, the angle ABC is also equal to 7 points ; therefore (by AH. 39, Geometry) the sides AC, CB, are equal, beuig opposite to the equal angles ABC, BAC. If these angles had not been equal, the side AC might have been calculated in the same manner as we shall now calculate the side AB, To find the side AB. As sine BAC 7 points Arith. Comp. 0.00843 Is to BC 15 miles 1.17609 So is sme ACB 2 points 9.58284 To AB 5.85 0.76736 This problem, or the first, may be used for finding the distance of a ship from any headland, &c., when taking a departure from the land. PROBLEM IIL Tivo ships sail from the same port; the first sails JV*. E. h E. 16 miles ; the second sails easterly 20 miles, and then finds that the first bears JV. JV*. W. : required the course of the second ship, and the distance between the two ships. BY PROJECTION. Draw the compass ESW, and let its centre A represent the port sailed from ; draw the N. E. J E. line AB equal to 16 miles ; also through B, the line BC, parallel to the N. N. W. line, and continue it indefinitely ; take a distance repre- senting 20 miles in your compasses, and putting one foot in A, describe with the other an arc cutting the line BC in C, and join AC. Then B will be the place of the first sliip, C that of the second, and AC the course steered by the second ship, which will be nearly E. S. E. ^ E., and BC the distance of the sliips 17i miles. BY LOGARITHMS. The course from B to C is S. S. E, (opposite to N. N. W.), and from B to A is S. W. h W. (opposite to N. E. i E.) ; the difference between these bearings is 6h points, equal to 73° 7', equal to the angle ABC ; having this angle and tlie sides AB, AC, we may find the other angles and side by Cases II. and III. of 01>lique Trigonometry, as follows • — PROBLEMS USEFUL IN NAVIGATION AND SURVEYING. 91 To find the angle C. As the side AC 20 miles 1.30103 Is to sine ABC 73° 7' 9.98087 So is side AB 16 miles 1.20412 11.18499 Subtract 1.30103 TosineaneleC 49° 57' 9.88396 For N. N. W., add 22 30 Sum is N. 72° 27' W., the bearing of A from C ; whence the course of the ship from A towai-ds C, is S. 72° 27' E., or E. S. E. i E., nearly. To find the distance of the ships BC. Add the angle C = 49° 57', to tlie angle B 73° 7', we obtain the sum 123° 4' ; subtracting this from 180°, leaves the angle CAB 56° 56'. As sine angle ABC 73° 7', Ai-. Co. 0.01913 Is to the side AC 20 miles 1.30103 So is sme CAB 56° 56' 9.92326 To the side BC 17.5 mUes 1.24;M2 PROBLEM IV. Two ships sail from the same port, the one JV. W. 30 miles, and the other JV*. E. by JV. 40 miles ; required the bearing and distance of the ships from each other. _^.C BY PROJECTION. Draw the compass NESW, and let its centi-e A represent the port sailed from ; draw the N. W. line AB equal to 30 miles, and the N. E. by N. line AC equal to 40 miles ; jom BC, which will be the bearing and distance of the two ships ; whence the bearing will be found to be W. S. W. h W-, and the distance 45.1 miles, neai-ly. BY LOGARITHMS, (by Cases IV. V. of Oblique Trigonometry.) Between the N. W. line AB, and the N. E. by N. line AC, there are 7 pohits, equal to angle BAC ; half the supplement of this to 180° is 50° 37^', equal to half sum of the angles C and B. To find the angfes B, C. As sum of AB and AC 70. . Ar. Co. Log. 8.154-90 Is to llieir difference 10 1.00000 So is tangent half sum angles 50° 37^'.. 10.08583 To tangent half diff. of angles 9 5 % . . 9.24073 Sum is angle B 60 30 Difference is angle C 40 45 To find the distance BC. As sine angle B.... 60° 30'Ar.Co.Log. 0.06030 Is to side AC 40 1.60206 So is sine angle A.. 78 45 9.'J9157 To the distance BC 45.1 1.65393 To the angle C, equal to 40° 45', add the angle representing the course from C to A, equal to 33° 45', the sum is 74° 30', which is the bearing of B from C, namely, S. 74° 30' W., or W. S. W. h W., nearly. PROBLEM V. Two ports bear from each other E. by JV. and W. by S., distance 400 miles : a ship from the easternmost sails northerly 450.7 miles ; another from the westernmost sails 300 miles, and meets thefrst : required the course steered by each ship. C BY PROJECTION. Draw the compass ESW, and let the centre B represent the westernmost port ; draw the E. by N. line BD equal to 400 miles, and D will be the eastern- most port ; with 300 in your compasses, and one foot in B, describe an arc ; witli 450.7 in your compasses, and one foot in D, describe another arc, cutting the former in C ; join DC, BC. Then BC will be the course sailed by the westernmost ship, and DC the course sailed by the easternmost ship. c(]>' 92 PROBLEMS USEFUL IN NAVIGATION AND SURVEYING. BY LOGARITHMS. To find the angle CBD By Theorem IV. Trigonometiy. Divide the triangle BCD into two right-angled triangles by means of the perpendicular CA, and bisect BD in a ; then As the base BD 400 Ar. Co. Log. Is to the sum of BC, CD 750.7 So is difference of BC, CD... 150.7 To twice Aa , 282.8. 7.39794 2.87547 2.17811 2.45152 Half, orAa 141.4 HalfBD = Ba= 200 Difference is BA 58.6 Then, in the triangle ACB, As hypotenuse DC 300 2.47712 Is to radius 90° 10.00000 So is AB 58.6 1.76790 To cosine CBD 78° 44' 9.29078 By Theorem V. Trigonometry. CD = 450.7 BD = 400 . . Ar. Co. Log. 7.39794 BC =300 . . Ar. Co. Loa-. 7.52288 Sum 1150.7 Half sum 575.35 Log. 2.75993 Half sum less CD 124.65 Log. 2.09569 Sum 2)19.77644 Half sum.. 39° 22', 2 .Cosine 9.88822 Doubled is 78 44= angle CBD. Having found this angle, we may find either of the others, thus : To find the angle CDB. As CD 450.7 Arith. Comp. 7.34611 Is to sine CBD 78° 44' 9.99155 So is BC 300 2.47712 To sine CDB 40° 45' 9.81478 As the angle CBD is 78° 44', or 7 points nearly, and the course fi-om B to D is E. by N., the course fi'om B to C must be north. The course fi"om D to B being W^. by S., or W. 11° 15' S., and the angle BDC equal to 40° 45', the bearing of C fiom D must be W 29° 30' N., because 40° 45' — 11° 15' = 29° 30' PROBLEM VL Coasting along shore, ice saio two headlands ; the first lore from us JV*. £., the secona E. JV. E. ; after sailing E. by S. 10 miles, the first hore JV. by E., and the second JV. E. by JV. : required the hearing of the two headlands from each other, and their distance. BY PROJECTION. Draw the compass NESW, and let its centre A represent the place of the ship at the first station ; draw the E. by S. line AB equal to 10 miles, and B Avill be the place of the ship at the second station; draw the N. E. line AC, and the E. N. E. line AD ; through the point B di-aw the lines BC, BD, parallel to the N. by E. and N. E. by N. lines, and the points C and D, where they intersect the lines drawn from A to the same headlands, will be the points representing them respectively ; jom the points C and D ; then will CD be the distance of the two headlands, and a line drawn through A parallel to CD will repre- sent the beai-ing of those places from each other on the compass. ^ BY LOGARITHMS. In the triangle ABC, we have all the angles and the side AB to find BC ; for the bearings of B and C from A are E. by S., and N. E., the difference being 5 points, equal to BAC ; and the bearings of B and A from C are S. by W., and S. W., the difference being 3 points, equal to the angle ACB. To find the side BC. As sine of ACB 3 points Arith. Comp. 0.2552G Is to the side AB 10 1.00000 So is sme angle BAC 5 points 9.91985 To BC 14.97 1.17511 In the triangle ABD, we have all the angles and the side AB to find BD ; fi)r the bearings of B and A from D are S. W. by S.j aiid ^Y. S. W., the difference being PROBLEMS USEFUL IN NAVIGATION AND SURVEYING. 93 3 points, equal to BDA ; aiid the bearings of B and D from A are E. by S., and E. N. E., the difference being also 3 points, equal to the angle BAD ; therefore the angle BAD = BDA, and (by ^rt. 39, Geometry) BD = AB = 10 miles. If these angles had not been equal, we might have calculated the side BD in the same manner asBC. Now, in the triangle CBD, we have BD = 10, BC = 14.97, and the angle CBD =: 22" 30' ; for the bearings of C and D from B are N. by E., and N. E. by N., differing 2 points, or 22^ 30' ; hence we may find tlie other angles and side CD as in Case IV. of Oblique Trigonometry. To find the angles BCD, BDC. To find the distance CD. As sum of BC, BD, 24.97, Arilh. Comp. 8.6025S As sine angle BCD 33° 44', Arith. Comp. 0.2534o Is to their diflerence 4.97 0.69636 Is to side BD 10 1.00000 So is tang-, half sum op. angles 78° 4o'.. 10.70134 So is sine angle CBD 22° 30' 9.58284 To tangent half diff. of angle s 45 1 . . 10.00023 To the distance CD G.89 0.83829 Sum is angle BDC = 123 46 Diflerence is angle BCD = 33 ^l, or nearly 3 points ; and as tiie bearing of B from C is S. by W., the bearing of D from C must be S.S.E. PROBLEM VII. Jlie bearings and distances of three points of land, A, B, C, being given, together tvith the horizontal angles ADC, CDB, measured in a boat placed over a shoal at the point D ; required the bearing and distance of the shoal from any one of the points A, B, C. BY PROJECTION. The sum of the two angles ADC, CDB, is equal to the angle ADB. Make the angles BAF, ABF, each equal to the complement of the angle ADB, and di"aw the lines AF, BF, which will intersect each other in the point F. Upon js F, as a centre, with the radius FA, equal to FB, describe the circle AEBD. Then any pomt D, of this circumference ADB, may be taken as the vertex of a triangle, whose base is AB, form- ing an angle ADB, which will satisfy the condition of being equal to the sum of the meas- ured angles ADC, CDB. In the same marmer we may find the centre G, of a circle BCD, whose circumference will contain the vertex D, of the triangle DCB, foiTning an angle at the vertex equal to the measured angle CDB. The point of intersection of these two circles is the place of the shoal at D; whence we easily obtain the distances AD, BD, CD; also the bearings of the shoal from the points A, B, C. Continue the line DC to meet the ch-cle ADB in E. BY LOGARITHMS. We have the bearings and distances of the points A, B, C, given from the map, or by previous observations, so that all parts of the triangle ABC are known. In the ti-iangle AEB, we have the angle EAB equal to the observed angle CDB {Art. 41. Geometry), also angle EBA equal to the observed angle CDA ; the smn of these tvvo angles subtracted from 180°, leaves the remaining an'gle AEB of the triangle AEB ; hence we have all the angles, and the bass AB of this triangle, to find AE, by Case I. of Oblique Trin■onometr}^ In the triangle AEC, we have AE by the preceding calcula- tion, and AC from the map, also the angle EAC =. CAB -f EAB =r CAB -f- EDB ; «y> that we have the two sides AE, AC, and the included angle EAC, to find the anglo 94 PROBLEMS USEFUL IN NAVIGATION AND SURVEYING. ACE, by Case IV. of Oblique Trigonometiy ; the supplement of this angle is the value of the angle ACD ; adding this angle to the observed angle CDA, and subti-acting the sum from 180°, we get the angle CAD. Then in the triangle CAD we have all the angles, and the side AC, to find CD, AD, by Case I. of ObUque Trigonometiy. In like manner we may find BD, in the triangle CBD. EXAMPLE. Suppose we have given, by the map, AB zn 3200 feet, BC = 1330 feet, AC — 1990 feet, angle BAC=12° 34' ; also, by observation, CDB=EAB=:25°, CDA=EBA=:28° ; requued ACD, CAD, AD, CD, BD. To find AE, in the triangle AEB. EAB = 25° EBA= 28 . .Sine 9.67161 Sum 63 180 AEB= 127 Arith. Comp. Sine 0.09765 AB =3200 Log-. 3.50515 AE =1881 Log. 3.27441 To find AEC, ACE, in the triangle AEC. BAG = 12°34.' EAB = 25 00 Sum 37 34=EAC 180 00 AC+AE=3871 AC— AE = 109 AEC+ACE =142 26 i(AEC-(-ACE)= 71°13' Tangent 10.46839 AC -j-AE = 3871... Ar. Co. Log. 6.41218 • AC — AE = 109 Log. 2.03743 i(AEC— ACE)= 4'' 44' ....Tangent 8.91800 |(AEC+ACE)=71 13 SumAEC=75 67 Difference ACE= 66 29; the supplement of this angle ACE is equal to ACD = 113° 31' CDA= 28 GO Sum CAD: 141 31 180 00 : 38 29 To find AD, CD, in the triangle ACD. As CDA = 28°. . . .Arith. Comp. Sine 0.32839 Is to AC = 1990 Log. 3.29885 So is ACD = 113° 31' Sine 9.96234 3887 Lo?. 3.58958 To AD As CDA = 28° . . . .Arith. Comp. Sine 0.32839 Is to AC =1990 Log. 3.29885 So is CAD = 38° 29' Sine 9.79399 To CD =2638 Log. 3.42123 To find BD, in the ti-iang.o BAD. BAC = 12° 34' CAD = 38 29 CDB = 25° 00' CDA = 28 00 Sum BAD 51 03 ADB = 53° 00 As ADB = 53° ... . Arith. Comp. Sine 0.09765 Isto AB =3200 Log. 3.50515 SoisBAD = 51° 03' Sine 9.89081 To BD =3116 Log-. 3.49361 This method becomes defective when the points F, G, approach very near to each other ; to avoid this, we must be careful not to take for the place of observation any point which approaches near to the cu-cumference of a circle which passes through the observed points A, C, B ; because a very small error in the observed angles might then produce a very great eiTor in the result, or place of the observer. Care must also be taken to have both the angles observed at the same point, without allowing tlae boat to drift, m which the observations are made. PROBLEM Vin. Being 96 fathoms from the bottom of a tower, I found Us altitude above the horizontal line draivn from my eye was 15° 10' ; required the elevation above that line. BY PROJECTION. Draw the horizontal line AB equal to 96 fathoms, and peqiendicular thereto, the line BC; make the angle BAC equal to 15° 10', and draw AC to cut BC in C ; then will BC be the height of the tower, 26 fathoms. PROBLEMS USEFUL IN NAVIGATION AND SURVEYING. 95 BY LOGARITHMS. As radius 90° 10.00000 Is to the distance AB 96 fathoms L98227 So is tangent angle A 15° 10' 9.43308 To the height BC 2C.0 fathoms 1.41535 When an object, whose elevation above the horizon is to be detennined, is at a very great distance, it will be necessary to notice the correction arising from tiie curvature of the earth and the refraction, and apply that con-ection to the height estimated by the above method. Thus, if the angular elevation of a mountain whose base was more distant than the limit of the visible horizon, was observed by an instrument of reflexion, the approximate lieight must first be obtained, as in the preceding example, and then the correction of that approximate height for the curvature of the earth, refraction, and dip, must be calculated by the following rule, and added to that height ; tlie sum will be the true height above the level of the sea. Rule. Find in Table X. the number of miles coiresponding to the height of the observer above the level of the sea, and take the difference between that number and the distance of the mountain from the observer in statute miles ; with that difference enter the same table, and find the height in feet corresponding, which will be the correction to be added to the approximate height to obtain the true height of the mountain above the level of the sea. Example. Suppose the distance was 32 statute miles (or 168960 feet), and the obsei-ved altitude 1° 2', the observer being 18 feet above the level of the sea ; required the height of the mountain above the same level. As radius Log. 10.00000 Is to distance 168960 Log. 5.22779 So is elevation 1° 2' Tang. 8.25616 Approximate height 3048 . Correction 398 .Log. 3.4a395 Distance of mountain. 32 Table X. 18 feet 5.61 Difference 26.39 Corresponding Corr. Table X. . . . . 398ft. Sum 3446 is the true height above the level of the sea. PROBLEM IX. I observed the altitude of the top of a lower above the level sand on the sea-shore to be 59°; then, measuring directly from it 98 yards, its elevation loas found to he 44° : requirea the height of the toiver. Let AB represent the height of the tower, C the first station, and D the second ; then we have the angle ACB equal to 59°, the angle ADB equal to 44°, the angle DAC — 59° — 44° = 15°. To fiind the side AC. As DAC 15° Sme 9.41300 Is to DC 98 Log. 1.99123 So is ADC 44° Sine 9.84177 11.83300 9.41300 To AC 263.0 Log. 2.42000 To find the height AB. As radius Log. 10.00000 Is to AC 263.0* Log. 2.42000 So is ACB 59° Sine 9.93307 12.35307 10.00000 To AB 225.5 Log. 2.35307 * The log. AC, by tlie preceding operation waa found to be 2.42000, differing but a little from the log. of 363. 9« PROBLEMS USEFUL IN NAVIGATION AND SURVEYING. PROBLEM X. By observation, I found the angle of elevation of a monument, at one station, to be 21'', and the horizontal angle, at this station, betiveen the spire of the monument and the second station, ivas I'd'' ; the horizontal angle, at the second station, between the spire and the first station, ivas 69° ; the distance between the two stations being 139 yards . required the height of the monument. ji Let AD represent the monument, C the first station, B tlie second ; then the vertical angle DCA is 21°; and the horizontal angles BCD equal to 79°, CBD equal to 69° ; the sum of these two angles being subtracted from 180°, leaves BDC equal to 32°. To find the side CD. As BDC 32° Sme 9.72421 Is to BC 199 Log. 2.14.301 So is CBD 69° Sine 9.97015 12.11316 9.72421 To CD 244.9 2.38895 c To find the height AD. As radius Log. Is to CD 244.9* Log. So is ACD 21° Tang. To AD 94 Log. 10.00000 2.38895 9.58418 1.97313 PROBLEM XL Sailing towards the land, I discovered a light-house just appearing in the horizon, my eye being elevated 20 feet above the sea; it is required to find the distance of the light-house, supposing it to be elevated 200 feet above the surface of the sea. The solution of tliis problem depends on the uniform curvature of the sea, by means of which all terrestrial objects disappear at certam. distances from the observer. These distances may be computed by means of Table X., in which the elevation in feet is given in one column, and the distance at which it is visible is expressed in statute miles in the other column. If the place from which you view the object be elevated above the horizon, you must add together the distances corresjponding to the height of the observer and the height of the object ; the sum will be the gi'eatest distance at which that object is visible from the observer; this process being similar to that in Problem VIII. In the present example, the height of the observer was 20 feet, and the height c.f the oljjcct 200 feet. In Table X. opposite 20 feet is 5.92 miles. 200 feet 18.71 Distance 84.63 statute miles, of about 69^ to a degi'ee ; t!ie distance in nautical leagues, of 20 to a degree, being about 7. PROBLEM XII. Jl man, being on the main-top-gallant-mast of a man-of-icar, 200 feet above the tvater, sees a 100 gun ship she had engaged the day before, hull to ; how far were those ships distant from each other ? A ship of 100 guns, or a first-rate man-of-war, is about 60 feet from the keel to the rails, from which deduct about 20, leaves 40 for the height of her quarter-deck above water. Now, a ship is seen hull to when her upper works just appear. In Table X. opposite 200 feet stand 18.71 40 feet 8.37 Distance 27.08 miles. TliC log. of CD, by the preceding operation, was found to be 2.38895, differing but a little from the los. of 0)1 !' PROBLEMS USEFUL IN NAVIGATION AND SURVEYING. 97 PROBLEM XIII. Upon seeing the Jlash of a gun, I counted 30 seconds, hy a watch, before I heard the report ; how far ivas that gun from me, supposing that sound moves at the rate of 1142 feet per second ? The velocity of light is so gi'eat, that the seeing of any act done, even at the distance of a number of miles, is instantaneous ; but, by observation, it is found that sound moves at the rate of 1142* feet per second, or about one statute mile in 4.6 seconds ; consequendy the number of seconds elapsed between seehig the flash and hearing the report being divided by 4.6, will give the distance in statute miles. In the present example, the distance was about Gh miles, because 30 divided by 4.6 gives Qh nearly. PROBLEM XIV. To find the difference between the true and apparent directions of the wind. Suppose that a ship moves in the direction CB from C to B, while the A. wind moves in its true direction from A to B ; the effect on the shij) Avill be the same as if she be at rest, and the wind blow iu the direction AC with a velocity represented by AC ; the velocity of the ship being represented by BC. In this case, the angle BAG will represent the difference between the true and the apparent directions of the wind ; the apparent beuig more ahead than the true, and the faster the vessel goes, the more ahead the wind will appear to be. We must, however, excejjt the case where the wind is directly aft, in which case the dii-ection is not altered. ^*=^^ It is owing to the difference between die true and apparent directions of the wind, that it appears to shift its direction by tacking ship ; and if the difference of the direc- tions be observed when on different boards (the wind on both tacks being suj^posed to remain constant, and the vessel to have the same velocity and to sail at tlie same distance from the wind), the half difference will be equal to tlie angle BAC. By knowing this, together with the velocity of the ship BC, and the angle BCA, we may obtain the true velocity of the wind ; or, by knowmg the velocity of the wind and of the ship, and the apparent direction of the wind, we may calculate the difference between the true and the apparent directions of the wind. Thus, if the velocity of a ship represented by BC be 7 miles per hour, that of the wind represented by AB 27 miles per hour, and the angle of the vessel's course with die apparent direction of the wind BCA equal to 7^ points ; the difterence between the true and apparent directions of the wind will be obtained by drawing the line BC equal to 7 miles, taken from any scale of equal parts, and making the angle BCA equal to 7h points ; then, with an extent equal to 27 miles, taken from the scale, and Avirh one foot in B, describe an arc to cut the line AC in A ; join AB ; then the angle BAC, being measured, will be the required difference between tlie true and apparent directions of the wind. BY LOGARITHMS. As AB 27 miles Arith. Comp. Log. 8.56864 Is to BCA 7h points Log. Sine 9.99790 So is BC 7 miles Log. 0.84510 To BAC 14= 57' Log. Sine 9.41164 So diat, in diis case, the difference between the true and apjjarent directions of the wind is about li points ; and, by tacking ship and sailing on the other board, as above mentioned, the wmd will appear to cliange its dnections above 2^ points. PROBLEM XV. To measure the height of a mountain by means of the heights of two barometers, taken at the top and bottom of the mountain. Procure two barometers, with a thermometer attached to each of them, in order to ascertain the temperature of the mercury in the barometers, and two other thermome- ters, of the same kind, to ascertain the temperature of the air. Then one observer at the top of the momitaiu, and another at the bottom, must observe, at the same time, * The velocity of sound at 2/2" Fahrenheit is 1090 feet per second, and for cacli additional degree of li'-at add 0.9G to ttiis velocity. 13 98 PROBLEMS USEFUL IN NAVIGATION AND SURVEYING. the heig^ats of the bai'ometers, and the theiinometers attached thereto, and the heights of the detached thermometers, placed in the open au-, but sheltered from tie sun. Having taken these obsei-vations, the height of the upper observer, above the lower, may be determined by the foUowuig rule, which is adapted to a scale of English inches and to Fahrenheit's thermometer : — Rule. Take the difference of the logarithms of the observed heights of the barom- eters at the two stations, considermg the first four figures, exclusive of the index, as whole numbers, the remainder as decimals ; to this difference must be applied the product of the decimal 0.454, by the diffei-ence of the altitudes of the two attached thermometers, by subtracts , if the thermometer be highest at the lowest station, otlierwise adding : the sun n- difference will be the approximate height in English fatlioms. Multiply this b; he decimal 0.00244, and by the difference between the mean of the two altitudes the detached thermometers and 32° ; the product will be a correction, to be added to .le approximate height when the mean altitude of the two detached thermometers exceeds 32°, otherwise subtracted : the sum or difference will be the true height of tlie upper above the lower observer in EngUsh fathoms, which, being multiplied by 6, will be the height in feet. EXAMPLE. Suppose the following observations were taken at the top and at the bottom of a •nountain ; required its height in fathoms. Attached Thermometer. Jbs. at lower station 57° upj)er station 43 Difference 14 Detached Thermometer. .56 .42 Mean. 49 32 Difference ... 17 Barometer. 29.68 inches Log. 14724.6 25.28 Log. 14027.8 Difference 696.8 0.454 X 14 ..6.4 Approximate height 690.4 690.4 X 17 X 0.00244 28.6 Height in fathoms .719.0 99 MENSURATION. PROBLE3I I. To find the area of a parallelogram. Rule. Multiply the base by the peqiendiculai* height ; the product will be the area. JVote. If both dimensions are given in feet, inches, &c., the product will be the nrea, expressed in square feet, scpiarc inches, &c., respectively. If one of tlie dimen- sions be given in feet and the other in inches, the product, divided by 12, will be the answer in squm-e feet. If both dimensions are given in inches, the proiluct will be square inches, which, being divided by 144, will be the answer in square feet. The same is to be miderstood in finding the area of other surfaces. Example I. Suppose the base BC of the rectangular parallelogi-am /i ABCD is 7 feet, and the perpendicular AB 3 feet ; required the area. [ The product of the base 7 feet by the perpendicular 3 feet gives the L area 21 square feet. Example II. Suppose ABCD is a board whose length BC is 22 feet, and breadth AB is 14 inches ; requii'ed the niunber of square feet. The product of the base 22 feet by the l)readth 14 inches is 308 ; this, divided by 12, gives 25} square feet, the sought area. Example III. If BC be 25 inches, and AB 20 mches, requu-ed the area in square feet. The product of the base 25 inches by the perpendicular 20 inches gives 500, which, divided by 144, gives the area 3.47 or 3^^^^^^ squai-e feet. Example lA''. Given the base AD of the oblique angular J?^ parallelogram ABCD, equal to 30 feet, and the perpendiculai' / \ height BE 15 feet ; required the area of the parallelogram. / \ Alultiply the base 30 feet by the perpendicular 15 feet ; /. L the pi'oduct 450 is the area in square feet. -^ -^ PROBLEM II. To find the area of a triangle. Rule. Multiply the base by half the pei-pendicular height, and the product will be the area required. Example. Given the base AC 30 feet, and the peq^endicular BD 20 feet ; required the area of the triangle. The base 30 multiplied by half the perpendicular 10 gives the area 300 squai'e feet. Rule. PROBLE3I III. To find the area of any regular right-lined figure. Reduce the figure to triangles, by drawing diagonals therein ; then find the area of each triangle, and the sum of tliem will be the area of the proposetl figure. O., instead of finding the area of each triangle sej)aratelv, you may find, at one operation, the area of two triangles, having the same diagonal, by nndtiplying the diagonal by half the sum of the perpendiculars let fall thereon. 100 MENSURATION. Example. Required the area of the figure ABCDE, in which CE BE = 22 feet, and the perpendicular AF z= 13 feet, BG = 14 feet, and DH r= 12 feet. The diagonal BE, 22 feet, multiplied by half the pei-pen- dicular AF, 6.5 feet, gives the area of the triangle ABE, 143 squai-e feet ; and the diagonal CE, 33 feet, multiplied by half the sum of the perpendiculars BG, DH, 13 feet, gives the ai-ea of the figure BCDE, 429 feet; this, added to the triangle ABE, 143 feet, gives the whole area 572 square feet. A'K--- 4. -^ PROBLEM IV. To find the area of a circle. Rule. Multiply the square of the diameter of the circle by the quantity 0.7854, and you will have the souglit area. A^oie. Instead of multiplying by 0.7854, you may multiply by 11 and divide by 14 ; the quotient will be the area nearly. This quantity, 0.7854, represents the area of a circle whose diameter is 1 ; the circumference of the same circle being 3.1416 nearly. The proportion of the diameter to the circumference is expressed in whole numbers by the ratio of 7 to 22 nearly, or more exactly by 113 to 355.* r Example. Required the area of a circle ABCD, whose diameter BD is 10.6 feet. The diameter 10.6 multiplied by itself and by 0.7854 gives the sought area, 88.247544 squai-e feet. PROBLEM V. To find the area of an ellipsis. Rule. Multiply the longest diameter by the least, and the product by 0.7854 ; this last product will be the area requu-ed. Example. Required the area of an ellipsis ABCD, whose longest diameter AC is 12 feet, and the shortest diameter BD 10 feet. The product of the two diametei-s is 12 X 10=: 120; this, multiplied by 0.7854, gives the sought area, 94.2480 square feet. The area of a sector of a circle may be found by means of die whole area of the circle obtained in Problem IV., by saying. As 360 degrees is to the angle contained KotHreen the two legs of the sector, so is the whole area of the cuxle to the area of the sector. There are various regular solids. The most noted are the following: — (1.) A Cube, whicli is a figure bounded by sLx equal squares. (2.) A Parallelopiped, which is a solid terminated by six quadiilateral figures, of which the opposite ones are equal and parallel. (3.) A Cylinder, which is a figure formed by the revolution of a rectangular parallelogram about one of its sides. (4.) A Pyramid, which is a solid decreasing gradually from the base till it comes tc a point. There are various kinds of pyramids, accordmg to the figure of their bases. Thus, if the base be a triangle, the solid is called a triangidar pyramid ; if a parallelogram, a parallelogramic pyramid ; and if a circle, a circular pyramid, or simply a cone. The point in which the pyramid ends ia called the vertex, and a line di'a^vn from the vertex perpendicular to the base is called the height of the pyramid. * This ratio may be easily remembered by observing tiiat, if the first three odd numbers, 1, 3, 5, are repeated twice, they will produce the quantity 113355 5 the three first figures of which make tl>e first term of the ratio, and the three last the last term of the ratio. MEiNSUIlATlON. 101 Rule. PROBLEM VI. To find the solidity of a cube. Multiplying the length of a side of the cube by itself, and the product by the *ame length, gives the solidity required ; which will be expressed in cubic feci if the diuiensions be given in feet, but iu cubic inches if the dimensions be given m inches, &:c, Example. If the side AB of the cube be G.3 feet, it is required to determine the solidity. The product of G.3 l)y G.3 is 39.G9 ; this, muUiplied again by 6.3, gives the solidity 250.047 cubic feet. JD A'^ PROBLEM VIL To fuid the solidity of a redangidar parallelopiped.: • :'!^ ', .. Rule. Muhiply the length, breadth, and depth, into each othc ; the product be the solidity required. ' o ' ' ' ' will JBy- EXAMFLE. Suppose, in the parallelopiped ABCDFGHE, the length EF is 3G feet, the breadth EG IG feet, and the depth DF 12 feet ; it is required to find the solidity. The product of the length 36 by the breadth 16 is 576 ; this, multiplied by the depth 12, gives the solidity 6912 cubic feet. PROBLEM VIII. To find the solidity of a cylinder. Rule. Multiply the square of the diameter of the base by the length, and tliia product by the constant quantity 0.7854 ; the last product will be the solidity required. Example. Required the solidity of a cylinder ADHF, whose length DH is 13 feet, and diameter of the base AD 11 feet. The diameter 11, multiplied by itself and by the length 13, gives 1573, which, being muhiiilied by 0.7854, gives the solidity in cubic feet 1235.4342. F A I' jwmmmnwmmm. ]/ ' B PROBLEM IX. \ To find the solidity of a grindstone. Grindstones, in the form of cylinders, are sold by the stone of 24 inches diameter, and 4 inches thick. The number of stones that any one contains, may be obtained by the following rule. Rule. ]\Iultiply the square of the diameter in inches by the thickness in inches, and divide the product by 2304, and you will have the number of stones required. Exaimfle. Required the number of stones in a grindstone whose diameter is 30 inches, and thickness 8 inches. The square of the diameter 36 is 1296, which, being multiplied by the thickness 8 gives 10368. This, divided by 2304, gives 4.5, or 4i stones, the solidity required. This problem may be solved by means of the line of numbers on Gunter's Scale, in a ver}' expeditious manner, by the following rule. Rule. Extend from 48 to the diameter; that extent, turned over twice the same way, from the thickness, will reach to the number of stones required. Thus, in the preceding example, the extent from 48 to the diameter 36, turned over twice, from the thickness 8, will reach to 4.5, or 4i, which is the number of stones sought. 102 MENSURATION. PROBLEM X. To find the. solidity of any pyramid or cone. RuLK. Multiply the area of the base by one thu'd of the perpen- dicular height of the pyramid or cone; the product will be the solidity required. Example I. If the j)yramid have a square base, the side of which is 4 feet, and the perpendicular height 6 feet, it is required to determine the solidity. The area of the base is 4 X 4 rr 16 square feet ; this, being mul- tiplied by one thhd of the height, or 2 feet, gives 32 feet, the solidity requu'ed. E\AMPLK .II. If the diameter of the base of a cone be 10.6 feet, and, tlie .p^ji-peijdicular height 30 feet, it is required to find the splidity. .' .The. area of" this ^ base was found in Problem IV. equal to 88.347544 ; this, muftii)lied by one tliird of the height, or 10 feet, gives the solidity required, equal to 882.47.544 cubic feet. Having obtained, by the foregoing rules, the number of cubic feet in any body, you may find the corresponding number of tons by dividuig the number of cubic feet l)y 40, which is the number of cubic feet contained in one ton. Thus, the solidity of the above- mentioned cone, 882.47544, being divided by 40, gives 22.06188C, which is the number of tons in that cone. PROBLEai XI. To find the tonnage ofi a $hip. By a law of the Congress of the United States of America, the tonnage of a ship is to be found in the following manner : — If the vessel be double-decked, take the length tliereof from the fore part of the main stem to the after pai-t of the stern-post above the uj)per deck ; the breadth thereof at the broadest i)art above the main wales ; half of this breadth shall be accounted the depth of such vessel ; then deduct from the length three fifths of the breaddi, multiply the remainder by the breadth, and the product by the depth ; divide this last product by ninety-fivCj and the quotient will be the true content or tonnage of such vessel. If the vessel be single-decked, take the length and breadth as above directed in respect to a double-decked vessel, and deduct from the length three fifths of the breadth, and taking the depth from the uuder'side of the deck-plank to the ceiling of the hold, multiply and divide as aforesaid ; the quotient will be the true content or tonnage of such vessel. Example. Suppose the length of a doul)le-decked vessel is 80 feet, and the breaddi 24 feet, what is her tonnage ? Three fifths of the breadth, 24 feet, is 14.4 feet, which, being siditracted from tlie length, 80 feet, leaves 65.6. This, multii)lied by the breadth, 24 feet, gives 1574.4 ; this multiplied by the dcptli, 12 feet (half of 24), gives 18892.8, which, being divided by 95, gives the tonnage 198.9. Car]>enters, in finding tlie tonnage, midtiply the length of the keel by the breadth of the main beam and the depth of the hold in feet, and divide the product by 95; the quotient is the number of toi^s. In doidile-decked vessels, half the breadth is taken for the depth. (^ 103 GAUGING. Having found the number of cubic inches m any body, by the preceding rules, you may thence determine the content in gallons, bushels, &c., by dividing that number of cubic niches by the number of cubic inches in a gallon, bushel, &c., respectively. A tvine gallon, by which most liquors are measured, contains 231 cubic inches. A beer gallon, by which beer, ale, and a few other liquors, are measured, contains 282 cubic inches. A bushel of corn, malt, &c., contains 2150.4 cubic inches; this measure is subdivided into 8 gallons, each of which contains 2G8.8 cubic inches. In all the folloiviyig i-ules, it ivill be supposed that the dimensions of the body are given in inches, and decimal parts of an inch. PROBLEM I. To find the number of gallons or bushels in a body of a cubic form. Rule. Divide the cube of the sides by 231, the quotient wHl be tlie answer in wine gallons ; or by 282, and the quotient will be tiie answer in beer gallons ; or bj 2150.4, and the quotient will be the number of bushels. Example. Required the number of wine gallons contained in a cubic cistern, the length of whose side is G2 inches. Multii)lying G2 by itself, and the product again by G2, gives the solidity 238328 which, being divided by 231, gives the content 1031| wine gallons. PROIJLEM II. To find the number of gallons or bushels contained in a body of the form of a rectangular parallelopiped. (See figure of Problem VII. of Mensuration.) Rule. Rlultiply the length, breadth, and depth, together ; divide this last product by 231 for wine gallons, by 282 for beer gallons, or by 2150.4 for bushels. Example. Required the number of wine gallons contained in a cistern ABCDFGHE (see fig. Prob. VII. of Mensuration) of the form of a parallelopiped, whose length EF is GG inches, its breadth FG 35 inches, and its depth DF 24 inches. Multiplying the length G6 by the breadth 35 gives 2310 ; multiplying this by the depth 24"gi\'es the solidity 55440, which, being divided by 231, gives 240 wine gallons. PROIJLEM III. To find the mmiber of gallons or bushels contained in a body of cylindrical form. Rule. Multijjly the square of the diameter by the height of the cylinder, and divide the product by 294.12 ; the quotient will be the number of wine gallons. If you divide l)y 359.05, tlie quotient will be the number of ale gallons ; and if you divide by 2738, the quotient will be the number of bushels. mte. These divisors are found by dividing 231, 282, and 2150.4, by 0.7854 respectively. Example. Required the number of wine gallons contained in the cylinder AFHD (see the fig. of Problem VIII. of Mensuration), the diameter AD of its base being 26 inches, and length DH 18 inches. The diameter 2G multiplied by itself gives G7G ; multiplying this by the length 18 gives the solidity 121G8, which, being divided by 294.12, gives the answer 41 wine gallons nearly. liJ4 GAUGING. PROBLEM IV. To find the number of gallons or bushels contained in a body of the form of a pyramid or cone. (See figures of Problem X. of Mensuration.) Rule. Multiply the area of the base of the p}Tamid or cone by one third of its perpendicular height ; the product, divided by 231, will give the answer in wine gallons. If it be divided by 282, the quotient will be the number of beer gallons ; or by 2150.4, the quotient will be the number of bushels. Example. Requii'ed the number of beer gallons contained in a pyramid DEFGK (see fig. Prob. X. Example I.), whose base is a square EFGK, a side of which, as EF, is equal to 30 inches, and the perpendicular height of the pyramid is 60 mches. The square of 30 is the area of the base 900 ; this, being multiplied by one thu'd of the ahitu'de 20, gives the soUdity 18000, which, being divided by 282, gives the answer in beer gallons 63.8. PROBLEM V. To find the number of gallons or busliels contained in a body of the form of a frustum of a cone. (See the figure below.) Rule. Multiply the top and bottom diameters together, and to the product add one third of the square of the difference of the same diameters ; multiply this sum by the perpendicular height, and divide the product by 294.12 for wine gallons, by 359.05 for ale gallons, or by 2738 for bushels. a£- •■ -^-B Example. Given the diameter CD of the bottom of a fifustum of a cone 36 inches, the toj) diameter AB :z= 27 inches, and the perpendicular height EF 50 inches ; required the contents in wine gallons. The product of the two diameters, 36 and 27, is 972 ; their difference is 9, which, being squared and divided by 3, gives 27 ; adding this to 972 gives 999, which, being multiplied by the height 50, gives the solidity 49950; dividing this by 294.12 gives the content in wine gallons 169.8. PROBLEM VI. To gauge a cask. To gauge a cask, you must measure the head diameters, AF, CD, and take the mean of them when they differ ; measure also the diameter BE at the bung (taking the measure within the cask); then measure the length of the cask, making due allowance for the thickness of the heads. Having these dimensions, you may calcu- late the content, in gallons or bushels, by the following rule : — Rule. Take the difference between the head and bung diameters ; multiply this by 0.62, and add the product to the head diameter ; the sum will be the mean diameter ; multi])ly the square of this by the length of the cask, and divide the product by 294.12 for wine gallons, by 359.05 for beer gallons, or by 2738 for bushels. The quantity 0.62 is generally used by gangers in finding the mean diameter of a cask. But if the staves are nearly straigiit, it will be more accurate to take 0.55, or less ; * if, on the contraiy, the cask is full on the quarter, it will be best to take 0.64 or 0.65. Example. Given the bung diameter EB m 34.5 inches, the head diameter AF z= CD =r 80.7 inches, and the length 59.3 hiches ; required the number of wine gallons this cask will hold. The difference of the two diametere, 34.5 and 30.7, is 3.8 ; this being multi])lied by 0.62, gives 2.4 nearly, to be added to the head diameter 30.7 to obtain the mean diameter 33.1. The square of 33.1 is 1095.61 ; multiplying this by the length 59.3, gives the solidity 64969.673 ; dividing this by 294.12, gives the content in wine gallons 220.9. * In the example to Problem V. preceding (which may be esteemed as the half of a hog-shead with staves perfectly straight), the multiplier is only 0.51. For this, being multiplied by 9 (thrf difference between AB and CU), produces 4.59 or 4.6 nearly ; adding this to 27 gives 31.G, whose square, being luultiplied by 60, and the product divided by 294.12, gives 170 gallons nearly. GAUGING. 105 To gauge a cask hy means of the. line of numbers on Gunler^s Scale, or that 07i the Callipers used by gangers. Make marks on the scale at the points 17.15, 18.95, and 52.33, whicli are the square roots of 294.12, 359.05, and 2738, respectively. A brass pin is generally fixed on the callipers at each of tliese points, which are called the gauge points. Having prepared the scale in this manner, you may calculate the number of gallons or bushels by the Allowing rule : — Rule. Extend from 1 towards the left hand to 0.62 (or less, if the staves be nearly iti'aight) ; that extent will reach from the difference between the head and bung diameters to a number to the left hand, which is to be added to the head diameter to get the mean diameter ; then put one foot of the compasses upon the gauge point (which is 17.15 for wuie gallons, 18.95 for ale gallons, and 52.33 for bushels), and extend the other to the mean diameter ; this extent, turned over twice the same way, from the length of the cask, will give the number of gallons or bushels respectively. In the preceding example, the extent from 1 to 0.62 will reach from 3.8 to 2.4 nearly, which, being added to 30.7, gives the mean diameter 33.1 ; then the extent from the gauge point 17.15 to 33.1, being turned over twice from the length 59.3, will reach to 220.8 wine gallons. If we use the gauge point 18.95, the answer will be in ale gallons ; and if we use 52^, the answer will be in bushels. 14 106 SURVEYING. Land is generally measured by a chain of 66 feet in length, divided mto 100 equal parts called link^, each link being 7.92 inches. A pole or rod is 16^ feet, or 25 links, m length. Hence a square pole contains 272^ square feet, or 625 square links. An acre of land is equal to 160 squai'e poles, and therefore contams 43560 square feet, or 100,000 square links. To find the number of square poles in any piece of land, you may take the dimen- sions of it in feet, and find the area in square feet, as in the preceding problems ; then divide this area by 43560, and the quotient will be the number of acres ; or by 272.25, and the quotient wWl be tlie number of square poles. If the dimensions be taken in links, and the area be found in square links, you may obtain the numl)er of acres by dividing by 100000 (that is, by crossing off tlie five right-hand figures), and the number of square poles may be obtamed by dividing by 625. PROBLEM L '"o Jlnd the numher of acres and poles in a piece of land in the form of a rectangular parallelogram. Rule. Multiply the base by the pei-pendicular height, and divide by 625 if the dimensions be taken in Ihiks, or by 272.25 if they be taken in feet ; the quotient will be the number of poles. Dividmg tliis by 160, we get the number of acres. Example. Suppose the base BC (see the figure of Ex. I. Pro!). I. of Mensuration) of the rectangular parallelogi-am ABCD is 60 feet, and the perpendicular AB 25 feet required the area m poles. The product of the base 60 by the perpendicular 25, gives the content 1500 square feet ; and by dividuig it by 272.25, we obtain the answer m square poles 5.5, nearly. PROBLEM II. To find the numher of acres and poles in a piece of land in the form of an ohlique-angidar parallelogram. (See the figure of Prob. I. Ex. IV. of Mensuration.) Rule. This area may be found in exactly the same manner as in the preceding problem, by midtiplying the base AD by the perpendicular height BE, and dividing ijy 625 when the dimensions are taken in links, or by 272.25 when taken in feet ; the quotient will be the answer in poles, which, being divided by 160, will give the answer in acres. Example. Suppose the base AD is 632 links, and the perpendicular BE 326 luiks; requii'ed the number of poles. Multiiily tlie base, 632 links, by the perpendicular, 326 links ; the product 206032, divided by 625, gives the answer in poles 329.7. PROBLEM III. To find the numher of acres and poles in a piece of land of a triangxdar form. Rule. ]Multij)ly the base by the pei-])Pndicular height, and divide the product by 1250 when the dimensions are given in luiks, or by 544.5 when tliey are given in feet ; the quotient will be the answer in poles. JVote. Instead of dividing by 1250, you may multiply by 8 and cross oflF the four right-hand figures. SURVEYliXG. 107 Example. Given the base AC (see figure of Problem II. of Mensuration) equal to JOO feet, and the perpendicular BD 150 feet ; required the area in poles. Multiply the base 300 by the perpendicular 150; the product 45000, divided by 544.5, gives tlie answer m poles 82.G. PROBLEM IV. To Jind the number of acres and poles in a piece of land of any irregular right-lined figure. Rule. Fuid the area, as in Pi-oblem III. o( Mensuration, by drawing diagonals, and reducing the figure to triangles ; the base of each triangle being nndtii)lied by the perpendicular (or by the sum of the perpendiculars falling on it), antl the sum of all these products divided by 1250 when the dimensions are given in links, but by 544.5 when in feet, will give the area of the figure in poles. Example. Suppose that a ])iece of land is of the same form as the figure in Prob. III. of Memuration, and that BE = 23 feet, CE = 33 feet, AF =: 13 feet, BG = 14 feet, and Dll zz: 12 feet; it is required to find the area in poles. The product of BE 22 feet, by AF 13 feet, gives double the triangle ABE 286 square feet ; and the diagonal CE 33 feet, multii)lied by the sum of the i)eri)endicular8 BG, DH, 26 feet, gives double the figure BCDE, 858 square feet ; the sum of this and 286, being divided by 544.5, gives the ai'ea 2.1 or 2-tj. poles. To find the content of afield by the Table of Difference of Latitude and Departure. This method is simple, and much more accurate than by projection, the boundaries being straight Imes whose bearings and lengths are known. The rule for making these calculations is as follows : — RULE. I. Begin at the western point of the field, as at the point A in the figure Prob. III. of Mensuration, for a point of departure ; and mark down, in succession, the bearings and lengths of the boundary lines AB, BC, &.C., as courses and distances in a traverse table. Fhid the corresponding differences of latitude and departure by Table I. or II. (or by logarithms), and enter them in their respective columns N. S. E. W. as in the adjoined table. Courses. Dist. N. S. E. W. Mcr. Dist. M. Korlh Jircas. South Jircas. N. 58° E. E. 6 S. S. 17 W. \V. N. 42° 35' W. 19. 20. 20. 20. 15.1 10.1 11.1 2.1 19.1 16.1 19.9 5.8 20.0 10.2 10.1 36.0 30.2 10.2 00 16.1 52.1 0G.2 40.4 10.2 1G2.61 113.22 109.41 1264.42 21.2 21.2 36.0 36.0 275.83 Half, 1373.83 275.83 1098. 549. 2. Find the departures or meridian distances of the points B, C, &c. from the point A, by ailding the departures when east, but subtracting when west, and mark them respectively against the bearings, hi the column of meritlian distance. 3. Place in the fii-st line of the column M the first meridian distance 16.1, and, in the following lines, the sum of the meridian distance which stands on the same line and that innnediately above it. Thus on the second line, I put 52.1, which is equal to the smn of 16.1 and 36.0. On the third line, 66.2 = 36.0 + 30.2, &c. 4. jMultiply the numbers in the column M by the differences of latitude in the same horizontal line, and place the product in the column of areas marked north or south, according as the difference of latitude is north or south. Thus in the first number in the column M is 16.1, which, being multiplied by the corresponding difference latitude 10.1 N., produces the north area 162.61. The second value of M .52.1, multii)lied by the second difference of latitude 2.1 S., produces the south area 109.41. The third values 66.2 and 19.1 S. produce the south area 1264.42. The fourth difference of i08 SURVEY liS'O. latitude is 0, wliicli, being multiplied by the foiu-th nieridiun distance 40.4, ])rocluces for the corresponding area, as is the case whenever the bearing is east or west, &c. 5. Add up all the north and all the south areas ; half their difference will be the area of the field in square measures of the same name as those made use of in meas- uring the lines, whether feet, links, or chains, &c. Thus the sum of all the nortli areas is 275.83, that of the south 1373.83 ; their difference is 1098, half of which is 549 square feet, the area of the given field. It may be obsei-ved that the hearings and lengths of the boundary lines in this example, are not exactly the same as those in Problem III. of Mensuration, which is the reason of the difference between the area above calculated and that found in Problem III. by dividing the field into triangles. If it be necessary, the differences of latitude and departure may bo taken to one decimal place farther, by entering the table with ten times the length 19, 20, &c., and taking one tenth of the corresponding differences of latitude and departure. In the above calculations we have supposed the survey to have been made with accuracy, in which case the simis of the differences of latitude in the columns N. S. must be equal to each other; also the sums of the departures in the columns E. W. This is the case in the above example, Avhere the sum of the differences of latitude is 21.2, and the sum of the departures 36.0 : but it most frequently happens that the numbers do not agree ; in which case the work must be carefully examined, and if no mistake be found, and the error be great, the place must be surveyed again ; but if the eiTor be small, it ought to be apportioned among all the differences of latitude and departure, in such manner as to produce the required correction with the least possible changes in the given numbers. The method of doing this was explained by me in the fourth number of the Analyst, in answer to a prize question of Professor Patterson, and is as follows : — Find the error in latitude, or the difference between the sums of southing and northing ; also the sum of the boundary lines, AB, BC, &c. Then say, As this sum is to the error in latitude, so is the length of any particular boundary to the correction of the corresponding difference of latitude, additive if in the column whose sum is the least, otherwise subtractive. The corrections of the departure are found by the same rule, except changing difference of latitude into departure. Thus, in the adjoined exam})le, the sum of the boundary lines is 161.6, the error of latitude is 0.10, and of departui-e 0.08 ; Bearings. Lengths. N. S. E. W. Corrections. Corrected Values. N. 0.02 .02 .02 .02 .02 E. N. S. E. W. N. 45° E. S. 30 W. S. 5 E. W. N.20 E. 40. . 25. 3G. 29.6 31. 28.28 29.13 21.65 35.86 28.23 3.14 10.60 12.50 29.60 0.02 .01 .02 .01 .02 23.30 0.02 29.15 21.63 35.84 28.30 3.16 10.62 42.08 12.49 29.59 1G1.6 57.41 57.51 57.41 42.02 42.10 42.02 0.10 0.08 57.47 57.47 42.03 Error, .10 Error, .08 and the corrections of the difference of latitude and departure are found by the following proportions : — 161.6 titudc. :: 40 0.02 :: 25 0.02 :: 36 0.02 :: 29.3 0.02 :: 31 0.02 Departure. 161.6 : 0.08 40 0.02 25 0.01 36 0.02 29.6 0.01 31 0.02 The first correction of latitude 0.02 is to be added to the first latitude 28.28, because it is in the column whose sum 57.41 is less than the other 57.51, so that the first * The boundary lines in this example are so nearly of an equal length, that the corrertion of the differenoe of latitude (taken to the nearest decimal) is 0.02 for each of them ; but in general they will be different. The table of difference of latitude and departure may be made use of in (inding: these corrections, thus : — Seek in the table till the first term IGl.G (or 1G2) is found in the distance column to correspond to the second term 0.10 (or 10) in the departure column ; thus opposite the third term 40 i'i, 36, (Sic, will be the sought corrections, as is evident. SURVEYING. 109 corrected difference of latitude is 28.30. The second is the difference between 21.65 and tlie second correction 0.02, because 21.65 is in the greatest column ; the corrected value is therefore 21.63. The third is found in the same manner to be 35.86 — 0.02 := 35.84. The fourth corrected difference of latitude is simply the fourth correction 0.02 placed in the colunni N, because tlie sum in tliat column, 57.41, is the least, and the fourth difference of latitude in the original table is 0. The fifth is the sum of 21).13, and the fifth correction 0.02, making 29.15. These are placed in their proper colunms in the corrected vahies. In a similar manner the first departure is equal to the sum of 28.23 and the first correction 0.02, which is equal to 28.30. The second is the difference between 12.50 and the second correction 0.01, making 12.49 ; and so as for the others, taking the sum when the departure is in the cohunn whose sum is the least (which, in the present case, is the east), and the difference when in the other column. In the traverse table thus corrected, the sum of the differences of latitude is 57.47 in both columns, and the sum of the departures 42.08. Having corrected the vahies of this traverse table, you must find the meridian distances, the column M, the north and south areas, &c., as in tlie former example. In projecting a survey of this kind, where there is a small error, you must plot off as usual the boundary lines AB, BC, CD, &.C., and it will be found that the termination of the last line AE will not fall exactly in the point A, but will be at a point near it, which we shall call a. To correct this error, you must draw through the points B, C, D, &;c., lines parallel to aA, in the du-ection from a to A, of such lengths as to be to Aff, as the distances of those points respectively from A (measm-ed on the boundary ABCD, &c.) are to the whole length of the boundary line ; through tiiese points draw the corrected lines terminating on A. The 3Ianncr of Surveying Coasts and Harbors. From what has been said in the preceding problems, the intelligent reader will readily perceive the method of surveying a coast or harbor. But as this is an impor- tant subject, we shall enter more fully into an explanation of the different methods which may be used. To take a draught of a coast in sailing along shore. Having brought the ship to a convenient place, from which the principal points of the coast or bay may be seen, either cast anchor, if it is convenient, or lie-to as steady as possible ; or, if the coast is too shoal, let the observations and measures be taken in a boat. Then, while the vessel is stationarj', take, with an azimuth com[)ass, the bearings, in degrees, of such points of the coast as form the most material projections or hollows.* Write down these bearings, and make a rough sketch of the coast, observing carefully to mark the points, whose bearings are taken, with letters or numbers, for tlie sake of reference. Tlien let the ship or boat run in a direct line (which must be very carefully meas- ured l)y the log, or otherwise) one, two, or three miles, until she comes to another situation, from which the same points, before observed, can be seen again with quite different bearings. Then let the vessel lie steady, as at the former station, and observe again the bearings of the same points, and make a rough sketch of the coast. This sketch may !)e made more accurately while the vessel is running the base line. To describe the chart from these observations, you must, in some convenient part of a sheet of paper, draw the magnetic meridian, and lay off the several bearings taken at the first station, marking them with their proper letters or numbers. Lay down also the bearings taken from the second station. Draw a line to represent the ship's run both in length and course, and from that end of the line expressing the fii-st station, draw lines parallel to the respective bearings taken from that end ; also from the other enil draw lines parallel to the bearings taken at that end, and note the intei-section of each pair of lines directed to the same point ; and through these intersections draw by hand a ciu'ved line, observing to wave it in and out as near as can be like the trending of the coast itself Then mark off the variation of the compass from the north end of the magnetic meridian, towards the right hand if it be west, or towards the left hand if it be east, and draw the ti-ue meridian through that point and the centre of the circle. * Tn taking the bcarinn^s, if the vessel has much motion, the mean of several observations should be taken. 110 SURVEYIiNG. eacli part draw tlie appearance of the land marked in the sketclies, distiii- lic rocky sliore, highland, l)each. &c., as in Plate V. or VIII. Thus the sand Against guishing the ^ , ,. . beaches may be marked asin Plate VIII. ngure 8, and the rocky shore as in figure 9, &c. Put in the saveral soundings, at low water,* in small figures, distinguishing whether they are fathoms or feet. Show the time of high water, on the full and change days, by Roman figures, and note the rise of the tide in feet. The direction and velochy of the flood tide are to be observed ; which may be done by heaving the log when the ship or boat is at anchor, and the direction is to be represented by an arrow. Insert a compass and a scale of miles or leagues, such as the vessel's run was laid domi by. Add the name of the place, and the latitude and longitude, as true as can be obtained. If there are shoals or sands on the coast, let them be observed in a boat, sailuig round them, keeping account of the courses, distances, and soundings, f But to put them in tlie draught, the observer in the boat must take the bearings of two pohits on the coasts (tlie bearings of which have been taken from the ship) from souie part of each sand or shoal so sailed round ; or the bearing of the boat at some part of the shoal, or of some beacon in that place, must be taken by the ship at each of the stations where the bearings of the shore Avere taken from the ship ; for by either of these means, one point of the sand being obtained, the rest of it can be laid doAvn from the observations taken in the boat. Rocky shoals may be marked on the chart as in Plate Vlll.figure ll,and sand-banks as in figure 10. If tlie coast be a bay or harbor, winding in such manner that all its parts cannot be seen at two stations, let as many bases or lines be run and measured exactly as may be found necessary, observing that the several distances run should join to one another, in the nature of atraverse, that each new set of objects or points observed should be taken from two stations at the ends of a known distance, and that the objects whose Dearin"-s are taken do not so much extend beyond the Umits of the base as to make ano-les'^vith it less than about h or % of a point, but rather resei-ve such objects for the next measured base line; for when lines lie very obliquely to one another, their intersections are not easily ascertained. If any particular parts of the harbor cannot be conveniently seen from either of the stations, take the boat into those places ; having well examined them, and made sketches thereof, estimating tlie lengths and breadths of the several inlets, either by the rowing or sailing of the boat, take as many bearings, soundings, and other notes, as may be thought necessary; then annex these particular views, in then- proper places, in the general draught. If there are any dangerous sands or rocks, besides inserting them in their proper places, you must see if there be any two objects ashore (such as a church, mill, house, noted cliff. Sec.) which appear in the same right line when on the shoal, and these objects must be noted on your chart. If none can be found, you must take the bearings of some remarkable points, and note them on your chart. By this means we may know how to avoid the danger. We must mark in the draught the kind of bottom obtained in sounding, whether mud, sand, shells, coral, rocky ground, &c. ; and where there is good anchorage, draw the figure of au anchor ; also, if there is any particular channel more convenient than another, it is to be pointed out by Unes drawn to its entrance from two or more noted marks ashore. The ])ositioiis of objects, taken by a magnetic compass, being liable to great uncer- tainties, as is well known to those who have had any experience, especially at sea, it has been recommended to observe only the bearings of the station-lines by the compass, and then measure the angles which the other objects make with these lines by a quadrant or sextant, which, for this purpose, must be held in a horizontal position. EXAMPLE I. (See Plate VII. fig. 1.) Suppose, in a sMp at A, we obsei-ve the bearings of the most remarkable points of a bay, C, D, E, F, G, II, and I, and then sail S. 04° E. li miles to B, and at B observe the beaiin"-s of the same points ; it is required to construct the chart. * If the soiiiuliii E. L E.N.E L N. N. E. h E. Draw the east line AB equal to 1200 fathoms ; from each end of this line draw the lines AC, AD, AE, AF, AG, BC, &c., at their respective bearings ; the points of intei-section will give the points C, D, E, F, and G. From the point D (which was found in this manner) draw the N. W. line DH equal to 680 fathoms, and through these points draw the lines DI, DK, DL, HI, &c., at their respective bearings ; the points of intersection of the corresponding lines will be the situation of the points I, K, L. Between these remarkable pomts, draw the outlines of the land, conformable to your rough draught. In order to determine the situation of the point M, which was seen too obliquely from the bases AB, DH, you may take the bearing of that point from B, and then from G (whose situation has been determined by bearings taken from the points A, B) ; the intersection of the lines BM, GM, will determme the situation of M. Method of surveying a small hank or shoal ichere great accuracy is required. The method of determining the extent and situation of shoal gi'ound by sailing round it, and keeping an account of the courses and distances sailed, is well adapted to the taking of an extensive survey, or to the exploring of a large bank, where great accuracy is not required. But the difficulty of ascertaining with precision the courses and distances sailed (which are liable to error on account of the tides, currents, and the different velocity of the boat at different times, owing to the unsteadiness of the wind) prevents this method from being sufficiently accurate to be used in exploring a dangerous shoal or bank at the entrance of a narrow channel of a harbor, or any other place where the exact form of the shoal is to be found ; and if to obtain the necessaiy degree of coirectness, the bearings of two remarkable objects are taken at every time of sounding, the time expended in taking the observations, if there be only one observer, v/ill be increased beyond all reasonable bounds. To obviate these difficulties, we may use either of die following methods, by which the necessaiy observations for determining the situation of the boat, can be made as fast as the soundings are taken. First Method. Procure a large sail-boat with a high mast, and a small row-boaL Bring the sail-boat to anchor on the bank which is to be exjjlored, and take accurately the bearings of two remarkable points of land, or other objects, whose situation has already been determined by ol «sei-vations taken on shore, or in sailing along the land. By this means the situation of the sail-boat may be accurately marked on the chart. Then enter the small boat, and row from the other in any particular direction, observing to keep the mast of the boat to bear upon any point of the compass, or (which is much more accurate) to keep the mast of tlie boat to range on any particular point of land, J' IT ji 1" :e :r J 3^ &. P7^ • 1 j 1 1 ^ 1 o,^ ^..^■^ 1 ! i 1 . 1 1 ■ 1 1 ' : 1 1 V -, '. M 1 i I 1 . 1 1 1 I i : I 1 1 i 1 ' ■ ! ► '■ . 1 1 ' Ml! ; 1 ' ^ • i ! ; ! 1 i • 1 III:. ' '• . j i \ \ ' \ 1 ! I : ! ' ' i vo •: : ! ' 1 t 1 ' ^ ! ; ' ' ' 1 I ! , ! 1 / ' 1 ' •■i : i : 1 , ^Mrl 1 1 ' \ i 1 ' ' ' i : 1 1 1 1 i \ ; 1 i 1 i ' 1 1 ' 1 1 ! / \ ' ■ ' M ' r / 1 \ ; 1 1 \ ' i 1 i \ ■ 1 j ■ III A 1 / . 1 ■ ' \ 1 ■ i 1 t ', \ \ ' 1 \ ! i 1 IS-? 11 1 1 . : ' 1 \ i 1 1 1 i ■' 1 ' 1 \ ' I : ; 1 ': 1 1 \ 1 ■ ' ! 1 1 1 I ' \ i LTvra^f L'OOIi / ' ' 1 • \ ; ; 1 '/ 1 ' 1 ! ' 1 \ [ ; 1 1 I ' 1 j ' / j 1 1 \ 1 / i I i 1 ' / 1 \ ! ' / 1 : 1 ID ■ j / ' ■- i ; \ J ! 1 1 J i t i \ 1 ' ! 1 1 1 1 1 l\ ' 1 1 1 1 j \ L 1 i 1 , 1 1 \ ' 1 1 1 . i_ _ V i ' ' 1 ' 1 1 1 1 1 MM Mm ] V 1 1 ; III'' 1 \ ' 1 ■ / 1 ■' hr^''-- 1 1 /. V 1 1 ' ; i i \ I 1 / ■ ! ' 1 1 1 1 l\ / 1 s. 1 1 ^ i 1/ " \ xi r ' 1 ~ M \1 1 i/i 1 t -^ ' i\i ' f ! ^. M^ 1 1 V, / i , '• .\ . ' ' ' .1 ' ' ■ i' : i i =■ ( 1 ' ■' i' -■- ' 1 ; ] 1 ' 1 1 1 ill i 1 L i 1 1 1 P 1 ' t 1 1 ■ ' ■ ' 1 t : ~^ 1 _ K.x- G^v.l^ r.i 'XT, i a ci TIDES. 121 TABLE A. Showing the rate of rise and fall of the tide at New York and Liverpool. Ilours before Height of tide. or after Low water. New York. Liverpool. brs. ft. ft. (G 4.2 18.9 5 3.7 1G.2 |. 4 2.9 10.4 3 1.8 6.2 P 2 0.9 3.0 1 0.2 0.9 0.0 0.0 ri 0.5 1.8 2 1.6 5.6 t^ 3 2.7 11.0 <5 4 . 3.6 16.1 5 4.1 19.7 6 4.4 20.7 opposite to each hour, from low water is shown the height which the level of the water would mark upon a staff the of Avhich was at low water. This curve or this table will enable the navi- gator to conjecture the probable rise and fall from low or high water at ports where the rise and fall is about the same as at New York or at Liverpool, but will not apply to others. If watching this tide staff from day to day in some port upon our coast we should note the time of high and low water, and the height be- ginning wilh, say, two days after change day of the moon, and continuing for a lunar month or twenty-eight days, we should find that on that day the lunitidal interval was, nearly the average of all which we would obtain in the course of the month, and that the water rose higher and fell lower than at any otlier high and low water. These are spring tides. The interval would go on decreasing until two days before the first quarter, when it would reach its least value. The height of high water would de- crease, and of low water increase, until one day after the first quarter, when the one would reach its least and the other its greatest height, corre- sponding to neap tides^ or least rise and fall of the water. From the period of its least value to three days before the full the lunitidal interval would increase and then decrease, and so onward to two days af'ter the full, when the interval would have its average value again, and the heights would again correspond to spring tides. The corresponding changes in the lunitidal intervals and heights take place from the full to change, passing through the moon's third quarter. It is hardly necessary to remind the navigator that at change the moon and sun cross the meridian together, or the hour of transit is hrs., and that at the first quarter the hour of transit (moon's southing) is 6 hrs., at the full, 12 hrs. This change in the lunitidal interval runs its course from change to full or full to change, that is, m a half lunar month ; it is 'hence called the half monthly inequality, and is in general the largest of the changes in the lunitidal interval, wliich must be taken into account. If there were no changes in the lunitidal interval, it would be very simple to deter- mine the time of high or low water at a place. A table of intervals and an almanac showing the time of transit, or as it is sometimes called in the almanacs the time of the moon's southing, would be all that is necessary. Suppose we wish to determine the time of high water at Boston on the 12 th of December, 1859. From the table of es- tablishments. No. LV., we take that of Boston, llh. 27m.; from a Boston almanac, the time of the moon's upper transit on that day Ih. 59m., A. M., adding the two numbers we have 13h. 26i-ri., or Ih. 26m., P. M., as the time of high water. The corresponding low water is 6h. after, or more exactly 6h. 13m. So, if the heights did not change, one number in the table would give us the rise and fall. This supposes that we had an almanac of the port at which we desired to know the time of high water, but as this would usually not be the case, we must take our result from the Nautical Almanac, with which we are provided. This referring to the time of transit of the moon over the meridian of Greenwich, and to the same meridian for the longitude, 2m. must be added to the time of transit at Greenwich for every hour of west longitude, and subtracted for every hour of east longitude. The same result may be had from the table B, where the numbers to be added to the time of the moon's transit are given for every ten degrees of longitude. Rule I.— Find the time of the moon's coming to the meridian of Greenwich, on the given day in the Nautical Almanac. Enter Table B and find the longitude of tiie given plaee in the left han;i column, corresponding to which is a number of minutes to be ap- plied to the time of passing the meridian at Greenwich, by adding when in luest longi- tude, but subtracting Avhen in east longitude ; the sum or difference will be nearly the time that the moon passes the meridian of the given place. To this corrected time add the time of high water or full sea from Table LV. The sum will bo the time of high water on that day. Example I. — Required the time of high water at Charleston (S. C.), November 19, 1859, in the afternoon, civil account. From the Nautical Almanac we find the moon's meridian passage at Greenwich, November 18, at 19h. 26ui., which corresponds to 7h. 16 122 TIDES. 26m., A. M., of the 19th day by civil account. From Table LIV. we have the longitude of Charleston 79° 54' W., Avhich, for this purpose, may be assumed as 80°. Entering Table B with 80°, we find the correction of the moon's passing the meridian to be 11 minutes, which is to be added as the longi- tude is west. The moon's meridian passage at Charleston is therefore at 7h. 3Tm., A. M. Adding to this the lunitidal in- terval 7h. 13m. from Table LV. we obtain 14:h. 5Qm., or 2h, 50m., P. M., as the time of high water at Charleston in the afternoon of November 19, 1859. Example II. — Required the time of high water at Portland (]\Iaine), Deceml^er 13, 1859, in the afternoon, civil account. The jSTautical Almanac gives the moon's meridian passnge at 14h. 47m. on the 12th, corresponding to 2h. 47m, A. M., on the 13th. The longitude of Portland is 70° 12' W., in time (Table XXI.) 4h. 41m. At the rate of two minutes for every hour of west longitude we should add 9m. to the Greenwich time of the moon's meridian passage, giving it for Portland at 2h. 56m. Adding the lunitidal interval from Table LV. llh. 25m., gives 14h. 21m., or 2h. 21m.j P. M., for the time of high water on December 13th. These results would be the time of high water, did not the lunitidal interval vary. If the changes of lunitidal interval Irom half monthly ine- quality were the same for all ports, it would be easy by a table of a single column to apply the required correction to the time of high water when the moon was not at full or change but this is not the case. It has been found, however, that the general law of this change' is the same, and that by knowing the greatest and least lunitidal interval for any port we can determine by computation the change of interval. The ports having nearly the same difference of greatest and least interval are grouped together, and the correction to be applied to the establishment, according to the age of the moon, is given in Table C. The ports Avhich may thus be classed together are the following : a. The ports of Encfland and of the western coast of Europe in general, b. The ports on tiie eastera or Atlantic coast of the United States, c. The ports of the western coast of Florida and of the western or Pacific coast of the United States. This table is arranged on the supposition that the corrected estabhshment is used, ■which is the case for the more important ports in Table LV. In other parts of TABLE B. Longitude of the place. CoiTc'Ction of moon's passini the iiieriiiian. dea. inin. 10 1 20 3 30 4 40 5 56 7 60 8 70 9 SO 11 90 12 100 14 110 15 120 16 130 18 140 19 150 20 100 22 170 23 180 24 TABLE C. Time of Moon's transit. pioup (a.) group group (^'■) Oh add 41m add 19m Om 1 " 17 " 6 subt. 17 2 sul;t. 11 subt. 8 " 32 3 " 27 " 16. " 44 4 " 40 " 22 " 47 5 " 47 " 24 '• 35 " 41 " 19 " 7 u 17 « 6 add 17 8 add 11 a a transparent slit, through which the horizon may be seen. These two glasses are set in brass frames, similar to that of the index glass, and fixed on movable bases, which are adjusted by screws so as to set the glasses in their true positions. In general there are three dark glasses or screens, I ; two red ones, of different shades, and one green. Each is set in a brass frame, which turns on a centre, that they may be used sei)arately or together. They serve to defend the eye from the rays of the sun during an obser- vation. The green glass is peculiarly adapted to take off the glare of the moon, but may be useil for the sun when much obscured by clouds. When these glasses are used for a fore observation, they are to be fixed as in figure 1 ; but when used for a back observation, they are to be placed at O. The sight vanes, K and L, are pieces of brass, standing perpendicular to the j)lane of the instrument. The vane K is called the fore sight vane, and L the back sight vane. There are two holes in the fore sigiit vane, the lower of which and the u|)pcr odire of the silvered })art of the fore horizon glass are equidistant from the plane of the instrument, and the other hole is opposite to the middle of the transparent part of that glass. The back sight vane has one perforation, which is exactly opposite to the middle of the traus])arent slit in the back horizon glass, , g n/. The ailjusli)}g lever (fig. 2), which is fixed on the back of the nuadrant, serves to Ju./t^.r^ ' <^ ^ adjust tlie horizon glass, by placing it parallel to the index glass. .^Vhen this lever is /^^q^jc. to be made use of, the screw B must be first loospned ; and when, by the adjuster A» 7^ ^ the horizon glass is sufficiently moved, the screw (I3)must be fastened agaui ; by this J^' "^ means the horizon glass will be kept from changing its position. To adjust a quadrant. As the quadrant, from various accidents, is liable to be out of order, it is necessary that the mariner should be able to ascertain the errors, and re-adjust the several parts, before he proceeds to make his observations. For this ])urpose, he must examine whether the index glass and the horizon glasses be ])er|)endicular to the plane of th.e instnnnent, and whether the plane of the fore horizon glass be parallel, and that of the back horizon glass perpendicular to the plane of the index glass, when on the veniier stands against on the limb. 1 St. To ascertain whether the index glass be perpendicidar to the plane of the quadrant. Place the index on the middle of the arc, and hold the index glass near the eye. Look into it, in a direction ])arallel to the plane of the instrument, and see if the reflected arc appear exactly in a line with the arc seen direct, or if the image of any point of the arc near B ap[)ear of the same height as the corresjionding \rdvt of the arc near C seen direct; if so, the index glass is perpendicular to the ])lane of the quadrant; if not, the error nuist be rectified by the screws on the base, behind the frame, by loosening the screw M, and tightening the screw N, or by loosening the screw N, and tightening the screw IM. 2d. To ascertain ivhether the fore horizon glass be perpendiculnr to the plane of the quadrant. Having adjusted the index glass, hold the instrument in a veitical position Look through tlie tore sight vane, and move the index till the reflected and direct images of the horizon, seen in the horizon glass, coincide. Then incline the instrument till its plane is nearly parallel to the horizon ; if the images still coincide, the horizon glass stands perj)endicular ; otherwise it does not, and must be adjusted by the screws placed before and behind it, loosening one of them, and tightening the other. Tliis adjustment may be made by the sun, moon, or a star, by holding the quadrant in a veitical position, and observing if the object seen by reflection appears to tiie right or left of the object seen direct, and moving the screws, as above, till both image? coincide. After having made the horizon and index glasses parallel, according to the directions in the following article, it will be best to re-examine this ailjustment. 3d. To make the horizon glass parallel to the index glass, ivhen on the vernier stands on on the arc. Having fixed the index, so that on the veniier stands on on the arc, look at any distant obiect, and see if the hnage of it coincides with the object itself; if it does, the 17 130 USE OF A QUADRANT OF REFLECTION. acljustment is complete ; if not, tliey must be made to coiiiciile by means of the adjusting level*. The horizon may be used for this purpose in the following manner: — [{old the plane of the instrument vertical ; look through the lower hole in the vane K, and direct the sight through the transparent part of the glass G to the horizon ; then if the horizon line, seen in the silvered and transparent part, coincides, or makes one straight line, the horizon glass is said to be adjusted ; but if the horizon lines do not coincide, slacken the screw B (fig. 2) in the middle of the adjusting lever, and turn the Jiorizon glass on its axis until the horizon lines coincide ; then fix the lever firmly bj"^ tightening the screw B. If this adjustment be again examined, it will perhaps be found impei-fect. In this case, therefore, it remains either to repeat the adjustment, or find the error of it (usually called the index error), which may be done thus: — Let the hoi'izon glass remain fixed, and move the index till the image and object coincide ; then the difference between on the vernier and on the arc is the index error, which is to be added to the angle or altitude observed, if tlie on the vernier be to the right hand of on the arc, otherwise to be subtracted. Thus, if the horizon is used, the instrument being held in a vertical position, you must look through the lower hole of the vane K, towards the horizon ; then move the index till the reflected and direct images of the horizon coincide ; the difference between on the vernier and on the arc will be the index error. 4th. To adjust the back horizon glass, that it may be perpendicidar to the plane of the. index glass, when on the vernier stands on on the. arc. Set the index as fiir to the right of on the arc, as twice the dip of the horizon (talien from Table XIII.) ; hold the quadrant in a vertical position ; look towards the horizon through the hole in the back horizon vane L, and the transparent slit of the back horizon glass II ; then, if the reflected horizon, which will appear inverted, coincide with that seen direct, the glass is truly adjusted ; otherwise the screw, in the centre of the lever on the imder side of the quach-ant, must be slackened, and the glass timied on its axis till both horizons coincide, when the lever should be fixed by tightening the screw. 5th. To adjust the back horizon glass, that it may be perpendicular to the plane of tJit quadrant. Put the index on ; hold the quadrant nearly parallel to the horizon ; look through the hole on the back sight vane, and if the true and reflected horizons appear in the same straight line, tlie glass is perpendicular to the plane of the instrument ; but if they do not coincide, the sunk screws, before and behmd the glass, must be turned till both appear to form one straight line. To take an altitude of the sun by a fore observation. If the sun is bright, turn down one or more of thft dark glasses; hold the instrument in a vertical position ; apply the eye to the upper hole in the fore sight vane, when the image is so bright as to be seen in the transj)arent part of the fore horizon glass, otherwise to the lower hole ; direct the sight to that part of the horizon beneath the sun, and move the index till you bring the image of his lower limb to touch the horizon directly under it ; but as this point cannot be exactly ascertained, the observer should move the instrument round to tlie right and left a little, keejjing, as nearly as possible, the sun always in that part of the horizon glass which is at the same distance as the eye from the plane of the quadrant;* by this motion the sun will appear to sweep the horizon, and must be made to touch it at the lowest j;art of the arc ; the degrees and minutes pointed out by the hidex, will be the observed altitude of the sun's lower limb at that instant. To take an altitude of the moon by a fore observation. In the night, Avhen the moon is bright, her image may be seen in the trans.parcni part of the fore horizon glass, and the observation may be taken exactly in the same * In common qiiadrants, if ihe upper hole lie looked throiigli, the sun's imai^'c must he made to appeal in the middle ol" me transparent part of the horizon glass ; but if (he louor hole be looked Uirovi<;h, tl!3 imag'e must be made to appear on the line joiniiig the silvered and Ira.isparent parts of the horizon fjlass, as these parts of the horizon glass are at llie same distances from the plane of the instrument, as (lie holes of the sight vanes respectively. USE OF A QUADRANT OF REFLECTION. 131 manner as an observation of the sun. If tlie image is so faint as not to l)e seen in tlie transparent part of the horizon glass, you must set the index to ; hold the plane of tlie quadrant in a vertical position ; direct the sight to the moon, and, at the same time, look for her reflected image in the silvered part of the horizon glass ; move the index forward till the moon's image (which will appear to descend) just touches the horizon ; then sweep the quadrant as in observing the sun, and bring iier round limb in contact with the horizon, whether it be her upper or lower. The degrees and minutes pointed out by the index, will be the observed altitude of that limb which was brought in contact with the horizon. To ta1servation is but little used, on account of the difficulty of adjusthig and observing. Various remedies have been proposed for these defects, but none have yet been generally adopted. The back observation of the altitude of any object, is useYid only when there is not an open horizon for the fore observation ; but even in that case, the fore observation may often be used, if the distance of the horizon be known, as will be explained hereaftei". To observe the meridian altitude of any celestial object by a fore observation. Wlien the object rises and sets, it comes to the meridian above the horizon only once in 24 liours, and is then at its greatest altitude ; and by observing it, the latitude may be easily determined. The sim comes to tlie meridian exactly at noon, or 12 o'clock apparen t time ; the moon and stars at various hours. To observe the meridian" altitude, begin TL fewniinutes before the time of passing the meridian ; bring the object to sweep the horizon, according to tlie preceding directions ; this operation must be repeated until the object begins to descend below the edge of the sea; the degrees and minutes then sho^vn by the index will be the meridian altitude. If the object does not set, it comes to the meridian below the pole, and is then at its least altitude ; this altitude may be observed as above directed, with this diflerence, that you must continue sweeping till the object begins to rise above the edge of the sea, instead of descending below it. The meridian altitude of any object may be taken in a similar manner by a back observation. Strictly speaking, this method of finding the meridian altitude is not absolutely accurate, except the ship be at rest, and the sun's declination constant. For if the ship is sailing towards the sun, the altitude will be increased ; but the altitude will be decreased in sailing from the sun. The correction of altitude arising from this source is generally very small, and it may be neglected in most cases, as will be shown hereafter. Advice to seamen in the choice of a quadrant. The joints of the frame must be close, without the least opening or looseness, and the ivory on the arc inlaid and fixed, so as not to rise in any place above the plane of the instrument ; all the divisions of the arc and vernier must be exceedingly fine and straight, so that no two divisions of the vernier (except the first and last) coincide, at the same time, with the divisions of the arc. All the glasses belonging to the quadrant 132 USE OF A QUADRANT OF REFLECTION should liave tlunr surfaces perfectly [)l;uie, and their fore and l)ack surfaces exactly parallel ; this may be verified, iu the horizon glass and index glass, l)y means of two distant objects, in the following manner: — IMove the index till both objects are exactly in contact, at the u})per edge of the silvered i)art of the horizon glass; then move llie quadrant in its omi [)lane, so as to make the united images move along the line, separating tjie silvered from^the transparent part of the horizon glass; and if, in this motion, the images continue united, the reflecting surfaces are good jilanes, otherwise the ])lanes are imi)erfect. To examine the dark glasses, we must bring the image of a distant object to coincide witli the object si.'en directly ; tlien turn the colorcil glass so that the |)Iane which was next to the uidex glass may now be next to the horizon ghiss, antl if the direct and reflected images still coincide, the surfaces of the ghiss are |>arallf-l. 133 DESCRIPTION AND USE OF A SEXTANT OF REFLECTION. A Sexta.nt is constnictecl on tlie same princlijles, and may be used for measuring altitudes in tlie same manner, as a quadrant.* Tlie arc of a sextant, as its name nuplii'S, contains 60°, l)ut, hy reason of the double reflection, is divided into 120'^. Tbis instrument is ]»articularly intended to measure the distance of the moon from the sun, a ])lanet, or a rixcd star ; and as that distance is wanted as accurately as possible, to determine tlie longitude of the place of observation, the instrument is constrnctccl with more care, and is provided with some adflitional appendages that are not in the fjUadrant. Plate IX., figure 3, re|)rcsents a sextant, the frame behig generally made of brass, or otlier hard metal; the handle at its back is made of wood. When observing, the instrument is to be held with one hand, by the handle, while the other hand moves the index. Tiie arc AA is dividetl into 120°, each degree into 3 parts of 20 minutes each, and the vernier scale is in general so divided as to show lialf or a quarter of a mimite. In some sextants, the degree is divided into six equal parts, of 10' each, and the vernier shows 10". In order to observe- witii accuracy, and make the images come precisely in contact, a tangent screw B is fixed to the index, and !)y this it can be moved with greater regularity than it can be by iiand ; bnt the screw IJ does not act until the index is fixed by the screw C, at the back of the sextant. Care must be taken not to force the tangent screw, when it arrives at either extremity of its arc. When the index is to be moved any consideraI)le quantity, the screw C nnist be loosened ; and when the index is brought nearly to the division requiretl, the back screw C must be tightened, and then the index moved gradually by tlie tangent screw. In many sextants, die lower ])art of the index glass, or that next tlie jilane of the instrument, is silvered as usual, and the back surface of the upper part painted black ; a screen, painted black, is fixed by its axis to the base of the index glass, and may be placed over the silvere(l ])art when the rays are strong; in this case, the image is to be reflected from the outer surface of the u|)per part, aiui the error which might possibly arise from the planes of the glass not being |)ara]li'l, is thereby avoided. The colored glasses are similar to those ai)])!ied to a common (piadrant, and are nsually fi)ur in number, jdaced at D, to screen tlie eye from the solar rays, and the glare of the moon ; they may be used separately or together, as occasion re(pures. In addition to tlicso, there are three similar glasses, placed behind the horizon glass, to be used in finding the index error by means of tlie sun, and in observing the sun's altitude, by an artificial horizon on land. The paler glass is sometimes used in observing altitudes at sea, to take off the strong glare of the horizon below the sun, arising from the smi's light, reflected iiTcgnlarly from the small rii)i)ling waves — an appearance which has lately been called kumatnge. A sextant is generally fnrnished with a tube without glasses, and two telescopes, the one representuig the objects erect or in their natural situation, the other inverting them, * There is not, in general, any apparatus for ilie back oljservalion fixed lo a sextant; but if the altitude of any celestial object be greater than (iO°, the sui>plcmcnl of liie altitude may be obtainee telescope from the horizon glass, without passing the colored glass. 'I'm eflect tills, the renlelle must be placed at I), and hnvcrcd so as to intercept the direct light ejitircly ; then place the colored gl.-ss at C, and direct the telescope to the silvered part of tne horizon glass ; move the central index, and if no nncolored images appear (reflected from the central glass), but all have the same tinge as tliat of the colored glass used, the horizon glass is in its proper position; otherwise it must Be turned on its axis till the uncolored images disappcnr. Times of otx. 4li. 20in. Os. 4 21 10 4 q.T) 1j 4 23 4 21 45 4 25 30 Angle. 6)26 16 40 6 ) G0° 24' 4 22 47 10 4 CIRCLE OF REFLECTION. 141 tnat the method is applicable to an object not on tlie nieriilian ; bnt, in tiiis case, the cross ol)servations, wliicJi give to tlie circle all its advantages, may be used, and the mean of the altitudes taken instead of a single altitude. This method is peculiarly adapted to the taking of altitntles for regulating a watch ; for this reason it will be particularly explained in the following article : — To take altitudes of the sun, or any crlcstiol object, hy cross observations, for rrgulatiuff a watch. Fix the central index on 0, and if the object be bright, and the altitude between 5° and 35°, place a large colored glass jjefore the central glass at a, a, otherwise a small one at C; hold the instrument m the Itsft hand, in a vertical position ; move the horizon index till the image of the reflected object be brought in com])lete contact with the horizon, in the middle between the two parallel wires of the telescope, as directed in the preceding article, and note the time of ol)servation by the watch ; then fasten the horizon index; hold the instrument in the right hand, in a vertical position ; move the central index according to the order of the divisions, till the reflect- ed image be again brought into complete contact with the horizon * as above, and note the time of observation. Then half the sum of the times, and half the angle shown by the index, will be a mean time, and a mean altitude corresponding thereto. If greater accuracy be required, the observation must be repeated, setting out from the points where the indices then are, and observing in the same manner by moving first the horizon index, then the central one ; continue taking as many of these cross observations as are judged necessary, and note the times of each observation; then the sum of the times, divided by the whole inmibcr of observations, will be a mean time ; and the angle shown by the central index, divided by the number of observations, will be a mean altitude corresponding thereto. Thus, if sixf observations were taken, and the times noted as in the adjoined table, the angle shown by the index being G0° 24', the mean time would be obtained by dividing the sum of the times, 26h. 16m. 40s., by 6, and the mean altitude by dividing 60° 24' by 6 ; therefore the mean time would be 41). 22m. 47s., and the mean altitude corresponding 10° 4'. To measure the distance bctioecn the sun and moon hy a circular instrument. The instrimient being well adjusted, fix the central index on 0, and, if the object be bright, place a small dark glass at C ; hold the instrument so that its plane may be directed to the objects with its face tiownwards when the sun is to the I'ight of the moon ; otherwise, with its face upwards ; direct the sight through the telescope to the moon; move the horizon index, according to the order of the divisions of the limb, till the reflected image of the stm appears in the telescope, and the nearest limbs of the sun and moon are almost in contact ; fasten the index, and make the coincidence of the limbs perfect, in the middle between the two jjarallel wires of the telescope, by means of the tangent screw of the horizon glass, and note the time of observation; * The arc described on ihe limb by Ihe central index, will be equal to twice Ihe altitude of the object, or twice the angle passed over by the oilier index ; if more cross observations be taken, each of the indices, when moved, will describe an arc equal to double the allitude of the object; the same is to be observed in measuring any other angular distance. If the instrument is furnished with the arc WSll, and sliding pieces U, X, you must bring the slide X to the central index, after taking the first observa- tion to the left, and place the slide U at the same degree, on the arc SW, that X is on the arc PR; then, in the next observation, tlie central index is to be brought to touch the slide U ; in the next observation to the left, the slide X is to be brought to the central index, and so on for the other observations. Thus, by means of the slides, the indices may be placed at nearly their proper angles with each other at the beginning of the observation, which will save considerable time. After being thus fixed, the contact must be completed by means of the tangent screw of the index, which is to be moved. t The number G is a convenient number to use, because the remainder of the division of the hours by 6 gives the first figure of the minutes ; and the remainder of the division of the minutes by G gives the first figure of the seconds. Thus, in the above example, in dividing 2Gh. by 6, we get 4h., and the remainder 2 is set down immediately for the first figure of the minutes ; the second figure of the minutes is the quotient 2, found by dividing 16m. by 6, and tlie remainder 4 of this last division is the first figure of the seconds. We may remark that, as the term 4h. 20ni. is common to all the 6 observations, it maybe neglected ; then adding the minutes in the column of units, and the seconds, the sum becomes 16m. JOs ; ■I'viding this by 6 gives 2m. 47s., to be connected with 4h. 20m., making, as above, 4h. 22n3 4.T'J 143 CIRCLE OF REFLECTlvON. then invert the instrument, and move the central index, according to tlie ord-er of the divisions of the Hmb, by a quantity equal to twice the arc passed over by the hoi'izon index (or twice the distance of the sun and moon);* direct the plane of the instrument to the objects ; look directly at the moon, and the sun will be seen in the field of the telescope ; fasten the central index, and make the contact of their nearest limbs complete, in the middle between the two jiarallel wires of the telescope, by means of the tangent screw of the central index, and note the time of observation ; then half the arc shown by the central index will be tlie distance of the nearest liuibs of the sun and moon, and half the sum of the times will be the mean time of observation. Having finished these two observations, two others may be taken in the same manner, setting out from the points where the indices then are, and moving first the horizon index, then the central index : proceed thus till as many observations as are judged necessary be taken, alwcys observing that the number of them be even ; then the angle shown by the central index (or that angle increased by 720^ or 1440°, &c., if the index has been moved once or twice, &.c., roinid the limb), bciUg divided by the whole number of observations, will give the mean distance ; and the sum of all the times, divided in like manner, will be the mean time of observation. 7^0 measure the distance between the moon and star hy a circular instrumint. Fix the central index on 0, and, if the moon be bright, and the distance between 5° and 35°, place a large green glass before the central mirror at a, a, otherwise a small one at C ; hold the instrimient so that its plane may be directed to the objects with its face downwards when the moon is to the right of the star, otherwise with its face upwards ; direct the sight through the telescope to the star; move the horizon index, according to the order of the divisions of the limb, till the reflected image of the moon appears in the telescope, and the enlightened limb of the moon be nearly in contact with the star ; fasten the index, and make the coincidence perfect, in the middle between the parallel wires of the telescope, by means of the tangent screw belonging to that index, and note the time of observation ; then invert the instrument, and move the central index, according to the order of the divisions of the limb, by a quantity equal to twice the arc passed over by the horizon index ; * direct the pl^ne of the instrument to the oljjects ; look directly at the star, and the moon will be seen in the field of the telescope ; fasten the central index, and make the contact of the enlightened limb of the moon and the star complete, in the middle between the two parallel wires of the telescope, by means of the tangent screw of that index, and note the time ; then half the arc sliown by the central index will be the distance of the star from the enlight- ened limb of the moon, and half the smn of the times will be the mean time of observation ; these two observations being completed, others may be taken in the same manner, according to the directions above given for measuring the distance of the sun fioin the moon. In continuing to take these cross observations by a circle furnished with the arc VVSR, and slides U, X, it will be very easy to bring the reflected image into the field of tlie telescope ; but if the instrument is not thus furnished, it will be often difiicult to bring the image into the field of the telescope, and much time will be lost, and the observations rendered tedious by that means; to remedy this, a small table of the angles, at which each index should be ])laced, ought to be made before beginning the observation ; this table is easily formed, as follows: — Find roughly, according to the directions heretofore given, the point at which tlie horizon glass must be placed to be parallel to the central glass, when tlie central index is on ; then find what point of the arc the horizon index stands upon, after measuring the first distance, as directed above; the difference between these two points will be the angular distance of the objects ; the doidile of this distance, being successively added to 0°, and to th.e aii'/l" i)ointed out by the horizon index after the first observation, w ill give the points of the arc where the indices must be placed at the 2d, 3d, 4th, &c. observations. Thus, if the point of parallelism is 471°, and the point where the horizon index is at the first observation is 525°, the difference, or 54°, will be the angular distance ; the double of this, or 108°, being added to 525°, gives 033°, which is the point of the arc where that index must be placed at the third observation ; 633° added to 108° gives 741° or 21° (because the divisions recommence at 720°), which is the point where the index must be y)laced at the fifth observa- tion, &c., as in the adjoined table. The central index being at * This may be clone expeditiously by means of the slides U, X. as Is explained in the precediii" no*e. Ccyitral Horizon Index. Index. 0° 525 108 033 21G 21 324 |..>I» 432 237 540 &c. &c. CIRCLE OF REFLECTION. 143 first on 0°, after the second observation it will be on 108°, at the fourth on 108° -|- 108° = 216°, at the sixth on 216° -|- 108° = 324°, &c. Thus, by constantly adding 108°, or twice the distance of the objects, the angles at which the indices must be placed will be obtained ; and by fixing them at these angles, the reflected image will be brought into the field of view without any trouble.* Having explained the methods of adjusting and using tlie circle of reflection, it remains to show how to calculate the error arising from not observing the contact of the objects in the middle between the parallel wires of the telescope, and also to estimate the eiTors arising from the want of parallelism of the mirrors and colored glasses. These verifications are much more necessary in a sextant than in a circle, and they may be m general neglected in a circle. To estimate the error arising from not observing the contact of the objects in the middle between the parallel wires of the telescope. To estimate this error, it is necessary to know the angular distance of the wires of the telescope, which may be thus determined : — Turn round the eye-j)iece of the telescope till the wires are perpendicular to the plane of the instrument, and put the central index on ; direct the telescope to any well-defined object, at least 12 feet distant, and move the horizon index till the direct and reflected image of the object coincide ; then make one of the wires coincide with the object, and turn the central index till the reflected image of the object coincides with the other wire — and the arc passed over by that index, will be the angular distance between the wires. This angle being obtained, the observer must, by means of it, estimate, at each obsen'ation, how much the place where the contact is observed is elevated above, or depressed below, the plane passing through the eye and the middle line between the two parallel Avires of the telescope : the con-ection in Table XXXV., con-esponding to this angle, is to be subtracted from the observed angular distance of the objects : thus, if the distance between the wires is 2°, one of them will be elevated above that plane 1°, and the other depressed below it, by the same quantity ; if, in taking an observation, the point of contact is estimated to be one thu-d part of the distance from the middle towards either wire, the angle of elevation or depression will be one third part of 1°, or 20' ; and if the observed distance is 120°, the correction m Table XXXV. will be 12", subtractive from the observed distance. The correction for each observed distance being ascertained, in the above manner, the sum of them must be subtracted from the whole angle shown by the central index, and the remainder, divided by the whole number of observations, will be the mean distance. Verification of the parallelism of the surfaces of the central mirror. This verification is to be made ashore, by observing the angular distance of two well-defined objects, whose distance exceeds 90° or 100°, having previously well adjusted the instrument : after taking several cross observations, and finding the mean distance, take out the central mirror, and turn it so that the edge which was formerly uppermost may now be nearest the ])laue of the instrument ; rectify its position, and take an equal number of cross observations of the angular distance of the same two objects; half the difference betv/een the mean of these and that of the former, will be the error of the observed angle, arising from the defect of parallelism of the central mirror. If the first mean exceeds the second, the error is subtractive, otherwise additive, the mirror being in its first position ; but the contrary when in its second position. Thus, if 10 observations are taken at each operation, and in the fii-st the angle shoAAii by the mdex is 1199° 53.V, and in the second 1200° 6^', by dividing bv 10 tlie mean angles are found to be lf9° 59' 21" and 120° 0' 39", and their diflference is 78" ; the half of it, or 39", is the error of the muTor, additive when it is in its first position, subtractive in the second. The error for any other angle may be found by Col. 4, Table XXXIV., when the inclination of the plane of the horizon glass to the axis of the telescope is 80°, by saying. As the tabular error corresponding to 120°, that is, 1' 30", is to the en-or found in the glass 39", so is the tabular error for any * If the distance of the object vanes durino; the observation, these angles will require correction as you proceed with the observations. Thus, if Ihe distance was increasing, and at tiie sixth observation il was found tiiat the central index was on 3'2G° instead of 324°, the increase being 2°, you must add 2° to the rest of the numbers in the table, and place the horizon index, at the seventh observation, on 129° 4- 2° = 131°, and the central index, a\. the eighth observation, at 432° -j- 2° = 43i°, &c. 144 CIRCLE OF REFLECTION. other angle 85°, which is (X 28", to tlie error of the glass coiTesi)onding 12" ; and in this manner a tahle of eiTors may be made, not only for the cross observations, but also for observations to the right or to the left.* It may be remarked that the errore are much less in the cross observations than in the observations to the right, which are those made witli a quadrant or sextant ; so that the circle has, in this respect, greatly the advantage of those instruments. The angle between the plane of the horizon glass and axis of the telescope produced being nearly the same in all observations and adjustments of the circle, no sensible error can arise from the want of parallelism in the surfaces of that glass. Verification of the paralleUsm of the colored glasses. Place one of the dark-colored glasses at C, and another at D ; fix the central index at 0, direct the telescojie to the sim, and move the horizon index till the limbs of the direct and reflected image coincide ; then turn the dark glass placed at C, so that the sui-face which was farthest from the horizon glass n)ay now be nearest to it, and if the contact of the same two limbs be complete, the surfaces of the glass y^laced at C are parallel ; but if the limbs lap over or separate, the central index must be moved to brhig them again in contact; then half the arc passed over by that index will be the error arising from the want of parallelism of the glass C. If great accuracy is required, the operation may be repeated by setting out from the point where the indices then are, and taking 4 or 6, &c., obsen^ations ; then the arc passed over by the central index, being divided by 4, 6, &c., will be the sought error. The other small glasses may be verified in the same manner ; and, by placing one of the larger glasses before the central index at a, a, and one of the smaller ones at D, the former may be verified as above. The gi-een glasses may be verified by observing the diameter of the full moon, or by some bright terrestrial object. It may be remarked, as one of the greatest advantages of the circle, that, in measur- ing an angle by the cross observations, no error can arise from the want of parallelism in the surfaces of the smaller dark glasses ; for if these glasses give too gi'eat an angle by an observation to the right, they will give too little by the same quantity by an observation to the left. It is not so with the large glasses placed at a, a, because the incidence of the rays on these glasses is more oblique in one observation than in the other, so that the errors do not wholly balance each other; .however, as these glasses are used only in measuring angles less than 35°, where the errore are nearly the same as if the incidence of the rays wei'e perpendicular, the errors of these glasses will also nearly compensate each other in the cross obsen^ations ; and if such observations only are used, it v/ill be unnecessary to verify the dark glasses. Even when taking observations to the right, or observations to the left, the error of the dark glasses will be destroyed, if the glass is turned at each observation, and the number of observa- tions is even ; but there are some cases in which an angle can only be measured by one observation ; then it will be necessary to allow for the error of the dark glass, if the distance is required to be found withiii a few seconds. * If ilie incliiiaiion of llie plane of the horizon glass and the axis of tlie telescope fliffer from 80°, you may fiiKJ the tabular numbers by the method given in the explanation of Table XXXIV. aflixed to the table*. Plate XI Fiffl Fig.6 Figvr SI I HOI DESCRIPTION AND USE OF A PORTABLE TRANSIT INSTRUMENT. A Transit Instrument is of no service on board of a vessel, but is much used arbore, in seaports, for regulating chronometers for sea voyages, and in making observations to determine the longitude. We have, therefore, thouglit it would be useful to give a brief description of it, with the methods of adjustment ; particularly as it may be considered as a valuable accession to the apparatus of a good navigator, who, while remaining in port a few days, can, by means of it, adjust and fix the rate of going of his chronometer with ease and accuracy, and also obtain the best data for determining the longitude of the place, by observing the times of tlie moon's transit or passage over the meridian. The figure in Plate XI., figure 1, represents this instrument, according to the usual construction of Mr. Troughton, with a telescope of about twenty inches foca length. The telescope tube AA is in two parts, connected together by a sphere B which also receives the larger ends of the two axes C, C, placed at right angles to the direction of the telescope, and forming the horizontal axis. This axis terminates in two cylindi-ical jjivots, which rest in Y's fixed at the upper end of the vertical standards D, D. One of the Y's possesses a small motion in azimuth, communicated I)y tuniing the azimuth screw a. In these Y's, the telescope turns upon its pivots ; but, that it may move in a vertical circle, the pivots must be precisely on a level with each other ; otherwise the telescope will revolve in a plane oblique to the horizon, instead of being perpendicular to it. The levelling of the axis, as it is called, is there- fore one of the most important adjustments of the instrument, and is effected by the aid of a spirit level E, which is made, for this purpose, to stride across the telescope, and rest on two pivots. The standards DD are fixed by screws upon a brass circle F, which rests on three screws b, c, d, forming the feet of the instriunent, by the motion of which the operation of levelling is performed. The two oblique braces GG are for the purpose of steady- ing the supports, it being essential for the telescope to have not only a free but a steady motion. On the extremity of one of the pivots, which extends beyond its Y, is fixed a circle II, which turns with the axis, while the double vernier, ee, remain.s stationary in a horizontal position, and shows the altitude to which the telescope is elevated. The verniers are set horizontal by means of a spirit level f, which is attached to them, and they are fixed in their position by an arm of brass g, clamped to the supports by a screw h; the whole of this apparatus is movable with the telescope, and, when the axis is reversed, can be attached, in the same manner, to the opposite standard. Near the eye-end, and in the principal focus of the telescope, is placed the diaphragm, or ivire-plate, which has five vertical and two horizontal wires. The centre vertical wire ought to be fixed in the optical axis of the telescope, and perpendicular to a line drawn through the pivots of the axis. It will be evident, upon consideration, that these wires are rendered visible, in the day-time, by the rays of light passing down the telescope to the eye ; but at night, except when a very luminous object (as the moon) is observed, they cannot be seen. Their illumination is therefore effected by piercing one of the pivots, and admitting the light of a lamp fLxed on the top of one of the standards, as sho\\ii at I. This light is directed to the wires by a reflector placed diagonally in the sphere B. The reflector, having a large hole in its centre, does not interfere with the rays passing down the telescope from the object, and thus the observer sees distinctly the wires and the object at the same time. When, however, the object is very faint (as a small star), the light from the lamp would overpower its feeble rays. To remedy this inconvenience, the lamp is so constructed that, by turning a screw at its back, or inclining the o];)ening of the lantern, more or less light may be admitted to the telescope, to suit the circumstances of the case. The telescope is furnished with a diagonal eye-piece, by which stars near the Kenith may be observed without inconvenience. 19 14G PORTABLE TRANSIT INSTRUMENT. Acljusiments of a transit insirumen' In fixing the instrument, it should be so placed that the telescope, when level, should point north and south as near as can possibly be ascertained. This can at first be done only in an approximate manner, as the correct determination of the meridian can only be obtained by observation, after the other adjustments are completed. To adjust the line of coUimation. The first adjustment is that of the line of collimation, or line of sight. Direct the telesco]ie to some distant, well-defined object (the more distant the better), and bisect it Avitii the middle of the central wire ; tlien lift the telescope veiy carefully out of its angular bearings or Y's, and replace it with the axis reversed ; ])oint the telescope again to the same object, and, if it be still bisected, the collimation adjustment is correct; if not, move the wires one half the error, by turning the small screws which hold the diaphragm near the eye-end of the telescope, and tlie adjustment will be accomplished ; but as half the deviation may not be correctly estimated in moving the wires, it becomes necessary to verify the adjustment by moving the telescope the other half, which is done by turning the azimuth screw a ; this gives the small azimutlial motion to the Y, before spoken of, and consequently to the pivot of the axis which it carries. Having thus again bisected the object, reverse the axis as before, and, if half the error was correctly estimated, the object will be bisected upon the telescope being directed to it ; if not quite correct, the operation of reversing and correcting half the error, in the same manner, must be gone through again, until, by successive approximations, the object is found to be bisected in both positions of the axis ; the adjustment will then be perfect. To adjust the ivires in the telescope. It is desirable that the central or middle wire (as it is usually termed), should be truly vertical, as we shall then have the power of observing the transit of a star on any part of it, as well as the centre. We may ascertain whether it is so, by elevating and depressing the telescope ; for when directed to a distant object, if it is bisected by every part of the wire, the wire is vertical ; if otherwise, it should be adjusted by turning the inner tube carrying the wire-plate until the above test of its being vertical be obtained, or else care must be taken that observations are made near the centre only. The other vertical wires are ;>laced, by the maker, equidistant from each other and parallel to the middle one ; therefore, when the middle one is adjusted, the others are so too ; he also places the two transverse wires at right augles to the vertical middle wire. These adjustments are always ])erformed by the maker, and are but little liable to derangement. When, however, they happen to get out of order, and the observer wishes to correct them, it is done by loosening the screws which hold the eye-end of tlie telescope in its place, and turning the end round a small quantity, by the hand, until the error is removed. But this operation requires very delicate handling, as it is liable to remove the wires from the focus of the object-glass. To Jix the axes or arms, upon which the telescope revolves, in a ho -izontal position The axes on which the telescope tui-ns, must then be set horizontal. To do this, ai)])ly the level to the pivots ; bring the air-bubble to the centi-e of the glass tube, by turning the foot-screw h, which raises or lowers that end of the axis, and consequently the level resting upon it ; then reverse the level, by turning it end for end, and, if the air- bubble still remain central, the axes will be horizontal ; but if not, half the deviation nnist be corrected by the foot-screw h, and the other half by turning the small screw i, at one end of the level, which raises or lowers the glass tube (containing the air- bubble) relative to its sn])i)orts, which rest upon the pivots. This, like most of the adjitstmmls, freqnentb/ requires several repetitions before it is accomplished, on account of the dificidti/ of estimating exactly half the error. This adjustment may also be made by means of the polar star ; first by observing direcdy its transit over any one of the vertical wires of the telescojje, and immediately afterwards observing the reflected image of the same star from a basin of quicksilver. For if the star a])pear on the same wire, the axis is properly adjusted ; if not, you must bring the wire half way towards it by the small screw i,and then, by the azimuth screw a, bring it upon the wire again. This being completed, you nnist, as soon rORTABLE TRANSIT INSTRUMENT. 147 as possible, loolc directly towards the star, and if it appear on the same wire, the adjustment is arcurate ; if not, repeat the operation till it is so ; observing that the motion of the pole-star is so very slow, that it will not be sensibly altei-ed in the interval of taking its ti'ansit directly and by reflection. The farther, however, you observe the star from the meridian, the more accurate will the observation be, since the motion of the star in a direction parallel to the horizon will then be the least ; and when it is at its greatest azimuth, the horizontal motion is nothing. To fix the inslrume.nl so thai the line of collimalion of the telescope may move accurately in the plane of the meridian. Having set llie axis, on which the telescope turns, parallel to the horizon, and proved the correct position of the central wire, or line of collimation, making it descril)e a vertical great circle, perpendiculai* to the axis, we must, in the last place, fix the instrument so that this vertical circle may be the meridian of the place of observation. We have supposed the instrument to be nearly in the meridian. It may bo so placed, with a great degi'ce of accuracy, at the v^ery first operation, by means of a well-regulated* and accin-ate time-keeper, by whii.'i we can detei'mine very nearly the exact instant of the transit of the pole-star over the meridian, either above or below the pole. A few minutes before the time of the transit, we must direct the telescope towards the star, and, by turning the azimuth screw a, bring the star upon the middle wire of the telescope. The apparent motion of this star is so very slow, that we can, by a very small and gentle motion of the azimuth screw a, keep the star constantly bisected on the middle vertical wire of the telescope, till the moment of this ^•ansit, as indicated by the time-keeper, has arrived ; then the instrument will be very learly in the plane of the meridian, and the final con-ections must be made in the . ollowing manner : — First Method. Make the observations of the transits of the pole-star, above and lelow the pole, at three successive transits, and note the times of observation by an iccurate time-keeper. Then, if the interval of time between the first and second 'ransits is equal to the interval between the second and third transits, the instrument will be truly fixed in the plane of the meridian. In this case, each of the intervals will be equal to 12 hours, sideral time, corresponding nearly to ll** 58'" 2% as shown by an accurate chronometer, regidated to mean solar time, f It is very impor- tant, in this operation, that the rate of the time-keeper should be perfectly imiform during both intervals ; but it is not necessary that its I'ate or regulation should be previously known. For, in the preceding example, if the time-keeper move too fast for mean solar time, and gain, for example, 10' in each of the above intervals, making them equal to 11'' 58"" 12^ by the time-keeper, their equality would prove the accuracy of the adjustment to the plane of the meridian, with the same degree of certainty as if the time-keeper were regulated to mean solar or sideral time. However, it is much more convenient to have it well regulated. Sujipose, now, that the intervals, instead of being equal to each other, are found to differ. In this case, tlie instrument is not placed accurately in the plane of the meridian ZM??tII (Plate XI. fig. 2, 3), but the motion of the telescope is in some vertical circle, as ZSsT, which cuts the horizon in the point T, situated to the west of the meridian H, in figure 2, or to the east, in figure 3 ; the distance from the meridian being measured on the horizon by tlie arc of azimuth HT. If we now su])i)ose that MWmE is the small circle described by the star in its diurnal motion, M will be the place of the star at its upper transit over the meridian, and ?n its place at the lower transit, when well adjusted ; but when the vertical motion of the telescojie is in the vertical circle ZSsT, the upper observed transit will be at S, and the lower observed transit at s ; the observed intervals of times being propor- tional to the arcs SWs, sES. Now, it is evident, from the inspection of figures 2, 3, * Tliis reg^ulation can be made by equal altitudes of the sun, observed with a sextant ; or by a siiifjle altitude, when the latitude of the place is known ; or by similar observations with a known star. Tlie method of obtaining the time from such observations, will be explained hereafter. t If the sun be supposed to move uniformly in the plane of the equator, rtie interval of two suscessive transits of the sun over the upper meridian, will be equal to 24 hours, mean solar time, and it is for this mean solar time that chronometers are usually adjusted. The interval between two successive transits of a fixed star over the same meridian, is very nearly equal to SS"* oG™ 4', mean solar time ; but it is found very convenient, in many astronomical and nautical calculations, to divide it into 24 hours, which are called hours of sideral time, and they are divided, as usual, into minutes and seconds. We iiave pvcn, in our collection of tables, two tables for facilitating the reduction of the one of these limes to the other. 148 PORTABLE TRANSIT INSTRUMENT. that t\.^ deviation is alwaj's towards that side of the jneridian where the least interval is observed ; as, for example, in figure 2, wliere the telescope describes the vertical circle ZSsT, to the west of the meridian, the western interval SWs is the least The correction of this adjustment is made by means of a slight motion of the azimuth screw a; and the quantity of this motion depends on the difference of the two intervals. Suppose, for example, that one of the intervals is ll** 58"' 2% and the other ll*" 58'" 22", which differ 20 seconds of time ; the half-difference, 10 seconds, represents the time required by the star to pass over both the small arcs MS, sm ; and, in the case of the pole-star, where the polar distance PM, or P?n, is very small, the arcs IMS, sm, are very nearly equal to each other, so that each of these arcs will be described in about one half of 10', or 5' ; or, in other words, the time required to describe the arc MS, or sm, is very nearly equal to one quarter part of the difference between the two intervals, which, in the present example, is ^ X 20' := 5'. To correct this, we must watch the pole-star, as it approaches towards tlie lower transit s, if the deviation be to the west of the meridian, or as it approaches towards the upper transit S, if the deviation be to the east of the meridian ; and, the moment the star is bisected by the middle wire of the telescope, we must begin to count these five seconds of time, and, by a very gentle motion of the azimuth screw a, keep the star constantly bisected by the wire until the expiration of the time of 5 seconds, or the quarter of the difference of the intervals. Then, if every part of the operation has been done accurately, and the time-keeper be perfectly coiTcct, the instrument will be accurately adjusted in the plane of the meridian ; but as this is one of the most important and delicate adjustments, it will be best to repeat again the observations of the three transits, to ascertain whether the first and second inteiTals of the successive transits are equal ; and, if a slight difference should still be found, it must be corrected by repeating the operation in the manner Ave have already explained. This method of adjusting the transit instrument (by means of the pole-star) is preferable to any other whatever. Delambre, who had much practical experience, says there is no advantage in using two stars ; and that, with a single star, the prefer- ence is to be given to the pole-star ; after this he recommends the stars d, §, y, of the Little Bear, and y Cephei. These stars being more distant from the Jjole, it may become necessary to make a small correction in the quarter part of the difference of the intervals, to correct for the difference of the arcs SM, sm. This correction is made by means of the Table A, page 151, which gives the correction for the pole- star, and for other stars where polar distance is le^s than 40'', supposing the difference of the two intervals to be 1000 seconds of time. Thus, if the polar distance of the star be 20°, and the latitude 42°, the tabular correction is 82% which is to be applied to one quarter part of the assumed difference of the intervals, 1000% that is, 250% making 250' -f- 82' = 332' for the distance of the star from the meridian, at the time of the lower transit, and 250' — 82' = 168' for the distance of the star from the meridian at the upper transit. These times, 332', 168', must be reduced in the same ratio as the actual difference of the two intervals bears to the tabular difference 1000'. Thus, if the observed difference of the two intervals were 205', instead of 1000', you must say. As 1000' is to 205% so is 332' to 68', and so is 168' to 34' , so that the correction to be applied to the lower transit is 68% and to the upper transit 34'. Tlierfifore, if the star be approaching towards the meridian at the time of the lower ti'ansit, you must proceed according to the former direction relative to the pole star, and keep the star constantly bisected by the middle wire of the telescope, by a slight and gentle motion of the azimuth screw a, from the time of its first transit by that wire, till you have counted 68' by the time-keeper. But if the star be a])in-oach- iiig towards the meridian at the upper, transit, you must adjust the instrmnent by means of the next upper transit, making an allowance of 34' for the distance from the meridian, and keeping the star constantly bisected, from the time of its transit by the middle wire, by means of the azimuth screw a, until the termuiation of the time of 34'. Before closing our remarks on this method of adjustment, we may observe, that if the angular value of one revolution of the azinuith screw be knouii, or the instrument possess an azimuth circle, by which the motion of the telescope may be accurately estimated, we may correct the adjustment by estimating the correction in azimiuh by means of Table B, whei'e the variations of azinuith, in seconds of a degree, are given for a supposed variation of 1000 seconds in the difference of the two intervals. Thus, in the i)receding example, where the polar distance of the star is 20°, latitude 42° diffeiT'uce of the two intervals 205' ; the tabular coirection for 1000' (difference of the two intervals) being 30' 42", we have 1000' : 205' : : 30' 42" : & 18" ; therefore the correction of azimuth is 6' 18", to bring it iiUo the plane of the meridian After the instriunent has been completely adjusted to the plane of the meridian, it PORTABLE TRANSIT INSTRUMENT. 149 is usual to fix a meridian mark on some distant object to the north, and another to the soutii ; and, by mean of these marks, the observer can ascertain, with much certainty, whelher the instrumjnt has been altered in its adjustments, from any accidental cause, since the last time it was used. Sometimes, with an additional glass to correspond to the distance of the mark, and a scale of seconds of azimuth made near the meridian mark, we may correct the instrument for a few seconds' motion in azimuth, when correcting the adjustment in the manner we have just been speaking of. We may here n-niark, that the instrument ought to be fixed on some very stable support (as, for exaniiiie, a stone block, imbedded in the ground five or six feet), and in as retired a situation as possible, to avoid the tremulous action from the motion of carriages, &c. It will also be extremely convenient, as well as conducive to accuracy, to have the instrument covered by a low building, with slits in the roof on the north and south, fixed with movable shutters, so that the particular part of the northern or southern sky, where the observed star is situated, may be visible, while the rest is covered over, to prevent the entrance of too much stray light to the eye, when obsei-ving in the twilight, or in the day-time. As a greater security from the interference of this kind of light, th^ observer may j)lace a thick cloth over his head, with a part of it very near the eye end of the telescope, which will serve very well to protect the eye from any other light except that which passes through the telescope. Second Method. This method of adjusting the instrument to ;he plane of the meridian, is by means of two well-known circnmpolar stars, of nearly the same declination, and differing nearly twelve hours in right ascension, by observing the one above, and the other below the pole. Then it is evident that any deviation in the instrument from the meridian, will produce contrary effects upon the observed times of transit, exactly as in the upper and lower transits of the same star. Here the time which ehijjses between the two observations, will differ from the time which would elapse according to the catalogue, by the sum of the effects of the deviation upon the two stars. We have given, in Table C, at the end of this article, the corrections in the times of the upper and lower transits of stars, for various declinations, and in dif- ferent latitudes, supposing the instrument to be 16' 40", or 1000", in azimuth from the plane of the meridian. Thns, if, in the latitude of 40°, we make an observation of the upper transit of a star whose polar distance is 25°, and, at about the same time, the lower transit of another star whose polar distance is 30°, we shall find from the table that the correction of the ujiper transit is 66% and of the lower 131% for 1000" of azimuth. If the deviation of the instrument were to the east of the meridian, by the quantity 1000", the upper transit would be observed too early by GQ^, and the lower too late by 13P ; consequently, the difference between the observed transits, and the times of passing the meridian given by the tables, would be G'6' -|- 131' = 197^ Suppose, now, by actual observation it was found that this difference, instead of being 197% was only 50' ; we should obtain the corresponding correction of azimuth by saying. As 197" is to 50% so is 1000'' to 254" ; and, to correct this error, we must move the azimuth screw a so as to give the instrument an increase of 254" north- westerly azimuth. In like manner we find the corrections of the times of the transit, by saying. As 197' is to 50% so is 66' to 17', the correction of the upper transit ; or. As 197' is to 50', so is 131' to 33% the correction of the lower transit ; and we must use these numbers for correcting the position of tlie instrutnent, in the same manner as we have before directed. Thus, in the above example, the star which was observed approaching towards the meridian, at the up])er transit, was 17' from the meridian in time ; therefore, at the next upper transit of the same star, we must observe it passes the middle wire of the telescope, and then, by means of the azimuth screw a, keep the star constantly bisected by the wire during 17 seconds of time, and then, if the observation has been accurately made, the instrutnent will be in the plane of the meridian. In determining the direction of the deviation, it must be recollected, that when the deviation is to the east, the star above the pole passes too eariy, and that below the pole too late ; therefore, if the upper star precedes, the interval by oliservation will exceed the trne interval, between the passages of the two stars ; but if tlie lower star precedes, the interval by obsei-vation will be less than the true intei-val. The con- trary takes i)lace when the deviation is to the west of the meridian. This method may be used advantageously with 8 Ursae Minoris, and Cephei 51 Hev., which are given in the Nautical Almanac. In like inanner, the pole-star may be combined with the stars of the Great Bear. Third Method. This method consists in observing the transits of any two stars, differing from each other considerably in declination, and but little in right ascension. The nearer the obsei-vations of the stars are to each other, the better, as this prevents 150 PORTABLE TRANSIT INSTRUMENT. the possibility of any eiTor arising from a change in the rate of the time-IceL'j)er And, as the ai)parent places of one hundred j)rincipal stars are now given in the Nautical Almanac, for every tenth day, it will be better to select two stars from that work. The principle upon which this third method is grounded, is, tliat a high star is less affected by a deviation of the instrument from the plane of the meridian, than a low star ; hence it is evident that if the obsei'ved differences of the transits, reduced to sideral time, be exactly equal to the difference of the computed right ascensions, the instrument will be correctly placed in the plane of the meridian ; if not, by repeated operations, by methods similar to those before explained, the adjustments must be completed. The restricted limits of this article do not allow us to go into many minute details which are used in large observatories. What we Irave here given will be sufficient for all the purposes to which a portable transit instrument is usually applied. To observe the transit of any heavenly body over the meridian. Having, by means of the previous adjustments, made the line of coUimation describe a great circle, passing through the zeniUi of the place, and the north and south points of the horizon, the instrument will be in a fit state for making the observations. We have said that the telescope contains five vertical and two horizontal wires, placed a short distance from each other. These last are intended to guide the observer in bringing the object to pass across the middle wire of the field, by moving the telescope till it appear between them. The central vertical wire is in the meridian, and the instant of passing this wire will be the time of the [)assage on the meridian by that star : but as, in noting the time, it will not often ha])pen that an exact second will be sho\Mi by the clock, when the star is bisected by the wire, but it will pass the wire in the interval between two successive seconds ; the observer must, therefore, whilst watching the star, listen to the beats of his clock, and count the seconds as they elapse : he will then be able to notice the space passed over by the star in every second, and, conse- quently, its distance from the wire at the second before it arrives at, and the next second after it has passed, the wire ; and, with a little practice, he will be able to estimate the fraction of a second at which the star was on the wire, to be added to the previous second. Thus, if the observation of passing the wire was midway between the 7th and 8th seconds, the time of the transit would be 7". 5; but if it ajjpeared more distant on the one side than on the other, it would be 7'. 3, or 7\7, &c., accord- ing to its apparent relative distance from the wire. This kind of observation must be taken at each of the five wires, and a mean of the whole taken, which will represent the time of the star's passage over the mean or meridional wire. The utility of having five wires, instead of the central one only, will be readily understood from the consideration that a mean result of several observa- tions is deserving of more confidence than a single one. In observing the su7i, the times of passing of both the first and second limbs over the wires, are to be observed and set down as distinct observations ; the mean of both observations gives the time of the passing of the centre across the meridian. The wires of the instrument are generally placed, by the maker, at such a distance from each other, that the first limb of the sun passes all of them before the second limb arrives at the first wire, and the observer can thus take the observations without hurry or confusion. The round Ihnh only of the moon can be obsei-ved, except ivithin an hour or two of the fidl moon. In observing the larger planets, the first and second limb may be obsei-vcd alternately over the five ivires ; that is, the first limb must be observed at the 1st, 3d, and 5th wires, and the second limb at the 2d and 4th wires ; and, by reducing these observations in the same manner as those of the sun, we obtain the meridional passage of the centre. When an obsei-vation at one or more of the wires has been lost, it is impossible to take the mean in the same way as in a perfect observation. If the centre wire is the one that is deficient, the mean of the other four may be taken as the time of the meridional passage ; or the mean of any two, equally distant on each side of the centre, supposing the intervals of the wires to be equal. But when any of the side wires are lost, and, indeed, under any circumstances of deficiency in the observation, the most correct metho/1 of proceeding is as follows : — Find, by a consider- able number of careful observations, over all the wires, the equatorial interval between each side wire, and the central one. These intervals are to be set down for future use. Then, when part of the wires only are observed, each wire is to be reduced to the mean, l)y adding to, or subtracting from, the time of observation, as the case may be, the equatorial interval between that wire and the centre wire, multipUed by the secant of the declination of the star PORTABLE TRANSIT INSTRUMENT. 151 We shall hereafter show the use of the transit instrument in regulating a chronom- eter ; and for determining the longitude, by means of the observations of the transits of the moon and moon-culminating stars. TABLE A. Correction, in seconds of time, to be applied to one fourth part of the difference of the two intervals, supposing the whole difference to be 1000' of time. This correction is suhlraclive from the quarter interval, at the upper transit; additive to the quarter interval at the lower transit. Lat. o 10 20 30 40 42 44 40 48 50 52 54 50 58 60 Pole Star. Polar Distance of the Star. 0° 5° 10° 15° 20° 25° 30° 35° 40° s. 1 2 4 5 6 6 7 7 8 s. s. 4 8 13 8 16 25 s. 12 24 34 s. 16 33 53 s. 21 42 67 s. 25 53 83 s. 31 64 101 s. 37 76 121 15 18 20 31 37 40 47 56 60 64 76 82 ■ 82 98 105 101 121 130 123 147 158 147 176 189 21 23 24 43 46 49 65 69 74 88 94 101 113 121 130 139 149 160 169 181 194 8 9 9 26 28 30 52 56 60 80 86 92 108 116 125 139 149 160 172 185 199 10 11 12 33 35 38 65 70 76 99 107 116 135 146 158 173 187 The difference of the two intervals actually observed, is to be multiplied by the number given by this table, and the product divided by 1000 (which is the same as to cross olil' the llirce righi-haud figures) ; the quotient is the correction to be applied to one fourth part of the difference of the intervals. TABLE B. Correction of tlie azimuth, in minutes and tenths of a minute of space, corre- sponding to a difference in the two intervals of 1000 seconds in time. Lat. o 10 20 30 35 40 42 44 46 48 50 52 54 56 58 60 Pole Star. Polar Distance of the Star. 0° 5° 10° 15° 20° 25° 30° 35° 40° 1 II 1.00 1.43 1.48 1.57 / II 5.29 5.34 5.50 6.20 / // 11.02 11.13 11.46 12.46 / // 16.47 17.03 17.52 19.23 / II 22.47 23.10 24.16 26.20 1 II 29.13 29.40 31.06 33.45 1 II 36.11 36.44 38.30 41.47 / // 43.53 44.33 46.42 50.40 1 II 52.35 53.23 57.16 60.43 2.04 2.12 2.16 6.42 7.09 7.23 13.29 14.25 14.52 20.30 21.55 22.36 27.51 29.46 30.42 35.40 38.09 39.19 44.10 47.14 48.41 53.34 57.17 59.03 64.11 08.38 70.45 2.21 2.26 2.31 7.37 7.. 54 8.12 15.22 15.54 16.31 23.21 24.10 25.06 31.42 32.50 34.05 40.37 42.04 43.40 50.18 52.05 54.04 61.00 63.10 05.35 2.38 2.45 2.52 8.32 8.54 9.20 17.11 17.57 18.48 26.07 27.16 28.34 35.29 37.03 38.48 45.28 47.28 49.43 56.17 58.46 61.33 3.01 3.11 3.23 9.48 10.21 10.58 19.46 20.51 22.06 30.02 31.04 33.35 40.47 43.03 45.37 52.16 55.09 58.27 152 PORTABLE TRAiNSlT INSTRUMENT. TABLE C. Con'ection, in seconds of time, for 1000 seconds of space of deviation in azimuth from the plane of the meridian, to be applied to the time of the transit of the star observed by the transit instrument. Upper Transit. Lower Transit. Lat. o 5 10 15 20 25 30 35 40 45 50 55 60 Pole Star. s. 24C7 2451 2418 Polar Distance. Pole Star. Polar Distance. Lat. 5 10 15" 20 25 30 35 40 45 50 55 60 s. 763 751 737 717 691 66] 625 584 539 490 438 381 10° 383 370 360 347 332 314 293 271 246 220 191 162 15° s. 257 241 233 222 210 197 182 165 147 128 109 88 20° 194 176 168 159 149 137 125 HI 97 "82 66 50 25° s. 157 136 129 121 111 101 ~90 79 66 ~54 41 27 30° 133 109 102 "94 85 76 ~66 56 45 "34 23 12 35° s. 116 89 82 m 58 49 40 30 20 10 40° s. 103 73 66 59 52 44 ~35 27 18 ~9 5° s. 764 760 751 737 717 691 661 625 584 539 490 438 10° s. 333 381 377 370 360 347 332 314 293 271 246 15° s. 257 256 253 248 241 233 222 210 197 182 20° s. 194 194 191 188 183 176 168 159 149 25° 5. 157 157 155 152 148 143 136 129 30° s. 133 132 131 128 125 121 US 35° s. 116 115 Il4 112 109 105 40° s. 103 103 102 100 ~97 s. 2463 2441 2365 2295 2208 2103 1982 1847 1697 1535 1360 .1176 2400 2341 2264 2170 2059 1932 1791 1637 1469 322 131 66 34 14 1291 FlateYIL 1861 153 ON PARALLAX, REFRACTION, AND DIP OF THE HORIZON. Parallax (or diurnal parallax) is tJie difference between tht true altitude of the sun, moon, or star, if it loere observed at the centre of the earth, and the apparent altitude observed, at tlie same instant, by a spectator, at any point on the surface of the earth. Thus, in Plate XII., figure 3, let ABC be the earth, C its centre, A the place of a spectator, ZAK a vertical plane, passing through the place D of the moon, or tlie place d of a planet; EDF, edG, circular arcs draAvn about C as a centre, and KZ part of the starry heavens. Then, if at any time the moon be at D, she will be referred to the point H, by a spectator supposed to be placed at tlie centre of the earth, and this is called the true place of the moon ; but the spectator at A will refer the moon to the point b, and tliis is called the apparent place of the moon ; the difference H6 (or the angle HD6i= ADC) is called the moon's ^;ara/Zar in cdtitude, which is evidently greatest wlien the moon is in the horizon at E, being tlien equal to the arc KI, and it decreases from the horizon to the zenith, and is there nothing. The parallax is less as the objects are farther from the earth : thus the parallax of a planet at d is represented hylah, being less than that of the moon at D ; and the horizontal parallax Kyof the planet is less than the horizontal parallax KI of the moon. As the parallax makes the objects appear lower than they really are, it is evic^nt tliat the parallax must be added to the apparent altitude to obtain the true altitude. Having the hori- zontal parallax, the parallax in altitude is easily found by the following rule: — ^s radius is to the cosine of tlie apparent altitude, so is Vie horizontal parallcux to the parcdlax in cdtitude. This rule may be easily proved ; for in the triangle CAE we have CE : CA :: radius : sine CEA ; and in the triangle CDA we have CD (or CE) : CA :: sine CAD : sine CDA ; hence we have radius : sine CEA : : sine CAD : sine CDA ; but CEA r= horizontal parallax, CDA :=z parallax in altitude, and sine CAD = cosine app. alt. Hence we have radius : cosine app. alt. :: sine hor. par. : sine par. in alt. ; but the parallaxes of the heavenly bodies being very small, the sines are nearly proportional to the parallaxes; hence we may say. As radius : cosine app. alt. :: hoi*. j)ar. : par. in alt. The sun's mean parallax in altitude is given in Table XIV., for each 5^ or 10° of altitude. The moon's horizontal parallax is given in the Nautical Almanac, for every noon and midnight at the meridian of Greenwich ; also that of the sun for every ten days, and the parallaxes of Venus, Mars, Jupiter, and Saturn, for every five days, throughout the year. Refraction of the heavenly bodies. It is known, by various exjieriments, that the rays of light deviate from their rectilinear course, in passing obliquely out of one medium into another of a different density; and if the density of the latter medium continually increase, the rays of light, in passing through it, will deviate more and more from the right lines in which they were projected towards the perpendicular to the surface of the medium. This may be illustrated by the following experiment : — Make a mark at the bottom of any basin, or other vessel, and place yourself in such a situation that the hitlier edge of the basin may just hide the mark from your sight ; then keep your eye steady, and let another person fill tlie basin gently with water ; as the basin is filled, you will perceive the mark come into view, and appear to be elevated above its former situation. In a similar manner, the light, in passing from the heavenly bodies through the atmosphere of the earth, deviates from its rectilinear course. By this means the objects appear higher than they really are, except when in the zenith. This ap;)arent elevation of the heavenly bodies above their true places, is called the refraction of those bodies To illustrate this, let ABC (Plate XII., fig. 2) represent the atmosphere suiTounding 154 PARALLAX, REFRACTION, &c. the eartli DEF, and let an observer be at D, and a star at a ; then, if there were no refraction, the observer would see the star according to the direction of the right line Da; but as the light is i-efracted, it will, when entering the atmosphere near A, be bent from its rectilinear course, and will describe a curve line from A to D, and, at entering the eye of the observer at D, will appear in the line D h, which is a tangent to the cm"ve at the point D, and the arc ab will be the refraction in altitude, or, simply, the refraction, which must be subtracted from the observed altitude to obtain the true. At the zenith, the refraction is nothing ; and the less the altitude, the more obliquely the rays will enter the atmosphere, and the greater will be the refraction : at the horizon, the refraction is greatest. In consequence of the refraction, any heavenly body im y be actually below the horizon when appearing above it. Tims, when the sun is at T below the horizon, a ray of light TI, proceeding from T, conies in a right line to I, and is there, on entering the atmosphere, turned out of its rectilinear course, and is so bent down towards the eye of the observer at D, that the sun appears in the direction of the I'efracted ray above the horizon at S. The mean quantity of the refraction of the heavenly bodies is given in Table XII All observed altitudes of the sun, moon, planets, or other heavenly bodies, must be decreased by the numbers taken from that table corresponding to the observed altitude of the object. The refraction varies Avith the temperature and density of the air, increasing by cold or greater density, and decreasing by heat and rarity of the atmosphere. The corrections to be ap])lied to the numbers taken from Table XII., for the different heights of Fahrenheit's Thermometer and the Barometer, are given in Table XXXVL* Thus, if the refraction be required for the apparent altitude 5°, when the thermometer is at 20°, and the barometer at 30. G7 inches, we shall have the mean refraction by Tal)le XII. equal to 9' 53", and by Table XXXVI. the correction corresponding to the height of the thermometer 20° equal to -(- 48'', and for the barometer 30.67 equal to -\- 22"; hence the true refraction will be 9' 53" + 48" -1- 22" = 11' 3". There is sometimes an irregular refraction near the horizon, caused by the vapors near tlie surface of the earth; the only method of avoiding the error arising from this source, which is sometimes very great, is to take the obsei'vations at a time when the object which is observed is more than 10° above the horizon. The refraction makes any terrestrial object appear more elevated than it really is. The quantity of this elevation varies, at different times, from ^ to -i^ of the angle formed, at the centre of the earth, between the object and the observer ; but, in general, this refraction is about J^ of that angle. Dq} of the horizon. Dip of the horizon is the angle of depression of the visible horizon below the true or sensible horizon (touching the earth at the observer), arising from the elevation of the eye of the observer above the level of the sea. Thus, in Plate XII., figure 1, let ABC represent a vertical section of the earth, whose plane, being produced, passes through the observer and the object, and let AE be the height of the eye of the observer above the surface of the earth ; then FEG, drawn parallel to the tangent to the surface at A, will represent the true horizon, and EIH, touching the earth at I, will represent the apparent horizon ; therefore the angle FEH will be the dip of the horizon. Let M be an object whose altitude is to be observed by a fore observation by bringing the image in contact with the apparent horizon at H ; then will the angle flIEH be the observed altitude, which is greater than the angle MEF (the altitude independent of the dip) by the quantity of the angle FEH ; so that, in taking a fore observation, the dip must be subtracted from the observed altitude to obtain the altitude corrected for the dip. In a back observation, the apparent horizon is in the dii'ection EK ; and, by continuing this line in the direction EL, Ave shall have the observed altitude MEL ; and it is evident that to this the dip LEF (= KEG) must be added to obtain the altitude corrected for the dip. In Table XIII. is given the dip, for every probable height of the observer, expressed in feet. In calculating this table, attention is paid to the terrestrial refraction, which decreases the dip a little, because IE becomes a cune line instead of a straight one, and EH is a tangent to tliat curve in the point E. * This talile is to be entered with the height of tlie thermometer or barometer at the tof), and the apparent altitude at the side ; under tlie former, and opposite the latter, will be the correction corre- sponding- to the thermometer or barometer, uhicli is to be apphed to the mean refraction, by addition o» subtraction, according to tlie signs at the top of the columns rcsDectively. PARALLAX, REFRACTION, &c. 155 What has been said concerning the dip of the liorizon, suj)poses it free from all encumbrances of land or other objects; but, as it often happens, when sliips arc sailing along shore, or at anchor in a harbor, that an oljservation is vwiiitcd when the sun is over the land, and the shore nearer the shij) tlian the visible horizon would be if it were imconfined, in this case, the dip of tlie hoi-izon will be different from what it otherwise would have been, and greater the nearer the ship is to that part of the shore to which the sun is brougiit down. For this reason Table XVI. has been inserted, which contains the dip of the sea at different heights of the eye, and at diftl'i-ent distances of the ship from the land. This table is to be entered at the top with the height of the eye of tiie observer above the level of the sea in feet; and in the left-hand side column, with the distance of the ship from the laml in sea miles and parts. Under the former, and oj)])osite the latter, stands the dip of tiie horizon, which is to be subtracted from the altitude observed by a fore observation, instead of the numbers in Table XIII. The distance of the land requisite in finding the dip from Table XVI., may be found nc'arly in the following manner: — Let two observers, one placed as high on the main-mast as he can conveniently be, and the other on the deck immediately beneath him, observe, at the same instant, the altitude of the sun or other object that may be wanted, and let the height of the eye of the upper observer above that of the lower be measured in feet, and multiplied by 0.56; then the product, being divided by the difference of the observed altitudes of the sun in minutes, will be tlie distance in sea miles, nearly. Thus, if the eye of the upper observer was G8 feet higher than that of the loAver, and the two observed altitudes of the sun 20° 0' and 20° 12', the distance of the land, in sea miles, would be 3.2. For C8 X 0.5G z=: 38.08, and this, being divided by the difference of the two observed altitudes of the sun 12', gives 3.2, nearly. Now, if the lower obsei-ver be 25 feet above the level of tiie sea, the dip corresponding to this height and the distance 3.2 miles will l)e C, which, being subtracted from 20° O', leaves 19° 54', the altitude corrected for the dip. The dip may be calculated, in this kind of observations, to a sufficient degree of accuracy, without using Table XVI., in the following manner: — Divide the difference of the heights of the two observers in feet, by the difference of the observed altitude in minutes, and reserve the quotient. Divide the height of the lower observer in feet by tills reserved number, and to the quotient add one quarter of the reserved number, and the sum will be the dip in minutes corresponding to the lower observer. Thus, in the above example, £S-=z5'.C is the reserved number, and ^z=4.4; to this add one fourth of 5'.G or l',4, and the sum will be the dip 5'.8, or nearly 6', corresponding to the lower observer, being the same as was found by the table. 156 TO FIND THE SUN'S DECLINATION. The declination of the sun is given, to the nearest minute, in Table IV.. for every noon, at Greenwich, from the year 1833 to 1848 ; and this table will answer for some years beyond that period, without any material error. If great accuracy is required, the declination may be taken from the Nautical Almanac* This declination may be reduced to any other meridian, by means of Table V., in the following manner :— To find the sun's declination, at noon, at any place. RULE. Take out the declination at noon, at Greenwich, from Table IV., or from the Nautical Almanac ; then find the longitude from Greenwich in the top column of Table V., and the day of the month in the side column ; under the former, and opposite to the latter, is a correction, in minutes and seconds, to be applied to the declination taken from Table IV. ; to know whether this correction be additive or subtractive, you must look at the top of the column where you found the day of the month, and you Avill see it noted whether to add or subtract, according as the lon- citude is east or west. This correction being applied, you will have the declination at noon at the given place. EXAMPLE I. Required the declination of the sun, at the end of the sea day, October 10, 1848, in the longitude of 130° E. from Gi-eenwich. Sun's declination, October 10, at Greenwich, at the end of the sea day, or beginning of die day in the N. A., by Table IV G° 48' S. Variation of dec, Table V., October 10, in 130° E. long sub. _0 8 True dec noon, October 10, in long. 130° E Aj^O. ^• EXAMPLE II. Required the sun's declination at noon ending the sea day of March 12, 1848, in the longitude of 65° W. from Greenwich. Sun's declination, March 12, by Table IV 3° 9' S Var. Table V., March 12, long. C5° W sub. 4 True declination, noon, March 12, long. 65° W 3 5 S The preceding correction ought always to be applied to the declination used in working a meridian observation to determine the latitude, though many mariners are in the habit of neglecting it. * In finding the declination, or any oilier quantity, in the Nautical Almanac, you must be careful to note the difference between the civil, nautical, and astronomical account of time. The civil day begins Ht midnight, and ends the following midnight, the interval being divided into 2-1 hours, and is reckoned in numeral succession from 1 to 12, then beginning again at 1 and ending at 12. The nautical or sea day begins at noon, 12 hours before the civil day, and ends the following noon ; the first 12 hours are mark- ed P. 31., the latter A. M. The astronomical day begins at noon, 12 hours after the civil day, and 24 hours after the sea day, and is divided into 2-1 hours, numbered in numeral succession from 1 to 2-1, beginning at noon, and ending the following noon. All the calculations of the Nautical Almanac are adapted to astronomical time; the declination marked in the Nautical Almanac, or in Table IV., is adapted to the beginning of the astronomical day, or to the end of the sea day, it being at the end of the sea day when mariners want the declination to determine their latitude. It would be much better if seamen would adopt the astronomical day, and wholly neglect the old method of counting by the Eca day riatc xnL .1^. Sf' w> ^.. .-y 5 tO Gj ^>^ \ >^^ Hsl. MJEFRA.riTI-im'^ ■ ■■Aa TAJiALLJnL. Fi O^ Fi^.o. • K 1861 TO FIND THE SUN'S DECLINATION. 157 To find the sun's declination, at any time, under any meridian. RULE. Reduce tlie sun's declination at noon at Greenwich, to noon under the given me- ridian, by the preceding rule ; then enter Table V. with the time from noon at the top, and the day of the month in the side column; under the former, and opposite the latter, will be the correction to be api)lied to tliat reduced declination. To know whether this correction be additive or subtractive, you must look at the top of the colunni Aviiere you found the day of the month, and you will find it noted whether to add or subtract, according as the time is before or after noon. EXAMPLE in. Required the sun's declination October 10, 1848, sea account, at S** 21"" in the forenoon, in the longitude of 130° E, from Greenwich. Sun's declination Oct. 10, at Greenwich, at noon, by Table IV 6° 48' S. Variation for 130° E. long subtract 8 Declination at noon, October 10, in long. 130° E 6 40 S. Variation of dec. for S*" 39"" from noon,* Oct. 10, subtract 3 True dec. Oct. 10, sea ace. in long. 130° E. at Si^ 21">, A. ]M 6 37 S. EXAMPLE IV. Required the sun's declination May 10, 1836, sea account, at 5'' 30"', P. M., in the longitude of 35° 45' E. from Greenwich. Variation of declination, May 10, in long. 35° 45' E subtract V 38" Variation of declination for 5^ 30™, P. M add 3 44 Dift". is additive, because the greatest number is so 2 00 May 10, sea account, is May 9, by N. A., at which time the sun's declination 17° 26 27 True dec. May 10, 5'' 30", P. M., sea account, in long. 35° 45' E 17 28 33 N. EXAMPLE V. Required the sun's declination March 26, 1836, sea account, at 3'', P. jM., in the longitude of 140° W. from Greenwich. Vm-iation of declination, March 26, in long. 140° W add 9' 08" Variation for 3", P. M , add 2 50 Sum 12 04 March 26, sea account, is iMarch 25, by N. A., .it v/hich time the sun's declination 1° 56 41 N Ti-ue dec. March 26, 3^ P. M., sea account, 2 08 45 N * In tlie present example, Ihe time is Oct. 10, StZl™, A, M., which evidently \vant5 3'' SO"" of the emi o*" tire spa day, Oct. 10, for which time the declination is marked in Table IV. 158 VARIATION OF THE COMPASS. It was many years after the discovery of the compass, before it was suspected that the magnetic needle did not point accurately to the north pole of the world ; but, about the middle of the sixteenth century, obsen'ations were made in England and France, which fully proved that the needle pointed to the eastward of the true north. This difference is called the variation of the coinpass, and is named east when the north point of the compass (or magnetic north) is to tlie eastward of the true north, but icest when the north point of the compass is to the westward of the true north. The quantity of the variation may be found by observing, with a compass, the bearing of any celestial object when in the horizon, (or, as it is called, the magnetic amplitude ;) tiie difference between this and the true amplitude, found by calculation, will be the variation. Tlie same may be obtained by observing the magnetic azimuth of any celestial object, (that is, its bearing by a compass when elevated above the liorizon ;) the difference between tbis and the true azimuth, found by calculation, will be the variation. Some years after the discovery of the variation, it was found that it did not remain constant ; for the easterly variation, observed in England, gradually decreased till the needle pointed to the true north, and then increased to the westward, and is now above two points. As all the com'ses steered by a compass must be corrected for the variation, to obtain the true courses, it is of great importance to the navigator to know how to find the vai'iation at any time. To do this, it is necessary to find the magnetic amplitude or azimuth of a celestial object, which may be done as follows : — To observe an amplitude by an azimuth compass* When the centre of the sun is about one of his diameters f above the horizon, turn the compass round in the box, until the centre of the sun is seen through the naiTOW slit which is in one of the sight-vanes, exactly on the thread which bisects the slit in the other : J at that instant pus'h the stop, which is in the side of the box, against the edge of the card, and the degi-ee and parts of a degree which stand against the middle line on the top, will be the magnetic aini)litude of the sun at that time, which is gen- erally reckoned from the east or west point of the com})ass. To observe an azimuth by an azimuth compass. Turn the compass round in the box until the centre of the sun is seen through the narrow slit which is in onelbf the sight vanes, exactly on the thread which bisects the slit on the other, or until the shadow of the thread falls directly along the line of the horizontal Itar ; f the card is then to be stopped, and the degree and parts of a degree which stand against the middle line of the stop, will lie the magnetic azimuth of the sun at that time, which is generally reckoned from the north in north latitude, and from the south in south latitude. § At the time of making this observation, you must also observe the altitude of the sun, in order to obtaiji the true azimuth. What is here said of the sun, is alike a])plicable to the moon, planets, and stars. * The figure of an azimuth compass, furnished wiih sight-vanes, is given in Plate VI., figure 5. Tlie card of this compass is similar to that of a common compass. t The observation is to be taken at that altitude on account of the dip, refraction, and parallax, the correction of altitude depending on these causes being, in general, nearly equal to the sun's diameter. X If the instrument is furnished with a magnifying glass fixed to one of the vanes, you may (instead of proceeding as above) turn tlie compass box until the vane is directed towards the sun, and when the bright speck (or rays of the sun collected b}- the magnifying glass) falls upon the slit of the other vane, or upon the line in the horizontal bar, the card is to t)e slopped, and the divisions read oft' as above. § If the compass vibrate considerably at the time of making the observations, it would be conducive VARIATIOiN OF THE COMPASS. 159 To find the true amplitude. RULE. By Logarithms. — To the log. secant of the latitude {rejecting 10 in the index) add the log. sine of the siui's declination; * the sum will he the log. sine of the true amplitude, or dista7ice of the su7i from the east or west point, towards the north in north declination, but towards the soidh in soidh declination. By Lvspectiox. — Find the declination at the top of Table VII., and the latitude in the side column ; under the former, and opposite the latter, will he the true amplitude. When great accuracy is required, you may proportion for the minutes of latitude and decli- nation. EXAMPLE L Required the sun's true amplitude, at risin":, in tlie hititude of 39° 0' N., on the 22d of December, 1848 when his decHnation was 23° 28'. BY LOGARITHMS. Latitude 39° 0' Log. Sec. 0.10950 Sun's dcclin. 23 28 Log. Sine, 9.000 12 True ampli. 30 49 Log. Sine, 9.709G2 BY INSPECTION. Under tlie declination 23° 28', and op- posite the latitude '39°, stands the true amplitude 30° 49'. Hence the true bearing or amplitude of the sun at rising is E. 30° 49' S., and at setting it is W. 30° 49' S. EXAMPLE II. Required the moon's true amplitude at setting, in the latitude of 35° 8' N., when her declination is 13° N. BY LOGARITHMS. Latitude 35° 8' Log. Sec. 0.08734 Moon's declin. 13 Log. Sine, 9.35209 True ampli. 15 58 Log. Sine, 9.43943 BY INSPECTION. Under the declination 13°, and opposite the latitude 35°, stands 15° 56', which ia nearly the true amplitude ; the exact value may be found by finding the amplitude for 36° latitude, and proportioning the differ- ence for the miles in the latitude. Hence the true amplitude at setting is W. 15° 58' N., and at rising E. 15° 58' N. EXAMPLE III. Required the sun's true ami)litude in the latitude of 42° 30' N., when his declinatiou was 20° S. BY LOGARITHMS. Latitr.de 42° 30' Log. Sec. 0.13237 Sun's declin. 20 00 Log. Sine, 9..53405 True ampli. 27 38 Log. Sine, 9.66642 BY INSPECTION. Under the declination 20°, and opposite the latitudes 42° and 43°, stand 27° 24' and 27° 53'; the mean of these gives the true ami^Iitude for the latitude of 42° 30' z= 27° 38'. Hence the amplitude at setting is W. 27° 38' S., and at rising E. 27"" 38' S. To find the true azimuth at any time. At the time of observing the magnetic azimuth, you must also observe tlie altitude of the object ; this altitude must be corrected as usual for the dip, parallax, refraction,f &c., in order to obtain the true altitude ; you must also find the declination of the to accurac3' to lake several azimuths and altitudes, and to lake the mean of all the azimuths and all the latitudes, and work the observation with the mean azimuth and altitude. The same is to be observed in taking an amplitude. * The declination of the sun at noon is given in the Nautical Almanac, and in Table IV., and must be corrected for the longitude of the ship and the hour of the day, by means of Table V. t In observations of the altitude of the sun's lower limb by a fore observation, it is usual to add 12' for the effect of dip, parallax, and semi-diameter. The refraction is to be subtracted from the sum, and the remainder will be the true altitude, nearly. 100 VARIATION OF THE COMPASS. object,* and the latitude of the place of observation, and then the true azimuth may be calculated by the following rule : — RULE. Add together the polar distance, f the latitude, and the true altitude ; take the dif- ference between the half-sum and the j)olar distance, and note the remainder. Then add together the log. secant of the latitude, tlie log. secant of the altitude, (rejecting 10 in each index,) the log. cosine of the half-sum, and the log. cosine of the remainder ; half the sum of these four logarithms will be the log. cosine of half the true azimuth, which, being doubled, will give the true azimuth, reckoned from the north in north latitude, but from the south in south latitude. EXAMPLE I. fn latitude 51° 32' N., the sun's true altitude was found to be 39° 28', his declination being then 16° 38' N. ; required the true azimuth ? Polar distance 73° 22' Latitude 51 32 Secant 0.20617 Altitude 39 28 Secant 0.11239 Sum 164 22 Half-sum 82 11 ' Cosine 9.13355 Polar distance , . 73 22 Remainder 8 49 Cosine 9.99484 2)19.44695 Half-sum Log. Cosine 58° 4' 9.72347 2 True azimuth IIG 8 from the north. The logarithm 9.72347 of this example is also the cosine of 121° 56', which, being doubled, gives another azimuth 243° 52', die former being 116° 8'. One of these corresponds to an obseiTation in the forenoon, the other to an afternoon observation. EXAMPLE II. In latitude 42° 16' S., the sun's true altitude was found to be 18° 40', his declination being then 7° 38' N. ; required the true azimuth. Polar distance 97° 38' Latitude 42 16 Secant 0.13076 Altitude 18 40 Secant 0.02347 Sum 158 34 Half-sum 79 17 Cosine 9.26940 Polar distance 97 38 Remainder 18 21 Cosine 9.97734 Sum 19.40097 Half-sum . . . Log. Cosine 59° 53' 9.70048 Q True azimuth 119 46 from the south. QUESTIONS TO EXERCISE THE LEARNER. Question I. Given the sun's altitude, corrected for dip, refraction, &c., 20° 46', his declination 17° 10' S., and the latitude of the place 40° 38' N. ; required the true azimuth. Ansiver. 137° 50' from the north. * The (leclinntion is to be found according- to the directions in the note in the last page. t The polar distance of tiie sun, moon, or star, is tiie distance from the elevated pole, and is found by subtracting the declination of tlie object from 90° when the latitude and declination are of the same name, but bj' adding the declinaljon to 90° when of different names. VARIATION OF THE COMPASS. 161 Q^uesf. II. What is the sun's azimuth in the jatitude of 20° 30' N. in the forenoon, when his correct central altitude is 24° 28', and his declination 22° 40' N. ? Ans. 75° 44' from the north. Q_uest. III. At the island of St. Helena, the sun's true central altitude was found to be 30° 23' in the forenoon, his declination being then 22° 58' S. ; required the azimuth at that tiuie. Ans. 72° 21' from the south. Quest. IV. What point of the compass did the star Aldel)aran bear on, in the latitude of 34° 23' S., on January 1, 1836, when the correct altitude of that star was 22° 26' ? .^ns. 130° 23' from the south. Having fhe true and the magnetic amplitude or azimuth, to find the variation. Having found the true and magnetic amplitude or azimuth, the variation may be easily deduced therefrom by the following rule, in which the amplitude is reckoned from the east or west point of the horizon, and is called north when to the northward of those points, but south when to the southward. The azimuth is reckoned from the north in north latitudes, but from the south in south latitudes, and is named east when falling on the east side of the meridian, otherwise west. If the observed and true amplitudes be both north or both south, their difference loill be the variation ; but if one be north and the other south, their sum will be the variation. If the true and observed azimuths be both east or both xoest, their difference loill be the variation, otherwise their sum ; and the variation will be easterlij when the point representing the true bearing is to the right hand of the point representing the magnetic bearing, but westerly when to the left hand ; Vie observer being supposed to look directly towards the point representing the magnetic bearing. EXAMPLE I. Suppose the sun's magnetic amplitude at rising is E. 26° 12' N., and the true amplitude E. 14° 20' N. ; required the variation. From the greater E. 26° 12' N. Take the less E. 14 20 N. Remains variation 11 52 E. The variation in this example is easterly, because the true amplitude falls to the jight of tlie magnetic. EXAMPLE II. The moon's true amplitude at rising was found to be E. 15^ 20' N., and her magnetic amplitude E. 10° 0' S. ; recjuired the variation. True amplitude E. 15° 20' N. Magnetic amplitude E. 10 S. Sum is the variation 25 20 W. EXAMPLE III. The sun's true azimuth being N. 80° E., and his magnetic azimuth N. 60° E., it is required to find the variation. True azimuth N. 80° E. Magnetic azimuth N. 60 E. DifF. is the variation 20 E. EXAMPLE IV. The star Aldebaran was observed at rising to bear by compass E. N. E., when the true amjilitude was N. E. by E. ; re- quired the variation. True amp.. .N. E by E. or E. 33° 45' N. Mag. amp E. N. E. or E. 22 30 N. DifF. is the variation 11 15 W. EXAMPLE V. The true amplitude of the planet Jupiter was E. 10° N. when his magnetic ami)li- tude was E. 20° S. ; required the variation. True amplitude E. 10° N. Magnetic amplitude E. 20 S. Sum is the variation 30 W To calculate the variation hy observing the swi's azimuth lohen at equal altitudes in the forenoon and afternoon. The variation of the compass may also be determined by observing the magnetic azimuths of the sun, in the mornhig and evening, when at the same altitude, the 162 VARIATION OF THE COMPASS. observer being suy)posed to be at the same place at both observations ; for it is evident that if the dechnation of the sun do not vary during the time elapsed between the observations, the middle point of the compass between the two bearings will be the bearing of the true north or south point of the horizon, at the place of observation, and the difference between that bearing and the north or south point of the comj)ass will be the variation. In this kind of observations, it will be convenient always to estimate the magnetic azimuths from the south point of the compass, calling them east or west, as before directed ; and this method is supposed to be made use of in the following rule. Then, if one azimuth be east and the other west, half their difference will be the variation, otherwise their half-sum, and the variation will be of the same name as their gi-eater azimuth, excepting, however, where the half-sum is taken and exceeds 90°, in which case its supjjjemcnt will be the variation, of a different name from the azimuth ; the variation being always supposed less than 90°. If the declination of the sun varies during the elapsed time betvv'een the observations (as is generally the case), an allowance may be made for that variation by apjdying a correction to the afternoon azimuth, calculated by the following rule : — RULE. Find, from Table IV., the daily variation of the sun's declination on the day of observation. Then to the constant logarithm 9.1249 add the log. cosine of the latitude of the place, the log. sine corresponding to the elapsed time between the observations found in the column P. M., the Prop. Log. of the daily variation of the sun^s declination, and the Prop. Log. of the elapsed time,* estimating hours and minutes as mimdes and seconds ; the sum, rejecting 30 in the index, ivill be the Prop. Log. of the correction to be applied to the western azimuth, by subtracting lohtn the sun is approaching toivards the noHhem hemisphere, otherwise by adding.^ The azimuth, thus corrected, is to be used in estimating tlie variation instead of the observed azimuth. It is not necetjsary, in this calculation, to find the latitude or declination to any great degree of accuracy, which is the greatest advantage of the method ; another of the advantages consists in being able to take a great number of observations, and apply- ing the correction at one operation to the variation deduced from the mean of all the observations, so that, when gi'eat accuracy is required (as in taking observations ashore), this method may be used with success ; and it is evident that it is alike applicable to the moon or any heavenly body ; but the observations must be taken in the same place, as it would increase the calculation considei'ably to make an allow- ance for the change of place, as well as for tlie cliange of declination ; and it would be better, in this case, to calculate each observation separately by the rules before given. • EXAINIPLE. Su])pose that, on the 10th of April, 1848, in the latitude of 42° 29' N., longitude 50° W., the sun's morning azimuth is observed to be S. .54° 24' D., and in the evening', when the sun is at the same altitude, is S. 39° 46' W., the elajjsed time between the observations being C 20™ ; required the variation. Constant logarithm 9.1249 Latitude 42° 29' Cosine 9.8677 Elapsed time 6^ 20™ Sine 9.8676 Daily variation of declination 22' P. L 9128 Elapsed time 6" 20", taken as 6' 20" P. L.. . . 1.4536 Con*, western azimuth 11' nearly P. L. — 1.2266 Western azimuth S. 39 46 W. Corrected azimuth S. 39 35 W. Morning azimuth S. 54 24 E. Difference S. 14 49 The half of which, 7° 24', is the varia- tion, which IS easterly, because the gi-eater azimuth S. 54° 24' E. is easterly. * The elapsed time may be tletennined by any common watch ; but if none be used in the observa- tions, it maj' be determined as follows : — If one of the observed azimuths be east and the other west, take half their sum, otherwise half their diflerence, and to the log. sine of this half-sum (or half-din"ercnce) add the log', secant of the sun's declination, and the log. cosine of the sun's correct allitude at the time of taking the azimuth ; the sum, rejecting 20 in the index, will be the log. sine to be used in the abo\e calculation, and this logarithm will correspond to the elapsed time marked in the column P. M. of Table XXVII. t In this rule it is supposed that the bearing of the sun by the afternoon observation, i.« to the west- VARIATION OF THE COMPAS?. 163 The variation, tliiis found, is to be allowed on all courses steered by the con pass, to obtain the true courses. To tnake this allowance, you must, look towards tlie point of the conii)ass tiie ship is sailing upon, and allow the variation from it ioivards the rxs;ht hand if the variniion be east, hut to the left hand if the variation be ivest. Tlius, if a shij) steer S. E. with one point westerly variation, the true course will be S. E. by E If tl.e variation be one point easterly, the course will be S. E. by S. The variation in Cambridge (Mass.), in 1708, was 9° W. ; in 1742, 8° W. ; in 1782, [>^ 4(5 W. ; decreasing aljout IJ minutes per year. At Salem (Mass.), in 1808, it was 5° 20 W. : in London, in 1580, 11° 15' E. ; in 1672, 2° 30' W. ; in 1780, 22° 41' W. : at Paris, in 1.550, 8° E. ; in 1660, 0° ; in 1769, 20° W. Hence it appears that, at London and Paris, the \ariation formerly increased 10 or 11 mimites per year ; but, by some late observations made in London, it ai)pears to be nearly stationar3^ Off the Cape of Good Hope, the annual increase is about 7 minutes. Besides this annual change of the variation, there is also a small diurnal change, which, at London, Paris, and Cambridge (Mass.), is from 10' to 15' By this quantity the absolute variation, at tliose ])laces, increases from about 8, A. IM., to 2, P. ]\1., when the needle becomes stationary for some time ; after that, the variation decreases, and the needle comes back again to its former situation, or nearly so, in the night, or by the next morning. In addition to the observations contained in the preceding table, it may be obsei'ved that the variation, which, at present, is less than ^ ])oint W. near Cape Cod, decreases in going to the westward along the coast of the United States of America, so that near Cape Hatteras it is scarcely sensible, and farther to the westward becomes easterly. In the leeward West India Islands it is about ^ point E. ; and in the windward islands i point E. Along the noi'thern shore of the Brazils there is a small easterly variation, which decreases in proceeding to the eastward, and becomes westerly neai" Cape Roque, where it is ^ point AV. In proceeding farther to the southward, along the coast of America, the easterly variation increases so as to be ai)ove 2 points E. near Cape Horn, and from thence gradually decreases along the coast of Chili and Peru, so as to be about 1 point E. under the equator near Quito ; but in proceeding to the nortiiward towards the N. W. coast of America, the easterly variation increases to more than 2 points. On the contrary, in proceeding to the eastward of the United States of America, the westerly variation increases, being nearly 1 point W. a little to the eastward of Cape Sable (Nova Scotia), and about 2| points W on the E. part of Newfoundland, and at the Western Islands. At the Orkney Islands it is 2^ ])oints westerly, and is nearly the same in the English Channel, and on the coasts of England, Scotland," and Ire- land. On the coast of Holland, it is from 1:^ to 2 points W. ; in the Cattegat and Sound, about 1-^ points W. ; in the western part of the Baltic, about 1:^ points; at the entrance of the Gulf of Finland, 1 ])oint W. ; in the Bay of Biscay, about 2i [loints W. ; near Cajie St. Vincents, 2 points W. ; in the Mediterranean, from 1 to II pohits W. ; near Cape Verd (Africa), 1^ points W. ; and from thence gradually increases along the western shore of Africa towards the Cape of Good Ilope, and is there a!)ove 2 points W., and from thence increases towards Cape Lagullas, and a little to the eastward, to 2i points or 25 points W., and then decreases in proceeding along the eastern shore of Africa, and is about | point westerly at the entrance of the Red Sea. In the Arabian Sea, Bay of Bengal, Java Sea, China Sea, and off the coast of Sumatra, it is very small, and on the S. E. part of New Holland is about I point E. Before the introduction of the method of finding the longitude by lunar oliservatjons, and the improvements in the construction of chronometers, and their introduction into connnon use, it was profjosed to find the longitude by means of the observed variation, and charts were constructed for tliis purjjose ; but this method is now wholly given uj), because there is always a gi-eat uncertainty in observations of the variation, since it is not uncommon to find 2 or 3 degrees difference between an azimuth in the morning and evening, when the shi]), during that time, has been nearly stationary ; the same difference will sometimes be found merely from making the observation when tlie ship is on a different tack. This is owing to the iron in the ship, \\hich attracts the compass by a force which is generally situated in a point near the centre of the ship. When this point and the compass are in the magnetic ward of the ineridian by compass ; but il there be a great variation, that bearing mig-ht be to the eastward of the meridian by tiie compass, and, in that case, the correction of the western azimuth must be app'ied in i contrary manner to the above directions 164 VARIATION OF THE COMPASS. meridian of .ne compass, the true variation is obtained ; but as soon as the position of the ship is changed, so as to bring this point to th-^. eastward or westward of the magnetic meridian passin" through the compass, a corresponding change or altera- tion in the variation to the eastward or westward is immediately perceived. This deviation sometimes amounted to 8° or 9° in the surveys of New Holland. This has since been confirmed by various observations in different places, particularly in the voyages towards the north pole, lately made by order of the English government. The method wliich v/as at first used to correct this error, which is sometimes of considerable importance in nautical surveys where great accuracy is required, was to -place the compass always in the same part of the ship, and to find, by actual observa- tion, the greatest deviation arising from tiiis local attraction, which is when the ship's head is directed east or west. The deviation, when the ship's head is in any other direction, is found by entering Tahle I. or Table II. in the page corresponding to that direction as a course, and witli that greatest error in minutes in the distance column, the corresponding number in the dei>arture colunui will be the required correction nearly. Thus, if the deviation was 2° 8' (or 128') when the ship's head was directed towards the east, the deviation, when in the direction of one point from the meridian, (that is, N. by E., N. by W., S. by E., or S. by W.), would be found by entering Table I. in the page for one point, or with the distance 128', the corresponding de])art- ure 25' would be the correction to be applied on all bearings taken by the compass when in that situation. Mr. Barlow has invented a method of correcting this error, making use of a curious property of the attractive force of iron on the compass, it having been found that this force depends on the attractive surface, and not wholly on the quantity of iron ; so that a solid globe of iron, 30 inches in diameter, would affect the coni])ass exactly in the same maimer as a holloiu shell of the same diameter, made of sheet iron only one tenth of an inch in thickness, though this shell could not contain but one hundredth part the quantity of iron which the globe does. Mr. Barlow therefore proposed to have a sheet of iron placed abaft the compass, cf such dimensions, and at such a distance, as should be found by experiment to bring the needle back to the magnetic meridian when the ship's head was east or west ; then, keeping the iron in that position, it would correct the error of the local attraction of the ship in every direction of the ship's head. This method has been tested by e.xperiment, and found to succeed admirably. It has also been attended with the great advantage of leaving the compass free to act by the natural magnetism of the earth in high latitudes, where the force is much enfeebled by the oblicpiity of its direction on account of the greatness of the dip. In the voyages above named, it was found that the compasses thus furnished traversed freely and accurately, when those of the conmion form moved very irregularly, and were, in some cases, almost useless. The Transactions of the Royal Society of London for the year 1833, contain a valuable chai't, by P. Barlow, upon which are marked the magnetic lines of equal variations, as they have been observed in late vojages of discovery, surveys, &c. We expect to give, in the collection of tables, a few numerical results from this chart On the dip of the magnetic needle. If the needle of a compass be exactly balanced on its point in a horizontal position, and then the magnetic virtue be communicated, the needle will point towards the north, and will also be inclined to the horizon, the north point of the needle tending down,\\anls, and the south point upwards, in northern climates, and the contrary in southern climates. This inclination of the needle to the iiorizon is called the dip of the magnetic needle, which is different in different places, though it has bewi found to remain nearly the same in the same place, since its discovery in the year 157(5, in which year, at London, the dip was 71° 50' ; in 1723, it was 74° or 75° ; and, ut i)resent, is about 72-^°. 31essrs. Humboldt and Biot published a method by which the dip may be calculated for any given ])lace, in north latitudes, to a considerable degree of accuracy. This method is explained in the 22d vol. of Tilloch's Magazine, and is in substance as follows : — According to their theory, there are two magnetic poles, one in the latitude of 79° 1' N., and in the longitude of 27° 42' W.* from Greenwich, the other diametrically opposite, in the latitude of 79° 1' S., and in the longitude of 152° 18' E. The great * Capt. Ross, in liis voyage to the north, found the northern pole to be in the latitude of 70° 5' 17 N., and in the longitude of yG° 46' 45" VV. VARlATlOiN OF THE COMPASS 165 circle of the earth 90^ distant from tliese poles is called the magnetic equator. On tho ma.^netic equator the dij) is nothing, and at the poles is 90°; at any other point on the surface of the earth, the dip varies with the distance from the magnetic pole. This distance may be calculated by connnon si)herical trigonometrj', or (which is much more siinj)Ie, and sufficiently accurate for this purpose) by measuring the tlistance on a terrestrial globe from the magnetic pole to the place for which the dip is to be calculated ; then to the log. cotangent of this distance add the constant logarithm 0.3010:j ; tiie sum will be the log. tangent of the dip. The dip was calculated, on these prinr-iples, for twenty-eigiit places in Euro])e, Asia, Africa, and America, and in ten places the theory did not differ 1° from actual observations, and in five places did not differ T , but at Spitzborgen the difference wad between 4° and 5°. (See page 459.) / THE MARINER'S COMPASS'.??. 166 TO FIND THE LATITUDE BY OBSERVATION. The latitude of a place, being its distance from the equator, is measured by an arc of the meridian contained between the zenith and the equator ; hence, if the distance of any heavenly botly from the zenith when on the meridian, and the declination of the object, be given, the latitude may be thence found. The meridian zenitli distance of any object may be found by observing its altitude when on the meridian, or by oljserving one altitude taken at a given hour from pass- ing the meridian, or by two altitudes taken out of the meridian and the elapsed time between the observations. Each of these methods will be explained by proper examples. Altitudes of the sun and moon, taken at sea, require four corrections in order to obtain the true altitude of their centres ; these are for semidiameter, dip, refraction, and parallax.* When a planet or star is observed, the corrections for dij) and refrac- tion only are to be applied, as the semidiameter and parallax of a planet are but a few seconds, and may be neglected in finding the latitude at sea. In a fore ohservation ivith a quadrant, sextant, or circle, the scmidiaineter is to be added if the lower limb is observed, but subtracted if the upper limb is observed. The dip and refraction are to be subtracted, and the parallax to be added, and the true central altitude will be thus obtained, which, being subtracted from 90°, will give the true zenith distance. In a back ohsei-vation ivith a quadrant, the semidiameter is to be subtracted if the lower liud) is observed, but added if the upper limb is observed. Tlie dip and paral- lax are to be added, and the refraction subtracted, and the central altitude will be obtained, which, being subtracted from 90°, will give the true zenith distance. In a hack observation ivith a sextant or circle, by measuring the su])i)lement of the altitude, (by bringuig the lower limb of the image of the object to touch the back horizon,) the senfidiameter and refraction must be added to the true altitude given by the instrument, and the dij) and jiarallax subtracted therefrom, and, by subtracting 90° from the remainder, the true zenitli distance will be obtained. To find the latitude by the meridian altitude of any object. Having obtained the true meridian zenith distance by either of these methods, you must then find the declination of the object at the time of observation. This may be found for the sun by the Nautical Almanac, or by means of Tables IV. and V., in the manner before explained. The declination of a fixed star may be easily found by inspection in Table VIII., or from the Nautical Almanac. The declination of the moon or a planet may be found, in the Nautical Almanac, in a manner wliich will be hereafter explained. Having the meridian zenith distance and declination, the latitude is to be found by the following rules. CASE I. Jflicji the object rises and sets. RULE. If the object bear south when upon the meridian, call the zenith distance noTth ; | but if the bearing be noiih, you must call the zenith distance south. Place the zenith * The semidiameter of the sun may be found in the Nautical Ahnanac, and is nearly 16'. The sun's parallax is found in Table XIV. ; the refraction in Table XII. ; the dip in Table XIII." The semidiam- eter and parallax of the moon may be found from the Nautical Almanac, as will be explained ncreafler. It may also be observed, tlial it is usual to add 12' for the correction for semidiameter, dip, and parallax, in a fore observation of the sun's lower limb, taken upon the deck of a common-sized vessel ; and, by subtractinsT the retraction from the sum, the true altitude will be obtained, nearly; and it ouj:ht alwaj'S to he kept in mind, that the refraction at low altitudes is of too much importance to be neglected. t In tliis rule, the sun is supposed to be the fixed point, and the zenith is referred to it. Thus, if the sun bears south from an observer (or from his zenith). Uie zenith bears north from the sun ; and it is this Intfer bearing which is used in the rule. TO FIND THE LATITUDE BY OBSERVATION. 167 distance under the declination, and, if they are of the same name, add them together but if they are of different names, take their difference; this sum or dificreuce will be the latitude, which will be of the same name as the greatest number. CASE 11. J f Tien the object does not set, hut comes to the meridian above the horizon ttvice in 24 hours. IMany stars are always above the horizon of certain places of the earth, and, in high latitudes, t!ie sun is sometimes above the horizon for several days, in which case the meridian altitude may be observed twice in 24 hours ; that is, once at the greatest height above the pole, and again at the lowest height upon the meridian below the pole. In the former case, the latitude is to be found by the preceding rule, but in the latter by the following : — RULE. Add the complement of the declination to the meridian altitude ; the sum will be the latitude, of the same name as the declination. Note. — When, the sun or star is on the equator, or has no declination, the zenith distance will be equal to the latitude of the {)lace, which will be of the same name as the zenith distance. When the sun or star is in the zenith, tlie declination will be equal to the latitude, and it will be of the same name as the declination. To find the latitude bi/ the meridian altitude of the sun or star. * EXAMPLE I. Suppose that, at the end of the sea day, June 21, 1848, in the longitude of G0° W., the meridian altitude of tiie sun's lower limb, bearing south, was found by a fore observation to be 40° 6' ; required the latitude, supposing the correction of the observed altitude for pai'allax, dip, and semidiameter, to be twelve miles. Observed altitude 40° OC Par., dip, and semidiam. . . .add 12 Sum 40 18 Refraction subtract 1 True altitude 40 17 Subtract from 90 00 True zenith distance 49 43 N. Sun's declination, Table IV. . . 23 27 N. Latitude 73 10 N. EXAMPLE II. Suppose that, at the end of the seadaj^, April 14, 1848, in tiie longitude of 140° E. from Greenwich, the altitude of the sun's lower limb, by a fore observation, was 00° 2.3' when on the tneridian and bearing south, the coi-rection for dip, semidiameter, and parallax, being twelve miles; reciuired the latitude. Observed altitude G0° 2;7 Correction add 12 True altitude* CO 37 Subtract from 90 00 True zenith distance 29 23 N. Sun's declination. Table IV. cor. by Table V. for long. 9 25 N. Latitude 38 48 N. EXAMPLE 111. Suppose that, at the end of the sea day. May 15, 1848, in the meridian of Green- wich, the meridian altitude of the sun's lower limb, bearing north, was found by a fore observation to be 30° GG', the cor- rection for parallax, dip, and semidiameter, being twelve miles ; required the latitude. Observed altitude 30° 06' Par., dip, and semidiam.. . .add 12 Sum 30 18 Refraction subtract 2 True altitude 30 16 Subtract from 90 00 True zenith distance 59 44 S. Sun's declination 18 58 N. Latitude 40 46 S. EXAMPLE IV. Suppose that, at the end of the sea day, Nov. 17, 1848, in the longitude of 80° E. from Greenwich, by a fore observation, the meridian altitude of the sun's lower limb was 50° OG', bearing south, the eye of the observer being seventeen feet above the surface of the sea ; required the latitude. Observed altitude 50° 06' Sun's semidiam add 16 , 50 22 Subtract dip and refraction ... 5 True altitude! 50 17 Subtract from 90 00 True zenith distance 39 43 N Sun's dec. cor. by Table V.. . . 19 03 S. Latitude 20 40 N * The refraction, being- small, is here neglected. t The parallax, being small, is here neglected, and the sun's semidiameter is supposed to be 16'. 168 TO FIND THE LATITUDE BY OBSERVATION. EXAMPLE V. By a fore obsei-vation, the meridian altitude of tlie sun's lower limb was found to be 40° 20', bearing south of the ob- server, the declination being 9" 56' N., and the eye twenty-six feet above the horizon ; — required the latitude of the place. Observed altitude 40° 20' Semidiameter add 16 40 30 Dip 5', refraction 1'. . .subtract 6 True alt. of the sun's centre * 40 30 Subtract from 90 00 Zenith distance 49 30 N. Declination 9 56 N. Latitude 59 26 N. EXAMPLE VL By a back observation with a quadrant of reflection, the meridian altitude of the sun's lower limb was 25° 12', when the declination was 21° 14' S., and the eye of the observer forty feet above the hori- zon, the sun bearing south ; required the latitude of the place of observation. Obsei-ved altitude 25° 12' Semidiameter subtract 16 24 56 Dip add 06 25 02 Refz'action subtract 02 True alt. of the sun's centre * 25 00 True zenith distance 65 00 N. Declination 21 14 S. Latitude 43 46 N. EXAMPLE VII. Suppose that, on January 1, 1830, an observer, seventeen feet above the water, finds by a fore observation that the alti- tude of Sirius is 53° 33' when passing the meridian to the southward ; required the latitude of the place of observation. Observed altitude 53° 33' Dip of the horizon . . . .subtract 4 53 29 Refraction subtract 01 53 28 True zenith distance 36 32 N. Sirius declin. Table VIILf. . . 16 29 S. Latitude 20 03 N. EXAMPLE VIII. Suppose that, on the 13th June, 1848, sea account, an observer, in a high north- ern latitude, and in the longitude of 65° W. from Greenwich, his eye l>eiug twenty feet above the surface of the water, ob- served by a fore observation the altitude of the sun's lower limb on the meridian below the pole 8° 14' ; required tlie lat' tude. The sun being below the pole at 12 hours before the end of the sea day June 13, the correction of declination corre- sponding in Table V. is — 1' 46", and the correction in 65° W. long, is -|- 0' 38'' ; hence both corrections make nearly 1', to be subtracted from the declination at noon 23° 15' N., which gives the declina- tion at the time of observation 23° 14' N., the comp. of which is 06° 46'. Observed alt. sun's lower limb 8° 14' Semidiameter add 16 • 8 30 Dip subtract 04 8 26 Refraction subtract 00 True alt. of the sun's centre * 8 20 Comiilement of declination ... 00 40 N. Latitude 75 00 N. EXAMPLE IX. Suppose that, by a back observation with a sextant, the lower limb of the sun's image was brought to the back horizon, and the angle shown by the index was 110° 10', the sun being then on the meridian and bearing south, tlte declination being 20° 5' N., the sun's semidiameter 10', and the observer 20 feet above the horizon ; required the lat- itude. Observed angle 110° 10' Semidiameter add 10 110 20 Dip sub. 4 11 22 Subtract 90 00 Zenith distance J 20 22 IN Declination 20 05 N. Latitude 40 27 N * The parallax, being small, is here neglected, ami tlic sun's semifliameter is supposed to be IG'. t The declinations ofthese bright stars are given for every 10 days in the Nautical Almanac. Wlien great accuracy is required, these declinations slio4jld be used instead of the numbers in Table VIII. i The refraction and parallax, being only a few seconds, are neglected. TO FIND THE LATITUDE BY OBSERVATION. IGO EXAMPLE X. Suppose tJiat, on January 10, 1830, an obsen'er, eighteen feet above the water, finds the altitude of the north star, when on the meridian below the pole, to be 36° 23' by a fore obsei-vation ; required the latitude of the place of observation. Observed altitude 36° 23' Subtract dip 4', ref. 1' 5 True altitude 36 18 Comp. declin. Table VIII. *. . . 1 36 N. Latitude 37 54 N. EXAMPLE XI. Suppose that, by a back ooservation with a sextant, the lower limb of the sun's image was brought to the back horizon, and the angle' sliown by the index was 106° 12', the altitude of the observer being twenty-two feet, and the correction for semidiameter, jiarallax, and dip, being (as usual) about 12'; required the true latitude, supposing the declination to be 20° S., and that the sun bore north at the time of observation. Observed angle 106° 12' Dip and seniidiam add 12 106 24 Subtract 90 00 Zenith distance f 16 24 S. Sun's declination 20 00 S. Latitude 36 24 S. We have observed, in the directions for finding the meridian altitude of an object, that an error will arise if the slii^i be in motion, or the sun's declination vary. The amount of this correction may be estimated in the following manner: — Find the number of miles and tenths of a mile northing or southing made by the ship in one hour, and also the variation of the sun's declination in an hour, expressed also in miles and tenths. Add these together, if they both cons{)ire to elevate or depress the sun; otherwise take their difference, which call the arc A. Find, in Table XXXII., the arc B, expressed in seconds, corresponding to the latitude and declination ; then the arc A, divided by twice the arc U, will express the time in minutes from noo?r, when the greatest (or least) altitude is observed. Moreover, the square of the arc A, divided by four times tlie arc B, will be the number oi' seconds to |ie applied to the observed altitude to obtain the true altitude, which would have been observed if the ship had been at rest. Thus, if the ship sail towards the sun south 11 miles per hour, and the declination increases northerly 1' per hour, we shall have A = 11 -[- 1 = 12. If the latitude is 42° N., and the declination 2° S., we shall have by Table XXXII. B=:2". In this case, tlie time from noon is Jt^-=z2 minutes, and the correction of altitude -l|4 =r 18 seconds only. * Tlie declination of tills star is given for every clay in the Nautical Almanac; when great accuracy is required, this declination should be used instead of that in Table VIII. t Tl^e refraction, being small, is neglected. 22 170 TO FIND THE LATITUDE BY A MERIDIAN ALTITUDE OF THE MOON. The latitude may be found at sea, by the moon's meridian altitude, more accurately tiian by any other method, excej't by the meridian altitude of the sun; but to do tliis, it is necessary to find the tiJne of her passing the meridian, and her declination at that time. To lacilitute these calculations, we have given the Tables XXVIII. and XXIX., the uses of which will evidently appear i'rom the following rules and examples. To find the mean time of the 7iioon's passing the meridian. Find, in the Nautical Almanac, the time of the moon's coming to the meridian of Greenwich for one day earlier than the sea account,* and also the time of her coming to the meridian of Greenwich the next day, when you are in west longitude, but the preceding day when in cast longitude ; take the difference between these times, w-ith which you nnist enter the top column of Ta!)Ie XXVIII., and against the ship's longitude in the side cohnnn will be a number of minutes to be a])plied to the time taken from the Nautical Almanac, for the day immediately preceding the sea account, by adding when in west longitude, but subtracting when in east longitude; the sum or difference will be the true time of passing the meridian of the given })lace. EXAMPLE. Required the time of me moon's passing the meridian of Philadelphia, April 19, 1836, sea account. The day preceding the sea account is April 18 ; on this day, the moon passed the meridian of Greenwich at 1'' 55'".6, and, being in west longitude, we find the time of her passing the meridian the next day 2^ 43'".0. The difference between these two times is 47™.4, which is to be found at the top of Table XXVIII. ; the nearest tabular nmnher is 48'" ; under this, and opposite 75°, (the longitude of Philadelphia,) is the con-ection 10™, nearly, to be added to 1'' 55^.6, to obtain the time of passing the meridian at Philadelphia, April 19' 2'' OS^.G, sea account, or April 18'' 2'' 05"\(), P. M., civil account. To find the mooti's declination ivhen on the meridian. Find the time of the moon's coming to the meridian as above; turn the ship's longitude into time by Table XXI.,f and add it thereto if in west longitude, but subtract it in east ; the sum or difference will be the time at Greenwich. Take out the moon's declination from the Nautical Almanac, for the nearest hour preceding the Greenwich time, | and also the variation for 10 minutes in the next cohunn. * Takiiiij llie time one day earlier than ihe sea arcount, reduces it to astronomical lime used in t!ie Nautical Almanac. Wc may observe tiiat llie time of the moon's coming' to the meridian, is <;ivcn in the Nautical Almanac to tenths of a minute, instead of seconds of time. This is done to facilitate the calculation of the right ascension and declination, by using common decimal fractions instead of se.\a- gesimals. t Longitude may be turned info time, without the help of Tabic XXL, by multiplying llie degrees and minutes of the longitude by 4, and considering the product as minutes and seconds of time respec- tively ; and, by the inverse process of dividing liy 4, we may turn time into degrees, &c. Thus, 80° X 4 =: 320" = 511 20'" ; and 15° IG' x 4 = GI-" Ol' = li^ ff" 4'. In like manner, l^ SO-" or 80"', being divided by 4, gives 20°, and IDH'", being divided by 4, gives 49°, which agree witli the fable If the ship be furnished with a chronometer, regulated for mean time at Greenwich, we may avoid the labor of tliis part of the operation by taking the time at Greenwich, as shown by the chronometer, at the very moment when the meridian altitude of the moon is observed. } If the time at Greenwich fall exactly upon any hour, the declination can then be taken /rom the Nautical Almanac, by mere inspection, without any reduclion. We may also remark, that the reduc- tion of the declination for the minutes and tenths of a minute of time, can be found by means of Table XXX : but il is better to do it by the process of muUiplicalion, as in the rule ffiven above. TO FIND THE LATITUDE BY THE MOOiN. 171 This variation is to be imiltiplieil by the niiiuites and tenths of a minute which oecuf in the time at Greenwicli ; the product, being divided by 10, gives tiie correction of the declination taken from the Nautical Almanac, additive if that declination be increasing, subtractive if decreasing ; the sum or difference will be the true declina- tion at the time of passing the meridian. NOTES. 1. By the above rule, the day of the month on which the moon passes the merid- ian must be taken one less than the sea account. When the longitude, turned into time, is added to the time of passing the meridian, and the hours of the same exceed 24'', you nnist subtract 24'', and add one to the day of the month ; if the longitude be sul)tractive, and greater than the time of passing the meridian, you nuist, before the subtraction, add 24 hours to the time of passing the meridian, and subtract one from the day of the month ; the sum or difference will be the time at Greenwich. 2. When the declination, taken from the Nautical Almanac for the nearest hour preceding the time at Greenwich, is decreasing, and the correction to be subtracted exceeds this declination, the difference of the two quantities will be the required declination, with a different name from that of the declination taken from the Nau- tical Almanac. 3. In the same manner we may find the declination for any other tiuie of the day, by making use of the given time instead of the time of the moon's passing the merid- imi. In all these rules, the second differences of the moon's motion are neglected. EXAMPLE. Required the moon's declination at the time of her passing the meridian of Phila- delphia, April 19, 1836, sea account. The time of passing the meridian, of Philadelphia was found, in the preceding example, to be April 19' 2^ 5"" .6 sea accoiuit, or x\pril 18'' 2'' 5"'.G by astronomical account; adding this to the longitude of Philadelphia, in time 5'' 1'" nearly, we obtain the time at Greenwich, April 18' 7^ G"\G. The declination in the Nautical Almanac for April 18^ 7^ is 21° 13' 52" N., and the variation 89" for 10 minutes of time nearly ; multiplying this by 6'".6, and dividing by 10'", we get 59", to be added to 21° 13' 52", because the declination is increasing, and we obtain 21° 14' 51" N. for the required declination at the time of the moon's passing the meridian of Philadelphia. To Jiiid the latitude by tlie nnon's rtieridian altitude, obtained hij a. fore observation. At the time of the moon's passing the meridian, the altitude of her round limb must be observed, whether it be the upper or lower limb. This altitude must be corrected for the semidiameter, dip, parallax, and refraction, in order to obtain the central altitude ; with which, and tlie declination, we may find the latitude by tlie same rules as we have used in finding die latitude from the sun's meridian altitude. In making these calculations, we must find, from the Nautical Almanac, the moon's semidiameter and horizontal parallax, corresponding to the time of ol)servation, reduced to the meridian of Greenwich, which was used in computing the declination. The moon's semidiameter is to be increased by the correction in Tal)le X\^, and this augmented semidiameter is to be added to the observed altitude, if the moon's lower limb be observed; but if the upper limb be observed, we must subtract this augment- ed semidiameter from the moon's observed altitude, to obtain the central altitude. I'roin this central altitude you must subtract the dip of the horizon, found in Table XIII., to obtain the apparent altitude. The correction for parallax and refraction is likewise to be added ; this correction is easily found by means of Ta!)le XIX., by subtracting the tabidar number corresponding to the moon's altitude and horizontal parallax from 59' 42" ; the remainder will be the correction for ])arallax and refrac- tion,* which is to be added to the ai)parent central altitude, to obtaiii the true, altitude ; and, by subtracting this true altitude from 90°, we obtain the true zenith distance. With this and tlie declinat'.on, we deduce the latitude by the usual rules, similar to those given for the sun in pages 1G6, 1G7. * 111 computing' this tabic, the mean refraction is used ; but, wlien very great accuracy' is required, ilie true rofraction ought to be used. The corrections arising from this cause may be obtained from Table XXXVI., and are to be applied to the above-found zenith distance, with the same signs as in tiiis table. 172 TO FIND THE LATITUDE BY THE MOON. EXAMPLE L Suppose tliaf, on the 27th of June, 183G, sea account, in the longitude of 80° W. from Greenwich, the meridian altitude of the moon's upper liujb was observed to be 40° 0', bearing south, the eye of the observer being elevated nineteen feet above the surface of the sea; retpiired the true latitude. June 27th, sea account, is June 2Gth by the Nautical Almanac ; on this day tlte moon passes the meridian of Greenwich at 9'' 55"'.9, mean time, and the next day at 10'' 59"'.8, the daily difference being G3'".9. In Table XXVIIL, umler 64'", (which is the nearest number in the table to G3'".9,) and opposite to the longitude 80°, stand 14'" ; adding tiiis to 9'' 55'".9, we get 10'' 09'".9 for tiie time of passing the meridian at the place of observation. J) passes the merid June 20' 10'' 10"" Ship's long. 80° W., in time, 5 20 Time at Greenwich. .. .June 26 15 30 :D's decli. June 26'' 15" 23° 37' 43''.2 S. Cor. for 30 "' is 30 X 8".798 -f 4 23 .9 Required declination 23 42 07 .1 S. Here the variation of the declination f(3r 10"" is, bv the Nautical Almanac, 87".98, or 8".798Yor 1'". ]\Iultiplying this l)y 30, we 's central altitude 50 15 41 Dip, Ta. XIII., for 17 feet, 4 03 2)'s aj>parcnt altitude 50 11 38 59' 42" Cor. T. XIX.— 24 7 diff.add 35 35 ;])'s true.ahitude 50 47 13 5's zenith distance 39 12 47 N. j)'s declination 8 51 33 N. Latitude 48 04 20 N. The latitmle may also be obtained from the moon's meridian altitude, by the following ap|)roximative method, which will vary but very little Irom the truth, except when the horizontal parallax and scmidiameter are very large or very small : — AhriJged approximative method of Jinding the latitude by the ?noon's meridian altitude, obtained by afore observation. To the observed altitude of the moon's lower limb add 12'; but if her upper limb be obsei'V'ed, subtract 20'. With this corrected altitude enter Table XXIX., and TO FIND THE LATITUDE BV THE MOON. 173 take out the corresponding number of minutes, which are to be added to tlie cor- rected ahitude ; the sum will be nearly e(iiial to the true altitude of the moon ; its complement is the zenith distance, which is to be used, as before, with the moon's decUnation, in finding the latitude, as by a meridian altitude of the sun. EXAMPLE III. Suppose that, on the 29th of November, 1836, sea account, in the longitude of 150" VV., tlie meridian altitude of the moon's upper limb was observed 00° 2G', bearing north ; recjuired tlie true latitude. Nov. 29th, sea account, is Nov. 28th by the Nautical Almanac ; on this day the moon j)asse(i the meridian of Greenwich at 16'' 33'".1, and the next day at 17'' 18'".6, differing 45'".5. In Table XXVIII., under 46"", (the nearest tabular nundicr,) and opposite the longitude 150°, stands l^"" ; adding this to 16'' 33'", we get 16'' 52'" for the time of i)assing the meridian of tiic ])lace of observation nearly. ]) passes the meridian 28' 16'' 52" Long. 150° W., in time 10 00 Time at Greenwich .... Nov. 29 02 52 3)'s dec, Nov. 29', 2^ . . . 20° 41' 06" N. Cor. for 52'" is 52 X 9".6, — 8 19 Required declination 20 32 47 N. Obs. alt. ])'s upper limb 60° 26' Subtract 20 Apparent altitude 60 06 Cor. Table XXIX add 28 ])'s true altitude 60 34 J)'ti zenith distance 29 26 S. J)'8 declination 20 33 N. Latitude 8 53 S. In this example, the moon's horizontal parallax is 54' 23" ; with this, and the altitude 60° 6', we lind the correction in Table XIX. is 33' 8" ; subtracting this from 59' 42", we get the correction of altitude 26' 34", instead of 28' found above from Table XXIX., making the corrected latitude 8° 54' 26" S. We shall now work Exami)les I. and II. by this approximative method. EXAMPLE IV. [Su.mo as Example I.] Alt. 3)'s upper limb 40° 00' Subtract 20 ])'s central altitude 39 40 Cor. Table XXIX add 43 3)'s true altitude 40 23 2)'s zenith distance 49 37 N. 2)'s declination 23 42 S. Latitude 25 55 N. Differing about 2' from the correct method "of calculation in Example I. EXAMPLE V. [Same as Example II.] Alt. D's lower limb 50° 00- Add 12 J)'s central altitude 50 12 Cor. Table XXIX add 36 ])'s true altitude 50 48 3)'s zenith distance 39 12 N ;])'s declination 8 52 N. Latitude 48 04 N IJeinj pie II nearly the same as in Exam- 174 TO FIND THE LATITUDE BY A MERIDIAN ALTITUDE OF A PLANET. The latitude may frequently be obtained, with great accuracy, (particularly in tlie morning and evening, when the horizon is well defined,) by observing the meridian altitude of Venus, Mars, Jupiter, or Saturn. From these altitudes we may find the latitude by similar methods to those we have already given for the sun. The times of passing the meridian of Greenwich, and the declinations of these planets, are inserted in the Nautical Almanac, at every noon, at Greenwich ; and, as the daily variations of these quantities are small, we can find, by inspection, to a sufficient degree of exactness for most nautical purposes, the corresponding times of transit and declijiations at the place of observation, and thence the latitude, as in the follow- ing rule : — RULE. Find, in the Nautical Almanac, the time of passing the meridian en the day nearest to that in which the observation is made ; this will be nearly the tiniC of passing the meridian at tlie place of observation.* Turn the ship's longitude into time, and add It to the time of passing the meridian, Avhen in west longitude, but subtract it in east ; the sum or difterence will be the time at Greenwich, ncarly.f Talce, from the Nau- tical Almanac, tiie planet's declination for tiie noon immediately preceding, and for that immediately following, the time of observation, and note the difference of the declinations when they are of the same name, but their sum when of different names; this sum or diflercnce will be the daily variation of declination. Then say. As 24 hours are to the daily variation of declination, so are the hours and minutes of the time at Greenwich to the correction of the declination ; to be applied to the first dec- lination taken from the Nautical Almanac, additive if the declination be increasing, subtractive if decreasing ; the sum or difference will be the declination of the planet at the time of observation. But you must observe that, if the correction of declination be greater than the declination first marked in the Nautical Almanac, their difference will be the sought declination, wJiich will be of a different name from the first declination. From the observed altitude of the planet, taken by a fore observation, subtract the refrartion and dip, the latter being, in general, about 4'. The remainder, being subtracted from U0°, will give the true zenith distance nearly, t with which, and the declination, we may find the latitude, as by an observation of the sun. EXAMPLE L Suppose that, on the 2:3d of October, 1836, sea account, in the longitude of C5° \Y., J ujjiter passed the meridian to the southward; the meridian altitude of his centre, being observed, was 45° 20', and the dip 4'; required the true latitude. Oct. 23d, sea account, is Oct. 22d by the Nautical Almanac ; and on that day Jupiter passed the meridian at 19'' 5'", nearly; adding the longitude G5°, timied into * If we wish to find the lime of passing' the meridian more accurately, we must take a proportional part of the dillerence of the times of coming to the meridian given in the Nautical Almanac, in like manner as in finding the declination of the planet ; always keeping in mind, that the time, according to the astronomical compulation, is used in the Nautical Almanac, and is one day less than the sea account. t This part of the operation may he avoided, if we have a chronometer regulated for Greenwich time, and note hy it the time of observation. t To be strictly accurate, we ought to subtract the parallax in altitude from this zenith distance. This is found in Table" X. A. Thus, if die horizontal parallax of the jilanet be 20", and the altitude 60°, the parallax in altitude bj- this table is 10", to be added to tlie observed altitude, or subtracted from the observed zeiiilh liislaiicc. 'i'hc centre of the planet being observed there is no correction for the semi- diameter of the planet TO FliND THE LATITUDE BY A I'LAiNET. 175 time, (that is, 4'' 20"",) we get the correspoiuling time at Greenwich, l)y the Nautical Ahnanac, Oct. 22' 23'' 25'" ; and, for tliis time, we tind tlie declination of the planet, by mere inspection of the Nautical Almanac, to be 1G° 45' N., nearly. From Jupiter's observed altitude 45° 2(y Subtract dip 4', refraction 1' 5 Leaves the true altitude 45 15 Whence the true zenith distance is 44 45 N. Jupiter's declination 16 45 N. Latitude 61 30 N. In this example we have found, by inspection, the time of passing the meridian, or the declination. If greater accuracy is required, we must take proportional parts of the daily variations, corresponding to the longitude of the place, and the time of o!)servation. Tiius, the time of passing the meridian on Oct. 22, by tiic Nautical Almanac, is 19'' 5"'.4, and on Oct. 23 is 19" 2"'.0, decreasing 3"'.4 daily, or fur 360° ol longitude. Then, by proportion, we have 360° : 3"'.4 : : 65° : 0"'.6 ; so that the cor rection of the time of passing the meridian for 65° W. longitude is 0'".6, to be subtracted from 19" 5"' .4, to o!)tain tlie time of passing the meridian in the place o) observation, 19" 4"'.8. Adding to this the longitude, turned into tiiue, 4" 20'", we get the corresponding time at Greenwich, 22' 23" 24 '".S. Now, by the Nautical Almanac, tlie declination, Oct. 22d, is 16° 47' 17".2 N., at noon, and the next dav, 16° 45' 18''.1 N., at noon, differing 1' 59".l, or 119".l. Then say, As 24" : 119".! :': 23" 24'".8 : 116 ' or 1' 56", to be subtracted from 16° 47' 17".2, to obtain the true declination, 16° 45' 21' nearly, at the time of observation. The horizontal parallax, by the Nautical Almanac is 1".56, which is wholly insensible ; and the semidiameter is 18", whicli must b< neglectetl because the central altitude was observed. Hence we see that these correc. tions in the calculations produce but very little change in the resulting latitude, an. that die process by inspection is sufliciently accurate ; and this will be found generail; to be the case with the planets Jupiter and Saturn. EXAMPLE II. Suppose that, on the 17th of September, 1836, sea account, in the longitude o* 75° E., Venus passed the meridian to the northward ; the meridian central altitude, being observed, was 26°, and the dip 4' ; required the true latitude. Sept. 17th, by sea account, is Sept. 16th by the Nautical Almanac ; and on this day Venus passed the meridian at 20" 59'", nearly; subtracting the longitude 75°:= 5", we get Sept. 16^^ 15" 59"" for the corresponding time at Greenwich. Now, by the Nau- tical Almanac, the declination of Venus, at noon, Sept. 16', was 14° 49' 33".7 N., and the next day 14° 44' 22".l N., differing 5' 11".6. Then we have 24" : 5' 11".6 : : 15" 59'" : 3' 27".5 ; su1)tractiiig this from 14° 49' 33".7, we get 14° 46' 06" N., nearly, fcr the planet's ieclination at the time of observation. From the observed central altitude of Venus. .. 26° OO' Subtract dip 4', refraction 2' 6 Leaves the true altitude nearly 25 54 Whence the true zenith distance is 64 06 S. Declination of Venus 14 46 N. Latitude 49 20 S. 176 TO FIND THE LATITUDE BY DOUBLE ALTITUDES. Form I. — Hy double altitudes of the sun. When (by reason of clouds, or from other causes) a meridian altitude cannot be olitained, the latitude may be found by two altitudes of the sun, taken at any time of the day, the interval or elapsed time between the obsei-vations being measured by a good watch or ciu-onometer, noticing the seconds, if possible, or estimating the times to a third or a quarter of a minute, if the watch is not furnished with a second-hand. The observed altitudes of the sun must be corrected, as usual, for the semidiameter, dip, refraction, and parallax, in the same manner as in finding the latitude by a merid- ian altitude. When great accuracy is required, the declination must be found at tlie time of each observation, using the third method of solution hereafter given ; but when the sun's declination varies slowly, or the elapsed time is small, it will in general be sufliciently accurate to find the sun's declination for the middle time between two observfttio7is, and to consider it as invariable during the observations, computing the latitude by the first or second method. Tills manner of finding the latitude is, in general, most to be depended upon where the sun's meridian zenith distance is great. If the sun passes the meridian near to he zenitli, much greater care must be taken in measuring the altitudes and noting the times, than would be necessary under other circumstances. The nearer the sun is to the meridian, at the time of one of the observations, the more correct the result will connnoiily be. In general, the elajised time ought to be as great, or greater, than the time of the nearest observations from noon. Similar remarks may be made upon every one of the following forms. In all these observations it is supposed that the watch moves uniformly according to apparent time, measuring twenty-four hours from the time of the sun's passing the meridian on two successive days at the same place of observation. If the watch gain or lose on apparent time, supposing the observer to be at rest, a correction must be apjilied for the gain or loss during the time elapsed between the observations, so as to ol)tain accurately the elapsed time or hour angle. It is not required that the watch shoidd be regulated so as to give precisely the ^our of observation ; the only thing ree^uired is to find the elapsed time with all possible accuracy. Form TI. — Double altitudes of a star. Doulde altitudes of a fixed star may be used in finding the latitude, and the calcu- lation is almost identical with that of'double altitudes of the sun ; the only difference consists in adding a small correction to the elapsed mean solar time between the observations, on account of the daily acceleration of 3' 5G" in the time a star comes to the meridian on successive days ; in other words, the elapsed time (or hour angle) must be reckoned in sideral time, of which we have already spoken in the second note on page 147. Now, as a chronometer is usually adjusted to mean solar time, and the observations marked by it, we must add to the mean time, elapsed between the observations, the correction given in Table LI., to reduce it to sideral time. Thus, if the interval in mean solar time be 3'', the corresponding correction in this table is -{- 2'J\G, making the interval in sideral time (or the correct hour angle) 3'' 00"" 2[)'.G, which is to be used in the rest of the calculation. In observations of a fixed star, the altitudes are to be corrected for dip and refrac- tion, as in finding the latitude by a meridian altitude. The declination of the star is to be found in Table VIII.* With these altitudes, the declination, and the hour * Or more acciiraicly in the NaiUical Almanac, if any one of the bright stars is observed whosa place is given in liial worL'. TO FIND THE LATITUDE BY DOUBLE ALTITUDES. 177 aii."-lc, the calculation is to be made by cither of the three mctliods hereafter given. The chief difficulty, in observations of this kind, with a fixed star, is the want of a good horizon in tlie night-time. The method, however, migiit sometimes be used with success, soon after the dawn of day, or hue in the evening twilight, at a time when the horizon is well defined, and the star sufficiently bright to bring its reflected image lo the horizon. Sqinetimes a good horizon is produced by the aurora borealis, \n which case a good observation might be made with stars in the northern horizon ; but a single observation of the polar star will answer the same purpose, and will be nmch more simjjle. Form III. — Double altitudes of a planet. Double altitudes of a planet (particularly Jupiter and Venus, on account of their great brightness) may sometimes be used with success. The observed altitudes must be corrected for dip and refraction. The parallax and scmidiameter, being small, may be neglected, except in cases where extreme accuracy is required. The declina- tion of the planet is to be found, in the Nautical Almanac, ibr the sujiposcd time at Greenwich. The daily variation of the time of coming to the meridian is also to be found in the same page ; and thus the time elapsed between the passage of the ])lanet over the meridian on two successive days is found ; then the corrected elaiKsed time, or hour angle, is obtained by the following rule : — Rule, ^s the interval of time between two successive passages of the ohjed over the mendian is to twenlijfour hours, so is the elapsed mean time between the observations lo the corrected elapsed time, or hour angle. With this hour angle, the declination, and corrected altitudes, the latitude may be found by either of the three following methods of calculation. Form IV. — Douhle altitudes of the moon. Double altitudes of the moon may also be used in finding the latitude. These observations may be easily and very accurately made ; but the calculation is much more complex than any of the preceding methods, on account of the great change in the moon's declination and right ascension during the elapsed time betw^een the observations. If, however, by the times of observation, and the longitude of the ship, (or else by a chronometer,) the time at Greenwich can be obtained within a few minutes, we may, from the Nautical Almanac, find the corresponding declination, semidiamcter and horizontal parallax of the moon for each of these observations. With the horizontal parallax, and the moon's apparent altitude, find the correction in Table XIX., which, being subtracted from 59' 42", leaves the correction of the moon's altitude for parallax and refraction ; * this is to be added to the corresponding observ- ed altitude, corrected for scmidiameter and di]), to obtain the moon's correct central altitude. This is to be done at each observation. Lastly, the time of the moon's passing the meridian on successive days, given in the Nautical Almanac, shows the interval of tune between two successive passages of the moon over the meridian,! and this time is to twenty four hours as the elapsed time between the observations is to the corrected elapsed time or hour angle. With this hoiH* angle, the correct central altitudes, and tiie declinations, the latitude may be found by the fourth of the folJovviiiic methods of calcidation, it being very rare that the other methods can be used, on account of the great change in the moon's declination. FoRJi V. — Uy altitudes of two different objects, taken at the same time. The latitude may be obtained by observing, at the same moment of time, the altitudes of two heavenly bodies ; as, for example, (I) The sun and moon ;| (2) The moon and a fixed star or planet ; J (3) A planet and a fixed star ; (4) Two planets ; (5) Two fixed * When cxtipmc accurac}' is not required, we may find the correction for parallax and refraction from Table XXJX., wliicli, if the altitudes are large, will not var^' much from the truth. t This time is "(iven to tenths of a minute, which in g-encral is sufficient, because, if the elapsed time be small, the ellect of tliis correction will be only a few seconds. It might be obtained more accurately by means of the right ascensions of the sun and moon, using the second differences, as taught in the Appendix. X A particular case of this method occurs in taking n lunar observation, which will be treated of sep arately, because, the distance of the two bodies being known, liie calculation becomes more simple. 23 178 TO FIND TPIE LATITUDE BY DOUBLE ALTITUDES stars. Ill tliese metliods tlie altitudes are to he corrected, as in the preceding Forms, for dip and refraction ; also for parallax and seniidiameter when necessary, as is always tlis case in observations of the moon and sim. The declinations of t!ie bodies are to be found for the supposed time of observation, reduced to the meridian of Greenwich, by means of the Nautical Almanac, or by Table VIII. for the fixed stars, as before taught. Then the difference of the right ascensions of the bodies (or that difference subtracted from 24 hours, if it exceed 12 hours) will bo the hour angle, which is to be used, with these declinations and coirected altitudes, in finding the latitude, by either of the three first methods, if the declinations shoidd be equal, or differ but one or two minutes ; otherwise by the fourth method, which, in fact, njay be considered as the only method to be used in this^ kind of observations, because, in almost all cases, the declinations of the objects differ considerably. For.M VI. — JBi/ altitudes of two different ohjccts, talccn within aflio ?.':iniitcs of each other, hy one observer. It may sometimes happen, for want of two good instrun)ents, or from not having two observers, that the preceding Form V. cannot be employed. In this case the whole of the observations may be made by one person, noticing the interval between the observations, and making the calculation as in the following Form VII. But it is in general much better to make the observations as near to each other as possible, and then, by a very simple process, the calculation may be reduced to that of Form V., in which the observations are taken at the sftme momtnt. This is done by observing tlie first object twice, before and after observing the second object. For if the intervals of time between these three observations be equal, (as, for example, one minute, or two minutes,) the half-sum of the two altitudes of the first object jnay be taken for the altitude corresponding to the time of observing the second altitude, and the calculation may then be made as in Form V. Thus, suppose at 10'' 2"', A. M., per watch, tlie altitude of Sirius was 17° 54', at 10'' 4™ per watch the altitude of Capella GO- 45', and at 10'' G™ per watch the altitude of Sirius was again observed and found to be 17° 58'. In this case, the intervals of time are exactly two minutes ; therefore tlie hall-sum of the altitudes of Sirius is to be taken 17° 5G', and combined witli the altitude of Ca])e]la C0° 45', supposing both to have been observed at 10'' 4'" per watch. This is the most simple form in which an observation of this kind can be made by one observer. If, from any cause whatever, the observations cannot be taken at exactly equal intervals, the altitude of the first object, at the time of observing the second object may be found by proportion, supposing the altitudes to vary uniformly during the few minutes of the observations. Thus, in the preceding example, supjiose the altitudes and the two first-noted times to remain unaltered, but the last observation of Sirius to have been at 10'' 10"" per watch, instead of 10'' G"". In this case, during the eight minutes of time elapsed between lO"* 2'" and 10'' 10"", Sirius would have risen 4', (from 17° 54' to 17° 58' ;) tlierefore, by proportion, it is found that in two minutes (the time elapsed between 10'' 2'" and 10'' 4"') the star would have risen 1', and the altitude would liave increased from 17° o4' to 17° 55'; therefore, at the time 10'' 4"' per watch, the altitude of Sirius must be taken at 17° 55', the altitude of Ca- pella G0° 45', and with these quantities, considered as observed at this last-mentioned time 10'' 4'", the calculation must be made as iji Form V. There are several advantages attending these tv/o last forms V., Yl., since no allowance is necessary for the change of place of the ship; the observations can be immediately made, in a short interval of fair weather, when the common method of dou!)le altitudes might fail from the intervention oC clouds ; the time can also be obtained at the same operation, &c. . Form VII. — Bj/ altitudes of tiro diffrre:it ohjcrts, inkcn at different times. This method differs but very litde from the two last. The altitudes are to be corrected, in the same manner, for dip and refraction ; also for parallax ami semi- diameter, when necessary. The right ascension and declination of each object is to be found for the supposed time of observing that object reduced to the mci'idian of Greenwich. Then the apparent elapsed time between the observations, is to be turned into sidcral time, which may be done, as in Form II., by adding the correct tion in Table LI. corresponding to this time ; add this sidereal time to tlie right ascension of the body first observed ; the difference between this sum and the right ascension of the body last observed is the hour angle.* This, witli the * If tlijs Qifference exceed 1*2 hours. s:iMr:iri ii frmn "^l !:oi;rs. nnd u?e the remainder as in Form V TO FIND THE LATITUDE BY DOUBLE ALTITUDES. 179 declinations and corrected altitudes, is to be vised in finding the latitude by the third or fourth of the following methods of calculation, it being very rarely the case that the first or second methods can be used, on account of the ditference of the ileclina tions. These three last forms, when a fixed star or planet is used, are restricted very much from the want of a good horizon in the night ; they are best adapted to the morning and evening twilight. GENERAL RE3IARKS. Having thus explaine-d several of tlie different forms of making tliese ol)servation;», and the manner of finding in each Ibrm tlic hour migle, the dedincUions, v.m\ tJie correct central altitudes, we shall now give foiu* difterent methods of calculating the latitude, and shall illustrate the rules by proper examples. In the frst and second methods, the declination is supposed to be the same at both observations, which is true as it respects observations of a fixed star, and is in general sufficiently correct for common observa lions of double altitudes of the sun. The first of these methods is direct and simple, not embarrassed with much variety of cases, requiring only ten openings of the Table XXVIL, without any hah ing or doubling of the logarithms, or the use of natural or versed sines. This method is in fact nearly, if not fully, as short as the second or approximative method invented by IMr. Douwcs, and which was exclusively used in the former editions of this work. Tiiis second (or Douwes') method is liable to the objection that the calculation must sometimes be repeated several times befoi-e a true solution can be obtained, and then it becomes extremely troublesome. This difficulty does not occur in the first method ; and on this account, as well as for its remarkable simplicity, the first method is always to be preferred. The third method is ai)plicable to cases where there is a small variation in the declination of the object, during the elapsed time between the observations, as most commonly happens when the sun is used. This mctliod is short and simple, and is much facilitated by the use of Table XLVI., which I have computed. The fourth method embraces the general solution of the problem in the case where any variation whatever of declination is noticed. This increases the labor consid- erably, and renders the solution more complex in its cases. It i?:, however, believed, that this method, drawn up in its present form by the author of this work, will be easily understood by navigators, and that they will thus be enabled to determine the latitude with considerable accuracy in cases where it might be of the utmost impor- tance to know it, and where other methods could not be resorted to on account of bad weather. This method is nearly, if not quite, as short as that published by Dr. Brinkley in the Nautical Almanac of 1825, and does not require, like his method, a second or third (or even a greater number) of operations. If the observer should change his place or station, during the elapsed time between the observations, a correction must be applied to one of the altitudes on this account. The manner of doing tliis is shown in the following examples. It may be observed that in like manner as there are two latitudes corresponding to tiie same meridian altitude of the sim, according as the zenith is north or south of the Sim when on the meridian, so in double altitudes there are generally two latitudes, corres]>onding to the proposed altitudes, according as the zenith and north pole are on the same side, or on different sides, of the arc or great circle passing through the two observed bodies, or through the two places of the game bpdy; and it therefore becomes necessar}^ to notice, at the time of observation, how tlie zenith and north j)ole are situated with respect to this great circle. To estimate the effect of small errors in the observations- When running in with the land, or crossing a dangerous parallel with no other means of obtaining the latitude than by double altitudes, it becomes a matter of great imjiortance to ascertain the possible error of the latitude thus coniputed, arising from suj)posed errors in the observed altitudes, or in the elapsed time. The differential expressions in spherical trigonometry afford mediods of doing this ; but they are not adapted to the nature of this work, on account of the complication and variety of cases. The following method, though long, is general and infallible, and was once used by the writer in a case of gi-eat anxiety and danger. Rule. After having computed the latitude by either of the four following methods, using the observed altitudes * and elapsed time, repeat the operation, varying * Tli:il is, the observed altitudes, corrected as usual for dip, refraction, parallax, and semidiametcr, If uecessarj-. 180 TO FIND THE LATITUDE BY DOUBLE ALTITUDES. the altitude you suspect may be erroneous by 2' or 3', (or whatever you suppose tlie limit of the error in that altitude msiy be ;) the difference between this second latitude and that first computed, is the effect of the suj)posed error in that altitude. If you suspect the second altitude also to be erroneous, the operation may be again repeated, varying this second altitude 2' or 3', (or whatever the limit may be sii|jposGd,) but using the first observed altitude and elajjsed time ; comparing this third comjjuted latitude with the Jirst, the difference is the effect of this supposed error in the second altitude. Finally, if the elapsed time is suj:)posed to be erroneous, the o])eration may be again repeated, using the observed altitudes and varying the elapsed time by 20 or 30 seconds, (or whatever the limit of this error may be supposed ;) the difference between this fowth latitude and that _^rsf computed is the effect of this su})j)0scd error of the elai)sed time. Thus, snjipose the first-computed latitude was 30^, the second 30° 1', the third 30° 3', the fourth 30° 2' ; tlie error arising from the first altitude would be V, tliat from the second altitude 3', and that from the elajjsed time 2'. If all these errors existed at the same time, the greatest limit of the error would be the sum of these quantities (or G'), so that the true latitude would be 30° ± 6', or between 21)° 54' and 30° G'. In this way the limit of the error may be obtained in any case, and the degree of confidence that may be placed in the observation obtained. This examination is sometimes very necessary, because the objects may be so situated, that a small error in the observa- tions might produce a considerable change in the comjJUted latitude. It may be observed tiiat tlie error of one. observation is frequently corrected, in whole or in part, by the error of the other ; the one tending to increase the latitude, the other to decrease it. FIRST METHOD. To find the latitude by double altitudes of the sun, or any other object, the declination being invariable. In this method, the log. sines, cosines, &c., of Table XXVII. are used ; atid, for brevity, the word log. is omitted in the rule. For the convenience of writing down at once, in the same line, all the logarithms which occur at the same opening of the book, they are arranged in three columns, as in the following formula ; and it will be very convenient to have one of these blanks prejiared at the connnencement of the operation, and then the logarithms may be written down, in their proper places, with great raj)idit3^ FORMULA. CoL. 1. Col. 2. Col. 3. Elapsed time, [p. ji.] Cosec. .Coscc. Declination Secant A Coscc. Cosine Half-sum alts Cosine Cosec. Half-diff. alts Sine Sec. • C Sine Cosine [Z less tliin 90° north or south, like the bearing of zenith.] SoC. \y. is the sum of B, Z, ifof the same name ; difference, ifof a different name.] . Cosine Cosec. (B less tlian 90°, liks liccliaaliun N. or S.] Latitude , Cosine Sine Sine RULE. (Using Table XXVII.) 1. Find the elaj)sed time* in column P. M. ; take out the corresponding cosecant, and put it in Col. I. 2. Put die secant of the declination in Col. 1 : its cosecant in Col. .3. 3. The sum of the logarithms in Col. 1 (rejecting 10 in the index) is the cosecant of the angle A, whose cosine is to be put in Col. 2 and Col. S.f 4. The sum of the logarithms in Col. 3 (rejecting 10 in the index) is the cosecant of the angle 13, (less than 1)0°,) which is to be named noHh or south, like the declination. * If any ollior ohjcct tlian llic sun is observed, the corrected elapsed time, or Jiour angle, found as before tnug^lii, is to i)e used. t Tlic cosines of A and C arc each wriUen down twice, which reduces the number of logarithms in eacji example from 17 to 13. TO FIND THE LATITUDE .BY DOUBLE ALTITUDES. 181 5. Find liulf tlie 5iim of tlie two altitudes; place its cosine in Col. 1, its cosecant in Col. 2. Find also half the difl'erence of the two altitudes ; place its sine in Col. 1, its secant in Col. 2. G. The sum of tlie three lower logarithms of Col. 1 (rejecting 20 in the index) is the sine of the angle C, whose cosine is to be placed in Col. 2 and Col. 3.* 7. The sum of the logm-iduns in Col. 2 (rejecting 30 in the index) is the secant of the zenith angle Z, which is to be taken out (less than 90°) and placed under B, in Qol. 3, naming it north if the zenith and north pole be situated on the same side of the arc or great circle pa.ssing through the two observed places (or objects), but south if the zenith and north jioie be situated on different sides of that great circlcf 8. The angle E is found by taking the sum of the angles B, Z, if they are of- the same name, or their difference il' of different names, marking E north or south, like the greatest of the two angles B or Z.| 9. Put the sine of E in Col. 3, and the sum of the two last-written logarithms of Col. 3 (rejecting 10 in the index) is the sine of the latitude, of the same name as E. If the time of observation were requu-ed, it might be foimd by the following rule, 6tJll using Table XXVII.:— Rule. Add the tangent of C to the secant of E ; the sum (rejecting 10 in the index) is the tangent of an angle. Take out half the corre.s])onding time in Col. P. M., (or in Col. A. M., increas,jd by 12 hours,) and this will represent the horary distance of the object from the meridian (uj)|)er or lower) at the nfiddle time between the two observations. Take the sum and dift'erence between this and half the elapsed time, or horn- angle, and they will be the hours and minutes distance from the meridian corresponding to both observations, expressed in apparent solar time if the sun be observed, sideral time if a star is observed, &c. EXAMPLE I. Being at sea, in latitude 40° 30' N. by account, when the sun's declination was 11° 17' N. at 10'' 2'" per watch, in the forenoon, the sun's correct central altitude was 46° 55', and, at ll*" 27'", per Avatch, in tlie forenoon, the correct central altitude was 54° y ; re(juired the true latitude. Subtracthig 10" 2'" from 11" 27™ gives the elapsed time 1" 25™. CoL. 1. El. time [p.m.] 1"25'", Cosec. 10.73429 Declination 11° 17' N. Sec. 10.00848 Col. 2. CoL 3. A Cosec. 10.74277 I sum alts. 50 32.. Cosine 9.80320 Adifllalts. 3 37 ...Sine 8. 79990 C Sine 9.34587 (Z less tl.an 90°, ai (1 N. or S. like bearing of zenilh.) ScCaUt 10.09509 Z 3G 33 N [K !s IIk' eum of B, Z, if of Ihc same name ; dij'erence if of a dlferent name.] Cosine 9.99278 Cosec. 10.11239 Secant 10.00087 Cosine 9.98905 .Cosec. 10.70850 .Cosine 9.99278 B 1 r 28' N. Cosec. 10.70128 [B Ifss ihan 90", named N. orS. likcdecliii.J Cosine 9.98905 E48 01 N. Sine 9.87119 Latitude 46 27 N. Sine 9.86024 If the sun had passed the meridian to the north of the observer, Z would have been 3(i° 33" S., and E =: 25° 5' S., whose sine 9.G2730, added to cos. C 9.98905, gives the sine of the latitude 9.GIG3.5, coi-responding to 24° 25' S. Li the iirst case, (in north latitude,) the tangent of C 9.35682, added to the secani E 10.174().3, gives 9.53145, which, in the tangents, corresponds to 2'' 30'" 12% nearly, whose half, 1" 15"' 6% is the time of the middle observation from noon ; adding and subtracting half the elap.sed time, 42'" 30% gives the times of the observations from noon 1" 57'" 36' and O" 32'" 36'. * 'J'lie cosines of A and C arc eacli writleii clown twice, which reduces the number of logarithms in each example from 17 to 15. t In observalions of the sun, the angle Z may in general be called north, if the zenith be north of the Eun when ou the meridian at its greatest altitude ; but south if the zenith be then south of the sun. When the object passes the meridian near the zenith, it may be doubtful whether il be noiili or south, ill wliicli case the latitude niay be computed upon both suppositions, and that one selected which agrees best with the estimated place of the ship; and this e.xira labor is very small. Rut observations on an object passing near the zenith are liable to great errors, and had better be rejected, X This case is easily remembered, because s is the first letter of same and staii, and d the first V ttcr of different aiid difference. 182 TO FIND THE LATITUDE -BY DOUBLE ALTITUDES. EXAMPLE n. At sea, in tlie latitude of 47° 19' N. by account, when the sun's declination was 12° IG' N., at 10'' 24'" A. M., per watch, the sun's correct central altitude was 49° tX ; at 1'' 14'" P. M., per watch, his correct central altitude was 51° 59' ; required the latitude. Subtracting lO*" 24"^ from l"" 14"' increased by 12'', leaves the elapsed time 2'' 50'". Col. 1. El. time [p.m.] 2" 50'^', Cosec. 10.44077 Declination 12° IC N. Sec. 10.01003 A Cosec. 10.45080 ^ sum alts. 50 34 . .Cosine 9.80290 h diff. alts. 1 25 . . . .Sine ^.39310 C Sine 8.G4G80 |Z Ies3 than 90° and N. or S., like bearing of zeuilh.] Col. 2. Col. 3. Cosine 9.97089 Cosec. 10.11218 Secant 10.00013 Cosine 9.99958 Secant 10.08278 Cosec. 10.67272 Cosine 9.97089 B13°08'N.Cosec. 10.G43G1 (Tl lesslhan Bli". mjn.i' N. orS. lilie declcu.l Cosine 9.99958 Z 34 16 N. '. of B, Z, if of the same name ; difference if of a different name.] E 47 24 N. Sine 9.8GG9 4 Latitude 47 20 N. Sine 9.8GG52 If the sun had passed the meridian to the north of the observer, Z would have been 34° 16' S., E r= 21° 08' S. ; its sine 9.55695, added to cosine C 9.99958, gives 9.55653, the sine of the latitude 21° 7' S. If the observed object, in this example, had been a fixed star, with the same dwli- nation 12° 16' N., the same altitudes 49° 9', 51° 59', but the elapsed time 2'' 49'" 32% the calculation would have been exactly as above. For, by adding, according to the rule in I'age 176, the correction in Table LI., 28% to reduce it to sidera! time, ^^■e shall ol)tain the corrected elapsed time, or hour angle, 2'' 50"', and every part of tiio work will be as above. If the ])lanet Venus had been observed, at the same corrected altitudes, on tlie I3th of ]\laroh, 1836, in a place where his declination at the middle time between the two ol)servations was, by the Nautical Almanac, 12° 16' N., and the elapsed time 2'' 50'" 03".5, the calculation would still be the same. For, by the Nautical Alma- nac, it appears that Venus passes the meridian on the 13th and 14th of March, at 2h 27m 12s a^j^jj 2'' 27'" 42' respectively, increasing 30% so that the interval of two successive transits is '24'' 00'" 30*. Then saying, As this interval is to 24'', so is tlie elaj)sed time 2'' 50"' 03'.5 to the corrected elapsed time, or hour angle, 2'' 50'" 00% which is to be used as above, all the rest of the work being the same. We may proceed in the same manner, if the moon be observed at a time when the declination varies but little. EXAMPLE III. IJenig at sea,- in latitude 50° 40' N. by account, when the sun's declination was 20° 0' S. at 10'' 17™ A. M., per watch, the sun's correct central altitude was found to be 17° 13', at 11'' 17"", per watch, the correct central altitude was found to be 19° 41'; required the latitude. Subtracting 10'' 17" from ll*" 17'", gives the elapsed time 1''. Col. 1. El. time [p.m.] 1" 0'", Cosec. 10.88430 Declination 20° 00' S. Sec. 10.02701 A Cosec. 10.91131 h sum alts, 18 27 Cosine 9.97708 h diff: alts. 1 14 , . ,Sine 8.33292 C Sine 9.22131 Col, 2, CoL. 3. [Z lem than 90", and N. or S. like bearing of zcniili.] Cosine 9.99670 Cosec. 10.49966 Secant 10.00010 Cosine 9.99390 Secant 10.49036 [E U the turn of B, Z, if of the same name ; dijercnce, if of a dijercnt name.] Cosec. 10.46595 Cosine 9.99G70 B20°10'S. Cosec. 10.46265 Z 71 08 N. [B less thnn 90°, mmed N. orS. likeUeclin.l .Cosine 9.99390 E 50 58 N, Sine 9.S903Q Latitude 50 00 N, Sine 9.88420 TO FIND THE LATITUDE BY DOUBLE ALTITUDES. 183 If the sun liad passed the meridian to the north of the observer, Z would have been 71° OS' S., and E=r:91° 18' S., whose sine 9.99989, added to 9.99390, gives the sine of tlie latitude 9.99379, corresponding to 80° 20' S. EXAMPLE IV. Being at sea, in the latitude of C0° 0' N. by account, when the sun was on the equator (or had no declination) at l"" 0"' P. M., per watch, hi.s correct central altitude was 28° 53', and at 3'> 0'" P. M., per watch, the correct central altitude was 20° 42' ; required the true latitude. CoL. 1. CoL. 2. CoL. 3. El. time [p.m.] 2'^ 0"^, Cosec. 10.58700 Declination Secant 10.00000 A 15° 00' Cosec. 10.58700 i sum alts. 24 47^ Cosine 9.95801 4 diff. alts. 4 5h Sine 8.85340 C Sine 9.39841 [Z less llun 90'', anj N. or S. like bearing of zenith.] Cosine 9.98494 Cosec. 10.37745 Secant 10.00110 Cosine 9.98594 Secant 10.34943 [E is the sum of B, Z, if of tlie same nam^ ; dtference, if of a dijerent came.] , [Cosec. Injinite.] [Cosine 9.98494] B 00° 00' [Cosec. I njinite.] [B less limn DO', named N. or S. lilie dtclin.] Cosine 9.98594 Z63_26 N. E G3_2G N. Sine 9.95154 Latitude 59 59 N. Sine 9.93748 • The calculations v/ould have been the same for south latitude, which would be 59° 59' S. The computation of A and B might have been dispensed with, for when the declination is nothing, B is notliing, and A is equal to half the elapsed time (P) turned into degrees by Table XXL, being, in this example, 15° ; in this case, all the logarithms included between the brackets [] may be omitted. In tRe preceding examples, both altitudes were supposed to be taken at the same place or station ; but as that is seldom the case at sea, the necessary correction for any change of place must be made in the following manner : — Let the bearing of the sun be observed, by the compass, at the instant of the first observation ; take the number of points between that bearing and the sliij)'s course, (cori-ectcd for lee-way, if she makes any,) with which, if less than eight, or with what it wants of sixteen points, if more than eight, enter the traverse table, and take out the difference of latitude corresponding to the distance run between the observations, Md this difference of latitude to the first altitude, if the number of points between the sun's bearing and the ship's course be less than eight ; but suhtrad the difference of latitude from the fi.-st altitude, if the number of points be more than eight, and that altitude will be reduced to what it would have bSen if observed at the same place where the second was.* This corrected altitude is to be used with the second observed altitude in finding the latitude by the above rule. The latitude resulting will be that of the ship at the time of taking the second altitude, and must be reduced to noon by means of tlie log. EXAMPLE V. In a ship, running N. by E. | E. per compass, at the rate of nine knots per hour, at 10'' 0'" A. M., per watch, the sun's correct central altitude was found to be 13° 18', bearing S. | E. by compass ; and at 1'' 40"" P. IM., per watch, the sun's central altitude was found to be 14° 15' ; the latitude by account being 49° 17' N., and the sun's declination 23° 28' S. Required the true latitude. * This is the only correction necessary to make ful! allowance for the run of the ship ; and the inex- perienceil cnlculator must lake care not lo fall into the error of applying^ a correction to the elapsed time, as is directed in several works of note, particularly in the "Complete Navigator," by Dr. Mackay. This will appear evident by supposing, in the above Example V., that a second observer, with a watch, regulated exactly like that used by the first, was at rest at the place of the second observation. Thej, at the first observation, at the same moment of time by both watches, the first observer would find the sun's altitude 13° 18', and the second observer 12° 49'. At the second observation, the times and altitudes would be alike, so thai the elapsed time found by both observers would be the same, and the observations would require no correction, except what arises from reducing the altitude from 13° 18 to 12° "iy, because the second observer is supposed to be at rest, and his observation requires no cor- rection. 184 TO FIND THE LATITUDE BY DOUBLE ALTJTUUEa. The coiredion to the first altitude. The time elapsed between the observations was 3'* 40'", and in tliat time tlie ship sailed 33 iniles upon tlie course N. by E. \ E., which makes an angle of 13.J |)oints with the sun's bearing at the first observation S. \ E., the complement of wliich to 16 points is 2J points. Now, in Table I., the course 2^ points, and distance 33'", give 29 miLs dillerence of latitude, whicli must be subtracted from the first altitude 13° 18', ijecause the ship sailed above eight |)oints irom the sun ; therefore the first altitude corrected will be 12° 49', which inust be used in the rest of the work. CoL. 1. CoL. 2. Col. 3. El.timc[i>.;i.]3''40">,Cosec. 10.33559 Declination 23° 28' S. Sec. 10.03749 A Cosec. 10.37308 h sum alls. 13 32 Cosine 9.98777 i diff. alts. 43 . ..Sine 8.097 18 C Sine 8.45^03 \Z lees lliu lid X. or S. like bciring of zeuiih.] Cosine 9.95704 Cosec. 10.G307G Secant 10.00003 Cosine 9.99982 Z Sec. 10.587G5 .Cosec. 10.39988 Cosi'ne 9.95704 B 2G° 05' S. Cos3c. 1 0.35G92 Cosine 9.99982 Z 75 01 N. (E ;s i! e sum ofB, Z, if of the sanu i ! ; diffcTcnce, if of a different i E 48 5G N. Latitude 48 .54 N. Sine Sine 9.87734 9.87716 If the sun had passed the meridian to the north of the observer, Z would have been 75° or S., and E = 101° 00' S., Avhose sine 9.99180, added to 9.99982, gives the sine of the latitude 9.991G2 corresponding to 78° 47' S. EXAMPLE VL Sailing N. E. h E. by compass, at the rate of nine knots an hoin*, at 0- 31"" 40' P. J\L, per watch, the altitude of the sun's lower limb was 28^ 20' above the korizon of the sea, the eye being elevated twenty feet above the surface of the water, and the Sim's bearing by coni])ass S. by W. ; and at 2'' 58'" 20' P. M., by watch, the altitude of the sun's lower limb was 1G° 41' above the horizon, the eye being elevated as before, t!ie latitude by account, at the time of the last observation, 48° 0' N., and the dechnation 13° 17' S. Required the true latitude at taking the last observation. The correction of these altitudes for semidiameter, parallax, and dip, was twelve miles, (additive,) which makes tJiem 28° 32', and 1G° 53'. The refraction corre- .sponding to the first was 2 miles, and for the second 3 miles : and, by siil)tracting these quantities, we have the true central altitudes, 28° 30', and 10° 50'. Now, the elapsed time between the observations was 2'' 20'" 40^, diu'ing which the s'lip sailed tvveniy-two miles (at nine miles ])er hour) in the direction of N. E. h. E. per com})ass; the bearing of the sun at the first observation S. by W. being \2h points distant from the shi|)'s course ; and as 12^ [)oints want 3^ of IG points, we iriust enter Table I., and find the course 3^ ])oints, and distance 22, corresponding to which in the latitude column is 17 miles, which, being suinractcd from the first altitude 28° 30', leaves the corrected first altitude 28° 13' ; with this, and the second altitude 1G° 50', the latitude is found in the following manner: — • CoL. 1. EI.time[p.M.] 2"2G'M0%Cosec. 10.50232 Declination 13°17'S. Secant 10.01178 CoL. 2. CoL. 3. A Cosec. 10.51410 h sum alts. 22 3U ..Cosine 9.90553 A diff alts. 5 41i ... .Sine 8.99640 Cosine 9.97801 Cosec. 10.41070 Secant 10.00215 Cosine 9.97902 C Sme 9.47003 • SZ less ih:in 90°, ;in(l N. or ."5. like bearing of zeniili.] £ U the $um of }I. 7, if ol the tam lamc ; difference, if of a different name.] . Cosec. 10.0387 J .Cosine 9.978G1 B 13° 58' S. Cosec. 10.01732 [B less linn 90°, namc-l N. or S. like ilcclio.] Cosine 9.97962 Z Sec. 10.37708 Z G5 11 N. E 51 13 N. Sine 9.89183 Latitude 48 03 N. Sine 9.87145 TO FIND THE LATITUDE 1}Y DUUliLE ALTITUDES. 185 If tlie snn Iiad passed the meridian to tlie north of tlie observer, Z would have oeen (io° 11' S., and K = 79" OS)' S., whose sine U.!}'J217, added to cosine of C 9.97902. gives the sine of the latitude 9.97179, corresponding to (J9° 34' S. EXAMPLE VII. [Same as Dr. Briiiklcy's, in the Nautical Almanac for 1800.] The latitude by account* G° 30' N., sun's decfniation 5° 30' N., the siui's correct central altitudes 35° 21', and 70° 01', elapsed time between the ob.servations 2'' 20"" ; required tiie laiitude, the sun pas.sing the meridian south of the observer. El.time[p.M.]2'^20"',Cosec. 10.52186 Declination 5°30'N. Sec. 10.00200 A Coscc. 10.5238G i sum alts. 52 41 Cosine 9.782G3 h diff. alts. 17 20 Sine 9.47411 C Sine 9.780G0 [Z less lliaii 90", njul N. or S. like bearing of zenilh.] (E is the cam of B, Z, if of ihe same name, diferencc, i( of a dijferent n.iine.] Cosine 9.979G2 Cosec. 10.09947 Secant 10.02018 Cosine 9.90170 Z Sec. 10.00097 Co.scc. 11.01843 Cosine 9.979G2 B 5°4G'N. . V-ZW. iliW ». .< . t^v^/^ . .Cosec. 10.99805 [E less itiiTi 90°, iKinied N. or S. li!,e Jcxlin.l . Cosine 9.90170 2 3 50 N. E 9 3G N. . ....Sine 9.22211 Lat. 7J38 N Sine 9.12381 If the smi had passed to the meridian north of tlie observer, Z would have been 3° 50' S., and E =r: 1° 5G' N., whose sine 8.52810, added to the cosine of C 9.90170, is 8.42980, which is the sine of the other latitude 1° 32' N., so that in this example both latitudes are north. SECOND METHOD Of finding the latitude by double altitudes of the sun, tohcn the variation of declination is neglected. This method of finding the latitude depends on a set of tables (marked XXIII., in this collection,) (irst prepared by Mr. Douwes, containing three logariihins, titled half elapsed time, middle time, and log. risiiig. The two former are arranged together as far as six hours ; the latter is placed at the end of the table, and is extended, in the present edition, as fir as twelve hours. The table with the })roi)er title must be entered at the top with the hour, at the side with the minute, and in the colunm marUed at the top with the seconds; the corresponding number will be the sought logarithm, to whiidi must be prefixed the index of the log. under 0" in the same horizontal line. Thus, to the time 3'' 52'" 10^ corrcsi)oiid the log. half elapsed time 0.07138, log. middle time 5.229G5, and log. rising 4.G7274, In general it will be sufficiently exact to take these logarithms to the nearest 10 seconds, particularly when the sun's /eiiitii distance is great; but if the log. to the nearest second is required, it may be f)und by taking the difterence of the tabular logarithms coriespondiiig to the next greater and next less time, and saying. As 10' is to that diflerence, so are the odd seconds of time to the correction of the first tabular logarithm, additive if increasing, suhtractive if decreasing. Thus, if the log. half elapsed time correspond- ing to 3'' 52'" 18' were required, the logs, corresponding to 3'' 52'" 10' and 3'' 52'" 20' are 0.07138 and 0.07119, whose difference is 19; then 10' : 19:: S'- : 15; this, stib- trrK'tci! ti-oin 0.07138, leaves 0.07123, the sought log;n-ithm. By inverting the process, ■we may find the nearest second corres])onding to any given logaritiim. We shall now give the rule lor calculating the latitude, adapted to double altitudes of the sun. RULE. • To the log. secant of the latitude by account (Table XXVII.) add the log. secant of the sun's declination, (Table XXVII.,) rejecting 10 in each index ; the stun is to be called the log. ratio. 24 18G TO FliND THE LATITUDE BY DOUBLE ALTITUDES. From the natural sine of the greatest altitude (Table XXIV.) subtract the natural sine of the least altitude, (Table XXIV.;) find the logarithm* of their difference, (in Table XXVI.,) and place it undet* the log. ratio. Subtract the time of taking the first observation from the time of taking the second, having previously increased the latter by twelve hours when the observations are on different sides of noon by the watch ; take half the remainder, which call hall" tlie elapsed time. With half the elapsed time enter Table XXIII., and from the cokniin of half elapsed time take out the logarithm answering tliereto, apd write it under the log. ratio. Add these three logarithms together, and with their sum enter Table XXIII. in the coUunn of middle time, where, having found the logarithm nearest thereto, take out the time corresponding, and put it under half the elapsed time. The difierence between these times will be the time from noon when the greater altitude was taken. With this time enter Table XXIII., and, from the cojunm of log. rising, take out the logarithm corresponding, from which logarithm subtract tke log. ratio ; th(! remainder will be the logaritlim of a natural number, which, being found in Table XXVI.,f and adfled to the natural sine of the greater altitude, will give the natural cosine of the sun's meridian zenith distance, which may be found in Table XXIV. Hence the latitude inay be obtained by the rules of pages IGG, 167. 1. If this computed latitude should differ considerably from the latitude by account, it will be proper to repeat the opei-ation, using the latitude last found instead of the latitude by account, till the result gives a latitude nearly agreeing with the latitude used in the comi)utation. 2. This methoc< is best suited to situations where the sun's meridian zenitli distance is not much less than half the latitude ; for in latitudes where the sun approaches near to the zenith, the observations must be taken much nearer to noon ; and the pre- ceding rule, instead of approximating, will in some cases give the results of successive operations wider and wider from the truth. To remedy this difficulty, a set of tables was published, by Dr. Brinkley, at the end of the Nautical Almanac for 1799; but the great variety of cases incident to his metliod, will hinder it from being generally used. Instead of Dr. liriukley's method, we may generally use the method of arithmetical computation, called Double Position, which will frequently give, in a more simple manner, the required latitude, as will be shown in Example X.; and, in general, it may be observed, that where Douwes's rule does not approximate, the ol)ject is ujost commonly so situated as not to furnish the necessary observations to obtain a correct latitude, whatever method of computation might be used. 3. The operation is the same whether the sun has north or south declination ; and also whether the ship is in north or south latitude. When the sun has no declination, the log secant of the latitude (rejecting 10 in the index) will be the log. ratio ; and when the latitude by account is nothing, the secant of the declination (rejecting 10 in the index) will be the log. ratio. This rule, as well as the former, is iounded on the supposition that the declination is taken for the middle time between the o!)scrva- tions, and that it does not vary during the elajised time, which, however, rarely happens, and a correction ought to be ai)plied to the latitude on this account. F»ut this correction is generally small ; and if it is large, the third method must be used ; and when the declinations differ very much from each other, we must use the fourth method. * The index of this logarithm being, as usual, one less than the number of figures contained in tlie difference of these natural sines ; ol)serving, also, that tlie ahitudes to be used are the correci central eiltitudes j that is, the observed altitudes corrected for dip, semidiameter, parallax, and refraction. t Taking, as usual, a number of figures equal to the index of that logarithm increased by unity. TO FIND THE LATITUDE BY DOUBI-E ALTITUDES. 187 EXAMPLE VIIL [Same as Exaimple L, prccetling.] Being at sea, in latitude 46° 30' N. by account, when the sun's declination was 11° 17' N. at 10'' 2'" in the forenoon, the sun's correct central altitude was 4G" 55' , and at 11'" 27'" in the forenoon, his correct central altitude was 54° 9'; required the tnie latitude, and true time of the day when the greater altitude was taken. Times. Jilt. Nut. Si. Lat. by ace 46° 30' Sec. 0.16219 2obscr. IP 27'" 00" 54° 9' 81055 Dec ...11 17 Sec. 0.00848 1 obser. 10 2 0^ 46 55 73036 Log. ratio 0.17067 Elap. time 1 25 Diir. n^.. sine». 8019 Log. difT. Nat. Sines 3.90412 i elap. time 42 30 Log. ^ elap. time 0.73429 Middle time 1" 15"- 10* 4.80908 ^ elap. time 42 30 2 obs. from noon 32 40 Its log. rising 3.00608 Log. ratio sub 0.17067 Nat. numb 685 corresponding to log. 2.83541 Nat. sine greatest alt 81055 Sum is nat. cosine ©'s zen. dist. 8l740. . .equal to 35° 10' N. ©'s declination "11 17 N. Lat. in 46 27 N. The latitude 46° 27' (differing only 3' from the latitude by account) may be assumed as the true latitude. By means of the time of the second observation from noon above found 32'" 40', the error of the watch may be found ; for, in the i)resent example, by subtracting 32"^ 40^ from 12'', we have the time of the second observation 11'' 27'" 20^; but the time of the watch was 11'' 27'" 0^ ; tlierefore the watch was twenty seconds too slow; a small tlifference would be fouad in these numbers, if we were to jirojjortion the logarithms of Table XXIII. to seconds. In the same manner, the error of the watch may be found in the following examples.* EXAMPLE IX. [Same as Example V., before given.] In this example the latitude by account is 49° 17' N. ; the sun's declination 2-3° 28' S. the first altitude corrected, as before, 12° 49' ; the second altitude 14° 15', Ilequired the true latitude. .4//. A„t. SL Lat. l>y ace 49° 17' Sec. 0.1 8554 2 obsor. 13" 40'" 0' 14° 15' 24615 Declination ... .23 28 Sec. 0.03749 1 obser. 10 12 49 22183 Log. ratio 0.22303 Elap. time 3 40 DifT. nat. si. 243 2 Its log 3.38596 h elaj). time 1 50 Its log 0.33559 Mid. time 10 10 Time corresponding to 3.94458 5 obser. from noon, 1 39 50 Its log. iu col. of risiug is 3.97028 Log. ratio 0.22303 ,5588 Nat. number of Log. .3.74725 Nat. sine greatest alt 24615 Nat. cosine ©'s nier. zen. dist 30203 = 72° 2.5' N. Declination 23 28 S. Latitude 48 57 N. * When the middle time is greater than half the elapsed time, both observations are on the same side of the meridian ; ollicrwise, on dilTerent sides ; whence it is easy to determine wiieliier the greater altitude be observed before or after noon 188 TO FIND THE LATITUDE BY DOUBLE ALTITUDES. But as the latitude by computation differs considerably from that by account, the work must be repeated. Lat. last found. . . 48° 57' . . .Sec. 0.18262 Declination 23 28 ... Sec. 0.03749 Log. ratio 0.22011 Diti: N. sine 2432 Irs log. 3.38596 h elapsed time l"* 50™ 0» Its log. 0.33559 Middle time 10 Its log 3 .94166 Time from noon 1 40 Its log. in col. of rising 3.97170 Log. ratio 0.22011 5644 Nat. number of Log. 3.75159 Nat. sine greatest ahitude 24615 30259 Nat. cos. mer. zen. distance . . . 72=23' N. Declination 23 28 S. True latitude 48 55 N. This latitude (differing only two miles from that which is used in tlie compTitation) may be depended upon as the true latitude of the sliip, at the time of tlie second observation. If the first altitude had not been corrected, the computed latitude would liave been Ibuud = 48° 40' N. * EXAMPLE X. [Same as Example VII., before given.] The latitude liy account 6° 30' N., sun's declination 5° 30' N., the sun's correct central altitudes 35° 21' and 70° 01', elapsed time 2'' 20", are given to find the true latitude. Making the calcidations wuh the latitude by account 6° 30', the computed latitude by the fir^t operation will be 8° 16'. Repeating the operation with the latitude 8° 16', tlie second oiieration will give 7° 10'.* This must be used for a third operation ; and by repeating the calculation accurately to seconds, it Vili require more than a dozen operations to obtain the true latitude 7° 38', which was founil, by the first method, by a single operation. Dr. Brinkley made the latitude 7° 30', differing 8' from a strict calculation by spherical trigonometry. The detail of this calculation is not here given, but is left to exercise the learner. The object of the ])resent example is to show liow tlie number of operations might be decreased by the arithmetical method of ilouble position before mentioned. Take the error or difference between the fii-st assumed latitude 6° 30', and the first computed latitude 8° 16', equal to 106' ; also the error or difference between the second assumed latitude 8= 16', and second comput- ed latitude 7° 10', which is 66'. Multiply theni crosswise, as in the adjoined scheme, according to the usual rule of double position ,-f dividing the sum of the products 1305° 16', by the sum of the errors 172, gives the corrected latitude 7° 35' N. The sum of the products is taken in this case, because one of the assumed latitudes was greater, and the other less, than its corre- sponding comjjiited latitude. II" both computed latitudes had bi;en greater, or both less, than the corresponding assiuned latitudes, the differences of the errors and of the products ought to have been taken. It will rarely hajjpen that more than one pro- cess of this kind will be refjuircd to give a correct result. In the present instance, however, it will be necessary ; for, by repeating the operation with the assumed latitude 7° 35', the resulting computed latitude is 7° 41.i', and the third error 6h'. Repeating anew the compiUation, with this and the second latitude 8° \6', and second error 66', the resulting latitude is 7° 38', the same as was foimd by the direct compu- tation by the first method, and as accurately as could be obtained by repeating the operations aljout fourteen times by the second method. In general, when SMch a largo number of operations are required to produce a correct result, it is a sure proof that the situation of the ol)ject is not Avell adapted to I, lis. Errors. Prod, cs. 6" 30' X 106: = 876° 16' 8 16 66-. = 429 00 172) 1305 16 7° 35'. * Slioht clifTereDros will be found in tlie.se calculations, by using logarithms to seven places of figures, and making ilie calculation accurately to seconds. f If the degrees of both latitudes are alike, the minutes only may be retained in these multiplications. TO FIND THE LATITUDE BY DOUBLE ALTITUDES 189 obtain an accurate latitude; and it would be lost labor, and lead to great n)istakes, to attempt it. Thus, in tlie present e.vatnple, if the greatest altitude had been decreased only 12' 42", tnaUing it 1)lify the operation, and may prevent mistakes. In this formula it is said that C is of the same affection as B; the meaning of which is, that if B is less than 90°, C also is less than 90° ; and if B is greater than 90°, C also is greater than 90°. Likewise A is of the same affection as the hour angle H, meaning that if the hour angle is less than 6 hours or 90°, A will be less than 90° ; and if the hour angle exceed 6 houi-s, the angle A will exceed 90°. 25 194 TO FIND THE LATITUDE BY DOUBLE ALTITUDES. Col. 1. RoarangleH[ F.M.]..Sec. Decli. d [at gr. alti.J Tan. A [diff. name from d.] Tan. D.Dec.[at least alt.] B C Cosec. Least altitude.... Sec Greatest altitude.. Sum, 3 last num. J Sum S S— g. alt.=Rera. Sine Sum of 4 logs. 2) SZ Sine FORMULA. CoL. 2. Col. 3. Tan Sine A [same affection as HJCosec. ...Cosine F Z C [same affection as B] Cosine Cotan. [F les. than 90°,dlfl. G Sine G Sine [I less [I nau I Tan. than 90°] Sec led as G.] Dec. D [at least alt.] K Latitude Sine [Z named N. or S., like the bearing of the zenith.] In some late works on navigation, no notice is taken of tlie cases where the hour angle exceeds 90°, or the distance of the objects exceeds 90°, and on that account the rules appear less subject to different cases than the following rule, which embracee all possible cases, and the apparent simplicity of the rules referred to, arises from their imperfections and incompleteness. RULE. 1. Find the hour angle H,* and take out the corresponding secant, which put in Col. 1, and its tangent in Col. 3. 2. Take the declination d, coiresponding to the greatest altitude, place its tangent in Col. 1, its sine in Col. 2. 3. The sum of the two logarithms in Col. 1 (rejecting 10 in the index) is the tan- gent of the angle A, which is less than 90° if the hour angle is less than 6 hours, (or 90°,) but greater than 90° if the hour angle is greater than 6 hours. This angle is to be marked noHh and south, with a different name from the declination d, at the greatest altitude. The cosecant of A is to be placed in Col. 2, its cosine in Col. 3. 4. Place the declination D, corresponding to the least altitude, below the angle A, and if they are of the f same name, take their sum, but if of different names, take their difference, and call this sum,| or difference, the angle B, making it north or south, like the greatest of the two quantities A, D. The cosine of B is to be placed in Col. 2, its cosecant in Col. 3. 5. The sum of the three logarithms in Col. 3 (rejecting 20 in the index) is the cotangent of an angle F, (less than 90°,) which is to be taken out and marked north or soidh, with a different name from B. 6. The sum of the three logarithms in Col. 2 (rejecting 20 in the index) is the cosine of the angle C, which is to be taken less than 90° if B is less than 90°, but greater than 90° if B is greater than 90°. The angle C, and its cosecant, are to be placed in Col. 1. 7. Place the altitudes below C, take the half-sum of these three quantities, subtract the greatest altitude from the half-sum, and note the remainder. Place the secant of the least altitude in Col. 1, its cotangent in Col. 2, its sine in Col. 3 ; the cosine of the half-sum in Col. 1, and the sine of the remainder in Col. 1. The sum of the four * The hour aii^le is the same as the elapsed time in double altitudes of the sun. This time is turned mlo deoj-rees by Table XXI., but it is more simple to double the hour angle, and find it in Col. P. M., Table XXVII., and take out its corresponding tangent. If this double angle exceeds ISi", reject IS^, and find the remainder in Col. a. m., and take out its corresponding tangent. lu the following exam- ples this double angle is marked with the letters P. M. annexed. t This rule is easily remembered in three places in which it occurs, from the circumstance that s is the first letter o( sum and same, and d the first letter o( difference and different. X If the sum be taken to find B, and it exceed 180°, subtract it from 360°, and call the remainder I^ with a different name from that of A, D. TO FIND THE LATITUDE BY DOUBLE ALTITUDES. 195 last logarithms of Col. 1, (rejecting 20 in the index,) being divided by 2, gives the sine of an acute angle, which being found and doubled, gives the zenith angle Z, which is to be named Jiorth if the zenith and north pole are on the same side of the are or ^eat circle, passing through the two objects, (or the two observed places of the same object,) but south if the zenith and south pole ai'e on the same side of that great circle.* 8. Take the sum of the angles Z and F if they are of the same name, but their difference if of different names ; this sum or difference is the angle G, to be marked north or south, like the greatest of the angles Z, F.f The sine of G is to be placed in Col. 2. 9. The sum of the two lower logarithms of Col. 2 (rejecting 10 in the index) is the tangent of an angle I, which is to be taken out (less than 90°) and marked north or south, like G. The secant of I is to be placed in Col. 3. 10. Write the declination D, corresponding to the least altitude below I, take their sum if of the same names, their difference \{ of different names. This sum or difference is the angle K, of the same name as the greater of these two quantities. The sine of K is to be placed in Col. 3. 11. The sum of the three last logai-ithms in Col. 3 is the sine of the required lati- tude, of the same name as K. EXAMPLE XIII. Given the sun's correct central altitude 41° 33', and his declination 14° N. After an interval of l"* 30", by watch, his correct central altitude was 50°, and his declina- tion 13° 58' N. Required the latitude, the sun being south of the observer when on the meridian. Col. '.- Hour ang. H Ih 33m [p. m. 3h] Sec. 10.03438 Decli. d. [at gr. alti.] 13° 58' N. Tan. 9.395C9 A [dif.name from d.] 15 04 S. Tan . 9.43007 D Dec. [at least alt.] 14 00 N. B 1 04 S. C 21 49 Cosec. I0.420S8 Least altitude 41 33 Sec. 10.1258s Greatest altitude 50 00 Sum 113 22 JSum 56 41 Cosine 9.73978 J S.— gr. alt. = Rem. 6 41 Sine 9.0n589 Sum 4 logs. 2)19.36143 JZ 28 39 Sine 9.68071 CoL. 2. Sine 9.38266 A [same aff. as H.] Cosec. 10.58512 .Cosine 9.99992 C [same aff. as B.] Cosine 9.96770 G Sine 9.93738 Cotan. 10.0.5243 I 44''20'N. Tan. 9.98981 Dec.D. 14 00 N. [at least alt.] K 58 20 N. Latitude 52 CoL. 3. Tan. 9.61722 .Cosine 9.98481 .Cosec. 11.73012 F 2M0'N. Cotan. 11.33215 Z 57 18 N. [F lei G 59 53 N. 1 90°,diff. mB.] Sine 9.82169 llessthan90'']Sec.l0.14552 1 named as G.] .Sine 9.92999 7 N. Sine 9.89720. 57 18 N. [named like bearing of zenith.] If the latitude had been south, Z, instead of being 57° 18' north, would be 57° 18' south ; G = 54° 38' S., I = 42° 37' S., K = 28° 37' S., and the latitude 25° 34' S. The labor of making this extra calculation is but little, and where any doubt exists of the name of Z, it is best to make the computation both ways; this, however, will rarely happen. The calculations of this example, and most of the following ones, are made to the nearest minute ; where great accuracy is required, it will be proper to take the logarithms and angles corresponding to seconds. * This case occurs also in the first and second methods of solution, and it must be determined on the spot by the situation of the objects. In double altitudes of the sun, moon, or planets, when the elapsed time is not very great, the angle Z is generally to be marked with the bearin"^ of the zenith from the observed object, when at its greatest altitude on the meridian, which in north latitudes, without the trop- ics, is in general north; in south latitudes, without the tropics, south. Sometimes, when the sun passes the meridian near the zenith, it may be doubtful whether the zenith be north or south ; in which case the problem may be solved for both cases, (which increases the labor but little,) and that one of the two computed latitudes selected which agrees best with the ship's reckoning; but it is generally safest not to use observations of this kind, which are generally liable to great errors from small mistakes in the altitudes. t If the stim be taken to find G, and it exceed 180°, subtract it from 360°, and call the remainder G,. with a different name from Z or F 196 TO FIND THE LATITUDE BY DOUBLE ALTITUDES. EXAMPLE XIV. The sun's correct central altitude was 32° 25', his declination 17° 0' S. ; 8 hours afterwards, by a watch, the sun's correct central altitude was 30° 8', and declination 16° 55' S., the observer being in a high soutli latitude ; required the latitude. Col. L Hour H 8li [p.m. lGh=4ti a. m.] Sec. 10.30103 Decli.d.[atgr.alti.] 17'00 'S. Tan . 9.48534 A[dif.name from d.] 148 33 N. Tan . 9.78637 D Dec. [at least alt.] 16 55 S. B 131 33 N. C Ill 51 Cosec. 10.03238 Least altitude 30 08 Sec. 10.06305 Greatest altitude.... 32 25 Sum .174 24 iSum 87 12 Cosine 8.68886 ^S.— gr.alti.= Rem. 54 47 Sine 9.91221 Bum 4 logs. iZ 12 53 Z Col. 2. 2)18.69650 Sine 9.34825 Sine 9.46594 A [same aff.as H.] Cosec. 10.28253 . Cosine 9.82240 Cfsameaff. as B.] Cosine 9..57087 G Sine 9.90005 Cotan. 10.23623 I 53°51'S. Tan. 10.13628 Dec. D 16 55 S. [at least alt.] K 70 4 6 S Latitude 53 28 S Col. 3. Tan. 10.23856 .Cosine 9.93100 .Cosec. 10.12644 F 26°50'S. Cotan. 10.29600 Z25 46 S.[F less than 90°,cUff. name from B.] G 52 36 S. Sine 9.70072 llesstlian90°]Sec. 10.22922 I named as G.] .Sine 9.97506 Sine 9.90500 25 46 S. [named like bearing of zenith.] This latitude differs 3' from the calculation in Example XL, page 191, on account of not noticing the seconds in the angles. If the zenith had been north of the great circle passing through the sun and moon, we should have Z =:25° 4G' N., G = 1° 04' S., 1 = 1° 50' S., K =: 18° 45' S., and the latitude 9° 18' S. EXAMPLE XV. Suppose, at the same moment of time, the moon's correct central altitude was 55° 20', the moon's declination 0° 3G' N., the sun's correct central altitude 37° 40', the sun's declination 0° 17' S. ; the hour angle, or difference of the right ascensions of the sun and moon, being, by the Nautical Almanac, 5 hours, or 75°. Requii'ed the latitude, supposmg it to be north. Col. 1. Hour angle H 5h [p. m. lOh] Sec. 10.58700 Decli. d. [at gr. alt.] 0''36' N. Tan. 8.02004 A[dif.namefromd.] 2 19 S. Tan. 8.C0704 D Decl. [at least alt.] 17 S. B 236 S. C... 75 00 Cosec. 10.01506 Col. 2. Least altitude 37 40 Greatest altitude... 55 20 Sec. 10.10151 Sum 168 00 J Sum 84 00 Cosine 9.01923 4 S.— gr.alt.=Rem. 28 40 Sine 9.68098 Sum of 4 logs. 2) 18.81678 14 50 Sine 9.40839 Sine 8.02002 A [same aff.as II.] Cosec. 11.39338 .Cosine 9.999.55 C [same aff. as B.] Cosine 9.41295 G Sine 9.70375 . . . i Cotan. 10.11241 I 33"'13'N. Tan. 9.81616 Sine 9.78609 rilesstlian90°] Sec.10.07748 „ ». „ ._ ^ r . ■■ [InamedasG.l Dec. D. 17 S. [at least alt.] "• -" K 32 56 N. Sine 9.73533 Latitude 23° 24' N. Sine 9.59890 Col. 3. Tan. 10.57195 .Cosine 9.99904 .Cosec. 11.34330 F 0°42'N. Cotan. 11.91489 Z 29 40 N. [Flessllian90'',diff. uanw from B.] 4Z Z 29 40 N. [named like bearing of zenith.] This latitude agr-ces with the calculation in Example XIL, page 192 If the zenith had been south of the great cu-cle passing through the objects, we should have Zz=29° 40' S., G = 28°58' S., I = 32°G' S., K=:32°23' S, and the latitude 22° 44' S. TO FIND THE LATITUDE BY DOUBLE ALTITUDES. 197 EXAMPLE XVI. Given the moon's correct central altitude 47° 37', the moon's declhiation 17° 29' S, the sun's correct central altitude, at the same time, 27° 22', the sun's declination 8° 28' S., the hour angle, or difference of right ascensions of the sun and moon, 5h 40ni 28», or 85° 7' ; required the latitude, supposing it to be north. Col. L Hr. HSo"?' [p.m. Ill' 201 5C"] Sec. 11.0G993 Decli. d. [at gr. alt. ] 17° 29 ' S. Tan . 9.49828 A [dif.nariie from d.] 74 53 N. Tan . 10.56821 D Decl. [at least alt. ] 8 23 S. B 66 25 N. C 82 51 Cosec. 10.00339 Col. 2. Least altitude 27 22 Greatest altitude... 47 37 Sec. 10.05155 8um 157 50 Sine 9.47774 A [same aff. as H.] Cosec. 10.01529 Cosine 9.60215 C [sameaff.asB.] Cosin e 9.09518 G Sine 9.58497 Cotan. 10.28599 I 3G''37iN. Tan. 9.87096 Dec. D. 8 28 S. [at least alt.] K 28 09 N Col. 3. Tan. 11.00835 .Cosine 9.41628 .Cosec. 10.03788 F 16° 43' S. Cotan. 10.5*251 Z 39 20 N. [Flesstha.i90°,diB-. Dame from B.] G 22 37 N. Sine 9.06246 rilessthan90°]Sec.lO 09548 [I named asG.] .Sine 9.07374 Latitude 15° 41' N. Sine 9.43168 i Sum 78 55 Cosine 9.28384 J S,— gr. aIt.=Rem. 31 18 Sine 9.71560 Bum of 4 logs. 2)19.05438 \7. 19 40 Sine 9.52719 Z 39 20 N. [named like the bearing of zenith.] If the zenith had been south of the gi-eat circle ))assing through the objects, we should have Zrr39°20'S., G=:56°3' S., I = 58° 2' S., K = 66° 3(y S., and the latitude 52° 46' S. FIFTH METHOD. To Jiiid the latitude from the altitudes and distances fotind in taking a lunar observation. This is a particular case of Form V., and is more simple than the general solution, because the true distance of the objects, computed in working the lunar observation, may be used to shorten the calculation of the latitudes ; we shall therefore give a particular rule for this method. Having the apparent altitudes and distance of the objects, find, by any of the methods of working a lunar observation hereafter given, the true distance. Find also the true altitudes, by correcting the apparent altitudes for parallax and refraction. The correction of the moon's altitude is equal to the difference between 59' 42" and the correction already found from Table XIX., in working the lunar observation ; this difference, added to the moon's apparent altitude, gives her true altitude. In like man- ner the correction of the sun's altitude is equal to the difference between GO' and the correction already found in Table XVIII. (or in Table XVII. if a star or ])lanet is used) ; this difference is to be subtracted from the sun's (or star's) apparent altitude, to obtain its true altitude. The time at Greenwich, corresponding to the true distance, having been found in working the lunar observation, take from the Nautical Almanac, for this time, the declinations of the sun and moon, as is taught in pages 156, 171. If, instead of the sun, a star is used, its declination may be obtained from Table VIII., or more accurately from ttie Nautical Almanac, if it be one of the 100 bright stare whose places are now given for every ten days in that work. If a planet is used, its declination is to be found in the Nautical Almanac, From these declinations, the north polar distances must be found, by adding the declinations to 90° if south, or subtracting from 90° if north. Having thus obtained the true distance, the true altitudes, the declinations and north polar distances, the latitude may be computed by the following rule, adapted exclu- sively to Tal)le XXVII., writing, as before, sine, cosine, &c., for log. sine, log. cosine, &c., the logarithms being arranged in three columns, as in the former methods. RULE. 1. Place in Col. 1 the true distance and the polar distances. Take their half-sum, subtract from this half-sum the polar distance of the object which had the greatest altitude, and note the remainder. Put in the same column the cosecant of the true distance, the cosecant of the polar distance of the object having the least altitude, the sine of the half-sum, the sine of the remainder. The sum of tliese four logarithms (rejecting 20 in the index) being divided by 2, gives the sine of an acute angle, wliich being found and doubled, is to be called the angle F. j98 TO FIND THE LATITUDE BY DOUBLE ALTITUDES. 2. Place in Col. 1 the true distance and the true altitudes. Take their half-sum^ and also the remainder or difference between the half-sum and the gi-eatest altitude. Place in the same column the cosecant of the distance, (before found,) the secant of the least altitude, the cosine of the half-sum, the sine of the remainder. The sum of these four logarithms (rejecting 20 in the index) being divided by 2, gives the sine of an acute angle, which being found and doubled, is to be called the angle Z. 3. If the zenith and north pole be situated on the same side of the great circle, passing through the two objects,* take the sum\ of the angles F and Z for the angle G ; but if the zenith and nodh pole be situated on different sides of that great circle, take their difference for the angle G. Place the cosine of G in Col. 2. 4. Write in Col. 2 the cotangent of the least altitude, and its sine in Col. 3.]: The sum of the two logarithms in Col. 2, is the tangent of the angle I, which is to be taken less tlian 90°, and marked south if the angle G is less than 90°, but noHh if G is more than 90°. Place the secant of I in Col. 3. 5. Place the declination corresponding to the least altitude, below I ; take their sum if of the same name, but their difference if of diffehnt names ; call this sum or difference the angle K, and mark it with the same name as the greatest of the two quantities. Place the sine of K in Col. 3. 6. The sum of the three logarithms in Col. 3 (rejecting 20 in the index) is the sine of the latitude, of the same name as K. Having found the latitude, the hour may be obtained by means of the true altitude and declination of the sun, star, or planet, by any of the usual methods hereafter given for tliat purpose ; but, if the last of the observed altitudes was that of the sun, star, or planet, the horary distance of that object from the meridian might be obtained more simply by the following rule, adapted to Table XXVII. Rule. Add the tangent of the angle G, the sine of the angle I, the secant of the angle K ; the sum, rejecting 20 in the index, is the tangent of an angle ; take out the corresponding time in the column P. M. or in the column A. M. increased by 12 hours; half of either of these times is the horary distance of the lowest observed object from the upper or lower meridian, whence the hour may be obtained directly if it be the sun, but if it be the star, a planet, (or the moon,) it is obtained by apj)lying its horary distance to the hour of passing the meridian, according to the usual methods of finding the time from an altitude of a fixed star or the moon. EXAMPLE XVII. [Same as Dr. Brinkley's, in the N A., 1825.] May 19 ' S** 6™, P. M., in the longitude of 7" 23'" west, it was found, by working a lunar observation, that the correct distance of the centres of the sun and moon was 90° 57' 20"; ti-ue altitude of the sun's centre 11° 33' 12"; true altitude of the moon's centre 27° 32' 18". At the same time, by the Nautical Almana'*, the sun's declination was 19° 56' 48" N., the moon's declination 13° 55' 48 ' N Required the latitude and hour by this observation. Col. 1 Col. 2. Col. 3. True distance P. dist.at le. alt P. dist. at gr. alt 90° 57' 20" . 70 03 12 . 76 04 12 Cosec. 10.00006 Cosec. 10.02687 Sum ; Sum S — p.d.atgr.a 237 04 44 118 32 23 .42 28 10 52 36 00 Sine 9.94374 Sine 9.82943 G is sum of F, Z great circle Z 61° 36* 52" if north pole an , but their differe d zenith are on same side of ice if on dilTerenl sides. iF 2 ) 19.80010 Sine 9.90005 I is less than 90°, named south if G IS less than 90', north if G is more than 90°. Angle F 105 12 00 Cosec. 10.00006 Sec. 10.00888 F G 105 12 00 166 48 52 Cosine 9.98840 Cotan. 10.68947 Tan. 10.67787 Tnie distance. 90 57 20 11 33 12 27 32 18 . Sine 9 30163 Greatest alt.... I 78 08 33 N. Sec. 10.68723 130 02 50 Dec 19 56 48 N. (at least alt.) J Sum C5 01 25 37 29 07 Cosine 0.62557 Sine 9.78430 K 98 05 21 N Sine 9.99566 i Sum — gr. alt Lati lude74°48'N. Sine 9.98452 2)19.41881 4Z Angle Z 30 48 26 61 36 52 Sine 9.70940 * In places without the tropics, the sum is used generally in northern latitudes, and the difference in southern latitudes. t If this sum should exceed 180°, subtract it from .360°, and call the remainder the ansjlc G. X Both these logarithms may bo taken out at the same time when the sine of the euigie was found in the computation of the angle Z TO FIND THE LATITUDE BY DOUBLE ALTITUDES. 199 To fnd the hour. G Tan. 9.36974 I Sine 9.99063 K Sec. 10.85166 Hour P. M. 7h 47ra 42«, or A. M. + lOh = iGh 12m 18s Tan. 10.21203 Divided by 9, gives the horary distance of ) „^ -3^ -,, „^ „. „„, the lowest object from the meridian, j J" 3^™ »i'. or tjn Obm UJ«. The sun being at the lowest altitude, his distance from the upper meridian waa gh gm 9>^ being the hour of the day, and the sun's distance from the lower meridian, or midnight, was S*" 53™ 51'. '' ADDITIONAL QUESTIONS FOR EXERCISE. In the following questions the sun's semidiameter is supposed to be 16', and the pai'allax nothing. 1. Being at sea, in latitude by account 39° 28' N., when tlie sun's declination was 20° 41' N., at ll'' 30™ 15% A. M., per watch, the altitude of the sun's lower limb was observed to be 68° 18' 45", and at 12" 26™ 28» P. M. was 70° 58', the height of the eye beuig 21 feet above the surface of the sea. Requii-ed the true latitude of the ship. Answer, 39° 28' N. 2. Being at sea in latitude 50° 40' N. by account, at 10'^ 17™ 30% A. M., per watch, the altitude of the sun's lower limb was observed to be 17° A'\, and at 11'' 17™ 30^ was \Q° 31'i, the declination being 20° S., and the height of the eye 21 feet above the sea. Required the latitude in. Answer, 50° 1' N. 3. Suppose a ship at sea, in latitude 47° 34' N. by account, and that at 9'' 55™ 30% by watch, the jaltitude of tlie sun's lower limb was 17° 24', bearing by compass S. by E. k E., and at 12'' 54™ 10' the altitude of the same limb was 21° 45'i, the declination being 19° 30' S., the height of the eye 20 feet above the sea, and the ship's course by compass E. h S., at the rate of 7 knots per hour. What was the true latitude ? Answer, 47° 24 N. 4. At ll"* 28™ 20% A. M., per watch, the altitude of the sun's lower limb was 28° 18', the sun bearing S. by W. by compass. At 2'> 58™ 20% P. M., the altitude of the same limb was 16° 40', the height of the eye 20 feet, his declination 13° 17' S., and the latitude by accoimt 47° 50' N., the ship's course during the elapsed time N. E., with her larboard tacks on board,* sailing at the rate of 6 knots, and making half a point lee-way. What latitude was she in when the last altitude was taken ? Answer, 48° 9' N. * The larboard side of a ship is the left side, when the observer is aft, lookins;- towards Iier head, and the starboard is (he right side. When a ship is sailing with her larboard tacks on board, the lee-way is allowed to the right hand ; but if her starboard tacks are on board, to the left hand. In calculating the answers to these questions, proportional parts were taken for the seconds ; a small dlfibrence woukl be found if the nearest logarithms only were taken. 200 TO FIND THE LATITUDE BY ONE ALTI- TUDE OF THE SUN TAKEN NEAR NOON, HAVING THE TIME OF OBSERVATION. When the sun does not pass near tlie zenith, the meruhan ahitude and the latitude of the place may be accurately determined by observing his altitude when near the meridian, and noting the time by a watch regulated thejjreceding morning or follow- ing evening, by either of the methods given in this work.* To this time by the watch must he a])plied a correction equal to the difference of longitude made by the ship (tiumed into time) in the interval between the regulation and the observation near the meridian, by adding ivhen the place of regxdation is to the loestward of the, place of taking the other ohservntion, otherwise oij subtracting ; the sum or difference will be the time of taking the observation ; whence the time from noon will be obtained ; with which, and the observed altitude, (corrected for semidiameter, dip, &c., as usual,) the sim's declination, (found in Table IV., or in the Nautical Almanac, and corrected for the longitude oi' the ship,) and the latitude by account, the latitude by observation may be found as follows : — RULE. Jlddtogether the log, cosine of the latitude by account, {Table XXVII.) the log. cosine of the declination, [Table XXVII.) the logarithm in the column of rising, [Table XXIII.) cor- responding to the apparent time from noon lohen the observation ivas taken ; reject 20 in the index; the natural number of the remainder being foxmd, [in Table XXVI.) and added to the naiural sine of the observed altitude, ( Table XXIV.) the sum tvill be the natural cosine of Oie meridian zenith distance, from ivhich the latitude may be obtained by the common rules. If the computed latitude differs considerably from the latitude by account, it is best to rejjeut the operation, using the latitude last found instead of the latitude by account. This method of finding the latitude by a single altitude of the sun, may be applied to any other celestial object. EXAMPLE I. Being at sea, in latitude 49° 50' N. by account, when the sun's declination was 20° S., at 11'' 29'" 20% A. M., apparent time, per watch, regulated the preceding morn- ing, in a ]ilace 20 miles of longitude to the eastward, the sun's correct central a.titude was 19° 41',f bearing south. Required the true latitude. Time per watch 11" 29"- 20» 2(y in time by Tab.XXL 1 20 Time of observation.. 11 28 Latitude 49° SO' Cosine 9.80957 12 Declin. 20 Cosine 9.97299 App. time from noon . . 32 Log. rising 2.98820 Nat. Num. 590 log. 2.77076 Central altitude 19° 41' Nat. Sine 33682 Mer. zen. dist. 69 57 N. Nat. Cosine 34272 Declination 20 OS. Latitude.... 49 57 N. * Tlie best time for regulating a watch is when the sun bears nearly east or west, and is above 10° from the horizon. t The observed altitude of the lower limb being 19° 32', ©'s semidiameter 16', dip 4', refraction 3', parallax too small to be noticed. TO FIND THE LATITUDE BY AN ALTITUDE NEAR NOON. 201 EXAMPLE II. At sea ill die latitude of 60° N. by account, the sun being on the equator, at C 59" 0% P. M., per watch, regulated to apparent time the preceding morning in a place 15 miles in longitude to the westward, the sun's correct central altitude * was 28° 53', bearing south. RcquLced the latitude. App. time per watch 0^' 59™ 0' Latitude 60° N. Cosine 9.69897 15' long, in time 1 Declination Cosine 10.00000 App. time from noon 1 Corresponding log. rising 3.53243 Nat. Numb.. 1704 Log. 3.23140 Central altitude ... .28° 53' Nat. Sine . . . 48303 Rler. zenith distance 60 N. Nat. Cosine. 50007 Declination Latitude 60 ON. When the observation is taken a few minutes before or after noon, the correction to be applied to the altitude, to obtain the meridian altitude, may be found Ijy means of Tables XXXIL and XXXIU., the first of which contains the variation of the alti- tude for one minute from noon, expressed in seconds and tenths ; the other contains the square of the minutes and seconds of a minute contained in the top and the side columns. By these tables the correction of the observed altitude may be found by the following rule : — RULE. Eyiter Table XXXIL, and find the latitude hy account in the side column, and the declination at the top, opposite the former, and under the latter, ivill he the change of altitude in seconds and tenths for one minute from noon : then enter Table XXXIIL, and find the minides of the apparent time from noon in the top column, and the seconds in the side column; under the former, and opposite the latter, loill be a number lohichis to be midtiplied hi the number taken from Table XXXIL, and the product tvill be the sought change of altitude, expressed in seconds and decimals. In making use of Table XXXIL, proportional parts may, if necessary, be taken for the miles of latitude and declination. The numbers in both these tables are expressotl in Avhole numbers and tenths. EXAMPLE III. Being at sea in the latitude of 40° N. when the sun's declination was 21° N., at 8™ past noon, apparent time, the sun's correct central altitude f was 70° 58'. Required the meridian altitude and latitude. In Table XXXIL, opposite 40° lat., and under 21° dec, is 4".3, and the number in Table XXXIIL corresponding to 8™ is 64.0. Multiplying 64.0 by 4".3, we get the correction 275" .2 (or 5 nearly). This quantity, being added to 70° 58', gives the meridian altitude 71° 3' ; and the latitude deduced therefrom is 39° 57' N. By observing several altitudes of the sun when near the meridian, and noting the times, the meridian altitude may bo obtained, by the above method, to a great degree of accuracy. For by using this method, many observations may be taken on the same daj^, and the mean of the meridian altitudes deduced therefrom will in general be much more correct than that obtained by a single observation, by the usual method. To obtain the correction to be applied to the mean of all the observed altitudes, proceed thus : — Take from Tai)lc XXXIIL the number corresponding to each time from noon, (the minutes being found at the top and the seconds at the side, the correction being under the former and opposite the latter,) and divide the sum of these tabular num- bers by the number of observations ; the quotient, being multiplied by the number taken from Table XXXIL, will be the correction to be applied to the mean of the obsei"ved altitudes, to obtain the meridian altitude. EXAMPLE IV. Being at sea in the latitude of 50° N. by account, when the sun's declination was 22° N., observed with a sextant, the altitudes of die sun's lower limb (bearing nearly * The observed altitude of the sun's lower limb being 28° 43', ©'s S. D. 16', dip 4', refraction 2', parallax too small to be noticed. f The observed altitude of the sun's lower limb being 70° 46', semidlameler 16', dip 4', parallax and refraction too small to be noticed. 26 202 TO FIND THE LATITUDE BY AN ALTITUDE NEAR NOON. south) as in the following table ; the correction for semidiameter, dip, refraction, &c., being 12' additive. Required the meridian altitude and latitude. The mean of the numbers from Table XXXIII. is 17.5 ; this being multiplied by the number of seconds from Table XXXJI., viz. 2" .5, gives the correction 43''.75, or 44", which, being added to the mean of the observed alti- tudes, 61° 46', gives the meridian altitude of the sun's lower limb, 61° 46' 44", or 61° 4? nearly; to this add 12' for semidiameter, &c., and we get 61° 59' for the correct central meridian altitude, whence the latitude is 50° ]' N. If the above altitudes had been taken with a circle, the calculation would have > been exactly the same, except that each altitude would not have been given, but the sum of all of them, 247° 4', would have been shown by the central index after finishing tlie observations. Obs. Alt. ©L. L. App. Time from Noon. m » 6 10 4 15 3 2 2 10 Numbers Tab. 38.0 18.1 9.2 4.7 70.0 O 1 61.45 61.46 61.46 61.47 Sum 247.04 Mean 61.46 17.5 EXAMPLE V. Having regulated my watch, I found it to be 6"" 2» too slow for apparent time. I then sailed to the southward and eastward till the ship had made 60' difference of longitude, and was by account in the latitude of 40° N., the sun's declination being 20° S. The sun being then nearly on the meridian, I observed ten altitudes of his lower limb by a circle of reflection, and noted the times by the watch as in the follow- ing table ; and the sum of all the altitudes taken from the circle was 298° 20'. Required the true latitude, supposing the dip to be 4' and the semidiameter 16'. When it was 12 o'clock by the watch, it was 12'' 6"" 2' apparent time at the place where the watch was regulated, and 12'' 10™ 2* apparent time at the place where the altitudes were taken to determine the latitude, because the former place was 60' or 4"" in time to the westward of the latter; consequently the watch was 10"" 2° too slow for apparent time at the place of taking the altitudes for determining the lati- tude. Hence we may determine the time from noon of taking each obsei-vation, as in the second column of the adjoined table, and find the num- bers corresponding in Table XXXIII., the mean of which is 6.97; this, multiplied by the number in Table XXXII. corresponding to the latitude 40° N. and declination 20° S., viz. 1".6, will give 11".152 or 11", which is the correction to be added to the mean of the obseiTed altitudes to obtain the meridian altitude. Now the sum of all the altitudes 298° 20', being divided by 10, the number of observations, gives 29° 50' 0" Add semidiameter 16' and the above correction 11" -|- 1^ ^ Add parallax found in Tal)le XIV -)- § Subtract dip 4' and refraction 1' 39" — 5 39 Central altitude 30 40 Zenitli distance. 59 59 20 N. Declination 20 OS. Latitude 39 59 20 N. When the meridian altitude of the object is small, the correction of altitude may be found by this method, for 12 or 15 minutes from noon, to a great degree of accuracy; but when the sun passes near the zenitli, the time of obsei-vation must be proportion- ally nearer to noon. Thus, in Example I., preceding, the time from noon was 32', and as the numbers in Table XXXIIL are the squares of the number of minutes, it follows, that the number corresponding to 32'" would be the square of 32, or 1024.0. Thi.s, being multiplied by the number 1".3 of Table XXXII., corresponding to the latitude 50° N. and declination 20° S., will give the correction 1331".2, or neai-ly 22', Time per App. Time Numbers Watch. froniNoon. Tab. XXXIII. 18.1 11.4543 4' 15" 46.58 3 9.0 47.52 2 6 4.4 48.50 1 8 1.3 49.28 30 0.2 50.48 50 0.7 51.10 1 12 1.4 52.13 2 15 5.1 53. 8 3 10 10.0 54.23 4 25 19.5 Sum 69.7 Mean 6.97 TO FIND THE LATITUDE BY AN ALTITUDE NEAR NOON. 203 which, being added to 19° 41', will give 20° 3' for the meridian altitude, or G9° 57' for the zenith distance, being the same as in that example. It is very advantageous in this method to observe as many altitudes in the afternoon as before noon, and at nearly the same distances from noon ; for in this case a small error in the regu- lating of the watch will not materially affect the calculation. This will appear evident by supposing, in the preceding example, that the watch was 11™ 2' too slow, instead of 10™ 2' ; by this means the times and numbers will be as in the adjoined table, and the mean of all the numbers, taken from Table XXXIIL, will be 8.15, which, being multiplied by 1",6, will give 13' nearly, for the correction, instead of 11", so that in this case an error of one minute in the regulation of the watch would only cause an error of 2 seconds in the meridian altitude. But it must be carefully observed, that, in using this method, you must not take the observation more than 2 or 3 minutes from noon, wlM3n the sun passes within 10° or 12° of the zenith. Times. In Tab. XXXIIL 3.15 lO.G 2.00 4.0 1.06 1.2 0.08 0.0 0.30 0.2 1.50 3.4 2.12 4.8 3.15 10.6 4.10 17.4 5.25 29.3 Sum 81.5 Mean 8.15 204 TO DETERMINE THE LATITUDE ON SHORE BY MEANS OF AN ARTIFICIAL HORIZON. It frequently happens that the latitude of a place on shore cannot be determined by the usual methods, by a quadrant, sextant, or circle, on account of not having an open horizon. In this case it is customai-y to make use of an artificial horizon formed by the surface of a vessel filled with mercury, water, Barbadoes tar, very clear mo- lasses, or any other fluid of sufiicient consistency not to be affected by the wind.* With this apparatus an observation may be taken on shore when the altitude of the object does not exceed 60°, with as much ease as at sea. Thus, if an altitude of the sun was required to be taken, the observer must place the vessel containing the mer- cury (or other fluid) in a firm position on the ground, and in a few minutes the surface of the liquor will attain a horizontal situation ; the obsei'ver must then place himself in a situation so as to see the image of the sun, formed by the fluid, which image will evidently be depressed as much below the horizon as the sun is elevated above it, so that, to obtain the double of the sun's altitude, it is only necessary for the observer to bring the image of the sun, formed by tiie instrument, down to the image formed by the artificial horizon, and the angle then pointed out by the index will be double of the altitude of the sun ; the half of which will be the apparent altitude. If the nearest limbs of the two images are brought in contact, the half of the angle obtained by the instrument will be the altitude of the sun's lower limb, but if the farthest limbs are brought in contact, the half angle will be the altitude of the upper limb. The alti- tude thus obtamed must be corrected for semidiameter, parallax, and refraction, as usual, but not for dip, because a truly horizontal surface is obtained by means of the artificial horizon.f In this manner the altitude of the sun, or any other bright object may be obtained when the altitude is less than G0°; at higher altitudes the angle cor- responding would be above 120°, which cannot be measured by a sexlant on account of the length of the arc, nor by any other instrument of reflection, in a convenient mannci*, with a sufficient degree of accm^acy. To illustrate this method we shall here add the following examples : — EXAMPLE II. The angular distance of the farthest limbs of the two images of the sun, when on the meridian, was obtained by the above method, and found to be Si" 0', when the declination was 10° N., and the semi- diameter 16'; the sun bearing north of the observ- er. Required the latitude : — Half of 34° 0' is the obs. alt 17° Subtract semidiameter 16 EXAMPLE L The angular distance of the nearest limbs of the two images of the sun was found by the above method to be 68° 10', when the declination was 10° S., and the sun's semidiameter 16', the sun bearing south of the observer. Required the lati- tude : — Half of 68° 10' is the obs. alt 34° 5' Add semidiameter 16 34 21 1 Subtract refraction True altitude 34 20 Zenith distance 55 40 N. Declination 10 OS. Latitude 45 40 N. 16 44 3 Refraction sub True altitude 16 41 Zenith distance 73 19 S. Declination 10 N. Latitude 63 19 S. * In case the wind blows fresh, you must use a screen formed of two plates of talc or glass whose sur- faces are ground perfectly parallel, and connected together in a frame so as to make an angle of about 90° with each other. This frame is to be placed over the box containing the fluid, and the rays of the sun, passing through one of the plates, are reflected from the surface of the liquor, and pass through the other plate to the eye of the observer. The use of these plates is to be avoided, when it can possibly be done, on account of the defect of parallelism of the surfaces. This error is generally greatest near tlie border of the glass, so that it has been recommended to cover the edge of the glass with a paper or some paint, to the distance of ^ or ^ inch from the frame. If the surfaces of the glass are perfectly parallel, the observed angle will be the same as if the screen had not been used. Instead t>( using the screen we may place one of the glasses of the screen upon the surface of the fluid, which will prevent it from being agitated by the wind, or other similar causes. If the reflecting fluid is molasses, air-bubbles will sometimes rise on the surface by the sun's heat ; this ma^' in some measure be avoided by heating Ihe molasses before using it. t If the instrument has an index error, it must be applied to the observed angle, or the half of the index error must be applied to the sun's altitude TO DETERMINE THE LATITUDE OiN SHORE, &c. 205 The latitude may be determined on sliore by this method to a great degree of accu- racy by means of a circle of reflection, by taking several altitudes a few minutes before and after the sun passes the meridian, and estimating the correction to be applied to the altitude by means of Tables XXXII. and XXXllJ. Thus, if, in the exanjple page 202, the obseiTations had been taken in this manner, the number of degrees denoted by the circle after taking ten observations, woulil have been 595° 20' ; this, being divided by 20, (twice the number of observations,) will give for tlie observed altitude 29° 46, and by adding the semidiameter IG', parallax 8", and the correction Ibund by Tables XXXII. and XXXIII., viz. 1] seconds, and subtracting the refraction 1' 39", the cen- tral altitude will be obtained, 30° 0' 40", as in the page before mentioned. Altitudes may be observed in this way in taking an azimuth for determining the variation, or for regulating a watch, in the manner explained in this work ; observing, in all cases, that the half of the observed angle is to be corrected for refraction, parallax, and semidiameter, but not for the dip of the horizon, and that half the index error only is to be applied. TO FIND THE POSITION* OF A SHIP ON A LINE OF BEARING. CASE I. "When the position of a ship is unknown, the latitude by account being uncertain, assume two or more latitudes, and work out the longitudes corresponding thereto. A line drawn on a chart through the two points tlnis determined, will represent (he line of equal altitudes. The place of the' ship will be somewhere on this line ; and if it passes through the land, the bearing of the land will be known. If the coast should run parallel to this line, you will have the distance of the ship from the land, but of course not tlie absolute position. EXAMPLE. December I7th, 1837.— The latitude, by account, being 51° 37' N., the "GreeiAvich time lOh. 47ra. 13s. A. M., the true altitude of the sun's centre was found to be 12° 10'. Required the true bearing of the land. Let the assumed latitudes be 51° and 52°, sun's dechnation 23° 23' S., and the equation of time — 3m. 37s. — The longitude corresponding to 51° latitude will be about 8° 42' W. The longitude corresponding to 52° latitude will be about 4° 50' W. A line drawn througli these positions A A', will represent the line of equal altitudes, and will also pass through "Small Lights," and run parallel to the S. E. coast of Ireland. The light was seen in tlie course of an hour, and the error in latitude ascertained to tc 8', C being the position of the ship. CASE IL When a douZle altitude is taken, the jiosition of the shi]) may be found by Avorking the longitude for each altitude, au in Case I., and then drawing two lines of equal altitudes through the four points A A' and B B' thus determined. The point of intersection of said lines will give the position of the ship. The necessary correction for the change of poai'ioyi, when the second altitude was taken, must be made as explained on page 183, or by moving the line A A' projected (parallel to itself) along the course and distance made good oy the ship. Thus, suppose between the observations the ship had sailed E. N. E. 25 miles. Then move the first line A A' parallel to itself on this course 25 miles, and draw a line whose intersection with the second line B B' will give the position required. It is evident, that when the two lines cross each oth- er at about right angles, the point of intersection is more easily found. A line drawn perpendicu- lar to the line of equal alti- tude shows the direction of the sun, and consequently _^ the azimuth. ^^ B\ The assumed latitude must be near the truth, to give val je to this method. "When the altitude is high, an error in the assumed latitude is of greater importance than when it is low. • From SuiDDer's work. &fff TO FIND THE LATITUDE BY AN ALTI- TUDE OF THE POLE STAR. Find in the s de column, the sum of the apparent time ol observation, and the sun's right ascension; the corresponding number, in the middle co lumn, will be the correc- tion of the tiue Eiltitude, on account of the distance of the star from the meridian. If th B time is If the time is found ir eitherof Correc- found in either these columns, tion of of these columns, the correction is the alti- the correction is subtraclive. tude. additive. H. M. H. M. O 1 H. M. H. M. 1 08 1 08 1 26 13 08 13 08 1 13 1 03 1 26 13 03 13 13 1 23 53 1 26 12 53 13 23 1 33 43 1 25 12 43 13 33 1 43 33 1 25 12 33 13 43 1 53 23 1 24 12 23 13 53 2 03 13 1 24 12 13 14 03 2 13 8 1 23 12 03 14 13 2 23 23 53 1 21 11 53 14 23 2 33 23 43 1 20 11 43 14 33 2 43 23 33 1 19 11 33 14 43 2 53 23 23 1 17 11 23 14 53 3 03 23 13 1 15 11 13 15 03 3 13 23 03 1 14 11 03 15 13 3 23 22 53 1 12 10 53 15 23 3 33 22 43 1 09 10 43 15 83 3 43 22 33 1 07 10 33 15 43 3 53 22 23 1 05 10 23 15 53 4 03 22 13 1 02 10 13 16 03 4 13 22 03 59 10 03 16 13 4 23 21 53 57 9 53 16 23 4 33 21 43 54 9 43 16 83 4 43 21 33 61 9 33 16 43 4 53 21 23 48 9 23 16 53 5 03 21 13 45 9 13 17 03 5 13 21 03 41 9 03 17 13 5 23 20 53 38 8 53 17 23 5 33 20 43 35 8 43 17 33 5 43 20 33 31 8 33 17 43 5 53 20 23 28 8 23 17 53 6 03 20 13 24 8 13 18 03 6 13 20 03 20 8 03 18 13 6 23 6 33 19 53 17 7 53 18 23 19 43 13 7 43 18 33 6 43 19 33 9 7 33 18 43 6 53 19 23 6 7 23 18 53 1 7 03 1 7 08 19 13 o 7 13 19 03 19 08 n 7 08 19 08 In northern climates, the latitude may be determined by means of an observed altitude of the pole star ; provided the apparent time of observation can be ascertained within a few minutes.* This method might be frequently used at sea, -when the horizon is well defined, if that star were of the first magnitude ; but being only of the second or third magnitude, it is sometimes so dim that it is rather difficult to determine the altitude with precision. How- ever, as there are times when it would be of great importance to determine the latitude ■within 8 or 10 miles, it was thought advisable to explain this method, which may be used when observations of the sun or moon cannot be obtained. Having, therefore, the apparent time of ob- servation (which miist be reckoned from noon to noon in numerical succession, that is, 6'', A. M., must be called 18'', (fee), and the observed altitude of the star determined by a fore ob- servation, you mnst subtract from the altitude the dip, which is in general 4 minutes, and the refraction, and you will obtain the true alti- tude of the star. Then the sun's right ascen- sion corresponding to the given day, must be found in Table VI. ,t and added to the appa- rent time of observation (rejecting 24 hours when the sum exceeds 24 hours) ; with that sum enter the adjoined table, and take out the corresponding correction, which must be added to, or subtracted from, the true altitude, ac- cording to the directions in the table ; the sum or difference will be the latitude of the place of observation. * If the star be not far from the meridian, an error of half an hour in the time would not affect the altitude above 1 or 2 miles. t It is accurate enough to take the'numbers from Table VI.; but in strictness the right ascension ought TO FIND THE LATITUDE BY AN ALTITUDE OF THE POLE STAR. 2U7 EXAMPLE I. At 7" 9" P. M., June 3, 1848, the observed altitude of the polo star was IG" 10', tlie dip 4'. Required the latitude of the place of observation. Obsei-ved altitude 16° W Hour of observation 7" 9"° Sub. dip 4', refrac. 3' 7 ©'3 right ascension 4 M T,.^,e altitude 16 3 Sum 11 .53 Correction corresponding add 1 21 Latitude 17 24 N. EXAMPLE II. On the 14th September, 1848, at 2*' S™ A. M., the altitude of the pole star was 24° 16', wlien the dip was 4'. Required the latitude. Observed altitude 24° IC Hour of ol)s. 2^ 2"" A. M., or . . . 14" 2" Dip 4', refrac. 2', sub 6^ ©'s right ascension H 28 True altitude 24 10 Sum, rejecting 24'' 1 30 Corresponding correction sub. 1 25 Latitude 22 45 N. EXAMPLE III. At 5" P. M., December 5, 1848, the observed altitude of the pole stai- was 25° 15', the dip 4'. Required the latitude of the place of observation. Observed altitude 25° 15' Hour of observation 5" GO™ Sub. dip 4', refrac. 2' 6 ©'s right ascension 16 47 Xrue altitude 25 09 Sum 21 47 Correction con-espondingsub. 54 Latitude 24 15 N. to be taken from tlie Nautical Almanac, for the hour of observation, reduced to Greenwich time, by adding or subtracting the longitude turned into time. This table will require a correction after a few years, on account of the variation of declination, and right ascension of the star. It correspoiuis nearly to the year 1860 ; for every year after that time you must add one quarter of a minute to the times in the side columns, and decrease the tabular corrections of altitude about -^-^ part. Thus for the year 1872 the times must be increased 3"> for the 12 ysars, »o that l*" 0E"> must be called l"" 11">, and all the corrections of altitude must be decreased g^ part, M tbal 1° 15' must ke 1° 12' nearly, and 0° 35' must be 0" 33 7" IS"" 35', astro- nomical time, by a watch which was very nearly regulated for mean time at the ship, observed the central altitude of the planet Jupiter, by a fore observation, and found it to be 32° 16' 23" ; the planet being to the west of the meridian, and the dip 4' 8" Required the mean time of observation at the ship. Adding the longitude 4'' 41™ to the time by the watch, we get the mean time at Greenwich, May 26" ll'' 59™ 35'; and with this time we get, from the Nautical Almanac, the sun's right ascension 4'' 15™ 04' ; Jupiter's right ascension T" 8" 32' ; Jupiter's declination 22° 48' 39" N., or polar distance 67° 11' 21". * Wlien verj' gjreat accuracy is required, we may notice Ihe parallax in altitude, which is found in Table X. A., and is to be added to the correct altitude computed by the above rule. We may also find the correction of refraction and parallax, by entering Table XVII. in the pa^e corresponding to the horizontal parallax of the planet, and taking out the corresponding number, which, being subtracted from GO', gives the correction for parallax and refraction, at one operation. 2U\ VO FIND THE TIME AT SEA BY A PLANET'S ALTITUDE. Observed altitude 32° 16' 23" Dip 4' 8", ref. 1' 30". .sub. 5 3 8 Correct altitude 32 10 45 Latitude 42 22 00 Secant 0.13145 Polar distance. 07 11 21 Cosecant 0.03537 Sum 2 ) 14 1 44 06 Half-sum 70 52 03 Cosine 9.51555 Altitude 32 10 45 Remainder 38 41 18 Sine 9.795 94 Sum 2 ) 19.47831 Half-sum sine 9.73916 Corresponding to this, in the column P. M., is 4'' 20"" 06' Planet's right ascension 7 08 32 Sum (being west of meridian) gives right ascension of meridian 11 34 38 Subtract the sun's rijjht ascension 4 15 04 Gives the app^nrent time at tlie ship 7 19 34 Equation of time by Nautical Almanac sub. 3 14 Mean time at the ship 7 16 20 The time by the watch being 7^ 18"" 35% it is 59' too slow for apparent time ; and 2™ 15^ too fast for mean time. EXAMPLE II. January 5'' 15'" 40™ 24% 1836, astronomical mean time, at a place in the latitude of 24° 16' N., longitude 34° 56' W. of Greenwich, observed the central altitude of the planet Saturn, by a fore observation, and found it to be 28° 15'; the planet being east of the meridian, and the dip 3' 41". Required the mean time of observation at the ship. Adding the longitude 2'' 19™ 44' to the time by the watch, we get the mean time at Greenwich, Jan. 5 ' 18'' 00™ 08' ; and with this time we get from the Nautical Almanac the sun's right ascension 19'' 5™ 13' ; Saturn's right ascension 14'' 10™ 22' ; Saturn's declination 10° 36' 17" S., or polar distance 100° 36' 17". Observed altitude 28° 15' 00" Dip 3' 4 1 ", ref. 1' 46", sub. 5 2 7 Correct altitude 28 09 33 Latitude 24 16 00 Secant 0.04018 Polar distance 100 36 17 Cosecant 0.00749 Sum 2) 153 01 50 Half-sum 76* 30 55 Cosine 9.36770 Altitude 28 09 33 Remainder 48 21 22 Sine 9.87349 Sum 2 ) 19.28886 Half-sum sine 9.64443 Correspondiu;.; to this, in column P. ]M., is 3'' 29™ 20' Saturn's rigiit ascension 14 10 22 Difference (being east of meridian) gives right ascension of meridian. . 10 41 02 Add 24'', and subtract the sun's right ascension 19 05 13 Gives the apparent time at the ship 15 35 49 Equation of time by Nautical Almanac add 5 46 Mean time at the ship 15 41 35 The time by the watch being 15'' 40™ 24', it was 4™ 35' too fast for apparent time, and 1™ 11' too slow for mean time. 217 TO FIND THE APPARENT TIME BY A STAR'S ALTITUDE. Correct the observed altitude for the dip and refraction, (the dip being generally 4 minutes when the obsei-vation is taken on the deck of a common-sized vessel ;) find the ship's latitude at the time of observation, and the star's right ascension and ileclination in Table VIII.* Add together the star's correct altitude, the ship's lati- tude, and the polar distance ; from the half-sum subtract the star's altitude, an.d note the remainder. Then add together the log. secant of the latitude, the log. cosecant of the polar distance, (rejecting 10 in each index,) the log. cosine of the half-sum, and the log. t^ine of the remainder; half the sum of these four logarithms will he the log. sine of half the hour angle ; take out the corresponding time in the column marked 1'. M. (Table XXVII.) and apply it to the star's right ascension, by subtracting when the star is ea&t of the meridian, or adding when west of the meridian ; the sum or difference will be the right ascension of the meridian. From the right ascension of the meridian (increased by 24 hours if necessary) subtract the sun's right ascension, taken from the Nautical Almanac ; f the remainder will be the apparent time at the ship, and by applying to it the equation of time, we get the mean time at the ship. EXAMPLE I. Suppose that, on September 8'' 14'' 19" 20',1836, astronomical time, as shown by a clironometer, regulated to mean time at Greenwich, when in the latitude of 7^ 45' S., and longitude of 29° 12' E. from Greenwich, the altitude of the star Procyon, being then east of the meridian, was observed by a fore observation, and found to be 28° 16', and the dip 4'. Required the mean time of observation at the ship. By inspection in the Nautical Almanac, we find that, on the above-mentioned day, Procyon's right ascension was 7'' 30™ 44% and the declination 5° 39' N., or polar distance 95° 39' nearly, agreeing nearly with the result from Table VIII., corrected for the annual variations, &c. Sun's right ascension by Nautical Almanac, Sej)t. 8, at mean noon — 11'' 07"" 47' Correction, Table XXXI., for 14" 19™ 10% mean time add 2 09 Sun's right ascension at the time of observation 11 09 50 Star's observed altitude 28° 16' Dip 4', ref 2', Table XIL, sub. Star's correct altitude 28 10 Latitude 7 45 Secant 0.00,399 Polar distance 95 39 Cosecant 0.00212 Sum 2) 131 34 Half-sum 05 47 Cosine 9.01298 Altitude 28 10 Remainder 37 37 Sine 9.78560 Sum 2 ) 19.40469 Ilalf-sum 9.70234 * The right ascensions and declinations of the stars in 'fable VIIL are the mean values for January 1st. 1830, and must be reduced to the time of observation by means of the annual variation g^iven in thp sime table. When vcrj' great accuracy is refjuircd, the riglit ascensions and declinations, thus obtained, must be corrected for the aberration and nutation, as explained in the precepts of Tables XLIl XLIII. j but in general these corrections ma}- be neglected. These corrections are, iiowever, all noticed in the places of 100 of the most noted tixed stars, given in the Nautical Almanac since the year 1834, for every ten days in the year; and when any of these stars are u>c(l, the places must be taken out, to the nearest day, from the Nautical Almanac, without any further correction, because the varia lions in ten days are very small. Thus, on July 29, 183G, Procyon's right ascension was 1^ 30"> 43», north polar distance 84° 21' 29", or 5° 38' 31" N. declination, corresponding to 95° 38' 41" so7i/h polar distance. This additional table of the Nautical Almanac simplifies this kind of calculation considerably. t The sun's right ascension and the equation of time are to be taken from the Nautical Almanac, for 28 218 TO FIND THE APPARENT TIME BY A STAR'S ALTITUDE. Corresponding to this half-sum, in Table XXVIL, m column P. M., is 4'' 02™ 04' Star's right ascension 7 30 44 Right ascension of the meridian 3 28 40 Increased by 24'", it is 27 28 40 Subtract the sun's right ascension 11 09 56 Leaves the apparent time at the ship 16 18 44 Equation of time by the Nautical Almanac sub. 2 43 Mean time at the ship 16 16 01 Now, the time by the chronometer being 14'' 19™ 20', it was too slow for apparent time by l*" 59" 24% or l** 56™ 41' too slow for mean time. We have, in this example, supposed the time at Greenwich to be given by the chronometer, which is the most simple way of proceeding ; but if you have no chronometer, regulated to Greenwich time, you must, in the usual manner, estimate as nearly as you can tlie time at Greenwich, by adding the longitude, if west, to the time at the ship, or subtracting the longitude, if east ; and then use this time in finding the numbers from the Nautical Almanac. EXAMPLE II. Suppose that, on Aprill6* 12 ''13"'03% 1836. astronomical time, as shown by a chro- nometer, regulated to mean time at Greenwich, when in the latitude of 48° 57' N., and longitude of 67° 25' W., the altitude of Aldebaran, when west of the meridian, was 22° 25', and the dip 4'. Required the apparent time at the ship. In the Nautical Almanac, we find on that day that Aldebaran's right ascension was 4" 26'" aO% declination 16° 10' N., or polar distance 73° 50'. Sun's right ascension by Nautical Almanac, April 16'', at mean noon . . 1'' 38™ 20' Cor. Table XXXI., for 12" 13'" 03^ 1 53 Sun's right ascension at the time of observation 1 40 13 Star's observed altitude 22° 25' Dip 4', refraction 2', Tab. XII. 6_ Star's correct altitude 22 19 Latitude 48 57 Secant 0.18262 Polar distance 73 50 Cosecant 0.01752 Sum 2 ) 145 08 Half-sum 72 33 Cosine 9.47694 Altitude 22 19 Remainder 50 14 Sine 9.88573 Sum 2)19.56281 Half-sum 9.78140 Corresponding to this half-sum, in Table XXVIL, column P. M., is 4" 57"' 33' Star's right ascension 4 26 30 Right ascension of the meridian 9 24 03 Subtract the sun's right ascension 1 40 13 Leaves the apparent time at the ship 7 43 50 Equation of time by the Nautical Almanac sub. 25 Mean time at the ship 7 43 23 Now, the time by the chronometer being 12'' 13™ 03% it was too fast for apparent time 4'' 29™ 13% or 4'' 29™ 38' for mean time. This method of obtaining the time by the stars would be accurate, if a good horizon could be obtained ; but as that is not always the case, it is best to regulate your watch by the sun. the time at Greenwich given by a chronometer, or by applying the longitude to the estimated lime a> the ship, .n the usual manner. 219 TO REGULATE A CHRONOMETER BY EQUAL ALTITUDES OF THE SUN, A CHRONOMETER may be regulated on shore by observing in the morning and evening tlie times wlien the sun is at the same altitude,* for the middle between these times would be the apparent time of noon by the chronometer, if the declination of the sun remained the same during the observation ; but if the declination varies, as is generally the case, the apparent time of noon, determined in this manner, which, for distinction, we shall call the middle /iHie,)must be corrected for the change of declination by an equation, called the equation of equal altitudes, and the middle time thus corrected will be the correct time of apparent noon by the chronometer. For greater accuracy, several altitudes should be taken in the morning, and corre- sponding ones in the afternoon, and the mean of the times of the morning and evening observations should be res])ectively taken, and the equation of equal alti- tudes, corresponding to the mean of all the observations, must be calculated and applied to the middle time, as if a single set of observations only Iiad been taken. In noting the times of observation, we must count the hours in numeral succession, so that if some of the observations are taken before IQ"* by the chronometer, and others after 12'', the next hour to 12'' must be called IS*", the next 14", &c. Half the sum of the times of observation, corresponding to any set of observations, (or the mean of a number of observations,) will be the middle time, and the difference of the times of observation will be the elapsed time. The equation of equal altitudes ponsists of two parts, which may be calculated by the following rule : — RULE.' 1. To the constant log. 8.8239 add the log. cotangent of the latitude, the log. sine corresponding to the elapsed time found in the cohunn 1*. M. of Table XXVII., the proportional logarithm of the hours and minutes of the elapsed time, reckoned as minutes and seconds, and the proportional logarithm of the daily variation of the sun's declination ; the sum (rejecting .30 in the index) will be the proportional logaritbm of the first part of the equation of equal altitudes, reckoning minutes and seconds as seconds and thirds respectively. 2, To the constant log. 8.8239 add the log. cotangent of the sun's declination, the log. tangent corresponding to the elapsed time found in the column P.M. of Table XXVII., the proportional logarithm of the hoiu's and minutes of the elapsed time reckoned as minutes and seconds, and the proportional logarithm of the daily variation of the sun's declination ; the sum (rejecting 30 in the index) will be the proportional logarithm of the second part of the equation of equal altitudes, reckoning minutes and seconds as seconds and thirds respectively. The first part of tlie equation of equal altitudes is to be added to the middle time when the sun is receding from the elevated pole, otherwise subtracted ;f and the second part is to be added when the declination is increasing, but subtracted when decrei^ing ; J these two corrections, being a])i)Iied to the middle time, will give the apparent time of noon by the chronometer. * Tlie alliludes should be taken when the sun rises or falls fast. The best lime for observation is when the bearing of the sun is nearly east or west, if the altitude exceed 8° or 10°, so as to avoid the irregular refraction near the horizon. In general, two or three hours from noon will be suflicient. An artificial horizon, formed by a vessel filled with mercury, may be used in taking these altitudes. t Thus, in north latitudes, the first part is to be added from the summer to tlie winter solstice, when the polar distance is increasing, and subtracted the rest of the year, when the polar distance is decreasing. i It is here supposed that the elapsed time is less than 12 hours, which is generally the case ; but if that time exceeds 12 hours, the second part must be applied in a contrary manner to the above rule. 2'iO TO REGULATE A CHRONOMETER BY EQUAL ALTITUDES. EXAMPLE. Suppose that, on the 9th of May, 1836, civil account, in the latitude of 40° N., and longitude 10° W., the following observations were taken at equal altitudes of the sun; required the error of the watch. t^lt. 0'* loiver limh. Times per chron. Times per chron. A. M. p. M. 15° 35' GhaQ-^Sl' 17" 32" 18* 15 45 6 31 07 17 31 00 15 55 6 32 14 1 7 29 54 Sum 93 12 93 12 Mean 6 31 04 17 31 04 6 31 04 Difference is elapsed time 11 00 00 Sum 2)24 02 08 Middle time 12 01 04 Constant log 8.8239 8.8239 Latitude 40° cotancent 10.0762 Declination 17° 27'. cotangent 10.5026 Elapsed time 11" sine 9.9963 Tangent 10.8806 Elapsed time IP, or IF P. L. 1.2139 1.2139 Variation declin. 15' 46"* P.L. 1.0575 1.0575 1st part 12" 14"' P. L 1.1678 2d part 0" 36'" P. L 2.478[i The first part of tliis equation, 12" 14'", is subtractivc, because the sun is proceeding towards the elevated pole ; and the second part, 36"', is additive, because the declina- tion is increasing, so that the whole equation is about 12 seconds subtractive ; this, being ajiplied to .the middle time, 12'' 1'" 4', gives the time of apj)arent noon by the chronometer, 12'' 0'" 52% so tliat the chronometer is 52 seconds too fast for ap[iarent time. * On May 9, at noon, by the Nautical Almanac, the declination was 17° 26' 27", and on the follow- ing noon 17° 42' 13", the ditTerence 15' 4G", beinj the daily variation; the declination corresponding to the longitude of 10° W . heins 17° 27' N. ilearly. 221 TO REGULATE A CHRONOMETER BY MEANS OF A TRANSIT INSTRUMENT. This method excels all others in brevity and accuracy ; but it can only be used on shore, and with tlie transit instrument that has been adjusted with the greatest possi- ble care, so as to liave the motion of the line of collimation of the telescope perfectly in the plane of the meridian. We have already given, from pages 145 to 152, the methods of making these adjustments, and of observing these transits ; we shall now inseit several examples for illustration. To determine the time hy the sun's transit over the middle wire of the telescope. In obsei-vations of this kind, we must note, by the chronometer, the times of the transit of the first and second limbs of the sun over the meridian wire ; the mean of the two observations will be the time of apparent noon, by the chronometer. Then the equation of time is to be taken from the Nautical Almanac for the apj)arent noon at Greenwich, and the correction applied to it for the longitude of the place of obser- vation, which is easily obtained by the means of the horary variation given in the same work. Applying this equation to the apparent time, by adding or subtracting, according to the directions in tlie Nautical Almanac, we get the mean time of ap- parent noon. The difference between this time and the tune by the chronometer, will be the error of the chronometer in mean time; moreover the difference between the time by the chronometer and IS'', will be the error of the chronometer for apparent lime. EXAMPLE I. Near noon, at the commencement of the SOt" of January, 183G, according to the astronomical computation of time, in a place 30°, or 2'', west of Greenwich, observed the transits of the limbs of the sun over the meridian wire of the transit instrument, for the purpose of regulating a chronometer. It is required to find, from these observations, the error of the chronometer, either for apparent or mean time. Transit of the first limb by the chronometer ll** 56'"10'.5 Transit of the second limb by the chronometer 11 58 27 .0 Sum 2) 14 37.5 Half-sum is the time of apparent noon by the chronometer 11 57 18 .7 Equation of time by Nautical Almanac, at apparent noon, Greenwich 13'"21'.C3 Correction for longitude, 2" X 0.432 add M Equation of time at the place of observation add 13 22 .5 Apparent time of observation at noon 12 00 00 .0 Mean time of observation 12 13 22 .5 Hence it appears, that the chronometer is too slow for apparent time 2 41 .3 Chronometer too slow for mean time IG 03 .8 EXAMPLE II. In another observation of the sun's transit, similar to the preceding, made June 25, 183G, in the longitude of G0°, or 4'^, east, we shall suppose that the time of the 222 TO REGULATE A CHRONOMETER BY A TRANSIT INSTRUMENT. Transit of the first limb, by the chronometer, was lli*" 02'" lO'.O Transit of the second limb, by the chronometer 12 04 27.8 Sum 2 ) 6 37.8 Half-sum is time of apparent noon by the chronometer 12 03 IS .9 Equation of time by Nautical Almanac at apparent noon at Gceenwich 2"" 14'.S4 CoiTection for longitude, 4*^ X 0.529 2 .12 Equation of time at the place of observation add 2 12 .2 Ap-parent time of observation at noon 12 00 00 .0 Mean time of observation 12 02 12 .2 Hence it appears, that the chronometer is too fast for apparent time 3™ 18'.9 And too fast for mean time 1 06 .7 To determine the time hy the sun's transit, observed at the Jive wires of the telescope. If the telescope of the transit instrument be furnished, as usual, with five equidistant and parallel wires, two on each side of the meridian wire, we can, with very little extra time or trouble, make the observations of the transits of the first limb of the sun at all the Avires, and mark down the corresponding times by the chronometer, in five sejiarate columns, on tlie same horizontal line, from left to right. Immediately after- wards,* make the observations of the transits of the second limb of the sun, over the same wires, and mark these times below the former numbers respectively, taking them in a contrary order, or from right to left. The sums of the two numbers in each of tho five columns will be nearly the same,f and the mean of the whole will be the time of the transit of the sun's centre over the meridian, as shown by the chronometer. Comparing tliis with the time of apparent noon, 12'', we get the error of the chronom- eter for a])parent time ; or by comparing it with the mean time of noon, we get the error of the chronometer for mean time, as in the two preceding examples. EXAMPLE III. July 23, 183C, in the longitude of 74°, or 4'' 56™, W., the following observations of the times of the transit of the sun's limbs over tlie wires of the transit instrument were made. Required the error of the chronometer for mean time. First limb Second limb. . . , Sum I. ^ 05\0 09.3 II. 5-^ 32'.0 8 42.1 14 14.3 14 14.1 Sum Mean of all is transit by chronometer. Mean time of ai)parent noon Chronometer too fust for mean time. . Chronometer too fast for apparent time III. 12'> Oo-" 59S5 12 08 14 .3 24 14 13.8 J4.3 14.1 14.2 14.3 10)70.7 12" 07™ 07».07 12 06 07.61 ' 59^46 ' 07».07 IV. 6™27» 7 47 V. 6™ 54M 7 20.2 14 14.2 14 14.3 Equation of Time. Noon at Greenwich -}- 6"" 07'..32 Corr. 4" 56™ X 0.059 .29 Equation of time. 6 07.61 12 h 12 6 07.6] * We Iiavc already remarked, in penoe 150, (hat the wires are so fixed in the telescope, that the first limb of the siin ])asscs over all of thoni licfore the second limb arrives at the first wire. t This equality in the sums renders it unnecessary to write down the hours of the observation, except in the middle column ; and we may also neglect, in the column of minutes, the figures which stand for lens of minutes; retaining the full expression of the lime only in the middle column. TO REGULATE A CHRONOMETER BY A TRANSIT INSTRUMENT. 223 EXAMPLE IV. May 14, 1836, in the longitude of 45°, or S*", east, the following transits of the sun's limb over the wires of the transit instrument, were obsei'ved. Required the error of the chronometer for mean time. First limb. ... Second limb. . Sum I. 10'.5 14.0 IL 53"^ 37'.5 56 46.5 50 24 .5 50 24 .0 Sum Mean of all the transits by chronometer Mean time of ap|)arent noon Chronometer too slow for mean time. Chronometer too slow for app. time . . III. 1P54'"05'.0 II 56 19.7 23 50 24.7 24.5 24.0 24.6 24.3 10 ) 122 .1 11"55"'12^2I II 56 03.74 0'"5P.53 4"- 47'.79 IV. 54"'32».5 55 52.1 50 24.6 V. 54"'59».3 55 25.0 50 24.3 Equation of Time. Noon at Greenwich — 3" 56'.30 Corr. 3" X 0.014. ._ M Equation of time — 3 56 .26 Apparent noon 12 00 00.00 Mean noon II 56 03 .74 To determine the time hy the transit of a fixed star over the meridian. In observations with the transit instrument, it is most commonly the case, that the chronometer wliich is used in making the observations, will give the viean time at Greenwich within a few seconds ; * and for this time we must find, in the Nautical Almanac, the sun's right ascension and that of the star. Subtracting the former from the latter, (increased by 24'' when necessary,) we get the apparent time of the star's transit over the meridian ; and by applying to it the equation of time, taken from the Nautical Almanac, for the above time at Greenwich, we obtain the mean time at the place of observation. The difference between this and the time of tlie transit, as noted by the chronometer, will represent its error. We may, as in observations of the sun, use the middle wire only, and note the time of the transit, when the star is bisected by that wire ; or, with greater chance of accuracy, we may take the mean of the observed times of passing the five wires, as a more correct time of the actual transit. To illustrate this, we shall give the following examples : — EXAMPLE V. July 24, 1836, in the longitude of 44° 39', or 2'' 58" 36% east, observed the transit of the star Arcturus over tlie middle wire of the telescope, the time by the chronometer, which was snpposred to be regidated very nearly for mean time in the meridian of Greenwich, being 8'' 00™ 10'. Required the mean time of the transit at the place of observation. S" 15™ 05».79 29 .70 0's right ascension at noon, at Greenwich, by Nautical Almanac. . Correction for 3'' 00'" 10' X 9'.89I ©'s right ascension at the estimated time at Greenwich Star's right ascension at the same time, by Nautical Almanac 8 15 35.49 14 08 12.13 Subtract ©'s right ascension, gives the apparent time of observation 5 52 36 .64 Equation of time at noon, Greenwich -{-Q"" 08'.74 Correction for 3" 00™ 10' X 0'.035 Jl Corrected equation of time -|" ^ 08 .85 . Mean time of observation Time by the chronometer Error of the chronometer for mean time Error of the chronometer for apparent time + 6 08.85 5 58 45.49 3 00 10.00 2 58 35.49 2 52 26.64 ••When we have no good regulation of the chronometer, from Greenwich, we must estimate the lime at that place, from the supposed time at the place of observation, by applying to it the longitude ; adding when west, or subtracting when east ; repealing the operation if we should find, after calculating ilw observations of the transit, that any essential error was made in the time at the place of observation. 224 TO REGULATE A CHRONOMETER BY A TRANSIT Il^STKljMENT. EXAMPLE VL March 10, 1836, in the longitude of 17° 18', or 1" OO"" 12% east, observed the transit of the star Siriusover the five wires of the telescope, at the times by the chronometer as given below ; the chronometer being supposed to give very nearly the mean time at Greenwich. Required the mean tune of this ti'ansit at the place of observation. /- First wire G" 14'" 01 '.5 \ Second wire 14 28 .7 Time of transit by the clu-onometer. < Meridian wire 14 56 .0 V Fourth wire 15 23 .2 ^ Fifth wire 15 50 .6 Sum 5124 40--.0 Mean of all the times by the chronometer is G*" 14"" 56 .0 Co 2)'s right ascension at noon, at Greenwich, by the Nautical Almanac 23'' 23" 10 .85 ;orrection for 6" 14"' 56= X 9M86 57 .40 (v)'s right ascension at the estimated time at Greenwich 23 24 08 .25 Star's right ascension at the same time by the Naut. Almanac -j- 24'' 30 37 55.39 Subtract 0's i-ight ascension, gives the apparent time of obsei-vation 7 13 47 .14 Equation of time for noon at Greenwich 10"' 25'. 45 Correction for G" 14-" 56^ X 0^G68 4.17 Corrected equation of time 10 21 .28 add 10 21 .28 Mean time of observation 7 24 08 .42 Time by the chronometer G 14 56 .00 Error of the chronometer for mean time 1 09 12 .42 Error of the chronometer for apparent time 58 51 .14 We may in the same way find the time by a transit of the planet, either by taking the mean of the times of the transits of the two limbs of the planet across the middle wire, or the mean of the times of the limbs passing all the wires; then the calculation is to be made, as in Examples V. VI. ; taking from the Nautical Almanac, and using the right ascension of the planet, instead of that of the star. This method is so plain, that it will not be necessary to give any examples. The ti-ansit of the moon might also be used ; but the calculation becomes so complex, on account of the rapidity of her motion, that it is wholly inexpedient to use such observations for regulating a chi'onometer. 225 LUNAR OBSERVATIONS. Almost all the methods of determining the difference of longitude between .iny two places, depend on the general principle of finding the dilference between the times of taking any observation, estimated under tlie meridian of both those places. For. in any place, it is the time of apparent noon when the sun is on the meridian ; and as the sun, by his diurnal motion, appears on the meridian of Greenwich (from which the longitude is reckoned) one hour earlier than in a place in 15° west longi- tude,* and cue hour later than in a place in 15° east longitude, and in proportion for a greater or less longitude, it follows that, if, at the time of taking an observation, the corresponding time at Greenwich be known, the longitude of the place of observation will be found by alloAving 15° for every hour of diffei'ence betv\een those times, the longitude being east when the time at Greenwich is earlier than at the place of observation, otherwise west. It is immaterial whether the times at both places be estimated for apparent or mean time, as the interval is the same when both are apparent as when both are mean ; it is, however, universally the practice, at i)resent, to use mean time in all these calculations. Now, an observer, at any place, may determine the apparent or mean time at any moment, by a watch regulated by any of the preceding methods ; and if, at the same moment, the apparent or mean time at Greenwich could be obtained, nothing more would be necessary for determining the longitude. One method of determining the time at Greenwich is by a watch regulated to Greenwich time ; for it is evident that if a watch could be so constructed as to go uniformly at all times, and in all places, an observer, furnished with a watch thus regulated, would only have to compare the time at the place of observation with the time at Greenwich, shown by the watch, and the difference of the times would give the difference of longitude. This method is useful in a short run ; but in a long voyage, implicit confidence cannot be placed in an instrument of such a delicate con- struction, and liable to so many accidents. Another method of determining the longitude, is by observing the beginning or end of an eclipse of the moon, or the satellites of Jupiter, and taking the difference between the mean time of observation and the mean time given in the Nautical Almanac for the meridian of Greenwich ; it being evident that such an eclipse must be observed at both places at the same moment of absolute time ; consequently the difference of the times will be the differ- ence of longitude. An observation of an eclipse of the sun, or an occultation, afler making allowance for parallax, &c., as taught in the Appendix to this work, may be used in like manner ; and this is a very accurate method. However, observations of eclipses are but of small practical utility at sea ; for those of the sun and moon happen too seldom, and the difficulty of oliserving the eclipses of Jupiter's satellites prevents that method from being made use of In the present improved state of the Nautical Almanac, we may easily determine the longitude on shore, by means of a transit instrument, by observing the time of the moon's transit over the meridian, or by observing the difference between the time of the moon's transit and that of some well-known and near star. Other metliods of finding the longitude at sea have been proposed, but among them all there is not one of such practical utility, as that by measuring the angular distance of the moon from the sun, or from certain fixed stars situated near the ecliptic, usually called a hmar ohservation, or, more frequently, "a lunary For observations of this kind may be taken, in fair weather, at all times (except near the time of new moon) when the objects are more than 8° or 10° above the horizon ; and as the moon moves in her orbit about 1' in 2'" of time, it follows that, if her angular distance can be ascertained from the sun or star within 1', the time at Greenwich will be known within 2 minutes, and the longitude within 30 miles. * Because the sua, by his apparent diurnal molion, describes 360 degrees in 21 hours, wliich makes 15 degrees in an hour. 29 226 LUiNAR OBSERVATIONS. To facilitate tliis methotl, there is annually published, by the Commissioners of Lon- gitude in England, a Nautical Almanac, containijig the true angular distances of the moon from the sun, from the four large planets, and from nine bright fixed stars, for the beginning of every third hour of mean time for the meridian of Greenwich ; and the mean time corresponding to any intermediate liom- may be found by proportional parts : hence, an observation of these angular distances l)eing taken in any place, and the corresponding mean time at Greenwich being found by the Almanac, and com- pared with the mean tune at the ship, tlieir difference will be the longitude of the l)lace of observation. But before tlie observed angular distance is compared with those in the Nautical Almanac, the corrections for parallax and refraction must be applied to obtain the true distance ; for, the moon being seen always lower than her true i)lace, and the sun and stars higher, the true distance is almost always greater or less than the observed distance. The angular distances of the moon from the sun and proper fixed stars and planets, are generally given in the Nautical Almanac from one object on each side of her, to afford a greater number of opportunities of observation, and to enable the observer to correct, iu a great degree, the errors of the instnmient, the adjustments, or a faulty liabit of observing the contact of the limbs, because these errors have a natural ten- dency to correct each other, in taking the mean of observations made with objects on different sides of the moon. Before taking the observation, the Nautical Almanac must be examined, to see from what objects the distances are computed, and from them only must the distances be measured. There are only nine fixed stars and four planets from which the angular distances are computed in the Nautical Almanac ; and as it is of the greatest importance to be able to discover them easily, we shall here add a number of remarks which will be found useful for that ])urpose. Tlie best way of discovering any star or planet, is by means of a celestial globe ; observing that, when a planet is used, we must estimate roughly, by inspecting the Nautical Almanac, the right ascension and declination of the planet, and make a mark on the corresponding point of the globe with a pencil, or by attaching a small piece of moist jiaper, and this must be considered as the place of the planet. If a globe cannot be obtained, the time of passing the meridian, and the meridian altitude of the object, may be calculated ; and by observing at that time, the object may be easily discovered. The distances marked in the Nautical Almanac afford also to the observer an easy method of knowing the star or planet from which the moon's dis- tance is to be observed ; for he has nothing to do but to set the sextant or circle to the distance comi)uted roughly for the apparent time, estimated nearly for the meridian of Greenwich, and direct his sight to the east or west of the moon, accord- ing as the object is marked E. or W. in the Nautical Almanac ; and, having found the reflected image of the moon upon the horizon glass, sweep the instrument to the right or left, and the image will pass over the sought star or planet, if above the horizon, and the weather clear: the star or planet is always one of the brightest, and is situated nearly in the ai-c passing through the moon's centre, perpendicular to the line connecting the two horns. The computed distance made use of in sweeping for the star, may be found in this manner: — Reckon the apparent time at the ship in the manner of astronomers, (by counting 24 hours from noon to noon, and taking the day one less than the sea account;) to this time apply the longitude turned into time, by adding in west, or subtracting in east longitude; the sum or difference will be the apparent time at Greenwieii nearly. Take tlie distances from the Nautical Almanac for the time immediately {)receding and following this estimated time, and note the diffcTcnce of these distances; then say. As 3*", or 180"", is to the dilTerence of the distances, so is the difference between the a])parent time at Greenwich and the next preceding time, set down in the Nautical Almanac, to a pro]iortional part to be added to the next preceding distance taken from the Nautical Almanac, if the distance be increasing, but subtracted if decr(;asing ; the sum or difference will be the distance at which the quadrant or sextant is to be fixed. In sweeping for the stars by this method, it will often happen that two or more are swept ujion at once ; this might cause some difficulty to an inexperienced observer, who would be at a loss to know which to jnake use of. To remove this, the follow- ing description of these stars is added: — LUNAR OBSERVATIONS. 227 « ARIETIS. ■VV This star bears about vvest, distant 22P, from the Pleiades, or the Seven Stars ; it is of the second magnitude, and may be known by means of the star n, of the third magnitude, situated S. W. from a Arietis, at the distance of 3^ degrees. South from the star n, at the distance of 1^°, is the star r, of the fourth magnitude. The northernmost of these stars is a Arietis. ALDEBARAN. About 35° E. S. E. from « Arietis, and 14° S. E. from the Pleiades, or Seven Stars, is tiie bright star Aldebaran. Near this star, to the westward, are six or seven stars of tlie third or fourth magnitude, forming, willi Aldebaran, a figure resembling tlie let- ter V, as is represented in the adjoined figure, where Aldebaran is marked a. At the distance of 23° from this star, in a S. E. direction, are three \ery bright stars, situated in a straight line, near to each other, forming the belt of Orion. POLLUX. At the distance of 45^ from Aldebaran, in the direction of E. N. E., is the star Pollux, whicli,is a bright star, though not of the first magnitude. N. W. from it, distant 5°, is the star Castor, of nearly the same magnitude ; and you will almost always sweep both at once : the southernmost is the one used. REGULUS 4. "^ % ^ EfigiilxiS. ^ SPICA. E. by S. \ S. from Pollux, at the distance of 37i°, is the star Regulus, of the first magnitude ; to the northward of this star (at the distance of 8°) is a star of the second magnitude ; near to these are five stars of the third magnitude, the whole forming a cluster resembling a sickle, represented in the adjoined figure, Regulus being in the extremity of the handle. A line drawn from the northern polar star, through its pointers, passes about 12° to the eastward of Regulus. E. S. E. from Regulus, at the distance of 54°, is the star Spica, of the first magnitude, with no very bright star near it; S. W. from this star, at the distance of about 16°, are five stars of tlie third or fourth magnitude, situated as in the adjoined figure ; the two northernmost of these stars, ?;, v, form a straight line with Spica, and by this mark it may be easily discovered. A line drawn from the northern polar star, through the middle star of the tail of the Great Bear, will pass near to Spica. ANTARES. % a AQUILiE. ^ E. S. E. from Spica, at the distance of 4G°, is the star Afitares, in 2G° of south declination ; it is a remarkable star, of a reddish color ; on each side of it, to the W. N. W. and S. S. E., about 2° distant, is a star of the third or fourth magnitude, no very bright star bein? near. N. E. from Antares, at the distance of G0°, is the very bright star a AquiI(B ; N. N. W. from which, at 2° distance, is a star of the third magnitude, and, S. S. E., at 3° distance, another star of a less magnitude. These three stars appear nearly in a straight line. The star a Aquilse is nearly of the same color as Antares. FOMALHAUT. a PEGASI. M'r •■X- S. E. from a Aquilae, at the distance of 60°, is the star Fomalhaut, which is a bright star of high soutliern declination its altitude in northern latitudes being small, never exceeding 4U° m the lati- tude of 40-^ N. This star bears nearly south from the star a Peg- asi, distant 45°. A line drawn from the pointers, through the northern polar star, and continued to the opposite meridian, will pass very near to a Pegasi and Fomalhaut. E. by N. from u AquiltB, at the distance of 43°, and westward from a Arielis, at the distance of 44°, is the star a Pegasi, which inay be known by means of four stars of different magnitudes, situated as in the adjoined figure ; in which a represents a Pegasi, (i a star of the second magnitude, bearing north of it, distant 13° . the others are of less magnitudes, and two of them, ?;, u, form a straight line with the star a Pegasi ; and by this mark it may be easily discovered 228 LUNAR OBSERVATIONS. General Remarks on the taking of a Lunar Observation. The accuracy of a lunar obsei*vation depends chiefly on the reguhntion of the cljroiiometer, and on the exact measurement of the angular distance of tiie moon from the sun or star ; a small error in the observed altitudes of those objects, will not in general much affect the result of the calculation. The best method of regulating a clironometei: at sea, is by taking an altitude of the sun when rising or falling quickly, or when bearing nearly east or west, the alti- tude being sufficiently great to avoid the irregular refraction near the horizon, and noting the time by the chronometer. With this altitude, the latitude of the ])lace, and the sun's declination, find the mean time of observation by either of the I)receding methods ; the difference between this time and that shown by the chro- nometer will show how much it is too fast or slow. A single observation, taken with care, will generally be exact enough; but if greater accuracy is required, the :iiean of a number of observations may be taken. If the distance of the sun and moon be observed when the sun is three or four points distant from the mei-idian, the mean time of observation may be deduced from the altitude of the sun taken at the precise time of measuring the distance; this will render the use of a chronom- eter unnecessary, and will prevent any irregularity * in its going from affecting the result of the observation. If a night observation is to be taken, the chronometer should be regulated by an altitude of the sun taken the preceding evening, and its going examined by means of another observation taken the next morning ; for the time found by an altitude of a star cannot be so well depended upon, except in the morning and evening twilight, as the horizon is generally ill-defined ; but the altitude may be sufficiently exact for finding the correction used in determining the angular distance. Although all the instruments used in these observations ought to be well adjusted, 3'et particular care should be taken of the sextant or circle used in measuring the angular distance of the moon from the sun or star, since an error of 1' in this distance will cause an error of nearly 30' in the longitude deduced therefrom. When a great angular distance is to be measured, it is absolutely necessary to use a telescojie, and the ])arallelism of it, with respect to the plane of the instrument, must be carefully examined ; but in measuring small distances, the use of the telescope is not of such great importance, and a sight-tube may then be used, taking care, however, that the eye and point of contact of the objects on the horizon-glass be equally distant from the plane of the instrument. But it ought to be observed, that it is always conducive to accuracy to use a telescope, and, after a little practice, it is easily done. Whilst one person is observing the distance of the objects, two others ought to be observing the altitudes. The chronometer should be placed near one of the o!)servers, or put into the hands of a fourth person appointed to note die time ; the observer who takes the angular distance giving previous notice to the others to be ready with their altitudes by the time he lias finished his observation ; which being done, the time, altitudes, and distance,f should be carefully noted, and other sets of observations taken, which must be done within the space of 15 minutes, and the mean of all these observations must be taken and worked as a single one. When a ship is close-hauled to the wind, with a large sea, or when sailing before the wind, and rolling considerably, it is difficult to measure the distance of the objects ; but when the wind is enough upon the quarter to keep tlie ship steady, there is no difficulty, especially in small distances, which are much more easily measured than large ones, and are not so liable to error from an ill adjustment of the telesco])e : an observer would therefore do well to choose those times lor observation when the distance of the objects is less than 70" or 80°. An observation of the sun and moon is generally m^. 'i easier to take when the altitude of the moon is less than that of tlie sun, because the instrument will be held in a more natural and easy manner When the moon is near the zenith, the observation is generally difficult to take, and liable to be erroneous, because the observer is forced to place himself in a disagreea- ble posture. For the same reason, an observation of the moon and a star or planet * It is not unromnion to find a clifTerence in l!ic regulation of a chronnmelcr in the forenoon and afternoon; tliis dilferonce generally arises from the uncertainty in the estimated latitude, or some sJiglit error in the observation, and perliaps partly from the irregularity in the going of the chroiiometer. t If the distances are measured liy a circular ins.trumcnt, it will not be necessary to note the several distances measured, but only the times and altitudes, as the sum of all the distances measured by the circle will be given b)' the instrument at the end of the observations ; and if the aliiiudes of the objects are also measured by circular instruments, it will not be necessary to note the several altitudes, but only the times of observation. LUNAR OBSERVATIONS. 229 is generally much easier to take when the star or planet is lower than the moon. This situation of the objects may in most cases lie obtained by taking the observation at a |)ro|)er time of the day. But it nnist be observed, that neither of the objects, if possil)le, ought to be at a less altitude than 10^, upon account of the uncertainty of the refraction near the horizon ; fir the horizontal refraction varies from 133' to 3G' 40" only by an alteration of 40° in the tliermomcter. This alteration might cause an error of two degrees in the longitude, with an observer who uses the mean refraction. In measming the distance of the moon from the sun, we must bring the moon's round limb in contact with the nearest limb of the sun. In measuring the distance of the moon from a planet or fixed star, her round limb must be brought in contact with the centre of the star or planet; observing that, the scmidiameter of the planet being oidy a lew seconds, the centre of it can be estimated sufficiently near for all the purposes of this observation.* In taking the altitude of the moon, the round limb, whether it be the ujjpcr or lower, must be brought to the iiorizon. In damp weather, it is rather dirficult to observe the altitude cf the stars, on account of their dimness, particularly a Pegasi and u Arietis. Sometimes they are so dim that they cannot be seen through the holes of the sight-vane of a quadrant, particularly if the mirrors are not well silvered ; in this case, the vane must be turned aside, and the eye held in nearly the same place, or the altitude must be taken by a sextant furnished with a sight-tube. We have here sujiposed that there were obsei-vers enough to measure the altitudes when the distance was observed ; but if that is not the case, the altitudes may be estimated by either of the methods which will be hereafter given. Preparations necessary for working a Lunar Observation. Find the mean time of observation b}'^ astronomical account, reckoning tlie hours from noon to noon in numerical succession from 1 to 24, and taking the day one less than the sea account ; to this time apply the longitude turned into time by Table XXI.f by adding if in west longitude, but subtracting if in east; the sum or difference J will be the supposed time at Greenwich, or reduced time. In ])age III. of the month of the Nautical Almanac, find the moon's scmidiameter and horizontal parallax, for the nearest noon and midnight before and after the reduced time, and find the difference of the parallaxes and the difference of the semi- diameters ; then enter Table XI. with these differences respectively in the side column, and the reduced time at the top; opposite the former, and under the latter, will stand the corrections § to be ap])lied respectively to the semidiameter and hori- zontal parallax ;iiarked first in the Nautical Alman-'c, additive if increasing, subtractive if decreasing; the sum or difference will be the horizontal semidiameter and the horizontal parallax, res])ectively, at the time of observation. To this horizontal semi- diameter must be added the augmentation from Table XV. corresponding to tlie moon's altitude; the sum will be the true scmidiameter of the moon. The sun's true semidiameter is to be found in pagell. of the month of the Nautical Almanac. To the observed altitude of the sun's or moon's lower limb add 12' ; but if the up])er limbs were observed, subtract 20', ami fi-om the observed altitude of the star or planet subtract 4', and you will have nearly the apparent altitudes of those objects respec- tively.ll * If 3113' °ne wishes to proceed witli perfect accurac}', he may bring' the round lim!) of liie moon to the nearest limb of the planet, and l!ien apply the planet's semidiameter, taken from the Nautical Alma- nac, in ilie same manner as in observations of the sun. t Or by multiplying by 4 se.icageslmall}', in the manner directed in the note page 170. t \\nien the sum exceeds 24 hours, you must subtract 24 hours, and add one to the day of the month ; and when the time to be subtracted is greater than tlie mean time, the latter must be increased by 24 iiours, and one day taken from the day of the month, conformably to the usual rules of addition and subtraction. If the chronometer used in taking the observation be regulated to Greenwich time, this part of the calculation will be unnecessary, because the reduced time at Greenwich will be given direcll3' by the chronometer. § These corrections may be found easily without the table, by saying, As 12 hours are to the reduced time, (rejecting 12 hours when it exceeds 12.) so is the difference of semidiameter or parallax for 12 hours to the corresponding correction. If the reduced time cannot be found accurately in the table, you must use the nearest numbers, which will, in general, be sufficiently accurate. II These altitudes are supposed to be taken at sea by a fore observation ; and the application cf the above numbers will give the apparent altitudes corresponding to observations taken on the deck of a common-sized vessel (where the dip is about 4' or5') to a sufficient degree of accuracy ; if the observer was 'lO or 50 feet above the water, 1' or 2' might be taken from these altitudes. The propriety of using these numbers will appear by considering that every wave, by raising the ship above the level of the sea, will alter the dip, and that an error of 1' or 2' in the altitudes will in general cause but a 230 LUNAR OBSERVATIONS To the observed distance of the moon from a star or planet add the moon's true Bemidiameter, if her nearest limb was obsei-ved, but subtract that semidiameter if her farthest limb was observed ; the sum or difterence will be the apparent distance. But to the observed distance of the sun and moon^s nearest limbs, add their true semidiame- ters ; (he sum will be the apparent distance. These preparations are necessary in every method of woi-king a lunar observation The most noted methods are those of Dunthorne, Borda, Maskelyne, Rios, Witchell, L}'ons, &c., and improvements thereon by various authors. Dunthorne's and similar methods have one great advantage in not being liable to a variety of cases; but tliese methods are tedious, when tables of logarithms to min- utes only are used, by reason of tlie great exactness required in proportioning the iogaj-jthms to seconds. This is obviated in the excellent methods published by Rios and Stansbury ; but they require large and expensive tables, and on that account are not in very general use. Witchell's and Lyons's methods do not labor under the; inconvenience of requiring large tables, nor do they require any particular notice of the seconds in finding the log. sines and log. tangents ; but these methods, as they were originally published, are embarrassed with a variety of cases ; sometimes tlie corrections are additive, sometimes subtractive ; and learners find a difficulty in rightly applying tliem. To remedy this, a method was published in the first edition of this work, in which two corrections were constantly additive, two subtractive, and one small correction was additive when the distance was less than 90°, but subtractive when above 90°. This method was further improved in the Appendix to that edition, liy means of four new tables, whicii are inserted in this edition, and numbered XVII. XV^III. XIX. and XX., by means of which the work is considerably shortened, and ail tlie corrections rendered additive. This method will now be given, after making a few lemarks on the manner of taking the corrections and logarithms from these new tables. Table XVII. contains a correction and logarithm to be used when the moon's dis- tance from a star or planet is observed ; and Table XVIII. is a similar one, to be used when the moon's distance from the sun is observed. Table XVII. contains six pages, corresponding to the horizontal parallax of the planet, supposing it to be either 0'', 5", 10", 15", 20", 25", or 30", as at the top of the pages respectively ; and tha page is to be used which agrees the nearest with the horizontal parallax of the |)lane at tlie time of observation.* These tables are so extended, that no proportional parts are necessary in taking out the corrections and logarithms, except tiie altitude of tlie sun or star be less than 7° 30', and at such altitudes an observation is liable to error on account of the uncertainty of the refraction ; so that, in using these tables, it is suffi- ciently accurate to find the number nearest to the given altitude of the sun or star, and make use of the corresponding correction and logarithm. Thus, if the star's altitude be 12° 25', the nearest number in Table XVII. is 12° 24', corresponding to which are the correction 55' 45", and the logarithm 1.31G1. Taljle XIX. contains the corrections and logarithms corresponding to the moon's horizontal parallax and altitude, both being found at the same opening of the book. Tlie corrections for seconds of parallax and minutes of altitude are easily taken out by means of Tables A, B, C, placed in the margin. The method of finding these corrections is given at the bottom of the table : they are always additive. Besides the two logarithms taken from Table XVII. (or XVIII.) and XIX., this new rule requires only four logarithms to be taken from Table XXVII. to four jilaccs of figures, and to the nearest minute, it being in general unnecessary to proportion fiir the seconds. We shall now give the rule for correcting the distance, and shall, for brevity, use the words sine, secant, and cosecant, instead of Zog-.^'ne, log. secant, and log. cosecant, respectively, and the same ]iractice will be observed in the second, third, and Iburtii iiiLthods of" correcting the distance. small error in the result of the calculation of a lunar observation, so that for all practical purposes the above numbers may be esteemed as sufficiently exact. It may also be observed, that the error arising from this source will not generally be greater than that arising from neglecting the equations depending on the spheroidal form of tlie earth, and on the density and temperature of the air} equations which are almost alwa^'s neglected. If any one wishes to olitain the apparent altitudes strictly, he must, from the observed altitudes, subtract the dip of the horizon taken from Table XIII., anu add or subtract the semidiameter of the object, according as the lower or upper limb is observed. * In strictness, when the horizontal parallax diirors from those in the table, we ouHit to take the numbers for the next greater and the next less number, and take a proportional part of llie dillcrences but tills degree of accuracy is wholly unnecessary In nautical observations. LUNAR OBSERVATIONS. 231 FIRST METHOD Of correcting the apparent distance of the moon from the sun* in which there is no variety of cases, all the corrections being additive. Add the apparent distance of tlie moon from the sun to their a])parent altitudes, and note tlie half-sum. Tlie difference between tJie half-sum and the a])parent dis- tance call the first remainder; and the difference between the half-sum and the sun's apparent altitude call the second remainder. Take from Table XXVII. the following logarithms, which mark beneath each other in two columns, viz. the sine of the apparent distance, to be marked in both columns, the cosecant of the second remainder, to be marked also in both cohnnns, the secant of the first remainder to be placed in tlie first column, and the secant of the half-sum in the second column.f Enter Table XVIII. (or Table XVII. if a star or planet be used), and take out the correction corresj)onding to the sun's altitude (or star or planet's); take also from the same table the corresponding logarithm, which place in column 1st. Enter Table XIX. with the moon's ai)parent altitude and horizontal jiarallax ; find the corresjionding correction, which j)lace under the former correction, and the logarithm, which place in column 2d. The sum of the four logarithms f of column first will be the proportional logarithm of the first correction, and the sum of the logarithms of column second f will be the proportional logarithm of the second correction; these corrections being found in Table XXII. are to be ])laced under the former corrections. Enter Table XX., and find tlie numbers which most nearly agree with the observed distance and the observed altitudes of the objects, and take out the corresponding correction in seconds, which is to be placed under those already found. Then, by adding all these corrections to the apparent distance, decreased by 2^, we shall get the true distance nearly .| To determine the longitude from the true distance. if the true distance of the objects can be found in the Nautical Almanac, in either of the j)ages where the distances are marked, on the day of the observation, the time Vv'ill !)e found at the top of the page. If the tiaie distance cannot be found exactly, in the Nautical xA.lmanac, you must find the two which are nearest to it, the one greater and the other less than the true distance ; and take out that one which- corresponds with tlie earliest or first of these times, with the corresponding proportional logarithm. Find the difference between this first distance and the true distance, and take out its proportional logarithm from Table XXII. The difference between these two pro- portional logarithms will be the proportional logarithm of a jiortion of time, to be added to the time standing over the first distance in the Nautical Almanac, and the sum will be the mean time of the observation at Greenwich. The difference between this time and the mean time at the ship, being turned into degrees and minutes by Talile XXI., will be the true longitude of the ship from Greenwich, at the time of observation. This longitude will be east if the time at the ship be greater than that at Greenwich, otherwise west.§ To exemplify the preceding rules, we shall now give several examples of correcting the apjtarent distance, including also the preparation and the determination of tlie longitude from the true distance. * 'I'lils rule is the same as tlial for corrcctiiin; the distance of the moon from a star or a planet, except in reading star or planet for sun, and usin^ Table XVII. instead of Table XVIII. t Rejecting' always the tens in the indices. i The distance obtained by this rule is not perfectly correct, since several small corrections must be applied to obtain the true distance to the nearest second, viz. (1) The refraction taken from Talile XII. which is made use of in constructing Tables XVII. XVIII. and XIX., ought to be corrected for the dilTercnt heights of the barometer and thermometer, as directed in page 154. (2) A correction must be applied for the spheroidal figure of the earth. And (3) a very small correction ought to be made in the numbers of Table XX. when the D's horizontal paralla,\ varies from 57' 30". But to notice all these corrections would increase the calculation very much, and the result of a single observation, in which all these things were noticed, would probably not be so accurate as the mean of two or three observations, taken at different times of the day, in which these corrections were neglected ; and the time necessary to take and work the latter observations would not be much greater than to work a single observation, in which all the corrections were noticed. § It may be necessary to observe that, if the times at the ship and Greenwich fall on different days, the latest day is to be reckoned the greatest, though the hour of the day may be the least ; thus, ITiii lav 1 hour is to be esteemed greater than IGth day "22 hours. 232 LUNAR OBSERVATIONS. EXAMPLE I. Suppose that, on tlie 7th of January, 183G, sea account, at 11" 57' past midnight, mean time, in the longitude of 127° 30' E., by account, the observed distance of the farthest iinil) of the moon from the star Aidebaran, was G8° 36' 00", the observed altitude of tlie 8tar 32° 14', and tlie observed altitude of the moon's lower limb 3-1° 43' Required the true longitude. Preparation. Sea account, Jan. 7, is by N. A. Jan. 6^ ISh li >» 67» Longitude 127° 30' E 8 30 00 Reduced time Jan. G'i 3^ ilm 67» 3 scmidiam. Jan. G, noon 15' 05'' Jhor. par. Jan. G, noon. . 55' £0" tH^ observed alt.. . . 32° 14 midniv N. A. at 3i> 67 41 43 DiiTercnco 21 17 Oh 4lr Add 3 Prop, log 2872 Prop, log 9272 14' Prop. log. diff... 6400 Mean time at Greenwich 3 41 14 Mean time at the sliip 12 11 57 Difference is longitude in time 8 30 43 == 127° 40' 45" E. from Greenwich. * This corr. = Corr. Tab. XIX. 15' 05" + Corr. Tab. A. 29"+ Corr. Tab. C. 3" : t Tliis log. = Log. Tal). XIX. 2231 + Log. Tab. C. 7 =2238. 15' 37 LUNAR OBSERVATlOlfS. 233 EXAMPLE n. Suppose, 1836, April 2^ 2^ 03™ 50' A. M., mean time, sea accoimt, in the longitude of 172^ E., by account, the observed distance of the moon's farthest limb irorn Antares, Avas 01° 04' 00", the observed altitude of the star G8° 29', the observed alti- tude of tlie moon's lower limb 45° 23'. Re(|uired the true longitude. Preparation. Sea account, April 2, or by N. A., April I'l l-J-i' OSmSO" Longitude 172° E 11 28 00 Reduced time April 1 J 021' 35m 50^ ]) semidiam. April 1, noon 13' 59" midnight 16 4 DitTercncc 5 TableXI 1^ Sum IG 00 Aug. Table XV H D semidiameter IG' 11'' D horizontal par. noon 58' 38" niidiilHit 58 -SG DlfTcrcnce TableXI D horizontal parallax 58' 42" 18 4 * observed a\\ G8° 29' Subtract 4 ^ apparent alt G8° 23' D obs. alt. L. L. .. 43° 23' Add 12 y> apparent alt 45° 35 Observed distance * K F. L 61° 04' 00" Subtract ]> semidiameter IG 11 . Apparent distance * D C0° 47' 49" App. dist. 60° 4B' J(f app.alt. C8 25 ]) app. alt. 45 35 Sum.... 174 48 Half-sum 87 24 1st Rem.. 26 36 Sd Rem. . 18 59 To Jind the true distance. Col. 1. Sine 9.9410 2dRein. lS°59'.Cosec. 0.4877 1st Rem .26 36 ...?ec. 0.0186 Table XVII Loe. 1.9438 1st Corr 0' 41". P. L. 2.4-211 Col. 2. Same 9.9410 Same Half-Sinn 87" 24'. Sec. Talile XIX. t Log. 2d Corr. 1'56"..P.L. App. dist. less 2" = 58° 47< 49' Talile XVII 59 37 Table XIX.* 19 32 1st Corr 41 2d Corr 1 56 Table XX 19 True distance 60° 09' 6A! To Jind the true longitude. True distance G0° 09' 54" Distance by N. A. at Oh. DifTerence Gl 40 13 Prop.Iog. 1 30 19 Prop, log 23^18 2995 2h 35m 0G» Prop. log. diff . 0G47 Add no 00 Mean time at Greenwich 2 35 06 Mean time at the ship 14 03 50 Difference is longitude in time 11 23 41-=172° 11' E. from Greenwich. * This corr. = Corr. Tab. XIX. 19' 17" + Corr. Tab. A. 12" + Corr. Tab. B. 3"= 19' b'2". t This log.= Log. Tab. XIX. 1915+ Log. Tab. C. 9= 1954 30 2.34 LUNAR OBSERVATIONS. EXAMPLE III. SupjKjse that, on the 30th of Oct. 1836, sea account, in the forenoon, in the longitude of 80° W., by account, the following observations of the sun and moon were taken; the times being noted by a chronometer which was 3™ 47' too slow for mean time at the place of observation. Required the true longitude. Preparation. Time per IVatch. Observed Distance © d N. L. Observed Altitude QL.L. Observed Altitude 5 L. L. H. M. S. 9 38 01 9 39 04 9 40 06 9 41 00 9 41 49 5 ) 200 00 9 40 00 Error + 3 47 Mean ^,,3,, O 1 II 111 35 49 35 13 34 47 34 20 33 66 174 10 111 34 50 D S. D. 14 63 © S. D. 16 09 App.dist. 112 05 52 O 1 24 47 24 51 24 55 24 59 25 03 124 35 24 55 Add 12 1 26 17 21 25 29 33 125 26 25 Add 12 ©App. alt.25 07 D App. alt. 26 37 Sea account, 30lh October, or N. A., October 29'! 21i> 43n'47» or 9>'43m47' A.M. Longitude 80° \V 5 20 00 Reduced time Ootober 30^ 3'' 03"'47» D semidiameter, Oct. 30, noon.... 14' 46" midnight 14 46 Difference TableXI Sum I'l 46 Aug.TableXV 7 J) semidiameter 14' 53" ]) horizon, par. Oct. 30, noon. ... 54' 10" midnignt 54 10 Difference TableXI ]) horizontal parallajc 54' 10 App.dist. 112° 06' ©app. alt. 25 07 D app. alt. 26 3 7 Sum 1G.3 50 Half-sum. 81 55 1st Rem.. 30 11 SiIRem... .'J6 48 To fond the true distance. Col. 1. Sine 9.9G69 2dRem.56°48'.Co3ec. 0.0774 lstRem.30 11... Sec. 0.0633 Table XVllI....Log. 1.6336 lstCorr.3'16"..P.L Col. 2. Same 9.9669 Same 0.0774 Half-sum 81° 55'. Sec. 0.8520 Table XIX. t Log. 0.2376 1.7412 2d Corr. 13' 14". P.L. 1.1339 To find the true longitude True distance 111°33' 53" Distance by N. A. at Oh 112 54 10 Difference App. dist. less 2° = 110° 05' 59" Table XVlll.... Table XIX.* 1st Corr 2d Corr Table XX True distance ... 58 06 13 10 3 16 13 14 13 Prop, iog 3458 1 20 17 Prop, log 3506 Add 2h 58m 01» Prop. log. diff. 004-8 Mean time at Greenwich.... Oct. 30'' 02h 58'T'01» Mean time at the ship Oct. 29 21 43 47 Difference is longitude in time 5 14 14 = 78° 33' 30'' W. from Greenwich. * This corr. = Corr. Tab. XIX. 12' 23" + Corr. Tab. A. 44"+ Corr. Tab. B. 8 ''=13' 10". t This log. = Log. Tab. XIX. 2364 + Log. Tab. C. 12 = 2376. LUNAR OBSERVATIONS. •23,- EXAMPLE IV. Suppose that, on the 12th of May, 183G, sea account, at about l** P. M., in the latitude of 30° S., and in tlie longitude of 4° 00' E., by account, the following obser- vations of the sun and moon were taken ; the sun being so situated that the apparent time could be observed by her altitude. Reciuired the true longitude. Preparation. Observed Distance m C N.L. Obsen-ed Altitude m L.L. Observed Altitude D U.L. o / // 46 07 09 06 01 04 56 3 ) 18 06 Mean 46 06 02 o / 39 59 45 31 135 39 45 — 2 39 43 Add 12 © app. alt. 39 55 O 1 28 32 28 07 27 42 84 21 28 07 + 01 28 08 Subtract. . 20 D app. alt. 27 48 Index errors .... — 03 Corrected dist... 46 05 59 © semidiameter. 15 51 D semidiameter. 15 25 Apparent dist... 46 37 15 Sea account, May 12, or N. A., May lid ih Qm 00» Longitude 4° 00' E 16 00 Reduced time May ll^ Oi> 44m OO' D semidiameter. May 11, noon ... 15' 17' midnight 15 13 Difference Table XI. 15 17 Aug. Table XV 8_ D semidiameter 15' 25" D horizontal p-irallax, noon 56' 04" midniirht ... 55 49 Difference. Table XI.. I) horizontal parallax 56' 03" App. dist. 46° 37' ©app. alt. 39 .55 Dapp. alt. 27 48 Sum 114 20 Half sum 57 10 1st Rem. . 10 33 2(1 Rem... 17 15 Tojlnd the true distance. Col. 1. Sine 9.8G14 2cl Rem. 17° lo'.Cosec. 0.5279 1st Rem. 10 33... Sec. 0.0074 Table XVni....Log. 1.8307 1st Corr. 1' 4 P. L. 2.2274 Col. 2. Same 9.&314 Same 0..5279 Half-sum 57° 10'. Sec. 0.2C58 Talile XIX.t....Log. 0.2215 2(1 Corr. 23' 5,= P. L. 0.87n(i App. dist. less 2° = = 44 '37' 15' TaMe XVIII.... 58 59 Table XIX.* U 51 1 04 2(1 Corr 23 55 Table XX 34 True distance 4G° 13' 41" correct altitude 39° 54' Latitude of ship.. 30 00 Polar distance... 1 07 58 Sum 177 52 Tojind the mean time and the true longitude. True distance 46° 13' 41" By N. A. at Oh .... 46 34 02 . Difference Secant.. 10.06247 Cosecant 10.02171 Half-sum 88 56 . . Cosine Half-sum alt. 49 02 . . Sine.. Sum.. Apparent time. . l^ 0"> 3» . .. Sine.. Eq. of time . sub. 3 54 Mean time .... 56 09 8.2G9S8 9.87800 18.23206 9.11603 20 21 . Difference 0°41'32". Add Prop. log. 3097 Prop. log. 9467 Prop. log. 6,370 Mean time at Green. 41 32 Mean time at ship. . 56 09 Longitude in time.. 14 37= :3°39'15"E.froro Greenwich. * This corr. = Corr. Tab. XIX. ll'02" + Corr. Tab. A.49" + rorr. Tab. B. 3"=n'.'>4". ' This log = Log. Tab. XIX. 2207 + Log. Tab. C. 8 == 221 5. 236 LUNAR OBSERVATIONS. EXAMPLE V. Supj)Ose that, on the 13th of Februaiy, 1830, sea account, at 8'' 36™ 00', mean time, A. M., in the longitude of 1G° W. from Greenwich, by account, six distances of tlie Sim and moon's nearest Ihnbs were observed, by a circle of reflection, to bo 273" 09' 06", the corresponding times and altitudes being as in the following table. Kenuired the true longitude. Preparation. Mean Time per Watch, A. M. H. M. s. 8 33 24 W3f. 35 18 36 36 37 04 39 02 Sums 6)36 00 Mean time 8 36 00 Observed Distance Q d N.L. Sum of the dis- tances taken from the circle at the end of the obser- vations. 273° 9' 06' 45 31 31 ©S.D. 16 13 5 S. D. 16 29 App.dist. 46 ai 13 Observed Altitude ®L.L. Add. 27 42 27 M is 02 28 12 28 21 28 44 55 28 09 12 app. alt. 28 21 ObseT^-ed Altitude D U. L. 42 24 42 42 42 51 43 01 43 11 43 21 17 30 42 55 Subtract. . 20 ]) app. alt. 42 35 February 13, sea account, or by N. A., February 12' 40™ 00> ]) semidiamcter, Feb. 12, midnight 16' 17' Feb. 13, noon... 16 18 Diflerence 1 TableXl 1 16 13 II Aug. Table XV > semidiametcr 16^29" J) luir. parallax, Feb. 12^ midnight 59' 46" Feb.l3,nooc.. . 59 47 Difference 1 TableXl 1^ D horizontal parallax 59' 47" To f.nd the true distance. \pp. dist. 46° 04' ©app. alt. 28 21 D app. alt. 42 V.5 Sum 117 00 Half-sum 58 30 1st Rem. 12 26 2d Kern. 30 09 Col. 1. Sine 9.8574 2d Kein. 30° 09' Cosec. 0.2991 1st Rem. 12 23 ..Sec. 0.0103 Table XVIII. ..Log. 1.G874 1st Corr. 2* 31". P. L. 1.8542 Col. 2. 9.8574 0.2991 0.2819 0.1878 0.G2G2 App. dist. less 2° Table XVIII.... Table XIX.* li;t Corr = 44 04' ly Same 58 22 Half-sum 58° 30' Sec Table XIX. t ...Log 2d Corr. 42' 34". P. L. 16 41 2 31 0(1 Corr 42 34 Table XX 34 True distance... .. 4(3 04-65' To find the true longitude. True distance 46°04' 55" Distance by N. A., Feb. 12^ 21^ . . . . 4^) 28 04 Prop, log 2551 Difference 23 09 Prop.log 8907 0h41'n39' Prop. log, diff. 6356 Add 21 Mean time at Greenwich 21 41 39 Mean time at the ship 20 36 00 Difference is lon-itude in time 1 05 39 = 16° 24' 45" W. from Greenwich. This corr. := Corr. Tab. XIX. 16' 29" -f Corr. Tab. A. 9"-t-Corr. Tab. B. 3"= 16' 41". This log. = Log. Tab. XIX. 1875+ Log. Tab. C. 3= 1878. LUNAR OBSERVATIONS. 2:17 EXAMPLE VL Suppose that, on the 21st of June, 183G, sea account, at C' 50™ 40' P. J>L, mean time, in the longitude of Gl° W., by account, tlie observed distance of the nearest limb of the moon from the centre of tlie planet Venus, was 35° 59' 57", the observed altitude of the planet 23° 00', and the observed altitude of the moon's lower limb 37° 31' Required the true longitude. Preparation. Sea account, June 21st, is by N. A. June SOJ Gi" 50™ 40' Long-itude 61° W. in lime 4 04 00 Reduced time June 20J lOi" 51"i AQ^ J> semidiam. June 20, noon 15' 10" midniffht 13 15 Difference 5 TaJbleXI 5 Sum 15 15 Au?. TableXV 10 D semidiaineter 13' 25" D hor.par. June 20, noon 55' 38" midniglit 55 59 Difference 21 Table XI 19 D horizontal parallax 55' 57" 5 observed alt. Subtract 23° 00' 4 $ apparent alt 22° 56' D obs. a!t. L. L... 37° 31 Add 12 » apparent alt 37° 43' Observed distance 5 ? N. L 35° 59' 57' I) scmidiameter add 15 25 Apparent distance 5 9 36° 15' 22" To find the true distance. App. tli?t. 36° to' ? app. alt 22 56 5 app. alt 37 43 Sura 9G 54 Half-sum 48 27 1st Rem.. 12 12 od Rem. . 25 31 Col. 1. Sine 9.7718 2dRem.25''31'.Cosec. 0.3658 1st Rem. 12 12... Sec. 0.0099 Table XVII. .. Log.* 1.6348 lstCorr.2'53"..r. L. 1.78^3 Col. 2. Same 9.7718 Same 0.3658 Ilalf-sum 48° 27'. Sec. 0.178:3 Table XIX Log. 0.2185 2d Corr. 52' 35" . P L. 0..53 14 App. dist. less 2° = 34° W 22* Table XVII.* 58 05 Table XIX 16 40 1st Corr 2 58 2d Corr 52 35 Table XX 41 True distance 36° 26' 2J" To find the true longitude. True distance 36°26'21" Distance by N. A. at 911 35 28 17 Prop. log 2985 Prop, log 4913 Prop. log. diff. 1928 Difference 58 04 Ih 53™ 28s Add 9 Mean time at Greenwich 10 55 28 Mean time at the ship 6 50 40 Difference is longitude in time 4 04 48 = 61° 12' W. from Greenwich. * The horizontal parallax of Venus being 20" by the Nautical Almanac, we must, in finding from Table XVIl. the correction and logarithm, use that table which is marked at the top, " Parallax 20," being the 93d page. 238 LUNAR OBSERVATIONS. EXAMPLE Vn. Suppose that, on the 27th of August, 1836, sea account, at 0'' 50"" 08' A. M., mean time, in the longitude 25° W., by account, tlie observed distance of the farthest limb of the moon from the centre of the planet Mars, was 114° 05' 17", tlie observed altitude of the planet 10° 30', and the obsei-ved altitude of the moon's upper limb 22° 51' Requii'ed the true longitude Preparation. Sea account, August 27. is by N. A. August 20^ 12h SOLOS' Longitude 25° W. in time 1 40 00 Reduced time August 26(1 14*> 30" 08» D semidiam. Aug. 26,mid. 16' 10" Aug. 27, noon 16 5 Difference 5 TableXI 1 IG Aug. Table XV. D semidiamcler IG' 15" ]) her. par. Aug. 26, mid. 59' 20" Aug. 27, noon 59 00 Difference 20 TableXI 4^ D horizontal paral!a,x. . 59' 16" (f observed alt 10° 30- Subtract. 4 cf apparent alt. ... 10° 26' D observed alt. U.L. 22° 51' Subtract 20 D apparent ah 22° 31 App. dist. 113M9I (f app. alt. 10 2G D app. alt. 22 31 Sum 14G 46 Half-sum. 73 23 1st Rem. . 40 2fi SdRem... 62 57 Observed distance J 5 F. L 114° 05' 17" 5 scmidiameler subtract 16 15 Apparent distance d" ]) 113° 49' 02" To find the true distance. Col. 1, Sine 9.9614 2aRem.62°57'.Cosec. 0.0503 lstReni.40 26... Sec. 0.1185 Table XVII.... Log.* 1.2525 lstCorr.7'27"..P. L. 1.3827 Col. 2, Same 9.9614 Same 0.0503 Half-sum 73° 23'. Sec. 0.5437 Table XIX Log. 0.1998 2d Corr. 31' 38". P.L. 0.7552 To find the true longitude. App. dist. less 2' = 111" 49' 02" Table XVII.*.... 55 03 Table XIX 7 12 1st Corr 7 27 2dCnrr 31 38 Table XX 14 True distance 113''30'36' True distance 113°30' 3G" Distance by N. A. at I2h 1'.4 55 06 Prop, log 2455 Difference 1 24 30 Prop, log 3284 2i> 28ni 44" Prop. log. diff. 08^9 Add 12 Jlean time at Greenwich 14 28 44 Mean time at the ship 12 50 08 Difference is lona-itude in time . . . 1 38 36 = 24° 39' W. from Greenwich. • Tiie horizontal parallax of Mars being 4".93, by the Nautical Almanac, we ma^'find the correction and logarithm in Table XVII., page 90, rorrpsponding to the nearest parallax 5'' LUNAR OBSERVATIONS. 239 SECOND METHOD Of Jinding the true distance of the moon from a ,tar* This method is giounded on that which was first published by Mr. Lyons, and afterwards improved by various persons by the introduction of tables similar to Tables XLVIL, XLVIIL, of the present collection. In Lyons's method there are four principal corrections, and several small ones, like those which are included in Table XX.; the first and second of these corrections depend on the refraction; the third and fourth, on the moon's parallax. These two last corrections correspond very nearly to the first and second of the present improved method. The first and seconti corrections of Lyons's method, with all the smaller corrections, are given very nearly by means of Table XLVIIL, under the name of the tlurd correction of the present method ; the numbers in this table are liable to an error of a few seconds in conse- quence of using the moon's mean horizontal parallax in computing the numbers. Several of the quantities in each page of die table have been compared by means of Shcpard's tables with the correct results, for the extreme values of the moon's horizontal parallax ; and it has been found that an error exceeding 5" will rarely occur in computing the distance from the numbers in the table, if the process of interpola- tion be carefully attended to, when the proposed distance and the altitudes are not expressly given in the table, as most commonly happens. Wheii this tabular form was first adopted in finding this third correction, the inter- vals were much longer than they now are, and the table contained only one page ; the process of interpolation was then difiicult, and liable to a considerable degree of inaccuracy, sometimes amounting to more tlian half a minute. This som-ce of error has been successively diminished by increasing the number of pages in the table ; and it was finally published by ]Mr. Thompson, in nearly the same form as in Table XLVIIL of the present collection, which is so extended that we can, without much error, neglect wholly the process of interpolation, and take out, by mere insi>ection, the tabular correction for the nearest degrees in the tal)le corresponding to the distance and altitudes. Thus, if the a})parent distance be 29° 10', the moon's apparent altitude 21° 15', and the star's apparent altitude 18° 25', we must enter the table in jiage 278, corresponding to the a|)parent distance 28°, moon's altitude 21°, star's altitude 18°, and take out the corresponding correction 1' 19" ; which differs but very little from tlie true value, found by interpolation. This second method has not the same advantage as the first method, of being wholly free from cases, for the second correction is found at the top of Table XLVIL when the distance is greater than 90°, and at the hoitom when less tlian 90° ; moreover the effect of the parallax of the sun, or that of a planet, is sometimes additive, and at other times subtraclive. In this, as well as in the third and fourth methods, the preparation is the same as in the first method ; and the process of finding the longi- tude from the true distance is also the same : it will therefore be unnecessary to repeat the rules for these calculations, which we have given in pages 229, 231, and we shall restrict ourselves to the explanation of the process for computing the true distance, which is done in the following manner : — RULE. ' To the proportional logarithm of the moon's horizontal parallax, (Table XXII.) arid the log. cosecant of the star's apparent altitude, (Table XXVII.) the log. sine of the star's apparent distance, (Table XXVII. ;) the sum (rejecting the tens in the indices) will be a logarithm which is to be found in Table XLVIL ; and the corresponding number of degrees, minutes, and seconds, taken at the top of the page, is the first correction. To the proportional logarithm of the moon's horizontal parallax, (Table XXII.) add the log. cosecant of the moon's apparent altitude,* (Table XXVII.) and the log. tangent of the apparent distance, (Table XXVII. ;) the sum (rejecting the tens in the indices) will be a logarithm which is to be found in Table XLVIL; and the corre- sponding second con-ection is to be found at the top of the table, if the apparent distance exceed 90° ; but the second correction is to be found at the bottom of the table, if the apparent distance be less than 90°. * 1 he same rule may be used for the sun or a planet, correcting for the parallax by means of Tables XLIX. an! L., as will be shown hcrcalier, 240 LUNAR OBSPmVATIONS. Take the third correction, by in?^">ection, from Table XL VIII., for tlie nearest degrees corresponding to the apparent distances and altitudes. Add these tliree corrections to the apparent distance ; the sum, decreased by 10° gives the ti'ue distance of the moon from the star. When the sun is used, instead of a star, we must take out the correction for the sun's parallax, in the part P, of the same page of Table XLVIII. in which the third correction is found ; and this correction is to be applied, by addition or subtraction, according to its sign in the table, to the true distance above computed, as for a star. When a planet is used, we can find the correction of the distance for the planet's parallax, by means of Tables XLIX., L. The first of these tables, lieing entei-ed with the nearest degrees of the distance and altitudes, gives the correction, with its sign, supposing the horizontal parallax to be 100". This is reduced to the actual parallax, by means of Table L. We may also find this correction very nearly by the table marked P, on the same page of Table XLVIII. where the third correction is found ; which gives tlie correction of the distance, with its sign, supposing the horizontal parallax to be equal to the sun's mean parallax, 8".6 ; if the horizontal parallax of the planet be greater or less than 8".G, this correction must be increased or decreased in the same proportion, always retaining the same sign. The coiTection thus found is to be apjjlied to the true distance, above computed tor a star. EXAMPLE VIII. [Being the same as Example III., page 234.] Suppose that, on the 30th of October, sea account, in the forenoon, in the longitude of 80° W., by account, at O'^ 43™ 47% mean time, the observed distance of the nearest limb of the sun and moon was 111° 34' 50", the altitude of the sun's lower limb 24° 55', and the altitude of the moon's lower limb 26° 25'. Requh'ed the true longitude. The preparation is the same as in page 234, which, for want of room on this page, we shall not repeat, but merely give the results, namely: — A]iparent distance 112° 05' 52" ; {v)'s apparent altitude 25° 07' ; D's apparent altitude 20° 37' ; ])'s semi- diameter 14' 53" ; 2)'s horizontal parallax 54' 10". To find the true distance. Dhor.par... 0°5i'10" Prop. log. 0.5215 ©app.alt... 25 07 00 Cosec 10.3722 App.clist...ll2 03 52 Sine 9.9GG9 IstCorr. ... 4 35 ll..Tab.XLVII. Lo g.0.8G0G SdCorr 4 50 09 SdCorr 2 45 Sum— 10°=111 33 57 ©par.Tab.P. —6 Same 0.5215 D apparent altilude 26° 37' Cosec. . 10.3487 Tangent 10.3914 2d Corr. Tab. XL VII Los 1.2616 111 33 51 = True distance, diflering 2" from the first method in page 234. To find the true longitude. True distance 111°33'51" Distance by M. A. at O'' 112 54 10 Prop, log 3458 Difiercnce 1 20 19 Prop, log 3505 2i»58n>033 Prop. log. diff. 0017 * Add Mean time at Greenwich. . . .Oct. SQJ Sh 58™ 03a Mean time at the ship Oct. 29 21 43 47 DifTerence is longitude in time 5 14 16 = 78° 34' V/. from GreenHicli. LUNAR OBSERVATIONS. 241 EXAMPLE IX. [Same as Example I., page 232.] Suj»pose that, on the 7th of January, 183G, sea account, at 11™ 57' mean time, past midnight, in the longitude of 127° 30' E., by account, tlie observed distance of the fai-thest Hmb of the moon from the star Aldebaran, was G8° 3G' 00", the observed altitude of the star 32° 14', and the observed altitude of the moon's lower limb 34° 43'. Required the true longitude. Preparation. Sea account, Jan. 7, is by N. A. Jan. C^ ISh llm 57» Longitude 127° 30' E 8 30 00 Reduced time Jan. C 57' Jsemidiam. Jan. G, noon 15' 05'' ]) hor. par. Jan. G, noon.. 55' 20" -5^ observed alt.... 32° 14 midniarbt 15 09 Difference Table XI. Auff. Table XV. 15 06 9 5 semidiameter 15' 15" midnight 55 34 Subtract. Difference. Table XI.. 14 4 I) horizontal parallax.... 55' 24" B obs. alt. L. L. Add 4 * apparent alt.... 32 10 34° 45' 12 D apparent alt. . . 34° 65' Observed distance * D F. L 68° 36' 00" D semidiameter subtract 15 15 Apparent distance* D 68° 20' 45" To find the true distance. Bhor.par 0°55'24" Prop. log. 0.5118 *app. alt 3210 00 Cosec. 0.2738 App. dist G3 20 '15 Sine 9.9632 IstCorr 4 28 16. .Tab. XLVII. Log. 0.7538 2dCorr 5 12 35 SdCorr.Tab.XLVI II. 1 25 Sum — 10°= 68° 03' 01" = True distance, differing 1" from the first method, in page 232. Same 0.5113 B apparent altitude 34° 55'.. Cosec. 0.2423 Tangen t 0.4012 2dCorr. Tab. XLVII Log. 1.1553 To find the longitude. True distance 68° 03' 01" Distance by N. A. at 3'> G7 41 43 Prop. lo"-. . . . 2872 Difference 21 18 Prop, log 9269 0'' 41m 16> prop_ lo^ ji(y 5397 Add 3 Mean time at Greenwich 3 41 16 Mean time at the ship 12 11 57 Difference is longitude in time. ... 8 30 41 = 127° 40' 15" E. from Grocnwiclj. 3 242 LUNAR OBSERVATIONS. THIRD METHOD Of finding the true distance of the moon from the sun, a planet, or a star. RULE. From the sun's refraction (Table XH.) take his parallax in altitude, (Table XIV.;) the remainder call the correction of the swi's altitude. In like manner, if a planet be used, we must find the planet's refraction, (in Table XII.) and subtract from it the parallax in altitude, (Table X. A.;) the remainder will be the correction of theplaneVs altitude. Knt if a star be used, we must find the refraction, (Table XII.) and that will be the correction of the star''s altitude.* From the proportional logarithm of the moon's horizontal parallax, (increasino^ the index by 10,) take the sine of the moon's apparent zenith distance, (Table XXVII. ;) the remainder will be the prop. log. of the parallax in altitude, which must be found in Table XXII., and the moon's refraction (Table XII.) subtracted therefrom ; the remainder will be the correction of the moon's altitude.f Add together the api)arent distance of the sun and moon, ([ilanet and moon, or star and moon,) and their apparent zenith distances, (or complement of their apparent altitudes,) and note the half-sum of these numbers ; the difference between the half- sum and the moon's apparent zenith distance call t\\e first remainder ; and the differ- ence between the half-sum and the sun's (planet or star's) apparent zenith distance, call the second remainder. To the constant log. 9.C990 add the cosecant of the half-sum, and the sine of the ap{)arent distance, (both taken from Table XXVII. ;) the sum (rejecting 20 from the index) will be a reserved logarithm. To the reserved logarithm add the sine of the sun's (planet or star's; apparent zenith distance, the cosecant of the first remainder, (both taken from Table XXVII.) and the ]M-op. log. of the correction of the sun's (planet or star's) altitude, (Table XXII.;) the sum (rejecting 30 from the index) will be the prop. log. of the fiist cor- rection, to l)c found in Table XXII. To the reserved logarithm add the sine of the moon's apparent zenith distance,! the cosecant of the second remainder, (Table XXVII.) and the prop. log. of the correction of the moon's altitude, (Table XXII. ;) the sum (rejecting 30 fi-om the index) will be the ])rop. log. of the second correction, to be found in Table XXII. Then, to the a])parent distance add the correction of the moon's altitude, and the fii'st correction, and subtract the smn of the second correction and the correction of the sun's (|)lanet or star's) altitude; the remainder will be the corrected distance. Enter Table XX., and find the numbers which most nearly agree with the observed distance, and the observed altitudes of the objects, and take out the corresponding correction in seconds, which is to be added to the corrected distance, and then 18' subtracted from the sum ; *the remainder will be the true distance.J We shall now give an example of this third method of correcting the distance ; but it will be unnecessary to repeat the preparation and the process to find the longittide, as it is very nearly the same as in page 232. EXAMPLE X. [Same as Example I., preceding.] Suppose the apparent distance of tlio centre of the moon from the star Aldebaran was G8° 20' 45", the apparent altitude of the star 32^ 10', the apparent altitude of tlie * We may also find this correction by means of Table XVII., or Table XVIII.j taking the difTerence ticlwccn tlie taljular number and GO' for the correction ; using Table XVIII, for the sun, and Table XVII. for a planet, or a fixed star. t Tliis correction may very easily be found by means of Table XIX., by subtracting tlie tabular number from 59' 42"; for tiie remainder will be the correction of the moon's altitude for parallax and refraction. t N(!glecting the small corrections mentioned in a note marked i, in page 231. LUNAR OBSERVATIONS. 243 moon's centre 34° 55', and the moon's horizontal parallax 55' 24". Required the true distance of the moon from the star. WOO' 3 app. alt.... 34 55 ]) zenith dist. 55 05 90° 00' ^ app. alt.... 32 10 ^ zenith dist. 57 50 Hor. par. 55' 24" P. L. 10.5118 ]) zenith dist. 55° 05' Sine 9.9138 45' 26"... P. L. 0.5980 5 refraction .... 1 21 Corr. D altitude 44 05 ^refraction l'31f App. dist G8°21' 5 zenith dist. 55 05 ^ zenitli dist. 57 50 Sum 181 16 Ifalf-sum 90 38 I> zenith dist. 55 05 1st Rem 35 33 llalf-snm .... 90 33 ^ zenith dist. 57 50 Qd Rem 32 48 Constant log 9.6990 Half-sum 90° 38' Cosec. 10.0000 Dist. C8° 21' Sine 9.9682 Reserved log 9.6G72 * zenith dist. 57° 50' ...Sine 9.9276 1st Rom. 35° 33' Cosec. 10.2355 * Corr. 1' 31" P. L. 2.0744 1st Corr. 2' 15" P. L. 1 .9047 Reserved log 9.6G72 5 zenith dist. 55° 05'... Sine 9.9138 2d Rem. 32° 48' Cosec. 10.26G2 ]) Corr. 44' 05" P. L. 0.6110 2d Corr. 1° 2' 40" P. L. 0.4582 Apparent distance G8° 20' 45" First correclioii add 2 15 Correction ]) altitude 44 05 69 07 05 Second correction. .. 1° 2' 40" Correction ^ altitude 1 31 sub. 14 11 Corrected distance 68 02 54 Correction Table XX. — 1 8" 7 68° 03' 01" agreeing within 1" of the first melliod. This method, as well as the first, was invented hy the autlior of this work, who also improved Witchell's metliod, and reduced considerably the number of cases. These improvements were made in consequence of a suggestion of the late Cliief Justice Parsons, (a gentleman eminently distinguished for his mathematical acquire- ments,) who had somewhat simplified Witchell's process; and it was found, upon e.xainination, that this improvement could be extended fartlicr than he had done it, and that the number of cases, with the manner of ap|dying the corrections, could be rendered more simple and symmetrical. This improvement of Witchell's process we shall now insert as the fourth method of computation. FOURTH METHOD Of finding the true distance of the moon from the sun, a planet, or a star. RULE. From the sun's refraction (Table XII.) take his parallax in altitude, (Table XIV. ;) the remainder will be the correction of the sun's allilude. In like manner, if a planet be used, we must find the planet's refraction, (in Table XII.) and subtract from it the parallax in altitude, (Table X. A. ;) the remainder will he the correction of the planet's alliludp. But if a star be observed, we must find the refraction, (Table XII. ;) and that will be the correction of the starts altitude* From the proportional logarithm of the moon's horizontal parallax, (increasing the index by 10,) take the cosine of the moon's apparent altitude, (Table XXVII.;) the remainder will be the proportional logarithm of the moon's parallax in altitude ; from wlilch subtracting the moon's refraction, (Table XII.) the remainder will be the cor- rection of the moon's allilude.f * This correction may be found in Table XVII. or XVIII., as is shown in a note to the third methofl, in page 242. t This correction may be found by Table XIX., as is sho'mi in a note to the tliird method,, m page 242. 244 LUNAR OBSERVATIONS 1. Add together the apparent aUitudes of the moon and sun, (planet or star,) and take the halt-sum ; subtract the least altitude from the greatest, and take the half- difference ; then add together The tangent of the half-sum, The cotangent of the half-difference. The tangent of half llie apparent distance ; The sum (rejecting 20 in the index) will be the tangent of the angle A, 'which must be sought for in Table XXVII., and taken out less than 90° when the sun's altitude is less than the moon's, otherwise greater than 90°, * The difference of the angle A, and half the apparent distance, is to be called the first angle, and their sum the second angle. 2. Add together the tangent of the first angle, The cotangent of the sun, planet, or star's apparent altitude, The prop. log. of the correction of the sun, planet, or star's altitude; The sum (rejecting 20 in tlie index) will be the prop. log. of the first correction. Or the refraction (Table XIJ.) corresponding to the first angle, or its sup])lement, will be the first correction nearly; particularly if the altitude of the sun, planet, or star, be great, and the first angle be near 90°. 3. Add together the tangent of the second angle. The cotangent of the moon's apparent altitude. The prop. log. of the correction of the moon's altitude ; The sum (rejecting 20 in the index) will be the prop. log. of the second correction. 4. The first correction is to be added to the apparent distance when the first angle is less than 90°, otherwise subtracted ; and in the same manner the second correction .is to be added when the second angle is less than 90°, otherwise subtracted. By applying these two corrections, we shall obtain the corrected distance. Enter Table XX., and find the numbers which most nearly agree with the observed distance and the observed altitudes of the objects, and take out the corresponding third correction in seconds, which is to be added to the corrected distance, and then 18" subtracted from the sum ; the remainder will be the true distance. We shall now give an example of this fourth method of correcting the distances omitting, as before, the preparation and the computation of the longitude from the tru» distance. EXAMPLE XL [The same as Example I., preceding.] Suppose the apparent distance of the centre of the moon from the star Aldebara.! was 68° 20' 45", the apjiarent altitude of the star 32° 10', the ai)parent altitude of thri moon's centre 34*^55', and the moon's horizontal parallax 55' 24". Required the trv.e distance of the moon from the star. D app. alt. 34° 55' * app. alt. 32 10 Sum G7 05 Half-sum .. 33° 33'. . Tang. 9.82IG1 Ilor. par. 55' 21" ... .P. L. 10.51 13 Difference _2_15 Ilalf-diff. . . 1° 23' Cotang. 11.G1711 D app. alt. 3t°55' ... Cosine 9.J 138 Half-di.st.. . 34° 10' . .Tang. 9.83171 45' 2G" .... P. L. 0.5'JSC Angle A.. . 86° 5G' . .Tang. 1 1.27043 1' 21" 5 refraction. Difference is 1st angle.. 52° 4G' . .Tang. 10.1192 4-1. 05 Corr. J) altiHulc. * app. alt 32° 10' Cotang. 10.2014 Corr. *alt. 1'31"..P. L. 2.0744 Apparent distance GS°20'45" IstCorr. .. 0'44"..P. L. 2.3950 1st correction add U Sumis 2d angle.. 121° OG'.. Tang. 10.2195 ' 68 2129 p app. alt. 34° 55' Cotang. 10.15G1 2d correction sub. 18 31 Corr. D alt. 44' 05" . . P. L. O.GllO 3d angle G8 02 55 2dCorr.... 18' 34".. P. L. 0.98G6 3d corr. Table XX.— 18" 7_ Trnedis'anee C8 03 02 Agreeing within 2" of the first method * Every cotangent in Table XXVII. corresponds to two angles, the one greater than 90°, the otiiei less than 90°. LUNAR OBSERVATIONS. 245 Mdhod of correcting for the second differences of the motions of the bodies in computing a lunar observation. In all the preceding calculation?, we have neglected the second difFerences of the moon's motion, in the intervals of 3 hours, between the times in which the distances are marked in the Nautical Almanac. The correction arising from this soiiue is o'cnerally quite small, and mav, in most cases, be neglected, as coming within the Umits of the usual errors of such observations. It is, however, very easy to find tins correction by means of the following table, which is similar to that in page 484 of the Nautical Almanac for 183G. In using this table, we must find the difierence between the two projjortional logarithms, conesjionding to the distances in the Nautical Alma- nac, which include the given distance. This difierence is to be sought tor at the top of the table ; and at the side we must find the interval which is calculated m the last part of the process of compiuiiig the true longitude, being the tiiue between the hour marked first in the Nautical Almanac, and the mean time of observation atGreenwicJi. Tlie number of seconds in the table corresponding to these two arguments is to be applied, according to the directions in tlie table, as a correction to the time at (Greenwich, computed by either of the preceding methods. Example 1. Thus, in the example page 232, we find that the two proportional lo<^arithms corresponding, on January tith, to 3'' and C', are 2872, 2864, whose difference is 8; and the interval past 3", computed in page 232, is 0" 41" 14'. Entering the table with 8 at the top, and 0'' 40'" at the side, (which is the nearest mimbor°to the interval 0" 41™ U%) we get the correction 2% to be added to die time at Greenwich, Q^ 41'" 14% (computed in page 232,) because the logarithms are decreasing ; hence the corrected time at Greenwich is 3'' 41"' 16". example page 237, we find that the two ])roportional 1 )n June 20th, to 9" and 12", are 2985 and 29G9, whose differ loga- ence Example 2. In the lithms corresj)onding, on , , is IG. Under this, and 0])i)osite the interval 1'' 55'" 28% computed in page 237, (or the nearest tabular nunilier 2'' 0'",) we find a correction 4= to be added to the time at Greenwich 10" 55'" 23% computed in i)age 237, making the corrected time at Green- wich 10" 55™ 32'. Table, showing the Correction required on account of the Second Differences of the Distances in the jYautical Almanac, in ivorkhig a Lunar Observation. Find at the top of the table the diflcrence between the proportional logarithm taken from the iNantical Almanac, in working a kuiar observation, and that which immediately follows it, and at the side the interval between the hour marked in the Nautical Almanac, and the mean time of the observation of the meridian at Greenwich. The corresponding namber is a correction, in seconds, which is to be added to the time at Greenwich, deduced from either of the preceding methods of working a lunar observation if the proportional logarithms are decreasing, but stihtracled if the pro- portional logarithms are increasing ; the sum or difference will be the corrected time at (Jreenwich. Approxi- mate Interval. n . M . 1(1 20 n. M. :i •2 50 •2 40 3(1-2 ?,() 40]'2 20 502 10 roo;27)o 1 lOll 50 1 201 40 1 301 30 Difference of tlie Proportional Logarithms in the NaiUical Almanac. 4 I 8 :i2|l6l20|24|28|32!3()|40|44i48|52|5G|G0!64J()8|72|76!80|84|8S|SJ2|nG Correction of the Time at Greenwich for Second Differences. s.\ s. 11 12 2'2 2 3 2 3 2 3 2 4 3 4 3i4 3 3 3I4 4 5 6 7 G 7 G 7 Gl8 7 8 010 lO'll s. I s. " 3 6 91011 12 13 9;ill2'l314 9 101111214 15 9 !lO 11112 14 15 10 10 12 13 1415 111213 14 1511G 1C1718 14 IG 17 1849 20 15 17il8 19 20,21 lG]7!l9 20'21,22 lG18il9'20>2ll23 10 1414 1718 20:21 G 12 151G17 19 2021 22 2324 21 22 23124 25127 22 24 25 2G27 2S 2r?25 2G,27 28j29 24'25',2G27,29l30 Ajypvo.ri- mate Interval. n. M. H.M. 03 ]0]2 50 20 2 40 30^2 30 40|2 20 50|2 JO 246 LUNAR OBSERVATIONS. Mctliod of taking a lunar observation by one observer. Three obsei^vers are requii-ed to make the necessary obsenations for determining rne longitude ; one to measure the distance of the bodies, and the others to take the altitudes. In case of not having a sufficient number of instruments or observers to take the altitudes, it has been customary to calculate them ; there being given the latitude of the place, the apparent time, the right ascensions, and the declinations of the objects. These calculations are long, when an altitude of a star is to be com- puted, and much more so when that of the moon is required ; and a considerable degree of accuracy is required in finding, from the Nautical Almanac, the moon's right ascension and declination, which must be liable to some error on account of the uncertainty of the ship's longitude. The following method of obtaining those alti-- tudes is far more simple, and sufliciently accurate. This method depends on the supposition that the altitudes increase or decrease uniformly. Before you measure the distance of the bodies, take their altitudes, and note the times by a chronometer ; then measure the distance, and note the time, (or you may measure a immber of distances, and note the corresponding times, and take the mean of all the times and distances for the time and distance respectively ;) after you have measured the distances^ again measure the altitudes, and note the times ; tiien, from the two observed altitudes of either of the objects, the sought altitude of that object may be fouud in the following manner: — Add together the proportional logarithm (Table XXII.) of the variation of altitude* of the object between the two times of observing the altitudes, and the prop. log. of the time elapsed between taking the first altitude and measuring the distance; from the sum subtract the prop, log.f of the time elapsed between obsei'vmg the two altitudes of that object ; the remainder will be the prop. log. of the correction, to be api)lied to the first altitude, additive or subtractive, according as the altitude was increasing or decreasing ; to the altitude, thus corrected, apply the correction for dip of the horizon and semidiameter, as usual. EXAMPLE. Suppose the distances and altitudes of the sun and moon were obseiTcd, as in the following table ; it is required to find the altitudes at the time of measuring the mean distance. Observations. Mcjui. Times by chronometer. 2h3m20' 2 4 20 2 5 50 ..2 4 30 Dist. and a JV. L. 40'= 0' 00" 40 30 40 1 30 40 40 Times by chrunvmcter, 2h 2m 0» 2 6 10 Difference , 4 10 Obs. alt. ])'sL.L. 20° 4G' 21 20 Times by chronometer. 2h2m30' 2 7 00 Difference. . 4 30 Ohs. alt. O^sL.L. 40° 20' 39 12 1 8 Variation ]) 's altitude. . Time 1st observation J) 34' Prop. log. 7238 oh 2m Qa Mean time of observing } a a tn distance > ^ * JU Difference. 2 30 Prop. log. 1.8573 Elapsed time between the two observations Correction of altitude. . . First altitude of moon . . Alt. D 's L. L. at time of the mean obs. of dist. 2.5811 4m 10' Prop. log. 1.G355 0° 20' Prop, lo! 20 46 add. 9456 IzL V^ariation 0's altitude. 1° 8' Prop. log. 4223 Time 1st observation 2ii2ra30» Time mean observation 2 4 30 Difference 2 00 Prop. lo g. 1.9542 Sum 2.3770 Elapsed time between the two observations 4 30 Prop. log. 1.6021 Correction of altitude... 0° 30' Prop. log. 7749 Sub. from ©'s 1st altitude 40 20 Alt. 0's L. L. at time of~) the mean observation > 39 50 of ihe distajices j Thus, at the time S"* 4™ 30% the mean observed distance of the sini and moon's nearest limbs was 40° 0' 40", the altitude of the moon's lower limb 21° &, and the altitude of the sun's lower limb 39° 50' ; these altitudes must be corrected for dip and semidiameter as usual. * Table XXII is only calculated as far as 3°, and if the variation of altitude exceed that quantiiy, you must enter the table with minutes and seconds, instead of degrees and minutes ; and the correction of altitude taken out in minutes and seconds must be called degrees and minutes respective! v i Or add its arithmetical complement, neglecting 10 in the index of the .sum LUiNAR OBSERVATIONS. 247 In tliis manner I have often obtained the altitudes in much less time than they could have been obtained by other calculations. The same method may be used for finding the sun's altitude, when taking an azimuth, by noting the times of taking the observations by a chronometer, and taking two altitudes, the one before, the other after the observation, and jiroporlioning the altitudes as above. Any person who wishes to calculate strictly the apparent altitudes, may proceed according to the following rules : — The apparent time,* the ship's latitude and longitude, and the sun's declination given, to find the apparent altitude of his centre. RULE. With the apparent time from noon, enter Table XXIII., and from the column of rising take out the logarithm corresponding, to which add the log. cosine of the latitude, and the log. cosine of the sun's declination ; their sum (rejecting 20 in the index) will be the logarithm of a natural number, which being suijtracted lioin the natural cosine of the sum of the declination and latitude, when they are of difterent names, or the natural cosine of their difference, when of the same name, will leave the natural sine of the sun's true altitude at the given time. The refraction, less parallax, being added to the true altitude, will give the apparent altitude. In general, it will be near enough to take out the refraction only from Table XII., and neglect the parallax. EXAMPLE I. Ilequired the true altitude of the sun's centre, in latitude 49° 57' N., and longitude 75° W., July 20, 183G, at C' 50™ 30' in the morning, apparent time, sea account. 12" 0™0» Apparent time G 56 30 Apparent time from noon 5 3 30 Its log. in column of rising 4.87850 Latitude 49 57 ON. Its log. cosine 9.80852 Declination at that time . . 19 24 15 N. Its log. cosine 9.97460 Natural number 45880 Its log. —4.66162 Difference .• 30 32 45 Natural cosine 86123 True altitude 23 44 Natural sine . . 40243 Refraction add 2 Apparent altitude 23 4G EXAMPLE II. What will be the true altitude of the sun's centre, in the latitude of 39° 20' N., and the longitude of 40° 50' W., November 26, 1836, at 3" 21'" 30% apparent time, in the alternoon, sea account ? Apparent time from noon 3''21'^30' Its log. in column of rising 4.5.5900 Latitude 39 20 00 N. Its log. cosine 9.88844 Declination at that time 20 53 09 S. Its log. cosine 9.97048 Natural number 26177 Its log. = 4.41792 Sum 60 13 09 Natural cosine 49668 ' True altitude 13 35 Natural sine. . . 23491 Refraction add 4 Apparent altitude 13 39 * If the mean time be ^ven, we must dedtice from it the apparent time, by applying the equation Table IV. A., with a different sign from that in the table, as taught in the introduction to the tables remarking, however, this equation is found more correctly in page II. of the Nautical Almanac. 248 LUNAR OBSERVATIOISS. The apparent time, toith the latitude and longitude of the ship, given, to Jind thi apparent altitude of the moon's centre. Turn tlie longitude into time, (by Table XXI.) and if in west longitude add it to, but in east longitude subtract it from, tlie apparent time * at the ship ; tlie sum or diflerence will be the apparent time at Greenwich. From this we may deduce the ini-an time at Greenwich, which is wanted in finding the moon's right ascension and declination. Tak" the sun's right ascension from tlie Nautical Almanac for the preceding noon at Greenwich, and add thereto the correction taken from Table XXXI. corresponding to the hours and minutes of the time at Greenwich ; the sum will be the sun's right ascension, which, being added to the ajjparent time at the ship, will give tlie right ascension of the meridian, rejecting 24 hours when the sum exceeds 24 hours. Take Irom the Nautical Almanac the moon's right ascension and declination ibr the lime at (jreenwich ; then the diflerence between the moon's right ascension and the right ascension of the meridian, will be the moon's distance f from tlie meridian, with which enter Table XXIIL, and take out the corres])onding logarithm from the column of rising, and add thereto the log. cosine of the latitude of the ship, and the log. cosine of the declination of the moon; the sum (rejecting 20 in the index) will be the logarithm of a natural number, (Table XXVI.) which, being subtracted from the natural cosine (Table XXIV.) of the sum of the declination and latitude when of different names, or the natural cosine of their difference when of the same name, will leave the natm-al sine of the moon's true altitude ; from which subtracting the correc- tion corresponding to the altitude in Table XXIX.| there will remain the apparent altitude nearly. EXAMPLE. What was the moon's apparent altitude, Ajiril 29, 1836, sea account, at 7" 55™ 52' P. M., in latitude 42° 34' S., longitude C5° 07' 30" W., from Greenwich ? April 29, sea account, or by astrononfical account April 28 7'' 55" 52' Longitude 05° 07' 30" W., in time 4 20 3 Apparent time at Greenwich April 28 12 16 22 Sun's right ascension, April 28^ 12'^ 10™ 22% by Nautical Almanac. . . 2'' 25'" 11 ' Api)arent time at the ship 7 55 52 Right ascension of the meridian 10 21 03 j)'s right ascension in time 12 33 27 2)'s distance from the meridian - 2 19 24 Corresponding, to which, in the column log. rising, \s 4.21027 Latitude 42° 34' S Cosine 9.80717 3)'s declination 10 N Cosine 10.00000 Natural number 11952 Log... 4.07744 Sum 42 50 Natural cosine . 73333 3)'s true altitude 37 52 Natural shie. . . 01381 Correction Table XXIX. 44 ])'s apparent altitude 37 08 nearly. This altitude would be decreased nearly 2', if the true correction of the altitude, corresponding to i!:e 3)'s horizontal parallax, 59', were used, as may be seen in note \, at the bo'.iom of the page. * The apparent lime is counted from noon to noon, marking the hours from 1 hour to 24 hours. We may remark, thai lliis process of findinj^ llic lime at Greenwich is unnecessary when you liavc a chronometer rcaulntcd for mean lime at Grcenwicli, because we can immediately obtain tlie appctrertl time, by a|>]ilying the equalion of lime, taken from the Nautical Almanac, or from Table IV. A., using a diflerent sign from ihat in tlie table. t When the distance exceeds 12 hours, you must enter Table XXIII. with the diflerence between that distance and 2l hours. I In slriclness you ought, instead of this correction, to use the correction of the moon's altitude, corresponding to Iier apparent altitude and horizontal parallax. This is easily found in Table XIX., using the D 's horizontal parallax and the apparent altitude found by the above process, a-id subtracting tlie tabular corrocti(m from 59' 42". Thus, if the )) 's horizontal parallax is 59', and the )) 's apparent altitude 37° 8', this correction would lie 59' 42"— 13' 65"=45' 47", instead of 44'. which is used above. LUNAR OBSERVATIONS. 249 Tkc apparent time, icith the latitiidc and longitude qf the ship, being given, to find the apparent altitude of the centre of a planet. Turn the longitude into time, (by Table XXI. ;) and if west, add it to, but if east longitude, subtract it from, the apparent time at llie ship; the sum, or ditlercnco, will be the apparent time at Greenwich. From this we may deduce the mean time at Greenwich, which is required in finding the right ascension and declination of the planet.* Take the sun's right ascension from tlie Nautical Almanac, for the preceding noon at Greenwich, and add thereto the correction taken from Table XXXI., corresi)ond- ing to the hours and minutes of the time at Greenwich ; the sum will bo the sun's right ascension, which, beuig added to the apparent time at the ship, will give the right ascension of the meridian, rejecting 24 hours when the sum exceeds 24 hours. Take from the Nautical Almanac the ])lanet's right ascension and declination for the time at Greenwich ; then the difference between the jjlanct's right ascension and the right ascension of the meridian, will be the ]»lanet's distance \ from the meridian ; with wiiich enter Table XXIII., and take out the corresponding logarithu), from the column of rising, and add thereto the log. cosine of the latitude of tiie ship, and the log. cosine of the declination of the j)lanet; the sum (rejecting 20 in tiie index) will be the logarithm of a natural Jiumbcr, (Table XXVI.) which, being subtracted from the natural cosine (Table XXIV.) of tlie sum of the declination ancl latitude when of dilierent names, or the natural cosine of tlieir difference when of the same name, will leave the natural sine of the planet's true altitude ; to which add the correction of altitude for parallax and refraction, and we shall get the a])parent altitude ; observing that this correction is found in Table XVII., in the page corresponding to the horizontal parallax of the planet; the difference between the tabular number and GO being the correction of the planet's altitude for refraction and parallax. EXAMPLE. What was the planet Jui)iter's apparent altitude, April 29, 183G, sea account, at 1^ 55'" 52^ P. AI., in latitude 42° 34' S., longitude 65° 7' 30'' W. fronr Greenwich .' April 29, sea account, is by astrononfical account April 28^ 7*" 55™ 52' Longitude G5° 07' 30" W., in time 4 20 30 Apparent time at Greenwich April 28 12 16 22 ©'s right ascension,|:April 28' 12'' IG™ 22' by Nautical Almanac 2 25 11 Apparent time at the ship 7 55 52 Right ascension of the meridian 10 21 03 J/'s right ascension, in time 6 47 08 ^'s distance from the meridian 3 33 55 Corresponding to which, in the colinnn of log. rising, is 4.G0733 Latitude 42° 34' S Cosine 9.86717 Declination 23 IG N Cosine 9.96316 Natural number 27395 Log... 4.43766 Sum 65 50 Natural cosine 40939 ^'s true altitude 7° 47' Natural sine . . . 13544 Correction Taiile XVII. add 7 § J^'s apparent altitude 7 54 * TIr.s is more easily obtained by a cliroiiometer ref;uhited to Greenwich time, as in tlic ]irerci!ing example of finding llie altitude of the moon. t When the distance exceeds 12 hours, you must enter Table XXIII. with the dilTerence betweou ihat distance and 24 liours. X The sun's right ascension at noon, April 28, is 2i> 23 "> IS^and the horary motion 9^484, which, for 12h IG-n 2i^ givcs,by Table XXXI., 1 16" =: 1' 60" nearly ; adding this to^ 23"> 15', we get the ©'s right ascension 2h 25'" \\». The planet's right ascension and declination are found by inspection in 'he Nautical Almanac. ^ This correction is found in page 89, Jupiter's parallax being only 1".5. The tabular correction corresponding to the apparent altitude 7° 64' is 53' 26" ; subtracting this from CO', we get d' 31". or nearly 7', for the correction arising from the refraction and parallax 250 LUNAR OBSERVATIONS. The apparent time, the Iqfitude and longitude, given, to find the apparent altitude of a fixed star. RULE. Turn the longitude into time, and add it to, or subtract it from, the apparent time * at the ship, according as tlie longitude is west or east ; the sum or difference Avill be the time at Greenwich. The apparent time at Greenwich may also be found by means of a chronometer, as in the preceding example, page 248. Find, in the Nautical Almanac, the sun's right ascension for the noon preceding the time at Greenwich, and add thereto the correction corresponding to the hours and minutes of the time at Greenwich, (using Tables XXX. XXXI. if necessary ;) the sum will be the sun's right ascension, which being added to the apparent time at the ship, will give the right ascension of the meiidian, rejecting 24 hours when the sum exceeds 24 hours. Find the star's right ascension and declination in the Nautical Almanac, or by means of Table VJII., as taught in page 217. The difference between the star's right ascension and the right ascension of the meridian, will be the distance of the star from the meridian. Find in the cokmni of rising of Table XXIIL the logarithm corresponding to the star's distance from the meridian,! and add thereto tlie log. cosine of the latitude of the ship, and the log. cosine of the declination of the star; the sum (rejecting 20 in the index) will be the logarithm of a natural number, (Table XXVI.) which being subtracted from the natural cosine (Table XXIV.) of the sum of the declination and latitude when of different names, or the natural cosine of their difference when of the same name, will leave the natural sine of the star's true altitude. The refraction being added to the true altitude, will give the apparent altitude. EXAMPLE. What was the apparent altitude of Aldebaran, at Philadelphia, April 12, 1836, sea account, at 5'' 57'" 18' in the afternoon, apparent time ? The star's right ascension and declination are found by inspection in the Nautical Almanac, as below ; this being the shortest and most accurate method of finding them. App. time by astronomical account, April 11'' 5''57'" 18' Longitude 75^ 9' W 5 36 Time at Greenwich April 11 10 57 54 ©'s right ascension, April 11, at noon,bvN.A. 1 19 54 Variation for 10" 57'" 54^ by Table XXXI. 1 41 0's right ascension at the time of observation 1 21 35 Jlpparcnt time of observation 5 57 18 Right ascension of the meridian 7 18 53 H<'s right ascension by Nautical Almanac. . 4 26 30 :^^'s distance from the meridian f 2 52 23 Its log. in col. rising 4.43102 Latitude of Philadelphia . . 39° 57' N 77 Cosine 9.88457 #'s declination 16 10 N Cosine 9.98248 Natural number 19864 Its log. 4.29807 Difference 23 47 Natural cosine . 91508 " True altitude 45 46 Natural sine .. . 71644 Refraction add 1 Apparent altitude 45 47 * The apparent time must be taken (as usual) one day less than the sea account, and the hour must be reckoned from noon to noon in numerical succession from 1 to 2-1. It may also be observed that, if tlie observer be fiiriiislicd willi a chronometer, regulated to mean Greenwich time, this part of the operation ma}' be saved, reducing the mean time to apparent, by applying the equation Table IV. A. ^ or that found in the Nautical Almanac, as in the preceding rules. t If die distance from the meridian exceed ]2 hours, vou must subtract it from 21 hours, before entering Table XXIII. LUNAR OBSERVATIONS. 251 Method of combining several lunar observations together. As a lunar observation is liable to some degree of uncertainty, on account of the imperfections of the instruments, the unavoidable errors of the observations, and the imperfections in the reductions, it will generally be conducive to accuracy to combine together several observations, taken on the same day, or on two or three successive days; and this may be done in the following manner: — After working the lunar observation, and finding the mean time of the observation on the meridian of Greenwich, by either of the ])ieceding methods, we must compare this time with the corresponding time of observation, as shown by the chronometer, and the difference will be the error of the chronometer for mean time at Greenwich, as shown by that lunar observation. Other observations, being taken on the same, or on successive days, and computed in the same manner, will also give the errors of the chronometer, corresponding to these observations respectively. The mean of all these errors, being found, will represent very nearly the error of the chronometer, relative to the mean time at Greenwich, and corres])onding to that moment of time which residts from taking the mean of all the times of observation at Greenwich, for all the lunar observations. Having obtained in this way the error of the chronometer relative to Greenwich time, and knowing its daily rate of loss or gain, we can determine at any moment the mean time at Greenwich, by the chronometer, as it is given by the mean of all these observations. Comparing this mean time widi the corresponding mean time at the same moment at the ship, as found by taking the sim's altitude, or by any other of the methods explained in pages 208 — 218, the difference will be the longitude of the ship, resulting fron^^the mean of all these observations. Tunes bij the Chronometer. April 6 10" 30 40 20 16 6 01 20 Simi 6)23 58 54 Mean, April 6' 3'> 59" >49s EXAMPLE I. Mean Times at Greenwich by Lunar Observations. April 6*1 2" 12™ 20- 2 32 38 3 42 05 4 22 25 5 18 34 6 03 16 6)24 11 18 Errors of the Chronometer for (Jreenwich Time. 2" 00' 2 20 1 40 2 10 2 18 1 56 12 24 2" 04= April 6'' 4" 01 "'53= Hence it appears, that, by the mean of the six lunar observations, when tlie time by tlie chronometer was, April 6', 3'' 59"' 49% it was 2™ 04= too slow for mean time at Greenwich. We shall now suppose, that, on April 6' 4'' 30™ 00% by the chronometer, an altitude of the sun was taken, and the mean time at the ship deduced therefrom, April 6' 6'' 24'" 56% and that it v.-as required to find the longitude of the ship ; the chronometer moving uniformly without gain or loss; we shall have 30"™ 00= 2 04 Time by the chronometer April 6'' Error of the chronometer by the lunar observations add Mean time at Greenwich April 6 Rlean time at the ship April 6 Longitude east of Greenwich 1 52 52 = 28^ 13 4 32 6 24 04 56 Times by the Chronometer. July 61 3^ IS™ 06= 7 4 16 15 3) July 8 5 17 12 21 12 48 33 ean 7 4' 16" '11^ EXAMPLE II. Mean Times at Greeiiwich by Lunar Obseri-atioiis. July 6-1 3" 17™ 16= 7 4 18 23 8 5 19 24 3)21 12 55 03 July 7' 4" IS" '21 Errors of the Chronometer for Greenwich Time. 2™ 10= 2 08 2 12 3)6 30 2"' 10= 252 TO FIND THE LONGITUDE BY ECLIPSES The mean of these three observations makes the chronometer too slow for Grcen- wicli time 2'" 10' ; and if we suppose the instrument to be well regulated for mean time, and on July 8'^' 4'' 10™ 15' by the chronometer, the mean time at the ship deduced from the sun's altitude, was July 8^ 2'' 15™ 25% we shall have, Time by chronometer July 8 ' 4'^ 10" 15' Error by the lunar observations add 2 10 Mean time at Greenwich July 8 4 12 25 Mean time at the ship July 8 2 15 25 Longitude west of G-reenwich 1 57 00 =: 20^ 15' This process may be used for regulating a chronometer when it has accidentally stopped, or h;is been allowed to run down. For, by comparing the two above examples, suj)i)osing them to have been taken by the same chronometer, The first set gives the error April 6' 3'" 59™ 49' equal to -|- 2™ 04' The second set gives the error July 7 4 IG 11 equal to -j- 2 10 Gain in 92 days -j- G= This is, however, an imperfect method of determining the daily gain or loss of the chronometer, on account of the imperfection of the observations ; and is only to be used in cases of absolute need. Tojind tlic longitude hy the eclipses of Jupiter's satellites. The eclipses of the satellites are given in the Nautical AlnAnac for mean time at Greenwich, and also for sideral time. There are two kinds of these eclipses — an immersion, denoting the instant of the disappearance of the satellite by entering into the shadow of Jupiter, and an emersion, or the histant of the appearance of the satellite in coming from the shadow. The immersions and emersions generally happen when the satellite is at some distance from the body of Jupiter, excejit. near the opposition of Jupiter to the sun, when the satellite approaches to his body. Before the opposi- tion, they liai)pen on the west side of Jupiter, and after the oi)position, on the east side. But if an astronomical telescope is used, which reverses the objects, the appear- ance will be directly the contrary. The configurations, or the positions in which Jupiter's satellites appear at Greenwich, are given, in the Nautical Almanac, every night, when visible. As these eclipses hapi)en almost dail}', they afford the most ready means of deter- mining the longitude of jjiaces on land, and might also be ap|)]ied at sea, if the obser- vations could be taken with sufficient accuracy in a ship under sail, which can hardly be done, since the least motion of a telescope which magnifies sufficiently to make these observations, would throw the object out of the field of view. Having regulated your chronometer for mean time at the jilace of observation, you must then find nearly the mean time at which the eclipse will begin at that place ; this may be done as follows : — Find from the Nautical Almanac the 77iean time of an immersion, or emersion, and apply thereto the longitude turned into time, by adding when in east, Init subtracting when in west longitude; the sum or difference will be nearly the 7ncff?i time when the eclipse is to be observed at the given place. If there be any uncertainty in the longitude of the place of observation, you must begin to look out for the eclipse at an earlier period ; and when the eclipse begins, you must note the time by the chronometer, and after ap])lying the correction for the error of the chronometer, if there be any, you will have the mean time of the eclipse at the place of observation ; the difference between this and the mean time in the Nautical Almanac, being turned into degrees, will be the longitude from Greenwich. EXAMPLE. Supj)Ose that, on the 21st of August, 183G, sea account, in the longitude of 127° 5.5' W., I)y account, an innuersion of the first satellite of Jui)iter was observed, at 10'' 24™ 47' P. M. mean time. Required the longitude. By Nautical Almanac, the time of innnersion is, . . August 20th 19'' 0™ 7' •By observation, August 21, sea account, or by N A August 20th 1 24 47 Longitude in time ° "^5 20 which, being turned into degrees, gives 128° 50' W. for the longitude of the place of observation. TO FIND THE LONGITUDE BY A CHRONOMETER. 253 To find the longitude by an eclipse of the moon. The determination of the longitude by an eclipse of the moon, is performed by comparing the times of the beginning or ending of tlie eclipse, as also the times when any number of digits are eclipsed, or when the earth's shadow begins to touch or leave any remarkable spot in the moon's face ; the dift'erence of these times Ijetween the like observations made at different places, turned into degrees, will be the difference of longitude of those places. When the beginning or end of an eclijise of the moon is observed at any place, the longitude of that place may be easily found by comparing the time of observation with the time given in the Nautical Almanac ; for the difference between the observed mean time of beginning or ending, and the mean time given in the Nautical Almanac, will be the shi])'s longitude in time, which may be turned into degrees by Table XXI. Thus, if the beginning of an eclipse of the moon was observed October 25, 1836, sea account, at 5'' 21 "\ mean time ; the mean time at Greenwich by the Nautical Almanac being October 24, or October 25, sea account, at 0'' 38'", their difference, 4'' 43'", is the longitude of the place of observation =: 70° 45', which is east from Greenwich, because the time at the place of oijservation is greatest. To find the longitude by a perfect time-keeper or chronometer. ■ It was before observed, that if a chronometer could be made in so perfect a man- ner as to move imifbrmly in all places, and at all seasons, the longitude might easily be deduced therefrom, by comj»aring the mean time shown by the chronometer, regulated to the meridian of Greenwich, (or some other known meridian,) with the mean time at the place of observation ; for tlie difference of these times would be the difference of longitude between that meridian and the \Aacc of observation. The moderate prices of good chronometers now, in comparison with their values many years since, together with the various im])rovements in their construction, have caused this method of determining the longitude to be very much used within a few years ; we shall therefore explain fully the use of this instrument, the methods of regulating and ascertaining its rate of going, and give examples of the calculations for finding the longitude. If a chronometer is to be used on a voyage, it must be adjusted, and its rate of going ascertained, before sailing. This is most conveniently done on shore by observ- ing, with a transit instnnnent, the times of the transits of the sim, or some lixed star, over the meridian, as is taught in pages 221 — 224. If you have no instrun)enl of this kind, the regulation may be made by taking altitudes*of the sun or some other heavenly body, and finding therefrom the mean time of observation, by any of the methods before given in pages 208 — 218. The best way of making these last obser- vations on land, is by an artificial horizon of quicksilver; finding and correcting the altitudes in exactly the same way as in computing the latitude in page 204. Comjiaring the mean time of observation, obtained in this way, with the time by the chronometer, shows how much it is then too fast or too slow for the meridian of the pla-ce of obser- vation ; and by repeating the ojieration on a future d.ay, the rate of going may be ascertained. If it is- found to gain or lose a few seconds, or parts of a second, per day, that allowance must be made on all future observations at sea. Thus if, on the 1st of June, 1836, at 5'' 10'" 20% by the chronometer, the mean time, deduced from an observation of the sun's altitude, was 5'' 12'" 40% the chronometer would then be too slow by the tiifference of those times, 2'" 20'; and if, on the 21st of June following, the time by the chronometer was 4'' 15" 35% when the mean time was 4'' 18'" 17% the chronometer would then be too slov/ by the difference of those times, or 2'" 42''; and the rate would have varied, in 20 days, from 2"' 20% to 2"' 42% which is a difference of 22' in 20 daj's, being IM per day; and this rate must be allowed on all futm-e observations at sea, until a new regulation can be obtained, at some place whose longitude is known. It is best to have a considerable number of days' interval between the two observations for fixing the rate, since by this means it may be determined to tenths of a second ; the absolute error of the observations being reduced, in finding the daily rate, by dividing by the number of days. Thus, if the above difference of 22' had been erroneous 2% and the true value 20% the daily rate would be one second, instead of I'.l, varying only one tenth of a second, notwithstanding the observations on which the rate was established contained an error of two seconds. Having regulated a chronometer, in the manner first mentioned, at a place whose 'ongitude from Greenwich is known, it is easy to find how much it is too fltst or too * See Tab. LVII. 254 TO FIND THE LONGITUDE BY A CHRONOMETER. slow for the meridian of Greenwicli, by reducing the mean time at the i)lace of the observer, as found by observations, to the meridian of Greenwich, by adding the longitude if west, subtracting if east ; the sum or difference will be the mean time of observation in the meridian of Greenwich ; the difference between this and the time given by the chronometer, shows how much it is too fast or too slow for Greenwich meaii time. Thus, by adding the longitude, which we shall suppose to be 4'' 5<3"', to the mean time of the above observation, 5'' 12'" 40% we get 10'' 8"^ 40' for the mean time at Greenwich ; from which subtracting the time by the chronometer, 5'' 10™ 20% we obtain 4" 58"^ 20' for the error of the chronometer relative to mean time at Green- wich ; being too sloiv for that time. * The chronometer having been thus regulated to Greenwich time, and the daily rate of its going ascertained, if this rate should remain unaltered, the time at Greenwich will be known by it, at any UToment at sea ; and if at tluit moment, by any observation of the sun, moon, planet, or a fixed star, the mea7i time at the ship be found by any of the methods explained in pages 208, &c., the difference between this 7nea?i time at the ship, and the mean time at Greenwich, shown by the chronometer, will be the longitude, which may be turned into degi'ees and minutes by Table XXI. EXAMPLE I. Wishing to regulate a cliroiiometer, in a place whose latitude is 51° 30' N., and longitude 130° E. from Greenwich, I observed, October 10, 1848, at 8^ 21"' A. 31., sea account, by a cln-onometer, the altitude of the sun's lower limb, by a fore observation, 13° 32', the correction for semidiameter, parallax, and dip, being 12'. It is required to find the error of the chronometer for mean time at Greenwich. The mean time of this observation, at the meridian of the ship, computed as in Example I., jiage 209, is 7^ 54'^ 18= A. M., or October 9' 19" 54'" 18% astronomical account. From this subtract * the longitude 130°, turned into time 8'' 40"^, (by Table XXL) we get the corresponding mean time at Greenwich, Oct. 9% 11'' 14"' 18"; and as the time by the chronometer is, October 9'', 20'' 21'" 00% it is too fast for mean time at Greenwicli by the difference of those two quantities, or 9^ 6'" 42'. EXAMPLE II. May 10, 183G, at 5^ 30"" P. M., sea account, by a chronometer, in latitude 39° 54' N,, in a place whose longitude was known to be 35° 45' E. from Greenwich, the altitude of the sun's lower limb by a fore observation was 15° 45', the correction for dip, pai-allax, and semidiameter, being 12'. It is required to find the error of the chro- nometer for mean time at Greenwich. The mean time of this observation, computed as in Example II., page 210, is May 9> 5'' 30™ 39% astronomical computation. From this subtract * the longitude, 35° 45', turned into time, 2" 23'", bv Table XXL; the remainder. May 9' 3" 7"' 39', is the mean time at Greenwich. The difference between this and the time by the chronometer, 5'' 30'", is 2'' 22'" 21% which expresses how much the chronometer is too fast for Greenwich mean time. EXAMPLE III. Suppose tliat, on July 27, 1836, sea account, the mean time was found, by an altitude of the sun, to be 1" 11'" IG' P. M., when, by a chronometer well regulated to mean time at Greenwich, it was 4" 3'" 8' P. M. Reciuired the longitude. JNIean lime at the place of observation P 11"" IG' Time at Greenwich by chronometer. . 4 3 8 Difference in the longitude 2 51 52 — 42° 58' W.,the longitude being west, because the time at Greenwich is the greatest. EXAMPLE IV. Suppose that, on I\Iay 14, 183G, sea account, the mean time was found, by an altitude of the sun, to be 3'' 59"' 09' P. M., when the time by the chronometer was * This is to be added, if the ship's long^ilude is west. TO FLND THE LOiN'GlTUDE BY A CHRO>fOMETER. 255 2'' P, M., tlie chronometer being too slow for mean Gi'ecnwich time 11'" 9'. Rciinired the longitude. Time by chronometer 2^ 0"" 00" Chronometer too slow for jnea?i lime at Greenwich 11 9 J\Iea7i time at Greenwich 2 11 09 P. M. Blean time at the ship 3 59 09 Difference is the longitude 1 48 00 = 27= 00' E. EXAMPLE V. Suppose that, on June 14, 183G, sea account, in a place whose longitude from Greenwich was known, a number of observations were taken to ascertain the going of the chronometer ; and it was found, that, on that day, it was 10' too slow for mean Greenwich time, and lost time 2^ per day ; and that, on July 14, 1836, sea account, the time per chronometer was G"" 0™ G' P. M., when, by an observed altitude of the sun, the viean time was 1'' 21'" 32' P. 31. Required the longitude. Error of chronometer, June 14 0'' 00™ 10' slow. 30 days, at 2' 1 slow. Error July 14 1 10 slow. Time per chronometer G G Time at Greenwich G 1 IG Me-an time at place of observation • 1 21 32 Longitude 4 39 44 = G9° 5G' W. EXAMPLE VI. Sui)pose that, on June 15, 1836, in tlie afternoon, astronomical account, at Boston, in the latitude of 42° 21' 15" N., and longitude 71° 04' 09" VV., several angular dis- tances of the sun's lower limb, from its reflected image in a basin of quicksilver, were observed, and the times noted by a chronometer, which was supposed to be very nearly regulat'jd for mean time at Greenwich ; the times and altitudes being as below ; the thermometer standing at 7G°, and the barometer at 30°.05. Required the error of the chronometer I'elative to mean time at Greenwich. Times by the chronometer, June 15' 7'' 55'" 20' Observed angle 91° 16' 20" 56 12 . DO 57 40 57 01 90 39 48 57 46 90 22 54 58 36 90 04 36 59 24 89 46 42 Sum . . . .-44 19 Sum. . . .6)543 08 00 Mean of the times June 15' 7'' 57" 23'.2 Mean angle 90 31 20 Half the mean angle is equal to altitude ©'s lower limb 45 15 IQ Refraction, Table XFL— 57"— Parallax, Table XIV.+ 6"=— 51" Table XXX VL, Thermometer 76°, correction — 3" ( ^ Barometer 30°.05, correction -j- 1" ^ — Correction for refraction and parallax 53 sub. 53 45 14 47 0's semidiameter * by Nautical Almanac 15 46 ©'s true altitude 45 30 33 * 111 fiiuling the suii's declir.ation, semidiamcter, &lc.. from the Nautical Almnnaf , llie time at Grceiiwicii is supiiosed to be tlie same as the mean time of the observation by the chronometer 7ti 57m 238.2, whivh is supnosed to be very nearly regulated to mean time at Greenwich. If you have no chronometer reg-ujated for that meridian, j^bu must estimate the time at Greenwich in'tlic usual manner, by jidding to the mean lime at the ship the longitude if west, or subtracting it if east 256 TO FIND THE LONGITUDE BY A CHRONOMETER. (v)'s true altitude 45° 30' 33" Latitude 42 21 15 Secant.. 0.13136 Polar distance. . 66 38 49 Cosecant 0.03712 Sum 2 ) 1.54 30 37 Half-sum 77 15 19 Cosine.. 9.34362 Remainder . . . . 31 44 4 6 Sine .... 9.72111 Sum 2 ) 19.23321 Sine of half-sum 9.61660 corresponds to 3'' 15"27'.5 ajiip.time. Equation of time by the Nautical Almanac -|- 9'.4 Rlean time at the place of observation 3 15 36'.9 Add the longitude of Boston, in time 4 44 16'.6 3Iean time at Greenwich 7 59 53'.5 Time by the chronometer, as above 7 57 23'.2 Chronometer, error slow 2"" 30^3 Hence it appears that, on the 15th of June, 1836, astronomical time, at 7'' 57™ 23'.2, by the chronometer, it was too slow for Greenwich time 2'" 30'.3. Suppose, now, that, a few days afterwards, as an example, on June 25, at about the same hour in the afternoon, a similar set of altitudes were observed, and the times noted by the same chronometer, the result of the calculation making the chronometer too slow by 2"' 45'.6 ; then we shall find that, in the interval of 10 days, from June 15 to June 25, it has varied by the quantity 2'" 45'.6 — 2™ 30'.3 = 15'.3. Dividing this variation by 10, (the nun)!)er of days in the interval,) we get 1^53 for the daily rate of loss in the chi-ononieter. If other sets of observations are made, which give results differing a little from P.53, we can use the mean of the different sets, as the most probable value of tije rate of the chronometer. EXAMPLE VII. On the 15th of Jime, 1830, astronomical account, at about S"" 45™ P. ]M., in the meridian of Cape Cod, which bore south, distant about 9 miles, took four altitudes of the sun, and noted the times by the chronometer, as in the table below ; the eye being 19 feet above the level of the sea, the thermometer at 65°, and the barometer at 29 inches. It is required to determine the error of the chronometer for mean time at Greenwich. Times by the chronometer 8^^ 25™ 36^ Observed angle 40° 00' 07" 26 32 39 50 18 27 22 3940 14 Sum 3) 79 30 Sum 3) 119 30 3 9 Mean of the tlu-ce observations. . . 8 26 30 ©'s altitude 39 50 13 Su{)poscd error of the chronome- > i i on t^- -n i i vtti i 4 i-r t,' r .■ , /-I • , > + 1 30 Dip, Table XIll sub. 4 17 ter for mean tune at Greenwich ^ ' ■ ' Estimated mean time at Greenwich 8 28 00 39 45 56 Refraction, Ta!)le XII —V 8" Parallax, Talile XIV 4-7" Table XXXVL Thermometer —2" Barometer — 1" sub. 1 04 39 44 52 ©'s sen^.idiamcter 15 46 ©'s true altitude 40 00 3 8 With the above estiniated time at Greenwich, we find, from the Nautical Almanac, the sun's declination 23° 21' 14'' N., the sim's semidiainetor 15' 46'', and the equation of time -|- 9\7. The latitude of Cape Cod being 42° 3' N., and as it is distant 9', in a south direction, the latitude of the ship is* 42° 12' N. * The ship being' on the meridian, we must adil the whole distance 9' to the latitude of Cape Cod, to get tlie latitude of tiic ship; but if the bearing be in any other direction, we must calculate by mesui'j TO FUND THE LONGITUDE BY A CHRONOMETER. 257 (v)'s true altitude 40" 00' 38" Latitude 42 12 00 Secant. . O.L^030 Polar distauce . ■ GG 38 46 Cosecant 0.03712 Sum 2 ) 148 51 24 [Lalf-sum 74 25 42 Cosine . . 9.42885 Ileanaiuder 34 25 04 Sine .... 9.75222 Sum 2)19.34849 Sine of liall-sum 9.G7424 corresponds to 3'' 45'" 29'. npp. time Equation of time by tlie Nautical Almanac -f- 9'.7 ftlean time at the place of observation 3 45 38'.7 Add the longitude of Cape Cod, 70° 4' — 4 40 1(^0 Rlean time at Greenwich 8 25 54°.7 Time by the chronometer 8 2G 30'.0 Error of the chronometer fur mean time at Green wicli. . . .fast 35".3 EXAMPLE VIII. At New York, on the 5th of June, 183G, by a transit of the sun over the meridian, it was found that a chronometer was too fast for mean time at Greenwicli, by 2'" 8\5; and by another transit, on the next day, June Gth, it was too fast 2'" lO'.O. From these observations it follows, that the daily gain of the chronometer at that time was 1'.5. The instrument was then taken on board a ship, which sailed immLdiutc'ly on a voyage along tlie seacoast, and, after a passage of 10 days, arrived at a place whose longitude from Greenwich had been well ascertained. There, by o'jsorvation, it was (bund, that at noon, June 16, 183G, the chronometer was 2'" 30'.5 too fast for mean time at Greenwich ; having gained 22'.0 in 11 days; or at the mean daily rate of 2^0, instead of 1S5, which was the rate at the commencement of the voyage. Now, the chronometer being a new one, and it being generally found that the daily rate of such an instrument is constantly increasing, it is required to find the error of the chro- nometer at noon on every day of the voyage, supposing the daily rate of gain to increase uniformly; the object in thus finding the actual error on each day, being for the purjjose of ascertaining the longitudes of several capes and places which were observed during the voyage. The calculation of this example is made as in the annexed table. Its first column contains the days of tlie month. The second column contains the estimated error of the chronometer, supposing its daily gain to be 1\5, as at die commencement of the voyage. The third column contains the gain of the chronometer on every successive day, supposing this uniform daily increment of the rate to be a fraction of a second, which is represented by /. The fourth column contains the error of the chronometer on each day, e.\i)ressed in terms of/; the numbers in this column are found by adding succes.-iively the daily gain in column 3, to the error of the chronometer on the [)reced- ing noon. Thus, on June 13, the error at noon is 2™ 20'.5-|-28 t, and the daily gain between June 13th and 14th is 1^5-|-8 t ; adding together these two quantities, we obtain 2™ 22'.0-j-3G t, for the error of the chronometer, June 14, at noon ; being the same as in coUunn 4. Proceeding in this way, by successive additions, we obtain the error of the chronometer, June 16, at noon, equal to 2'" 25^0-|-55<,• and as this was found by observation to be 2'" 30'..5, wc shall have 2™ 25'.0-|-55<=::2"' 30'.5 ; whence we get 55 < = 2™ 30'.5 — 2"' 25'.0 =r 5'.5. Dividing this by 55, the coefiicient of<, we get /:^0'.l. Hence tlie daily gain in the acceleration is izr: O'.l ; and by substi- tuting this value of t in the errors at noon on the difterent days, given in column 4, we get the corresponding numbers in column 5, which represent how much the chronometer is too fast for mean time at Greenwich at each noon, from June 5 to June 16 ; supposing the daily acceleration of the rate to be 0^1, or yV of a second. Taking the successive daily differences of these errors, we get, as in colun^.n 6, the daily gain of the chronometer, which increases from 1^5 to2\5 during tlie voyage. of the table of difference of latitude and departure, the latitude and longitude of the ship, at tlie time of observation, in the same manner as when ta-king a departure from tlie land. Thus, if the true hear- ing of the cape, in the above example, were S. S. \V. 9', tlicdiflcreiicc of latitude will hi 8'..3, departure 3'.4., dillc-roiicc of longitude 4'.() ; lience the latitude of the ship will be 42° 3' + 8'. 3 = 42° ll'.3 = 42° 11' 18", and the longitude 70° 4' — 4'.G = Gy° 59'.4 = G9° 59' 24'' ,; which must, in this case, be »6ed instead of the above values. 33 258 TO FIND THE LOxXGITUDE BY A CHRONOMETER Col. 1. Col. 2. Col. 3. Col. 4. Col. 5. Col. 6. Error of the chronometer, Daily gain, sup- posing the rate Error of the chro- Error of ike jDai/)/ ™iH Daies. siipposing the daily cra'ui tu be ls.5. to be uniformly increasing by the quantity t. day, expressed as terms of t. noon each day in time. in seconds June 5. 2'" 08».5 1«.5 2m 08'.5 2™ 08^5 1^5 " C. 2 10.0 1 .5 + « 2 10.0 2 10.0 1 .G " 7. 2 11.5 1 .5 + 2 t 2 11 .5 + < 2 11 .6 1 .7 " 8. 2 13.0 1 .5 -f 3 < 2 13.0 + 3< 2 13.3 1 .8 " 9. 2 14.5 1 .5 + 4 < 2 14 .5 + « 2 15.1 1 .9 " 10. 2 IG.O 1 .5 + 5< 2 IG .0 + 10« 2 17.0 2.0 " 11. 2 17.5 1 .5 + 6 i 2 17 .5 4- 15 « 2 19.0 2.1 " 12. 2 19.0 1 .5 4- 7 < 2 19.0 + 2U 2 21.1 2.2 " 13. 2 20.5 1 .5 4- 8 f 2 20 .5 + 23 < 2 23.3 2.3 " 14. 2 22.0 1 .5 + 9 < 2 22 .0 + 30 « 2 25.6 2.4 " 1.5 2 23.5 1 .5 + 10 « 2 23.5 + 45^ 2 23.0 2 .5 " 1(1. 2 25.0 o 25 .0 -f 55 f 2 30 .5 EXAMPLE IX. We shall suppose, as in the preceding example, that at noon June 5, 1836, the chronometer ^vas too fast 2™ 8".5, and at noon June C, 1836, it was too last '2'" lO'.O; indicating a daily gain of I'.5. In proceeding on a voyage, the vessel stopped, on the 10th of June, 1830, at a port whose longitude was unknown ; and, witli a view to determine this longitude by the chronometer, observations were made, by wiiich it was found, that between the successive noons of June 10th and June 11th, 18.3(3, the daily gain was 2".0. It is req^uired to determine the error of the chronometer on the different days, supposing the daily gain to ije uniform. The actual rate of the chronometer is i)articularly required on the 10th and 11th of June, so that we may use the rate of the chronometer in finding the longitude of the place arrived at. In the intervals of the two days, commencing June 5 and June 10, the daily gams were respectively 1".5 and 2*.0 ; having increased 0^.5, in tlie daily rate, in an interval of 5 days; being at the rate of O-.l per day. With this daily increase, we can comjjute the daily gain, as in column 2 of the following table ; and from these numbers we can deduce successively the errors of the chronometer, as in column 3. Col. 1. Col. 2. Col. 3. Dates. Daily rate of Chronometer too gain. fast. June 5. 1^5 2'" 08«.5 " 6. 1 .6 2 10.0 " 7. 1 .7 2 11 .6 " 8. 1 .8 2 13.3 " 9. 1 .9 2 15.1 " 10. 2.0 2 17.0 " 11. 2 19.0 Hence it ai)pears, that on June 10, the chronometer was 2™ IV'.O too fast tbr Greenwich mean time; and on June 11, it was 2™ 19'.0; which can be used in determining the longitude. TO FIND THE LONGITUDE BY A CHRONOMETER. 259 Precautions in using a chronometer. We shall close this article on chronometers, by the following directions relative ta the manner of taking care and using them, published in a small tract on this subject, by Mr. Stansbury : — In carrying a chronometer to and from a ship, you must secure the gimbals by the stay, to keep it steady ; and by all means avoid giving the instru- ment a quick circular motion. A chronometer should be placed so as to expose it as little as possible to sudden shocks, from tlie sea striking the ship, or from the shutting of doors, &c. It ought not to be exposed to a current of air. Nothing magnetic should be allowed near it. When the chronometer is on board a ship, free the staj', let the instrument swing horizontally, and place it securely, and so that it may be dis- turbed as little as possible during the voyage ; using for deck-observations a common watch, which must be compared with the chronometer before and after any obser- vation. In winding up a chronometer, turn it over gently ; jnit the valve back, a{)ply the key, turn it moderately, and avoid sudden jerks. A pocket chronometer must be held inunovable in the one hand, whilst winding with the other, in order to avoid a circular motion, which may not only alter the rate, but injure the instrument. If a chronometer should happen to run down, or stop, it nnist, when wound up, have a quick circular motion in the plane of the dial to set it agoing. Never touch the hands to set the chronometer, but wait till the time arrives at which they point. Be regular in winding. Get an observation as soon as you leave a port, to ascertain if you have the correct difference from Greenwich time ; and in case it should happen to stop, or to run down, during the passage, it may be corrected by lunar observations, by the method explained in pages 251, 252. It has been found that chronometers gain by an increase of the density of the air, and lose by a decrease of the density. The firing of guns on board a vessel will sometimes alter the rate of going, unless the instriunent be well suspended, or held in the hand during the fii-ing. Any sudden jar will sometimes alter the rate. The imperfection of the oil used wiii, after some time, impair the instrument. The mechanism for correcting the changes in the temperature may not do it com- pletely, and some error may arise from this source. Notwithstanding these various causes of error, it is wonderful to observe how accurately some of these chronometers perform their office. The manner of using a chronometer in finding the longitude by means of observa- tions of the moon's transits over the meridian, with a transit instrument, will be given in the Appendix to this work. On a variation chart. In the year 1700, Dr. Ilalley published a chart, in which the lines of the variation of the conipass were drawn, for the purpose of determining the longitude by means of the observed variation ; and, since that time, several charts of this kind have been published for the same purpose ; but the method is not sufficiently accurate to be of any practical use. A variation chart is, however, useful, as a subject of scientific intiuiry, and for the purpose of correcting a ship's course. The latest and by far the best work of this kind, is that of the Admiralty, published in 185'J, and repub- lished by E. &, G. W. Blunt, in 1860. Every navigator should have it for daily use. 2G0 METHOD OF KEEPING A SHIP'S RECKONING OR JOURNAL AT SEA. A ship's regkoning is that account, by which it can be known at any time where the ship is, and on what course or courses she must steer to gain her port. Dead RECKONING is that account deduced from the ship's run from tlie last observation. THE LOG-BOARD. H. K. F. Courses. Winds. Lce- icay. Transactions. 2 6 S. W. N. E. 4 5 5 G 5 N.W.byW. 8 5 Moderate gales 10 4 5 E. N. E. N. W. and fair weather. 12 4 5 At 8 A. M., saw 2 4 5 a ship to tlie 4 4 5 northward. G 4 5 8 5 S. W. W. N. W. 1 No observation. 10 4 5 12 4 The daily occuiTences on board a ship are marked on a board or slate, called the log-board or log-slate, kept in the steerage for that purpose, being usually divided into seven columns : the first contaiss the hours from noon to noon, being marked by some for every two hours, but usually for every single hour; in the second and third columns are the knots and fathoms the ship is found to run per hour, set against the hours when the log was hove. Some navigators do not divide the knot into ten fathoms, but into half-knots only, making the tliird column H. K. The fourth colimin contains the courses steered bj^ compass ; the fifth, the winds ; the sixth, the lee- way ;* and the seventh, the alteration of the sails, the business done aboard, and what other remarks the oflicer of the watch thinks proper to insert. For it should be obsrrvetl, that it is usual to divide a ship's company into two parts, called the starboard and larboard watches, who do the duty of the ship for four hours and four hours, alter- nately, except from 4 to 8 P. M., which is divided into two watches. The i-emarks made on the log-board are daily copied into a book, called the Log-Book, which is ruled like the log-board. This book contains an authentic record of the shi])'s trans- actions ; and the persons who keep a i-eckoning, transcribe them into thc\i- journals, and thence make the necessary deductions relative to the ship's ])lace, every day at noon ; this o])eration is called working a dajfs loork. While a ship is in port, the reinarlv5 entered in the Log-I>ook m-e c-,\\\cAharhor-u'ork,OY harhor-journal ; and the day is then estimated according to the civil comjuUation, as on shore; that is, from midnight to midniglit; but at sea, the da3's work ending at noon is dated the same as the civil day, so that the day's work marked Blonday begins on Sunday noon, anrl ends on Monday at noon; the day thus marked is called a nautical day; the first 12 hours oemg marked P. M., the latter A. M. There are various ways of kecjiing journals at sea, according to the different tastes of navigators. Some keep only an abstract of each day's transactions, specifying the weather, what sliips or lunds were seen, The cause of the lee-way, and manner of allowing for it, are c.xi)laiiiecl in llic following- i)age. METHOD OF KEEPING A JOURNAL AT SEA. 261 accidents on board, the latitude, longitude, course, and run ; these particulars being drawn from the ship's Log-Book. Others keep a full copy of the Log-13ook, and the deductions drawn therefrom, arranged in proper columns ; this is the most satisfactory method to tliose who may have occasion to uispect the Journal ; and we have adopted it in the following, but shall give an abstract, at the end, conformable to the other method. When a ship is about losing sight of the land, the bearing of some noted place (whose latitude and longitude are known) must be observed, and its distance estima- ted and marked on the Log-Book ; this is called takiiig a departure. In working tiiis first day's work, the calculation is to be made in the same manner as if the ship had sailed that distance from that j)lace upon a course opposite to that bearing, and that course and distance are to be entered accordingly into the traverse table, after allow- ing for the variation. To allow for the variation. We have already taught the methods of finding the variation, which must be allowed on all courses steered, and on all bearings taken with the compass; to the right hand, if the varialion be east, but to the left hand, if ivest ; the observer being supposed to be placed in the centre of the comi)ass, looking towards the point from which the variation is to be allowed. EXAMPLES. Courses bij compass. Variation, in points. True courses. N. E. by E. 2 W. N. E. by N. N. E. U E. N. E. by E. i E N. W. 3 W. W. by IN'. S. K 3 E. S. by E. s. s. w. 1^ W. S. i W. E. S. E. u vv. E. 1 S. S. W. i w. h w. S. W. i S. N. N. E. IE. U E. N. E. i E. To find the he-icay, and alloio for it. The courses must likewise be cori-ected for lee-way ; the nature of which may be thus explained : — When a sliip sails upon a wind, in a fresh gale, that part of the wind which acts upon the hull and rigging, together with a considerable part of the force exerted on the sails, tends to drive her immediately from the direction of the wind, or, as it is termed, to leeward. But since the !)ow of a ship exposes less surface to tlie water than the side, the resistance will be less in the first case than in the second ; the velocity, therefore, in the direction of her head, will, in most cases, be greater than the velocity in the direction of her side, and the ship's course will be between the two directions ; and the angle contained between the course towards which the ship's head is directed, and the course she really describes through the water, is termed her let- way. The quantity of lee-way to be allowed will depend upon a variety of circum- stances ; as the mould and trim of the ship ; the quantity of sail she carries ; her velocity through the water, &o. : hence no general rules can be laid down with accui-acy that will determine the quantity of lee-way in all cases. The following have, however, been usually given by most writers on navigation : — 1. When a ship is close-haided, with jdl her sails set, the water smooth, and a light breeze of wind, she is then supposed to make little or no lee-way. 2. When the top-gallant sails are handed, allow 1 point. 3. When under close-reefed topsails, allow 2 points. 4. When one topsail is handed, allow 2^ points. 5. When both topsails are handed, allow 3^ points. 6. When tlie fore-course is handed, allow 4 points. 7. When under the mainsail only, allow 5 points. 8. When under a balanced mizzen, allow 6 points. 9. When under bare poles, allow 7 points. As these allowances depend entirely on the quantity of sail set, without regard to any other circumstance, it is evident that they can be considered only as probable 262 METHOD OF KEEPING A JOURNAL AT SEA. conjectures, and may indeed sei-ve to work up the day's work of a Journal that has been neglected. But smee the computation of a ship's way depends mucli upon the accuracy of this allowance, it would be proper for the officer of the watch to mark the lee-way on the log-board, in the column reserved for tllat purpose. The lee-way may be estimated by observing the angle which the wake of the ship makes with the point right astern, by means of a semicircle marked on the tafterel, and divided into points and quarters ; by means of which the angle contained between the direction of the wake and the point of the compass directly astern, may be easily as- certained. The lee- way, thus determined, is to be allowed on all courses steered, to the right hand of the course steered, when the larboard tacks ait aboard,* but to the left hand, ivhen the starboard tacks are aboard ; the person making the allowance being supposed to be looking towards the point of the compass the ship is sailing upon. Courses steered. N. W. E. N. E. E. S. E. W. by N. E. N. E. h E. Winds. N. N. E. North. South. N. by W. S. E. EXAMPLES. Lee-way. 1 point. 2 1 h 3 True courses. N. W. by W. East. E. by S. W. h N. N. E. h N. When the variation and lee-way are both to be allowed on a course, you may do it at once, by allowing their sum when they are both the same way, or their difference when the allowance is to be made in differentways, taking care to make the allowance in the same way as the greater quantity ought to be, whether it be the variation or lee-way. EXAMPLE L A ship steers W. by N., with her lar- board tacks aboard, and makes one point lee-way, there being two points westerly variation. Required the true course. Lee- way to the right-liand 1 point. Vai-iation to the left 2 points. Difference allowed to the left . . 1 point. Whence the course is west. EXAMPLE n. A ship steers E. S. E., with her star- board tacks aboard, and makes two ))oint3 lee-vvay, there being one point westerly variation. Required the true course. Lee-way to the left 2 points. Variation to the left 1 point. Sum allowed to the left 3 points. Whence the coui-se is E. by N. In a violent gale, with a head wind and heavy sea, when it woidd be dangerous to carry sail, it is usual to lie to under sufficient sail to prevent the vessel from rolling so much as to endanger the masts and rigging. When a ship is lying to, the tiller is put over to leeward, and when the ship has head-way, the rudder acts upon her to bring her to the wind ; the ship then loses her way in the water, which ceasing to act on the rudder, her head falls off from the wind, and the sail which is set fills and gives her fresh way through the water, which acting on the rudder, brings her head again to the wind. Thus the ship is kept continually falling off and coming to. In tliis case, you must observe the points on which she comes iqi and falls off, and take the middle between the two points for the apparent course, from which allow the vai'iation and lee-way, and you will obtain the true course. EXAMPLE. A ship, lying to under lier mainsail, with her starboard tacks aboartl, comes uj) E. by S., and Ihlls off N. E. by E., there being one point westerly variation, and she makes 5 jioints lee-way. What course does she make good ? The middle between E. by S. and N. E. by E. is E. by N. ; and by allowing ti points to the left hand (viz. 5 for lee-way and 1 for variation) tlie true com-se will be obtained, N. by E. I ec llie note, page 199. METHOD OF KEEl'IXG A JOURNAL AT SEA. 2G3 To exercise the learner, we sliall add tlie examples ol' correcting for variation and lee-way contained in the following table : — THE TABLE. Courses steered. N.W. A W. W. w. s. w. w. W. by N. s. w. s. s. s. w. s. w. w. W. by N. S. E. by S. E. N. E. E. E. S E. S. E. VV. S. W. W. by N. N. W. S. N. bv E. N. W.'by N. N. W. by W. W. by S. Winds. N. N. E. N. N. W. S. s. s. w. N. by W. W. N. W. W. S. V/. W. N. W. by W. S. S. VV. N. by W. E. S. E. S. h E. N. N. b^ E. E. S. E. N. E. S. S. W. by S. W. S. VV. w. s. w. N. W. by W. W. by S. N. by E. N. W. by N. Lee- Varia- way tion points. points. h 1 W. i 1 W. 1 1 W. 5 1 W. H k\v. U 5 W. 1 14 w. 1 li w. i H w. 11 li w. 1 li w. o li w. 1 li w. n li w. 1 li VV. liW. h 1:1 W. h 15 VV. a li| w. 1 Yi W. 1 V\ W. 1 I E. k 1 E. li 1 E. l.-t liE. n 2i E. Courses corrected. N. r^ W. S. 6h W. S. (\\ W. \V. S. 7 W. S. 11 W. S. S. E. S. i E. S. S. W. i W. W. h N. W. S. W. I W. S. i W. E. by N. E. N. E. i E. E. i N. E. N. E. I E. S. by E. i E. E. I S. S. W. by W. W. i N. N. W. I W. S. i E. N. N. E. 5 E. N. s w. N. W. by W. i W. W. .i S." If the ship has been acted npon by a current or a heave of the sea, yon must allow tlie sf^t and drift as a course and distance in the Traverse Table, as directed in l)age 125. Having corrected the courses for lee- way and variation, and estimated the dititancea sailed, the latitude and longitude in at noon are to be found by either of the preceding methods of sailing. Tiie latitude and longitude, thus calculated, are called the latitiide and longitude by (/ea(/-?-ecA'oni/?n;-; and if the real course and distance made good by the ship could be e^j the sex-t>xy, and adopt ASTKO^o:siiCAi. time, it would lessen their labor and tend to much greater accuracy in their daihj works. Why cannot this be done ? 8. Find on a general chart the spot corresponding to the latitude and longitude by observation, and that place will represent the situation of the ship, whence the bearing and distance of the intended port may be found. The same may be obtained by middle latitude sailing, by inspection of Table II., thus: Find the middle latitude between the place of the ship and the proposed place, and seek for that latitude as a course in Table 11., and find, in the corresponding page of the table, the difference of longitude (between the ship and the proposed place) in the distance column, opposite to which, in the latitude column, will be the departure. Seek in Table II. for this departure and the difference of latitude (between the ship and the proposed place) till they are found to agree ; corresponding thereto will be the bearing and distance required. If the magnetic bearing be required, the variation must be allowed on the true Ijearing ; to the right hand if the variation is westerly, or to the left hand if easterly. 9. When the latitude, by account, is uncertain, the known position of the ship " on a line of bearing" may be of very great importance. In this case the mode of proceeding is shown on page 205, or in Sumner's work. We shall now proceed to exemplify the above rules; first by a few examples of separate days' works, and then by a Journal from Boston to Madeira, kept in the usual form. O-k 266 METHOD OF KEEPING A JOURNAL AT SEA. EXAMPLE I. Yesterday, at noon, we were in the latitude of 48° 21' N., and the longitude of 36° 28' W., and have sailed till this day at noon, as per log-board. Required the course and distance made good, with the latitude and longitude in. LOG-BOARD. H. 2 K. f) F. Courses. Winds. N. Lee- icaij. Remarks. S. W. by W. I W. These 24 hours, moderate gales 4 5 5 and cloudy weather. f) 5 N. W. At 4 P. M., spoke ship Washing- 8 5 ton, from New York, bound to 10 3 G S. W. 1 W. Cork. 12 3 4 2 3 4 4 4 5 6 4 6 At 6 A. M., stowed the anchors, 8 5 S. W. i S. W.N.W. and unbent the cables, and coiled 10 4 5 them between decks. 12 4 Variation 2^ points westerly.* Courses. Dist. N. S. E. W. S. W. 4 S. S. S. VV. h w. S. by W.'i W. 43 39 27 33.2 34.4 aj.8 27.3 18.4 7.8 DifF. Latitude 93.4 Dep. 53.5 TRAVERSE TABLE. By examining the log-board, it api)ears that the siiip goes large, and makes no lee-way ; there- fore, by allowing the variation on each of the courses, they will stand as in the adjoined Traverse Table. Then the distances marked on the log-board must be summed u]) and doubled, because they are marked only for eveiy two hours.f In allowing for the knots, we must reckon 10 to a mile ; and when the tenths are above 5, we must add 1 mile to the distance. Having found the distances, we must find the corresponrling differences of latitude and departures, in Table I. or II., and then, with the whole difference of latitude and departure, we must find the course and distance made good, and the difference of longitude, by Case II. of Middle Latitude Sailing. In the ])resent e.xample, the difference of latitude is 93' = 1° 33' S. Yesterday's latitude 48 21 N. The difference is the latitude in 40 48 N. Sinn of the latitudes 95 9 Middle latitude 47 34 Widi the difference of latitude made good, 93.4 S., and the departure, 53.5 W., We must enter Table II., and we shall find they corresjiond nearly to a course of S. 30° W., and distance 108 miles. Then, with the middle latitude 47° 34', or 48°, we must enter Table II., and we shall find the departm'e 53.5 in the latitude column; ojiposite to which, in the distance column, is die Difference of longitude 80' z=z 1° 20' W. Longitude left 36 28 W. Sum is the lon and ll^ 15'", and found, by the first method, on page 180, the latitude at the time of the second observation to be 41° 44' N. Variation 5 of a point westerly. Course. S.80"15'E. Dlst. 162 Diff. Lat. S. 27 Dep. E. IGO Lat. by D. R. N. 41° 43' Lat. by Obs. N. 41° 43' Diff. Lonsr. E. 3° 35' Longitude in, by D. R. Lun.Obs. Chron. W. 64° 22' 64° 20' TRAVERSE TABLE. Courses. Dist. N. S. E. W. E. iS. E. 1 S. E.S.E4E. 42 42 79 2.1 6.2 19.2 41.9 41.5 76.G DifF. Lat. 27.5 160.0 Dep. The variation being allowed on each course, and the distances summed up, they will stand as in the adjoining traverse table ; hence, by means of Table I., we find the difierence of latitude 27.5, and the departure IGO.O, which correspond to the course of nearly S. 80° 15' E., and the distance 1G2 miles. Yesterday's latitiulc 42°10'N. Difference of latitude 27 S. Latiludein 41 43 N. Sum of latitudes 83 53 Middle latitude 41 6G With the middle latitude 41° 5G', or 42°, as a course, we must enter Table IL, and seek for the dcjiarture IGO.O in the latitude column; the nearest number to which is 159.8, correspond- ing:^ to 'I'e distance 215, which is therefore the dillercnce of longitude, equal to. . . . 3° 35' E. Yesterday's longitude C7 57 W. Longitude in G 4 22 W. To find Ike hearing and distance ofFunchal, Latitude in 4I°43'N. Funchal's latitude.. 32 38 N. Difference of latitude 9 5 60 meridional parts 2759 IMcridional parts 2073 Meridional diff. latitude G8G Longitude in 64°22'\V. Funchal's longitude. . 16 54 W. Difference of longitude 47 28 60 In miles 545 In miles 2848 By Case I. of Blercator's Sailing, we find the bearing of Fimchal to be S. 7G° 27' E., and its distance 2326 miles. When the sun was upon the meridian, the altitude of his lower limb was observed, and found to be 50° 31', to which add 12' for the semidiameter, parallax, and the dip of the horizon ; the refraction (given in Table XII.) for this altitude, being small, is neglected; hence the correct central altitude was 50° 43', which, being subtracted from 90°, leaves the zenith distance 39° 17', which must be called north, because the sun bore south when on the meridian ; then, in Table IV., we find the sun's declina- tion at noon at Greenwich =2° 22' N.; to this add the correction 4' taken from Table V., correspond- ing 1.0 the ship's longitude ; the sum is 2°26'N.=:the correct declination; and since the declination and zenith distance are both north, we must add them together, and the sum will be the latitude by observation = 41° 4-3' N., which agrees with the latitude bv account. 372 JOURNAL OF A VOYAGE H. Courses. E. S. E. Winds. N. by E. N. N. E. N.E.byN. Lee- ivay. Remarks on board, Sunday, March27, 1860. All these 24 hours, fresh breezes and clear. Observed the distance of the sun from the moon. The longitude at noon, by ihejirst method, on page 231, was found to be 60° 16' W. Meridional alt. sun's lower limb. . 51° 52' Add for semidiameter, dip, &c.. . 12 Sun's correct altitude 52 04 Subu-act from 90 00 Sun's zenith distance 37 56 N. Sun's correct declination 2 50 N. Latitude observed 40 46 N. Thermometer at noon 47o do. at midnight , 45o Barometer at noon 29.70 do. at midnight 29.62 Variation | of a point westerly, per amplitude. Course. E.S.E.IE. Dist. 192 Dif. Lat. S. 47 Dcp. E. 186 Lat. Inj D. R. N. 40° 5fi' Lat. bij Ohs. N. 40° 40' Diff. Lons- E. 4° 8' D. R. Longitude in, by W. G0° 14' Lun.Obs. o D 60° 16' Chron. TRAVERSE TABLE. Course. Dist. I9'i N. S. DiiF. Lat. 4G7 E. 1 W. E.S.E.5E. 18G.2 Dep. The ship sailed all day ujjon tiie same course, which, being corrected for the variation, is E. S. E. % E. ; the whole distance sailed is 192 miles, and the ditierence of latitude is 47 miles= 0'47'S. Yesterday's latitude 41 43 N. Latitude by daily reckoning 40 .56 N. So that the latitude by account difTers 10 miles from the latitude by observation. Latitude yesterday by observation 41° 43' N. Latitude by observation this day 40 46 N. Difference of lat. by observation.. 57 SSum of latitudes 82 29 Middle latitude 41 14 With the middle latitude 41° 14' as a course, and the departure 1SG.2 as diiference of latitude, we find the corresponduig distance 218, wliich is equal to the ditference of longitude 4° 8' E. Yesterday's longitude 04 22 W. Lontritude in GO 14 W. Tojind (he heating and distance of Funckal. Latitude in 4()°40'N. Funchal's latitude 32 33 N. Ditf. of latitude.. 8 8 GO In miles 488 Meridional parts 2G83 Meridional parts 2073 Mer. diff. lat. 010 Longitude in 00° 14' W. Funchal's longitude 10 54 W. DifTerpnce longitude 43 20 GO In miles 2G0O With tiie meridional difference oflatitude and difference of longitude, the bearing is found to be S. 7()° 48' E. ; witli that, and the proper difference oflatitude, the distance is found to be 2137 miles,* by Case I. Mercator. * If llie course was calculated to seconds, and the mcridioiml parts taken to one or two places of decimals, it would sometimes make a dilTerciice of a few miles in the calculated distance. We may here remark, that, as this Journal is only designed to exemplify the rules of navigation, we have not endeavored to give the tnic variation. FROM BOSTOxN TO .MADEIRA. 273 H. K. F. 1 ~i 2 7 3 6 G 4 G 6 5 G G 6 7 5 4 8 5 4 9 5 6 10 5 G 11 5 6 12 5 6 1 5 3 2 5 3 3 5 5 4 5 5 5 G 6 G 7 6 8 G 9 G 10 6 11 5 12 5 Courses. S.E.byE. S. E. S.E.byS. Winds. N.E.byE. E. N. E. E. by N. Lee- way. 1 Remarks on board, JVonday, March 28, 18G0. Fresh gales, with rahi. At 4 A. J\I., spoke the ship Franklin, from Philadelphia, boimd to Lisbon. At noon, observed meridian alti- tude S's lower limb 53° 57' Add for semidiameter, &c 12 0's correct altitude 54 9 Subtract from 90 00 #'s zenith distance 35 51 N. 0's correct declination 3 13 N. Latitude observed 39 4 N. By an altitude of the pole star taken at 9"* P. M., the latitude was found, by the rule on page 206, to be 38o 32' N. Thermo, at noon 48° do. at midnight 44^ Barom. at noon 29.60 do. at midnight 29.88 Variation, 5 of a point westerly. Course. S.42°29'E Dist. 138 Diff. Lat. S. 102 Dep. E. 93 Lat. by D. R. N. 39° 4' Lat. by Obs. N. 39° 4' Diff. Lon 25' W. 26° 17' W. 26° 2G' TRAVERSE TABLE. Courses. Dist. N. S. E. W. East. S. E. by E. N.E.byE.^E. 50 80 GO 25.7 44.4 50.0 6G.5 54.2 25.7 44.4 25.7 170.7 Dep. Diff : Lat. 18.7 The lee-way and variation being allowed on the courses, they will stand as in the adjoined traverse table ; then, with the dif- ference of latitude 18.7, and the departure 170.7, the course is found to be S.83° 45' E., and the distance 172 miles. Yesterday's latitude 33° 13' N. Difference of latitude 19 S. Latitude in 32 54 N. Sum of the latitudes 06 7 Middle latitude 33 3 With the middle latitude 33° 3', and tlie de- parture 170.7, we find the difK of long, to be nearly 204 miles . . = 3° 24' E. Yesterday's longitude 29 49 W. Longitude in 26 25 W. To find live beaiing and distance of Funchal. Latitude in 32° 54' N. Funchal's latitude 32 38 N. Difference of latitude IG Sum of latitudes 65 32 Middle latitude 32 46 Longitude in 26°25' W. Funchal's longitude 16 54 W. Difference of longitude. 9 31 GO In miles 571 Hence the bearing of Funchal is found to be S. miles. 3° 5' E., and its distance 480 FROM BOSTON TO MADEIRA. 285 II. K. F. 1 7 5 2 7 5 3 8 4 8 5 8 5 6 8 5 7 9 8 9 9 9 10 9 11 9 12 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 9 10 9 11 9 12 9 Courses. E.byS4S. E. by S. JFinds, South. Lee- way. Remarks on board, Saturday, April 0, 18G0. Fine breezes, with variable weatlxn-. Meridian altitude gi's lower limb 64° 37' Correction for dip, &c 12 0's correct altitude 64 49 Subtract from 90 GO ^'s zenith distance 25 UN. @'s declination 7 45 N. Observed latitude 35 56 N. Observed the meridian altitude of the planet Mars ; and the latitude at noon, by the rule on page 174, was found to be 32° 54' North. Took a lunar observation, by observing the distance,of the moon from Spica : the lon- gitude at noon, deduced therefrom, was 22° 09' W. Thermometer at noon G4° do. at midnight 61° Barometer at noon 29.85 do. at midnight 29.80 Variation, li point westerly. Course N.89°12'E, Dist. 210 Diff. Lat. Dep. E. 209 Lat. by D.R. N. 32° 57' Lat. by Ohs. N. 32° 5G' Diff- Lonrr. E. 4° 10' Longitude in, by D. R. D * Chron. W. 22° 15' W. 22° CD' W. 22° 17' TRAVERSE TABLE. Courses. Dist. N. S. E. W. E. i S. E. <^N. 120 90 8.8 5.9 119.9 89.C Diff. Lat. 8.8 5.9 2.9 5.9 209.5 Dep. The variation being allowed on the courses, they will stand as in the adjoined table; then, with the difference of latitude 2.9, and the departure 209.5, the course is found to be N. 89° 12' E., and the distance 210 miles, nearly. Yesterday's latitude 32° 54' N. Difterencc of latitude 3 N. Latitude by account 32 57 N. With the middle latitude 32° 55', and the departure 209.5, the difference of longi- tude is found 250 miles. . . =r 4° 10' E. Yesterday's longitude 2G 25 W. Longitude in 22 15 W. To find the hearing and distance of Funchal. Latitude in 32°5G'N. Funchal's latitude 32 38 N. Difference of latitude 18 Sum of latitudes 65 34 Middle latitude 32 47 Longitude in 22° 15' W. Funchal's lonj^itude 10 54 W. Difference of longitude . 5 21 60 In miles 321 Hence the bearing of Funchal is found to be S. 86° 11' E., and its distance 270 miles. 2S6 JOURNAL OF A VOYAGE FROM BOSTON TO MADEIRA. H. K. F. 1 ~9" T 2 9 5 3 9 5 4 9 5 5 9 5 6 9 5 7 9 8 9 9 9 10 9 11 9 12 9 1 9 2 9 3 9 4 9 5 9 9 7 9 8 9 9 9 10 9 11 12 Courses. S.KbyK E.byS.| S. Winds. s. s.w. Lee- way. Remarks on board, Sunday, April 10, 1860. All this day, fine breezes, with very clear weather. At 10 A. M., made the land ; the southern part of Madeira bearing per compass E. by S. I S., distant 19 leagues. Observed the distance of the moon from the star Fomalhaut; the longitude at noon was 17° 15' W. Thermometer at noon 65° do. at midnight 63° Barometer at noon 29.94 do. at midnight 29.90 Variation, \% point westerly. Course. S.83°57'E. Disl. 256 Diff. Lat. S. 27 D> ep. E. 255 Lat. by D. R. N. 32° 29' Lat. btj Obs. Diff. Loner. E. 5° 3' D.R. Longitude in, bij W. 17° 12' 17° 15' D * Chron. TRAVERSE TABLE. Courses. E.S. E.|E. East. East. Dist. Ill 90 57 N. S. E. W. 27.0 107.7 90.0 57.0 DifF. Lat. 27.0 254.7 Dep. In the traverse table are placed the bearing and distance of the land at 10 A. M., (after allowing the variation.) Hence the whole difference of latitude is 27 miles, the depart- ure 254.7, the course S. 83° 57' E., and the distance 256 miles. Yesterday's latitude 32° 56' N. Difference of latitude 27 S. Latitude by account 32 29 N. Sum of latitudes 65 25 Middle latitude 32 42 With the middle latitude 32° 42', and the departure 254.7, the dift^ of longitude is found to be 303 miles = 5° 3' E. Yesterday's longitude 22 15 W. Longitude of S.partof Madeira 17 12 W. Therefore the latitude of the southern point of Madeira, by account, is 32° 29' N., and its longitude 17° 12' W. These values differ but little from those in the Table of Latitudes and Longitudes ; we may, therefore, conclude that the Journal is nearly correct, and the latitude and longitude of tliat part of IMadeira well laid down. Monday, April 11, 1836. — Pleasant gales and fair weather. At 4 P. M., came to off" Funclial. At 8 P. M., went on shore. A« ABSTRACT OF THK FOREGOING JOURNAL. 287 o o g o o 0:5 O O H O <1 c:5 H CO "f to u 1 ■g m 3 be CO * CO o . O C5 IC n r-H Mn< 1"^ o c^ ■* i-( IC rH lO iO LO r-l CO 1-1 o 1— ( m o (^ o Cl —( o o cri o C5 -f rf in CO 00 CO >-, t- i>. t^ i> 00 00 00 i^ i^ i^ 00 QO 00 GO QO 00 ,F J" ai ai m aj 02 ai Yi m m m 02 CO «: CO CO CO i ^ s o 00 o CO -^ M CO lO O rH CO CO l^ o 00 CO i^ S ;; lO (N 1—1 1— 1 I— 1 CO Ol 1-1 O CO 1—1 1—1 lO rt< CM T— I ^ 5=^ o t^ •^ o QO 00 t^ «* O CO i-t O GO CO Ci CO CM -d" o o o lO lO O lO »/5 Tf -^ Tf CO CO CM CM (M -S^ — "f ^ ■s° is t~ 1^ O rt i^^ \n i^ Oi <0 '3 1— 1 f-l CO o •* I— 1 I— 1 o I— ( o •*2 S o o tri •P* S O CO '^ in —1 O 00 CO CO CM f^ O 13 lO ■r}< n< •<* CO CO CM (M l-^ ^ 13^ p. o? Tf OJ O? '^ M 00 o ^ CO tc i-» Ci lo LO (M ^ lO a 1—1 r-i r-l CO (M in CO 1—1 r-l lO •^ CM 1— i 1^ 1-1 c5 o t o GO 00 l^ -^ o irj 1-1 O 00 CO a CO CM r- 1^ 'O CO o O UO o to iO -rj" Tj 'a* CO CO (M c^ C^ i-( 1-1 CO ^ -d O w CO ■^ no o l^ lO -H CO CO C3 p =^ •^ ■* ■^ CO 1—1 n !^ ^ ko •^ rO o 1-H o c^ t^ r^ CO •nf ^ CO c^J rt "o ■^ •^ CO CO CO CO CO CO CO CO ^ Q? ^ c C5 o CO r-^ "* 00 lo >n 3 1— j ■^ »o CO ic »o CO lO 1-1 lO (M '^ 1—1 lO in CM irt lO f^ • OJ 1— 1 o O l^ l^ J^ t^ CO CO rj -H CO CO CO CO CO c^^ (M CT a •^ ■^ 'I' CO CO CO CO CO CO CO CO CO CO CO P-i ^ Tj- "H ^ 01 2? QO CO rt .H CJ W i^ '^ 1-1 CO o (M o "^ « Q Ci o o CO 00 CO lO O O —1 o o -^ l-N. I— 1 in iT 1— 1 '"' 1—1 I— 1 Ol Ql Ol 1-1 rH OJ Oi CM CM o c4 H* y c4 H H p4 h W H W w w W ^1 Co a ^ -f lo CO S ~ '^ "■ lO Ci o li^ CO -* o lO CM CO 2 ,5 2-. s? 0{ 2 1-1 (fi ■^ CO t^ **^ ■^ CO CO t^ CO — 1 CM o •<* CO 1—1 C5 O 00 GO l>. 00 i> r^ CO 00 00 00 00 00 00 a ^ ai ai m ;d rd !» CO 03 OJ CO c/i CO _^ . 3 CO O ^ QO "^ ^ ** ^ " -* ** -* ^ ^ 4 ^ ii ^ ^ a 3 1—1 l-O r^ ivT go" cT o -T r-^ C^ CO" rjJ" LO" co~ jvT CO d^ cd" ^ o< OJ !M (M 0< CO CO -* ^ j^ ^ rH ^ r- o "p « "p "p "p p t:^ 7Z TZ 7:2 7:3 n; 772 r3 72 — -a =o 3 s 3 3 3 3 3 s s s s Si L> ;_ C^ &- c i ^ i iH I, IL, Pi r-> ^ p ^ <1 < <5 <1 < a- a- <; Q •< a ai < Q s ^ s ° ? s 1 s § g § B S H ^ H • <: ^ ■ man-of-war would go in smooth wa- 5 tcr and clean full. 1 to 2 knots. 3 " 4 « 5 " 6 " J Royals. Top-gallant sails over single reefa Double-reefed topsails. I Triple-reefed topsails, t Close-reefed topsails and courses, under Close-reefed main-topsail and reefed foresaiL " Storm stay-sails. " Bare pole.s Letters to denote the state of the weather, b. Blue sky ; whether with clear or hazy atmdsphere. c. , Cloudy ; but detached opening clouds. d. Drizzling rain. f. Foggy — F, thick fog. g. Gloomy, dark weather, h. Hail. 1. Lightning. m. Misty, hazy atmosphere. o. Overcast ; the whole sky being covered with an impervious cloud. p. Passing temporary showers. q. Squally. r. Rain ; continued rain. s. Snow. t. Thunder. u. Ugly, threatening appearance of the weather. V. Visibility of distant objects whether the sky be cloudy or not. w. Wet dew. Under any letter, indicates an extraordinary degree. All tne ordinary phenomena of the weather may be easily recorded by the combination of these letters. Thus : g^ V. Gloomy, dark weather, but distant objects remarkably visible. TO FIND THE LONGITUDE BY A CHRONOMETER. 289 [Continued from page 258.1 EXA]MPLE X. Suppose a vessel in a port in west longitude on the Gth of June, 1836, finds the daily rate of the chronometer to be -{- 5% and after steering west, in 18 days arrives on the 24th of June in port, and finds by observation tliat the daily rate is -j- 8". Required the correction of the observed longitude on the 18th of June. The daily rate at tiie first port is 5' The daily rate at the second port is 8' Sum 13 Mean daily rate 6^5 Let the difference of longitude between the two places by the first daily rate be 23°5(y 00" And by tiie mean daily rate 23 43 15 DiflTerence is the correction of the longitude of the second port for 18 days, and is easterly 6 45 Log. of the correction of the second port, & 45'' = 405" 2.60746 Log. of 18 days by Table A. ar. co 7.76700 Constant Log 10.37446 Log. of 12 days (from 6th to 18th of June) by Table A 1.89209 Log. of correction, 184".7 = 3' 4" 7 2.26655 This correction, it is evident, gives the place of observation on the 18th more east- erly, because the second place of observation is to tlie eastward of the position given by the daily rate at tlie port of departure. Having the coiistant loganthm as above, the coiTections for the other days are readily found, by substituting for the log. of 12 days the log. from Table A. of the days elapsed since the rate was first ascertained. This method, by Rossel, is found in the 3d vol. of Biot. Astronomie Physique. The logarithms are used in the same manner as proposed by Galbraith. TAELE A. Days. 1 Log. 0.00000 Days. 21 Log. 2.36361 Days. 41 Log. Days. 61 Log. 3.27669 Days. 81 Log. 3.52127 Days. 101 Log. 3.71189 2.93500 2 0.47712 22 2.40312 42 2.95569 62 3.29070 82 3.53186 102 3.72041 .3 0.77815 23 2.44091 43 2.97589 63 3.30449 83 3.54233 103 3.72884 4 1.00000 24 2.47712 44 2.99564 64 3.31806 84 3.55267 104 3.73719 5 1.17609 25 2.51188 45 3.01494 65 3.33143 85 3.56289 105 3.74547 6 1.32222 26 2.54531 46 3.03383 06 3.34459 86 3.57299 106 3.75366 7 1.44716 27 2.57749 47 3.05231 67 3.35755 87 3.58297 107 3.76178 8 1.55630 28 2.60853 48 3.07041 68 3.37033 88 3.59284 108 3.76982 9 1.05321 29 2.63845 49 3.08814 69 3.38292 89 3.60260 109 3.77779 10 1.74036 30 2.66745 50 3.10551 70 3.39533 90 3.61225 110 3.78569 11 1.81954 31 2.69548 51 3.12254 71 3.40756 91 3.62180 111 3.79351 12 1.89209 32 2.72263 52 3.13925 72 3.41963 92 3.63124 112 3.80127 13 1.95904 33 2.74896 53 3.15564 73 3.43152 93 3.64058 113 3.80895 14 2.02119 34 2.77452 54 3.17173 74 3.44326 94 3 64982 114 3.81657 15 2.07918 35 2.79934 55 3.18752 75 3.45484 95 3.65896 115 3.82413 16 2.13354 36 2.82347 56 3.20303 76 3.46627 96 3.66801 116 3.83161 17 2.18409 37 2.84696 57 3.21827 77 3.47756 97 3.67697 117 3.83904 18 2.23300 38 2.86982 58 3.23325 78 3.48869 98 3.68583 118 3.84640 19 2.27875 39 2.89209 59 3.24797 79 3.49969 99 3.69461 119 3.85370 20 2.32222 40 2.91381 60 3.26245 80 3.51055 100 3.70329 120 3.86094 TABLE 1. [Pags-1 1 Difference of Latituc e and Departure for ^Paiirt , N.^E. N. iW. S.:5E. s.^w ! Dist. Lat. Dcp. Dist. Lat. Dep. Dist. Lat. Dep. 05.9 Dist. Lat. Dep. Dist. 241 Lat. 240.7 Dep. II. 8 I 01 .0 00.0 61 60.9 o3 121 120.9 i8i 180.8 08.9 2 02^ 3 o3# 00. r 62 6r .9 o3.c 22 121 .9 06.0 82 181.8 08.9 42 241.7 II. 9 00. 1 63 62.9 o3.] 23 122.9 06.0 83 182.8 09.0 43 242.7 II .9 4 04.0 00.2 64 63.9 o3.i 24 123.9 06.1 84 i83.8 09.0 44 243 7 12,0 5 c5.o 00.2 65 64.9 o3.2 2b 124.8 06.1 8b i84.8 09. 1 45 244.7 12.0 6 06.0 00.3 66 65.9 o3.2 20 125.8 06.2 86 i85.8 09.1 46 245.7 12.1 7 07.0 00.3 67 66.9 o3.3 27 126.8 06.2 87 186.8 09.2 47 246.7 12. 1 8 08.0 00.4 68 67.9 o3.3 28 127.8 06.3 88 187.8 09.2 48 247-7 12.2 9 09.0 00.4 69 68. 9 o3.4 29 128.8 06.3 89 188.8 09.3 49 248.7 12.2 10 10.0 00.5 70 69.9 o3.4 3o i3, 129.8 06.4 90 189.8 09.3 bo 25l 249.7 250.7 12.3 11 1.1 .0 00.5 71 70.9 o3.5 i3o.8 ob.4 191 190.8 09.4 12.3 12 12.0 00.6 72 71.9 o3.5 32 i3i .8 06.5 92 191 .8 09.4 52 25l .7 12.4 i3 i3.o 00.6 73 72.9 o3.6 ■33 i32.8 06.5 93 192.8 09.5 53 252.7 12.4 1 4 i4.o 00.7 74 73.9 o3.6 34 i33.8 06.6 94 193.8 09.5 54 253.7 12.5 1 5 i5.o 00.7 75 74.9 o3.7 3 b i34.8 06.6 95 194.8 09.6 5b 254.7 12.5 i6 16.0 00.8 76 75.Q o3.7 3(3 i35.8 06.7 96 195.8 09.6 56 255.7 12.6 17 17.0 00.8 77 76.9 o3.8 37 i36.8 06.7 97 196.8 09.7 57 256.7 12.6 i8 18.0 00.9 78 77-9 o3.8 38 137.8 06.8 98 197.8 09.7 58 257-7 12.7 19 19.0 00.9 79 78.9 03.9 39 i38.8 06.8 99 198.8 09.8 b9 258.7 12.7 20 20.0 01 .0 8g 79-9 03.9 40 139.8 06.9 200 199.8 09.8 bo 259.7 12.8 21 21 .0 01 .0 81 80.9 04.0 i4i 140.8 06.9 201 200.8 09.9 261 260.7 12.8 22 22.0 01 .1 82 81 .9 o4.o 42 i4i.8 07.0 02 201.8 09.9 b2 261 .7 12.9 23 23.0 01 .1 83 82.9 04. 1 43 142.8 07.0 o3 202.8 10. 63 262.7 12.9 24 24.0 01 .2 84 83.9 o4. 1 44 143.8 07.1 04 2o3.8 10. (!>4 263.7 i3.o 25 25.0 01.2 85 84.9 o4.2 Ab 144.8 07.1 o5 204.8 10. 1 65 264.7 l3.G 26 26.0 01.3 86 85.9 04.2 46 145.8 07.2 06 2o5.8 10. 1 66 265.7 i3.i 27 27.0 01.3 87 86.9 04.3 4i 146.8 07.2 07 S06.8 10-2 67 266.7 i3.i 23 28-0 01 .4 88 87.9 04.3 48 147-8 07.3 08 207.7 10.2 68 267.7 l3.2 29 29.0 01 .4 89 88.9 04.4 49 i48.8 07.3 09 208.7 10.3 69 268.7 l3.2 3o 3i 3o.o 3i.o 01 .5 01.5 90 89-9 04.4 bo 149.8 07.4 10 209.7 10.3 70 269.7 l3.2 91 90.9 o4.5 i5i i5o.8 07.4 211 210.7 10.4 271 270.7 i3.3 32 32. 01 .6 92 91.9 04. b b2 i5i.8 07.5 12 211 .7 70.4 72 271.7 i3.3 33 33.0 01 .6 93 92.9 04.6 53 i52.8 07.5 i3 212.7 10.5 73 272.7 i3.4 M 34.0 01.7 94 93.9 04.6 54 i53.8 07.6 i4 213.7 10.5 74 273.7 i3.4 3b 3b. 01.7 95 94.9 04.7 bb i54.8 07.6 i5 214.7 10.5 75 274.7 i3.5 36 36.0 01.8 96 95.9 04.7 bb i55.8 07.7 16 215.7 10.6 76 275.7 i3.5 37 37.0 01.8 97 96.9 04.8 57 1 56.8 07.7 17 216.7 10.6 77 276.7 i3.6 38 38.0 01 .9 98 97-9 04.8 b8 157.8 07.8 18 217.7 10.7 78 277.7 i3.6 39 39.0 01 .9 99 98.9 04.9 b9 i58.8 07.8 19 218.7 10.7 79 278.7 i3.7 40 4o.o 02.0 TOO 99.9 04.9 o5.o bo 159.8 07.9 20 219.7 10.8 80 279.7 i3.7 4i 4 1 .0 02.0 lOI 100.9 161 160.8 07.9 221 220.7 10.8 281 280.7 i3.8 42 41.9 02.1 02 mi .9 o5.o b2 i6t.8 07.9 22 221 .7 10.9 82 281.7 i3.8 43 42.9 02 . 1 o3 102.9 o5.i 63 162.8 08.0 23 222.7 10.9 83 282.7 i3.9 44 43.9 02 .2 o4 103.9 o5.i 64 i63.8 08.0 24 223.7 II .0 64 283.7 i3.9 45 44.9 02.2 o5 104.9 05.2 65 164.8 08.1 25 224.7 II .0 85 284.7 .'4-0 46 45.9 02.3 06 105.9 o5.2 66 i65.8 08.1 26 225.7 II. I 86 285.7 14.0 47 46-9 02.3 07 106.9 o5.3 67 166.8 08.2 27 226.7 II .1 87 286.7 14.1 48 47.9 02.4 08 107.9 o5.3 68 167.8 08.2 28 227.7 II .2 88 287.7 14.1 49 48.9 02.4 09 108.9 o5.3 69 168.8 08.3 29 228.7 11.2 89 288.7 l4.2 5o 49.9 02.5 10 109.9 o5.4 70 169.8 08.3 3o 229.7 II. 3 90 289.7 l4.2 5i 50.9 02.5 III 110-9 o5.4 171 170.8 08.4 23l 23o.7 II. 3 291 290.6 i4.3 b2 bi.9 02.6 12 III .9 o5.5 72 171. 8 08.4 32 23i .7 II. 4 92 291 .6 i4.3 bJ 52.9 02.6 i3 112. 9 o5.5 73 172.8 08.5 33 232.7 II. 4 93 292.6 14.4 b4 53.9 02.6 i4 113.9 o5.6 74 173.8 08.5 34 233.7 II. 5 94 293.6 i4-4 bb 54.9 02.7 lb 114. 9 o5.6 75 174.8 08.6 35 234.7 II. 5 95 2^4.6 i4.b bb bb.9 02.7 16 115.9 05.7 76 175.8 dS.6 36 235.7 II. 6 96 295.6 i4.5 b7 b6.9 02.8 17 116. 9 o5.7 77 176.8 08.7 37 236.7 II. 6 97 296.6 i4.6 b8 b7.9 02.8 18 117. 9 o5.8 78 177.8 08.7 38 237.7 II. 7 98 297.6 14.6 ^9 b8.9 02.9 19 118.9 o5.8 79 178.8 08.8 39 238.7 11.7 99 298.6 14.7 fin b9.9 02.9 20 119.9 05.9 80 179.8 08.8 40 239.7 II. 8 3oo 299.6 14.7 Oisi. \^op. Lat. Df^t. nop. Lat. Di.l. Dnp. I,at. Dist. Dop. Lnt. Dist. Dep. Lat. E. :{N. E.riS. W. .4N. w. \ S. [For 75 Points., j Fjge 2] TABLE L DilTereace of Latitude and Departure for J Point. IV. iE JS.iVV S.^E. S.iW. Dist. Lat. Dep. 00. 1 Dist. Lat. Dep. Dist. 121 Lat. 120.4 Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 01 .0 61 60.7 06.0 11.9 181 180.1 17-7 241 239.8 23.6 2 3 02.0 o3.o 00.2 00.3 62 63 61 .7 62.7 06.1 06.2 22 23 121 .4 122.4 12.0 12. I bo iSi.i 182.1 17.8 17.9 42 43 240.8 24%B 23.7 23.8 4 o4.o 00.4 64 63.7 06.3 24 123.4 12.2 84 i83.i 18.0 AA 242.8 23.9 5 o5.o 00.5 65 64.7 06.4 25 124.4 12.3 85 184. 1 18.1 45 243.8 24.0 6 06.0 00.6 66 65.7 06.5 26 125.4 12.4 86 ■85.1 18.2 46 244.8 24.1 7 07.0 00.7 67 66.7 06.6 27 126.4 12.4 87 1S6. 1 18.3 47 245.8 24.2 8 u8.o 00.8 68 67.7 06.7 28 127.4 12.5 88 1S7.1 18.4 48 246.8 24.3 9 09.0 0D.9 69 68.7 06.8 29 128.4 12.6 89 188.1 18.5 49 247-8 24.4 lO 10. 01 .0 70 69.7 06.9 3o 129.4 12.7 90 189. 1 18.6 5o 248.8 24.5 II 10.9 01 .1 71 70.7 07.0 i3i i3o.4 12.8 191 190.1 18.7 25l 249.8 24.6 la n.9 01 .2 72 71-7 07.1 32 i3i.4 12.9 92 191 .1 18.8 52 2 5o.8 24.7 i3 12. Q 01 .3 73 72.6 07.2 33 i32.4 l3.0 93 193. 1 18.9 53 251.8 24.8 i4 i3.9 01 .4 74 73.6 07.3 34 i33.4 i3.i 94 193. 1 19.0 51 252.8 24.9 i5 14.9 01.5 75 74.6 07.4 35 i34.3 l3.2 95 194.1 19. 1 55 253.8 25.0 i6 i5.9 01 .6 76 75.6 07.4 36 i35.3 i3.3 96 195. 1 19.2 56 254.8 25.1 17 16.9 01.7 77 76.6 07.5 37 i36.3 i3.4 97 196.1 19.3 57 255.8 25.2 i8 17.9 01.8 78 77.6 07.6 38 137.3 i3.5 98 197.0 19-4 58 256.8 25.3 19 18.9 01 .9 79 78.6 07.7 39 i38.3 i3.6 99 198.0 .9.5 59 257.8 25.4 20 19.9 02.0 80 79.6 07.8 4o 139.3 13.7 200 199.0 .9.6 60 258.7 25.5 21 20.9 02.1 81 80.6 07.9 i4i i4o.3 i3.8 201 200.0 19.7 261 259.7 25.6 22 21 .9 02.2 82 81.6 08.0 42 i4i.3 13.9 02 201 .0 19.8 62 260.7 25.7 23 22.9 02.3 83 82.6 oS.i 43 142.3 14.0 o3 202.0 19.9 63 261 .7 25.8 24 23.9 02.4 84 83.6 08.2 AA 143.3 i4-i 04 2o3.o 20.0 64 262.7 25.9 25 24.9 02.5 85 84.6 08.3 45 144.3 l4-2 o5 204.0 20.1 65 263.7 26.0 26 25.9 02.5 86 85.6 08.4 46 145.3 14.3 06 2o5.o 20.2 66 264.7 26.1 27 26.9 02.6 87 86.6 08.5 47 146.3 14.4 07 206.0 20.3 67 265.7 26.2 28 27.9 02 .7 88 87.6 08.6 48 147.3 14.5 08 207 . 20.4 68 266.7 26.3 29 28.9 02.8 89 88.6 08.7 49 148.3 14.6 09 208 . 20.5 69 267.7 26.4 3o 29.9 02.9 90 89.6 08.8 5o 149-3 14.7 10 209.0 20.6 70 268.7 26.5 3i 3o.9 o3.o 91 90.6 08.9 i5i i5o.3 14.8 211 210.0 20.7 271 269.7 26.6 32 3i.8 o3.i 92 91 .6 09.0 52 i5i.3 14.9 12 21 I .0 20.8 72 270.7 26.7 33 32.8 03.2 93 02.6 09.1 53 i52.3 i5.o i3 2 12.0 20.9 73 271.7 26.8 34 33.8 o3.3 94 93.5 09.2 54 i53.3 i5.i i4 2l3.0 21 .0 74 272.7 26.9 35 34.8 o3.4 95 94.5 09.3 55 i54.3 l5.2 i5 214.0 21. 1 7b 273.7 27.0 36 35.8 o3.5 96 95.5 09.4 u6 i55.2 i5.3 16 2l5.0 21 .2 7b 274.7 27.1 3? 36.8 o3.6 97 96.5 09.5 57 i56.2 i5.4 17 2:6.0 21.3 77 275.7 27.2 38 37.8 o3.7 98 97.5 09.6 58 157.2 i5.5 18 217.0 21 .4 78 276.7 27.2 39 38.8 o3.8 99 98.5 09.7 59 i58.2 lb. 6 19 217.9 21.5 79 277.7 27.3 4o 39.8 03.9 100 99.5 09.8 60 159.2 i5.7 20 218.9 21.6 80 278.7 27.4 4i 40.8 04.0 lOI 100.5 09.9 161 160.2 i5.8 221 219.9 21.7 281 279.6 27.5 42 41.8 o4.i 02 loi .5 10. 62 161 .2 15.9 22 220.9 21.8 82 280.6 27.6 43 42.8 04.2 03 102.5 10. 1 63 162.2 16.0 23 221 .9 21 .9 83 281.6 27.7 M 43.8 04.3 04 io3.5 10.2 64 i63.2 16.1 24 222.9 22.0 84 282.6 27.8 45 44.8 04.4 o5 104.5 10.3 65 164.2 16.2 25 223.9 22.1 85 283.6 27.9 46 45.8 04.5 06 io5.5 10.4 66 i65.2 16.3 26 224.9 22.2 86 284.6 28.0 47 46.8 04.6 07 106.5 10.5 67 166.2 16.4 27 225.9 22.2 87 285.6 28.1 48 47.8 04.7 08 107.5 10.6 68 167.2 16.5 28 226.9 22.3 88 286.6 28.2 49 48.8 04.8 09 108.5 10.7 69 168.2 16.6 29 227.9 22.4 89 287.6 28.3 5o 49-8 '.4.9 10 109.5 10.8 70 169.2 16.7 JO 228.9 22.5 90 2S8.6 28.4 5i 5o.8 OD.O III 110.5 10.9 171 170.2 16.8 23l 229.9 22 6 291 289.6 28.5 52 5i .7 ' o5.i 12 III. 5 1 1 .0 72 171 .2 ib.9 32 230.9 22 ; 92 290.6 28.6 53 52.7 t)5.2 i3 112.5 11 .1 73 172.2 17.0 33 23l .9 22.8 93 291 .6 28.7 54 53.7 o5.3 i4 ii3.5 II .2 74 1 73 . 2 17. 1 M '32.9 22.9 94 292.6 28.8 55 54.7 o5.4 i5 114.4 II. 3 75 174.2 17.2 35 233.9 23. 95 293.6 28.9 56 55.7 o5.5 16 ii5.4 II. 4 76 175.2 17.3 36 234.9 23.1 96 294.6 29.0 57 56.7 o5.6 17 116.4 II. 5 77 176. 1 17.3 37 235.9 23.2 97 295.6 29.1 58 57.7 ()5.7 18 117.4 II. 6 78 177-1 17-4 38 236.9 23 3 98 296.6 29.2 59 58.7 o5.8 19 118.4 II. 7 79 178. 1 17.5 39 237.8 ^i.A 99 297.6 29.3 60 59.7 05.9 20 119.4 II. 8 80 179. 1 17.6 4o 238.8 23.5 3oo 298.6 29.4 i Oop. i I.nt. Dlsl Dep. 1 Lat. Dist. Dep Lat. Dist. Den. Lat. Disi.l Dep. Lat. E.iN. E.^S. W. h N. W. h S. [For 7.i Points. J TABLE L [Page 3 Difference of Latitude and Departure for f Point N.|E. N.| W S.|E. S.5W • Disi. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. 35.4 I 01 .0 00. 1 61 60.3 09.0 121 119. 7 17.8 181 179.0 26.6 241 238.4 2 02.0 00.3 62 61.3 09.1 22 120.7 17.9 82 180.0 26.7 42 239.4 35.5 3 o3.o 00.4 63 62.3 09.2 23 121 .7 18.0 83 181.0 26.9 43 240.4 35.7 4 o4.o 00.6 64 63.3 09.4 24 122.7 18.2 84 1 182.0 27.0 44 241.4 35.8 5 04.9 00.7 65 64.3 09.5 25 123.6 18.3 85 i83.o 27.1 45 242.3 35.9 6 05.9 00.9 66 65.3 09.7 26 124.6 18.5 86 184.0 27.3 46 243.3 36.1 7 06.9 01 .0 67 66.3 09.8 27 125.6 18.6 8- i85 27.4 47 244.3 36.2 8 07.9 01.2 68 67.3 10.0 28 126.6 18.8 88 186 27.6 48 245.3 36.4 9 08.9 01 .3 69 68.3 10. 1 29 127.6 18.9 89 187.0 27-7 49 246.3 36.5 10 09.9 or .5 70 69.2 10.3 3o 128.6 19.1 9" 191 187.9 18S.9 27.9 28.0 bo 25? 247-3 248.3 36.7 36.8 1 1 10.9 01 .6 71 70.2 10.4 i3i 129.6 19.2 12 11.9 01.8 72 71.2 10.6 32 i3o.6 19.4 92 189.9 28.2 52 249.3 37.0 i3 12.9 01 .9 73 72.2 10.7 33 i3i.6 19.5 93 190.9 28.3 53 25o.3 37.1 i4 i3.8 02.1 74 73.2 10.9 34 i32.5 19.7 94 191. 9 28.5 54 25i.3 37.3 i5 14.8 02.2 75 74.2 11 .0 35 i33.5 19.8 95 192.9 28.6 55 252.2 37.4 i6 i5.8 02.3 76 75.2 11 .2 36 i34.5 20.0 96 193.9 28.8 5b 253.2 37.6 17 16.8 02.5 77 76.2 11.3 37 i35.5 20. 1 97 194.9 28.9 i)7 254-2 37.7 i8 17.8 02.6 7« 77.2 II. 4 38 i36.5 20.2 98 193.9 29.1 58 255.2 37-9 '9 18.8 02.8 79 78.1 11.6 39 137.5 20.4 99 196.8 29.2 59 256.2 38.0 2C) 19.8 02.9 80 79.1 11.7 4o i38.5 20.5 200 197.8 29.3 bo 257.2 38.1 21 20.8 o3.i 81 80.1 II. 9 i4i 139.. 5 20.7 201 198.8 29.5 261 258.2 38.3 22 21.8 03.2 82 81. 1 12.0 42 i4o.5 20.8 02 199.8 29.6 62 259.2 38.4 23 22.8 o3.4 83 82.1 12.2 43 i4i.5 21 .0 o3 200.8 29.8 63 260.2 38.6 24 23.7 o3.5 84 83.1 12.3 44 142.4 21 .1 04 201.8 29.9 64 2bi .1 38.7 25 24.7 03.7 85 84.1 12.5 A5 143.4 21.3 o5 202.8 3o.i 65 262.1 38.9 26 25.7 o3.8 86 85.1 12.6 46 144.4 21.4 06 2o3.8 3o.2 bb 263.1 39.0 27 26.7 04.0 87 86.1 12.8 4i 145.4 21 .0 07 204.8 3o-4 67 264.1 39.2 28 27.7 04.1 88 87.0 12.9 48 146.4 21.7 08 205.7 3o.5 b8 265.1 39.3 29 28.7 04.3 89 88.0 i3.i 49 147-4 21 .9 09 206.7 3o.7 69 266.1 39.5 3o 29.7 04.4 90 89.0 l3.2 i3.4 5o 148.4 22.0 10 207.7 3o.8 70 267.1 39.6 39.8 3i 3o.7 04.5 91 90.0 i5i 149.4 22.2 211 208.7 3i .0 271 268.1 32 3i.7 04.7 92 91 .0 i3.5 52 i5o.4 22.3 12 209.7 3i.i 72 269.1 39.9 33 32.6 04.8 93 92.0 i3.6 53 i5i.3 22.4 i3 210.7 3i.3 73 270.0 4o. I 34 33.6 o5.o 94 93.0 i3.8 54 i52.3 22.6 i4 211 .7 3i.4 74 271 .0 40.2 35 34.6 o5.i 95 94.0 i3.9 55 i53.3 22.7 i5 212.7 3i.5 7^ 272.0 40.4 36 35.6 o5.3 q6 95.0 i4.i 56 i54.3 22.9 16 2i3.7 dr. 7 76 273.0 40.5 37 36.6 o5.4 97 96.0 l4.2 57 i55.3 23. 17 214.7 3i.8 77 274.0 40.6 38 37.6 o5.6 q8 96.9 14.4 58 i56.3 23.2 18 2i5.6 32. 78 275.0 4o.8 39 38.6 05.7 QQ 97-9 i4.5 59 157.3 23.3 19 216.6 32.1 79 276.0 40.9 4o 4i 39.6 4o.6 05.9 06.0 100 98.9 14.7 6c) i58.3 23.5 20 217.6 32.3 80 277.0 4i.i 101 99.9 i4.8 161 159.3 23.6 221 218.6 32.4 281 278.0 4l -2 42 4i.5 06.2 02 100.9 i5.o 62 160.2 23.8 22 219.6 02.6 8,2 278.9 4i.4 43 42.5 06.3 o3 101 .9 i5.i 63 161.2 23.9 23 220.6 32.7 83 279.9 41.5 44 43.5 06.5 o4 102.9 i5.3 ^4 162.2 24.1 24 221 .6 32.9 84 280.9 4i .7 45 44.5 06.6 o5 103.9 i5.4 65 i63.2 24.2 25 222.6 33.0 85 281.9 4i.8 46 45.5 06.7 06 104.9 i5.6 66 164.2 24-4 26 223.6 33.2 8b 282.9 42 .0 47 46.5 06.9 07 io5.8 .5.7 67 i65.2 24.5 27 224.5 33.3 «7 283.9 42.1 48 47.5 07.0 08 106.8 1 5. 8 68 166.2 24.7 28 225.5 33.5 88 284.9 42.3 49 48.5 07.2 09 107.8 16.0 69 167.2 24.8 29 226.5 33.6 89 285.9 42.4 5o "57 49-5 07.3 10 108.8 16.1 70 168.2 24.9 3o 227.5 33.7 90 286.9 42.6 5o.4 07.5 1 1 1 109.8 16.3 171 169. 1 25.1 23l 228.5 33.9 291 287.9 42.7 52 5i.4 07.6 12 IJ0.8 16.4 72 170. 1 25.2 32 229.5 34.0 92 288.8 42.8 53 52.4 07.8 i3 III. 8 16.6 73 171. 1 25.4 33 23o.5 34.2 93 289.8 43.0 54 53.4 07.9 i4 112. 8 16.7 74 172. 1 25.5 34 23i.5 34.3 94 290.8 43.1 55 54.4 08.1 i5 ii3.8 16.9 75 173. 1 25.7 35 232.5134.5 9'J 291.8 43.3 56 55.4 08.2 16 114.7 17.0 76 174. 1 25.8 36 233.4; 34.6 9b 292.8 i 43.4 57 56.4 08.4 17 115.7 17.2 11 175. 1 26.0 37 234.4 34.8 97 295.8 U3. 6 58 57.4 08.5 18 116. 7 17.3 78 176. 1 26.1 38 235.4 34 9 98 294.8 43.7 59 58.4 08.7 19 117. 7 17.5 79 177-1 26.3 39 236.4 35.1 99 295.8 43.9 60 59.4 08.8 20 118.7 17.6 80 178.1 26.4 40 237.4 35.2 3oo 296.8 44.0 Disl. Dop. Lat. i);.t Dep. Lat. Dist Dep. Lat. Dist Dep. Lat. Dist. Dep. Lat. L. E.$N. E.|S. W.| N. W.3 8. [iFor "li Points. | Page 4] TABLE L Difference of Latitude and Departure for 1 Point. NbyE. N.byW. S.byE. S byW. Dist. Lat. Dep. 00.2 Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. 47-0 I 01 .0 61 59.8 II. 9 121 118. 7 23.6 181 177.5 35.3 241 236.4 2 02.0 00.4 62 60.8 12. 1 22 1 19. 7 23.8 82 178.5 35.5 42 237.4 47.2 3 02.9 00.6 63 61.8 12.3 23 120.6 24.0 83 179.5 35.7 43 238.3 47-4 4 o3 . 9 00.8 64 62.8 12.5 24 121 .6 24.2 84 180.5 35.9 M 239.3 47.6 5 04.9 01 .0 65 63.8 12.7 25 122.6 24.4 85 181.4 36.1 45 240.3 47.8 b o5.9 01 .2 66 64.7 12.9 26 123.6 24.6 86 182.4 36.3 46 241.3 48.0 7 06.9 01 .4 67 65.7 i3.i 27 124.6 24.8 87 i83.4 36.5 47 242.3 48.2 8 07.8 01 .6 68 66.7 i3.3 28 125.5 25. 88 184.4 36.7 48 243.2 48.4 9 08.8 01.8 69 67.7 i3.5 29 126.5 25.2 89 i85.4 36.9 49 244.2 48.6 10 09.8 02.0 70 68.7 13.7 3o 127.5 25.4 90 186.3 37.1 5o 245.2 48.8 II 10.8 02.1 71 69.6 i3.9 i3i 128.5 25.6 191 187.3 37.3 25l 246.2 49.0 12 II. 8 02.3 72 70.6 14.0 32 129.5 25.8 92 188.3 37.5 52 247.2 49.2 iJ 12.8 02.5 73 71 .6 14.2 33 1 3o . 4 25.9 93 189.3 37.7 53 248.1 49-4 i4 13.7 02.7 74 72.6 i4.4 M i3i.4 26.1 94 190.3 37.8 54 249.1 49.6 li) 14.7 02.9 7i> 73.6 14.6 35 i32.4 26.3 95 191 .3 38.0 55 25o.i 49.7 lb i5.7 o3.i 76 74.5 i4.8 36 i33.4 26.5 96 192.2 38.2 56 25l .1 49.9 I? lb. 7 o3.3 77 75.5 i5.o 37 i34.4 26.7 97 193.2 38.4 57 252.1 5o.i i8 17.7 o3.5 78 76.5 l5.2 38 i35.3 26.9 98 194.2 38 6 58 253.0 5o.3 19 18.6 o3.7 79 77.5 i5.4 39 i36.3 27.1 99 195.2 38.8 59 254-0 5o.5 20 19.6 03.9 80 78.5 i5.6 4o 137.3 27.3 27.5 200 196.2 39.0 60 255. 50.7 21 20.6 04.1 81 79-4 i5.8 i4i i38.3 201 197. 1 39.2 261 256. 5o.Q 22 21.6 04.3 82 80.4 16.0 42 139.3 27.7 02 198.1 39.4 62 257.0 5i.i 2ci 22.6 04.5 83 81.4 16.2 Ai i4o.3 27.9 OJ 199. 1 39.6 63 257.9 5i.3 24 23.5 04.7 84 82.4 16.4 M i4i .2 28.1 04 200.1 39.8 64 255.9 5i.5 2b 24.5 04.9 85 83.4 16.6 45 142.2 28.3 o5 201 .1 4o.o 65 259.9 5i.7 2b 25.5 o5. 1 86 84.3 16.8 46 143.2 28.5 ob 202.0 4o.2 66 260.9 5i.9 27 26.5 o5.3 87 85.3 17.0 47 144.2 28.7 07 203.0 40.4 67 261 .0 52.1 28 27.5 o5.5 88 86.3 17.2 48 145.2 28.9 08 204.0 40.6 68 262.9 52.3 29 28.4 05.7 89 87.3 17.4 49 i46.i 29.1 09 2o5.0 40.8 69 263.8 52.5 Jo 29.4 05.9 90 88.3 17.6 5o i47-i 29.3 10 206.0 4i .0 70 264.8 52.7 3i 3o.4 06.0 91 89.3 17.8 i5i 148.1 29.5 211 206.9 4i .2 271 265.8 52.9 J2 3i.4 06.2 92 90.2 17.9 52 149.1 29.7 12 207.9 41.4 72 266.8 53.1 33 32.4 06.4 93 91 .2 18.1 53 i5o. I 29.8 i3 208.9 41.6 73 267.8 53.3 M 33.3 06.6 94 92.2 18.3 54 i5i .0 3o.o i4 209.9 41.7 74 268.7 53.5 35 34.3 06.8 95 93.2 18.5 55 l52.0 3o.2 i5 210.9 41.9 75 269.7 53.6 36 35.3 07.0 96 94.2 18.7 56 i53.o 3o.4 16 211.8 42. 1 76 270.7 53.8 37 36.3 07.2 97 95.1 18.9 57 1 54.0 3o.6 17 212.8 42.3 77 271.7 54.0 38 37.3 07.4 98 96.1 19.1 58 i55.o 3o.8 18 2i3.8 42.5 78 272.7 54.2 39 38.3 07.6 99 97.1 19.3 59 155.9 3i .0 19 214.8 42.7 79 273.6 54.4 40 39.2 07.8 100 98.1 19.5 60 i56.9 3l.2 20 2i5.8 42.9 80 274.6 54.6 4i 40.2 08.0 lOI 99.1 19.7 161 157.9 3i.4 221 216.8 43.1 281 275.6 54.8 42 41.2 08.2 02 100. 19.9 62 i58.9 3i.6 22 217.7 43.3 82 276.6 55.0 43 42.2 08.4 o3 loi .0 20. 1 63 159.9 3i.8 23 2IS.7 43.5 83 277.6 55.2 4A 43.2 08.6 04 102.0 20.3 Q>^ 160.8 32.0 24 219.7 43.7 84 278.5 55.4 4b 44.1 08.8 o5 io3.o 20.5 65 161.8 32.2 25 220.7 43.9 85 279.5 55.6 46 45.1 09.0 06 104.0 20.7 66 162.8 32.4 26 221.7 44.1 86 280.5 55.8 47 46.1 09.2 07 104.9 20.9 67 i63.8 32.6 27 222.6 44.3 87 281.5 56.0 48 47.1 09.4 08 105.9 21 .1 68 164.8 32.8 28 223.6 44.5 88 282.5 56.2 49 48.1 09.6 09 106.9 21.3 69 i65.8 33.0 29 224-6 44.7 89 283.4 56.4 bo 49.0 09.8 10 107.9 21.5 70 166.7 33.2 3u 225.6 44.9 45". I 90 284.4 56.6 5i 5o.o 09.9 III 108.9 21.7 171 167.7 33.4 23l 226.6 291 285.4 56.8 b2 5i .0 lO.I 12 109.8 21 .9 72 168.7 33.6 32 227.5 45.3 92 286.4 57.0 53 52.0 10.3 i3 110.8 22.0 73 169.7 33.8 33 228.5 45.5 q3 287.4 57.2 54 53.0 10.5 i4 III. 8 22.2 74 170.7 33.9 34 229.5 45.7 94 288.4 57.4 55 53.9 10.7 i5 112.8 22.4 73 171 .6 34.1 35 23o.5 45.8 95 289.3 57.6 56 54.9 10.9 16 ii3.8 22.6 76 172.6 34.3 36 231.5 46.0 96 290.3 i)7.7 ^7 55.9 II. I 17 114.8 22.8 77 173.6 34.5 37 232.4 46.2 97 291 .3 57.9 58 5b. 9 II. 3 18 ii5.7 23.0 78 174.6 34.7 38 233.4 46.4 g8 292.3 58.1 59 57 9 II. 5 19 116.7 23.2 79 175.6 34.9 39 234.4 46.6 99 293.3 58.3 bo 58 8 .11.7 20 117.7 23.4 80 176.5 35.1 4o 235.4 46.8 3oo 294.2 58.5 IVp. Lat. Dist. Dop. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. E.byN. E.byS. VV.byN. W.byS. [For 7 Points. TABLE I. LPage 5 Difference of Latitude and Departure for 1| Points. N.byE.iE. N.byW.^W. S.byE.iE. S byW.:iW Dist. Lat. Dep. Dist. 61 Lat. Dep. Dist. Lat. Dep. 29.4 Dist. Lat. Dep. Dist. Lat. Dep. I 01 .o 00.2 59.2 14.8 121 1 17.4 i8i 175.6 44.0 24 1 233.8 58.6 2 01 .9 00.5 62 60.1 i5.i 22 118. 3 29.6 82 176.5 44.2 42 234.7 58.8 3 02.9 00.7 63 61. 1 i5.3 23 119. 3 29.9 83 177.5 44.5 43 235.7 59.0 4 o3.9 01 .0 64 62.1 i5.6 24 120.3 3o.i 84 178.5 44.7 44 236.7 59.3 5 04.9 01.2 65 63.1 i5.8 23 121 .3 3o.4 85 179.5 45.0 45 237.7 59.5 6 o5.8 01 .5 66 64. 16.0 26 122.2 3o.6 86 180.4 45.2 46 238.6 59.8 7 06.8 01.7 67 65.0 16.3 27 123.2 30.9 87 181. 4 45.4 4i 239.6 60.0 8 07.8 01 .9 68 66.0 16.5 28 124.2 3i.i 88 182.4 45.7 48 240.6 60.3 9 08.7 02.2 69 66.9 16.8 29 125. I 3i.3 89 i83.3 45.9 49 241.5 60.5 10 09.7 02.4 70 67.9 17.0 3o 126. 1 3i.6 90 184.3 46.2 5o 242.5 60.7 1 1 10.7 02.7 71 68.9 17.3 i3i 127. 1 3i.8 191 i85.3 46.4 231 243.5 61 .0 12 II. 6 02.9 72 69.8 17.5 32 128.0 32.1 92 186.2 46.7 52 244.4 61.2 1 i3 12.6 03.2 73 70.8 17.7 33 129.0 32.3 93 187.2 46.9 53 245.4 61.5 i4 i3.6 o3.4 74 71.8 18.0 34 i3o.o 32.6 94 18S.2 47.1 54 246.4 61.7 i5 14.6 o3.6 75 72.8 18.2 35 i3i.o 32.8 95 189.2 47-4 55 247-4 62 .0 if) i5.5 03.9 76 73.7 18.5 36 i3i .9 33.0 96 190. 1 47-6 56 248.3 62.2 17 16.5 04.1 77 74.7 18.7 37 132.9 33.3 97 191 .1 47.9 57 249.3 62.4 i8 17.5 04.4 7» 75.7 19.0 38 133.9 33.5 98 192. 1 48.1 58 25o.3 62.7 '9 1S.4 04.6 79 76.6 19.2 39 i34.8 33.8 99 193.0 48.4 59 25l .2 62.9 2() 19.4 04.9 80 77.6 19.4 40 i4i i35.8 i36.8 34.0 34.3 200 194.0 48.6 60 252.2 63.2 2 1 20.4 o5.i 81 78.6 19.7 201 195.0 48.8 26! 253.2 63.4 22 21.3 o5.3 82 79.5 19.9 42 137.7 34.5 02 195.9 49.1 62 254.1 63.7 23 22 3 o5.6 83 80.5 20.2 43 i38.7 34.7 o3 196.9 49.3 63 255.1 63.9 24 23.3 o5.8 84 81.5 20.4 44 139.7 35.0 04 197.9 49.6 64 256.1 64.1 25 24.3 06.1 85 82.5 20.7 45 140.7 35.2 o5 198.9 49.8 65 257. 1 64.4 26 25.2 06.3 86 83.4 20.9 46 i4i.6 35.5 06 199.8 DO. I 66 258. 64.6 27 26.2 06.6 87 84.4 21 .1 4i 142 .6 35.7 07 200.8 5o.3 67 259.0 64.9 ' 28 27.2 06.8 88 85.4 21 .4 48 143.6 36. 08 201.8 5o.5 68 260.0 65.1 1 ^9 28.1 07.0 89 86.3 21 .6 49 144.5 36.2 09 202.7 5o.8 69 260 . 9 65.4 3o 29.1 07.3 90 87.3 21 .9 22.1 5o 145.5 36.4 10 2o3.7 5i.o 70 271 261 .9 262.9 65.6 3i 3o. I 07.5 91 88.3 i5i 146.5 36.7 211 204.7 5i.3 65.8 32 3i .0 07.8 92 89.2 22.4 52 147.4 36.9 12 2o5.6 5i.5 72 2.63.8 66.1 33 32.0 08.0 93 90.2 22.6 53 148.4 37.2 i3 206.6 5i.8 73 264.8 66.3 34 33.0 08.3 94 91 .2 22.8 54 149.4 37.4 i4 207.6 52.0 74 265.8 66.6 35 34.0 08.5 95 92.2 23.1 55 i5o.4 37.7 i5 208.6 52.2 7^ 266.8 66.8 36 34.9 08.7 96 93.1 23.3 56 i5i.3 37.9 16 209.5 52.5 76 267.7 67.1 37 35.9 09.0 97 94.1 23.6 57 i52.3 38.1 17 210.5 52.7 77 268.7 67.3 38 36.9 09.2 98 96.1 23.8 58 i53.3 38.4 18 211.5 53.0 78 269.7 67.5 39 37.8 09.5 99 96.0 24.1 59 i54.2 38.6 19 212.4 53.2 79 270.6 67.8 4o 38.8 09.7 100 97 -o 24.3 60 1 55. 2 38.9 20 2i3.4 53.5 80 271 .6 68.0 68.3 4i 39.8 10. lOI 98.0 24.5 161 i56.2 39.1 221 214.4 53.7 281 272.6 42 40.7 10.2 02 98.9 24.8 62 I57.I 39.4 22 2i5.3 53.9 82 273.5 68.5 43 41.7 10.4 o3 99.9 25.0 63 1 58. 1 39.6 23 216.3 54.2 83 274.5 68.8 44 42.7 10.7 04 100.9 25.3 64 159. 1 39.8 24 2,7.3 54.4 84 275.5 69.0 45 43.7 10.9 o5 lOI .9 25.5 65 160. 1 4o.i 25 218.3 54.7 85 276.5 69.2 46 44.6 II .2 06 102.8 25.8 66 161 .0 40.3 26 219.2 54.9 86 277.4 69.5 47 45.6 II. 4 07 io3.8 26.0 67 162.0 40.6 27 220.2 55.2 87 278.4 69.7 48 46.6 II. 7 08 104.8 26.2 68 i63.o 40.8 28 221.2 55.4 88 279.4 70.0 49 47.5 II. 9 09 105.7 26.5 69 163.9 4t.i 29 222. 1 55.6 89 280.3 70.2 5o '57 48.5 12. 1 10 106.7 26.7 70 164.9 4i.3 3o ^3? 223.1 53.9 90 281.3 70.5 49-5 12.4 1 1 1 107.7 27.0 171 165.9 41.5 224. 1 56.1 291 282.3 70.7 52 5o.4 12.6 12 108.6 27.2 72 166.8 4i.8 32 225.0 56.4 92 283.2 71.0 53 5i.4 12.9 i3 109.6 27.5 73 167.8 42.0 33 226.0 56.6 93 284.2 71.2 54 52.4 i3.i i4 no. 6 27.7 74 168.8 42.3 34 227.0 56.9 94 285.2 71.4 55 53.4 i3.4 i5 III .6 27.9 75 169.8 42.5 35 228.0 57.1 95 286.2 71.7 56 54.3 i3.6 16 112. 5 28.2 76 170.7 42.8 36 228.9 57.3 96 287. 1 71.9 !>7 55.3 i3.8 17 1x3.5 28.4 77 171.7 43.0 37 229.9 57.6 97 288.1 72.2 58 56.3 i4.i 18 114.5 28.7 78 172.7 43.3 38 23o.9 57.8 98 289.1 72.4 59 57.2 i4.3 19 ii5.4 28.9 79 173.6 43.5 39 231.8 58.1 99 290.0 72.7 ()0 58.2 i4.6 I.at. 20 116. 4 29.2 80 174.6 43.7 4o Dist. 232.8 Dep. 58.3 Lat. 3oo 291 .0 72.9 l>is(.| Dop. Dist. Dop. i,a'. Dist. Dop. Lat. Dist. Dep. Lat. E.N.E.5E. E.S.E.fE. W.N.W.sW. W.S.W.5W. [Far 6^ Points. Page 6] TABLE L Difference of Latitude and Departure for li Points. N.byEiE. N.byW.iW. S.byEdE S.byW.^W. Disi. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. 241 Lat. 23o.6 Dep. I 01 .0 00.3 61 58.4 17-7 121 ii5.8 35.1 181 173.2 52.5 70.0 2 01 .9 00.6 62 59.3 18.0 22 - -^ _ 35.4 82 174.2 52.8 42 231.6 70.2 3 02.9 00.9 53 60.3 18.3 23 117 7 35.7 83 175. 1 53.1 4'i 232.5 70.5 4 o3.8 01 .2 64 61.2 1? u 24 118. 7 36.0 84 176. 1 53.4 44 233.5 70.8 5 o4.8 01.5 65 62.2 18.9 25 119.6 36.3 85 177.0 53.7 45 234.5 71. 1 6 o5.7 01.7 66 63.2 19.2 26 120.6 36.6 86 178.0 54.0 46 235.4 71.4 7 06.7 02.0 67 64.1 19.4 27 121 .5 36.9 87 178.9 54.3 47 236.4 71.7 8 07.7 02.3 68 65.1 19.7 28 122.5 37.2 88 179.9 54.6 48 237.3 72.0 9 08.6 02.6 69 66.0 20.0 29 123.4 37-4 89 180.9 54.9 49 238.3 72.3 lO 09.6 02.9 70 67.0 20.3 3o i3i 124.^ 37.7 90 181.8 55.2 5o 251" 239.2 72.6 72.9 II 10.5 o3.2 71 67.9 20.6 125.4 38.0 191 182.8 55.4 240.2 12 II. 5 o3.5 72 68. 9 20.9 32 126.3 38.3 92 183.7 55.7 52 241 .1 73.2 i3 12.4 o3.8 73 69.9 21.2 33 127.3 38.6 93 184.7 56. 53 242 . 1 73.4 U i3.4 o4.i 74 70.8 21.5 ■M 128.2 38.9 94 i85.6 56.3 54 243.1 73.7 li) 14.4 04.4 75 71.8 21.8 35 129.2 39.2 95 186.6 56.6 55 244-0 74.0 i6 i5.3 04.6 76 72.7 22.1 36 i3o.i 39.5 96 187.6 56-9 56 245. u 74.3 17 16.3 04.9 77 73.7 22.4 37 i3i .1 39.8 97 188.5 57.2 57 245.9 74.6 i8 17.2 o5.2 78 74.6 22.6 38 l32.I 4o.i 98 189.5 57.5 58 246.9 74.9 19 18.2 o5.5 79 75.6 22.9 39 i33.o 4o.3 99 IVC.4 57.8 59 247.8 75.2 20 19.1 o5.8 80 76.6 23.2 40 i34.o 40.6 200 191-4 58.1 60 261 248.8 75.5 21 20.1 06. 1 81 77.5 23.5 i4i 134.9 40.9 201 192.3 58.3 249.8 22 21 .1 06.4 82 78.5 23.8 42 135.9 4i .2 02 193.3 58.6 62 250.7 76.1 23 22.0 06.7 83 79-4 24.1 43 i36.8 4i.5 o3 194.3 58.9 63 25l .7 76.3 24 23.0 07.0 84 80.4 24.4 44 137.8 4i.8 04 195.2 59.2 64 262.6 76.6 2b 23.9 07.3 85 81.3 24.7 45 i38.8 42.1 o5 1 96 . 2 59-5 65 253.6 76.9 2b. 24.9 07.5 86 82.3 25.0 46 139.7 42.4 06 197.1 59.8 66 264.5 77-2 27 25.8 07.8 87 83.3 25.3 47 140.7 42.7 07 198.1 60.1 67 255.5 77-5 2S 26.8 08.1 88 84.2 25.5 48 141.6 43.0 08 199.0 60.4 68 256.5 77-8 29 27.8 08.4 89 85.2 25.8 49 142.6 43.3 09 200.0 60.7 69 267.4 78.1 3o 28.7 08.7 90 86.1 26.1 5o 143.5 43.5 10 201 .0 61 .0 61.3 70 271 268.4 78.4 78.7 3i 29.7 09.0 9' 87.1 26.4 i5i 144.5 43.8 211 201 .9 269.3 32 3o.6 09.3 92 88.0 26.7 52 145.5 44.1 12 202.9 61.5 72 260.3 79.0 33 3i.6 09.6 93 89.0 27.0 53 146.4 44.4 i3 2o3.8 61.8 73 261 .2 79.2 S4 32.5 09.9 94 90.0 27.3 54 147.4 44.7 i4 204.8 62.1 74 262.2 79-5 6t> 33.6 10.2 95 90.9 27.6 55 148.3 45.0 i5 205.7 62.4 75 263.2 79-8 35 34.4 10.5 96 91.9 27.9 56 149-3 45.3 16 206.7 62.7 76 264.1 80.1 ^7 35.4 10.7 97 92.8 28.2 57 l5o.2 45.6 17 207.7 63. 77 265.1 80.4 d^ 36.4 II. 98 93.8 28.4 58 i5i .2 45.9 18 208.6 63.3 78 266.0 80.7 39 37.3 II. 3 99 94.7 28.7 59 1 52. 2 46.2 iq 209.6 63.6 7C/ 267.0 81.0 40 38.3 II. 6 100 95.7, 29.0 60 i53.i 46.4 20 221 210.5 63.9 64.2 ho 267.9 81.3 Si. 6 4i 39.2 II. 9 101 96.7 29.3 161 1 54. 1 46.7 211 .5 281 268.9 42 40.2 12.2 02 97.6 29.6 62 i55.o 47-0 22 212.4 64-4 82 269.9 81.9 i4J 4i.i 12.5 o3 98.6 29.9 63 i56.o 47-3 23 2i3.4 (XI.7 83 270.8 82.2 44 42.1 12.8 04 99.5 30.2 64 i56.9 47-6 24 214.4 65. 84 271 .8 82.4 4S 43.1 i3.i o5 100.5 3o.5 65 157.9 47-9 25 2i5.3 65.3 85 272.7 82.7 46 44.0 i3.4 06 loi .4 3o.8 66 1 58. 9 48.2 26 216.3 65.6 86 273.7 83. 47 45.0 i3.6 07 102.4 3i.i 67 159.8 48.5 27 217.2 65.9 87 274.6 83.3 4« 45.9 13.9 08 io3.3 3i.4 68 160.8 48.8 ?^S 2,8.2 66.2 88 276.6 83.6 49 46.9 l4.3 09 104.3 3i.6 69 161 .7 49.1 21} -19.1 66.5 89 276.6 83.9 bo 47.8 14.5 10 io5.3 3i .9 70 171 162.7 i63.6 49.3 49.6 3o 23l iA).l 66.8 90 277.5 84.2 84.5 5i 48.8 i4.8 11 ! 106.2 32.2 221 .1 67.1 291 278.5 52 49.8 i5.i 12 107.2 32.5 72 164.6 49-9 32 222.0 67.3 92 279.4 84.8 53 5o.7 i5.4 l3 108. 1 32.8 73 165.6 5o.2 33 223.0 67.6 93 280.4 85.1 54 5. .7 i5.7 i4 109. 1 33.1 74 I&6.5 5o.5 34 223.9 67.9 94 281.3 85.3 55 52.6 16.0 i5 IIO.O 33.4 75 167.5 5o.8 35 224.9 68.2 q5 282.3 85.6 56 53.6 16.3 16 11 1 .0 33.7 76 168.4 5i.i 36 225.8 68.5 96 283.3 86.9 57 54.5 16.5 17 112. 34.0 7^ 169.4 5i.4 37 226.8 68.8 97 284.2 86.2 58 55.5 16.8 18 112. 9 34.3 78 170.3 5i.7 38 227.8 69.1 98 285.2 86. 6 59 56.5 17. 1 19 113.9 34.5 79 171 .3 52.0 39 228.7 69.4 99 2S6.1 86.8 bo 57.4 17-4 20 114. 8 34.8 80 172.2 52.3 40 Dist. 229.7 Dep. 69.7 3 00 287.1 87.1 Dis. Dep. Lat. Uisl. Dop. Lat. Dist. Dep. Lat. Lat. l^ist.l Dep. Lat. E.N.E.AE. E.S.E.iE. VV.N.W.^W. W.S.W..JIW. [For G.i Points. r~ 1 TABLE L 11 'age 7 1 Difference of Latitude and Departure for If Points. N.byE.; [E. N.byW.^VV. S.byE.^E. S.byW.^W. ir.si. Lai. Dep. Disl. Lat. Dcp. Uist. Lat. Dep. 40.8 Dist. Lat. Dep. Disl. L.ii. Dep. I 00.9 GO. 3 61 57.4 20.6 121 113.9 181 170.4 61.0 241 226.9 81.2 2 101 .9 00.7 62 58.4 20.9 22 4i.i 82 I7'.4 61.3 42 227.9 81.5 3 02.8 01 .0 63 59.3 21 .2 23 iij.8 41.4 83 172.3 61.7 43 228. s 81.9 4|o3.8 01 .3 64 60.3 21 .6 24 116.8 4i .S 84 173.2 62.0 44 299.7 82.2 5 04.7 01.7 65 61.2 21.9 25 117.7 42.1 85 174.2 62.3 45 23o.7 82.5 6 ob.6 02.0 66 62.1 22.2 26 118.6 42.4 86 175.1 62.7 46 23 1 .6 82.9 7 06.6 02.4 67 63.1 22.6 27 119.6 42.8 87 176.1 63 .0 47 232.6 83.2 8 07.5 02.7 68 64. 22.9 28 120.5 43.1 88 177.0 63.3 48 233.5 83.5 9 08.5 o3.o 69 65. 23.2 29 121.5 43.5 89 17S.0 63.7 49 234. i 83.9 10 09.4 o3.4 70 65.9 23.6 3o 122.4 43.8 90 178.9 64.0 5o 235.4 84.2 1 1 10.4 03.7 71 66.8 23.9 i3i 123.3 44.1 •91 179-8 64.3 25l 236.3 84 6 12 11.3 04.0 72 67.8 24.3 32 124.3 44.5 92 180.8 64.7 52 237.3 84.9 i3 12.2 04.4 73 68.7 24.6 33 125.2 44.8 93 181. 7 65.0 53 238.2 85.2 i4 (3.2 04.7 74 69.7 24.9 34 126.2 45.1 94 182.7 65.4 54 239.2 85.6 lb I4.I o5.i 7b 70.6 2b. 3 35 127.1 45.5 95 i83.6 65.7 55 240. 1 85.9 i5 i5.i o5.4 76 71.6 25.6 36 128.0 45.8 96 184.5 66.0 56 241 .0 86.2 17 16.0 o5.7 77 72.5 25.9 37 129.0 46.2 97 i85.5 66.4 57 242.0 86.6 18 16.9 06.1 78 73.4 26.3 38 129.9 46.5 98 186.4 66.7 58 242.9 86.9 87.3 19 17.9 06.4 79 74.4 26.6 39 i3o.9 46.8 99 187.4 67.0 59 243.9 20 18.8 06.7 80 75.3 27.0 4o i3i.8 47.2 200 18S.3 67.4 60 244.8 87.6 21 19.3 J07.1 81 76.3 27.3 i4i i32.8 47-5 201 189.3 67.7 261 245.7 87.9 22 2v0.7 07.4 82 77.2 27.6 42 133.7 47-8 02 1 90 . 2 68.1 62 246.7 88.3 23 21.7 07.7 83 78.1 28.0 43 i34.6 48.2 o3 191 . 1 68.4 63 247.6 88.6 24 22.6 08.1 84 79.1 28.3 44 i35.6 48.5 04 192. 1 68.7 64 248.6 8S.9 25 33.5 08.4 85 80.0 28.6 45 i36.5 48.8 o5 1 93 . 69.1 65 249.5 89.3 26 24.5 08.8 86 81.0 29.0 i3-.5 49.2 06 194.0 69.4 66 250.5 89.6 27 25.4 09.1 87 81.9 29.3 47 i38.4 49-!^ 07 194.9 69.7 67 25i.4 89.9 28 26.4 09.4 88 82.9 29.6 48 139.3 49.9 08 195.8 70.1 68 252.3 90.3 ^9 27.3 09.8 89 83.8 3o.o 49 i4o.3 5o.2 09 196.8 70.4 69 253.3 90.6 3o 28.2 10. 1 90 84.7 3o.3 bo l4l .2 5o.5 10 197.7 70.7 70 254-2 91.0 3i 29.2 10.4 91 85.7 3o.7 i5i 142.2 50.9 211 198.7 71.1 271 255.2 91.3 32 3o. I 10.8 92 86.6 3i .0 52 143. 1 5i .2 12 199.6 71-4 72 256. 1 91.6 33 3i.i 1 1 . 1 93 87.6 3i.3 53 I44.I 5i.5 i3 200.5 71.8 73 257.0 92.0 34 32. II. 5 94 88.5 3. .7 54 145.0 51.9 i4 201.5 72.1 74 258. 92.3 35 33.0 II. 8 9b 89.4 32. 55 145.9 52.2 i5 202.4 72.4 7b 258.9 92.6 36 33.9 12. 1 96 90.4 32.3 56 146.9 52.6 16 2o3.4 72.8 76 259.9 93.0 37 34.8 12.5 97 91 .3 3a. 7 57 147.8 52.9 17 204.3 73.. 77 260.8 93.3 38 35.8 12.8 98 92.3 33.0 58 148.8 53.2 18 2o5.3 73.4 78 261 .7 93.7 39 36.7 i3.i 99 93.2 33.4 59 i49-7 53.6 19 206.2 73.8 79 262 .7 94.0 4o 4i 37.7 38.6 i3.5 100 94.2 33.7 60 i5o.6 53.9 20 207.1 74.1 80 263.6 94.3 i3.8 lOI 95.1 34.0 161 i5i.6 54.2 221 208.1 74.5 281 264.6 94.7 42 39.5 i4.i 02 96.0 34.4 62 i52.5 54.6 22 209.0 74.8 82 265.5 95.0 4i 40. b 14.5 o3 97.0 34.7 63 i53.5 54.9 23 210.0 75.1 83 266.5 95.3 44 41.4 i4.8 04 97-9 35.0 64 i54.4 55.2 24 210.9 75.5 84 267.4 95.7 4^ 42.4 l5.2 ob 98.9 35.4 65 i55.4 55.6 25 211. 8 75.8 85 268.3 96.0 46 43.3 i5.5 06 99.8 35.7 66 i56.3 55.9 26 212.8 76.1 8() 269 . 3 96.4 47 44.3 i5.8 07 K)0 . 7 36. 67 157.2 56.3 27 213.7 76.5 87 270.2 9*'-7 48 45.2 16.2 08 101.7 56.4 68 i58.2 56.6 28 214.7 76.8 88 271 .2 97.0 49 46.1 16.5 09 102.6 36.7 69 159.1 56.9 29 2i5.6 77.1 89 272 . 1 97.4 bo 47.1 16.8 10 io3.6 37.1 37.4 70 160. 1 57.3 3o 216.6 77.5 90 273.0 97.7 98.0 5( 48. 17.2 III 104.5 171 161 .0 57.6 23l 217.5 77.8 291 274.0 b2 4'9.o 17. b 12 io5.5 37.7 72 161 .9 57.9 3p 218.4 78.2 92 274.9 98.4 b3 49.9 17.9 i3 106.4 38.1 73 162.9 58.3 33 219.4 78.5 93 275.9 98.7 b4 bo. 8 18.2 i4 107.3 38.4 74 i63.8 58.6 34 220.3 78.8 94 276.8 99.0 bb bi.8 18.5 lb 108.3 38.7 75 164.8 59.0 35 221 .3 79.2 95 277.8 99.4 bb 52.7 18.9 16 109.2 39.1 76 165.7 59.3 36 222.2 79.5 96 278 7 99-7 b7 b3.7 19.2 17 110.2 39.4 77 166.7 59.6 37 223.1 79.8 97 279.6 100. 1 b8 54.6 19. b i8 I II .1 39.8 78 167.6 60.0 38 224.1 80.2 98 280.6 100.4 D9 bb.6 19.9 19 1 12.0 4o.i 79 168.5 60.3 39 225. 80.5 99 281.5 100.7 bo bb.b 20.2 20 it3.o 40.4 80 169.5 60.6 40 Dist. 226.0 Drp. 80.9 3oo 282.5 101. 1 Dist. Dcp. Lat. Dist. Dpp. Lat. Dist. Drp. Lat. Lni. Di4.6 01.9 65 60.1 24.9 25 ii5.5 47-8 85 170.9 70.8 45 226.4 93.8 6 u5.5 02.3 66 61.0 25.3 26 116.4 48.2 86 171. 8 71.2 46 227.3 94-1 7 06.5 02.7 67 61.9 25.6 27 117.3 48.6 87 172.8 71.6 47 228.2 94.5 8 07.4 o3.i 68 62.8 26.0 28 118. 3 49.0 88 173.7 71.9 48 229.1 94.9 Q 08.3 o3.4 69 63.7 26.4 29 119. 2 49-4 89 174.6 72.3 49 23o.O 95.3 lO 09.2 o3.8 70 64.7 26.8 3o i3i 120. 1 J21 .0 49.7 5o.i 90 175.5 72.7 5o 23l .0 95.7 1 1 10.2 04.2 71 65.6 27.2 191 176.5 73.1 25l 23l .9 96.1 13 II .1 04.6 72 66.5 27.6 32 122.0 5o.5 92 177-4 73.5 52 232.8 96.4 i3 12.0 o5.o 73 67.4 27.9 33 122.9 50.9 93 178.3 73.9 53 233.7 96.8 i4 12.0 o5.4 74 68.4 28.3 34 123.8 5i.3 94 179.2 74-2 54 234.7 97-2 i5 I3.q 05.7 75 69.3 28.7 35 124.7 5i.7 95 180.2 74.6 55 235.6 97.6 i6 i4.8 06. 1 76 70.2 29.1 36 125.6 52. 96 181. 1 75.0 56 236.5 98.0 17 l5.7 06.5 77 71. 1 29.5 37 126.6 52.4 97 182.0 75.4 57 237.4 98.3 iS 16.6 06.9 78 72.1 29.8 38 127.5 52.8 98 182.9 75.8 58 238.4 98.7 19 17.6 07.3 79 73.0 3o.2 39 128.4 53.2 99 183.9 76.2 59 23q.3 99.1 20 18.5 07.7 80 73.9 3o.6 3i .0 4o i4i 129.3 53.6 200 184.8 76.5 60 240 . 2 99.5 21 19.4 08.0 81 74.8 i3o.3 54.0 201 185.7 76.9 261 241 .1 99.9 22 20.3 08.4 82 75.8 3i.4 42 i3i .2 54.3 02 186.6 77.3 62 242.1 100.3 ?.3 21 .2 08.8 83 76.7 3i.8 43 l32.I 54.7 o3 187.5 77-7 63 243.0 100.6 24 22.2 09.2 84 77.6 32.1 AA i33.o 55.1 04 188.5 78.1 64 243.9 101.0 25 23.1 09.6 85 78.5 32.5 45 i34.o 55.5 o5 189.4 78.5 65 244.8 101.4 26 24.0 09.9 86 79-5 32.9 46 134.9 55.9 06 190.3 78.8 66 245.8 101.8 27 24.9 10.3 87 80.4 ii.i 47 i35.S 5'j.3 07 191 .2 79.2 67 246.7 102.2 28 25.9 10.7 88 81.3 33.7 48 i36.7 56.6 08 192.2 79.6 68 247.6 102.6 ^9 26.8 1 1 . 1 8q 82.2 34.1 49 137.7 57.0 09 193.1 80.0 69 248.5 102.9 3o 27.7 II. 5 90 83.1 34.4 5o i38.6 57 .-4 10 194.0 80.4 80.7 70 249.4 io3.3 3 1 28.6 II. 9 91 84.1 34.8 i5i 139.5 57.8 21 1 194.9 271 25o.4 103.7 3?. 29.6 12.2 92 85.0 35.2 52 140.4 58.2 12 195.9 81. 1 72 251.3 104.1 33 3o.5 12.6 93 85.9 35.6 53 141.4 58.6 i3 196.8 81.5 73 252.2 104.5 34 3i.4 i3.o 94 86.8 36. 54 142.3 58.9 i4 197-7 81.0 74 253.1 104.9 35 32.3 i3.4 95 87.8 36.4 55 143.2 59.0 i5 19S.6 82.3 7^ 254.1 105.2 36 33.3 i3.8 96 88.7 36.7 56 144. 1 ^9-7 16 199.6 82.7 76 255. io5.6 37 34.2 l4.2 97 89.6 37.1 57 145.0 60.1 17 200.5 83.0 77 255. Q 106.0 38 35.1 14.5 98 90.5 37.5 58 i46.o 60.5 18 201 .4 83.4 78 256.8 106.4 39 36.0 14.9 99 91.5 37.9 59 146.9 60.8 '9 202.3 83.8 79 257.8 106.8 40 37.0 i5.3 100 92.4 38.3 60 i6x 147.8 148.7 61 .2 61.6' 20 2o3.3 84.2 80 258.7 107.2 41 37.9 l5.7 lOI 93.3 38.7 221 204 . 2 84.6 281 259.6 107.5 42 38.8 16. 1 02 94.2 39.0 62 149.7 62.0 22 2o5.i 85.0 82 260 . 5 107.9 43 39.7 16.5 o3 95.2 39.4 63 i5o.6 62.4 23 206.0 85.3 83 261.5 108.3 AA 40.7 16.8 04 96. 1 39.8 64 i5i.5 62.8 24 206.9 85.7 84 262.4 108.7 45 41.6 17.2 o5 97.0 40.2 65 i52.4 63.1 2 5 207.9 86.1 85 263.3 109.1 46 42.5 17.6 06 97-9 40.6 66 i53.4 63.5 26 208.8 86.5 86 264.2 109.4 47 43.4 18.0 07 98.9 40.9 67 154.3 63.9 27 209.7 86.9 87 265.2 109.8 48 44.3 18.4 08 9Q.8 41.3 68 155.2 64.3 28 210.6 87.3 88 266. 1 1 10.2 4q 45.3 18.8 09 100.7 41.7 69 i56.i 64.7 29 21 I .6 87.6 89 267.0 110.6 5o 46.2 19. 1 10 loi .6 42.1 70 157. 1 65.1 3o 212.5 88.0 90 267.9 11 1.0 111.4 5r 47.1 19.5 III 102.6 42.5 171 i5&.o 65.4 23l 2i3.4 88.4 291 268.8 52 48. 19.9 r? io3.5 42.9 72 i58.9 65.8 32 214.3 88.8 92 269.8 111.7 53 49.0 20.3 i3 104.4 43.2 73 159.8 66.2 33 2i5.3 89.2 93 270.7 112.1 54 49.9 20.7 i4 105.3 43.6 74 160.8 66.6 M 2l6.2 89.5 94 271 .6 112. 5 55 5o.8 21 .0 i5 106.2 44.0 7^ 161 .7 67.0 35 217. 1 89.9 9^ 272.5 112. 9 56 5i.7 21.4 16 107.2 U.A 76 162.6 67.4 36 218.0 90.3 96 273.5 ii3.3 57 52.7 21.8 17 108. 1 44.8 77 i63.5 67.7 37 219.0 90.7 97 274.4 !i3.7 58 53.6 22.2 18 109.0 45.2 78 164.5 68.1 38 219.9 91.1 9« 275.3 1 i4o 5q 54.5 22.6 19 109.9 45.5 79 i65.4 68.5 39 220.8 91.5 99 276.2 114.4 60 55.4 23. 20 no. 9 45.9 80 166.3 68.9 4o 221 .7 91.8 3 00 277-2 11 4-8 Di^t. Dop. Lat. Dist. Drp. Lat. Dist. Dep. I>at. Dist.i Dep. Lat. Dist. Dep. Lat. E.N.E. E.S.E. W.N.W. w.s.w. [For 6 Points. TABLE L [Page 9 Diflference of Latitude and Departure for 2|- Points. iX.N.E. ^E. N.N.W.iW. S.S.E.AE. S.S.W.^W. Dlst. Lai. Dcp. Dist. Lat. Dep. Dist Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dcp. I 00.9 00.4 61 55.1 26.1 121 109.4 5. .7 181 i63.6 77-4 241 217.9 ic3 c 2 01.8 00. c b2 56. 26.5 22 no. 3 52.2 82 164.5 77.8 42 218.8 io3.5 3 02 .7 01 .3 63 57.0 26.9 23 III. 2 52.6 83 i65.4 78.2 43 219.7 103.9 4 o3.6 01.7 b4 57-9 27.4 24 112. 1 b3.o 84 166.3 78.7 44 220.6 104.3 5 o4.5 02. 1 6b 58.8 27.8 25 ii3.o 53.4 85 167.2 79.1 45 221.5 104.8 o5.4 02.6 6b 59.7 28.2 2b 113.9 53.9 86 168. 1 79.5 46 222.4 io5,2 7 06.3 o3 67 60. G 28.6 27 114.8 54.3 87 169.0 80.0 47 223.3 105.6 8 07.2 o3.4 68 61.5 29.1 28 11D.7 54.7 88 169.9 80.4 48 224.2 106.0 9 08.1 o3.8 69 62.4 29.5 29 116. 6 bb.2 89 170.9 80.8 49 225.1 106.5 10 09.0 04.3 70 71 63.3 64.2 29.9 3o.4 3o 117. 5 bb.6 90 171. 8 81.2 5o 226.0 106.9 II 09.9 04.7 i3i 118. 4 56.0 191 172.7 81.7 25l 226.9 107.3 12 10.8 ob.i 72 65.1 3o.8 32 119. 3 56.4 92 173.6 82.1 52 227.8 107.7 i3 II. 3 o5.b li 66.0 3l.2 ■6-^ 120.2 56.9 93 174.5 82.5 53 228.7 10S.2 i4 12.7 06.0 l4 66.9 3i.b 3h 121 . 1 57.3 94 175.4 82.9 54 229.6 1 108.6 1 lb i3.b 06.4 7b 67.8 32.1 3b 122.0 57-7 95 176.3 83.4 55 230.5 109.0 16 i4.5 06.8 76 68.7 32.b 36 122.9 58.1 96 177.2 83.8 56 23i.4 109.5 17 lb. 4 07.3 77 69.6 32.9 37 123.8 58.6 97 178.1 84.2 57 232.3 109.9 18 lb. 3 07.7 7» 70.5 ii.>, 38 124.8 59.0 98 179.0 84.7 58 233.2 1 10.3 19 17.2 oS.i V 71.4 33.8 39 125.7 59.4 99 179.9 85.1 59 234.1 110.7 20 18. 1 08.6 80 72.3 34.2 4o 126.6 59.9 200 180.8 85.5 60 235. III. 2 21 19.0 09.0 81 73.2 34.6 i4i 127.5 60.3 201 181. 7 85.9 261 235.9 111.6 22 19.9 09.4 bii 74.1 3d. I 42 128.4 60.7 02 182.6 86.4 62 236.8 112.0 2J 20.8 09.8 83 75.0 3b.b 43 129.3 61. 1 o3 i83.5 86.8 63 237.7 112.4 24 SI. 7 10.3 84 75.9 3b.9 •44 i3o.2 61.6 04 184.4 87.2 64 238.7 112.9 2b 22.6 10.7 8b 76.8 3b.3 4':^ i3i . I 62.0 o5 i85.3 87.6 65 239.6 ii3.3 2b 23.5 II .1 8b 77-7 3b.8 46 l32.0 62.4 06 186.2 88.1 66 240.5 II3.7 27 24.4 II. 5 B7 78.6 37.2 47 132.9 62.9 07 187.1 88.5 67 241.4 114.2 28 25.3 12.0 88 79.6 37.b 48 i33.8 63.3 08 188.0 88.9 68 242.3 114.6 29 26.2 12.4 89 80.5 38.1 49 134.7 63.7 09 188.9 89.4 69 243.2 ii5.o Jo 27.1 12.8 90 81.4 38. b bo i35.6 64.1 10 189.8 89.8 70 244.1 1 1 5.4 3i 28.0 i3.3 91 82.3 38.9 i5i i36.5 64.6 211 190.7 90.2 271 245.0 115-9 32 28.9 i3.7 92 83.2 39.3 52 137.4 65.0 12 191 .6 90.6 72 245.9 116.3 3i 29.8 i4.i 9^ 84.1 39.8 53 i38.3 65.4 i3 192.5 91. 1 73 246.8 116.7 34 3o.7 i4.b 94 85. 4o.2 b4 109.2 65.8 i4 193.5 91.5 74 247.7 117.2 3b 3i.b ib.O 9b 85.9 4o.b bb i4o.i 66.3 i5 194.4 91.9 75 248.6 II 7.6 36 32. b lb. 4 96 86.8 4i.o bb i4i .0 66.7 16 195.3 92.4 76 249.5 118.0 37 33.4 lb. 8 97 87.7 4i.b b7 141.9 67.1 17 196.2 92.8 77 25o.4 118.4 38 34.4 16.2 98 88.6 41.9 b8 142.8 67.6 18 197.1 93.2 78 25i.3 1 18.9 39 ^0.3 lb. 7 99 89.5 42 3 b9 143.7 68.0 19 198.0 93.6 79 252.2 119.3 40 4i 3b. 2 17. 1 100 90.4 42.8 bo 144.6 6S.4 20 198.9 94.1 80 253.1 119-7 37.1 17.5 lOI 91 .3 43.2 161 145.5 68.8 221 199.8 94.5 281 254.0 1 20. 1 42 38. 18.0 02 92.2 43.b 62 146.4 69.3 22 200.7 94-9 82 254.9 120.6 43 38.9 18.4 o3 93.1 44.0 63 147.4 69.7 23 201 .6 95.3 •83 255.8 1 21.0 44 39.8 18.8 o4 94.0 44.b 64 148.3 70.1 24 202.5 95.8 84 256.7 121.4 4b 40.7 19.2 Ob 94.9 44.9 bb 149.2 70.5 25 2o3.4 96.2 85 257.6 121. 9 46 41.6 19.7 Ob 9b.8 4b.3 bb i5o.i 71.0 26 204.3 96.6 86 258.5 122.3 47 42. b 20.1 07 96.7 4b.7 67 i5i .0 71-4 27 205.2 97.1 87 259.4 122.7 48 4i.4 20.5 08 97.6 46.2 b8 i5i .9 71.8 28 206.1 97.5 88 260.3 123. 1 ^9 44. i 21.0 09 98.5 4b.b 69 i52.8 72.3 29 207.0 97.9 89 261.3 123.6 bo 45.2 21.4 10 99.4 47-0 70 153.7 72.7 3o 207.9 98.3 90 262.2 124.0 5i 46.1 21.8 III 100.3 47-5 171 i54.6 73.1 23l 208.8 98.8 291 263.1 124.4 b2 47 -o 22.2 12 101.2 47-9 72 i55.5 73.5 32 209.7 99.2 92 264.0 124.8 b3 47-9 22.7 i3 102.2 48.3 73 i56.4 74.0 33 210.6 99.6 93 264.9 125.3 ^4 48.8 23.1 i4 io3.i 48.7 74 157.3 74.4 34 211 .5 lOO.O 94 265.8 125.7 bb 49.7 23. b lb 104.0 49.2 75 i58.2 74.8 35 212.4 100.5 95 266.7 126.1 1 Db 5o.6 23.9 lb 104.9 105.8 49.6 76 159. 1 75.2 36 2i3.3 100.9 q6 267.6 1266 57 5i.5 24.4 17 5o.o 77 160.0 75.7 37 214.2 101.3 97 268.5 127,0 58 b2.4 24.8 18 106.7 bo.b 78 160.9 76.1 38 2l5.1 101.8 q8 269.4 127.4 b9 b3 3 2b. 2 19 107.6 50.9 79 161. 8 76.5 39 216. I 102.2 QQ 270.3 127.8 bo 54.2 25.7 20 108.5 bi.3 80 162.7 77.0 4o 217.0 102.6 3oo 271 .2 128.3 Dist. Dpp. 1 Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dcp. Lat. Dist. Dnp. Lat. N.E.byE4E. S.E.byE.^E. N.W.byW.sW. S.W.byW.^W. [For h\ Points. Page 10] TABLE L Differ ence of Latitude and Departure for 2^ Points. N.N.E.iE. N.N.W.^W. S.S.E.^E. S.S.W.JW. D!St.[ Lat. Deo. Dist. Lat. D«p. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Let. Dep. 1 1 3.6 I 1 00 . 9 00.5 61 53.8 28.8 121 106.7 57.0 181 159.6 85.3 241 212.5 2 01.8 00.9 62 54 7 29.2 22 107.6 57.5 82 160.5 85.8 42 2i3.4 114.1 3 02.6 01 .4 63 55.6 29.7 23 108.5 58.0 83 161 .4 86.3 43 214.3 1 14.5 4 o3.5 01 .9 64 56.4 3o.2 24 109.4 58.5 84 162.3 86.7 44 2l5.2 1 1 5.0 5 o4.4 02.4 65 57.3 3o.6 25 no. 2 58.9 85 i63.2 87.2 45 216. I 1 1 5.5 6 o5.3 02.. 8 66 58.2 3i.i 26 III .1 59.4 86 164.0 87.7 46 217.0 1 16.0 7 o6.2 o3.3 67 59.1 3i.6 27 [12.0 59.9 87 164.9 88.2 47 217.3 116.4 8 07.1 o3.8 68 60.0 32.1 28 112. 9 60.3 88 i65.8 88.6 48 218.7 116.9 9 07.9 04.2 69 60.9 32.5 29 ii3.8 60.8 89 166.7 89.1 49 219.6 117.4 lO 08.8 04.7 70 61.7 33.0 3o 114. 6 61.3 90 167.6 89.6 5o 220.5 117.8 1 1 09.7 05.2 71 62.6 33.5 i3i- ii5.5 61.8 191 168.4 90.0 25l 221 .4 118.3 12 10.6 o5.7 72 63.5 33.9 32 116. 4 62.2 92 169.3 90.5 52 222.2 118.8 i3 II. 5 06.1 73 64.4 34.4 33 117. 3 62.7 93 170.2 91.0 53 223.1 119.3 i4 12.3 06.6 74 65.3 34.9 M 118. 2 63.2 94 171. 1 91.5 54 224.0 119-7 i5 l3.2 07.1 75 66.1 3b.4 35 119. 1 63.6 95 172.0 91.9 55 224.9 120.2. i6 14. 1 07.5 76 67.0 35.8 36 119. 9 64.1 96 172.9 92.4 56 225.8 120.7 17 i5.o 08.0 77 67.9 3bJ 37 120.8 64.6 97 173.7 92.9 57 226.7 121. 1 i8 l5.9 08.5 78 68.8 36.8 38 121 .7 65.1 98 174.6 93.3 58 227.5 121.6 19 16.8 09.0 79 69.7 37.2 39 122.6 65.5 99 175.5 93.8 59 228.4 122. 1 20 17.6 09.4 80 70.6 37.7 4o 123.5 66.0 200 176.4 94.3 60 229.3 122.6 21 18.5 09.9 81 71.4 38.2 i4i 124.4 66.5 201 177.3 94.8 261 23o.2 123.0 22 19.4 10.4 82 72.3 38.7 42 I2D.2 66.9 02 178. 1 95.2 62 23l .1 123.5 23 20.3 10.8 83 73.2 39.1 43 126. 1 07.4 00 179.0 95.7 63 23l .9 124.0 24 21.2 II. 3 84 74.1 39.6 44 127.0 67.9 %4 179.9 96.2 64 232.8 124.4 25 22.0 II. 8 85 75.0 a'o.i 45 127.9 68.4 ob 180.8 96.6 65 233.7 124.9 26 22.9 12.3 86 75.8 40.5 46 128.8 68.8 06 181. 7 97.1 66 234.6 125.4 27 23.8 12.7 «7 76.7 4i.o 47 129.6 69.3 07 182.6 97.6 671 235.5 125.9 28 24.7 l3.2 88 77.6 41.5 48 i3o.5 69.8 08 i83.4 98.1 68 236.4 126.3 29 25.6 .3.7 89 78.5 42.0 49 i3i.4 70.2 09 184.3 98.5 69 237.2 126.8 3o 26.5 i4.i 90 79-4 42.4 bo 1 32. 3 70.7 10 i85.2 99.0 70 238.1 127.3 3i 27.3 14.6 91 80.3 42.9 i5i i33.2 71.2 21 1 186.1 99.5 271 239.0 127.7 32 28.2 i5.i 92 81. 1 4i.4 b2 i34.i 71.7 12 187.0 99-9 72 239.9 128.2 33 29.1 i5.6 93 82.0 43.8 53 134.9 72.1 i3 187.8 100.4 73 ■2A0.S 128.7 34 3o.o 16.0 94 82.9 44.3 54 i35.8 72.6 i4 188.7 100.9 74 241.6 129.2 35 3o.9 16.5 95 83.8 44.8 bb 1 36. 7 73.1 i5 189.6 101.4 75 242.5 129.6 36 3l.7 17.0 96 84.7 45.3 b6 137.6 73.5 16 190.5 101.8 76 243.4 i3o.i 37 32.6 17-4 97 85.5 45.7 57 i38.5 74.0 17 191 .4 102.3 77 244.3 i3o.6 38 33.5 17.9 98 86.4 46.2 b8 139.3 74.5 18 192.3 102.8 78 245.2 i3i.o 39 34.4 .8.4 99 87.3 46.7 59 l40.2 75.0 19 193. 1 io3,2 79 246.1 i3i.5 4o 35.3 18.9 100 88.2 47-1 60 i4i .1 75.4 20 194.0 103.7 80 246.9 l32.0 4i 36.2 .9.3 lOI 89.1 47.6 161 142.0 75.9 221 194.9 104.2 281 247.8 i32.5 4s 37.0 19.8 02 90.0 48.1 62 142.9 76.4 22 195.8 io4;7 82 248.7 132.9 43 37.9 20.3 o3 90.8 48.6 63 143.8 76.8 .23 196.7 io5.i 83 249.6 i33.4 44 38.8 20.7 04 91.7 49.0 64 144.6 77.3 24 197.6 io5.6 84 25o.5 133.9 45 J9.7 21.2 o5 92.6 49.5 65 145.5 77.8 25 198.4 1 06. 1 85 25i.3 1 34.3 40 40.6 21.7 06 93.5 5o.o 66 146.4 78.3 26 199.3 106.5 86 252.2 134.8 47 41.5 22.2 07 94.4 5o.4 67 147.3 78.7 27 200.2 107.0 87 253.1 1 35.3 48 42.3 22.6 08 95.2 50.9 68 i48.2 79.2 28 201. 1 107.5 88 254.0 i3b.8 49 43.2 23.1 09 96.1 5i.4 69 149.0 79-7 29 202.0 107.9 89 254.9 i36.2 bo 44.1 23.6 10 97.0 51.9 70 149.9 80.1 3o 202.8 108.4 90 255.8 1 36.7 5i 45 24.0 III 97-9 52.3 171 i5o.8 80.6 23l 203.7 108.9 291 256.6 137.2 52 45 9 24.5 12 98.8 52.8 72 i5i .7 81. 1 32 204.6 109.4 92 257.5 137.6 53 46.7 25.0 i3 99-7 53.3 73 i52.6 81.6 33 2o5.5 109.8 93 258.4 i38.i 54 47.6 25.5 i4 100.5 53.7 74 i53.5 82.0 34 206.4 110.3 94 259.3 i38.6 55 48.5 25.9 i5 101.4 54.2 75 i54.3 82.5 35 207.3 1 10.8 95 260.2 139.1 56 49-4 26.4 16 102.3 54.7 76 i55.2 83. 36 208.1 III. 2 96 261 .0 139.5 57 50.3 26.9 17 I03.2 55.2 77 i56.i 83.4 37 209.0 1 1 1.7 97 261 .9 i4o.o 58 5l.2 27.3 18 104. 1 55.6 78 157.0 83.9 38 209.9 112. 2 98 262.8 i4o5 59 52 .0 27.8 19 104.9 56.1 79 157.9 84.4 39 210.8 112.7 99 263.7 140.9 6o Dist 52.9 28.3 20 io5.8 56.6 80 i58.7 84.9 40 21 1 .7 ii3.i 3oo 264.6 i4i4 Dep. l.iit. Dist. Pep. I.ai. Dlsl. Dop. Lat. Dist. Dep. Lat. Dist. Dep. Lai. N.E.byE.AE. S E.byE.^E. N.W.byW.^W. S.W.byW.^W. [For 5^ Points. TABLE L [Page U Difference of Latitude and Departure for 2| Points. N.N.E4E. N.N.W.|W. S.S.E.|E. S.S.W.|W. Disl. Lat. Dop. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. ! Dep. Dist. 241 Lat. 206.7 Dep. 123.9 I 00.9 00.5 61 52.3 3i.4 121 io3.8 62.2 181 i55.2 93.1 2 01.7 01. 62 53.2 J 1. 9 22 104.6 62.7 82 i56.i 93.6 42 207.6 124.4 3 02.6 01 .5 63 54.0 32.4 23 io5.5 63.2 83 157.0 94.1 43 208.4 124.9 4 o3.4 02.1 64 54.9 32.9 24 106.4 63.7 ^4 157.8 94.6 44 209.3 125.4 5 04.3 02.6 65 55.8 33.4 25 107.2 64.3 85 i58.7 95.1 45 210. 1 126.0 6 o5.i o3.i 66 56.6 33.9 26 108. 1 64.8 86 159.5 95.6 46 21 1 .0 126.5 7 06.0 o3.6 67 57.5 34.4 27 108.9 65.3 87 160.4 96.1 47 211 .9 127.0 8 06.9 04.1 68 58.3 35.0 28 109.8 65.8 88 161. 3 96.7 48 212.7 127.5 9 07.7 04.6 69 59.2 35.5 29 no. 6 66.3 89 162.1 97.2 49 2i3.6 128.0 lO 08.6 o5.i 70 60.0 36.0 3o III .5 66.8 90 i63.o 97-7 5o 214.4 128.5 II 09.4 o5.7 71 60.9 36.5 i3i 112.4 67.3 191 i63.8 98.2 25l 2i5.3 1 29.0 12 10.3 06.2 72 61.8 37.0 32 Il3.2 67.9 92 164.7 98.7 52 216. 1 129.6 i3 II. 2 06.7 73 62.6 37.5 33 114.1 68.4 93 i65.5 99.2 53 217.0 i3o.i iS 12.0 07.2 74 63.5 38.0 34 114.9 68.9 94 166.4 99-7 54 217.9 i3o.6 i5 12.9 07.7 75 64.3 38.6 35 ii5.8 69.4 95 167.3 100.3 55 218.7 i3i.i i6 i3.7 08.2 76 65.2 39.1 36 116. 7 69.9 96 168.1 100.8 56 219.6 i3i.6 17 14.6 08.7 77 66.0 39.6 37 117.5 70.4 97 169.0 101.3 57 220.4 l32.I i8 i5.4 09.3 7« 66.9 4o.i 38 118.4 70.9 98 169.8 101.8 58 221 .3 1 32.6 19 16.3 09.8 79 67.8 40.6 39 119. 2 71.5 99 170.7 102.3 59 222.2 1 33.2 20 17.2 10.3 80 68.6 4i.i 4o i4i 120.1 72.0 200 171 .5 102.8 60 223.0 i33.7 21 18.0 10.8 81 69.5 4t.6 120.9 72.5 201 172.4 io3.3 261 223.9 i34.2 22 18.9 11.3 82 70.3 42.2 42 121.8 73.0 02 173.3 io3.8 62 224.7 134-7 23 19.7 II. 8 83 71.2 42.7 43 122.7 73.5 o3 174.1 104.4 63 225.6 i35.2 24 20.6 12.3 84 72.0 43.2 44 123.5 74.0 04 175.0 104.9 64 226.4 135.7 25 21 .4 12.9 85 72.9 43.7 45 124.4 74.5 o5 175.8 io5.4 65 227.3 1 36.2 26 22.3 i3.4 86 73.8 44.2 46 125.2 75.1 06 176.7 105.9 66 228.2 1 36.8 27 23.2 13.9 87 74.6 44.7 47 126. X 75.6 07 177.5 106.4 67 229.0 137.3 26 24.0 14.4 88 75.5 45.2 48 126.9 76.1 q8 178.4 106.9 68 229.9 137.8 29 24.9 14.9 89 76.3 45.8 49 127.8 76.6 09 179.3 107.4 69 230.7 1 38.3 3o 25.7 i5.4 90 77.2 46.3 5o 128.7 77-1 10 180. 1 108.0 70 231.6 i38.8 3i 26.6 i5.9 91 78.1 46.8 i5i 129.5 77.6 21 1 181. 108.5 271 232.4 139.3 32 27.4 16.5 92 78.9 47.3 52 i3o.4 78.1 12 181. 8 109.0 72 233.3 139.8 33 28.3 17.0 93 79.8 47.8 53 i3i .2 78.7 i3 182.7 109.5 73 234.2 140.4 M 29.2 17.5 94 80.6 48.3 54 l32.1 79.2 i4 i83.6 IIO.O 74 235.0 140.9 35 3o.o 18.0 95 81.5 48.8 55 132.9 79-7 i5 184.4 110.5 75 235.9 141.4 36 3o.9 18.5 96 82.3 49-4 56 i33.8 80.2 16 i85.3 III.O 76 236.7 141.9 37 3i.7 19.0 97 83.2 49.9 57 134.7 80.7 17 186.1 11 1.6 77 237.6 142.4 38 32.6 19.5 9S 84.1 5o.4 58 i35.5 81.2 18 187.0 112. 1 78 238.4 142.9 39 33.5 20.1 99 84.9 50.9 59 i36.4 81.7 19 187.8 112.6 79 239.3 143.4 4o 34.3 20.6 100 85.8 5i.4 60 137.2 82.3 20 188.7 ii3.i 80 240 . 2 143.9 4i 35.2 21 .1 lOI 86.6 51.9 161 i38.i 82.8 221 189.6 ii3.6 281 241 .0 144.5 42 36.0 21 .6 02 87.5 52.4 62 139.0 83.3 22 190.4 114.1 82 241.9 145.0 43 36.9 22.1 o3 88.3 53.0 63 139.8 83.8 23 191 .3 114.6 83 242.7 145.5 44 37.7 22 .6 o4 89.2 53.5 64 i4o.7 84.3 24 192. 1 ll5.2 84 243.6 1 46.0 45 38.6 23.1 o5 90.1 54.0 65 i4i.5 84.8 25 193.0 II5.7 85 244.5 146.5 46 39.5 23.6 06 90.9 54.5 66 142.4 85.3 26 193.8 116.2 86 245.3 147-0 47 40.3 24.2 07 91.8 55.0 67 143.2 85.9 27 194.7 116.7 87 246.2 147.5 48 4t .2 24.7 08 92.6 55.5 68 144.1 86.4 28 195.6 117.2 88 247.0 I48.I 49 42.0 25.2 09 93.5 56.0 69 i45.o 86.9 29 196.4 117-7 89 24-7.9 148.6 be 42.9 25.7 10 94.4 56.6 70 145.8 87.4 3o 197.3 118. 2 90 248.7 1 49. 1 5i 43.7 26.2 III 95.2 57.1 171 146.7 87.9 23l 198.1 118.8 291 249.6 149.6 52 44. b 26.7 12 96.1 57.6 72 147.5 88.4 32 199.0 1 19.3 92 250.5 i5o.i 53 45.5 27.2 i3 96.9 58.1 73 148.4 88.9 33 199.9 1 19.8 93 251.3 i5o.6 54 46.3 27.8 i4 97.8 58.6 74 149.2 89.5 M 200.7 120.3 94 252.2 i5i.i 55 47.2 28.3 i5 98.6 59.1 75 i5o.i 90.0 35 201 .6 120.8 q5 253. i5i.7 56 48.0 28.8 16 99.5 59.6 76 i5i.o 90.5 36 202.4 121.3 96 253.9 l52.2 i)7 48.9 29.3 17 100.4 60.2 77 i5i.8 91.0 37 2o3.3 121.8 97 254-7 i52.7 58 49.7 29.8 18 lOI .2 60.7 78 i52.7 91.5 38 204.1 122.4 98 255.6 i53.2 59 5o.6 3o.3 19 102. 1 61.2 79 i53.5 92.0 39 2o5.o 122.9 99 256.5 i53.7 bo 5i.5 3o.8 20 102.9 61.7 80 154.4 92.5 40 205.9 123.4 3 00 257.3 154.2 Dist-i Dcp. I. at. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. N.E.byE.iE. S.E.byE.iE. N.W.byW.^W. S. W.by W.^W . [For 5^ Points. rage 12] TABLE I. DilTerenca of Latitude and Departure for 3 Points. N.E.byiN. N.W.byN. S.E.byS. S.W.byS. Dist. Lai. Dep. Disi. Lat. Dep. Dist. Lat. Dep. Dist. 181 82 83 84 85 86 87 88 89 90 Lat. Dep. Dist. Lat. Dep. I 2 3 4 5 6 7 8 9 lO 00. 8 01.7 02.5 o3.3 o4.2 o5.o o5.S 06.7 07.5 08.3 00.6 01 .1 01.7 02.2 02.8 o3.3 03.9 04.4 o5.o o5.6 61 62 63 64 65 66 67 68 69 70 5o.7 5i.6 52.4 53.2 54.0 54.9 55.7 56.5 57.4 58.2 33.9 34.4 35.0 35.6 36.1 36.7 37.2 37.8 38.3 38.9 121 22 23 24 25 26 27 28 3? 100.6 101.4 102.3 io3.i 103.9 104.8 io5.6 106.4 107.3 108. 1 67.2 67.8 68.3 68.9 69.4 70.0 70.6 71. 1 71-7 72.2 i5o.5 i5i.3 l52.2 I53.0 i53.8 154.7 i55.5 i56.3 157. 1 i58.o 100.6 lOI.I 101.7 102.2 102.8 io3.3 103.9 104.4 io5.o io5.6 241 42 43 44 45 46 47 48 ii 200.4 201 .2 202.0 202.9 203.7 204.5 2o5.4 206 . 2 207.0 207.9 133.9 134.4 i35.o i356 1 36 I 1 36.7 137.2 137.8 i38.3 i38.9 1 1 12 i3 i4 i5 i6 17 iS 19 20 09. 1 10. 10.8 II. 6 12.5 i3.3 i4.i i5.o 1 5. 8 16.6 06. 1 06.7 07.2 07.8 08.3 08.9 09.4 1 0.0 10.6 11. 1 7' 72 73 74 75 76 77 78 79 80 59.0 59.9 60.7 61.5 62.4 63.2 64.0 64.9 65.7 66.5 39.4 4o.o 4o.6 41.1 41.7 42.2 42.8 43.3 43.9 44.4 i3i 32 33 34 35 36 37 38 39 4o 108.9 109.8 1 10.6 III .4 112. 2 ii3.i 113.9 114.7 ii5.6 116. 4 72.8 73.3 73.9 74.4 75.0 75.6 76.1 76.7 77.2 77.8 191 92 93 94 95 96 97 98 99 200 i58.8 159.6 160.5 161. 3 1 62 . 1 i63.o i63.8 164.6 i65.5 166.3 106. 1 106.7 107.2 107.8 108.3 108.9 109.4 1 1 0.0 110.6 111. 1 25l 52 53 54 55 56 57 58 60 208.7 209.5 210.4 211 .2 212.0 212.9 213.7 214.5 2i5.4 216.2 139.4 i4oo 1 40.6 i4i.i 141.7 142.2 142.8 143.3 143.9 144.4 21 22 23 24 25 26 27 28 19 3o 17.5 18.3 19. 1 20.0 20.8 21 .6 22.4 23.3 24.1 24.9 II. 7 12.2 12.8 i3.3 13.9 14.4 i5.o i5.6 16. 1 16.7 81 82 83 84 85 86 87 88 89 90 67.3 68.2 69.0 69.8 70.7 71.5 72.3 73.2 74.0 74.8 45.0 45.6 46.1 46.7 47-2 47.8 48.3 48.9 49.4 5o.o i4i 42 43 44 45 47 48 49 5o 117. 2 118. 1 118. 9 119.7 120.6 121. 4 122.2 123. I 123.9 124.7 7S.3 78.9 79-4 80.0 80.6 81. 1 81.7 82.2 82.8 83.3 201 02 o3 o4 o5 (}6 07 08 09 10 167. 1 168.0 168.8 169.6 170.5 171 .3 172. 1 172.9 173.8 174.6 1 1 1.7 112. 2 112.8 11 3.3 1 13.9 114.4 1 1 5.0 1 1 5.6 116.1 1 16.7 261 62 63 64 65 66 67 68 69 70 217.0 217.8 218.7 219.5 220.3 221 .2 222.0 222.8 223.7 224.5 145.0 i45.6 146.1 146.7 147.2 147-8 148.3 148.9 149.4 i5o.o 3i 32 33 34 35 36 37 38 39 40 25.8 26.6 27.4 28.3 29.1 ^9-9 3o.8 3i.6 32.4 33.3 17.2 17.8 18.3 18.9 19.4 20.0 20.6 21. 1 21.7 22.2 91 93 94 95 96 97 98 99 TOO 75.7 76.5 77.3 78.2 79.0 79.8 80.7 81.5 82.3 83.1 5o.6 5i.i 51.7 52.2 52.8 53.3 53.9 54.4 55.0 55.6 i5i 52 53 54 55 56 57 58 59 60 125.6 126.4 127.2 128. c 128.9 129.7 i3o.5 i3i.4 l32.2 133.0 83.9 84.4 85.0 85.6 86.1 86.7 87.2 87.8 88.3 88.9 211 12 i3 i4 i5 16 17 18 '9 20 175.4 176.3 177-1 177-9 178.8 179.6 180.4 181. 3 182.1 182.9 117.2 1 17.8 118.3 118.9 119.4 120.0 120.6 121. 1 121.7 122.2 271 72 73 74 75 76 77 78 79 80 225.3 226.2 227.0 227.8 228.7 229.5 23o.3 23l .1 232. 232.8 i5o.6 i5i.i 1 5 1. 7 l52.2 i52.8 i53.3 153.9 154.4 i55.o 1 55.6 1 56.1 1 56.7 157.2 1578 i58.3 1 58.9 1 59.4 160.0 160,6 161. 1 4i 42 43 44 45 46 47 48 49 5o 34.1 34.9 35.8 36.6 37.4 38.2 39. 1 39.9 40.7 4r.6 22.8 23.3 23.9 24.4 25.0 25.6 26.1 26.7 27.2 27.8 lOI 02 o3 04 o5 06 07 08 09 10 84.0 84.8 85.6 86.5 87.3 88.1 89.0 89.8 90.6 91.5 56.1 56.7 57.2 57.8 58.3 58.9 59.4 60.0 60.6 61. 1 161 62 63 64 65 66 67 68 69 70 171 72 73 74 75 7fi 77 78 79 80 133.9 134.7 i35.5 i36.4 137.2 i38.o i38.9 139.7 i4o.5 i4i.3 89.4 90.0 90.6 91. 1 91.7 92.2 92.8 93.3 93.9 94.4 95.0 95.6 96.1 96.7 97.2 97.8 98.3 98.9 99.4 100. 221 22 2 3 24 25 26 27 28 3o i83.8 184.6 i85.4 186.2 187. 1 187.9 18S.7 189.6 190.4 191 .2 122.8 123.3 123.9 124.4 I25.0 125.6 126. 1 126.7 127.2 127.8 281 82 83 84 85 86 87 88 89 90 233.6 234.5 235.3 236.1 237.0 237.8 238.6 239.5 240.3 241 .1 5i 52 53 54 55 56 57 58 60 Disi. 42.4 43.2 44.1 44.9 45.7 46.6 47.4 48.2 49.1 .49.9 I)C|). 28.3 28.9 29.4 3o.o 3o.6 3i.i 3i.7 32.2 32.8 33.3 Lat. II I 12 i3 i4 i5 16 17 18 '9 20 92.3 93.1 94.0 94.8 95.6 96.5 97.3 98.1 98.9 99.8 61.7 62.2 62.8 63.3 63.9 64.4 65.0 65.6 66.1 66.7 142.2 143.0 143.8 144.7 145.5 i46.3 l47-2 i48.o i48.8 149.7 23l 32 33 34 35 36 37 38 39 4o Dist. 192. 1 192.9 193.7 194.6 195.4 196.2 197-1 197.9 198.7 199.6 128.3 128.9 129.4 i3o.o i3o.6 i3i.i i3i.7 l32.2 1 32.8 i33.3 291 9^ 93 94 96 97 98 3oo 242.0 242.8 243.6 244.5 245.3 246.1 246.9 247.8 24s. 6 249.4 161.7 162.2 162.8 i63.3 163,9 164.4 i65.o i65.6 166.1 166.7 Hist. Dep. Lat. Disi. Dep. Lat. Dep. Lat. Dist. Dep. Lat. N.E.byE. S.E.byE. N.W.byW. S.W.by W. [For .5 Points. ; *" ^ TABLE L frase 13 Difference of Latitude and Dep irture for 3^ Points. N.E.^N. N.W4N. S.E.IS. S.W.5S. Dist Lat. Dep. Disl. 61 Lat. 49.0 Dep. Disl 121 Lat. 97.2 Dep. Disl Lat. Dep. Disl. Lat. Dep. I GO . 8 00.6 36.3 72.1 181 145.4 107.8 241 193.6 143.6 2 01 .6 01 .2 62 49-8 30.9 22 98.0 72.7 82 i46.2 10S.4 42 194.4 i44-2 3 02.4 or.t 63 5o.6 37.5 23 98.8 73.3 83 i47-o 109.0 43 195.2 144.S 4 ()3.2 02.4 64 5i.4 38.1 24 99.6 73.9 84 147-8 109.6 44 1 9b . 145.4 5 o4.o o3.o 63 52.2 38.7 2 5 100.4 74.5 85 148.6 110.2 45 196.8 145.9 6 o4.8 o3.6 66 53.0 39.3 26 lOI .2 75.1 86 149.4 1 10.8 46 197.6 146.5 7 o5.6 04.2 67 53.8 39.9 27 102.0 75.7 87 i5o.2 1 1 1.4 4i 198.4 i47-i S,o6.4 04.8 68 54.6 40.5 28 102.8 76.2 88 i5i .0 1 1 2.0 48 199.2 147-7 9 07.2 o5.4 69 55.4 4i.i 29 io3.6 76.8 89 i5i.8 112.6 49 200 . 1 48. 3 10 08.0 06.0 70 56.2 41.7 3o 104.4 77.4 90 i52.6 Il3.2 5o 200.8 148.9 1 1 0S.8 06.6 71 57.0 42.3 i3i I05.2 78.0 191 153.4 ii3.8 25l 201 .6 149-5 12 09.6 07.1 72 57.8 42.9 32 106.0 78.6 92 i54.2 1 14.4 52 202.4 i5o.i i3 10.4 07.7 73 58.6 43.5 33 106.8 79.2 93 i55.o 1 1 5.0 53 2o3.2 i5o.7 i4 1 1.2 08.3 74 59.4 44.1 M 107.6 79.8 94 155.8 1 1 5.6 54 204.0 i5i.3 /5 12.0 08.9 75 60.2 44.7 35 108.4 80.4 9b 156.6 116. 2 55 204.8 i5i.9 i6 12.9 .09.5 76 61 .0 45.3 36 109.2 81.0 96 i5- 4 116.8 56 2o5.6 i52.5 I? ■ 3.7 10. 1 77 61.8 45.9 37 IIO.O 81.6 97 i58.2 117.4 57 206.4 i53.i i8 i4.5 10.7 78 62.7 46.5 38 no. 8 82.2 98 159.0 1 17.9 58 207.2 153.7 IP i5.3 H.3 79 63.5 47-1 39 III .6 82.8 99 159.8 11S.5 59 20S.0 154.3 20 16. 1 II. 9 80 81 64.3 47.7 4o 112. 4 83.4 ~847o~ 200 1 60 . 6 1 19.1 6(, J08.8 1 54 9 21 16.9 12.5 65.1 48.3 i4i ii3.3 201 161. 4 119.7 261 209.6 i55.5 22 17-7 i3.i 82 65.9 48.8 42 114. 1 84.6 02 162.2 120.3 62 210.4 1 56. 1 23 18.5 l3.7 83 66.7 49.4 43 114. 9 85.2 OJ i63.i 120.9 63 211. 2 1 56.7 24 19.3 i4.3 84 67.5 5().o 44 115.7 85.8 04 163.9 121. 5 64 212.0 1.57.3 25 20. 1 r4.9 85 68.3 5o.6 45 116. 5 86.4 o5 164.7 122. 1 65 212.8 157.9 26 20.9 i5.5 86 69.1 5l.2 46 117. 3 87.0 06 1 65. 5 122.7 66 2.3.7 I5S.5 27 21 .7 16. 1 87 69.9 5i.8 4i 118. 1 87.6 07 166.3 123.3 67 214.5 1 59. 1 28 22.5 16.7 88 70.7 52.4 48 118. 9 88.2 08 167.1 123.9 68 2i5.3 159.6 29 23.3 17.3 89 71.5 53.0 49 119.7 88.8 09 167.9 124.5 69 216.1 160.2 3o 24.1 17.9 90 72.3 53.6 5o 120.5 89.4 10 168.7 125. 1 70 216.9 160.8 3r 24.9 18.5 91 73.1 54.2 i5i 121 .3 90.0 21 1 169.5 125.7 271 217.7 161.4 32 23.7 19. 1 92 73.9 54.8 52 122. 1 90.5 12 170.3 126.3 72 218.5 162.0 33 26.5 19.7 93 74.7 55.4 53 122.9 91. 1 i3 171. 1 126.9 73 219.3 163.6 34 27.3 20.3 94 75.5 56.0 54 123.7 91.7 i4 171. 9 127.5 74 220. 1 i63.2 35 28.1 20.8 95 76.3 56.6 55 124.5 92.3 i5 172.7 128. 1 75 220.9 163.8 36 28.9 21.4 96 77.1 57.2 56 125.3 92.9 16 173.5 128.7 76 221 .7 164.4 3? 29.7 22.0 97 77-9 57.8 b7 126. 1 93.5 17 174.3 129.3 77 222.5 i65.o 38 3o.5 22.6 98 78.7 58.4 58 126.9 94.1 18 175. 1 129.9 78 223.3 i65.6 39 3t.3 23.2 99 79-5 59.0 59 127.7 94.7 19 175.9 i3o.5 79 224. 1 16b. 2 4o 4i' 32.1 23.8 100 80.3 5o.b bo 128.5 95.3 20 176.7 i3i.i 80 224.9 ibb.S 32.9 24.4 lOI 81. 1 60.2 161 129.3 959 221 i77-b i3r.6 281 225.7 167.4 42 33.7 25.0 02 81.9 60.8 62 i3o.i 96.5 22 178.3 l32.2 82 226.5 168.0 43 34.5 25.6 o3 82.7 61.4 63 1 3o . 9 97.1 23 179.1 i32.8 83 227.3 168.6 44 35.3 26.2 04 83.5 62.0 64 i3i.7 97-7 24 179.9 i33.4 84 228.1 169.2 45 3b I 26.8 o5 84.3 62.5 65 i32.5 98.3 25 180.7 1 34.0 85 228.9 169.8 46 3(3 9 27.4 06 85.1 63.1 ()b i33.3 Q8.9 26 181. 5 1 34.6 86 229.7 170.4 47 37.8 28.0 07 85.9 63.7 67 i34.i 99.5 27 182.3 i35.2 87 23o.5 1 71.0 48 38.6 28.6 n8 86.7 64.3 ()8 134.9 100. 1 28 i83.i i35.8 88 231.3 1 7 1 .6 49 39.4 29.2 09 87.5 64.9 69 .35.7 100.7 29 183.9 1 36.4 89 232. I 172.2 5o 4o.2 29.8 10 88.4 65.5 70 i36.5 IOI.3 IUI.9 3u 184.7 137.0 90 291 232.9 233.7 172.8 173.3 5i 4i .0 3o.4 I II 89.2 66.1 171 137.3 23 I i85.5 137.6 52 4i.8 3i.o 12 90.0 66.7 72 i38.2 102.5 32 186.3 i38.2 92 234.5 173.9 53 42.6 3i.8 i3 90.8 67.3 73 139.0 io3.i 33 187. 1 1 38.8 93 235.3 174.5 M 4'i.4 32.2 i4 91 .6 67.9 74 139.8 io3.7 34 188.0 139.4 94 236.1 175.1 bb 44.2 32.8 i5 92.4 68.5 7b 140.6 104.2 35 18S.8 i4o.o 95 236-9 175.7 5b 45.0 33.4 16 93.2 69.1 76 i4i.4 104.8 36 189.6 i4o.6 96 237.7 176.3 b7 45.8 34.0 17 94.0 69.7 77 142.2 io5.4 37 190.4 l4l.2 97 238.6 176.9 58 46.6 34.6 18 94.8 70.3 78 143.0 106.0 38 191 .2 i4i.8 98 239.4 177-5 59 47.4 35.1 19 95.6 70.9 79 143.8 1 06.6 39 192.0 142.4 99 240 2 178.1 bo 48.2 35.7 20 96.4 71.5 80 144.6 107.2 40 192.8 143.0 3oo 241 .0 178.7 DIst. Dop. Lat. Dist. Dcp> Lat. Dsi. Dep. 1 Lai. l>i.st. Dop. Lat. 1 Disl. Dop. Lat. N.E.3E. S.E3E. N.W.s W. S.W.^V/. [For 43 Points, j Page 141 TABLE L Difference of Latitude and Departure for 3^ Points. n.e; ^N. N.W.^N. S.E.iS. S.W.iS. 1 Dist.l Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Up. I 00.8 00.6 61 47-2 38.7 121 93.5 76.8 181 139.9 1 14.8 241 186.3 if-wi.9 2 01. b 01 .3 62 47-9 39.3 22 94.3 77-4 82 140.7 1 1 5.5 42 187. 1 ib3.5 6 02.3 01 .9 63 48.7 4o.o 23 95.1 78.0 83 i4i.5 1 1 6. 1 43 187.8 154.2 4 o3.i 02.5 64 49-5 40.6 24 95.9 78.7 84 142.2 1 16.7 AA 188.6 1 54.8 b 03.9 03.2 65 5o.2 4l.2 2b 96.6 79.3 85 143.0 117.4 45 189.4 i55.4 b 04. b o3.8 66 5i.o 41.9 2b 97-4 79-9 86 143.8 118.0 46 190.2 i56.i 1 o5.4 04.4 67 5i.8 42.5 27 98.2 80.6 87 144.6 118.6 47 190.9 1 56.7 a 06.2 o5.i 68 52.6 43.1 28 98.9 81.2 88 145.3 1 19.3 48 191-7 157.3 9 07.0 Ob. 7 69 53.3 43.8 29 99-7 81.8 89 146.1 1 19.9 49 192.5 1 58.0 10 07.7 06.3 70 54.1 AA-A 3o 100.5 82.5 90 146.9 120.5 5o 193.3 1 58.6 II 08.5 07.0 71 54.9 45.0 i3i loi .3 83.1 191 147.6 121.2 25l 194.0 159.2 12 09.3 07.6 72 55.7 4b.7 32 102.0 83.7 92 148.4 121.8 52 194.8 .59.9 i3 10. 08.2 73 56.4 46.3 33 102.8 84.4 93 149.2 122.4 53 195.6 160.5 i4 10.8 08.9 74 57.2 46.9 34 io3.6 85.0 94 i5o.o 123. 1 54 T96.3 161.1 lb II. b 09.5 75 58. 47 -b 35 104.4 85.6 95 i5o.7 123.7 55 197.1 161.8 lb 12.4 10.2 76 58.7 48.2 36 io5.i 86.3 96 i5i.5 124.3 56 197-9 162.4 17 i3.i 10.8 77 59.5 48.8 37 105.9 86.9 97 i52.3 125. 57 198.7 i63.o i8 13.9 II. 4 7B 60.3 49-i) 38 106.7 87.5 98 i53.i 125.6 58 199.4 163.7 19 14.7 12. 1 79 bi.i bo. I 39 107.4 88.2 99 i53.8 126.2 59 200.2 164.3 20 ib.b 12.7 80 61.8 5o.8 40 108.2 88.8 200 i54.6 126.9 60 201 .0 164.9 21 16.2 i3.3 81 62.6 5i.4 i4i 109.0 89.4 201 i55.4 127.5 261 201 .8 1 65.6 22 17.0 i4.o 82 63.4 52.0 42 109.8 90.1 02 i56.i 128.1 62 202.5 166.2 2j 17.8 14.6 83 64.2 b2.7 A^ no. 5 90.7 o3 i56.9 128.8 63 2o3.3 166.8 24 18. b l5.2 84 64.9 b3.3 AA III .3 91.4 04 157.7 129.4 H 204.1 167.5 25 19.3 lb. 9 85 65.7 b3.9 45 112. 1 92.0 o5 i58.5 i3o.i 65 204.8 168.1 2b 20.1 16.5 86 66.5 54.6 46 112. 9 92.6 06 159.2 1 30.7 66 2o5.6 168.7 27 20.9 17. 1 87 67.3 bb.2 47 ii3.6 93.3 07 160.0 i3i.3 67 206.4 169.4 28 21 .b 17.8 88 68.0 bb.8 48 114.4 93.9 08 160. S l32.0 68 207.2 170.0 29 22.4 18.4 89 68.8 bb.b 49 ll5.2 94.5 09 161. 6 132.6 69 207.9 170.7 Jo 23.2 19.0 90 69.6 b7.i bo 116.0 95.2 10 162.3 i33.2 70 208.7 171.3 3i 24.0 19.7 91 70.3 D7.7 i5i 116.7 95.8 21 1 i63.i 133.9 271 209.5 171.9 32 24.7 20.3 Q2 71. 1 58.4 b2 117.5 96.4 12 163.9 i34.5 72 210.3 172.6 33 2b. b 20.9 93 71.9 59.0 53 118. 3 97.1 i3 164.7 i35.i 73 21 1 .0 173.2 34 26.3 21 .6 94 72.7 b9.b M 119. 97-7 i4 i65.4 i35.8 74 211.8 173.8 3b 27.1 22.2 95 73.4 bo.3 bb 119. 8 98.3 i5 166.2 i36.4 75 212.6 174.5 3o 27.8 22.8 96 74.2 60.9 5b 120.6 99.0 16 167.0 137.0 76 2i3.4 175. 1 ^7 28 .b 23.5 97 75.0 bi.b !)7 121 .4 99.6 17 167.7 137.7 77 214. 1 175.7 38 29.4 24.1 98 75.8 62.2 58 122. 1 100.2 18 168.5 i38.3 78 214.9 176.4 39 3o. I 24.7 99 76.5 62.8 59 122 .9 100.9 19 169.3 i38.9 79 2i5.7 177.0 40 30.9 25.4 100 77.3 63.4 60 123.7 IOI.5 20 170. 1 139.6 80 216.4 177.6 4i 3i.7 26.0 lOI 78.1 64.1 161 124.5 102. 1 221 170.8 i4o.2 281 217.2 178.3 42 32. b 26.6 02 78.8 64.7 62 125.2 102.8 22 171 .6 i4o.8 82 218.0 178.9 Ai 33.2 27.3 o3 79.6 65.3 63 126.0 io3.4 23 172.4 i4i.5 83 218. 8 179.5 AA 34.0 27.9 o4 80.4 66.0 64 126.8 104.0 24 173.2 142.1 84 219.5 180.2 4b 34.8 28.5 o5 81.2 66.6 65 127.5 104.7 25 173.9 142.7 85 220.3 180.8 4b 35.6 29.2 06 81.9 67.2 66 128.3 io5.3 26 174.7 143.4 86 221 .1 181.4 47 3b. 3 29.8 07 82.7 67.9 68.5 67 129.1 105.9 27 175.5 144.0 87 221 .9 1S2.1 48 37.1 3o.5 08 83.5 68 129.9 106.6 28 176.2 144.6 88 222.6 182.7 f9 37.9 3i.i 09 84.3 69.1 69 i3o.6 107.2 29, 177.0 145.3 89 223.4 183.3 bo 38.7 3. .7 10 85.0 69.8 70 i3i.4 107.8 3o 177.8 145.9 90 224.2 184.0 bi 39.4 32.4 1 II 85.8 70.4 •71 I 32. 2 108.5 23 I 178.6, 146.5 291 224.9 184.6 b2 4o.2 33.0 12 86.6 71.1 72 i33.o 109.1 32 179.3 147.2 92 225.7 i85.2 b3 4i .0 33.6 i3 87.4 71-7 73 133.7 109.8 33 180.1 147.8 93 226.5 185.9 ^4 41.7 34.3 i4 88.1 72.3 74 134.5 110.4 34 1 80 . 9 148.4 94 227.3 186.5 bb 42.5 34.9 i5 88.9 73.0 7^) i35.3 1 1 I.O 35 181 7 149.1 95 228.0 187.1 bb 43.3 35.5 16 89.7 73.6 7b i36.o 1 1 1.7 36 182.4 149.7 96 228.8 187.8 ^7 44.1 36.2 17 90.4 74.2 77 i36.8 I 12.3 37 i83.2 i5o.4 97 229.6 18S.4 b8 AA-^ 36.8 18 91 .2 74.9 78 137.6 1 1 2.9 38 184.0 1 5 1.0 98 23o.4 189.0 b9 45.6 J7.4 19 92.0 75.b 79 i38.4 11 3.6 39 184.7 i5i.6 99 23 1. 1 189.7 bo A^'^.A 38.1 20 93.8 7b. I 80 139. 1 Il4.2 40 I)i.;t. i85.5 Dop. i52.3 3oo 231.9 190.3 Dist. I)('|). l.iU. Dist. Dep. Lnl. Dist. Dpp. Lat. Lnt. Dist. Dep. Lat. N.EAE. S.E.AE. N.W..i^ V. S.W.AW. [For 4.i Points. 1 TABLE L 1 fP.ige 15 Difference of Latitude and Departure for 3f Points. N.E-iN. N.W4N. S.E.^S. S.W.iS. Dist Lai. ' Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 00 . 7 00 . 7 61 45.2 4i.o 121 89.7 81.3 181 i34.i I 21.6 241 178.6 161.8 2 oi.5'oi.3 62 4'i.9 4i.6 22 90.4 81.9 82 134.9 122.2 42 179.3 162.5 .1 02.2 02.0 63 46.7 42.3 23 91. 1 82.6 83 i35.6 122.9 Ai 180. 1 i63.2 ^ ()3 .0 02.7 64 47.4 43.0 24 91.9 83.3 84 i36.3 123.6 AA 180.8 J 63.0 ^. o3.7 o3.4 65 48.2 43.7 25 92.6 83.9 85 137. 1 124.2 45 181. 5 164.6 f) o4.4 04.0 66 48.9 A^:^ 26 93.4 84.6 86 137.8 124.9 46 182.3 i65.2 7 o5.2 04.7 07 49-6 45.0 27 94.1 85.3 «7 i38.6 125.6 47 i83.o 166.0 « 05.9 o5.4 68 5o.4 45.7 28 94.8 86.0 88 139.3 126.3 48 i83.8 166.5 9 06 . 7 06.0 69 5i.i 46.3 29 95.6 86.6 89 i4o.o 126.9 49 184.5 167.2 lO 07.4 06.7 70 51.9 47-0 3o 96.3 87.3 90 140.8 127.6 5o iS5.2 167.9 168.6 II 08.2 07.4 71 52.6 47-7 i3i 97.1 88.0 191 141.5 128.3 25l 186.0 12 08.9 08.1 72 53.3 48.4 32 97.8 88.6 92 142.3 128.9 62 186.7 169.2 i3 09.6 08.7 73 54.1 49.0 33 98.5 89.5 93 143.0 129.6 53 187.5 169.9 (4 10.4 09.4 74 54.8 49.7 M 99.3 90.0 94 143.7 i3o.3 64 188.2 170.6 i5 1 1 .1 10. 1 75 55.6 5o.4 35 100. 90.7 95 144.5 i3i.o 66 188.9 171-2 if) II. 9 10.7 76 56.3 5i.o 36 100.8 91.3 96 145.2 i3i.6 66 189.7 171-9 '7 12 .6 II. 4 77 57.1 5i.7 37 loi .5 92.0 97 146.0 i32.3 67 190.4 172.6 i8 i3.3 12. 1 78 57.8 52.4 38 102.3 92.7 98 146.7 i33.o 68 191 .2 173.3 ■9 14.1 12.8 79 58.5 53.1 39 io3.o 93.3 99 147-4 i33.6 69 191. 9 173.9 JO 14.8 i3.4 80 59.3 53.7 4o io3.7 94.0 200 148.2 134.3 60 192.6 174.6 21 i5.6 14.1 81 60.0 54.4 i4i 104.5 94-7 201 148.9 i35.o 261 1 193.4 176.3 22 16.3 j4.8 82 60.8 55.1 42 io5.2 9U 02 149.7 i35.7 62 194. 1 176.9 2 3 17.0 i5.4 83 61.5 55.7 43 106.0 96.0 o3 i,5o.4 i36.3 63 194-9 176.6 24 17.8 16. 1 84 62.2 56.4 AA 106.7 96.7 04 i5i .2 137.0 64 196.6 177.3 25 18.5 16.8 85 63.0 57.1 45 107.4 97-4 OD i5i .9 137-7 65 iy6.4 178.0 26 19.3 17.5 86 63.7 57.8 46 108.2 98.0 06 i52.6 i38.3 66 197-1 178.6 27 20.0 18.1 87 64.5 58.4 47 108.9 98.7 07 1 53. 4 139.0 67 197.8 179.3 28 20.7 18.8 88 65.2 59.1 48 109.7 99.4 08 I54.I 139.7 68 198.6 180.0 29 21.5 19.5 89 65.9 59.8 49 1 10.4 100. 1 09 154.9 i4o.4 69 199.3 180.6 3o 22.2 20.1 90 66.7 60.4 5o III . I 100.7 10 i55.6 i4i-o 70 271 200 . 1 181.3 3i 23.0 20.8 91 67.4 61. 1 i5i 1 1 1 .9 10 1. 4 211 i56.3 141.7 182.0 32 23.7 21.5 92 68.2 61.8 52 112. 6 102. 1 12 157.1 142.4 72 201 .5 182.7 33 24.5 22.2 93 68.9 62.5 53 ii3.4 102.7 i3 157.8 143.0 73 202.3 i83.3 34 25.2 22.8 94 69.6 63.1 M 1 14. 1 io3.4 i4 i58.6 143.7 74 2o3.0 184.0 35 25.9 23.5 95 70.4 63.8 55 ii4.8 104. 1 i5 159.3 144.4 76 2o3.8 184.7 36 26.7 24.2 96 71. 1 64.5 56 ii5.6 104.8 16 160.0 145.1 76 204.5 186.4 37 27.4 24.8 97 71 .9 65.1 57 116. 3 105.4 17 160.8 145.7 77 2o5 .2 186.0 38 28.2 2 5.5 98 72.6 65.8 58 117. 1 1 06. 1 18 161.5 146.4 78 206 186.7 39 28.9 26.2 99 73.4 66.5 59 117. 8 106.8 19 162.3 i47-i 79 206 7 187.4 4- 4i 29.6 26.9 TOO 74.1 67.2 6g 118. 6 107.4 20 i63.o i47-7 80 281 207.5 208,2 188.0 3o.4 27.5 lOI 74.8 67.8 161 1 19.3 108.1 221 i63.8 148.4 188.7 42 3i.i 28.2 02 75.6 68.5 62 120.0 108.8 22 164.5 149. 1 82 208.9 189.4 43 3, .9 28.9 o3 76.3 69.2 63 120.8 109.5 23 i65.2 149.8 83 209.7 1 90. 1 4i 32.6 29.5 o4 77.1 69.8 64 121. 5 1 10. 1 24 166.0 i5o.4 84 210.4 190.7 4'') 33.3 3o.2 o5 77.8 70.5 65 122.3 1 10.8 25 166.7 i5i.i 86 211. 2 191.4 46 34.1 3o.9 06 78.5 71.2 66 123. 111.5 26 167.5 i5i.8 86 21 1 .9 192.1 i- 34.8 3i.6 07 79.3 71.9 72.5 67 123.7 112.2 27 168.2 i52.4 87 212.7 192.7 48 35.6 32.2 08 80.0 68 124.5 1X2.8 28 168.9 i53.i 88 2i3.4 193.4 49 36.3 32.9 09 80.8 73.2 69 125.2 ii3.5 29 169.7 i53.8 89 214.1 194. 1 5o 5i 37.0 37.8 33.6 34.2 10 81.5 73.9 70 126.0 114.2 3o 170.4 i54.5 90 291 214.9 2i5.6 _^i8 196.4 III 82.2 74.5 171 126.7 114.8 23l 171 .2 i55.i 52 38.5 34.9 12 83.0 75.2 72 127.4 ii5.5 32 171-9 i55.8 92 216.4 1 96. 1 53 39.3 35.6 i3 83.7 75.9 73 128.2 1 16.2 33 172.6 i56.5 93 2 1 7 . 1 196.8 64 4o.o 36.3 i4 84.5 76.6 74 128.9 1 16.9 34 173.4 157. 1 94 217.8 197.4 55 40.8 36.9 i5 85.2 77.2 75 129.7 II7-5 35 I74-I 157.8 96 218.6 .98., 56 4i.5 37.6 16 86.0 77-9 76 i3o.4 118. 2 36 174.9 1 58.5 96 219.3 198.8 67 42.2 38.3 17 86.7 78.6 77 i3i.i 118.9 37 175.6 169.2 97 220.1 199.5 58 43.0 39.0 18 87.4 79.2 78 i3i .9 119.5 38 176.3 169.8 98 220.8 200.1 59 43.7 39.6 19 88.2 79-9 79 i32.6 120.2 39 177. 1 160.5 ,99 221 .5 200.8 60 44.5 40.3 20 88.9 80.6 80 133.4 120.9 40 177.8 161. 2 3oo 222. d 20 1. £ DUi. I),-P. l.at. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. 1 Lat. Dist. Dep. 1 Lnt. N.E.iE. S.E.iE. N.W.^W. S.W.AW. [For 4i Po nts. Page 16] TABLE I. Difference of Latitude and Departure for 4 Points. N.E. N.W. S.E. S.W. Dist. Lai. 1 Dep. Dist. Lai. Dep. Dist. Lai. Dep. 85.6 Dist. Lat. Dep. Disi. Lat. Dep. I no . 7 00.7 61 43.1 43.1 121 85.6 181 128.0 128.0 241 170.4 170.4 2 OI.4 01 .4 62 43.8 43.8 22 86.3 86.3 82 128.7 128.7 42 171. 1 171. 1 3 02. 1 02. 1 63 44.5 44.5 23 87.0 87.0 83 129.4 129.4 A'i 171.8 171.8 4 02.8 02.8 64 45.3 45.3 24 87.7 87.7 84 i3o.i i3o.i AA 172.5 172.5 5 o3.5 o3.5 65 46.0 46.0 25 88.4 88.4 85 i3o.8 i3o.8 45 178.2 178.2 6 04.2 04.2 66 46.7 46.7 26 89.1 89.1 86 i3i.5 i3i.5 46 178.9 178.9 7 04.9 04.9 67 47.4 47-4 27 89.8 89.8 87 l32.2 l32.2 47 174.7 174.7 8 03.7 o5.7 68 48.1 48.1 28 90.5 90.5 88 132.9 182.9 48 175.4 175.4 9 06.4 06.4 69 48.8 48.8 29 91 .2 91.2 89 i33.6 i33.6 49 176.1 176.1 lO 07.1 07.1 70 49.5 49.5 3o 91.9 91.9 90 i34.4 134-4 5o 176.8 176.8 II 07.8 07.8 71 50.2 5o.2 i3i 92.6 92.6 191 i35.i i35.i 25l 177.5 177-5 12 08.5 08.5 72 50.9 50.9 J2 93.3 93.3 92 i35.8 i35.8 52 178.2 178.2 i3 09.2 09.2 73 5i.6 5i.6 33 94.0 94.0 93 i36.5 i36.5 53 178.9 178.9 i4 09.9 09.9 74 52.3 52.3 M 94.8 94.8 94 187.2 187.2 54 179.6 179.6 i5 TO. 6 10.6 75 53.0 53.0 35 95.5 95.5 95 187.9 187.9 55 180.8 180.8 i6 II. 3 II. 3 76 53.7 5J.7 36 96.2 96.2 96 i38.6 1 38.6 56 181. 181.0 17 12.0 12.0 77 54.4 54.4 37 96.9 96.9 97 139.3 189.3 57 181. 7 181.7 i8 12.7 12.7 78 55.2 55.2 38 97.6 97.6 98 i4o.o i4o.o 58 182.4 182.4 ^9 i3.4 i3.4 79 55.9 55.9 39 98.3 98.3 99 140.7 140.7 59 i83.i i83.i 20 i4.i 14.1 80 81 56.6 56.6 4o 99.0 99.0 200 i4i.4 i4i-4 60 i83.8 iS3.8 21 i4.8 14.8 57.3 57.3 i4i 99-7 99-7 201 142. 1 142. 1 261 184.6 184.6 22 1 5. 6 i5.6 82 58. 58 .0 42 100.4 100.4 02 142.8 142.8 62 i85.3 i85.3 9 3 16.3 16.3 S3 58.7 58.7 43 lOI . I 101. 1 o3 143.5 143.5 63 186.0 186.0 24 17.0 17.0 84 59.4 59.4 A^ 101.8 101.8 04 144.2 144.2 64 186.7 186.7 25 17.7 17.7 85 60. 1 60.1 45 102.5 102.5 o5 145.0 145.0 65 187.4 187.4 26 1 18.4 18.4 86 60.8 60-8 46 103.2 io3.2 06 145.7 145.7 66 188. 1 1S8.1 2" IQ.I 19. 1 87 61.5 61.5 47 103.9 103.9 07 146.4 146.4 67 188.8 188.8 28 iq.8 19.8 88 62.2 62.2 48 104.7 104.7 08 i47-i 147-1 68 189.5 189.5 29 20.5 20.5 8q 62.9 62.9 49 io5.4 105.4 09 147-8 i47-« 69 190.2 190.2 3o 21 .2 21 .2 90 qi 63.6 63.6 5o 106. 1 1 06. 1 10 148.5 148.5 70 190.9 190.9 3i 21.9 21 .9 64.3 64.3 i5i 106.8 106.8 211 149-2 149.2 271 191 .6 191.6 32 22.6 22.6 92 65.1 65.1 52 107.5 107.5 12 149-9 149-9 72 192.3 192.8 33 23.3 23.3 q3 65.8 65.8 53 T08.2 108.2 li i5o.6 i5o.6 73 198.0 198.0 34 24.0 24.0 94 66.5 66.5 54 108.9 I0S.9 i4 i5i.3 i5i.3 74 198.7 198.7 35 24.7 24.7 95 67.2 67.2 55 109.6 109.6 i5 1D2.0 l52.0 1^ 194.5 194.5 36 25.5 25.5 96 67.9 67.9 56 no. 3 1 10.3 16 i52.7 152.7 76 195.2 195.2 37 26.2 26.2 97 68.6 68.6 57 III.O III.O 17 i53.4 i53.4 77 195.9 193.9 38 26.9 26.9 9S 69.3 69.3 58 III. 7 1 1 1.7 18 i54.i i54.i 78 196.6 196.6 39 27.6 27.6 99 70.0 70.0 59 112. 4 112.4 '9 154.9 154.9 79 197.3 197.3 40 28.3 28.3 1 00 70.7 70.7 60 ii3.i ii3.i 20 i55.6 1 55.6 80 198.0 198.0 4t 29.0 29.0 lOI 71.4 71-4 161 ii3.8 ii3.8 221 i56.3 1 56.3 281 198.7 198.7 42 29.7 29.7 02 72.1 72.1 62 114.6 114.6 22 157.0 157.0 82 199-4 199-4 43 3o.4 3o.4 o3 72.8 72.8 63 ii5.3 ii5.3 23 157.7 157.7 83 200.1 200.1 A/\ 3l.T 3i.i o4 73.5 73.5 64 1 16.0 1 16.0 24 i58.4 i58.4 84 200.8 200.8 45 3i.8 3T.8 o5 74.2 74.2 65 116. 7 1 16.7 25 159. 1 159. 1 85 201 .5 201.5 46 32.5 32.5 06 75.0 75.Q 66 117-4 117.4 26 159.8 159.8 86 202.2 202.2 47 33.2 33.2 07 75.7 75.7 67 118. 1 118. 1 27 160.5 160.5 87 202.9 202.9 48 33. Q 33. p 08 76.4 76.4 68 118.8 1 18.8 28 161 .2 161.2 88 2o3.6 208.6 49 34.6 34.6 09 77.1 77.1 69 119.5 1 19.5 29 161.9 161.9 89 204.4 204.4 5o 5t 35.4 36.1 35.4 36.1 10 77.8 77.8 70 120.2 120.2 3o 162.6 162.6 90 205.1 205.1 II I 78.5 78.5 171 120.9 120.9 23l i63.3 i63.3 291 2o5.8 2o5.8 52 36.8 36.8 12 79.2 79.2 72 121 .6 1 2 1. 6 32 164.0 164.0 92 206.5 206.5 53 37.5 37.5 i3 79-9 79-9 73 122.3 122.3 ii 164.8 164.8 93 207.2 207.2 54 38.2 38 . 2 i4 80.6 80.6 74 123.0 123.0 34 i65.5 i65.5 94 207.9 207-9 55 38. Q 38. Q lb 81.3 81.3 75 123.7 123.7 35 166.2 166.2 95 208.6 208.6 56 39.6 39.6 16 82.0 82.0 76 124.5 124.5 3b 166.9 166.9 96 209.3 209.3 57 40.3 40.3 17 82.7 82.7 77 125.2 125.2 37 167.6 167.6 97 210.0 210.0 58 4i.o /r.o 18 83.4 83.4 78 125.9 125.9 38 16S.3 i68.3- 98 210.7 210.7 59 41.7 4l.7 19 84.1 84.1 79 126.6 126.6 39 169.0 169.0 99 211 .4 2 1 1. 4 60 42.4 42.4 20 84-9 84.9 80 127.3 127.3 40 169.7 1 169.7 3oo 212. 1 212.1 Dist. Dep. Lr,t. Dist Dep. Lat. Dist. Dep. Lat. Dist. Dep. 1 Lat. Dist. Dep. Lat. N.E. N.W. S.l -■ S.W. [For 4 Points. TABLE IL 1 [Page 17 Difference of Latitude and Departure for 1 Degree. DIst. Lat. Dcp. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 01 .0 00.0 61 61 .0 01 .1 121 121 .0 02.1 181 181. o3.2 241 241 .0 04.2 2 02.0 00.0 62 62.0 01 .1 22 122.0 02. I 82 182.0 o3.2 42 242.0 04.2 3 o3.o 00. 1 63 63.0 01 . 1 23 123.0 02.1 83 i83.o 03.2 43 243.0 o4.2 4 o4.o 00 I 64 64.0 01 .1 24 124.0 02.2 84 184.0 o3.2 M 244.0 o4.3 5 o5.o 00. 1 65 65.0 01 .1 25 125.0 02.2 85 i85.o o3.2 45 245.0 04.3 6 06.0 00. 1 66 66.0 01 .2 26 126.0 02.2 86 186.0 o3.2 46 246.0 o4.3 7 07.0 00. 1 67 67.0 01 .2 27 127.0 02.2 87 187.0 o3.3 47 247.0 04.3 8 08.0 00. I 68 68.0 01 .2 28 128.0 02.2 88 188.0 o3.3 48 248.0 04.3 9 09.0 00.2 69 69.0 01 .2 29 129.0 02.3 89 189.0 o3.3 49 249.0 04.3 10 10. 00.2 70 70.0 01 .2 3o i3o.o 02.3 90 190.0 o3.3 5o 25o.O 04.4 1 1 II.O 00.2 71 71.0 01 .2 i3i i3i .0 02.3 191 191 .0 o3.3 25l 25l .0 04.4 12 12.0 00.2 72 72.0 01 i 32 l32.0 02.3 92 192.0 o3.4 52 252.0 04.4 i3 i3.o 00.2 73 73.0 01 i 33 i33.o 02.3 93 193.0 o3.4 53 253.0 04.4 i4 i4-o 00.2 74 74.0 01 .3 34 i34.o 02.3 94 194.0 o3.4 54 254.0 04.4 ID i5.o 00.3 75 75.0 01 .3 35 i35.o 02.4 95 195.0 o3.4 55 255.0 04.5 i6 16.0 00.3 76 76.0 01 .3 36 i36.o 02.4 96 196.0 o3.4 56 256.0 04.5 17 17.0 00.3 77 77-0 01 .3 37 137.0 02.4 97 197.0 o3.4 57 257.0 04.5 i8 18.0 00.3 7S 78.0 01 .4 38 i38.o 02.4 98 198.0 o3.5 58 258.0 04.5 19 19.0 00.3 79 79.0 01 .4 39 139.0 02.4 99 199.0 o3.5 59 259.0 o4.5 20 20.0 00.3 80 80.0 01 .4 40 i4o.o 02.4 200 200.0 o3.5 60 260.0 04.5 21 21 .0 00.4 81 81.0 01 .4 i4i i4i.fr 02.5 201 201 .0 o3.5 261 261 .0 04.6 22 22.0 00.4 82 82.0 01 .4 42 142.0 02.5 02 202.0 o3.5 62 262.0 04.6 23 23. 00.4 83 83.0 01 .4 43 143.0 02.5 03 203.0 o3.5 63 263. 04.6 24 24.0 00.4 84 84. 01 .5 U i44.o 02.5 04 204.0 o3.6 64 264.0 o4.6 25 25.0 00.4 85 85. 01.5 45 145.0 02.5 o5 205.0 o3.6 65 265.0 04.6 26 26.0 00.5 86 86.0 01.5 46 146.0 02.5 06 206.0 o3.6 66 266.0 04.6 27 27.0 00.5 87 87.0 01 .5 47 i47-o 02.6 07 207.0 o3.6 67 267.0 04.7 28 28.0 00.5 88 88.0 01 .5 48 148.0 02.6 08 208.0 o3.6 68 268.0 04.7 29 29.0 00.5 89 89.0 01 .6 49 i49-o 02.6 09 209.0 o3.6 69 269.0 04.7 Jo 3o.o 00.5 90 90.0 01 .6 5o i5o.o 02.6 10 210.0 o3.7 70 270.0 04.7 3i 3i.o 00.5 91 91 .0 01 .6 i5i i5i .0 02.6 211 211 .0 o3.7 271 271 .0 04.7 32 32.0 00.6 92 92.0 01 .6 52 l52.0 02.7 12 212.0 o3.7 72 272.0 04.7 33 33.0 00.6 93 93.0 01 .6 53 i53.o 02.7 i3 2l3.0 o3.7 73 273.0 o4.8 34 34.0 00.6 94 94.0 01.6 54 1 54.0 02.7 i4 214.0 o3.7 74 274.0 04.8 35 35.0 00.6 95 95.0 01.7 55 155.0 02.7 i5 2l5.0 o3.8 75 275.0 04.8 36 36.0 00.6 96 96.0 01.7 56 1 56.0 02.7 16 216.0 o3.8 76 276.0 04.8 37 37.0 00.6 97 97.0 01.7 57 157.0 02.7 17 217.0 o3.8 77 277.0 04.8 38 38. 00.7 98 98.0 01.7 58 i58.o 02.8 18 218.0 o3.8 78 278.0 04.9 39 39.0 00.7 99 99.0 01.7 59 159.0 02.8 19 219.0 o3.8 79 279.0 04.9 40 4o.o 00.7 roo lOO.O 01.7 60 160.0 02.8 20 220.0 o3.8 80 280.0 04.9 4i 4i .0 00.7 lOI lOI .0 01.8 161 i6i .0 02.8 221 221 .0 03.9 281 281 .0 04.9 ■42 42.0 00.7 02 102.0 01.8 62 162.0 02.8 22 222.0 03.9 82 282.0 04.9 43 43.0 00.8 o3 io3.o 01.8 63 1 63.0 02.8 23 223.0 03.9 83 283. 04.9 ^■^ 44.0 00.8 04 104.0 01.8 64 164.0 02.9 24 224.0 03.9 84 284.0 o5.o 45 45.0 00.8 o5 io5.o 01.8 65 i65.o 02.9 25 225.0 03.9 85 285.0 o5.o 46 46. 00.8 06 106.0 01.8 66 1 66 . 02.9 26 226.0 03.9 86 286.0 o5.o 47 47.0 00.8 07 107.0 01 .9 67 167.0 02.9 27 227.0 04.0 87 287.0 o5.o 48 48.0 00.8 08 108.0 01.9 68 168.0 02.9 28 228.0 04.0 88 288.0 o5.o 49 49.0 00.9 09 109.0 01 .9 69 169.0 02.9 29 229.0 04.0 89 289.0 o5.o 5o 5o.o 00.9 10 IIO.O 01 .9 70 170.0 o3.o o3.o Jo 2 3o.O 04.0 90 290.0 o5.i 5i 5i.o 00.9 II I II I .0 01.9 171 171. 23l 23l.O 04.0 291 291 .0 o5.i 52 52.0 00.9 12 112. 02.0 72 172.0 o3.o 32 232.0 04.0 92 292.0 o5.i 53 53.0 00.9 i3 ii3.o 02.0 73 173.0 o3.o 33 233. 04. 1 93 293.0 o5.i H 54.0 00.9 i4 ii4.o 02.0 74 174.0 o3.o 34 234.0 04.1 94 294.0 o5.i 55 55.0 01. i5 ii5.o 02.0 75 175.0 o3.i 35 235.0 04.1 95 295.0 o5.i 56 56.0 01 .0 16 116.0 02.0 76 176.0 o3.i 36 236. 04.1 96 296.0 o5.2 ^7 57.0 01 .0 17 117. 02.0 77 177.0 o3.i 37 237.0 04.1 97 297.0 o5.2 58 58. 01 .0 18 118.0 02.1 78 178.0 o3.i 38 238.0 04.2 98 298.0 05.2 59 59.0 01 .0 19 /19.0 02.1 79 179.0 o3.i 39 239.0 04.2 99 299.0 o5.2 bo 60.0 01 .0 20 120.0 02.1 80 180.0 o3.i 4o 240.0 04.2 3(K) 3()0.o o5.2 I»ist Dnp. Lai. Dist. Dop. Lat. Dist. Dep. Lat. Disi.| Dep. 1 Lni. Dist. Dcp. Lat. [ •'or 89 Degrees. Page 18] TABLE n. Difference of Latitude and Departure for 2 Degrees. Dist Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist.[ Lit. Dep. 08.4 I 01 .0 00.0 61 61 .0 02.1 121 120.9 04.2 181 180.9 06.3 241 240.9 2 02.0 00. 1 62 62.0 02.2 22 12'. 9 04.3 82 181.9 06.4 42 241.9 08.4 3 o3.o 00. 1 63 63. 02.2 23 12... 9 04.3 83 182.9 06.4 43 242.9 08.5 4 04.0 00. 1 64 64.0 02.2 24 123.9 04.3 84 183.9 06.4 44 243.9 08.5 5 o5.o 00.2 65 65.0 02.3 25 124.9 04.4 85 184.9 06.5 45 244.9 08.6 6 06.0 00.2 66 66.0 02.3 26 125.9 o4-4 86 185.9 06.5 46 245.9 08. e 7 07.0 00.2 67 67.0 02.3 27 126.9 04.4 87 186.9 06.5 47 246.8 08.6 8 08.0 00.3 68 68.0 02.4 28 127.9 04.5 88 187.9 06.6 48 247.8 08.7 9 09.0 00.3 69 69.0 02.4 29 128.9 04.5 89 188.9 06.6 49 248.8 08.7 10 10. 00.3 70 70.0 02.4 3o 129.9 04.5 04.6 90 189.9 06.6 5o 249.8 08.7 II II. 00.4 71 71.0 02.5 i3i 1 30.9 191 190.9 06.7 25l 250.8 08.8 12 12.0 00.4 72 72.0 02.5 32 i3i .9 04.6 92 191.9 06.7 52 251.8 08.8 i3 i3.o 00.5 73 73.0 02.5 33 132.9 04.6 93 192.9 06.7 53 252.8 08.8 i4 i4-o 00.5 74 74.0 02.6 34 133.9 04.7 94 193.9 06.8 54 253.8 08.9 i5 i5.o 00.5 7^ 7b. 02.6 35 134.9 04.7 95 194.9 06.8 55 254.8 08.9 i6 16.0 00.6 76 76.0 02.7 36 135.9 04.7 96 195.9 06.8 56 255.8 08.9 t? 17.0 00.6 77 77.0 02.7 37 i36.9 o4.8 97 196.9 06.9 57 256.8 09.0 i8 18.0 00.6 78 78.0 02.7 38 137.9 04.8 98 197.9 06.9 58 257.8 09.0 19 19. c 00.7 79 79.0 02.8 39 i38.9 04.9 99 198.9 06.9 59 258.8 09.0 20 20.0 00.7 80 80.0 02.8 40 139.9 04.9 04.9 200 199.9 07.0 60 259.8 09.1 21 21 .0 00.7 81 8i.o 02.8 i4i 140.9 201 200.9 07.0 261 260.8 09.1 22 22.0 00.8 82 82.0 02.9 42 i4i.9 o5.o 02 201 .9 07.0 62 261.8 09.1 23 23. 00.8 83 82.9 02.9 43 142.9 o5.o 00 202.9 07.1 63 262.8 09.2 24 24.0 00.8 84 83.9 02.9 44 143.9 o5.o o4 2o3 . 9 07.1 64 263.8 09.2 25 23. 00.9 85 84.9 o3.o 45 144.9 o5.i o5 204.9 07.2 65 264.8 09.2 26 26.0 00.9 86 85.9 o3.o 46 145.9 o5. 1 06 205.9 07.2 66 265.8 09.3 27 27.0 00.9 87 86.9 o3.o 47 146.9 o5.i 07 206.9 07.2 (^7 266.8 09.3 28 20.0 01. 88 87.9 o3.i 48 i47-9 o5.2 08 207.9 07.3 68 267.8 09.4 29 29.0 01 .0 89 88.9 o3.i 49 148.9 05.2 09 208.9 07.3 69 268.8 09.4 3o Ji 3o.o 01 .0 90 89.9 o3.i bo i5i 149-9 o5.2 10 209.9 07.3 70 269.8 09.4 3i .0 01. 1 9f 90.9 o3.2 i5o.9 o5.3 211 210.9 07.4 271 270.8 09.5 32 3i.o 01. 1 92 91.9 o3.2 62 ibi.9 o5.3 12 21 1 .9 07.4 72 271.8 09.5 33 33.0 01 .2 93 92.9 o3.2 53 Ib2.9 o5.3 i3 212.9 07.4 73 272.8 09.5 34 3'(.o 01 .2 94 93.9 o3.3 54 ib3.9 o5.4 i4 213.9 07.5 74 273.8 09.6 35 35.0 01 .2 95 94.9 o3.3 b") 154.9 o5.4 i5 214.9 07.5 7^ 274.8 09.6 36 36.0 01.3 96 95.9 o3.4 56 ibb.9 o5.4 16 215.9 07.5 76 275.8 09.6 37 J7.0 01.3 97 96.9 o3.4 ^7 i56.9 o5.5 17 216.9 07.6 77 276.8 09.7 38 38. 01.3 98 97-9 o3.4 58 157.9 o5.5 18 217.9 07.6 78 277.8 09.7 39 39.0 01 .4 99 98.9 o3.5 b9 i58.9 o5.5 '9 218.9 07.6 79 278.8 09.7 40 4o.o 01 .4 100 99.9 o3.5 60 159.9 ob.6 20 219.9 07.7 80 279.8 09.8 4i 4i .0 01 .4 lOI 100.9 o3.5 161 160.9 o5.6 221 220.9 07.7 281 280.8 09.8 45 42.0 01.5 02 lOI .9 o3.6 62 161 .9 o5.7 22 221 .9 07.7 82 2S1.8 09 . 8 43 43.0 01 .5 o3 102.9 o3.6 63 162.9 o5.7 23 222.9 07.8 Hi 282.8 09.9 44 44.0 01.5 o4 103.9 o3.6 64 163.9 o5.7 24 223.9 07.8 84 983.8 09.9 45 45.0 01 .6 o5 104.9 o3.7 65 164.9 o5.8 25 224.9 07.9 8b 284.8 09.9 46 46. 01 .6 06 105.9 o3.7 66 165.9 o5.8 26 225.9 07.9 86 285.8 1 10. 1 47 47-0 01.6 07 106.9 o3.7 67 166.9 o5.8 27 226.9 07.9 87 286.8 1 10. ■48 48. 01.7 08 107.9 o3.8 68 167.9 05.9 28 227.9 08.0 88 287.8 i 10. 1 49 49.0 01.7 09 108.9 o3.8 69 168.9 o5.9 29 228.9 08.0 89 288.8 ■ 10.1 5o 5o.o 01 .7 10 109.9 o3.8 70 169.9 05.9 06.0 3u 229.9 08.0 90 289.8 10. 1 ! 5i 5i.o 01.8 III no. 9 03.9 171 170.9 23l 230.9 08.1 291 290.8 10.2 52 52.0 01.8 12 III .9 03.9 72 171-9 06.0 32 23l .9 08.1 92 291.8 10.2 1 53 53.0 01.8 i3 II 2 . 9 03.9 73 172.9 06.0 33 232.9 08.1 93 292.8 10.2 1 54 54.0 01.9 i4 II3.9 04.0 74 173.9 06.1 34 233.9 08.2 94 293.8 10.3 i 5? 55.0 01 .9 i5 ii4-9 04.0 75 174.9 06.1 35 234.9 08.2 951294.8 10.3 56 56. 02.0 16 115.9 04.0 76 175.9 06.1 36 235.9 08.2 96^295.8 10 3 57 57.0 02.0 17 no. 9 04.1 77 176.9 06.2 37 236.9 08.3 97 296.8 10.4 58 58. 02.0 18 117. 9 04.1 78 177-9 06.2 38 237-9 08.3 98 ^9Z-" 10.4 5q 59.0 02.1 19 118.9 04.2 79 178.9 06.2 39 238.9 08.3 .99 298.8 10.4 Go Dist. 60.0 Hep. 02.1 Lat. 20 119. 9 04.2 Lat. 80 179.9 06.3 40 239.9 08.4 3oo 299.8 10.5 L.it. Dist. Dep. Dist. Dep. Lat. Dist. ])ep. Lril. Dist. Dep. [For 83 Degrees. TABLE IL [Page 19 Difference of Latitude and Departure for 3 Degrees. Dist. Lat. Dcp. Dist. Lat. Dcp. o3.2 Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat Dep. { 01 .0 00. 1 61 60.9 121 120.8 06.3 181 180.8 09.5 24 1 240.7 12.6 2 02.0 00. 1 62 61 .9 o3.2 22 121. 8 06.4 82 181.8 09.5 42 241.7 12.7 ,3 o3.o 00.2 C3 62.9 o3.3 23 122. 8 06.4 83 182.7 09.6 43 242.7 12.7 4 o4.o 00 2 64 63.9 o3.3 24 123.8 06.5 84 183.7 09.6 44 243.7 12.8 ") o5.o 00.3 65 64.9 o3.4 25 124.8 06.5 85 184.7 09.7 45 244.7 12.8 6 06.0 00.3 66 65.9 o3.5 26 125.8 06.6 86 185.7 09.7 46 245.7 12.9 7 07.0 00.4 67 66.9 o3.5 27 126.8 06.6 87 186.7 09.8 47 246.7 12.9 8 08.0 00.4 68 67.9 o3.6 28 127.8 06.7 88 187.7 09.8 48 247.7 i3.o 9 09.0 00.5 69 68.9 o3.6 29 128.8 06.8 89 188.7 09.9 49 248.7 i3.o 10 10. 00.5 7" 69.9 o3.7 3o 129.8 Ob. 8 90 189.7 09.9 5o 249.7 i3.i 1 1 1 1 .0 00.6 71 70.9 03.7 i3i i3o.8 06.9 191 190.7 10.0 25l 250.7 i3.i 12 12.0 CK).6 72 71.9 o3.8 32 i3i.8 06.9 92 191.7 10. 62 25l .7 l3.2 i3 i3.o 00.7 73 72.9 o3.8 33 i32.8 07.0 93 192.7 10. 1 53 252.7 l3.2 1 4 i4.o 00.7 74 73.9 03.9 34 i33.8 07.0 94 193.7 10.2 54 253.7 i3.3 i5 i5.o 00.8 75 74.9 03.9 35 i34.8 07.1 95 194.7 10.2 55 254.7 i3.3 i6 16.0 00.8 76 75.9 04.0 36 i35.8 07.1 96 195.7 10.3 5b 255.6 i3.4 17 17.0 00.9 77 76.9 04.0 37 i36.8 07.2 97 196.7 10.3 J)7 256.6 i3.5 i8 18.0 00.9 78 77-9 04.1 38 137.8 07.2 08 197-7 10.4 58 257.6 i3.5 19 ly.o 01. 79 78.9 04.1 39 i38.8 07.3 99 198.7 10.4 59 258.6 i3.6 20 20.0 01 .0 80 79-9 04.2 40 139.8 07.3 200 199.7 10.5 bo 259.6 i3.6 21 21 .0 01 .1 81 80.9 04.2 i4i i4o.8 07.4 201 200.7 10.5 261 260.6 i3.7 22 22.0 01 .2 82 81 .9 04.3 42 i4i.8 07.4 02 201 .7 10.6 b2 261 .6 i3.7 23 23.0 01 .2 83 82.9 04.3 43 142.8 07.5 o3 202.7 10.6 b3 262.6 i3.8 24 24.0 01 .3 84 83.9 04.4 44 143.8 07.5 04 203.7 10.7 64 263.6 i3.8 2 5 25.0 01 .3 85 84.9 04.4 45 144.8 07.6 o5 204.7 10.7 b5 264.6 13.9 26 26.0 01 .4 86 85.9 04.5 46 145.8 07.6 06 205.7 10.8 bb 265.6 13.9 27 27.0 01 .4 87 86.9 04.6 47 i46.8 07.7 07 206.7 10.8 67 266.6 14.0 28 28.0 01.5 88 87.9 04.6 48 147.8 07.7 08 207.7 10.9 b8 267.6 14.0 29 29.0 01.5 89 88.9 04.7 49 148.8 07.8 09 208.7 10.9 69 268.6 14.1 3o 3i 3o.o 3i .0 01 .6 01 .6 90 89.9 04.7 5o 149-8 07.9 10 209.7 1 1 .0 70 271 269.6 270.6 i4.i 91 90.9 04.8 i5i i5o.8 07.9 211 210.7 11. 14.2 32 32.0 01.7 92 91.9 04.8 52 i5i.8 08.0 12 211 .7 II. I 72 271.6 14.2 33 33.0 01.7 g3 92.9 04.9 53 i52.8 08.0 i3 212.7 11 .1 73 272.6 i4.3 34 34.0 01.8 94 93.9 04.9 54 i53.8 08.1 i4 213.7 11 .2 74 273.6 14.3 35 35.0 01.8 95 94-9 o5.o 55 i54.8 08.1 i5 214.7 11.3 7^ 274.6 i4.4 3G 36.0 01 .9 96 95.9 o5.o 56 i55.8 08.2 16 215.7 II. 3 7b 275.6 i4-4 37 36.9 01.9 97 96.9 o5.i 57 i56.8 08.2 17 216.7 11.4 77 276.6 14.5 38 37.9 02.0 98 97-9 o5.i 58 157.8 08.3 18 217.7 11.4 78 277.6 14.5 39 38.9 02 .0 99 98.9 o5.2 59 i58.8 08.3 19 218.7 11.5 Z9 278.6 i4.6 4o 39.9 02. 1 100 99.9 o5.2 o5.3 60 159.8 08.4 08.4 20 219.7 11.5 80 279.6 14.7 4i 40.9 02.1 lOI 100.9 161 160.8 221 220.7 11.6 281 280.6 14.7 42 4! .9 02.2 02 lOI .9 o5.3 62 161.8 08.5 22 221 .7 11.6 82 281.6 14.8 43 42.9 02.3 o3 102.9 o5.4 63 162.8 08.5 23 222.7 II. 7 83 282.6 i4.8 44 43.9 02.3 o4 103.9 o5.4 64 i63.8 08.6 24 223.7 11.7 84 283.6 14.9 45 44.9 02.4 o5 104.9 o5.5 65 164.8 08.6 25 224.7 II. 8 85 284.6 14.9 46 45.9 02.4 06 105.9 o5.5 66 i65.8 08.7 26 225.7 II. 8 8b 285.6 i5.o 47 46.9 02.5 07 106.9 o5.6 67 166.8 08.7 27 226.7 11.9 87 286.6 i5.o 48 4-7.9 02.5 08 107.9 05.7 68 167.8 08.8 28 227.7 11.9 88 287.6 i5.i 49 48.9 02.6 09 108.9 o5.7 69 168.8 08.8 29 228.7 12.0 89 288.6 i5.i 5o 49.9 02.6 10 109.8 o5.8 70 169.8 08.9 3o 229.7 12.0 90 289.6 l5.2 5i 50.9 02.7 III no. 8 o5.8 171 170.8 08.9 23l 2 3o.7 12. 1 291 290.6 l5.2 52 5. .9 02.7 12 III. 8 05.9 72 171. 8 09.0 32 23l .7 12.1 92 291 .6 i5.3 53 52.9 02.8 i3 112. 8 05.9 73 172.8 09.1 33 232.7 12.2 93 =9'-6 i5.3 54 53.9 02.8 i4 ii3.8 06.0 74 173.8 09.1 34 233.7 12.2 94 293.6 i5.4 55 54.9 02.9 i5 114.8 06.0 75 174.8 09.2 35 234.7 12.3 95 294.6 i5.4 56 55.9 02.9 16 ii5.8 06.1 76 175.8 09.2 36 235.7 12.4 9b 295-6 i5.5 57 56.9 o3.o 17 116.8 06.1 77 176.8 09.3 37 236.7 12.4 97 296.6 i5.5 58 57.9 o3.o 18 117. 8 06.2 78 177.8 09.3 38 237.7 12.5 98 297.6 i5.6 59 58.9 o3.i 19 118. 8 06.2 79 178.8 09.4 39 238.7 12.5 99 298.6 i5.6 bo 59.9 o3.i 20 119. 8 06.3 80 179-8 09.4 4o 239.7 12.6 3oo 299.6 i5.7 nist. Dcp. Lat. |l)is, Dcp. Lat. Dist Dop. Lat. Dist.j Dep. Lat. Dist Dep. Lat. [For 87 Degrees. Page 20] TABLE IL Difference of Latitude and Departure for 4 Degrees. Dist Lat. Dep. Dist. Lat. Dep. 04.3 Dist. Lat. Dep. Dist Lat. Dep. 12.6 Dist. Lat. Dep. I 01 .0 00. 1 61 60.9 121 120.7 08.4 181 180.6 241 240.4 16.8 2 02.0 00. 1 62 61.8 04.3 22 121. 7 08.5 82 181. 6 12.7 42 241.4 16.9 3 o3.o 00.2 63 62.8 04.4 23 122.7 08.6 83 182.6 12.8 43 242.4 17.0 4 o4.o 00.3 64 63.8 o4.5 24 123.7 08.6 84 i83.6 12.8 M 243.4 17.0 5 o5.o 00.3 65 64.8 04.5 25 124.7 08.7 85 184.5 12.9 45 244.4 17. 1 6 06.0 00.4 66 65.8 o4.6 26 125.7 08.8 86 i85.5 i3.o 46 245.4 17.2 7 07.0 00.5 67 66.8 04.7 27 126.7 08.9 87 186.5 i3.o 47 246.4 17.2 8 08.0 00.6 68 67.8 04.7 28 127.7 08.9 88 187.5 i3.i 48 247.4 17.3 9 09.0 00.6 69 68.8 04.8 29 128.7 09.0 89 188.5 l3.2 49 2.48.4 17.4 lO 10. 00.7 70 69.8 04.9 3o 129.7 09.1 90 189.5 i3.3 5o 249.4 17.4 II II. 00.8 71 70.8 o5.o i3i i3o.7 09.1 191 190.5 i3.3 25l 25o.4 17.5 12 12.0 00.8 72 71.8 o5.o 32 i3i.7 09.2 92 191. 5 i3.4 b2 251.4 I7.b i3 i3.o 00.9 73 72.8 o5.i ■6-6 132.7 09.3 93 192.5 i3.5 53 262.4 17.6 i4 lA.o 01 .0 74 73.8 05.2 M 133.7 09.3 94 193.5 i3.5 54 253.4 17.7 i5 i5.o 01 .0 75 74.8 o5.2 35 134.7 09.4 95 194.5 i3.6 55 254.4 17.8 i6 16.0 01. 1 76 75.8 o5.3 ■i6 1J5.7 09.5 96 195.5 13.7 56 255.4 17.9 17 17.0 01 .2 77 76.8 o5.4 37 136.7 09.6 97 196.5 i3.7 57 256.4 17.9 i8 18.0 01.3 78 77.8 o5.4 38 137.7 09.6 98 197.5 i3.8 68 267.4 18.0 19 19.0 01.3 79 78.8 o5.5 39 i38.7 09.7 99 198.5 i3.9 59 258.4 18. 1 20 20.0 01.4 80 79.8 o5.6 40 139.7 09.8 200 199.5 14.0 bo 269.4 18. 1 2-1 20.9 01.5 81 80.8 o5.7 i4i 140.7 09.8 201 200.5 14.0 261 260.4 18.2 22 21.9 01.5 82 81.8 05.7 42 141.7 09.9 02 201.5 14.1 62 261.4 18.3 23 11. C) 01 .6 83 82.8 o5.8 43 142.7 10. OJ 202.5 14.2 63 262.4 18.3 24 23.9 01.7 84 83.8 05.9 M 143.6 10. o4 2o3.5 14.2 64 263.4 18.4 25 24.9 01.7 85 84.8 05.9 45 144.6 10. 1 o5 204.5 14.3 65 264.4 18.5 26 25.9 01.8 86 85.8 06.0 46 145.6 10.2 06 2o5.5 14.4 66 265.4 18.6 27 26.9 01 .9 87 86.8 06.1 47 i46.6 10.3 07 206.5 14.4 67 266.3 18.6 28 27.9 02.0 88 87.8 06.1 48 147-6 10.3 08 207.5 14.5 68 267.3 18.7 29 28.9 02.0 89 88.8 06.2 49 i48.6 10.4 09 208.5 14.6 69 268.3 18.8 3o 29.9 02. 1 90 89.8 06.3 5o 149.6 10.5 10 209.5 14.6 70 269.3 18.8 3 1 3o.9 02.2 91 90.8 06.3 i5i i5o.6 10.5 an 210.5 14.7 271 270.3 18.9 32 3i.9 02.2 92 91.8 06.4 52 i5i.6 10.6 12 211 .5 i4.8 72 271.3 19.0 33 32. 9 02.3 93 92.8 06.5 53 i52.6 10.7 i3 212.5 14.9 73 272.3 19.0 34 33.9 02.4 94 93.8 06.6 54 i53.6 10.7 i4 2i3.5 14.9 74 273.3 19. 1 35 34.9 02.4 95 94.8 06.6 65 i54.6 10.8 i5 214.5 i5.o 75 274.3 19.2 36 35.9 02.5 96 95.8 06.7 56 i55.6 10.9 16 2i5.5 i5.i 76 275.3 19.3 37 36.9 02.6 07 96.8 06.8 57 i56.6 II .0 17 216.5 i5.i 77 276.3 19.3 38 37.9 02.7 98 97.8 06.8 58 157.6 II. 18 217.5 l5.2 78 277.3 19-4 39 38.9 02.7 99 98.8 06.9 59 i58.6 II. I 19 218.5 i5.3 79 278.3 19.6 4o 4i 39.9 02.8 100 99.8 07.0 bo 159.6 II. 2 30 219.5 i5.3 80 281 279.3 280.3 19. D 19.6 40.9 02.9 lOI 100.8 07.0 161 160.6 II. 2 221 220.5 i5.4 42 41.9 02.9 02 101.8 07.1 62 161. 6 II. 3 22 221 .5 i5.5 82 281.3 19.7 43 42.9 o3.o o3 102.7 07.2 63 162.6 1 1. 4 23 222.5 i5.6 83 282.3 19.7 Ai 43.9 o3.i o4 103.7 07.3 64 i63.6 II. 4 24 223.5 i5.6 84 283.3 19.8 45 44.9 o3.i o5 104.7 07.3 65 164.6 II. 5 25 224.5 i5." 85 284.3 19.9 46 45.9 o3.2 06 105.7 07.4 66 i65.6 II. 6 26 225.4 i5.& 86 285.3 20.0 47 46.9 o3.3 07 106.7 07.5 67 166.6 II. 6 27 226.4 i5.8 87 2S6.3 20.0 48 47-9 o3.3 08 107.7 07.5 68 167.6 II. 7 28 227.4 i5.9 88 287.3 20.1 49 48.9 o3.4 09 108.7 07.6 69 168.6 II. 8 29 228.4 16.0 89 2S8.3 20.2 5o 5i 49.9 o3.5 10 109.7 07.7 70 169.6 II. 9 3o 229.4 16.0 90 289 3 20.2 50.9 o3.6 III 110.7 07.7 171 170.6 II. 9 23 1 23o.4 16.1 291 290.3 20.3 52 5i .9 o3.6 12 III. 7 07.8 72 171. 6 12.0 32 23i.4 16.2 92 291 .3 30.4 53 52.9 o3.7 i3 iii.7 07.9 73 172.6 12. 1 33 232.4 16.3 93 292.3 20.4 54 53.9 o3.8 i4 113.7 08.0 74 173.6 12.1 34 233.4 16.3 94 293.3 20.5 55 54.9 o3.8 i5 114.7 08.0 75 174.6 12.2 35 234.4 16.4 95 204.3 20. D 56 55.9 03.9 16 115.7 08.1 76 175.6 12.3 36 235.4 16.5 96 295.3 20.6 57 56.9 04.0 17 116. 7 08.2 77 176.6 12.3 37 236.4 16.5 97 296.3 20.7 58 57.9 04.0 18 117. 7 08.2 78 177.6 12.4 38 237.4 16.6 98 297.3 20.8 59 58.9 04.1 19 118. 7 08.3 79 178.6 12.5 39 238.4 16.7 99 298.3 20.9 60 59.9 04.2 20 119. 7 08.4 80 179.6 [2.6 4o 239.4 16.7 3 00 299.3 20.9 Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. 1 Lat. Dist. Dep. Lat. Dist. Dep. Lat. [For 86 Degrees. TABLE II. f!'H='e2i Difference of Latitude and Departure for 5 Degrees. Dist. Lat. Dep. Dist. 6i 62 63 64 65 66 67 68 69 70 Lat. 60.8 61.8 62.8 63.8 64.8 65.7 66.7 67.7 68.7 69.7 Dep. o5.3 o5.4 o5.5 o5.6 05.7 o5.8 o5.8 05.9 06.0 06.1 Dist. Lat. 1 Dep. Dist. Lat. 1 Dep. Dist.l Lat. Dep. I 2 3 4 5 6 7 8 9 10 01 .0 02.0 o3.o 04.0 o5.o 06.0 07.0 08.0 09.0 10. 00. 1 00.2 00.3 00.3 00.4 00.5 00.6 00.7 00.8 00.9 121 22 23 24 25 26 27 28 =9 3o 120.5 121 .5 122.5 123.5 124.5 125.5 126.5 127.5 128.5 129.5 10.5 10.6 10.7 10.8 10.9 1 1 .0 1 1 .1 II .2 II .2 II. 3 181 82 83 84 85 86 87 88 89 90 180.3 181. 3 182.3 i83.3 184.3 i85.3 186.3 187.3 188.3 189.3 i5.8 15.9 15.9 16.0 16. 1 16.2 16.3 16.4 16.5 16.6 241 42 43 A^ 45 46 47 4^ 240.1 241 .1 242. 1 243.1 244.1 245.1 246.1 247-1 248.1 249.0 21 .0 21 .1 21.2 21.3 21 .4 21.4 21.5 21 .6 21.7 21.8 II I 2 i3 i4 i5 i6 17 i8 19 20 II .0 12.0 i3.o 13.9 14.9 15.9 16.9 17.9 18.9 19.9 01 .0 01 .0 01. 1 01.2 01 .3 01 .4 01.5 01 .6 01.7 01.7 01.8 01 .9 02.0 02.1 02.2 02.3 02.4 02.4 02.5 02.6 71 72 73 74 75 76 77 78 79 80 70.7 71-7 72.7 73.7 74.7 75.7 76.7 77-7 78.7 79-7 06.2 06.3 06.4 06.4 06.5 06.6 06.7 06.8 06.9 07.0 i3i 32 33 M 35 36 37 38 39 4o i3o.5 i3i.5 i32.5 i33.5 i34.5 i35.5 i36.5 137.5 i38.5 139.5 11. 4 11. 5 11. 6 11. 7 11. 8 11. 9 II. 9 12.0 12. 1 12.2 191 92 93 94 96 97 98 99 200 190.3 191 .3 192.3 193.3 194.3 195.3 196.3 197.2 19S.2 199.2 16.6 16.7 16.8 16.9 17.0 17. 1 17.2 17.3 17.3 17.4 25l 52 53 54 55 56 57 58 60 25o.O 25l .0 252.0 253. 254.0 255. 256. 257.0 258. 259.0 21 .9 22.0 22.1 22.1 22.2 22.3 22.4 22.5 22.6 22.7 21 22 23 24 25 26 27 28 3o 20.9 21.9 22.9 23.9 24.9 25.9 26.9 27.9 28.9 29.9 81 82 83 84 85 86 87 88 89 90 80.7 81.7 82.7 83.7 84.7 85.7 86.7 87.7 88.7 89.7 07.1 07.1 07.2 07.3 07.4 07.5 07.6 07.7 07.8 07.8 i4i 42 A'i 45 46 47 48 49 5o i4o.5 i4r.5 142.5 143.5 144.4 145.4 146.4 147-4 148.4 149.4 12.3 12.4 12.5 12.6 12.6 12.7 12.8 12.9 i3.o i3.i 201 02 o3 o4 o5 06 07 08 09 10 200.2 201 .2 202.2 2o3.2 204.2 2o5.2 206.2 207.2 208.2 209.2 17.5 17.6 '7.7 17.8 17.9 18.0 18.0 18.1 18.2 18.3 261 62 63 64 65 66 67 68 69 70 260.0 261 .0 262.0 263.0 264.0 265.0 266.0 267.0 268.0 269.0 22.7 22.8 22.9 23. 23.1 23.2 23.3 23.4 23.4 23.5 3i 32 33 34 35 36 37 38 39 4o 30.9 3i .9 32.9 33.9 34.9 35.9 36.9 37.9 38.9 39.8 02.7 02.8 02.9 o3.o o3.i o3.i o3.2 o3.3 o3.4 o3.5 91 92 93 94 95 96 97 98 99 100 90.7 91 .6 92.6 93.6 94.6 95.6 96.6 97.6 98.6 99.6 07.9 08.0 08.1 08.2 08.3 08.4 08.5 08.5 08.6 08.7 i5i 52 53 54 55 56 57 58 60 i5o.4 i5i.4 i52.4 i53.4 i54.4 i55.4 i56.4 157.4 i58.4 159.4 l3.2 l3.2 i3.3 i3.4 i3.5 i3.6 i3.7 i3.8 13.9 13.9 211 12 i3 i4 i5 16 17 18 19 20 210.2 211 .2 2 12.2 2l3.2 2l4.2 2l5.2 216.2 217.2 218.2 219.2 18.4 18.5 18.6 18.7 18.7 18.8 18.9 19.0 19. 1 19.2 271 72 73 74 75 76 77 78 Z9 80 270.0 271 .0 272.0 273.0 274.0 274.9 275.9 276.9 277.9 278.9 23.6 23.7 23.8 23.9. 24.0 24.1 24.1 24.2 24.3 24.4 4i 42 43 4^ 45 46 47 48 49 5o 40.8 4i.8 42.8 43.8 44.8 45.8 46.8 47.8 48.8 49-8 o3.6 03.7 o3.7 o3.8 03.9 04.0 04. 1 04 . 2 04.3 04.4 lOI 02 o3 04 o5 06 07 08 09 10 100.6 loi .6 102.6 io3.6 104.6 105.6 106.6 107.6 108.6 109.6 08.8 08.9 09.0 09. 1 09.2 09.2 09.3 09.4 09.5 09.6 161 62 63 64 65 66 67 68 69 70 160.4 161. 4 162.4 i63.4 164.4 t65.4 166.4 167.4 168.4 169.4 14.0 i4.i 14.2 i4.3 i4.4 i4.5 i4.6 14.6 14.7 i4.8 221 22 23 24 25 26 27 28 3? 220.2 221 .2 222.2 223.1 224.1 225.1 226.1 227.1 228.1 229.1 19.3 19.3 19-4 19.5 19.6 '9-7 19.8 19.9 20.0 20.0 281 82 83 84 85 86 87 88 89 90 279.9 280.9 281.9 282.9 283.9 284.9 285.9 286.9 287.9 288.9 24.5 24.6 24.7 24.8 24.8 24.9 25.0 25.1 25.2 25.3 5i 52 53 54 55 56 57 58 59 60 5o.8 5i.8 5s. 8 53.8 54.8 55.8 56.8 57.8 58.8 59.8 04.4 04.5 04.6 04.7 04.8 04.9 o5.o o5.i o5.i o5.2 III 12 i3 i4 i5 16 17 18 19 20 1 10.6 II 1 .6 112. 6 ii3.6 ii4.6 ii5.6 116. 6 117. 6 118. 5 119. 5 09.7 09.8 09.8 09.9 10. 10. 1 10.2 10.3 10.4 10.5 171 72 73 74 75 76 77 78 79 80 170.3 171. 3 172.3 173.3 174.3 175.3 176.3 177.3 178.3 179.3 r4.9 i5.o i5.i l5.2 i5.3 i5.3 i5.4 i5.5 i5.6 i5.7 23l 32 33 M 35 36 37 38 39 4o 23o.I 23l .1 232.1 233.1 234.1 235.1 236.1 237.1 258.1 239.1 20. 1 20.2 20.3 20.4 20.5 20.6 20.7 20.7 20 8 20.9 291 92 93 94 9j 96 97 98 99 JdO 289.9 290.9 291.9 292 .9 293.9 294.9 295.9 296.9 297.9 298.9 25.4 25.4 25.5 25.6 25.7 25.8 25.9 26.0 26.1 26.1 Disi. nop. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. 1 [ForK Degre es. Page S22] TABLE IL Difference of Latitude and Departure for 6 Degrees. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 01 .0 00. 1 61 60.7 06.4 121 120.3 12.6 181 180.0 18.9 241 239.7 25.2 2 02.0 00.2 62 61.7 06.5 22 121 .3 12.8 82 181. 19.0 42 240.7 25.3 3 o3.o 00.3 63 62.7 06.6 23 122.3 12.9 83 182.0 19. 1 43 241.7 25.4 4 o4.o 00.4 64 63.6 06.7 24 123.3 i3.o 84 i83.o 19.2 AA 242.7 I'^.'J 5 o5.o 00.5 6b 64.6 06.8 25 124.3 i3.i 85 184.0 19.3 45 243.7 -5.6 6 06.0 00.6 66 65.6 06.9 26 125.3 l3.2 86 i85.o 19.4 46 244.7 25.7 7 07.0 00.7 67 66.6 07.0 27 126.3 i3.3 87 186.0 19.5 47 245.6 25.8 8 08.0 00.8 68 67.6 07.1 28 127.3 i3.4 88 187.0 19.7 48 246.6 25.9 9 09.0 00.9 69 68.6 07.2 29 128.3 i3.5 89 188.0 19.8 49 247-6 20.0 10 11 09.9 10.9 01 .0 01 .1 _i2_ 71 69.6 07.3 3o 129.3 i3.6 90 189.0 19-9. 5o 25l 248.6 249.6 26.1 26.2 70.6 07.4 i3i i3o.3 l3.7 191 190.0 20.0 12 II. 9 01 .3 72 71.6 07.5 32 i3i.3 i3.8 92 190.9 20.1 52 25o.6 26.3 Ij 12.9 01 .4 73 72.6 07.6 33 i32.3 i3.9 93 191. 9 20.2 53 25i.6 26.4 i4 .3.9 CI .5 74 73.6 07.7 34 i33.3 14.0 94 192.9 20.3 54 252.6 26.6 i6 14.9 01 .6 7!) 74.6 07.8 35 i34.3 14. 1 95 19J.9 20.4 55 253.6 26.7 lb ,b.9 01.7 7b 75.6 07.9 36 i35.3 14.2 96 194.9 20.5 56 254.6 26.8 17 16.9 01.8 77 76.6 08.0 37 i38.2 i4.3 97 195.9 20.6 57 255.6 26.9 i8 17.9 01 .9 78 77.6 08.2 38 137.2 14.4 98 196.9 20.7 58 256.6 27.0 19 18.9 02.0 79 78.6 08.3 39 i38.2 i4.5 99 197.9 20.8 59 257.6 27.1 20 19.9 02.1 80 79.6 08.4 40 139.2 14.6 200 198.9 20.9 60 258.6 I'! .1 27.3 21 20.9 02.2 81 80.6 08.5 i4i i4o.2 14.7 201 199.9 21 .0 261 259.6 22 21 .9 02.3 82 81.6 08.6 42 l4l -2 i4.8 02 200.9 21 .1 62 260.6 27.4 23 22.9 02.4 83 82.5 08.7 43 142.2 14.9 o3 201 .9 21 .2 63 261 .6 27.5 24 23.9 02.5 84 83.5 08.8 AA 143.2 i5.i 04 202.9 21.3 ■ H 262.6 27.6 2b 24.9 02.6 8b 84. b 08.9 45 144.2 l5.2 o5 203.9 21 .4 65 263.5 27.7 2b 2b. 9 02.7 86 85.5 09.0 46 145.2 i5.3 06 204.9 21.5 66 264.5 27.8 27 26.9 02.8 87 86.5 09.1 47 146.2 i5.4 07 205.9 21 .6 67 265.5 27.9 28 27.8 02 .9 88 87.5 09.2 48 147.2 ib.b 08 206.9 21.7 68 266.5 28.0 29 28.8 o3.o 89 88.5 09.3 49 148.2 i5.6 09 207.9 21.8 69 267.5 28.1 do 29.8 o3.i 90 89.5 09.4 5o 149.2 lb. 7 10 208.8 22.0 70 268.5 28.2 3i 3o.8 o3.2 91 90.5 09.5 i5i i5o.2 i5.8 211 209.8 22. 1 271 269.5 28.3 02 3i.8 o3.3 92 91. b 09.6 52 i5i .2 ib.9 12 210.8 22.2 72 270.5 28.4 33 32.8 o3.4 93 92.5 09.7 53 l52.2 16.0 i3 211. 8 22.3 73 271.5 28.5 M 33.8 o3.b 94 93.5 09.8 54 i53.2 16. 1 i4 212.8 22.4 74 272.5 28.6 '3b 34.8 03. 7 95 94.5 09.9 55 i54.2 16.2 i5 2i3.8 22.5 75 273.5 28.7 3b 3b. 8 o3.8 96 95.5 lO.O 56 i55.i 16.3 16 214.8 22.6 76 274.5 28.8 ^1 3b. 8 03.9 97 96.5 10. 1 57 i56.i 16.4 17 2i5.8 22.7 77 275.5 29.0 38 37.8 04.0 ^/» 97.5 10.2 58 157. 1 16.5 18 216.8 22.8 78 276.5 29.1 39 38.8 04.1 99 98.5 10.3 59 i58.i 16.6 19 217.8 22.9 79 277.5 29.2 40 4i 39.8 oA-1 100 99.5 10.5 60 159. 1 16.7 20 218.8 23.0 23.1 80 278.5 29.3 40.8 04.3 lOI 100.4 10.6 161 160. 1 16.8 221 219.8 281 279.5 29.4 42 4i.8 04.4 02 101.4 10.7 62 161 .1 16.9 22 220.8 23.2 hi 280.5 29.5 Ai 42.8 04. b o3 10.2.4 10.8 63 162. 1 17.0 23 221.8 23.3 83 281.4 29.6 A^ 43.8 04.6 o4 io3.4 10.9 64 i63.i 17. 1 24 222.8 23.4 84 282.4 29.7 45 44.8 04.7 ob 104.4 II .0 65 164. 1 17.2 25 223.8 23.5 85 283.4 29.8 46 45.7 04.8 06 io5.4 II .1 66 i65.i 17-4 26 224.8 23.6 86 284.4 29.9 47 4b. 7 04.9 07 106.4 II .2 67 166. 1 17.5 27 225.8 23.7 87 285.4 3o.o 48 47.7 o5.o 08 107.4 11.3 68 167. 1 17.6 28 226.8 23.8 88 286.4 3o.i 49 48. 7 Ob. I 09 108.4 II. 4 69 168. 1 17.7 29 227.7 23.9 89 287.4 3o.2 bo 5i 49.7 5o.7 05.2 o5.3 10 109.4 II. 5 70 169. 1 17.8 17.9 3o 228.7 24.0 90 288.4 3o.3 III 110.4 11.6 171 170. 1 23 I 229.7 24. 1 291 289.4 3o.4 b2 bi.7 o5.4 12 III .4 11.7 72 171. 1 18.0 32 230.7 24.3 92 290.4 3o.5 b3 52.7 ob.b i3 112. 4 11.8 73 172. 1 18. 1 33 23l .7 24.4 93 291 .4 3o.6 54 b3.7 ob.fc i4 ii3.4 II. 9 74 173.0 18.2 34 232.7 24.5 94 292.4 3o.7 bb ^4.7 ob.7 lb 114.4 12.0 75 174.0 18.3 35 233.7 24.6 95 293.4 3o.8 bb bb.7 05.9 lb ii5.4 12. 1 76 175.0 18.4 36 234.7 24.7 96 294.4 30.9 37 bb.7 06.0 17 116. 4 12.2 77 176.0 18.5 37 235.7 24.8 97 295.4 3i.o b8 b7.7 06. 1 18 117.4 12.3 78 177.0 18.6 38 236.7 24.9 98 296.4 3i.i b9 b8.7 06.2 19 118. 3 12.4 79 178.0 18.7 39 1*37.7 25.0 Q9 297.4 3i.3 bD 59.7 Ob. 3 20 119.3 12.5 80 179.0 18.8 40 Disi. 238.7 Dep. 25.1 3oo 298.4 3i.4 Dist. l)c.p. Lat. Dist Dep. Lat. Dist. Dep. Lat. L;il. Dist. Dep- Lat. [For 84 Degrees. TABLE IL [Page 23 Difierence of Latitude and Departure for 7 Degrees. Disl Lat. Dep. Disl. Lai. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lai. Dep. 29.4 I 01 .0 00. 1 61 60.5 07.4 121 120.1 14.7 181 1-79.7 22.1 241 239.2 2 02.0 00.2 62 61.5 07.6 22 121 .1 14.9 82 180.6 22.2 42 240.2 29.5 3 o3.o 00.4 63 62.5 07.7 23 122.1 i5.o 83 181.6 22.3 43 241 .2 29.6 4 o4.o 00.5 64 63.5 07.8 24 123.1 iS.i 84 182.6 22.4 AA 242.2 29.7 5 ()5.o 00.6 65 64.5 07.9 25 124.1 l5.2 85 i83.6 22.5 45 243.2 29.9 6 06.0 00.7 66 65.5 08.0 26 125.1 i5.4 86 184.6 22.7 46 244.2 3o.o 7 06.9 00.9 67 66.5 08.2 27 126.1 i5.5 87 i85.6 22.8 47 245.2 3o. I 8 07.9 0£ .0 68 67.5 08.3 28 127.0 i5.6 88 186.6 22.9 48 246.2 3o.2 9 08.9 01 .1 69 68.5 08.4 29 128.0 i5.7 89 187.6 23.0 49 247.1 3o.3 10 09.9 01 .2 70 69.5 08.5 3o 129.0 i5.8 90 188.6 23.2 5o 248.1 3o.5 1 1 10.9 01 .3 71 70.5 08.7 i3i 1 3o . 16.0 191 189.6 23.3 25l 249.1 3o.6 12 II. 9 01.5 72 71.5 08.8 32 i3i .0 16. 1 92 190.6 23.4 52 25o.i 3o.7 i3 12.9 01 .6 73 72.5 08.9 ■d-^ l32.0 16.2 93 191 .6 23.5 53 25l .1 3o.8 i4 ■ 3.9 01.7 74 73.4 09.0 34 i33.o 16.3 94 192.6 20.6 54 252. I 3i .0 i5 14.9 01.8 75 74.4 09.1 35 i34.o 16.5 95 193.5 23.8 55 253.1 3i.i i6 .5.9 ot .9 76 75.4 09.3 36 i35.o 16.6 96 194.5 23.9 56 254.1 3l.2 17 16.9 02.1 77 76.4 09.4 37 i36.o 16.7 97 195.5 24.0 57 255.1 3i.3 i8 17.9 02.2 78 77.4 09.5 38 137.0 16.8 98 196.5 24.1 58 256.1 3i.4 19 18.9 02.3 79 78.4 09.6 39 i38.o 16.9 99 197.5 24.3 59 257.1 3i.6 20 .9.9 02.4 80 79-4 09.7 4o 139.0 17. 1 200 198.5 24.4 60 258.1 3. .7 3i.8 21 20.8 02.6 8i 80.4 09.9 i4i 139.9 17.2 201 199.5 24.5 261 259. 1 22 21.8 02.7 82 81.4 10. 42 140.9 17.3 02 200.5 24.6 62 260.0 3i.9 23 22.8 02.8 83 82.4 10. 1 43 141.9 17.4 00 201 .5 24.7 63 261 .0 32.1 24 23.8 02.9 84 83.4 10.2 Ai 142.9 17.5 o4 202.5 24.9 64 262.0 32.2 25 24.8 o3.o 85 84.4 10.4 45 143.0 17.7 o5 2o3 . 5 25.0 65 263.0 32.3 26 25.8 03.2 86 85.4 10.5 46 144.9 17.8 06 204.5 25.1 66 264.0 32.4 27 26.8 o3.3 87 86.4 10.6 47 145.9 17.9 07 205.5 25.2 67 265.0 32.5 28 27.8 o3.4 88 87.3 10.7 48 146.9 18.0 08 206.4 25.3 68 266.0 32.7 29 28.8 o3.5 89 88.3 10.8 49 i47-9 18.2 09 207.4 25.5 69 267.0 32.8 3o 3i 29.8 o3.7 90 69.3 1 1 .0 5o 148.9 18.3 10 208.4 25.6 70 271 268.0 269.0 32.9 33 .0 3<).8 o3.8 91 90.3 1 1 .1 i5i 149.9 18.4 211 209.4 25.7 32 3i.8 03.9 92 91.3 II .2 52 1 50.9 18.5 12 210.4 25.8 72 270.0 33.1 33 32.8 04.0 93 92.3 II. 3 53 i5i .9 18.6 i3 211 .4 26.0 7-J 271 .c 33.3 34 33.7 o4.i 94 93.3 II. 5 54 152.9 18.8 i4 212.4 26.1 74 272. c 33.4 35 31.7 04.3 95 94.3 II. 6 55 i53.8 18.9 i5 2i3.4 26.2 75 273.0 33.5 36 35.7 04.4 90 95.3 II. 7 56 154.8 19.0 i6 214.4 26.3 76 273.9 33.6 37 36.7 04.5 97 96.3 II. 8 57 i55.8 19.1 17 2.5.4 26.4 77 274.9 33.8 38 37.7 04.6 98 97.3 II. 9 58 i56.8 .9.3 18 216.4 26.6 78 275.9 33.9 39 38.7 04.8 99 98.3 12. 1 59 157.8 19-4 19 217.4 26.7 79 276.9 34.0 4(. 3y.7 04.9 100 99.3 12.2 bo i58.8 19.5 20 218.4 26.8 26.9 80 277.9 34. 1 4i 40.7 o5.o lOI 100.2 12.3 161 159.8 19.6 221 219.4 281 278.9 34.2 42 41.7 o5.i 02 lOI .2 12.4 62 160.8 19.7 22 220.3 27.1 82 279.9 34.4 43 42.7 05.2 o3 102.2 12.6 63 161. 8 19.9 23 221 .3 27.2 83 280.9 34.5 44 43.7 o5.4 04 I03.2 12.7 64 162.8 20.0 24 222.3 27.3 84 281.9 34.6 45 44.7 o5.5 o5 104.2 12.8 65 i63.8 20. 1 25 223.3 27.4 85 282.9 34.7 46 45.7 o5.6 06 I05.2 12.9 66 164.8 20.2 26 224.3 27.5 86 283.9 •^4.9 Si 46.6 05.7 07 106.2 i3.o C7 i65.8 20.4 27 225.3 27.7 87 284.9 35.0 48 47-6 o5.8 08 107.2 l3.2 68 166.7 20.5 28 226.3 27.8 88 285.9 35.1 49 48.6 06.0 09 108.2 i3.3 69 167.7 20.6 29 227.3 27.9 89 286.8 35.2 5o 49.6 06.1 10 109.2 i3.4 70 168.7 20.7 3u 228.3 28.0 90 287.8 35.3 35.5 5f 5o.6 06.2 I II I 10.2 i3.5 171 169.7 20.8 23 1 229.3 28.2 291 288.8 52 5i.6 06.3 12 I I I .2 i3.6 72 170.7 21 .0 32 23o.3 28.3 92 289.8 35.6 53 52.6 06.5 i3 112. 2 i3.8 73 171. 7 21 .1 33 23i.3 28.4 93 290.8 35.7 54 53.6 06.6 i4 1X3.2 i3.9 74 172.7 21 .2 34 232 3 28.5 94 291.8 35.8 55 54.6 06.7 i5 Il4.I i4.o 75 173.7 21.3 35 233.2 28.6 95 292.8 36. 56 55.6 06.8 16 iiS.i i4.i 76 174.7 21.4 36 234.2 28.8 96 293.8 36.1 ''7 56.6 06 . 9 17 116. 1 14.3 77 175.7 21 .6 37 235.2 28.9 97 294. 8 36.2 58 57.6 07.1 18 117. 1 14.4 78 176.7 21.7 38 1 236.2 29.0 98 295.8 36.3 59 58.6 07.2 19 118. 1 i4.5 79 177-7 21.8 39 237.2 29. 1 99 296.8 36.4 b(. 59.6 07.3 20 1 19. 1 i4.6 80 178.7 21 .9 40 238.2 29.2 3oo 297.8 36.6 Dist. Dep. Lat. Disl. Dep. Lat. Dist. Dep. Lat. Disl. Dep. Lat. Dist. Dep. Lat. [For 83 Degrees. Page 24] TABLE IL Difference of Latitude and Departure for 8 Degrees. Dist. 1 Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 01 .0 00. 1 61 60.4 08.5 121 1 19.8 16.8 181 179.2 25.2 241 238.7 33.5 2 02.0 00.3 62 61.4 08.6 22 120.8 17.0 82 180.2 25.3 42 239.6 33.7 3 o3.o 00.4 63 62.4 08.8 23 121. 8 17. 1 83 181.2 25.5 43 240.6 33.8 4 04.0 00.6 64 63.4 08.9 24 122.8 17.3 84 182.2 25.6 44 241 .6 34.0 5 o5.o 00.7 65 64.4 09.0 25 123.8 17-4 85 i83.2 25.7 45 242.6 34.1 6 05.9 06.9 07.9 00.8 66 65.4 09.2 26 124.8 17.5 86 184.2 2D. 9 46 243.6 34.2 7 01 .0 67 66.3 09.3 27 125.8 17-7 87 i85.2 26.0 4i 244.6 34.4 8 01. 1 68 67.3 09.5 28 126.8 17.8 88 186.2 26.2 48 245.6 34.5 9 08.9 01 .3 69 68.3 09.6 29 127.7 18.0 89 187.2 2b. 3 49 246.6 34.7 10 09.9 01 .4 70 69.3 09.7 3o 128.7 18.1 90 188.2 26.4 5o 247.6 34.8 II 10.9 01.5 71 70.3 09.9 i3i 129.7 18.2 191 189.1 26.6 25 1 248.6 34.9 12 II. 9 01 .7 72 71.3 10. 32 1 30.7 18.4 92 190.1 26.7 52 249.5 35.! 1 3 12.9 01.8 73 72.3 10.2 33 i3i .7 18.5 93 191 .1 26.9 53 25o.5 35.2 1/1 I3.Q 01 .9 74 73.3 10.3 34 i32.7 18.6 94 192. 1 27.0 54 25i.5 35.3 1 5 i4.q 02.1 75 74.3 10.4 35 133.7 18.8 95 193.1 27.1 55 252.5 35.5 ifi i5.8 02.2 76 75.3 10.6 36 134.7 18.9 96 194. 1 27.3 5b 253.5 35.6 17 16.8 02.4 77 76.3 10.7 37 135.7 19. 1 97 195. 1 27.4 57 254.5 35.8 18 17.8 02.5 78 77-2 10.9 38 i36.7 19.2 98 196. 1 27.6 58 255.5 35.9 19 18.8 02 .6 79 78.2 II .0 39 137.7 19.3 99 197.1 27.7 59 256.5 36.0 20 19.8 02.8 80 79.2 II .1 4o i38.6 19.5 19.6 200 198. 1 27.8 bo 257.5 .36.2 21 20.8 02.9 81 80.2 II. 3 i4i 139.6 201 199.0 28.0 261 258.5 36.3 22 2T,8 0-3.1 82 81.2 II. 4 42 i4o.6 19.8 02 200.0 28 1 62 259.5 36.5 23 22.8 o3.2 83 82.2 II. 6 43 i4i.6 19.9 o3 201 .0 28.3 63 260.4 36.6 •?4 23,8 o3.3 84 83.2 II. 7 44 142.6 20.0 04 202.0 28.4 64 261.4 36.7 9.5 24.8 o3.5 85 84.2 n.8 45 143.6 20.2 o5 203.0 28.5 65 262.4 36.9 26 25.7 o3.6 86 85.2 12.0 46 144.6 20.3 06 204.0 28.7 66 263.4 37.0 27 26.7 o3.8 87 86.2 12. 1 47 145.6 20.5 07 205.0 28.8 67 264.4 37.2 28 27.7 03.9 88 87.1 12.2 48 146.6 20.6 08 206.0 28.9 68 265.4 37.3 29 28.7 04.0 8q 88.1 12.4 49 i47-5 20.7 09 207.0 29. I 69 266.4 37.4 JO 29.7 04.2 90 89.1 12.5 5u i5i i48.5 149.5 20.9 10 208.0 29.2 70 267.4 37.6 3i 3o.7 04.3 91 90.1 12.7 21 .0 211 208.9 29.4 271 268.4 37.7 32 3l.7 04.5 92 91. 1 12.8 52 i5o.5 21 .2 12 1 209.9 29.5 72 269.4 379 33 32.7 04.6 93 92.1 12.9 53 i5i.5 21.3 i3 210.9 29.6 73 270.3 38.-. 34 33.7 04.7 94 93.1 i3.i 54 i52.5 21 .4 i4 2 1 1 . 9 29.8 74 271 .3 38.1 35 34.7 04.9 95 94.1 l3.2 55 i53.5 21 .6 i5 212.9 29.9 7^ 272.3 •38.3 36 35.6 o5.o 96 95.1 i3.4 5b i54.5 21.7 lb 213.9 3o.i 76 273.3 38.4 37 36.6 o5.i 97 96.1 i3.5 57 i55.5 21 .9 17 214.9 3o.2 77 274.3 38 6 38 37.6 o5.3 98, 97.0 i3.6 58 i56.5 22.0 18 215.9 3o.3 78 275.3 38.7 39 38.6 o5.4 Q9 98.0 i3.8 59 157.5 22.1 19 210.9 3o.5 79 276.3 38.8 4o 39.6 o5.6 100 99.0 13.9 bo i58.4 22.3 20 217.9 3o.6 80 277.3 39.0 4i 40.6 o5.7 lOl 1 00.0 14.1. 161 159.4 22.4 221 218.8 3o.8 281 278.3 39.1 4? 41.6 o5.8 02 101 .0 14.2 62 160.4 22.5 22 219.8 30.9 82 279.3 39.2 43 42.6 06.0 o3 102.0 i4.3 63 161.4 22.7 23 220.8 3i .0 83 280.2 39.4 44 43.6 06.1 04 io3.o 14.5 64 162.4 22.8 24 221.8 3l.2 84 281.2 39.5 45 44.6 06.3 o5 104.0 14.6 65 i63.4 23. 25 222.8 3i.3 85 282.2 39.7 46 45.6 06.4 06 io5.o 14.8 66 164.4 23.1 2b 223.8 3i.5 8b 283.2 39.8 47 46.5 06.5 07 106.0 14.9 67 i65.4 23.2 27 224.8 3i.6 87 284.2 39.9 48 47.5 06.7 08 106.9 i5.o 68 166.4 23.4 28 225.8 31.7 88 285.2 4o.i 49 48.5 06.8 oq 107.9 l5.2 69 167.4 23.5 29 226.8 3i .9 89 286.2 40.2 5o 49-5 07.0 10 108.9 lb. 3 70 168.3 23.7 Jo 227.8 32.0 90 287.2 40.4 5i 5o.5 07.1 II I 109.9 i5.4 171 169.3 23.8 23l 228.8 32.1 291 288.2 40.5 5?. 5t.5 07.2 12 110.9 i5.6 72 170.3 23.9 32 229.7 32.3 92 289.2 40.6 03 52.5 07.4 i3 III .9 i5.7 73 171 .3 24.1 33 23o.7 32.4 93 290.1 40.8 54 53.5 07.5 i4 112. 9 15.9 74 172.3 24.2 34 23l .7 32.6 94 291 .1 40.9 55 54.5 07.7 i5 113.9 16.0 7^ 173.3 24.4 35 232.7 32.7 95 292.1 4i.i 56 55.5 07.8 16 114.9 16. 1 7b 174.3 24.5 3b 233.7 32.8 9b 293.1 !4i.2 1 57 56.4 07.9 17 115.9 16.3 77 175.3 24.6 37 234.7 33.0 97 294.1 4i.3 58 57.4 08.1 18 116. 9 16.4 78 176.3 24.8 38 235.7 33.1 98 295.1 41.5 59 58.4 08.2 19 117. 8 16. 6 79 177.3 24.9 39 236.7 33.3 99 296.1 4i 6 60 59.4 j Dep. 08.4 Lat. 20 118. 8 lb. 7 80 dTsI. 178.2 25.1 Lat. 40 Dist. 237.7 33.4 3oo Dist. 297. 1 Dep. 4i.6 Dist. Dep. Lat. Dep. Dep. Lat. [For 82 Degrees. TABLE II [Page 25 DifFerence of Latitude and Departure for 9 Degrees. Dist. Lat. Dep. Dist. Lat. Dep. 09.5 Dist. Lat. Dep. Dist.| Lat. Dep. Dist. Lai. Dep. I 01 .0 00.2 61 60.2 121 119. 5 18.9 181 178.8 28.3 24 1 238.0 37.7 2 02.0 00.3 62 61.2 09.7 22 120.5 19. 1 82 179.8 28.5 42 239.0 37.9 3 o3.o 00.5 63 62.2 09.9 23 121 .5 19.2 83 180.7 28.6 43 240 . 38. 4 o4.o 00.6 64 63.2 lO.O 24 122.5 '9-4 84 181 .7 28.8 44 241 .0 38.2 '■> 04.9 00.8 65 64.2 10.2 25 123.5 19.6 85 182.7 28.9 45 242.0 38.3 5 o5.9 00.9 66 65.2 10.3 26 124.4 19.7 86 183.7 29.1 4b 243.0 38.5 7 06.9 01 .1 67 66.2 10.5 27 125.4 19.9 87 184.7 29.3 47 244.0 38.6 .S 07.9 01 .3 68 67.2 10.6 28 126.4 20.0 88 185.7 29.4 48 244.9 38.8 9 08.9 01 .4 69 68.2 10.8 29 127.4 20.2 89 186.7 29.6 f9 245.9 39.0 10 09.9 01 .6 70 69. 1 1 1 .0 3o 128.4 20.3 90 187.7 29.7 5o 246.9 39.1 II 10.9 01.7 71 70.1 II .1 i3i 129.4 20.5 191 188.6 29.9 25l 247-9 39.3' 12 II. 9 01 .9 72 71. 1 II. 3 32 i3o.4 20.6 92 189.6 3o.o b2 248.9 39.4 i3 12.8 02.0 73 72.1 II. 4 33 i3i.4 20.8 93 190.6 3o.2 53 249-9 39.6 1 4 i3.8 02.2 74 73.1 II. 6 34 i32.4 21 .0 94 191 .6 3o.3 54 250.9 39.7 i5 i4.8 02.3 75 74.1 II. 7 35 i33.3 21 .1 95 192.6 3o.5 55 25l .9 39.9 1 6 i5.8 02.5 76 75.1 II. 9 36 i34.3 21.3 96 193.6 3o.7 5b 252.8 4o.o '7 16.8 02.7 77 76.1 12.0 37 i35.3 21.4 97 194.6 3o.8 i>7 253.8 40.2 i8 17.8 02.8 78 77.0 12.2 38 i36.3 21.6 98 195.6 3i.o 58 254.8 40.4 '9 18.8 o3.o 79 78.0 12.4 39 137.3 21.7 99 196.5 3i.i 59 255.8 40.5 20 19. 8 o3.i 80 79.0 12.5 4o i38.3 21.9 200 197.5 3i.3 bo 256.8 40.7 21 20.7 o3.3 81 80.0 12.7 i4i 139.3 22.1 201 19S.5 3i.4 261 257.8 40.8 22 21.7 o3.4 82 81.0 12.8 42 i4o.3 22.2 02 199.5 3i.6 b2 258.8 41 .0 23 22.7 o3.6 83 82.0 i3.o 43 i4i .2 22.4 o3 200.5 3i.8 b3 259.8 4i.i 24 23.7 o3.8 84 83. i3.i 44 142.2 22.5 04 201 .5 31.9 64 260.7 41.3 25 24.7 03.9 85 84.0 i3.3 45 143.2 22.7 o5 202.5 32.1 65 261.7 41.5 2fi 25.7 04.1 86 84.9 i3.5 46 144.2 22.8 06 2o3.5 32.2 bb 262.7 4i.6 27 26.7 04.2 87 85.9 i3.6 47 145.2 23. 07 204.5 32.4 67 263.7 4i.8 28 27-7 o4.4 88 86.9 i3.8 48 146.2 23.2 08 2o5.4 32.5 68 264.7 4i .9 29 28.6 04.5 89 87.9 i3.9 49 l47-2 23.3 09 206.4 32.7 69 265.7 42.1 3o 29.6 04.7 90 88.9 i4.i 5o 148.2 23.5 10 207.4 32.9 70 271 266.7 42.2 42.4 3i 3o.6 04.8 91 89.9 l4.2 i5i 149.1 23.6 211 208.4 33.0 267.7 32 3i.6 o5.o 92 90.9 14.4 52 i5o.i 23.8 12 209.4 33.2 72 268.7 42.6 33 32.6 o5.2 93 91.9 14.5 53 i5i .1 23.9 i3 210.4 33.3 73 269.6 42.7 34 33.6 o5.3 94 92.8 14.7 54 ID2.I 24.1 i4 211 .4 33.5 74 270.6 42.9 35 34.6 o5.5 95 93.8 14.9 55 i53.i 24.2 i5 212.4 33.6 7^ 271 .6 43.0 36 35.6 o5.6 96 94.8 i5.o 56 i54.i 24.4 16 2i3.3 33.8 76 272.6 43.2 37 36.5 o5.8 97 95.8 l5.2 57 i55.i 24.6 17 214.3 33.9 77 273.6 43.3 38 37.5 05.9 98 96.8 i5.3 58 i56.i 24.7 18 2i5.3 34.1 78 274.6 43.5 3g 38.5 06. 1 99 97.8 i5.5 §9 157.0 24.9 19 216.3 34.^ 79 275.6 43.6 4" 39.5 06.3 100 98.8 i5.6 60 i58.o 25.0 20 221 217.3 218.3 34.4 80 276.6 43.8 4i 40.5 06.4 lOI 99.8 i5.8 161 159.0 25.2 34.6 281 277.5 44.0 42 41.5 06.6 02 100.7 16.0 62 160.0 25.3 22 219.3 34.7 82 278.5 44.1 43 42.5 06.7 o3 lOI .7 16. 1 63 161 .0 25.5 23 220.3 34.9 83 279.5 44.3 44 43.5 06.9 o4 102.7 16.3 64 162.0 25.7 24 221 .2 35.0 84 280.5 44.4 45 44.4 07.0 o5 103.7 16.4 65 i63.o 25.8 25 222.2 35.2 85 281.5 44.6 46 45.4 07.2 06 104.7 16.6 66 164.0 26.0 26 223.2 35.4 8b 282.5 44.7 47 46.4 07.4 07 105.7 16.7 67 164.9 26.1 27 224.2 35.5 87 283.5 44.9 48 47.4 07.5 08 106.7 16.9 68 165.9 26.3 28 225.2 35.7 88 284.5 45.1 49 48.4 07.7 09 107.7 17. 1 69 166.9 26.4 29 226.2 35.8 89 285.4 45.2 5o 49.4 07.8 ID 108.6 17.2 70 167.9 26.6 3o 227.2 36.0 90 286.4 45.4 5i 5o.4 08.0 1 I I IC9.6 17-4 171 168.9 26.8 23l 228.2 36.1 291 287.4 45.5 52 5i.4 08.1 12 110.6 17.5 72 169.9 26.9 32 229.1 36.3 92 288.4 45.7 53 52.3 08.3 l3 III .6 17-7 73 170.9 27.1 33 23o.i 36.4 93 289.4145.8 1 54 53.3 08.4 i4 112. 6 17.8 74 171-9 27.2 34 23l .1 36.6 94 290.4 46.0 55 54.3 08.6 i5 ii3.6 18.0 75 172.8 27.4 35 232.1 36.8 95 291 .4 46.1 56 55.3 08.8 16 114.6 18.1 76 173.8 27.5 36 233.1 36.9 96 292 .4 46.3 57 56.3 08.9 17 ii5.6 18.3 77 174.8 27.7 37 234.1 37.1 97 293.3 46.5 58 57.3 09.1 18 116.5 18.5 78 175.8 27.8 38 235.1 37.2 98 294.3 46.6 59 58.3 09.2 19 117. 5 18.6 79 176.8 28.0 39 236.1 37.4 99 295.3 46.8 bo Dist. 59.3 1 09.4 Dep.l Lat. 20 118. 5 18.8 80 177.8 28.2 40 237.0 37.5 3oo J96.3 46.9 Dist.' Dep. 1 Lat. Dist. Dep. Lat. Di^f Dep. Lat. Dist Dep. Lat. [For 81 Degrees. Page 26] TABLE IL Difference of Latitude and Departure for 10 Degrees. Disl. Lat. Dep. Dist. Lat. Dep. Dist. Lai. Dep. 21 .0 Disl. 181 Lat. 178.3 Dep. 3i.4 Disl. Lat. Dep. I 01 .0 00.2 61 60.1 10.6 121 1 19.2 24 1 237.3 4i.8 2 02.0 00.3 62 61. 1 10.8 22 120.1 21.2 82 179.2 3i.6 42 238.3 42.0 3 o3.o 00.5 63 62.0 10.9 23 121 .1 21.4 83 180.2 3i.8 43 289.3 42.2 4 o3.9 00.7 64 63. 1 1 . 1 24 122. 1 21.5 84 181.2 32.0 44 240.3 42.4 5 04.9 00.9 65 64.0 II. 3 25 123. I 21.7 85 182.2 32.1 45 241.3 42.5 6 05.9 01 .0 66 65.0 II .0 26 124. 1 21 .9 86 i83.2 32.3 46 242.3 42.7 7 06.9 01 .2 67 66.0 II. 6 27 125. I 22.1 87 184.2 32.5 47 243.2 42.9 8 07. Q DI.4 68 67.0 II. 8 28 126. 1 22.2 88 i85.i 32.6 48 244.2 43.1 9 08.9 01 .6 69 68.0 12.0 29 127.0 22.4 89 186.1 32.8 49 245.2 43.2 lO 09.8 01.7 70 68.9 12.2 3o 128.0 22.6 22.7 90 187.1 33.0 5o 246.2 43.4 II 10.8 01 .9 71 69.9 12.3 i3i 129.0 191 188. 1 33.2 25l 247.2 43.6 12 II. 8 02.1 72 70.9 12.5 32 i3o.o 22.9 92 189. 1 33.3 52 248.2 43.8 i3 12.8 02.3 73 71.9 12.7 33 i3i .0 23.1 93 190. 1 33.5 53 249.2 43.9 i4 i3.8 02.4 74 72.9 12.8 34 l32.0 23.3 94 191 .1 33.7 54 25o.i 44.1 i5 14.8 02.6 75 73.9 i3.o 35 132.9 23.4 95 192.0 33.9 55 25l . 1 44.3 i6 i5.8 02.8 76 74.8 l3.2 36 133.9 23.6 96 193.0 34.0 56 252.1 44.5 17 16.7 o3.o 77 75.8 i3.4 37 134.9 23.8 97 194.0 34.2 57 253.1 44.6 i8 17-7 o3. 1 78 76.8 i3.5 38 135.9 24.0 98 195.0 34.4 58 254.1 44.8 19 18.7 o3.3 79 77.8 .3.7 39 186.9 24.1 99 196.0 34.6 59 255.1 45.0 20 19.7 o3.5 80 78.8 13.9 4o 137.9 24.3 200 197.0 34.7 60 256.1 45.1 21 20.7 o3.6 81 79.8 i4.i i4i i38.9 24.5 201 197.9 34.9 261 257.0 45.3 22 21.7 o3.8 82 80.8 14.2 42 139.8 24.7 02 198.9 35.1 62 258.0 45.5 23 22.7 04.0 83 81.7 14.4 Ai 140.8 24.8 OJ 199.9 35.3 63 259.0 45.7 24 23.6 04.2 84 82.7 i4.6 44 i4i.8 25.0 o4 200.9 35.4 64 260 . 45.8 25 24.6 o4.3 85 83.7 i4.8 45 142.8 25.2 o5 201 .9 35.6 65 261 .0 46.0 26 25.6 04.5 86 84.7 14.9 46 143.8 j 25.4 06 202.9 35.8 66 262.0 46.2 27 q6.6 04.7 87 85.7 i5.i 47 144.8 25.5 07 203.9 35.9 67 262. G 46.4 28 27.6 04.9 88 86.7 i5.3 48 145.8 25.7 08 204.8 35. 1 68 263.9 46.5 29 28.6 o5.o 89 87. 6 i5.5 49 146.7 25.9 09 2o5.8 36.3 69 264.9 46.7 3o 29.5 o5.2 90 88.6 i5.6 5o 147-7 26.0 10 206.8 36.5 70 265.9 46.9 3i 3o.5 o5.4 91 89.6 i5.8 i5i 148.7 26.2 211 207.8 36.6 271 266.9 47.1 32 3i.5 o5.6 92 90.6 16.0 52 149.7 26.4 12 208.8 36.8 72 267.9 47-2 33 32.5 o5.7 93 91 .6 16. 1 53 i5o.7 26.6 i3 209.8 37.0 73 268.9 47 4 34 33.5 05.9 94 92.6 16.3 54 i5i .7 26.7 i4 210.7 37.2 74 269.8 47 G 35 34.5 06.1 95 93.6 16.5 55 i52.6 26.9 i5 21 1. 7 37.3 75 270.8 47.8 36 35.5 06.3 96 94.5 16.7 56 i53.6 27.1 16 212.7 37.5 76 271.8 47.9 37 36.4 06.4 97 95.5 16.8 57 i54.6 27.3 17 213.7 37.7 77 272.8 48.1 38 37.4 06.6 98 96.5 17.0 58 i55.6 27.4 18 214.7 37.9 7a 273.8 48.3 39 38.4 06.8 99 97.5 17.2 59 i56.6 27.6 19 2j5.7 38. 79 274.8 48.4 4o 39.4 06.9 100 98.5 17-4 60 157.6 27.8 PC 216.7 38.2 80 275.7 48.6 4i 40.4 07.1 ipi 99.5 17.5 161 i58.6 28.0 221 217.6 38.4 2Sr 276.7 48.8 42 41.4 07.3 02 100.5 17-7 62 159.5 28.1 22 218.6 38.5 82 277.7 49.0 43 42.3 07.5 o3 101 .4 17.9 63 160.5 28.3 23 219.6 38.7 83 278.7 49.1 M 43.3 07.6 04 102.4 18.1 64 161. 5 28.5 24 220.6 38.9 84 279.7 49.3 45 A^.6 07.8 o5 io3.4 18.2 65 162.5 28.7 25 221 .6 39.1 85 2S0.7 49.5 46 45.3 08.0 06 104.4 18.4 66 i63.5 28.8 26 222.6 39.2 86 281.7 49.7 47 A^.i 08.2 07 io5 4 18.6 67 164.5 29.0 27 223.6 39.4 87 282.6 49.8 48 47-3 08.3 08 106.4 18.8 68 i65.4 29.2 28 224.5 39.6 88 283.6 5o.o 49 48.3 08.5 ■ 09 107.3 18.9 69 166.4 29.3 29 225.5 39.8 89 284.6 5o.2 5o 5i 49.2 08.7 10 108.3 19.1 70 167.4 29.5 3u 226.5 J9.9 291 285.6 So. 4 5o.2 08.9 III 109.3 19.3 171 168.4 29.7 23l 227.5 4o. 1 286.6 5o.5 52 5l.2 09.0 12 1 10.3 19-4 72 169.4 29.9 32 228.5 40.3 92 287.6 50.7 53 52.2 09.2 i3 III. 3 ig.6 73 170.4 3o.o 33 229.5 40.5 93 288.5 50.9 54 53.2 09.4 i4 1 12.3 19.8 74 171 .4 3o.2 34 23o.4 40.6 94 289.5 5i.i 55 54.2 09.6 i5 ii3.3 20.0 75 172.3 3o.4 35 23i.4 40.8 9i 290.5 5l.2 56 55.1 09.7 16 Il4-2 20. 1 7^ 173.3 3o.6 36 232.4 4i .0 96 291 .5 bi.4 57 56.1 09 . 9 17 Il5.2 20.3 77 174.3 3o.7 37 233.4 41.2 97,292.5 5i.6 58 57.1 10. 1 18 116. 2 20.5 78 175.3 3o.9 38 234.4 4i.3 98 293.5 5. .7 59 58.1 10.2 19 117. 2 20.7 79 176.3 3i.i 39 235.4 4i.5 99 294.5 5.. 9 6o 59.1 10.4 20 118. 2 20.8 Lat. 80 177.3 3i.3 40 236.4 41.7 3 00 295.4 52.1 Disl. Di.p. Lat. Dist. Dep. Dist. Dep. Lat. Dist. Dep. Lat. Disl. Dep. i Lat. [I ^orSC ) Degrees. TABLE XL [Paf.f 27 Difference of Latitude and Departure for 11 Degrees. Dist. Lat. Dep. Dist. Lat. Dcp. Dist. Lat. Dpp. Dist. Lat. ^ep. Dist. Lat 46.0 I 01. 00.2 61 59.9 11 .6 121 118. 8 23.1 181 177-7 34.5 241 236.6 2 02.0 00.4 62 60.9 II. 8 22 119. 8 23.3 82 178.7 34.7 42 237.6 46.2 3 02.9 00.6 63 61.8 12.0 23 120.7 23.5" 83 179.6 34.9 43 238.5 46.4 4 03.9 00.8 (^A 62.8 12.2 24 121.7 23.7 «4 180.6 35.1 AA 239.5 46.6 5 04.9 01 .0 65 63.8 12.4 25 122.7 23.9 *5 181.6 35.3 45 240.5 46.7 6 05.9 01 .1 66 64.8 12.6 26 123.7 24.0 86 182.6 35.5 46 241.5 46.9 7 06.9 01 .3 67 65.8 12.8 27 124.7 24.2 87 i83.6 35.7 47 242.5 47.1 8 07.9 01 .5 68 66.8 i3.o 28 125.6 24.4 88 184.5 35.9 48 243.4 47.3 9 08.8 01.7 69 67.7 l3.2 29 126.6 24.6 89 i85.5 36.1 49 244.4 47.5 10 09.8 01 .9 70 68.7 i3.4 3o 127.6 24.8 90 186.5 36.3 Soj 245.4 47.7 11 10.8 02,1 71 69.7 i3.5 i3i 128.6 25.0 191 187.5 36.4 25i 246.4 47-9 la II. 8 |o2.3 72 70.7 i3.7 32 129.6 25.2 92 188.5 36.6 52 247.4 48.1 i3 12.8 02.5 73 71-7 13.9 6i i3o.6 25.4 93 189.5 36.8 53 248.4 48.3 14 i3.7 02.7 74 72.6 i4.i M i3i.5 25.6 94 190.4 37.0 54 249.3 48.5 i5 I4.-7 02.9 75 73.6 14.3 35 i32.5 25.8 9^ 191. 4 37.2 55 250.3 48.7 iG i5.7 o3.i 76 74.6 14.5 36 i33.5 26.0 96 192.4 37.4 56 251.3 48.8 17 16.7 o3.2 77 75.6 14.7 37 1 34. 5 26.1 97 193.4 37.6 57 252.3 49.0 i8 17.7 o3.4 78 76.6 14.9 38 i35.5 26.3 98 194.4 37.8 58 253.3 49.2 19 18.7 o3.6 79 77.5 i5. 1 39 i36.4 26.5 99 195.3 38.0 59 254.2 49-4 20 19.6 o3.8 80 78.5 i5.3 4o 137.4 26.7 200 196.3 38.2 60 255.2 49-6 21 20.6 04.0 81 79-5 i5.5 i4i i38.4 26.9 201 197.3 38.4 261 256.2 49.8 C2 21.6 04.2 82 80.5 1 5. 6 42 139.4 27.1 02 198.3 38.5 62 257.2 5o.o 23 22.6 04.4 83 81.5 i5.8 A'i 140.4 27.3 o3 199.3 38.7 63 258.2 5o.2 24 23.6 04.6 84 82.5 16.0 AA 141.4 27.5 04 200.3 38.9 64 259.1 5o.4 25 24.5 04.8 85 83.4 16.2 45 142.3 27-7 o5 201.2 39.1 65 260. 1 5c.6 26 25.5 o5.o 86 84.4 16.4 46 143.3 27.9 06 202.2 3q.3 66 261 .1 5o.8 27 26.5 o5.2 87 85.4 16.6 47 144.3 28.0 07 2o3.2 39.5 67 2rj2 . 1 5o.9 28 27.5 o5.3 88 86.4 16.8 48 145.3 28.2 08 204.2 39.7 68 263.1 5. I 29 28.5 o5.5 89 87.4 17.0 49 i46.3 28.4 09 205.2 39.9 69 264.1 5i.3 3o 3i 29.4 o5.7 90 88.3 17.2 5o l47-2 28.6 28.8 10 206 . 1 4<5. 1 70 265.0 3i.5 2,0. A 05.9 91 89.3 17-4 i5i I4S.2 211 207.1 4o.3 271 266.0 5i.7 32 3i.4 06.1 92 90.3 17.6 52 149.2 29.0 12 208 . I 40.5 72 267.0 5i .9 33 32.4 06.3 93 91.3 17-7 53 l5o.2 29.2 i3 209.1 40.6 73 268.0 52.1 34 33.4 06.5 94 92.3 17.9 54 i5i .2 29.4 i4 210.1 40.8 74 269.0 52.3 35 34.4 06.7 95 93.3 18. 1 55 l52.2 29.6 i5 21 1 .0 4i .0 75 269.9 52.5 36 35.3 06.9 96 94.2 18.3 56 I53.I 29.8 lb 2 12.0 4i .2 76 270.9 52.7 37 36.3 07.1 97 95.2 18.5 57 154.1 3o.o 17 2l3.0 4i.4 77 271.9 52.9 38 37.3 07.3 98 96.2 18.7 58 i55.i 3o. I 18 214.0 4i.6 78 272.9 53.0 39 38.3 07.4 99 97.2 18.9 59 i56.i 3o.3 19 2l5.0 4i.8 79 273.9 53,2 4<) 39.3 07.6 100 98.2 19. 1 6f. 157.1 3o.5 20 216.0 42.0 80 281 274.9 53.4 4i 40.2 07.8 lOI 99.1 19.3 161 i58.o 3o.7 221 216.9 42.2 275.8 53.6 42 4i .2 08.0 02 100. 1 .9.5 62 159.0 30.9 22 217.9 42.4 82 276.8 53.8 43 42.2 08.2 o3 lOI . I 19.7 63 160.0 3i.i 23 218.9 42.6 83 277.8 54.0 AA 43.2 08.4 04 102. 1 19.8 64 161 .0 3i.3 24 219.9 42.7 84 278.8 54.2 45 44.2 08. 6 o5 io3.i 20.0 65 162.0 3i.5 25 220.9 42.9 85 279.8 54.4 ^{6 45.2 08.8 06 104. 1 20.2 66 i63.o 3i.7 26 221.8 43.1 86 280.7 54.6 47 46.1 09.0 07 io5.o 20.4 67 163.9 3i .9 27 222.8 43.3 87 281.7 54.8 48 47.1 09.2 08 1 06 . 20.6 68 164.9 32.1 28 223.8 43.5 88 282.7 55.0 49 48.1 09.3 09 107.0 20.8 69 165.9 32.2 29 224.8 43.7 89 283.7 55.1 5o 49.1 09.5 10 108.0 21 .0 70 166.9 32.4 3o 225.8 43.9 90 284.7 55.3 5i 30.I 09.7 HI 1 09 . 21.2 171 167.9 32.6 23l 226.8 44.1 291 285.7 55.5 52 5i.o 09.9 12 109.9 21 .4 72 168.8 32.8 32 227.7 44.3 92 286.6 55.7 53 52. 10. 1 i3 1 10.9 21.6 73 169.8 33.0 Si 228.7 44.5 93 287.6 55.9 54 53.0 10.3 lA III. 9 21.8 74 170.8 33.2 34 229.7 44.6 94 288.6 56.1 ^^ 54.0 10.5 i5 112. 9 21.9 75 171. 8 33.4 35 230.7 44.8 95 289.6 56.3 56 55.0 10.7 lb 113.9 22.1 76 172.8 33.1. 36 23l .7 45.0 9b 290.6 56.5 57 56.0 10.9 17 114.9 22.3 77 173.7 33. K 37 232.6 45.2 97 291 .5 56.7 5b 56.9 ! 1 1 . 1 18 ii5.8 22.5 78 174.7 34.0 38 233.6 45.4 98 292 .5 56.9 ^9 57.9 II. 3 19 116. 8 22.7 79 175.7 34.2 39 234.6 45.6 99 293.5 37.1 bo Disi. 58.9 11.4 l>.|.. I Lat. 20 117. 8 22.9 8c. 176.7 34.3 4o 235.6 45.8 3t.o 294.5 57.2 Dist. Dep. Lat. Dist. Drp. 1 Lat. Dist. Dcp. Lat. Di.st.j Dep. Lat. [T "or 79 Deirrees. Page 28] TABLE IL Difference of Latitude and Departure for 12 Degrees. Disl. Lat. Dep. Dist. Xat. Dep. Dist. 121 Lat. 118. 4 Dep. 25.2 Dist. Lat. Dep. Dist. Lat. Dep. I 01 .o 00.2 61 5o.7 12.7 181 177.0 37.6 241 235.7 5o.i 2 02.0 00.4 62 60.6 12.9 22 119. 3 25.4 82 178.0 37.8 42 236.7 5o.3 3 02.9 00.6 63 61.6 i3.i 23 120.3 25.6 83 179.0 38.0 43 237.7 5o.5 4 03.9 00.8 64 62.6 i3.3. ► 24 121 .3 25.8 84 180.0 38.3 44 238.7 50.7 b 04.9 01 .0 65 63.6 i3.5 25 122.3 26.0 85 181. 38.5 45 239.6 5o.Q b 05.9 01.2 6b 64.6 i3.7 26 123.2 26.2 86 181. 9 38.7 46 240.6 5..i 7 Ob. 8 01.5 67 65.5 13.9 27 124.2 26.4 87 182.9 38.9 47 24i .6 5i.4 8 07.8 01.7 68 66.5 14. 1 28 125.2 26.6 88 183.9 39.1 48 242.6 5i.6 9 08.8 01 .9 69 67.5 14.3 29 126.2 26.8 89 184.9 39.3 49 243.6 5i.8 10 09. S 02.1 70 68.5 14.6 3o 127.2 27.0 90 i85.8 39.5 5o 244.5 52.0 II 10.8 02.3 71 69.4 i4.8 i3i 128.1 27.2 191 186.8 39.7 25l 245.5 52.2 12 II. 7 02.5 72 70.4 i5.o 32 129. 1 27.4 92 187.8 39.9 52 246.5 52.4 iJ 12.7 02.7 7^ 71-4 l5.2 33 iSo.i 27.7 93 188.8 4o.i 53 247.5 52.6 i4 i3.7 02.9 74 72.4 i5.4 34 i3i . I .27-9 94 189.8 4o.3 54 248.4 52.8 lb 14.7 o3.i 75 7-:! -4 i5.6 35 l32.0 28.1 95 190.7 4o.5 55 249.4 53.0 lb lb. 7 o3.3 76 74.3 i5.8 36 i33.o 28.3 96 191. 7 4o.8 56 25o.4 53.2 17 16.6 o3.5 77 75.3 16.0 37 i34.o 28.5 97 192.7 4i .0 57 25i.4 53.4 18 17. b 03.7 78 76.3 16.2 38 i35.o 28.7 98 193.7 4i .2 58 252.4 53.6 ^9 18. b 04.0 79 77.3 ib.4 39 i36.o 28.9 99 194.7 41.4 59 253.3 53.8 20 19.6 04.2 80 81 78.3 16.6 4o 1 36. 9 29. 1 200 195.6 4i.6 60 254.3 54.1 21 20.5 04.4 79.2 16.8 i4i 137.9 29.3 201 196.6 4i.8 261 255.3 54.3 22 21. b 04.6 82 80.2 17.0 42 i38.9 29.5 02 197.6 42 .0 62 256.3 54.5 23 22.5 04.8 83 81.2 17.3 43 139.9 29.7 o3 198.6 42.2 63 257.3 54.7 24 23. b o5.o 84 82.2 17.5 44 140.9 29.9 04 199.5 42.4 64 258.2 54.9 2b 24.5 o5.2 8b 83.1 17.7 45 i4i.8 3o.i o5 200.5 42.6 65 259.2 55.1 26 25.4 o5.4 86 84.1 17.9 46 142.8 3o.4 06 201 .5 42.8 66 260.2 55.3 27 26.4 o5.6 87 85.1 18.1 47 143.8 3o.6 07 202.5 43.0 67 261 .2 55.5 28 27.4 o5.8 88 86.1 18.3 48 144.8 3o.8 08 2o3.5 43.2 68 262.1 55.7 29 28.4 06.0 89 87.1 18.5 49 145.7 3i .0 09 204.4 43.5 69 263.1 55.9 60 29.3 06.2 90 88.0 18.7 5o 146.7 3l.2 10 2o5.4 43.7 70 271 264.1 265.1 56.1 56.3 3i 3o.3 06.4 91 89.0 18.9 i5i 147-7 3i.4 211 206.4 43.9 32 3i.3 06.7 92 90.0 19. 1 52 148.7 3i.6 12 207.4 44.1 72 266.1 56.6 33 32.3 06.9 93 91 .0 19.3 53 149.7 3i.8 i3 208.3 44.3 73 267.0 56.8 M 6.i.6 07.1 94 91.9 19.5 54 i5o.6 32. i4 209.3 44.5 74 268.0 57.0 3b 34.2 07.3 9b 92.9 19.8 55 i5i.6 32.2 i5 210.3 44.7 75 269.0 57.2 3b 3b. 2 07.5 96 93.9 20.0 56 i52.6 32.4 16 21 1 .3 44.9 76 270.0 b7.4 ^7 3b. 2 07.7 97 94.9 20.2 57 i53.6 32.6 17 212.3 45.1 77 270.9 57.6 38 37.2 07.9 98 95.9 20.4 58 i54.5 32.9 18 2l3.2 45.3 78 271.9 57.8 39 38.1 08.1 99 96.8 20.6 59 155.5 33.1 19 214.2 45.5 79 272.9 58.0 40 39.1 08.3 100 lOI 97.8 98.8 20.8 60 i56.5 33.3 33.5 20 2l5.2 45.7 80 273.9 58.2 58.4 4i 40. 1 08.5 21 .0 161 157.5 221 216.2 45.9 281 274.9 42 4i.i 08.7 02 99.8 21 .2 62 i58.5 33.7 22 217. 1 46.2 82 275.8 58.6 43 42.1 08.9 o3 100.7 21 .4 63 159.4 33.9 23 218. I 46.4 83 276.8 58.8 44 43.0 09.1 o4 ior.7 21 .6 64 1UO.4 34.1 24 219.1 46.6 84 277.8 59.0 4b 44.0 09.4 o5 102.7 21.8 65 161.4 34.3 25 220.1 46.8 8b 278.8 59.3 4b 4b. 09.6 06 io3.7 22.0 66 162.4 34.5 26 221 .1 47-0 86 279.8 59.5 47 46.0 09.8 07 104.7 22.2 67 163.4 34.7 27 222.0 47.2 87 280.7 59.7 48 47-0 10. 08 105.7 22.5 68 164.3 34.9 28 223.0 47-4 88 281.7 59.9 49 47-9 10.2 09 106.6 22.7 69 i65.3 35.1 29 224.0 47-6 89 282.7 60.1 bo 48.9 10.4 10 107.6 22.9 70 166.3 35.3 3o 225.0 47-8 90 283.7 60.3 5i 49.9 10.6 III 108.6 23.1 171 167.3 35.6 23 1 226.0 48.0 291 284.6 60.5 b2 bo. 9 10.8 12 109.6 23.3 72 168.2 35.8 32 226.9 48.2 92 285.6 60.7 b3 bi.8 II .0 i3 110.5 23.5 73 169.2 36.0 33 227.9 48.4 93 286.6 60.9 b4 b2..8 II. 2 i4 III. 5 23.7 74 170.2 36.2 34 228.9 48.7 94 287.6 61.1 bb b3.8 II. 4 lb 112. 5 23.9 75 171 .2 36.4 35 229.9 48.9 95 288.6 61.3 bb b4.8 II. b 16 ii3.5 24.1 76 172.2 36.6 36 230.8 49.1 96 289.5 61.5 ^7 bb.8 II. 9 17 114. 4 24.3 77 173. 1 36.8 37 231.8 49-3 97 290.5 6r .7 58 56.7 12. 1 18 ii5.4 24.5 78 174. 1 37.0 38 232.8 49.5 98 291 .5 62.0 59 57.7 12. J 19 116.4 24.7 79 175. 1 37.2 39 233.8 49-7 99 292.5 62.2 60 58.7 12.5 20 117.4 24.9 Lat. 80 176.1 37.4 _4o Dist. 234.8 Dep. 49.9 Lat. 3oo 293.4 62.4 Lat. Dist. Dep. Lat. DIst. Dep. Disl. Dep. Lat. Dist. Deu. '"or 78 Degrees TABLE IL [Page 29 Difference of Latitude and Departure for 13 Degrees. Disl. Lat. Dep. Dis.. Lat. i Dcp. Dist. Lat. Dep. Dist. Lat. Dep. Dist. 241 Lat. Dep. | I ot .0 00.2 61 59.4' !3, 7 121 117. 9 27.2 181 176.4 40.7 234.8 54.2 2 CI. 9 CO. 4 62 5o.4 1 i3.9 22 118. 9 27.4 82 177.3 40.9 42 235.8 54-4 3 02.9 00.7 63 5i.4'i4.2 23 119. 8 27.7 83 178.3 41.2 43 236.8 54.7 4 08.9 00.9 64 62.4 i4.4 24 120.8 27.9 84 179-3 41.4 44 237.7 54.9 5 04.9 01 .1 65 63.3 14.6 25 121. 8 28.1 85 180.3 4i.6 45 238.7 55.1 6 o5.8 01.3 66 64.3 i4.8 26 122.8 28.3 86 181.2 4i.8 46 239.7 55.3 } 06.8 01 .6 67 65.3 i5.i 27 123.7 28.6 87 182.2 42.1 47 240.7 55.6 s 07.8 01.8 68 66.3 i5.3 28 124.7 28.8 88 i83.2 42.3 48 241 .6 55.8 9 08.8 02.0 69 67.2 i5.5 29 125.7 29.0 89 184.2 42.5 49 242.6 56.0 10 09.7 02.2 70 68.2 l5.7 3o 126.7 29.2 90 i85.i 42.7 5o 243.6 56.2 56.5 1 1 10.7 02.5 71 69.2 16.0 i3i 127.6 29.5 191 186.1 43.0 25l 244.6 12 II. 7 f.2.7 72 70.2 16.2 32 ■128.6 29.7 92 187.1 43.2 52 245.5 56.7 i3 12.7 02.9 73 71. 1 16.4 33 129.6 29.9 93 188.1 43.4 53 246.5 56.9 i4 i3.6 o3.i 74 72.1 16.6 34 i3o.6 3o.i 94 189.0 43.6 54 247-5 57.1 i5 14. b o3.4 75 73.1 16.9 35 i3i.5 3o.4 95 190.0 43.9 55 248.5 57.4 i6 i5.6 o3.6 76 74.1 17. 1 36 i32.5 3o.6 96 191 .0 44.1 56 249.4 57.6 17 16. b o3.8 77 73. 17.3 37 i33.5 3o.8 97 192.0 44.3 57 25o.4 57.8 i8 17.5 04.0 7» 76.0 17.5 38 i34.5 3i .0 98 192.9 44.5 58 25i.4 58.0 19 18.5 04.3 79 77.0 17.8 39 i35.4 3i.3 99 193.9 44.8 59 252.4 58.3 20 19.5 04.5 80 77-9 18.0 40 i36.4 3i.5 200 194.9 45.0 60 253.3 58.5 21 20.5 04.7 81 78.9 18.2 i4i 137.4 3i.7 201 195.8 45.2 261 254.3 58.7 22 21.4 04.9 82 79-9 18.4 42 i38.4 3i .9 02 196.8 45.4 62 255.3 58.9 23 22.4 o5.2 83 80.9 18.7 43 139.3 32.2 o3 197.8 45.7 63 256.3 59.2 24 23.4 o5.4 84 81.8 18. Q 44 i4o.3 32.4 04 198.8 45.9 64 257.2 59.4 25 24.4 o5.6 85 82.8 19. I 45 i4i.3 32.6 o5 199.7 46.1 65 258.2 59.6 26 25.3 o5.8 86 83.8 19.3 46 142.3 32.8 06 200.7 46.3- 66 259.2 59.8 27 26.3 06.1 87 84.8 19.6 47 143.2 33.1 07 201 .7 46.0 67 260.2 60.1 28 27.3 06.3 88 85.7 19.8 48 144.2 33.3 08 202.7 46.8 68 261 .1 60.3 29 28.3 06.5 89 86.7 20.0 49 145.2 33.5 09 2o3.6 47-0 69 262,1 60.5 60 29.2 06.7 90 «7.7. 20.2 5o 146.2 33.7 10 204.6 47-2 70 263.1 60.7 3i 3o.2 07.0 91 88.7 20.5 1.5 1 i47-i 34.0 211 2o5.6 47.5 271 264.1 61.0 32 3l.2 07.2 92 89.6 20.7 52 I48.I 34.2 12 206.6 47-7 72 265.0 61 .2 33 32.2 07.4 93 90.6 20.9 53 149. 1 34.4 i3 207.5 47-9 73 266.0 1 61 4 34 33.1 07.6 94 91.6 21 .1 54 i5o.i 34.6 i4 208.5 48.1 74 267,0 ; 61 .6 35 34.1 07.9 95 92.6 21.4 55 i5i.o 34.9 i5 209.5 48.4 75 268.0 61.9 36 35.1 08.1 96 93.5 21 .6 56 l52.0 35.1 16 210.5 48.6 76 268.9 62.1 37 36.1 08.3 97 94.5 21.8 57 i53.o 35.3 17 211 .4 48.8 77 269.9 62.3 38 37.0 08.5 98 95.5 22.0 58 i54.o 35.5 18 212.4 49.0 78 270.9 62.5 39 33.0 08.8 99 96.5 22.3 59 154.9 35.8 19 2i3.4 49-3 79 271.8 62.8 4o 39.0 09.0 100 97-4 22.5 60 155.9 36. 20 214.4 49.5 80 272.8 63.0 4i 39.9 09 . 2 lOI 98.4 22.7 161 i56.9 36.2 221 2i5.3 49-7 281 273.8 63.2 43 40.9 09.4 02 99.4 22.9 62 157.8 36.4 22 216.3 49.9 82 274.8 63.4 43 41.9 09.7 o3 100.4 23.2 63 i58.8 36.7 23 217.3 5o.2 83 275.7 63.7 44 42.9 09.9 04 loi .3 23.4 64 159.8 36.9 24 218.3 5o.4 84 276.7 63.9 45 4i.8 10. 1 o5 102.3 23.6 65 160.8 37.1 25 219.2 5o.6 85 277-7 64.1 46 44.8 10.3 06 io3.3 23.8 66 161 .7 37.3 26 220.2 5o.8 86 278.7 64.3 47 45.8 10.6 07 104.3 24.1 67 162.7 37.6 27 221 .2 5i.i 87 279.6 164.6 48 46.8 10.8 08 io5.2 2'i.3 68 163.7 37.8 28 222.2 5i.3 88 280.6 ,64.8 49 47.7 II .0 09 106.2 24.5 69 164.7 38.0 29 223. I 5i.5 89 281.6 65.0 5o 48.7 II. 2 10 107.2 24.7 7" 171 i65.6 166.6 38.2 3o 224.1 51.7 90 282.6 65.2 65.5 5i 49-7 11.5 III I03.2 25.0 38.5 23l 225.1 52.0 291 283.5 32 50.7 II. 7 12 109.1 25.2 72 167.6 38.7 32 226.1 52.2 92 284.5 65.7 53 5i.6 II. 9 i3 no. I 25.4 73 168.6 38.9 33 227.0 52.4 93 285.5 65.9 54 52.6 12 I i4 III. I 25.6 74 169.5 39.1 34 228.0 52.6 94 286.5 66.1 55 53.6 12.4 i5 112. 1 25.9 75 170.5 3q.4 35 229.0 52.9 95 287.4 66.4 56 54.6 12.5 16 ii3.o 26.1 76 171. 5 39.6 36 23o.o 53.1 96 288.4 66.6 57 55.5 12.8 37 114.0 26.3 77 172.5 39.8 37 230.9 53.3 97 289.4 66.8 58 56.5 i3.o 18 i:5.c 26.5 78 173.4 4o.o 38 23l .9 53.5 98 290.4 67.0 59 57.5 i3.3 19 :i6.o 25.5 79 174.4 4o.3 39 232.9 53.8 99 291 .3 67.3 bu 5S.5,i3.5 20 116.9 27.0 80 175.4 4o.5 4:) 233.8 54.0 3oo 292.3 67.5 Dist. Dpp. ' Lat. Dist, Dop Lai. Dist. Dep. Lnt. Disl. D.'p. Lat. Disl. Dep. Lat. V For 77 Degrees. Page 30] TABLE n. Difference of Latitude and Departure for 14 Degrees. Dist. Lat. Dep. 00.2 Dist. Lat. Dep. Dist. Lat. Dep. Dist Lat. Dep. Dist. Lat. Dep. .58.3 I 01. 61 59.2 14.8 121 117. 4 29.3 181 175.6 43.8 241 233.8 2 01.9 00.5 62 60.2 i5.o 22 118.4 29.5 82 176.6 44-0 42 234.8 58.5 3 02.9 00.7 63 61.1 l5.2 23 119. 3 29.8 83 177-6 44-3 43 235.8 58.8 4 03.9 01 .0 64 62.1 i5.5 24 120.3 3o.o 84 178.5 44.5 A^ 236.8 59.0 5 04.9 01.2 65 63.1 i5.7 25 121 .3 3o.2 85 179.5 44.8 45 237-7 59.3 6 o5.8 01 .5 66 64.0 16.0 26 122.3 3o.5 86 180.5 45.0 46 238.7 59.5 7 06.8 01.7 67 65. 16.2 27 123.2 3o.7 87 181. 4 45.2 47 239.7 59.8 8 07.8 01.9 68 66.0 16.5 28 124.2 3i .0 88 182.4 45.5 48 240.6 60.0 9 08.7 02.2 69 67.0 ,6.7 29 125.2 3l.2 89 i83.4 45.7 49 241 .6 60.2 lO 09.7 02.4 70 67.9 16.9 3o 126. I 3i.4 3i.7 90 184.4 46. 5o 242.6 60.5 II 10.7 02.7 71 68.9 17.2 i3i 127. I 191 i85.3 46.2 25l 243.5 60.7 12 II. 6 02.0 72 69.9 17-4 32 128. I 3i .9 92 186.3 46.4 52 244.5 61.0 i3 12.6 o3.i 73 70.8 17-7 33 129.0 32.2 93 187.3 46.7 53 245.5 61 .2 i4 i3.6 o3.4 74 71.8 17.9 34 i3o.o 32.4 94 188.2 46.9 54 246.5 61.4 i5 i4.6 o3.6 75 72.8 18. 1 35 i3i.o 32.7 95 189.2 47-2 55 247.4 61.7 i6 i5.5 03.9 76 73.7 18.4 36 l32.0 32.9 96 190.2 47-4 56 248.4 61.9 17 16.5 04. 1 77 74-7 18.6 37 132.9 33.1 97 191 .1 47-7 57 249.4 62.2 i8 17.5 04.4 78 75.7 18.9 38 i33.9 33.4 98 192.1 47-9 58 250.3 62.4 19 18.4 04.6 79 76.7 19. 1 39 134.9 33.6 99 193.1 48.1 59 251.3 62.7 20 19.4 04.8 80 77.6 19.4 40 i35.8 33.9 200 194. 1 48.4 60 252.3 62.9 21 20.4 o5.i 81 78.6 19.6 i4i i36.8 34.1 201 195.0 48. G 261 253.2 63.1 22 21.3 o5.3 82 79.6 19.8 42 137.8 34.4 02 196.0 48.9 62 254.2 63.4 23 22.3 o5.6 83 80.5 20. 1 43 i38.8 34.6 OJ 197.0 49.1 63 255.2 63.6 24 23.3 o5.8 84 81.5 20.3 M 139.7 34.8 o4 197.9 49-4 64 256.2 63.9 25 24.3 06.0 85 82.5 20.6 45 140.7 35.1 o5 198.9 49-6 65 257.1 64.1 26 25.2 06.3 86 83.4 20.8 46 141.7 35.3 06 199.9 49-8 66 258.1 64.4 27 26.2 06.5 87 84.4 21 .0 47 142.6 35.6 07 200.9 5o.i ^7 2:j9. I 64.6 28 27.2 06.8 88 85.4 21.3 48 143.6 35.8 08 201.8 5o.3 68 260.0 64.8 29 28.1 07.0 89 86.4 21.5 49 144.6 36.0 09 202.8 5o.6 69 261 .0 65.1 3o 29.1 07.' 90 87.3 21.8 5o 145.5 36.3 10 2o3.8 5o.8 70 262.0 65.3 3i 3o. I 07.5 9' 88.3 22.0 i5i 146.5 36.5 211 204.7 5i.o 271 263.0 65.6 32 3i .0 07,7 92 69.3 22.3 52 147-5 36.8 12 205.7 5i.3 72 263.9 65.8 33 32.0 08.0 93 90.2 22.5 53 148.5 37.0 i3 206.7 5i.5 73 264.9 66.0 34 33.0 08.2 94 91 .2 22.7 54 149.4 37.3 i4 207.6 5i.8 74 365.9 66.3 35 34.0 08.5 95 92.2 23.0 55 i5o.4 37.5 i5 208.6 52.0 7^ 266.8 66.5 36 34.9 08.7 96 93.1 23.2 56 i5i.4 37.7 16 209.6 52.3 76 267.8 66.8 37 35.9 09.0 97 94.1 23.5 57 i52.3 38. 17 210.6 52.5 77 268.8 67.0 38 36.9 09.2 98 95.1 23.7 58 i53.3 38.2 18 211 .5 52.7 78 269.7 67.3 39 37.8 09.4 99 96. 1 24.0 59 i54.3 38.5 19 212.5 53.0 79 270.7 67.5 40 38.8 09.7 100 97.0 24.2 60 i55.2 38.7 20 2i3.5 53.2 80 271.7 67-7 4i 39.8 09.9 lOI 98.0 24.4 161 i56.2 38.9 221 214.4 53.5 281 272.7 68.0 42 40.8 10.2 02 99.0 24.7 62 157.2 39.2 22 2i5.4 53.7 82 273.6 68.2 43 41.7 10.4 o3 99.9 24.9 63 i58.2 39.4 23 216.4 53.9 83 274.6 68.5 U 42.7 10.6 04 100.9 25.2 64 159. 1 39.7 24 217.3 54.2 84 275.6 68.7 45 43.7 10.9 o5 lOI .9 25.4 65 160. 1 39.9 25 218.3 54.4 85 276.5 68.9 46 44.6 II .1 Otj 102.9 25.6 66 161 .1 40.2 26 219.3 54.7 86 277.5 69.2 47 45.6 II. 4 07 io3.8 25.9 67 162.0 40.4 27 220.3 54-9 87 278.5 69.4 48 46.6 II. 6 08 104.8 26.1 68 i63.o 40.6 28 221 .2 55.2 88 279.4 69.7 49 47.5 II. 9 09 io5.8 26.4 69 164.0 40.9 29 222.2 55.4 89 280.4 69.9 5o 48.5 12. 1 10 106.7 26.6 70 i65.o 4i.i 3o 223.2 55.6 90 281.4 70.2 5i 49. "i 12.3 III 107.7 26.9 171 165.9 41.4 23 I 224.1 55.9 291 282.4 70.4 52 5o.5 12.6 12 108.7 27.1 72 166.9 41.6 32 225.1 56.1 92 283.3 70.6 53 5i.4 12.8 i3 109.6 27.3 73 167.9 41.9 33 226.1 56.4 93 284.3 70.9 54 52.4 i3.i i4 no. 6 27.6 74 168.8 42.1 34 227.0 56.6 94 285.3 71. 1 55 53.4 i3.3 1 5 III .6 27.8 75 169.8 42.3 35 228.0 56.9 95 286.2 71.4 56 54.3 i3.5 16 112. 6 28.1 76 170.8 42.6 ■66 229.0 57-1 96 287.2 71.6 57 55.3 i3.8 17 ii3.5 28.3 77 171-7 42.8 37 23o.o 57.3 97 288.2 71.9 58 56.3 i4-o 18 114.5 28.5 78 172.7 43.1 38 230.9 57.6 98 289.1 72.1 59, 57.2 i4.3 19 ii5.5 28.8 79 173.7 43.3 39 231.9 57.8 99 290. 1 72.3 60: 58.2 Dist.l Dcp. i4.5 Lat. 20 116. 4 29.0 Lat. 80 174.7 43.5 4o 232.9 58.1 000 29 1 . 1 72.6 Dist. Dcp. Dist. Dep. Lai. Dist. Dep. Lat. Dist. Dep. Lat. [For 7G Degre es. TABLE n [Page 31 Dinference of Latitude and Departure for 15 Degrees. Disl. Lai. 1' Dop. Uisl. Lat. Dep. Dist.j Lat. Dep. Dist. Lat. Dep. Dist. Lat. D*.-p. I ul .0 00.3 61 58.9 i5.8 121 116. 9 3i.3 181 174-8 46.8 241 232.8 62.4 2 01.9 00.5 b2 59.. 9 16.0 22 117. 8 3i.6 82 175.8 47.1 42 233.8 62.6 3 02.9 00.8 63 60.9 16.3 23 118.8 3i.8 83 176.8 47.4 43 234.7 62.9 4 o3.9 01 .0 b4 61.8 16.6 24 119. 8 32.1 84 177-7 47.6 44 235.7 63.2 5 04.8 01 .3 bb 62.8 16.8 2b 120.7 32.4 8b 178.7 47-9 45 236.7 63.4 6 o5.8 01 .6 6b 63.8 17. 1 26 121 .7 32.6 86 '79-7 48.1 46 237.6 63.7 7 Ob. 8 01.8 67 64.7 17.3 27 122.7 32.9 87 180.6 48.4 47 238.6 63.9 64.2 8 07.7 02. 1 68 65.7 17.6 28 123.6 33.1 88 181.6 48.7 48 239.5 9 08.7 02.3 69 66.6 17.9 29 124.6 33.4 89 182.6 48.9 49 240. L 64.4 lO 09.7 02.6 70 67.6 18. 1 3o 125.6 33.6 90 i83.5 49.2 5o 241.5 64.7 II 10.6 02.8 71 68.6 18.4 i3i 126.5 33.9 191 184.5 49.4 25l 242.4 65.0 12 11. b o3.i 72 69.5 18.6 32 127.5 34.2 92 185.5 49.7 52 243.4 65.2 i3 12.6 o3.4 73 70.5 18.9 ^i 128.5 M.4 93 186.4 5o.o 53 244.4 65.5 i4 i3.b o3.6 74 71.5 19.2 M 129.4 34.7 94 187.4 5o.2 54 245.3 65.7 lb i4.b 03.9 7!) 72.4 19.4 3b i3o.4 34.9 95 188.4 5o.5 55 246.3 66.0 i6 ib.b o4.i 76 73.4 19.7 36 i3i.4 35.2 96 189.3 5o.7 56 247.3 66.3 J7 lb. 4 04.4 77 74.4 19.9 37 i32.3 35.5 97 190.3 5i.o 57 248.2 66.5 i8 17-4 04.7 7» 7b. 3 20.2 38 133.3 35.7 98 .91.3 5l.2 58 249.2 66.8 '9 18.4 04.9 79 76.3 20.4 39 i34.3 36. 99 192.2 5i.5 59 25o.2 67.0 20 .9.3 05.2 80 77.3 20.7 4o i35.2 36.2 200 193.2 5i.8 60 25l .1 67.3 2! 20.3 o5.4 81 78.2 21 .0 .i4i i36.2 36.5 201 194.2 52.0 261 252.1 67.6 22 21.3 o5.7 82 79.2 21.2 42 137.2 36.8 02 195.1 52.3 62 253.1 67.8 23 22.2 06.0 83 80.2 21.5 43 i38.i 37.0 o3 196. 1 52.5 63 254.0 68.1 24 23.2 06 . 2 84 81. 1 21.7 44 139. 1 37.3 04 197.0 52.8 64 255.0 68.3 25 24.1 06.5 8b 82.1 22.0 4b 1 40 . 1 37.5 o5 198.0 53.1 65 256. 68.6 2b 2b. I 06.7 8b 83.1 22.3 46 i4i .0 37.8 06 199.0 53.3 66 256.9 68.8 27 2b. I 07.0 87 84.0 22.5 47 142.0 38. 07 199.9 53.6 67 257.9 69.1 2« 27.0 07.2 88 85. 22.8 48 143.0 38.3 08 200.9 53.8 68 258.9 69.4 29 28.0 07. D 89 86.0 23.0 49 143.9 38.6 09 201 .9 54.1 69 259.8 69.6 Jo 29.0 07.8 90 80.9 23.3 bo 144.9 38.8 10 202.8 54.4 70 260.8 69.9 3i 29.9 08.0 9' 87.9 23.6 i5i 145.9 39.. 21 1 2o3.8 54.6 271 261.8 70.1 32 30.9 08.3 92 88.9 23.8 52 146.8 39.3 12 204.8 54.9 72 262.7 70.4 iS 31.9 08.5 9i 89.8 24.1 b3 147.8 39.6 i3 2()5.7 55.1 73 263.7 70.7 34 32.8 0S.8 94 90.8 24.3 b4 148.8 39.9 i4 206 . 7 55.4 74 264.7 70.9 jt) 33.8 09. I 9-^ 91.8 24.6 bb 149.7 4o. 1 i5 207.7 55.6 75 265.6 71.2 3b 34.8 09.3 9b 92.7 24.8 b6 1 5o . 7 40.4 16 208.6 55.9 7t) 266.6 71-4 ^7 3^.7 09.6 97 93.7 25.1 i)7 i5i .7 4o.6 17 209.6 56.2 77 267.6 71-7 38 3b. 7 09.8 98 94.7 2b. 4 b8 1 52 .6 40.9 18 210.6 56.4 78 268.5 72.0 39 37.7 10. I 99 95.6 25.6 b9 i53.6 4i .2 19 211.5 56.7 79 269.5 72.2 40 38. b 10.4 i(i() 96.6 25.9 60 154.5 41.4 41.7 20 212.5 56.9 80 270. D 72.5 4i 39.6 10.6 101 97.6 26. 1 161 i55.5 221 213.5 57.2 281 271.4 72.7 42 40.6 10.9 02 98.5 26.4 62 i56.5 4i .9 22 214.4 57-5 82 272.4 73.0 43 4i.b I I . I o3 99.5 26.7 63 157.4 42.2 23 2i5.4 57-7 83 273.4 73.2 44 42. b II. 4 04 100.5 26.9 64 158.4 42.4 24 216.4 58. 84 274.3 73.5 4b 43.5 1 1. 6 ob loi .4 27.2 65 159.4 42.7 25 217.3 58.2 85 275.3 73.8 4f) 4^.4 11.9 Ob 102.4 27.4 66 160.3 43.0 26 218.3 58.5 86 276.3 74.0 47 4':). 4 12.2 07 io3.4 27.7 67 161. 3 43.2 27 219.3 58.8 87 277.2 74.3 48 4b. 4 12.4 08 104.3 28.0 68 162.3 43.5 28 220.2 59.0 88 278.2 74.5 i:'9 47.3 12.7 09 io5.3 28.2 69 i63.2 43.7 29 221 .2 59.3 89 279.2 74.8 bo 48.3 12.9 10 106.3 2S.5 70 164.2 44.0 3o 222.2 59.5 90 280.1 75.1 bi 49.3 1 3. 2 1 1 1 1 1 07 . 2 28.7 171 i65.2 44.3 23l 223.1 59.8 291 281. 1 75.3 b2 bf > . 2 i3.b 12 108.2 29.0 72 166. 1 44.5 32 224.1 60.0 92 282.1 75.6 33 bi.2 13.7 i3 109. I 29.2 73 167.1 44.8 33 225.1 60.3 93 283. 75.8 b4 b2.2 14.0 i4 1 10. 1 29.5 74 168.1 45.0 34 226.0 60.6 94 284.0 76.1 bb b3.r l4.2 lb III .1 29.8 75 -169.0 45.3 35 227.0 60.8 95 284.9 76.4 bb b4.i i4.b lb 112.0 Bo.o 76 170.0 45.6 36 228.0 61.1 96 285.9 76.6 b7 bb.i i4.8 17 ii3.o 3o.3 77 171 .0 45.8 37 228.9 61.3 97 286.9 76.9 b8 bb.o i5.o 18 114.0 3o.b 78 171.9146.1 1 38 229.9 61.6 98 287.8 11 ■' b9 b7.o i5.3 '9 114.9 3o.8 79 172.9 46.3 39 230.9 61 .9 99 288.8 11-4 fx) b8.o i5.5 l.at. 20 1 15.9 3i.i 80 173.9 46.6 4o 231.8 62.1 3()o 289.8 77-6 Dist. nop. I)ist.| Dep. Lat. Dist. Dep. Lai. Dist. Dep. Lat. Dist. Dep. Lat. I [> "or 75 Defrre es. Page 32j TABLE IL Difference of Latitude and Departure for 16 Degrees Dist Lat. Dep. Dist. Lat. Dcp. Dist. Lat. Dep. Dist Lat. Dep. Dist. Lat. Dep. I 01 .0 00.3 61 58.6 16.8 121 116.3 33.4 181 174.0 49.9 241 23l .7 66.4 2 01 .9 00.6 62 59.6 17. 1 22 117.3 33.6 82 174.9 5o.2 42 232.6 66.7 3 C 2 . 9 1 DO . 8 63 60.6 17-4 23 Ij8.2 33.9 83 175.9 5o.4 43 233.6 67.0 4 o3.8 01 .1 64 61.5 17.6 24 119. 2 34.2 84 176.9 50.7 44 234.5 67.3 5 04.8 01 .4 65 63.5 17.9 25 120.2 34.5 85 177.8 5i.o 45 235.5 67.5 6 o5.8 01.7 66 63.4 18.2 26 121 .1 34.7 86 178.8 5i.3 46 236.5 67.8 7 06.7 01 .9 67 64.4 18.5 27 122.1 35.0 87 179.8 5i.5 47 237.4 68.1 8 07.7 02.2 68 65.4 .8.7 28 123.0 35.3 88 180.7 5i.8 48 238.4 68.4 9 08.7 02.5 69 66.3 19.0 29 124.0 35.6 89 181.7 52.1 49 239.4 68.6 10 09.6 02.8 70 67.3 19.3 3o 125. 35.8 90 182.6 52.4 52.6 5o 240.3 68.9 II 10.6 o3.o 71 68.2 19.6 i3i 125.9 36.1 191 i83.6 25l 241.3 69.2 12 II. 5 o3.3 72 69.2 19.8 32 126.9 36.4 92 184.6 52.9 52 242.2 69.5 i3 12.5 o3.6 73 70.2 20.1 33 127.8 36 7 93 185.5 53.2 53 243.2 69.7 i4 i3.5 03.9 74 71. 1 20.4 34 128.8 36.9 94 186.5 53.5 54 244.2 70.0 i5 14.4 04.1 75 .72.1 20.7 35 129.8 37.2 95 187.4 53.7 55 245.1 70.3 i6 1 5. 4 04.4 76 73.1 20.9 36 1 30.7 37.5 96 188.4 54.0 56 246.1 70.6 17 i6.3 04.7 77 74.0 21 .2 37 i3i.7 37.8 97 189.4 54.3 57 247.0 70.8 i8 17.3 o5.o 78 75.0 21.5 38 l32.7 38. 98 190.3 54.6 58 248.0 71.1 19 18.3 o5.2 79 75.9 21.8 39 i33.6 38.3 99 191 .3 54.9 59 249.0 71.4 20 19.2 o5.5 80 76.9 22.1 4o i34.6 38.6 200 192.3 55.1 60 249.9 71.7 21 20.2 o5.8 81 77-9 22.3 i4i i35.5 38.9 201 193.2 55.4 261 250.9 71.9 22 21. 1 06.1 82 78.8 22.6 42 i36.5 39.1 02 194.2 55.7 62 25l .9 72.2 23 22.1 06.3 83 79.8 22.9 43 137.5 3q.4 00 195.1 56.0 63 252.8 72.5 24 23.1 06.6 84 80.7 23.2 44 i38.4 39.7 o4 196. 1 56.2 64 253.8 72.8 25 I 24.0 06.9 85 81.7 23.4 45 139.4 4o.o o5 197.1 56.5 65 254.7 73.0 26)25.0 07.2 86 82.7 23.7 46 140.3 40.2 06 198.0 56.8 66 255.7 73.3 27 •■ 26.0 07.4 87 83.6 24.0 47 i4i.3 40.5 07 199.0 57.1 67 256.7 73.6 28 1 26.9 07.7 88 84.6 24.3 48 142.3 4o.8 08 199.9 57.3 68 257.6 73.9 29127.9 08.0 89 85.6 24.5 49 143.2 4i.i 09 200.9 57.6 69 258.6 74.1 30J28.8 08.3 90 86.5 24.8 5o 144.2 4i.3 10 201 .9 57.9 70 259.5 74.4 3r 129.8 08.5 91 87.5 25.1 i5i 145.2 4i.6 211 202.8 58.2 271 260.5 74.7 32 130.8 08.8 92 88.4 25.4 52 146.1 41.9 12 2o3.8 58.4 72 261 .5 75.0 33i3i.7 09.1 93 89.4 25.6 53 i47-i 42.2 i3 204.7 58.7 73 262.4 75.2 34'32.7 09.4 94 90.4 25.9 54 148.0 42.4 i4 205.7 59.0 74 363.4 75.5 35.33.6 09.6 95 91.3 26.2 55 149.0 42.7 i5 206.7 59.3 75 264.3 75.8 36 134.6 09.9 96 92.3 26.5 56 i5o.o 43.0 16 207.6 59.5 76 265.3 76.1 37 35.6 10.2 97 93.2 26.7 57 i5o.9 43.3 17 208.6 59.8 77 266.3 76.4 38 36.5 10.5 98 94.2 27.0 58 i5i .9 43.6 iS 209.6 60.1 78 267.2 76.6 3q 37.5 10.7 99 95.2 27.3 59 i52.8 43.8 19 210.5 60.4 79 268.2 76.9 40 38.5 II .0 100 96.1 27.6 60 i53.8 44.1 20 211 .5 60.6 80 269.2 77.2 4i 39.4 II. 3 lOI 97.1 27.8 161 154.8 44.4 221 212.4 60.9 281 270. 1 77.5 42 40.4 II. 6 02 98.0 28.1 62 155.7 44.7 22 2i3.4 61 .2 82 271 .1 77-7 43 4i.3 II. 9 o3 99.0 28.4 63 i56.7 44.9 23 214.4 61.5 83 272.0 78.0 44 42.3 12. 1 04 100. 28.7 64 157.6 45.2 24 2i5.3 61.7 84 273.0 78.3 45 43.3 12.4 o5 100.9 28.9 65 i58.6 45.5 25 216.3 62.0 85 274.0 78.6 46 44.2 12.7 06 lOI .9 29.2 66 159.6 45.8 26 217.2 62.3 86 274.9 78.8 47 45.2 i3.o 07 102.9 29.5 67 160.5 46. 27 218.2 62.6 87 275.9 79.1 48 46.1 l3.2 08 io3.8 29.8 68 161. 5 46.3 28 219.2 62.8 88 276.8 79-4 49 47.1 i3.5 09 104.8 3o.o 69 162.5 46.6 29 220.1 63.1 89 277.8 79-7 5o 48.1 i3.8 10 io5.7 3o.3 70 i63.4 46.9 3o 221 .1 63.4 90 278.8 79-9 5i 49.0 i4.i III 106.7 3o.6 171 164.4 47-1 23 I 222.1 63.7 291 279.7 80.2 52 5o.o i4.3 12 107.7 3o.9 72 i65.3 47.4 32 223.0 63.9 92 280.7 80.5 53 50.9 14.6 i3 108.6 3i.i 73 166.3 47.7 33 224.0 64.2 93 2S1.6 80.8 54 31.9 14.9 i4 109.6 3i.4 74 167.3 48. 34 224.9 64.5 94 282.6 81.0 55 52.9 l5.2 i5 110.5 31.7 75 168.2 48.2 35 225.9 64.8 95 283.6 81.3 56 53.8 i5.4 16 III. 5 32.0 76 169.2 48.5 36 226.9 65.1 96 284.5 81.6 57 54.8 i5.7 17 112.5 32.2 77 170.1 48.8 37 227.8 65.3 97 285.5 81.9 58 55.8 16.0 18 ii3.4 32.5 78 171. 1 49.1 38 228.8 65.6 98 286.5 82.1 5q 56.7 16.3 19 114.4 32.8 79 172. 1 49-3 39 229.7 65.9 99 287.4 82.4 60 67.7 16.5 20 ii5.4 33.1 Lat. 80 173.0 49.6 40 23o.7 66.2 3oo 288.4 82.7 Uist. Dop. I,at. Dist. Dep. Dist. Dep. Lat. | Dist. Dep. Lat. Dist.j Dep. Lat. For 74 Degrees. TABLE IL 1 Page 33 Difference of Latitude and Depart ure for 17 Degrees. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 01 .0 00.3 61 58.3 17.8 121 115.7 35.4 181 173. 1 52.9 241 23o.5 70.5 2 01 .9 00.6 62 59.3 18.1 22 116.7 35.7 82 174.0 53.2 42 231.4 70.8 3 02.9 00.9 63 60.2 18.4 23 117. 6 3b. 83 175.0 53.5 4-^ 232.4 71.0 4 o3.8 01 .2 64 61 .2 18.7 24 118.6 36.3 84 176.0 53.8 44 233.3 71.3 5 04.8 01.5 65 62.2 19.0 25 119.5 36.5 85 176.9 54.1 45 23^1.3 71.6 6 00.7 01.8 66 63.1 19.3 26 120.5 36.8 86 177-9 54.4 46 235.3 71.9 7 06.7 02.0 67 64.1 19.6 27 121 .5 37.1 87 178.8 54.7 47 236.2 72.2 8 07.7 02.3 b8 65.0 19.9 28 122.4 37.4 88 179.8 55.0 48 237.2 72.5 9 08.6 02.6 69 66.0 20.2 29 123.4 37.7 89 180.7 55.3 49 238.1 72.8 10 09.6 02.9 70 66.9 20.5 3o 124.3 38. 90 181.7 55.6 5o 239.1 73.1 II 10.5 o3.2 71 67.9 20.8 i3i 125.3 38.3 191 182.7 55.8 25l 240.0 73.4 12 II. 5 o3.5 72 68.9 21 .1 32 126.2 38.6 92 i83.6 56.1 52 241 .0 73.7 i3 12.4 o3.8 73 69.8 21.3 33 127.2 38.9 93 184.6 56.4 53 241.9 74.0 i4 i3.4 04.1 74 70.8 21 .6 34 128. 1 39.2 94 iS5.5 56.7 54 242.9 74.3 i5 i4.3 04.4 76 71-7 21.9 35 129.1 39.5 95 186.5 57.0 55 243.9 74.6 i6 i5.3 04.7 76 72.7 22.2 36 i3o.i 39.8 96 187.4 57.3 56 244.8 74.8 17 16.3 o5.o 77 73.6 22.5 37 i3i .0 4o.i 97 188.4 57.6 57 245.8 75.1 i8 17.2 o5.3 7» 74.6 22.8 38 l32.0 40.3 98 189.3 57.9 58 246.7 75.4 19 18.2 o5.6 79 75.5 23.1 39 132.9 40.6 99 190.3 58.2 59 247-7 75.7 20 19.1 o5.8 80 76.5 23.4 4o 133.9 40.9 4i .2 200 201 191 .3 192.2 58.5 58.8 60 248.6 76.0 21 20. 1 06.1 81 77.5 23.7 i4i i34.8 261 249.6 76.3 22 21 .0 06.4 82 78.4 24.0 42 i35.8 4i.5 02 193.2 59.1 62 25o.6 76.6 23 22.0 06.7 83 79-4 24.3 43 i36.8 4i.8 o3 194.1 59.4 63 251.5 76.9 24 23.0 07.0 84 80.3 24.6 44 137.7 42.1 04 195. 1 59.6 64 252.5 77.2 25 23.9 07.3 85 81.3 24.9 45 i38.7 42.4 o5 196.0 59.9 65 253.4 77.5 26 24.9 07.6 86 82.2 25.1 46 139.6 42.7 06 197.0 60.2 66 254.4 77.8 27 25.8 07.9 87 83.2 25.4 47 i4o.6 43.0 07 198.0 60.5 67 255.3 78. 1 28 26.8 08.2 88 84.2 25.7 48 i4i.5 43.3 08 198.9 60.8 68 256.3 78.4 29 27-7 08.5 89 85.1 26.0 49 142.5 43.6 09 199.9 61.1 69 257.2 78.6 3o 28.7 08.8 90 86.1 26.3 5o 143.4 43.9 10 200.8 61.4 70 258.2 78.9 3i 29.6 09.1 91 87.0 26.6 i5i 144.4 44.1 211 201.8 61.7 271 259.2 79.2 32 3o.6 09.4 92 88.0 26.9 52 145.4 44.4 12 202.7 62.0 72 260.1 79.5 33 3i.6 09.6 93 88.9 27.2 53 i46.3 44.1 i3 2o3.7 62.3 73 261. 1 79.8 34 32.5 09.9 94 89.9 27.5 54 147.3 45.0 i4 2o4.6 62.6 74 262.0 80.1 35 33.5 10.2 95 90.8 27.8 55 148.2 45.3 i5 2o5.6 62.9 75 263.0 80.4 3G 34.4 10.5 96 91.8 28.1 56 149.2 46.6 16 206.6 63.2 76 263.9 80.7 37 35.4 10.8 97 92.8 28.4 57 1 5o . I 45.9 17 207.5 63.4 77 264.9 81.0 38 36.3 II .1 98 93.7 28.7 58 i5i.i 46.2 18 208.5 63.7 78 265.9 81.3 39 37.3 II. 4 99 94-7 28.9 59 l52.1 46.5 19 209.4 64.0 79 266.8 81.6 4o 38.3 II. 7 100 95.6 29.2 60 i53.o 4b. 8 20 210.4 64.3 80 267.8 81.9 4i 39.2 12.0 101 96.6 29.5 161 1 54.0 47.1 221 211 .3 64.6 281 268.7 82.2 42 40.2 12.3 02 97.5 29.8 62 154.9 47.4 22 212.3 64.9 82 269.7 82.4 43 4i.i 12.6 o3 98.5 3o.i 63 155.9 47.7 23 2i3.3 65.2 83 270.6 82.7 44 42.1 12.9 04 99.5 3o.4 64 i56.8 47-9 24 214.2 65.5 84 271 .6 83.0 45 43.0 l3.2 o5 100.4 30.7 65 157.8 48.2 25 2l5.2 65.8 85 272.5 83.3 46 44.0 i3.4 06 101.4 3i .0 6-6 i58.7 48.5 26 216. I 66.1 86 273.5 83.6 47 44.9 i3.7 07 102.3 3i.3 67 159.7 43.8 27 217. I 66.4 87 274.5 83.9 48 45.9 14.0 08 io3.3 3i.6 68 160.7 49.1 28 218.0 66.7 88 275.4 84.2 49 46.9 14.3 09 104.2 3i .9 69 161.6 49-4 29 219.0 67.0 89 276.4 84.5 5o 47.B i4.b 10 io5 2 32.2 70 162.6 49.7 3o 220.0 67.2 90 277.3 84.8 5i 48.8 14.9 III ic6. 1 32.5 171 163.5 5o.o 23l 220.9 67.5 291 278.3 85.1 52 49-7 l5.2 12 107. 1 32.7 72 164.5 5o.3 32 221 .9 67.8 92 279.2 85.4 53 5o.7 i5.5 i3 108.1 33.0 73 i65.4 5o.6 33 222.8 68.1 93 280.2 85.7 54 5i.6 [5.8 i4 109.0 33.3 74 166.4 50.9 34 223.8 68.4 94 281.2 86.0 53 52. f) 16.1 i5 IIO.O 33.6 75 167.4 5l.2 35 224.7 68.7 95 282.1 86.2 56 53.6 16.4 16 110.9 33.9 76 168.3 5i.5 36 225.7 69.0 g6 283.1 86.5 b7 54.5 16.7 17 III .9 34.2 77 169.3 5i.7 37 226.6 69.3 97 284.0 86.8 d8 t)5.3 17.0 18 112.8 34.5 78 170.2 52.0 38 227.6 69.6 98 285.0 87.1 59 5b. 4 17.2 19 ii3.8 34.8 79 171 .2 52.3 39 228.6 69.9 99 285.9 87.4 bo I)ist. 57.4 17.5 20 114.8 35.1 80 Disl. 172. 1 52.6 40 2?q.5 70.2 3oo 286.9 87.7 Dep. Lr.t. Disl. Dep. Lat. Dop. Lat. Disl. ■Dep. Lat. Dist. Dep. Lat. [1 ''or 73 Degrees. I";i;,'e :U TABLE IL Dilference of Latitude and Departure for 13 Degrees. Dist Lai. Oep. 00.3 Dlsl. Lat. D-.p. Dist.] Lat. Dep. Disl. Lat. Dep. Dist. Lat. 1 D,.p. I 01 .0 61 58. 18.9 121 1 1 15. 1 37.4 181 172. 1 55.9 241 229.2 74.5 2 01.9 00.6 b2 59.0 19.2 22 116.0 37.7 82 173. 1 56.2 42 23o.2 74.8 d 02.9 00.9 M 59.9 i9.b 23 117. 38. 83 174.0 56.6 43 23l .1 75.1 4 o3.8 01 .2 b4 60.9 19.8 24 117. 9 38.3 84 175.0 56.9 44 232.1 75.4 i) o4.8 01. b bb 61.8 20.1 2b 118. 9 38.6 85 175.9 57.2 45 233.0 75.7 b Ob. 7 01 .9 bb 62.8 20.4 26 119.8 38.9 86 176.9 57.5 46 234.0 76.0 7 06.7 02.2 b7 63.7 20.7 27 120.4 39.2 87 177.8 57.8 47 234.9 76.3 8 07.6 02. b b8 64.7 21 .0 28 121 .7 39.6 88 178.8 58.1 48 235.9 76.6 9 08.6 02.8 b9 65.6 21.3 29 122.7 39.9 89 179-7 58.4 49 236.8 76.9 lO 09.5 o3. 1 70 66.6 21. b 21.9 3o 123.6 4o.2 90 191 180.7 181. 7 58. 7 59.0 5o 237.8 77-3 II 10.5 o3.4 71 67.5 i3i 124.6 40.5 25l 238.7 77-6 12 II. 4 o3.7 72 68.5 22.2 32 125.5 4o.8 92 182.6 59.3 52 239.7 77-9 i3 12.4 04.0 7'i 69.4 22.6 33 126.5 4i.i 93 i83.6 59.6 53 240.6 78.;. i4 i3.3 04.3 74 70.4 22.9 34 127.4 41.4 94 184.5 59-? 54 241.6 78.5 15 14.3 04. b 7b 71.3 23.2 35 128.4 41.7 95 185.5 60.3 55 242.5 78.8 i6 lb. 2 04.9 76 72.3 23.5 36 129.3 42.0 96 186.4 60.6 56 243.5 79.1 17 16.2 ob.3 77 73.2 23.8 37 i3o.3 42.3 97 187.4 60.9 57 244.4 79-4 i8 17. 1 ob.b 78 74.2 24.1 38 i3i .2 42.6 98 188.3 61 .2 58 245.4 79-7 19 18. 1 05.9 79 7b. I 24.4 39 \3i.i 43.0 99 189.3 61.5 59 246.3 80.0 20 19.0 06.2 06.5 80 7b. I 24.7 40 i33.i 43.3 200 190.2 61.8 60 247-3 80.3 21 20.0 81 77.0 25.0 i4i i34.i 43.6 201 191 .2 62.1 261 248.2 80.7 22 20.9 06.8 82 78.0 2b. 3 42 i35.i 43.9 02 192. 1 62.4 62 249.2 81.0 23 21 .9 07.1 83 78.9 23.6 43 1 36.0 44.2 o3 193.1 62.7 63 25o.i 81.3 24 22.6 07.4 84 79-9 26.0 44 137.0 44.5 04 194.0 63. H 25l . I 81.6 2b 23.8 07.7 8b 80.8 26.3 45 137.9 44.8 o5 195.0 63.3 65 252.0 81.9 2b 24.7 08.0 86 81.8 26.6 46 13S.9 45.1 06 195.9 63.7 66 253.0 82.2 27 2b. 7 08.3 87 82.7 26.9 4i 139.8 45.4 07 196.9 64.0 67 253.9 82.5 28 26.6 08.7 88 83.7 27.2 48 140.8 4b. 7 08 197-8 64.3 68 254.9 82.8 29 27.6 09.0 89 84.6 27.5 49 141.7 46.0 09 198.8 64.5 69 255.8 83.1 Jo 28. b 09.3 90 85.6 27.8 5o 142.7 46.4 10 199.7 64-9 70 256.8 83.4 3i 29.5 09.6 91 86.5 28.1 i5i 143.6 46.7 211 200.7 65.2 271 257.7 83.7 32 3o.4 09.9 92 87.5 28.4 52 144.6 47-0 12 201 .6 65.5 72 258.7 84.1 :i6 3i.4 10.2 93 88.4 28.7 53 145.5 47.3 i3 202.6 65.8 73 259.6 84.4 34 32.3 10.5 94 89.4 29.0 54 146.5 47-6 i4 2o3.5 66.1 74 260.6 84.7 3!) dd.d 10.8 9b 90.4 29-4 55 147-4 47-9 i5 204.5 66.4 75 261.5 85.0 3b 34.2 II .1 9b 91.3 29.7 56 148.4 48.2 16 2o5.4 66.7 76 262.5 S5.3 ^7 3b. 2 II. 4 97 92.3 3o.o 57 149-3 48.5 17 206.4 67.1 77 263.4 85.6 38 3b. I II. 7 98 93.2 3o.3 58 i5o.3 48.8 18 207.3 67-4 78 264.4 85.9 39 37.1 12. 1 99 94.2 3o.6 59 i5i .2 49. 1 19 208.3 67.7 79 265.3 86.2 40 38. 12.4 100 9b.; 30.9 3l.2 6(j l52.2 49-4 20 209.2 68.0 68.3 80 266.3 86.5 86.3 4i 39.0 12.7 lOI 96.1 161 153.1 49.8 221 210.2 281 267 .2 42 39.9 i3.o 02 97.0 3i.5 62 i54.i 5o.i 22 211.1 68.6 82 268.2 87.1 4i 40.9 i3.3 o3 98.0 3i.8 63 i55.o 5o.4 23 212.1 68. 9 83 269.1 87.5 44 4i.8 i3.6 04 98.9 32.1 64 i56.o 50.7 24 2l3.0 69.2 84 270. 1 87.8 4b 42.8 i3.9 ob 99.9 32.4 65 i56.9 5i.o 25 214.0 69.5 85 271 .1 88. 1 4b 43.7 14.2 06 100.8 32.8 66 .57.Q 5i.3 26 214.9 69.8 86 272.0 88.4 47 44.7 14.5 07 101.8 33.1 67 i5S.8 5i.6 27 215.9 70.1 87 273.0 88.7 48 4b. 7 14.8 08 102.7 33.4 68 159.8 5i.9 28 216.8 70.5 88 273.9 89.0 49 4b. b 16. 1 09 103.7 33.7 69 160.7 52.2 29 217.8 70.8 89 274.9 89.3 bo 5i 47.b ib.b 10 104.6 34.0 70 161 .7 52.5 52.8 3o 23 I 218.7 71. 1 90 291 275.8 276 8 89.6 48.5 i5.8 II I io5.6 34.3 171 162.6 219.7 71.4 89.9 ba 49-i) lb. I 12 106.5 34.6 72 i63.6 53.2 32 220.6 71.7 93 277-7 90.2 bd 5o.4 lb. 4 i3 107.5 34.9 73 164.5 53.5 33 221 .6 72.0 93 278.7 90.5 64 bi.4 lb. 7 i4 108.4 35.2 74 i65.5 53.8 34 222.5 72.3 94 279.6 90.9 bb b2.3 17.0 lb 109.4 35.5 75 166.4 -M-i 35 223.5 72.6 95 280.6 91.2 bb b3.3 .7.3 lb no. 3 35.8 76 167.4 54.4 36 224.4 72.9 96 281.5 91.5 b7 b4.2 .7.b 17 III .3 36.2 77 168.3 54.7 37 225.4 73.2 97 282.5 91.8 b8 bb.2 17.9 18 1 12.2 36.5 78 169.3 55.0 38 226.4 73.5 98 283.4 92.1 b9 b6.i 18.2 •9 I l3.2 36.8 79 170.2 55.3 39 227.3 73.9 99 284.4 92.4 bo b7.i 18. b 20 114.1 37.1 80 171 .2 55.6 4o Dist. 228.3 Dep. 74-2 Lat. 3 00 285.3 92.7 Dist. De,,. Lat. Dist Do p. Lai. Dist. Dop. Lat. Dist. Dep. Lat. [1 ^or 72 Degrees. TABLE II. [Page 35 Difference of Lat tude and Departure for 19 Degrees. Dist Lat. Dep. 00.3 Dist. Lat. Do p. Dist. Lat. Dcp. 39.4 Dist. Lat. Dep. Dist. Lat. Dep I 00.9 61 57-7 19.9 121 114.4 181 171. 1 58.9 241 227.9 78.5 2 01.9 00.7 62 58.6 20.2 22 ii5.4 39.7 82 172. 1 59.3 42 228.8 78.8 3 02.8 01 .0 63 59.6 20. D 23 116. 3 4o.o 83 173.0 59.6 43 229.8 79.1 4 o3.8 01 .3 64 60.5 20.8 24 117. 2 40.4 84 174.0 59.9 AA 230.7 79-4 5 04.7 01 .6 65 61.5 21 .2 25 118.2 40.7 85 174.9 60.2 45 23l .7 79.8 6 o5.7 02.0 66 62.4 21.5 26 119. 1 4i .0 86 175.9 60.6 46 232.6 80.1 7 06.6 02.3 67 63.3 21.8 27 120. 1 4i.3 87 176.8 60.9 47 233.5 80.4 8 07.6 02.6 68 64.3 22.1 28 121 .0 41.7 88 177.8 61 .2 48 234.5 80.7 9 08.5 02.9 69 65.2 22.5 29 122.0 42.0 89 178.7 61.5 49 235.4 81. 1 10 09.5 o3.3 70 66.2 22.8 3o 122.9 42.3 90 179.6 61 .9 5o 236.4 81.4 81.7 II 10.4 o3.6 71 67.1 23.1 i3i 123.9 42.6 191 180.6 62.2 25l 237.3 12 II. 3 03.9 72 68.1 23.4 32 124.8 43.0 92 181.5 62.5 52 238.3 82.0 i3 12.3 o4.2 73 69.0 23.8 33 125.8 43.3 93 182.5 62.8 53 239.2 82.4 i4 l3.2 04.6 74 70.0 24.1 M 126.7 43.6 94 i83.4 63.2 54 240.2 82.7 i5 l4.2 04.9 75 70.9 24.4 35 127.6 44.0 95 184.4 63.5 55 241 .1 53.0 i6 i5.i 05.2 76 71.9 24.7 36 128.6 AA.'i 96 i85.3 63.8 56 242.1 83.3 17 16. 1 o5.5 77 72.8 25.1 37 129.5 44.6 97 186.3 64.1 57 243.0 83.7 i8 17.0 05.9 78 73.8 25.4 38 i3o.5 44-9 98 187.2 64.5 58 243.9 84.0 '9 18.0 06.2 79 74.7 25.7 39 i3i.4 45.3 99 188.2 64.8 59 244.9 84.3 20 18.9 06.5 80 75.6 26.0 40 i32.4 45.6 200 189. 1 65.1 60 245.8 84.6 85. 21 19.9 06.8 81 76.6 26.4 i4i i33.3 45.9 201 190.0 65.4 261 246.8 22 20.8 07.2 82 77.5 26.7 42 134.3 46.2 02 191 .0 65.8 62 247.7 85.3 23 21.7 07.5 83 78.5 27.0 A'i i35.2 46.6 OJ 191. 9 66.1 63 248.7 85.6 24 22.7 07.8 84 79-4 27.3 AA i36.2 46.9 o4 192.9 66.4 64 249.6 86.0 25 23.6 08.1 85 80.4 27.7 45 137. 1 47-2 o5 193.8 66.7 65 25o.6 86.3 26 24.6 08.5 86 81.3 28.0 46 i38.o 47.5 06 194.8 67.1 66 251.5 86.6 27 25.5 08.8 87 82.3 28.3 47 139.0 47-9 07 195.7 67.4 67 252.5 86.9 38 26.5 09.: 88 83.2 28.7 48 139.9 48.2 08 196.7 67.7 68 253.4 87.3 29 27.4 09.4 89 84.2 29.0 49 140.9 48.5 09 197.6 68.0 69 254.3 87.6 3o 28.4 09.8 90 85.1 29.3 5o i4i.8 48.8 10 198.6 68.4 70 255.3 87.9 88.2 3i 29.3 10. 1 9' 86.0 29.6 i5i 142.8 49.2 211 199.5 68.7 271 256.2 32 3o.3 10.4 92 87.0 3o.o 52 143.7 49.5 12 200.4 69.0 72 257.2 88.6 33 3l.2 10.7 93 87.9 3o.3 53 144.7 49-8 i3 201 .4 69.3 73 258.1 88.9 34 32.1 II .1 94 88.9 3o.6 54 145.6 5o.i i4 202.3 69.7 74 259.1 89.2 35 33.1 II. 4 95 89. 8 3o.9 55 1.46.6 5o.5 i5 2o3.3 70.0 75 260.0 89.5 36 34.0 II. 7 96 90.8 3i.3 56 147-5 5o.8 16 204.2 70.3 76 261.0 89.9 3? 35.0 12.0 97 91.7 3i.6 57 148.4 5i.i 17 205.2 70.6 77 261.9 90.2 38 35.9 12.4 98 92.7 3i .9 58 149.4 5i.4 18 206. 1 71.0 78 262.9 90.5 39 36 9 12.7 99 93.6 32.2 59 i5o.3 5i.8 19 207.1 71.3 79 263.8 90.8 40 4i 37.8 i3.o 100 94.6 32.6 60 i5i.3 52.1 20 208.0 71.6 80 264.7 91.2 38.8 i3.3 lOI 95.5 32.9 161 l52.2 52.4 221 209.0 72.0 281 265.7 91.5 42 39.7 i3.7 02 96.4 33.2 62 i53.2 52.7 22 209.9 72.3 82 266.6 91.8 43 40.7 i4.o o3 97-4 33.5 63 i54.i 53.1 23 210.9 72.6 83 267.6 92.1 ^i 4i.6 14.3 o4 98.3 33.9 64 i55.i 53.4 24 211. 8 72.9 84 268.5 92.5 45 42.5 14.7 o5 99.3 34.2 65 i56.o 53.7 25 212.7 73.3 85 269.5 92.8 46 43.5 i5.o 06 100.2 34.5 66 157.0 54.0 26 213.7 73.6 86 270.4 93.1 ^1 44.4 i5.3 07 101 .2 34.8 67 157.9 54.4 27 214.6 73.9 87 271 .4 93.4 48 45.4 i5.6 08 102. 1 35.2 68 i58.8 54.7 28 2i5.6 74.2 88 272.3 93.8 49 46.3 16.0 09 io3.i 35.5 69 159.8 55.0 29 216.5 74.6 89 273.3 94.1 5o 47.3 16.3 10 104.0 35.8 70 160.7 55.3 3o 217.5 74.9 90 291 274.2 275.1 94.4 94-7 5i 48.2 16.6 III loS.o 36.1 171 161 .7 55.7 23l 218.4 75.2 52 49.2 16.9 12 105.9 36.5 72 162.6 56.0 32 219.4 75.5 92 276.1 95.1 53 5o.i 17.3 i3 106.8 36.8 73 i63.6 56.3 33 220.3 75.9 93 277.0 95.4 54 5i.i 17.6 i4 107.8 37.1 74 164.5 56.6 34 221 .3 76.2 94 278.0 95.7 55 52.0 17.9 i5 108.7 37.4 75 i65.5 57.0 1 35 222.2 76.5 95 278.9 96.0 56 52.9 I«.2 16 109.7 37.8 76 166.4 57.3 36 223.1 76.8 96 279.9 96.4 ■^1 5J.9 18.6 17 no. 6 38.1 77 167.4 57.6 37 224.1 77.2 97 280.8 96.7 58 54.8 18.9 18 II 1. 6 38.4 78 168.3 58.0 38 225.0 77.5 98 281.8 97.0 59 55.8 19.2 •9 112. 5 38.7 79 169.2 58.3 39 226.0 77.8 99 282.7 97.3 ho 56.7 19.5 20 ii3.5 39.1 80 170.2 58.6 40 226.9 78.1 3oo 283.7 97-7 Disi. ])e|). I.at. nisi. Dcp. Lat. Dist. Dcp. Lat. Dist. Dcp. Lat. Dist. Dep. Lat. [For 71 Degrees. I'age 3G] TABLE IL Difference of Latitude and Departure for 20 Degrees. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. D.st. Lat. Dep. Dist. Lat. Dep. I 00.9 00.3 61 57.3 20.9 121 ii3.7 4i.4 181 170.1 1 61.9 241 226.5 82.4 2 01 .9 00.7 62 58.3 21 .2 22 114.6 41.7 82 171. 1 62.2 42 227.4 82.8 3 02.8 01 .0 63 59.2 21.5 23 ii5.6 42.1 83 172.0 1 62 6 43 '228.3 83.1 4 o3.8 01 .4 64 60.1 21 .9 24 116. 5 42.4 84 172.9 62.9 AA 229.3 83.5 5 04.7 01.7 65 61. 1 22.2 25 117. 5 42.8 85 173.8 63 3 45 23o.2 83.8 6 o5.6 02.1 66 62.0 22.6 26 118. 4 43.1 86 174.8 63.6 46 23l .2 84.1 7 06.6 02.4 67 63.0 22.9 27 1 19.3 43.4 87 175.7 64.0 47 232.1 84.5 8, f.7.5 02.7 68 63.9 23.3 28 120.3 43.8 88 176.7 64.3 48 233.0 84.8 P 08.5 o3.i 69 64.8 23.6 29 121 .2 44.1 89 177.6 64.6 49 234.0 85.2 10 09.4 o3.4 70 65.8 23.9 3o 122.2 U.b 90 178.5 65.0 65.3 5o 234.9 85.5 II 10.3 o3.8 71 66.7 24.3 i3i 123. 1 44.8 191 179.5 25l 235.9 85.8 12 II. 3 04.1 72 67.7 24.6 32 124.0 45.1 92 180.4 65.7 52 236.8 86.2 i3 12.2 04.4 73 68.6 25.0 33 125.0 45.5 93 181.4 66.0 53 237.7 86.5 14 l3.2 o4.8 74 69.5 25.3 34 125.9 45.8 94 182.3 66.4 54 238.7 86.9 i5 i4.i o5.i 75 70. D 25.7 35 126.9 46.2 95 i83.2 66.7 55 239.6 87.2 16 i5.o o5.5 76 71.4 26.0 36 127.8 46.5 96 184.2 67.0 56 240.6 87.6 17 16. G o5.8 .77 72.4 26.3 37 128.7 46.9 97 i85.i 67.4 57 241.5 87.9 18 16.9 06.2 78 73.3 26.7 38 129.7 47.2 98 186.1 67.7 58 242.4 88.2 19 17.9 06.5 79 74.2 27.0 39 i3o.6 47-5 99 187.0 68.1 59 243.4 88.6 20 18.8 06.8 80 75.2 27.4 4o i3i.6 47-9 200 187.9 68.4 60 244.3 88.9 21 19.7 07.2 81 76.1 27.7 i4i i32.5 48.2 201 188.9 68.7 261 245.3 89.3 22 20.7 07.5 82 77.1 28.0 42 i33.4 48.6 02 189.8 69.1 62 246.2 89.6 23 21.6 07.9 83 78.0 28.4 43 134.4 48.9 o3 190.8 69.4 63 247-1 90.0 24 22.6 08.2 84 78.9 28.7 ^A i35.3 49-3 04 191.7 69.8 64 24s. I 90.3 25 23.5 08.6 85 79-9 29. 1 45 i36.3 49.b o5 192.6 70.1 65 249.0 90.6 26 24.4 08.9 86 80.8 29.4 46 137.2 49.9 06 193.6 70.5 66 25o.o 91.0 27 25.4 09.2 87 81.8 29.8 47 i38.i 5o.3 07 194.5 70.8 67 250.9 91.3 28 26.3 09.6 88 82.7 3o.i 48 139. 1 5o.6 08 195.5 71.1 68 25i.8 91.7 29 27.3 09.9 89 83.6 3o.4 49 i4o.o 5i.o 09 196.4 71.5 69 252.8 92.0 3o 28.2 10.3 90 84.6 3o.8 5o i4i .0 5i.3 10 197.3 71.8 70 253.7 92.3 3x 29.1 10.6 91 85.5 3i.i i5i i4i.9 5i.6 211 198.3 72.2 271 254.7 92.7 32 3o.i 10.9 92 86.5 3i.5 52 142.8 52.0 12 199.2 72.5 72 255.6 93.0 33 3i .0 II. 3 93 87.4 3i.8 53 143.8 52.3 i3 200.2 72.9 73 256.5 93.4 34 3i .9 II. 6 94 88.3 32.1 54 144.7 52.7 i4 201 . 1 73.2 74 257.5 93.7 35 32.9 12.0 95 89.3 32.5 55 145.7 53.0 i5 202.0 73.5 75 258.4 94.1 36 33.8 12.3 96 90.2 32.8 56 146.6 53.4 16 203.0 73.9 76 259.4 94.4 37 34.8 12.7 97 91 .2 33.2 57 147.5 53.7 17 203.9 74.2 77 260.3 94-7 38 35.7 i3.o 98 92.1 33.5 58 148.5 54.0 18 204.9 74.6 78 261.2 95.1 39 36.6 i3.3 99 93.0 33.9 59 149.4 54.4 19 2o5.8 74.9 79 262.2 95.4 40 37.6 i3.7 100 94.0 34.2 60 i5o.4 54.7 55.1 20 206.7 75.2 80 263.1 95.8 4i 38.5 i4.o lOI 94.9 34.5 161 i5i.3 221 207.7 75.6 281 264.1 96.1 42 39.5 14.4 02 95.8 34.9 62 l52.2 55.4 22 208.6 75.9 82 265.0 96.4 43 40.4 14.7 o3 96.8 35.2 63 153.2 55.7 23 209.6 76.3 83 265.9 96.8 M 4r.3 i5.o o4 97-7 35.6 64 I54.I 56.1 24 210.5 76.6 84 266.9 97.1 45 42.3 i5.4 o5 98.7 35.9 65 i55.o 56.4 25 211 .4 77.0 8b 267.8 97.b 46 43.2 i5.7 06 99.6 36.3 66 i56.o 56.8 26 212.4 77.3 86 268.8 97.8 47 AA.'i 16. 1 07 100.5 36.6 67 i56.9 57.1 27 2i3.3 77.6 87 269.7 98.2 48 45.1 16.4 08 loi .5 36.9 68 157.9 57.5 28 214.2 78.0 88 270.6 98.5 49 46. 16.8 09 102.4 37.3 69 i58.8 57.8 29 2l5.2 78.3 89 271 .6 98.8 5o 47-0 17.1 10 io3.4 37.6 70 159.7 58.1 3o 216.1 78.7 90 272.5 99.2 5i 47-9 17.4 III 104.3 38. 171 160.7 58.5 23l 217.1 79.0 291 273.5 99.5 52 48.9 17.8 12 io5.2 38.3 72 161. 6 58.8 32 218.0 79.3 92 274.4 99-9 53 49-8 18. 1 i3 106.2 38.6 73 162.6 59.2 ■Si 21S.9 79-7 93 275.3 100.2 54 50.7 1S.5 i4 107. 1 39.0 74 i63.5 59.5 34 219.9 80.0 94 276.3 100.6 55 5i.7 18.8 i5 108. 1 39.3 75 164.4 59.9 35 220.8 80.4 95 277.2 100.9 56 52.6 19.2 16 109.0 39.7 76 i65.4 60.2 36 221.8 80.7 96 278.1 101.2 57 53.6 iq.5 17 109.9 4o.o 77 166.3 60.5 37 222.7 81.1 97 279.1 101.6 58 54.5 19.8 18 1 10.9 40.4 78 167.3 60.9 38 223.6 81.4 98 280.0 101.9 59 55.4 20.2 19 III. 8 40.7 79 168.2 61.2 39 224.6 81.7 99 281.0 102.3 60 56.4 20.5 20 112. 8 4i .0 80 169. 1 61.6 40 Dist. 225.5 Dep. 82.1 Lat. 3oo 281.9 102.6 Dist Dop. Lai. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. For 70 Degress. TABLE IL [Page 37 DilTercnce of Latitude and Departure for 21 Degrees. Uist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 00.9 00.4 61 56.9 21 .9 121 ii3.o 43.4 181 169.0 64.9 24 1 225.0 86.4 2 01 .9 00.7 62 57.9 22.2 22 113.9 43.7 82 169.9 65.2 42 225.9 86.7 3 02.8 01. 1 63 58.8 22.6 23 114.8 44.1 83 170.8 65.6 4^ 226.9 87.1 4 03.7 or. 4 64 59.7 22.9 24 ii5.8 44.4 84 171.8 65.9 44 227.8 87-4 5 04.7 01.8 65 60.7 23.3 25 116. 7 44.8 85 172.7 bb.3 45 228.7 87.8 f) o5.6 02.2 66 6i.6 23.7 26 117. 6 45.2 86 173.6 66.7 46 229.7 88.2 7 06.5 02.5 67 62.5 24.0 27 118.6 45.5 87 174.6 67.0 47 23o.6 88.5 8 07.5 02.9 68 63.5 24.4 28 119. 5 45.9 88 175.5 67.4 48 23i.5 88.9 9 08.4 03.2 69 64.4 24.7 29 120.4 46.2 89 176.4 67.7 49 232.5 89.2 lO 1 1 09.3 o3.6 70 65.4 25.1 25.4 3o 121 .4 46.6 46.9 90 177.4 68.1 5o 233.4 89.6 10.3 03.9 71 66.3 i3i 122.3 191 178.3 68.4 25l 234.3 90.0 12 tr .2 04.3 72 67.2 25.8 32 123.2 47.3 92 179.2 68.8 52 235.3 90.3 1 3 12.1 04.7 73 68.2 26.2 33 124.2 47-7 93 180.2 69.2 53 236 2 90.7 i4 1 3. 1 o5.o 74 69.1 26.5 34 125. I 48.0 94 181.1 69.5 54 237. 1 91.0 i5 i4.o o5.4 75 70.0 26.9 35 126.0 48.4 95 182.0 69.9 55 238.1 91.4 ifj 14.9 o5.7 76 71 .0 27.2 36 127.0 48.7 96 i83.o 70.2 5b 239.0 91.7 17 .5.9 06.1 77 71.9 27.6 37 127.9 49.1 97 183.9 70.6 57 239.9 92.1 i8 16.8 06.5 78 72.8 28.0 38 128.8 49.5 98 184.8 71.0 58 240.9 92.5 "9 17-7 06.8 79 73.8 28.3 39 129.8 49-8 99 i85.8 71.3 59 241.8 92.8 20 18.7 07.2 80 74.7 28.7 4o i3o.7 5o.2 200 186.7 71.7 60 242.7 93.2 21 19.6 07.5 81 75.6 29.0 i4i i3i.6 5o.5 201 187.6 72.0 261 243.7 93.5 22 20.5 07.9 82 76.6 29.4 42 i32.6 30.9 02 188.6 72.4 62 244.6 93.9 23 21 .5 08.2 83 77.5 29.7 43 i33.5 5i .2 o3 189.5 72.7 63 24X5 94.3 24 22.4 08.6 84 78.4 3o. I 44 134.4 5i.6 04 190.5 73.1 64 246.5 94.6 25 23.3 09.0 85 79-4 3o.5 45 i35.4 52.0 o5 191 .4 73.5 65 247-4 q5.o 26 24.3 09.3 86 80.3 3o.8 46 i36.3 52.3 o5 192.3 73.8 66 248.3 95.3 27 25.2 09.7 87 81.2 3l.2 47 137.2 52.7 07 193.3 74.2 67 249.3 93.7 28 26.1 10. 88 82.2 3i.5 48 i38.2 53.0 08 194.2 74.5 68 25o.2 96.0 29 27.1 10.4 89 83.1 3i.9 49 139. 1 53.4 09 195.1 74-9 69 25 1. I 96.4 JO 28.0 10.8 90 84.0 32.3 5o i4o.o 53.8 10 196.1 75.3 70 252.1 96.8 97.1 3i 28.9 II. I 91 85.0 32.6 i5i i4: .0 54.1 211 197.0 75.6 271 253.0 32 29.9 11.5 92 85.9 33.0 52 141.9 54.5 12 197.9 76.0 72 253.9 97.5 33 3o.8 IT. 8 93 86.8 33.3 53 142.8 54.8 i3 198.9 70.3 73 254.9 97.8 34 3i.7 12.2 94 87.8 33.7 54 143.8 55.2 i4 199.8 76.7 74 255.8 98. 2 35 32.7 12.5 95 88.7 34.0 55 144.7 55.5 i5 200.7 77.0 75 256.7 98.6 36 33.6 12.9 i3.3 96 89.6 34.4 56 145.6 55.9 16 201 .7 77-4 70 257.7 98.9 37, 34.5 07 00.6 34.8 57 146.6 56.3 17 202.6 77.8 77 258.6 99.3 38 35.5 i3.6 98 91.5 35.1 58 147-5 56.6 18 2o3.5 78.1 78 259.5 99.6 39 36.4 i4.o 9Q 92.4 35.5 59 148.4 57.0 19 204.5 78.5 79 260.5 100.0 4o 37.3 i4.3 100 93.4 35.8 60 149.4 57.3 20 2o5.4 78.8 80 261 .4 100.3 4 1 38.3 14.7 lOI 94.3 36.2 161 i5o.3 57.7 221 206.3 79.2 281 262.3 100.7 42 39.2 i5.i 02 95.2 36.6 62 l5l.2 58.1 22 207.3 79.6 82 263.3 lOI.I 43 4o.i i5.4 o3 96.2 36.9 63 l52.2 58.4 23 208.2 80.3 83 264.2 101.4 . 44 4i.i i5.8 04 97.1 37.3 64 i53.i 58.8 24 209.1 84 265.1 101.8 45 42.0 16. 1 o5 98.0 37.6 65 i54.o 59.1 25 210.1 bo.b 85 266.1 102.1 46 42.9 16.5 06 99.0 38.0 66 i55.o 59.5 26 211 .0 81.0 8b 267.0 102.5 47 43.9 16.8 07 99.9 38.3 67 155.9 59.8 27 211 .0 81.3 87 267.9 102.9 48 44.8 17.2 08 100.8 38.7 68 i56.8 60.2 28 212.9 81.7 88 268.9 103.2 49 45.7 17.6 09 101.8 39.1 69 157.8 60.6 29 2i3.8 82.1 89 269.8 io3.6 5o 4(5.7 17.9 10 1.02.7 39.4 70 i58.7 60.9 3o 214.7 82.4 90 270 .'7 103.9 5i 47.6 18.3 III io3.6 39.8 171 159.6 61.3 23l 215.7 82.8 291 271.7 104.3 52 48.5 18.6 12 104.6 4o.i 72 160.6 61.6 32 216.6 83.1 92 272.6 104.6 53 49.5 19.0 i3 io5.5 4o.5 73 161.5 62.0 33 217.5 83.5 93 273.5 io5.o 54 5o.4 19.4 i4 106.4 40.9 74 162.4 62.4 34 218.5 83.9 94 274.5 1 io5.4 1 55 5i.3 19.7 i5 107.4 4l.2 75 i63.4 62.7 35 219.4 84.2 95 275.4 105.7 56 52.3 20. 1 16 108.3 4i.6 76 164.3 63.1 36 220.3 84.6 96 276.3 106. 1 57 53.2 20.4 17 109.2 41.9 77 i65.2 63.4 37 221 .3 84.9 97 277.3 106.4 58 54.1 20.8 18 no. 2 42.3 78 166.2 63.8 38 222.2 85.3 98 278.2 106.8 59 55.1 21 .1 19 III. I 42.6 79 167.1 64.1 3q 223.1 85.6 99 279.1 107.2 60 56.0 21.5 20 112. 43.0 80 168.0 64.5 40 224.1 86.0 3oo 280.1 107.5 Disl. Dep. Lat. Dist. Dep. Lat. Disl Dep. Lat. Dist Dep. Lat. Dist Dep. Lat. For G9 Degi ees. Page 38J TABLL II. 1 DifTerence of Latitude and Departure for 22 Degrees. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 00.9 00.4 61 56.6 22.9 121 112. 2 45.3 181 167.8 67.8 241 223.5 90.3 2 01 .9 00.7 62 57.5 23.2 22 Ii3.i 45.7 82 168.7 68.2 42 224.4 90.7 3 02.8 01. 1 63 58.4 23.6 23 ii4.o 46.1 83 169.7 68.6 43 225.3 91.0 4 o3.7 01.5 64 59.3 24.0 24 ii5.o 46.5 84 170.6 68.9 AA 220.2 91.4 5 04.6 01 .9 65 60.3 24.3 25 115.9 46.8 85 171. 5 69.3 45 227.2 91.8 6 o5.6 02.2 66 61 .2 24.7 26 116.8 47-2 86 172.5 69.7 46 228.1 92.2 7 06.5 02.6 67 62.1 25.1 27 117. 8 47-6 87 173.4 70.1 47 229.0 92.5 8 07.4 o3.o 68 63. 25.5 28 118. 7 47-9 88 174.3 70.4 48 229.9 92.9 9 08.3 o3.4 69 64. 25.8 29 119. 6 48.3 89 175.2 70.8 49 230.9 93.3 10 09.3 03.7 70 64-9 26.2 3o 120.5 48.7 90 176.2 71.2 5o 23i.8 93.7 II 10.2 04.1 71 65.8 26.6 i3i 121 .5 49.1 191 177. 1 71.5 25l 232.7 94.0 12 II . X o4 . 5 72 66.8 27.0 32 122 4 49.4 92 178.0 71.9 52 233.7 94-4 i3 12. 1 04.9 73 67.7 27.3 33 123.3 49-8 93 178.9 72.3 53 234.6 94-8 r4 i3.o 05.2 74 68.6 27.7 34 124.2 5o.2 94 179-9 72.7 54 235.5 95.2 1 5 13.9 o5.6 75 69.5 28.1 35 125.2 5o.6 95 180.8 73.0 55 236.4 95.5 i6 i4.8 06.0 76 70.5 28.5 36 126. I 50.9 96 181. 7 73.4 56 237.4 95-9 I? i5.8 06.4 77 71.4 28.8 37 127.0 5i.3 97 182.7 73.8 i)7 238.3 96.3 i8 16.7 06.7 78 72.3 29.2 38 128.0 5. .7 98 i83.6 74.2 58 239.2 96.6 19 17.6 07.1 79 73.2 29.6 39 128.0 52.1 99 184.5 74-5 59 240.1 97-0 20 18.5 07.5 80 81 74.2 3o.o 4o 129.8 52.4 200 i85.4 74-9 60 241 .1 97-4 21 19.5 07.9 75.1 3o.3 i4i i3o.7 52.8 201 186.4 75.3 261 242.0 97-8 22 20.4 08.2 82 76.0 3o.7 42 i3i .7 53.2 02 187.3 75.7 62 242.9 98.1 23 21.3 08.6 83 77.0 3i.i 43 i32.6 53.6 OJ 188.2 76.0 63 243.8 98.5 24 22.3 09.0 84 77-9 3i.5 M i33.5 53.9 o4 189. 1 76-4 64 244.8 98.9 25 23.2 09.4 85 78.8 3i.8 45 i34.4 54.3 o5 190. 1 76.8 65 245.7 99.3 26 24.1 09.7 86 79-7 32.2 46 i35.4 54.7 06 191.0 77.2 66 246.6 99.6 27 25.0 10. 1 87 80.7 32.6 47 i36.3 55.1 07 191.9 77-b 67 247-6 1 00.0 28 26.0 10.5 88 81.6 33.0 48 137.2 55.4 08 192.9 77-9 68 248.5 100.4 29 26.9 10.9 89 82.5 33.3 49 i38.2 55.8 09 193.8 78.3 69 249.4 100.8 3o 27.8 II .2 90 83.4 33.7 5o 139. 1 56.2 10 194.7 78.7 70 25o.3 lOI.I 3i 28.7 II. 6 Qi 84.4 34.1 i5i i4o.o 56.6 211 195.6 79.0 271 25i.3 101.5 32 29.7 12.0 92 85.3 34.5 52 140.9 56.9 12 196.6 79-4 72 252.2 101.9 33 3o.6 12.4 93 86.2 34.8 53 141.9 57.3 i3 197.5 79.8 73 253.1 102.3 34 3[.5 12.7 94' 87.2 35.2 54 142.8 57.7 i4 198.4 80.2 74 254.0 102.6 35 32.5 i3.i 95, 88.1 35.6 55 143.7 58.1 i5 199.3 80.5- 75 255. io3.o 36 33.4 i3.5 g6 89.0 36.0 56 144.6 58.4 16 200.3 80.9 76 255.9 io3.4 37 34.3 i3.9 97 89.9 36.3 57 145.6 58.8 17 201 .2 81.3 77 256.8 io3.8 38 35.2 14.2 98 90.9 36.7 58 146.5 59.2 18 202.1 81.7 78 257.8 104. 1 39 36.2 i4.6 99 91.8 37.1 59 147-4 59.6 19 203.1 82.0 79 258.7 104.5 40 37.1 i5.o 100 92.7 37.5 60 148.3 59.9 60.3 20 204.0 82.4 80 259.6 104.9 4i 38. i5.4 lOI 93.6 37.8 161 149.3 221 204.9 82.8 281 260.5 io5.3 42 38.9 l5.7 02 Q4.6 38.2 62 i5o.2 60.7 22 205.8 83.2 82 261 .5 io5.6 43 39.9 16. 1 o3 95.5 38.6 63 i5i.i 61. 1 23 206.8 83.5 83 262.4 106.0 U 4o.8 16.5 04 96.4 39.0 64 l52.I 61.4 24 207.7 83-9 84 263.3 106.4 45 41.7 16.9 o5 97.4 39.3 65 I53.0 61.8 25 208.6 84.3 85 264.2 106.8 46 42.7 17.2 06 98.3 39.7 66 153.9 62.2 26 209.5 84-7 86 265.2 107.1 47 43.6 17.6 07 99.2 4o.i 67 i54.8 62.6 27 210.5 85.0 87 266.1 107.5 48 44.5 18.0 08 1 00. 1 40.5 68 i55.8 62.9 28 211. 4 85.4 88 267.0 107.0 49 45.4 18.4 09 lOI . I 4o.8 69 i56.7 63.3 29 212.3 85.8 89 268.0 108.3 5o ^^.^ IS. 7 10 102.0 41.2 70 157.6 63.7 3o 2i3.3 86.2 90 268.9 108.6 5i 47.3 19.1 III 102.9 4i.6 171 i58.5 64.1 23 I 214.2 86.5 291 269.8 109.0 52 48.2 19.5 12 io3.8 42.0 72 159.5 64.4 32 2 1 5 . 1 86.9 92 270.7 109.4 53 49.1 19.9 i3 104.8 42.3 73 160.4 64.8 33 216.0 87.3 93 271.7 109.8 54 5o.i 20.2 i4 105.7 42.7 74 161. 3 65.2 U 217.0 87-7 94 272.6 1 10. 1 55 5i.o 20.6 i5 ic6.6 43.1 73 162.3 65.6 35 217.9 88.0 95 273.5 1 10.5 56 5i.9 21 .0 16 107.6 43.5 76 i63.2 65.9 36 218.8 S8.4 96 274.4 110.9 57 52.8 21 .4 17 108.5 43.8 77 164.1 66.3 37 219.7 88.8 97 275.4 1 1 1.3 58 53.8 21.7 18 109.4 44.2 78 i65.o 66.7 38 220.7 89.2 98 276.3 1 1 1.6 59 54.7 22.1 19 no. 3 44.6 79 166.0 67.1 39 221 .6 89.5 99 277.2 112.0 60 55.6 22.5 TaT 20 II 1 .3 45.0 80 166.9 67.4 40 222.5 89.9 3oo 278.2 112.4 Dist. De). Dist Dep. Lat. Dist. Dep. Lai. Dist. Dep. Lat. Dist. Dep. Lat. [For G8 Deg rees. TABLE IL [Page 33 Difference of Latitude and Departure for 23 Degrees. Dist. Lat. Dfp. Dist. Lai. Di-p. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 00.9 00.4 61 56.2 23.8 1 21 III. 4 47.3. iSi 166.6 70.7 241 221 .8 94.2 2 01 .8 GO. 8 62 b7.. 24.2 22 112.3 47.7 82 167.5 71.1 42 222.8 94.6 3 02.8 01 .2 63 58. 24.6 23 Il3.2 48.1 83 168.5 71.5 43 223.7 94-9 4 o3.7 01 .6 64 58.9 25. 24 114.1 48.5 84 169.4 71.9 44 224.6 95.3 5 04.6 02.0 bb 59.8 2b. 4 25 ii5.i 48.8 85 170.3 72.3 45 225.5 9D.7 6 ob.5 02.3 66 60.8 25.8 26 116.0 49.2 86 171 .2 72.7 46 226.4 96.1 7 06.4 02.7 67 6,. 7 26.2 27 116. 9 49.6 87 172.1 73.1 47 227.4 96.5 8 07.4 o3.i 68 62.6 26.6 28 117.8 5o.o 88 173. 1 73.5 48 228.3 96.9 9 08.3 o3.5 69 63.5 27.0 29 118.7 5o.4 89 174.0 73.8 49 229.2 97.3 10 09.2 03.9 70 64.4 27.4 3o 119.7 bo. 8 90 174.9 74.2 bo 23o.i 97-7 II 10. 1 04.3 71 65.4 27.7 i3i 120.6 5l.2 191 175.8 74.6 25l 23l .0 98.1 12 II .0 04.7 72 66.3 28.1 32 121 .5 5i.6 92 176.7 75.0 52 232.0 98.5 i3 12.0 oS.i 73 67.2 28.5 33 122.4 52.0 93 177.7 75.4 53 232.9 98.9 i4 12.9 o5.5 74 68.1 28.9 34 123.3 b2.4 94 178.6 75.8 54 233.8 99.2 lb i3.8 o5.9 7b 69.0 29.3 35 124.3 52.7 95 179.5 76.2 55 234.7 99.6 lb 14.7 06.3 76 70.0 29.7 36 125.2 53.1 96 180.4 76.6 56 235.6 100.0 17 ib.b 06.6 77 70.9 3o.i 37 126. I b3.5 97 181.3 77.0 57 236.6 100.4 i8 ib.b 07.0 7S 71.8 3o.b 38 127.0 53.9 98 182.3 77.4 58 23-7.5 100.8 19 17. b 07.4 79 72.7 30.9 39 128.0 b4.3 99 i83.2 77.8 59 238.4 101.2 20 18.4 07.8 80 73.6 3i.3 40 128.9 54.7 200 184.1 78.1 60 239.3 101.6 21 19.3 08.2 81 74.6 3. .6 i4i 129.8 55.1 201 i85.o 78.5 261 240.3 102.0 22 20.3 08.6 82 75.5 32. 42 i3o.7 55.5 02 185.9 78.9 62 241 .2 102.4 23 21.2 09.0 83 76.4 32.4 43 i3i.6 55.9 o3 186.9 79.3 63 242.1 102.8 24 22.1 09.4 84 77.3 32.8 44 i32.6 56.3 04 187.8 79-7 64 243.0 103.2 2b 23.0 09.8 85 78.2 33.2 45 i33.5 56.7 o5 188.7 80.1 65 243.9 103.5 26 23.9 10.2 86 79.2 33.6 46 i34.4 57.0 «5 189.6 80.5 66 244.9 103.9 27 24.9 10.5 87 80.1 34.0 47 i35.3 57.4 07 190.5 80.9 67 245.8 104.3 28 2b. 8 10.9 88 81.0 'M.4 48 i36.2 b7.8 08 191 .5 81.3 68 246.7 104.7 29 26.7 II .3 89 81.9 34.8 49 137.2 58.2 09 192.4 81.7 69 24-'. 6 io5.i 3o 27. b II. 7 90 82.8 3b. 2 5o i3S.i 58.6 10 193.3 82.1 70 248.5 105.5 3i 28.5 12. 1 91 83.8 35.6 i5i 139.0 59.0 211 194.2 82.4 271 249.5 105.9 32 29.5 12.5 92 84.7 35.9 52 139.9 59.4 12 19D.1 82.8 72 25o.4 106.3 6:i 3o.4 12.9 93 85.6 36.3 53 140.8 59.8 i3 196.1 83.2 73 251.3 106.7 M 3i.3 i3.3 94 86.5 36.7 54 i4i.8 60.2 i4 197.0 83.6 74 252.2 107.1 6b 32.2 i3.7 95 87.4 37.1 55 142.7 60.6 i5 197.9 84.0 75 253.1 107.5 3b 33.1 i4.i 96 88.4 37.5 56 143.6 61 .0 16 198.8 84.4 76 254.1 107.8 ^7 34.1 i4.5 97 89.3 37.9 57 144.5 61.3 17 199.7 84.8 77 255. 108.2 38 3b. i4.8 98 90.2 38.3 58 145.4 61.7 18 200.7 85.2 78 255.9 108.6 39 3b. 9 l5.2 99 91. 1 38.7 59 146.4 62. 1 19 201 .6 85.6 79 256.8 109.0 40 36.8 i5.6 100 92.1 39.1 6(3 147-3 62.5 20 202.5 86.0 80 257.7 109.4 4 1 37.7 16.0 lOI 93.0 39.5 161 148.2 62.9 221 2o3.4 86.4 281 258.7 109.8 42 38.7 16.4 02 93.9 39.9 62 149. 1 63.3 22 204.4 86.7 82 259.6 110.2 4d 39.6 16.8 o3 94.8 40.2 63 i5o.o 63.7 23 2o5.3 87.1 83 260.5 1 1 0.6 44 40.5 17.2 04 95.7 40.6 64 i5i.o 64.1 24 206.2 87.5 84 261 .4 III.O 4b 41.4 17.6 o5 96.7 4i .0 65 .51.9 64.5 25 207.1 87.9 85 262.3 1 1 1.4 4b 42.3 18.0 06 97.6 41.4 66 152.8 64.9 26 208.0 88.3 86 263.3 1 11.7 47 43.3 18.4 07 98.5 4i.8 67 153.7 65.3 27 209.0 88.7 87 264.2 1 12. 1 48 44.2 18.8 08 99.4 42.2 68 154.6 65.6 28 209.9 89.1 88 265.1 112. 5 49 4b. I 19. 1 09 100.3 42.6 69 i55.6 66.0 29 210.8 89.5 89 266.0 1 12.9 bo 46. 19.5 10 loi .3 43.0 7" 156.5 66.4 3o 21 1 .7 89.9 90 291 266.9 ii3.3 5i 46.9 19.9 III 102.2 43.4 171 157.4 66.8 23l 212.6 90.3 267.9 1 13.7 b2 47.9 20.3 12 io3. 1 43.8 72 i58.3 67.2 32 2i3.6 90.6 92 268.8 114.1 bi 48.8 20.7 i3 104.0 44.2 73 159.2 67.6 33 214.5 91.0 93 269.7 114.5 b4 49.7 21. 1 i4 104.9 44.5 74 160.2 68.0 34 2i5.4 91.4 94 270.6 1 14.9 bb 5o.6 21.5 lb 105.9 44.9 75 161. 1 68.4 35 216.3 91.8 95 271 .5 ii5.3 bb 5i.5 21 .9 22.3 16 106.8 45.3 76 162.0 68.8 36 217.2 92.2 96 272.5 nb.7 b7 b2.b 17 107.7 45.7 77 162.9 69.2 37 218.2 92.6 97 273.4 ' 16.0 b8 b3.4 22.7 18 108.6 46.1 78 i63.8 69.6 38 219.1 93.0 98 274.3,116.4 1 b9 b4.3 23.1 19 109.5 46.5 79 164.8 69.9 39 220.0 93.4 99 275.2 1 16.8 bo bb.2 23.4 20 no. 5 46.9 80 ibb.7 70.3 40 Dist. 220.9 Dep. 93.8 Lat. 3oo 276.2 117.2 Dist. Dep. Lat. Dist. Dcp. Lat. Dist. Dep. Lat. Dist. Den. Lat. [For C7 Degrees. Page 40] TABLE IL Difference of Latitude and Departure for 24 Degrees. Dist. I Lat. 00.9 Dep. 00.4 Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. 61 55.7 24.8 121 no. 5 49.2 181 i65.4 73.6 241 220.2 98.0 2 01.8 00.8 62 56.6 25.2 22 III .5 49 -(3 82 166.3 74.0 42 221 .1 98.4 3 02.7 01 .2 63 57.6 25.6 23 112. 4 5o.o 83 167.2 74.4 43 222.0 98.8 4 o3.7 01 .6 64 58.5 26.0 24 ii3.3 5o.4 84 168.1 74.8 M 222.9 99.2 5 04.6 02.0 65 59.4 26.4 25 Il4-2 5o.8 85 169.0 75.2 45 223.8 99-7 6 o5.5 02.4 66 60.3 26.8 26 ii5.i 5i.2r 86 169.9 75.7 46 224.7 100. 1 7 06.4 02.8 67 61.2 27.3 27 116. 5i.7 87 170.8 76.1 47 225.6 100.5 8 07.3 o3.3 68 62.1 27.7 28 116.9 52.1 88 171.7 76.5 48 226.6 100.9 9 08.2 03.7 69 63.0 28.1 29 117. 8 52.5 89 172.7 76.9 49 227.5 101.3 10 09. 1 04.1 70 63.9 28.5 3o 118.8 52.9 90 173.6 77.3 5o 228.4 101.7 1 1 lO.O 04.5 71 64.9 28.9 i3i 119-7 53.3 191 174.5 77-7 25l 229.3 102.1 12 II .0 04.9 72 65.8 29.3 32 120.6 53.7 92 n^.A 78.1 52 23o.2 102.5 i3 II. 9 o5.3 73 66.7 29.7 33 121 .5 54.1 93 176.3 78.5 53 23l.I 102.9 i4 12.8 o5.7 74 67.6 3o. I M 122.4 54.5 94 177.2 78.9 54 232.0 io3.3 i5 i3.7 06.1 75 68.5 3o.5 35 123.3 54.9 95 178. 1 79-3 55 233.0 103.7 i6 14.6 06.5 76 69.4 30.9 36 124.2 55.3 96 179-1 79-7 56 233.9 104.1 17 i5.5 06.9 77 70.3 3i.3 37 125.2 55.7 97 180.0 80.1 57 234.8 104.5 18 16.4 07.3 78 71.3 3i.7 38 126. I 56.1 98 180.9 80.5 58 235.7 104.9 19 17.4 07.7 79 72.2 32.1 39 127.0 56.5 99 181. 8 80.9 59 236.6 105.3 20 18.3 oS.i 80 73.1 32.5 40 127.9 56.9 200 182.7 81.3 60 237.5 io5.8 21 19.2 08.5 81 74.0 32.9 i4i 128.8 57.3 201 i83.6 81.8 261 238.4 106.2 22 20. 1 08.9 82 74.9 33.4 42 129.7 57.8 02 184.5 82.2 62 239.3 106.6 23 21.0 09.4 83 75.8 33.8 43 i3o.6 58.2 o3 i85.4 82.6 63 240.3 107.0 24 21.9 09.8 84 76.7 34.2 A-i i3i.6 58.6 o4 186.4 83.0 64 241.2 107.4 25 22.8 10.2 85 77-7 34.6 45 i32.5 59.0 o5 187.3 83.4 65 242.1 107.8 26 23.8 10.6 86 78.6 35.0 46 i33.4 59.4 06 188.2 83.8 66 243.0 108.2 27 24.7 II. «7 79.5 35.4 47 i34.3 59.8 07 189.1 84.2 67 243.9 108.6 28 25.6 II. 4 88 80.4 35.8 48 i35.2 60.2 08 190.0 84.6 68 244.8 109.0 29 26.5 II. 8 89 81.3 36.2 49 i36.i 60.6 09 190.9 85.0 69 245.7 109.4 3o 3i 27-4 28.3 12.2 12.6 90 82.2 36.6 5o i37.o 61 .0 10 191.8 85.4 70 246.7 109.8 91 83.1 37.0 i5i 137.9 61.4 211 192.8 85.8 271 247.6 110.2 32 29.2 i3.o 92 84.0 37.4 52 i38.9 6r.8 12 193.7 86.2 72 248.5 1 10.6 33 3o.i i3.4 93 85. 37.8 53 139.8 62.2 i3 194.6 86.6 73 249.4 in.o 34 3i.i i3.8 94 85.9 38.2 54 140.7 62.6 i4 195.5 87.0 74 250.3 1 1 1.4 35 32.0 14.2 95 86.8 38.6 55 i4i.6 63. i5 196.4 87.4 75 25l.2 III. 9 36 32.9 14.6 96 87.7 39.0 56 142.5 63.5 16 197.3 87.9 76 252.1 112. 3 37 33.8 i5.o 97 88.6 39.5 57 143.4 63.9 17 198.2 88.3 77 253.1 112.7 38 34.7 i5.5 98 89.5 39.9 58 144.3 64.3 18 199.2 88.7 78 254.0 ii3.i 39 35.6 15.9 99 90.4 40.3 59 145.3 64.7 19 200.1 89.1 79 254.9 ii3.5 40 4i 36.5 16.3 100 91.4 40.7 60 i46.2 65.1 20 201 .0 89.5 80 255.8 113.9 37.5 16.7 lOI 92.3 4i.i 161 i47-i 65.5 221 201 .9 89.9 281 256.7 114.3 •42 38.4 17. 1 02 93.2 4i.5 62 148.0 65.9 22 202.8 90.3 82 257.6 114-7 43 39.3 17.5 o3 94.1 41.9 63 148.9 66.3 23 203.7 90.7 83 258.5 ii5.i Ai 4o.2 17.9 04 95.0 42.3 64 149-8 66.7 24 204.6 91. 1 84 259.4 ii5.5 45 4i.i 18.3 o5 95.9 42.7 65 1 50.7 67.1 25 2o5.5 91.5 85 260.4 1 1 5.9 46 42.0 18.7 06 96.8 43.1 66 i5i.6 67.5 26 206.5 91.9 86 261.3 116.3 47 42.9 19. 1 07 97-7 43.5 67 i52.6 67.9 27 207.4 92.3 87 202.2 116.7 48 43.9 19.5 08 98.7 43.9 68 153.5 68.3 28 208.3 92.7 88 263.1 117. 1 49 4'i.8 19.9 09 99.6 44.3 69 1 54. 4 68.7 29 209.2 93.1 89 264.0 1 1 7.5 5o 45.7 20.3 10 1 00 . 5 44.7 70 i55.3 69.1 3o 210. 1 93.5 90 264.9 1 18.0 5i 46.6 20.7 III loi .4 45.1 171 1 56. 2 69.6 23l 21 1 .0 94.0 291 265.8 118.4 52 47 ^ 21.2 12 102.3 45.6 72 1 57. 1 70.0 32 211 .9 94.4 92 266.8 118.8 53 48.4 21.6 1 3 I03.2 46. 73 i58.o 70.4 33 212.9 94.8 93 267.7 1 19.2 54 49.3 22.0 i4 104.1 A6.A 74 159^0 70.8 ■M 2i3.8 95.2 94 268.6 119.6 55 5o.2 22.4 i5 io5.i 46.8 75 159.9 71.2 35 214.7 95.6 95 269.5 120.0 5b 5l.2 22.8 16 1 06 . 47.2 76 160.8 71.6 36 2i5.6 96.0 96 270.4 120.4 !)7 52. I 23.2 17 106.9 47-6 77 161 .7 72.0 37 216.5 96.4 97 271 .3 120.8 5b 53.0 23.6 18 107.8 48. 78 162.6 72.4 38 217.4 96.8 98 272.2 1 2 1. 2 59 53.9 24.0 19 108.7 48.4 79 i63.5 72.8 39 218.3 97.2 99 273.2 1 2 1. 6 bo 54.8 24.4 20 109.6 48.8 80 164.4 73.2 4o 219.3 97.6 3oo 274.1 122.0 Disi. De'j. Lat. Dist. Dc.p. Lnt. Dist. Dtp. Lat. Dist. Dep. Lat. Dist.l Dep. 1 Lat. [For GO Degrees. TABLE IL [Page 41 Difference of Latitude and Departure for 25 Degre !es. Uist. Lat. Dcp. Dist. Lat. Dep. 25.8 Dist. Lat. Dep. Dist. Lat. Dcp. Dist. Lat. Dep. I 00.9 00.4 61 55.3 121 109.7 5i.i 181 164.0 76.5 241 218.4 101.9 2 01 .8 00.8 62 56.2 26.2 22 1 10.6 5i.6 82 164.9 76.9 42 219.3 102.3 3 02.7 01 .3 63 57.1 26.6 23 III. 5 52. 83 165.9 77.3 43 220.2 102.7 4 o3.6 01.7 64 58.0 27.0 24 112.4 52.4 84 166.8 77-8 AA 221.1 io3.i 5 04.5 02.1 65 58.9 27.5 25 ii3.3 52.8 85 167.7 78.2 45 222.0 103.5 6 o5.4 02.5 66 59.8 27.9 26 Il4.2 53.2 86 168.6 78.6 46 223.0 104.0 7 06.3 o3.o 67 60.7 28.3 27 ii5.i 53.7 87 169.5 79.0 47 223.9 104.4 8 07.3 o3.4 68 61.6 28.7 28 116.0 54.1 88 ,170.4 79-^ 48 224.S 104.8 9 0S.2 o3.8 69 62.5 29.2 29 1 16.9 54.5 89 171.3 79-9 49 225.7 105.2 10 09.1 04.2 7" 63.4 29.6 3o.o 3o 117. 8 54.9 90 172.2 80.3 5o 226.6 105.7 II 10. 04.6 71 64.3 i3i 118.7 55.4 191 173.1 80.7 25l 227.5 106. 1 12 10.9 o5.j 72 65.3 3o.4 32 119.6 55.8 92 174.0 81. 1 52 228.4 106.5 i3 II. 8 o5.5 73 66.2 30.9 33 120.5 56.2 93 174.9 81.6 53 229.3 106.9 ]4 12.7 05.9 74 67.1 3i.3 34 121 .4 56.6 94 175.8 82.0 54 23o.2 107.3 i5 i3.6 06.3 75 68.0 3i.7 35 122.4 57.1 95 176.7 82.4 55 23l.I 107.8 i6 i4.5 06.8 76 68.9 32.1 36 123.3 57.5 96 177.6 82.8 56 232.0 108.2 17 i5.4 07.2 77 69.8 32.5 37 124.2 57.9 97 178.5 83.3 57 232.9 108.6 i8 16.3 07.6 7S 70.7 33.0 38 125.1 58.3 98 179.4 83.7 58 233.8 109.0 19 17.2 08.0 79 71.6 ■6i.A 39 126.0 58.7 99 180.4 84.1 59 234.7 109.5 20 18.1 08.5 80 72.5 33.8 40 126.9 59.2 59.6 200 181.3 84.5 60 235.6 109.9 21 19.0 08.9 81 73.4 34.2 i4i 127.8 201 182.2 84.9 261 236.5 1 10.3 22 19.9 09.3 82 74.3 34.7 42 128.7 60.0 02 i83.i 85.4 62 237.5 1 10.7 23 20.8 09.7 83 75.2 35.1 43 129.6 60.4 o3 184.0 85.8 63 2 38.4 III. I 24 21.8 lO.I 84 76.1 35.5 AA i3o.5 60.9 04 184.9 86.2 64 239.3 111.6 25 22.7 10.6 85 77.0 35.9 45 i3i.4 61.3 o5 i85.8 86.6 65 240.2 1 12.0 26 23.6 II. 86 77-9 36.3 46 i32.3 61.7 06 186.7 87.1 66 241.1 1 1 2.4 27 24.5 II. 4 87 78.8 36.8 47 133.2 62.1 07 187.6 87.5 67 242.0 1 1 2.8 28 25.4 II. 8 88 79.8 37.2 48 i34.i 62.5 08 188.5 87-9 68 242.9 ii3.3 29 26.3 12.3 89 80.7 37.6 49 i35.o 63. 09 189.4 88.3 69 243.8 113.7 3o 27.2 12.7 90 81.6 38. 5o 135.9 63.4 10 190.3 88.7 70 244.7 114.1 3i 28.1 i3.i Qi 82.5 38.5 i5i i36.9 63.8 211 191.2 89.2 271 245.6 114.5 32 29.0 i3.5 92 83.4 38.9 52 137.8 64.2 12 192.1 89.6 72 246.5 1 1 5.0 33 29.9 i3.9 93 84.3 39.3 53 i38.7 64.7 i3 193.0 90.0 73 247-4 n5.4 34 3o.8 14.4 94 85.2 39.7 54 139.6 65.1 i4 193.9 90.4 74 248.3 ii5.8 35 3i.7 14.8 95 86,1 4o. I 55 140.5 65.5 i5 194.9 90.9 75 249.2 1 16.2 36 32.6 l5.2 96 87.0 40.6 56 141.4 65.9 16 195.8 91.3 76 25o.i 116.6 37 33.5 i5.6 97 87.9 4i .0 57 142.3 66.4 17 196.7 91.7 77 25l.O 117.1 38 34.4 16. 1 98 88.8 41.4 58 143.2 66.8 18 197.6 92.1 78 252.0 117.5 39 35.3 16.5 99 89.7 4i.8 59 144. 1 67.2 19 198.5 92.6 79 252.9 117.9 40 36.3 16.9 100 90.6 42.3 60 145.0 67.6 20 199-4 93.0 80 253.8 118.3 4i 37.2 17.3 lOI 91.5 42.7 161 145.9 68.0 221 200.3 93-4 281 254.7 118.8 42 38.1 17.7 02 92.4 43.1 62 146.8 68.5 22 201.2 93.8 82 255.6 119.2 43 39.0 18.2 o3 93.3 43.5 63 147-7 68.9 23 202.1 94.2 83 256.5 119.6 A/\ 39.9 18.6 04 94.3 44.0 QA 148.6 69.3 24 2o3.o 94-7 84 257.4 120.0 45 40.8 19.0 o5 95.2 aA-A 65 149.5 69.7 25 203.9 95.1 85 258.3 120.4 46 41.7 19.4 06 96.1 44.8 66 i5o.4 70.2 26 2o4-8 95.5 86 259.2 120.9 47 42.6 19.9 07 97.0 45.2 67 i5i.4 70.6 27 205.7 95.9 87 260.1 12 1. 3 A^ 43.5 20.3 08 97-9 45.6 68 i52.3 71.0 28 206.6 96.4 88 261.0 1 2 1. 7 49 u.^ 20.7 09 98. 8 46.1 69 i53.2 71.4 29 207.5 96.8 89 261.9 122. 1 bo 5i 45.3 21 .1 10 99-7 46.5 70 154.1 71.8 3o 208.5 97-2 90 262.8 122.6 46.2 21 .6 III 100.6 46.9 171 i55.o 72.3 23l 209.4 97.6 291 263.7 123.0 52 47.1 22.0 12 Id .5 47.3 72 155.9 72.7 32 210.3 98.0 92 264.6 123.4 53 48.0 22.4 i3 102.4 47-8 73 i56.8 73.1 33 211. 2 98.5 93 265.5 123.8 54 48.9 22.8 i4 io3.3 48.2 74 157.7 73.5 M 212. 1 98.9 94 266.5 124.2 55 49.8 23.2 i5 104.2 48.6 75 i58.6 74.0 35 2l3.0 99.3 95 267.4 124.7 56 5o.8 23.7 lb io5.i 49.0 76 159.5 74.4 36 2l3.9 99-7 96 268.3 125. 1 57 51.7 24.1 17 106.0 49.4 77 160.4 74.8 37 214.8 100.2 97 269.2 125.5 58 52.6 24.5 18 106.9 49.9 78 161. 3 75.2 38 215.7 100.6 98 270.1 125.9 59 53.5 24.9 19 107.9 5o.3 79 162.2 75.6 39 216.6 lOI.O 99 271.0 126.4 bo 54.4 25.4 20 108.8 50.7 8g i63.i 76.1 4o 217.5 101.4 3oo 271.9 126.8 Dist. Pep. Lat. Dist. Dcp. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. [1 ^-or G.^ Degr ees. Page 421 TABLE IL 1 Difference of Latitude and Departure for 26 Degrees. Disi. Lat. Uep. Disl. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 00.9 00.4 61 54.8 26.7 121 108.8 53.0 181 162.7 79.3 241 216.6 io5.6 2 01.8 00.9 62 55.7 27.2 22 109.7 53.5 82 i63.6 79.8 42 217.5 106.1 3 02.7 01 .3 63 56.6 27.6 23 no. 6 53.9 83 164.5 80.2 43 21S.4 106.5 4 o3.6 01.8 64 57.5 28.1 24 III. 5 54.4 84 i65.4 80.7 44 219.3 107.0 5 04.5 02.2 65 58.4 28.5 25 112. 3 54.8 85 166.3 81.1 45 220.2 107.4 6 o5.4 02.6 66 59.3 28.9 26 Il3.2 55.2 86 167.2 81.5 46 221.1 107.8 7 06.3 o3.i 67 60.2 i!9.4 27 114.1 55.7 87 168.1 82.0 47 222.0 108.3 8 07.2 o3.5 68 61. 1 29.8 28 ii5.o 56.1 88 169.0 82.4 48 222.9 108.7 9 08.1 03.9 69 62.0 3o.2 29 115.9 56.5 89 169.9 82.9 49 223.8 109.2 10 1 09.0 04.4 70 62.9 3o.7 3o 116. 8 57.0 90 170.8 83.3 5o 224.7 109.6 II . 09 . 9 04.8 71 63.8 3i.i i3i 117. 7 57.4 191 171.7 83.7 25l 225.6 1 1 0.0 12 10.8 o5.3 72 64.7 3i.6 32 118. 6 ^7.9 92 172.6 84.2 52 226.5 110.5 i3 II. 7 05.7 73 65.6 32. 33 119. 5 58.3 93 173.5 84.6 53 227.4 IIO.O i4 12.6 06.1 74 66.5 32.4 U 120.4 58.7 94 174.4 85.0 54 228.3 111.3 i5 i3.5 06.6 75 67.4 32.9 35 121.3 59.2 95 175.3 85.5 55 229.2 11 1.8 16 14.4 07.0 76 68.3 33.3 36 122.2 59.6 96 176.2 85.9 56 23o.l 112.2 17 i5.3 07.5 77 69.2 33.8 37 123. I 60. 1 97 177.1 86.4 57 23l.O 112.7 18 16.2 07.9 7S 70.1 34.2 38 124.0 60.5 98 178.0 86.8 58 231.9 ii3.i 19 17. 1 08.3 79 71.0 34.6 39 124.9 60.9 99 '78.9 87.2 59 232.8 ii3.5 20 18.0 08.8 80 71.9 35.1 4o 125.8 61.4 200 I'/ 9-8 87.7 60 233.7 114.0 21 18.9 09.2 81 72.8 35.5 i4i 126.7 61.8 201 180.7 88.1 261 234.6 1 14.4 22 19.8 09.6 82 73.7 35. q 42 127.6 62.2 02 181.6 88.6 62 235.5 114.9 23 20.7 10. 1 83 74.6 36.4 43 128.5 62.7 o3 182.5 89.0 63 236.4 115.3 24 21.6 10.5 84 75.5 36.8 44 129.4 63.1 04 i83.4 89.4 H 237.3 115.7 25 , 22.5 II .0 85 76.4 37.3 45 i3o.3 63.6 o5 184.3 89.9 65 238.2 116. 2 26 23.4 II. 4 86 77.3 37.7 46 l3l.2 64.0 06 185.2 90.3 66 239.1 116.6 27 24.3 II. 8 87 78.2 38.1 47 l32.I ^4.4 07 186.1 90.7 67 240.0 117.0 28 25.2 12.3 88 79.1 38.6 48 i33.o 64.9 08 186.9 91.2 68 240.9 1 17.5 29 26.1 12.7 89 80.0 39.0 49 133.9 65.3 09 187.8 91.6 69 241.8 117.9 3o 27.0 l3.2 90 80.9 39.5 39.9 5o i34.8 65.8 10 188.7 92.1 70 242.7 118.4 3i 27.9 i3.6 Qi 81.8 i5i 135.7 66.2 211 189.6 92.5 271 243.6 118.8 32 28.8 i4.o 92 82.7 4o.3 52 i36.6 66.6 12 190.5 92.9 72 244.5 119. 2 33 29.7 i4.5 93 83.6 40.8 53 137.5 67.1 i3 191.4 93.4 73 245.4 119.7 34 3o.6 14.9 94 84.5 41.2 54 i38.4 67.5 i4 192.3 93.8 74 246.3 120.1 35 3i.5 i5.3 95 85.4 4i.6 55 139.3 67.9 i5 193.2 94.2 75 247.2 120.6 36 32.4 i5.8 96 86.3 4^.1 56 140.2 68.4 16 194.1 94.7 76 248.1 1 21.0 37 33.3 16.2 97 87.2 42.5 ^7 i4i.i 68.8 17 195.0 95.1 77 249.0 121.4 38 34.2 16.7 98 88. 1 43. c 58 142.0 69.3 18 195.9 95.6 78 249.9 121.9 39 35.1 17. 1 99 89.0 43.4 59 142.9 69.7 19 196.8 96.0 79 25o.8 122.3 4o 36. 17.5 18.0 100 89.9 43.8 bo 143.8 70.1 20 197.7 96.4 80 251.7 122.7 4. 36.9 lOI 90.8 44. i 161 144.7 70.6 221 198.6 96.9 281 252.6 123.2 42 37.7 18.4 02 91.7 44.1 62 145.6 71.0 22 199.5 97.3 82 253.5 123.6 43 38.6 18.8 o3 92.6 45.2 63 i46 5 71.5 23 200.4 97.8 83 254.4 I24.I 44 39.5 19.3 04 93.5 45.6 ^4 147-4 71-9 24 201.3 98.2 84 255.3 124.5 45 40.4 19.7 o5 94.4 46.0 65 148.3 72.3 25 202.2 98.6 83 256.2 124.9 46 41.3 20.2 06 95.3 46.5 66 149.2 72.8 26 203.1 99.1 86 257.1 125.4 47 42.2 20.6 07 96.2 46.9 67 i5o.i 73.2 27 204-0 99.5 87 358.0 125.8 48 43.1 21 .0 08 97.1 47.3 68 i5i.o 73.6 28 204.9 99.9 88 258.9 126.3 49 44.0 21.5 09 98.0 47-8 69 i5i.9 74.1 29 2o5.8 10U.4 89 259.8 126.7 bo 44.9 21.9 10 98.9 48.2 48.7 70 152.8 74.5 3o 206.7 100.8 90 260.7 127.1 5. 45.8 22.4 III 99.8 171 153.7 75.0 23l 207.6 101.3 291 261.5 127.6 52 46.7 22.8 12 100.7 49.1 72 i54.6 75.4 32 208.5 101.7 92 262.4 128.0 53 47.6 23.2 i3 loi .6 49.5 73 155.5 75.8 33 209.4 102.1 93 263.3 128.4 54 43.5 23.7 .4 102.5 5o.o 74 i56.4 76.3 34 210.3 102.6 94 264.2 128.9 55 49-4 24.1 i5 io3.4 5o.4 75 157.3 76.7 35 21 1.2 io3.o 95 265.1 129.3 56 5o.3 24.5 16 104.3 50.9 76 i58.2 77.2 36 212.1 io3.5 96 266.0 129.8 57 5l.2 25.0 17 105.2 5i.3 77 159.1 77.6 37 2l3.0 103.9 97 266.9 l3o.2 58 52.1 25.4 18 106. 1 5i.7 78 160.0 78.0 38 213.9 io4.3 98 267.8 i3o.6 59 53.0 25.9 19 107.0 52.2 79 160.9 78.5 39 214.8 104.8 99 268.7 i3i.i 60 53.9 26.3 20 107.9 52.6 80 161.8 78.9 4o 215.7 io5.2 3 00 269.6 i3i.5 Dist. IK.p. Lat. Dist. Dcp. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dcp. Lat. [ ■"or 64 Degrees. "1 TABLE II. [ Pago 43 Differe nee of Lati tude and Departure for 27 Degre es. Dist. Lai. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 00.9 00.5 61 54.4 27-7 121 107.8 54.9 181 161.3 82.2 241 214.7 109.4 3 01.8 00.9 62 55.2 28.1 22 108.7 55.4 82 162.2 82.6 42 21 5.6 109.9 3 02.7 01 .4 63 56.1 28.6 23 109.6 55.8 83 i63.i 83.1 43 216.5 110.3 4 o3.6 01 .8 64 57.0 29.1 24 1 10.5 56.3 84 163.9 83.5 AA 217.4 110.8 5 o4.5 02.3 65 57.9 29.5 25 III. 4 56.7 85 164.8 84.0 45 218.3 III. 2 6io5.3 02.7 66 58.8 3o.o 26 112.3 57.2 86 i65.7 84.4 46 219.2 111.7 7 06.2 o3.2 67 59-7 3o.4 27 Il3.2 57.7 87 166.6 84.9 47 220.1 112.1 8 07.1 o3.6 68 60.6 30.9 28 114.0 58.1 88 167.5 85.4 48 221.0 112.6 9 08.0 04.1 69 61.5 3i.3 29 114.9 58.6 89 168.4 85.8 49 221.9 I i3.o 10 II 08.9 09.8 04.5 o5.o 70 62.4 3i.8 3o ii5.8 59.0 90 169.3 86.3 5o 222,8 ii3.5 71 63.3 32.2 i3i 116.7 59.5 191 170.2 86.7 25l 2236 114.0 12 10.7 o5.4 72 64.2 32.7 32 117.6 59.9 92 171. 1 87.2 52 224.5 1 14.4 i3 11.6 05.9 73 65.0 33.1 33 118. 5 60.4 93 172.0 87.6 53 225.4 1 14.9 i4 12.5 06.4 74 65.9 33.6 34 119. 4 60.8 94 172.9 88.1 54 226.3 ii5.3 i5 i3.4 06.8 75 66.8 34.0 35 120.3 61.3 95 173.7 88.5 55 227.2 ii5.8 i6 i4.3 07.3 76 67.7 34.5 36 121 .2 61.7 96 174-6 89.0 56 228.1 116.2 17 i5.i 07.7 77 68.6 35.0 37 122. 1 62.2 97 175.5 89.4 57 229.0 1 16.7 i8 16.0 08.2 78 69.5 35.4 38 123.0 62.7 98 176.4 89.9 58 229.9 117.1 19 16.9 08.6 79 70.4 35.9 39 123.8 63.1 99 177-3 90.3 59 23o.8 117.6 20 17.8 09.1 80 71.3 ib.6 4o 124.7 63.6 64.0 200 178.2 90.8 60 231.7 118.0 21 18.7 09.5 81 72.2 36.8 i4i 125.6 201 179.1 91.3 261 232.6 118.5 22 19.6 10. 82 73.1 37.2 42 126.5 64.5 02 180.0 91.7 62 233.4 118.9 23 20.5 10.4 83 74.0 37.7 43 127.4 64.9 o3 180.9 92.2 63 234.3 1 19.4 24 21.4 10.9 84 74.8 38.1 ^^ 128.3 65.4 04 181.8 92.6 64 235.2 1 19.9 25 22.3 II. 3 85 75.7 38.6 45 129.2 65.8 o5 182.7 93.1 65 236.1 120.3 26 23.2 II. 8 86 76.6 39.0 46 i3o.i 66.3 06 i83.5 93.5 66 237.0 120.8 27 24.1 12.3 87 77.5 39.5 47 i3i .0 66.7 07 184.4 94.0 67 237.9 121.2 28 24.9 12.7 88 78.4 4o.o 48 i3i.9 67.2 08 i85.3 94.4 68 238.8 121.7 29 25.8 l3.2 89 79.3 40.4 49 i32.8 67.6 09 186.2 94-9 69 239.7 122. 1 3o 26.7 i3.6 90 80.2 40.9 5o i33.7 68.1 10 187.1 95.3 70 271 240.6 122.6 3i 27.6 14.1 91 81. 1 41.3 i5i i34.5 68.6 211 188.0 95.8 241.5 I23.0 32 28.5 i4.5 92 82.0 4i.8 52 i35.4 69.0 12 188.9 96.2 72 242.4 123.5 33 29.4 i5.o 93 82.9 42.2 53 i36.3 69.5 i3 189.8 96.7 73 243.2 123.9 34 3o.3 i5.4 94 83.8 42.7 54 137.2 69.9 i4 190.7 97.2 74 244.1 124.4 35 3l.2 i5.9 95 84.6 43.1 55 i38.i 70.4 i5 191.6 97.6 75 245.0 124.8 36 32.1 16.3 96 85.5 43.6 56 139.0 70.8 16 192.5 98.1 76 245.9 125.3 37 33.0 16.8 97 86.4 44.0 57 139.9 71.3 17 193.3 98.5 77 246.8 125.8 38 33.9 17.3 98 87.3 44.5 58 140.8 71.7 18 194.2 99.0 78 247-7 126.2 39 34.7 '7-7 99 88.2 44.9 59 141.7 72.2 19 195.1 99-4 79 248.6 126.7 40 4i 35.6 ~3'6:y 18.2 18.6 100 89.1 45.4 bo 142.6 72.6 20 196.0 99.9 80 249.5 127.1 lOI 90.0 45.9 161 143.5 73.1 221 196.9 100.3 281 2 5o.4 127.6 42 37.4 19. 1 02 90.9 46.3 62 144.3 73.5 22 197.8 100.8 82 251.3 128.0 43 38.3 19.5 o3 91.8 46.8 63 145.2 74.0 23 198.7 101.2 83 252.2 128.5 U 39.2 20.0 04 92.7 47-2 64 I46.I 74.5 24 199.6 101.7 84 253.0 128.9 45 4o. I 20.4 o5 93.6 47-7 65 i47-o 74.9 25 2CJ0.5 102. 1 85 253.9 129.4 46 4i.o 20.9 06 94.4 48.1 66 147.9 7^.4 26 201.4 102.6 86 254.8 129.8 47 4i .91 21.3 07 95.3 48.6 67 i48.8 75.8 27 202.3 io3.i 87 255.7 i3o.3 48 42.8 21.8 08 96.2 49.0 68 149.7 76.3 28 203.1 io3.5 88 256.6 i3o.7 49 43.7 22.2 09 97.1 49.5 69 i5o.6 76.7 29 204.0 104.0 89 257.5 l3l.2 5o 44.6 22.7 10 98.0 49-9 70 i5i.5 77.2 3o 204.9 104.4 90 2 58.4 i3i.7 5i 45.4 23.2 III 98.9 5o.4 171 i52.4 77.6 23l 2o5.8 104.9 291 259.3 i3i.i 52 46.3 23.6 12 99.8 5o.8 72 i53.3 78.1 32 206.7 io5.3 92 260.2 i32.6 53 47-2 24.1 i3 100.7 5i.3 73 i54.i 78.5 33 207.6 io5.8 93 261. 1 i33.o 54 48.1 24.5 i4 loi .6 5i.8 74 i55.o 79.0 34 208.5 106.2 94 262.0 i33.5 55 49.0 25. i5 102.5 52.2 75 155.9 79-4 35 209.4 106.7 95 262.8 133.9 56 49.9 1 25.4 16 io3.4 52.7 76 i56.8 79-9 36 210.3 107.1 96 263.7 134.4 i)7 5o.8 25.9 17 104.2 53.1 77 157.7 80.4 37 211.2 107.6 97 264.6 134.8 58 5i.7 26.3 18 io5.i 53.6 78 i58.6 80.8 38 212. 1 108.0 98 265.5 i35.3 59 52.6 26.8 19 106.0 54.0 79 159.5 81.3 39 2l3.0 108.5 99 266.4 135.7 bo 53.5I27.2 20 106.9 54.5 80 160.4 81.7 40 2i3.8 109.0 3oo 267.3 i36.2 Disi. Dop. i Lat Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist.J Dep. Lat. [For C3 Degrees. Piige 4-1] TABLE IL Difference of Latitude and Departu re for 23 Degrees. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist.J Lat. 1 Dep. I 00.9 00.5 61 53.9 28.6 121 ]o6.8 56.8 181 159.8 85.0 241 212.8 ii3,i 2 01.8 00.9 62 54.7 29. 1 22 107.7 57.3 82 160.7 85.4 42 213.7 ii3.6 3 02.6 01.4 63 55.6 29.6 23 108.6 57.7 83 161.6 85.9 43 214.6 ii4.i 4 o3.5 01 .9 64 56.5 3o.o 24 109.5 58.2 84 162.5 86.4 44 2i5.4 114-6 5 04.4 02.3 65 57.4 3o.5 25 no. 4 58.7 85 i63.3 86.9 45 216.3 ii5.o fi o5.3 02.8 66 58.3 3i .0 26 III .3 59.2 86 i64-2 87.3 46 217.2 ii5.5 7 06.2 o3.3 67 59.2 3i.5 27 112. 1 59.6 87 i65.i 87.8 47 218.1 1 16.0 8 07.1 o3.8 68 60.0 3i.9 28 ii3.o 60.1 88 166.0 88.3 48 219.0 1 16.4 9 07.9 04.2 69 60.9 32.4 29 113.9 60.6 89 166.9 88.7 49 219.9 116.9 10 08.8 04.7 70 61.8 32.9 33.3 3o 114. 8 61.0 90 167.8 89.2 5o 220.7 1 17.4 1 17.8 II 09.7 o5.2 71 62.7 i3i 115.7 61.5 191 168.6 89.7 25l 221.6 12 10.6 o5.6 72 63.6 33.8 32 116. 5 62.0 92 169.5 90.1 52 222.5 118.3 t3 II. 5 06.1 73 64.5 34.3 33 117. 4 62.4 93 170.4 90.6 53 223.4 118.8 i4 12.4 06.6 74 65.3 34.7 34 118. 3 62.9 94 171.3 91.1 54 224.3 119.2 i5 l3.2 07,0 75 66.2 35.2 35 119. 2 63.4 95 172.2 91.5 55 225.2 1 19.7 i6 i4.i 07.5 76 67.1 35.7 36 120.1 63.8 96 173.1 92.0 5b 226.0 120.2 17 i5.o 08.0 77 68.0 36.1 37 121 .0 64.3 97 173.9 92.5 57 226.9 120.7 i8 i5.9 08.5 78 68.9 36.6 38 121. 8 64.8 98 174.8 93.0 58 227.8 121. 1 19 16.8 08.9 79 69.8 37.1 39 122.7 65.3 99 175.7 93.4 59 228.7 1 2 1. 6 20 17.7 09.4 80 70.6 37.6 40 123.6 65.7 200 176.6 93.9 60 261 229.6 23o.4 122.1 21 18.5 09.9 81 71.5 38. i4i 124.5 66.2 201 177.5 94.4 122.5 22 19.4 10.3 82 72.4 38.5 42 125.4 66.7 02 178.4 94.8 62 231.3 123.0 23 20.3 10.8 83 73.3 39.0 43 126.3 67.1 OJ 179.2 95.3 63 232.2 123.5 24 21 .2 II. 3 84 74.2 39.4 44 127. 1 67.6 o4 180.1 95.8 64 233.1 123.9 25 22.1 II. 7 85 75.1 39.9 45 128.0 68.1 o5 181.0 96.2 65 234.0 124.4 26 23. 12.2 86 75.9 40.4 46 128.9 68.5 06 181.9 96.7 66 234.9 124-9 27 23.8 12.7 87 76.8 40.8 47 129.8 69.0 07 182.8 97.2 67 235.7 125,3 28 24.7 i3.i 88 77-7 4i.3 48 i3o.7 69.5 08 183.7 97-7 68 236.6 125.8 29 25.6 i3.6 89 78.6 4i.8 49 i3i.6 70.0 09 184.5 98.1 69 237.5 126.3 3o 26.5 i4.i 00 79-i> 42.3 5o i32.4 70.4 10 i85.4 98.6 70 238.4 126.8 3i 27.4 i4.6 91 80.3 42.7 i5i i33.3 70.9 211 186.3 99.1 271 239.3 127.2 32 28.3 i5.o 92 81.2 43.2 52 i34.2 71-4 12 187.2 99.5 72 240.2 127.7 33 29.1 i5.5 93 82.1 43.7 53 i35.i 71.8 i3 188.1 100. 73 241.0 128.2 34 3o.o 16.0 94 83. 44.1 54 1 36.0 72.3 i4 189.0 100.5 74 241.9 128.6 35 30.9 16.4 95 83.9 44.6 55 1 36. 9 72.8 i5 189.8 100.9 75 242.8 129.1 36 3i.8 16.9 96 84.8 45.1 56 137.7 73.2 16 190.7 101.4 lb 243.7 129.6 37 32.7 17.4 97 85.6 45.5 57 i38.6 73.7 17 191.6 101.9 77 244.6 i3o.o 38 33.6 17.8 98 86.5 46.0 58 139.5 74.2 18 192.5 102.3 78 245.5 i3o.5 3q 34.4 18.3 99 87.4 46.5 59 i4o.4 74.6 19 iy3.4 102.8 79 246.3 i3i.o 40 35.3 18.8 100 88.3 46.9 60 i4i.3 75.1 20 194.2 io3.3 80 247.2 i3i.5 4i 36.2 19.2 lOI 89.2 47-4 161 142.2 75.6 221 195. 1 io3.8 2S1 248.1 1 3 1. 9 42 37.1 19.7 02 90.1 47-9 62 143.0 76.1 22 196.0 104.2 82 249.0 i32.4 43 38.0 20.2 o3 90.9 48.4 63 143.9 76.5 23 196.9 104.7 83 249.9 132.9 44 38.8 20.7 o4 91.8 48.8 64 144.8 77.0 24 197.8 io5.2 84 25o.8 i33.3 45 39.7 21 . 1 o5 92.7 49-3 65 145.7 77.5 25 198.7 105.6 85 25l.b 133.8 46 4o.6 21 .6 06 93.6 49-8 66 146.6 77-9 26 199.5 106. 1 8b 252.5 1 34-3 47 4i.5 22.1 07 94.5 5o.2 67 147-5 78.4 27 200.4 106.6 87 253.4 134.7 48 42.4 22.5 08 95.4 50.7 68 148.3 78.9 28 201.3 107.0 88 254.3 i35.2 49 43.3 23.0 09 96.2 5l.2 69 149.2 79.3 29 202.2 107.5 89 255.2 i35.7 5o 5 1 44.1 45.0 23.5 23.9 10 97.1 5i.6 70 i5o.i 79.8 3o 203.1 108.0 90 256.1 1 36. 1 III 98.0 52.1 171 iSi.o 80.3 23l 204.0 108.4 291 256.9 1 36.6 52 45.9 24.4 12 98.9 52.6 72 i5i .9 80.7 32 204.8 108.9 92 257.8 137.1 53 46.8 24.9 i3 99.8 53.1 73 i52.7 81.2 33 205.7 109.4 93 258.7 137.6 54 47.7 25.4 i4 100.7 53.5 74 i53.6 81.7 34 206.6 109.9 94 259.6 )38.o 55 48.6 25.8 i5 loi .5 54.0 75 i54.5 82.2 35 207.5 no 3 9b 260.5 i38.5 56 49-4 26.3 16 102.4 54.5 76 i55.4 82.6 36 208.4 1 10.8 96 261.4 139.0 57 5o.3 26.8 17 io3.3 54.9 77 i56.3 83.1 37 209.3 111.3 97 262.2 139.4 58 5i .2 27.2 18 104.2 55.4 78 157.2 83.6 38 210. 1 1 1 1.7 98 263.1 139.9 5q 52.1 27.7 19 io5.i 55.9 "9 i58.o 84.0 39 211.0 112.2 99 264.0 140.4 6o_ Dist" 53.0 28.2 20 106.0 56.3 80 i58.9 84.5 40 21 1.9 112.7 3oo 264.9 i4o.8 Dep. 1 Lat. Dist Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Tor C2 Deg rees. TABLE IL rr;,;.;.- I.", Differe nee of Latitude and Departure for 29 Degrees. Disl. Lat. Dcp. Dist. Lat. Dep. Dist. Lat. Dcp. 58.7 Dist. Lat. Dep. Dist. Lai. Dep. I 00.9 00.5 61 53.4 29.6 121 io5.8 iSi i58.3 S7.8 241 210.8 1 16.8 2 01.7 01 .0 62 54.2 3o.i 22 106.7 59.1 82 159.2 88.2 42 211.7 117.3 3 02.6 01 .5 63 55.1 3o.5 23 107.6 59.6 83 1 60. 1 88.7 43 212.5 117-8 4 o3.5 01 .9 64 56.0 3i .0 24 108.5 60.1 84 160.9 89.2 44 2 1 3.4 118.3 5 o4.4 02.4 65 56.9 3i.5 25 109.3 60.6 85 161.8 89-7 45 214.3 118.8 6 o5.2 02.9 66 67.7 32.0 26 no. 2 61.1 86 162.7 90.2 46 2l5.2 119.3 7 06.1 o3.4 67 58.6 32.5 27 III. I 61.6 87 i63.6 90.7 47 2l6.0 119.7 8 07.0 03.9 68 59.5 33.0 28 112. 62.1 88 164.4 91. 1 48 216.9 120.2 9 07.9 04.4 69 60.3 33.5 29 112. 8 62.5 89 i65.3 91.6 49 217.8 120.7 10 08.7 04.8 70 61 .2 33.9 3o II3.7 63.0 90 166.2 92.1 5o 218.7 121.2 II 09.6 o5.3 71 62. 1 34.4 i3i 114.6 63.5 191 167.1 92.6 25l 219.5 121.7 12 10. b o5.8 72 63.0 34.9 32 II5.4 64.0 92 167.9 93.1 52 220.4 122.2 i3 U.4 06.3 73 63.8 35.4 33 116.3 64.5 93 168.8 93.6 53 22 1.3 122.7 i4 12.2 06.8 74 64.7 35.9 34 117. 2 65. 94 169.7 94.1 54 222.2 123.1 ID i3.i 07.3 75 65.6 36.4 35 118. 1 65.4 95 170.6 94-5 55 223.0 123.6 i6 i4-o 07.8 76 66.5 36.8 36 118. 9 65.9 96 I7I-4 95.0 56 223.9 1 24. 1 17 14.9 q8.2 77 67.3 37.3 37 119.8 66.4 97 172.3 95.5 57 224.8 124.6 i8 lb. 7 08.7 7S 68.2 37.8 38 120.7 66.9 98 173.2 96.0 58 225.7 125.1 '9 lb. 6 09.2 79 69.1 38.3 39 121.6 67.4 99 174-0 96.5 59 226.5 125.6 20 17. b 09.7 80 70.0 .-.8.8 39.3 4o T41 122.4 67.9 200 174-9 97.0 60 261 227.4 228.3 126.1 21 18.4 10.2 81 70.8 123.3 68.4 201 175.8 97-4 126.5 22 19.2 10.7 82 71.7 39.8 42 124.2 68.8 02 176.7 97-9 62 229.2 127.0 23 20.1 II .2 83 72.6 40.2 43 125. 1 69.3 o3 177.5 98.4 63 23o.O 127 5 24 21 .0 II. 6 84 73. b 40.7 44 125.9 69.8 04 178-4 98.9 64 230.9 128.0 2b 21 .9 12. I 8b 74.3 4l.2 4b 126.8 70.3 o5 179.3 99.4 65 231.8 128.5 26 22.7 12.6 86 75.2 41.7 46 127.7 70.8 06 180.2 99.9 66 232.6 129.0 27 23.6 i3.i ^7 7b. I 42.2 47 128.6 71.3 07 181.0 100.4 67 233.5 129.4 28 24. b i3.6 88 77.0 42.7 48 129.4 71.8 08 181.9 100.8 68 234.4 129.9 29 2b. 4 i4.i 89 77.8 43.1 49 i3o.3 72.2 09 182.8 101.3 69 235.3 i3o.4 3o 26.2 i4.5 90 78.7 43.fi bo l3l.2 72.7 10 183.7 101.8 70 236.1 i3o.9 3i 27.1 i5.o 91 79.6 44.1 :5i I32.I 73.2 211 i84-5 102.3 271 237.0 i3i.4 32 28.0 i5.5 92 80.5 44.6 52 132.9 73.7 12 i85.4 102.8 72 237-9 i3i.9 33 28.9 16.0 93 81.3 45.1 53 i33.8 74.2 i3 186.3 io3.3 73 238.8 i32.4 34 29.7 16.5 94 82.2 45.6 54 134.7 74.7 i4 187.2 io3.7 74 239.6 i32.8 3i) 3o.b 17.0 95 83.1 46.1 55 i35.6 75.1 i5 188.0 104.2 75 240.5 i33.3 36 3i.5 17.5 96 84.0 46.5 56 i36.4 75.6 16 188.9 104.7 76 241.4 1 33.8 37 32.4 17.9 97 84.8 47-0 57 i37.3 76.1 17 189.8 10D.2 77 242.3 1 34.3 38 33.2 iS.4 98 8b. 7 47-5 58 i38.2 76.6 18 190.7 105.7 78 243.1 i34.8 39 34.1 .8.9 99 86.6 48.0 b9 139.1 77-1- 19 191. 5 106.2 79 244.0 i35.3 4o 4i 3b. 19.4 100 87. b 48. b bo 139.9 77.6 20 192.4 106.7 80 244-9 i35.7 35.9 19.9 lOI 88.3 49.0 161 140.8 78.1 221 193.3 107.1 281 245.8 i36.2 42 36.7 20.4 02 89. 2 49.5 62 i4i.7 78.5 22 194.2 107.6 82 246.6 i36.7 43 37.6 20.8 o3 90.1 49.9 63 142.6 79.0 23 195.0 108.1 83 247.5 137.2 44 38. b 21.3 04 91 .0 5o.4 64 143.4 79.5 24 195.9 108.6 84 248.4 1 37.7 4b 39.4 21.8 OD 91.8 50.9 65 144.3 80.0 25 196.8 109.1 85 249.3 i38.2 4(> 4o . 2 22.3 06 92.7 bi.4 66 145.2 80.5 26 197-7 109.6 86 25o.I i38.7 47 4i.i 22.8 07 93.6 5i ;9 67 I46.I 81.0 27 198.5 no. I 87 25l.O 139.1 48 42.0 23.3 08 94.5 52.4 68 i46.9 81.4 28 199-4 no. 5 88 251.9 139.fi 49 42.9 23.8 09 9b. 3 52.8 69 i47-8 81.9 29 200.3 in.o 89 252.8 i4o.i bo 43.7 24.2 10 96.2 b3.3 53.8 70 148.7 82.4 82.9 3o 201.2 in. 5 90 253.6 i4o.6 5i 44.6 24.7 III 97.1 171 149-6 23l 202.0 112.0 291 254.5 i4i.i b2 4b. b 25.2 12 98.0 54.3 72 i5o.4 83.4 32 202.9 112. 5 92 255.4 i4i-6 b3 46.4 2b. 7 i3 98.8 54.8 73 i5i.3 83. q 33 2o3.8 it3.o 93 256.3 142.0 b4 47.2 t 26.2 1 4 99-7 55.3 74 l52.2 84.4 34 204.7 n3.4 94 257.1 142.5 bb 48.1 2D. 7 lb 100.6 bb.8 75 i53.i 84.8 35 2o5.5 1 13.9 95 258.0 143.0 bb 49.0 27.1 16 loi .5 56.2 76 1 53. 9 85.3 36 206.4 n4.4 96 2 58.9 143.5 t)7 49.9 27.6 17 102.3 56.7 77 i54.8 85.8 37 207.3 n4-9 97 259.8 144.C b8 5o.7 28.1 18 io3.2 57.2 78 155.7 86.3 38 208.2 n5.4 98 260.6 144.5 b9 bi.6 28.6 19 104. 1 37-7 79 i56.6 86.8 39 209.0 1 15.9 99 261.5 145.0 bo b2.b 29.1 20 io5.o 58.2 8n 157-4 87-3 4o 209.9 116.4 3 00 262,4 145.4 1 Oist. Dop. I.nl. Dist. Dop. Lat. Dist. ])cp. Lat. Disl.l Dcp. Lnt. Dist. Dep. Lat. ! [ ^or Gl Degrees. Page 40] TABLE II Difference of Lati tude and Departure for 30 Degrees. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist.j Lat. Dep. Dist. Lat. ( Dep. I 00.9 00.3 bi 52.8 3o.5 121 104.8 60.5 18, 1 56.8 90.5 241 208.7 120.5 2 01.7 01 .0 62 53.7 3i .0 22 105.7 61 .0 82 157.6 91.0 42 209.6 I 21.0 3 02.6 01 .5 63 54.6 3i.5 23 106.5 61.5 83 i58.5 91.5 43 210.4 121.5 4 o3.5 02.0 64 55.4 32.0 24 107.4 62.0 84 159.3 92.0 44 211. 3 122.0 5 04.3 02.5 65 56.3 32.5 25 108.3 62.5 85 160.2 92.5 45 212.2 122.5 6 05.2 o3.o 66 57.2 33.0 26 109. 1 63.0 86 161. 1 93.0 46 2l3.0 123.0 7 06.1 o3.5 67 58. 33.5 27 IIO.O 63.5 87 161.9 93.5 47 213.9 123.5 8 06.9 04.0 68 58.9 34.0 28 no. 9 64.0 88 162.8 94.0 48 214.8 124.0 9 07.8 04.5 69 59.8 34.5 29 III. 7. 64.5 89 163.7 94.5 49 215.6 124.5 lO 08.7 o5.o 70 60.6 35.0 35.5 3o 112. 6 65. 65.5 90 164.5 95.0 5o 216.5 125.0 II 09.5 o5.5 71 61.5 i3i ii3.4 191 i65.4 95.5 25l 217-4 125.5 12 10.4 ob.o 72 62.4 36.0 32 114.3 66.0 92 166.3 96.0 52 218.2 126.0 i3 II. 3 ob.5 73 63.2 36.5 33 Il5.2 66.5 93 167. 1 96.5 53 2 1 9. 1 12fi.5 i4 12. 1 07.0 74 64.1 37.0 34 116. 67.0 94 168.0 97.0 54 220.0 1 27.0 lb i3.o 07.5 75 65.0 37.5 35 lib. 9 67.5 95 168.9 97.b 55 220.8 127.5 lb i3.9 08.0 76 65.8 38. 36 117. 8 68.0 96 169.7 98.0 56 221.7 128.0 17 14.7 08. b 77 66.7 38.5 37 118. 6 68.5 97 170.6 98.5 57 222.6 128.5 i8 ib.b 09.0 78 67.5 39.0 38 119. 5 69.0 98 171.5 99.0 58 223.4 129.0 19 ib.b 09.5 ■79 68.4 39.5 39 120.4 69.5 99 172.3 99.5 59 224.3 129.5 20 17.3 10. 80 69.3 4o.o 4o 121 .2 70.0 200 173.2 100. 60 225.2 i3o.o 21 18.2 10.5 81 70.1 40.5 i4i 122. 1 70.5 201 1 74. 1 100.5 261 226.0 i3o.5 22 19. 1 II .0 82 71.0 4i .0 42 123.0 71.0 02 174.9 lOI.O 62 226.9 i3i.o 23 19.9 II. 5 83 71.9 4i.5 43 123.8 71.5 o3 175.8 101.5 63 227.8 i3i.5 24 20.8 12.0 84 72.7 42.0 44 124.7 72.0 04 176.7 102.0 64 228.6 i32.o 2b 21.7 12. b 85 73.6 42.5 45 125.6 72.5 o5 177.5 102.5 65 229.5 i32.5 2b 22.5 i3.o 86 74.5 43.0 46 126.4 73.0 06 178.4 io3.o 66 23o.4 i33.o 27 23.4 i3.5 87 75.3 43.5 47 127.3 73.5 07 179.3 io3.5 67 23l.2 1 33.5 28 24.2 14.0 88 76.2 44.0 48 128.2 74-0 08 180.1 104.0 68 232.1 i34.o 29 2b. I 14.5 89 77-1 44.5 49 129.0 74.5 09 181.0 104.5 69 233.0 i34.5 So 3i 26.0 ib.o 90 91 77.9 78.8 45.0 45.5 5o 129.9 75.0 10 181.9 io5.o 70 233.8 1 35.0 i5.5 i5i i3o.8 75.5 211 182.7 io5.5 271 234.7 i35.5 32 27-7 16.0 92 79-7 46.0 52 i3i.6 76.0 12 i83.6 106.0 72 235.6 1 36.0 33 28. b 16.5 93 80.5 46.5 53 i32.5 76.5 i3 184.5 106.5 73 236.4 136.5 34 29.4 17.0 94 81.4 47-0 54 i33.4 77.0 i4 i85.3 107.0 74 237.3 137.0 3b 3c. 3 .7.5 95 82.3 47. b 55 i34.2 77.5 i5 186.2 107.5 75 238.2 137.5 3b 3l.2 18.0 96 83.1 48. 56 i35.i 78.0 16 187. 1 108.0 76 239.0 i38.o J7 32.0 18.5 97 84.0 48. b 57 1 36.0 78.5 17 187.9 10S.5 77 239.9 138.5 38 32 .9 19.0 98 84.9 49.0 58 i36.8 79.0 18 188.8 109.0 78 240.8 139.0 39 33.8 .9.5 99 85.7 49-^ 59 137.7 79.5 19 189.7 109.5 79 241.6 139.5 40 34. b 20.0 100 86.6 5o.o 60 i38.6 80.0 20 190. D IIO.O 80 242.5 i4o.o 4i 35.5 20.5 lOI 87.5 5o.5 161 139.4 80.5 221 191. 4 1 10.5 281 243.4 i4o.5 42 3b. 4 21 .0 02 88.3 bi.o 62 i4o.3 81.0 22 192.3 II 1. 82 244.2 i4i-o 4S 37.2 21 .5 o3 89.2 5i.5 63 i4i .2 81.5 2 3 193. 1 1 1 1.5 83 245.1 i4i.5 44 38.1 22.0 04 90.1 52.0 64 142.0 82.0 24 194.0 1 1 2.0 84 246.0 142.0 4b 39.0 22.5 o5 90.9 52. b 65 142.9 82.5 25 194.9 112. 5 85 246.8 142.5 4b 39.8 23.0 06 91.8 b3.o 66 143.8 83.0 26 195.7 ii3.o 86 247-7 143.0 47 4o . 7 23.5 07 92.7 b'3.b 67 144.6 83.5 27 196.6 ii3.5 87 248.5 143.5 48 4i.6 24.0 08 93.5 54.0 68 145.5 84.0 28 197.5 ii4-o 88 249.4 i44-o ^9 42.4 24.5 09 94.4 b4.b 69 146.4 84.5 29 198.3 114.5 89 25o.3 144.5 bo 43.3 25.0 10 95.3 bb.o 70 i47-2 85. 3o 199.2 ii5.o 90 25l.I 145.0 bi 44.2 25.5 1 1 1 9b. I 55.5 171 I48.I 85.5 23l 200.1 ii5.5 291 252.0 145.5 b2 45.0 26.0 12 97 -o bb.o 72 149.0 86.0 32 200.9 1 16.0 92 252.9 1 46.0 b3 4b. 9 2b. 5 i3 97.9 56.5 73 149-8 86.5 33 201.8 1 16.5 93 253.7 146.5 b4 4b. 8 27.0 i4 98.7 b7.o 74 1 5o . 7 87.0 34 202.6 1 17.0 94 254-6 147.0 bb 47. b 27.5 i5 99.6 bv.b 75 i5i.6 87.5 35 2o3.5 117.5- 95 255.5 147-5 bb 48. b 28.0 lb 100.5 b8.o 76 152.4 88.0 36 204.4 118.0 96 2 56.3 i48.o b7 49.4 28.5 17 loi .3 58.5 77 i53.3 88.5 37 205.2 118.5 97 257.2 148.5 b8 5().2 29.0 18 102.2 59.0 78 154.2 89.0 38 206.1 1 19.0 98 258.1 149-0 b9 bi.i 29.5 >9 io3.i 59.5 79 i55.o 89.5 39 207.0 119. 5 99 258.9 149.5 60 52.0 3o . f> 20 103.9 60 . 80 Dist. 155.9 90.0 40 207.8 120.0 3oo 259.8 i5o.o Dist. n.-p. l.nt. Dist. Dcp. I^at, Dcp. Lat. Dist.j Dep. Lat. Dist. Dep. Lat. [ For GO Degrees. TABLE IL [!• lye 47 Difference of Latitude and Departure for 31 Degrees. Dist. Lai. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. [ Dep. I 00.9 00.5 61 52.3 3i.4 121 io3.7 62.3 181 i55.i 93.2 241 206.6 124. 1 2 01 .7 01 .0 62 53.1 3i .9 22 104.6 62.8 82 1 56.0 93-7 42 207.4 ; 12-4 6 3 02.6 01.5 63 54.0 32.4 23 105.4 63.3 83 1 56.9 94.3 43 208.3 ■ 125.2 4 o3.4 02.1 64 54.9 33.0 24 106.3 63.9 84 157.7 94.8 AA 209.1 1 125.7 S 04.3 02.6 65 55.7 33.5 25 107. 1 bA.A 85 1 58.6 95.3 45 210.0 126.2 6 o5.i o3.i 66 56.6 34.0 26 108.0 64-9 86 159.4 95.8 46 210.9 126.7 06.0 o3.6 67 57.4 34.5 27 108.9 65.4 87 160.3 96.3 47 21 1.7 127.2 8 06.9 04.1 68 58.3 35.0 28 109.7 65.9 88 161. 1 96.8 48 212.6 127.7 9 07.7 04.6 69 59.1 35.5 29 no. 6 66.4 89 162.0 97-3 49 2 13.4 128.2 10 08.6 o5 .2 70 60.0 3b. I 3o III .4 67.0 90 191 162.9 763".7 97-9 5o 214.3 128.8 1 1 09.4 05.7 71 60.9 36.6 i3i 112. 3 67.5 98.4 25l 2l5.I 129.3 12 10.3 06.2 72 61.7 37.1 32 u3.i 68.0 92 164.6 98.9 52 216.0 129.8 i3 II .1 06.7 73 62.6 37.6 33 ii4-o 68.5 93 1 65. 4 99.4 53 216.9 i3o.3 i4 12.0 07.2 74 63.4 38.1 M 114-9 69.0 94 166.3 99.9 54 217.7 i3o.8 i5 12.9 07.7 75 64.3 38. b 35 115.7 69.5 95 167. 1 100.4 55 218.6 i3i.3 i6 i3.7 08.2 76 65.1 39.1 36 116. 6 70.0 96 168.0 100.9 56 219.4 i3i.8 17 i4.6 08.8 77 66.0 39.7 37 117.4 70.6 97 168.9 101.5 57 220.3 i32.4 i8 i5.4 09.3 78 66.9 4o.2 38 118. 3 71. 1 98 169.7 102.0 58 221.1 i32.9 19 16.3 09.8 79 67.7 40.7 39 119. 1 71.6 99 170.6 102.5 59 222.0 1 33.4 20 17. 1 10.3 80 68.6 4i .2 4o 120.0 72.1 200 171.4 io3.o 60 261 222.9 223.7 133.9 21 18.0 10.8 81 69.4 41.7 i4i 120.9 72.6 201 172.3 io3.5 134.4 22 18. 9 II. 3 82 70.3 42.2 42 121 .7 73.1 02 173.1 104.0 62 224.6 1 34.9 23 19.7 II. 8 83 71. 1 42.7 43 122.6 73.7 o3 174.0 104.6 63 225.4 i35.5 24 20.6 12.4 84 72.0 Ai.i AA 123.4 74.2 04 174-9 io5.i 64 226.3 i36.o 25 21 .4 12.9 85 72.9 43.8 45 124.3 74.7 o5 175-7 io5.6 65 227.1 i36.5 i6 22.3 i3.4 86 73.7 AA.'^ 46 125. 1 75.2 06 176-6 106.1 66 228.0 137.0 27 23.1 r3.9 87 74.6 44.8 47 126.0 75.7 07 177.4 106.6 67 228.9 137.5 28 24.0 14.4 88 7b. 4 45.3 48 126.9 7b. 2 08 178.3 107.1 68 229.7 i38.o 99 24.9 [4.9 i5.5 89 76.3 45.8 49 127.7 7b. 7 09 1 79. 1 107.6 69 2 3o.6 i38.5 3o 3i 25.7 90 77-1 4b. 4 5o 128.6 77-3 10 180.0 108.2 70 23i.4 139. 1 56.6 16.0 91 78.0 46.9 i5i 129.4 77.8 211 180.9 108.7 271 232.3 139.6 32 27.4 16.5 92 78.9 47-4 52 i3o.3 78.3 12 181.7 109.2 72 233.1 1 40. 1 33 28.3 17.0 93 79-7 47-9 53 i3i.i 78.8 i3 182.6 109.7 73 234.0 i4o.6 34 29. 1 17.5 94 80.6 48.4 54 l32.0 79-3 i4 i83.4 110.2 74 234.9 i4i-i 35 3o.o 18.0 95 81.4 48.9 55 i32.9 79.8 i5 184.3 110.7 75 235.7 i4i-6 3G 3o.Q 18.5 96 82.3 49-4 56 i33.7 80.3 16 i85.i 111.2 76 236.6 142.2 3? 3. .7 19. 1 97 83.1 5o.Q 57 i34.6 80.9 17 1S6.0 111.8 77 ' 237.4 142.7 38 32.6 19.6 08 84.0 5o.5 58 i35.4 81.4 18 1S6.9 112.3 78 238.3 143.2 39 33.4 20.1 99 84.9 5i.o 59 i36.3 81.9 '9 187.7 112.8 79 239.1 143.7 4o 34.3 20.6 100 85.7 5i.5 60 137. 1 82.4 20 188.6 113.3 80 240.0 144.2 4i 35.1 21 .1 101 86.6 52.0 161 i38.o 82.9 221 189.4 ii3.8 281 240.9 144.7 42 36. 21.61 03 87.4 52.5 62 i38.9 83.4 22 190.3 114.3 82 241.7 145.2 43 36.9 22. I o3 88.3 53.0 63 139.7 84.0 23 191. 1 1 14.9 83 242.6 145.8 44 37.7 22.7 04 89.1 53. b 64 1 40.6 84.5 24 192.0 1.5.4 84 243.4 146.3 45 38. 6 23.2 o5 90.0 54.1 65 141.4 85. 25 192.9 115.9 85 944.3 146.8 46 39.4 23.7 c6 90.9 54. b 66 142.3 85.5 26 193.7 116.4 86 245.1 i47-3 47 4o:3 24.2 07 91.7 55.1 67 143. 1 86.0 27 194.6 116.9 87 246.0 147-8 48 4i.i 24.7 08 92.6 55. b 68 144.0 8t3.5 28 195.4 1 17-4 88 246.9 i48.3 49 42.0 25.2 09 93.4 5b. I 69 144.9 87.0 29 196.3 "7-9 89 247-7 i48.8 bo 5i 42.9 25.8 10 94.3 5b. 7 70 145.7 87.6 3o 197-1 1 18.5 90 248.6 149.4 43.7 26.3 II I 95.1 57.2 171 i46.6 88.1 23l 198.0 1 19 291 249.4 149.9 52 44.6 26.8 12 96.0 b7.7 72 147-4 88.6 32 198.9 1 1 9-5 92 250.3 i5o.4 53 145.4 27.3 i3 96.9 58.2 73 i48.3 89.1 33 199.7 120.0 93 25l.2 1 50.9 54146.3 27.8 i4 97-7 58.7 74 149. 1 89.6 34 200.6 120.5 94 252.0 .51.4 55 47.1 28.3 i5 98.6 59.2 75 i5o.o 90. 1 35 201.4 121. 95 252.9 .51.9 56 48.0 28.8 16 99.4 59.7 76 1 50.9 90.6 36 202.3 121.5 96 253.7 i52.5 ^7 48.9 29.4 17 100.3 bo. 3 77 i5i.7 91 .2 37 2o3.i 122. 1 97 254.6 1 53.0 58 49.7 99.9 18 lOI .1 bo. 8 78 i52.6 91.7 38 204.0 122.6 98 255.4 i53.5 59 5n.6 3o.4 19 102.0 bi.3 79 1 53.4 92.2 39 204-9 123.1 99 256.3 i54.c bo 5i.4 30.9 90 102.9 bi.8 80 i54-3 92.7 40 2o5.7 123.6 3oo 2D7.1 .54.5 1 ).■,.. l.nt. nist. Dop. Lat. Disi. Dop. Lat. Dist. Dep. Lat. Dist. Dep. Lat. [For 59 Degr ees. Page 4tj] TABLE 11. '1 1 1 Differe nee of Latitude and Departure for 32 Degrees. Dist. Lat. Dep. 00.5 Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. 1 27.7 I GO. 8 61 5i.7 32.3 121 102.6 64.1 181 i53.5 95.9 241 204.4 2 01.7 01 . 1 62 52.6 32.9 22 io3.5 64-7 82 i54.3 96.4 42 205.2 12S.2 3 02.5 01 .6 63 53.4 33.4 23 104.3 65.2 83 i55.2 97.0 43 206.1 128.8 4 o3.4 02.1 64 54.3 33.9 24 io5.2 65.7 84 i56.o 97.5 44 206.9 129.3 5 04.2 02.6 65 55.1 34.4 25 106.0 66.2 85 156.9 98.0 45 207.8 129.8 6 o5.i o3.2 66 56. 35.0 26 106.9 66.8 86 157.7 98.6 46 208.6 i3o.4 7 05.9 o3.7 67 56.8 35.5 27 107.7 67.3 87 i58.6 99.1 47 209.5 iSo.i; 8 06.8 04.2 68 57.7 36.0 28 108.6 67.8 88 159.4 99.6 48 210.3 i3i.4 9 07.6 04.8 69 58.5 36.6 29 109.4 68.4 89 160.3 100.2 49 211.2 1 3 1. 9 10 08.5 o5.3 70 59.4 37.1 3o no. 2 68.9 90 161. 1 100.7 5o 212.0 i32.5 II 09.3 o5.8 7' 60.2 37.6 i3i III .1 69.4 191 162.0 101.2 25l 212.9 i33.o 12 10.2 06.4 72 61. 1 38.2 32 III .9 69.9 Q2 162.8 101.7 52 213.7 i33.5 i3 II .0 06.9 73 61 .9 38.7 33 112. 8 70.5 93 163.7 102.3 53 214.6 i34.i i4 11. 9 07.4 74 62.8 39.2 34 ii3.6 71.0 94 164.5 102.8 54 215.4 i34.6 i5 12.7 07.9 75 63.6 39.7 35 114.5 71.5 ■ 95 i65.4 io3.3 55 216.3 i35.i i6 i3.6 08.5 76 64.5 40.3 36 ii5.3 72.1 q6 166.2 103.9 56 217.1 i35.7 17 14.4 09.0 77 65.3 4o.8 37 116.2 72.6 97 167. 1 104.4 ^7 217.9 i36.2 i8 i5.3 09.5 78 66.1 4i.3 38 117. 73.1 98 167.9 104.9 58 218.8 i36.7 I 19 16. 1 10. 1 79 67.0 41.9 39 117.9 73.7 99 168.8 io5.5 59 219.6 137.2 20 17.0 10.6 80 67.8 42.4 42.9 4o i4i 118. 7 74.2 200 169.6 106.0 60 220.5 137.8 21 17.8 II .1 81 68. 7 119. 6 74.7 201 170.5 106.5 261 221.3 i38.3 22 18.7 II. 7 82 69.5 43.5 ■ 42 120.4 75.2 02 171.3 107.0 62 222.2 i38.8 23 19.5 12.2 83 70.4 44.0 43 121.3 75.8 OJ 172.2 107.6 63 223.0 139.4 24 20.4 12.7 84 71.2 44.5 44 122. 1 76.3 o4 173.0 108. 1 64 223.9 1399 25 21 .2 l3.2 85 72.1 45.0 45 123.0 76.8 o5 173.8 108.6 65 224.7 i4o.4 26 22 .0 i3.8 86 72.9 45.6 46 123.8 77.4 06 174.7 109.2 66 225.6 i4i.o 27 22.0 i4.3 87 73.8 46.1 47 124.7 77-9 07 175.5 109.7 67 226.4 i4i.5 28 23.7 i4.8 88 74.6 46.6 48 125.5 78.4 08 ilb.4 no. 2 68 227.3 142.0 29 24.6 i5.4 89 75.5 47.2 49 126.4 79.0 09 177.2 no.8 69 228.1 142.5 3o 25.4 i5.9 90 76.3 47-7 5o 127.2 79.5 10 211 178. 1 178.9 111.3 70 229.0 i43.i 3i 26.3 16.4 91 77.2 48.2 i5i 128. 1 80.0 111.8 271 229.8 143.6 32 27.1 17.0 92 78.0 48.8 52 128.9 80.5 12 179.8 112.3 72 230.7 144.1 33 28.0 17.5 93 78.9 49.3 53 129.8 81. 1 i3 180.6 112. 9 73 23i.5 144-7 34 28.8 18.0 94 79-7 49-8 i.4 i3o.6 81.6 i4 181.5 113.4 74 232.4 145.2 35 29.7 18.5 95 80.6 5o.3 55 i3i.4 82.1 i5 182.3 113.9 75 233.2 145.7 36 3o.5 19. 1 96 81.4 50.9 56 i32.3 82.7 16 i83.2 1 14.5 76 234.1 146.3 37 3i.4 19.6 97 82.3 5i.4 57 i33.i 83.2 17 184.0 n5.o 77 234.9 146.8 38 32.2 20. 1 98 83.1 51.9 58 i34.o 83.7 18 184.9 n5.5 78 235.8 147-3 3q 33.1 20.7 9Q 84. 52.5 59 i34.8 84.3 19 185.7 116.1 79 236.6 147-8 4o 33.9 21 .2 100 84.8 53.0 60 i35.7 84.8 85.3 20 186.6 116.6 80 281 237.5 238.3 i48.4 148.9 4! 34.8 21.7 lOI 85.7 53.5 161 i36.5 221 187.4 117. 1 4a 35.6 22.3 02 86.5 54.1 62 137.4 85.8 22 188.3 117.6 82 239.1 i49-4 43 36.5 22.8 o3 87.3 54.6 63 i38.2 86.4 23 189.1 118.2 83 240.0 i5o.o 44 37.3 23.3 04 88.2 55.1 64 139. 1 86.9 24 190.0 118.7 84 240.8 i5o.5 45 38.2 23.8 o5 89.0 55.6 65 139.9 87.4 25 190.8 119.2 85 241.7 i5i.o 46 39.0 24.4 06 89.9 56.2 66 i4o.8 88.0 26 191.7 119.8 86 242.5 i5i.6 ^1 39.9 24.9 07 90.7 56.7 67 i4i.6 88.5 27 192.5 120.3 87 243.4 l52.I 48 40.7 25.4 08 91 .6 57.2 68 i42.5 89.0 28 193.4 120.8 88 244.2 i52.6 49 4i.6 26.0 09 92.4 57.8 69 143.3 89.6 29 194.2 1 2 1. 4 89 245.1 i53.i 5o 42.4 26.5 10 93.3 58.3 58.8 70 171 i44.2 90. 1 90.6 3o 1 95. 1 121. 9 90 245.9 i53.7 5i 43.3 27.0 1 1 1 94.1 145.0 23l 195.9 122.4 291 246.8 i54.2 52 44.1 27.6 12 95.0 59.4 72 145.9 91. 1 32 196.7 122.9 92 247.6 154.7 53 44. q 28.1 i3 95.8 59.9 73 146.7 91.7 33 197.6 123.5 93 248.5 i55.3 54 45.8 28.6 i4 96.7 60.4 74 i47-C 92.2 34 198.4 124.0 94 249.3 i55.8 55 46.6 29.1 i5 97. b 60.9 7^ 148.4 92.7 35 199.3 124.5 95 250.2 i56.3 56 47.5 29.7 16 98.4 61.5 7^3 149.3 93.3 36 200.1 I25.I 96 2DI.0 1 56.9 5? 48.3 3o.2 17 99.2 62 .0 77 1 5o . I 93.8 37 201.0 125.6 97 251.9 157.4 58 49-2 3o.7 18 100. 1 62.5 78 1 5 1 .0 94.3 38 201.8 1 16. 1 98 252.7 157.9 59 5o.o 3[.3 19 100.9 63.1 79 i5i.8 94.9 39 202.7 126.7 99 253.6 i5S.4 6o IV-sl. 50.9 3i.8 20 1 1 . 8 63.6 80 i52.6 9^-4 40 2o3.5 127.2 3oo 254.4 159.0 Dep. Lat. Oisl Dep. 1 Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. [For 58 Degrees. TABLE IL [I'ugp '19 Difterence of Latitude and Departure for 33 Degrees. i)isl. Lat. Dep. Disl. Lat. Dep. Dist. Lat. Qcp. Dlst. Lat. Dep. Disl. Lat. Dep. I 00.8 00.5 61 5i .2 33.2 121 ioi.5 65.9 iSi i5i.8 98.6 24 1 202.1 i3i.3 2 01.7 01 .1 62 52.0 33.8 22 102.3 66.4 82 1 52.6 99.1 42 2o3.o i3i.8 3 02.5 01 .6 63 52.8 34.3 23 I03.2 67.0 83 i53.5 99-7 43 2o3.8 i32.3 4 o3.4 02.2 64 53.7 34.9 24 104.0 67.5 84 154.3 100.2 A4 2o4-6 132.9 5 o4.2 02.7 65 54.5 35.4 25 104.8 68.1 85 i55.2 100.8 45 2o5.5 i33.4 6 o5.o o3.3 66 55.4 35. Q 26 105.7 68.6 86 i56.o 101.3 46 206.3 1 34.0 7 05.9 o3.S 67 56.2 36.5 27 106.5 69.2 87 i56.8 101.8 47 207.2 134-5 8 06.7 04.4 68 57.0 37:0 28 1 07 . 3 69.7 88 i57.7 102.4 48 208.0 i35.i 9 07.5 04.9 69 57.9 37.6 29 108.2 70.3 89 i58.5 102.9 49 208.8 135.6 10 08.4 o5.4 70 58.7 38.1 3o 109.0 70.8 90 159.3 io3.5 60 209.7 i36.2 u 09.2 06.0 71 59.5 38.7 i3i 109.9 71.3 191 160.2 io4-o 25l 210.5 136.7 12 10. 1 06.5 72 60.4 39.2 32 1 10.7 71-9 92 161.0 104.6 52 211.3 137.2 i3 10.9 07.1 73 61 .2 39.8 33 III. 5 72.4 93 161.9 io5.i 53 212.2 137.8 i4 II. 7 07.6 74 62.1 4o.3 34 112. 4 73.0 94 162.7 105.7 54 21 3.0 i38.3 13 12.6 0S.2 75 62.9 4o.8 35 I l3.2 73.5 95 i63.5 106.2 55 213.9 138.9 i6 i3.4 08.7 76 63.7 41.4 36 ii4.i 74-1 96 164.4 106.7 56 214.7 139.4 17 i4.3 09.3 77 64.6 41.9 37 114.9 74.6 97 i65.2 107.3 b7 2i5.5 i4o.o i8 I3.I 09.8 78 65.4 42.5 38 u5.7 73. 2 98 166. 1 107.8 58 216.4 140.5 19 i5.9 10.3 79 66.3 43.0 39 116. 6 75.7 99 166.9 108.4 59 217.2 i4i-i 20 16.8 10.9 80 67.1 43.6 40 117. 4 76.2 200 167.7 108.9 60 21S.1 i4i.6 21 .7.6 II. 4 81 67.9 44.1 i4i 118.3 76.8 201 168.6 109.D 261 218.9 142.2 22 18.5 12.0 82 68.8 44.7 42 119. 1 77.3 02 169.4 IIO.O 62 219.7 142.7 23 19.3 12.5 83 69.6 45.2 43 119. 9 77-9 o3 170.3 110.6 63 220.6 143.2 24 20.1 i3.i 84 70.4 45.7 4^ 120.8 78.4 04 171. 1 III. I 64 221.4 143.8 23 2! .0 i3.6 85 71.3 46.3 45 121. 6 79-0 o5 171.9 111.7 65 222.2 144.3 26 21.8 14.2 86 72.1 46.8 46 122.4 79.5 06 172.8 112.2 66 223.1 144.9 27 22.6 14.7 87 73.0 47.4 47 123.3 80.1 07 173.6 112. 7 67 223.9 145.4 28 23.5 l5,2 88 73.8 47-9 48 124. 1 80.6 08 174.4 ii3.3 68 224.8 146 29 24.3 i5.8 89 74.6 48.5 49 125.0 81.2 09 175.3 ii3.8 69 225.6 146.5 3o 25.2 16.3 90 75.5 49.0 49- (3 5o 125.8 81.7 10 1 76. 1 1 14.4 70 226.4 i47-i 3i 26.0 16.9 91 76.3 i5i 126.6 82.2 211 177.0 114.9 271 227.3 147.6 32 26.8 17.4 92 77.2 .50.1 52 127.5 82.8 12 177-8 ii5.5 72 22S.I 1 48. 1 33 27.7 18.0 93 78.0 50.7 53 128.3 83.3 i3 178.6 1 16.0 73 229.0 148.7 34 28.5 18.5 94 78.8 5l.2 54 129.2 83.9 i4 179.5 116.6 74 229.8 149-2 35 29.4 19. 1 95 79-7 5l.7 55 i3o;o 84-4 i5 180.3 117.1 7b 23o.6 149.8 36 3o.2 19.6 96 80.5 52.3 56 i3o.8 85.0 16 181.2 1 1 7.6 76 23i.5 i5o.3 37 3i.o 20.2 97 61.4 52.8 57 i3i .7 85.5 17 182.0 118.2 77 232.3 i5o.9 38 3, .9 20.7 98 82.2 53.4 58 i32.5 86.1 18 182.8 1 18.7 78 233.2 i5i.4 09 32.7 21.2 99 83. 53.9 59 i33.3 86.6 19 183.7 119.3 79 234.0 l52.0 40 33.5 21.8 100 83.9 54.5 60 i34.2 87.1 20 184.5 119.8 80 234.8 i52.5 4i 34.4 22.3 lOI 84.7 55.0 161 i35.o 87.7 221 i85.3 120.4 281 235.7 i53.o 42 35.2 22.9 02 85.5 55.6 62 135.9 88.2 22 186.2 120.9 82 236.5 1 53.6 43 36.1 23.4 o3 86.4 56.1 63 i36.7 88.8 23 187.0 121. 5 83 237.3 i54-i 44 36.9 24.0 o4 87.2 56.6 64 137.5 89.3 24 187.9 122.0 84 238.2 1 54-7 45 37.7 24.5 o5 88.1 57.2 65 i38.4 89.9 25 188.7 122.5 85 239.0 i55.2 46 38.6 25.1 06 88.9 57.7 66 139.2 90.4 26 189.5 123. 1 86 239.9 i55.8 47 3q.4 25.6 07 89.7 58.3 67 i4o. I 91 .0 27 190.4 123.6 87 240.7 1 56.3 48 40.3 26.1 08 90.6 58.8 68 140.9 91.5 28 191. 2 124.2 88 241-5 i56.9 49 4i.i 26.7 09 91.4 59.4 69 i4i .7 92.0 29 192. 1 124.7 89 242.4 157.4 5o 5. 4i .9 27.2 10 92.3 59.9 70 142.6 92.6 3o 192.9 125.3 90 243.2 157.9 42.8 27.8 I II 93.1 60.5 171 143.4 93. 1 23l 193.7 125.8 291 244.1 i58.5 52 43.6 28.3 12 93.9 61 .0 72 144.3 93-7 32 194.6 126.4 92 244.9 159.0 53 4i.4 28.9 i3 94.8 61.5 73 145.1 94 . 2 33 19^-4 126.9 93 245.7 159.6 54 45.3 29.4 1 4 95.6 62. 1 74 145.9 94.8 ■34 196.2 127.4 9^ 246.6 160.1 55 46.1 3o .0 i5 96.4 62.6 75 i46.8 95.3 35 197-1 128.0 95 247-4 160.7 56 47-0 3o.5 16 97.3 63.2 76 147-6 95.9 36 197.9 128.5 96 248.2 161.2 !)7 47.8 3i .0 '7 98.1 63.7 77 148.4 96.4 37 198.8 129.1 97 249-1 i6i.8 58 48.6 3i.6 18 99.0 64.3 78 149-3 96.9 38 199.6 129.6 98 249.9 162.3 59 49.5 32.1 19 99.8 64.8 79 1 5o . 1 97 . 5 39 200.4 l3o.2 99 25o.8 162.8 bo 5o.3 32.7 20 100.6 65.4 80 1 5 1 . 98 . 4o 201.3 1 30.7 3 00 25i.6 1 63.4 r-ist.i Dcp.i Lai. DIst. Hop. Lat. Dist-l Dop. 1 Lat. Disl. Dop. Lai. Disl Dep. Lat. [For 57 Degrees. Page 50J TABLE IL Difference of Latitude and Departure for 34 Degrees. Disl. I Lai. Dcp. 00.6 Disl. ~67 Lat. Dep. 34.1 Disl. Lat. Dep. Disl. Lat. Dcp. Dist. Lat. Dep. 00.8 5o.6 121 100.3 67.7 181 i5o.i 101.2 241 199.8 134.8 2 01.7 01 .1 62 5i.4 34.7 22 lOI .1 68.2 82 i5o.9 101.8 42 200.6 i35.3 3 02.5 01.7 63 52.2 35.2 23 102.0 68.8 83 i5i.7 102.3 43 201.5 135.9 4 o3.3 02.2 64 53.1 35.8 24 102.8 69.3 84 i52.5 102.9 AA 202.3 1 36.4 5 04. 1 02.8 65 53. q 36.3 2 5 io3.6' 69.9 85 i53.4 io3.5 45 203.1 137.0 6 o5.o o3.4 66 54.7 36.9 26 IC4.5 70. b 86 i54.2 104.0 46 203.9 137.6 7 o5.8 o3.9 67 55.5 37.5 27 io5.3 71.0 87 1 55.0 104.6 47 204.8 i38.i 8 06.6 o4.5 68 56.4 38.0 28 !06.I 71.6 88 i55.9 io5.i 48 20b.6 i38.7 9 07.5 o5.o 69 57.2 38.6 29 106.9 72.1 89 i56.7 105.7 49 206.4 139.2 lO 08.3 ob.6 70 58. 39.1 3o 107.8 72.7 90 ib7.b 106.2 bo 207.3 139.8 1 1 09.1 06.2 71 58.9 39.7 i3i 108.6 73.3 191 i58.3 106.8 25l 208. 1 140.4 12 09.9 06.7 72 59.7 4o.3 32 109.4 73.8 92 159.2 107.4 52 208.9. 140.9 IJ 10.8 07.3 73 60.5 4o.8 33 no. 3 74.4 93 160.0 107.9 53 209.7 i4i.5 i4 II. 6 07.8 74 61.3 4i.4 34 III. I 74.9 94 160.8 108.5 54 210.6 142.0 lb 12.4 08.4 75 62.2 41.9 35 III .9 7b. b 95 161.7 109.0 55 211.4 142.6 lb i3.3 08.9 76 63.0 42.5 36 112.7 7b. I 96 Ib2.b 109.6 56 212.2 143.2 17 14.1 09.5 77 63.8 43.1 37 ii3.6 76.6 97 i63.3 110.2 57 2l3.I 143.7 i8 14.9 10. 1 78 64.7 43.6 38 114. 4 77.2 98 164.1 1 10.7 58 213.9 144-3 19 lb. 8 10.6 79 65.5 44.2 39 Il5.2 77.7 99 i65.o 1 1 1.3 59 2i4-7 1 44 -8 20 lb. 6 II .2 80 66.3 44.7 45.3 40 116. 1 78.3 200 i65.8 1 1 1.8 60 2i5.5 145.4 21 17.4 II. 7 81 67.2 i4i 116. 9 78.8 201 166.6 112.4 261 216.4 145.9 22 18.2 12.3 82 68.0 45.9 42 117. 7 79-4 02 167.5 ii3.o 62 217.2 146.5 23 19. 1 12.9 83 68.8 46.4 43 118. 6 80.0 o3 168.3 ii3.5 63 218.0 i47-i 24 19.9 i3.4 84 69.6 47-0 M 119. 4 80.5 04 169.1 ii4.i 64 218.9 147-6 2!) 20.7 i4.o 85 70.5 47.5 45 120.2 81.1 o5 170.0 114.6 65 219.7 148.2 2b 21 .6 14.5 86 71.3 48.1 46 121 .0 81.6 06 170.8 Il5.2 66 220.5 148.7 27 22.4 i5.i 87 72.1 48.6 47 121 .9 82.2 07 171.6 ii5.8 67 221.4 149.3 28 23.2 l5.7 88 73.0 49.2 48 122 .7 82.8 08 172.4 116.3 68 222.2 149-9 29 24.0 16.2 89 73.8 49-8 49 123.5 83.3 09 173.3 116.9 69 223.0 i5o.4 3o 24.9 16.8 90 74.6 5o.3 5o 124.4 83.9 10 174. 1 1 17-4 70 223.8 i5i.o 3i 25.7 .7.3 91 75.4 5o.9 i5i 125.2 84.4 211 174.9 1 18.0 271 224.7 i5i.5 32 26.5 17.9 92 76.3 5i.4 52 126.0 85.0 12 175.8 118.5 72 225.5 l52.I 33 27.4 18.5 93 77-1 52.0 53 126.8 85.6 i3 176.6 119. 1 73 226.3 i52.7 M 28.2 19.0 94 77-9 52.6 54 127.7 86.1 14 177-4 119.7 74 227.2 i53.2 3b 29.0 19.6 q5 78.8 53.1 55 128.5 86.7 i5 178.2 120.2 75 228.0 i53.8 36 29.8 20. 1 96 79.6 53.7 56 129.3 87.2 16 179-1 120.S 76 228.8 i54.3 37 3o.7 20.7 97 80.4 54.2 57 i3o.2 87.8 17 179.9 1 2 1. 3 77 229.6 1 54-9 38 3i.5 21.2 98 81.2 54.8 58 i3i .0 88.4 18 180.7 1 2 1. 9 78 230.5 i55.5 39 32.3 21.8 99 82.1 55.4 59 i3i.8 88.9 19 181.6 122.5 79 231.3 1 56.0 4o 4i 33.2 34.0 22.4 22 .9 IOC) 82.9 55.9 56.5 60 i32.6 89.5 20 182.4 123.0 80 232.1 i56.6 lOI 83.7 161 i33.5 90.0 221 i83.2 123.6 281 233.0 I57-I 42 34.8 23.5 02 84.6 57.0 62 i34.3 90.6 22 184.0 124. 1 82 233.8 157.7 43 35.6 24.0 o3 85.4 57.6 63 i35.i 91. 1 23 i84-9 124.7 83 234.6 i58.3 44 36.5 24.6 04 86.2 58.2 64 1 36.0 91.7 24 185.7 125.3 84 235.4 1 58.8 45 37.3 25.2 o5 87.0 58.7 65 i36.8 92.3 25 186.5 125.8 85 236.3 159.4 46 38.1 25.7 06 87.9 59.3 66 137.6 92.8 26 187.4 126.4 86 237.1 159.9 47 39.0 26.3 07 88.7 59.8 67 , 1 38. 4 93.4 27 188.2 126.0 87 237.9 160.5 48 39.8 26.8 08 89.5 60.4 68 139.3 93.9 28 189.0 127.5 88 238.8 1 61.0 f9 40.6 27.4 09 90.4 61 .0 69 i4o.i 94.5 29 189.8 128.! 89 239.6 i6r.6 bo TT 4i.b 42.3 28.0 28.5 10 1 1 1 91.2 61. b 70 140.9 9b.. 3o 190.7 128.6 90 240.4 162.2 92.0 62.1 171 i4i.8 95.6 23l i9'-5 129.2 291 241.2 162.7 b2 43.1 29. T 12 92.9 62.6 72 142.6 96.2 32 ,92.3 129.7 92 242.1 i63.3 b3 43.9 29.6 i3 93.7 63.2 73 143.4 96.7 33 193.2 i3o.3 93 242.9 i63.8 b4 44.8 3<). 2 i4 94.5 63.7 74 144.3 97.3 34 194.0 i3o.9 94 243.7 164.4 bb 45.6 3o.8 i5 95.3 64.3 75 145.1 97-9 35 194.8 i3i.4 95 244.6 i65.o bb 46.4 3i.3 16 96.2 64.9 76 145.9 98.4 36 195.7 l32.0 96 245.4 i65.5 b7 47.3 31.9 17 97.0 bb.4 77 146.7 99.0 37 196.5 i32.5 97 246.2 166.1 b8 48.1 32.4 18 97.8 66.0 78 147-6 99.5 38 197.3 i33.i 98 247-1 166.6 b9 48.9 33.0 '9 98.7 bb.b 79 148.4 1 00 . 1 39 198.1 i33.6 99 247-9 167.2 bo Disl. 49.7 33.6 Dep. I, at. 20 99.5 b7.i 80 149.2 1 00 . 7 40 199-0 i34.2 3oo 248.7 167.8 Disl. Dcp. Lai. Disl. Dop. Lat. Disl. Dep. Lat. Disl. Dep. Lat. 1 [ '^cr 5G Degrees. TABLE IL [Page 5] 1 Difference of Latitude and Departure for 35 Degrees. Disi. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 00.8 00.6 61 5o.o 35.0 121 99.1 69.4 181 148.3 io3.8 241 197.4 i38.2 2 01 .6 01. 1 62 5o.8 35.6 22 99.9 70.0 82 i49-i 104.4 42 198.2 i38.8 3 02.5 01.7 63 5i.6 36.1 23 100. 8 70.5 83 i49-9 io5.o 43 199.1 139.4 4 o3.3 02.3 64 52.4 36.7 24 loi .6 71.1 84 i5o.7 io5.5 M 199.9 1 40.0 5 o4.i 02.9 65 53.2 37.3 25 102.4 71.7 85 i5i.5 106. 1 45 200.7 i4o.5 6 04.9 o3.4 66 54.1 37.9 26 I03.2 72.3 86 152.4 106.7 46 201.5 i4i.i 7 o5.7 o4.o 67 54.9 38.4 27 104.0 72.8 87 i53.2 107.3 47 202.3 141.7 8 06.6 o4.6 68 55.7 39.0 28 104.9 73.4 88 i54.o 107.8 48 2o3.I 142.2 9 07.4 05.2 69 56.5 39.6 29 105.7 74.0 89 1 54.8 108.4 49 2040 142.8 10 08.2 o5.7 70 57.3 4o.2 3o 106.5 74.6 90 i55.6 109.0 5o 204.8 143.4 II 09.0 06.3 71 58.2 40.7 i3i 107.3 75.1 191 i56.5 109.6 25l 2o5.6 144.0 12 log. 8 06.9 72 59.0 4i.3 32 108. 1 75.7 92 157.3 1 10. 1 52 206.4 144.5 i3 10.6 07.5 73 59.8 4i .9 33 108.9 76.3 93 i58.i 110.7 53 207.2 145.1 i4 II. 5 08.0 74 60.6 42.4 M 109.8 76.9 94 i58.9 111.3 54 208.1 1457 i5 12.3 08.6 75 61.4 43.0 35 110.6 77-4 95 159.7 111.8 55 208.9 i46.3 if) i3.i 09.2 76 62.3 43.6 36 III. 4 78.0 96 160.6 112.4 56 2097 i46.8 17 .3.9 09.8 77 63.1 44.2 37 112. 2 78.6 97 161.4 1 i3.o 57 210.5 147.4 i8 14.7 10.3 78 63.9 44.7 38 ii3.o 79.2 98 162.2 ii3.6 58 211.3 i48.o '9 i5.6 10.9 79 64.7 45.3 39 113.9 79-7 99 1 63.0 114.1 59 212.2 1 48 .6 20 16.4 11.5 80 81 65.5 66.4 45.9 46.5 40 114.7 80.3 200 1 63.8 114.7 60 2l3.0 i49-i 21 17.2 12 .0 i4i ii5.5 80.9 201 164.6 n5.3 261 2i3.8 149.7 22 18.0 12.6 82 67.2 47 -o 42 116. 3 81.4 02 i65.5 1.5.9 62 214.6 i5o.3 23 18.8 l3.2 83 68.0 47.6 43 117. 1 82.0 o3 166.3 116.4 63 21 5.4 i5o.9 24 19.7 i3.8 84 68.8 48.2 M 118. 82.6 04 167.1 1 17.0 64 216.3 i5i.4 25 20.5 i4.3 85 69.6 48.8 i5 118. 8 83.2 o5 167.9 117.6 65 217.1 1 52.0 26 21.3 14.9 86 70.4 49-3 46 119.6 83.7 06 168.7 118. 2 66 217.9 i526 27 22.1 i5.S 87 71.3 49.9 47 120.4 84.3 07 169.6 118.7 67 218.7 i53.i 28 22.0 16,1 88 72.1 5o.5 48 121 .2 84.9 08 170.4 1 19.3 68 219.5 153.7 29 23.8 16.6 89 72.9 5i .0 49 122. 1 85.5 09 171.2 119.9 69 220.4 1 54.3 3o 24.6 17.2 90 73.7 5i.6 5o 122.9 86.0 10 172.0 120.5 70 221.2 154.9 3i 25.4 17.8 91 74.5 52.2 i5i 123.7 86.6 211 172.8 121.0 271 222.0 i55.4 32 26.2 18.4 92 75.4 52.8 52 124.5 87.2 12 173.7 1 2 1. 6 72 222.8 i56.o 33 27.0 18.9 93 76.2 53.3 53 125.3 87.8 i3 174.5 122.2 73 223.6 i56.6 34 27.9 19.5 94 77.0 53.9 54 126.1 88.3 14 175.3 122.7 74 224.4 157.2 35 28.7 20.1 95 77.8 54.5 55 127.0 88.9 i5 176.1 123.3 75 225.3 j57.7 36 29.5 20.6 96 78.6 55.1 56 127.8 89.5 16 176.9 123.9 124.5 76 226.1 i58.3 37 3o.3 2! .2 97 79.5 55.6 57 128.6 90.1 17 177.8 77 226.9 1 58.9 38 3i.i 21.8 98 80.3 56.2 58 129.4 90.6 18 178.6 125.0 78 227.7 159.5 39 3, .9 22.4 99 81. 1 56.8 59 i3o.2 91.2 19 179.4 125.6 79 228.5 160.0 40 4i 32.8 22.9 100 81.9 57.4 60 i3i.i 91 .8 20 180.2 126.2 80 229.4 160.6 33.6 23.5 lOI 82.7 57.9 161 i3i .9 92.3 221 181.0 126.8 281 23o.2 161.2 42 34.4 24.1 02 83.6 58.5 62 132.7 92.9 22 181.9 127.3 82 23l.O 161.7 43 35.2 24.7 o3 84.4 59.1 63 i33.5 93.5 23 182.7 127.9 83 23i.8 162.3 U 36.0 25.2 o4 85.2 59.7 64 i34.3 94.1 24 i83.5 128.5 84 232.6 162.9 i63.5 45 36.9 25.8 o5 86.0 60.2 65 i35.2 94.6 25 184.3 129.1 85 233.5 46 37.7 26.4 06 86.8 60.8 66 i36.o 95.2 26 i85.i 129.6 86 234.3 164.0 47 38.5 27.0 07 87.6 61.4 67 i36.8 95.8 27 165.9 l3o.2 87 235.1 164.6 48 39.3 27.5 08 88.5 61.9 68 i37.6 96.4 28 186.8 i3o.8 88 235.9 i65.2 49 4o.i 28.1 09 89.3 62.5 69 i3S.4 96.9 29 187.6 i3i.3 89 236.7 i65.8 5o 4i.o 28.7 10 90.1 63.1 70 139.3 97.5 3o 188.4 i3i.9 90 237.6 166.3 5i 41.8 29.3 III 90.9 63.7 171 i4o.i 98.1 23 I 189.2 i32.5 291 238.4 166.9 52 42.6 29.8 12 91.7 64.2 72 140.9 98.7 32 190.0 i33.i 92 239.2 167.5 53 Ai.A 3o.4 i3 92.6 64.8 73 141.7 99.2 33 190.9 i33.6 93 240.0 168. 1 54 44.2 3i.o i4 93.4 65.4 74 142.5 99.8 34 191.7 1 34.2 94 240.8 168.6 55 45.1 3i.5 i5 94.2 66.0 75 143.4 100.4 35 192.5 i34.8 95 241.6 169.2 56 45.9 32.1 16 95.0 66.5 76 144.2 100.9 36 193.3 1 35.4 96 242.5 169.8 67 46.7 32.7 17 95.8 67.1 77 145.0 loi .5 37 194. 1 135.9 97 243.3 170.4 58 47. b ii.i 18 96.7 67.7 78 145.8 102.1 38 195.0 i36.5 98 244.1 170.9 D9 48.3 33.8 19 97.5 68.3 79 i46.6 102.7 39 195.8 i37.i 99 244.9 171.5 Oo 49.1 M.A 20 98.3 68.8 80 147-4 I03.2 4o 196.6 137.7 3oo 245.7 172.1 Dist. Dep. Lat. Din. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. i Lat. [ For 55 Degrees. Page 52] TABLE IL Difference of Latitude and Departure for 36 Degrees. Dibl. Lai. Dep. 00.6 Dist. Lat. Dep. Dist. Lat. Dep. Dist.| Lat. i Dep. Dist. 241 Lat. D?p. I 00.8 61 49.4 35.9 121 97-9 71. 1 181 146.4 106.4 195.0 141.7 2 01.6 01 .2 62 50.2 36.4 22 98.7 71.7 82 147.2 107.0 42 ^95. 8 1 :42.2 1 3 02.4 01.8 63 5i.o 37.0 23 99.5 72.3 83 148.1 107.6 43 196.6 142.8 4 03.2 02.4 64 5i.8 37.6 24 100.3 72.9 84 14S.9 108.2 44 1974 143.4 5 o4.o 02.9 65 52.6 38.2 25 lOI .1 73.5 85 i49-7 108.7 45 198.2 144.0 6 04.9 o3.6 66 53.4 38.8 26 loi .9 74.1 86 i5o.5 109.3 46 , 199.0 144.6 7 o5.7 o4.i 67 54.2 39.4 27 102.7 74.6 87 i5i.3 109.9 47 199.8 145.2 B 06.5 04.7 68 55.0 4o.o 28 io3.6 75.2 88 l52.I 110.5 48 200.6 145.8 9 07.3 o5.3 69 55.8 40.6 29 io4.4 75.8 89 152.9 III. I 49 201.4 146.4 10 08.1 o5.9 70 56.6 4i.i 3o io5.2 76.4 90 1.9 1 153.7 154.5 111.7 5o 202.3 146.9 II 08.9 06.5 71 57.4 4i.7 i3i 106.0 77.0 1 12.3 25l 2o3.I 147-5 12 09.7 07.1 72 58.2 42.3 32 106.8 77.6 92 i55.3 112. 9 52 203.9 I48.I i3 10.5 07.6 73 59. 1 42.9 33 107.6 78.2 93 i56.i ii3.4 53 204.7 148.7 i4 11. 3 08.2 74 59.9 43.5 34 108.4 ■ 78.8 94 i56.9 ii4.o 54 2o5.5 149-3 lb 12. 1 08.8 7^ 60.7 44.1 35 109.2 79-4 95 157.8 1 14.6 55 206.3 149.9 i6 12.9 09.4 76 61.5 44.7 36 IIO.O 79-9 96 1 58.6 Il5.2 56 207.1 i5o.5 17 i3.8 lO.O 77 62.3 45.3 37 no. 8 80.5 97 159.4 II5.8 57 207.9 i5i.i la 14.6 10.6 78 63.1 45.8 38 III .6 81. 1 98 160.2 116.4 58 208.7 i5i.6 19 i5.4 II .2 79 63.9 46.4 39 112. 5 81.7 99 161.0 1 17.0 59 309.:) l52.2 20 16.2 II. 8 80 64.7 47-0 40 ii3.3 82.3 200 161.8 117.6 60 210.3 i52.8 21 17.0 12.3 81 65.5 47-6 i4i 1 14. 1 82.9 201 162.6 118.1 261 211.2 i53.4 22 17.8 12.9 82 66.3 48.2 42 114.9 83.5 02 163.4 118.7 62 212.0 i54-o 23 18.6 i3.5 83 67.1 48.8 43 ii5.7 84.1 OJ 164.2 1 19.3 63 212.8 i54.6 24 19.4 i4.i 84 68.0 49-4 44 116. 5 84.6 o4 i65.o 119.9 64 2i3.6 i55.2 25 20.2 14.7 85 68.8 5o.o 45 117. 3 85.2 o5 i65.8 120.5 65 214.4 1 55.8 26 21 .0 i5.3 86 69.6 5o.5 46 118. 1 85.8 06 166.7 121. 1 66 2l5.2 1 56.4 27 21.8 i5.9 87 70.4 5t.i 47 118. 9 86.4 07 167.5 121. 7 67 216,0 1 56.9 2» 22.7 16.5 88 71.2 5i.7 48 119-7 87.0 08 168.3 122.3 68 216.8 157.5 29 23.5 17.0 89 72.0 52.3 49 120.5 87.6 09 169.1 122.8 69 , 217.6 i58.i 3o 24.3 17.6 90 91 72.8 52.9 5o 121 .4 88.2 10 169.9 123.4 70 ' 218.4 1 58.7 3i 25.1 18.2 73.6 53.5 i5i 122.2 88.8 211 170.7 124.0 271 219.2 159.3 32 25.9 18.8 92 74.4 54.1 52 123. 89.3 12 171.5 124.6 72 220.1 159-9 33 26.7 19.4 93 75.2 54.7 53 123.8 89.9 i3 172.3 125.2 73 220.9 160.5 34 27.5 20.0 94 76.0 55.3 54 124.6 90.5 i4 173. 1 125.8 74 221.7 161. 1 3b 28.3 20.6 95 76.9 55.8 55 125.4 91. 1 i5 173.9 126.4 75 222.5 161.6 36 29.1 21 .2 96 77-7 56.4 56 126.2 91.7 16 174.7 127.0 76 223.3 162.2 37 29.9 21.7 97 78.5 57.0 67 127.0 92.3 17 175.6 127.5 77 224.1 162.8 38 3o.7 22.3 98 79-3 57.6 58 127.8 92.9 18 176.4 128. 1 78 224.9 i63.4 39 3i.6 22.9 99 80.1 58.2 59 128.6 93. b 19 177.2 128.7 79 225.7 164.0 40 32.4 23.5 100 80.9 58.8 6c 129.4 94.0 20 178.0 129.3 80 226.5 164.6 4i 33.2 24. 1 lOI 81.7 59.4 161 i3o.3 94.6 221 178.8 129.0 281 227.3 i65.2 42 34.0 24.7 02 82.5 60.0 62 i3i.i 95.2 22 179.6 i3o.5 82 228.1 i65.8 43 34.8 25.3 o3 83.3 60.5 63 i3i.9 95.8 23 180.4 i3i.i 83 229.0 166.3 44 35.6 25.9 o4 84.1 61. 1 64 132.7 96.4 24 181.2 i3i.7 84 229.8 166.9 45 36.4 26.5 o5 84.9 61.7 65 i33.5 97.0 25 182.0 i32.3 85 23o.6 167.5 46 37.2 27.0 06 85.8 62.3 66 i34.3 97.6 26 182.8 i32.8 86 23 1. 4 168.1 47 38. 27.6 07 86.6 62.9 67 i35.i 98.2 27 i83.6 i33.4 87 232.2 168.7 48 38.8 28.2 08 87.4 63.5 68 135.9 98.7 28 184.5 i34.o 88 233.0 169.3 49 39.6 28.8 09 88.2 64.1 69 i36.7 99.3 29 i85.3 i34.6 89 233.8 169.9 bo 40.5 29.4 10 89.0 64.7 70 137.5 99.9 3o 186.1 i35.2 90 234.6 170.5 5i 4i.3 3o.o III 89.8 65.2 171 i38.3 100.5 23l 186.9 i35.8 291 235.4 171.0 b2 42.1 3o.6 12 90.6 65.8 72 139.2 101 .1 32 187.7 i36.4 92 236.2 171.6 53 42.9 3l.2 i3 91.4 66.4 73 i4o.o lOI .7 33 188.5 137.0 93 237.0 172.2 54 43.7 3l.7 i4 92.2 67.0 74 i4o.8 102.3 34 189.3 137.5 94 237.9 172.8 55 44.5 32.3 i5 93.0 67.6 7i> i4i.6 102.9 35 190. 1 i38.i 95 238.7 173-4 56 45.3 32.9 16 93.8 68.2 76 142.4 io3.5 36 190.9 i38.7 96 239.5 174.0 57 46.1 33.5 17 94-7 68.8 77 143.2 104.0 37 191.7 139.3 97 240.3 174.6 58 46.9 34.1 18 95.5 69.4 78 i44.o 104.6 38 192.5 139.9 98 241 .1 175.2 59 47.7 34.7 19 9().3 69.9 79 144.8 I05.2 39 193.4 i4o.5 99 241.9 i7b-7 60 48.5 35.3 20 97.1 70.5 80 145.6 io5.8 4o 194.2 i4i.i 3oo 242.7 176.3 Hop. Lnt. Dist. Dop, Lat. Dist. Dep. Lat. Dist. Dep. i Lat. Dist. Dep. Lat. [For 54 Degrees. — \ TABLE 11. [i'''s«s3 . Difference of Latitude and Departure for 37 Degrees. l)is>t. Lai. Dep. Disi. Lai. Dep. 36.7 37.3 37.9 38.5 39.. 39.7 40.3 40.9 4i.5 42.1 42.7 43.3 43.9 44.5 45.1 45.7 46.3 46.9 47.5 48.1 Dist. Lat. Dep. Disi. Lat. Dep. Dist. Lat. Dep. 145.6 146.2 146.8 i47-4 i48.o 148.6 149-3 ,49.9 i5o.5 i5i.i i5i.7 i52.3 152.9 i53.5 i54.i i54.7 i55.3 155.9 i56.5 I 2 3 4 5 6 7 8 9 10 12 i3 i4 i5 iG 17 iS '9 20 00.8 oi .6 02.4 o3.2 o4.o o4.8 o5.6 o6.4 07.2 08.0 08.8 09.6 10.4 11 .2 12.0 12.8 i3.6 14.4 l5.2 16.0 00.6 01 .2 01.8 02.4 o3.o o3.6 04.2 04.8 o5.4 06.0 06.6 07.2 07.8 08.4 09.0 09.6 10.2 10.8 II. 4 12.0 61 62 63 64 65 66 67 68 69 70 48.7 49-5 5o.3 5i.i 5i .9 52.7 53.5 54.3 55.1 55.9 121 22 23 24 25 26 27 28 =9 3o 96.6 97.4 98.2 99.0 99.8 lOO.D loi .4 102.2 io3.o I03.8 72.8 73.4. 74.0 74.6 75.2 75.8 76.4 77.0 77.6 78.2 181 82 83 84 85 86 87 88 89 90 1 44 -6 145.4 i46.2 146.9 i47-7 i48.5 i49-3 i5o.i i5o.9 i5i.7 108.9 109.5 no. I 110.7 11 1.3 11 1.9 112.5 ii3.i 113.7 114.3 114.9 ii5.5 116.2 116.8 II7-4 118.0 118.6 119.2 119.8 120.4 241 42 43 44 45 46 47 48 it 25l 52 53 54 55 56 57 58 59 60 192.5 193.3 194.1 194.9 195.7 196.5 197.3 198.1 198.9 199.7 200.5 201.3 202.1 202.9 2o3.7 204.5 205.2 206.0 206.8 207.6 71 72 73 74 73 76 77 78 79 80 56.7 57.5 58.3 59.1 59.9 60.7 61.5 62.3 63.1 63.9 i3i 32 33 34 35 36 37 38 39 4o 104.6 105.4 106.2 107.0 107.8 108.6 109.4 110.2 II 1 .0 III. 8 78.8 79-4 80.0 80.6 81.2 81.8 82.4 83.1 83.7 84.3 191 93 94 95 96 97 98 99 200 i52.5 i53.3 1 54.1 154.9 155.7 i56.5 157.3 i58.i 158-9 159.7 2t 2 2 23 24 25 26 27 28 =9 3o 16.8 17.6 18.4 19.2 20.0 20.8 21.6 22.4 23.2 24.0 12.6 l3.2 i3.8 14.4 i5.o i5.6 16.2 16.9 17.5 18. 1 81 82 83 84 85 86 87 88 89 90 64-7 65.5 66.3 67.1 67.9 68.7 69.5 70.3 71. 1 71.9 48.7 49-3 5o.o 5o.6 5l.2 5i.8 52.4 53.0 53.6 54.2 i4i 42 43 44 45 46 47 48 49 5o 112. 6 ii3.4 Il4-2 ii5.o ii5.8 116.6 II7-4 118. 2 119. 119.8 84.9 85.5 86.1 86.7 87.3 87.9 83.5 89.1 89.7 90.3 201 02 o3 o4 o5 06 07 08 09 10 160.5 161.3 162.1 162.9 163.7 164.5 i65.3 166.1 166.5 167.7 121.0 1 2 1. 6 122.2 122.8 123.4 124.0 124.6 125.2 125.8 126.4 261 62 63 64 65 66 67 68 69 70 208.4 209.2 210.0 210.8 211.6 212.4 2l3.2 2l4.0 214.8 2i5.6 157.1 157.7 i58-3 1 58-9 ,59.5 1 60. 1 160.7 161.3 161.9 162.5 3i 32 33 34 35 36 37 3S 39 4o 24.8 25.6 26.4 27.2 28.0 28.8 29.5 3o.3 3i.i 3i .9 i8.7 19.3 19.9 20.5 21. 1 21.7 22.3 22.9 23.5 24.1 9' 92, 93 94 9? 96 97 98 99 ICO 72.7 73.5 74.3 75.1 75.9 76.7 77.5 78.3 79.1 79-9 54.8 55.4 56. 56.6 57.2 57.8 58.4 59.0 59.6 60.2 i5i 52 53 54 55 56 57 58 59 6g 120.6 121 .4 122.2 123.0 123.8 124.6 125.4 126.2 127.0 127.8 90.0 91.5 92.1 92.7 93.3 93.9 94.5 95.1 95.7 96.3 211 12 i3 i4 i5 16 17 18 19 20 168.5 169.3 170. 1 170.9 171.7 172.5 173.3 174.1 174.9 175.7 127.0 127.6 128.2 128.8 129.4 i3o.o i3o.6 l3l.2 i3i.8 i32.4 271 72 73 74 75 76 77 78 79 80 216.4 217.2 218.0 218.8 219.6 220.4 221.2 222.0 222.8 223.6 i63.i 163.7 164.3 164.9 i65.5 166.1 166.7 167.3 167.9 168.5 4i 42 43 44 45 46 47 -■H 49 So 32.7 33.5 3i.3 35.1 35.9 36.7 37.5 38.3 39. 1 39.9 24.7 25.3 25.9 26.5 27.1 27.7 28.3 28.9 29.^ 3o. I 101 02 o3 04 o5 06 07 08 09 ID III 12 l3 ■i4 i5 16 17 18 19 20 Dist. 80.7 81.5 82.3 83.1 83.9 84.7 85.5 86.3 87.1 87.8 88.6 89.4 90.2 91 .0 91.8 92.6 93.4 94.2 95.0 95.8 6u.8 6r.4 62.0 62.6 63.2 63.8 64.4 65. 65.6 66.2 66.8 67.4 68.0 68.6 69.2 69.8 70.4 71.0 71.6 72.2 161 62 63 64 65 66 67 68 69 70 128.6 129.4 i3o.2 i3i .0 i3i.8 i32.6 i33.4 i34.2 i35.o i35.8 96.9 97.5 98.1 98.7 99.3 99.9 100.5 roi .1 101 .7 102.3 221 22 23 24 25 26 27 28 It 176.5 1-77.3 178.1 178.9 179-7 180.5 181.3 182.1 182.9 183.7 i33.o i33.6 i34.2 i34.8 i35.4 i36.o 1 36.6 137.2 i37.8 i38.4 281 82 83 84 85 86 87 88 89 90 224.4 225.2 226.0 226.8 227.6 228.4 229.2 23o.o 23o.8 23i.6 169.1 169.7 170.3 170.9 171.5 172.1 172.7 173.3 173.9 174.5 5 1 52 53 54 55 56 57 58 6() 40.7 41.5 42.3 43.1 43.9 44.7 45.5 46.3 47-1 47.9 3o.7 3i.3 3.-9 32.5 33.1 33.7 34.3 34.9 35.5 36.1 171 72 73 74 75 76 77 78 Z9 80 i36.6 137.4 i38.2 139.0 139. S i4o.6 i4i.4 142.2 143.0 143.8 102.9 io3.5 io4.i 104.7 io5.3 i()5.9 106.5 107. 1 107.7 108.3 23l 32 33 34 35 36 37 38 39 40 184.5 i85.3 186.1 186.9 187.7 188.5 189.3 190.1 190.9 191.7 139.0 139.6 l40.3 140.8 i4i.4 142.0 142.6 143.2 143.8 144.4 291 92 93 94 95 96 97 98 99 3oo 232.4 233.2 234-0 234-8 235-6 236-4 237.2 238.0 238.8 239.6 175.1 175.7 1763- 176.9 177.5 178. 1 178.7 179.3 179-9 180.5 Dist. D,.p. Lnt. Dep. Lat. Dist. Dep. Lat. Disi. Dep. Lat. Dist. Dep- I Lat. [For .53 Degrees. Pago 5-1] TABLE IL Difference of Latitude and Departure for 38 Degrees. Dist. Lat. Dep. Dist. Lat. Dep. 37.6 Dist. Lat. Dep. Dist. 181 Lat. 142.6 Dep. Dist. Lat. Dep. I 00.8 00.6 6i 48.1 121 95.3 74.5 ni.4 241 189.9 148.4 2 01 .6 01 .2 62 48.9 38.2 22 96.1 7b. I 82 143.4 112. 1 42 190.7 149.0 3 02.4 01.8 63 49.6 38.8 23 96.9 7b.7 83 144.2 112.7 43 191. 5 i49-6 4 o3.2 02.5 64 bo. 4 39.4 24 97-7 76.3 84 145.0 n3.3 A4 192.3 i5o.2 6 03.9 o3.i 65 bl.2 4o.o 25 98.5 77.0 85 145.8 113.9 45 193.1 i5o.8 b o4-7 03.7 66 52.0 4o.6 26 99.3 77-6 86 i46.6 ii4-b 46 193.9 i5i.5 7 ob.b 04.3 67 52.8 4i .2 27 100. 1 78.2 87 \4iA Il5.! 47 194.6 l52.I « Ob. 3 04.9 68 b3.6 41.9 28 100.9 78.8 88 i48.i 115.7 48 195.4 152.7 9 07.1 ob.b 69 b4.4 42.5 29 lOI .7 79-4 89 148.9 116.4 49 196.2 i53.3 10 07.9 06.2 70 bb.2 43.1 3o 102.4 80.0 90 149.7 1 17.0 5o 197.0 153.9 II 0S.7 06.8 71 55.9 43.7 i3i io3.2 80.7 191 i5o.5 117.0 25l 197-8 154.5 12 09.5 07.4 72 bb.7 44.3 32 104.0 81.3 92 i5i.3 118.2 52 198.6 i55.i i3 10.2 08.0 73 b7.b 44.9 33 104.8 81.9 93 l52.1 118.8 53 199.4 i55.8 i4 II .0 08.6 74 58.3 45.6 34 io5.6 82.5 94 152.9 119.4 54 200.2 i56.4 lb II. 8 09.2 75 b9.i 46.2 35 106.4 83.1 9b i53.7 120. 1 55 200.9 .57.0 i6 12.6 09.9 76 ^9-9 46.8 36 107.2 83.7 96 ib4.b 120.7 56 201.7 157.6 17 i3.4 10.5 77 60.7 47-4 37 1 08 . 84.3 97 i55.2 121.3 57 202. D 1 58.2 i8 l4.2 11. 1 78 61.5 48.0 38 108.7 85. 98 i56.o 1 2 1. 9 58 2o3.3 i58.8 19 ib.o II. 7 79 62.3 48.6 39 109.5 85.6 99 1 56.8 122.5 59 204.1 159.5 20 lb. 8 12.3 80 63. 49-3 49.9 4o no. 3 86.2 200 1 57.6 123.1 60 204.9 1 60. 1 21 16.5 12.9 81 63.8 i4i III .1 86.8 201 i58.4 123.7 261 2o5.7 160.7 22 17.3 i3.5 82 64.6 5o.5 42 1 1 1 .9 87.4 02 159.2 124-4 62 206.5 161.3 23 18. 1 14.2 83 65.4 5i.i 43 112. 7 88.0 o3 160.0 125.0 63 207.2 161.9 24 18.9 i4.8 84 66.2 5l.7 44 ii3.5 88.7 04 160.8 125.6 H 208.0 162.5 25 19.7 i5.4 85 67.0 52.3 45 114.3 89.3 o5 161.5 126.2 65 208.8 i63.2 26 20.5 16.0 86 67.8 52.9 46 ii5.o 89.9 06 162.3 126.8 66 209.6 1 63 .8 27 21.3 16.6 87 68.6 53.6 47 ii5.8 90.5 07 i63.i 127.4 67 210.4 164.4 28 22.1 17.2 88 69.3 54.2 48 116. 6 91.1 08 163.9 128.1 68 211.2 i65.o 29 22.9 17.9 89 70.1 54.8 49 117-4 91.7 09 164.7 128.7 69 212.0 i65.6 Jo 23.6 18. b 90 70.9 55.4 5o 118. 2 92.3 10 ibb.5 129.3 70 212.8 166.2 3i 24.4 19.1 91 71-7 56.0 i5i 119. 93.0 211 166.3 129.9 271 2i3.6 166.8 32 2b. 2 19.7 92 72.5 56.6 52 119. 8 93.6 12 167.1 i3o.5 72 214.3 167,5 33 26.0 20.3 93 73.3 57.3 53 120.6 94.2 i3 167.8 i3i.i 73 2l5.1 168.1 34 2b. 8 20.9 94 74.1 57.9 54 121 .4 94.8 i4 168.6 i3i.8 74 215.9 168.7 35 27.6 21.5 95 74.9 58.5 55 122. 1 95.4 i5 169.4 i32.4 75 216.7 169.3 36 28.4 22.2 96 75.6 59.1 56 122.9 96.0 16 170.2 i33.o 76 217.5 169.9 37 29.2 22.8 97 76.4 59.7 57 123.7 96.7 17 171.0 J33.6 77 218.3 170.5 38 29.9 23.4 98 77.2 60.3 58 124.5 97.3 18 171.8 i34.2 ■ 78 219.1 171. 2 39 30.7 24.0 99 78.0 Ol .0 59 125.3 97-9 19 172.6 i34.8 79 219.9 171.8 4o Ji.b 24.6 100 78.8 61.6 60 126. 1 98. b 20 173.4 i3b.4 80 220 6 172.4 4i 32.3 25.2 lOI 79.6 62.2 161 126.9 99.1 221 174.2 1 36.1 281 221.4 173.0 42 33.1 25.9 02 80.4 62.8 62 127.7 99-7 22 174.9 i36.7 82 222.2 173.6 43 33.9 26.5 o3 81.2 63.4 63 128.4 100.4 23 175.7 137.3 83 223.0 174.2 44 34.7 27.1 04 82.0 64.0 64 129.2 101 .0 24 176.5 137.9 84 223.8 174.8 45 3b. b 27.7 o5 82.7 64.6 65 1 3o . 101 .6 25 177.3 i38.5 85 224.6 i7b.b 46 36.2 28.3 06 83.5 65.3 66 i3o.8 102.2 26 178.1 139.1 86 225-4 1 76. 1 47 37.0 28.9 07 84.3 65.9 67 i3i.6 102.8 27 178.9 139.8 87 226.2 176.7 48 37.8 29.6 08 8b. I 66.5 68 i32.4 io3.4 28 179-7 140.4 88 226.9 '77-3 49 38.6 3o.2 09 85.9 67.1 69 i33.2 104.0 29 180.5 i4i.o 89 227.7 1779 bo 39.4 3o.8 10 86.7 67.7 70 i34.o 104.7 3o 181. 2 141.6 90 228.5 178.5 5i 40.2 3i.4 II I 87.5 68.3 171 134.7 io5.3 23 I 182.0 142.2 291 229.3 179.2 52 4i .0 32.0 12 88.3 69.0 72 i35.5 105.9 32 182.8 142.8 92 23o.l 179-8 53 4i.8 32.6 i3 89.0 69.6 73 i36.3 106.5 33 i83.6 143.4 93 230.9 180.4 54 42.6 33.2 i4 89.8 70.2 74 137. 1 1 07 . 1 34 184.4 144.1 94 23i.7 181.0 bb 43.3 33.9 i5 90.6 70.8 75 137.9 107.7 35 i85.2 144-7 95 232.5 181.6 b6 44.1 34.5 16 91 .4 71.4 76 i38.7 108.4 36 186.0 145.3 96 233.3 182.2 57 44.9 35.1 17 92.2 72 .0 77 139.5 1 09 . 37 186.8 145.9 97 234-0 182.9 b8 4b. 7 35.7 18 93.0 72.6 78 i4o.3 109.6 38 187.5 146.5 98 234.8 i83.5 59 46.5 36.3 19 9J.8 73.3 79 i4i . I 1 10.2 39 188.3 147.1 99 235.6 184.1 bo 47.3 36.9 20 94.6 73.9 80 i4i.8 no. 8 4o 189.1 147-8 3 00 236.4 184.7 Dist. Dep. Lat. Disl. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist.l Dep. Lat. [1 Por 52 Degrees. TABLE IL JPuge 5i) Difference of La itude and Departure for 39 Degrees. Dist. Lai. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 00.8 00.6 61 47-4 38.4 121 94.0 76.1 181 140.7 113.9 24 1 18-7.3 i5i.7 2 01 .6 01 .3 62 48.2 39.0 22 94.8 76.8 82 141.-^ 114.5 42 188. 1 i52.3 3 02.3 01 .9 63 49.0 39.6 23 95.6 77.4 83 142.2 Il5.2 43 188.8 i52.9 4 oJ.i 02.5 64 49-7 4o.3 24 96.4 78.0 84 143.0 11 5.8 44 189.6 i53.6 b o3.9 o3.i 6b 5o.5 40.9 25 97.1 78.7 85 143.8 116.4 45 190.4 i54.2 6 04.7 o3.8 66 bi.3 4i.b 26 97-9 79.3 86 144.5 117.1 46 191.2 1 54.8 7 o5.4 04.4 67 b2.i 42.2 27 9».7 79-9 «7 145.3 117.7 47 192.0 i55.4 8 06.2 o5.o 68 b2.8 42.8 28 99.5 80.6 88 I46.I 118.3 48 192.-' 1 56. 1 9 07.0 Ob. 7 69 b3.6 4^.4 29 100.3 81.2 89 146.9 118.9 49 193.5 i56.7 lO 07.8 06.3 70 54.4 44.1 3o lOI .0 81.8 90 •47-7 1 19.6 5o 194.3 157.3 1 1 08.5 06.9 71 55.2 44.7 i3i I0I.8 82.4 191 148.4 120.2 25l 195.1 i58.o 12 09.3 07.6 72 bb.o 4b. 3 32 102.6 83.1 92 i49-2 120.8 52 195.8 i58.6 i6 10. 1 08.2 73 b6.7 45.9 33 io3.4 83.7 93 i5o.o 121.5 53 196.6 159.2 i4 10.9 08.8 74 b7.b 46.6 34 io4. 1 84.3 94 i5o.8 122.1 54 197.4 159.8 lb II. 7 09.4 75 b8.3 47-2 35 104.9 85.0 95 i5i.5 122.7 55 J98.2 160.5 lb 12.4 10. 1 76 59. 1 47.8 36 io5.7 85.6 96 i52.3 123.3 56 198.9 161.1 17 l3.2 10.7 77 b9.8 48. b 37 106.5 86.2 97 i53.i 124.0 57 199.7 161.7 i8 i4-o II. 3 78 bo.b 49.1 38 107.2 86.8 98 153.9 124.6 58 200.5 162.4 19 i4.8 12.0 79 61.4 49.7 39 108.0 87.5 99 154.7 125.2 59 201.3 i63.o 20 21 ib.b 16.3 12.6 l3.2 80 62.2 5o . 3 4o 108.8 88.1 200 1 55.4 125.9 60 202.1 1 63.6 81 62.9 5i.o i4i 109.6 88.7 201 i56.2 126.5 261 202.8 164.3 22 17.1 i3.8 82 b3.7 bi.b 42 110.4 89.4 02 157.0 127.1 62 2o3.6 164.9 2J 17.9 14.5 83 b4.b 52.2 4'^ III .1 90.0 o3 157.8 127.8 63 204.4 ,65.5 24 18.7 lb. I 84 bb.3 52.9 44 HI .9 90.6 04 1 58.5 128.4 64 2o5.2 166.1 2!) .9-4 lb. 7 85 bb.i 53.5 45 112.7 91 .3 o5 159.3 129.0 65 205.9 166.8 2b 20.2 16.4 86 66.8 b4.i 46 ii3.5 91.9 06 160.1 129.6 66 206.7 167.4 27 21 .0 17.0 H7 67.6 b4.8 47 Il4-2 92.5 07 160.9 i3o.3 67 207.5 168.0 28 21.8 17.6 88 b8.4 bb.4 48 ii5.o 93.1 08 161.6 1 30.9 68 208.3 168.7 29 22 .5 18.3 89 69.2 56. 49 ii5.8 93.8 09 162.4 i3i.5 69 209.1 169.3 Jo 23.3 18.9 90 69-9 bb.b 5o 116.6 94.4 10 i63.2 l32.2 70 209.8 169.9 3i 24.1 19.5 91 70.7 57.3 i5i 117. 3 95.0 211 164.0 i32.8 271 210.6 170.5 32 24.9 20.1 92 71.5 57.9 D2 118. 1 95.7 12 164.8 i33.4 72 2 1 1.4 171.2 33 25.6 20.8 93 72.3 58.5 53 118.9 96.3 i3 i65.5 1 34.0 73 212.2 171.8 34 26.4 21.4 94 73.1 59.2 54 119.7 96.9 i4 166.3 i34.7 74 212.9 172.4 3 b 27.2 22.0 95 73.8 59.8 55 120.5 97.5 i5 167.1 i35.3 75 2l3.7 173.1 db 28.0 22.7 96 74.6 60.4 56 121 .2 98.2 16 167.9 135.9 76 214.5 173.7 37 28.8 23.3 97 75.4 61 .0 57 122.0 98.8 17 168.6 1 36.6 77 2i5.3 174.3 38 29.5 23.9 98 76.2 bi.7 58 122.8 99.4 18 169.4 137.2 78 216.0 175.0 39 3o.3 24. b 99 76.9 62.3 59 123.6 100. 1 19 170.2 137.8 79 216.8 175.6 4o 3i.i 25.2 100 77-7 6:2.9 60 124.3 100.7 20 1 71.0 i38.5 80 217.6 176.2 4i 31.9 25.8 lOI 78.5 63.6 161 125.1 101 .3 221 171.7 139.1 281 218.4 176.8 42 32.6 2G.4 02 79.3 64.2 62 125.9 lOI .9 22 172.5 139.7 82 219.2 177-5 43 i^.4 27.1 o3 80.0 b4.8 63 126.7 102.6 23 173.3 i4o.3 83 219.9 178.1 44 34.2 27.7 04 80.8 bb.4 64 127.5 103.2 24 1 74. 1 i4i.o 84 220.7 178.7 4^) 35.0 28.3 o5 81.6 6b. I 65 128.2 io3.8 25 174.9 141.6 85 221.5 179.4 46 3b. 7 28.9 06 82.4 bb.7 66 129.0 104.5 26 175.6 142.2 86 222.3 180.0 47 36.5 29.6 07 83.2 67.3 67 129.8 io5.i 27 176.4 142.9 87 223.0 180.6 48 37.3 3o.2 08 83.9 68.0 68 1 3o . 6 io5.7 28 177.2 143.5 88 223.8 181.2 ^9 38.1 3o.8 09 84.7 68.6 69 i3i.3 106.4 29 178.0 1 44. 1 89 224.6 181.9 bo 38.9 3i.b 10 8b. b 69.2 70 l32.I 1 07 . 3o 178.7 144.7 90 225.4 182.5 5, 39.6 32.1 II I 86.3 69.9 171 132.9 107.6 23l 179.5 145.4 291 226.1 i83.i b2 40.4 32.7 12 87.0 70.5 72 i33.7 108.2 32 180.3 i46.o 92 226.9 i83.8 b3 41.2 33.4 i3 87.8 71. 1 73 i34.4 ^108.9 33 181. 1 i46.6 93 227.7 184.4 t)4 42.0 34.0 i4 88.6 71-7 74 1 35. 2 109.5 34 181.9 147.3 94 228.5 i85.o bb 42.7 34.6 lb 89.4 72.4 75 i36.o no. 1 35 182.6 147.9 95 229.3 i85.6 bb 43. b 35.2 lb 90.1 73.0 76 i36.8 110.8 36 i83.4 148.5 96 23o.O 186.3 i)7 44.3 3b. 9 17 90.9 73. b 77 137.6 III. 4 37 184.2 149.1 97 230.8 166.9 58 4b. I 3b. b 18 91.7 74.3 78 i38.3 112.0 38 i85.o 149.8 98 23].6 187.5 b9 45.9 37.1 19 92.5 74.9 79 139. 1 I 12.6 3q 1^.5.7 i5o.4 99 232.4 188.2 bo 45.6 37.8 20 93.3 75.5 80 139.9 II3.3 40 i86.5 i5i.o 3 00 233.1 i ib8.» i)ist. Do p. I.nt. Dlsi. Dep. Lat. Disi. Dep. Lat. Dist. Dep. Lat. Dist. Dep. 1 Lat. [1 ^or 51 Decrees Page 5G] TABLE IL 1 Difference of Latitude and Departure for 40 Degrees. Dist. Lat. Dep. Dist. I Lai. Dep. Dist. Lat. Dep. Dist. 181 Lat. i38.7 Dep. 116.3 Dist. Lat. Dep. I 01J.8 00.6 61 46.7 39.2 121 92.7 77.8 241 184.6 154.9 2 01.5 01.3 62 47. b 39.9 22 93.5 78. 4 82 139.4 117. 42 185.4 i55.6 3 02 .J 01 .9 63 48.3 40.5 23 94.2 79.1 83 i4o.2 1 17.6 43 1 86. 1 i56.2 4 o3.r 02.6 64 49.0 41.1 24 95.0 79-7 84 i4i.o 1 18.3 AA 186.9 1 56.8 3 o3.8 03.2 65 49.8 4i.8 25 95.8 80.3 85 141.7 118.9 45 187.7 1575 6 ()4.b 03.9 66 5o.6 42.4 26 96.5 81.0 86 142.5 1 19.6 46 188.4 i58.i 7 o5.4 o4.5 67 5i.3 43.1 27 97.3 81.6 87 143.3 120.2 47 189.2 i58.8 8 06.1 o5.i 68 52.1 43.7 28 98.1 82.3 88 144.0 120.8 48 190.0 159.4 9 06.9 o5.8 69 52.9 44.4 29 98.8 82.9 89 144.8 121. 5 49 190.7 1 60. 1 10 11 07.7 06.4 70 53.6 45.0 3o 99.6 83.6 84.2 90 145.5 122. 1 5o 25l 191. 5 192.3 160.7 08.4 07.1 71 54.4 45.6 i3i 100.4 IQI 146.3 122.8 161.3 12 09.2 07.7 72 55.2 46.3 32 lOI . I 84.8 92 147-1 123.4 52 193.0 162.0 i3 10. 08.4 73 55.9 46.9 3:^ 101.9 85.5 q3 147-8 124. 1 53 193.8 162.6 i4 10.7 09.0 74 56.7 47.6 34 102.6 86.1 94 148.6 124.7 54 194.6 i63.3 i5 II. 5 09.6 75 07.5 48.2 35 io3.4 86.8 g5 149.4 125.3 55 195.3 163.9 i6 12.3 10.3 76 58.2 48.9 36 io4.2 87.4 96 i5o.i 126.0 56 1 96. 1 164.6 17 i3.o 10.9 77 59.0 49.^ 37 io4-9 88.1 97 i5o.9 126.6 57 196.9 i65.2 i8 i3.8 II. 6 78 59.8 5o.i 38 io5.7 88. 7 98 i5i.7 127.3 58 197.6 i65.8 19 14.6 12.2 79 60.5 5o.8 39 106.5 89.3 99 i52.4 127.9 59 198.4 166.5 20 i5.3 12.9 80 61.3 5i.4 52.1 4o 107.2 90.0 200 i53.2 128.6 60 261 _i99^ 199.9 167.1 21 16.1 i3.5 81 62.0 i4i ■ 108.0 90.6 201 1 54.0 129.2 167.8 22 16.9 14.1 82 62.8 52.7 42 108.8 91.3 02 154.7 129.8 62 200.7 168.4 23 17.6 14.8 83 63.6 53.4 43 109.5 91.9 o3 i55.5 i3o.5 63 201.5 169.1 24 18.4 i5.4 84 64.3 54.0 AA no. 3 92.6 o4 1 56.3 i3i.i 64 202.2 169.7 25 19.2 16. 1 85 65.1 54.6 45 III. I 93.2 o5 157.0 i3i.8 65 2o3.o 170.3 26 19.9 16.7 86 65.9 55.3 46 III. 8 93.8 06 157.8 i32.4 66 2o3.8 171. 27 20.7 17.4 87 66.6 55.9 47 1 12.6 94.5 07 1 58.6 1 33. 1 67 204.5 171.6 28 21.4 18.0 88 67.4 56.6 48 ii3.4 Q5.I 08 159.3 133.7 68 2o5.3 172.3 29 22.2 18.6 89 68.2 57.2 49 ii4-i 95.8 09 160. 1 i34.3 69 206.1 172.9 ■50 23.0 19.3 90 68.9 57.9 5o 114. 9 96.4 10 160.9 i35.o 70 206.8 173.6 3i 23.7 19.9 91 69.7 58.5 i5i ii5.7 97.1 211 161.6 i35.6 271 207.6 174.2 32 24.5 20.6 92 70.5 59.1 52 116. 4 97-7 12 162.4 i36.3 72 208.4 174.8 33 23.3 21 .2 93 71.2 59.8 53 117. 2 98.3 i3 i63.2 i36.9 73 209.1 175.5 34 26.0 21 .9 94 72.0 60.4 64 118. 99.0 i4 163.9 137.6 74 209.9 176. 1 35 26.8 22.5 95 72.8 61. 1 55 118. 7 99.6 i5 164.7 i38.2 75 210.7 176.8 36 27.6 23.1 96 73.5 61.7 56 119. 5 100.3 16 i65.5 i38.8 76 2 1 1.4 177-4 37 28. 3 23.8 97 74.3 62.4 57 120.3 100.9 17 166.2 139.5 77 212.2 178. 1 38 29. 1 24.4 98 75.1 63. 58 121. 101.6 18 167.0 i4o.i 78 2l3.0 178.7 39 29.9 25.1 99 75.8 63.6 59 121. 8 102.2 19 167.8 i4o.8 79 2i3.7 179.3 4o 3o.6 25.7 100 76.6 64.3 60 122.6 102.8 20 168.5 i4i.4 142. 1 80 214.5 180.0 Ai 3i.4 26.4 lOI 77-4 64.9 161 123.3 io3.5 221 169.3 281 2 1 5.3 180.6 42 32.2 27.0 02 78.1 65.6 62 1 24. 1 104. 1 22 170.1 142.7 82 216.0 181.3 43 32.9 27.6 o3 78.9 66.2 63 124.9 104.8 23 170.8 143.3 83 216.8 1 8 1. 9 M 3J.7 28.3 04 79-7 66.8 64 125.6 io5.4 24 171.6 i44-o 84 217.6 182.6 45 34.5 28.9 o5 80.4 67.5 65 126.4 106. 1 25 172.4 144.6 85 218.3 i83.2 46 35.2 29.6 06 81.2 68.1 66 127.2 106.7 26 173. 1 145.3 86 219. 1 1 83.8 47 3b. 3o.2 07 82.0 68.8 67 127.9 107.3 27 173.9 145.9 87 219.9. 184.5 4S 36.8 3o.9 08 82.7 69.4 68 128.7 108.0 28 174-7 146.6 88 220.6 i85.i 49 37.5 3i.5 09 83.5 70.1 69 129.5 108.6 29 175.4 l47-2 89 221.4 i85.8 5o 38.3 32.1 10 84.3 70.7 70 l30.2 109.3 3o 176.2 147-8 90 222.2 186.4 5i 39., 32.8 III 85.0 71.3 171 i3i .0 109.9 23l 177.0 148.5 291 222.9 187.1 52 39.8 S6.A 12 85.8 72.0 72 i3r.8 no. 6 32 177.7 149.1 92 223.7 187.7 53 40.6 34.1 i3 86.6 72.6 73 i32.5 II 1 .2 33 178.5 149.8 93 224.5 188.3 54 41.4 34.7 i4 87.3 73.3 74 i33.3 III. 8 34 179.3 i5o.4 94 225.2 189.0 55 42. 1 35.4 i5 88.1 73.9 75 i34.i 112. 5 35 180.0 i5i.i 95 226.0 189.6 56 42.9 36.0 16 88. q 74.6 76 i34.8 ii3.i 36 180.8 i5i.7 96 226.7 190.3 57 43.7 1 36.6 17 89.6 75.2 77 i35.6 ii3.8 37 181.6 i52.3 97 227.5 190.9 58 44.4 37.3 18 90.4 75.8 78 i36.4 114.4 38 182.3 i53.o 98 228.3 191.6 59 45.2 37.9 19 91.2 76.5 79 137. 1 ii5.i 39 i83.i 1 53.6 99 229.0 192.2 b.i 46. 38.6 20 91.9 77-1 80 137.9 ii5.7 40 183.9 1 54.3 3oo 229.8 192.8 ni.i. 1),.,,. Lat. Dist. Dep. Lat. Dist. Dep. 1 Lat. Dist. Dep. Lat. Dist. Dep. Lat. | [For 50 Degrees. TABLE IL [Page 57 Difference of Latitude and Departure for 41 Degrees Disl. Lat. Dep. Dist. Lat. Dep. Dist. Lat. D.'p. I Dist "i^i Lai. Dep. Disl. Lat. I Dep. ] I 00.8 00.7 61 46. 4o.o 121 91.3 79-4 1 36.6 118.7 241 181.9 i58.i 2 or. 5 01 .3 62 46.8 40.7 22 92. 1 80.0 82 137.4 119.4 42 182.6 i58.8 3 02.3 02.0 63 47.5 4i.3 23 92.8 80.7 83 i38.i 1 20. 1 43 i83.4 159.4 4 o3.o 02.6 64 48.3 42.0 24 93.6 81.4 84 i38.9 120.7 44 184.1 160.1 fi o3.8 o3.3 65 49.1 42.6 25 94.3 82.0 8b 139.6 1 2 1. 4 45 1S4.9 160.7 6 04.5 o3.9 66 49.8 43.3 26 95.1 82.7 86 140.4 122.0 46 185.7 161.4 o5.3 04.6 67 5o.6 44.0 27 95.8 83.3 87 i4i.i 122.7 47 186.4 162.0 S 06.0 o5.2 68 5i.3 44.6 28 96.6 84.0 88 141.9 123.3 48 187.2 162.7 9 06.8 o5.9 69 52.1 45.3 29 97-4 84.6 89 142.6 124.0 49 187.9 i63.4 10 1 1 07.5 osTS" 06.6 07.2 70 71 52.8 45.9 3o 98.1 85.3 90 143.4 124.7 5o 188.7 164.0 164.7 53.6 46.6 i3i 98.9 85-9 191 144.1 125.3 25l 189.4 12 09.1 07.9 72 54.3 47.2 32 99.6 86.6 92 144.9 126.0 52 190.2 1 65.3 1 3 09.8 08.5 73 55.1 47-9 33 100.4 87.3 93 145.7 126.6 53 190.9 166.0 1 4 10.6 09.2 74 55.8 48.5 34 loi .1 87.9 94 146.4 127.3 54 191-7 166.6 i') II. 3 09.8 75 56.6 49.2 35 loi .9 88.6 95 147-2 127.9 55 192.5 167.3 i6 12. 1 10.5 76 57.4 49.9 36 102.6 89.2 96 i47-9 128.6 56 193.2 168.0 17 12.8 II. 2 77 58.1 5o.5 37 io3.4 89.9 97 148.7 129.2 t>7 194.0 168.6 i8 i3.6 II. 8 78 58.9 5l.2 38 io4.i 90.5 98 1 49 -4 129.9 58 194.7 169.3 '9 i4.3 12.5 79 59.6 5i.8 39 104-9 91 .2 99 l5o.2 i3o.6 59 195.5 169.9 2(> i5.i i3.i 80 60.4 52.5 4o io5.7 91.8 200 1 50.9 l3l.2 60 196.2 170.6 21 i5.8 i3.8 81 61. 1 53.1 i4i 106.4 92.5 201 1 5 1.7 i3i.9 261 197.0 171.2 22 16.6 14.4 82 61 .9 53.8 42 107.2 93.2 02 i52.5 i32.5 62 197.7 171.9 23 17-4 i5.i 83 62.6 54.5 43 107.9 93.8 o3 i53.2 i33.2 63 198.5 172.5 24 18. 1 l5.7 84 63.4 55.1 44 10S.7 94.5 04 154.0 i33.8 64 199.2 173.2 25 18.9 16.4 85 64.2 55.8 45 109.4 95.1 o5 154.7 i34.5 65 200.0 173.9 26 19.6 17. 1 8& 64.9 56.4 46 no. 2 95.8 06 155.5 i35.i 66 200.8 174.5 27 20.4 17-7 87 65.7 57.1 47 no. 9 96-4 07 i56.2 i35.8 67 201.5 175.2 28 21 .1 18.4 88 66.4 57.7 48 III. 7 97.1 08 157.0 1 36.5 68 202.3 175.8 29 21.9 19.0 89 67.2 58. 4 49 112. 5 97.8 09 157.7 137.1 69 2o3.o 176.5 3o 3i 22.6 19.7 90 67.9 59.0 5o Il3.2 98.4 10 i58.5 137.8 70 271 2o3.8 177.1 23.4 20.3 91 68.7 59.7 i5i 114.0 99.1 211 159.2 i3S.4 204.5 177.8 32 24.2 21 .0 92 69.4 60.4 52 114.7 99-7 12 160.0 139.1 72 2o5.3 178.4 33 24.9 21 .6 93 70.2 61 .0 53 ii5.5 100.4 i3 160.8 139.7 73 206.0 1 79. 1 34 25.7 22.3 q4 70.9 61.7 54 116.2 101 .0 i4 161.5 i4o.4 74 206.8 179.8 35 26.4 23.0 q5 71.7 62.3 55 1 17.0 101 .7 i5 162.3 i4i-i 7^ 207.5 180.4 36 27.2 23.6 96 72.5 63. 56 117. 7 102.3 16 i63.o 141.7 7b 20&.3 181. 1 37 27. Q 24.3 97 73.2 63.6 57 118.5 io3.o 17 i63.8 142.4 77 209.1 181.7 3.H 28.7 24.9 98 74.0 64.3 58 119. 2 io3.7 18 164.5 i43.o 78 209.8 182.4 39 29.4 25.6 9Q 74.7 64.9 59 120.0 104.3 19 i65.3 143.7 79 210.6 i83.o 4o 3o.2 26.2 100 75.5 65.6 60 120.8 ' io5.o 20 221 166.0 144.3 80 211.3 183.7 4i 3o.9 26.9 101 76.2 66.3 161 121.5 io5.6 166.8 145.0 281 212.1 184.4 42 3. .7 27.6 02 77.0 66.9 62 122.3 106.3 22 167.5 145.6 82 212.8 i85.o 43 32.5 28.2 o3 77-7 67.6 63 123.0 106.9 23 168.3 i46.3 83 21 3.6 :85.7 44 33.2 28.9 o4 78.5 68.2 64 123.8 107.6 24 169.1 147.0 84 214.3 186.3 45 34.0 29.5 o5 79.2 68.9 65 124.5 108.2 25 169.8 147.6 85 2l5.I 187.0 46 34.7 3o.2 06 80.0 69.5 66 125.3 108.9 26 170.6 i48.3 86 2 1 5.8 187.6 47 35.5 3o.8 07 80.8 70.2 67 126.0 109. 27 171. 3 148.9 87 216.6 188.3 48 36.2 3i.5 08 81.5 70.9 68 126.8 no. 2 28 172.1 149.6 88 217.4 188.9 49 37.0 32.1 09 82.3 71.5 69 127.5 110.9 29 172.8 l5o.2 89 218. 1 1896 5u 5i 37.7 38.5 32.8 33.5 10 83.0 72.2 70 128.3 in .5 3o 173.6 i5o.9 90 218.9 190.3 II I 83.8 72.8 171 129.1 112.2 23l 174.3 i5i.5 291 219.6 190.9 52 39.2 34.1 12 84.5 73.5 72 129.8 112.8 32 175.1 l52.2 92 220.4 191.6 53 4o.o 34.8 i3 85.3 74.1 73 i3o.6 ii3.5 33 175.8 152.9 93 221.1 192.2 54 40.8 35.4 i4 86.0 74.8 74 i3i.3 n4-2 34 176.6 i53.5 94 221.9 192.9 55 4i.5 36.1 i5 86.8 75.4 75 l32.1 114. 8 35 177.4 i54.2 95 222.6 193.5 56 42.3 36.7 i6;87.5.76.i 76 i32.8 n5.5 36 178.1 i54.8 96 223.4 194.2 57 43.0 37.4 17I88.3 76.8 77 i33.6 116.1 37 178.9 i55.5 97 224.1 194.8 58 43.8 38.1 18 89.1 77-4 78;i34.3 116. 8 38 179.6 i56.i 98 224.9 195.5 59 44.5 38.7 19 89.8 78.1 79 i35.i II7-4 39 180.4 i56.8 99 225.7 '9^^ 60 45.3 39.4 20 90.6 78.7 80 i35.8 118. 1 4o 181. 1 157.5 3oo 226.4 196.8 Dcp. Lat. Dist Dcp. Lat. Dist Dop. 1 Lat. Dist Dep. Lat. Dist. Dep. Lat. 1 [For 49 Degrees. Page 58J TABLE II. Difference of Latitude and Departure for 42 Decrees. • 1 Dlst. I Lai. Dep. Disi. Lai. Dep. Dist. 121 Lat. Dep. Dist. Lat. Dep. Dist. Lat. [ D> p. 1 00.7 00.7 61 45.3 40.8 89.9 81 .0 181 i34.5 121.1 241 179-1 161 3 2 01.5 01.3 62 46.1 4i.5 22 90.7 81.6 82 i35.3 121.8 42 179-8 161.0 3 02.2 02.0 63 46.8 42.2 23 91.4 82.3 83 i36.o 122.5 43 1806 162.6 4 o3.o 02.7 64 47-6 42.8 24 92.1 83.0 84 i36.7 123.1 44 181.3 i63.3 5 o3.7 o3.3 65 48.3 43.5 25 92.9 83.6 85 i37.5 123.8 45 182.1 163.9 6 o4.5 o4.o 66 49.0 44.2 26 93.6 84.3 86 i38.2 124.5 46 182.8 164.6 7 05.2 04.7 67 49-8 44.8 27 94.4 85.0 87 139.0 12D.I 47 i83.6 i65.3 8 05.9 o5.4 68 5o.5 45.5 28 95.1 85.6 88 139.7 125.8 48 184.3 165.9 9 06.7 06.0 69 5i.3 46.2 29 95.9 86.3 89 i4o.5 126.5 49 i85.o 166.6 lO II 07.4 08.2 06.7 07.4 70 52.0 46.8 3o 96.6 87.0 90 l4l.2 127. 1 5o i85.8 167.3 71 52.8 47.5 i3i 97-4 87.7 191 i4i.9 127.8 25l 186.5 168.0 12 08.9 08.0 72 53.5 48.2 32 98.1 88.3 92 142.7 128.5 52 187.3 168.6 i3 09.7 08.7 73 54.2 48.8 33 98.8 89.0 93 143.4 129.1 53 188.0 169.3 i4 10.4 09.4 74 55.0 49-5 34 99.6 89.7 94 i44.2 129.8 54 188.8 170.0 lb II .1 10. 75 55.7 5o.2 35 100.3 90.3 95 144.9 i3o.5 55 1S9.5 170.6 i6 11.9 10.7 76 56.5 50.9 36 lOI.I 91 .0 96 145.7 i3i.i 56 190.2 171.3 17 12.6 11.4 77 57.2 5i.5 37 101.8 91.7 97 146.4 i3i.8 57 191.0 172.0 i8 i3.4 12.0 78 58.0 52.2 38 102.6 92.3 98 i47-J i32.5 58 191.7 172.6 19 14. 1 12.7 79 58.7 52.9 39 103.3 93.0 99 147-9 i33.2 59 192.5 173.3 20 14.9 i3.4 80 59.5 53.5 40 104.0 93.7 200 148.6 1 33.8 60 193.2 174.0 21 i5.6 14. 1 81 60.2 54.2 i4i 104.8 94.3 201 149-4 i34.5 261 194.0 174.6 22 lb. 3 14.7 82 60.9 54.9 42 io5.5 95.0 02 i5o.i i35.2 62 194.7 175.3 23 17. 1 i5.4 83 61.7 55.5 43 106.3 95.7 o3 1 50.9 i35.8 63 195.4 176.0 24 17.8 16. 1 84 62.4 56.2 44 107.0 96.4 04 i5i.6 i36.5 64 196.2 176.7 25 18. b 16.7 85 63.2 56.9 45 107.8 97.0 o5 i52.3 137.2 65 196.9 177-3 26 19.3 17.4 86 63.9 57.5 46 108.5 97-7 06 i53.i i37.8 66 197-7 178.0 27 20.1 18.1 87 64.7 58.2 47 109.2 98.4 07 i53.8 i38.5 67 198-4 178.7 28 20.8 18.7 88 65.4 58.9 48 IIO.O 99.0 oS 1 54 .6 139.2 68 199.2 179.3 29 21 .6 19.4 89 66.1 59.6 49 no. 7 99-7 09 i55.3 139.8 69 199.9 180.0 3o 22.3 20.1 90 91 66.9 67.6 60.2 60.9 5o III. 5 100.4 ID i56.i i4o.5 70 200.6 180.7 3i 23. 20.7 i5i 112.2 101 .0 211 i56.8 l4l-2 271 201.4 181.3 32 23.8 21.4 92 68.4 61.6 52 ii3.o 101 .7 12 157.5 i4i-9 72 202.1 182.0 33 24.5 22. 1 93 69.1 62.2 53 1 13.7 102.4 ■ l3 i58.3 142.5 73 202.9 182.7 34 25.3 22.8 94 69.9 62.9 54 114.4 io3.o i4 159.0 i43.2 74 2o3.6 i83.3 35 26.0 23.4 95 70.6 63.6 55 ll5.2 io3.7 i5 159.8 143.9 75 204.4 184.0 36 26.8 24.1 96 71.3 64.2 56 1 15.9 104.4 16 160.5 i44.5 76 205.1 184.7 37 27.5 24.8 97 72.1 64.9 57 116.7 io5.i 17 161.3 i45.2 77 205.9 i85.3 38 28.2 25.4 98 72.8 65.6 58 II7-4 io5.7 18 162.0 145.9 ■ 78 206.6 186.0 39 29.0 26.1 99 73.6 66.2 59 118.2 106.4 19 162.7 146.5 79 207.3 186.7 40 29.7 26.8 100 74.3 66.9 60 118.9 107. 1 20 i63.5 147-2 80 208.1 187.4 4i 3o.5 27.4 101 75.1 67.6 161 119.6 107.7 221 164.2 147-9 281 208.8 188.0 42 3l.2 28.1 02 75.8 68.3 62 120.4 10S.4 22 i65.o 148.5 82 209.6 188.7 43 32. 28.8 o3 76.5 68.9 63 121.1 109. 1 23 165.7 149.2 83 210.3 189.4 U 32.7 29.4 o4 77.3 69.6 64 121.9 109.7 24 166.5 149.9 84 211. 1 190.0 45 33.4 3o.i o5 78.0 70.3 65 122.6 lie. 4 25 167.2 i5o.6 85 211.8 190.7 46 34.2 3o.8 06 78.8 70.9 66 123.4 III .1 26 168.0 l5l.2 86 212.5 191.4 47 34.9 3i.4 07 79.5 71.6 67 124. 1 III. 7 27 168.7 1 5 1.9 87 2i3.3 192.0 48 35.7 32.1 oS 80.3 72.3 68 124.8 112.4 28 169.4 i52.6 88 214.0 193.7 49 36.4 32.8 09 81.0 72.9 69 125.6 1 i3.i 29 170.2 i53.2 89 214.8 193.4 60 5i 37.2 33.5 10 81.7 73.6 70 126.3 ii3.8 3o 23l 1709 17) 7 153.9 1 54.6 90 2i5.5 194.0 37-9l34.i III 82.5 74.3 171 127. 1 114. 4 291 216.3 194-7 52 38.6 34.8 12 83.2 74.9 72 127.8 ii5.i 32 172.4 i55.2 92 217.0 195.4 53 39.4 35.5 i3 84. 75.6 73 128.6 ii5.8 33 173.2 155.9 93 217.7 196.1 54 4o.i 36.1 i4 84.7 76.3 74 129.3 116.4 34 173.9 i56.6 94 218.5 .96.7 \ Kr, 40.9 36.8 i5 85.5 77.0 75 i3o.i 117. 1 35 174.6 157.2 95 219.2 .97-4 56 4i.6 37.5 16 86.2 77.6 76 i3o.8 117.8 36 175-4 157-9 96 220.0 198.1 5? 42.4 38.1 17 86.9 78.3 77 i3i.5 118.4 37 176.1 1 58.6 97 220.7 198.7 58 43.1 38.8 18 87.7 79.0 78 i32.3 119. 1 38 176.9 159.3 98 221.5 199.4 59 43.8 39.5 19 88.4 79-6 79 i33.o 119.8 39 177.6 159.9 99 222.2 200.1 bo 44.6 4o. I 20 89.2 80.3 80 i33.8 120.4 4o 178.41 160.6 3uo 222.9 200.7 Dist. Dcp. I.al. Disl.| Dep. Lat. Dist.l Dep. ! Lat. Dist. Dep. 1 Lat. Dist Dep. Lat. [ For 48 Degrees. TABLE II. [P-Se59 Difference of Latitude and Departure for 43 Degrees. Dist. Lat. Dep. Dist. Lai. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist 241 42 43 44 45 46 47 48 5o 25l 52 53 54 55 56 57 58 60 Lat. 176.3 177.0 1777 178.5 179.2 179.9 180.6 181.4 182.1 182.8 1 83 .6 184.3 i85.o i85.8 186.5 187.2 188.0 188.7 189.4 190.2 Dep. 164.4 i65.o 165.7 166.4 167.1 167.8 168.5 169.1 169.8 170.5 171.2 171.9 172.5 173.2 173.9 174.6 175.3 176.0 176.6 177-3 178.0 178.7 1-9.4 180.0 180.7 181.4 182.1 182.8 i83.5 1 84.1 184.8 i85.5 186.2 186.9 187.5 188.2 188.9 189.6 190.3 191.0 I 2 3 4 5 6 7 8 9 10 00.7 01 .5 02.2 02.9 o3.7 o4.4 o5.i 05.9 06.6 07.3 00.7 01 .4 02.0 02.7 o3.4 o4. 1 04.8 o5.5 06.1 06.8 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 Z9 80 44.6 45.3 46.1 46.8 47-5 48.3 49.0 49-7 5o.5 5l.2 5i .9 52.7 53.4 54.1 54.9 55.6 56.3 67.0 57.8 58.5 4i.6 42.3 43.0 43.6 44.3 45.0 45.7 46.4 47.1 47.7 48.4 49.1 49.8 5o.5 5i.i 5i.8 52.5 53.2 53.9 54.6 55.2 55.9 56.6 57.3 58.0 58.7 59.3 60.0 60.7 61.4 121 22 2 3 24 25 26 27 28 =9 3o 88.5 89.2 90.0 90.7 91.4 92.2 92.9 93.6 94.3 95.1 82.5 83.2 83.9 84.6 85.2 85.9 86.6 87.3 88.0 88.7 181 82 83 84 85 86 87 88 89 90 i32.4 i33.i i33.8 1 34.6 i35.3 i36.o i36.8 137.5 i38.2 139.0 123.4 124.1 124.8 125.5 126.2 126.9 127.5 128.2 128.9 129.6 12 i3 i4 i5 i6 17 i8 19 20 08.0 08.8 09.5 10.2 11 .0 11.7 12.4 l3.2 .J. 9 i4.6 07.5 08.2 08.9 09.5 10.2 10.9 II. 6 12.3 i3.o i3.6 i3i 32 33 34 35 36 37 38 39 4o 95.8 96.5 97.3 98.0 98.7 99.5 100.2 100.9 101 .7 102.4 89.3 90.0 90.7 91.4 92. 1 92.8 93.4 94.1 94.8 95.5 191 9^ 93 94 95 96 97 98 99 200 .39.7 140.4 l4l.2 I4..9 142.6 143.3 144.1 144.8 145.5 i46.3 i3o.3 i3o.9 i3i.6 i32.3 i33.o i33.7 i34.4 i35.o ]35.7 i36.4 21 22 23 24 25 26 27 23 =9 3o i5.4 16.1 16.8 17.6 18.3 19.0 19.7 20.5 21 .2 21 .9 14.3 i5.o i5.7 16.4 17.0 17.7 18.4 19. 1 19.8 20.5 81 82 83 84 85 86 87 88 89 90 59.2 60.0 60.7 61.4 62.2 62.9 63.6 64.4 65.1 65.8 i4i 42 43 44 45 46 47 48 49 5o io3.i 103.9 104.6 io5.3 106.0 106.8 1 07 . 5 108.2 109.0 109.7 96.2 96.8 97.5 98.2 98.9 99.6 100.3 100.9 loi .6 102.3 201 02 o3 o4 o5 06 07 08 09 10 147.0 147-7 148.5 149.2 149.9 i5o.7 i5i.4 l52.I 152.9 1 53.6 137.1 137.8 i38.4 139. 1 139.8 140.5 l4l.2 141.9 142.5 143.2 261 62 63 64 65 66 67 68 69 70 190,9 1 9 1 .() 192.3 .93.1 193.8 194.5 195.3 196.0 1 96.7 197.5 3i 32 33 34 35 36 37 38 39 4o 22 .7 23.4 24.1 24.9 25.6 26.3 27.1 27.8 28.5 29.3 21. 1 21.8 22.5 23.2 23.9 24.6 25.2 25.9 26.6 27.3 9> 93 94 96 97 98 99 100 66.6 67.3 68.0 68.7 69.5 70.2 70.9 71-7 72.4 73.1 62.1 62.7 63.4 64.1 64.8 65.5 66.2 66.8 67.5 68.2 i5i 52 53 54 55 56 57 58 60 110.4 II 1 .2 III .9 112. 6 ii3.4 ii4-i 114. 8 ii5.6 116.3 117.0 io3 .0 io3.7 104.3 io5.o 105.7 106.4 107. 1 107.8 108.4 1 09 . 1 211 12 i3 i4 i5 16 17 18 19 20 i54.3 i55.o i55.8 i56.5 157.2 i58.o 158.7 159.4 160.2 160.9 143.9 144.6 i45.3 145.9 i46.6 147.3 i48.o i48.7 149.4 i5o.o 271 72 73 74 75 76 77 78 Z9 80 198.2 198.9 199.7 200.4 201.1 201.9 202. () 2o3.3 204.0 204.8 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 60 3o .0 00.7 3i.4 32.2 32.9 33.6 34.4 35.1 35.8 36.6 37.3 38.0 38.8 39.5 4o.2 4 1 .0 41.7 42.4 43.1 43.9 28.0 28. 6 29.3 3o.o 3o.7 3i.4 32.1 32.7 33.4 34.1 34.8 35.5 36.1 36.8 37.5 38.2 38-9 39.6 4o.2 40.9 lOI 02 o3 04 o5 06 07 08 09 ID 73.9 74.6 75.3 76.1 76.8 77.5 78.3 79.0 79-7 80.4 68.9 69.6 70.2 70.9 71.6 72.3 73.0 73.7 74.3 75.0 161 62 63 64 65 66 67 68 69 70 117.7 118.5 119. 2 119. 9 120.7 121 .4 122.1 122.9 123.6 124.3 109.8 110.5 III .2 111.8 112. 5 Il3.2 113.9 114.6 ii5.3 115.9 221 22 23 24 25 26 27 28 If. 161.6 162.4 i63.i 1 63.8 164.6 i65.3 166.0 166.7 167.5 168.2 1 50.7 i5i.4 l52.1 i52.8 i53.4 i54.i i54.8 i55.5 1 56.2 i56.9 281 82 83 84 85 86 87 88 89 90 2()5.5 206.2 207.0 207.7 208.4 209.2 209.9 210.6 211.4 212.1 191.6 192.3 193.0 193.7 194.4 195. 1 196.4 197.1 197.8 U I 12 i3 i4 i5 16 17 18 19 20 81.2 81.9 82.6 83.4 84.1 84.8 85.6 S6.3 87.0 87.8 75.7 76.4 77-1 77-7 78.4 79.1 79.8 80.5 81.2 81.8 171 72 73 74 75 76 77 78 Z9 80 125. I 125.8 126.5 127.3 128.0 128.7 129.4 l3o.2 1 30.9 i3i.6 116.6 117. 3 118.0 118.7 119. 3 120.0 120.7 121.4 122. 1 122.8 23 1 32 33 34 35 36 37 38 39 40 168.9 169.7 170.4 171. 1 171.9 172.6 173.3 1 74. 1 174.8 175.5 157.5 i58.2 i58.9 159.6 160.3 161.0 161.6 162.3 1 63.0 i63.7 291 92 93 95 96 9^ 98 ,99 3oo 212.8 2i3.6 214.3 2 1 5.0 2.5.7 216.5 217.2 217.9 218.7 219.4 198.5 199.1 199.8 200.5 201.2 201.9 202.6 203.2 203.9 2o4 6 Lni. Dist. Dq,. T>at. Dist. Dep. I.at. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. [For 47 Degrees. 1 Pago 60] TABLE IL Difference of Latitude and Departure for 44 Degree^;. Dist. I.at. Dep. Dist. Lat. Dep. Dist. Lat. Do p. Dist. Lat. Dep. Dist. Lat. Dep. I 00.7 00.7 61 43.9 42.4 121 87.0 84.1 181 i3o.2 125.7 241 173-4 '" 4 2 01 .4 CI .4 62 44.6 43.1 22 87.8 84.7 82 i3o.9 126.4 42 174.1 j>^.i 3 02.2 02.1 63 45.3 43.8 23 88.5 85.4 83 i3i.6 127. 1 43 174-8 168.8 4 02.9 02.8 64 46. 44.5 24 89.2 86.1 84 i32.4 127.8 A^ 175.5 169.5 5 o3.b o3.5 65 46.8 45.2 25 89.9 86.8 85 i33.i 128.5 45 176.2 170.2 6 o4.3 04.2 66 47.5 45.8 26 90.6 87.5 86 i33.8 129.2 46 177.0 170.9 7 o5.o 04.9 67 48.2 46.5 27 91.4 88.2 87 i34.5 129.9 47 177-7 1 7 1. 6 8 ob.8 o5.6 68 48.9 47.2 28 92. 1 88.9 88 i35.2 i3o.6 48 178.4 172.3 9 06.5 06.3 69 49-(3 47-9 29 92.8 89.6 89 i36.o i3i.3 49 179.1 173.0 10 07.2 06.9 70 5o.4 48.6 3o 93.5 90.3 90 i36.7 l32.0 5o 179-8 173.7 II 07.9 07.6 71 5i.i 49-3 i3i 94.2 91 .0 191 137.4 i32.7 25l 180.6 174.4 12 08.6 08.3 72 bi,8 5o.o 32 95.0 91.7 92 i38.i i33.4 52 181.3 175 1 i3 09.4 09.0 73 52.5 50.7 33 95.7 92.4 93 i38.8 i34.i 53 182.0 175.7 i4 10. 1 09.7 74 53.2 5i.4 34 96.4 93.1 94 139.6 i34.8 54 182.7 176.4 lb 10.8 10.4 75 54.0 52.1 35 97.1 93.8 95 i4o.3 i35.5 5b i83.4 177.1 i6 II. 5 II .1 76 54.7 52.8 .36 97.8 94.5 96 i4i.o i36.2 56 184.2 177.8 17 12.2 II. 8 77 55.4 53.5 37 98.5 95.2 97 141.7 1.36.8 57 184.9 178.5 18 12.9 12.5 78 56.1 54.2 38 99.3 95.9 98 142.4 137.5 58 i85.6 179.2 19 i3.7 l3.2 79 56.8 54.9 39 100. 96.6 99 143.1 i38.2 59 186.3 179.9 20 14.4 .3.9 80 57.5 55.6 40 100.7 97.3 200 143.9 i38.9 60 187.0 180.6 21 i5.i i4.6 81 58.3 56.3 i4i loi .4 97-9 201 144.6 139.6- 261 187.7 181.3 22 i5.8 i5.3 82 59.0 57.0 42 102. 1 98.6 02 145.3 i4o.3.^ 62 188.5 182.0 23 16.5 16.0 83 59.7 57.7 43 102.9 99.3 o3 146.0 i4i-o 63 189.2 182.7 24 17.3 16.7 a4 60.4 58.4 M io3.6 1 00.0 04 146.7 i4i.7 64 189.9 i83.4 2b 18.0 17.4 85 61. 1 59.0 45 104.3 100.7 o5 i47-5 142.4 65 190.6 184.1 26 18.7 18. 1 86 61 .9 59.7 46 io5.o 101.4 06 i48.2 143.1 66 191. 3 184.8 27 19.4 18.8 87 62.6 60.4 47 io5.7 102.1 07 148.9 i43.8 67 192. 1 i85.5 28 20.1 19.5 88 63.3 61. 1 48 106.5 102.8 08 i49-6 144.5 68 192.8 1S6.2 29 20.9 20. 1 89 64.0 61.8 49 107.2 io3.5 09 i5o.3 i45-2 69 193.5 186.9 Jo 21 .6 20 8 90 91 64.7 65.5 62.5 63.2 5o 107.9 104.2 10 i5r.i 145.9 70 194.2 1S7.6 3i 22.3 21.5 i5i 108.6 104.9 21 1 i5i.8 146.6 271 194.9 188.3 32 23.0 22.2 92 66.2 63.9 52 109.3 io5.6 12 i52.5 147.3 72 195.7 188.9 33 23.7 22.9 93 66.9 64.6 53 no. I 106.3 i3 i53.2 i48.o 73 196.4 189.6 M 24.5 23.6 94 67.6 65.3 54 no. 8 107.0 i4 153.9 148.7 74 197-1 190.3 3b 25.2 24.3 95 68.3 66.0 55 III .5 107.7 i5 154.7 149.4 7^ 197.8 191.0 36 2b. 9 25.0 96 69.1 66.7 56 112. 2 108.4 16 i55.4 i5o.o 76 198.5 191.7 ^7 26.6 25.7 97 69.8 67.4 57 112.9 109.1 17 i56.i 1 50.7 77 199.3 192.4 38 27.3 26.4 98 70.5 68.1 58 ii3.7 109.8 18 i56.8 i5i.4 78 200.0 193.1 39 28.1 27.1 99 71.2 68.8 59 114.4 110.5 19 157.5 l52.1 79 200.7 193.8 4o 28.8 27.8 100 71.9 69.5 60 ii5.i III .1 20 1 58.3 i52.8 i53.5 80 201.4 194.5 4i 29.5 28.5 lOI 72.7 70.2 161 ii5.8 HI .8 221 159.0 281 202.1 195.2 42 3o.2 29,2 02 73.4 70.9 62 116.5 112. 5 22 159.7 i54.2 82 202.9 195.9 43 3o.9 29.9 o3 74.1 71.5 63 117. 3 I l3.2 23 160.4 154.9 83 2o3.6 196.6 M 31.7 36.6 04 74.8 72.2 64 118. 113.9 24 161. 1 i55.6 84 2o4-3 197.3 ^^ 32.4 3i.3 o5 75.5 72.9 65 118.7 114.6 25 161.9 i56.3 85 2o5.o 198.0 46 33.1 32. 06 76.3 73.6 66 119.4 ii5.3 26 162.6 157.0 86 2o5.7 198.7 47 33.8 32.6 07 77.0 74.3 67 120.1 1 16.0 27 i63.3 157.7 87 206.5 199-4 48 34.5 33.3 08 77-7 75.0 68 120.8 116.7 28 164.0 i58.4 88 207.2 200.1 49 35.2 34.0 09 78.4 75.7 69 121 .6 117.4 29 164.7 159.1 89 207.9 200.8 bo 5i 3b. 36.7 34.7 35.4 10 1 1 1 79.1 76.4 70 122.3 118. 1 3o i65.4 159.8 90 208.6 201.5 79.8 77-1 171 123.0 118.8 23l 166.2 160.5 291 209.3 202.1 b2 37.4 36.1 12 80.6 77.8 72 123.7 1 19.5 32 166.9 161.2 92 210.0 202.8 53 38.1 36.8 i3 81.3 78.5 73 124.4 120.2 33 167.6 161.9 93 210.8 2o3.5 54 38.8 37.5 i4 82.0 79.2 74 125.2 120.9 M 168.3 162.6 94 2»JI.5 2o4.2 CK 39.6 38.2 i5 02.7 79-9 75 125.9 121 .6 35 169.0 i63.2 95 212.2 204.9 56 4c. 3 38.9 16 83.4 80.6 76 126.6 122.3 36 169.8 163.9 96 212.9 2o5.6 57 4i.o 39.6 17 84.2 81.3 77 127.3 123. 37 170.5 164.6 97 2l3.6 206.3 58 41-7 40.3 18 84.9 82.0 78 128.0 123.6 38 171.2 165.3 98 214.4 207.C 59 42.4 4i .0 19 85.6 82.7 79 128.8 124.3 39 171-9 166.0 99 2l5.I 207.7 60 43.2 41.7 20 86.3 83.4 80 129.5 125.0 40 172.6 166.7 3 00 2i5.8 208.4 Dist. Dep. I. at. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. [ ''or 46 Degrees. TABLE II. [Page 61 Difference of Latitud e and Departure for 45 Degrees. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. Dist. Lat. Dep. I 00.7 00.7 61 43.1 43.1 121 85.6 85.6 181 128.0 128.0 241 170.4 170.4 2 01 .4 01 .4 62 43.8 43.8 22 86.3 86.3 82 128.7 128.7 42 171.1 171. 1 3 02.1 02.1 63 44.5 44.5 23 87.0 87.0 83 129.4 129.4 43 171. 8 171.8 4 02.8 02.8 64 45.3 45.3 24 87.7 87.7 84 i3o.i i3o.i 44 172.5 1-2.5 5 o3.5 o3.5 65 46.0 46.0 25 88.4 88.4 85 i3o.8 i3o.8 45 173.2 173.2 6 o4.2 04.2 66 46.7 46.7 26 89.1 89.1 86 i3i.5 i3i.5 46 173.9 173.9 04.9 04.9 67 47-4 47-4 27 89.8 89.8 87 l32.2 l32.2 47 174.7 174.7 8 0D.7 o5.7 68 48.1 48.1 28 90.5 90.5 88 132.9 132.9 48 175.4 175-4 9 06.4 06.4 69 48.8 48.8 29 91 .2 91.2 89 i33.6 i33.6 49 170.1 176.1 10 07.1 07.1 70 49.5 49-t5 3o 91.9 91.9 90 1 34. 4 134.4 5o, 176.8 176.8 11 07.8 07.8 71 5o.2 5o.2 i3i 92.6 92.6 191 i35.i I35.I 25l 177-5 177.5 12 08.5 08.5 72 5o.9 50.9 32 93.3 93.3 92 i35.8 i35.8 52 178.2 1 178.2 i3 09.2 09.2 73 5i.6 5i.6 33 94.0 94.0 93 i36.5 i36.5 53 178.9 178.9 i4 09.9 09.9 74 52.3 52.3 M 94.8 94.8 94 137.2 137.2 54 179.6 179.6 i5 10.6 10. 75 53.0 53.0 35 95.5 95.5 95 137.9 137.9 55 180.3 180.3 i6 11.3 II. 3 76 53.7 53.7 36 96.2 96.2 96 i38.6 i38.6 56 181.0 181.0 J7 12.0 12.0 77 54.4 54.4 37 96.9 96.9 97 139.3 139.3 57 181.7 181.7 i8 12.7 12.7 78 55.2 55.2 38 97.6 97.6 98 i4o.o i4o.o 58 182.4 182.4 19 i3.4 i3.4 79 55.9 55.9 39 98.3 98.3 99 140.7 140.7 59 i83.i i83.i 20 14.1 i4.i 80 56.6 56.6 40 99.0 99.0 200 i4i.4 141.4 60 i83.8 i83.8 184.6 21 i4.8 i4.8 81 57.3 57.3 i4i 99-7 99-7 201 142.1 142.1 261 184.6 22 i5.6 i5.6 82 58.0 58.0 42 100.4 100.4 02 142.8 142.8 62 i85.3 i85.3 23 16.3 16.3 83 58.7 58.7 43 101 .1 101. 1 o3 143.5 143.5 63 186.0 186.0 24 17.0 17.0 84 59.4 59.4 44 101.8 101.8 04 144.2 144.2 64 186.7 186.7 25 17-7 17.7 85 60.1 60.1 45 102.5 102.5 o5 145.0 145.0 65 187.4 187.4 26 18.4 18.4 86 60.8 60.8 46 io3.2 io3.2 06 145.7 145.7 66 188. 1 188. 1 27 19. 1 19. 1 87 61.5 61.5 47 103.9 io3,9 07 146.4 146.4 67 188.8 188.8 28 19.8 19.8 88 62.2 62.2 48 104.7 104.7 08 i47-i 147-1 68 189.5 189.5 29 20.5 20.5 89 62.9 62.9 49 io5.4 io5.4 09 147-8 147.8 69 1 90 . 2 190.2 3o 21.2 21 .2 90 91 63.6 64.3 bi.b 64.3 5o 1 06 . 1 106.1 10 148.5 148.5 70 271 190.9 191 .6 190.9 1 9 1. 6 3i 21 .9 21 .9 i5i 106.8 106.8 211 149.2 149.2 32 22.6 22.6 92 65.1 65.1 52 107.5 107.5 12 149.9 149-9 72 192.3 192.3 33 23.3 23.3 93 65.8 65.8 53 108.2 108.2 i3 i5o.6 i5o.6 73 1 93 . 193.0 34 24.0 24.0 94 66.5 66.5 54 108.9 108.9 i4 i5i.3 i5i.3 74 193.7 193.7 35 24.7 24.7 95 67.2 67.2 55 109.6 109.6 i5 l52.0 l52.0 7^ 194.5 194.5 36 25.5 25.5 96 67.9 67.9 56 110.3 110.3 16 i52.7 152.7 1^ 195.2 195.2 37 26.2 26.2 97 68.6 68.6 57 III .0 III.O 17 i53.4 i53.4 77 195.9 19D.9 38 26.9 26.9 98 69.3 69.3 58 III. 7 111.7 18 i54.i 1 54.1 78 196.6 196.6 39 27.6 27.6 99 70.0 70.0 59 112.4 1 12.4 19 154.9 154.9 79 197.3 197.3 4o 28.3 28.3 100 70.7 70.7 60 ii3.i ii3.i 20 221 i55.6 i55.6 80 198.0 198.0 4i 29.0 29.0 lOI 71.4 71-4 161 ii3.8 ii3.8 i56.3 i56.3 281 198.7 198.7 42 29.7 29.7 02 72.1 72.1 62 114.6 114.6 22 157.0 157.0 82 199-4 199-4 43 3o.4 So. 4 o3 72.8 72.8 63 ii5.3 ii5.3 23 157.7 157.7 83 200. 1 2C0.I 44 3i.i 3i.i o4 73.5 73.5 64 116.0 116.0 24 i58.4 1 58.4 84 2(J0.8 200.8 45 3i.8 3i.8 o5 74.2 74.2 65 116.7 116.7 25 159.1 1D9.1 85 201 .5 201.5 46 32.5 32.5 06 75.0 75.0 66 117.4 117.4 26 159.8 159.8 86 202.2 202.2 47 33.2 33.2 07 75.7 75.7 67 118.1 118.1 27 160.5 160.5 87 202.9 202.9 48 33. 9 33.9 08 76.4 76.4 68 118. 8 118.8 28 161 .2 161.2 88 2o3.6 203.6 49 34.6 34.6 09 77.1 77-1 69 119.5 119.5 29 161 .9 161.9 89 204.4 204.4 5o 35.4 35.4 10 77.8 77.8 70 120.2 120.2 3o 162.6 162.6 i63.3 90 205.1 205.1 5i 36.1 36.1 III 78.5 78.5 171 120.9 120.9 23l i63.3 291 2o5.8 2o5.S 52 36.8 36.8 12 79.2 7Q.2 72 121 .6 121.6 32 164.0 164.0 92 206.5 206 5 53 37.5 37.5 i3 79-9 79-9 73 122.3 122.3 33 164.8 164.8 93 207.2 207 2 54 38.2 38.2 i4 80.6 80.6 74 123.0 I23.0 34 i65.5 i65.5 94 207.9 207.9 55 38. Q 38.9 i5 81.3 81.3 75 123.7 123.7 35 166.2 166.2 9i 208.6 2&8.6 56 39.6 39.6 16 82.0 82.0 76 124.5 124.5 36 166.9 166.9 96 209.3 209.3 57 4o.3 40.3 17 82.7 82.7 77 125.2 125.2 37 167.6 167.6 97 210.0 210.0 58 4i .0 4i .0 18 83.4 83.4 78 125.9 125.9 38 168.3 168.3 98,210.7 210.7 59 41.7 41.7 19 84.1 84.1 79 126.6 126.6 39 169.0 169.0 99 211 .4 211.4 bo 42.4 42.4 20 84-9 84-9 80 127.3 127.3 4o 169.7 169.7 3oo 212. 1 2 1 2.1 Dist. Dep. Lai. Dist. Dep. Lat. Dist. Dep. Lat. Dist. Dep. Lat. Dist. D..p. Lat. For 45 Degrees. Page 62] TABLE III. Meridional Parts. M. 0° 1° 2' 3° 40 5° 6° 7° 8° 9° 10° 11° 12° 13° M. 60 120 180 240 3oo 36 r 421 482 542 6o3 664 725 787 I I 6i 121 i8i 241 3oi 362 422 483 543 6o4 665 726 788 I 2 2 62 122 182 242 302 363 423 484 544 6o5 666 727 789 2 3 3 63 123 i83 243 3o3 364 424 485 545 606 667 728 790 3 4 5 4 64 124 184 244 3o4 3o5 365 425 486 546 607 668 729 791 4 5 5- 65 125 i85 245 366 426 487 547 608 669 73o 792 fj 6 66 126 186 246 3o6 367 427 488 548 609 670 73 ( 793 6 7 7 67 127 187 247 3o7 368 428 489 549 610 671 732 794 7 S 8 68 128 188 248 3o8 369 429 490 55o 611 672 734 793 8 10 9 69 129 189 249 25o 3o9 370 43o 491 55i 612 673 735 796 _9 ID 10 70 i3o 190 3io 371 43 1 4o2 552 6i3 674 736 797 II II 71 i3i 191 25l 3ii 372 432 493 553 6i4 675 737 798 :i '12 12 72 l32 192 252 3l2 373 4i^ 494 554 6i5 676 738 799 12 i3 i3 73 i33 193 253 3i3 374 4M 495 555 6i6 677 739 800 i3 i4 i5 i4 74 1 34 194 254 3i4 375 435 496 556 617 678 740 801 i4 i5 i5 75 i35 195 255 3i5 376 436 497 557 618 679 74 1 802 i6 16 76 1 36 196 256 3i6 377 437 498 558 619 680 742 8o3 16 17 17 77 i37 197 257 3i7 378 438 499 559 620 681 743 8o4 17 i8 18 78 1 38 198 258 3i8 379 439 5oo 56o 621 682 744 8o5 18 12 20 19 20 79 80 ,39 199 259 319 38o 44o 5oi 56 1 622 683 745 806 11 20 i4o 200 260 320 38 1 44 1 502 562 623 684 746 807 21 21 81 i4i 201 261 321 382 442 5o3 564 624 685 747 808 21 22 22 82 142 202 262 322 383 443 5o4 565 625 687 748 809 22 23 23 83 143 203 263 323 384 444 5o5 566 626 688 749 810 23 24 25 24 84 1 44 204 264 324 385 445 5o6 507 567 627 6S9 75o 811 24 25 25 85 i45 205 265 325 386 446 568 628 690 75i 812 26 26 86 1 46 206 266 326 387 44i 5o8 569 629 691 752 8i3 26 27 27 87 i47 207 267 327 388 44^ 509 570 63 1 692 753 8i5 27 28 28 88 1 48 208 268 328 389 449 5io 571 632 693 7^4 816 28 29 3o 29 89 149 i5o 209 210 269 270 33o 390 45o 5ii 572 633 694 755 817 29 3o 3o 90 33i 39. 45i 5l2 573 634 695 756 818 3i 3i 9' i5i 211 271 332 392 452 5i3 574 635 696 757 819 3i 32 32 92 I 52 212 372 333 393 453 5i4 575 636 697 758 820 32 33 33 93 1 53 2l3 273 334 394 454 5i5 576 637 698 759 821 33 34 35 34 94 1 54 2l4 274 275 335 395 455 5i6 5i7 i>77 578 638 699 760 761 822 823 34 35 35 95 1 55 2l5 336 396 456 639 700 36 36 96 1 56 216 276 337 397 457 5i8 ^79 64o 701 762 824 36 37 37 97 1 57 217 277 338 398 458 5.9 58o 64 1 702 763 825 37 38 38 98 1 58 218 278 339 399 459 520 58 1 642 7o3 764 826 38 4o 39 99 .59 219 279 280 340 341 4"o 460 J2I 582 643 704 765 827 39 40 4o 100 160 220 4ot 46 1 522 583 644 7o5 766 828 4r 4i lOI 161 221 281 342 402 462 523 584 645 706 767 829 4i 42 42 102 162 222 282 343 4o3 463 524 585 646 707 768 83o 42 43 43 io3 i63 223 283 344 4o4 464 525 586 647 708 769 83 r 4-i 44 45 44 io4 1 64 224 284 345 4o5 465 526 587 648 709 770 8J2 44 45 45 io5 i65 225 285 346 4o6 466 527 588 649 710 77' 833 46 46 106 166 226 286 347 407 467 528 589 65o 711 772 834 46 47 47 107 167 227 287 348 4o8 468 529 590 65 [ 712 773 835 47 48 48 108 168 228 288 349 409 469 53o 591 652 7i3 774 83b 48 49 5o 49 109 169 229 289 35o 4io 470 53 1 592 653 714 77b 837 49 5o 5o 110 170 23o 290 35i 4ii 471 532 593 654 7i5 777 838 5i 5i 1 1 1 171 23l 291 352 4l2 472 533 594 655 716 778 839 5i 52 52 112 172 232 292 353 4i3 473 534 595 656 717 779 84o 52 53 53 ii3 173 233 293 354 4i4 474 535 596 657 7.8 780 84 1 53 54 55 54 ii4 174 934 294 355 4i5 476 536 597 658 719 781 842 54 55 55 ii5 175 235 295 356 4i6 477 537 598 659 720 782 843 56 56 116 176 236 296 357 417 478 538 599 660 721 783 844 56 5- 57 117 177 237 297 358 4i8 479 539 600 661 722 784 845 ^7 5H 58 118 178 238 298 359 419 480 540 601 662 723 785 846 d8 59 119 179 239 299 36o 420 48 1 54 1 602 663 724 786 847 M. 0° 1° 2° 3° 40 5° G° 7° 8° 9° 10° 11° 12° 13° TABLE III. Meridional Parts. fPage 63 M. o 14° 15° 1G° 17° 18° 19° 20° 21° 22° 23° 24° 25° 2G° 27° 1684 M. 848 910 973 io35 109S 1161 1225 1289 1 354 1419 1484 i55o 1616 I 85o 911 974 36 99 63 26 1 90 55 20 85 5i 18 85 T ? 85i 9i3 975 37 IIOO 64 27 91 56 21 86 52 19 86 2 3 852 914 976 38 01 65 28 92 57 22 87 53 20 87 3 4 5 853 9.5 977 39 02 66 29 93 58 23 88 54 21 88 4 5 854 916 978 io4i no3 1167 I23o 1295 i359 1424 1490 1 556 1622 1689 6 855 9'7 979 42 o5 68 32 96 60 25 91 57 23 90 6 7 856 918 980 4i 06 69 33 97 61 26 92 58 24 9' 7 8 857 919 981 44 07 70 ■M 98 62 27 93 59 25 93 8 lO 858 ~85^ 920 982 45 08 71 35 99 63 28 94 60 26 94 _? 10 921 983 1046 1 109 1172 1236 i3oo 1 364 i43o 1495 i56i 1628 1695 1 1 860 922 9«4 47 10 73 37 01 66 3i 96 62 29 96 II I? 861 923 985 48 II 74 38 02 67 32 97 63 3o 97 12 i3 862 924 986 49 12 7i 39 o3 68 33 98 64 3i 98 i3 i4 i5 863 925 987 5o i3 76 4o o4 69 34 1435 99 65 32 99 1700 i4 i5 864 926 988 io5i iii4 1177 I24[ i3o5 1370 i5oo 1 567 i633 iti 8()5 927 989 52 i5 78 42 06 71 36 02 68 34 01 16 17 866 928 990 53 16 79 43 07 72 37 o3 69 35 o3 17 i8 867 929 991 54 17 81 44 08 73 38 o4 70 37 04 18 19 20 868 930 993 55 18 II 19 82 45 10 74 39 o5 71 38 o5 17(^6 19 20 8tJ9 93. 994 io56 ii83 1246 i3ii 1375 1 440 i5o6 1572 1639 21 870 932 995 57 20 84 48 12 76 4i 07 73 4o 07 21 22 871 933 996 58 21 85 49 i3 77 43 08 74 4i 08 22 23 S72 9^4 997 59 22 86 5o i4 79 44 09 7!) 42 09 23 24 25 873 9J5 998 60 23 87 5i i5 80 45 10 77 43 II 24 25 874 936 999 1061 1125 1188 1252 i3i6 i38i 1 446 i5ii 1578 1644 1712 26 «75 9^7 louo 63 26 89 53 17 82 47 i3 79 45 i3 26 2? 876 93s 01 64 27 90 54 18 83 48 i4 80 47 i4 27 28 «77 939 02 65 28 9' 55 19 84 49 i5 81 48 i5 28 29 3o 878 879 941 o3 66 29 92 56 20 85 5o 16 82 49 16 29 3o 942 1004 1067 ii3o 1 193 1257 l32I 1 386 i45i .517 i583 i65o 1717 3! 880 943 o5 68 3i 94 58 22 87 52 18 84 5i 18 3i 32 882 944 06 69 32 95 59 24 88 53 19 85 52 20 32 33 883 945 07 70 33 96 60 25 89 55 20 86 53 21 33 34 35 884 946 08 71 34 ii35 98 6, 26 90 56 21 88 54 22 1723 34 35 885 9-^7 1009 1072 1199 1262 1327 1392 1457 l522 1589 1 656 36 886 948 10 73 36 1200 64 28 93 58 24 90 i)7 04 36 37 8S7 949 1 1 74 37 01 65 29 94 59 25 91 58 25 37 38 888 95o 12 75 38 02 66 3o 95 60 26 92 59 26 38 39 4o 889 93. i3 76 39 o3 67 3i 96 61 27 93 60 27 39 4o 890 952 ioi4 1077 I i4o 1204 1268 i332 1397 1462 i528 1594 1661 1729 4i 89. 953 i5 7a 4i o5 69 33 98 63 29 95 62 3o 4^ 4.-- 892 9^)4 16 79 42 oG 70 34 99 64 3o 96 63 3i 42 43 893 9^5 18 80 44 07 71 35 i4oo 65 3i 98 64 32 43 44 45 894 895 956 19 81 45 08 72 36 01 67 32 99 66 33 44 45 g'-v 1020 1082 ii46 1209 1273 i338 l4o2 1 468 i533 1600 1667 1734 46 896 938 21 84 47 10 74 39 o3 69 35 01 68 35 46 47 897 959 22 85 48 1 1 75 4o o5 70 36 02 69 36 47 48 898 9fjt) 23 86 49 I? 76 4i 06 71 37 o3 70 38 48 49 5o 899 96 1 24 87 5o i3 77 42 07 72 38 04 71 39 49 5o 900 962 1025 1088 ii5i I2l5 1278 i343 i4o8 1473 1539 iCo5 1672 1740 5i 9f)i 963 26 89 52 16 80 44 09 74 4o 06 73 4i 5i 52 902 964 27 90 53 17 81 45 10 75 4i 08 7!5 42 52 53 903 9*) 5 28 9' 54 18 82 46 II 76 42 09 76 4^ 53 54 55 904 966 29 92 55 19 83 4i 12 77 43 10 77 44 54 55 9o5 968 io3o 1093 ii56 1220 1284 1 348 i4i3 i479 1 544 1611 1678 1746 56 906 969 3i 94 57 21 85 49 i4 80 46 12 79 47 56 :i7 9"7 970 32 95 58 22 86 5o i5 81 47 i3 80 48 57 d8 908 97' 33 96 59 23 87 52 16 82 48 i4 81 49 58 59 M. 909 972 34 97 60 24 88 53 18 83 49 i5 82 5o 27° 5? M 14° 15° 1G° 17° 18° 19° 20° 21° 22° 23° 24° 25° 2C° Page 64] TABLE III Meridional Parts. M. o 28° 29° 30° 3J° 32° 33° 34° 35° 36° 37° 38° 39° 40° 41° -M. 1 75 1 1819 1S88 1958 2028 2100 2171 2244 23i8 2393 2468 2545 2623 2702 I 52 21 90 59 3o 01 73 46 19 94 70 46 24 o3 I 2 53 22 91 bo 3i 02 74 47 20 95 71 48 25 o4 2 J 55 23 92 62 32 o3 75 48 22 96 72 49 27 06 3 4 5 56 24 93 63 33 04 76 49 23 98 73 5o 28 07 2708 4 5 1757 1825 1894 1964 2o34 2I05 2178 225o 2324 2399 2475 255 1 2620 b 58 2b 95 65 35 07 79 52 25 2400 76 53 3i 10 6 7 59 27 96 bb 37 08 80 53 27 01 77 54 32 II 7 b 60 29 98 b7 38 OQ 81 54 28 o3 78 55 33 12 8 _9 10 6; 1762 3o 99 b9 39 10 82 55 29 o4 80 57 34 i4 2715 _9 10 i83i 1900 1970 204o 2III 2184 2257 233o 24o5 2481 2558 2636 II 64 32 01 71 4i i3 85 58 32 06 82 59 37 16 n 12 65 33 02 72 43 i4 86 59 33 08 84 60 38 18 I? iJ 66 34 o3 73 44 i5 87 60 34 09 85 62 4o 19 1 3 i4 i5 67 35 o5 74 45 16 88 61 35 10 86 63 4i 20 i4 i5 1768 i837 1900 1976 2o46 2117 2190 2263 2337 241 1 2487 2564 2642 2722 lb 69 38 07 77 47 19 91 64 38 i3 89 66 44 23 16 17 70 39 08 7» 48 20 92 65 39 i4 90 67 45 24 17 i8 72 40 09 79 5o 21 93 66 40 i5 91 68 46 26 18 £9 20 li 4i 10 80 5i 22 94 68 42 16 92 69 48 27 £9 20 1774 1842 1912 1981 2o52 2123 2196 2269 2343 2418 2494 2571 2649 2728 21 7^ AS i3 83 53 25 97 70 44 19 95 72 5o 29 21 22 76 45 i4 84 54 26 98 71 45 20 96 73 5i 3i 22 2J 77 46 i5 85 56 27 99 72 46 22 98 75 53 32 23 24 25 7S 47 lb 86 67 28 2200 74 48 23 99 76 54 33 24 25 1780 1848 1917 1987 2o58 2129 2202 2275 2349 2424 25oo 2577 2655 2735 2b 81 49 18 88 59 3i o3 76 5o 25 01 78 57 36 26 27 82 5o 20 90 60 32 o4 77 5i 27 o3 80 58 37 27 28 83 52 21 91 61 33 o5 79 53 28 04 81 59 39 28 29 3o 84 53 22 92 63 34 07 80 54 2355 29 • o5 82 61 4o 29 3o 1785 i854 1923 1993 2064 2i35 2208 2281 2430 25o6 2584 2662 2742 3i 86 55 24 94 65 37 09 82 56 32 08 85 63 43 3i 32 H7 56 25 95 66 38 10 83 58 33 oq 86 65 44 32 33 89 57 27 97 67 39 II 85 59 34 10 88 66 46 33 34 35 90 ■ 58 28 98 69 4o i3 86 60 35 12 89 67 47 34 35 1791 i860 1929 1999 2070 2l4l 2214 2287 236i 2437 25i3 2590 2669 2748 3b 92 61 3o 2000 71 43 i5 88 63 38 i4 91 70 5o 36 37 93 62 3i 01 72 44 16 90 64 39 i5 93 71 5i 37 38 94 63 32 02 73 45 17 91 65 4o 17 94 73 52 38 39 40 95 64 34 04 75 46 19 92 66 42 18 95 74 54 39 40 1797 i865 1935 2005 2076 2147 2220 2293 2368 2443 25i9 2597 2675 2755 4i 98 66 36 06 77 49 21 95 69 44 21 98 76 56 4i 42 99 68 37 07 7» 5o 22 9(3 70 45 22 99 78 58 42 AS 1800 69 38 08 79 5i 24 97 71 47 23 2601 79 59 43 44 45 01 70 39 10 80 52 25 98 73 48 24 02 80 60 44 45 1802 1871 1941 201 1 2082 2i53 2226 2299 2374 2449 2526 2603 2682 2762 4fa o3 72 42 12 83 55 27 23oi 75 5i 27 o4 83 63 46 47 o5 73 AS i3 84 56 28 02 76 52 28 c6 84 64 47 48 06 7!) A4 i4 85 57 3o o3 78 53 3o 07 86 66 48 49 5o 07 76 45 i5 86 58 3i 04 79 54 2456 3i 08 87 67 49 5o 1808 1877 1946 2017 2088 2 1 59 2232 23o6 238o 2532 2610 2688 2768 5i 09 7S 48 18 89 61 33 07 81 57 33 II 90 70 5i :i2 10 79 49 19 90 62 35 08 83 58 35 12 91 71 52 d3 11 80 5o 20 91 63 36 09 84 59 36 i4 92 72 53 b4 55 i3 "I'sTJ 81 5i 21 92 64 2i65 37 2238 11 23l2 85 61 37 i5 94 74 54 55 i883 1952 2022 2094 2386 2462 2538 2616 2605 2775 56 i5 84 53 24 95 67 39 j3 88 63 40 17 96 76 56 57 16 85 65 25 96 68 4i i4 89 64 4i 19 98 78 57 58 17 86 56 26 97 69 42 16 90 66 42 20 99 79 58 59 M. 18 «7 57 27 98 70 33° 43 17 35° 9' 36° 67 44 21 2700 80 59 M. 28° 29° 30° 31° 32° 34° 37° 38° 39° 40° 41° TABLE III. [Page Meridional Parts. 65 M. I 2 3 4 5 6 7 8 9 10 11 12 i3 i4 i5 i6 17 i8 !9 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 4o 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 M. 42° 43° 44° 45° 46° 47° 48° 3292 93 95 96 98 49° 3382 84 85 87 88 50° 3474 76 78 79 81 51° 52° 3665 67 68 70 72 53° 54° 55° M. I 2 3 4 5 6 7 8 ^ 10 II 12 i3 r4 i5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3i 32 33 34 35- 36 37 38 39 4o 4i 42 43 AA 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 M. 27S2 83 84 86 87 2863 64 66 67 69 2946 47 5o 5i 3o3o 3i 33 36 3ii6 17 18 20 21 3i23 24 26 27 29 32o3 o4 06 07 09 3569 70 72 74 75 3764 65 67 69 70 3865 66 68 70 71 3968 70 71 73 75 2788 90 91 92 94 2870 71 73 74 75 2953 54 56 57 58 3o37 38 40 4i 3210 12 i3 i4 16 3^99 33oi 02 o3 o5 33o6 08 09 II 12 3390 93 96 3397 3400 02 o3 3482 84 85 87 88 3577 78 80 82 83 3673 75 77 78 80 3772 74 75 77 79 3780 82 84 85 87 3873 75 77 78 80 3977 78 80 82 84 2795 9Z 98 o99 2801 2877 78 80 81 82 2960 61 63 64 65 3o44 46 47 48 5o 3i3o 3i 33 36 3217 19 20 22 23 3490 9^ 93 95 96 3585 86 88 90 9' 3593 96 98 99 368 1 83 85 86 88 3882 83 85 87 89 3985 87 89 91 92 2802 o3 o5 06 07 2884 85 86 88 89 2967 68 70 71 72 3o5i 53 54 55 57 3i37 39 40 42 •43 3 144 t 3225 26 28 =9 3i 33i4 16 17 19 20 34o5 07 08 10 II 3498 35oi o3 04 3690 9^ 9^ 96 3789 90 92 94 95 3890 92 9i 95 97 3994 9^ 98 x99 4001 4oo3 o5 06 08 10 2809 ID II i3 i4 2891 96 2974 75 76 78 79 3o58 60 61 63 64 3232 M 35 37 38 3322 23 25 26 28 34i3 i4 16 17 19 35o6 07 09 10 12 35i4 i5 17 18 20 36oi 02 04 06 07 3609 10 12 i4 i5 3698 99 3701 o3 04 3797 .n99 38oo 02 o4 3899 3901 02 o4 06 2815 17 18 20 21 2897 99 2900 02 o3 2981 82 83 85 86 3o65 67 68 70 71 3i52 53 55 56 57 3i59 60 62 63 65 3240 4i 42 45 3329 3i 32 34 35 3337 38 4o 4i 43 3420 22 23 25 27 3706 08 09 II i3 38o6 07 09 II 12 3907 09 II i3 i4 4012 i4 i5 17 19 2822 24 25 26 28 2904 06 07 08 10 2988 89 91 95 93 3073 74 75 77 78 3247 48 5o 5i 53 3428 3o 3i 33 M 3521 23 25 26 28 0617 18 20 22 23 3714 16 17 19 21 38i4 16 17 19 21 3822 24 26 27 29 3916 18 19 21 23 3925 26 28 3o 32 4021 22 24 26 28 4029 3i 33 35 37 2829 3o 32 33 34 2911 i3 i4 i5 17 2995 96 98 3ooo 3o8o 81 83 84 85 3i66 68 69 71 72 3254 56 57 60 3344 46 47 3436 37 39 40 42 3443 45 47 48 5o 3529 3i 32 36 3625 26 28 3o 3i 3722 24 26 27 29 3731 32 34 36 37 2836 37 39 40 4i 2Ql8 '19 21 22 24 3002 o3 o5 06 07 3087 88 90 93 3173 75 76 78 79 3262 63 65 66 68 3352 53 55 56 58 3537 39 4o 42 43 3633 34 36 38 39 383 1 32 34 36 38 3933 35 37 38 4o 4o38 4o 42 AA 45 2843 45 47 48 2849 5i 52 54 55 2925 26 28 3? 3009 10 12 i3 i4 3094 95 97 98 3ioo 3i8i 82 84 85 87 3269 71 72 74 75 3359 61 62 64 65 345 1 53 54 56 57 3545 47 48 5o 5i 364 1 AA 46 47 3739 4i 42 AA 46 3839 4i 43 AA 46 3942 AA 45 47 49 4047 ^9 5i 52 54 2932 33 35 36 37 3oi6 17 19 20 21 3ioi o3 o4 o5 07 3i88 90 91 92 94 3277 78 80 81 83 3367 68 70 71 73 3459 60 62 64 65 3553 55 56 58 59 3649 5i 52 54 55 3747 ^9 5o 52 54 3848 ^9 5i 53 54 3951 52 54 56 58 4o56 58 60 61 63 a856 58 60 62 2939 4o 42 43 3o23 24 26 27 29 3io8 10 II i3 i4 3195 97 98 3200 01 3284 86 87 89 90 3374 76 78 3467 68 70 71 73 356i 62 64 66 67 3657 60 62 64 3755 57 ^.9 60 62 3856 58 60 61 63 3959 61 63 64 66 4o65 67 69 70 72 42° 43° 44° 45° 46° 47° 48° 49° 50° 51° 52° 53'^ 54° 55° P^g* 66] TABLE III. Meridional Parts. o 56° 57° 58° 59° 60° 61° 4649 62° 4775 63° 4905 64° 5o39 65° 66° 5324 67° 5474 68° 563 1 69-^ 5795 — 1 M.; 4074 4i83 4294 4409 4527 5i79 I 7b 84 9b II 29 5i 77 07 42 81 26 77 33 97 I 2 77 86 98 i3 3i 53 79 09 44 84 28 79 36 58oo 2 3 79 88 43oo i5 33 55 81 12 46 86 3i 82 39 o3 3 j4 5 81 90 02 17 35 57 84 i4 49 5o5i 88 33 84 42 06 5809 4 5 4o83 4192 43o4 4419 4537 4660 4786 4916 5191 5336 5487 5644 b 85 94 06 21 39 62 88 18 53 93 38 89 47 II 6 7 86 95 08 23 4i 64 90 20 55 95 4i 92 5o 14 7 8 88 97 09 25 43 66 92 23 58 98 43 95 52 17 8 lO 90 99 II 27 45 68 94 25 60 5200 46 97 55 20 _9 10 4092 4201 43i3 4429 4547 4670 4796 4927 5062 52o3 5348 55oo 5658 5823 II 94 o3 i5 3i 49 72 98 29 65 o5 5i 02 60 25 11 12 95 o5 17 33 5i 74 4801 3i 67 07 53 o5 63 28 12 i3 97 07 19 34 53 76 o3 34 69 10 56 07 66 3i i3 i4 i5 99 08 21 36 55 78 o5 36 71 12 5214 58 536i 10 55i3 68 5671 34 5837 14 i5 4ioi 4210 4323 4438 4557 4680 4807 4938 5074 lb o3 12 25 40 59 82 09 4o 76 .17 63 i5 74 39 16 17 04 14 27 42 62 84 II 43 78 19 66 18 76 42 17 i8 Ob 16 28 44 64 87 i4 45 81 22 68 20 79 45 18 £9 20 08 18 3o 46 66 89 16 47 83 24 71 23 5526 82 5685 48 585i !9 20 4iio 4220 4332 4448 4568 4691 4Si8 4949 5o85 5226 5373 21 12 21 M 5o 70 93- 20 5i 88 29 76 28 87 54 21 22 i3 23 36 52 72 95 22 54 90 3i 78 3i 90 56 22 2j i5 25 38 54 74 97 24 56 92 34 80 33 93 59 23 24 25 17 27 4o 56 76 99 26 58 95 36 83 36 95 62 5865 24 25 4u9 4229 4342 4458 4578 4701 4829 4960 5097 5238 5385 5539 56q8 2b 21 3i 44 60 80 o3 3i 63 99 4i 88 41 5701 68 26 27 22 32 46 62 82 o5 33 65 5l02 43 90 44 04 71 27 28 24 M 47 64 84 07 35 67 o4 46 93 46 06 74 28 29 3o 26 4128 3b 49 66 86 4588 10 37 69 06 48 95 49 09 76 29 3o 4238 435i 4468 4712 4839 4972 5io8 525o 5398 5552 5712 5879 3i 3o 40 53 70 90 i4 42 74 II 53 5401 54 i5 82 3i J2 32 42 55 72 92 16 44 76 i3 55 o3 57 17 85 32 33 33 44 57 74 94 18 46 78 i5 58 06 59 20 88 33 34 35 35 4b 59 76 96 20 48 81 18 60 08 54ii 62 23 91 34 35 4i37 4247 436 1 4478 4598 4722 485o 4983 5l20 5263 5565 5725 5894 3b 39 49 63 80 4600 24 52 85 22 65 i3 67 28 96 36 ^7 4i 5i 65 82 02 26 55 87 25 67 16 70 3i 99 37 38 42 53 67 84 o4 28 57 90 27 70 18 73 34 5902 38 39 4o 44 55 69 86 06 46o8 3i 59 92 29 72 5275 21 5423 75 36 o5 39 4o 4i46 4257 4370 4488 4733 4861 4994 5i32 5578 5739 5908 4i 48 59 72 90 10 35 63 96 ■34 77 26 80 42 II 4i 42 5o bo 74 92 12 37 65 99 36 80 28 83 45 i4 42 43 52 62 76 94 i4 39 68 5ooi 39 82 3i 86 47 17 43 44 45 53 4i55 64 78 95 16 4618 4i 70 o3 4i 84 33 88 5o 19 5922 44 45 4266 438o 4497 4743 4872 5oo5 5x43 5287 5436 5591 5753 4b 57 68 82 99 20 45 74 08 46 89 38 94 56 25 46 47 59 70 84 45oi 23 47 76 10 48 92 4i 06 58 28 47 48 61 72 86 o3 25 5o 79 12 5i 94 43 99 61 3i 48 49 5o 62 74 88 o5 27 52 81 i4 53 97 46 56o2 64 34 49 5o 4i64 4275 4390 4507 4620 4754 4883 5oi7 5i55 5299 5448 56o4 5767 5937 5i 66 77 92 09 3i 56 85 19 58 53oi 5i 07 70 40 5i 52 68 79 94 II 33 58 87 21 60 o4 54 10 72 43 52 53 70 81 96 i3 35 60 90 23 62 06 56 12 75 46 53 54 55 72 83 98 i5 37 4639 62 92 26 65 5167 09 59 i5 78 48 5951 54 55 417^ 4285 4399 45i7 4764 4894 5028 53ii 5461 56i7 5781 5b 7t) 87 4401 19 4i 66 96 3o 69 i4 64 20 83 54 56 57 77 89 o3 21 43 69 98 33 72 16 66 23 86 57 57 58 Z9 91 o5 23 45 71 4901 35 74 19 69 25 89 60 58 59 81 92 07 25 47 73 o3 37 76 21 71 28 92 63 M. 56° 57° 4 58° 1 59° 60° 61° 62° 63° 64° 65° 66° 67° 68° 1 69° TABLE III. rragoc? Meridional Parts. M. o 70° 71° 6i46 72° 73° 74° 75° 7G° 77° 78° 79° 8o46 80° 8375 81° 82° 83° 5966 6335 6534 6746 6970 7210 7467 7745 8739 9145 9606 I 09 49 33 38 49 74 i4 7'^ 49 5i 81 45 53 14 I •} 72 52 4i 4i 53 78 18 76 54 5b 87 52 bo 22 2 ^ 75 55 45 45 57 82 22 81 59 61 93 5b 67^ 3,| 3 4 78 5981 58 48 48 60 86 27 85 64 67 98 65 74 39 9647 4 5 6161 635 1 6552 6764 6990 723l 7490 7769 8072 84o4 8771 9182 ti 84 64 54 55 68 94 35 1 94 74 77 10 78 89 55 b 7 86 67 58 58 71 97 39 98 78 83 lb 84 .9b 64 7 8 89 70 61 62 75 7001 43 75o3 83 88 22 91 9203 72 b _? lO 92 5995 73' 64 65 79 6782 o5 47 07 88 93 27 97 8804 11 80 _9 u 6177 6367 6569 7009 7252 75i2 7793 8099 8433 9218 9(389 1 1 98 80 71 72 8b i3 56 16 98 8104 39 10 25 97 1 1 1? 6001 83 74 76 90 17 60 21 7803 09 45 17 33 9706 12 rl o4 86 77 79 93 21 64 25 08 i5 5i 23 40 14 i3 i4 i5 07 89 80 83 97 25 68 3o i3 20 57 do 48 23 i4 i5 60 I G 6192 6384 6586 6801 7029 7273 7535 7S17 8125 8463 8836 9255 973 1 i6 i3 95 87 90 04 33 77 39 22 3i 69 43 62 40 lb 16 98 90 93 08 37 81 44 27 3b 74 P 70 48 17 i8 19 6201 94 97 12 4i 85 48 32 4i 80 56 77 57 18 .'9 30 22 6025 o5 97 6600 i5 45 89 53 37 47 8b 63 85 65 £9 20 6208 6400 66o3 6819 7048 7294 7557 7842 8i52 8492 8869 9292 9774 21 28 II o3 07 23 52 98 62 47 58 98 7b 9300 83 21 }7 3i i4 07 10 26 56 73o2 66 52 63 85o4 83 07 91 22 23 34 17 10 i4 3o 60 06 7' 57 68 10 89 i5 9800 23 24 i5 37 6o4o 20 6223 i3 17 34 6838 64 II 76 62 7S67 74 lb 96 22 09 24 25 6417 6621 7068 73i5 7580 8179 8522 8903 9330 9817 26 43 26 20 24 4t 72 19 85 72 85 28 09 ^7 26 2b 27 46 3o 23 28 45 76 23 89 77 90 34 16 45 35 27 28 49 33 27 3i 49 80 28 94 82 96 40 23 53 44 2b 2y 3o 52 6o55 36 3o 35 53 84 32 99 «7 7892 8201 4b 3o 60 52 29 3o 6239 6433 6639 6856 7088 7336 7603 8207 8552 8936 9368 9861 3i 58 42 37 42 60 92 4i 08 97 12 58 43 7b 70 3i 32 61 45 4o 46 64 96 45 12 7902 18 65 5o 83 79 32 33 64 49 43 49 68 7100 49 17 07 23 71 57 91 88 33 34 35 67 52 47 53 71 04 53 22 12 29 77 63 99 97 34 35 6070 6255 645o 6656 6875 7108 7358 7626 7917 8234 8583 8970 9407 9906 3fi 73 58 53 60 79 12 62 3i 22 40 89 77 14 i5 3b 37 76 61 57 63 83 16 66 36 ■27 45 95 84 22 24 37 38 79 64 60 67 86 20 71 40 32 bi 8601 91 3o 33 3ii 39 4o 82 6o85 68 63 70 90 6894 24 75 45 37 56 07 98 38 42 i9 4o 6271 6467 6674 7128 7379 765o 7942 8^862 8614 9005 9445 9951 4i 88 74 70 77 98 32 84 54 48 67 20 12 53 60 4i 42 9' 77 73 81 6901 36 88 59 53 73 2b 18 61 69 42 43 94 80 77 85 o5 4o 92 64 58 79 32 25 69 78 43 An 4'i 97 83 80 88 09 45 97 68 63 84 08 32 77 87 44 45 6 1 00 6287 6483 6692 6913 7149 7401 7673 7968 8290 8644 9039 9485 9996 46 o3 90 87 95 17 53 06 78 73 95 5i 4b .9^ iooo5 4b 47 06 93 90 99 20 57 10 83 78 83oi 57 53 9501 iooi5 47 48 09 96 94 6702 24 61 i4 87 83 07 63 bo 09 10024 48 49 5o 12 61 1 5 99 97 06 28 65 19 92 89 12 69 67 17 ioo33 10043 49 5o 63o3 65oo 6710 6932 7169 7423 7697 7994 83i8 8676 9074 9525 5i 18 06 04 i3 36 73 27 7702 99 24 82 81 33 ioo52 5i 52 21 09! 07'. 17 4o 77 32 06 8004 29 88 88 4i 1 006 1 52 53 24 12 11' 20 43 81 36 II 09 35 95 9b 49 1 007 1 53 54 55 27 6i3o i5 i4 24 47 85 4i 16 i4 4i 8701 9103 57 10080 54 55 63i9 65i7 6728 6951 7189 7445 7721 8020 8347 8707 9110 9565 10089 56 33' 22 21 3i 55 94 49 25 25 52 i4 17 73 10099 5b 57 36 25' 24 35 59 98 54 3o 3o 58 20 24 81 10108 57 58 40 28 28 38 63 7202 58 35 35 64 26 3i 89 10118 58 59 M. 43 70° 32 3i 42 66 06 63 4o 4o 69 33 38 98 10127 55 M. 71° 72° 73° 74° 75° 76° 77° 78° 79° 80° 81° 82° 83° Page 68] TABLE IV TheS un's Declination for App irent Noon at Greenwich for the year 1848, 1 which wi 1 answer nearly for the years 1852, 1856, 1860. T JAN. FEB. WAR. APRIL MAY. JUNE. JULY. AUG. SEPT. OCT. NOV. DEC. 1 P South. South. South, Jforth. Jforth. JVorth. J\rorth. Jforth. Jforth. South Smith. South. 1 1 1 ( O 1 1 o / / O 1 O 1 o / / 21.53 23. 3 17.15 7.25 4.42 15.12 22. 7 23. 6 17.57 8. 9 3.20 14.35 2 22.59 16.58 7. 2 5. 15.30 22.15 23. 2 17.41 7.47 3.44 14.54 22. 2 2 3 22.53 16.41 6.39 5.28 15.48 22.22 22.57 17.26 7.25 4. 7 15.12 22.11 3 4 22.47 16.23 6.16 5.51 16. 5 22.29 22.52 17.10 7. 3 4.30 15.31 22.19 4 6 S 22.41 16. 5 5.53 6.14 16.22 22.36 22.46 16.54 6.41 4.53 15.49 16. 7 22.27 22.34 5 6 22.34 15.47 5.29 6.37 16.39 ^2.42 22.40 16.37 6.19 5.16 7 22.27 15.28 5. 6 6.59 16.56 22.48 22.34 16.20 5.56 5.39 16.25 22.41 7 8 22.19 15.10 4.43 7.22 17.12 22.53 22.27 16. 3 5.34 6. 2 16.42 22.47 8 <■) 22.11 14.51 4.19 7.44 17.28 22.59 22.20 15.46 5.11 6,25 17. 22.53 9 10 U 22. 3 14.31 3,56 8. 6 8.28 17.44 23. 3 22.13 15.28 4.48 6.48 17.17 22.58 10 TT 21.54 14.12 3.32 17.59 23. 7 22. 5 15.11 4.25 7.11 17.33 23. 3 12 21.44 13.52 3. 9 8.50 18.14 23.11 21.56 14.53 4. 2 7.33 17.49 23. 8 12 13 21.35 13.32 2.45 9.12 18.29 23.15 21.48 14.34 3.39 7.56 18. 5 23.12 13 U 21.24 13.12 2 21 9.33 18.44 23.18 21.39 14.16 3.16 8.18 18.21 23.15' 14 1 16 21.14 12.52 1.58 9.55 18.58 23.20 21.29 13.57 2.53 8.40 18.37 18.52 23.18 23.2r 15 16 21. 3 12.31 1.34 10.16 19.12 23.23 21.20 13.38 2,30 9. 2 17 20.51 12.10 1.10 10.37 19.25 23.24 21. 9 13.19 2. 7 9.24 19. 6 23,23 17 IS 20.39 11.49 0.47 10.58 19.38 23.26 20.59 13. a 1.44 9.46 19.21 23.25 18 19 20.27 11.28 0.23 11.19 19.51 23 27 20.48 12.40 1.20 10. 8 19.35 23.26 19 20 21 20.14 20. 1 11. 7 0. lA^. 11.39 20.04 23.27 20.37 12.20 0.57 10.30 19.48 23.27 23.27 20 21 10.45 0.24 12.00 20.16 23.27 20.25 12. 0.34 10.51 20. 2 22 19.48 10.23 0.48 12.20 20.28 23.27 20.13 11.40 0.10 11.12 20.14 23.27 22 23 19.34 10. 2 1.12 12.40 20.40 23.26 20. 1 11.20 0.13 S, 11.33 20.27 23.27 23 24 19.20 9.40 1.35 13.00 20.51 23.25 19.49 10.59 0.37 11.54 20.39 23.25 24 2-5 26 19. 6 9.17 1.59 13.19 13,39" 21. 1 23.24 19.36 10.39 1. 12.15 20.51 23.24 25 '26 18.51 8.55 2,22 21.12 23.22 19.23 10.18 1.23 12.36 21. 2 23.22 27 18.36 8.33 2.46 13.58 21.22 23.20 19. 9 9.57* 1.47 12.56 21.13 23.19 27 2.S 18.20 8.10 3. 9 14.17 21.32 23.17 18.55 9.36 2.10 13.16 21.24 23.16 28 20 18. 5 7.48 3.33 14.35 21.41 23.14 18.41 9.14 2.34 13.36 21.34 23.13 29 30 17.48 3.56 14.54 21.50 23.10 18.27 8.53 2.57 13.56 21.44 23. 9 30 3l 31 17.32 4.19 21.59 18.12 8.31 14.15 23. 5 Table ] V. A — Tl le Equ ation Df Time for j 4ppar€ ;nt No on at ( jrreenwich, for 1848, or ne arly fo r 1852 , 1856 I, 1860. To be ap plied 1 o the App. Time. JAN. FEB. MAR. APRIL. aiAY. JUNE. JULY. AUG. SEPT. OCT. NOV. DEC. Jldd to .Sdd to .ddd to Add to Sub.fr. Sub.fr. Add to Add to Sub.fr. Sub.fr. Sub.fr. Sub.fr. >. J3pp. . Jlpp. Jipp. Jlpp. Jlpp. Jlpp. App. App. App. App. App. App. >> P T Time. Time. Time. Time. Tune. Time. Time. Time. Time. Time. 'Time. Time. P M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. 3.36 13.50 12.31 3.51 3. 5 2.28 3.31 6. 0.14 10.25 16.16 10.36 1 2 4. 4 13.58 12.19 3.33 3.12 2.19 3.42 5.56 0.33 10.44 16.17 10.13 2 3 4.33 14. 5 12. 6 3.16 3.19 2. 9 3.53 5.52 0.52 11. 3 16.17 9.49 3 4 5. 14.11 11.53 2.58 3.25 1.59 4. 4 5.47 1.12 11.21 16.16 9.24 4 6 6 5.28 14.17 11.39 11.25 2.40 3.30 1.49 4.15 5.41 1.31 11.38 16.14 8.59 8.34 5 6 5.55 14.22 2.23 3.35 1.38 4.25 5.35 1.51 11.56 16.11 7 6.21 14.26 11.11 2. 6 3.39 1.27 4.35 5.28 2.11 12.13 16. 8 8. 8 7 8 6.47 14.29 10.56 1.49 3.43 1.16 4.44 5.20 2.32 12.30 16. 4 7.42 8 9 7.13 14.31 10.40 1.32 3.46 1. 4 4.53 5.12 2.52 12.46 15.58 7.15 9 10 11 7.38 14.33 10.25 1.15 0.59 3.49 0.53 5. 2 5. 3 3.13 13. 2 15.52 6.47 10 TI 8.02 14,33 10. 9 3.51 0.41 5.10 4.54 3.34 13.17 15.46 6.20 12 8.26 14.33 9,52 0.43 3.52 0.29 5.17 4.44 3,55 13.32 15.38 5.52 12 13 8.49 14.32 9.36 0.27 3.53 0.16 5.25 4.34 4.16 13.46 15.30 5.23 13 14 9.11 14.31 9.19 0.12 3.54 0. 4 5.31 4.23 4.37 14. 15.20 4.55 14 15 16 9.33 14.28 9. 2 .S.O. 3 3.54 .4.0. 9 5.38 4.11 4.58 14.13 15.10 4.26 15 To 9.54 14.25 8.44 0.18 3.53 0.21 5.43 3.59 5.19 14.26 14.59 3.56 17 10.15 14.21 8.26 0.32 3.52 0.34 5.48 3,46 5.41 14.38 14.47 3.27 17 18 10.34 14.16 8. 9 0.46 3.50 0.47 5.53 3.33 6.02 14.49 14.34 2,57 18 19 10.53 14.11 7.51 0.59 3.47 1. 5.57 3.20 6.23 15. 14.21 2.27 19 20 21 11.12 11.29 14. 5 7.32 1.13 3.45 1.13 6. 1 3. 6 6.44 15.10 14. 6 1,57 20 21 13,58 7.14 1.25 3.41 1.26 6. 4 2.51 7. 5 15.20 13.51 1.27 22 11.46 13.51 6.56 1.38 3.37 1.39 6. 6 2,36 7.26 15.29 13.35 0.57 22 23 12. 2 13.43 6.37 1.49 3.32 1.52 6. 8 2.21 7.46 15.37 13.18 0.27 23 24 12.17 13.35 6,19 2. 1 3.27 2. 4 6.10 2. 5 8. 7 15.44 13. .4.0. 3 24 25 26 12.31 12.45 13.25 6. 2.11 3.22 2.17 6.11 1.49 8.27 15.51 12.42 0.33 25 26 13.16 5.42 2 22 3.15 2.30 6.11 1.33 8.48 15,57 12 22 1. 3 27 12.58 13. 5 5.23 2.31 3. 9 2 42 6.11 1.16 9. 8 16. 2 12. 2 1 33 27 28 13.10 12.55 5. 5 2.41 3. 2 2,55 6.10 0.58 9.27 16. 6 11.42 2. 2 28 29 13.21 12.43 4.46 2.49 2.54 3. 7 6. 8 0.41 9.47 16.10 11.20 2.32 29 30 13.31 4.28 2.57 2.46 3.19 6. 6 0.23 10. 6 16.13 10.58 3. 1 30 31 13.41 4.10 2.37 6. 4 0. 5 16.15 3.29 1 31 j TABLE IV Page 69] The Sun's Declination for Apparent Noon It Greenwich, for the year 1849, 1 which \vi 1 answer nearly for the years 1853, 1857. 1861. a' O T JAN. FEB. MAR. APRIL MAY. JUNE. JULY. AUG. SEPT. OCT. NOV. DEC. Q 1 Soiit/t. South. South. JVort/i. J^ort/t. M'orth. JVorth. J\l'orth. J^orth. South. South. Smith. f / > . / o ( O 1 o / o / / o / / / 23. 17. 2 7.30 4.37 15. 8 22. 5 23. 7 18. 8.15 3.15 14.30 21.51 ? 22.54 16.45 7. 7 5. 15.26 22.13 23. 3 17.45 7.53 3.38 14.49 22. 2 3 22.49 16.27 6.44 5.23 15.43 22.20 22.58 17.30 7.31 4. 1 15. 8 22. 9 3 4 22.43 16.10 6.21 5.46 16. 1 22.28 22.53 17.14 7. 9 4.24 15.26 22.17 4 5 22.36 15.51 5.58 6. 8 16.18 22.34 22.48 16.58 6.46 4.48 15.45 22.25 5 6 22.29 15.33 5.35 6.31 16.35 22.41 22.42 16.41 6.24 5.11 16. 3 22.32 6 7 22.21 15.14 5.12 6.54 16.51 22.47 22.36 16.24 6. 2 5.34 16.21 22.39 7 S 22.13 14.55 4.48 7.16 17. 8 22 52, 22.29 16. 8 5.39 5.57 16.38 22.45 8 9 22. 5 14,36 4.25 7.38 17.24 22.57 22.22 15.50 5.16 6.20 16.56 22.51 9 10 11 21.56 14.17 4. 1 8. 1 17.40 23. 2 22.14 15.33 4.54 6.42 7. 5 17.13 17729 22.57 23. 2 10 11 21.47 13.57 3.38 8.23 17.55 23. 6 22. 7 15.15 4.31 12 21.37 13.37 3.14 8.45 18.10 23.10 21.58 14.57 4. 8 7.28 17.46 23. 7 12 13 21.27 13.17 2.51 9. 6 18.25 23.14 21.50 14.39 3.45 7.50 18. 2 23.11 13 14 21.16 12.57 2.27 9.28 18.40 23.17 21.41 14.21 3.22 8.13 18.17 23.14 14 15 21. 5 12.36 2. 4 9.50 18.54 23.20 21.32 14. 2 2.59 8.35 8.57 18.33 " 18^48 23.18 "2372r 15 16 20.54 12.15 1.40 10.11 19. 8 23.22 21.22 13.43 2.36 17 20.42 11.54 1.16 10.32 19.22 23.24 21.12 13.24 2.12 9.19 19. 3 23.23 17 18 20.30 11.33 0.52 10.53 19.35 23.25 21. 2 13. 5 1.49 9.41 19.17 23.25 18 19 20.18 11.12 0.29 11.14 19.48 23.26 20.51 12.45 1.26 10. 3 19.31 23.26 19 20 21 20. 5 19.51 10.50 0. 5 11.34 11.55 20. 1 20.13 23.27 20.40 20.28 12.25 1. 3 10.24 19.45 "19758" 23.27 23.27 20 21 10.29 0.19A^. 23.27 12. 5 0.39 10.46 •79 19.38 10. 7 0.42 12.15 20.25 23.27 20.16 11.45 0.16 11. 7 20.11 23.27 22 23 19.24 9.45 1. 6 12.35 20.37 23.27 20. 4 11.25 0. 8S. 11.28 20.24 23.27 23 24 19. 9 9.23 1.30 12.55 20.48 23.26 19.52 11. 4 0.31 11.49 20.36 23.26 24 2-5 2G 18.55 9. 1 1.53 13.15 20.59 23.24 19.39 10.44 0.54 12.10 20.48 23.24 25 26 18.40 8.38 2.17 13.34 21. 9 23.22 19.26 10.23 1.18 12.31 21. 23.22 27 18.24 8.16 2.40 13.53 21.20 23.20 19.12 10. 2 1.41 12.51 21.11 23.20 27 28 18. 8 7.53 3. 4 14.12 21.29 23.18 18.59 9.41 2. 5 13.11 21.21 23.17 28 29 17.52 3.27 14.31 21.39 23.14 18.45 9.19 2.28 13.31 21.32 23.14 29 30 17.36 3.50 14.49 21.48 23.11 18.30 8.58 2.51 13.51 21.42 23.10 30 31 17.19 4.14 21.57 18.15 8.36 14.11 23. 6 31 Table 1 V. A — T\ le Equ ation DfTim e for J "ipparf mt No on at Green\ vich, for 1849, or ne arly fo r 1853 , 185- ', 186] . To be ap plied 1 o the App. r rime. JAN.. FEB. MAR. APRIL. MAY. JUNE. JULY. AUG. SEPT. OCT. NOV. DEC. ^dd to Add to Add to Add to Siib.fr. Sub fr. Add to Add to Sub.fr. Sub.fr. Sub. fr. Sub.fr. ^ Jlpp. App. App. App. App. App. App. App. App. App. App. App. t-. T Time. Time. Time. Time. Time. Time. Time, Time. Tone. Time. Time. Time. P JM. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. 3.58 13.56 12.35 3.56 3. 3 2.31 3.27 6. 1 0.10 10.21 16.16 10.41 2 4.26 14. 4 12.22 3.38 3.11 2.22 3.39 5.57 0.29 10.39 16.17 10.18 ?. 3 4.54 14.10 12.10 3.20 3.17 2.13 3.50 5.52 0.48 10.58 16.17 9.54 3 4 5.21 14.16 11.56 3. 2 3.24 2. 3 4. 1 5.47 1. 7 11.16 16.16 9.30 4 ■5 "6 5.48 14.20 11.43 2.44 3.29 1.53 4.11 5.42 1.27 11.34 16.14 9. 5 5 6 6.15 14.24 11.28 2.26 3.35 1.42 4.21 5.35 1.47 11.52 1 16.12 8.40 7 6.41 14.27 11.14 2. 9 3.39 1.31 4.31 5.28 2. 7 12. 9 16. 8 8.14 7 8 7. 6 14.30 10.59 1.52 3.43 1.20 4.40 5.21 2.28 12.26 16. 4 7.48 8 9 7.31 14.31 10.43 1.35 3.47 1. 9 4.49 5.18 2.48 12.42 15.59 7.21 9 10 7,55 14.32 10.28 1.18 3.50 0.57 4 58 5. 4 3. 9 12.58 15.53 6.53 10 11 8.19 14.32 10.12 1. 2 3.52 0.45 5. 6 4.55 3.29 13.13 15.47 6.26 11 12 8.42 14.31 9.55 0.46 3.54 0.33 5.14 4.46 3.50 13.28 15.39 5.58 12 13 9. 5 14.30 9.39 0.30 3.55 0.21 5.22 4.35 4.11 13.42 15.31 5.29 13 14 9.27 14.28 9.22 0.15 3.55 0. 8 5.28 4.25 4.32 13.56 15.21 5. 14 15 16 9.48 14.25 9. 5 S.O. 1 3.55 AO. 4 5.35 4.13 4.53 14. 9 15.11 4.31 15 16 10. 9 14.21 8.47 0.15 3.54 0.17 5.41 4. 2 5.14 14.22 15. 4. 2 17 10.29 14.17 8.30 0.30 3.53 0.30 5.46 3.49 5.35 14.34 14.48 3.32 17 18 10.48 14.12 8.12 0.43 3.51 0.43 5.51 3.37 5.56 14.45 14.35 3. 3 18 19 11. 6 14. 6 7.54 0.57 3.49 0.56 5.56 3.23 6.17 14.56 14.22 2.33 19 20 21 11.24 14. 7.36 1.10 3.46 1. 9 6. 3.10 6.38 15. 6 14. 8 ■ 2. 3 20 21 11.41 13.53 7.18 1.23 3.42 1.22 6. 3 2.55 6.59 15.16 13.53 1.33 22 11.58 13.45 7. 1.35 3.38 1.35 6. 6 2.41 7.20 15.25 13.37 1. 3 22 23 12.13 13.37 6.42 1.47 3.34 1.48 6. 8 2.25 7.40 15.33 13.20 0.33 23 24 12.28 13.28 6.23 1.58 3.29 2. 1 6.10 2.10 8. 1 15.41 13. 3 0. 3 24 2.5 12.42 13.18 6. 5 2. 9 3.23 2.14 6.11 1.54 8.21 15.48 12.45 ^.0.27 25 26 26 12.55 13. 8 5.46 2.19 3.17 2.27 6.11 1.37 8.42 15.54 12.26 0.57 27 13. 7 12.58 5.28 2.29 3.10 2.39 6.11 1.20 9. 2 15.59 12. 6 1.26 27 28 13.19 12.46 5.10 2.38 3. 3 2.52 6.10 1. 3 9.22 16. 4 11,46 1.56 28 29 13.29 4.51 2.47 2.56 3. 4 6. 9 0.45 9.42 16. 8 11.25 2.25 29 30 13.39 4.33 2.55 2.48 3.16 6. 7 0.27 10. 1 16.11 11. 3 2.54 30 81 13.48 4.14 2.40 6. 4 0. 9 16.14 3.23 31 Page 70] TABLE IV I 1 The Sun's Declination for Apparent Noon at Greenwich, for the year 1850. 1 which wi I answer nearly for the years 1854, 1858, 1862. T JAN. FEB. MAR. APRIL. MAY. JUNE. JULY. AUG. SEPT. OCT. NOV. DEC. South. South. South. JVortA. JVorth. JVortA. JVorth. JVorth. JVorth. .^outh. Sonth. South. / o / O 1 / o ' / o / o / o / o / 1 1 23. 1 17. 6 7.36 4.31 15. 3 22. 3 23. 8 18. 4 8.20 3. 9 14.25 21.49 1 2 22.56 16.49 7.13 4.54 15.21 22.11 23. 4 17.49 7.58 3.32 14.44 21.58 9 3 22.50 16.32 6.50 5.17 15.39 22.19 22.59 17.33 7.36 3.56 15. 3 22. 7 3 4 22.44 16.14 6.27 5.40 15.56 22.26 22.54 17.18 7.14 4.19 15.22 22.15 4 5 22.38 15.56 6. 4 6. 3 16.14 22.33 22.49 17. 2 6.52 4.42 15.40 22.23 5 6 6 22.31 15.37 5.41 6.26 16.31 22.39 22.43 16.45 6.30 5. 6 15.59 22.30 7 22.23 15.19 5.18 6.48 16.47 22.45 22.37 16.29 6. 7 5.28 16.16 22.37 7 8 22.15 15. 4.54 7.11 17. 4 22.51 22.31 16.12 5.45 5.51 16.84 22.44 8 9 22. 7 14.41 4.31 7.33 17.20 22.56 22.24 15.55 5.22 6.14 16.51 22.50 9 10 11 21.58 14,21 4. 7 3.44 7.55 17.36 23. 1 22.16 15.37 4.59 6.37 17. 8 22.56 10 21.49 14. 2 8.17 17.51 23. 5 22. 9 15.19 4.36 7. 17.25 23. 1 11 12 21.39 13.42 3.20 8.39 18. 7 23. 9 22. 15. 2 4.14 7.22 17.42 23. 6 12- 13 21.29 13.22 2.57 9. 1 18.22 23.13 21.52 14.43 3.51 7.45 17.58 23.10 13 14 21.19 13. 2 2.33 9.23 18.. 36 23.16 21.43 14.25 3.28 8. 7 18.14 23.14 14 15 16 21. 8 12.41 12.20 2. 9 1.46 9.44 18.51 23.19 21.34 14. 6 3. 4 8.30 18.29 23.17 15 16 20.57 10. 6 19. 5 23.22 21.24 13.48 2.41 8.52 18.44 23,20 17 20.45 11.59 1.22 10.27 19.19 23.24 21.14 13.29 2.18 9.14 18.59 23.22 17 18 20.33 11.38 0.58 10.48 19.32 23.25 21. 4 13. 9 1.55 9.36 19.14 23.24 18 19 20.21 11.17 0.34 11. 9 19.45 23.26 20.53 12.50 1.32 9.58 19.28 23.26 19 20 21 20. 8 19.55 10.56 0.11 11.29 19.58 23.27 20.42 12 30 1. 8 10.19 "10741 19.42 "19755" 23.27 23.27 20 2T 10.34 0.13. V. 11.50 20.10 23.27 20.31 r..io 0.45 22 19.41 10.12 0.37 12.10 20.22 23.27 20.19 11.50 0.22 11. 2 20. 8 23.27 22 23 19.27 9.50 1. 12.30 20.34 23.27 20. 7 11.30 0. 2S. 11.23 20.21 23.27 23 24 19.13 9.28 1.24 12.50 20.45 23.26 19.55 11.10 0.25 11.44 20.33 23.26 24 25 26 18.58 9. 6 1.47 13.10 13.29 20.56 23.25 19.42 10.49 0.49 12. 5 20.45 23.25 25 26 18.43 8.44 2.11 21. 7 23.23 19.29 10.28 1.12 12.26 20.67 23.23 27 18.28 8.21 2.34 13.48 21.17 23.21 19.16 10. 7 1.35 12.46 21. 8 2.3.21 27 28 18.12 7.59 2.58 14. 7 21.27 23.18 19. 2 9.46 1.69 13. 6 21.19 2.3.18 28 29 17.56 3.21 14.26 21.37 23.15 18.48 9.25 2.22 13.26 21.29 23.15 29 30 17.40 3.45 14.45 21.46 23.12 18.34 9. 3 2.46 13.46 21.39 2,3.11 30 31 17.23 4. 8 21.54 18.19 8.42 14. 6 23. 7 31 Table I V. A — Tl le Equ ation of Tin e for . 4ppar6 mt No on at Green\ vich, for 1850 or ne arly fo r 1854 [, 1858, 186'^ J. To be ap plied \ o the App. ''. ^ime. JAN. FEB. MAR. APRIL. MAY. JUNE. JULY. AUG. SEPT. OCT. NOV. DEC. Md to Md to Add to Add to Sub.fr. Sub.fr. Add to Add to Sub.fr. Sub.fr. Sub.fr. Sub.fr. >> App. .Spp. App. App. App. App. App. App. App. App. App. App. >i o T Tims. Time. Time. Tune. Time. Time. Time. Time. Time. Time. Time. Time. M. S. U. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. M. S. 3.51 13.54 12.37 3.59 3. 3 2.35 3.23 6. 1 0. 6 10.17 16.16 10,47 ?. 4.19 14. 1 12.24 3.41 3.10 2.26 3.35 5.57 0.25 10.36 16.17 10.24 9 3 4.47 14. 8 12.12 3.23 3.17 2.16 3.46 5.53 0.44 10.64 16.17 10. 3 4 5.15 14.14 11.58 3. 5 3.24 2. 6 3.57 5.48 1. 3 11.12 16.16 9.. 36 4 5 5.42 14.19 11.45 2.47 3.29 1.56 4. 8 5.43 1.23 11.30 16.15 9.11 5 6 fi 6. 8 14.23 11.31 2.30 3.35 1.45 4.18 5.37 1.42 11.48 16.12 8.40 7 6.34 14.26 11. M 2.12 3.39 1.35 4.28 6.30 2. 2 12. 6 16. 9 8,20 7 8 7. 14.29 11. 2 1.55 3.43 1.23 4.38 5.23 2.22 12.22 16. 5 7.54 S 9 7.25 14.31 10.46 1.38 3.47 1.12 4.47 5.15 2.43 12.38 16. 7.27 9 10 TT 7.50 14.32 10.31 1.22 3.49 1. 4 56 5. 7 3. 3 12.54 15.55 7. 10 U 8.14 14.32 10.15 1. 5 3.52 0.48 5. 5 4.. 58 3.24 13. 9 15.48 6.32 n 8.37 14.32 9.59 0.49 3.53 0.36 5.13 4.48 3.45 13.24 15.41 6. 4 12 13 9. 14.31 9.43 0.34 3.54 0.24 5.20 4.38 4. 6 13.39 15.33 5,36 13 u 9.22 14.29 9.26 0.18 3.55 0.11 5.27 4.28 4.27 13.53 15.24 5, 7 14 15 16 9.44 14.26 9. 9 0. 3 3.55 .4.0. 2 5.34 4.17 4.48 14. 6 15.14 4.38 15 16 10. 5 14.23 8.52 8.0.12 3.54 0.14 5.40 4. 5 6. 9 14.19 15. 3 4. 9 17 10.25 14.18 8.34 0.26 3.53 0.27 5.46 3.53 6.30 14.31 14.52 3.40 17 18 10.45 14.14 8.17 0.40 3.52 0.40 5.51 3.40 6.51 14.43 14.39 3.11 IS iq 11. 3 14. 8 7.59 0.54 3.49 0.53 5.55 3.26 6.12 14.64 14.26 2.41 19 20 11.21 14. 2 7.41 1. 7 3.47 1. 6 6.59 3.13 6.33 16. 5 14.12 2.11 20 21 11.33 13.55 7.23 1.20 3.43 1.19 6. 2 2.58 6.55 15.15 13..57 1.41 ?,■?, 11.55 13.47 7. 4 1.32 3.40 1.32 6. 5 2.44 7.15 16.24 13.42 1.11 22 93 12.10 13.39 6.46 1.44 3.35 1.45 6. 7 2.28 7.36 16.32 13.25 0.41 23 ''4 12.25 13.30 6.27 1.56 3.31 1.57 6. 9 2.13 7.57 15.40 13. 8 0.11 24 2.5 12.39 13.20 6. 9 2. 7 3.25 2.10 6.10 1.57 8.18 15.47 12.50 .1.0.18 2.3 26 9f\ 12.52 13.10 6.50 2.17 3.19 2.23 6.10 1.40 8.38 15.54 12,32 0,48 V 13. 4 13. 5.32 2.28 3.13 2.35 6.10 1.23 8.58 15.59 12.12 1.18 27 ?,8 13.16 12.48 5.13 2.37 3. 6 2.47 6. 9 1. 6 9.18 16. 4 11.52 1.48 28 ^9 13.27 4.54 2.46 2.59 3. 6. 8 0.49 9.38 16. 8 11.31 2.17 29 30 31 13.36 4.36 2.55 2.51 3.11 6. 6 0.31 9.57 16.12 11. 9 2.46 30 13.45 4.17 2.43 6. 4 0.13 16.14 3.15 31 TABLE IV [Page 7] The Sun's Declination for Anna rent Noon at Greenwich, for the year 1851, 1 wh ich \vi 1 answer nearl}' for the years 1855, 1859, 1863. 1 P JAN. FEB. aiAR. APRIL. l\L\v. JUNE. JULY. AUG. SEPT. OCT. NOV. DEC. P Soxah. South. South. JVurth. jVurth. Xorlh. JVorth. JVorth. A'orlh. South. South. South. ' 1 O 1 O 1 o / / o / ' o / ' 1 O 1 1 23. 2 17.11 7 41 4.25 14.59 22. 1 2.3. 9 18. 8 8.25 3. 3 14.21 21.47 I f. 22..57 16..53 7.19 4.49 15.17 22. 9 2.3. 5 17.53 8. 3 3.27 14.40 21.56 2 3 22..i;J 16.36 6.56 .5.12 15.. 35 22.17 23. 1 17.37 7.42 3.-50 14.59 22. 5 3 4 22.46 16.18 6..33 5.35 15.52 22 24 22..56 17.21 7.19 4.13 15.17 22.13 4 5 6 22.39 16. 6.10 5.57 16.10 22.31 22.50 17. 5 6.57 6.35 4.36 4..59 15.36 15.54 22.21 "22729" 5 6 22.32 15.42 5.46 6.20 16.27 22.38 22.45 16.49 7 92 2.5 15.23 5.23 6.43 16.44 22.44 22.39 16.33 6.13 5.23 16.12 22.36 7 8 22.17 1.5. 5 5. 7. 5 17. 22.50 22.32 16.16 5.-50 5.46 16.30 22.42 8 22. 9 14.46 4.36 7.28 17.16 22.55 22.25 15.59 5.27 G. 8 16.47 22.49 9 10 11 22. 14.26 4.13 7.50 17.32 23. 23. 4 22.18 15.41 5. 5 6.31 17. 4 22.54 23. 10 IT 21..51 14. 7 3.49 8.12 17.48 22.10 15.24 4.42 6..54 17.21 12 21.42 13.47 3.26 8.34 18. 3 23. 9 22. 2 15. 6 4.19 7.17 17.38 23. 4 12 13 2 1.. 32 13.27 3. 2 8.-56 18.18 23.12 21..54 14.48 3.-56 7.39 17..54 23. 9 13 14 21.22. 13. 7 2.39 9.18 18.33 2-3.16 21.45 14.30 3.33 8. 2 18.10 23.13 14 1.5 16 21.11 12.46 2.15 9.39 18.47 23.19 21.36 14.11 3.10 8.24 18.25 23.16 15 16 21. 12.25 1.51 10. 1 19. 2 23.21 21.27 13.52 2.47 8.46 18.41 2.3.19 17 20.43 12. 5 1.28 10.22 19.15 23.23 21.17 13.33 2.24 9. 8 18..56 23 22 17 18 20.. 36 11.44 1. 4 10.43 19.29 23.25 21. 7 13.14 2. 1 9.30 19.10 23.24 18 19 20.24 11.22 0.40 11. 4 19.42 23.26 20..56 12..55 1.37 9.52 19.24 23.25 19 20 21 20.11 19..58 11. 1 0.17 11.24 19.55 23.27 20.45 12.35 1.14 10.14 19.38 23.27 20 21 10.39 0. IN. 11.45 20. 7 23.27 20.34 12.15 0.51 10.35 19.52 23.27 22 19.44 10.18 0.31 12. 5 20.19 23.27 20.22 11.-55 0.27 10..57 20. 5 23.27 22 23 19.. 31 9..56 0.,54 12.25 20.31 23.27 20.10 11.35 0. 4 11.18 20.18 23.27 23 24 19.16 9.34 1.18 12.45 20.43 23.26 19.58 11.14 0.20 S. 11. .39 20.30 23.26 24 2,5 20 19. 2 9.12 1.42 2. 5 13. 5 20.54 23.25 19.45 10.54 0.43 12. 20.42 23.25 25 26 18.47 8.49 13.25 21. 4 23.23 19.32 10.33 1. 6 12.21 20.54 23.23 27 18.32 8.27 2.29 13.44 21.15 23.21 19.19 10.12 1.30 12.41 21. 5 23.21 27 28 18.16 8. 4 2..52 14. 3 21.25 23.19 19. 5 9.51 1..53 13. 1 21.16 23.19 28 29 18. 3.16 14.22 21.34 2.3.16 18.51 9.30 2.17 13.22 21.27 23.16 29 30 31 17.44 3.39 14.40 21.44 23.13 18.37 9. 9 2.40 13.41 21.37 23.12 30 17.27 4. 2 21.52 18.23 8.47 14. 1 23. 8 31 Table 1 V. A — Tl le Equ at ion of Time for . A.ppar( 5nt No on at Green) mch, for 1851 or ne arly fo r 1855 ., 185f ), 1863. To be ap plied 1 o the App.l ^ime. JAN. FEB. MAR. APRIL. MAY. JUNE. JULY. AUG. SEPT. OCT. NOV. DEC. Ji'ld to .Bdd to .add to Add to Sub. fr. Sub.fr. Add to Add to Sub.fr. Siib.fr. Sub.fr. Sub.fr. Jlpp. Jipp. j]pp. App. App. App. App. App. App. App. App. App. >^ T Time. Time. Time. Tune. Time. Time. 'lime. Time. Time. Time. Time. Time. P M. S. M. S. M. S. HL S. M. S. M. S. M. S. 3.23 M. S. M. S. M. S. M. S. 16.15 M. S. 10.52 3.44 13.52 12.40 4. 4 2.-59 2.34 6. 3 0. 10.12 2 4.12 14. 12.28 3.46 3. 7 2.25 3.34 6. 0.19 10.31 16.17 10.30 2 3 4.40 14. 7 12.15 3.28 3.14 2.16 3.46 5.56 0.38 10.49 16.17 10. 6 3 4 5. 8 14.13 12. 2 3.11 3.20 2. 6 3.57 5.51 0..57 11. 8 16.17 9.42 4 5 6 5.3.5 14.18 11.49 2.53 3.26 1.-56 1.46 4. 8 5.46 1.17 11.26 16.16 9.18 5 6 6. 2 14.23 11.35 2.35 3.31 4.18 5.40 1.37 11.44 16.14 8.53 7 6.28 14.26 11.21 2.18 3.36 1..35 4.28 5.33 1.-57 12. 1 16.11 8.27 7 8 6.-54 14.29 11. 6 2. 1 3.40 1.24 4.38 5.26 2.17 12.18 16. 7 8. 1 8 9 7.20 14.31 10.51 1.44 3.44 1.13 4.47 5.18 2.38 12.35 16. 3 7.35 9 10 7.44 14.32 10..36 1.27 3.47 1. 1 4.56 5.10 2.-58 12.51 15.-57 7. 8 10 11 8. 8 14.33 10.20 1.10 3.49 0.49 5. 4 5. 1 3.19 13. 6 15.51 6.40 11 12 8.32 14.32 10. 4 0.54 3.51 0..37 5.12 4.51 3.40 13.21 15.44 6.12 12 13 S..55 14.31 9.47 0.38 3.53 0.25 5.20 4.41 4. 1 13.36 15.36 5.44 13 14 0.17 14.29 9.30 0.23 3.-54 0.13 5.27 4.30 4 22 13.50 15.27 5.16 14 15 16 9.. 38 14.26 9.13 0. 7 3.54 0. 5.33 4.19 4.43 14. 4 15.18 4.47 15 16 9.-59 14.23 8..56 S.O. 8 3..54 A 0.12 5.39 4. 8 5. 4 14.17 1-5. 7 4.18 17 10.20 14.19 8.. 39 0.22 3..53 0.25 5.45 3.-55 5.26 14.29 14.-56 3.48 17 18 10.39 14.14 8.21 0.36 3.51 0.38 5.-50 3.43 5.47 14.41 14.44 3.19 18 19 10..58 14. 9 8. 3 0.50 3.49 0.51 5.-54 3.30 6. 8 14.52 14.30 2.49 19 20 21 11.16 14. 2 7.45 1. 4 3.47 1. 4 5..59 3.16 6.29 15. 3 14.17 2.19 20 2T 11..33 13..56 7.27 1.16 3.44 1.17 6. 2 3. 2 6.-50 15.13 14. 2 1.49 22 11.50 13.48 7. 8 1.29 3.40 1.30 6. 5 2.47 7.11 15.22 13.46 1.19 22 23 12. 5 13.40 6.-50 1.41 3.36 1.43 6. 7 2.32 7.31 15.31 1-3.30 0.49 23 24 12.20 13.31 6.32 1.53 3.31 1..56 6. 9 2.17 7.52 15.39 1.3.13 0.19 24 25 26 12.35 13.22 6.13 2. 4 3.25 2. 8 6.11 2. 1 8.12 15.46 12..55 ^.0.11 25 26 12.48 1.3.12 5.55 2.14 3.20 2.21 6.11 1.45 8.33 15..52 12.36 0.41 27 13. 1 13. 2 5.36 2.24 3.13 2.34 6.12 1.28 8.53 15..58 12.17 1.11 27 28 1.3.13 12.51 5.18 2.34 3. 6 2.46 6.11 1.11 9.13 16. 3 11..57 1.41 28 29 13.24 4.-59 2.43 2..59 2..59 6.10 0..54 9.33 16. 7 11.36 2.10 29 30 31 13.34 13.4'^ 4.41 2.51 2.51 3.11 6. 8 0.36 9.-52 16.11 11.14 2.40 30 31 4.23 2.43 6. 6 0.18 1 16.13 3. 9 1 I'age 72J TABLE V. 1 For reducing the Sun's Declination, as given in the Nautical Almanac for Noon at Greenwich, to any other Time under any other Meridian. Add aft. N. Sub. aft. N. H.M H.M H.M H.M H.M H.M H.M H.M 1 Sub. aft. N. Add aft. N. Sub. bef. N. Add bef. N. 0.2c 5 0. 4C 10 1. 15 1. 20 20 1. 40 25 2. 30 2. 20 35 2. 40 40 Add bef. N. Sub. bef. N. Add in W. Sub. in W. Sub. in W. Add in W. Sub. in E. Add in E. Deg m"!s" 0. c Deg. M.S 0. Dcg M.S 0. Deg. 3I.S. 0. Deg. M.S. 0. Deg. M.S. 0. Deg. M.S. 0. Deg. M.S. 0. Add in E. Sub. in E. Days. Days. Days. Days. i Decemb. 21 Decemb. 21 21 June 21 June 20 22 0. 0. I 0. I 0. I 0. 2 2 0. 2 0. 3 ?2 20 19 23 0. 0. I 0. 2 0. 2 0. 3 0. 4 0. 5 0. 6 23 19 18 24 0. I 0. 2 0. 3 0. 4 0. 6 0. 7 0. 8 0. q 24 18 17 25 0. I 0. 3 0. 4 0. 6 0. 7 0. 9 O.II 0.12 25 17 i6 26 0. 2 0. 4 0. 5 0. 7 0. 9 O.II o.i3 o.i5 26 r6 i5 27 0. 2 0. 5 0. 6 0. 8 O.II o.i3 o.i5 0.18 37 i5 i4 28 0. 3 0. 6 0. 7 O.IO 0. 12 o.i5 0.18 0.21 28 14 i3 ^9 0. 3 0. 7 0. 9 0.12 o.i5 0.18 0.21 0.24 29 i3 12 3o 0. 3 0. 4 0. 7 0. 8 O.IO O.II o.i3 o.i5 0.17 0. 19 0.20 0.22 0.23 0.26 0.27 o.3o 3o June 12 II Decemb. 3i I July 11 10 January i 0. 4 0. 8 0.12 0.16 0.20 0.24 0.28 0.32 2 10 9 2 0. 4 0. 8 o.i3 0.17 0.21 0.26 o.3o 0.35 3 9 8 3 0. 6 0. 9 o.i4 0.19 0.24 0.29 0.33 0.38 4 8 7 4 0. 5 O.IO O.lb 0.21 0.26 o.3i 0.36 o.4i 5 7 6 5 0. 5o.ii 0.16 0.22 0.28 0.33 0.38 0.44 6 6 5 6 0. 60.12 0.17 0.24 o.3o 0.35 0.41 0.47 7 5 4 7 0. 60.12 0.18 0.25 o.3i 37 0.43 0.49 8 4 3 8 0. 60. i3 0.19 0.26 0.33 0.39 0.45 0.52 9 3 2 9 0. 7 0. 7 o.i4 o.i4 0.20 0.21 0.27 0.29 0.34 oT36 0.41 0.43 0.48 o.5o 0.55 0.57 10 2 Decemb. i 10 II I June Novemb. So II 0. 7 o.i5 0.22 o.3o 0.37 0.45 0.52 I. 12 3 1 May 29 12 0. 8 0.16 0.23 o.3i 0.39 0.47 0.55 I. 3 i3 3o 28 i3 0. 8 0.16 0.24 0.33 o.4i 0.49 0.57 1. 6 i4 29 27 i4 0. 8 0.17 jo. 25 0.34 0.42 o.5i 0.59 I. 8 i5 28 26 i5 0. 9 0.18 0.26 0.35 0.44 0.53 1 . 2 1 .11 16 27 25 16 0. 9 0.18 0.27 0.37 0.46 0.55 I. 4 i.i3 17 26 24 17 0. 9 0.19 0.28 0.38 0.47 0.57 I. 6 1. 16 18 25 23 18 o.io 0.20 0.29 0.39 0.49 0.58 I. 9 1. 19 19 24 22 19 O.IO O.IO 0.20 0.21 o.3o o.3i 0.40 0.41 o.5o o.5i 1 . 1 . 2 1 .10 1 .12 1.20 1 .22 20 23 21 20 21 22 20 21 O.II 0.22 0.32 0.43 0.53 I. 4 i.i4 1.25 22 21 19 22 O.II 0.22 0.33 0.44 0.55 I. 6 1. 17 1.28 23 20 18 23 O.II 0.23 0.34 0.45 0.56 I- 7 i.ig i.3o 24 19 17 24 0.12 0.23 0.34 0.46 0.57 I. Q 1 .21 1.32 25 18 16 25 0.12 0.24 0.35 0.47 0.59 I .11 1.23 1.35 26 17 i5 26 0.12 0.24 0.36 0.48 I. 1. 12 1.24 1.36 27 16 i4 27 0.12 0.25 0.37 0.49 I. 2 i.i4 1 .26 1 .39 28 l5 i3 . 28 o.i3 0.26 0.38 o.5i 1. 4 1. 16 1.28 r.4T 29 i4 If January 3o o.i3 o.i3 0.26 0.27 0.39 0.41 0.53 0.55 I. 6 I. 9 1. 19 r .22 1.33 1.36 1.45 i.5o 3 1 July 12 9 February i 2 August 10 7 3 o.i4 0.28 0.42 0.57 1 .1 1 1.25. 1.39 1.53 4 8 5 5 o.i4 0.29 0.43 0.58 i.i3 1.27 1.42 1.56 6 6 3 7 o.ib o.3o 0.45 I. i.i5 i.3o 1.44 i.5g 8 4 Novemb. i 9 o.i5 o.3i 0.46 1 . 2 1. 17 1.32 1.47 2. 3 10 2 May October 3o II 0.16 0.32 0.47 I. 3 1. 19 1.35 i.5o 2. 6 12 3o April 28 i3 0.16 0.32 0.48 I. 5 1 .21 1.37 1.53 2. Q i4 28 26 i5| 0.16 0.33 0.49 I. 6 1 .22 I.3q 1.56 2.12 16 26 24 17 0.17 0.34 o.5o I- 7 1.24 1.41 1.58 2.l5 18 24 21 20 0.17 0.17 c. 34 0.35 0.52 0.53 I. 9 1 .11 1.27 1 .20 1.44 1.46 2. I 2. 4 2.19 2.22 21 21 18 23 24 18 i5 February 26 0.18 0.36 0.54 i.i3 i.3i r.4o 2. 7 2.25 27 i5 12 March i 0.18 0.37 0.55 i.i4 1.32 1.5. 2. q 2.28 3o August 12 9 4 0.19 0.38 0.56 i.i5 1.34 1.53 2.1; 2.3o 2 Sept. 9 6 7 0.19 0.38 0.57 1. 16 1.35 1.54 2.ii 2.32 5 6 October 3 10 0.19 0.38 0.57 1. 17 1.36 1.55 2.14 2.34 8 3 April Septem. 3o i3 0.19 0.39 0.58 1. 17 1.37 1.56 2.l5 2.35 II 3 1 March 27 16 0.19 0.39 T.5M 1. 18 1.38 1.57 2.16 2.36 i4 28 24 i; 0-20 0.39 j.5i 1. 18 1.38 I., 57 2.16 2.36 17 25 After Before 0.20 o.4o 0.59 I.IQ i.3q 1.58 2.17 2.36 Before After Equinox. Equinox. 1 1 Equinox. Equinox. TABLE V. [Pago 73 For reducing the Sun's Declination, as given in the Nautical Ahnanac for Noon at Greenwich, to any other Time under any other Merit] ian. Add aft. N. Sub. aft N. H.M H.M H.31 H.M H.M H3I H.M Sub. aft. N. Add aft. N. Sub. bef. N. Add bef. N. 3. C 45 3. 20 50 3. 40 55 4. CO 4. 20 65 4. 40 70 5. 75 Add bef: N. Sub. bef. N. Add in W. Sub. in W. Sub, in W. Add in W. Sub. in E. Add in E. Deg M.S 0. c Ucg. M.S 0. c De- M.S 0. Dcg. M.S. 0. Deg M.S. 0. Deg M.S. 0. Deg. M.S. 0. Add in E. Sub. in E. Days. Days. Daj's. Days. December 21 December 21 21 June 21 June 20 22 0. 3 0. 3 0. A 0. 4 0. 4 0. 5 0. 5 22 20 19 23 0. 6|0. 7 0. 8 0. 9 0. 9 o.io O.II 23 19 18 24 O.IO 0,11 0.12 o.i3 o.i4 o.i5 0.16 24 18 17 25 o.iii O.ID 0.16 0.18 0.19 0.20 0.22 25 '7 16 26 0.16 0.18 0.20 0.22 0.24I0.26 0.27 26 16 i5 27 0.20 0.22 0.24 0.26 0.29 o.3i 0.33 27 i5 i4 28 0.23 0.25 0.28 o.3i 0.34 0.36 0.38 28 i4 i3 2Q 0.26 0.29 0.32 0.35 0.38 o.4i 0.44 29 i3 12 3o o.3o 0.33 0.33 0.37 0.36 0.40 0.40 0.44 0.43 0.48 0.46 o.5i o.5o 0.55 3o June 12 II Decembe r 3i I July II 10 January- I 0.36 0.40 0.44 0.48 0.53 0.57 I. I 2 10 9 2 0.39 0.44 0.48 0.53 0.67 I. 2 I. 6 3 9 8 3 0.43 0.48 0.53 0.57 I. 2 I- 7 I. II 4 8 7 4 0.46 o.5i 0.56 I. 1 I- 7 1. 12 1. 17 5 7 6 5 0.49 0.55 I. I. 6 I. II 1. 17 1 .22 6 6 5 6 0.52 0.58 I. 4 1. 10 1.16 1 .22 1.27 7 5 4 7 0.55 I. I I. 7 1. 14 1 .20 1 .26 1.32 8 4 3 8 0.58 I. 5 I. II 1. 18 1 .24 i.3i 1.37 9 3 2 9 I. I I. 4 I. 8 1. 12 i.i5 I.iq 1.22 1 .26 i.29]i.36 i.33|i.4i 1.43 1.48 10 2 December i 10 II I June November 3o II I- 7 i.i5 1.23 1 .3o 1.37 1.45 1.52 12 3 1 May 29 12 1. 10 1. 18 1.26 1.34 1.4^ i.5o 1.57 i3 3o 28 i3 i.i3 1 .22 i.3o 1.38 1.46 1.54 2. 2 i4 29 27 i4 1. 16 1.25 1.34 1 .42 i.5o 1.58 2. 7 i5 28 26 i5 1. 19 1.28 1.37 1.46 1.55 2. 3 2.12 16 27 25 16 1.22 i.3i i.4o 1.49 1.59 2. 8 2.17 17 26 24 17 1.25 1.35 1.44 1.53 2. d 2.12 2.21 18 25 23 18 1.28 1.38 1.47 1.57 2. 7 2.16 2.26 19 24 22 19 i.3o 1.33 i.4i 1.44 i.5i 1.54 2. I 2. 4 2. II 2.21 2.3l 2.35 20 23 21 20 2.l5 2.25 21 22 20 21 1.36 1.47 1.57 2. 8 2.192.29 2.40 22 21 19 22 1.39 i.5o 2. 2. II 2.22 2.33 2.44 23 20 18 23 1.41 1.53 2. 4 2.l5 2,26 2.37 2.48 24 19 17 24 1.43 1.55 2. 7 2.18 2.3o 2.41 2.52 25 18 16 25 1.46 1.58 2.10 2.21 2.33 2.45 2.56 26 17 i5 26 1.48 2, I 2.l3 2.25 2.37 2.49 3. I 27 16 i4 27 i.5i 2. 4 2.16 2.28 2.40 2.52 3. 5 28 l5 i3 28 1.54 2. 7 2.19 2.3l 2.44 2.56 3. 9 29 i4 II January 3o 1.58 2. 3 2. II 2.17 2.24 2.3o 2.37 2.43 2.5l 2.57 3. 4 3. II 3.17 3.24 3 1 July 12 9 February I 2 August 10 7 3 2. 7 2.21 2.35 2.49 3. 3 3.17 3.32 4 8 5 5 2. II 2.25 2.40 2.54 3. 9 3.23 3.38 6 6 3 7 2.14 2.29 2.44 2.59 3.i4 3.29 3.44 8 4 November i 9 2.18 2.33 2.49 3. 4 3.19 3.35 3.5o 10 2 May October 3o II 2.22 2.38 2.53 3. 9 3.25 3.4i 3.56 12 3o April 28 i3 2.25 2.41 2.58 3.14 3.3o 3.46 4. 3 i4 28 ^ 26 i5 2.29 2.45 3. 2 3.18 3.35 3.5i 4. 8 16 26 24 17 2.32 2.49 3. 5 3.22 3.39 3.56 4.i3 18 24 21 20 2.36 2.40 2.53 2.58 3. II 3.i5 3.s8 3.33 3.45 3.5i 4. 3 4. 8 4.20 4.26 21 21 18 23 24 18 i5 February 26 2.43 3. I 3.20 3.38 3.56 4.14 4.32 27 i5 12 March I 2.46 3. 5 3.23 3.42 4. I 4.19 4.38 3o August 12 9 4 2.49 3. 8 3.26 3.45 4. 4 4.23 4.4i 2 SepL 9 6 7 2.5l 3.10 3.29 3.48 4. 7 4.26 4.45 5 6 October 3 10 2.53 3.i3 3.32 3.5i 4.10 4.215 4.49 8 3 April September 3o i3 2.55 3.14 3.33 3.53 4.i3 4.32 4.5i II 3 1 March 27 16 2.56; 3.i5 3.34 3.54 4.14 4.33 4.52 i4 28 ^^ 19 2.56' 3.i5 3.35 3.55 4.i5 4.33 4.52 17 25 After [Before 2.56'3.i5 3.35 3.55 4.i5 ^.34 4.53 Before After Equinox. .Equinox. 1 1 Equinox. Kquinox. 10 Page 74] TABLE V. For reducing the Sun's Declination, as given in the Nautical Almanac for Noon at Greenwich, to any other Time under any other Merid ian. Add afl. N. Sub. aft. N. H.M H.M H.M HM H.M H.M H.M Sub. afl. N. Add aft. N. Sub. bef. N. Add bef. N. 5. 20 80 5. 40 86 6. 90 6. 20 95~ 6. 40 100 7. 10.5 7. 20 110 Add bef. N. Sub. bef N, Add in W. Sub. in W. Sub. in W. Add in W Sub. in E. Add in E. Deg^. mTs^ 0. Deg. M.S. 0. Deg-. M.S. 0. Deg. M.S. 0. Deg. M.S. 0. Deg. M.S. 0. Deg. M.S. 0. Add in E. Sub. in E Days. Days. Days. Days. December 21 Decembei 21 21 June 21 June 20 22 0. 5 0. 6 0. 6 0. 7 0. 8 0. 8 0. 8 22 20 19 23 O.II 0.12 o.i3 o.i4 o.i5 o.i5 0.16 23 19 18 24 0.17 0.19 0.20 0.2] 0.22 0.23 0.24 24 18 17 25 0.23 0.25 0.26 0.28 0.2910.31 0.32 25 17 16 26 0.29 o.3i 0.33 0.35 0.37 0.38 0.40 26 .6 i5 27 0.35 0.38 0.40 0.42 0.44 0.46 0.49 27 .5 i4 28 0.41 0.43 0.46 0.49 o.5i 0.54 0.57 28 14 i3 29 0.47 o.5o 0.53 0.56 0.59 I. 2 I. 5 29 i3 12 3o 0.53 0.59 0.56 1 . 2 0.59 I. 6 I. 3 1 .10 I. 6 i.i3 I. 9 1. 17 1 .12 1 .21 3o June 12 II Decembei 3i I July II 10 January I I. 5 I. 9 i.i3 1. 17 1 .21 1.25 1 .29 2 10 9 2 1 .11 i.ib 1. 19 1 .24 1.28 1.32 1.37 3 9 8 3 I. lb 1. 21 1 .26 I .'i 1. 3b 1.40 1.45 4 8 7 4 1 .22 1.27 1.32 1.3" 1.42 1.47 1.53 5 7 6 5 1.27 1.33 1.38 1.44 1.49 1.54 2. 6 6 5 6 1.33 1 .39 1.45 i.5i 1.57 2. 2 2. 8 7 5 4 7 1.39 1.45 i.5i 1.57 2. 3 2. 9 2.16 8 4 3 8 1.44 1 .5o I bi 2. 4 2.10 2.16 2.23 9 3 2 9 i.5o 1.55 1.56 2. 2 2. 3 2. 9 2.10 2.16 2.17 2.23 2.23 2.3o 2.3o 1738 10 2 December i 10 II I June November So II 2. 2. 7 2.l5 2.22 2.3o 2.37 2.45 12 3 1 May 29 12 2. b 2.l3 2.21 2.29 2.37 2.44 2.52 i3 3o 28 i3 2.10 2.19 2.27 2.35 2.43 2.5. 3. i4 29 27 i4 2.16 2.25 2.33 2.42 2.5o 2.58 3. 7 i5 28 26 i5 2.21 2.3o 2.38 2.47 2.56 3. 5 3.i3 16 27 25 16 2.26 2.35 2.44 2.53 3. 2 3. II 3.21 17 26 24 17 2.3l 2.4o 2.5o 2.59 3. 9 3.18 3.28 18 25 23 18 2.36 2.46 2.55 3. 5 3.i5 3.24 3.34 19 24 22 19 2.41 2.46 2.5l 2.56 3. I 3. 6 3.11 3.17 3.21 3.27 3.3i 3.37 3.4\ 3.48 20 23 21 20 21 22 20 21 2.5o 3. 2 3.12 3.23 3.33 3.44 3.55 22 21 19 22 2.55 3. 6 3.17 3.28 3.39 3.5o 4. I 23 20 18 23 3. 3. II 3.22 3.33 3.45 3.56 4. 7 24 19 17 24 3. 4 3.16 3.27 3.3q 3.5o 4. I 4.i3 25 18 16 25 3. 8 3.20 3.32 '•i-44 3.56 4. 7 4.19 26 17 i5 26 3.i3 3.25 3.37 3.4q 4. I 4.i3 4.26 27 16 i4 27 3.17 3.29 3.42 3.54 4. 6 4.1Q 4.3. 28 i5 i3 28 3.22 3.34 3.47 4. 4.12 4.25 4.38 29 i4 II January 3o 3.3o 3.38 3.43 3.5i 3.56 4. 5 4. 9 4.18 4.22 4.32 4.36 4.46 4.49 4.59 3 1 July 12 9 February I 2 August 10 7 3 3.46 4. 4.14 4.28 4.42 4.56 5. 10 4 8 5 5 3.52 4. 6 4.21 4.36 4.5o 5. 5 5.19 6 6 3 7 3.59 4.14 4.29 4.44 4.59 5.i4 5.29 8 4 November i 9 4. 5 4-21 4.36 4.52 i>. 7 5.23 5.38 10 2 May October 3o II 4.12 4.28 4.44 5. 5.16 5.3i 5.47 12 3o April 28 i3 4.194.35 4.5i 5. 7 5.23 5.40 5.56 i4 28 26 i5 4.244.41 4.57 5.14 5.3o 5.47 6. 3 16 26 24 17 4.304.47 5. 3 5.21 5.38 5.55 6.12 18 24 21 20 4.37 4.44 4.55 5. 2 5.12 5.19 5.29 5737 5.47 5.55 6. 4 6.i3 6.21 6.3i 21 21 18 23 24 18 i5 February 26 4.5o 5. 8 5.26 5.44 6. 2 6.20 6.38 27 i5 12 March I 4.56 5.i5 5.33 5.52 6.10 6.29 6.47 3o August 12 9 4 5. 5.19 5.38 5.57 6.16 6.34 6.53 2 Sept. 9 6 7 5. 4 5.23 5.42 6. I 6.20 6.3q 6.58 5 6 October 3 10 5. 8 5.27 5.46 6. 5 6.25 6.44 7. 3 8 3 April September 3o i3 5. II 5.3o 5.4g 6. 8 6.28 6.47 7. 6 II 1 3 1 March 27 16 5.12 5.3i 5.5. 6. II 6.3i 6.5o 7. 9 i4 28 24 19 5.12 5.32 5.52 6.12 6.32 6.5i 7.H 17 25 After Before ^ 5.i3 5.33 5.53 6.i3 6.33 S.52 7. II Before After Equinox. Equinox. Equinox. Equinox. TABLE V. f Pago 75 For reducing the S an's Declination, as given in the Nautical Almanac for Noon at Greenwich, to any o;her Time under any other Merit ian. Add aft. N. Sub. aft N. H.M H.31 H.M H.M H.Rl H.M H.M Sub. aft. N. Add aft. N. Sub. bef. N. Add bef. N. 7. 40 115 8. 120 3. 20 125 8. 40 130 9. C 135 9. 20 140 9. 40 145 Add bef. N. Sub. bef. N. Add in W. Sub. in \V. Sub. in W. Add in W. Sub. in E. Add in E. Deg M.S 0. Deg. M.S. 0. Deg M.S 0. Deg. M.S. 0. Deg- M.S 0. Deg M.S 0. Deg. 31. S. 0. Add in E. Sub. in E. Days. Days. Days. Days. December 21 JDeceinber 21 21 June 21 June 20 22 0. 9 0. 9 0. 9 O.IO o.io|o.io o.io 22 20 19 23 0.17 o.ib 0.18 0.19 0.19 0.20 0.21 23 19 10 24 0.25 0.26 0.27 0.28 0.29 o.3o o.3i 24 18 17 25 0.34 0.35 0.36 0.38 0.39 o.4i 0.43 25 17 16 26 0.42 0.44 0.46 0.48 0.49 o.5i 0.53 26 16 ID 27 o.5i 0.53 0.55 0.57 0.59 I . I I. 3 27 i5 i4 28 0.59 I. 2 I. 5 I- 7 I. 9 1. 12 i.i4 28 14 i3 29 1 . 8 I. II 1. 14 1. 17 1. 19 1 .22 1.25 29 i3 12 3o 1. 16 1.24 1. 19 1.28 1.23 1.02 1 .26 1.35 1 .29 1 .39 1.32 1.43 1.35 1.46 3o June 12 II December 3i I July 1 1 10 January I 1.33 1.37 1.41 1.45 1.49 1. 53 1.57 2 10 9 2 1.42 1. 4b i.5i 1.55 1.59I2. 3 2. 7 3 9 8 3 1.49 1.54 1.59 2. 4 2. 9 2.l3 2.18 4 8 7 4 1.68 2. 3 2. 8 2.l3 2.19 2.23 2.28 5 7 6 5 2. 5 2. II 2.16 2.22 2.28 2.33 2.39 6 6 5 6 2. i4 2.20 2.26 2.32 2.38 2.43 2.49 7 5 4 7 2.22 2. 28 2.34 2.4l 2.47 2.53 2.59 8 4 3 8 2.29 2.36 2.43 2.49 2.56 3. 3 3. 9 9 3 2 9 2.37 2.45 2.44 2.52 2.5l 2.59 2.58 3. 6 3. 5 3.14 3.12 3.21 3.19 3.28 10 2 December i 10 II I June November So TI 2.52 3. 3. 7 3.i5 3.23 3.3o 3.38 12 3 1 May 29 12 3. 3. 8 3.16 3.24 3.32 3.39 3.47 i3 3o ^ 28 I 3 3. 8 3.16 3.24 3.32 3.40 3.49 3.57 i4 29 27 i4 3.i5 3.24 3.32 3.4i 3.49 3.58 4. 6 i5 23 26 t5 3.22 3.3i 3.40 3.49 3.58 4. 7 4.16 16 27 25 16 3.3o 3.39 3.48 3.57 4. 7 4.16 4.25 17 26 24 17 3.37 3.46 3.56 4. 6 4.16I4.24 4.'M 18 25 23 18 3.44 3.54 4. 4 4.14 4.24 4.33 4.43 19 24 22 19 3.5i 3.58 4. I 4. 8 4. II 4.19 4.21 4.29 4.3i 4.39 4.41 4.5o 4.5i 5. 20 23 21 20 21 22 20 21 4. 5 4.16 4.27 4.37 4.484.59 5. 9 22 21 10 22 4.12 4.23 4.M 4.45 4.56 5. 7 5.18 23 20 18 23 4.19 4.3o 4.41 4.53 5. 4 5.i5 5.26 24 19 17 24 4.25 4.36 4.48 5. 5.12 5.23 5.34 25 18 16 25 4.3i 4.43 4.55 5. 7 5.19 5.3o 5.42 26 17 i5 2fi 4.38 4.5o 5. 2 5.14 5.26 5.38 5.5o 27 16 i4 27 4.43 4.56 5. 8 5.21 5.33 5.46 5.58 28 l5 i3 28 4.5o 5. 3 5.16 5.28 5.40 5.54 6. 6 29 i4 1 1 January 3o 5. 2 5.i3 5.i5 5.27 5.28 5.4o 5.41 5.54 5.54 6. 8 6. 8 6.22 6.21 6.35 3 1 July 12 9 February I 2 August 10 7 3 5.24 5.38 5.52 6. 6 6.20 6.35 6.49 4 8 5 5 5.34 5.49 6. 4 6.18 6.33 6.47 7. 2 6 6 3 7 5.44 5.59 6.14 6.29 6.44 6.59 7.14 8 4 November i 9 5.53 6. q 6.24 6.40 6.55 7. II 7.26 10 2 May October 3o II 6. 3 6.18 5.34 6.5o 7. 6 7.21 7.37 12 3o April 28 i3 6.12 5.28 (3.44 7. 7.16 7.32 7.48 i4 28 26 i5 6.20 5.36 6.53 7.10 7.26 7.42 7.58 16 26 24 17 6.29 5.45 7. 2 7.19 7.36 7.52 3. 9 18 24 21 20 6.39 6.4s 3.56 7. 6 7.i3 7-24 7.3i 7.42 7.48 8. 8. 5 8.17 3.22 3.34 21 21 18 23 24 18 i5 February 26 6.57 7.i5 7.34 7.52 8.10 3.28 3.46 27 i5 12 March I 7- 6 7.24 7.42 3. I 8.20 3.38 3.57 3o August 12 9 4 7-12 7.3. 7.5o 3. 9 8.28 3.46 7. 6 2 Sept. 9 6 7 7-17 7.36 7.55 ^.14 8.33 3.53 5.12 5 6 October 3 10 7-237.42 i. I 3.20 8.39 3.59 ,.i8 8 3 April September 3o i3 7-267.45 i. 4 i.24 8.43 9. 3 ^.22 II 3 1 Maroh 27 16 7.29,7.48. i. 7 3.27 8.47 7. 6 9.25 i4 28 24 19 7.3o 7.5oi 3.10 3.29 8.49 9. 8 7-27 17 25 After Before 7.3il7.5oi 3.10 3.3o 8.5o ?. Q 7.28 Before After Equinox. Equinox. 1 1 V-, Equinox. Equinox. Page 76] TABLE V. For reducing the Sun's Declination, as given in the Nautical Almanac foi Noon at Greenwich, to any other Time under any other Meridian. Add aft. N. Sub. aft. N. H.IM H. M. H. M. li. M. H. M. H. M. H. M. Sub. aft. N. Add aft. N. Sub. bef. N. Add bef. N. 10 10. 20 10. 40 11. 11. 20 170 11. 40 12. Add bef N. Sub. bef N. Add in W. Sub. in W. 130 155 160 165 175 ISO Sub. in W. Add inTvT Sub. in E. Add in E. Dejx Deg. Deg. Deg-. Beg. Deg. Deg. Add in E. Days. Sub. in E. Days. Days. Daj's. M.S. M. S. M. S. M. S. M. S. M. S. M. S. Decemb.21 Decemb. 21 0. 0. 0. 0. 0. 0. 0. 21 June 21 June 20 22 O.II O.ll 0.12 0.12 0.12 o.i3 o.i3 22 20 19 23 0.22 0.23 0.24 0.24 0.25 0.26 0.26 23 19 18 24 0.33 0.34 0.35 0.36 0.37 0.38 0.39 24 18 17 25 0.44 0.46 0.47 0.48 o.5o o.5i 0.53 25 '7 16 26 0.55 0.57 0.58 I. I. 2 I. 4 I. 6 26 16 i5 27 I. 6 I. 8 1 .11 i.i3 i.i5 1. 17 1. 19 27 i5 i4 28 1. 17 1 .20 1.23 1.25 1.27 i.3o 1.32 28 i4 i3 29 1.28 i.3i 1.34 1.37 i.4o 1.43 1.46 29 i5 12 3o 1.39 i.5o 1 .42 1.45 1.49 I .52 1.55 1.59 3o June 12 II Decemb. 3i 1.54 1.57 2. I 2. 5 2. 8 2.12 I July 11 10 January i 2. I 2. 5 2. 9 2,l3 2.17 2.21 2.25 2 10 n 2 2.12 2.16 2.20 2.25 2.3o 2.34 2.38 3 9 g 3 2.23 2.27 2.32 2.37 2.42 2.47 2.5l 4 8 7 4 2.34 2.39 2.44 2.49 2.54 2.59 3. 4 5 7 6' 5 2.44 2.5o 2.55 3. 3. 6 3.12 3.17 6 6 5i 6 2.55 3. I 3. 6 3.12 3.18 3.24 3.3o 7 5 4 7 3. 5 3. II 3.17 3.23 3.29 3.36 3.42 8 4 3 8 3.i5 3.21 3.28 3.34 3.4i 3.48 3.54 9 3 2 9 3.25 3.35 3.32 3.38 3.45 3.56 3.52 3.59 4. 6 10 2 Decemb. i 10 3.42 3.49 4. 4 4.11 4.18 II I June Novemb.So 11 3.45 3.52 3.59 4. 7 4.i5 4.22 4.3o 12 3 1 Mav 29 12 3.55 4. 3 4.10 4.18 4.26 4.34 4.42 i3 3o 28 i3 4. 5 4.i3 4.21 4.29 4.38 4.46 4.54 i4 29 27 i4 4.i5 4.23 4.3i 4.40 4.49 4.57 5. 5 i5 28 26 i5 4.24 4.33 4.4i 4.5o 4.59 5. 8 5.17 16 27 25 16 4.34 4.43 4.52 5. I 5.10 5.19 5.28 17 26 24 17 4.43 4.53 5. 2 5. II 5.21 5.3o 5.4o 18 25 23 18 4.52 5. 2 5.12 5.22 5.32 5.41 5.5i 19 24 22 19 5. I 5.12 5.22 5.32 5.42 5.53 5.52 6. 2 20 23 21 20 5.10 5.21 5.3i 5.42 6. 3 6.i3 21 22 20 21 5.20 5.3i 5.41 5.52 6. 3 6.14 6.24 22 21 19 22 5.29 5.4o 5.5i 6. 2 6.i3 6.24 6.34 23 20 18 23 5.37 5.49 6. 6. II 6.23 6.34 6.44 24 19 17 24 5.45 5.5? 6. 9 6.20 6.32 6.43 6.54 25 18 / \t 25 5.54 6. 6 6.17 6.29 6.4i 6.53 7. 4 26 17 26 6. 2 6.i4 6.26 6.38 6.5i 7. 3 7-i4 27 16 i4 27 6.10 6.22 6.34 6.47 7. 7.12 7.24 28 i5 i3 28 6.19 6.3i 6.43 6.56 7- 9 7.22 7-34 29 i4 II January 3o 6.34 6.47 7. 7.i3 7.26 7-4o 7.53 3i July 12 9 February i 6.49 7. 3 7.16 7.3o 7-43 7.57 8. II 2 August 10 7 3 7. 3 7-17 7.3. 7.45 7.59 8.i3 8.28 4 8 5 5 7.16 7.3i 7.45 8. 8.14 8.28 8.43 6 6 3 7 7.29 7-44 7.59 8.i4 8.28 8.43 8.58 8 4 Novemb. i 9 7-4i ".56 8.12 8.27 8.42 8.58 9.13 10 2 May October 3o 11 7.53 8. 8 8.24 8.40 8.56 9. 12 9.28 12 3o April 28 i3 8. 4 8.20 8.36 8.53 9. 9 9.23 9.42 i4 28 26 i5 8.i5 8.32 8.48 9. 5 9.21 9-38 9-54 16 26 24 17 8.26 8.43 9. 9.17 9-34 9.5o 10. 7 18 24 21 ■ 20 8.40 8.57 9.14 9.32 9.49 10. 6 10.24 21 21 18 23 8.52 9.10 9.28 9.46 10. 3 10.21 10.39 24 18 i5 February 26 9- 4 9.22 9.40 9.58 10.16 10.34 10.53 27 i5 12 March i 9.15 9.33 9.5i 10.10 10.29 10.47 II. 6 3o August 12 9 4 9.24 9.43 10. I 10.20 10.39 10.58 1 1 .16 2 Sept. 9 6 7 9.30 9.50 ro. 9 10.28 10.47 II. 6 11.24 5 6 October 3 10 9.37 9.56 10.16 10.35 10.54 11. i3 II .32 8 3 April Septem. 3o i3 9.41 10. 10.21 10.40 10.59 11.18 11.38 II 3 1 March 27 16 9-45 10. 4 10.24 10.44 II . 3 11.22 II .42 i4 28 24 19 9-47 10. 6 [O.26 10.46 II. 5 11.24 11.44 17 25 After Before 9-48 10. 7 10.27 10.47 II. 6 11.25 11.45 Before After Equinox. Equinox. Equinox. Equinox. TABLE VI. [P=^ge77 Sun's Right Ascension. A JAN. FEB. .MAU. APR. .AIAY. JUNE. JULY AUG. SEPT. OCT. NOV. DEC. Q 1 A. in. h. m. h. in. h. m. h. TO. k. m. h. m. h. TO. h. TO. h. TO. II. TO. h. m. i8.46 20. 58 22.48 0.42 2.33 4.36 6.40 8.45 10.41 12.29 14.25 16.29 1 2 i8.5o 21 . 2 22.52 0.46 2.37 4.40 6.44 8.49 10.45 12.33 14.29 16.33 2 3 1 8. 55 21.6 22.56 0.49 2.41 4.44 6.48 8.53 10.48 12.36 14.33 16. 38 3 4 10.59 21 . 10 23.00 0.53 2.45 4.48 6.53 8.57 10.52 12.40 14.37 16.42 4 5 6 19. 4 21 .14 23. 3 0.57 2.48 4.52 6.57 9. 10.56 12.44 14. 4i 16.47 5 6 19. 8 21.19 23. 7 1 . 2.52 4.56 7- I 9. 4 10.59 12.47 14.45 16. 5i 7 19. 12 21 .23 23. 11 I. 4 2.56 5. I 7. 3 9. 8 II. 3 12. 5i 14.49 16.55 7 8 19.17 21.27 23.14 I- 7 3. 5. 5 7- 9 9.12 11. 6 12.55 14.53 17. 8 iJ 19.21 21 .3o 23.18 I . II 3. 4 5. 9 7.i3 9.16 II .10 12.58 14.57 17- 4 9 10 11 19.25 21 .34 23.22 i.i5 3. 8 5.i3 7.17 9.20 11.14 i3. 2 i5. 1 17. 8 10 11 19.30 21.38 23.25 1.18 3.12 5.17 7.21 9.23 11.17 i3. 6 i5. 5 17.13 12 19.34 21.42 23.29 1 .22 3.16 5.21 7.25 9.27 11.21 i3. 9 i5. 9 17.17 12 13 19.38 21.46 23.33 1 .26 3.20 6.25 7.29 9.3i II .24 i3.i3 i5.i3 17.22 13 14 19.43 21 .5o 23.36 1 .3o 3.24 5.29 7.33 9.35 11.28 i3. 17 15.17 17.26 14 15 16 19.47 21.54 23.40 1.33 3.27 5.34 7.37 9.38 11.32 1 3 . 2 1 l5.22 17. 3i 17.35 15 16 19.51 21.58 23.44 1.37 3.3i 5.38 7.41 9.42 11.35 i3.24 15.26 17 19.56 22. 2 23.47 i.4i 3.35 5.42 7-46 9-46 11 .39 13.28 i5.3o 17.39 17 18 20. 22. 6 23. 5i 1.44 3.39 5.46 7.5o 9.50 11.42 i3.32 i5. 34 17-44 18 19 20. 4 22.10 23.55 1.48 3.43 5.5o 7.54 9.53 11.46 i3.35 i5.38 17-48 19 20 21 20. 8 22. i3 23.58 1.52 3.47 5.54 7.58 9.57 II. 5o 13.39 i5.42 17-53 20 20. i3 22.17 0. 2 1.55 3.5i 5.59 8. 2 10. I 11.53 13.43 15.47 17.57 21 22 20.17 22.21 0. 6 1.59 3.55 6. 3 8. 6 10. 4 11.57 13.47 i5.5i 18. 2 22 23 20.21 22.25 0. 9 2. 3 3.59 6. 7 8.10 10. 8 12. i3.5i i5.55 18. 6 23 24 20.25 22.29 o.i3 2. 7 4. 3 6.11 8.14 10.12 12. 4 i3.54 1 5. 59 18.10 24 25 26 20.29 22.32 0.17 2. 10 4. 7 6.i5 8.18 10.16 12. 7 i3.58 16. 3 18. i5 25 ~26" 20.34 22.36 0.20 2.14 4. II 6. 19 8.21 10.19 12. II i4. 2 16. 8 18.19 27 20.38 22.40 0,24 2.18 4.i5 6.24 8.25 10.23 12. i5 i4. 6 16.12 18.24 27 28 20.42 22.44 ■0.27 2.22 4.20 6.28 8.29 10.27 12.18 14.10 16.16 18.28 28 29 20.46 22.46 0.3! 2.26 4.24 6.32 8.33 io.3o 12.22 14. i4 16.21 18.33 29 30 31 20. 5o 0.35 2.29 4.28 6.36 8.37 10.34 12.26 14. 18 16.25 18.37 30 31 20.54 0.38 4.32 8.41 10.37 l4.2I 18.4 H This Table gives nearly the Sun's Right Ascension Co r the ^-ears 18.33, 1834, 1835, and 1836, and is sufficiently exact for finding when any Star comes to th e meridian. But in all calculations for deter- niinlna;- the longitude by celestial observations, the Sun's Right Ascension must be taken from the Nauti- cal Almanac, where it is calculated to a greater degree o; accuracy. Table VI. A. Correction for the daily variation of the Equati on of Time found in Table IV. A. Find the daily variation of Equation of Time at the top, the hour at Greenwich at the side. 3 n // // 1 / // II ;/ // ; II 1 1 II // // // '/ // II II 1 II II II II // // II // // II ti 1 3 A I 5 G 7 8 £ 101 112 13 14 15 16 17 18 192 121 22 23 24 25 26 27 28 29 30 Q 1 o c ) c 1 I I I 1 1 I I I I I I 1 I 1 1 I I I 15 2 c ) 1 I I 1 1 I 1 1 I I 1 I 2 2 2 2 2 2 2 2 2 2 2 2 3 30 3 1 1 I 1 1 I I 1 2 2 2 2 9 2 2 2 3 3 3 3 3 3 3 3 4 4 4 45 4 o I 1 1 1 I I 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 60 5 1 1 1 I I 2 2 2 9 3 3 3 3 3 4 4 4 < i 4 5 5 5 5 5 6 6 6 6 75 6 I 1 2 2 2 2 3 3 3 ; 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 90 7 o I 1 2 2 2 3 3 3 4 4 4 4 5 5 5 6 3 6 6 7 7 8 8 8 8 9 105 8 o I 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 120 CI 2 2 3 3 3 4 4 5 5 5 6 6 6 7 7 3 8 8 g Q Q 10 10 1 1 1 1 II 135 10 o I : 2 3 3 3 A 4 5 5 5 6 6 7 7 8 8 3 Q 9 10 10 10 10 1 1 1 1 12 12 i3 150 11 o 2 3 3 4 A 5 5 6 6 6 7 7 8 8 Q ? 10 11 II 1 1 12 12 i3 i3 i4 1G5 12 2 : 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 I J 1 1 II 12 12 i3 i3 i4 i4 i5 i5 180 13 2 : 3 3 4 4 E 5 6 7 7 8 8 9 9 10 10 I I 1 1 12 12 i3 i4 i4 i5 i5 16 16 195 14 2 2 3 4 4 5 5 6 6 -7 8 8 Q 9 10 II II I 2 12 i3 r3 i4 i5 i5 16 16 17 18 210 15 2 3 4 4 5 e 6 7 8 8 Q 9 10 II II 12 I 3i3 i4 i4 i5 16 16 17 18 18 19 225 u; 2 ^ 3 4 5 5 e 7 7 8 9 9 10 II 1 1 12 i3 I 3i4 i5 i5 16 17 17 18 19 ■9 20 240 17 2 I 4 4 5 6 C 7 8 p 9 10 11 1 1 12 i3 i3 I ^i5 16 16 17 18 18 19 20 21 21 255 18 2 1 i 4 5 5 6 7 8 8 9 10 11 II 12 i3 i4 i4i 5 16 17 17 18 19 20 20 21 22 23 270 19 2 2 .: 4 5 6 6 7 8 9 10 10 II 12 i3 i3 i4 i5 I 3 17 17 18 >9 20 21 21 22 23 24 285 20 2 3 I 4 5 6 7 8 8 9 10 II 12 i3 i3 i4 i5 161 718 18 •9 20 21 22 23 23 24 25 300 21 2 3 ^ 4 5 6 7 8 9 I 11 11 12 i3 i4 i5 16 171 3ia 19 20 21 ^■^ 23 24 25 25 26 315 22 2 3 I 5 6 6 7 8 91 II 12 i3 14 i5 16 17 17 1 3lQ 20 21 22 23 24 25 26 27 28 330 23 2 3 A 5 6 7 8 9 10 I I 12 12 i3 i4 i5 16 17 18 I 5!2o 21 22 23 24 25 26 27 28 29 345 24 2 3 I 5 6 7 8 9 io|i I 12 i3 i4 i5 16 17 18 19 2 ^21 22 23 24 25 26 27 28 29 3o 360 1 Page 78J TABLE VII. Amplitudes. O < 1— 1 O Q Lat. « « CO NJin !£) r~-co On O " CS no NsTiD O t--00 OnO 11 CM no N^ ID o r~oo On CM CM CN m CM CO ^ a CO o 00 CO Q loo O O « N5-^o CN CN ro ro CO ro Onoo t~- -' O -I r^Nq- ►, 00 o in « CM CM no NJ- N^NJu-, X) OO " ID « CM no ID NTOOnooOinnoi- O n -no -^ o |nt CN ro oo ro no ro CO CM CN CT C-) « Ol no no no no no Nq- m ni « c^ m n ^^•^N^^N^ Ng-ID ID ID ID ID to to to to l> t^ \^'?.i o ^ O " CN xq-^O CO ►-. -^00 no ooo M « w oi o no CO ID CM On t^iD no Nq-LD ID >-■ ID NTin ID Cn On M no NTID n 1 ni to 1- to CM On CO N-> CN OO LO CO O ^vT r^ - O - M « « ni o no 1 r^CO On^D NT CN CO Nq-N^ID l-l O On O - CO cN CO CO in " in CO CM to "CO CM no in CM no liD no ni ID 1 no C-iCSCSCSWOilCSDnMtNCI i Nq-iD CO to ID no no no M « no NTLD no NT to CO 1 in ONin ►. ►1 CM CO in - no in O On O Om> O On On On On On O OnCnOnO-OnD o' o' d d t^r^t^r^r^r^lr^t^r^r-r~-r^ cooooocococo ODCO OnOnOnOn OnO CO I— I O f-i -1 n ^m r^ O ni ID CO m O c in o ID NM CM CO CO NsT NT ID f^no O CO. ID NX ID w w CM no nicMnMCTNTiincO'- Ni O O O O O O O O O O O O O O O ^ O 'O to r^ i> t^ 1^ i> t^ r~-00 OO CO CO CO CO On o I— 1 o M M (N Nyj-uo r^ On « Ng. (^ „ NT CO CO r~- CM r^ 1 no onvo no o r- cMCMnonoNTNTiniD mcmcm •O"^noconono IXTto On no NTin 11 CM no NTin lT) ID tn uo uo m urj ir> iT^ i^ ^D ^^ IDLDlDiniDlD IDUOO^OOO tototo r~-t^r-ir^i^i> 1—1 S n O 1-1 1-1 cs CO in (O 00 "1 no to On no (O O ID On-^ CM o CO no no NT CNID M r^Nq. „ NTlD n ni ONr^lDNTCOCO |ntnt^ ninoNTiD wniCONT xr Njj- Nq- v^ N^ vq- ■^•^^N^a-^N^ NT ^ NT Nq- -^ Nq. N^NTiD ID ID ID IDIDIDID(0^ ItOtOtO o CO S 00"«ro'<3-!r)030cNinoO mntcom^q^ Onions Onid omcMnonoNg-^iDiD n CM C r^to NTCO |nO no NT mnonoNTiD -ic^co mocnnorono noronononono nononoconono nononoNjjNq-vq. NT Nq- NT NT NT ID in in in o r-i O O " Ci no NT ID r^ On« N^yD On CN O Onco CO « CM CM o no CO CM i^ CM t^no ON|tO ni o t^iD no 1 M n w NTNTiDiD " niConoNjiD « o « d cs n) cs cs oi M « n) ni CM CM n) CM CM ni CM Oi ni CM CMCono conocononono NTNq-Nq- o I— I 1—1 S Q O O - c\ fv) Nq- >D r^oo O CO in 00 O NT r- « Ng- M m CM m CO no ONno 00 no 00 CO no NT NT ID ID ONin « cnO NT « CM CM CO Nq- ni O On ID CM CM m CM CM « « no no o o O O I-" I-' « no in O CO On -. "^ (O On -1 NT CO - 1 -1 ni CM CM no uo ONno CO CM I-- CO no ^T' NT ID ID no 00 NT « r^N:^ .- CM CM no 1- On r- NqNTiD O O O O O O o* o* o o d o' O O O O O o d o o d d - — 1- ». 1_ 11 ►-■..- o O O « " CN no •^LD C^-CO O O) -T [^ Cn m in CO M « « ni CM CM - in Onco t^ cm CO no CO NT «^ ID to - r^ ni 00 NT ID -1 n M I 00 ID CO CO Nq- On 0> CN Ov O O On On On On On On On On On On On On On On On On On On OnC o c o o O O O 00 S O O « w CN no N^riD On ~ no in r^ On CM ID 00 - NTCO « to m no CO no NT XT O NT On NT Onid ID ID ID — - r~no CM o no Q CO 00 00 00 00 00 00 00 00 00 00 CO CO CO CO 00 OO CO 00 CO 00 CO CO CO CO CO OO On On On On On On o t^ S O O " " cs o no Nq-iD r-oo On ■-1 CO ID r^ On CM 1 •-•«•- 1-1 CM NT t— O no to O CM CM no no CO NT NTtO CM to " in NT NT ID in OtO 1 n - CM Q r-- r^ t^ r^ t^ 1^ t^ t^ t^ r-- t^ r- r^ r-~ r~- t-- t^ t-- r^ f^ r^ r^ 1^ r^ r^ r^ r-- r^co oo 00 00 00 o CO S O O '-' >- « no NsjiD o r-co O " no ID C^ On - C1-) O CO 1 Nq- CM CM CM CM CO no r-- - xrco CM to CO NT Nq- NTin in ID O Q ^O vo ^ O O O o o o o o o o to o o o o to ^o to to o to to to to ^o o o r- r^ r^ o lO S O O " >- CN ct no N^iD o r- 00 On — CM NT ID t^ On - NTtO 00 «" CM CM CM CM - NT r-~ c no r^ no no no NT Nq- NT O NTOO ID ID ID Q ID LD to CO m in in in ID in in ID ID ID ID ID ID ID ID ID ID ID ID ID ID ID in ID ID ID ID ID in o '^ ^ O O O " " " CM cs fo NsTNq-iD O oco O 11 m NfiD O On 1 no „ « M « m CM ID r^ONCMNTt^ ocoto CM m n: no no CO ntntn no no no no no no no no no no no CO no no CO CO "O CO no CO no CO CO CO no no no no no CO CO o (M S O O O O - « « « d CM DO V) NT -T ID ID <0 t--co On OnO « CM NT ID <0 I"- On CN ni ni Q m (N « cs « « CT CM CM « n) m CM CM CM ng CM CM ni n< ni CM CM o CM m CM CM CM n CM CM CM o 1—1 S O O O O O O O I 1 1 1 -1 m CM CM CM no no -O NTNq-iD ID O 1 o r- r--co On On C "1 ni L it. O ~ CM no ^^n o r-OO On O " m •ONTiDtO t^cOONQ 1-c ntnoN^ D to t~-CO On O 11 CM no 1 . CM dcn.no |no CO ! TABLE VII Amplitudes. [Page 7;i Lat. 1^00 o o •- S5 iro^oo^ro \cr> en m cr> CO en ^ en en en en ^-H d ~n ° ;=; o O o >- « « -I O^ r^ro CO ro O 00 in o in "^^ -q-in in !D r^ r-- ° ^ OO O CTv O O O ° S ^CC CO CO O ooo o o « o '-^ Q ooo r^ r^ t-^ I ^O CN I in - ro I 00 Ov o ■ I Ooo cs cN o in ■ jfo ro x: in « ro OOO""" ° ^ ^^•^xrininminino fo ^ "nT ^ ^ 'j t^"^oo m o o r^ t-~- r--oo in in in in o O CO 00 O On O C ^ o om o r- in -< ro o >n o r^ t^co CO OO ° ^ o -= Q cN m n ro rooo-q-Njxrin lu-iinooo I ro o — I r~in o < r^oo OS o o o irororo ro^^q-N^inin io in CM in « Ng- in o O r^ r~co CO O O O " « o o o o o o cs cs cs ro ro ro N^ -^t ^ m in o SOOO OOOO OnOOO O O mom t^ r^ r^ r-- r^ t^ r^ r-oc co co oo co oo oo O O " I- ■- " OnOOnQnOO OO'-'^'-'CN O t^ r~-CO On o ^ i-i X r- -. tS Vr ro xr CO o o m ro Njj „ N^rm m o o t^ r-- CN OnCO Onn^ cs in I- Nj I-. m O O O O o O oot--t-^r-r^ r-~t^ r^co cooo coco onOnQno O O " >- - in in in in in uo m m m m in m ' o O O O O O O r^r^r^r-oo'cocooo CnqsOn CO en en rrt m ^^ '^^^'^^^^^^ "^ ^^ "^-r ^^ "^ u^ mmminmo ' O cN N^ r^ On cN NT* r^oooo ONro r^^in On^ o^m c r^ro o ronrorofy^vq-Ng'-'q-u-jininin i-.-h«im oro"^N^m o o o o t^ r^ rn m rn en rn en en en en en en ^^ \~-> c .2 CO 0) Q fl a 0) 3 0) -G ■u s 2 "1 fcD (:4 O 0) S H 2 ^2 T3 C a O o Lat. o •1 M ro Hrm r-00 o^ ►- « m 10 1:2 ■-6<£'6 oooooo OOOOOO oooooo OOOOOO 000 °?; S COO OOO O^- CN^sT t^OvC 01-^ OOO'-'-— -c>H«OICNCT int^ONOr^N^OCOON-enLO t^ON- pi cs picnenen cnroenNq-N^vq. -^jn^i^ ddyD OOOOOO oooooo to 00000 OOOOOO 000 °^ -a o O I^ Cs n rv-> - « - LO CO On - c< N^O r^ On C cs CN CN CN CN m en xro r^ON-en;o r^o cncn^ CN en Nq-in K ^ o o ^^6^^^--6 d d d d d oooooo oooooo 000 o 2 2 OOO en N!T -^ IT) OOOOOO r-CO On OnO - c - « cs cs en NjiT) r^OO On - CN cs CO «_--,eNCN CNCSCN 3 ■^<~i^ 0000 OOOOOO oddooo 000 O s ooo|oooooo r^co CO On C — cNeocoNq-jLnino r^CO On On - ■- rs cs s OOO 1 O O O O O d d d d O' 'd '6^:'^ O 00 s OOO cooooo r^ r-co On oooooo '^zS.ZZZ en^NTNg-inooir^oo On s OOO oooooo oooooo oooooo oooooo lo 00 o t^ s o - -- OOO « CN 0-) ro Ncj vq. OOOOOO U-) in r ~ CO oooooo CO On ON - c - -^ ^ - CN en en Nq-^]- in H OOO oooooo oooooo oooooo OOOOOO 000 O ^ O - " OOO « « ro ro ro xr OOOOOO N^uTinooo r^r^cocoONCNOC"-cNCN oooooo COCCOC --.-.«-- en en NT h- tsi^o^D oooooo oooooo oooooo oooooo 000 o lO S O " - OOO - c-i CN tN ro m OOOOOO Nq-v-i-Nq-n-iu-ju-, .^0 r^r--r--CO co OnCnCnO C OOOOOO cooooo 0000-- « « « mh OOO d d d d 000000 oooooo oooooo d d d O "-^ g S5S — I-H M (M CS en OOOOOO tn en m N^ Nq. vj- oooooo iniotnoooiO r^t^p^ooco oooocooooooo On On On a OOO OOOOOO oooooo oooooo 10 00000 do o CO a OO- ----t> r^r^r-r-r^t^j t^ i^ i> i> r^ r^ -» 2 ^ P m vn CO LD UT LT) r-^r~-t^t^t~-r-^lr^i^r~r^c^r^ So-o(olr^r-~r^r^i~^f~ r^t^r^-r-r^f- C^ ^ i ^lO oi _ ^ r^ o c t- t^ t^OO CO OO r- t^ r~- t^oo 00 r- r-- r-- r^ t-~co 00 CO CO CO CO 00 00 CO 00 OO 03 OO OO CO OO 00 CO CO t-^ t^ r^ r^ r' r~- r^oo co oo co o o 1 in osro r^ I - o oovo oo r^r-r-r^lr-r-^r^t^r^r^ r^r~i-^r^t~-t^lr^r^oooococo oooooo CO oo co_ ro ro -^ ° ^ CO C ol . _ _ _ ~^Lnm Ln in lo coo oi _o_o_o io_o o ooot-^t^t~- r^t-^i^r^r^r- r^ r- t-- r- r- r^ r^ r^ (^ t^co 00 00 oo co o o oo O 1-^ r- r^ r^ r^ r^ r^ r- r- r- 1^ r^ 1^ r- r^ r-- r- r^oo co co oo 0_0_OlO O OOP o Ovr^ -^o o - o r^ r^ r^ r-- r^ N:r r^ o cN in o ^ « N- CN CN O) t^ t^ i-^ r^ t^ r- ° ^ ~ lo o o ^C O 00 o oo oooo oo t--r-~r--r^ t^ r- t^ r- r- r- r-r~-t~-r-~t~-r~-lr~-r^r^ ° Hi o o o o o o 'o o o o r- r^ r^ t^ t^ r^ r^ r- r^t^r^i~~t^r~- r-~i~^i^ o ;:4 I o o o o o ^ oooooo r~- r^ r-- r-~ r^ r- - ■•Si o o o o o r-^ r- r-~ r~- r^ r^ r^ r^ r^ r~- O ro O CS CS CN o "^^ O O O OOP OOOOOO OOOOOO o [^ t~~ r^ r- t-~- oooooo'oooooo m in in in o O ooooooloooo r~-c-~ Oi OJ cs 'o o o OJ O) O) o o o r^co o^ CN 04 O) o o o oooooo oooooo oooooo in o O o r^ r^ I w; I ^ 3 I ro ro ^^ in O — lo O O I o o o o o o o o oooo 'oooooo oooooo oooooo I o o o o o o oooooo ! O OjOOO' I _• l-^in O I n 'o o o o r~ CO oo o o o o o ooo in o r^oc O^ o oooooo oooo o_oJo oo vjNTininoOlt^r^oo On^O S I o O O O O O O O O I o o o o o o oooooo O I^CO On O - . oo ro ro oo CO OOOOOO iOOO ojoooo on~^^^ininO|r^r--ooco OvOI— >-■ oioo'-^m o o o oooooo 'ooo ooo oooooo ooo '^ '-O o o o o OOOOOO O o -■ - oooooo ifOooN^^iniino r^r>co C^ o o o o o o lo o o o o o ooo I ^ lin m un So o o CO On On On O C' 12 = - « 0{ lOO ro ■ OOOO O O lO o o o o o ~.\ o o o oooooo oooo oo I = i o o o OOOOOCIOCOOCC OOOOOOOOOOOO oooooo 'o o oooo oooooo iQooooo ooo "-sr^^^^a-mininiinmooooio t^r^ OOOOOCOOOOOOOOO oooooo ^oooooo Iqoo lOCOOOO OOOOOOIOOO OOOOOO OOOOCOOOO loooooo OOOOOO'OOP Lat. 00 1^ 00 Pago I TABLE X. For finding the Distance of Terrestrial Objects at Sea, in Statute Miles. Heighl Distance. Height Distance. Height Distance. Heiglit Distance. Heiglit Distance. Height Distance. Height Distance. infeeu MU. Dec. in feet Mil. Dec. in feel. MU. Dec. in feeu MU. Dec. in feet. MU. Dec. in feet. MU. Dec. in feet. MM. Dec. I 1.32 26 6.75 55 9.81 210 19.17 460 28.37 920 4o.i3 3l00 73.7 2 1.87 27 6.87 60 10.25 220 19.62 470 28.68 940 40.56 3200 74.8 3 2.29 28 7.00 65 10.67 23o 20.06 480 28.98 960 40.99 33oo 76.0 4 2.65 29 7.12 70 11.07 240 20. 5o 490 29.29 980 41.42 3400 77-1 b 2.96 3o 7.25 7b 11.46 25o 20.92 5oo 29.58 1000 41.80 35oo 78.3 6 3.24 3i 7-37 80 11.83 260 21.33 520 3o.i7 IIOO 43.90 36oo 79-4 7 3.5o 32 7.48 85 12.20 270 21.74 54o 30.74 1200 45.80 3700 80.5 8 3.74 33 7.60 90 12.55 280 22.14 56o 3i.3i i3oo 47.70 38oo 81.6 9 3.97 34 7.71 95 12.89 290 22.53 58o 31.86 1 400 49.50 3900 82.6 lO 4.x8 35 7.83 100 i3.23 3oo 22.91 600 32. 4i i5oo 5l.20 4000 83.7 II 4.39 36 7-94 io5 i3.56 3io 23.29 620 32.94 1600 52.90 4ioo 84.7 12 4.58 37 8.o5 no i3.88 320 23.67 64o 33.47 1700 54.50 4200 85.7 i3 4.77 38 8.16 ii5 14.19 33o 24.03 660 33.99 1800 56.10 43oo 86.8 i4 4.95 39 8.26 120 14.49 340 24.39 680 34.50 1900 57.70 44oo 87.8 i5 5.12 40 8.37 125 14.79 35o 24.75 700 35.00 2000 59.20 45oo 88.7 i6 5.29 41 8.47 i3o i5.o8 36o 25.10 720 35. 5o 2100 60.60 4600 89.7 17 5.45 42 8.57 i35 15.37 370 25.45 740 35.99 2200 62.10 4700 90.7 i8 5.61 43 8.68 i4o i5.65 J80 25.79 760 36.47 23oo 63.40 4800 91.7 19 5.77 44 8.78 i45 15.93 390 26.13 780 36.95 2400 64.80 4900 92.6 20 5.92 45 8.87 i5o 16.20 4oo 26.46 800 37.42 25oo 66.10 5ooo 93.5 21 6.06 46 8.97 160 16.73 4io 26.79 820 37.88 2600 67.50 IraUe 96.1 22 6.21 47 9.07 170 17.25 420 27.11 84o 38.34 2700 68.70 23 6.34 48 9.17 180 17.75 43o 27.43 860 38. 80 2800 70.00 24 6.48 49 9.26 190 18.24 440 27.75 880 39.25 2900 71 .20 25 6.61 5o 9.35 200 18.71 45o 1 28.06 900 39.69 3ooo 72.50 TABLE X. A. Parallax in Altitude of a Planet. Horizontal Parallax of a Planet. II II 1 II // // II II II // ;/ II // II II // // II /; // II // // II // n // // II // 1 2 3 4 5 6 7 8 9 10 1] 12 13 14 15 IC 17 18 19 20 21 22 23 24 25 26 27 28 30 35 2 3 4 5 6 7 8 9 ^ II 12 i3 i4 i5 16 17 18 iq 20 21 22 23 M 25 26 27 28 3o 35 2 3 4 5 6 7 8 9 10 II 12 i3 1 4 i5 16 17 18 IQ 20 21 22 23 24 25 26 27 28 3o 35 2 3 4 5 6 7 8 8 9 10 II 12 i3 i4 i5 16 17 18 19 20 21 22 23 23 24 a5 26 28 33 2 3 3 4 5 6 7 8 9 10 10 II 12 i3 i4 i5 16 16 17 18 19 20 21 22 23 2.3 24 26 3o 2 2 3 4 5 6 7 7 8 9 10 II II 12 i3 i4 i5 16 16 17 18 19 20 20 21 22 23 25 29 2 2 3 4 5 5 6 7 8 8 9 10 II u 12 i3 i4 i5 i5 16 17 18 18 19 20 21 21 23 27 2 3 4 4 5 6 7 7 8 10 10 II 12 12 i3 14 i5 i5 16 17 18 18 19 20 20 22 26 2 3 3 4 5 6 6 7 8 8 9 lO 10 II 12 i3 i3 14 i5 i5 16 17 17 18 19 19 21 24 2 3 3 4 5 5 6 7 7 8 9 10 10 II 12 12 i3 i4 i4 i5 16 16 17 18 18 20 23 2 2 3 4 4 5 6 6 7 7 8 9 9 10 10 II 12 12 i3 i4 i4 i5 i5 16 17 17 18 22 2 2 3 3 4 5 5 6 6 7 7 8 9 9 10 10 II II 12 i3 i3 i4 i4 i5 i5 16 17 20 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 10 10 II 11 12 12 i3 i3 i4 i4 i5 16 19 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 II II 12 12 i3 i3 i4 i5 J7 2 2 3 3 4 4 4 5 5 6 6 7 7 7 8 8 Q 9 10 10 II 11 II 12 12 i3 i5 2 2 2 3 3 4 4 4 5 5 5 6 6 7 7 7 8 8 9 9 9 10 10 II II 12 i4 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 8 8 8 9 Q 9 10 10 12 2 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 7 7 7 ^ 8 8 g 9 II 2 2 2 2 3 3 3 4 4 4 4 5 5 5 6 6 6 6 7 7 7 7 8 10 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 8 I 2 2 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 7 I I 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 6 I I 1 I 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 5 I I I I I I I I 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 4 I I I I I I I I I I I 1 I I 2 2 2 2 2 2 2 2 2 I I I I I I 1 I I I I I I I I I .90 90 TABLE XI. fi'-s«67 Seek the nearest number to the reduced time in the top column, and the difference of parallax, proportional logarithm , or semi-diameter for 12 hours in the side column ; under 1 the former, and opposite the latter, is the correction to be applied to the number, marked first in the Nautical Almanac, additive if increasing, subtraclive if decreasing. 5 S Reduced Time. h h h h h h h h h h h h h h h h h h h h h h h h £i 1 ii 2 _^i 3 _31 4 ii 5 §_ G i^ 7 3_ 8 3l 9 _^1 10 m 11 i^l 12 S25 h 1^ h h h "h h ~h h h h h h h h h h h h h h h" h h >• 12i 13 13i 14 14i 15 15i IG lOi 17 l!i 18 18;^ 19 19^ 20 20i 21 21d 22 22h 23 2^ 24 I I I I I I I I ] I 1 1 1 2 I I I 1 I I I I I I I I 2 2 2 2 2 2 3 I I I I 1 I I 2 2 2 2 2 2 2 2 3 3 3 3 4 I I I 1 I 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 5 I I I 2 2 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 6 I 1 2 2 2 2 ~T ~3 3 3 4" "4 4 4 5 5 5 5 "6 6 7 I 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6 6 6 7 7 8 I 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 9 I 2 2 3 3 3 4 4 4 5 5 6 6 6 7 7 7 8 8 9 9 10 11 2 2 2 3 3 4 4 4 5 5 5 5 5 5 6 6 6 7 _7 7 7 8 7 8 8 9 8 9 _9. 10 _9 10 10 II 10 11 2 2 ^ 12 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 II 12 i3 2 2 3 3 4 4 5 5 6 6 7 8 8 9 9 10 10 11 II 12 12 i3 1 4 2 2 3 3 4 5 5 6 6 7 8 8 9 9 10 10 II 12 12 i3 i3 i4 i5 2 2 3 4 4 5 6 6 7 7 8 _9 _9_ 10 II II 12 12 i3 i4 i4 i5 i6 2 T 3 4 5 5 6 7 7 8 9 9 10 1 1 II 12 ~iT 73 T4" i5 i5 16 17 2 3 4 4 5 6 6 7 8 8 9 10 II 1 1 12 i3 i3 i4 i5 16 16 17 18 2 3 4 4 5 6 7 7 8 9 10 ID II 12 i3 i3 i4 i5 16 16 17 18 '9 2 2 3 4 5 6 6 7 8 9 9 10 11 12 '^ i3 i4 i5 16 17 17 18 19 20 21 2 2 2 3 T 4 4 5 "5 6 _7 7 8 9 10 1 1 12 12 i3 i4 i5 16 17 12 17 17 "18 18 '9 19- 20 20 21 6 7 8 9 10 10 II 12 Ti T4 i5 76 22 2 3 4 5 5 6 7 8 9 10 II 12 l3 i4 i5 16 16 17 18 19 20 21 22 23 2 3 4 5 6 7 8 9 10 II II 12 l3 i4 i5 16 17 18 •9 20 21 22 23 24 2 3 4 5 6 7 8 9 10 II 12 i3 i4 i5 16 17 18 19 20 21 22 23 24 25 2 3 4 5 6 7 8 9 10 II 12 i4 i5 16 n 18 19 20 21 22 23 24 25 26 T 3 4 5 ~6 8 9 10 1 1 12 73 i4 i5 ~T6 17 18" '9 21 22 u 24 25 26 27 2 3 4 6 7 8 9 10 1 1 12 i3 i5 16 17 18 19 20 21 22 24 25 26 27 28 2 3 5 6 7 8 9 10 12 i3 i4 i5 16 17 '9 20 21 22 23 24 26 27 28 29 2 4 5 6 7 8 10 II 12 i3 i4 16 17 18 19 21 22 23 24 25 27 28 29 3o 2 4 5 6 7 _9_ 10 11 12 i4 i5 16 17 J-9. 20 21 22 24 25 26 27 29 3o 3i T 4 T "6" 8 9 10 12 73 14 i5 17 iS '9 21 22 ^ 25 26 27 28 3o 3i 32 3 4 5 7 8 9 r I 12 i3 i5 16 17 '9 20 21 23 24 25 27 28 29 3i 32 33 3 4 5 7 8 10 1 1 12 i4 i5 16 18 '9 21 22 23 25 26 27 29 3o 32 33 34 3 4 6 7 8 10 u i3 14 16 17 18 20 21 23 24 25 27 28 3o 3i 33 34 35 3 4 6 7 _9 10 12 i3 i5 16 17 _'9. 20 22 23 25 26 28 29 3i 32 34 35 36~ 3 4 6 7 9 ID 12 i3 15 16 18 19 21 22 ^ 25 27 28 3o 3i 33 34 36 37 2 3 5 6 8 9 II 12 i4 i5 •7 18 20 22 23 25 26 28 29 3i 32 34 35 37 38 2 3 5 6 8 9 II i3 i4 16 '7 '9 21 22 24 25 27 28 3o 32 33 35 36 38 39 2 3 5 6 8 10 II i3 i5 16 18 '9 21 23 24 26 28 29 3i 32 34 36 37 39 4o 2 3 5 7 8 10 12 i3 i5 17 18 20 22 23 25 27 28 3o 32 33 35 37 38 40 4i 2 T 5 7 9 10 12 ~4 lb 17 '9 20 22 I4 26 27 ^ 3i 32 34 l6 38 1^ 4i 42 2 3 5 7 9 10 12 i4 16 17 '9 21 23 24 26 28 3o 3i 33 35 37 38 40 42 43 2 4 5 7 9 1 1 l3 i4 16 18 20 21 23 25 27 29 3o 32 34 36 38 39 4\ 43 M 2 4 5 7 9 II i3 i5 i5 18 20 22 od 26 27 29 3i 33 35 37 38 4o 42 44 45 2 4 6 7 9 11 i3 i5 17 '9 21 22 24 26 28 3o 32 34 36 37 39 4i 43 45 TABLE XVII [Page 83 When a Star, or either (»f the Planets Jupiter or Saturn, is observed. Parallax 0" ►A p. Alt. DM "■). Cor. Log'. *Ap. Alt. Cor. Log-. A.r.-|Cor. Log. 'Ai>. Alt. Cor. Log. -Vp. Alt. Cor. Log. 1.7517 D ftl M S U M M S D M M S D M ftl S 5o. 8 0.9581 ;o. 54.45 1.2277 i3.i5 56. 2 1.3433 19. 57.16 1.4925 36.3o 58.43 lO 5o.24 0.9700 3 54.47 1.2297 20 56. 3 1.3459 1.3485 10 57.17 1 .4960 37. 58.45 1.7568 20 50.39 0.9S17 6 54.48 1.23.7 25 56. 5 20 57.19 1.4996 3o 58-46 1.7618 3o 5o.54 0.9930 9 54. 5o 1.2337 3o 56. 6 1.3511 3o 57.20 i.5o3i 38. 58-47 r.7668 4o 5 1. 8 1. 004 1 12 54.5 1 1.2357 35 56. 8 1.3537 4o 57.22 i.5o66 3o 58-49 1.7717 5o 6. 5l.2I 5 1. 34 i.oi5o i5 54.53 1.2377 4o i3.45 56. 9 1.3562 5o 57.23 i.Sioi 39. 58. 5o 1.7766 1.02 55 10.18 54.54 1.2397 56. 10 1.3587 20. 57.25 i.5i36 39.30 58. 5i I -78 10 10 51.46 i.o36o 21 54.56 1.2417 5o 56.12 i.36i2 10 57.26 1.5170 4o. 58.52 1.7854 20 51.57 1.0462 24 54.57 1.2437 55 56. 1 3 1.3637 20 57.27 1.5204 3o 58.53 1 .7900 3o 52. 9 i.o562 27 54.58 1.2457 14. 56.15 1.3662 3o 57.29 1.5238 4i. 58.55 1 .7946 4o 52.19 1 .0660 3o 55. 1.2476 5 56.16 1.3687 4o 57.30 1.5271 3o 58.56 1-7987 5o D2.29 1.0755 33 33. I 1.2496 10 56.17 1. 3711 1.3735 5o 21. 57.31 57.33 i.53o4 42. 58.57 1.8028 7- 52. 3o 1.0849 10.36 55. 3 i.25i5 i4.t5 56.19 1.5338 42-3o 58-58 1.8070 10 52.49 1 .094 1 39 55. 4 1.2534 20 56. 20 1.3759 10 57.34 1.5370 Ai. 58-59 1.8112 20 52.58 I.I032 42 55. 5 1.2553 25 56.21 1.3783 20 57.35 1.5401 3o 59. i.8i52 3o 53. 6 1.1120 45 55. 7 1.2572 3o 56.22 1.3807 3o 57.36 1.5432 4A. o|59. 1 1.8192 35 53.11 1.1164 48 55-. 8 1.2591 35 56 24 1.383 1 4o 57.37 1.5463 3o 59. 2 1.8230 4o 7.45 53. i5 53.19 I.I 207 5i 55.9 1.2610 4o 56.25 1.3855 5o 57.39 1.5494 45. 59. 3 1.8268 I.I25o 10.54 55.11 1.2629 14.45 56.26 1.3878 22. 57.40 1.553 5 46. 59. 5 1.8338 48 53.2 1 1. 1275 57 55.12 1.2648 5o 56.28 1.3901 10 57.41 1.5556 47- 59. 7 1.841 1 31 53.24 i.i3oi 11. 55.13 1.2667 55 56.29 1.3924 20 57-42 1.5586 48. 59. 9 r-8478 54 53.26 1. 1 326 3 55.14 1.26S6 1 5. 56.3o 1.3947 3o 57.43 i.56i6 49- 39.11 1-8547 57 53.28 i.i35i 6 55.16 1.2705 5 56.3i 1.3970 4o 57.44 1.5646 5o. 59.12 1.8611 8. 873 53.3o 53.33 I.I3-6 i.i4oi _9 I i.i;- 55.17 1.2724 10 i5.i5 56.32 1.3993 5o 57-45 1.5676 5i. 59.14 1-8676 5'5.i8 1.2742 56.33 1.4016 23. 57-46 1.5706 52. 59.16 1.8734 6 53.35 1.1425 ID 55.19 1 .2760 20 56.34 1.4039 10 57.47 1.5736 53. 59.17 1-8794 9 53.37 i.i45o 18 55.21 1.2778 25 56.36 1.4061 20 57.49 1.5765 54. 59.19 1.8846 12 53.39 1. 1474 21 55.22 r.2796 3o 56.37 i.4o83 3o 57.50 1.5794 55. 59.20 1 .8900 i5 f3.42 1. 1499 24 55.23 1.28:4 35 56.38 i.4io5 4o 57.51 1.5822 5b. 59.22 1.8956 i8 8.21 53.44 53.46 i.i523 I.I547 27 55.24 1.2832 r.^57; 4o 56.39 1.4127 5o 57-52 i.585o 57. 59.23 1 .9003 ii.3o 55.25 15.45 56.40 1.4149 24. 57.53 1.5879 58. 59.24 1.9050 24 53.48 1. 1 57 1 33 55.27 1.2S68 5o 56.41 1.4171 10 57-54 1.5907 59- 59.26 1.9102 27 53.5o 1. 1595 ■66 55.28 1.2886 55 56.42 1.4193 20 57-55 1.5935 60. 59-27 1.9142 3o 53.52 1.1619 39 55.29 1.2904 16. 56.43 1.4214 3o 57.56 1.5963 61. 59.28 1.9183 33 53.54 1. 1 642 4? 55. 3o 1.2922 5 56.44 1.4236 40 57.56 1 .5990 62. 59.30 1.9226 36 8.39 53.56 53.58 1. 1 666 45 55. 3i 1.2940 10 56.45 1.4258 5o 57.57 1.6017 63. 59.3 r 1.9270 1. 1 689 11.48 55.32 1.2957 i6.i5 56.46 1.4279 25. 57-58 1 -6044 64. 59.32 1.9302 42 54. 1.1713 5i 55.34 1.2974 20 56.47 1.4300 20 58. 1-6097 65. 59-33 1-9335 45 54- 2 1.1735 54 55.35 1.2991 25 56.48 1.4321 4o 58. 2 1.6149 66.- 59.35 1 .9369 48 54. 4 1.1758 57 55.36 i.3oo8 3o 56.49 1.4342 26.00 58. 4 1.6201 67. 59.36 1 .9404 5i 54. 6 1.1781 1 1. 55.37 i.3o25 35 56. 5o 1.4363 20 58. 5 1.6251 68. 59.37 ..9438 54 8.57 54.08 54^ 1.1804 3 12. 6 55.38 i.3o42 4^^ 56.5 1 1.4384 40 58. 7 i.63oi 69. 59.38 1 .9471 1. 1826 55.39]i.3o59 16.45 56.52 i.44o5 27. 58. 9 I.6350 70. 59.39 1.9501 9. 54.12 1. 1. 849 9 55.40 1 .3076 5o 56.53 1.4425 20 58.10 1 .6400 71- 59.40 1.9528 3 54.13 1.1871 12 55.41 1.3093 55 56.54 1.4445 4o 58.12 1.6449 72. 59.42 1.9553 6 54.15 1. 1 893 ID 55.42 i.3iio 17. 56.55 1.4465 28. 58- 1 3 1 .6498 73. 59.43 ..9578 9 54.17 1. 1916 18 55.43 1.3127 5 56.56 1.4486 • 20 58-i5 1.6545 74. 59.44 1.9603 12 54.19 1. 1938 21 55.44 i.3i44 10 17.. 5 56.57 1.4506 Ao 58.16 1. 659 1 75. 59.45 1.9625 V.i5 54.21 1.1960 12.24 55.45 i.3i6i 56.58 1.4526 29. 58.18 1.6635 76. 59.46 1.9643 18 54.22 1.1982 27 55.46 1.3178 20 56.59 1.4546 3o 58. 20 1 .6702 77- 59.47 1 .9660 2J 54.24 i.atx)3 3o 55.47 1.3194 2 5 57.0 1.4566 3o. 58.22 1 .6769 78. 59.48 1 .9676 24 54.26 1. 2025 33 55.48 1.32II 3o 57. I 1 .4586 3u 58.24 1 .6833 79. 59.49 1 .9692 27 54.28 I.2o4~ 36 55.49 1.3227 35 57. 2 1 .4606 3i. 58.25 1 .6896 80. 59.50 1 .9706 3o 9.33 54.29 1.2068 39 55. 5o 1.3243 40 57. 31.4626 3o 58.27 1.6957 81. 59.51 1.9714 54.3 1 1.2089 12.42 55. 5i 1.3259 17.45 57. 4 1 .4646 32. 58.29 1.7018 82. 59.52 .9722 36 54.33 1. 2 1 10 45 55.52 1.3273 5o 57. 4 1.4665 3o 58.3 1 1-7079 83. 59.53 1.9729 39 'MM r.2i32 48 55.53 1.3291 55 57. 5 1 .4684 33. 58.33 1.7140 84. 59.54 1-9734 42 54-36 I.2I53 5i 55.54 i.33o7 18. 57. G 1.4703 3o 58.34 1.7202 85. 59.55 1-9737 45 54.37 1.2173 54 55.55 1.3323 10 57. 81.4741 34. 58.36 1-7263 86. 59.56 1.9739 48 54.3^ 1. 2194 57 55.56 1.3339 20 57. 9 1.4778 3o 58.37 I -73 1 2 87. 59.57 1.9741 9.51I54.41 I.22[5 i3. 55.57 1.3355 i8.3o 57.11 1.481 5 35. cj< 58.39 1.7362 88. 59.58 1.9742 54 54.42 1.2236 5 55.59 1.3381 4o 57.13 1.4852 3o 58. 4o 1.7414 89. 59.59 1.9742 57 54.44 1.2257 10 56. 1.3407 5o 57.14 1.4888 36. 058.42I1. 74661 90. 60. 1.9742 12 Page 90] TABLE XVII. ! When the Planet Venus or Mars is used, and the i'arallax is nearly equal to 5". Parallax 5". -Ap. All. Cor. Log *Ap. Alt. Cor. Log. *Ap. Alt. D M Cor. M S Log. *Ap. All. Cor. Log. *Ap. Alt. D 31 36.3o 37. 3o 38. 3o 39. "~3^ 40. 3o 4i. 3o 42. Cor. 31 S 58.47 58.49 58.5o 58.5i 58.52 58.54 58.55 58.56 58.57 58.58 59.00 59.01 Log. 1.775c 1 .7803 1.7855 1.7907 1 .7953 1 .8008 DiM M S U 31 M S D M 31 S 5. 10 20 3o 4o 5o 5o.i3 50.29 5o.44 50.59 5i.i3 51.26 0.961/^ 0.9733 0.9851 0.9966 1 .0078 1. 01 88 10. 3 6 9 12 i5 54.5o 54.52 54.53 54.55 54.56 54.58 3342 2363 2383 2{o4 2,)25 2445 i3.i5 20 25 3o 35 4o 13.45 5o 55 i4. 5 ic 56 .07 56.08 56.10 56.11 56.12 56.14 I.352I 1.3547 1.3574 i.36oo 1.3626 1.3652 19. 10 20 3o 4o 5o 57.20 57.22 57.23 57.25 57.26 57.28 1.5049 i.5o86 I.5I23 i.5i59 1.5195 1. 5201 6. 10 20 3o 4o 5o 51.39 5i.5i 52.03 52.14 52.24 52.34 1.0295 I .o4oo i.o5o3 1 .0604 1 .0703 1.0800 10.18 21 24 27 3o 33 54.59 55.00 55.02 55.03 55.05 55.06 2465 2485 2 5o5 2525 2545 2565 56.15 56.17 56.18 56.19 56.21 56.22 1.3678 1 .3704 1.3729 1.3754 1.3779 i.38o4 20. 10 20 3o 4o 5o 57.29 57.31 57.32 57.33 57.35 57.36 1.5266 I.5301 1.5336 1.5371 i.54o5 1.5439 1 .8o56 1. 8 104 i.8i5i 1.8198 1.8244 1.8289 7- o lO 20 3o 35 4o 52.44 52.54 53.03 53.11 53.16 53.20 1.0895 1 .0988 1. 1079 I.I 169 I.I2l3 1. 1257 I0.36 39 42 45 48 5i 55.07 55.09 55.10 55.12 55.13 55.14 2585 2605 2624 2643 2662 2681 14. i5 20 25 3o 35 4o 56.23 56.25 56.26 56.27 56.29 56.3o 1.3829 1.3854 1.3878 1.3902 1.3927 1.3951 21. 10 20 3o 4o 5o 57.37 57.39 57.40 57.41 57.42 57.43 1.5473 1.5507 1.5540 1.5573 i.56o5 1.5637 42.3o 43. 3o 44. 3o 45. 59.02 59.03 59.04 59-05 59.06 59.07 1.8333 1.8376 1.8419 1.84G1 i.85o2 1.8543 7-45 48 5i 54 57 8. 53.24 53.26 53.28 53. 3o 53.32 53.35 i.i3oo 1.1326 i.i352 1. 1378 i.i4o4 1. 1429 10.54 57 II. 3 6 9 55.15 55.17 55.18 55.19 55.21 55.22 2701 2720 2739 2758 2777 2796 14.45 5o 55 i5. 5 10 56.3 1 56.32 56.34 56.35 56.36 56.37 1.3975 1.3999 1.4023 1 .4o46 1 .4070 1 .4093 22. 10 20 3o 4o 5o 57.44 57.46 57.47 57.48 57.49 57.50 1.5669 1. 5701 1.5733 1 .5764 1.5-95 1.5826 1.5857 1.5887 1.5917 1 .5947 1.5977 1 .6007 46. 47. 48. 49. 5o. 5i. 59.09 59.10 59.12 59.14 59.15 59.17 1.8622 1 .8699 1.8773 1 .8844 1. 8914 1. 898 1 8. 3 6 9 12 i5 18 53.38 53.40 53.42 53.44 53.47 53.49 1. 1454 1. 1479 i.i5o4 1. 1529 I.I554 1. 1578 11.12 i5 18 21 24 27 55.23 55.?4 55.26 55.27 55.28 55.29 2815 2833 2852 2871 2889 2907 i5.i5 20 25 3o 35 40 1 5.45 5o 55 16. 5 10 56.38 56.39 56.40 56.41 56.43 56.44 1.4116 1.4139 1.4162 i.4i85 1.4208 1.4231 23. 10 20 3o 40 5o 57.51 57.52 57.53 57.54 57.55 57.56 32. 53. 54. 55. 56. 57. 59.19 59.20 59.22 59.23 59.24 59.26 1 .9045 1. 9107 I. 9167 1.9226 1.9282 1.9336 8.21 24 27 3o 33 36 8.39 42 45 48 5i 54 53. 5i 53.53 53.55 53.57 53.59 54.01 54.03 54.05 54.07 54.09 54.11 54.13 i.i6o3 1. 1627 i.i65i 1. 1675 1. 1 699 1.1722 ii.3o 33 36 39 42 45 55. 3o 55.32 55.33 55.34 55.35 55.36 2925 2944 2962 298c =998 3oi6 3^3 3o5o 3o68 3o86 3io4 3l22 56.45 56.46 56.47 56.48 56.49 56. 5o 1.4253 1.4276 1.4298 1.4320 1.4342 1.4364 24. 10 20 3o 40 5o 57.57 57.58 57.59 58.00 58.01 58.02 i.6o36 i.6o65 1 .6094 1.6122 i.6i5i 1.6179 58. 59. 60. 61. 62. 63. 59.27 59.28 59.30 59.31 59.32 59.33 1.9388 1.9438 1 .9486 1.9532 1.9577 1. 9619 1. 1746 1. 1770 ..1793 ,.1817 1.1840 1. 1 863 11.48 5i 54 57 12. 3 12. 6 9 12 i5 18 21 55.37 55.38 55.39 55.41 55.42 55.43 i6.i5 20 25 3o 4o 16.45 5o 55 17. 5 10 56.5 1 56.52 56.53 56.54 56.55 56.56 56.57 56.58 56.59 57.00 57.01 57.02 1 .4386 1 .4408 1.4430 I.445I 1.4472 1 ..^493 1.4:^4 1.4535 1.4556 1.4577 1.4597 1.4618 25. 20 4o 26. 20 4o 27. 20 40 28. 20 4o 58.03 58.05 58.06 58.08 58. 10 58.11 1 .6207 1.6262 1.6317 1.6371 1.6424 1.6476 64. 65. 66. 67. 68. 69. 59.34 5q.36 59.37 59.38 59.39 59.40 1 .9660 1 .9700 1.9738 1 .9773 1 .9807 1.9839 8.57 6 9 12 9.15 18 21 34 27 3o 54.15 54.17 54.18 54.20 54.22 54.24 54.25 54.27 54.29 54.3 1 54.32 54.34 1 . 1 886 1 . 1 908 1 . 1 93 1 1.1953 1.1975 1. 1998 1 .2021 1 .2043 i.2o65 1.2087 1.2109 1.2l3() 55.44 55.45 55.46 55.47 55.48 55.49 3i39 3i56 3173 3190 3207 3224 58.13 58.15 58.16 58.18 58.19 58.21 1.6527 1.6578 1.6629 1 .6678 1.6727 1.6775 70. 71. 72. 73. 74. 75. 59.41 59.42 59.43 59.44 59.45 59.46 1 .9870 1.9899 1.9927 1 .9953 1.9977 2.0000 12.24 27 3o 33 36 39 12.42 45 48 5i 54 57 55. 5o 55.5i 55.52 55.53 55.54 55.55 55.56 55.57 55.58 55.59 56.00 56.01 3242 3259 3276 3293 33io 3326 17.15 20 25 3o 35 40 57.03 57.04 57.05 57.06 57.07 57.07 1.4639 1 .4660 1.4680 1 .4700 1.4720 1 .4740 29. 3o 3o.oo 3o 3i. 3o 58.22 58.24 58.26 58.28 58.3o 58.32 1.6823 1.6893 1 .6962 1.7029 1.7095 1. 7160 76. 77. 78. 80. 81. 59.47 59.48 59.49 59.50 59.51 59.52 2.0022 2.0042 2.0060 2.0077 2.0092 2.0106 9.33 36 39 42 45 48 54.36 54.37 54.39 54.41 54.42 54.44 1.2l5l 1.2173 1. 2195 1.2216 1.2237 1.2258 3343 3359 3376 3392 3408 3424 17-45 5o 55 18. 10 20 i8.3o 40 5o 57.08 57.09 57.10 57.11 57.13 57.14 57.16 57.17 57.19 1 .4760 1.4780 1 .4800 1.4820 1.4859 1 .4898 32. 3o 33. 3o 34. 3o 58.33 58.35 58.37 58.38 58.4o 58.4i 1.7224 1.7287 1.7349 1 .7409 1.7468 1.7526 82. 83. 84. 85. 86. 87. 88. 89. 90. 59.53 59.54 59.55 59.55 59.56 59.57 59.58 5?.59i So. 00 2.0118 2.0129 2.0139 2.0147 2.oi53 2.0158 2.0162 2.C164 2.0164 9.51 54 57 54.45 54.47 54.49 1.2279 i.23oo 1.2321 i3. 5 TO 56.02 56.o3 56.o5 3440 3468 3495 1.4936 1 .4974 1.5oi2 35. 3o 36. 58.43 58.44 58.46 1.7584 1 .7640 1.7695 TABLE XVI] , [Page 91 When the Planet Venus or Mars is used, and the Parallax is nearly equal to KV. Parallax 10''' *AP. All. DM Cor. Log. *Ap. All. Cor. Log. *Ap. Alt. Cor. Log. *Ap. All. D I\l Cor. Log. -Ap. All. D IM Cor. Log. M i< D JVI M S D IM JM S M S M S 5. 5o.i8 0.9650 10. 54.55 1.2412 i3.i5 56.11 i.36i3 19. 57.25 1.5 1 80 36.3o 58.5 1 1.7996 10 50.33 0.9771 3 54.57 1.2433 20 56.13 1 .3640 10 57.27 1.5218 37. 58.53 1.8052 20 50.48 0.9890 6 54.58 1.2454 25 56.14 1.3667 20 57.28 1.5256 3o 58.54 1.8108 3o 5i.o3 1 .0006 9 55.00I1.2475 3o 56.16 1.3693 3o 57.30 1.5293 38. 58.55 i.8i63 4o 5l.I7 i.oi 19 12 55.01 1.2496 35 56.17 1.3720 40 57.31 i.533o 3t. 58.56 1.821-7 5(. 6. 5i.3i 51.43 I.023o 1.0338 i5 55.02 i.25i6 40 i3.45 56.19 1.3746 5o 57.33 1.5367 39. 58.58 58.59 1.8269 1.8321 10.18 55.04 1.2536 56.20 1.3773 20. 57.34 1.5404 39.30 10 5i.55 1.0444 21 55.05 1.2557 5o 56.22 1.3799 10 57.35 1.5440 4o. 59.00 1.8372 20 52.07 1 .0549 24 55.07 1.2577 55 56.23 1.3825 20 57.37 1.5476 3o 59.0. 1.8422 3o 52.18 i.o65i 27 55.08 1.2597 14. 56.24 i.385i 3o 57.38 1.5512 4i. 59.02 1.8472 4o 52.29 1. 075 1 3o 55.10 1. 2617 5 56.26 1.3877 40 57.39 1.5547 3o 59.03 1. 852 1 5o 52.39 1.0849 33 55.11 1.2638 10 14. i5 56.27 1.3902 5o 57.41 1.5582 42. 59.04 59.05 1.8569 1.8616 7- o 52.49 1 .0945 10.36 55.12 1. 2658 56.28 1.3927 21. 57.42 1.5617 42. 3o 10 52.58 1 . 1 039 39 55.14 1.2678 20 56.3o 1.3952 10 57.43 1.5652 43. 59.06 1.8662 90 53.07 i.ii3i 42 55.15 1 .2698 25 56.3 1 1.3977 20 57.44 1.568^ 3o 59.07 1.8708 3o 53.15 1. 1222 45 55.16 1.2718 3o 56.32 1 .4002 3o 57.45 1.5720 44. 59.08 1.8753 35 53.19 1. 1 267 48 55.18 1.2737 35 56.33 1.4027 4o 57.47 1.5754 3o 59.09 1.8797 4o 7-45 53.24 53.28 i.i3ii I.I355 61 55.19 1.2756 40 14.45 56.35 56.36 i.4o52 . 5o 57.48 1.5787 45. 59.10 1 .8840 10.54 55.20 1.2776 1 .4077 22. 57.49 1.5820 46. 59.12 1.8925 4« 53.3; i.i38i 57 55.22 1.2795 5o 56.37 1.4 lOI 10 57.5o 1.5853 47- 59.14 1 .9007 5i 53.33 1. 1 407 11. 55.23 1.2815 5:) 56.38 1.4125 20 57.51 1.5886 48. 59.15 1 .9087 54153.30 1. 1433 3 55.24 1.2835 i5. 56.39 1.4149 3o 57.52 1.5919 49. 59.17 1. 9 1 64 5753.38 1. 1459 6 55.26 1.2854 5 56.4 1 1.4173 40 57.54 1.5951 5o. 59.19 1.9238 8. 53.40 I.I485 9 II. 12 55.27 55.28 1.2873 1.2892 10 i5.i5 56.42 '■4197 1.4221 5o 23. 57.55 57.56 1.5983 5i. 59.20 1. 9310 8. 3 53.43 i.i5ii 56.43 1.601 5 52. 59.22 1.9380 6 53.45 I.I536 i5 55.29 1.291 1 20 56 44 1.4245 10 57.57 1 .6046 53. 59.23 1.9448 9 53.47 i.i56i 18 55.3o 1.2930 25 56.45 1.4269 20 57.58 1 .6077 54. 59.25 1.9513 12 53.49 1. 1 586 21 55.32 1.2949 3o 56.46 1.4292 3o 57.59 1.6108 55. 59.26 1.9576 i5 53.5i 1.1611 24 55.33 1.2968 35 56.48 i.43i5 40 58. 00 1. 6 1 39 56. 59.27 1.9637 i8 53.53 I.I636 27 55.34 1.2987 1 .3oo5 4o i5.45 56.49 1.4338 5o 58.01 1.6170 57. 59.28 59.30 _i^60 1.9751 8.21 53.56 1. 1 661 1 1 .3o 55.35 56. 5o 1.436] 24. 58.02 1 .6200 58. 24 53.58 1. 1686 33 55.36 1 .3o24 5o 56.5i 1.4384 10 58.03 1.6230 59. 59.3 1 1 .9806 27 54.00 1.1710 36 55.38 i.3o42 55 56.52 1 .4407 20 58. o4 1 .6260 60. 59.32 1.9858 3o 54.f)2 1. 1734 39 55.39 1 .3o6o 16. 56.53 1.4430 3o 58.05 1.6290 61. 59.33 1 .9909 33 54.04 1. 1758 42 55.40 1.3078 5 56.54 1.4453 4o 58. 06 I.6320 62. 59.34 1.9958 36 8.39 54.06 1. 1782 45 55.41 1.3096 10 56.55 1.4476 5o 58.07 1.6349 63. 59.36 2.0005 54.08 1. 1806 11.48 55.42 i.3ii4 i6.i5 56.56 1.4498 25. 58. 07 1 .6378 64. 59.37 2.0049 4i 54.10 i.iS3o 5i 55.43 i.3i33 20 56.57 1.4520 20 58.09 1.6435 65. 59.38 2.0092 45 54. 1 2 I.I854 54 55.44 i.3i5i 25 56.58 1.4542 4o 58.11 1 .6492 66. 59.39 2.oi33 48 54.14 1.1878 5? 55.45 1.3169 3o 56.59 1.4564 26. 58.1 3 1.6548 67. 59.40 2.0172 5i 54.16 1.1901 12. 55.47 i.3]87 35 57.00 1 .4586 20 58.14 1 .6604 68. 59.41 2.0209 54 54.18 1. 1 925 3 55.48 i.32o5 4o 37.01 1 .4608 4o 58.16 1 .6659 69. 59.42 2.0245 8.57 54.20 1. 1 948 12. 6 55.49 1.3223 16.45 57.02 1 .463() 27. 58.17 1 .67 1 2 70. 59.43 2.0279 9. (. 54.22 1-1971 9 35. 5o 1.3241 5o 57.03 1.4652 20 58.19 1 .6765 71- 59.44 2.o3l I 3 54.23 1. 1994 12 55.5 1 1.3258 55 57.04 1.4673 40 58.21 1.6818 72. 59.45 2.o34i (i 54.25 1.2017 i5 55.52 1.3275 17. 57.05 1.4694 28. 58.22 1 .6870 73. 59.46 2.0370 9 54.27 1 .2o4o 18 55.53 1 .3293 5 57.06 i.47ir> 20 58.24 1.6921 74. 59.46 2.0397 12 ^5 54.29 54.3(. 1.2063 21 5:).54 i.33io 10 57.07 • .4737 4o 58.25 1. 697 1 75. 76. 59.47 2.0422 1 .2085 12.24 55.55 1.3327 17.15 57.08 1.4758 29. 58.26 1. 702 1 59.48 2.0446 18 54.32 1.2 I 08 27 55.56 1.3345 20 57.09 1 .4779 3o 58.28 1 .7094 77- 59.49 2.0468 21 54.34 1.2I0I 3o 55.57 1.3362 25 57.10 1 .4800 3o. 58.3o 1.7166 78. 59.50 2.0488 1 24 54.36 1. 2 1 53 33 55.58 1.3379 3o 57.10 1.4821 3o 58.32 1.7237 79. 59.51 2.0 507 27 54.37 1. 2175 36 55.59 1.3396 35 57... 1.4842 3i. 58.34 1 .7307 80. 59.52 2.0524 3o 54.39 1. 2197 39156.00 i.34i3 4o 57.12 1 .4863 1.4883 3o 58.36 1.7375 81. 59.53 2.0539 Q.33 54-41 1. 2219 12.42 56.01 1 .3430 17.45 57.13 32. 58.38 1.7442 82. 59.53 2.0553 36 54.42 1.2241 45(56.02 1.3447 5o 57.14 1 .4904 3o 58.39 1.7508 83. 59.54 2.0565 39 54.44 1.2263 48 56.03 1 .3464 55 57.15 1.4924 33. 58.4 1 ..7573 84. 59.55 2.0575 42 54.46 1.2285 5 1 56.04 1. 34s I 18. 57.16 1.4944 3o 58.43 1.7637 85. 59.56 2.o58/i 45 54.47 i.23o6 54 56.o5 1.3498 10 57.17 1.4984 34. 58.44 1 .7699 86. 59.57 2.0592 48 54.49 1.2328 57 56.06 i.35i4 20 57.19 i.5o24 3o 58.46 i.776(; 87. 88. 59.58 59.58 2.0597 9.5. 54.5o 1.2349 1 3. 56.07 i.353o i8.3o 57.21 i.5o63 35. 58.47 1.7821 2.0601 54 54.52 1.2370 5| 56.08 1.3558 40 57.22 I.5l02 3o 58.49 1. 788 1 89. 59.59 2.o6o3 57 54.54 I.239I 10 56. 10 L_ 1.3586 5o 57.24 i.5i4i 36. 58. 5o 1.7939 90. 60.00 2.o6o3 Page 92] TABLE XVII When the Planet Venus or Mars is used, and the Parallax is nearly e qual to 15". Parallax 15' »Ap. Alt. DIM 5. o Cor. M S 5o.2 2 Log. *Ap. Alt. Cor. Log:. Alt. Cor. hog. *Ap. Alt. Cor. Log. -A p. Alt. Cor. Log. 1.8257 D M M S D M M S D 31 M S D M M te 0.9688 10. o!55,oo 1.2483 i3.i5 56.16 1.3706 19. 57.30 i.53i4 36.3o 58.55 10 5o.38 0.9810 3 55.C2 i.25o4 20 56.18 1.3734 10 57.31 1.5353 37. 58.57 :.83i7 20 50.53 0.9929 6 55.03 1.2525 25 56.iq 1.3762 20 57.33 1.5392 3o 58.58 1.8376 3o 5i.o8 1 .0046 9 5b.o5 1.2546 3o 56.21 1.3789 3o 57.34 1.543: 38. 58.59 1.S434 4o 5l.22 1. 0160 12 55.06 1.2567 35 56.22 i.38i6 40 57.36 1.5470 3o 59. 0( 1.849: 5o 51.35 1.0272 i5 55.07 1.2588 4o 13.45 56.24 1.3843 5o 57.37 i.55o8 39. 59.02 1 .8548 6. o 5 1. 48 I.0382 10.18 55.09 1.2609 56.25 1.3870 20. 57.39 1.5546 39.30 59.03 1.8603 10 5 2. CO 1 .0490 21 55.10 i.263o 5o 56.26 1.3897 10 57.40 1.5583 4o. 59.04 I.S658 20 52.12 1.0595 24 55.12 i.265i 55 56.28 1.3923 20 57.41 1.5620 3o 59.05 1.8712 3o 52.23 1.069S 27 55.13 1.2672 14. 56.29 1.3949 3o 57.43 1.5657 4i. 59.06 1.8765 4o 52.34 1 .0799 3o 55. i5 1.2693 5 56.3o 1.3975 40 57.44 1.5694 3o 59.07 :.88i7 5o 52.44 1.0898 36 55.16 1. 2713 10 56.32 1.4001 5o 57.45 1.5730 42. 59.08 1.8868 7- o 52.54 1.0995 I0.36 55.17 1.2733 i4.i5 56.33 1.4027 21. 57.47 1.5766 42.3o 59.09 1.89:8 ID 53.03 1. 1090 39 55.19 1.2754 20 56.34 i.4o53 10 57.48 i.58o2 43. 59.10 1 .8968 20 53.12 1. 1184 42 55.20 1.2774 25 56.36 1 .4079 20 57.49 1.5837 3o 59.1 : 1.90:7 3o 53.21 1. 1 276 45 55.21 1.2794 3o 56.37 i.4io5 3o 57.50 1.5872 44. 59.12 1 .9065 35 53.25 I.j322 48 55.23 1. 2814 35 56.38 i.4i3o 40 57.51 1.5907 3o 59.13 1.9:12 4o 7-45 53.29 53.34 I.I366 5i 55.24 1.2834 4o 56.4o i.4i55 5() 57.53 1.5942 45. 59.14 [.9159 i.i4io 10.54 55.25 1.2854 14.45 56.41 1.4180 22. 57.54 1.5977 46. 59.15 1.9251 4ti 53.36 1. 1437 57 55.27 1.2874 5o 56.42 1.4205 10 57.55 1 .601 1 47. 59.17 : .9340 5i 53.38 I.I464 II. 55.28 1.2893 55 56.43 1.4230 20 57.56 1 .6045 48. 59.19 1.9426 54 53.41 1. 1490 3 55.29 1.2913 i5. 56.44 1.4255 3o 57.57 1 .6079 49. 59.20 : .9509 57 53.43 i.i5i6 6 55.3o 1.2933 5 56.46 1.4279 40 57.58 1.6112 5o. 59.22 1.9590 8. o 8. 3 53.45 1.1542 9 55.32 1.2952 10 i5.i5 56.47 i.43o4 5o 57.59 1.6145 5i. 59.23 1 .9668 53.48 I.I567 1 1 . 1 2 55.33 1.2971 56.48 1.4329 23. 58.00 1.6178 52. 59.25 1.9744 6 53. 5o 1. 1593 i5 55.34 1.2990 20 56.49 1.4353 10 58.0I 1.62:1 53. 59.26 1.9817 Q 53.52 1. 1619 18 55.35 i.3oio 25 56. 5o 1.4377 20 58.02 1.6244 54. 59.27 1.9888 12 53.54 1. 1644 21 55.37 1.3029 3o 56.5i 1. 440 1 3a 58.03 1.6276 55. 59.29 1.9957 i5 53.57 1 . 1 669 24 55.38 i.3o48 35 56.52 1.4425 4o 58.04 i.63o8 56. 59.30 2.0023 i8 53.59 1. 1695 27 55.39 1.3067 4o 56.53 1.4449 5o 58.05 I.6340 57- 59.31 2.0087 8.21 54.01 1. 1720 ii.3o 55.40 i.3o86 i5.45 56.54 1.4473 24. 58. 06 1. 637 1 58. 59.32 2.0149 24 54.03 1. 1745 33 55.41 i.3io5 5o 56.56 1 .4497 10 58.07 1.6402 59. 59.34 2.0209 1 27 54.05 1.1770 36 55.42 i.3i24 55 56.57 1.4520 20 58.08 1.6433 60. 59.35 2.0267 3o 54.07 1-1795 39 55.44 i.3i43 16. 56.58 1.4543 3o 58.09 1.6464 bi. 59.3b 2.o322 33 54.09 1.1819 42 55.45 1.3:62 5 56.59 1.4566 4o 58.10 1.6495 62. 59.37 2.0375 36 54.11 1.1843 45 55.46 i.3i8o 10 57.00 1.4589 5o 58.11 1.6526 63. 59.38 2.0427 8.3g 54.13 1.1868 11.48 55.47 1. 3198 i6.i5 57.01 1.4612 25. 58.12 1.6556 64. 59.39 2.0476 42 54.15 1.1892 5i 55.48 1 .3217 20 57.02 1.4635 20 58.14 1.6616 65. 59.40 2.o523 45 54.17 1. 1916 54 55.49 1.3235 25 57.03 1.4658' 4o 58.i6 1.6675 66. 'J9-4i 2.0568 48 54.19 1. 1940 57 55.5o 1.3253 3o 57.04 1 .468 1 26. 58.17 1.6734 67. 59.42 2.0612 5i 54.21 1. 1964 12. 55.5i 1 .3271 35 57.o5 1.4704 20 58.19 1.6792 68. 59.-43 2.0654 54 54.23 1. 1988 3 55.53 1.3289 4o 57.06 1.4726 1.4748 40 58. 20 : .6849 69. 59.44 2.0693 8.57 54.25 1.201 2 12, 6 55.54 1.3307 16.45 57.07 27. 58.22 1 .6905 70. 59.44 2.0730 9- o 54.26 I.2035 9 55.55 1.3325 5o 57.08 1.4770 20 58.24 1 .6960 71. 59.4:) 2.0766 3 54.28 1 .2058 12 55.56 1.3343 55 57.09 1 .4792 40 58.25 1.7015 72. 59. 461 2. 0800 6 54.3o 1.2081 i5 55.57 I.336I 17. 57.10 1 .4814 28. 58.26 1.7069 73. 59.47 2.0832 9 54.32 I. 2104 18 55.58 1.3379 5 57.11 1.4836 20 58.28 1. 7123 74. 59.48 2.0862 12 54.34 1.2127 21 55.59 1.3397 10 57.12 1.4858 4o 58.29 1.7176 75. 59.49 2.0890 9.i5 54.35 1.2l5o 12.24 56. 00 i.34i5 17.15 57.12 1.4880 29. 58.3i 1.7228 76. 59.49 2.09:6 i8 54.37 1.2173 27 56. 01 1.3433 20 57.13 1.4902 3o 58.33 i.73o5 77- 59.50 2.0940 21 54.3q 1.2196 3o 56. 02 1.3450 25 57.14 1.4924 3o. 58.35 1.7381 78. 59.51 2-0962 24 54.41 1. 2219 33 56.03 1.3468 3o 57.15 1.4945 3o 58.37 1.7456 79- 59.52 2-0983 27 54.42 1.2242 36 56.04 1.3485 35 57.16 1.4966 3i. 58.38 1.7529 80. 59.53 2-ioo3 3o 54.44 1.2264 39 56.05 i.35o3 4o 57.17 i.49«7 3o 58 .40 1.7601 1 .7671 81. 59.53 2.1021 9.33 54.46 1.2286 12.42 56.06 1 .3520 17.45 57.18 i.5oo8 32. 58.42 82. 59.54 2.io36 36 54.47 i.23o8 45 56.07 1.3537 5o 57.19 1.5029 3o 58.44 1 .7740 83. 59.55 2.1049 39 54.49 I.2330 48 56.08 1.3554 55 57.19 i.5o5o 33. 58.45 1.7809 84. 59.56 2.1060 42 54. 5i 1.2352 5i 56.09 1.3571 18. 57.20 1.5071 3o 58.47 1.7876 85. 59-56 2.1070 45 54.52 1.2374 54 56.10 I.358S 10 57.22 I.5lI2 34. 58.48 1.7942 86. 59-57 2.1078 48 54.54 1.2396 57 56.11 i.36o5 20 57.24 1.5 1 53 3c 38.5o 1 .8007 87. 59-58 2.11)84 9.5i 54.55 1.2418 i3. 56.12 1.3622 18.30j57.25 1.5194 35. 58.5i 1 .8071 88. 59-5Q 2.1089 54 54.57 1.2440 5 56.13 I.3650 4057.27 1.5235 3o 58.53 i.8i34 89. 59.59 2.1092 57 54.58 1. 2461 io56.i5| 1 .3678 5o 57.29 1.5275 36. 58.54 1 .8106 90. So.oo 2.1093 TABLE XVII. ■ [Page 93 When the Planet Venus or Mars is used, and thi i Parallax is nearly equal to 20". Pakallax 20'' • Alt. DM Cor. M t: Log-. *Ap. Alt. Cor. Log. *Ap. Alt. Cor. Log. *Ap. Alt. D J\l Cor. 1 Log. *Ap. Alt. Dl\I 36.3c Cor. Log. 1.8535 D M M S D I\l M S 31 S 58.5c; 5. 50.27 0.9725 10. 55. o5 1.2554 i3.i5 56.2 1 i.38o2 19. 57.35 1.5453 10 50.43 0.9848 3 55.06 1.2576 20 56.22 i.383o 10 57.37 1.5494 i.55j4 37. 59.01 1.8599 20 50.58 0.9969 6 55.08 1.2597 25 56.23 1.3858 20 57.38 3o 59.02 1 .8662 3o 5i.j3 1 .0087 9 55.09 1.2619 3o 56.25 1.3886 3o 57.39 1.5574 38. 59.03 1.8724 4c) 51.27 1.0202 12 55.11 1.2640 35 56.27 1.3914 4o 37.41 i.56i4 3o 59.04 1.8785 5o 5 1.40 i.o3i5 i5 10.18 55.12 1.2662 4o 13.45 56.28 56.3o 1 .394 1 1.3969 5o 57.42 1.5653 39. 39 3o 59.05 1 .8846 6. 51.53 1.0426 55.14 1.2683 20. 57.43 1 .5692 59.1 !( 1 .8906 10 52. o5 1.0535 21 55.15 1.2704 5o 56.3 1 1 .3996 10 57.45 1.5731 4o. 59.07 1 .8964 20 52.17 1.0641 24 55.17 1.2725 55 56.32 1.4023 20 57.46 1.5770 3o Sg.og 1.9021 3o 62.28 1.0745 27 55.18 1.2746 i4- 56.33 1 .4o5o 3o 57.47 1 .58o8 4i. So.ic 1.9078 4o 52.39 1.0847 3o 55.20 1.2767 5 56.34 1 .4077 4o 57.49 1 .5846 00 59.11 1.9134 5o 7- o 02.49 52.59 1 .0947 1 . 1 046 33 55.21 1.2788 10 i4.i5 56.36 56.38 1.4104 1.4 i3i 5o 57.50 1.5884 42. 59.12 1.9189 1.9243 I0.36 55.22 1.2809 21. 57.51 1.5921 42.3o 59.13 10 53.08 1.1142 39 55.24 i.283o 20 56.39 i.4i57 10 57.53 1.5958 43. 59.1411.9297! 20 D3.I7 1.1237 42 55.25 i.285i 25 56.4o i.4i83 20 57.54 1.5995 3o 59.15 1.9350 3o 33.26 i.i33o 45 55.26 1.2871 3o 56.42 1.4209 3o 57.55 1 .6o3 1 44- 59.15 1.9402 35 53.3o 1. 1376 48 55.28 1. 2891 35 56-43 1.4235 4o 57.56 1 .6067 3o 59.16 1 .9454 4o 53.35 1.1422 5i 55.29 1.291 1 40 14.45 56-44 56.46 1.4261 1.4287 5o 57.57 i.6io3 45. 59.17 1.950.1 7-45 53.39 1.1467 10.54 55.3o 1.2932 22. 57.58 1.6139 46. 59.19 1 .9604 4« 53.41 1.1494 57 55.32 1.2952 5o 56.47 i.43i3 10 57.59 1.6175 47- 59.21 1 .9700 5i •)3.43 I.l520 11. 55.33 1.2972 55 56.48 1.4339 20 58.01 1.6210 48. 59.22 1 .9793 54 33.40 1.1547 3 55.34 1.2992 i5. 56.49 1 .4364 3o 58.02 1 .6245 49. 59.24 1 .9884 57 53.48 1.1574 6 55.35 1.3oi2 5 56.5o 1 .4389 4o 58.03 1.6280 5o. 59.25 1-9972 S. 53. 5o 1. 1 600 9 55.37 i.3o32 10 i5.i5 56.52 56.53 I -44 1 4 1.4439 bo 23. 58.04 58.o5 i.63i4 5i. 59.27 2.0057 8. 3 53.53 1. 1 626 11.12 55.38 i.3o52 1.6348 52. 59-28 2.0140 6 53.55 I.I652 i5 55,3q 1.3071 20 56.54 1.4464 . IC. 58. 06 1.6382 53. 59.29 2.0221 9 53.57 1.1678 18 55.40 1.3091 25 56.55 1.4489 20 58.07 1.6416 54. 59-30 2.0299 12 53.59 1. 1704 21 55.42 i.3iii 3o 56.56 i.45i4 3o 58.08 1 .645o 55. 59-32 2.0374 ID 5401 1.1729 24 55.43 i.3i3o 35 56.57 1.4539 40 58.09 1 .6483 56. 59.33 2-0447 l8 54.04 1. 1755 27 ii.3o 55.44 1 .3:49 i.3i68 40 1 5.45 56.58 56.59 1.4563 5o 58.10 1.65 1 6 57. 58. 59-34 59.35 2.c)5i8 2.0587 8.21 54.06 1. 1 780 55.45 1 .4587 24. 58.11 1 .6549 24 54.08 i.i8o5 33 55.46 1 .3187 5o 57.01 1.461 1 10 58.12 1.6582 59. 59.36 2.o653 27 54.10 i.i83o 36 55.47 1.3206 55 57.02 1.4635 20 58.13 1.661 5 60. 59.37 2.0717 3o 54.12 1.1855 39 55.49 1.3225 16. 57.03 1.4659 3o 58.14 1 .6647 61. 59.38 a.c>779 33 54.14 1.1880 42 55. 5o 1.3245 6 57.04 1.4683 4o 58.15 1 .6679 62. 59.39 2.0838 36 8.3q 54.16 54.18 1. 1 905 45 55.5i 1.3264 7T3r8'3 10 57.05 1 .4707 5o 58.16 1.6711 63. 59.402.08951 1.1930 11.48 55.52 16. i5 57.06 1 .4730 25. 58.16 1 .6742 64. 59-41 2.0950 42 54.20 I.iq55 61 55.53 I.3302 20 57.07 1.4754 20 58.18 i.68o5 65. 59-42 2 . 1 oo3 45 54-2 2 1.1979 54 55.54 1.3321 25 57.08 '.4777 40 58.20 1.6867 66. 59-43 2.io54 48 54-24 1.2004 57 55.55 1.3340 3o 57.09 1 .4800 26. 58.22 1.6928 07. 59-44 2.1102 5i 54.26 1.2028 12. 55.56 1.3358 35 57.10 1.4823 20 58.23 1 .6988 68. 59-45 2.ri48 54 8.57 54.2.8 54.3(. 1.2052 3 55.57 1.3377 40 57.11 1 .4846 40 58.25 1 .7047 69. 59.45 2.1192 1.2076 12. 6 55.59 1 .3396 16.45 57.12 1.4869 27. 58.26 i.7if)6 70. 59.46 2.1234 9. o 54-3i 1.2100 9 56.o(j 1 .3414 5o 57.13 1.4S92 20 58.28 1.7164 71. 59.47 2.1274 3 54-33 1.2124 12 56.01 1.3432 55 5714 1.4915 40 58. 3o 1.7222 72. 59-48 2.l3l2 ('. 54.35 1.2147 i5 56. 02 1 .3450 17. 57.14 1.4938 28. 58.3i 1.7279 1.7335 70. 59.48 2.134s 9 54-37 1.2170 18 56. o3 1.3468 5 57.15 1.4961 20 58.32 74. ^9-49 2.1382 12 54-39 1.2194 21 36-04 1 .3486 10 57.16 1.49S3 4o 58.34 1.7391 75. 59-5C, 2.i4i4 9.i5 54.40 1.2217 12.24 56. o5 1 .35o4 17. i5 57.17 i.5oo5 29. 58.35 1.7446 76. 159.51 2.1444 i8 '^4.4^ 1.2240 27 56.o6 1.3522 20 57.18 1.5028 3o 58.37 1.7527 77. i59-5i 2.1471 21 54.44 1.2263 3o 56.07 1.3540 25 57.19 i.5c5o 3o. 58.39 1 .7607 78. 59.52 2.1496 24 54.46 1.2286 33 56.08 1.3558 3o 57.20 1.5072 3o 58.4i 1.7685 79. 59.53 2.l520 27 54-47 1.2309 36 56.09 1.3576 35 57.21 1.5094 3i. 58.43 1.7762 80. 59.54 2.1542 3o 9.33 54-49 54.5i 1.2332 1.2355 39 56.10 1.3594 1 .3612 40 57.22 i.5ii6 3o 32. 58.44 1.7838 58.46 1.7913 81. |59.54 2.i56i 12.42 56.11 17-45 57.23 i.5i37 82. 59.55'2.i578 36 54-52 1.2377 45 56.12 1.3629 5o 57.23 i.5i59 3o 58.48^ 1.7987 83. 59.55I2.1594I 39 54.54 1.2400 48 56.13 1.3647 55 57.24 i.5i8i 33. 0158.49 1.8059 84. 59-56j2.i6o7 42 54.56 1.2423 5i 56.14 1.3664 18. 57.25 1.5202 3o58.5i i.8i3o 85. 59.57 2 1618 45 54.57 1.2445 54 56.15 1 .368 1 10 57.27 1.5245 34. o!58.52 1.8200 86. 59.57 2,1627 48 54.59 1.2467 57 56.16 1.3698 20 i8.3o 57.28 1.5287 3o!58.54 1.8269 87. 59-58 2.1634 9.5i 55.00! 1.2489 i3. 56.1- 1.3715 57.30 1.5329 35. 0^58.55 1.8337 88. 59-59 2.1639 54 55.02 I.25l I 5 56.18 1 .3744 40 57.39 1. 5371 3o;58.57i.84o4 36. 0,58.58,1.8470 89. 59.59 2.1642 57 55.03 1.2533 TO 56. 20] 1.3773 5o 57.33 1.5412 90. So.oo 2.1643 Page 94] TABLE XVII , When the Planet Venus or Mars is used, and the Parallax is nearly equal to 25^'. Parallax 25'' • •Ap Alt. Cor. Log. *Ap. Alt. Cor. Log. *Ap. Alt. Cor. Log-. *Ap. Alt. Cor. Log. *Ap. Alt. Cor. Log. DM M S D M M S D M M S D M M S D M M S 5. 5o.32 0.9763 10. 55.10 1.2628 i3.i5 56.26 1.3899 19. 67.39 1.5697 36.3o 69.03 1.8832 lO 5o.48 0.9887 3 55.11 i.265o 20 56.28 1.3928 10 67.41 1.5639 37. b9.ob 1.8901 20 5i.o3 1 .0009 6 55.13 1. 2671 25 56.29 1.3967 20 57.42 1. 668 1 3o 59.06 1.8968 Jo 5i.i8 1. 01 28 9 55.14 1.2693 3o 56.3 1 1.3986 3o 67.44 1.6722 38. 69.07 1.9035 4o bi.32 1.0245 12 55.16 1. 2715 35 56.32 1.4014 4o 67.45 1.5763 3o 69.08 1. 9100 5o 51.45 1.0359 i5 55.17 1.2737 40 56.33 1 .4042 5o 57.47 i.58o4 39. 69.09 1.9166 6. o 51.58 1. 047 1 10.18 55.19 1-2759 i3.45 56.35 1 .4070 20. 67.48 1.5844 39.30 69.10 1.9229 10 52.11 i.o58i 21 55.20 1. 2781 5o 56.36 I. ■1098 10 67.49 1.5884 4o. 69.11 1.9292 20 52.22 1.0688 24 55.22 1.2802 55 56.38 1. 4126 20 57.61 1.5924 3o 59.12 1.9354 3o 52.34 1 .0793 27 55.23 1.2823 i4- 56.39 i.4i54 3o 67.62 1.6964 4i. 69.13 1.94x6 4o 52.44 1.0896 3o 55.24 1.2844 5 56.4o 1.4182 40 67.53 1 .6oo3 3o b9.i4 1 .9477 bo 52.54 1 .0998 33 55.26 1.2865 10 56.42 1.4209 5o 21. 67.66 67.66 1 .6042 1.6081 42. 69.16 1.9537 7- 53.04 1. 1098 10.36 55.27 1.2886 i4-i5 56.43 1.4236 42.3o 69'. 1 6 1.9696 lO 53.13 1.1195 39 55.29 1.2907 20 56.44 1.4263 10 67.57 1.6119 43. 69.17 1.9654 20 53.22 1.1291 42 55.3o 1.2928 2b b6.4b 1.4290 20 67.68 1.6167 3o 59.18 1.971 1 3o 53.3i I.I385 45 55.3i 1.2949 3o 56.47 1.4317 3o 58 .00 1.6195 44. 69.19 1.9768 3h 53.35 r.i432 48 55.33 1.2970 3b 56.48 1.4344 4o 58.01 1.6233 3o 69.20 1.9824 4o 53.40 i.i47« 5i 55.34 1.2991 4o b6.49 1 .4370 60 22. 58.02 58.03 1.6271 45. 69.21 1.9879 7-45 53.44 I.l522 10.54 55.35 I.3oi2 14-45 56.5o 1.4397 i.63o8 46. 69.22 1 .9987 4» 53.46 i.i55o 57 55.36 i.3o32 5o 56.62 1.4423 10 68.04 t.6346 47. 69.24 2.0092 5i 53.48 1. 1 577 II. 55.38 i.3o52 55 56.53 1-4449 20 68.06 1.6382 48. 69.26 2.0194 54 53.5i 1.1604 3 55.39 1.3073 i5. 56.54 1-4475 3o 58.06 1.6418 49- 69.27 2.0294 67 53.53 i.i63i 6 55.4o 1.3093 5 56.55 1.4501 40 58.07 1.6454 60. 69.28 2.0391 8. o 53.55 I.I658 9 II. 12 55.41 i.3ii3 10 56.56 1.4627 60 23. 68.08 1 .6490 61. 59.3c 2.0485 8. 3 53.58 1. 1684 55.43 i.3i33 i5.i5 56.57 1.4552 68.09 1.6626 62. 69.31 2.0677 6 54.00 1.1711 i5 55.44 i.3i53 20 66.69 1.4578 10 58.10 1.6661 63. 69.32 2.0666 9 54.02 1-1737 18 55.45 1.3173 25 67.00 1 .4604 20 58.11 1.6696 54- 69.33 2.0762 12 b4.o4 1.1763 21 55.46 1.3193 3o 67.01 1.4629 3o 58.12 1. 663 1 55. 69.34 2.0836 i5 54.06 1. 1790 24 55.48 I.32I3 35 57 02 1.4654 4o 58.13 1.6666 66. 69.36 2.0918 i8 54.09 1. 1816 27 55.49 1.3233 40 67.03 1 .4679 5o 68.14 1. 670 1 67. 59.37 2.0997 8.21 54.11 1. 1842 ii.3o 55. 5o 1.3253 1 5.45 67.04 1 .4704 24. 68. i5 1.6735 68. 59.3s 2.1073 24 54.13 1. 1868 33 55. 5i 1.3272 5o 67.06 1.4729 10 68.16 1 .6769 59. 69.39 2.1 147 27 54.15 1. 1893 36 55.52 1.3292 bb 67.06 1-4754 20 58.17 1 .6803 60. 69.40 2.1219 3o 54.17 1.1918 39 55.53 i.33ii 16. 67.07 i-477« 3o 68.18 1.6837 61. 69.41 2.1288 33 54.19 1. 1943 42 55.55 I.333I 5 67.08 i.48o3 4o 68.19 1.6870 62. 59.41 2.1356 36 8.39 54.21 1. 1968 45 55.56 I.3350 10 i6.i5 67.09 1.4827 60 25. 68.20 68.21 1 .6903 7I3936 63. 69.42 2.1419 54.23 1. 1993 11.48 55.57 1.3370 67.10 1.4862 64. 59.43 2.1481 42 b4.2b 1.2018 bi 55.58 1.3389 20 67.1. 1.4876 20 58.23 1.7002 65. 59-44 2.1641 4b 54.27 r.2o43 54 55.59 1.3408 25 67.12 1 .4900 40 58.25 1.7067 66. 69.46 2.1699 48 54.29 1.2068 57 56. 00 1.3427 3o 67.13 1.4924 26. 68.26 i.7i3i 67. 69.46 2.1654 bi 54.3 1 1.2092 12. 56.01 1.3446 35 57-14 1.4948 20 68.28 1.7194 68. 69.46 2.1707 b4 8.57 54.33 r.2117 3 56.02 1.3465 4o 57-15 1.4972 4o 68.29 1.7266 69. 59.47 2.1767 54.35 I.2l4l 12. 6 56.03 1.3484 16.45 57.16 1.4996 27. 68.3 1 1.7318 70. 69.48 2.i8o5 9. 54.36 I.2I65 9 56.04 I.3502 5o 67.17 1.6019 20 58.32 1.7379 71. 69.49 2.i85i 3 54.38 r.2189 12 56.05 I.352I 55 67.18 i.5o42 40 58.34 1.7440 72. 69.49 2.1894 6 b4.4o r.22i3 i5 56.06 1.3539 17. 67.19 i.6o65 28. 58.36 1.7600 73. ,59.60 2.1935 9 b4.42 1.2237 18 56.08 1.3558 5 67.20 i.5o88 20 58.37 1.7669 74. 69.61 2.1973 I? 9.. 5 b4.43 1. 2261 21 56.09 1-3577 10 67.21 1.6111 4o 58.38 1.7617 75. 76. 59.6. 2.2009 54.45 1.2285 12.24 56. 10 r.3596 17-15 67.22 i.5i34 29. 68.39 1.7675 59.62 2.2043 18 54-47 1 .2309 27 56.11 r.36i4 20 67.23 1.6167 3o 58.4i 1 .7760 77. 69.53 2,2076 21 54.49 1.2333 3o 56.12 1.3632 25 67.24 i.5i8o 3o. 68.43 1.7845 78. 69.53 2.2Io5 24 54.51 1.2356 33 56.13 1.3650 3o 67.26 I.5203 3o 68.45 1.7928 79- 69.54 2.2l32 27 54.52 1.2379 36 56.14 1 .3668 35 67.261.5226 3i. 58.47 1. 8010 80. 69.54 2.2l56 3o 54.54 1.2402 39 56.15 1 .3686 40 67.27 1.6249 3o 58.49 1 .8090 81. 69.55 2.2178 9.33 54.56 r.2425 12.43 56.16 1.3704 17-45 57-27 1.6271 32. 58. 5o 1.8169 82. 59.66 2.2198 36 54.57 1.2448 45 56.17 1.3722 bo 67-28 1.5293 3o 68.62 1.82,47 83. 69.56 2.2216 39 54.59 1-2471 48 56.18 1.3740 55 67-29 i.53i6 33. 58.54 1.8324 84. 59.57 2.2232 42 55.00 1-2494 5i 56.19 f.3758 18. 67.30 1.5337 3o 58.55 1 .8400 85. 69.67 2.2245 45|55.02 i.25i6 54 56. 20 1.3776 10 57.32 1.5381 34. 58.66 1.8474 86. 59.68 2.2266 48 55.04 1.2538 57 56.2 1 1.3794 20 67.33 1.5426 3o 58.68 1.8547 87. 59.68 2.2264 9.5. 55.05 i.256i i3. 56.21 i.38ri i8.3o 67.35 1.5468 35. 068.59 1.8620 88. 69.69 2.2269 54 55.07 1.2584 5 56.23 1. 384 1 40 57.36 1.6611 3o 69.01 1.8692 89. 69.692.2273 57 55.08 1 .2606 10 56.25 1.3870 5o57.38ji.5554 36. 69.02 1 .8763 90. 60.00j2.2274 TABLE XVII [Page 95 When the Planet Venus is used, and the P arallax is nearly equal to 30". Parallax 30' All. D.M ("or. Log. 'A p. Alt. U M Cor. Log. 'iiS: Cor. Log. *Ap. All. Cor. Log. *Ap. Alt. Cor. Log. M S 1) M M S D M M S U M M S 5. o 5<).37 0.9801 10. o55.i5 1.2702 i3.i5 56.3 1 1 .3999 19. 57.44 1.5745 36.3o 59.07 I 91 5i lo 5o.53 0.9926 3 55.1b 1.2724 20 56.32 1 .4029 10 5746 1.5789 37. 59.08 1.9225 20 5 1. (.8 1.0049 6 55.18 1.2746 25 56.34 1 .4o58 20 57-47 1.5832 3o 59.10 1.9297 3(.- 51.23 1.0170 9 55.19 1.2769 3o 56.35 1.4087 3c 57.49 1.5875 38. 59.11 1.9369 4<) 51.37 1.0288 12 :)D.2i 1.2791 35 36.37 1.4116 40 57.50 1. 5918 3o 39.12 1.9440 5.- 5i.5<. 1 .o4o3 i5 DD.22 1.2813 i.28"35 40 56.38 1.4145 5o 57.52 1.5960 39. 59.13^1.9511 0. 52.03 i.o5i6 10.18 55.24 13.45 56.4o 1.4174 20. 57.53 1 .6002 39.30 59.14 1 .9580 10 52.15 1.0627 21 55.2^) 1.2857 5c 56.4 1 1.4203 10 57.54 1 .6044 4o. 59.15 1 .9648 n> 52.27 1.0735 24 55.26 1.2879 55 56-42 1. 423 1 20 57.56 i.6o85 3o 59.16 1.9716 3n 52.38 1 .0842 27 55.28 1.2900 14. 56.44 1.4259 3o 57.57 1.6126 4i. 59.17 1.9783 4o 52.49 1 .0947 3o 33.29 1.2922 5 5645 1.4287 4o 57.58 1.6167 3o 59.18 1 .9849 5(. 52.59 i.i(»l9 6i 53. 3i 1.2944 10 36.46 i.43i5 5o 57.59 1.6207 42. 39.19 1.9914 7. 53.09 1.1149 10.36 55.32 1.2966 14. i5 56.48 1.4343 21. 58.0I 1.6247 42. 3o 59.20 1 .9978 10 53.18 1. 1248 39 55.33 1.2987 20 56.49 1. 437 1 10 58.02 1.6287 43. 59.21 2.0042 PCI 53.27 I.I345 42 55.35 i.3oo8 25 56. 5o 1 .4399 20 58.03 1.6327 3o 39.22 2.oio5 3i. 53.36 i.i44i 45 55.36 1.3029 3o 56.52 1-4427 3o 58.04 1.6366 44. 59.23 2.0167 35 53.40 1.1488 48 55.37 i.3o5o 35 56.53 1-4454 40 58.05 1 .64o5 3o 59.24 2.0228 40 53.45 i.i534 61 55.39 1. 3071 4o 56.54 1. 448 1 5o 58.07 16444 45. 59.24 2.0289 7-45 53.49 I.I579 10.54 55.40 1.3092 14.45 56.55 1.4508 22. 58. 08 1 .6483 46. 59.26 2.o4o8 4« 53.5i 1. 1 608 57 55.41 i.3ii3 5o 56.57 1.4535 10 58.09 1.6522 47- 39.27 2.0524 5i 53.53 I.I636 II. 55.43 i.3i34 55 56.58 1.4562 20 58. 10 1 .656o 48. 39.29 2.0637 54 53.56 I.I663 3 55.44 i.3i55 i5. 56.59 1.4589 3o 58.11 1.6598 49. 59.30 2.0747 57 53.58 1 . 1 690 6 55.45 1.3176 5 57.00 1.4616 4o 58.12 1.6636 5o. 59.32 2.0855 8. 54.00 1-1717 9 55.46 1.3197 10 37.01 1.4643 5o 58.i3 1.6673 5i. 59.33 2.0960 8. 3 54.02 1. 1744 11.12 55.48 1. 3217 i5.i5 57.02 1.4669 23. 58.14 1.6710 52. 59.34 2.1062 6 54.05 1.1771 i5 55.49 1.3237 20 57.03 1.4695 10 58. i5 1.6747 53. 59.35 2.1162 9 54.07 1. 1798 18 55. 5o 1.3258 25 57.05 1. 472 1 20 58.i6 1.6784 54. 59.36 2.1259 12 54.09 1. 1824 21 55.5i 1.3278 3o 57.06 1-4747 3o 58.17 1.6820 55. 59.37 2.i353 i5 54.11 i.i85i 24 55.53 1.3298 35 57.07 1 .4773 4o 58.18 1.6857 56. 59.38 2.1445 18 54.13 1.1877 27 55.54 i.33i8 40 57.08 1.4799 5o 24. 58.19 1 .6893 57. 59.39 2.1534 8.21 54.16 1. 1 903 II. 3o 55.55 1.3338 i5.45 57.09 1.4824 58. 20 1.692Q 58. 59.40 2.1621 24 54.18 1. 1 929 33 55.56 1. 3358 5o 57.10 1.4850 10 58.21 1.6965 59. 59.41 2.I-705 27 54.20 1. 1955 36 55.57 1.3378 55 57.11 1.4876 20 58.22 1 .7000 60. 59.42 2.1787 3(. 54.22 1.1981 39 55.58 1.3398 16. 57.12 1.4901 3o 58.23 1.7035 61. 59.43 2.1866 3il 54.24 I; 2007 42 55.59 1.3418 5 57.13 1.4926 4o 58.24 1 .7070 62. 59.44 2.1942 36 8.39 54.26 54.28 I.2o32 45 56.01 1.3438 10 57.14 1. 495 1 5o 58.25 1.7105 63. 59.45 2.2016 i.2o57 11.48 56.02 1.3458 i6.i5 57.15 1.4976 25. 58.26 1 .7140 64. 59.45 2.2087 42 54.30 1.2082 5i 56.03 1.3478 20 57.16 i.5ooi 20 58.27 1.7208 65. 59.46 2.2 I 56 45 54.32 1.2107 54 56.o4 1.3498 25 57.17 1.5026 4o 58.29 1.7276 66. 59.47 2.222i 48 54.34 [.2l32 57 56.05 1.3517 3o 57.18 i.5o5o 26. 58.3 1 1.7343 67. 59.48 2.228(3 31 54.36 I.2l57 12. 56. 06 1.3536 35 57.19 1.5075 20 58.32 1.7410 68, 59.48 2.2347 54 54.38 1. 2182 3 56.07 1.3555 4o 57.20 1. 5 1 00 4o 58.34 1.7476 69. 59.49 2.24o5 8.57 54.39 1.2207 12. .6 56.08 1.3574 16.45 57.21 1.5124 27. 58.35 1 .754 1 70. 59.50 2.2461 9. 54.41 1.2232 9 56.09 1.3593 5o 57.22 i.5i48 20 58.37 1 .7605 71. 59.50 2.25l4 3 54.43 1.2257 12 56.10 I.36I2 55 57.23 1. 5172 40 58.38 1 .7669 72. 59.51 2.2 565 • () 54.45 1. 2281 i5 56.11 1. 363 1 17. 57.24 1.5196 28. 58 .40 1.7732 73. 59.51 2.2613 9 54.47 i.23o5 18 56.12 1.3650 5 57.25 1.5220 20 58.41 1.7794 74. 59.52 2.2658 12 54.48 r.2329 21 56.13 1.3669 10 57.26 1.5244 40 58.43 1.7856 75. 59.52 2.2701 54.5o 1.2353 12.24 56.14 1.3688 17.15 57.27 1.5268 29. 58.44 1.7917 76. 59.53 2.2741 18 54.52 1.2377 27 56. 1 5 1.3707 20 57.28 1.5292 3o 58.46 1 .8007 77. 59.54 2.2778 21 54.54 1. 240 1 3o 56.16 1.3725 25 57-29 i.53i5 3o. 58.48 1.8097 78. 59.54 2.2812 24 54.55 1.2425 33 56.17 1:3744 3o 57-29 1.5338 3o 58. 5o i.8i85 79. 59.55 2.2844 27 54.57 1.2449 36 56.18 1.3763 35 57.30 1.5362 3i. 58. 5i 1.8272 80. 59.55 2.2873 3o 54.59 1.2472 39 56.19 1.3782 . 4o 57.31 1.5385 3o 58.53 1.8357 81. 59.56 2.2899 9.33 55.00 1.2495 12.42 56. 20 i.38oo 17-45 57.32 1.5408 32. 58.54 1.8441 82. 59.562.29231 36 55.02 i.25i8 45 56.21 i.38i8 5o 57.33 1. 543 1 3o 58.56 1.8524 83. 59.57 2.2944 39 55.04 1.2541 48 56.22 1.3837 55 57.34 1.5454 33. 58.58 1.8606 84. 59.57 2.2962 42 55.05 1.2 564 5i 56.23 1.3855 18. 57.35 1.5477 3o 58.59 1.8687 85. 39.58 2.2977 45 55.07 1.2587 54 56.24 1.3873 10 57.36 1.5523 34. 59.01 1.S767 86. 59.58 2.2990 48 55.09 1. 2610 57 56.25 1.3891 20 57.38 1.5568 3o 59.02 1.8846 1.8924 87. 59.58 2 .3ooo 9.5, 55.10 1.2633 i3. 56.26 1.3909 18. 3o 57.39 i.56i3 35. 59.03 88. 59.59 2.3007 54 55.12 1.2656 5 56.28 1.3939 4o 57.41 1.5657 3o 59.05 1.9000 89. 59.59 2.301 I 1 57155. i3|i. 2679 10 56.29 1.3969 5o 57.43,1.5701 36. 59.06 1.9076 90. 5o.oo 2.3o3i Page 96] TABLE XVII When the Planet Venus is used, and the Parallax is nearly equal to 35''. Parallax 35'^ • *Ap. Alt. DM Cor. 31 S Log. *Ap. Alt. Cor. Log-. *Ap. Alt. Cor. Log. Alt. Cor. Log. A.?.- Cor. D M M S Log. D JM M S D mIM S D M M S 5. o 5o.42 0.9840 ID. 55.20 1.2777 i3.i5l56.36 1.4101 19. 57.49 1.5899 36.3o 59.114.9495 1 10 50.58 0.9966 3 55.21 1.2800 20I56.37 i.4i3i 10 57.50 1 .5944 :?7. 59.12 1.9575 20 5i.i3 1 .0090 6 55.23 1.2823 25 56.39 1.4161 20 57.52 1.5989 3o 59.14 1 .9654 3o 51.28 1.0212 9 55.24 1.2846 3o 56 .40 1.4191 3o 57.53 i.6o33 38. 59.15 1.9732 4o 5i.42 i.o33i 12 55.26 1.2860 35 56.42 1.4221 4o 57.55 1 .6077 3o 59.16 1 .9809 5o 5i.55 1.0447 i5 55.27 1.2891 4o 56.43 1.4251 5o 57.56 1.6121 39. 59.17 1 .9886 6. 52.o8 i.o56i 10.18 55.29 1.2913 i345 56.44 1.4281 20. 57.57 i.6i65 39.30 59.18 1. 996 1 10 52.2a 1.0673 21 55.3o 1.2936 5o 56.46 1.4310 10 57.59 1.6208 4o. 59.19 2.oo36 ■20 52.32 1.0783 24 55.3i 1.2958 55 56.47 1.4339 20 58.00 1.6251 3o 59.20 2.0110 3o 52.43 1.0891 27 55.33 1.2980 i4. 56.49 1 .43G0 3o 58.02 1.6294 4i. 59.21 2.0183 4o 52.54 1.0997 3o 55.34 1.3002 5 56.5o 1.4397 4o 5$.o3 1.6336 3o 59.22 2.0255 5o 53.04 I.I 100 33 55.36 i.3o24 10 56.5i 1.4426 5o 58. 04 1.6378 42. 59.23 2.0327 7- 53.14 1.1202 10.36 55.37 i.3o46 i4.i5 56.53 1.4455 21. 58.05 1.6420 42. 3o 59.24 2.0398 10 53.23 l.l302 39 55.38 i.3o68 20 56.54 1.4483 10 58.00 1.6462 43. 59.25 2.0468 20 53.32 i.i4oo 42 55.40 1.3090 25 56.55 1.4511 20 58.08 i.65o3 3o 59.26 2.0537 3o 53.4i 1. 1497 45 55.41 i.3iii 3o 56.56 1.4540 3o 58.09 1.6544 44. 59.26 2.0606 35 53.45 1.1545 48 55.42 i.3i32 35 56.58 1 .4568 4o 58.IO 1.6585 3o 59.27 2.0674 4o 53.49 1.1592 5i 55.44 i.3i54 4o 56.59 1.4596 5o 58.11 1.6626 45. 59.28 2.0741 7-45 53.53 I.I638 10.54 55.45 1.3176 14.45 57.00 1.4624 22. 58.12 1.6666 46. 59.29 2.0873 4» 53.56 1.1667 57 55.46 1.3197 5o 57.01 1.4652 .10 58.13 1 .6706 47. 59.31 2.ioo3 5i 53.58 1.1695 11. 55.48 1.3218 55 57.02 1 .4679 20 58.14 1.6746 48. 59.32 2.1 129 54 54-01 1.1753 3 55.49 1.3239 i5. 57.04 1 .4706 3o 58.i5 1.6785 49. 59.33 2.1253 57 54.03 1. 1751 6 55. 5o 1.3260 5 57.05 1.4734 4o 58.17 1.6824 5o. 59.35 2.1375 8. o 54.05 1.1778 9 55.5i 1.3281 10 57.06 1.4761 .50 58.18 1.6863 5i. 59.36 2.1493 8. 3 54.07 i.i8o5 11.12 55.53 i.33o2 i5.i5 57.07 1 .4788 23. 58.19 1 .6902 52. 59.37 2.1609 6 54.10 1.1832 i5 55.54 1.3323 20 57.08 i.48i5 10 58.20 1. 694 1 53. 59.38, 2.1723 Q 54.12 1.1859 18 55.55 1.3344 25 57.09 1.4842 20 58.21 1 .6980 54. 59.39 2.1833 12 54.14 1.1886 21 55.56 1.3365 3o 57.10 1.4869 3o 58.22 1.7018 55. 59.40 2.1940 i5 54.16 1. 1913 24 55.58 1.3386 35 57. 1 2I 1. 4896 40 58.23 1.7056 56. 59.41 2.2045 j8 54.18 1. 1939 27 55.59 1.3406 4o 57.13 1.4922 5o 58.24 1.7094 57. 59.42 2.2148 8.21 54.21 1 . 1 966 1 1 .3o 56.00 1.3426 i5.45 57.14 1.4948 24. 58.25 1.7132 58. 59.43 2.2248 24 54.23 1.1993 ■63 56.01 1.3447 5o 57.15 1.4975 10 58.26 1.7169 59. 59.44 2.2346 27 54.25 1. 2019 36 56.02 1.3467 55 57.16 i.Sooi 20 58.27 1.7206 60. 59.45 2.2440 3o 54.27 1.2045 39 56.03 1.3487 16. 57.17 1.5027 3o 58.28 [.7243 61. 59.45 2.2.532 33 54.29 1. 2071 42 56.04 1.3507 5 57.18 i.5o53 4o 58.29 1.7280 62. 59.46 2.2621 36 8.3q 54.3i 54.33 1.2097 45 56.06 1.3527 10 57.19 1.5079 5o 58.29 1.7317 63. 59.47 2.2708 1.2123 11.48 56.07 1.3547 16. i5 57.20 i.5io4 25. 58.3o 1.7353 64. 59.48 2.2792 42 54.35 1.2148 5i 56.08 1.3567 20 57.21 i.5i3o 20 58.32 1.7425 65. 59.48 2.2873 45 54.37 1.2173 54 56.09 1.3587 25 57.22 i.5i56 40 58.34 1.7496 66. 59.49 2.2951 48 54.39 1.2199 57 56. 10 1 .3607 3o 57.23 i.5i8i 26. 58.35 1.7567 67. 59.49 2.3026 5i 54.41 1.22Vf 12. 56.11 1.3627 35 57.24 1.5206 20 58.37 1.7637 68. 59.50 2.3099 54 54.43 1.2249 3 56.12 1.3647 1.3667 4o 57.25 I.523I 4o 58.38 1.7706 69. 59.51 2.3 1 68 8.57 54.44 1.2275 12. 6 56. 1 3 16.45 57.26 1.5256 27. 58.40 1.7775 70. 59.51 2.3235 9. 54.46 1 .23oo 9 56.14 1.3686 5o 57.27 1.5281 20 58.4 1 1.7843 71- 59.52 2.3299 3 54.48 1.2325 12 56.15 1.3706 55 5728 1 .53o6 40 58.43 1.7910 72. 59.52 2.3359 6 54.50 i.235o i5 56.16 1.3725 17. 57.29 1.5331 28. 58.44 1.7977 70. 59.53 2.3417 9 54.52 1.2374 18 56.17 1.3745 5 57.30 1.5356 20 58.46 1.8043 74. 59.53 2.3471 12 54.53 1.2399 21 12.24 56.18 56.19 1.3764 1.3783 10 17.15 57.31 1.5381 4o 58.47 1.8108 73. 59.54 2.3523 9.i5 54.55 1.2423 57.32 i.54o5 29. 58.48 1.8173 76. 59.55 2.3571 i8 54.57 1.2448 27 56.20 i.38o2 20 57.33 1.5430 3o 58.5o 1.8269 77- 50.55 2.36i6 21 54.59 1.2472 3o 56.21 I.382I 2 5 57.34 1.5454 3o. 58.52 1.8364 78. 59.55 2.3658 24 55.00 1.2496 33 56.22 1 .384o 3o 57.34 1.5478 3o 58.54 1.8457 79- 59.56 2.3697 27 55.02 1.2520 36 56.23 1.3859 35 57.35 i.55o2 3i. 58.55 1.8550 80. 59.56,2.3732 3o 55.04 1.2544 39 56.24 1.3878 40 57.36 1.5526 3o 58.57 1.8641 81. 59.57I2.3764 9.33 55.05 1.2568 12.42 56.25 1.3897 17.45 57.37 1.5550 32. 058.59 1.8732 82. 59.57,2.3793 36 55.07 1.2592 45 56.26 1.3916 5o 57.38 1.5574 3o|59.oi 1.8821 83. 59.572.3819 39 55.09 1.2615 48 56.27 1.3935 55 57.39 1.5598 33. 059.02 1 .8909 84. 59.582.3841 42 55.10 1.2639 5i 56.28 1.3954 18. 57.39 1.562 1 30|59.o3 1 .8996 85. 59.58 2.3859 45 55.12 1.2662 54 56.29 1.3973 10 57.41 1.5668 34. 0,59.05 1..9081 86. 59.582.3874 48 55.14 1.2685 57 56.3o 1.3991 20 57.43 1.5715 3059.06 1.9165 87. 59.59'2.3886 9.5i 55.15 1.2708 i3. 56.3 1 1 .4009 18. 3o 57.441.5761 35. 0J59.08 1.9249 88. 59.592.3895 54 55.17 1. 2731 5 56.33 i.4o4o 4o 57.461.5807 3o 59.09 1.9332 89. 59.592.3900 57 55.18 1.2754 iO|56.34'i.4o7i 5o 57.48 1.5853 36. 059.10I1.9414 90. 60.002.3903 TABLE XVIII. LPage 97 When the Sun is used. ©Ap Alt. Cor. Log. ©Ap All. Cor. Log. 0Ap. Alt. Cor. Log. 0Ap Alt. D 31 19. c Cor. M & Log. 0Ap Alt. Cor. Log. D M 5. o JI & D M M S D M M fe D IVI M S 5o.i6 0.9645 10. 54.54 1.2397 i3.i5 56.10 1.3592 57-24 i.5i49 36.3o 58.5o 1.7934 10 5o.32 0.9766 3 54.55 1.2418 20 56.12 1.3619 IC 57-25 1.5187 37. 58.5i 1 .7990 20 5o.48 0.9885 b 54-57 1.2439 25 56.13 1 .3646 20 57-27 1.5225 3o 58.53 1.8045 3o 5i. 3 1 .0000 9 54.58 1.2460 3o 56.15 1.3672 3o 57-28 1.5262 38. 58.54 x.8xoo 4o 5i.i6 I.QIl3 12 55. 1. 248 1 35 56.16 1 .3699 4o 57.30 1.5299 3o 58.55 1.8x54 5o 6. o 5i.3o I.0223 i5 55. I i.25oi 4o 56.17 1.3725 5o 57.31 1.5336 39. 58.57 1.8206 1.8257 51.42 i.o33o io.i8 55. 3 1.2522 i3.45 56.19 1.3751 20. 57.33 1.537.2 39.30 58.58 10 51.55 1 .0437 21 55. 4 1.2543 5o 56. 20 1.3777 10 57.34 1.5408 40. 58.59 x.83o7 20 52. 6 r.o54i 24 55. 6 1.2563 55 56.22 i.38o3 20 57.35 1.5444 3o 59. 1.8357 3o 52.17 1.0643 27 55. 7 1.2583 i4- 56.23 1.3828 3o 57.37 1.5480 4i- 59. I X.8406 4o 52.28 1 .0742 3o 55. 8 1.2603 5 56.24 1.3853 40 57.38 i.55i5 3o 59. 2 X.8454 5o 52.38 I .oS4o 33 55.10 1.2623 10 56.26 1.3878 5o 57.39 I.5550 42. 59. 3 i.85oo 7- 52.48 1.0935 I0.36 55.11 1.2643 i4-i5 56.27 1 .3904 21. 57-41 1.5585 42. 3o 59. 4 1.8546 ID 52.57 r.1029 39 55.13 1.2663 20 56.28 1.3929 10 57.42 1.5619 43. 59.5 1.8593 20 53. 6 1.1122 42 55.14 1.2683 25 56.3o 1.3954 20 57.43 1.5653 3o 59. 6 1.8638 3o 53.15 1.1212 45 55.15 1.2702 3o 56.3i 1.3979 3o 57-44 1.5686 44. 59. 7 1.8683 35 53.19 1. 1257 48 55.17 1.2722 35 56.32 1 .4004 4o 57-46 1.5719 3o 59. 8 1.8726 4o 53.23 i.i3oi 5i 55.18 1.2742 40 14.45 56.33 1.4029 5o 57.47 1.5752 45. 59. 9 1.8768 7-45 53.27 I.I345 10.54 55.19 1. 2761 56.35 i.4o53 22. 57.48 1.5784 46. 59.x X 1.8848 48 53.3o i.i37i 57 55.20 1.2780 5o 56.36 1-4077 10 57-49 1.5817 47. 59.x3 1.8928 6i 53.32 1. 1397 II. 55.22 1.2799 55 56.37 1.4101 20 57.50 1.5849 48. 59.X5 1 .9004 54 53.35 1. 1423 3 55.23 1. 2818 i5. c D6.38 1.4125 3o 57.51 i.588i 49. 59.16 1 .908 1 57 53.37 I.I448 6 55.24 1.2837 5 56.39 1.4149 4o 57.52 1. 5913 5o. 59.18 X.9X54 «. o 53.39 1. 1474 9 55.26 1.2856 10 56.4i 1.4173 5o 57.53 1.5945 5i. 59.19 1.9225 8. 3 53.41 1-1499 II. 12 55.27 1.2875 i5.i5 56.42 1-4197 23. 57-54 1.5976 52. 59.21 1.9294 6 53.44 i.i524 i5 55.28 1.2894 20 56.43 1.4221 10 57.56 1 .6008 53. 59.22 X.9362 9 53.46 i.i55o 18 55.29 1. 2913 25 56.44 1.4244 20 57-57 1.6039 54. 59.24 1.9424 12 53.48 1. 1575 21 55.3o 1.2932 3o 56.45 1.4267 3o 57-58 1.6070 55. 5Q.25 1.9484 i5 53. 5g 1. 1599 24 55.32 1. 2951 35 56 46 1.4290 4o 57-59 1.6101 56. 59.26 1 .9544 i8 53.52 1.1624^ 27 55.33 1.2970 4o 56.47 i.43i3 5o 58. 1. 61 3 1 57. 59.28 1 .9602 8.21 53.55 1. 1649 ii.3o 55.34 1.2988 15.45 56.49 1.4336 24. 58. I 1.6161 58. 59.29 X.9658 =4 53.57 1. 1673 33 55.35 1 .3007 5o 56.5o 1.4359 10 58. 2 1.6191 59. 59.30 X.9715 27 53. 5q 1. 1698 36 55.36 i.3o25 55 56.5i 1.4382 20 58. 3 1. 622 1 60. 59.31 1.976X 3o 54. I 1. 1722 39 55.38 i.3o43 16. 56.52 1 .4404 3o 58. 3 1.6250 61. 59.33 X.9807 33 54. 3 1. 1746 42 55.39 i.3o6i 5 56.53 1.4427 40 58. 4 1.6279 62. 59.34 1.9854 36 54. 5 1-1770 45 55.40 1 .3079 10 56.54 1.4449 5o 25. 58. 5 i.63o8 63. 59.35 1.9901 8.39 54. 7 1. 1794 11.48 55.41 1.3097 i6.i5 56.55 1. 447 1 58. 6 1.6336 64. 59.36 1 .9946 42 54. 9 1.1818 61 55.42 i.3ii5 20 56.56 1.4493 20 58. 8 1.6393 65. 59.37 1 .9986 45 54.11 i.i84i 54 55.43 i.3i33 25 56.57 i.45i5 40 58.10 1.6449 66. 59.38 2.0025 48 54.i3 I.I865 57 55.44 i.3i5i 3o 56.58 1.4537 26. 58.11 i.65o5 67. 59.39 2 .0064 5i 54.15 1.18S8 12. 55.45 1 .3169 35 56.59 1.4559 20 58.13 1.6559 68. 59.40 2.0x00 54 54.16 1.1912 3 D5.46 1 .3187 4o 57. 1. 458 1 4o 58.15 1. 661 2 69. 59.4X 2.0x36 8.57 54.18 1. 1935 12. 6 55.48 i.32o5 16.45 57- I J .4602 27. 58.16 1.6665 70. 59.42 2.0173 9. 54.20 1. 1958 9 D5.49 1.3223 5o 57. 2 1.4624 20 58.18 1 .6718 71. 59.43 2.0208 3 54.22 1.1981 12 55.5o 1.3240 55 57. 3 1.4646 40 58.19 1 .677 1 72. 59.44 2.0238 6 54.24 1.2004 i5 55.5t 1.3257 17. 57- 4 1.4667 28. 58.21 1.6824 73. 59.45 2.0268 9 54.26 1.2026 18 55.52 1.3275 5 57- 5 1 .4688 20 58.22 1.6874 74. 59.46 2.0296 12 54.27 1.2049 21 35.53 1.3292 10 37- 6 1 .4709 4o 58.24 1.6923 75. 59.47 ).0322 ..o343 9.. 5 54.29 1. 2071 12.24 55.54 1.3309 17-15 57. 6 1 .4730 29. 58.25 1.6972 76. 59.48 : 18 54.31 1.2094 27 55.55 1.3326 20 ^7. 7 1-4751 3o 58.27 1 .7046 77- 59.49 : ..o363 21 54.33 r.2ii6 3o 55.56 1.3343 25 57. 8 1.4772 3o. 58.29 1.71x7 78. 59.50 : .0382 24 54.34 1. 2139 33 55.57 i.336o 3o 57. 9 1-4793 3o 58.3i 1.7187 79- 59.51 : .o4oo 27 54.36 I.2l6l 36 55.58 1.3377 35 37. 10 1.4814 3i. 58.33 1.7255 80. 59.52 : .0417 3o 54.36 I.2J83 39 55.59 1 .3394 4o 37.11 1.4835 3o 58.35 1 .732 1 81. 82. 59.52 : 59.53 : .0432 .0446 9.33 54.39 1.2205 12.42 56. 1 .34 1 1 17-45 37.12 1.4855 32. 58.36 1.7387 36 54.41 1.2227 45 56. I 1.3427 5o 37.13 1.4876 3o 58.38 1.7454 83. 59.54 : .o45o 39 54.43 1.2248 48 56. 2 1-3444 55 37-14 1.4896 33. 58. 4o 1.7520 84. 59.55 : .0453 42 54.44 1.2270 5i 56. 3 1.3461 18. 37-i4 1.4916 3o 58.4i 1.7582 85. 59.56 : .0456 45 54.46 1. 2291 54 36. 4 1.3478 10 37.16 1.4956 34. 58.43 1 .7643 86. 59.57 : .0458 48 54.48 i.23i3 57 56. 5 1.3494 20 37.18 1 .4995 3o 58.44 1.7702 87. 59.57 s .0460 9.5i 54.49 1.2334 i3. 56. 6 1.35x0 i8.3o 37.19 i.5o34 35. 58.46 1.7762 88. 59.58 : .o46x 54 54.5i 1.2355 5 56. 7 1.3538 40 37.21 1.5073 3o 58.47 1.7821 89. 59.59 = .0462 57 54.52 1.2376 10 56. 9I 1.3565 5o 37.22 i.Siii 36. 58.49 1.7878 90. 60. 2 .0462 13 rsge98] TABLE XIX. Correction. ii r- Table A. Tai;leB. D 's Horizontal Parallax. Proportional part for Seconds of Parallax. For Min. of Alt. <^ Add. Add. D.A 5 I. 54' i4-35 I 55' "3T35 50' 12. 3f 57' 11.36 58' 59' 60' 8^36 61' S.0"l"2" 7.37^595857 3" 56 4" 55 5" 54 6' 5: 1711 T2 8"|9"| M. S. 12 10. 36 9.36 5i 5o I l4.20 I 3.20 12.2c II .20 10.21 9.21 8.21 7.21 10494847 46 45 44 4; 42 4i 4o 2 1 1 9 2 o i4. 5 I 3. 5 12. t II . 6 10. 6 9. 6 8. 7 7- 7 20 39 38 37 36 35 34 33 32 3i 3o 4 8 3 i3.5i I 2.52 II .52 10.52 9.52 8.53 7.53 6.53 3 29 2 827 26 25 24 2C 22 21 20 5 6 4 o 1 3. 38 1 2.39 II .3^ 10.39 9.39 8.4o 7.40 6.40 4 19 I 817 16 i5 i4 IC 12 11 10 7 4 3 5 6^ i3.26 I 2.26 11.27 10.27 9.27 8.27 7.28 6.28 5 9 8 7 6 5 4 54 53 2 5I I 57 5^ 8 9 2 9 r3.i7 I 2.18 ii.jfc 10.18 9.19 8.19 7.19 6.20 o59 58|57l 56 55 I i3. 6 I 2. 7 11. - 10. 7 9. 8 8. 8 7. « 6. 8 io49|48 47 46 45 44 43 42 4i 4o 2 7 2 12.55 I 1.56 10. 5e 9.57 8.57 7.57 6.58 5.58 20 39138 37 36 35 34 33 32 3i 3o 3 3 12.45 I 1.46 10. 4t 9.46 8.47 7-47 6.48 5.48 3 29 2 827 26 25 24 23 22 21 20 5 4 4 12.36 I 1.36 10.37 9-37 8.37 7.38 6.38 5.39 4o 19 I 817 16 i5 14 i3 12 II 10 7 3 2 5 7 12.27 I 12.18 I 1.27 10.2& 9.28 9.19 8.28 7.29 6.29 5.3o 5o 9 8 7 6 5 4 3 2 I 8 9 1 1.19 10.19 8.20 7.20 6.21 5.21 1 TABLE XIX. Logarithms. 1^ '■< Table C. Cor. for Sec Apparent Altitude of ]) 's Centre. of Par. «Ph Add. / 1 1 / / / ; / / / / / / • "54" S. 5 3o84 510 3o58 5 20 3o33 5 30 3oo9 5 40 2987 5 50 2966 6 2946 6 10 2926 6 20 2908 6 30 6 40 6 50 7 Sec. Cor. 2891 2874 2859 2844 i4 10 3o68 3o4i 3oi6 2993 2971 2950 2930 291 1 2892 2875 2859 2843 2828 I i3 20 3o5i 3o25 3ooo 2977 2955 2934 2914 2895 2877 2860 2843 2827 2813 2 II 3o 3o35 3009 2984 2961 2939 2918 2898 2879 2861 2844 2828 2812 2797 3 9 40 3019 299J 2968 2945 2923 2902 2883 2864 2846 2828 2812 2797 2782 4 8 5o 3oo3 2977 2952 2929 2907 2887 2867 2848 283o 2813 2797 2781 2767 5 6 55 2987 2961 2936 2913 2891 2871 285i 2833 2815 2798 2781 2766 2751 6 5 10 2971 2945 2921 2898 2876 2855 2836 2817 2799 2782 276b 2751 2736 7 ■ 3 1 20 2955 2929 2905 2882 2860 2840 2820 2802 2784 2767 2751 2736 2721 8 I 3o 2939 2918 2889 2866 2845 2824 2805 2786 2769 2752 2736 2720 2706 9 /\n 292J 2907 2891 2897 2882 2866 0H73 o85i 2829 2814 2798 2809 2790 2774 2759 2771 275'^ 2737 2721 2705 2691 2676 2661 56 5o 2858 2842 2835 2820 2793 2778 2756 2741 2738 2723 2721 2706 2706 2690 2690 2675 Sec. Cor. i4 10 2876 285i 2827 2804 2783 2763 2744 2725 2708 2691 2676 2660 2646 I i3 20 2860 2835 2811 2789 2768 2748 2729 2710 2693 2676 2661 2646 263 1 2 II 3o 2844 2820 2796 2774 2752 2732 2714 2695 2678 2661 2646 263 1 2617 3 10 40 2829 2804 2780 2758 2737 2717 2698 2680 2663 2647 263 1 2616 2602 4 8 5o 2813 2789 2765 2743 2722 2702 2683 2665 2648 2b32 2616 2601 2587 5 7 57 2798 2773 2750 2728 2707 2687 2669 265o 2633 2617 2601 2587 2573 6 5 10 2783 2758 2735 2713 2692 2672 2654 2636 2618 2602 2587 2572 2558 7 8 4 20 2767 2743 2720 2698 2677 2657 2639 2621 2604 2588 2572 2557 2543 3o 2752 2728 2705 2683 2662 2642 2624 2606 2589 2573 2558 2543 2529 9 I "58" 4o 5o 2737 2722 2707 2713 2698 2683 2690 2675 2660 2668 2653 2638 2647 2632 2618 2628 2613 2 5q8 2609 2595 2 58o 2591 2577 2562 2574 256o 2545 2558 2544 2543 2529 2528 25i4 25i5 2500 Sec. Cor. i4 2529 25i4 25oo 2486 10 2692 2668 2645 2623 2603 2 584 2 565 2548 253i 25i5 2 5oo 2485 2472 I i3 20 2677 2653 263o 2609 2588 2569 255i 2533 25i6 250I 2485 2471 2457 2 II 3o 2662 2638 2Ci5 2594 2574 2554 2536 2519 2502 2486 2471 2457 2443 3 10 40 2647 2623 2601 2579 2559 2540 2522 2 5o4 2488 2472 2457 2443 2429 4 8 5o 2632 2608 2 586 2 565 2 544 2525 25o7 2490 2473 2458 2443 2428 24i5 5 7 5q 2617 2594 2571 255o 253o 25ll 2493 2476 2459 2444 2429 2414 2401 6 5 4 10 2603 2579 2557 2536 25i6 2497 2479 2461 2445 2429 24i5 2400 2387 7 8 20 2588 2565 2542 2521 250I 2482 2465 2447 243i 24i5 2400 2386 2373 3o 2573 2550 2528 2507 2487 2468 245o 2433 2417 2401 2386 2372 235q 9 IkT 4o 5o 2559 2544 2 53o 2535 2521 2507 25i3 2499 2485 2492 2478 2464 2473 2458 2444 2454 2440 2426 2436 2422 2408 2419 24o5 2391 24o3 2389 2375 2387 2373 2373 2359 2358 2345 2345 233 1 Sec. Cor. 2359 2345 233i 23l7 i4 10 25i5 2492 2470 2450 243o 241 1 2394 2377 236i 2345 233i 23i7 23o4 I i3 20 25oi 2478 2456 2435 2416 2397 238o 2363 2347 2332 23i7 23o3 2290 2 II 3o 2487 2464 2442 2421 2402 2383 2366 2349 2333 23i8 23o3 2290 2276 3 10 40 2472 245o 2428 2407 2388 2869 2352 2335 2819 23o4 2290 2276 2 263 4 8 5o 2458 2435 24i4 2393 2374 2356 2339 2321 23o6 2290 2276 2262 2249 5 7 6i 2444 2421 2400 2379 236o 2342 2325 23o8 2292 2277 2262 2249 2236 5 b 10 2430 2407 2386 2 365 2 346 2328 23l I 2294 2278 2263 2249 2235 2222 7 8 4 3 20 2416 2393 2872 235i 2332 23(4 2297 2280 2265 225o 22J0 2222 2209 30 2402 2879 2358 2338 23i8 23oo 2283 2167 225l 2 236 2222 2208 2195 9 \ TABLE XIX. fPaseoo Correction. I ^ S Table A. Table B. D 's Horizontal Parallax. Proportional part for Seconds of Parallax. For Min. of Alt. ^^ Add. Add. D. 31. 54' 55' 56' 57' 5S' 8.23 59' 7.23 60' 6.24 61' 5.24 S. 0"l"2"3"4"j 59 58 5^56 551 5" 54 6" 7" 8" 9" M. 5i 5o S. c 7 o 12.2 11 .22 10.22 9.22 53 52 10 12. I. 5ii.i3 10. i4 9.14 8.i5 7.i5 6.16 5.16 10 4948474 645 ^A 43 42 4i4 ol 2 5 ?.o 12. J ) 11. 5 10. 6 9. 6 8. 7 7- 7 6. 8 5. 8 20 39 38 37 36|35| 34 33 32 3i 3o i 4 3 3o II. 5t 5 10.58 9.59 8.59 8. 7. 6. 1 5. I 3a 29 2f 3 27 2 625 24 23 22 21 20 s 4o II. 5c ) 10. 5i 9.5i 8.52 7.53 6.53 5.54 4.54 4o 19 I 3 17 I 6i5 i4 i3 12 II 10 I 1 5c 9 i 3 7 6 5 4 3 2 1 9 TABLE XIX. Logarithms. o '<' Table C. >2 i2 Apparent Altitude of j) 's centre. Correction for Seconds of Parallax. «A, Add. / / / / / / lo / / / / / / M. 54 S. 7 3 2841 7 6 2836 7 9 283i 7 12 2827 7 15 2823 7 187 21 7 24 281 1 7 27 7 30 7 33 7 30 Sec. Cor. 2819 2815 2807 2803 2799 2795 i3 10 2825 2821 2816 2812 2808 2804 2800 2796 2791 2787 2783 2780 I 12 20 28 UJ 2805 2800 2796 2792 2788 2784 2780 2776 2772 2768 2765 2 10 3o 2794 2790 2785 2781 2777 2773 2769 2765 2761 2757 2753 2749 3 9 4o 2779 2774 2770 2766 2762 2758 2754 2750 2746 2742 2738 2734 4 7 5o 2764 2759 2755 2751 2747 2743 2739 2735 2731 2727 2723 2719 5 6 55 2748 2744 2739 2735 2731 2727 2723 2719 2716 2712 2708 2704 6 4 3 lO 2733 2729 2724 2720 2716 2712 2708 2704. 2700 2696 2692 2689 7 8 20 2718 2714 2709 2705 2701 2697 2693 2689 2685 2681 2677 2674 I 3o 2703 2699 2694 2690 2686 2682 2678 2674 2671 2667 2663 2659 9 "56" 4o 5o 2688 2673 2658 2684 2669 2654 2679 2664 2649 2675 2660 2645 2671 2656 2641 2667 2653 2638 2663 2649 2634 2659 2645 263o 2656 2641 2626 2652 2637 2622 2648 2633 2618 2644 263o 26:5 Sec. Cor. i3 lO 2643 2639 2635 263 1 2627 2623 2619 261 5 2611 2607 2603 2600 I 12 20 2628 2624 2620 2616 2612 2608 2604 2600 2597 2593 2589 2 586 2 10 3o 2614 2610 2606 2602 2598 2594 2590 2586 2582 2578 2574 257. 3 9 4o 2599 2595 2591 2587 2583 2579 2575 2571 2567 2 563 2559 2556 4 7 5o 2 584 258o 2576 2572 2 568 2 564 256o 2556 2553 2549 2545 2542 5 6 57 2570 2 566 2562 2558 2554 255o|2546 2542 2538 2534 253o 2527 6 4 10 2555 255i 2547 2543 2539 2535 253i 2527 2524 2520 25i6 25i3 7 3 20 a 540 2536 2532 2528 2524 2521 25x7 25i3 25l0 25o6 2502 2499 8 I 3o 2526 2522 25i8 25i4 25lO 25o6 25o3 2499 2495 2491 2487 2484 9 ~5'8" 4o 5o o 25l2 2497 2483 25o8 2493 2479 25o4 2489 2475 25oo 2485 2471 2496 2481 2467 2492 2478 2463 2488 2474 2460 2484 2470 2456 2481 2467 2452 2477 2463 2448 2473 2459 2444 2470 2456 2441 Sec. Cor. i3 10 2469 2465 2461 2457 2453 2449 2445 2441 2438 2434 243o 2427 I 12 20 2454 245o 2446 2442 2438 2435 243 1 2427 2424 2420 2416 24i3 2 10 3o 2440 2436 2432 2428 2424 2421 2417 24i3 24lO 2406 2402 2399 3 9 7 4o 2426 2422 2418 2414 2410 2407 24o3 2399 2396 2392 2388 2385 4 5o 2412 2408 24o4 240D 2396 2393 2389 2385 2382 2378 2374 2371 5 6 59 o 2398 2394 2390 2386 2382 2379 2375 2371 2 368 2364 236o 2357 6 5 10 2384 23So 2376 2372 2368 2365 236i 2357 2354 235o 2346 2343 7 3 20 2370 2366 2362 2358 2354 235i 2347 2343 2340 2336 2332 2329 8 2 3o 2356 2352 2348 2345 234i 2337 2334 233o 2327 2323 23i9 23i6 9 I 6^ 4o 5o o 2342 2328 23i4 2338 2324 23ll 2334 2320 2307 233i 23i7 23o3 2327 23i3 2299 2323 2309 2296 2320 2 3o6 2292 23i6 23o2 2289 23i3 2299 2286 2309 2295 2282 23o5 2291 2278 2302 2288 2275 See. Cor. l3 10 23oi 2297 2293 2290 2286 22S2 2279 2275 2272 2268 2264 2261 I 20 2287 2283 2279 2276 2272 226S 2265 2261 2258 2254 225o 2247 2 3o 2273 2270 2266 2262 2258 2255 225l 2248 2245 2241 2237 2234 3 Ao 2260 2256 2252 2249 2245 2241 2238 2234 223l 2227 2223 2220 4 5o 2246 2243 2239 2235 223l 2228 2224 222T 2218 22l4 2210 2207 5 6 6i 2233 2229 2225 2222 2218 22l4 22II 2207 2 2o4 2200 2196 2193 6 5 10 2219 2216 2212 2209 22o5 2201 2198 2194 2I9I 2187 2i83 2180 7 4 20 2206 2202 2198 2195 2I9I 2187 2l84 2181 2178 2174 2170 2167 8 2 3o 2192 2189 2i85 2182 2I78I2I74 2i7i|2i67 2164 2 J 60 2l57 2i54 9 I P-^geioo] TABLE XIX. Correction. -• -• Table A. Table B. j> 's Horizontal Parallax. Proportional part for Seconds of Parallax. For Min. of Alt. Add. Add. D. 7 M. 3o !34' 55' 56' 57' 58' 8. 07 59' . 06 60' . I 61' 5. I S. 0"1"2"3" 1^5^58 5^56 4" 55 5" 6" 5453 7" 52 8" 57 3" 5o M. S. 6 II. 5 3 10.58 9.598 .59 4o II. 5 ■) 10. 5i 9.51 8 .52 7.536 .535.54 4.54 104948 4746 45 44 43 42 4i io 2 5 5o II. 4 i 10.44 9.458 .45 7.466.465 .47 4.47 20393837, 36 35 34 33 32 3j io i 4 8 II. 3 7 10.38 9.388 .39 7.406.40 5 .41 4.4i 3() 29 2 8 27 lb 25 24 23 22 21 20 5 6 7 2 lO II. 3 I 10.32 9.328 .33 7.336.345 .35 4.35 4o 19 I 8 17 6 i5 i4 i3 12 II !0 2 1 20 I I .2 5 10.26 9.268 • 27 7.286 .28|5 .29 4.3o 5o 9 8 7 6 5 4 3 2 I 9 TABLE XIX. Logarithms. o 2 Table C. ^1 Apparent Altitude of 5 's centre. Correction for Seconds of Parallax. «p: Add. / / / / / / / / / / 1 / M. S. o 7 39 2791 7 42 2787 7 45 2783 7 48 2780 7 51 2777 7 54 2774 7 57 2770 8 2766 8 3 8 6 8 9 8 12 Sec. Cor. 2762 2759 2756 2753 i3 lO 2776 2772 2768 2765 2762 275c; 2755 2751 2747 2744 2741 2738 I I? 20 2761 2757 2753 2750 2747 2743 2739 2736 2782 2729 2726 2723 2 10 3o i745 2741 2737 2734 2731 2727 2724 2721 2717 2714 2711 2708 3 9 4o 2780 2726 2722 2719 2716 2712 2709 2706 2702 2699 2696 2693 4 7 5o 2715 271 1 2707 2704 2701 2697 2694 2691 2687 2b84 268 1 2678 5 6 55 o 2700 2696 2692 2689 2686 2682 2679 2676 2672 2669 2666 2663 6 4 3 10 2685 2681 2677 2674 2671 2667 2664 2661 2657 2654 2b5i 2048 7 8 20 2670 2666 2662 2659 2656 2652 2649 2646 2642 2639 2636 2633 I JO 2655 265 1 2647 2644 2641 2637 2634 263i 2627 2624 2621 2bl8 9 ~5"6 4o 5o o 2640 2626 261 1 2637 2622 2607 2633 2618 2603 263o 2615 2600 2627 2612 2597 2623 2608 2593 2619 2605 2590 2616 2602 2587 2612 2598 2609 2595 2606 2592 2603 2589 Sec. Cor. 2583 258o 2577 2574 i3 10 2596 2592 2589 2586 2583 2579 2575 2572 2569 2566 2563 256o I 12 20 2582 2578 2574 2571 2568 2564 256i 2558 2554 255i 2548 2545 2 10 3o 2 567 2563 2559 2556 2553 2549 2546 2543 2540 2537 2534 253i 3 9 4o 2552 2548 2545 2542 2539 2535 2532 2529 2525 2522 25i9 25lb 4 7 5o 2538 2534 253o 2527 2524 2520 25i7 25i4 25ll 25o8 25o5 2502 5 6 57 o 2523 2519 25i6 25i3 25l0 25o6 25o3 25oo 2496 2493 2490 2487 6 4 10 2509 25o5 25C2 2499 2496 2492 2489 2486 2482 2479 2476 2473 7 8 3 20 24q5 2491 2487 2484 2481 2477 2474 2471 2468 2465 2462 2459 I 3o 2480 2476 2473 2470 2467 2453 2460 2437 2454 245i 2448 2445 9 T8 4o 5o 2466 2452 2438 2462 2448 2434 2459 2445 243 1 2456 2442 2428 2453 2439 2425 2449 2435 2421 2446 2432 2418 2443 2429 24i5 2440 2425 2411 2437 2422 2408 2434 2419 24o5 2431 2416 2402 Sec. Cor. i3 10 2424 2420 2416 24i3 2410 2407 24o4 2401 2397 2894 2391 2388 I 12 20 2410 2406 2402 2399 2896 2393 2390 2387 2383 ^So 2377 2374 2 10 3o 2396 2392 2388 2385 2382 2379 2376 2373 2369 2366 2363 236o 3 9 4o 2382 P.378 2375 2372 2369 2365 2362 2359 2356 2353 235o 2 347 4 7 5o 2368 2364 236 1 2358 2355 235i 2348 2345 2342 2339 2336 2333 5 6 5q o 2354 235o 2347 2344 234i 2337 2334 233i 2328 2325 2322 23i9 6 5 10 234o 2336 2333 233o 2827 2323 23 20 23i7 23i4 23l I 23o8 23o5 7 3 20 2326 2322 23i9 23i6 23 1 3 23lO 2807 23o4 23oi 2298 2295 2292 8 2 3o 23l2 23o8 23o5 2302 2299 2296 2293 2290 2287 2284 2281 2278 9 I "6^ 4o 5o. 2299 2285 2271 2295 2281 2268 2292 2278 2265 2289 2275 2262 2286 2272 2259 2282 2269 2255 2279 2266 2252 2276 2 263 2249 2273 2260 2270 2257 2267 2254 2264 225l Sec. Cor. 2 246 2243 2240 2237 l3 10 2258 2254 225l 2248 2245 2242 2239 2236 2233 223o 2227 2224 I 12 20 2244 2240 2237 2234 223l 2228 2225 2222 2219 2216 22l3 2210 2 'O 3o 223l 2227 2224 2221 2218 22l5 2212 2209 2206 22o3 2 200 2197 3 9 4o 2217 22l4 22II 2208 22o5 2 201 2198 2195 2192 2189 2186 2i83 4 8 5o 2204 2 200 2197 2194 2191 2l8b 2i8d 2182 2179 2176I2173 2170 5 6i 2190 2187 2184 2181 2178 2175 2172 2169 2166 2i63 2160 2l57 6 5 10 2177 2174 217I 2168 2i65 2I6I 2i58 2i55 2l52 2149I2147 2i44 7 8 4 20 2164 2160 2i57 2i54 2l5l 2i4b 2145 2142 2189 2i36|2i33 2i3o 2 3o 2l5l 2l47 2i44 2l4l 2i38 2i35 2l32 2129 2126 2123I2120 2II7 9 I 1 TABLE XIX. [i-^se 101 Correction. X' a Table A. Table B. 5 's Horizontal Parallax. Proportional part for Seconds of Parallax. For Min. of Alt. 's centre. Cor. Sec. of Par. A;S Add. 1 / / / 1 , 1 / / / / / M. 54 S. 13 13 5 1310 1315 13 20 13 25 13 30 13 35 1 3 40 13 45 13 50 253i 13 55 2529 Sec. Cor. i3 2552 255o 2548 2545 2543 2541 2539 2537 535 2533 10 2538 2536 2534 253i 2529 2527 2525 2523 2521 25i9 25i7 25i5 I 12 20 2523 2521 2519 25i7 25i5 25i3 25ll 2509 2507 25o5 25o3 25oi 2 10 3o 2509 25o7 25o5 2502 25oo 2498 2496 2494 2492 2490 2488 2486 3 9 4o 2494 2492 2490 2488 2486 2484 2482 2480 2478 2476 2474 2472 4 7 35" 5o 24S0 2478 2476 2474 \ 472 2470 2468 ,2466 2464 2462 2460 2458 5 6 6 5 2466 2464 2462 2460 458 2456 2454 2452 2450 2448 2446 2444 10 2452 2450 2448 2446 2444 2442 2440 2438 2436 2434 2432 2430 7 3 20 2437 2435 2433 243 1 2429 2427 2425 2423 2421 2419 2417 24i5 8 2 3o 2423 2421 2419 2417 24i5 24i3 241 1 2409 2407 24o5 24o3 2401 y 4o 2409 2407 24o5 24o3 2401 2399 2397 2395 2393 2391 2389 2387 "56" 5o o 2395 2393 2391 23S9 2387 2385 2383 238i 2379 2365 2377 2375 2373 Sec. Cor. i3 238i 2379 2377 2375 2373 2371 2369 2367 2364 2362 236o lO 2367 2365 2363 236 1 2359 2357 2355 2353 235i 235o 2348 2 346 I 12 20 2353 235i 2349 2347 2345 2343 234: 2339 2337 2336 2334 2332 2 ID 3o 2339 2337 2335 2333 233i 2329 2327 2325 2323 2322 2320 23i8 3 9 4o 2326 2334 2322 2320 23i8 23 1 6 23i4 23l2 23X0 23o8 23o6 23o4 4 8 ^ 5o o 23l2 23lO 23o8 23o6 23o4 2302 23oo 2298 2296 2295 2293 2291 5 6 6 5 2298 2296 2294 2292 2290 2288 2286 2284 2282 2281 2279 2277 10 2285 2283 22SI 2279 2277 2275 2273 2271 2269 2267 2265 2263 7 3 20 2271 2269 2207 2265 2263 2261 2259 2257 2255 2254 2252 225o 8 2 3o 2257 2255 2253 225l 2249 2247 2245 2243 2241 2240 2238 2236 9 I 4o 2244 2242 2240 2238 2236 2234 2232 223o 2228 2227 2225 2223 ^ 5o 223o 2228 2226 2224 2222 2220 221S 2216 22l4 22l3 22II 2209 Sec. Cor. i3 2217 22l5 22l3 221 t 2209 2207 2205 2203 2201 2200 2198 2196 lO 2 203 2201 2199 2198 2196 2194 2192 2190 2188 2187 2l85 2l83 I 12 20 2190 2188 2186 2184 2182 2180 2178 2176 2174 2173 217I 2169 2 10 3o 2177 2175 2173 217I 2169 2167 2i65 2i63 2161 2160 2i58 2i56 3 Q 4o 2i63 2l6l 21^9 2i58 2i56 2i54 2l52 2i5o 2i48 2l47 2145 2i43 4 s ^ 5o o 2l5o 2i48 2i46 2i45 2 2 143 2l4l 2 1 39 2 I 37 2i35 2i34 2l32 2l3o 5 6 6 5 2137 2i35 2i33 2l3l 129 2127 2125 2123 2122 2120 2118 2117 10 2124 2122 2120 2II8 2116 2Il4 2II2 2H0 2108 2107 2io5 2103 7 4 20 2111 2109 2107 2I05 2I03 2I0I 2099 2097 2095 2094 2092 2090 8 3 3o 2098 2096 2094 2092 2090 2088 2086 2084 2082 2081 2079 2077 y I 4o 2o85 2o83 2081 2079 2077 2075 2073 2071 2069 2068 2066 2064 "fc 5o o 2072 2070 2068 2066 2064 2062 2060 2o58 2o56 2o55 2o53 205l Sec. c Cor. i3 2059 2o57 2o55 2o53 205l 2049 2o47 2045 2o43 2042 2040 2o38 10 2o46 2o44 2042 2o4o 2o38 2o36 2o34 2o32 203l 2029 2027 2026 I 12 20 2o33 203l 2029 2027 2025 2023 202 1 2019 2018 2016 20l4 20l3 2 10 3o 2020 2018 2016 20l4 2012 2010 2008 2006 2005 2003 2001 2000 3 9 4o 2007 2005 2003 2002 2000 1998 1996 1994 1992 1990 1989 1987 4 8 Tf 5o 1994 1992 1990 1989 1987 1985 1983 I981 1979 1977 1976 1974 5 6 7 5 1 98 1 1979 1977 1976 1974 1972 1970 1968 1967 1965 1963 1962 10 1969 1967 1965 1963 1961 1959 1957 1955 1954 1952 1950 1949 7 8 4 3 20 1956 104 1952 1961 1949 1947 1945 1943 1942 iq4o 1938 1937 •^0 1943 1941 [939 1938 1936 I9M 1932 1930 1929 1927 1925 1924 y 2 Page 108] TABLE XIX. 1 > Correction. ■^ i Table A. TableB. ]) 's Horizontal Parallax. Proportional part for Seconds of Parallax. For Mill. of Alt. <'^ Add. Add. D. i4 M. o 54' 55' 56' 57' 58' 6.i3 59' 5.i5 GO' 61' S. 0" 1" 5^56 2" 55 3" 4" 5453 5" G" 5I57 7" 55 8" 49 9" 4F M. S. 10. 6 9. 88 • 9 7. II 4.17 3.18 10 10. 5 9. 7a • 9 7. II 6.i3 5.i5 4.16 3.18 10 47 46 45 4443 4 242 4i 40 39 2 20 lo. 5 9. 7a • 9 7. II 6.i3 5.14 4.16 3.18 20 38 57 36 3534 33|32 3i 3o 29 4 3o 10. 5 9. 7a • 9 7. II 6.i3 5.14 4.16 3. 18 3o 28 27 26 25 24 2 3 22 21 20 19 5 6 4o 10. 5 9. 7a • 9 7. II 6.i35.i5 4.16 3.18 40 18 7 16 i5 i4 I 3 12 II II 10 5o 10. 5 9. 78. 9 7. II 6.J35.I5 4.17 3.19 5o 9 8 7 6 5 41 3 9 I 8 9 TABLE XIX. Logarithms. o^ Tablk C. Apparent Altitude of D 's centre. Cor. Sec. of Par. «;ih Add. / / / / / / / / / , / M. S. o 14 14 5 1410 14 15 14 20 14 25 14 30 14 35 14 40 14 45 14 50 14 55 2 5o8 Sec. Cor. 18 2527 2525 2523 2521 2520 25i8 25i6 25i4 25i3 25ll 2509 10 2bi3 25ll 2509 2507 25o6 25o4 2502 25oo 2499 2497 24q5 2494 I 12 20 2499 2497 249b 2493 2492 2490 2488 2486 2485 2483 2481 2480 2 10 3o 2484 2482 2480 2478 2477 2475 2473 2471 2470 2468 2466 2465 3 9 4o 2470 2468 2466 2464 2463 2461 2459 2457 2456 2454 2452 245i 4 7 ~5T 5o o 24bb 2454 2452 2450 2449 2447 2445 2443 2442 2440 2488 2487 5 6 6 5 2442 2440 2438 2436 2435 2433 243 I 2429 2428 2.426 2424 2428 10 2428 2426 2424 2422 2421 2419 2417 24 1 5 24x4 2412 2410 2409 7 3 20 24i3 2412 2410 2408 2406 24o5 24o3 2401 2400 2898 2396 2895 8 2 3o 2399 2398 2896 2394 2892 2891 2389 2887 28S6 2384 2882 2881 9 4o 23«b 2384 2382 238o 2378 2377 2375 2873 2872 2870 2868 2867 1j6 bo 2871 2870 2368 2366 -^ 364 2363 236i 2359 2858 2856 2354 2358 Sec. Cor. t3 2358 2356 2354 2352 35i 2349 2347 2845 2844 2842 2840 2889 10 2344 2342 2340 2 338 2337 2335 2333 233i 2880 2828 2826 2825 I 12 20 233o 2328 2326 2324 2323 2321 23 I Q 2817 2816 23i5 2818 2812 2 10 3o 23i6 23i4 23i3 23ll 2809 23o8 2806 23o4 2808 2801 2299 2298 3 4o 2302 23oo 2299 2297 2295 2294 2292 2290 2289 2287 2285 2284 4 8 ^ bo o 2289 2287 228b 2283 2282 2280 2278 2276 2275 2274 2272 2271 5 6 6 5 2275 2273 2272 2270 2268 2267 2265 2268 2262 2260 2258 2257 10 2261 2259 22b8 2256 2254 2253 225l 2249 2248 2246 2244 2248 7 3 20 2248 2246 224b 2243 2241 2240 2238 2286 2235 2288 2281 2280 8 2 Jo 2234 2282 2 23 I 2229 2227 2226 2224 2222 2221 2219 2217 2216 9 I 4o 2221 2219 2218 2216 22l4 22l3 22II 2209 2208 2206 2204 2 203 ^ bo 2207 22o5 2204 2202 2 200 2199 2197 2195 2194 2198 2I9I 2190 Sec. Cor. i3 2194 2192 2191 2189 2187 2186 2184 2182 2181 2179 2177 2176 10 2181 2179 2178 2176 2174 2173 217I 2169 2168 2166 2164 2168 I 12 20 2167 2ibb 2164 2162 2160 2159 2 I 57 2i55 2i54 2i53 2l5l 2l50 2 10 3o 2i54 2162 2l5l 2149 ai47 2 1 46 2i44 2l42 2l4l 2l4o 2i38 2187 3 9 4o 2l4l 2139 21^8 2i36 2 1 34 2i33 2l3l 2129 2128 2126 2124 2123 4 8 ^ bo 2128 2126 2125 2123 2121 2120 2II8 21 16 2Il5 2Il3 2III 2II0 5 6 6 5 2Il5 2Il3 21 12 2II0 2108 2107 2I05 2io3 2102 2100 2098 2097 lO 2I0I 2099 2098 2096 2094 2093 2091 2089 2088 2087 2o85 2084 7 4 20 2088 2086 208b 2o83 2081 2080 2078 2076 2075 2074 2072 2071 8 3 3o 2075 2073 2072 2070 2068 2067 2o65 2o63 2062 2061 2059 2o58 9 I 4o 2062 2060 2059 2o57 2o55 2o54 2052 2o5o 2049 2o48 2046 2o45 1 &7 bo o 2049 2047 2o46 2o45 2043 2042 2o4o 2088 2087 2o35 2o33 2082 Sec. Cor. 2o36 2o34 2o33 2o32 2o3o 2029 2027 2025 2024 2022 2020 2019 i3 lO 2024 2022 2021 2019 2017 2016 20l4 2012 2CII 2010 2008 2007 I 12 20 201 1 2009 2008 2006 2004 2003 2001 1999 1998 1997 1995 1994 2 10 Jo 1998 199b 199b 1993 I99I 1990 1988 1986 1985 1984 1982 1981 3 9 40 1985 1983 1982 1980 1978 1977 1975 1978 1972 1971 1969 1968 4 8 ~67 bo 1972 i960 1970 1969 1968 1966 i9bb 1963 I 96 I i960 1959 1957 1956 b 6 7 5 1958 1957 ,955 1953 1952 1950 1948 1947 1946 1944 1943 lO 1947 194b 1 9 14 1942 1940 19^9 1987 193b 1934 1933 1981 1980 7 4 20 1935 1933 1982 1930 1928 1927 1925 1928 1922 1921 I9I9 1918 8 Jo 1922 1920 1919 I9I7 r9i5 I9I4 1 I9I2 1910 1909 1908 190b 1903 9 TABLE XIX. Correction. [Page 109 9 2126 2123 2120 2II7 21l4 2112 2110 2108 2io5 3 9 4o 2 I 21 2119 2116 2Il3 2110 2107 2I04 2101 2099 2097 2094 2092 4 8 '5^ 6o O 2108 2106 2I03 2 100 2097 2o84 2094 2091 2088 20S6 2084 2081 2079 5 6 6 5 4 2095 2092 2089 2086 2081 2078 2075 2073 2071 2068 2066 10 2082 2079 2076 2073 2071 2068 2o65 2062 2060 2o58 2o55 2o53 8 20 2069 2066 2o63 2060 2o58 2o55 2052 2049 2047 2045 2042 2o4o 3 3o 2o56 2o53 2o5o 2o47 2045 2042 2o39 2o36 2o34 2o32 2o3o 2027 y X 4o 2()43 2o4l 2o38 2o35 2o32 2029 2026 2023 2021 2019 20x7 20l4 &)" 5o o 2()3o 2028 2025 2022 2020 2017 20T4 20II 2009 2006 2004 2001 Sec, Cor.| 2017 201 5 2012 2009 2007 2004 2001 1998 1996 1993 1991 1988 i3 10 20o5 2002 1999 1996 1994 I99I 1988 I9S5 I9S3 1981 1979 1976 I 12 2() 1992 1989 1986 1983 1981 1978 1975 1972 1970 1968 1966 1903 2 10 3c. 1979 1977 1974 1971 1909 1966 1963 i960 1958 1955 1953 1950 3 9 4o 1966 1964 1961 1958 1955 1953 1950 1947 1945 1942 1940 19^7 4 8 "67 5o o 1954 I 95 I 1948 1945 1943 1940 1937 1934 1932 1930 1928 1925 5 6 ■7 5 4 1 94 1 1939 1936 1933 1931 1928 1925 1922 1920 I9I7 1915 I9I2 lO 1928 1926 1923 1920 1918 1916 1912 1909 1907 1905 1902 1900 7 8 20 I9I6 I914 1911 1908 1906 1903 1900 1897 1895 1892 1890 1887 3o 1903 1901 1898 1895 1893 1 1890 1887 i885 i883 1880 1878 ; 1875 y 2 P'tgeiio] TABLE XIX. Correction. | ^ c Table A. TableB. < S Proportional part for Seconds For Min. c.-" J) 's Horizontal Parallax. of Parallax. of A.lt. <^ Add. Add. 1 D. 17 M. 54' ! 55' 1 5G' 57' 7.19 58' 59' 60' 61' S. 0"1" 56 55 2" 3" 4" 54535^ 5' 5^ G" 5^ 7" 49 S" 48 9" 47 M. 0' S. 10.11 9.14s •17 6.22 5.24 4. -27 3.3o 10 10.12 9.ibfc .18 7.20 6.23 5.26 4.28 3.3i 10 46 i6 454. 143 42 4i 4o 39 38 20 10. i3 9.i6i .19 7.21 6.24 5.27 4.3o 3.32 20 37 36 35 34|33 3:j 3i 3o 29 28 4 :3o 10. i4 9.17^ .20 7.23 6.25 5.28 4.3i 3.34 3o 27 26 25 2 3 24 2,- 22 21 20 19 5 6 7 40 10. i5 9.1st .21 7.24 6.27 5.29 4.32 3:35 4o 18 17 161 5i4 IC 12 11 10 9 78" bo 10.16 9.i9t .22 7.25 6.28 6.29 5.3i 5.32 4.34 4.35 3.37 3.38 5o 8 56 7 55 6 5"4 5 5 5 3 5^ / 5" 3 "5o 2 4q I 48 47 8 9 10.18 9.21 t .23 7.26 10 10.19 9.22 f: i.25 7.28 6.3: 5.34 4.37 3.40 10 47 46 4544143 42I41 4o 39 38 •2. 20 10.20 9.23t .2b 7.29 6.32 5.35 4.38 3.4i 20 37 36 3534I33 3; 3i 3o 29 28 4 3o 10.22 9.25 i i.28 7.3. 6.34 5.37 4.40 3.43 3o 28 27 26 2 524 28 22 21 20 19 5 6 7 4o 10.23 9.26 J i.29 7.32 6.36 5.39 4.42 3.45 40 18 17 161 5i4 i3 12 II 10 9 bo 10.24 9.28t i.3i 7-34 6 37 5.40 4.44 3.47 5o 9 8 7 6 5 4 3 2 I 8 a TABLE XIX.- Logarithms. | ° i TaelkCI ^4 Cor Rpc Apparent Altitude of ]) 's centre. of Par, «a- Add. 1 / f / / / / / / / / 1 / M. '54 S. 17 1710 17 20 17 30 1 7 40 17 50 18 1810 18 20 18 30 2450 18 40 2448 18 50 2446 Sec. Cor. l3 2471 2469 2466 2464 1462 2459 2457 2455 2452 10 2457 2454 2452 2449 2447 2444 2442 2440 2438 2436 2434 2431 I 12 20 2443 2440 2438 2435 2433 2430 2428 2426 i4i4 2422 2420 2417 2 10 3o 2429 2426 2424 2421 2419 2416 2414 2412 2410 2408 2406 24o3 3 9 4o 24i5 2412 2410 2407 24o5 2402 2400 2898 2896 2894 2892 2889 4 7 "55" 5o 2401 2398 2396 2393 2391 2388 2386 2384 2882 2368 2880 2878 2875 5 6 6 5 2387 2384 2382 2379 2377 2374 2372 2870 2366 2364 2861 10 2373 2370 2368 2365 2363 236o 2358 2356 2354 2352 235o 2 348 7 3 20 2359 2356 2354 235i 2349 2346 2344 2342 2340 2338 2336 2334 8 2 3o 2345 2342 2340 2337 2335 2333 233i 2829 2826 2024 2822 2820 9 4o 233r 2329 2326 2324 2322 2319 23 1 7 23i5 23l2 2810 2808 23o6 56 5o 23i7 23i5 23l2 23ro 23o8 23o5 23o3 2801 2299 2297 2295 2298 Sec. Cor. l3 23o3 23oi 2298 2296 2294 2291 2289 2287 2285 2283 2281 2279 10 2290 2287 2285 2282 2280 2278 2276 2274 2271 2269 2267 2265 I 12 20 2276 2274 2271 2269 2267 2264 2262 2260 2258 2256 2254 2252 2 10 3o 2262 2260 2257 2255 2253 225o 2248 2246 2244 2242 2240 2238 3 ^ 4o 2249 2247 2244 2242 2240 2237 2235 2233 2281 2229 2227 2225 4 8 "57" 5o 2235 2233 223o 2228 2226 2 2 23 22 21 2219 2217 22l5 22l3 22II b 6 6 5 2222 2220 2217 22l5 22l3 2210 2208 2206 2204 2202 2200 2198 10 2208 2206 22o3 2201 2199 2197 2195 2193 2190 2188 2186 2i85 7 3 20 2195 2193 2190 2188 2186 2l83 2181 2179 2177 2175 2x78 2171 8 2 3o 2182 2180 2177 2175 2173 2170 2168 2166 2164 2162 2160 2i58 9 I 40 2168 2166 2i63 2161 2 I 59 2i57 2i55 2i53 2l5l 2149 2l47 2i45 Is" 5o 2i55 2i53 2l5o 2i48 2 1 46 2i43 2l4l 2189 2187 2i35 2i33 2l3l Sec. Cor. i3 2142 2l4o 2i37 2i35 2i33 2i3o 2128 2126 2 124 2122 2120 2118 10 2129 2127 2124 2122 2120 2II7 2Il5 2Il3 2111 2109 2107 2I05 I 12 20 2116 2114 211 1 2109 2107 2I04 2102 2 100 2098 2096 2094 2092 2 10 3o 2102 2100 2097 2095 2093 2091 2089 2087 2o85 2o83 2081 2079 3 9 4o 2089 2087 2084 2082 2080 2078 2076 2074 2072 2070 2068 2066 4 8 "5^ 5o 2076 2074 2071 2069 2067 2o65 2o63 2061 2059 2057 20f 5 2o53 2o4o 5 6 6 5 2o63 2061 2o58 2o56 2o54 2o52 2o5o 2048 2046 2o44 ■xoi 10 2o5o 2o48 2046 2044 2042 2039 2o37 2o35 2o33 2o3r 2029 2027 7 4 3 20 2o37 2o35 2o33 so3i 2029 2026 2024 2022 2020 2018 2016 20l5 8 3o 2025 2023 2020 2018 2016 20l3 20II 2009 2007 2005 2003 2002 y 4o 2012 2010 2007 2oo5 2003 2001 1999 1997 1995 1993 I99I 1989 6^ 5o 1999 1997 1994 1992 1990 19S8 1986 1984 1982 1980 1978 1976 Sec. Cor. l3 1986 1984 1981 1979 1977 1975 i973 1 9-' I 1969 1967 1965 1964 10 1973 1971 1969 1967 1965 1962 i960 1958 1956 1954 1932 1951 I 12 20 1 96 1 1959 1956 19^4 1952 1950 1948 19-16 1944 1942 1940 1988 2 ID 3g 1948 1946 1943 1941 1939 1937 1935 1933 1981 1929 1927 1926 3 9 4o 1935 : 1933 1 93 1 1929 1927 1925 1923 1921 1919 1917 1915 1918 4 8 eT bo .923 1921 1918 1916 1904 1914 1902 I9I2 1899 1910 1908 1906 1904 1902 I90I b 6 7 5 4 1910 1908 1906 1897 1895 1894 1892 1890 1888 10 1898 1896 1893 1891 1889 1887 i885 i883 1881 .879 1877 1876 7 8 20 i885 1 883 1881 1879 1877 1875 1873 1871 1869 1867 i865 i8b3 .3 3o 1873 1871 , >868 1866 1864 1862 i860 i858 i856 i854 i8b2 i85i 9 2 TABLE XIX. Correction. [Page in ^ r. Tabli: a. TableB. •< 4) Proportional part for Seconds For Min. a. M D '3 Horizontal Parallax. of Parallax. of Alt. <'^ Add. Add. D. 19 M. 54' 55' 56' 57' 58 6.3 59' 60' 95.424.46 61' 3.49 S. 0" 56 1" 5'5 2" 3 545 '4"}5 3 5"^ 15 'G" 1 5o 7// 49 d"9" 48'48" M. 1 S. 10.26 9.29 S .32 7.36 10 10.28 9.3it .34 7-37 6.4 i5.44'4.47 J.5i 10 47 46 45 44!43|4 241 4o 39138 2 20 12.29 9.33i .36 7.39 6.4 35. 46'4. 49 J. 53 20 37 36 35 3 4 33 3 2J1 3i 3029 4 '^ 3o 10. 3i 9.34i .38 7.41 6.4 55. 484.51 J.bb 3o 28 27 26 2 5 24 2 J 22 21 20 19 5 6 7 4o 10.33 9.366 .3q 7-43 6.4 65.5o|4.53 3.57 40 18 17 i6i5li4|i 4 1-3 12 11 10 20 5o 10.34 9.38 6 .41 7.45 6.4 85.52 4.56 J. 59 5o 9 55 8 54 7 53 5 6 5 4 3 2 5i 5o49 2 48 I 4^ 47 9 2 10.37 9-41 6 ■ 44 7.48 6.5 I 5.55 4.59 4. 2 10 10.39 9.43 6 .46 7.5o 6.5 45.57 5. 1 4. 5 10 4b 4b 444342 4 1 40 39 38 37 2 20 10. 4i 9-456 .46 7.52 6.5 65.59 5. 3 4. 7 20 36 35 34 33 33 3 2J1 3o 29 28 4 1 3o 10.43 9.476 .50 7.54 6.5 86. 2 5. 5 4. 9 3o 27 26 25 2 4 2-3 22I21 20 •9 18 5 1 4o 10.45 9.496 .52 7.56 7- 06. 4 5. 8 4.12 40 17 17 161 ^1' 3 12 1 1 10 9 7 5o 10.47 9.516 .55 7-59 7- 26. 6 3. 10 4.14 5o 8 7 6 3 2 2 1 8 9 2 2 TABLE XIX. Logarithms. c ^ Table C. W^rt Cor. Sec. Apparent Altitude of D 's Centre. of Par. «a, Add. ■ / / / / / / / / / 1 / 1 M. 31 S. 19 19 10 19 20 19 30 19 40 19 50 20 20 10 20 20 20 30 20 40 20 50 Sec. Cor. 2445 2443 2441 2439 2437 2435 2433 243i 2429 2427 2426 2424 12 10 243o 2428 2427 2425 2423 2421 2419 2417 24i5 24i3 2412 2410 I II 20 2416 2414 2412 2410 2408 2407 24o5 24(33 2401 2399 2398 2396 2 3o 2402 2400 2398 2396 2J94 239J 2391 2389 2387 2385 2384 2382 3 8 40 2388 2386 2384 2382 2J80 2J79 2377 2J7b 2373 2371 2370 2368 4 6 ^5" 5o 2374 2372 2370 2368 2366 2353 2 36 5 2JbJ 236i 2359 2357 2344 2356 2342 2354 234 1 5 6 5 4 236o 2358 2357 2355 235i 2349 2335 2347 2346 10 2347 2345 2343 234 1 2339 2337 2333 2332 233o 2328 2327 7 2 20 2333 233 1 2329 2327 2325 2323 2321 23i9 23i8 23i6 23i4 23i3 8 I 3o 23i9 23i7 23i5 23i3 23ll 23lO 23o8 23o6 23o4 2302 23oi 2299 9 4o 23o5 2jo3 23oi 2299 2297 2296 2294 2292 2291 2289 2287 2286 "56" bo 229a 2290 2288 2286 22S4 2282 2280 2278 2277 2275 2273 2272 Sec Cor. 2278 2276 2274 2272 2270 2269 2267 2265 2264 2262 2260 2259 12 10 2264 2262 2261 2259 2257 2255 2253 225l 22 5o 2248 2246 2245 I 11 20 225l 2249 2247 2245 2243 2242 2240 2238 2 236 2234 2233 223l 2 9 3o 2237 2235 2233 223l 2229 2228 2226 2224 2223 2221 2219 2218 3 8 4o 2224 2222 2220 2218 2216 22l5 22l3 22II 2210 2208 2206 2205 4 7 ^ DO 2210 2208 2207 22o5 2 203 2201 2199 2197 2196 2194 2192 2191 5 6 5 4 3 2197 2195 2193 219I 2189 2188 2186 2184 2i83 2181 2179 2178 10 2l84 2182 2180 2178 2176 2175 2173 2171 2170 2168 2166 2i65 7 20 2170 2168 2167 2i65 2i63 2161 2159 2i57 2i56 2i54 2l52 2l5l 8 I 3o 2 I 57 2i55 2i53 2l5l 2149 2i48 2i46 2i44 2143 2l4l 2139 2i38 9 4o 2i44 2l42 2l40 2i38 2i36 2i35 2i33 2l3l 2i3o 2128 2126 2125 "58" 5o 2i3o 2128 2127 2125 2123 2122 2120 2118 2117 2Il5 2Il3 2112 Sec. Cor. 2II7 2Il5 2Il4 2II2 2110 2109 2107 2io5 2Io4 2102 2 100 2099 12 10 2I04 2102 2IOI 2099 2097 2095 2093 2091 2090 2088 2087 2086 I II 20 2091 2089 2088 20S6 2084 2082 2080 2078 2077 2075 2074 2073 2 9 Jo 2078 2076 2075 2073 2071 2069 2067 2o65 2064 2062 2061 2060 3 8 40 2o65 2o63 2062 2060 2o58 2o56 2o54 2052 205l 2049 2o48 2047 4 7 5? 5o 2052 2050 2049 2047 2045 2043 204l 2039 2o38 2o36 2o35 2o34 5 6 6 4 3 2039 2037 2o36 2o34 2032 2o3i 2029 2027 2026 20^4 2022 2021 10 2026 2024 2023 2021 2019 2018 2016 20l4 20l3 2011 2009 2008 8 20 20 1 4 2012 2010 2008 2006 2003 2003 2001 2000 1998 1996 1995 3o 2001 1999 1997 1995 1993 1992 1990 1988 1987 1985 1984 1983 y 4o 1988 198b 198b 1983 1981 1979 1977 1975 1974 1972 1971 1970 "6^ bo 1975 1973 1972 1970 1968 1967 1965 1963 1962 i960 1958 1957 Sec. Cor 1963 I 96 I 1959 1957 1955 1954 1952 1950 1949 1947 1946 1945 12 10 I9D0 1948 1940 1944 1942 1941 J9J9 1937 1936 1934 1933 1932 I II 20 1937 1935 I9J4 1932 1930 1929 1927 1925 1924 1922 1920 1919 2 Jo 192b 1923 I 92 I I9I9 1917 1916 1914 I9I2 I9II 1909 1908 1907 J 8 40 I9I2 I9IO 1909 1907 I90D 1904 1902 1900 1899 1897 1895 1894 4 7 67 5o 1900 1898 1887 i885 1896 1884 1894 1882 1892 I89I 1889 1887 1886 1884 i883 1882 5 6 6 5 3 1880 1879 1877 1875 1874 1872 1871 1870 10 1875 1873 187I 1869 1867 1866 1864 1862 1 86 1 i859 i858 i857 / « 20 1862 i860 .859 i857 i8b5 i854 i852 i85o 1849 1 847 i846 1845 2 3o i85o 1 848 1847 1845 1843 1842 i84o 1 838 i837 i835 i834l i833 9 1 ij f Page 112] TABLE XIX. Correction. D. M. 23 24 D 's Horizontal Parallax. 54' 10.49 10. 5i 10.53 10.55 10.58 II . 9.53 9.55 9.57 10. 10. 7 10. 9 10.12 10. 1 10.17 10. 19 10.23 10.26 10.29 10. 3i 10.34 10.37 10.40 10.43 10.46 10.49 10.52 10.55 5G' 8.57 8.59 9' 9- 4 9. 6 9. 9 9' 9.14 9.16 9.19 9.21 9.24 9.28 9.31 9.33 9-36 9.39 9 -42 9-45 8. 3 8. 6 8. 8 8. 10 8.i3 8.i5 8.23 8.26 8.29 8.33 8.35 8.38 8.41 8.44 8.47 8.5o 8.53 8.56 9. o 9. 3 9. 6 58' 59' GO' 61' i5 7-17 7.20 7.22 7.25 7.28 7.3i 7-34 7.37 7.40 7.43 7-46 7-49 7.52 7T55 7.59 8. 2 8. 5 8. 8 8.12 .16 .14 Table A. Proportional part for Seconds of Parallax. Add. Table B For Min of alt. Add. 1// -2" 3" 4" 6" 5o 8" 19" M. S. Explanation of Table XIX. This table consists of two parts, for finding a correction of the moon's distance and a loga- rithm corresponding : they are both in the same page from the beginning of the t^able to the altitude of 21 degrees, after which the correction is on the left hand page, and the logarithm on the right, both being found at the same opening of the book, in the following manner. To find the Correction of Table XIX. 1. Enter the table marked Correction, and find in the side column the moon's apparent altitude, or the altitude next less, if there be any units of miles in the altitude ; opposite to this, and under the minutes of the moon's horizontal parallax, will be the approximate cor- rection. 2. Enter table A, abreast of the approximate correction, and find the seconds of the moon's horizontal parallax, viz. the tens of seconds at the side, and the units at the top, under the latter, and opp(5site the former will be the correction of table A. 3. Enter table B, abreast of the approximate correction, and find the units of miles in the moon's apparent altitude (neglected above), opposite to which will be a number of seconds, which, being added to the corrections found from table XIX. and from table A, will give the sought correction. To find the Logarithm of Table XIX. Enter the table marked Logarithms, in the column titled at the top with the degrees and minutes nearest to the moon's apparent altitude, and find the logarithm corresponding to the moon's horizontal parallax in the side column, or the next less parallax, if there be units of seconds in it. Abreast of this in the table C, opposite the units of seconds of par- allax neglected, will be a correction, to be added to the former logarithm, to obtain the logarithm sought. It was observed in a former part cf his work, that in fij.mg these tables so as to render the corrections of the tables A, B, C, additive, it had been found necessary to make the great- est corrections correspond to 0" of parallax and 0' of altitude, so that ichen you find the ex- act parallax and altitude in the side and top columns of table XIX. it icill still be necessary to refer to the tables .4, B, or C, to take out the corrections corresponding to 0" of parallax or 0' of altitude. This is evident from the inspection of the tables, but it was proper to make this remark as a caution to prevent mistakes. To illustrate these rules, the following ex- amples are given, in which all the corrections are put down and added together; but after a little practice it will be very easy to take the numbers from the table by inspection and add them together without the trouble of writing them down separately. TABLE XIX. [Page 1:3 Logarithms. s i Table C. k| Cor. Sec -"^ rt Apparent Altitude of 5 's centre. of Par. '^■Ch Add. / / / 1 / / / / / /[ / 1 1 M. T4 S. 21 2120 214022 22 '-20 22 40 23 C 23 20 23 40 24 C 24 2C 24 40 2891 Sec. Cor. 12 2422 2419 2416 24i3 2410 2407 2404 2401 2399 2896 2894 10 2408 24o5 2402 2399 2896 2893 2890 2887 2385 2382 2880 2877 I II 20 2394 2391 2388 2385 2382 2879 2876 2878 23-1 2368 2 366 2363 2 9 3o 238.) 2377 2374 2371 2368 2365 2862 2860 2357 '2355 2352 235o 3 H 4o 2 366 2363 236o 2357 2354 2352 2349 2346 2344 2341 2889 2325 2336 4 6 T-r 5o 2352 233o 2349 2335 2346 2343 2340 2338 2335 2882 233o 2327 2822 5 6 5 4 2332 2329 2326 2824 2321 2818 2816 2818 2811 2808 10 2325 2322 23i9 23i6 23i3 2810 2807 2804 2802 2299 2297 2294 7 2 20 23ll 2j0« 23o5 2302 2299 2296 2293 2291 2288 2286 2284 2281 8 I 3o 2297 2294 2291 2288 2285 2283 2280 2277 2275 2272 2270 2267 9 4o 2284 2281 2278 2275 2272 2269 2266 2268 2261 2 258 2 256 2253 l6 5o 2270 2267 2264 2261 2258 2256 2253 2250 2248 2245 2243 2240 Sec. Cor. 12 2257 2253 2250 2247 2244 2242 2289 2286 2234 2281 2229 2226 10 2243 2240 2237 2234 223l 2229 2226 2228 2221 2218 22IO 22l3 I II 20 2229 2226 2223 2220 2217 22l5 2212 2210 2207 22o5 2203 2200 2 9 3c, 2216 22l3 2210 2207 2204 2202 2199 2196 2194 2191 2189 2186 3 8 4o 2 2o3 2200 2197 2194 2191 2188 2i85 2l83 2180 2178 2176 2178 4 7 ^ 5o 2189 2186 2l83 2180 2177 2.75 2172 2170 2167 2 1 54 2i65 2l5l 2i63 2149 2160 2i46 5 6 5 4 2176 2173 2170 2167 2164 2162 2i59 2i56 10 2i63 2160 2i57 2i54 2l5l 2149 2i46 2143 2l4l 2i38 2i36 2i33 7 3 20 2149 2i46 2i44 2l4l 2i38 2i35 2182 2l3o 2128 2125 2123 2120 8 I 3o 2i36 2i33 2l30 2127 2 124 2122 2119 2117 2Il5 2112 2no 2107 9 4" 212.3 2120 2II7 2Il4 2III 2109 2106 2104 2IOI 2099 2097 2094 "58" bo 21 10 2107 2I04 2I0I 2098 2o85 2096 2098 2091 2078 2088 2086 2078 2084 2071 2081 2068 Sec. Cor. 12 2097 2094 2091 2088 2o83 2080 3075 10 2084 2081 2078 2075 2072 2070 2067 2o65 2062 2060 2o58 2o55 I II 20 2071 2068 2065 2062 2o59 2o57 2o54 2052 2049 2o47 2045 2042 2 9 3c 2o58 2o55 2o52 2049 2046 2044 204 1 2089 2o36 2o34 2082 2029 3 8 4o 2o45 2042 2o39 2o36 2o33 2o3l 2028 2026 2028 2021 2019 2016 4 7 59" 5o 2o32 2029 2026 2023 2020 2018 20l5 20l3 2010 2008 2006 2oo3 5 6 6 4 2019 2016 20l3 2010 2008 2006 2oo3 2000 1998 1995 1993 '99' 10 2006 2oo3 2001 1998 1995 1993 1990 1987 1985 T982 1980 1978 7 8 3 ?o 1993 1990 1088 1985 1982 1980 1977 1975 1972 1970 1968 1965 2 3o 1981 1978 1975 1972 1969 1967 1964 1962 1959 1957 1955 1953 y 4o 1968 1955 1962 1959 19^7 19^4 1952 1949 1947 1944 1942 1940 "6^ 60 1955 1952 1950 1947 1944 1942 1939 1926 1937 1924 1934 1982 1980 1927 Sec. Cor. 12 1943 1940 1937 1934 1 93 1 1929 1921 1919 1917 1915 10 1930 1927 1925 1922 1919 1917 I9I4 I9I2 1909 1907 1905 1902 I II 20 I9I7 1914 I9I2 1909 1906 1904 I90I 1899 1896 1894 1892 1890 2 10 3o 1905 1902 1900 1897 1894 1892 1889 1887 1884 1B82 1880 1877 3 8 4o 1892 1889 1887 1 884 1881 1879 1876 1874 1871 1869 1867 ib65 4 7 61 5.) 1880 1877 1 865 1875 1862 1872 1859 1869 "i85f 1867 1864 1862 1859 1857 i855 1843 i853 i84o 5 6 ft 5 1 868 1 854 i852 i85o 1847 1845 10 1 855 i852 i85o 1847 1844 1842 1889 1887 i834 1882 i83o 1828 7 3 20 i843 i84o i838 1 835 i832 i83o 1827 1825 1822 1820 1818 1816 8 2 3o i83r 1828 1 1825 1822 1 1820 1817 i8i5 i8i3 1810 1808 1 1806 i8o3 9 I 15 Page 114] TABLE XIX. Correction. it Table A. Tab leB. ]) 's Horizontal Parallax. Proportional part for Seconds of Parallax. For Min. of alt. <;'=* Add. Add. D ^5 ai. 54 55' 56' 57' 58' 59' CO' 61' S. 0" 53 1" 5i 2" 57 3" 53 4" 49 5" 48 6" 48 7" 47 8" 46 9" 45 JM. "~o~ S. 11.53 10.59 to. 5 9.10 8.16 7.22 6.27 5.33 i(j 11.57 II. 2 10. 8 9.14 8.i97.25j6.3i 5.36 10 ^A Ai 42 4i 4o 39 39 38 37 36 2 1 20 12. II. b lO.II 9.17 8.237.286.34 5.40 20 Jb M 33 32 3i 3o 3o 20 28 27 4 I Jo 12. J II. 9 io.i5 9.20 8.26 7.32 6.3fc 5.44 3o 26 25 24 23 22 21 20 20 19 lb 5 6 7 2 2 40 12. 6 II .12 10.18 9.24 8.3o 7.36 6.41 5.47 4o 17 16 i5 i4 i3 12 11 II 10 9 26 bo 12. 9 1 1 .i5 10.21 9.27 8.33 7.39 7.43 6.45 5.5i 5.55 5o 8 53 7 52 6 57 5 5o 4 49 3 4^ 2 48 I I 46 45 8 9 1 3 3 1 12. ]3 II. 19 10.25 9.3. 8.37 6.4c; 10 12.16 II .22 10.28 9-M 8.40 7.47 b.bJ b.59 K) 44 4J 42 4i 4o 4o 39 38 37 36 20 12.19 II .25 10.32 9.38 8.44 7.5o 6.5e 6. 3 20 35 J4 33 32 32 3i 3o 2Q 28 27 3 1 Jo 12.23 II .29 10.35 Q.41 8.48 7.54 7. c 6. 7 3o 26 25 24 23 23 22 21 20 19 18 5 2 4o 12.26 I I .32 10.39 Q.45 8.5i 7.58 7. 4|6.ii 4o '7 16 i5 i4 14 i3 12 II 10 9 7 3 27 bo 12.29 11.36 10.42 9.49 8.55 8. 2 7. 8 6.i5 6.20 5o 8 52 7 5i 6 5^ 6 5 48 4 48 3 2 46 45 8 9 3 3 12.34 II .40 10.47 9.53 9. 8. 6 7.i3 ID 12.37 11.44 10. 5i 9.57 9- 4 8.10 7-17 6.24 lo 4J 42 4i 40 40 39 38 37 36 35 1 20 12. 4l 11.48 10.54 10. I 9. 8 8.i4 7.21 6.28 20 M 33 :ii 32 3i 3o 29 28 27 26 3 4 5 1 2 Jo 12.44 II. 5i 10.58 10. 5 9. 11 8.18 7.25 6.32 Jo 2 b 24 24 23 22 21 20 19 18 17 2 2 3 4o 12.4s 11.55 II . 2 10. 9 9.i5 8.22 7.29 6.36 4o 17 16 i5 i4 i3 12 II 10 9 9 6 ^8 bo t2.52 II .59 II. 5 10.12 9.19 8.26 7.33 6.40 6.44 5o 8 5^ 7 5i 6 5o 5 49 4 48 3 48 2 47 I 46 I 45 8 9 3 3 12.55 12. 2 II. 9 10.16 9.23 8.3o 7.37 10 12.59 12. 6 II. i3 10.20 9.27 8.34 7.42 6.49 10 A'i 42 4i 4i 40 39 38 37 36 35 h 1 20 iJ. 3 12.10 II. 17 10.24 9.3i 8.39 7.46 6.53 20 M 34 JJ 32 3i 3o 29 28 27 26 4 2 Jo i3. 6 12. i4 II. 21 10.28 9.36 8.43 7.5o 6.57 3o 26 25 24 23 22 21 20 19 t9 18 5 2 2 3 40 i3.io 12.18 II .25 10.32 9.40 8.47 iM 7. 2 4o 17 16 i5 i4 i3 12 12 II 10 9 7 29 bo !3.i4 12.22 11.29 10. 36 9-44 8.5i 7.59 7. 6 5o 8 7 6 5 5 4 47 3 46 2 45 I 43 9 4 i3. 19 12.26 11.34 10.42 9.49 8.57 8. 4 7.12 5i 5o 49 48 48 (O 1J.2J 12. 3o 11.38 10.46 9.53 9. I 8. 8 7.16 10 42 41 4i 4o 39 38 37 36 35 34 2 1 20 i3.27 12.34 11.42 io.5o 9.57 9. 5 8.i3 7.21 20 34 33 32 3i 3o 29 28 27 27 26 4 1 2 Jo iJ.Ji 12.38 11.46 10.54 10. 2 9.10 8.17 7.25 Jo 25 24 23 22 21 21 20 19 18 17 5 2 4o i3.35 12.43 [i.5o 10.58 10. 6 9.148.22 7.3o 4o 16 i5 14 i4 i3 12 II 10 9 8 7 3 bo 13.39 12.4- 11.55 II. 3 10.10 9.188.26 7.34 5o 7 7 6 5 4 3 2 I 8 9 3 4 EXAMPLE I. Given the moon's apparent a titude 44° 27', and her horizontal parallax 5G' 55", Rec pircd the correction and log-arithm 1 For the Correctior [. For the Logarithm. In Tab. xix. to alt. 44° 20' and pa r. 6G' is 19' 54" In Tab. xix. to nearest alt.44.i° and par. 5C ' .^n" 20RR . , Tab. A. 55" parallax 3 .. Tab. C. 5" parallax .. . . 5 ..Tab. B. 7' altitude 5 Sought correction Sought logarithm 2093 . 20' 2" EXAMPLE II. Given the moon's apparent altit ude 50° IG', and horizontal parallax 59' 0". Required the corre( tion and log-arithm ? For the Correction . For the Logarithm. In Tab. xix. to aii. 50° 10' and pa r. 69'is 22' 3" In Tab. xix. to alt. 50° and par. 59' 0". . ... 913 .. Tab. A.O"paral . . Tab B 6' altitu ax 38 4 . . Tab. G. 0" parallax . 12 Je ... 1925 0)0)/ 45"| AMP LE III. EX Given the moon's apparent altiti ide 28° 27', and horizontal parallax 54' 10". Required the correc tion and log-arithm ? For the Correctior . For the Loerarithm. In Tab. xix. to alt. 28'^ 20' and pa r. 54'is 13' 3" Tab. xix. to nearest alt. 28° 30' and par. 54' 10" i J354 .. Tab. A. 10" pnrallav d.-^ Table C. 0" parallax 12 .. Tab. B. 7' altituc e 3 ;3G6 Sought logarithm Sr>noh» rorrPfllr n' 49" 1 / TABLE XIX. [rise us Logarithms. o ts Table C. y (0 Apparent Altitude of D 's Centre. Cor. Sec. of Par. CiCn Add. / / ? I / / / / / / / / / M. i4 S. 25 •25 2'J 25 4C 2G 2387 2d 20 •2G40 27 27 30 28 2871 28 30 2368 29 2865 29 30 2862 Sec Cor. 12 2J89 2387 2384 2880 2878 2876 2874 10 2375 2J7J 2871 2369 2867 2364 2862 2860 2357 2854 235i 2348 I 1 1 20 2861 2359 2jb7 2355 2jbJ 235i 2849 2846 2848 2841 2388 2335 2 3o 2J47 2345 2J4J 234i 2889 2887 2335 2332 2829 2827 2824 2821 8 40 2334 23J2 2329 2827 2825 2828 2821 2818 2815 23i3 2810 2807 4 6 55 5o 2820 23i8 23lb 23i3 28 i I 2809 23o7 2804 2801 2299 2296 2298 5 6 5 4 23o6 23o4 23oi 2299 2297 2295 2298 2291 2288 2286 2283 2280 10 2292 2290 2288 22S6 2284 2282 2280 2277 2274 2272 2269 2266 7 2 20 2279 2277 2274 2272 2270 2268 2266 2264 2261 2258 2255 2 258 8 I Jo 2265 2 263 2261 2269 2267 2255 2258 225o 2247 2245 2242 2289 9 40 225[ 2249 2247 2245 2248 2241 2289 2286 2233 223l 2228 2226 ItT bo 2 2 38 2206 22J4 2282 2280 2228 2226 2228 2220 2218 22l5 22l3 Sec. Cor. 2224 2222 2220 2218 2216 2214 2212 2210 2207 2205 2202 2199 12 10 22II 2209 2207 2205 22o3 2201 2199 2196 2198 2I9I 2188 2i85 I II 20 2198 2190 2193 2I9I 2189 2187 2i85 2188 2180 2178 2175 2172 2 9 8 Jo 2l84 2182 2180 2178 2176 2174 2172 2170 2167 2i65 2162 2169 8 40 2I7I 2169 2167 2i65 2168 2161 2159 2i56 2i58 2l5l 2148 2i46 4 7 ^ bo 2ib8 2 I bo 2ib3 2l5l 2149 2147 2i46 2148 2l4o 2i38 2i85 2182 5 6 5 4 2i44 2142 2l40 2i38 2i36 2184 2182 2l3o 2127 2125 2122 2119 10 2l3l 2129 2127 2125 2128 2I2I 2119 2II7 2ll4 2II2 2109 2106 7 3 20 2118 2I16 2Il4 2112 2110 2108 2106 2I04 2I0I 2099 2096 2093 8 I Jo 2io5 2103 2I0I 2099 2097 2095 2093 2091 2088 2086 2088 2080 9 40 2092 2090 2088 2086 2084 2082 2080 2078 2075 2078 2070 2067 ^ bo 2079 2077 2075 2078 2071 2069 2067 2o65 2062 2059 2057 2o54 Sec. Cor 2066 2064 2062 2060 2o58 2o56 2o54 2052 2049 2046 2044 204 1 12 10 20bJ 2o5l 2q49 2047 2045 2048 204l 2089 2o36 2o38 2081 2028 I II 20 2o4o 2o38 2o36 2084 2082 2080 2028 2026 2028 2020 2018 20l5 2 g Jo 2027 2025 2023 2021 2019 2017 20l5 20l3 2010 2007 2005 2003 3 s 40 20l4 2012 2010 2008 2006 2005 2008 2000 1997 1994 1992 1990 4 7 "57 bo 2001 1999 1997 199b 1 99 J 1992 1990 1987 1984 1982 1969 1980 1967 '977 1964 5 6 6 4 3 1989 1987 1985 1988 1981 1979 1977 1975 1972 10 1976 1974 1972 1970 1968 1966 1964 1962 i9b9 1956 1954 1952 7 8 20 1 9b J 1 96 1 i9b9 1967 1955 19^4 1952 1949 1946 1944 1942 1989 2 Jo I95I 1949 19^7 1945 1943 I94I 1939 1987 1934 1981 1929 1927 9 40 i9i8 1936 1934 1982 1980 1928 1926 1924 1921 1918 I9I6 1914 "6^ bo 1925 1923 I92I I9I9 1917 I9I6 1914 I9I2 1909 1906 1904 1902 Sec. Cor. 1913 I9I1 1909 1907 1905 1908 1 90 1 1899 1896 1898 I89I 1889 12 10 1900 1898 189b 1894 1892 1891 1889 I8S7 1 884 1881 1879 1877 I II 20 1888 1886 1884 1882 1880 1878 1876 1874 1871 1869 1867 1864 2 10 3o 187b 1873 I87I 1869 1867 1866 1864 1862 i859 i856 1 854 1 852 3 8 4o 1 863 I86I i8b9 1857 i855 i8b4 i852 1849 1846 i844 1842 l88q 4 7 61 5o i85i 1849 1847 1845 1 1843 i84r 1889 1887 1884 1882 1880 1827 5 6 6 5 i838 i836 !834 1882 i83o 1829 1827 1825 1822 1819 1817 i8i5 10 1826 1824 1822 1820 1818 1817 i8i5 I8I8 1810 1807 i8o5 1808 7 3 20 i8i4 1812 1810 1808 1806 i8o5 i8o3 1800 1797 179b 1798 1791 8 2 3o 1801 1799 1798 1796 1794 [ 1792 1790 1788 1785 1788 1 1781 1778 1 9 1 1 P«?«ii6] TABLE XIX. Correction. ■^ = Table A. Tab.B. J> 's Horizoi^tal Parallax. Proportional part for Seconds of Parallax. ForM. of alt. <'^ Add. Add. D. 3^ M. 54' 55' 56' 57' 58' 59' GO' Gl' S. 0" 5i 1" Vo 2" 49 3" 48 4" 48 5" 47 6" 46 7" 45 8" '9" M. ? s. i3.43 12. 5i tl.5q II. 7 10. i5 9.23 8.3. 7.39 AA 43 lO i3.47 12.55 12. C II. II 10.19 9.27 8.35 7-44 10 42 42 4i 4o 39 38 37 3b ib 35 2 1 20 i3.5i 12.59 12. 7 II. 16 10.24 9.32 8.40 7.48 20 34 33 32 3. 3o 29 29 2b 27 26 4 2 3o i3.55 i3. 3 12.12 II .20 10.29 9.37 8.45 7.53 3o 2b 24 23 23 22 21 20 19 18 17 b 6 3 4o i3.5q i3. 8 12.16 11.24 10.33 9.41 8.49 7.58 4o 17 16 lb 14 .3 12 II .0 10 V 7 3 37 bo o i4. 3 l3.I2 12.20 1 1 .29 10.37 9-46 8.54 8. 3 bo 8 7 6 b 4 4 46 3 45 2 I Ai u Ao- _9_ 4 i4. 9 i3.i7 12.26 11.34 10.43 9.5i 9. 8. 9 5o 49 48 47 47 10 i4.i3 l3.22 12. 3o II .39 10.47 9.56 9. 5 8.i3 .0 4i 4i 40 39 38 37 36 ib 35 M 1 20 I4.I7 i3.26 12.35 11.43 10.52 10. I 9.10 8.18 20 Si 32 3. Jo 3o 29 28 27 26 25 4 3o l4.2I i3.3o 12,39 11.48 10.57 10. 6 9.14 8.23 ^0 24 24 23 22 21 20 19 .8 18 17 5 5 4o 14.26 i3.35 12.44 11.53 II. 2 10.10 9.19 8.28 4o 16 lb 14 i3 12 12 11 .0 9 8 7 4 4 3l bo i4.3o i3.39 12.48 11.57 II. 6 10. i5 9.24 8.33 bo 7 6 6 5 4 3 46 2 45 I AA 43 4^ H 9 14.35 i3. 44 12.53 12. 2 II. II 10.20 9.29 8.38 5o 49 48 47 47 lO 14.39 i3.48 12.57 12. 7 II. 16 10.25 9-34 8.43 lo 42 4i 4o 39 38 37 36 36 35 M 2 1 20 i4.43 i3.53 i3. 2 12. II II. 21 10. 3o 9.39 8.48 20 33 32 3i 3i 3o 29 28 27 26 26 4 2 3o i4.48 13.57 i3. 7 12.16 11.25 10.35 9-44 8.54 3o 25 24 23 22 21 20 20 19 18 17 5 6 2 40 i4.52 i4. 2 i3.ii 12.21 II .3o 10. 4o 9.49 8.59 4o 16 i5 i5 14 i3 12 II 10 9 9 3 33 bo .4.57 14. 7 i3.i6 12.26 11.35 10.45 9-54 9- 4 DO 8 7 6 5 4 4 45 3 AA 2 43 1 4^ ~A\ 9 4 i5. 2 l4.I2 l3.22 12. 3i II .41 10.5. ID. I 9.10 49 48 47 46 46 10 i5. 7 14.17 13.27 12.36 11.46 10. 56 ID. 6 9.15 .0 41 4o 39 38 37 36 36 35 34 33 2 3 4 1 20 lb. 12 14.22 i3.3i 12.41 11.5. II. I 10. II 9.21 20 32 3i 3i 3o 29 28 27 26 26 25 2 3o lb. 16 14.26 i3.36 12.46 11.56 .1. 6 10.16 9.26 3o 24 23 22 21 21 20 19 18 17 .6 5 2 3 40 lb. 21 14.3, i3.4i .2.5. 12. I II .11 10.21 9.3. 4o .6 i5 1 4 .3 .2 II I . 10 9 8 3 bo lb. 26 14.36 i3.46 .2.56 12. 6 II. 17 10.27 9.37 5o 7 6 6 5 4 3 45 2 Ta I 43 I 9 4 1 1 34 i5.3i 14. 4i i3.5i .3. I 12. II II .22 10.32 9.42 49 48 47 47 46 4242 10 lb. 3b 14.46 i3.56 .3. 6 12.17 11.27 .0.37 9-48 .0 4i 4o 3q 38 37 37 36 35 3433 2 20 ib.4o i4.5o i4. I .3.11 12.22 II .32 .0.43 9.53 20 32 02 3. ■Jo 29 28 28 27 26'25 3 4 2 3o lb. 45 14.55 i4. 6 i3.i6 12.27 11.38 .0.48 9.59 3o 24 23 23 22 21 20 19 18 18 .7 5 6 7 3 4o lb. bo i5. i4.n .3.22 .2.32 11.43 .0.54 10. 4 4o .6 i5 i4 .4 .3 12 1 1 10 9 9 4 35 bo o lb. 55 i5. 5 14.16 .3.27 12.38 11.48 10.59 .0.10 5o 8 7 6 5 4 4 3 43 2 4^ i| 4T'4T 8 9 1 2 4 5 16. i5.ii 14.22 .3.33 12.44 11.55 11. 5 10.16 48 47 46 46 45 10 16. 5 i5.i6 14.27 i3.38 12.49 .2. II .11 .0.22 10 4o 39 38 37 37 36 35 34 33 33 1 20 16.10 l5.2I 14.32 i3.43 12.54 12. 6 II. 17 10.28 20 32 3. 3o 29 28 28 27 26 2524 4 2 3o 16. i5 i5.26 r4.38 .3.49 i3. .2. II . 1 .22 10.33 3o 24 23 22 21 20 19 19 18 17!. 6 5 6 7 4o 16.20 i5.32 14.43 .3.54 i3. 5 12.17 11.28 10.39 4oi5 i5 i4 .3 12 II II 10 98 4 DO 16.25 15.37I 14.48 .3.59 l3.IT 12.22 11.33 10.45 5o 7 6 6 5 4 3 2 2 i| U 9 4 5 EXAMPLE IV. Given Ihe moon's apparent altitude 76° 36', and her horizontal parallax 56' 18". Required the correction and logarithm ? Fo)- the Correction. For the Logarithm. In Tab. xix. to alt. 76° 30' and par. 6G' is 46' 37" In Tah.xix. to nearest alt. 77° and par. 56' 10" 2110 ..Tab. A. 18" parallax 10 ..Tab. C. 8 parallax 2 ..Tab. B. 6' altitude 6 Soug-ht logarithm 2112 Souffht correction 46' 53" EXAMPLE V. Given the moon's apparent altitude 16° 25', and her horizontal parallax 58' 45' . Required the correc- tion and logarithm 1 For the Correction. i For the LogaHthm. In Tab. xix. to alt. 16° 20' and par. 58' is 6' 17" Tab. xix. to nearest alt.l6° 20' and par .58' 40" is 2099 .. Tab. A. 45" parallax 14 ..Tab.B. 5' altitude Tab. C. 5" parallax 6 Sought logarithm 9Ifl'i .. fi' 31" TABLE XIX. [P .i,c 117 Logarithms. i^ Table C. Apparent Altitude of 5 's centre. Cor. Sec. of Par. «a. Add. M. ~54 s. 30 2 3Go 30i 2358 31 2356 3U 2354 32 S2h 33 33i 34 34i 35 35^ 2338 Sec. Cor. 2J52 2349 2347 2345 2344 2342 234o 12 JO 2 346 2344 2342 2340 2JJ8 2336 2334 2332 233o 2328 2326 2324 1 11 ■f.O 23 J J 233o 2328 2326 2J24 2322 2320 23i8 23i6 23i4 23 1 3 23ll 2 9 8 3o 23 10 23i6 23i4 23l2 2J10 23o8 23o6 23o4 23o2 23oo 2299 2297 3 40 23o5 2 3o2 23oO 2298 2296 2294 2292 2290 2289 2287 2285 2283 4 7 "55" bo 2291 2289 2287 2285 2283 2281 2279 2277 2275 2274 2272 2270 5 6 5 4 2278 2275 2273 2271 2269 2267 2265 2 263 2262 2260 2258 2256 10 2264 2262 2260 2258 2 2 56 2254 2252 225o 2 248 2246 2245 2243 7 2 20 225l 2248 2246 2244 2242 2240 2238 2236 2234 2232 223l 2229 8 I Jo 2237 22J5 223J 223l 2229 2227 2225 2223 2221 2219 2218 2216 9 "56" 4o 5o 2224 22r ! 2221 32gS 2219 2206 2217 2204 22l5 2202 22l3 2200 221 I 2198 2209 2196 2208 2194 2206 2192 2204 2I9I 2202 2189 Sec. Cor. 2197 2194 2192 2190 2188 2186 2184 2182 2181 2179 2177 2175 12 10 2iS3 2181 2179 2177 2175 2173 2171 2169 2168 2166 2164 2162 I II 20 2170 2lb8 2166 2164 2162 2160 2168 2x56 2i54 2 I 52 2l5l 2149 2 9 8 Jo 2i57 2i55 2lbJ 2l5o 2i48 2i46 2145 2143 2l4l 2139 2i38 2i36 3 4o 2i44 2l4l 2139 2i37 2lJ5 2IJJ 2l3l 2129 2128 2126 2124 2122 4 7 ^ bo 2i3o 2128 2126 2124 2122 2120 2118 2116 2Il5 21l3 2111 2109 5 6 5 4 2117 2Il5 211J 2111 2109 2107 2io5 2I03 2102 2100 2098 2096 10 2104 2102 2100 2098 2096 2094 2092 2090 2089 2087 2085 2083 7 3 20 2091 2089 2087 2o85 208J 2081 2079 2077 2076 2074 2072 2070 8 I Jo 2078 2076 2074 2072 2070 2068 2066 2064 2o63 2061 2059 2o57 9 Ts" 40 5o 2o65 2o63 2o52 2o5o 2061 2048 2o59 2o46 2o57 2044 2o55 2042 2o53 2o4o 2o5l 2o38 2025 2o5o 2o37 2048 2o35 2o46 2o33 2o44 2o3l Sec. Cor. 2u39 2o37 2o35 2o33 203l 2029 2027 2024 2022 2020 2018 13 10 202fj 2024 2022 2020 2018 2016 20l4 20l3 20 I I 2009 2008 2006 1 II 20 20l3 2011 2009 2007 2005 2oo3 2002 2000 1998 1996 1995 1993 2 9 Jo 2001 1998 199b 1994 1 99 J 1991 1989 1987 1985 1983 1982 1980 3 8 4o I9SS 1986 19S4 1982 1980 1978 197b 1974 1973 1971 1969 1967 4 7 ^ bo •97^ •97^ 1 97 1 1 909 1967 1965 i9b3 1 96 1 1900 1958 1957 1955 5 6 6 4 1962 i960 1958 1956 1954 1952 1 95 1 1949 1947 1945 1944 1942 10 1950 194s 1946 1944 1942 1940 19J8 1936 1935 1933 1 93 1 1930 7 3 20 1937 ig.ib 1933 1931 1929 1927 1925 1923 1922 1920 1919 19.7 8 2 Jo 1925 1923 1921 1919 1917 1915 1913 I9II I9I0 1908 1906 1904 y I "60" 4o 5o 1912 1900 1887 1910 1898 1908 1896 1906 1894 1881 1904 1892 1902 1890 1900 1888 1875 1898 1886 1873 1897 i885 1895 i883 1894 1881 1892 1880 Cor. Sec. i885 i883 1879 1877 1872 1870 1869 1867 12 10 187b 187J 1871 i860 1867 i865 1 863 I86I i860 i858 1 857 i855 I 11 20 1862 i860 i858 1 856 1 854 i852 i85r 1849 1847 1845 1 844 1842 2 10 Jo i85o 1 848 1 846 i844 1842 1840 1 838 1 8 36 i835 1 833 i832 i83o 3 8 4o .837 i835 iS33 i83i i83o 1828 1826 1824 1823 1821 1820 i8r8 4 7 1)T 5o 1825 1823 i8u 1821 1809 1819 'I'Sof 1817 i8i5 i8i4 1812 1 800 1811 1809 1807 1806 5 6 6 5 i8o5 i8o3 1802 1798 1 796 1795 1793 10 1801 1799 1797 1795 1793 1791 [789 1787 1786 1784 1783 1781 7 3 20 17H9 • 7»7 1783 1 78 J 1781 1779 1777 177b 1774 1772 1771 1769 8 2 1,, ^^1 ■ 77^) '774 1772 1770 1769 .767 1765 1 1763 1 . 1762 1760 j 1759 1757 I PaseliS] TABLE XIX. Correction. -J _• Table A. Tab.B. J) 'a Horizontal Parallax. Proportional part for Seconds of Parallax. ForM. of alt. Add. Add. D. 36 M. 54' 55' 56' 57' l4~6 58' 59' 60' 61' S. 0" 4"7 1" 4"6 2" 45 3" 45 4" 44 5" 6" 434^ 7 ■' 4i d" 4"i 9" 40 M. "o" 1 S. 1 iG.3i (5.43 14.54 I3.I7 12.29 11.40 10. 5i lO i6.3(i i5.48 i5. 14. 11 i3.23 12.34 11.46 10.57 10 39 38 37 37 36 35 34 33 J3 32 1 2 2 20 16.42 i5.53 i5. 5 14.17 i3.28 12.40 II. 5i II. 3 20 Jl 3o 29 28 28 27 26 25 24 24 4 3o 16.47 i5.58 iS.io 14.22 i3.34 12.45 11.57 II. 9 3o 23 22 21 20 20 19 18 17 16 16 5 6 3 3 40 16.52 16. 4 i5.i6 14.27 i3.3q 12. 5i 12. 3 II. i5 4o lb i4 i3 12 12 II 10 Q c 8 7 4 3? bo 16.57 16. 9 l5.2I 14.33 i3.45 12.67 12. 9 II .21 5o 7 6 5 4 4 43 2 4^ I 47 47 >o 4(1 9 5 17. 2 16.14 15.26 14.38 i3.5o i3. 3 12. i5 11.27 47 46 45 45 44 10 17. 7 16.20 i5.32 14-44 i3.56 i3. 8 12.20 11.33 10 J9 38 37 37 J6 35 34 34 JJ 32 2 1 2 2 20 i-.i3|i6.25 15.37 14. 5o i4. 2 i3.i4 12.26 11.39 20 3i 3o 3o ■iq 28 27 26 26 2b 24 4 3o 17.18 16. 3o 15.43 i4.55 i4. 8 l3.20 12.32 11.45 3o 23 22 22 21 20 iq 18 18 17 16 5 6 7 4o .7.23 16. 36 1 5. 48 i5. I i4.i3 i3.26 12.38 II. 5i 4o i5 i4 i4 i3 12 II 10 10 9 8 4 38 bo 17.29 16.41 i5.54 i5. 6 .14.19 14.26 i3.32 12.44 11.57 12. 4 5o 7 46 7 45 6 44 5 44 4 3 42 3 47 2 47 1 40 3^ 9 5 .7.35 16.48 16. i5.i3 i3.38 12. 5i lO 17.40 16.53 16. 6 15.19 i4.32 i3.44 12.57 12.10 10 38 37 37 36 35 34 33 33 J2 3i 2 1 20 17-46 16.59 16.12 i5.25 14.37 i3.5o i3. 3 12.16 20 30 3o 29 28 27 26 26 2 5 24 23 ■J 4 2 2 do 17. bi 17. 4 .6.17 i5.3o 14.43 i3.56 l3. q 12.22 3o 22 22 21 20 19 IQ iS 17 16 i5 5 6 7 a 3 4 4o 17. b7 17.10 16.23 i5.36 14.49 i4. 2 i3.i5 12.20 4o i5 i4 i3 12 12 1 1 10 9 8 8 3^ bo o 18. 2 18. 8 17.15 17.21 16.29 16.34 15.42 1 5. 48 i4.bb i4. 8 l3.22 13.28 12.35 5o 7 46 6 45 5 44 4 4 43 3 42 2 4"i I 47 I 4^ 3q H 9 5 5 i5. I 14. i4 12.41 10 18. i3 17.27 16.40 i5.54 i5. 7 l4.20 13.34 12.47 10 38 37 37 36 35 34 34 33 J2 3 1 2 1 20 18.19 17.32 16. 46 i5.5q ib.i3 14.27 i3.4o 12.54 20 3i 3o 2Q 28 27 27 26 2b 24 34 4. 2 Jo 18.24 17.3s 16.52 16. 5 i5.i9 14.33 1 3. 46 i3. 3o 23 22 21 20 20 IQ 18 17 17 16 5 3 4 4 40 18. 3o 17-44 16.57 16.11 i5.25 i4.3q i3.53 i3. 6 4o i5 i4 i4 i3 12 1 1 10 10 9 8 7 4^ bo 18. 36 17.49 17- 3 16.17 i5.3i 14.45 13.59 i3.i3 5o 7 7 6 5 4 3 47 3 4^ 2 4^ 1 B 9 5 5 18.42 17.56 17.10 16.24 i5.38 14.52 14. 6 l3.20 45 u 43 ^i 42 39 38 10 18.48 18. 2 17.16 16. 3o i5. 44 14.59 14. i3 13.27 10 37 37 36 35 M 34 33 32 3i;3i 20 18.54 18. 8 17.22 16. 36 i5.5i i5. 5 14.19 13.33 20 3o 2q 28 ■il ■21 26 25 24 2423 y 2 2 3 Jo 18. b9 18. i4 17.28 16.42 i5.57 i5.ii 14.25 i3.4o 3o 22 21 21 20 iQ 18 18 17 i6|,5 5 40 .9. b 18.19 17.34 16.48 16. 3 I5.I7 14.32 i3.46 4o i5 i4 i3 12 II 11 10 9 8 8 7 4 4 4F bo o 19.11 18.25 17.40 16.55 16. 9 i5.24 i4.38 i3.53 5o 7 6 43 5 42 5 4^ 4 4i 3 40 2 3^ 2 3^ I 38 3^ 8 9 5 6 19.18 18.32 17-47 17. 2 16.16 i5.3i 14.46 14. 10 19.23 18. 38 17. b3 .7. 8 16.23 ■ 5.37 .4.52 14. 7 ID 36 36 35 M 33 33 32 3i 3o 3o 2 1 20 19.20 i«.44 17.59 17-14 16.29 i5. 44 14.59 i4.i4 20 29 28 27 27 26 25 24 24 23 22 4 2 Jo 19.35 18. bo 18. b 17.20 16.35 i5.5o i5. 5 14.20 3o 21 21 20 IQ 18 18 17 16 i5i5 5 3 4o 19.41 18. 56 18.11 17.26 16.42 i5.57 l5.12 14.27 4o i4 i3 12 12 1 1 ID Q Q 8 7 7 4 DO 19.47 19. 2 18.17 17.33 16.48 16. 3 .5.19 14.34 5o 6 6 5 4 3 3 2 I 8 9 5 6 EXAMPLE VI. Given the moon's apparent altitude 11° 20', and horizontal parallax GO' 43''. Required the correction and logarithm 1 Tojind the Correction. Tojind the Logarithm. In Tab. xix. to alt. 11° 20' and par. GO is 4' 30" Tab. xix. to nearest alt. 1]°20' and par.G0'40" 2052 ..Tab. A. 43" parallax 16 Tab. C. 3"parallax 9 ..Tab. B.O' altitude 2 Sought logarithm 2061 48" EX AMPLE VII. Given the moon's apparent altitude 8° 40', and horizontal parallax 5G' 20' . Required the correc- tion and logarithm ? To find the Correction. i Tojind the Logariih7n. InTab. xix. to alt. 8° 40' and par. 5G' is 9' 18" Tab. xix. to nearest alt. 8° 39' and par. 5C' 20 ' 2518 ..Tab. A.20" parallax 38 Tab. C. 0" parallax 13 .. Tab. B. 0' altitude 5 Sought logarithm 2531 Smin-ht />nrrf.nt on 10' 1" TABLE XIX. rpa.,- l:;.l Logarithms. 5 i Tablk C. 1 Apparent Altitude of 5 's centre. Cor of Sec. Far. «2- Add. 1 IM. '5T s. u 33 ^37 3Gi 2335 37 37i 38 38i 39 39i 40 2326 m 2324 41 2323 41^ 232 2 Sec. Cor. 2334 2332 233i 2329 2328 2327 12 ' 10 232.3 2321 2320 23i8 23i7 23i5 23 1 4 23i3 23 12 23 10 23o9 23o8 1 II 20 2309 23o7 23o6 23o4 23o3 23o2 23oi 23oo 2298 2297 2296 2294 2 l3„ 229(3 2294 2293 2291 2290 2288 2287 2286 2285 2283 2282 2281 3 4o 2282 2280 2279 2277 2276 2274 2273 2272 2271 2270 2268 2267 4 7 "5T 5o 2 2b8 2266 2 265 2264 2263 2261 2260 2258 2257 2256 2255 2254 5 6 5 4 2255 2253 2252 225o 2249 2247 2240 2245 2244 2242 2241 2240 10 2241 2239 2238 2236 2235 2234 2233 2232 2230 2229 222S 2227 7 2 ao 2228 2226 2225 2223 2 22 2 2220 2219 2218 2217 22l5 2214 22l3 8 I 3o 2214 2212 22II 2210 2209 2207 2206 2204 22o3 2202 2201 2200 9 3(3" 4o 220I 2188 2199 2186 2198 2l85 2196 2l83 2195 2182 2193 2180 2192 2179 219I 2178 2190 2177 2189 2175 2188 2174 2186 2173 Sec. Cur 2174 2173 217I 2169 2168 2167 2166 2164 2i63 2162 2161 2160 12 10 2161 2159 2i58 2i56 2lb5 2i53 2l52 2l5l 2l5o 2149 2i48 2l47 I II 20 2l48 2i46 2i45 2i43 2l42 2l4o 2139 2i38 2i37 2i35 2i34 2i33 2 9 8 3o 2i3b 2i33 2l32 2l3o 2129 2127 2126 2125 2124 2122 2121 2120 3 4o 2I2T 2119 2118 2117 2I16 21 l4 2Il3 2II2 2111 2109 2108 2107 4 7 5? bo 2108 2106 2I05 2I04 2103 2:01 2100 2099 2097 2084 2096 2o83 2095 2082 2094 2081 5 6 5 4 2095 2093 2092 2090 2089 2088 2087 2086 lO 2082 2080 2079 2077 2076 2075 2074 2073 2071 2070 2069 2068 7 3 20 2069 2067 2066 2064 2o63 2062 2061 2060 2o58 2o57 2o56 2o55 8 2 3o 20b6 2o54 20b3 2o5l 2o5o 2049 2048 2o47 2o46 2044 2043 2042 9 Ts" 4" 5o 2043 2o3o 204 r 2028 2040 2039 2o38 2025 2o36 2023 203b 2022 2009 2o34 2021 200S 2o33 2020 2007 203l 2o3o 2029 2016 2oo3 2027 1 2026 2018 2017 2006 2oo5 Sec. Cor. 2017 2016 20l5 20 1 3 2012 2010 12 10 2oo5 2oo3 2002 2000 1999 1997 1996 1995 1994 1993 1992 1991 I II 20 1992 1990 1989 1987 1986 1985 1984 1982 1981 1980 1979 1978 2 Zk) ■y/y 1977 1976 1975 1974 1973 1Q7I 1970 1969 1967 1966 1965 3 8 i 4(. 1966 1965 1964 1962 I 96 I i960 1958 1957 1966 1955 1954 1953 4 7 1 59 bo 104 I94I 1952 1939 1 95 1 ^1938" 1949 1948 1936 1947 1934 1946 1933 1944 1932 1943 1942 1941 1940 5 6 6 4 1937 1 93 1 1929 1928 1927 H) 1929 1927 1926 1924 1923 I92I 1920 1919 1918 I9I7 1916 i9.b 7 3 20 I9I6 1914 I9I3 I9II I9IO 1909 1908 1907 1906 1904 1903 1902 8 2 3o 1903 1902 1 90 1 1899 1898 1896 189b 1894 1893 1892 i8qi 1890 9 I 60 4o 5o 1891 1879 r866 1889 1877 1888 1876 1 88b 1874 1862 1 885 1873 1884 I87I i8S3 1870 1882 1S69 i88i 1868 1879 1867 1878 1866 1877 1 865 Sec. Cor. 1864 i863 1861 1859 i858 1857 1 856 i855 1 854 i853 12 10 1 854 i852 i85i 1849 1 848 1847 i8-f6 1845 1844 1843 i84i i84o I II ao i84i i84o 1839 i837 i836 i834 1833 i832 i83i i83o 1S29 1828 2 10 3o 1829 1827 1826 182b 1824 1822 1 82 1 1820 1819 18)8 1817 1816 3 8 4o 1817 i8i5 1814 1812 1811 1810 1809 1S08 1807 i8o5 i8o4 i8o3 4 7 So i8o5 i8o3 1802 1800 1 79V 1798 1797 1796 1795 1793 1792 179' 5 6 61 1792 1791 1790 1788 1787 1785 1784 1783 1782 ,78. 1780 1779 6 5 10 1780 1778 1777 1776 1775 1773 1772 1771 1770 1769 1768 1767 7 20 1768 1766 1765 1764 1763 1 76 1 1700 1759 1758 1757 1756 1755 8 2 00 1756 1754 1753 1752 1751 1749 1748 1747 j 1746 1745 1 n44 r743 '^ 1 I Page 120] TABLE XIX. Correction. ii s Table A. Tab.B- < g D 's Horizontal Parallax. Proportional part for Seconds of Paralla.x. ForM. of Alt. Add. Add. D 4^ M. 54' 55' 5G' 57' 53' 59' 60' 01' S. 0" 1" 44 4'3 2" 43 3" 4^ 4" 47 5" 4) 6" 4^ 7" 8" 30 38 9" 37 M. s. IQ.53 rg. 8 18.24 17.39 16.54 16.10 i5.25 i4.4i lO 19.59 19.14 18. 3o 17.45 17. I 16.16 i5.32 14.47 10 37 36 35 34 M 33 32 3i 3i 3o 2 1 20 20. 5 19.20 i8'.36 17.52 17. 7 16.23 i5.39 14.54 20 29 28 28 27 2b 2b 25 24 23J23 3 3 3o 20.11 19.26 18.42 17.58 17.14 16.29 i5.45 i5. 1 3o 22 21 20 20 19 18 17 17 ibi5 5 3 4 4 .jo 20.17 19.33 18.48 18. 4 17.20 16. 36 i5.52 i5. 8 40 i4 i4 i3 12 11 1 1 10 9 9 8 I _9 5o 2D 2 3 19.39 18.55 18.11 17.27 16.43 15.59 [5.i5 5o 7 6 6 5 4 3 39 3 39 2 38 1 3^36 5 6 43 20. 3o 19.46 19. 2 18.18 17.34 16. 5o 16. 6 i5.23 43 42 42 4i 40 10 20 36 19.52119. 8 18.25 .7-41 16.57 16. i3 i5.3o 10 36 35 M 34 33 3^ 3i 3i 3o 29 1 20 20.42 19.58119.15 18. 3i 17-47 17- 4 16.20 ,5.37 20 28 28 27 26 26 25 94 23 23 29 4 3 3o 20.48 20. 5 19.21 18. 38 17.54 I7-II 16.2- i5.44 3o 21 21 20 19 18 18 17 lb i5 i5 6 7 3 4o 20.54 20.11 19.28 18.44 18. I 17.17 16.34 i5.5i 4o 14 i3 10 12 II 10 10 9 8 7 5 44 5o 21. 20.17 19.34 18. 5i .8. 7 17.24 16. 4i i5.58 16. 6 5o 7 42 6 4i 5 4~i 5 4o 4 39 3 38 2 38 2 3^ I 36 35 B 9 5 6 21. 8 20.25 19.41 18. 58 18. i5 17.32 16.49 10 21. i4 20. 3l 19.48 19. 5 18.22 17.39 16. 56 16. i3 10 35 34 33 33 32 3i 3i 30J29 28 2 1 20 21 .20 20.37 19.54 19. 11 18.28 17-46 17. 3 l6.20 20 28 27 2b 26 2 5 24 93 2322 21 4 ^ 3o 21.26 20.44 20. I 19.18 18.35 17.52 17.10 16.27 3o 21 20 19 18 18 17 lb ibi5 i4 5 4 4o 21.33 20. 5o 20. 7 19.25 18.42 17.59 17.17 16.34 4o i3 i3 12 11 1 1 10 9 8 8 7 7 5 45 5o 21.39 20.56 20.14 19.32 18.49I18. 6 17.24 16. 4i 5o 6 6 5 4 3 3 37 2 37 1 1 9 6 21.46 21. 4 20.21 19.39 18.57 18.14 17-32 16.49 4i 4o 4o 39 38 36;35|35 10 21.53 21 . 10 20.28 19.46 19. 3 18.21 17.39 16.57 10 M 33 33 32 3i 3o^3o 99 28 28 2 1 20 21.59 21.17 20.35 19.52 19.10 18.28 17-46 17- 4 20 27 26 2b 25 24 23 23 99 21 21 4 3 3o 22. 5 21.23 20.41 19.59 19.17 18.35 .7.53 17.11 3o 20 19 19 18 17 16 lb i5i4 '4 5 3 4o 22.12 21.30 20.48 20. 6 19.24 18.42 18. 17. .8 4o i3 12 12 11 10 9 9 8 7 7 7 5 46 5o o 22.18 21.36 20.55 20. 1 3 ,9.31 19.38 18.49 18. 7 18. i5 17.26 .7.33 Do 6 4i 5 4^ 5 40 4 3o 3 38 2 38 2 3^ 1 35 8 9 6 22.25 21.43 21, I 20.20 18. 56 36!35 10 22. 3l 2I.5o 21. 8 20.27 19-45 19. 3 18.29 17.40 10 34 33 33 32 3i 3i 3o 9929 28 3 4 1 20 22.38 21.56 21. i5 20.33 19.52 19. II 18.29 17-48 20 27 27 26 95 24 24 23 22 92 21 3 3o 22.44 22, 3 21 .22 20.40 19.59 19.18 18. 36 17.55 3o 20 20 19 18 18 17 lb i5i5 i4 6 4 4 4o 22. 5l 22.10 21.28 20.47 20. 6 19.25 18.44 18. 2 4o iJ i3 12 II II 10 9 9 8 7 7 5 47 5o o 22.57 22.16 21.35 20.54 20. 1 3 19.32 18. 5i 18.10 5o 7 4o 6 39 5 39 4 38 4 37 3 37 2 36 2 I 35'35 34 9 _6_ 23. 5 22.24 21.43 21 . 2 20.21 19.40 18.59 18.18 10 23.11 22. 3l 21.50 21.9 20.28 19-47 19. 7 18.26 10 33 33 32 3i 3i 3o|29 2828 27 2 I ?f) 23.18 22.37 21 .57I21 .16 20.35 19-55 19.14 18.33 20 26 26 25 24 24 23 99 22 21 20 4 3 So 23.25 22.44 22. 4 2 1.23 20.43 20. 2 19.21 18.41 3o 20 19 18 18 17 16 lb i5 M 14 5 4 4 4o 23. 3i 22. 5l 22.11 2 1 .3o 20. 5o 20. 9 19.29 18.48 4o i3 12 12 11 10 10 9 8 8 7 7 6 6 1 48 5o o 23.38 22.58 22.18 21 .37 20.57 20.17 19.36 i3.56 5o 6 3^ 5 38 5 38 4 37 3 36 3 36 2 35 I 3"4 I 34 33 9 1 23.46 23. 6 22.25 21.45 21. 5 20.25 19-45 19. 5 10 23.52 23.12 22.32 21.52 21.12 20.32 19.52 19.12 10 J2 32 3i 3o 3o 29 28 9« 27 2b 2 3 4 1 2 3 20 23.59 23.19 22.39I22. 21.20 20.4o|20. 19.20 20 2b 25 24 24 23 22 2 9 2 1 20 20 3o 24. 6 23.26 22.46 22. 7 21.27 20.47 20. 7 19.28 3o 19 18 18 17 16 ibiib i4 14 i3 b 6 4 4 io 24.13 23.33 22.53 22.14 21.34 20.55 20. j5 19.35 40 12 12 11 10 10 9; 8 8 7 b 7 5 4g 5o 24.20 23.40 23. I 22.2! 22.29 21.42 21. 2 20.23 19.43 5o 6 38 5 3^ 4 37 4 36 3 35 2 35 2 34 1 33 8 9 7 1 24.27 23.48 23. 9 21 .5o 21 .10 20. 3l 19.52 33|32 10 24.34 23.55 23. 1622. 36 2 I . 57 21.18 20.39 20. 10 3i 3i 3o 3099 28 28 27 2696 2 1 2 3 Ut 24.41 24. 2 23.93 22.44 22. 5 21 .26 20.46 20. 7 20 25 24 24 23 22 29 21 20 20 19 4 3o 24.48 24. 9 23. 3o 22. 5 1 22.12 21 .33 20.54 20 . 1 5 3o 18 18 17 17 i6 l5 i5 i4 i3 i3 b 4 io 24.55 24.16 23.3722.58 22.19 21.41 21. 2 20.23 4o 12 1 1 1 1 10 9 9 8 7 7 6 7 5 5o 25. 2 24.23 23.4423. 6 22.27 21.48 21 . 9 20. 3 1 5o 5 5 4 4 3 2 ' I 9 7 TABLE XIX. [ra. e 101 Logarithms. i TablkcI K=5 Cor. '•'" a Apparent Altitude of D 's centre. of Pnr Ai, Add. 1 , M. s. o 42 m 43 43i . 44 m 45 45i 46 47 48 49 23T)"&' Sec. Cor. 12 2321 2320 23i9 23i8 23i7 23i6 23i5 23l4 23i3 23ll 2309 lO 23o7 23o6 23o5 23o4 23o3 23o2 23oi 2 3 00 2299 2297 2296 2294 I II 20 2293 2292 2291 2290 2289 2288 2287 2286 2285 2284 2282 2280 2 9 3o 2280 2279 2278 2277 2276 2275 2274 2273 2272 2270 2268 2267 3 8 4() 2266 2265 2264 2263 2262 2261 2260 2259 2258 2256 2255 2253 4 7 55" 5o 2253 225l 2250 2249 2248 2247 2246 2246 2245 2243 224l' 2240 5 6 5 4 2239 2238 2237 2236 2235 2234 2233 1 2232 223l 2229 2228 2226 10 2226 2224 2223 2222 2221 2220 2219 2219 2218 2216 22l4 22l3 7 3 2o 2212 22II 2210 2209 2208 2207 2206 2205 2204 2 203 2201 2199 8 I 3o 2199 2198 2197 2196 2195 2194 2193 2192 2 I 91 2189 2188 2186 y 4o 2i85 2l84 2l83 2182 2181 2180 2179 2 '79 2178 2 1 76 2174 2173 "56 5o o 2172 2I7I 2170 2169 2168 2i55 2.67 2 1 54 2166 2i53 2i65 2164 2i63 21C1 2159 ■Sec. Cor. 12 2159 2i58 2i57 2i56 2l52 2l5l 2149 2i48 2146 10 2146 2144 2143 ' 2142 2l4l 2i4o 2139 2i38 2i38 2i36 2i34 2 1 33 I II 20 2l32 2l3l 2i3o 2129 2128 2127 2126 2125 2125 2123 2121 2120 2 9 3o 2119 2II8 2117 2116 2Il5 2Il4 2Il3 2II2 21 I I 2II0 2io8 2107 3 8 4o 2106 2I05 2104 2I03 2102 2IOI 2100 2099 2098 2097 2095 2093 4 7 5? 5o o 2093 2092 2091 2090 2089 2088 2087 2086 2o85 2.072 2084 2082 2080 5 6 5 4 2080 2079 2078 2077 2076 2075 2074 2073 2071 2069 2067 lO 2067 2066 2o65 2064 2o63 2062 2061 2060 2059 2o58 2o56 2o54 7 3 20 2o54 2o53 2052 205l 2o5o 2049 2o48 2o47 2046 2o45 2043 2042 8 2 So 204l 2o4o 2039 2o38 2o37 2o36 2o35 2o34 2o33 2032 2o3o 2029 9 4o 2028 2027 2026 2025 2024 2023 2022 2021 2021 2019 :.TI7 2016 l8 5o o 20 r 5 20l4 20l3 2012 2011 2010 2009 2008 1996 2008 2006 2005 2003 Sec. Cor. 12 2002 200 I 2000 1999 1998 1998 1997 1995 1993 1992 1990 ID 1990 1989 1988 1987 1986 1985 1984 1983 1982 1981 1979 1977 I II 20 1977 197b 1975 '974 1973 1972 1971 1970 1970 1968 1966 1965 2 Q 3o 1964 I9b3 1962 1 96 1 i960 1959 19^)8 1957 '9^7 1955 19^4 1952 3 8 4o 1952 1 95 1 1950 1949 1948 1947 1946 1945 1944 1943 1 94 1 1939 4 7 59 DO o 1939 1938 1937 1936 1935 19^4 1933 1932 1932 1930 I917 1928 1927 5 6 6 4 3 1926 1925 1924 1923 1922 1922 1 92 1 1920 1919 I9I6 1914 10 1914 1913 I9I2 1911 1910 1909 1908 1907 1906 1905 1 903 1902 7 8 20 1 90 1 1900 1899 1898 1897 1897 1896 1895 1894 1892 I89I 1889 2 3o 1889 1888 1887 1886 i885 1 884 1 883 1882 18S2 1880 1878 .877 y I 4o 1876 1875 1874 1873 1872 1872 1871 1870 1869 1867 1866 1864 r>:> 5o o 1 864 1 852 1 863 1862 1861 i860 1859 i858 1846 i857 1845 i857 1 855 i854 i852 i84o Sec. Cor. 12 i85i i85o 1849 i848 1847 1844 1843 i84i lO 1839 1338 1837 i836 1 835 i835 1 834 i833 i832 i83o 1829 1827 I II 20 1S27 1826 1825 1824 1823 1822 1821 1820 1820 1818 ,8.7 i8i5 2 10 3o i8i5 1814 i8i3 1812 1811 1810 1809 1808 1808 1806 i8o4 i8o3 3 8 4o 1802 i8or iSoo 1800 1799 1798 1797 1796 '79'i 1794 1792 1791 4 7 "67' 5o o 1790 1789 1788 1787 1786 ,78b 1 785 1784 1783 1782 1780 1779 5 6 6 5 1778 1777 1776 1775 1774 1774 1773 1772 177' 1769 1768 1766 10 1766 1765 1764 1763 1762 1 76 1 1760 1759 1759 .757 1756 1754 I 20 1754 1753 1752 I75i 1750 1749 1748 1747 1747 1745 1744 1742 2 3o 1742 1 74 1 1740 1739 1738 1737 1736 1735 1735 1733 1732 1730 y I 16 .f*« Page 122] TABLE XIX > Correction. ^ a Table A. Tab.B. J) 's Horizontal Parallax. Proportional part for Seconds of Parallax. For M. of Alt. ^"^ ^ Add. Add. D. 5o M. 54' 55' 56' 57' 53' 59' GO' 61' S. 0" 1'^ 3'7 2" 3^ 3" 36 4" 35 5" 35 G" 34 7" 8" 9" 34 33 3^ M. ^0 s. IT" 25. 9 24.30 23. 5i 23.13 22.34 21.56 21.17 20.39 10 25. i6 24.37 23.59 23.20 22.42 22. 3 21 .25 20.47 10 32 3i 3o 3o 29 28 28 27 27 26 2 1 20 25.23 24.44 24. 6 23.28 22.49 22.11 21.33 20.54 20 25 25 24 23 23 22 21 2i|2:- 19 4 5 3 3o 25. 3o 24. 5i 24.13 23.35 22.57 22.19 21.41 21 . 2 3o IV 18 18 17 16 16 i5 i4ii4 i3 4 4o 25.37 24. 5q 24 . 2 J 23.42 2.3. 4 22.26 21.48 21 .10 4o 12 12 11 II 10 9 9 «' 7 / 6 7 5 57 5o 25.44 25. 6 24.28 24.36 23. 5o 23.12 23.21 22.34 22.43 21.56 21.18 DO 6 37 5 36 5 36 4 35 4 35 M 2 33 2 33 I 8 9 b 7 1 2 25.52 25.14 23.58 22. 5 21 .27 32[3i 10 25.59 25.21 24.43 24. 6 23.28 22. 5i 22. i3 21 .35 10 3i 3o 3o 29 28 28 27 26 26,25 2 20 26. 6 25.28 24. 5i 24.13 23.36 22.58 22.21 21.43 20 25 24 23 23 22 21 21 20 20 ig 4 3 3o 26.13 25.36 24.58 24.21 23.43 23. 6 22.29 21 .5i 3o 18 18 17 16 16 i5 i5 i4 i3 i3 5 6 7 4 5 5 4o 26.20 25.43 25. 6 24.28 23. 5i 23.14 22.37 21 .59 4o 12 II 11 10 10 9 8 8 7 6 5^ 5o o 26.27 ^."35 25. 5o 25.58 25.13 25.21 24.36 23.59 24. 7 23.22 23. 3i 22.45 22.54 22. 8 22. 17 5o 6 36 5 35 5 35 4 34 3 34 3 33 2 ¥2 I 3l 1 37 37 ~0~ 6 7 24.44 10 26.42 26. 6 25.29 24.52 24. i5 23.38 23. 2 22.25 10 3o 29 29 28 27 27 26 26 25|24 2 2 2() 26.50 26.13 25.36 25. 24.23 23.46 23.10 22.33 20 24 23 23 22 21 21 20J20 14 i3 I9I8 4 3 3o 26.57 26.20 25.44 25. 7 24.31 23.54 23.18 22.41 3o 18 17 16 16 i5 i5 l3 12 5 4 4o 27. 4 26.28 25. 5i 25.15 24.39 24. 2 23.26 22.49 40 12 II 10 10 9 9 8 7 7 6 7 5 53 DO O 27.11 26.35 25.59 25.23 24.46 24.55 24.10 24.19 23.34 22.58 DO 6 35 5 34 4 34 4 33 33 2 3I 2 37 I 37 I 3^ 3o 8 9 7 1 27.20 26.43 26. 7 25. 3i 23.43 23. 7 10 27.27 26. 5 1 26.15 25.39 25. 3 24.27 23. 5i 23.15 10 29 28 28 27 27 26 25 25 24 24 2 2 2 3 20 27.34 26.58 26.22 25.47 25.11 24.35 23. 5q 23.23 20 23 22 22 21 21 20 20 19 18 18 4 3o 27.41 27. 6 26.30 25.54 25.19 24.43 24. 7 23.32 3o 17 17 16 i5 i5 i4 i4 i3 12 12 5 6 7 4 5 4o 27.49 27.13 26.38 26. 2 25.27 24.51 24.15 23. 4o 4o II II 10 9 9 8 8 7 6 6 5 54 5o 27.56 27.21 26.45 26.10 25.34 25.43 24.59 25. 8 24.24 24.33 23.48 23.58 5o 5 34 5 33 4 33 3 32 3l 2 37 2 37 1 35 29 29 U 9 7 28. 4 27.29 26.54 26.19 10 28.12 27.87 27. 2 26.26 25. 5i 25.16 24.41 24. 6 10 28 28 27 26 26 25 25 24 24 23 2 3 4 2 2 3 20 28.19 27.44 27. 9 26.34 25.59 25.24 24.49 24.14 20 22 22 21 21 20 19I19 18 18 17 3o 28.27 27.52 27.17 26.42 26. 7 25.32 24.58 24.23 3o 17 16 i5 i5 i4 i4 i3 12 12 II 6 4 4o 28.34 27.59 27.25 26.50 26.15 25.41 25. 6 24.31 4o II 10 10 9 8 8 7 7 *; 5 6 55 5o 28.42 28. 7 27.32 26.58 26.23 26.32 25.49 25.14 24.40 24.49 5o 5 33 4 32 4 32 3 3i 3 3i 2 3^ I 3^ I 29 9 6 7 28.50 28.16 27.41 27. 7 25.58 25.24 lO 28.58 28.23 27.49 27.15 26.4026. 6 25.32 24.58 10 27 27 26 26 25 24 24 23 23 22 2 2 20 29. 5 28.31 27.57 27.23 26.49 26. 1 4 25.40 25. 6 20 22 21 21 20 19 1918I18 17 17 4 3 3-0 29. i3 28.39 28. 5 27.31 26.67 26.23 25.49 25.15 3o 16 i5 i5 i4 i4 i3i3 12 11 II 5 6 7 4 4o 29.20 28.46 28.12 27.39 27. 526.31 25.57 25.23 4o 10 to 9 9 8 7 7 6 b 5 6 56 5o 29.28 29.35 28.54 28.20 27.47 27.13 26.39 26. 5 25.32 DO 5 4 3 3 2 2 33 I 3^ I 29 29 9 7 29. 2 28.28 27.55 27.21 26.47 26.14 25.40 33 32 32 3i 3i 10 29.43 29. 9 28.36 28. 3 27.29 26.56 26.22 25.49 10 27 27 26 26 25 25 24 24 23 22 2 3 4 2 2 3 20 29.50 29.17 28.44 28.11 27.3727. 4 26. 3i 25.58 20 22 21 21 20 20 19 19 18 18 17 3o 29.58 29.25 28.52 28.19 27.4627.12 26.3926. 6 3o 16 16 i5 i5 i4 i4 i3 i3 12 II b 6 4 4o 3o. 6 29.33 29. 28.27 27.54 27.21 26.4826.15 4o II 10 £0 9 9 8 8 7 b b 7 6 57 5o o 3o.i3 29.40 29. 8 28.35 28.44 28. 2 27.29 26.56 26.23 27. 626.33 5o 5 3^ 5 3i 4 3i 4 3o 3^ 3 29 2 29 I 1 21 27 9 7 3o.22 29.49 29.17 28.11 27.39 10 3o.3o 29.57 29.25 28.52 28.19 27.47 27.14 26.42 10 27 26 26 25 24 24 23 23 22 22 2 2 20 30.37 3o. 5 29.33 29. 28.28 27.55 27.23 26.51 20 21 21 20 20 19 19 18 17 17 16 4 3 3o 3o.45 3o.i3 29.41 29. 8 28.36 28. 4 27.32 26.59 3o 16 i5 i5 i4 14 i3 i3 12 12 11 b 6 4 4o 30.53 3o.2I 29.49 29.16 28.44 28.12 27.40 27. 8 4o 10 10 9 9 8 8 7 7 6 6 7 6 58 5o 3i. I 3i. 9 30.29 29.57 29.25 28.53 28.21 27.49 27.17 5o 5 5 4 3 3 2 2 Is I 27 I 27 8 9 30.37 3o. 6 29.34 29. 2 28.30 27.58 27.27 3i 3o 3o 29 29 10 3i .17 30.45 3o.i4 29.42 29.10 28.39 28. 7 27.35 10 26 25 25 24 24 23 23 22 22 21 2 3 4 2 2 3 20 3i.25 30.53 30.22 29.50 29.19 28.47 28.16 27.44 20 21 20 19 19 18 18 17 17 lb lb 3o 3i.33 3i. I 3o.3o 29.59 29.27 28.56 28.24 27.53 3o i5 i5 i4 i4 i3 10 12 12 11 II b 4 4o 3 1.40 3i. 930.38 3o. 7 29.36 29. 4 28.33 28. 2 4o 10 10 9 8 8 7 7 6 6 5 7 6 59 5o o 3 1. 48 31.57 3i .1730.46 31.26 30.55 3o.i5 3o.24 29.44 29.13 28.42 28.11 5o 5 3o _4 29 29 3 28 3 28 2 27 2 27 I 1 9 7 29.53 29.23 28.52 28.21 10 32. 5 3i.34 3i. 3 30.33 3o. 2 29.31 29. 28.30 10 25 24 24 23 23 22 22 21 21 20 2 2 20 32. i3 3i. 4231.12 3o.4i 3o. 10 29.40 29. 9 28. 3q 20 20 19 19 18 18 17 17 i6 lb i5 4 3 3o 32.21 3i.5o|3i .20 30.49 30.19 29.48 29.18 28.48 3o i5 i4 14 i3 i3 12 12 II II 10 5 4o 32.29 31.58 31.28 30.58 3o . 27 29.57 29.27 28.56 4o 10 9 9 8 8 7 7 6 b 5 7 6 5o 32.37 32. 6 3i.36 3r. 6 30.36 3o. 629.36 29. 5 5o 5 4 4 3 3 2 I 1 8 9 8 •■" ' TABLE XIX. [Page 123 Logarithms. Apparent Altitude of ]) 's centre. Table C. Cor. for Seconds of Parallax- Add. M. "5T s. o lO 20 3o 4o 5o o lO 20 3o 4o 5o 10 20 3o 4o 5o lO 20 3o 4o 5o lO 20 3o 4o 5o o 10 20 3o 4o 5o lO 20 3o 4o 5o o lO 20 3o 50 51 52 53 54 55 56 57 58 59 Sec. Cor. 23u6 2293 2279 2265 2252 2238 23o5 2291 2277 2264 225o 2237 23o3 2290 2276 22G2 2249 2235 2302 2288 2275 2261 2248 2234 23oi 2287 2274 2260 2246 2233 23uO 2286 2272 2259 2245 2232 2298 2285 2271 2258 2244 2230 2297 2284 2270 2257 2243 2230 2296 2283 2269 2256 2242 2229 22o5 2202 2268 2254 2241 2227 I 2 3 i 6 7 8 9 12 II 9 8 7 5 4 3 I 2225 22II 2198 2184 217I 2i58 2223 2210 2196 2l83 2x70 2i56 2222 2208 2195 2182 2168 2i55 2221 2207 2194 2180 2167 2i54 2219 2206 2193 2179 2I()6 2i53 2218 2205 219I 2178 2i65 2l52 2217 2204 2190 2177 2164 2l5o 2216 2203 2189 2176 2i63 2149 22l5 2202 2188 2175 2162 2i48 2214 2201 2187 2174 2l6l 2l47 Sec. Cor. 2i45 2l3l 2II8 2io5 2092 2079 2i43 2l3o 2II7 2104 2091 2078 2142 2129 2II6 2102 2089 2076 2l4l 2127 2Il4 2IOI 2088 2075 2i39 2126 2Il3 2100 2087 2074 2i38 2125 2112 2099 2086 2073 2i37 2124 2III 2098 2o85 2072 2i36 2123 2II0 2097 2084 2071 2i35 2122 2109 2096 2o83 2070 2i34 .2121 2108 2095 2082 2069 I 2 3 4 5 6 7 8 9 12 II 9 8 7 5 4 3 2 20G6 2o53 2o4o 2027 20l4 2002 2o65 2o52 2o39 2026 2ol3 2000 2o63 2o5o 2o37 2025 2012 1999 2062 2049 2o36 2023 2010 1998 2061 2048 2o35 2022 2009 1997 2060 2047 2o34 2021 2008 1996 2059 2o46 2o33 2020 2007 1994 2o58 2045 2o32 2019 2006 1993 2057 2044 203l 2018 2005 1992 2o56 2043 2o3o 2017 2004 1991 Sec. Cor. 1989 1976 1963 1 95 I 1938 1926 1987 1975 1962 1949 1937 1924 1986 1973 I96I 1948 1936 1923 1985 1972 i960 1947 1934 1922 1984 1971 1958 1946 1933 1921 1983 1970 1957 1945 1932 1920 1982 1969 195(3 1944 1931 I9I8 1981 1968 1955 1943 1930 1917 1980 1967 1954 1942 1929 1917 1979 1966 1953 1941 1928 1916 1 2 3 4 5 6 7 8 9 12 II 7 6 4 3 2 I913 I 90 1 1888 1876 i863 i85i 7838" 1826 i8i4 1802 1789 1777 I9I2 1899 18S7 1874 1862 1849 I9IO 1898 i885 1873 1861 1848 1909 1897 1 884 1872 1859 1847 1908 1896 J 883 1871 i858 1846 1907 1895 1882 1870 1857 1845 1833 1820 1808 1796 1784 1771 1906 1893 1881 1869 i855 i844 i832 1819 1807 1795 1783 1770 1905 1892 i8«o 1868 i855 1843 1904 1892 1879 1867 i854 1842 1903 I89I 1878 1866 i853 i84i Sec. Cor. 1837 1825 i8i3 1800 1788 1776 i836 1824 1811 1799 1787 1775 i835 1822 1810 1798 1786 1774 1834 1821 1809 1797 1785 1773 i83i 1818 1806 1794 1782 1770 i83o 1817 i8o5 1793 1781 1769 1829 1816 1804 1792 1780 1768 I 2 3 4 5 6 7 8 9 12 II 10 8 7 6 5 3 2 I 1765 1753 1741 1729 1764 1752 1740 1728 1763 i75i 1739 1727 1761 1749 1737 1725 1760 1748 1736 1724 1759 1747 1735 1723 1758 1746 1734 1722 1757 1745 1733 1 72 1 1757 1744 1732 1720 1756 1743 1731 1719 Pageiaj] TABLE XIX. Correction. iS a Table A. Tab.B. < S ]) 's Horizontal Parallax. P''oportional part for Seconds of Parallax. For xM. of Alt. i!^ Add. Add. D 60 M. 54' 55' 56' 3i.45 57' 3i.i5 53' 3o.45 59' I 60' 1 61' 1 S. 0" 29 [II 2" 2928 3" 4"! 27 G" l6 7" i6 8" 25 9" 3'5 1 s. 1 32.45 32. i5 3o. 1 5 29.45 29.15 ^ 27 10 32.53 32. 24 3 1. 54 3i.24 3o.54 3o. 24 29.54 29.24 1024 2423 23 22 22 21 21 20 30 2 3 20 1 33. I 32.3232. 2 31.32 3i. 3 3o.33 3o. 3 29.34 2019 1918 18 17 17 16 16 i5 i5 I 3 3o 33. 9 32.40 32.10 3i.4i 3i . II 30.42 30.12 29.43 3o i4 i4 i3 i3 12 12 II II 10 It) 5 4 5 4o 33.1? 32.48 32.19 31.49 3l.20 3o.5o 3o.2I 39.52 4o 9 9 8 8 7 7 6 6 5 5 ? 6 57 5o 33.25 33.34 32.5632.27 3i.58 32. 7 31.28 30.59 3o.3o 3o. I 5o 4 28 4 28 3 27 3 27 2 2 I I e 24 i 7 8 1 33. 5 32.36 3 1. 38 3i. 9 3o.4o 3o.ii 10 33. 4i 33.14 32.45 32.16 3i.47 3i .18 3o.49 3o . 20 10 23 23 22 22 21 21 20 20 19 '9 S 2 3 3 20 33. 5i 33.22 32.53 32.24 3i.55 3i .27 30.58 3o.39 20 18 18 17 17 '7 16 16 i5 i5 14 4 '3o 33. 5o 33. 3o 33. 1 32.33 32. 4 31.35 3i. 7 3o.38 3o 14 i3 i3 12 12 11 II 10 10 9 5 4 5 ;4o 34. 7 33.38 33.10 32. 4i 32. i3 31.44 3i.i6 30.47 4o 9 8 8 7 7 6 6 5 5 5 7 6 5o 34. i5 33.46 33.18 32. 5o 32.21 31.53 3 1. 25 3o.56 5o 4 _4 3 3 i 2 I I 8 9 7 8 6l 34.24 33.56 33.27 32.59 32. 3 1 32. 3 31.35 3i. 7 27 27 26 26 ^ ^ 24 T/i ^ ^ 1 1 10 34.32 34. 4 33.36 33. 8 32.40 32.12 31.44 3i.i6 10 22 22 21 21 21 20 20 19 '9 18 3 4 3 3 20 34.40 34.12 33.44 33.-I7 32.49 32.21 31.53 3i.25 20 18 17 17 16 16 i5 i5 i5 i4 1 4 3o 34.48 34.21 33.53 33.25 32.57 32. 3o 32. 2 3 1 . 34 3o i3 i3 12 12 II II 10 10 9 9 5 6 4 5 4o 34.56 34.29 34. I 33.34 33. 6 32.39 32.11 31.44 4o 8 8 8 7 7 6 6 5 5 4 7 6 63 5o 35. 5 35.14 34.37 34.40 34.10 34.19 33.42 33.52 33.15 33.25 32.48 33.58 32.20 31.53 5o 4 26 3 26 3 25 2 2"5 2 2 I I l3 22 i_ 32 7 8 1 32. 3o 32. 3 10 35.22 34.55 34.28 34. I 33.34 33. 7 32.39 32.12 10 22 21 21 20 20 19 19 18 18 17 2 2 3 20 35. 3o 35. 3 34.36 54. 9 33.42 33.16 32.49 32.22 20 17 17 16 i6 i5 i5i4 i4 i3 i3 4 4 3o 35.38 35.12 34.45 34.18 33. 5i 33.25 32.58 32. 3l 3o i3 12 12 II II 10 10 9 5 Q 4 40 35.47 35.20 34.53 34.27 34. 33.34 33. 7 32.40 4o 8 8 7 7 6 6 5 5 5 4 7 6 5o 35.55 35.28 35. 2 34.36 34. 9 33.43 33.16 32. 5o 5o 4 3 3 2 2 I I 8 7 8 64 36. 4 35.38 35.12 34.45 34.19 33.53 33 . 26 33. 25 ^ ^ ^ ^ ^ 22 22 32 21 ' 1 1 10 36.12 35.46 35.20 34.54 34. 28 34. 2 33.36 33. 9 10 21 20 20 19 ^9 19 18 18 17 17 ^ 3 4 20 36.21 35.55 35.29 35. 3 34.37 34.11 33.45 33.19 20 16 16 16 i5 i5 14 i4 i3 i3 13 4 3o 36.20 36. 3 35.37 35.12 34.46 34.20 33.54 33.28 3o 12 12 II II TO 10 9 9 9 8 5 6 4 5 4o 36.37 36.12 35.46 35.20 34.55 34.29 34. 3 33.38 40 8 7 7 6 6 6 5 5 4 4 7 8 9 1 6 65 5o 36.46 36.55 36.20 36. 3o 35.55 36. 4 35.29 35.39 35. 4 35.14 34.38 34.13 34.23 33.47 3"3.57 5o 3 24 3 24 3 2 l3 2 22 I 22 I 22 21 21 20 7 8 1 34.48 10 37. 3 36.38 36.13 35.48 35.23 34.57 34.32 34. 7 10 20 19 i5 19 19 18 18 17 17 17 16 2 3 4 3 4 20 37.12 35.47 36.22 35.57 35.32 35. 6 34.41 34.16 20 16 i5 14 i4 i4 i3 i3 12 13 3o 37.20 36.55 36. 3o 36. 5 35.41 35.16 34. 5i 34.26 3o 12 II II 10 10 9 9 9 8 8 5 4 5 40 37.28 37. 4 36.39 36.14 35.5ol35.25 35. 34.35 4o 7 7 7 6 6 5 5 4 4 4 7 6 66 5o 37.37 37.45 37.12 37.2, 36.48 36.5(i 36.23 36.32 35.5935.34 36. 8135.43 35. 9 34.45 5o 3 T4 3 2 i3 2 i3 2 22 I 22 I 22 21 21 30 8 9 1 7 8 1 35.19 34.54 10 37.54 37.29 37. 5 36.41 36.1735.52 35. 28 35. 4 10 20 20 19 19 18 18 i8 17 n 16 2 2 3 4 20 38. 2 37.3s 37.14 36. 5o 36.2636. 2 35.38 35.13 20 16 16 i5 i5 14 i4 i4 i3 i3 13 4 3o 33.11 37.47 37.23 36.59 36.35'36.ii 35.47 35.23 3o 12 12 II II 10 10 10 9 9 8 5 6 4 5 4o 38.10 37.55 37.31 37. 8 36.44 36. 20 35.56 35.33 4o 8 8 7 7 6 6 6 5 5 4 7 6 5o 38.27 38. 4 37.40 37.17 36.53 36.29 36. 6 35.43 5o 4 4 3 3 3 2 2 I I 8 9 7 8 67 38.37 38.13 37 . 5o 37.27 37. 3 36.40 36. 16 35.53 23 23 22 22 21 21 21 20 30 30 1 1 10 38.45 38.22 37.59 37.36 37. i2|36.49 36.26 36. 2 10 19 19 18 18 18 17 17 16 16 16 2 3 4 2 20 38.54 38. 3i 38. 8 37.45 37.31 36.58 36.35 36.12 20 i5 i5 i5 1 4 i4 i3 i3 i3 12 12 4 3o 3.9. 2 38.39 38. T7 37.54 37.3i!37. 8 36.45 36.22 3o II II II 10 10 10 9 9 8 8 5 6 5 4o 39. u 38.48 38.2 5 38. 3 37.40 37.17 36.54 36. 3i 4o '8 7 7 6 6 6 5 5 5 ) 7 6 68 5o 39.19 38.57 38.34 38.12 38.22 37.49 37.26 37. 4 36. 4r 5o _4 3 3 3 2 2 20 2 20 I 19 I 19 19 8 9 1 7 8 1 39.29 39. 7 38.44 37.59 37.37 37.14 36.52 22 22 21 21 21 10 39.38 39.15 38.53 38. 3 1 38. 8 37.46 37.24 37. I 10 18 18 18 17 17 16 16 16 i5 i5 2 20 39.46 39.24 39. 2 38. 4o 38. 18 3^.55 37.33 37.11 20 i5 i4 ■ 4 i4 i3 i3 12 12 12 11 4 3 4 3o 39.55 39.33 39.11 38.49 38.37 38. 5 37.43 37.21 3o 1 1 II 10 10 9 9 9 8 8 8 5 G 5 5 4o 4o. 3 39.41 39 . 20 38.58 38.36 38.14 37.52 37.30 4o 7 7 7 6 6 5 5 5 4 4 7 6 69 5o 40.12 39.50 39.29 39. 7 38.45 38.55 38.3.4 38. 2 37.40 5o 4 3 3 2 2 2 19 I I '9 • 18 78 8 9 I 7 8 1 2 4o.3I 40. 39.3s 39.17 38.34 38.12 37.5. 21 2 1 20 20 30 10 4o.3o 4o. 9 39.47 39.26 39. 5 38.43 38.2 2 38. 1 10 17 17 17 16 16 16 ID i5 i5 i4 20 40.39 4o.i8 39.56 39.35 39.1^ 38.53 38.32 38.10 20 i4 i4 i3 i3 i3 12 12 12 II 1 1 3 4 3 4 3o 40.47 40.26 4o. 5 39.44 39.33 39. 2 38. 4 1 38. 20 3o to 10 10 g 9 9 8 8 8 7 5 5 4o 40.56 40.35 4o.i4 39.53 39.33 39. 12 38. 5 1 38. 3o 4o 7 7 6 6 6 5 5 4 4 4 7 6 5o 4i. 5 4o.4'i 4n.23 4o. 3 39.43 39. 21 39. , 38. 40 5o 3 3 3 2 2 2 I I I 8 9 7 8 TABLE XIX. [Page 125 Logarithms. 5 2 Tablk C. 1 Cor. for Seconds Apparent Altitude of ]> 's centre. of Pa rallax. '='Sh Add. 1 c c ^5T s. GO Gl 62 G3 64 G5 66 67 68 69 Sec. Cor. 2295 2294 2293 2292 2291 2291 2290 2289 2289 2288 12 10 22«I 2280 2279 2278 2278 2277 2276 2276 2273 2274 I II 20 2267 2266 2266 2265 2264 2263 2263 2262 2261 2261 2 9 «0 2254 2253 22D2 225l 225o 225o 2249 224s 2248 2247 3 8 4o 2240 2239 2238 2238 2237 2236 2236 2235 2234 2234 4 7 55" 5o o 2227 222t) 2225 2224 2223 2223 2222 2221 2221 2S20 5 6 5 4 22l3 2212 2212 2211 2210 2209 2209 2208 22CJ7 2207 10 2200 2199 2198 2197 2197 2196 2195 2195 2194 2194 2180 7 3 20 2186 2186 2185 2184 2i83 2i83 2182 2181 2181 8 I Jo 2173 2172 217I 217I 2170 2169 2169 2168 2167 2167 9 "56 4o 5o o 2160 2l47 2159 2 1 46 2i58 2145 2i57 2144 2 I 57 2143 2i56 2i43 2l55 2142 2i55 2l4l 2 I 54 2l4l 2i54 2l4o Sec. Cor. 2i33 2l33 2l32 2l3l 2l3o 2i3o 2129 2128 2128 2127 12 10 2120 2119 2119 2118 2117 2116 2I16 2Il5 21l4 21l4 I II 20 2107 2106 2io5 2105 2I04 2io3 2X03 2102 2101 2101 2 9 8 3o 2094 2093 2092 2092 20'91 2090 2090 2089 2088 2088 3 4o 20«1 2080 2079 2078 2078 2077 2077 2076 2075 2075 4 7 ^ 5o o 2068 2067 2066 2o65 2o65 2064 2064 2o63 2062 2062 5 6 5 4 2o55 2o54 2o53 2o53 2052 205l 205l 2o5o 2049 2049 lO 2042 2o4l 2040 2o4o 2039 2o38 2o38 2o37 2o36 2o36 7 3 30 2029 2028 2028 2027 2026 2025 2025 2024 2024 2023 8 2 3o 2016 2016 20 1 5 20l4 20l3 20l3 2012 20 I I 2011 2010 9 T8 4o 5o 2004 1991 2oo3 1990 2002 1989 2001 1988 2000 1988 2000 1987 1999 1986 1999 1986 1998 1985 1998 1985 Sec. Cor. 1978 1977 1976 1976 1975 1974 1974 1973 1972 1972 12 lO 1965 1965 1964 1963 1962 1962 1961 i960 i960 1959 I II 2o 1953 1952 1 95 1 1950 1950 1949 1948 1948 1947 1947 2 9 3o 1940 1939 1938 1938 1937 1936 1936 1935 1934 1934 3 8 4o 1927 1927 1926 1925 1924 1924 1923 1923 1922 1921 4 7 ^ bo o 1915 1914 1913 1912 1912 I91I 1911 1910 1909 1909 5 6 6 4 1902 1902 I 90 1 1900 1899 1899 1898 1898 1897 1896 10 1890 1889 1888 1887 1S87 1886 1886 1885 1884 i884 7 3 20 1877 1877 1876 1875 1874 1874 1873 1873 1872 1872 8 2 3o i865 1864 i863 i863 1862 i8bi 1861 i860 i860 1859 9 I ()0 4o 5o i853 i84o i852 i84o i85i 1839 i85o 1 838 i85o 1837 1849 1837 1848 i836 i848 i836 1847 i835 1847 i834 Sec. Cor. 1S28 1827 1826 1826 1825 1824 1824 1823 1823 1822 12 lO 1816 i8i5 i8i4 i8i3 i8i3 1812 1812 1811 1810 1810 1 II 20 i8o3 i8o3 1802 1801 1801 1800 1799 1799 1798 1798 2 10 3o 1791 1791 1790 1789 1788 1788 1787 1787 1786 1786 3 8 4o 1779 1778 1778 1777 1776 1776 1775 1774 1774 1773 4 7 ""bT 5o o 1767 1755 1766 1754 1765 ""i"753" 1765 1764 1763 1763 "1762 1762 1761 5 6 6 5 4 1753 1752 1751 1751 1750 1750 1749 10 1743 1742 1741 1740 1740 1739 1739 1733 1737 .737 7 8 20 i73i 1730 1729 1728 1728 1727 1727 1726 1725 1725 JO 1719 1718 1717 1716 1716 17x5 I7i5 1714 17.3 1713 9 I Pageias] TABLE XIX. Correction. ij c Table A. Tab.B. < 8 D 's Horizontal Parallax. Correction for Seconds of Parallax. For M. of Alt. -=;'■* Add. Add. D. M. 54 55 56' 57 58' 39.52 59' 60' 61' S. 0" 20 1" 20 2" 19 3" 19 1" 19 5" 78 G" 78 7" 78 8" 17 9" 17 M. s. 70 4i.i4 40.54 40.33 4o.i3 39.32 39.11 38. 5i 10 41.28 4i. 3 40.42 4o.22 40. 2 39.41 3o.2I 39. I 10 17 16 lO 16 i5 i5 i5 i4 1 4 i4 1 2 2 20 4i.32 41.12 4o.bi 40. 3i 4o.li 39.51 39.31 39.10 20 i3 i3 i3 12 12 12 II 11 II 10 3 4 3o 4i.4o 41.20 4i. 4o.4o 40.20 40. 39.40 39.20 3o 10 10 9 9 9 8 8 8 7 7 .=. • 5 6 6 40 41.49 41.29 4i. 9 4o.5o 4o.3o 40.10 39.50 J9.30 4o 7 6 6 6 5 5 5 4 4 4 6 71 bo 41.58 41.38 4i.i8 40.59 40.39 40.19 4o. 39.40 5o 3 3 3 2 2 2 17 1 17 I 17 1 76 76 7 8 42. 8 41.48 41.28 4i- 9 40.49 4o.3o 4o.io 39.51 19 19 18 18 18 10 42.16 41.57 41.J8 41-18 40.59 40.39 40.20 4o. I 10 16 lb i5 lb i5 i4 i4 i4 ]3 i3 2 20 42.25 42. 6 41.47 41.27 4i. 8 40.49 4o.3o 40.11 20 i3 12 12 12 II 11 II 10 10 10 3 4 4 Jo 42.34 42.1b 4i.bfa 41.37 41.18 40.59 40.39 40.20 3o 9 9 9 8 8 8 8 7 7 7 5 5 40 42.43 42.24 42. 5 41.46 41.27 4i. 8 40.49 40. 3o 4o 6 6 6 b 5 5 4 4 4 3 7 7 72 bo 42. 5i 42.33 42.14 41-55 4i.36 41.18 40.59 40. 4o 5o 3 3 78 2 17 2 17 2 17 I 76 I 76 I 76 I 76 75 8 9 7 8 43. I 42.43 42.24 42. 5 41-47 41.28 4i.io 4o.5i 18 10 43.10 42. 5i 42. JJ 42.15 41.56 4i.38 4i .20 4i. I 10 i5 lb i4 i4 i4 i3 i3 i3 i3 12 k 2 20 43.19 43. 42.42 42.24 42. 6 41.48 41.29 4i.ii 20 12 12 II 11 11 10 10 10 10 Q 4 4 Jo 43.27 4':^. 9 42. 5i 42.33 42.15 41.57 4i .39 4l.21 3o 9 9 8 8 8 7 7 7 7 6 5 6 7 a 40 43.36 4J-18 43. 42.43 42.25 42. 7 41.49 41-J1 40 6 6 5 b 5 4 4 4 3 3 7 73 bo 43.45 43.55 4J-27 43.37 43.10 43.20 42.52 43. 2 42.34 42.17 41.59 4i-4i 5o 3 3 2 2 2 I 76 1 75 I 75 75 74 8 9 8 8 42.45 42.27 42.10 41.52 17 17 16 16 16 10 44. 4 43-46 43.29 43.12 42.54 42.37 42.19 42. 2 10 i4 i4 14 i3 i3 i3 12 12 12 12 2 2 20 44. iJ 43. bb 43.38 43.21 43. 4 42.47 42.29 42.12 20 11 11 11 10 10 10 10 9 9 9 4 4 Jo 44-21 44. 4 43.47 43. 3o 43.13 42.56 42.39 42.22 3o 8 8 8 8 7 7 7 6 6 6 5 5 40 44. 3o 44.13 43.57 43.40 43.23 43. 6 42.49 42.32 4o 6 5 5 5 4 4 4 4 3 3 7 7 74 bo 44.39 44.22 44. 6 43.49 43.32 43.43 43.16 43.26 42.59 43.10 42.42 42.53 5o 3 16 2 76 2 73 2 75 2 75 I 75 I 74 I 74 74 74 8 9 8 44.49 44-32 44-16 43.59 10 44.58 44.42 44-25 44. 9 43.52 43.36 43.20 43. 3 10 i3 i3 i3 12 12 12 12 II u II 2 20 45. 7 44. 5i 44. S4 44-18 44. 2 43.46 43. 3o 43.13 20 11 10 10 10 10 9 9 9 8 8 4 4 Jo 45.16 45. 44.44 44.28 44-12 43.56 43.40 43.23 3o 8 8 7 7 7 7 6 6 6 5 5 6 7 5 4o 45.25 45. 9 44.53 44.37 44-21 44. 5 43.49 43.34 4o 5 5 5 4 4 4 4 3 3 3 7 75 bo 45.34 45.43 45.18 45. 2 44-46 44-57 44. Ji 44-41 44.15 44-26 43.59 44.10 43.44 43.55 5o 3 Is 2 75 2 74 2 74 I 74 I 74 1 73 1 73 73 73 8 9 8 9 45.28 45.12 10 45.52 45.37 45.22 45. 6 44-5i 44-36 44-20 44. 5 10 12 12 12 12 II II 11 II 10 10 2 2 20 46. I 45.46 45.31 45.16 45. 1 44-45 44 -3o 44.15 20 10 10 9 9 9 8 8 8 8 4 4 Jo 46.10 4b. 55 45.40 45.25 45.10 44-55 44-40 44-25 3o 7 7 7 7 6 6 6 6 5 5 5 5 6 7 40 46.19 46. 4 45.50 45.35 45.20 45. 5 44. 5o 44-35 4o 5 5 4 4 4 4 3 3 3 3 7 76 bo 46.28 46.38 46.14 46.24 45.59 46. 9 45.44 45.55 45.29 45.40 45. i5 45.26 45. 44-45 5o 2 I4 2 74 2 74 2 73 I 73 I 73 I 71 I 12 12 13 8 9 8 9 45.11 44-57 10 46.47 46.33 46.18 46. 4 45. 5o 45.35 45.21 45. 7 10 12 11 11 11 11 10 IC 10 10 10 2 2 20 46.56 46.42 46.28 46.14 45.59 45.45 45. 3i 45.17 20 9 9 9 9 8 8 8 8 7 7 4 4 Jo 47. 5 4b. 5i 46. J7 46.23 46. 9 45.55 45.41 45.27 3o 7 7 7 6 6 6 e 5 5 5 5 6 7 5 6 7 4o 47.14 47. 46.46 46.33 46.19 46. 5 45. 5i 45.37 4o 5 4 4 4 4 3 3 3 3 3 77 bo 47-23 47- 9 46.56 47- 6 46.42 46.53 46.28 46.39 46. i5 46.26 46. I 46.12 45.47 45.59 5o 2 73 2 73 2 73 2 12 I 12 I 12 I 12 1 11 u 1 1 8 9 8 9 47-33 47.20 lO 47-42 47-29 47-15 47- 2 46-49 46.35 46.22 46. 9 10 II II 10 10 10 10 IC 9 9 Q 2 2 20 47-5i 47-38 47-25 47-12 46.59 46.45 46.32 46.19 20 9 8 8 8 8 8 7 7 7 7 3 4 3 4 Jo 48. 47-47 47 -J4 47-21 47- 8 46.55 46.42 46.29 3o 6 6 6 6 6 5 5 5 5 5 5 5 4o 4B- 9 47-56 47-44 47-31 47.18 47. 5 46.52 46.39 4o 4 4 4 4 3 3 3 3 3 2 7 7 78 bo 48.18 48.28 48. 6 48.16 47-53I47-40 48. 3I47-51 47-28 47-15 47- 2 46. 5o 47. 1 5o 2 12 2 12 2 12 I 1 1 1 II I 11 I II I 11 10 10 8 9 8 9 47-38 47-26 47-i3 10 48 37 48.25 48. i3 4«. 47-48 47-36 47-24 47-11 10 10 10 10 9 9 9 c 9 8 8 2 20 48.46 48.34 48.22 48.10 47-58 47-46 47-34 47-21 20 b 8 8 7 7 7 7 7 6 6 4 4 Jo 48.55 48.44 48.32 48.20 48. 8 47-56 47-44147-32 3o 6 6 6 5 5 5 5 5 4 4 5 4o 49. 5 48.53 48.41 48.29 48.17 48. 6 47-54 47-42 4o 4 4 4 3 3 3 3 3 2 2 7 7 79 5o 49.14 49. 2 48. 5o 48.39 48.27 48.38 48.16 48.26 48- 4 48. i5 47-52 48. 4 5o 2 1 1 2 11 2 11 1 10 I 10 1 10 1 10 I 10 10 9 8 9 U 8 9 49.24 49.12 49. 1 48.49 ID 49.33 49.22 49-10 48.59 48.48 48.36 48.25 48.14 10 9 9 9 9 8 8 8 8 8 8 2 2 20 49-42 49-3i 49.20 49- 9 48.57 48.46 48.35 48.24 20 7 7 7 7 7 6 6 6 6 6 3 4 3 4 Jo 49-5i 49.40 49-29 49-18 49- 7 48.56 48.45 48.34 3c 5 5 5 5 5 5 4 4 4 4 5 6 7 5 4o 5o. 49.49 49-39 49.28 49.17 49- 6 48.55 48.45 4o 4 3 3 3 3 3 3 2 2 2 7 li)o 5o. 9 49-59 49.48 49.37 49.27 49-16 49- 6 48.55 5o 2 2 ' I I I I 1 8 9 8 9 t TABLE XIX. Logarithms. [Page 127 s g Tablk C. 1 33^ Cor. for Seconds 1 Jf a Apparent Altitude of 5 's Centre. of Parallax. aCU Add. o M. IT s. c 70 71 72 73 2286 74 2285 75 2 285 76 2285 77 78 79 Sec. Cor. 2287 2287 2286 2284 2284 22S4 12 lO 2274 2273 2273 2272 2272 2272 2271 2271 2271 2271 I II 20 2260 2260 2259 2259 2258 2258 2258 2257 2257 2257 2 9 3o 2247 2246 2246 2245 2245 2245 2244 2243 2243 2243 3 8 4o 2233 2233 2232 2232 223l 223l 223l 2230 223o 223o 4 7 Is" 5o o 2220 2219 2219 2218 2218 2218 2217 2217 2216 2216 5 6 5 4 2206 2206 2205 2205 2204 2204 2204 2 2o3 22u3 2 2o3 lO 2193 2192 2192 2191 2191 219I 2190 2190 2190 2190 7 8 3 20 2179 2179 2179 2178 2178 2178 2177 2176 2176 2176 I 3o 2166 2166 2lb5 2i65 2lb4 2i64 2164 2i63 2i63 ' 2i63 9 "56" 4o 5o 2i53 2l40 2l52 2139 2l52 2i39 2l52 2i38 2l5l 2i38 2l5l 2i38 2l50 2i37 2i5o 2 1 37 2) 5c 2:5o 2i37 2i37 Sec. Cor. 2 I 26 2126 2126 2125 2125 2125 2124 2123 2123 2123 12 lO 2Il3 2Il3 2II2 2II2 2U2 2111 2III 2110 2110 2II0 I II 20 2100 2100 2099 2099 2098 2098 2098 2097 2097 2097 2 Jo 20S7 2087 2086 20S6 2o85 2o85 2o85 2084 2084 2084 3 6 4o 2074 2074 2073 2073 2072 2072 2072 2071 2071 2071 4 7 37 5o o 206 1 2061 2060 2060 2059 2059 2o59 2o58 2o58 2o58 5 6 5 4 2048 2048 2047 2047 ao46 2o46 2o46 2o45 2045 2045 10 2o35 2o35 2o34 2o34 2o34 2o33 2o33 2o32 2o32 2032 7 3 20 2022 2022 2022 2021 2021 2021 2020 2019 2019 2019 8 2 3o 2010 2009 2009 2008 2008 2008 2007 2007 2007 2007 9 4o 1997 1996 1996 1995 1995 1995 1994 1982 1994 1994 1994 I981 "sF 5o 1984 1984 1983 1983 19S2 1982 I98I 1968 I981 Sec. Cor. 1971 I97I 1970 1970 1970 1970 1969 1968 1968 12 ID '9^9 i9b8 1958 1957 1957 1957 1956 1956 1956 1956 I II 20 1946 1946 1945 1945 1944 1944 1944 1943 1943 1943 2 9 Jo 19J3 1933 1933 1932 1932 1932 I93I 1931 1930 1930 3 8 40 1921 1920 1920 1920 1919 1919 1919 1918 1918 1918 4 7 5^ bo o 1908 1896 1908 78"^ 1907 1895 1907 1895 1907 1894 1907 1894 1906 1893 1905 1905 1893 1905 1893 5 6 6 4 1893 ID i883 1883 1882 1882 1882 1882 I88I 1880 1880 1880 7 3 20 1871 1870 1870 1870 1869 1869 1869 1868 1868 1868 8 2 3o i858 i858 i8b8 i857 1857 j857 1 856 i856 1 856 i856 9 I "fc 4o 5o 1 846 i834 1 846 1 833 1845 i833 1845 i833 1844 i832 1844 i832 1844 i832 1843 i83i 1843 i83i 1843 i83i Sec. Cor. 1822 1821 1821 1820 1820 1820 1819 1819 1819 1819 12 lO 1809 1809 1808 1S08 1808 1808 1807 1806 1806 1806 I II 20 1797 1797 1796 1796 1795 1795 1795 1794 1794 1794 2 10 3o 1783 17S4 1784 1784 1783 1783 1783 1782 1782 1782 3 8 4o 1773 1772 1772 1771 1771 1771 1770 1770 1770 1770 4 7 "gF 5o 176. 1760 1760 1759 n^9 1759 1758 1758 1758 1758 5 6 6 5 1719 1748 1748 1747 1747 1747 !746 1746 1746 1746 lO 1736 1736 1736 1735 1735 1735 1734 1734 1734 1734 8 3 20 I7'4 1724 1724 1723 1723 1723 1722 1722 1722 1722 2 3o I7I2 1712 1712 1711 1711 1711 1710 1710 1710 1710 9 1 r-ige 123] TABLE XIX. Correction. . ^ c Table A. Tab. B. < S ]) 's Horizontal Parallax. Proportional part for Seconds of Parallax. ForM of Alt. <" Add. Add. D 8^ M. 54' 55' 56' 57' 58' 59' 60' CI' S. 0" 10 1" 10 2" 10 3" 10 4" 5" 6" ~9 711 9 9 9^ 9 M. s. 1 So.iySo. Q 49-58 49-48 49.38 49.27 49.17 49. 6 10 5o. 2850.18 5o. 8 49-58 49.47 49.37 49.27 49-17 10 8 8 8 8 8 8 7 7 7 7 2 1 20 50.38 50.27 5o.i7 5o. 7 49.57 49-47 49.37 49.27 20 7 7 b 6 6 6 6 6 5 5 a 4 3a 50.47 50.37 10.27 50.17 5o. 7 49-57 49.47 49.37 3o 5 5 5 5 4 4 4 4 4 4 5 6 40 50.56 5o.46 5o.36 50.27 50.17 5o. 7 49.57 49.48 4o 3 3 3 3 3 3 2 2 2 2 I 9 7 87 10 5i. 5 50.55 5i. 6 5o.46 50.56 5o.36 5o.47 50.27 5o.i7 5o. 8149. 58 5o 2 2 I I I I "8 I ~8 I ~8 ~8 9 5i.i5 50.37 50.28 50.19 5o. 9 9 9 9 9 8 10 51.24 5i.i5 5i. 6 5o.57 50.47 50.38 00.29 5o.20 10 8 7 7 7 7 7 7 6 6 6 2 ).<) 51.33 51.24 5t.i5 5i. 6 5o.57 50.48 50.39 5o.3o 20 6 6 6 6 5 5 5 5 5 5 4 4 37 58 270C 9615 783o 6568 5592 4795 4l22 3540 3026 58 S. .2626 9579 7806 655o 5578 4783 4lI2 353i 3oi8 59 S. 0° (y 0° V 0° 2' 0° ?,' 0° 4' 0" 5' 0° (V 0° 7' 0° B' TABLE XXIL [Pag e 133 Proportional Logarithms. s. o A m h m h m h m h VI /( VI h m A m A 711 0° 9' 0° 10' 0° 11 0° 12' 0° 13' 0° 14' 0° 15' 0° 16' 0° 17' S. 1 .3(110 1.2553 I .2r39 I .1761 I .i4i3 I. 1091 I .0792 I .o5i2 I .0248 I 3on2 2545 2l32 1755 1 408 1086 0787 o5o7 0244 I 2 3094 2538 2126 1749 l402 108 1 0782 o5o2 0240 2 J 29156 2 D 3 1 2119 1743 1397 1076 0777 0498 0235 3 4 5' 2978 2524 2Il3 1737 I39I 1 07 1 0773 0493 023l 4 5 1 . 2970 i.25i7 I . 2 1 06 I.I73I 1.1386 1 .1066 I .0768 1 .0489 1 .0227 6 2962 25lO 2099 1725 i38o 1061 0763 o484 0223 6 7 2954 25o2 2093 1719 1 374 io55 0758 0480 0219 7 8 2946 2495 20S6 1713 i369 io5o 0753 0475 02l4 8 _9_ lO 2939 2488 2080 1707 1 363 1045 0749 1.0744 0471 0210 9 10 1 .?93i I .2481 I .2073 I .1701 I.I358 I . io4o I .0467 I .0206 II 29^3 2474 2067 1695 i352 io35 0739 G.562 0202 II 12 P915 2467 2061 1689 1 347 io3o 0734 o458 0197 12 i3 2907 2460 2o54 1 683 1 342 1025 0730 0453 0193 i3 i4 i5 2899 2453 2o48 1677 1 336 1020 0725 0449 0189 14 i5 1 .2S91 1.2445 1 . 204 1 1.1671 i.t33i I .ioi5 1 .0720 1 .0444 I.OI85 i6 2883 2438 20 3 5 1 665 1 325 1009 0715 o44o 0181 16 17 2876 243i 2028 1660 1 3 20 1004 071 1 0435 0176 17 i8 28(18 2424 2022 i654 i3i4 0999 0706 043 1 0172 18 '9 20 2860 24-17 2016 1 648 1 309 0994 0701 0426 0168 ''9 20 I .2852 I .2410 1 .2009 1 . 1642 I .i3o3 I .0989 I . 0696 I .0422 1 .0164 21 2845 24o3 2003 i636 1298 0984 0692 o4i8 0160 21 22 2837 2396 1996 i63o 1292 0979 0687 o4i3 oi56 22 2j 2829 2389 1990 1624 1287 0974 0682 0409 oi5i ^3 24 25 2821 2382 1984 1619 1282 0969 0678 o4o4 0147 24 25 1. 2814 I .2375 1-1977 i.i6i3 1 .1276 1 .0964 1 .0673 I .o4oo I .0143 2b 2806 2368 I97I 1607 1271 0959 0668 0395 0139 26 27 2798 2362 1965 iGoi 1266 0954 o663 0391 oi35 27 28 2791 2355 1955 1595 1200 0949 0659 o387 oi3i 28 29 3o 2783 2348 1952 1 589 1255 0944 o654 o382 0126 29 3o 1.2775 1 .2341 I .1946 1. 1 584 1 .1249 1 .0939 I .0649 1.0378 I .0122 Ji 3768 2334 1939 1578 1244 0934 0645 o374 0118 3i J 2 2760 2327 1933 1572 1239 0929 o64o 0369 oii4 32 ciJ 2753 2320 1927 1 566 1233 0924 o635 o365 0110 33 M 35 2745 23i3 1 92 1 i56i 1228 0919 o63i o36o 0106 34 35 1.2738 I .23o7 I.T9I4 I.I555 I .1223 I. 0914 1 .0626 I .o356 1 .0102 36 2730 23oo 1908 1 549 I217 0909 0621 o352 0098 36 ^7 2722 2293 1902 1 543 I2I2 0904 06 1 7 o347 0093 37 38 27t5 2286 1896 i538 1207 0899 06 1 2 o343 0089 38 39 40 2707 2279 1889 i532 I20I 0894 0608 0339 008 5 39 40 I .2700 I .2272 I.I883 1.1526 1 . I I 96 1.08S9 I .o6o3 i.o334 1 .0081 4i 2692 2266 1877 I 520 M91 0884 0598 o33o 0077 4i ^2 2685 2259 1871 i5i5 1 1 86 0880 0594 0326 0073 42 43 2678 2252 i865 1 509 1 180 0875 0589 032I 0069 43 44 45 2670 2245 1859 i5o3 1175 0870 o585 o3i7 oo65 44 45 1.2663 1 .2239 I.I852 1. 1498 1 . 1 1 70 I.0865 i.o58o I .o3i3 1 .0061 46 2655 2232 1 846 1492 1164 0860 0575 o3o8 00 5 7 46 47 2648 2225 i84o i486 1 1 59 o855 0571 o3o4 oo53 47 48 2640 2218 i834 i48i ii54 o85o o566 o3oo 0049 48 49 5o 2633 2212 1828 1475 1 149 0845 o562 0295 0044 49 5o I .2626 I .2205 1. 1822 1. 1469 1. 1 143 i.o84o 1.0557 1 .0291 1 . 0040 5i 2618 2198 1816 1464 ii38 c)835 o552 0287 oo36 5i 52 2611 2192 1809 1458 ii33 o83i o548 0282 oo32 52 53 2604 2i85 7 8o3 i452 1128 0826 o543 0278 0028 53 54 55 2596 2178 1797 1447 I I 23 0821 0539 0274 0024 54 55' 1.2589 r .2172 1.1791 1.1441 1.1H7 I. 0816 1.0534 1.0270 1 .0020 56 2582 2i65 1785 i436 1112 081 1 o53o 0265 0016 56 5-; 2 574 2159 1779 i43o 1107 0806 o525 0261 0012 57 58 2567 2l52 1773 1424 1102 0801 0D2I 0257 0008 58 59 S. 2 56o 2r45 1767 1419 1097 0797 o5i6 0252 ooo4 59 S. 0° 9' 0° 10' 0° 11' 0° 12' 0° 13' 0° 14' 0^ 15' 0° 16' 0° 17' P''g«i34] TABLE XXII. Proportional Logarithms. S. o I 2 3 4 5 6 7 8 9 lO II 12 i3 i4 i5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 35 37 38 39 40 4i 42 43 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 S. h m 0° 18' h m 0°19' h m 0°20' h m 0°21' h rn 0°22' h m 0°23' h m 0°24' h m 0°25' h VI 0°26' h m 0°27' h m 0°28' h m 0°29' S. 1 0000 9996 9988 9984 9765 9761 9758 9754 9750 9542 9539 9535 9532 9528 9331 9327 9324 9320 9317 9128 9125 9122 9119 9115 8935 8932 ^9=? 8926 8923 8751 8748 8745 8742 8739 8573 8570 8568 8565 8562 84o3 84oo 8397 8395 8392 8239 8236 8234 823i 8228 8081 8079 8076 8073 8071 7929 7926 7924 7921 7919 I 2 3 4 ' 5 6 8 9 10 II 12 i3 i4 i5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 A^ 45 A(> 47 48 49 5o 5i 52 53 54 99S0 9976 9972 996S 9964 9746 9742 9739 9735 973 1 9524 9521 9517 95i4 9510 93i3 9310 9306 93o3 9300 9112 9109 9106 9102 9099 8920 8917 8913 8910 8907 8904 8901 8898 8895 8892 8736 8733 8730 8727 8724 8559 8556 8553 855o 8547 8389 8386 8384 838 1 8378 8226 8223 8220 8218 8215 8068 8066 8o63 8061 8o58 7916 7914 791 1 7909 7906 9960 9956 9952 9948 9944 9727 9723 9720 9716 9712 9506 95o3 9499 9496 9492 9296 9293 9289 9286 9283 9096 9092 9089 9086 9083 8721 8718 8715 8712 8709 8544 8542 8539 8536 8533 8375 8372 8370 8367 8364 8212 8210 S207 8204 8202 8o55 8o53 8o5o 8o48 8045 7904 7901 7899 7896 7894 9940 9936 9932 9928 9924 9920 9916 9912 9908 9905 9708 97o5 9701 9697 9693 9488 9485 9481 9478 9474 9279 9276 9272 9269 9266 9079 9076 9073 9070 9066 8888 8885 8882 8879 8876 8706 8703 8700 8697 8694 853o 8527 8524 8522 85i9 836i 8359 8356 8353 835o 8199 8196 8194 8191 8188 8043 8o4o 8037 8o35 8o32 7891 7889 7887 7884 7882 9690 9686 9682 9678 9675 9471 9467 9464 9460 9456 9262 9259 9255 9252 9249 9063 9060 9057 9053 90 5 8873 8870 8867 8864 8861 8691 8688 8685 8682 8679 85i6 85i3 85io 85o7 85o4 8348 8345 8342 8339 8337 8186 8i83 8181 8 1 78 8175 8o3o 8027 8025 8022 8020 7879 7877 7874 7872 7869 9901 9897 9893 9889 9885 9671 9667 9664 9660 9656 9453 9449 9446 9442 9439 9245 9242 9238 9235 9232 9047 9044 9041 9037 9034 8857 8854 885 1 8848 8845 8676 8673 8670 8667 8664 85o2 8499 8496 8493 8490 8334 833 1 8328 8326 8323 8173 8170 8167 8i65 8162 8017 &oi4 8012 8009 8007 8004 8002 7999 7997 7994 7867 7864 7862 7859 7857 7855 7852 785o 7847 7845 9881 9877 9873 9869 9865 9652 9649 9645 9641 9638 9435 9432 9428 9425 9421 9418 9414 9411 9407 9404 9228 9225 9222 9218 9215 903 1 9028 9024 9021 9018 8842 8839 8836 8833 883o 8661 8658 8655 8652 8649 8487 8484 8482 8479 8476 8320 83i8 83i5 83i2 83o9 8159 8i57 8i54 8i52 8149 9861 9358 9854 9850 9846 9634 9630 9626 9623 96'9 9212 9208 9205 9201 9198 9015 9012 9008 9005 9002 8827 8824 8821 8817 8814 8646 8643 864o 8637 8635 8632 8629 8626 8623 8620 8473 8470 8467 8465 8462 83o7 83o4 83oi 8298 8296 8i46 8i44 8i4i 8i38 8i36 7992 7989 7987 7984 7981 7842 7840 7837 7835 7832 9842 9838 9S34 983o 9827 9615 9612 9608 9604 9601 9400 9397 993 9390 9386 9195 91 91 9188 9185 9181 9178 9175 9171 9168 9165 8999 8996 S992 8989 8986 89S3 8980 8977 8973 8970 881 1 8808 88o5 8802 _8799_ 8796 8793 8790 8787 8784 8459 8456 8453 845 1 8448 8293 8290 8288 8285 8282 8i33 8i3i 8128 8125 8123 7979 7976 7974 7971 7969 7830 7828 7825 7S23 7820 9823 9819 9815 9811 9807 9597 9593 9590 9586 9582 9383 9^79 9376 9372 9369 8617 86i4 8611 8608 86o5 8445 8442 8439 8437 8434 8279 8277 8274 8271 8269 8120 8117 8ii5 8112 8110 7966 7964 7961 7959 7956 7818 7815 7813 781 r 7808 9803 9800 9796 9792 9788 9579 9575 9571 9568 9564 9365 9362 9358 9355 9351 9162 91 58 9155 9152 9148 8967 8964 8961 8958 8954 8781 8778 8775 8772 8769 8602 8599 8597 8594 8591 843 1 8428 8425 8423 8420 8266 8263 8261 8258 8255 8107 8104 8102 8099 8097 7954 7951 7949 7946 7944 7806 7803 7801 7798 7796 9784 9780 9777 9773 9769 9561 9557 9553 9550 9546 0°19' 9348 9344 9341 9337 9334 9145 9142 9 1 38 9 35 9132 8951 8948 8945 8942 8939 8766 8763 8760 8757 8754 8588 8585 8582 8579 3576 8417 84i4 84ii 8409 8406 8253 8250 8247 8244 8242 8094 8091 8089 8086 8084 7941 7939 7936 7934 7931 7794 7791 7789 7786 7784 55 56 57 58 59 0° 18' 0°20^ 0°21' 0° 22' 0° 23' 0°24' 0°25' 0°26' 0°27' 0°28'0°29'| S. TABLP XXII. li'='S'='35 Proportional Logarithms. S. o I 2 3 4 5 6 7 8 9 10 II 12 i3 i4 i5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 4o 4i 42 43 A^\ 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 S, h m 0°30' k m 0° 31' h m 0° 32' h m 0° 33' h m 0°34' h m 0° 35' h m 0°3G' h m 0°37' h in 0°3S' h m 0° 39' h VI 0° 40' h m 0°41' S. I 2 3 4 5 6 7 8 9 10 1 1 12 i3 i4 i5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 40 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 .1 S. 7782 7779 7777 7774 7772 7639 7637 7634 7632 763o 75oi 7499 7497 7494 7492 7368 7365 7363 7361 7359 7238 7236 7234 7232 7229 7112 7110 7108 7106 7104 6990 6988 6986 6984 6982 6871 6869 6867 6865 6863 6755 6753 6751 6749 6747 6642 6640 6638 6637 6635 6633 663 1 6629 6627 6625 6532 653o 6529 6527 6525 6523 6521 65i9 65i8 65i6 6425 6423 6421 6420 64i8 64 16 64i4 64i3 64n 6409 7769 7767 7765 7762 7760 7627 7625 7623 7620 7618 7490 7488 7485 7483 7481 7357 7354 7352 7350 7348 7227 7225 7223 7221 7219 7102 7100 7098 7096 7093 6960 6978 6976 6974 6972 6861 6859 6857 6855 6853 6745 6743 6742 6740 6788 7757 7755 7753 775o 7748 7616 7613 76 1 1 7609 7607 7479 7476 7474 7472 7470 7346 7344 7341 7339 7337 7217 7215 7212 7210 7208 7091 7089 7087 7085 7083 6970 6968 6966 6964 6962 685 1 6849 6847 6845 6843 6736 6734 6732 6730 6728 6624 6622 6620 6618 6616 65i4 65i2 65 10 65o9 65o7 6407 6406 64o4 6402 6400 7745 7743 774 1 7738 7736 7604 7602 7600 7597 7595 7467 7465 7463 7461 7458 7335 7333 7330 7328 7326 7206 7204 7202 7200 7198 7081 7079 7077 7075 7073 6960 6958 6956 6954 6952 684 1 684o 6838 6836 6834 6726 6725 6723 6721 6719 66i4 6612 661 1 6609 6607 65o5 65o3 65oi 65oo 6498 6398 6397 6395 6393 6391 7734 773 1 7729 7726 7724 7593 7590 7588 7586 7583 7456 7454 7452 745o 7447 7324 7322 7320 7317 73i5 7196 7193 7191 7189 7187 7071 7069 7067 7065 7063 6950 6948 6946 6944 6942 6832 683o 6828 6826 6824 6717 6715 6713 67 II 6709 66o5 66o3 6601 6600 6598 6496 6494 6492 6491 6489 6390 6388 6386 6384 6383 7722 7719 7717 7714 7712 7581 7579 7577 7574 7572 7445 7443 744 1 7438 7436 73i3 73ii 7309 7307 73o4 7i85 7i83 7181 7179 7177 7061 7059 7057 7055 7052 6940 6938 6936 6934 6932 6822 6820 6818 6816 68 1 4 6708 6706 6704 6702 6700 6596 6594 6592 6590 6589 6487 6485 6484 6482 648o 633 1 6379 6377 6376 6374 7710 7707 7705 77o3 7700 7570 7567 7565 7563 7560 7434 7432 7429 7427 7425 73o2 73oo 7298 7296 7294 7175 7172 7170 7168 7166 7o5o 7048 7046 7044 7042 6930 6928 6926 6924 6922 6812 6810 6809 6807 68o5 6698 6696 6694 6692 6691 6587 6585 6583 658 1 6579 6478 6476 6475 6473 6471 6372 6371 6369 6367 6365 6364 6362 636o 6358 6357 7698 7696 7693 7691 7688 7558 7556 7554 755 1 7549 7423 7421 7418 74i6 74i4 7291 7289 7287 7285 7283 7164 7162 7160 7i58 7i56 7040 7o38 7o36 7034 7o32 6920 6918 6916 6914 6912 68o3 6801 6799 6797 6795 6689 6687 6685 6683 6681 6678 6576 6574 6572 6570 6469 6467 6466 6464 646a 7686 7684 7681 7679 7677 7547 7544 7542 7540 7538 74 1 2 7409 7407 74o5 74o3 7281 7279 7276 7274 7272 7i54 7i52 7149 7147 7145 7o3o 7028 7026 7024 7022 6910 6908 6906 6904 6902 6793 6791 6789 6787 6785 6679 6677 6676 6674 6672 (>5()S 6567 6565 65fi3 656 1 ()559 6558 6556 6554 6552 6460 6459 6457 6455 6453 6355 6353 635i 635o 6348 7674 7672 7670 7667 7665 7535 7533 753i 7528 7526 74oi 7398 7396 7394 7392 7270 7268 7266 7264 7261 7143 7i4i 7139 7137 7i35 7020 7018 7016 7014 7012 6900 6898 6896 6894 6892 6784 6782 6780 6778 6776 6670 6668 (■6f;6 6663 645 1 645o 6448 6446 6444 6346 6344 6343 634 1 6339 6338 6336 6334 6332 633i 7663 7660 7658 7655 7653 7524 7522 7519 7517 75i5 7390 7387 7385 7383 738i 7259 7257 7255 7253 725i 7133 7i3i 7129 7127 7124 7010 7008 7006 7004 7002 6890 6888 6886 6884 6882 6774 6772 6770 6768 6766 6661 6659 6657 6655 6653 655o 6548 6547 6545 6543 6443 644 1 6439 6437 6435 765i 7648 7646 7644 7641 75i3 75io 7508 7506 75o3 7379 7376 7374 7372 7370 7249 7246 7244 7242 7240 7122 7120 7118 7116 7ii4 7000 6998 6996 6994 6992 6881 6879 6877 6875 6873 6764 6763 6761 6759 6757 665 1 665o f648 6646 6644 654 1 6539 6538 6536 6534 6434 6432 643o 6428 6427 6329 6327 632 5 6324 6322 0°80' 0°31' 0° 3-2' 0° 33'|0° 34 0°35' 0° 3G'iO° 37' 0°38' 0° 39'i0° 40' 0° 41' i'^?- '3^] TABLE XXII. Proportional Logarithms. S. h m ! Ii 111 h m A m h VI 1 h m h m h m h m h m h VI h m 0° 42' 0^ 43' 0044/ 0°45' 0° 4G' 0° 47' 0°48' 0°49' 0°50 0°51' 0=52' 0°53' S. 6320 6218 6118 602 X 5925 5832 5740 565 1 5563 5477 5393 53x0 I 63i9 6216 6xx7 6019 5924 583o 5739 5649 5562 5476 5391 5309 I 2 63i7 6215 6ix5 6017 5922 5829 ^737 5648 556o 5474 5390 53o7 2 3 63 1 5 6213 6ii3 6016 5920 5827 5736 5646 5559 5473 5389 53o6 3 4 5 63i3 6211 6x12 6ox4 5919 5826 5734 5645 5557 5471 5387 5386 53o5 53o3 5 63 1 2 6210 6110 60 1 3 5917 5824 5733 5643 5556 5470 6 63io 6208 6108 60 X I 5916 5823 573 X 5642 5554 5469 5384 53o2 6 7 63oS 6206 6107 6009 5914 5821 5780 564o 5553 5467 5383 53oo 7 8 63o6 6205 6io5 6008 59x3 58x9 5728 5639 555i 5466 5382 5299 8 > 9 10 63o5 6203 6io3 6006 59XX 58x8 5727 5637 555o 5464 5463 538o 5298 "5l^6 9 1 in 1 63o3 6201 6102 6oo5 5909 58i6 5725 5636 5549 5379 II 63oi 6200 6100 6oo3 5908 58x5 5724 5635 5547 546x 5377 5295 1 1 ; 12 63oo 6198 6099 6001 5906 58x3 5722 5633 5546 5460 5376 5294 X2 i3 6298 6196 6097 6000 5905 58x2 5721 5632 5544 5459 5375 5292 i3 i4 i5 6296 6195 6095 5998 5997 5908 58io 57x9 563o 5543 5457 5373 529X 5290 1 4 x5 6?94 6193 6094 5902 5809 57x8 5629 554i 5456 5372 i6 6293 6191 6092 5995 5900 58o7 5716 5627 5540 5454 5370 5288 x6 17 6291 6190 6090 5993 58q8 58o6 57x5 5626 5538 5453 5369 5287 17 i8 6289 6188 6089 5992 5897 58o4 57x3 5624 5537 5452 5368 5285 18 19 20 6288 6186 6087 6o85 5990 5895 58c3 57x2 5623 5536 5450 5366 5284 5283 20 62S6 6i85 5989 5894 58ox 57!0 5621 5534 5449 5365 21 6284 6i83 6084 5987 5892 58oo 5709 5620 5533 5447 5364 5281 21 22 6282 6181 6082 5985 5891 5798 5707 56x8 553 X 5446 5362 5280 22 23 62S1 6179 6081 5984 5SS9 5796 5706 56x7 553o 5445 536i 5279 23 24 25 6279 6178 6079 5982 5888 5795 5704 56x5 5528 5443 5442 535o 5358 5277 5276 24 25 6277 6176 6077 5981 5886 5793 5703 56i4 5527 26 6276 6174 6076 5979 5884 5792 5701 56x3 5526 5440 5357 5275 2/j 27 6274 6173 6074 5977 5883 5790 5700 56ix 5524 5439 5355 5278 •-7 28 6272 6171 6072 59-b 588 1 5789 5698 56x0 5523 5437 5354 5272 28 29 3o 6271 6169 6071 5974 588o 5787 5697 56o8 5521 5520 5436 5353 D271 5269 29 3o ■ 6269 6168 6069 5973 5878 5786 5695 5607 5435 535x 3i 6267 6166 6067 5971 5877 5784 5694 56o5 55x8 5433 535o 5268 3i 32 6265 6i65 6066 5969 5875 5783 5692 56o4 55x7 5432 5348 5266 32 33 6264 6i63 6064 5968 5874 5.78 X 5691 56o2 55x6 543o 5347 5265 33 34 35 6262 6161 6o63 5966 5S72 5780 5689 56ox 55x4 5429 5346 5264 34 35 6260 6160 6061 5965 5S70 5778 5688 5599 55x3 5428 5344 5262 36 62 5g 61 58 6059 5963 5869 5777 5686 5598 55x1 5426 5343 5261 36 37 6257 6i56 6o58 596 X 5867 5775 5685 5596 55x0 5425 534 X 5260 37 38 6255 6i55 6o56 5960 5866 5774 5683 5595 55o8 5423 5340 5258 38 39 40 6254 6x53 60 5 5 5958 5864 5772 5682 5594 55o7 5422 5339 5257 39 40 6252 6i5i 6o53 5957 5863 5771 568o 5592 55o6 542 X 5337 5256 4i 6250 6i5o 6o5x 5955 586i 5769 5679 559X 55o4 54x9 5336 5254 4i 42 6248 6i48 6o5o 5954 586o 5768 5677 5589 55o3 54x8 5335 5253 42 43 6247 6i46 6o48 5952 5858 5766 5676 5588 55oi 5416 5333 5252 Ai 45 6245 6145 6046 5950 5856 5765 5674 5586 55oo 54x5 54x4 5332 525o AA 45 6243 6143 6045 5949 5855 5763 5673 5585 5498 533i 5249 46 6242 6i4i 6043 5947 5853 5761 5671 5583 5497 54x2 5329 5248 4b 47 6240 6i4o 6042 5946 5852 5760 5670 5582 5496 54x1 5328 5246 47 48 6233 6x38 6n4o 5944 585o 5758 5669 558o 5494 5409 5326 5245 48 49 5o 6237 6x36 6o38 6037 5942 5849 5757 5667 5579 5493 5408 5325 5244 49 5o 6235 6x35 5941 5847 5755 5666 5578 5491 5407 5324 5242 5 1 6233 6x33 6o35 5939 5846 5754 5664 5576 5490 54o5 5322 524x 5x 52 6232 6x3x 6o33 593s 5844 5752 5663 5575 5488 54o4 5321 5240 52 53 6230 6i3o 6o32 5936 5843 575x 566 X 5573 5487 5402 5320 5238 53 54 55 6228 6x28 60 3o 5935 584 1 5749 566o 5572 5486 540 X 5400 53x8 5237 54 55 6226 6126 6029 5933 5839 5748 5653 5570 5484 53x7 5235 56 6225 6x25 6027 5931 5838 5746 5657 5569 5483 5398 53x5 5234 56 57 6223 6x23 6025 5930 5836 5745 5655 5567 5481 5397 53i4 5233 57 53 6221 6l2X 6024 5928 5835 5743 5654 5566 5480 5395 53x3 523i 58 , 59 S. 6220 0°42' 6x20 6022 5927 5833 5742 5652 5564 5478 5394 53x1 523o S. 0°43' 0°44' 0° 45' 0° 4G'!0° 47' 0°48' 0° 4u' 0° 50' 0° 51'|0° 52' 0°53' TABLE XXII. [!'' .0.137 Proportional Logarithms. S. o h m k m /( 'm h w ll VI It m h m h m \ li m h m /i m h m 0°54' 0°55' 0°56' 0"57' 4994 0°58' 4918 O'' 5!)' 4844 1°0' 1°1' 1^2' 1°3' 104/ 1°5' S. 5229 5i49 5071 4771 4699 4629 4559 4491 4424 I 5227 5i48 5070 4993 4917 4843 4770 4698 4628 4558 4490 4422 I 2 5226 5i46 5o68 4991 491(3 4842 4769 4697 4626 4557 4489 442 1 2 3 5225 5i45 5067 4990 491^ 484 1 4768 4696 4625 4556 4488 4420 3 4 5 5223 5 144 5ob6 4989 4913 4839 4766 4695 4624 4555 4486 4419 4 5' 5222 5r43 5o64 4988 4912 4838 4765 4693 4623 4554 4485 4418 6 5221 5i4i 5o63 4986 491 1 4837 4764 4692 4622 4552 4484 4417 6 7 5219 5i4o 5062 4985 4910 4836 4763 4691 4C21 455i 4483 4416 7 8 5218 5i39 5o6i 4984 4908 4834 4762 4690 4619 455o 4482 44i5 8 9 lO 5217 5i37 5o59 49S3 ^907 4833 4760 4689 4618 4549 4548 4481 44j4 10 52i5 5i36 5o58 4981 49116 4832 4759 1 4688 4617 4480 44^^- II 52i4 5i35 5o57 4980 4905 483 1 4758 4686 46 1 6 4547 4479 44 1 1 II 12 52i3 5 1 33 bob:j 4979 4903 48 3o 4757 4685 46i5 4546 4477 44io 12 iJ 52II 5i32 5o54 4977 4902 4828 47^6 4684 46i4 4544 4476 4409 i3 i4 i5 5210 5i3i 5o53 4976 4901 4827 4754 4683 4612 4543 4475 44o8 i4 i5 5209 5129 5o5i 4975 4900 4826 4753 4682 4611 4542 4474 4407 i6 5207 5 1 28 5o5o 4974 4899 4825 47^2 46So 4610 4541 4473 4406 16 17 5206 5i27 5o49 4972 4897 4823 475i 4679 4609 4540 4472 44o5 17 i8 52o5 5i25 5o48 4971 4896 4822 475o 4678 4608 4539 4471 4404 18 19 20 52o3 5i24 5o46 4970 4895 4821 4748 4677 4607 4'3d\i 4469 4468 44o2 4401 '9 20 5202 5i23 5o45 4969 4894 4820 4747 4676 4606 4536 21 5201 5l22 5o44 4967 4892 4819 4746 4675 46o4 4535 4467 44oo 21 22 5.99 5l20 5o43 4966 4891 4817 4745 4673 46o3 4534 4466 4399 22 23 5198 5ii9 5o4i 4965 4890 48i6 4744 4672 4602 4533 4465 4398 2 3 24 25 5i97 5ri8 5o4o 4964 4889 48 1 5 4742 4671 4601 4532 453i 4464 4463 4397 24 25 5195 5ii6 5o39 4962 4887 48 1 4 4741 4670 4600 4396 25 5194 5ii5 5o37 4961 4886 4812 4740 4669 4599 453o 4462 4395 26 27 5193 5ii4 5o36 4960 4885 4S11 4739 4668 4597 4528 4460 4394 27 28 5191 5i 12 5o35 4959 4884 48 10 4738 4666 4596 4527 4459 4393 28 29 3o 5190 5iii 5()34 49!)7 4882 4809 4736 4665 4595 4526 4458 4391 29 3o 5189 5iio 5o32 4956 488i 4808 4735 4664 4594 4525 4457 4390 3i 6187- 5io8 5o3i 49'j5 488o 4806 4734 4663 4593 4024 4456 4389 3i 32 5i86 5i07 5o3o 4954 4S79 48o5 4733 4662 4592 4523 4455 4388 32 33 5i85 5io6 5028 4952 4877 48o4 4732 4660 4590 4522 4454 4387 33 35 5i83 5io5 5027 4951 4876 48o3 4730 4659 4589 4520 4453 4386 34 35 5182 5io3 5026 4950 4875 4801 4729 4658 4588 45i9 4452 4385 36 5i8i 5l02 5o2 5 4949 4874 4S00 4728 4657 4587 45i8 445o 4384 36 37 5i79 5ioi 5o23 4947 4873 4799 4727 4656 4586 45i7 4449 4383 37 38 5178 5099 5022 4946 4871 4798 4726 4655 4585 45i6 4448 438 1 38 39 4o 5i77 5098 5o2i 4945 4943 4870 4797 4724 4653 4584 45i5 45i4 4447 4446 438o 43^ 39 40 5175 5097 5019 4869 4795 4723 4652 4582 4i 5i74 5095 5oi8 4942 4668 4794 4722 465 1 458i 45i2 4445 4378 4i 42 5.73 5094 5oi7 4941 4866 4793 4721 465o 458o 45ii 4444 4377 42 43 5172 5093 5oi6 4940 4865 4792 4720 4649 4579 45io 444'i 4376 43 45 5170 5092 5oi4 4938 4937 4864 4863 4791 4789 4718 4648 4578 4509 45o8 444 1 4375 44 45 5169 5090 5oi3 4717 4646 4577 4440 4374 46 5 1 68 50S9 5oi2 4936 4861 4788 4716 4645 4575 45o7 4439 4373 46 47 5r66 5o88 5oii 4935 4860 4787 47i5 4644 4574 45u6 4438 4372 47 48 5 1 65 5o86 5009 4933 4859 4786 4714 4643 4573 45o5 4437 4370 48 49 5o 5 1 64 5o85 5oo8 4932 4858 4785 4783 4712 4642 4572 45o3 4436 4369 4368 49 5o 5162 5o84 5007 4931 4856 471 1 464o 4571 45o2 4435 5i 5i6i 5082 5oo5 4930 4855 4782 4710 4639 4570 45oi 4434 4367 5 1 52 5 1 60 5o8i 5oo4 4928 4854 4781 470Q 4638 4569 45oo 4433 4366 52 53 5i58 5o8o 5oo3 4927 4853 4780 4708 4637 4567 4499 443 1 4365 53 54 55 5i57 5079 5o02 4926 4852 4778 4707 4705 4636 4566 4498 443o 4364 54 55 5i56 5o77 5ooo 4925 485o 4777 4635 4565 4497 4429 4363 56 5i54 5076 4999 4923 4849 4776 4704 4633 A'M 4495 4428 4362 56 57 5i53 50-75 4998 4922 4848 4775 4703 4632 4563 4494 4427 436i 57 58 5 1 52 5073 4997 4921 4847 4774 4702 463 1 4562 4493 4426 4359 58 59 S. 5i5o 5072 4995 4920 4845 4772 4701 463o 456o 4492 4425 4358 59 S. 0^^ 54' 0° 5.^' 0° 56' 0° 57' 0° 58' 10° 59' 1°0' 1°I' 1°2' 1°3' 1°4' i°5' IH r»g«i38j TABLE XXII. Proportional Logarithms. S. o h m k m h m h m A m h m h m h m h m h m h VI h m F6' 1°7' 1°8' 1°9' 1°10' 1°11' 1°12' 1° 13' 1°14' 1°15 1°1G' V IT S. 4357 4292 4228 4i64 4102 4o4o 3979 3919 386o 38o2 3745 3688 I 4356 4291 4227 4i63 4ioi 4039 3978 3919 3859 38oi 3744 8687 I 2 4355 4290 4226 4162 4ioo 4o38 3977 3918 3858 38oo 3743 3686 2 3 4354 4289 4224 4i6i 4099 4o37 3976 3917 3857 3799 3742 3685 3 4 5 4353 4288 4223 4i6o 4098 4o36 3975 3916 3856 3798 3741 3684 4 5 4352 4287 4222 4i59 4097 4o35 3974 3915 3856 3797 3740 3683 6 435i 4285 4221 4i58 4096 4o34 3973 3914 3855 3796 3739 8682 6 7 435o 4284 4220 4i57 4095 ' 4o33 3972 3913 3854 3795 3788 368 1 7 8 4349 4283 4219 4i56 4093 4o32 3971 3912 3853 3794 3787 368o 8 9 lO 4347 4282 4218 4i55 4092 4o3i 3970 391 1 3852 3793 3786 8735 8679 8678 9 10 4346 4281 4217 4i54 4091 4o3o 3969 3910 385i 3792 1 1 AM^ 4280 4216 4i53 4090 4029 3968 3909 385o 3792 3734 3677 II 12 4344 4279 42i5 4i52 .4089 4028 3967 3908 3849 3791 8733 8677 12 i3 4343 4278 4214 4i5i 4088 4027 3966 3907 3848 3790 8782 8676 i3 i4 i5 4342 4277 42i3 4i5o 4087 4026 3965 3906 3847 3789 8781 8675 8674 i4 i5 4341 4276 4212 4149 4o86 4025 3964 3905 3846 3788 8780 i6 4340 4275 4211 4i47 408 5 4024 3963 3904 3845 3787 3729 8673 16 17 4339 4274 4210 4i46 4o84 4o23 3962 3903 3844 3786 8728 8672 17 [8 4338 4273 4209 4i45 4o83 4o22 3961 3902 3843 3785 8727 8671 18 19 20 4336 4271 4207 4i44 4082 402I 3960 3901 3842 3784 8727 8670 19 20 4335 4270 4206 4i43 4081 4020 3959 3900 384 1 3783 8726 8669 21 4334 4269 42o5 4i42 4080 4019 3958 3899 384o 3782 8725 3668 21 22 4333 4268 4204 4i4i 4079 4018 3957 3898 3839 3781 8724 8667 22 23 4332 4267 4203 4i4o 4078 4017 3956 3897 3838 3780 3728 3666 23 24 25 433 1 4266 4202 4201 4i39 4077 4oi6 3955 3896 3837 3779 8722 8665 24 25 433o 4265 4i38 4076 4oi5 3954 3895 3336 3778 8721 8664 26 4329 4204 4200 4i37 4075 4oi4 3953 3894 3835 3777 0720 3663 26 27 4328 4263 4199 4i36 4074 4oi3 3952 3893 3834 3776 8719 3663 27 28 4327 4262 4i9« 4i35 4073 40I2 3951 3892 3833 3775 8718 3662 28 29 3o 4326 4261 4197 4i34 4072 4oii 3950 3891 3832 3774 8717 8661 29 3o 432 3 4260 4196 4i33 4071 4oio 3949 3890 383 1 3773 8716 366o 3i 4323 4259 4195 4i32 4070 4009 3948 3889 383o 3772 3715 3659 3i 32 4322 42 58 4194 4i3i 4069 4008 3947 3888 3829 3771 3714 3658 32 33 4321 4256 4193 4i3o 4068 4007 3946 3887 3828 3770 8718 3657 3^ 34 35 4320 4319 4255 4192 4129 4067 4oo6 3945 3886 3827 3769 8768 8712 8711 3656 34 35 4254 4191 4128 4066 4oo5 3944 3885 3826 3655 :ib 43i8 4253 4189 4127 4o65 4oo4 3943 3884 382 5 3768 8710 3654 36 37 43i7 425a 4i88 4126 4o64 4oo3 3942 3883 3824 3767 8709 3653 37 38 43i6 425i 4187 4i25 4o63 4002 3941 3882 3823 3766 3709 3652 38 39 4o 43i5 42 5o 4i86 4i85 4124 4062 4001 3940 388i 3822 3821 3765 8708 365i 39 40 43i4 4249 4l22 4o6i 4ooo 3939 388o 3764 8707 365o 41 43i3 4248 4i84 4l2I 4o6o 3999 3938 3879 3820 3763 8706 3649 4i 42 43ii 4247 4i83 4l20 4059 3998 3937 3878 3820 3762 3705 3649 42 43 43io 4246 4182 4II9 4o58 3997 3936 387^ 3819 3761 3704 3648 Ai 44 45 4309 4245 4i8i 4ri8 4o56 3996 3935 3876 38i8 3760 8708 3647 44 45 43o8 4244 4 1 80 4117 4o55 3995 3934 3875 3817 3759 3702 3646 4t) 4to7 4243 4179 4116 4o54 3993 3933 3874 38i6 3758 8701 3645 46 47 43o6 4241 4i7« 4ii5 4o53 3992 3932 3873 38i5 3757 8700 3644 47 48 43o5 4240 4i77 4ii4 4o52 3991 3931 3872 38i4 3756 8699 3643 48 49 5o 43o4 4239 4176 4ii3 4o5i 3990 3930 3871 38i3 3755 8698 3642 49 5o 43o3 4238 4175 4lI2 4o5o 3989 3920 3870 38i2 3754 8697 364 1 5i 43o2 4237 4174 4iii 4o49 3988 3928 3869 38ii 3753 8696 364o 5i 5 1 43oi 4236 4173 4iio 4o48 3987 3927 3868 38io 3752 8695 8689 52 53 43cc I 4235 4172 4109 4o47 3986 3926 3867 3809 375i 8694 8638 53 54 55 429S 42J4 4233 4171 4 1 08 4o46 3985 3925 3866 38o8 3750 8698 8687 54 55 4297 4169 4107 4045 3984 3924 3865 3807 374Q 8698 3636 56 4296 4232 4168 4 1 06 4044 39S3 3923 3864 38o6 3748 8692 3635 56 57 4295 423 1 4167 4io5 4043 3982 3922 3863 38o5 3747 8691 8635 57 58 4294 423o 4i66 4io4 4042 3981 8921 3862 38o4 3746 8690 3634 58 59 S. 4293 4229 4i65 4io3 4o4i 3980 3920 386i 38o3 3746 3689 3638 59 S. 1°(3' 1°7' 1°8' 1°9' 1°10' Pll' 1°12' 1° 13' 1°14' 1 15' 1° 16'1°17'| TABLE XXII. iP^^ge 139 Proportional Logarithms. S. o I 2 3 4 5 6 7 8 9 10 1 1 12 i3 i4 i5 i6 17 i8 _L9_ 20 31 22 23 24 25 26 ^7 28 29 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 ' 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 J9_ S. h m P18' h m 1°19' h m 1°20' h m 1°21' h m 1° 22'' h m 1°23' h vi I°24' h m 1°25' h m 1°26' h m 1°27' h m 1°28' h m 1° 29' S. I 2 3 4 5 6 8 10 1 1 12 ]3 i4 i5 . 16 '7 18 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35^ 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 69 S. 3632 363 1 363o 3629 362S 3576 3576 3575 3574 3573 3522 3521 3520 3519 35i8 3468 3467 3466 3465 3464 34i5 34i4 34i3 3412 3411 3362 336i 336o 3359 3358 33io 3309 33o8 33o7 33o6 33o6 33o5 33o4 33o3 33o2 3259 3258 3257 3256 3255 3208 3207 3206 32o5 32o4 3i58 3,57 3i56 3 1 55 3i54 3i53 3i53 3i52 3i5i 3i5o 3io8 3i07 3 106 3io5 3io5 3io4 3io3 3 102 3ioi 3ioi 3o59 3o58 3o57 3o56 3o56 3o55 3o54 3o53 3o52 3o52 3627 3626 3625 3624 3623 3572 3571 3570 3569 3568 35i7 35i6 35i5 35i5 35i4 3463 3463 3462 3461 3460 34io 3409 34 o8 34d8 3407 3358 3357 3356 3355 3354 3254 3253 3253 3252 325i 32o4 32o3 3202 3201 3200 3623 3622 3621 3620 3619 3567 3566 3565 3565 3564 35i3 35i2 35ii 35io 3509 3459 3458 3457 3456 3455 3406 34o5 3404 34o3 3402 3353 3352 335i 335i 335o 33oi 33oo 33oo 325o 3249 3248 3247 3247 3'99 3198 3198 3197 3196 3i49 3i48 3i48 3i47 3 1 46 3i45 3i44 3i43 3i43 3i42 3 1 00 3099 3098 3097 3096 3096 3095 3094 3093 3092 3o5i 3o5o 3o49 3o48 3o47 3o47 3o46 3 04 5 3o44 3o43 36i8 36i7 36 16 36i5 36i4 3563 3562 356 1 356o 3559 35o8 3507 35o6 35o6 35o5 3454 3454 3453 3452 345 1 3401 3400 3400 3399 3398 3349 3348 3347 3346 3345 3297 3296 3295 3294 3294 3246 3245 3244 3243 3242 3195 3194 3193 3193 3192 36i3 36i2 36ii 36 10 36 10 3558 3557 3556 3555 3555 35o4 35o3 35o2 35oi 35oo 345o 3449 3448 3447 3446 3397 3396 339J 3394 3393 3345 3344 3343 3342 334i 3293 3292 3291 3290 3289 3288 3288 3287 3286 3285 3242 3241 3240 3239 3238 3237 3236 3236 3235 3234 3191 3190 3189 3i88 3i88 3187 3i86 3i85 3]84 3i83 3i4i 3i4o 3i39 3i38 3i38 8091 3091 3090 3089 3o88 3o43 3o42 3o4i 3o4o 3089 3609 36o8 3607 36o6 36o5 3554 3553 3552 355i 355o 3499 3498 3497 3497 3496 3446 3445 3444 3443 3442 3393 3392 3391 3390 3389 3340 3339 3338 3338 3337 3i37 3i36 3i35 3i34 3i33 3i33 3i32 3i3i 3i3o 3129 3087 3087 3o86 3o85 3o84 3o83 3082 3082 3o8i 3o8o 8089 3o38 3o37 3o36 3o35 3o34 3o34 3o33 3o32 3o3i 36o4 36o3 36o2 36oi 36oo 3549 3548 3547 3546 3545 3495 3494 3493 3492 3491 3490 3489 3488 3488 3487 3441 3440 3439 3438 3438 3388 3387 3386 3386 3385 3336 3335 3334 3333 3332 3284 3283 3282 3282 3281 3233 3232 323i 323i 323o 3i83 3i82 3i8i 3i8o 3179 3599 359S 3598 3597 3596 3545 3544 3543 3542 3541 3437 3436 3435 3434 3433 3384 3383 3382 338r 338o 3332 333i 333o 3329 3328 3280 3279 3278 3277 3276 3229 3228 3227 3226 32 25 3178 3178 3i77 3176 3.75 3129 3i28 3i27 3i26 3i25 8079 3078 3078 3077 3076 3o3o 3o3o 8029 8028 8027 8026 3o?6 3o25 3o24 3o23 3595 3594 3593 3592 3591 3540 3539 3538 3537 3536 3486 3485 348 i 3483 3482 3432 343 1 343 1 343o 3429 3379 3379 3378 3377 3376 3327 3396 3325 3325 3324 3276 3275 3274 3273 3272 3225 3224 3223 3222 3221 3i74 3173 3173 3172 3171 3i24 3i24 3i23 3l22 3l2I 3075 3074 3073 3073 3072 3590 3589 3588 3587 3587 3535 3535 3534 3533 3532 3481 3480 3480 3479 3478 3428 3427 3426 3425 3424 3375 3374 3373 3372 3372 3323 3322 3321 3320 3319 3271 3270 3270 3269 3268 3220 3220 3219 3218 3217 3170 3169 3i68 3 1 68 3167 3l20 3i 19 3119 3II7 3071 8070 3069 8069 3c68 3022 3022 302I 3020 8019 3586 3585 3584 3583 3582 353i 353o 3529 3528 3527 3477 3476 3475 3474 3473 3423 3423 3422 3421 3420 3371 3370 3369 3368 3367 3319 33i8 33i7 33i6 33i5 3267 3266 3265 3265 3264 3216 32i5 32i4 32i4 32i3 3 1 66 3 1 65 3 1 64 3i63 3 1 63 3ii6 3ii5 3ii4 3ii4 3ii3 8067 3o66 3o65 3o65 3o64 3o63 8062 3o6i 3o6o 3oDo 3oi8 3oi8 3oi7 3oi6 3(>i5 358 1 35So 3579 3578 3^77 3526 3525 3525 3524 3523 3472 3471 3471 3470 3469 3419 3418 3417 3416 34i5 3366 3365 3365 3364 3363 33i4 33i3 33i3 33i2 33 1 1 3263 3262 3261 3260 3259 3212 32II 3210 3209 3209 3162 3i6i 3i6o 3i59 3i58 3lI2 3iii 3iio 3iio 3 1 09 3oi4 3oi4 3oi3 3oi2 3oii 1° J8'; 1° 19' 1° 20' l°2ri°22' r23' I°24' 1° 25'il° 26' 1° 27';!"^ 28'|1° 29^ Page 140] TABLE XXII. Proportional Logarithms. S. I 2 3 4 5 6 7 8 9 lO II 12 1 3 i4 i5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 . 34 35 36 37 38 39 4o 4r 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 S. h m 1°30 h m r 31' 1° 32' k m 1° 33' h VI 1°34' h m 1°35' h m 1°36' h m 1°37' h m 1°38' h m F39' h m r40' li m 1°41' 25lO 2509 25o8 25o7 25o7 S. I 2 3 4 5 6 7 8 9 10 II 12 i3 i4 i5 16 17 18 19 3oio 3009 3009 3oo8 3oo7 2962 2962 2961 2960 2909 2915 2914 2913 2912 2912 2868 2867 2866 2866 2865 2821 2821 2820 2819 2818 2775 2775 2774 2773 2772 2730 2729 2729 2728 2727 2685 2684 2684 2683 2682 2640 2640 2639 2638 2638 2596 2596 2595 2594 2593 2553 2552 255i 255i 255o 3 006 3oo5 3oo5 3oo4 3on3 2958 2958 2957 2955 2955 2911 2910 2909 2909 2908 2864 2863 2862 2862 2861 2818 2817 2816 28[5 2815 2772 2771 2770 2769 2769 2726 2725 2725 2724 2723 2681 2681 2680 2679 2678 2637 2636 2635 2635 2634 2593 2592 2591 2591 2590 2549 2548 2 548 2547 2 546 2 5o6 2 5o5 2 5o4 25o4 25o3 3oo2 3ooi 3ooi 3ooo 2999 2954 2954 2953 2952 2951 2907 2906 2905 2905 2904 2860 2859 2859 2858 2857 2814 2813 2812 2811 2811 2768 2767 2766 2766 2765 2722 ■i.-jii 2721 2720 2719 2678 2677 2676 2675 2675 2633 2632 2632 263 1 263o 2589 2588 2588 2587 2586 2545 2545 2544 2543 2543 25(12 25(J2 25oi 2 5oC) 2499 299S 2997 2997 2996 2995 2950 2950 2949 2948 2947 2903 2902 2901 2901 2900 2856 2855 2855 2854 2853 2810 2809 2808 2808 2807 2764 2763 2763 2762 2761 2719 2718 2717 2716 2716 2674 2673 2672 2672 2671 2629 2629 2628 2627 2626 2585 2585 2584 2583 2583 2542 254i 2540 2540 2539 2499 2498 2497 2497 2496 2994 2^93 2993 2992 2991 2946 2946 2945 2944 2943 2898 2897 2896 2852 2852 285i 285o 2849 2806 2805 2805 2804 2803 2760 2760 2759 2758 2757 2715 2714 2713 2713 2712 2670 2669 2669 2668 2667 2626 2625 2624 2624 2623 2582 258 1 258o 258o 2579 2538 2538 2537 2536 2535 2495 2494 2494 2493 2492 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 2990 2989 2989 2988 2987 2942 2942 2941 2940 2939 2895 2894 2894 2893 2892 2848 2848 2847 2846 2845 2802 2801 2801 2800 2799 2756 2756 2755 2754 2753 2711 2710 2710 2709 2708 2666 2666 2665 2664 2663 2622 2621 2621 262c 2610 2578 2577 2577 2576 2575 2535 1 2492 2534 1 2491 2533 1 2490 2533 2489 2532 1 2459 2986 29S5 2985 2984 2983 2939 2938 2937 2936 2935 2891 2891 2890 2889 2888 2845 2844 2843 2842 2842 2798 2798 2797 2796 2795 2753 2752 2751 2750 2750 2707 2707 2706 2705 2704 2663 2662 2661 2660 2660 2618 2618 2617 2616 2615 2574 2574 2573 2572 2572 253i 253o 253o 2529 2528 24S0 24S7 2487 2486 2485 2982 2981 2981 2980 2979 2935 2934 2933 2932 2931 28S7 2887 28S6 2885 2884 2841 2840 2839 2838 2838 2795 2794 2793 2792 2792 2749 2748 2747 2747 2746 2704 2703 2702 2701 2701 2659 2658 2657 2657 2656 26i5 2614 2613 2612 2612 261 1 2610 2610 2609 2608 2571 2570 2569 2569 2568 2527 2527 2526 2525 2525 2485 2484 2483 2482 2482 35 36 37 38 39 40 4i 4-2 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 S. 2978 2977 2977 2976 2975 2931 2930 2929 2928 2927 2883 2883 2882 2881 28S0 2837 2836 2835 2835 2834 2791 2790 2789 2788 2788 2745 2744 2744 2743 2742 2700 2699 2698 2698 2697 2655 2655 2654 2653 2652 2567 2566 2566 2565 2 564 2524 2523 2522 2522 2521 2481 2480 2480 2479 2478 2974 2973 2973 2972 2971 2927 2926 2925 2924 2924 2880 2879 2878 2877 2876 2833 2832 283 r 2S3i 283o 2829 2828 2828 2827 2826 2787 2786 2785 2785 2784 2741 2741 2740 2739 2738 2696 2695 2695 2694 2693 2652 265 1 265o 2649 2649 2607 2607 2606 2605 2604 2 564 2563 2562 256i 256i 2520 252Q 2519 25l8 25i7 2477 2477 2476 2475 2475 2970 2969 2969 2968 2967 2923 2922 2921 2920 2920 2876 2875 2S74 2873 2873 2783 2782 2782 2781 2780 2738 2737 2736 2735 2735 2692 2692 2691 2690 2689 2648 2647 2646 2646 2645 2604 2603 2602 2601 2601 256o 2559 2559 2 558 2557 25i7 25i6 25i5 25i5 25i4 2474 2473 2472 2472 2471 2966 2965 2965 2964 2963 2919 2918 2917 2916 2916 2872 2871 2870 28G9 2869 2825 2825 2824 2823 2822 2779 2779 2778 2777 2776 2734 2733 2732 2732 2731 2689 2688 2687 2687 2686 2644 2643 2643 2642 2641 2600 2599 :|? 2597 2556 2556 2555 2554 2553 25i3 25l2 25l2 25ll 25 1 C 2470 2470 2469 2468 2467 1°30' 1° 31' 1° 32' 1° 33' r34' 1°35' 1°3G' 1°37' 1°38' 1°39' 1°40'1°41' TABLE XXII. [I'afe^- 141 Proportional Logarithms. s. //. m h m A 7ft h VI h m h VI h VI \h m h m h VI h m h VI 1° 42' 1° 43 '1°44 1"45 '1°4G ' 1° 47' 1°48 1°49 1°50' P51 '1°52 1°53' s. o 2467 1 2424 2382 2341 23oo 2259 22X8 2x78 2x39 2099 2061 2022 I 2466 2424 1 2382 2340 2299 2258 2218 2x78 2x38 2099 2060 2021 I 2 2455 2423 1 238i 2339 2298 2258 23X7 2x77 2x37 2098 2059 2021 2 3 2465 2422 238o 2339 2298 2257 33X6 2x76 2i37 2098 2059 2020 3 4 2464 2422 338o 2338 2297 3356 33l6 2x76 2x36 2097 2C.58 20x9 4 5 2463 2421 2379 2337 2296 3356 33l5 2175 2x36 2096 2o57 2019 5 6 2462 2420 2378 2337 2296 2255 33X4 2174 2x35 2096 2057 2018 6 7 2462 2419 2378 2336 2295 2254 22X4 2x74 2x34 2095 2o56 20x7 7 S 2461 2419 3377 2335 2294 2253 22x3 2x73 2i34 2094 3o55 20x7 8 9 2460 2418 3376 2335 2294 2253 22X2 23X2 2x72 2172 2x33 2X32 2094 3o55 ;oi6 9 lO 2460 2417 2375 2334 2393 2252 2093 3o54 2016 10 II 2459 2417 2375 2333 3392 225l 23X1 217X 2X32 2092 3o53 20 X 5 II 12 2458 2416 2374 2333 2291 225x 23(0 2x70 2x3l 2092 3o53 20X4 12 i3 2458 24i5 2373 2332 2291 2250 22X0 2170 2x3o 2091 2052 20X4 i3 i4 24'i7 24 1 5 3373 233i 2390 2249 2209 23()5 2 169 2169 2x3o 2090 3053 20X3 i4 i5 i5 2456 24 1 4 2373 233i 2289 2249 2x29 2090 2o5x 2012 i6 2455 24i3 2371 233o 2289 3248 2208 2168 2X38 2089 2o5o 20X2 16 17 2455 2412 2371 2339 2388 3347 2307 3167 3138 2088 2o5o 301 1 17 i8 2454 2412 3370 3328 2287 2247 2206 2167 2x27 2088 2049 30X0 18 _i9_ 20 2453 241 1 2369 2368 2338 2287 2346 3206 2166 2126 2087 2o48 20X0 19 30 2453 3410 3327 2386 3345 32o5 2x65 2x26 2086 2048 2009 31 2452 2410 2368 3326 2285 3245 3 2o4 2x65 2X25 3086 2047 2009 2X 22 2.45 1 2409 3367 2326 2285 2344 2204 2x64 2X34 3o85 2046 3008 33 23 245o 2408 3366 2335 2284 2343 2 2o3 2x63 3X34 3o85 2o46 3007 23 24 25 2450 241 .8 3366 3334 3283 3343 2202 2x63 2123 3o84 2045 2007 34 25' 2449 2407 3365 2334 3283 3342 2202 2162 2X22 2 08 3 2044 2006 26 3448 2406 2364 2323 3283 224x 220X 2x6x 3X32 3 08 3 2044 2oo5 26 27 2448 24o5 2364 2 33 2 3381 2241 2200 2x61 3I2I 208 '^ 3043 20o5 27 28 2447 24o5 2363 2333 3381 2240 2200 2160 2120 2f;8l 1 3042 2004 38 29 2446 2404 2362 2321 2280 2239 2199 2x59 3X20 3081 2043 2oo3 39 3o 3o 244^ 24o3 2363 2320 2279 2339 2198 2x59 2x19 3080 3o4x 2003 3i 2445 24o3 236i 2 3 20 2279 2338 219S 3x58 21X8 2079 304x 2002 3x 32 2444 2402 2 36o 2319 2278 2237 2x97 2X37 2I18 2079 2o4o 2001 32 33 2443 2401 2359 23i8 2277 2237 2196 2x57 .2117 2078 2039 2001 33 34 ~35 2443 2401 2359 3358 23i7 33i7 2277 2236 2196 3 1 56 21X6 2077 3039 2000 34 35 2442 2400 2376 2235 2195 2x55 21X6 20"'7 3o38 2000 36 2441 2399 2357 33i6 2275 2235 2x94 2x55 21x5 2076 3o37 '999 36 37 244 r 2398 2357 23i5 2274 2234 2x94 2x54 2XX5 2075 2o37 1998 37 38 2440 2398 2356 23x5 2274 2333 2x93 2x53 2 1 14 2075 3o36 1998 38 39 2439 2397 2355 23x4 3373 2233 2192 2x53 2Xl3 3074 3o35 '997 39 4o 40 2438 2396 2355 23x3 3373 2232 2193 21 53 2X x3 3073 3o35 iv,6 4i 2438 2396 2354 33x3 2272 233l 2191 2l5x 2X12 3073 3o34 10^6 4i 42 2437 1 2395 2353 33X2 227X 223l 2190 3l5l 2XXX 3072 3o33 X995 42 43 2436 2394 2353 23X1 2270 333o 2190 3x5o 2IX I 2072 3o33 '994 43 44 2436 3394 2352 23X1 2270 3339 3x89 2149 21X0 2071 2032 1994 1993 44 45 45 3435 3393 235i 23 10 2369 3229 2188 3x49 2109 2070 3032 46 2434 2392 235o 2309 2268 2228 2x88 3x48 2x09 3070 2o3 X X993 46 47 2433 2391 335o 2309 2268 2337 2x87 3x47 2x08 3069 2o3o 1992 47 48 2433 2391 3349 23o8 2367 3227 2x86 2X47 2x07 3068 3o3o 199X 48 49 2432 2390 3 348 23o7 2266 2226 2186 2x46 2107 3068 2029 '99' 49 5o 243i 2389 3 348 23o7 2266 2235 2x85 2x45 2106 3067 2028 1990 5o 5i 243i 2389 1 3347 23o6 2265 2225 2184 2145 2I05 2066 2028 X989 5i 52 243o 2388 1 3346 23o5 2364 2224 2x84 2:44 2io5 3066 3037 igfic, 52 53 2439 2387 2 346 2 3o4 2264 2223 2x83 2i43 2104 2o65 3026 19S8 53 54 2429 2428 2387 2345 23o4 2263 2223 2x82 3i43 2xo3 2io3 3 064 3036 19S7 54 55 55 3 386 2344 23o3 2262 2232 2:82 2l42 3064 ■ 203 5 1 1987 56 2427 3385 2344 23o2 2262 2221 2181 2X4I 21C2 2o63 2025 1986 56 57 2426 2384 2343 2302 2261 2220 2x80 2l4x 2!0X 2062 2024 1986 57 58 2436 2384 2342 23oi 2260 2220 2180 2l4o 2IOI 2062 2023 X985 58 59 S. 2425 2383 2342 23oo 2260 22x9 1 2x79 2x39 2100 2o6x 2023 1984 D9 1°42' 1° 43' 1°44' r 45' 1°4G'|1°47'| r J 8 1° 49'| 1°50' Pol' 1°52'1°53'| S.l P-^gei42] TABLE XXII. Proportional Logarithms. S. o I 2 3 4 5 6 7 8 9 lO 1 1 12 i3 i4 i5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 "35 36 37 38 39 40 4i 41 43 44 45 46 47 48 49 5o 5i 52 53 54 ^5 56 f^i 58 59 h m 1°54' h m 1°55' A m 1°56' h m 1°57' A 7n 1°58' h m 1°59' h m 2° 0' k m 2° V h m 2° 2' h m 2° 3' h m 2° 4' S. I 2 3 4 5 6 7 8 9 10 II 12 i3 i4 [5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 , 3o '( 3i 32 33 34 35 36 37 38 39 4o 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55' 56 57 58 S. 1984 1983 1982 1982 1981 1946 1945 1944 1944 1943 1908 1908 1907 1906 1906 1871 1870 1870 1869 1868 1 834 i833 i833 i832 i83i 1797 1797 1796 1795 1795 1761 1760 1760 1759 1759 1725 1724 1724 1723 1722 16S9 1689 1688 16S7 1687 1 654 i653 i652 i652 i65i 1619 1618 1617 1617 1616 1981 1980 1979 1979 1978 1943 1942 1941 1941 1940 1905 1904 1904 1903 1903 1868 1867 1867 1866 1 865 i83i i83o i83o 1829 1828 1794 1794 1793 1792 1792 1758 1757 1757 1756 1755 1722 1721 1721 1720 1719 1686 1686 1 685 1684 1684 i65i i65o i65o 1649 1648 1616 i6i5 i6i4 i6i4 i6i3 1977 1975 1975 1939 1939 1938 1938 1937 1902 1901 1 90 1 1900 1899 1 865 1 864 1 863 i863 1862 1828 1827 1827 1826 1825 1791 1791 1790 1789 1789 1755 1754 1754 1753 1752 1719 1718 1718 17.7 i683 1 683 1682 1681 1681 1648 1647 1647 1 646 1645 i6i3 1612 1612 1611 1610 1974 1974 1973 1972 1972 1936 1936 1935 1934 1934 1899 1898 1898 1897 1896 r862 1861 i860 i860 1859 1825 1824 1823 1823 1822 1788 1788 1787 1786 1786 1752 1751 1751 1750 1749 1716 1715 1715 1714 1714 1680 1680 1679 1678 1678 1645 1644 1644 1643 1643 1610 1609 1609 1608 1607 1971 1970 1970 19O9 1068 1933 1933 1932 1 93 1 1 93 1 1896 1895 1894 1894 1893 1859 i858 i857 i857 i856 1822 1821 1820 1820 1819 1785 1785 1784 1783 1783 1749 1748 1748 1747 1746 1713 1712 1712 1711 1711 1677 1677 1676 1676 1675 i642 i64i i64i i64o i64o 1607 i6q6 1606 i6q5 i6o5 1968 1967 1967 1966 1965 1930 1929 1929 1928 1928 1893 1892 1891 1891 1890 i855 i855 1 854 i854 i853 1819 1818 1817 1817 1816 1782 1781 1781 1780 1780 1746 1745 1745 1744 1743 1710 1709 1709 1708 1708 1674 1674 1673 1673 1672 1639 i638 i638 1637 i637 1604 i6o3 i6o3 1602 1602 1965 1964 1963 1963 1962 1927 1926 1926 1925 1924 1889 1889 1888 1888 1887 i852 i852 i85i i85o i85o 1816 i8i5 i8i4 i8i4 i8i3 1779 1778 1778 1777 1777 1743 1742 1742 1741 1740 1707 1706 1706 1705 1705 1671 1671 1670 1670 1669 1 636 i635 i635 1 634 1 634 1601 1600 1600 1599 1599 1962 1961 19G0 i960 1959 1924 1923 1923 1922 1921 1886 1886 1 885 1884 1884 1849 1849 1848 1847 1847 1812 1812 1811 1811 1810 1776 1775 1775 1774 1774 1740 1739 1739 -1738 1737 1704 1703 1703 1702 1702 1668 1668 1667 1667 1666 i633 i633 i632 i63i i63i 1598 1598 1597 1596 1596 1958 195s 1956 1956 1919 1919 1918 i883 :88 3 1SS2 1881 18S1 1 846 1845 1845 1844 1844 1809 1809 1808 1808 1807 1773 1772 1772 1771 1771 1737 1736 1736 1735 1734 1701 1700 1700 1699 1699 1 665 1 665 1 664 1664 1 663 i63o i63o 1629 1628 1628 1627 1627 1626 1626 1625 1595 1595 1594 1593 1593 1592 1592 1591 1591 1590 1955 195D 1954 1953 1953 1952 i95i 1951 1950 1950 1918 1917 1916 1916 1915 1880 1880 1S79 1878 1878 1843 1843 1842 i84i i84i 1806 1806 i8o5 i8o5 1804 1770 1769 1769 1768 1768 1734 1733 1733 1732 1 73 1 1698 1697 1697 1696 1696 1 663 1662 1661 1661 1660 1914 1914 1913 1913 1912 1877 1876 1876 1875 1875 1840 1839 1839 i838 i838 i8o3 i8o3 1802 1802 1801 1767 1766 1766 1765 1765 1 73 1 1730 1730 1729 1728 1695 1694 1694 1693 1693 1660 1659 i658 i658 1657 1624 1624 1623 1623 1622 1589 1 589 1 588 1 588 1 587 1949 1948 1948 19^7 1946 1911 1911 1910 1909 1909 1874 1873 1873 1872 1871 1837 i836 i836 1 835 i835 1800 1800 1799 1798 1798 1764 1763 1763 1762 1762 1728 1727 1727 1726 1725 1692 1692 1691 1690 1690 1657 1 656 i655 i655 1 654 1621 1621 1620 1620 1619 1 587 1 586 1 585 1 585 1 584 S. 1° 54' l°55' rsG' 1°57' l°o8' 1° 59' 2° 0' 2° 1' 2° 2' 2° 3' 2° 4' TABLE XXII. [^»seH3 Proportional Logarithms. S. o I 2 3 4 5 (') 7 8 9 lO 1 1 12 i3 i4 i5 i6 17 i8 19 20 21 22 23 24 35 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 4o 4i 42 43 Ai ■45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 S. h m, 2° 5' h m 2= & /i VI 20 7, h VI 2° 8' h m 2° 9' k m 2° 10' h VI 2° 11' h TO 2° 12 h VI 2° 13' h VI 2° 14' h m 2° 15' s. I 2 3 4 5 5 7 8 _9_ 10 II 12 i3 i4 i5 16 17 18 19 20 21 22 23 24 25 26 27 23 29 3o 3i 82 83 34 35 36 37 38 39 40 4i 42 43 44 "45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 S. 1 584 i583 i582 i582 i58i 1 549 1 548 1 548 1 547 1 547 i5i5 i5i4 i5i4 i5i3 l5l2 i48i 1 480 1479 1479 1478 1 447 1 446 1 446 1445 1445 i4i3 i4i3 14:2 1412 i4ii i38o 1879 1879 1878 1378 i347 1 346 1 346 i345 1 345 i8i4 i3i4 i3i3 i3i3 l3l2 1282 1281 1281 1280 1280 1249 1249 1248 1248 1247 i58i i58o ! 5So i579 1578 1 546 1 546 i545 1 544 1 544 l5l2 i5ii i5ii i5io i5io 1478 i477 1 477 1476 1476 1 444 1443 1 443 1442 1442 i4ii i4io 1409 1409 i4o8 1877 1877 1876 1376 1875 1 344 1 344 1 343 1 343 1842 i3ii i3ii i3io i3io 1809 1279 1278 1278 1277 1277 1247 1246 1246 1345 1245 1578 1 577 1 577 1576 1576 1 543 1 543 1 542 1 542 i54i 1 509 i5o8 i5o8 i5o7 1 507 1475 1474 1474 1473 1473 1472 1472 1471 1470 1470 i44r i44i 1 440 1 440 1439 1 438 i438 1437 1437 i436 1 408 1407 1 407 i4o6 i4o6 i4o5 i4o4 i4o4 i4o3 i4o3 1874 1874 1878 1878 1872 1872 1871 1871 1870 1870 1869 1 368 1 368 1867 1867 1 366 1 366 i365 1 865 1864 1342 i34i 1 340 i34o 1889 1809 i3o8 1808 1807 1807 1276 1276 1275 1275 1274 1244 1 24'' 1243 1242 1242 1575 1 574 1 574 1S73 1573 1 540 1 540 1539 1539 1 538 i5o6 i5o6 i5o5 i5o4 i5o4 1339 1 338 i338 i337 i337 i336 i335 i335 1 334 1 334 i3o6 i3o6 i3o5 i3o4 i3o4 i3o8 i3o3 l302 l302 i3oi 1274 1273 1278 1272 1271 1241 1241 1240 1240 1289 1572 1571 1571 1670 1570 i538 1 537 1 536 1 536 i535 i5o3 i5o3 l502 l502 i5oi 1469 1469 1 468 1 468 1467 1 436 1435 i435 1434 i433 l402 l402 i4oi i4oi i4oo 1271 1270 1270 1269 1269 1289 1238 1238 1287 1287 1569 1 569 1 568 1 567 1 567 i535 1 534 1 534 1 533 i532 i5oo i5oo 1 499 1499 1498 1467 1 466 1 465 1 465 1 464 1 43 3 1432 i432 i43i i43i 1899 1399 1398 1398 1897 i338 i833 1882 1882 i33i i3oi i3oo i3oo 1299 1298 1268 1268 1267 1267 1266 1236 1235 1285 1284 1284 1 566 1 566 1 565 i565 1 564 i532 i53i i53i i53o i53o 1498 1497 1496 1496 1495 1 464 1 463 1 463 1462 i46i i43o 1429 1429 1428 1428 1897 1396 1396 1895 1894 1 368 1 363 1862 1862 i36i i33i i33o 1829 1829 1828 1298 1297 1297 1296 1296 1266 1265 1264 1264 1263 12 83 1233 1282 1232 1281 1 563 i563 1 562 1 562 i56i 1529 i528 i528 i527 i527 1495 1494 1494 1493 1493 i46i 1 460 1 460 1459 1459 1427 1427 1426 1426 1425 1394 1398 1893 1892 1892 i36i i36o i36o 1859 1359 1828 1827 1827 1826 1826 1295 1295 1294 1294 1298 1263 1262 1262 1261 1261 I23l 1280 1280 1229 1229 i56i i56o 1559 1559 1 558 i526 1 526 i525 i524 i524 1492 1491 1491 1490 1490 i458 i458 1457 i456 i456 1424 1424 142.3 1423 1422 1891 1391 1 390 i389 1889 i358 i357 i357 i356 1 356 i355 i355 i354 1 354 i353 i325 1825 1824 1823 i323 1822 1822 1821 1821 1820 1292 1292 1291 1 291 1290 1290 1289 1289 1288 1288 1260 1260 1259 1259 1258 1228 1227 1227 1226 1226 1 558 1 557 i556 1 556 i555 i523 i523 l522 l522 1 52 1 1489 1489 1 488 1487 1487 i455 i455 1454 1454 i453 1422 1421 1421 1420 1419 1 388 1 388 1 387 1887 1 386 1257 1257 1256 1256 1255 1255 1254 1254 1253 1253 1225 1225 122.4 1224 1228 1228 1222 1222 I22I I 221 1 555 1 554 1 554 i553 i552 I 520 l520 i5i9 i5i9 i5i8 i486 i486 i485 i485 i484 1452 1452 1 45 1 i45i i45o 1419 i4i8 i4i8 1417 i4i7 1 386 i385 1 384 i384 i383 i352 i352 i35i i35i i35o 1820 1819 i3i9 i3i8 1817 1287 1287 1286 1285 1285 i552 i55i i55i i55o i55o i5i8 i5i7 i5i6 i5i6 i5i5 i483 i483 1482 1482 i48i i45o 1449 1449 1 448 1447 i4i6 i4i6 i4i5 i4i4 i4i4 1 383 i382 1 382 i38i i38i i35o 1 349 1 349 1 348 1848 1817 i3i6 i3i6 i3i5 i3i5 1284 1284 1288 1283 1282 1252 1252 T25l I250 I25o X220 I219 1219 1218 13l8 2° 5' 2^^ 0' 2° 7' 2° 8' 2= 9' 2° 10' 2° 11' 2^12' 2"^ 13' 2M4' 2° 15' Page ]44] TABLE XXII. Proportional Logarithms. S. o I 2 3 4 5 6 7 8 9 10 1 1 12 i3 i4 i5 i6 17 18 19 20 21 22 23 24 25 26 27 28 29 3(' 3i 3-2 33 34 35 36 37 38 39 4o 4i 42 43 44 45 46 47 48 49 5o 5[ 52 53 54 55 56 57 58 59 2°iG' k m 2° 17' h m 2° 18' h m 2° 19' h m 2° 20' h m 2° 21' h m 2° 22' h m 2° 23' h m 2° 24' h m 2° 25' h m 2°2G' S. I 2 3 4 5 6 7 8 9 10 II 12 i3 i4 i5 i6- 17 18 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 4o 4i 42 43 A^ 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 S. 1217 1217 1216 1216 I2l5 I2l5 I2I4 I2l4 I2l3 I2l3 1 186 ii85 ii8'4 1 184 ii83 ii54 ii53 ii53 Il52 Il52 II23 1122 1122 1121 1 1 20 1091 1091 1090 1090 1089 1061 1060 1060 1059 io58 1000 1029 1029 1028 1028 0999 0999 0998 0998 0997 0969 0969 0968 0968 0967 0939 0939 oo38 0938 0937 0909 0909 0908 0908 0907 ii83 1182 1182 1181 1181 ii5i ii5i ii5o ii5o 1 149 1120 1119 1119 1118 1118 1089 1088 1088 1087 1087 io58 io57 io57 io56 io56 1027 1027 1026 1026 1025 0997 0996 0996 0995 0995* 0967 0966 0966 0965 0965 0937 0936 0936 0935 0935 0907 0906 0906 0905 0905 I2I2 I2II 121 I I2I0 I2I0 II»0 1 180 1 179 1179 1 1 78 1 149 ii48 I [48 ii47 1 147 1117 1117 1116 1116 iii5 1086 1086 io85 io85 1084 io55 io55 io54 io54 io53 1025 1024 1024 I023 1023 1022 1022 I02I I02I 1020 0994 0994 0993 0993 099" 0992 0991 0991 0990 0990 0964 C964 0963 0963 0962 0962 0961 0961 0960 0960 0934 0934 0933 0933 0932 0904 0904 0903 0903 0902 1209 1209 1208 1208 1207 1178 1177 1177 1 176 1175 1 1 46 ii46 1 145 1145 ii44 iii5 1114 iii4 iii3 iii3 1084 io83 io83 1082 1082 io53 io52 I052 io5i io5i 0932 0931 0931 0930 0930 0902 0901 0901 0900 0900 1207 1206 1206 I205 I2o5 1 175 1 174 1 174 1173 1 173 1 143 1 143 1142 1 142 ii4i 1112 1112 nil I III II 10 1081 1081 1080 1080 1079 io5o io5o 1049 1049 io48 1020 IOI9 1019 IO18 IO18 0989 0989 0988 0988 0987 0959 0969 0958 0958 0957 0929 0929 0928 0928 0927 0899 CS99 0S98 0898 0897 1204 1204 I203 1202 1202 1172 1172 1171 1 171 1 170 ii4i ii4o ii4o 1 1 39 1 1 39 mo 1 109 II 09 1 108 1 108 1079 1078 1078 1077 1076 1048 1047 1047 io46 io46 IOI7 IOI7 IO16 IO16 ioi5 0987 0986 0986 0985 0985 0957 0956 0956 0955 0955 0927 0926 0926 0925 0925 0897 0896 0896 0895 0895 I20I I 201 1200 1200 I 199 1 170 1 1 69 1 169 1 1 68 1168 ii38 ii38 ii37 ii37 ii36 1 107 1 106 1 106 iio5 iio5 1076 1075 1075 1074 1074 1045 1045 1044 1044 1043 ioi5 ioi4 . ioi4 10x3 ioi3 0984 0984 0983 0983 0982 0954 0954 0953 0953 0952 0924 0924 0923 0923 0922 0894 0894 0893 0893 0892 1199 I 198 II9S I 197 IP97 1167 1 167 1 166 ii65 ii65 ii36 ii35 ii35 ii34 ii34 1 104 iio4 iio3 II o3 1102 1073 1073 1072 1072 1071 1043 1042 1042 io4i io4t 1012 1012 lOI I ion lOIO 0982 0981 0981 0980 0980 0952 0951 0951 0950 0950 0922 0921 0921 0920 0Q20 0892 0891 0891 0890 0890 I I 96 I I 96 I 195 I 195 I 194 .193 I 193 I I 92 .1192 II9I 1 1 64 ii64 ii63 1 1 63 1 162 ii33 Il32 Il32 ii3i ii3r 1102 IIOI IIOI 1 100 1100 1071 1070 1070 1069 1069 io4o io4o 1039 1039 io38 1009 1009 1008 100& 1007 0979 0979 0978 0978 0977 0949 0949 0948 0948 0947 0919 0919 0918 0918 0917 0889 0889 0S88 0888 0887 1 162 1161 1161 1 1 60 1 1 60 ii3o ii3o 1129 1129 1 1 28 1099 1099 1098 1098 1097 1068 1068 1067 1067 1066 io37 io37 io36 io36 io35 1007 1006 1006 ioo5 100 5 0977 0976 0976 0975 0975 0947 0946 0946 0945 0945 0917 0916 0916 0915 0915 0887 0886 08S6 o885 oS85 H9I I I 90 : 190 1.89 1.89 ii58 ii58 ii57 1128 1127 1127 1 1 26 IT 26 1097 1096 1096 1095 1095 1066 io65 io65 1064 1064 io35 io34 io34 io33 io33 1004 ioo4 ioo3 ioo3 1002 0974 0974 0973 0973 0972 0944 0944 0943 0943 0942 0914 0914 0913 0913 0912 0884 0884 o883 o883 o883 n88 1 188 1187 1187 1 186 ii57 ii56 ii56 ii55 ii54 II25 II25 I I 24 I I 24 II23 1094 1094 1093 1092 1092 io63 io63 1062 1062 1061 io32 1032 io3i io3i io3o 1002 lOOI 100 1 1000 1000 0972 0971 0971 0970 0970 0942 0941 0941 0940 0940 0912 09-1 091 1 0910 0910 0882 0882 0881 0881 0880 9° 16' 2° 17' 2° 18' 2° 19' 2° 20' 2° 21' 2° 22' 2° 23' 2° 24' 2=25' 2° 96' 1 TABLE XXII. [l-age 115 1 Proportional Lc (garilhms s o /* 7n h m h m h VI k m h m A in A m h m A m A m 2° 27' 2° 28' 2° 29' 2^30' 2° 31' 2'' 32' 2° 33' 2° 34' 2° 35' 2° 3G' 0621 2° 37' 0594 S. 0880 08 5o 0821 0792 07()3 0734 0706 0678 0649 I 0879 o85o 0820 0791 07G2 0734 0705 0677 0649 0621 0593 I 2 0879 0S49 0820 0791 0762 0733 0705 0677 0648 0621 0593 2 3 0878 0849 0819 0790 0762 0733 0704 0676 0648 0620 0592 3 4 5 0878 o848 0819 0790 07()i 0732 0704 0676 0648 0620 0592 0591 4 5 0877 o848 0818 0789 0761 0732 0703 0675 0647 0619 6 0877 0847 0818 0789 0760 0731 0703 0675 0G47 0619 0591 6 7 0876 0847 0817 0788 0760 0731 0703 0674 o646 0618 0591 7 8 0876 o846 0817 0788 0759 0730 0702 0674 0646 0618 0590 8 9 10 0875 .0846 0816 0787 0759 0730 0702 0673 0645 0617 0617 0590 0589 9 10 0875 0845 0816 0787 0758 0730 0701 0673 0645 II 0874 0845 0816 0787 0758 0729 0701 0672 0644 0616 0589 II 12 0874 0844 08 1 5 0786 0757 0729 0700 0672 0644 0616 (,588 12 i3 0873 0844 o8[5 0786 0757 07^8 0700 0671 0643 061 5 o588 i3 i4 i5 0873 0843 o8i4 0785 0756 0728 0699 0699 0671 0643 06 1 5 i;6T5~ 0587 o587 i4 i5 0872 0843 0814 0785 0756 0727 0670 0642 i6 0872 0842 o8i3 0784 0755 0727 0698 0670 0642 06 1 4 c586 16 17 0871 0842 o8i3 0784 0755 0726 0698 0670 064 1 o6i4 o586 17 i8 0871 0841 0812 0783 0754 0726 0697 o6()9 064 1 06 1 3 o585 18 19 20 0870 084 1 0812 0783 0754 0725 0697 0669 oG4i 061 3 (.585 19 20 0870 0840 081 1 0782 0753 0725 0696 0668 o64o 0612 o585 21 0869 0840 081 1 0782 0753 0724 0696 0668 0640 0612 o584 21 22 0869 0809 0810 07S1 0752 0724 0695 0667 0639 06 1 1 o584 22 2j o8()8 0839 0810 0781 0752 0723 0695 0667 0639 061 1 o583 23 24 25 (.868 o838 0809 0780 07S0 0751 0723 0694 0666 o638 0610 o583 o582 24 25 0867 ()838 0809 0751 0722 0694 0666 o638 0610 26 0S67 0837 0808 0779 0751 0722 0694 o665 0637 0609 (>582 26 27 oSfifi 0S37 0808 0779 0750 0721 0693 o665 0637 0609 o58i 27 28 0S66 08 3G 0807 0778 0760 0721 0693 0664 o636 0609 ()58i 28 29 3o o865 08 36 o835 0807 0778 0749 0721 0692 0692 0664 o636 0608 0608 o58o o58o 29 3o' o865 0806 0777 0749 0720 oG63 0635 3i 0HG4 08 3 5 0806 0777 0748 0720 0691 o663 o635 0607 0579 3i 32 o8(34 o834 o8o5 0776 0748 0719 0691 o663 o634 0607 0579 32 33 o863 o834 o8o5 0776 0747 0719 0690 0662 o634 0606 o579 33 34 35 o863 oS34 0804 0775 0747 0718 0690 0DD2 o634 0606 0578 M 35 0862 o833 0804 0775 0746 0718 0689 0661 o633 06' >5 0578 36 0862 0833 080 3 0774 0746 0717 0689 0661 o633 oGm5 o577 36 37 0861 C.832 o8o3 0774 0745 0717 0688 ' 0660 o632 0604 o577 37 38 0861 o832 0802 0774 0745 0716 0688 0660 o632 0604 0576 38 39 4o 0860 o83i 0802 0773 0744 0716 0687 0669 o63i 060 3 0576 39 40 0860 o83i 0801 0773 0744 0715 0687 0659 o63i 060 3 o575 4i 0859 o83() 0801 0772 0743 0715 0686 o658 oG3o 0602 0575 4i 42 0859 o83c) 0801 0772 0743 0714 0686 06'") 8 o63o 0602 0574 42 43 08 58 0829 0800 0771 0742 0714 0686 0()*17 0629 0602 o574 43 44 45 C.858 0829 0800 0771 0742 0713 068 5 068 5 0657 0629 0601 0573 44 45 08 57 0828 0799 0770 0741 0713 0628 060 1 0573 46 08 57 0828 0799 0770 0741 0712 0684 o656 0628 o6uo 0573 46 47 08 56 0827 0798 0769 0740 0712 0684 06 5 5 0628 0600 0572 47 48 08 56 0827 0798 0769 0740 071 I o683 (,655 0627 0599 o5-'2 48 49 5o o855 0826 0797 0768 0740 07 II o683 0682 o655 "c.654" 0627 0G26 0599 0571 49 5o (xS55 0826 0797 076S 0739 07 I I 0598 (.571 5i oS55 0825 0796 0767 0739 0710 0682 o654 C626 0598 0570 5i 52 ()854 0825 0796 0767 0738 0710 0681 06 5 3 0625 0597 0570 b2 53 08 54 0824 0795 0766 0738 0709 0(58 1 o()53 0625 0597 o569 53 54 55 08 5 3 0824 0795 0766 0737 0709 0680 0652 ~c65"i' 0624 0624 0596 0596 0569 o5fi8 54 ■55" oS53 0823 0794 0765 0737 0708 0680 56 08 5 2 0823 0794 0765 0736 0708 0679 on5 1 0623 0596 o568 56 57 o852 0822 0793 0764 0736 0707 0679 o65i 0G23 0595 o568 57 58 o85i 0822 0793 0764 0735 0707 0678 o65() 0622 0595 o567 58 .A9_ S o85i 0831 0792 0763 0735 0706 0678 2° 33' 06 5o 2°"ji4' 0622 2^^ 35' 0594 o5G7 59 S. 2° 27' 2° 28' 9=29' 2° 30' 2° 31' 2° 33' 2°3()' 2° 37' 19 ^''seJ^G] TABLE XXII. Proportional Logaritbms. S. o I 2 3 4 5 6 7 8 9 10 II 12 i3 i4 i5 i6 I? i8 19 20 31 2 2 23 24 25 26 27 28 29 3o 81 32 33 34 35 36 37 38 39 40 4t 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 S. h m 2° 38' h m 2^39^ h m 2° 40' li m 2° 41' /i m 2° 42' h VI 2° 43' h m 2° 44' h m 2° 45' h m 2° 46' h m h m 2° 48' S. 1 2 3 4 5 6 7 8 9 10 1 1 12 i3 i4 i5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 80 3i 82 33 34 35 86 37 38 39 4o 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 S. o566 o566 o565 o565 o564 0589 o538 o538 o587 0537 o5i2 o5n o5ii o5io o5io o484 o484 o484 o483 o4S8 o458 0457 0457 04 56 0456 043 1 o43o 0480 o48o 0429 o4o4 o4o4 o4o3 o4o3 o4o8 0878 0877 0377 0877 0876 o852 o35i o85i o35o o35o 0826 0825 0825 0824 0824 o3oo 0299 0299 0298 0298 o564 o568 o563 o562 o562 o586 o536 o586 o585 o535 o5o9 o5o9 o5o8 o5o8 o5o7 0482 0482 0481 048 i 0480 0455 0455 0454 0454 0454 0429 0428 0428 0427 0427 o4o2 o4o2 o4oi o4oi o4oo 0876 0875 0875 0874 0874 0849 0349 0849 o348 0848 0828 0828 0828 0322 0822 0297 0297 0297 0296 0296 o562 o56i o56i o56o o56o o534 0534 o533 o533 o532 o5o7 o5o7 o5o6 o5o6 o5o5 0480 o48o 0479 0479 0478 0453 0458 0452 0452 o45i 0426 0426 0426 0425 0425 o4oo 0399 0899 0899 0898 0874 0878 0873 0872 0872 o347 o347 0846 o346 o346 o32i 0821 0820 0820 0819 0295 0295 0294 0294 0294 0559 0559 o558 o558 0557 o532 o53i o58i o53i o53o o5o5 o5o4 o5o4 o5o3 o5o3 0478 0477 0477 0476 0476 o45i o45o o45o o45o 0449 0424 0424 0428 0428 0422 0898 0897 0897 0896 0896 0895 0895 0895 0894 0894 0871 0871 0870 0870 0870 0845 0845 o344 o344 o348 0819 0819 o3i8 o3i8 0817 0298 0293 0292 0292 0291 0557 o557 o556 o556 o555 o53o 0529 0529 0528 0528 0502 o5o2 o5o2 o5oi o5oi 0475 0475 0475 0474 0474 0449 o448 0448 o44i o44i 0422 0422 0421 0421 0420 0869 0369 o368 0868 0867 o343 0842 0842 0842 0841 0817 o3i6 0816 o3i6 o8t5 0291 0291 0290 0290 0289 o555 0554 o554 o553 o553 0527 0527 0526 o526 0526 o5oo o5oo 0499 0499 0498 0473 0473 0472 0472 0471 o446 o446 o446 0445 0445 0420 0419 0419 o4i8 o4i8 0898 0893 0892 0892 0892 0867 0866 o366 o366 o365 0841 o34o o34o 0889 0889 o8i5 o8r4 o8i4 08 1 3 o3i3 0289 0288 0288 0288 0287 o552 o552 o552 o55i o55i o525 o525 o524 o524 o523 0498 0498 0497 0497 0496 0471 0471 0470 0470 0469 0444 0444 0443 0443 0442 o4i8 0417 0417 o4i6 o4i6 0891 0891 0890 0890 0889 o865 0864 o364 o368 o863 0339 o838 0338 0887 0887 0818 0812 03l2 o3ii o3ii 0287 0286 0286 0285 0285 o55o o55o 0549 0549 o548 o523 o522 0522 052I 052I 0496 0495 0495 0494 0494 0469 0468 o468 0467 0467 0.442 0442 044 1 044 1 o44o o4i5 •o4i5 o4i4 o4i4 o4i4 0889 o388 o388 o388 0887 o863 0862 0862 0861 o36i o386 0886 0886 o335 o335 o3io o3io 0810 0809 0809 0285 0284 0284 0283 0288 o548 o547 o547 o546 o546 052I o520 o520 o5i9 o5i9 0498 0493 0498 0492 0492 0467 0466 0466 o465 o465 o44o 0439 0489 0438 0488 o4i8 o4i3 04l2 04l2 • o4ii 0887 0886 0886 o385 o385 o36o o36o 0359 0359 0359 o334 0884 o833 o883 o388 0808 0808 0807 0807 0807 0282 0282 0282 0281 0281 o546 o545 0545 o544 o544 o5i8 o5i8 o5i7 o5i7 o5i7 0491 0491 0490 0490 0489 oi64 0464 0468 o463 0462 o488 0437 0437 o436 o436 o4ii o4io o4io o4io 0409 o384 o384 0884 o383 o383 o358 o358 0857 0857 o356 o332 0882 o33i o38i o33o 0806 o3o6 o3o5 o8o5 0804 0280 0280 0279 0279 0279 o543 o543 0542 0542 o54i o5i6 o5i6 o5i5 o5i5 o5i4 0489 0489 0488 o488 0487 0462 0462 0461 o46i 0460 0485 0435 0434 0434 0484 0409 o4o8 o4o8 0407 0407 o382 0882 0881 0881 o38i 08 56 o356 o355 o855 o354 o38o 0829 0829 0829 0828 o3o4 0804 o8o3 o8o3 o3o2 0278 0278 0277 0277 0276 o54i o54i o54o o54o 0539 o5i4 o5i3 o5i3 o5r2 o5r2 0487 o486 0486 o485 0485 o46o 0459 0459 o458 o458 o438 0488 0432 0482 0481 o4o6 o4o6 o4o6 o4o5 o4o5 0880 o38o 0879 0879 0878 o854 o858 o353 o353 o852 0828 o327 0827 0826 0826 0802 o3oi o3oi o3oo 0800 0276 0276 0275 0275 0274 2° 38' 2° 39' 2M0' 2° 41' 2° 42' 2° 43' 2° 44' 2° 45' 2° 46' 2° 47' 2° 48' ' TABLE XXIL [Page 147 Proportional Logarithms. A 771 h VI h m Ii in h in A 7/1 h m h m h m h m h m s. o 2° 49' 2° 50 2" 51 2^52 2° 53' 2=54 2=55 2° 50 2=5? 2=58' 2° 59' S. 0274 0248 0223 0197 0172 0147 0122 0098 0G73 0049 0024 I 0273 0248 0222 0197 0172 0147 0122 0097 0073 0048 0024 I 2 0273 0247 0222 0197 0171 CI 46 0122 0097 0072 0048 0023 2 3 0273 0247 0221 0196 0171 oi46 0121 0096 0072 0047 0023 3 _4_ 5 0272 0247 0221 0196 0171 oi46 0121 0096 0071 0047 0023 4 5 0272 0246 0221 0195 0170 0145 0120 0096 0071 0046 0022 6 0271 0246 0220 0193 0170 0145 0120 0095 0071 0046 0022 6 7 0271 0245 0220 0194 0169 0144 0U9 0095 0070 0046 0021 7 8 0270 0245 0219 0194 0169 oi44 0119 0094 0070 0045 0021 8 9 0270 0244 0219 0194 0169 0143 0119 0094 0069 0045 002f _9_ 10 lO 0270 0244 0219 0193 oi63 0143 0118 0093 0069 0044 0020 ti 0269 0244 02I& 0193 0168 0143 0118 C093 0068 oo44 0020 1 1 12 0269 0243 0218 0192 0167 0(42 0117 0093 0068 0044 0019 12 i3 0268 0243 0217 0192 0167 0142 0117 0092 0068 0043 0019 i3 i4 0268 0242 0217 0192 0166 oi4i 0117 0092 0067 0043 0019 i4 i5 i5 0267 0242 0216 0191 0166 oi4i 0116 0091 0067 0042 0018 i6 0267 0241 0216 0191 0166 oi4i 0116 0091 0066 0042 0018 16 17 0267 0241 0216 0190 oi65 oi4o oii5 0091 0066 0042 0017 17 i8 0266 0241 02l5 0190 oi65 oi4o oii5 0090 0066 oo4i 0017 18 '9 0266 0240 02l5 0189 0164 ot39 oii4 0090 oo65 004 1 0017 19 20 0265 0240 02l4 0189 0164 0139 oii4 0089 006 5 oo4o 0016 20 21 0265 0239 02l4 0189 oi63 0139 oii4 0089 0064 oo4o 0016 21 22 0264 0239 02l3 0188 oi63 oi38 oii3 0089 0064 oo4o ooi5 22 23 0264 0258 02l3 0188 oi63 oi38 oii3 0088 0064 0039 001 5 23 24 0264 0238 02l3 0187 0162 oi37 01 12 0088 006 3 0039 ooi5 24 25 0263 0238 0212 0187 0162 01 37 0112 0087 oo63 oo38 00 14 25 26 0263 0237 0212 0187 0161 01 36 01 1 2 0087 0062 oo38 00 1 4 26 27 0262 0237 02 I I o}86 0161 01 36 OIII 0087 0062 oo38 001 3 27 28 0262 0236 02II 0186 0161 oi36 OIII 0086 0062 0037 001 3 28 29 3o 0261 0236 02 11 oi85 0160 oi35 Olio 0086 0061 oo37 0012 29 0261 0235 0210 oi85 0160 oi35 OHO oo85 0061 oo36 0012 3o 3i 0261 0235 0210 0184 0159 oi34 OHO oo85 0060 oo36 0012 3i 32 0260 0235 0209 0184 0159 oi34 0109 0084 0060 oo36 001 1 32 33 0260 0234 0209 0184 01 58 oi34 0109 0084 0060 oo35 00 1 1 33 34 35 0259 0234 0208 oi83 oi58 oi33 0J08 0084 0059 oo35 0010 34 35 0259 .0233 0208 oi83 01 58 oi33 0108 oo83 0059 oo34 0010 36 0258 0233 0203 0182 oi57 Ol32 0107 oo83 oo5S oo34 0010 36 37 0258 0233 0207 0182 oi57 Ol32 0107 0082 oo58 oo34 0009 37 38 0258 0232 0207 0181 oi56 oi3i 0107 0082 oo57 oo33 0009 38 39 0257 0232 0206 0181 oi56 oi3i 0106 0082 co57 oo33 0008 J9_ 4o 40 0257 023l 0206 0181 oi56 oi3i 0106 0081 0057 oo32 0008 4i 0256 023l 020D 0180 oi55 oi3o oio5 0081 oo56 00 3 2 0008 4i 42 0256 023o 0205 0180 oi55 oi3o oio5 0080 00 5 6 oo3i 0G07 42 43 o2d:) 023o 0205 0179 oi54 0129 oio5 0080 oo55 oo3i 0007 43 44 0255 023o 0204 0179 oi54 0129 oio4 0060 oo55 oo3i 0006 44 45 45 0255 0229 0204 0179 oi53 0129 0104 0079 oo55 oo3o 0006 48 0254 0229 O203 0178 oi53 0128 oio3 0079 oo54 oo3o 0006 46 47 0254 0228 0203 0178 oi53 0128 oio3 0078 oo54 0029 ooo5 47 48 0253 0228 0202 0177 Ol52 0127 oio3 0078 oo53 0029 ooo5 48 49 ■50 0253 0227 0202 0177 Ol52 0127 0102 0077 oo53 0029 ooo4 49 5o 0252 0227 0202 0176 Gl5l 0126 0102 0077 oo53 0028 0004 61 0252 0227 0201 0176 oi5i 0126 OIOI 0077 0052 0028 0G04 5i 52 0252 0226 0201 0176 oi5i 0126 OIOI 0076 oo52 0027 ooo3 52 53 025i 0226 0200 0175 01 5o OI25 0100 0076 oo5i 0027 ooo3 53 54 55' 025l 0225 0200 0175 oi5o 0125 0100 0075 oo5i 0027 0002 54 '55 D25o 0225 0200 0174 0149 0124 0100 0075 oo5i • OC26 0G02 56 0260 0224 0199 0174 0149 0124 0099 0075 00 5o 0026 i 0002 56 5)7 035o 0224 0199 0:74 oi48 0124 0099 0074 oo5o 0025 0001 57 dS 0249 0224 0198 0173 oi48 OI23 0098 0074 0049 0025 0001 58 59 0249 0223 0198 0173 oi48 0123 0098 0073 0049 0025 0000 2°59^ 59 S. s. 2° 49' 2^50' i 2°5V 2° 52' 2^53 2° 54' 2° 55' 2=5G' 2° 57' f)0 KOI Page 148] TABLE XXIII To find the Latitude by two Altitudes of the Sun. HALF ELAPSED TIME. MIDDLE TLME. Hour. Hour. M. Iniinite. 3. 2. 2.36oi8 2.05916 I . i.883o7 * .75814 1 .60125 I .5820S I .5i5i5 I. 45718 I .4o6o5 23 li. 25 26 27 28 3o 3i 32 33 3, ~3T" 36 3- 38 39 40 4i 42 43 J4 45' 46 47 48 i9_ 5o 5i 52 53 55" j6 5? 58 59 I .36o32 I .31896 I .28120 I .24647 t .21432 I .18440 I .i5642 I .i3oi3 1 . io536 I .08193 56ii o35i5 o 1 5 1 6 99606 97777: I .05970 I.03857 I .01843 0.99918 0.98077 0.96310396023 0.9.(614194338 0.92982 92716 0.91411 91 154 0.89094189647 10" 13833 29324 02440 85959 74042 20" 04701 5701S 50494 • 823 398. 353 1 5 31243 27522 24095 20919 1 796 1 5192 12590 1 01 36 07814 8373o 23525 99221 83732 723^9 63322 5586i 49496 43946 39027 30" 40" 66121 153627 50" 43936 34609 3o6oo 6931 3549 204 1 2 7487 14748 12171 09740 7439 o5254 o3i75 01 192 99296 9748c o.8843o 87015 0.8 5644 0.84317 ).83o3o 0.81780 0.80567 793S7 0.78239 -77I2 2 0.76033 0.74972 0.73937 0.72927 -7194') o . 70976 0.70034 0.691 13:68962 0.68212 6806^1 0.6733067185 o.i:;6l')66'66324 o. 65620 6548 o.6479ij64655 o. 6397816 38 - o.63iSi 63(yjo 0.62400 (ii632 60879 o . 60 1 4o a8i9i 86783 85420 84 1 00 82819 87576 8o368 79193 7805 1 7693s 75854 74797 73767 72760 7^778 70818 5988 95738 94o63 92452 90899 89401 8-953 86553 85 197 83884 83609 81372 80170 9001 77863 76756 18409 1 3834 0969! 962 2 5 '9342 2 90790 81 613^79593 77663 7070069121J67597 61986,60690 54733:53634 4852047566 43(_i86'42243 38258 375o3 33915 33I3T 29967 26349 23oio 19910 17018 14307 11757 09348 07067 04901 02838 00870 98988 97184 95454 93790 29342 25774 22478 19415 16554 13872 1 1 346 08960 06699 o455o o2 5o4 oo55o 98682 96891 95172 93519' 92i89'9i928 90646:9039. 80156:88913 59431 5256i 46632 41417 36762 32558 28727 25207 21952 8925 0" 16096 1 3440 0939 08574 6333 04202 0217 00233 98378 96600 75676 74624 3597 87717,87481 86324:86096 84976184755 8366983455 82401 182193 81 169,80967 7997379777 78809178618 7767777491 76574176393 7549975323 7445 1 74279 734 0.59414 02271 6i5o(i 60755 600 1 8 59294 72595:7243 71616171455 70660 7o5o3 69725169571 ''-'■'°-- 68660 68811 67916 67040 66782 65342 6 ;5l9 63711 62919 62142 61 J80 606 3 1 59897 59175 67769. 66896 6604 1 653o4 64383 63578 62789 620 1"4 6788 61254 6o5o8 73261 72266 71295 70346 69418 685 TO 6762' 66752 65900 65o66 64248 63/t45 62659 94892 93250 91669 90143 88671 87247 85870 84535 83242 986 80767 79581 78428 773c6 76212 59775 59056 01 129 6o3S5 59654 5893- 75 1 47 74107 73093 72103 7 1 1 36 Jnf.lNeg 2. 2.94085 3. 3.24187 3.. 4 1 796 3.54289 3.63978 3.71895 3.78588 3.84385 3.89498 3.94071 3.98207 4- 4.01983 4 -05456 4.08671 7019c 69265 6836 67476 66609 65760 6 -(928 64ii3 633 1 3 62529 61759 61004 60262 59534 588i8j 23 25 26 27 28 ?9 3o 3i 32 33 35 36 37 38 39_ 4o 4i 42 43 _44_ 45 46 47 48 49_ 5o 5i 52 53 i^i 55 56 'J7 58 59 11663 4-i446i 17090 4-19567 4.21910 4-24i33 4-2624( 4.28260 4.3oi85 4-32026 _10" 16270 00779 27663 44 1 44 56061 654o2 78085 79609 85380 90294 20" 46373 06578 3oS82 46371 57764 947S8 98860 0258i 06008 09 1 84 12142 14911 i75i3 996 22289 4-3:^793 4.35489 4.37121 4.38692 40209 4.41673 43o88 4.44459 4.45786 4 .47073 4-48323 4-49536 4.50716 4.5i864 4.52981 24492 26588 28587 3o497 32026 06781 74242 80607 86157 91076 95494 995o3 o3i72 06554 09691 30" 40" 50" 63982 11694 33878 48490 09408 6877^ 75370 8i583 87017 91845 96188 76476 16269 36681 5o5io 60982 0013600761 03754 04329 694 1 3 76469 82537 S7860 9260c 96872 86167 2o4o8 39313 02440 o2 5o6 12616 15355 17932 2o363 22664 34080 35765 37387 38949 4o456 47^ 43320 44683 46oo3 7284 48527 49735 50910 52052 53i65 4.54070I54249 4.55i3i 4.56166 4.57176 4.58i63 4.59127 4 . 60069 4 • 6( 1990 4.61891 4.62773 53o6 56336 57343 58325 59285 60223 6ii4i 62039 62918 63779 64622 65448 662 58 67053 4.63637 .64483 4.65312 4.66125 4.66922 4.67703 67832 4.6847if6S597 '1.69224 4.69963 4.70689 24849 26928 28911 30807 32623 34365 36o4o 37651 39204 40702 42i5o 4355o 44906 46219 47494 8731 9933 5i 102 52240 53347 07093 10193 73oS5 1 5706 1 8346 20755 2 3o36 25202 27265 29233 3iii5 32919 34649 363x3 37914 39457 40947 42386 43779 45127 4643 47702 07625 10688 13549 1623 1S757 21 143 23404 553 27599 29553 3i42i 332 12 54427 55479 565o6 57508 58487 59443 60878 61292 62187 63o63 63921 6476 1 65584 66392 67184 G7961I 6S723: 6934869472} 70085 70206; 70809I709281 48934 5oi3o 5129.' 52426 53529 54604 55652 56674 57673 58648 59600 6o532 61443 62334 63207 64062 64899 65720 66525 67314 34931 36584 38175 39709 41190 42622 44007 45348 46648 47910 70672 77542 83471 8686 93341 97545 01376 081 5 1 178 14007 i6663 19164 21529 23770 5901 27931 29870 31725 335o3 49186 50826 5i485 52612 53710 352II 36853 38434 39960 1432 2856 44233 45568 46861 481 17 547S0 55824 56842 57837I58000 58So8 58967 49336 5o522 1675 52797 53891 54956 55996 57010 59757 60681 61593 62481 6335 1 68089 68849 69595 70338 71047 6420 3 65o37 65855 66658 C->7444 6821668344 59913 6o838 6 1 742 62627 63494 64343 65175 65990 66790 67574 6S974 69718 70449 7 1 1 66 19099 70569 71286 TABLE XXIII [Page 149 To find the Latitude by two Altitudes of the Sun. HALF ELAPSED TIME. MIDDLE TIME. 1 Houii. 1 Hour. M. o I 2 3 4 5 6 7 8 9 10 II 12 i3 i4 i5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 0" ]0' 20" 30" 40" 58232 57539 56857 56187 55528 54880 54242 536 1 4 52995 52387 51787 5.197 5o6i5 5oo42 49477 50" 58ii5 57424 56745 56076 55419 54773 541 36 535IO 52893 52286 5 1688 51099 5o5i9 49947 49383 M. I 2 3 4 5 6 7 8 9 10 II 12 i3 14 0" 10" 20" 30" 40" 50" 0. 58700 0.57999 0.57310 o.5ti633 0.55966 58583 57884 57196 56521 55856 552o3 54559 53926 533o3 52690 5208b 51491 50905 5o327 49758 58465 57768 57083 56409 55747 55095 54453 53822 53x00 52589 5 1 986 51393 5o8o8 5o232 49664 58348 57653 56970 56298 55637 4.71403 4.72104 4.72793 4.73470 4.74137 7i52o 72219 72907 -3582 74247 71638 72335 73020 73694 74356 71755 72450 73i33 738o5 74466 7.871 72564 73246 73916 74575 71988 72679 73358 74027 74684 75330 75967 76593 77210 77817 7841 5 79004 79584 801 56 80720 81275 81823 82 363 82896 83421 83940 8445i 84956 85454 85945 8643o 86909 87382 87849 883ii 88766 89216 89661 90100 90534 90963 91387 91806 92221 92630 o.553i I 0. 54666 . 54o3 1 0.53406 0.52791 54987 54347 53718 53098 52487 4.74792 4.75437 4.76072 4.76697 4.77312 74900 75544 76177 76800 7741 3 75008 75650 76281 76903 775i4 751 16 75756 76385 77005 77616 78217 78809 79392 799()6 8o533 81091 8i64i 82184 82719 83247 83768 84281 84788 85288 85782 86269 86750 87225 87694 88 1 58 75223 75861 76489 77108 77716 7S3i6 78906 79488 8006. 80626 8ii83 81732 82274 82808 83334 83854 84366 84872 85371 85864 86350 8683o 87304 87772 88234 0.52 1 86 0.51589 0.5 1 002 o.5o42 3 0.49852 5 1 886 51294 5071 1 5oi37 49570 4.779'7 4.78514 4.79101 4.796S0 4.8o25i 78017 78612 79198 79776 80345 78117 78710 79295 7987 1 80439 0.49290 0.48736 0.48189 0.47650 0.47119 49197 48644 48099 47561 47o3i 49104 48553 48009 47473 46944 46422 45907 45399 44898 444o3 49012 4846a 47919 47384 46856 48920 48371 47829 47295 46769 46249 45737 4523i 44732 44239 43753 43273 42799 4233i 41869 48828 48280 47740 47207 46682 461 63 45652 45 1 47 44649 44 1 58 43673 43194 42721 42254 41792 4i337 40887 40442 4ooo3 39569 39140 38716 38297 37882 37473 37068 36668 36272 35881 35494 35111 34732 34357 33987 33620 33257 32899 32543 32192 3 1844 3 1 5oo 3ii59 30822 3o48S 3oi58 i5 16 17 18 '9 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 U 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 4.80813 4.81367 4.81914 4.82453 4.82984 4.835o8 4.84025 4.84536 4.85o39 4.85536 80906 81459 82004 82542 83072 83595 84iii 84620 85 1 22 856i8 80999 8i55o S2094 82630 83i59 83681 84196 84704 85205 85700 0.46595 0.46078 0.45567 0.45064 0.44567 465oS 45992 45483 44981 44485 46335 45822 453i5 448 1 5 4432 1 43834 43353 42878 42409 41945 0.44077 0.43592 o.43ii4 0.42642 0.42176 43995 435i2 43o35 42564 42099 43915 43432 42956 42486 42022 4.86026 4.865n 4.86989 4.87461 4.87927 86108 86591 87068 87539 88004 88463 88917 89365 89808 90246 86188 86671 87147 87617 8S081 88539 88992 89439 89881 903 1 8 0.41716 0.41261 0.40812 o.4o368 0.39930 4i64o 41186 40738 40295 39857 39425 38998 38575 38 1 58 37745 4i564 4ii II 40664 40222 39785 39354 38927 385o6 380S9 37677 37270 36867 36469 36076 35687 4 1 488 4io36 40590 40149 39713 4i4i2 4096 1 4o5i6 40076 39641 4.88387 4.88842 4.89291 4.89735 4.90173 8861 5 89067 89513 89954 90390 90821 91247 91667 92083 92494 88691 89.42 89587 90027 90462 90892 9i3i7 91737 92152 92562 0.39497 0.39069 0.38646 0.38227 0.37814 39282 38856 38436 38o2o 37609 37203 36801 364o3 36oi I 3562 2 3921 1 38786 38366 37951 37541 37135 36734 36338 35946 35558 35174 34795 34420 340 -[S 33681 333i8 32958 32602 3225o 31902 3 1 2 1 6 30878 3o544 3o2i3 4 . 90606 4.91034 4.91457 4.91S76 4.92289 90678 91 io5 91528 91945 92358 90749 91176 91597 92014 92426 0.37405 0.37001 0.36602 0.36206 o.358r6 37338 36934 36535 36i4i 35751 4.92698 4.93102 4.93501 4.93897 4.94287 92765 93169 93568 93962 94352 92833 93236 93634 94027 94416 92900 93302 93700 94092 94481 92968 93369 93765 94 1 57 94545 93o35 93435 9383 1 94222 94609 0.35429 o.35o47 . 34669 0.34295 0.33925 35365 34984 34607 34233 33864 35302 34921 34544 34172 338o3 33438 33078 32720 32367 32018 35238 34858 34482 341 10 33742 33378 3^018 32661 32309 31960 3i6i4 31272 30934 3o599 3026S 4.94674 4.95o56 4.95434 4.95808 4.96178 94738 95119 95496 95870 96239 94801 95182 95559 95931 96300 94865 95245 95621 95993 96361 94929 95308 95683 96055 9642? 94992 95371 95746 96116 96483 96846 97204 97560 9:911 98259 98603 98944 99281 99615 99945 0.33559 0.33197 0.32839 0.32485 0.32 1 34 33499 33i37 32780 32426 32076 4.96544 4 . 96906 4.97264 4.97618 4.97969 96604 96966 97323 97677 98027 98374 98717 99057 99393 99725 966()5 97025 97383 97736 98085 98437 98774 991 1 3 99448 99780 96725 97085 97442 9779^ 98143 98489 98831 99169 99504 95835 96785 97145 97501 97853 98201 98546 98887 99225 99559 99S90 0.31787 o.3r443 o.3iio3 0.30766 o.3o433 3i729'3i672 3i386i3i329 3io46l3o99o 30710 3o655 3o378[3o323 4.98316 4.98660 4.99000 4.99337 4.99670 ^^seim TABLE XXIII. To find the Latitude by two Altitudes of the Sun. HALF ELAPSED TIME. MIDDLE TIME. 2 Hours. 2 Hours. 31. o I 2 3 4 5 6 7 8 9 lO II 12 i3 i4 i5 i6 17 iS 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 4o 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 0" 10" 20" 30" 40" 50" 31. 0" 10" 20" 30" 40" 50'' D.3oio3 0.29776 0.29453 0.29133 0.28816 3oo48 29722 29400 29080 28764 29994 29068 29346 29027 2871 1 29939 29614 29293 28974 28659 28346 28037 27731 27428 27127 29885 29561 29239 28921 28607 29831 29507 29186 28869 28554 I 2 3 4 5 6 7 8 9 10 II 12 i3 14 i5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 5 . ooooo 5.00327 5.oo65o 5.00970 5.01287 ooo55 oo38i 00703 0IO23 f)i339 00109 00435 00757 01076 01892 01705 02014 02821 02625 02926 00164 00489 00810 01129 01444 00218 00 54 2 00864 01 182 01496 00272 00596 009 1 7 01284 01549 0.28502 0.28191 0.27884 0.27579 0.27277 28450 28140 27833 27529 27227 269-'9 26633 26341 2605 1 25763 28398 28089 07782 27478 27177 26879 26584 26292 26003 25716 25432 25i5o 24872 24595 24322 28295 27986 27680 27378 27078 28243 27935 27630 27327 27028 2673, 26438 26147 25859 25573 5.01601 5.01912 5.02219 5.02524 5.02826 5.o3i25 5.03421 5.03714 5 . o4oo4 5.04292 5.04577 5.04859 5.o5i39 5.o54i6 5.o569Ci 5.05962 5.06232 5.06498 5.06763 5.07025 oi653 01963 02270 02574 02876 01757 02066 02872 02<375 02976 01808 021 17 02423 02725 08025 01860 02168 02473 02776 08075 08872 o3665 08956 t)4244 o453o 0.26978 0.26682 0.26389 0.26099 0.258(1 2683o 26535 26244 25955 25668 25385 25io4 24825 24550 24276 26781 36487 26195 25907 25621 o3i74 03470 08762 o4o52 04340 08224 o35i9 o38ii o4ioo 04387 08278 o3568 08859 o4i48 04435 08822 o36i6 08908 04196 04482 0.25526 0.25244 . 24964 0.24687 0.2441 3 25479 25197 24918 24641 24367 25338 25o57 24779 24504 24231 25291 250Ii 24733 24458 24186 04624 04906 o5i85 05462 o5736 04671 04953 o523i o55o8 05781 04718 04999 05278 o5553 05827 04765 o5o46 o5324 05599 05872 04812 (J5092 05370 o5645 05917 0.24141 0.23871 o.236o5 0.23340 0.23078 24096 23827 2356o 23296 23o35 22775 22519 22264 22012 21762 24o5i 23782 235i6 23253 22991 22732 22476 22222 21970 21720 24006 23738 23472 23209 22948 23961 23693 23428 23i65 22905 23916 23649 23384 23l22 22862 06007 06276 06543 06807 07068 o6o52 06821 06587 o685o 07 1 1 2 06097 06365 o663i 06894 07155 06142 06187 0641006454 0667506719 06938106981 0719SI07241 0.22819 0.22561 0.223o6 0.22054 o.2i8o3 22690 22433 22180 21928 21679 22647 22391 22138 21887 21 638 22604 22349 22096 21 845 21596 5.07284 5.07542 5.07797 5 . 08049 5 .o83oo 07328 07584 07839 (j8o9i 0834 1 07871 07627 078S1 081 33 08383 oS63o 08875 091 18 09359 09597 07418 07670 07928 08175 08424 07456 07712 07965 08216 o8465 07499 07754 08007 (.82 58 o85o7 08753 08997 09289 09478 09716 09951 10184 io4i6 10645 10872 1 1097 1 1820 1 1542 1 1761 11978 12194 12407 1 26 19 12829 i3(.37 i3243 .3447 13650 i3S5i i4n5o 0.21555 0.2 1 309 0.21 066 0.20824 0.20585 2i5i4 21269 21025 20784 20545 21473 21228 20985 20744 2o5o6 21432 21187 20945 20704 2o466 2023o 19996 19764 19534 19306 19081 18857 1 8635 184 1 5 18197 17981 17767 17554 17344 17135 21391 21147 20905 2o665 20427 2i35o 21 106 20864 20625 2o387 5.08548 5.08794 5.09037 5.09279 5.09518 085S9 08834 09078 09319 09558 08671 08916 09158 09899 09687 09878 1 1 07 10889 10569 10797 11022 1 1 246 1 1 468 11 688 1 1 906 (JS7 1 2 08956 09198 09438 09676 099 1 2 ioi46 10877 10607 io834 1 1 o63- i32o8 i3i79 i3i5o l3l2I 1 3093 5.16866 16S95 II 0. i3u64 i3o35 i3oo7 12978 12950 1 292 1 1 1 5. 17039 17068 17096 17125 17 1 53 17182 12 0.12S93 12864 12836 12808 12779 12731 12 5.17210 17239 17267 17295 17324 17352 i3 0. 12723 12695 12666 12638 I26IO 12582 i3 5.17380 17408 17437 17465 17493 17521 i4 i5 0. 12 554 12526 12499 12332 1 247 1 12443 i24i5 i4 5.17549 17577 17604 17632 1 7660 176S8 0.12387 i236o i23o5 12277 12249 i5 5.17716 17743 17771 17798 17826 17854 i6 0. 12222 12195 12167 I2l40 I21l3 i2o85 16 5.17881 17908 17936 17963 17990 18018 17 O.I2058 12o3l 12004 1 1977 1 1949 11922 17 5.18045 18072 18099 18126 i8i54 18181 i8 0.11S95 11868 11842 ii8i5 11788 11761 18 5.18208 18235 18261 18288 i83i5 ■ 8342 19 20 . 1 1 734 11708 11681 ii654 11628 1 1601 '9 5.18369 18395 18555 18422 i858i 18449 i86(j8 18475 i85o2 0. 1 1 575 11 548 1]522 1 1495 1 1469 11443 20 5.18528 18634 18660 21 0. ii4[6 1 1390 1 1 364 11 338 Il3l2 11285 21 5.186S7 18713 18739 18765 18791 18818 22 0. 1 1259 11233 1 1207 11181 iii56 1 1 1 3o 22 5.18844 18870 18896 18922 18947 18973 23 0. II 104 1 1078 I1052 11027 1 1 00 1 10975 23 5.18999 19025 19051 19076 19102 19128 24 25 0.10950 0. 10797 10924 10772 10899 10746 10873 10721 10848 1 0696 10822 10671 24 25 5.19,53 19179 19204 i923t 19255 19281 5.19306 19331 19357 19382 19407 19432 26 . 1 0646 10620 10595 105/0 10545 10520 26 i>. 19457 1948^ 19508 19533 19558 19583 27 0.10496 10471 io446 10421 10396 10371 27 5 . 1 9607 19632 19657 19682 19707 19732 28 0.10347 Io322 10298 10273 10248 10224 28 5.19756 19781 19805 19830 19855 19879 29 3o 0.1 01 99 IOI75 ioi5i 10126 10102 10078 29 5 . 1 9904 19928 19952 19977 20001 20025 0. ioo53 10029 iooo5 09981 09957 09933 3o 5.2oo5o 20074 20098 20122 20 1 46 20170 3i 0.09909 09885 09861 09837 09813 09789 3i 5.20194 20218 20242 20266 20290 2o3i4 3;! 0.0976509741 09718 09694 09670 09647 32 5.20338 2o362 2o385 20409 20433 2o456 33 0.0962309599 09576 09552 09529 09506 33 5.20480 2o5o4 20527 2o55i 20574 20597 34 35 0.094s 2 09459 09435 09412 o9389[o9366 34 5.20621 20644 20668 20691 20714 20737 20876 0.09343 093 I 9 09296 09273 09250 09227 35 5.20760 20784 20807 2o83o 20853 36 0.09204 09181 09158 09136 091 1 3 09090 36 5.20899 20922 20945 20967 20990 210l3 37 . 09067 09044 09022 08999 08977 0S954 37 5.2io36 21059 21081 21 104 21126 21149 38 0.08931 08909 08886 08S64 0884508819 38 5.21172 21194 21217 21239 21261 21284 39 40 0.08-97 08775 0864 I 08752 08619 08730 08597 08708 08686 39 40 5.21 3o6 21328 2i35i 21373 21395 21417 0.08664 08575 08553 5.21 439 21462 21 484 2i5o6 21528 2i55o 4i 0.0853 1 o85io 08488 08466 08444 08422 4i 5.21572 21593 2i6i5 216J7 21659 21681 42 . o84o 1 o837Q 08357 08336 o83i4 08293 42 5.21 702 21724 2 1746 21767 21789 21810 43 0.08271 0S250 0S228 oSao7 081 85 08164 43 5.2x832 2i853 21875 2189C 21918 21939 45 0.08143 08121 08100 08079 o8o58 o8o36 45 5.21960 21982 220o3 22l3o 22024 22045 22067 o.oSoi5 07994I07973 07952 07931 07910 5.22088 22109 22l5l 22172 22193 46 0.07889 07868 07848 07827 07806 07785 46 5.222l4|22235 22255 2227C 22297 223l8 47 0.07765 07744 07723 0770307682 07661 47 5.22338 22359|2238o 224OG 22421 22442 48 . 0764 1 07620 07600 0757907559 07539 48 5.22462 22483 2 2 5o3 22524 2 2544 22 564 49 5o 0.07518 07498 07478 07458 0-437 07417 49 5.22585 22605 22625 22645 22766 22666 22786 22686 22806 0.07397 07377 07357 07337 07317 0729- 5o 5.22706J22726 22746 5i 0.07277 07257 07237 0721707197 07178 5i 5.2282622846 22866 22886 22906 22925 52 f). 07 1 58 07 1 38 07119 07099 07079 O7o6f) 52 5.22945 22965 22984 23oo4 23o24 23o43 53 . 07040 07021 07001 06982 06962 06943 53 5.23o63 23o82 23l02 23121 23i4i 23 160 54 55 0.06923 06904 06885 06866 06846 06751 06731 06827 067 1 2 54 55 5.23i8o 23199 5.23595J23314 5.23410 23429 23218 23333 23237 23257 23276 23391 0.06808 06789 06770 23352 23372 56 . 0(5693 06674 o6656 0663706618 06599 56 23447 23466 23485 235o4 57 o.o658o o656i o6543 o6524 o65o5 06487 57 5.23523 23542 2356o 23579 23598 236i6 58 . 06468 06449 0643 1 064 12 06394 06375 58 5.23635123654 23672 23691 2370923728 59 0.06357 o6338 o632o o63o2 06283 06265 59 5.2374623765 2J783 238oiJ2382o|23838| "^'^^J TABLE XXIII To find the Latitude by two Altitudes of the Sun. HALF ELAPSED TIME. MIDDLE TIME. 4 Hours. 4 Hours. r.i. o 1 2 3 4 5 6 7 8 9 lO II 12 i3 i4 i5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 4o 4i 42 43 4^ 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 0" 1 10" 20" 30" 06192 06084 05977 05871 05766 o5662 05559 05457 05356 05257 o5i58 o5o6o 04964 04868 04774 04680 04588 04496 o44o6 043 17 40" 50" M. I 2 3 4 5 6 7 8 9 10 II 12 i3 i4 i5 16 17 18 19 0" 10" 2387'4 23983 24091 24197 24302. 20" 23892 24001 24108 24215 24320 30" 40" 50" 0.06247 0.061 3b . o6o3o 1;. 05924 o.o58i8 06229 06120 06012 05906 o58oi 06211 (j6i02 05995 o5888 05783 06174 06066 05959 o5853 05748 J5645 05542 o544o o534o o524o o5i42 o5o44 04948 04852 04758 0466 5 04573 04481 04391 o43o2 061 56 06048 05941 05836 o573i 05627 05525 05423 o5323 05224 o5i25 o5o28 04932 04837 04743 04649 04557 04466 04376 04287 5.23856 5.23965 5.24073 5.24179 5.24285 2391 1 24019 2412O 24232 24337 24441 24544 24646 24747 24846 24945 25o43 25i39 25235 25329 28929 24037 24144 24251-) 24355 24458 24561 24663 24763 24863 24961 25o59 25i55 2525l 25345 23947 24o55 24 1 62 24267 24372 24476 24578 24680 24780 24879 24978 25075 25171 25266 2 5360 0.05714 o.oSoio o.o55o& o54o7 o.o53o6 o56y6 05593 05491 05390 05290 05679 05576 05474 05373 05273 o5i74 o5o77 04980 04884 04789 5.24389 5.24493 5.24595 5.24696 5.24797 24407 24510 24612 24713 2481 3 24424 24527 24629 24730 2483o 0.0020/ o.o5io9 o.o5oi2 0.04916 o.(j482i 05191 05093 04996 04900 o48o5 5.24896 5 . 24994 5.25091 5.25187 5.25282 5.25376 5.25469 5.25561 5.25652 5.25742 24912 25oio 25107 252o3 25298 24929 25026 25l23 25219 253i4 '0. 04727 0.04634 0.04542 o.o445i 0.04361 04711 04619 04527 04436 04346 04696 o46o3 045 1 2 04421 04332 25392 25484 25576 25667 25757 25845 25933 26020 26105 26190 25407 255oo 25591 25682 25771 25860 25947 26034 26120 26204 25423 255i5 25607 25697 25786 25875 25962 26048 26134 26218 25438 2553o 25622 25712 258oi 25454 25546 25637 25727 258i6 0.04275 0.041 85 0.04098 . o4o 1 2 0.03927 04258 04170 o4o83 03998 03913 04243 o4i56 04069 03983 03899 04228 o4i4i o4o55 03969 o3bS5 o38o2 03719 03638 o355^ 03478 03399 o3322 o3245 o3i7o 03095 04214 04127 o4o4o 03955 03871 037S8 03706 o3624 o3544 o3465 o3386 o33o9 o3233 o3i57 o3o83 04199 04112 04026 03941 o3857 03774 03692 o36ii o353i 03452 03373 03296 03220 o3i45 o3o70 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 5.25831 5.25918 5.26005 5.26091 5.26176 5.26260 5.26343 5.26425 5.2C5o6 5.26586 5.26665 5.26743 5.26820 5.26896 5.26971 25889 25976 26063 26148 26232 25904 25991 26077 26162 26246 o.o3843 0.037G0 0.03678 0.03597 o.o35i7 o.o3438 o.o336o o.o3>83 o.o3;o7 o.o3i32 03829 03747 o3665 o3584 o35o4 o3425 03348 03271 03195 o3i2o o38i5 03733 o365 1 03571 03491 o34i2 03335 o3258 o3i82 o3i07 26274 26356 26438 26519 26599 26678 26755 26832 26908 26983 26288 26370 26452 26532 26612 26691 26768 26845 26921 26996 263oi 26384 26465 26546 26625 263 1 5 26397 26479 26559 2D638 26829 26411 26492 26572 2665 1 26704 26781 26858 26933 27008 26717 26794 26870 26946 27020 26780 26807 26883 26958 27033 o.o3o5S 0.029S5 0.02913 0.02841 0.02771 o3o46 02973 02901 02829 02759 o3o34 02961 02889 02818 0274s o3o2i 02949 02877 02806 02736 o3oo9 02937 02865 02794 02724 02997 02925 02853 02783 02713 35 36 37 38 39 40 4i 42 43 45 46 47 48 49 5.27045 5.27118 5.27190 5.27262 5.27332 27057 271 3o 27202 27274 27344 27069 27142 27214 27285 27355 27082 27154 27226 27297 27367 27436 27504 27571 27637 27703 27094 27166 27288 27309 27379 27447 275i5 27582 27648 27713 2710b 2717S 27250 27820 27890 27459 27526 27593 27659 27724 0.02701 0.02633 0.02565 0.02499 0.02433 02690 02622 02554 02488 02422 02357 02294 o223l 02169 02108 02678 02610 02543 02477 0241 1 02347 02283 02221 02159 02098 O2o38 01979 01921 1 864 01808 01752 01698 1 644 01591 01 540 02667 02599 02532 02466 o»4oo o233() 02273 02210 02149 020S8 02656 025S8 02521 02455 02390 02326 02262 02200 02139 02078 02644 02577 025lO 02444 02379 023 1 5 02 252 02190 02128 02068 5.27402 5 . 27470 5.27538 5.27604 5.27670 274i3 27481 27549 27615 27681 27425 27493 27560 27626 27692 0.02 368 O.02 3o4 0.02241 0.02179 0.021 18 5.27735 5.27799 5.27862 5.27924 5.27985 27746 27809 27872 27934 27995 27756 27820 27882 27944 28005 28065 28124 28182 28239 28295 2835T 284o5 28459 285l2 28563 27767 27830 27893 27954 28015 28075 28134 28191 28249 283o5 27777 27841 27903 27964 28025 28085 28143 28201 28258 283i4 27788 27851 27918 27975 28035 28094 28153 28211 28267 28823 28878 28432 28485 28538 28589 o.o2o58 0.01999 . 1 940 O.OI883 0.01826 02048 01989 01931 01S73 01817 OI761 01707 01 653 1 600 o:548 02028 01969 01912 01 854 01798 02018 01960 01902 01845 01789 02009 01950 01892 01 836 01780 5d 5i 52 53 54 55 56 57 58 59 5.28045 5.28104 5.28163 5.28220 5.28277 28055 28114 28172 28230 28286 2S342 2S396 28450 285o3 28555 0.01771 0.01716 0.01662 0.01609 0.01557 01743 1 6S9 oi635 01 583 oi53i 01734 1 680 01627 01574 oi523 01725 01671 01618 01 565 oi5i4 5.28332 5.28387 5.28441 5 . 28494 5.28546 2836o 28414 28468 28520 28572 28369 28423 28476 28529 28580 TABLE XXIJI. To find the Latitude by two Altitudes of the Sun. [I'jge 153 '1 HALF ELAPSED TLME. 5 HouHs. MIDDLE TIME. 5 Hours. i3 i4 i5 i6 17 i8 19 23 24 25 26 27 28 29 3o 3i 33 33 34 35 36 37 38 4o 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 0" . o I 5o6 .01455 .oi4o6 .01357 .oi3io 10' .01263 .01217 .01172 01 I28|0I 120 .0108401077 01497 01447 01398 01349 Ol302 OI255 01209 01 164 01489 01439 01390 01 34 1 01294 20' o 1 247 01202 OII57 oiii3 01070 .0104201035 .0100000993 .0096000953 .oo92o]oo9i3 .00881 00874 .ou843bo836 .oo8o5 00799 .00769 00763 .0073300728 . 0069900693 01028 00987 00946 00907 00868 oo83o 00793 00757 00722 00687 .00665 00659 .00632:00626 . 00600 .oo568 .00538 0059 oo563 oo533 .oo5oS .00480 .00452 .00425 .00399 oo5o4 oo475bo47o .00373 .00349 .0032.5 .oo3o2 .00280 .00259 .00239 .00219 10654 00621 00589 oo558 00528 00499 00447 00420 00394 00369 oo345 oo32i 00298 00276 00443 oo4i6 00390 00255 00235 00216 oo365 oo34i oo3i7 00295 00273 .0020000197 .00 1 83 00180 001 63 00147 00l32 00117 00104 .00166 .00149 .00 1 34 .00120 . 00 1 06 . 00093 .ooobi .00070 .00060 .ooo5o . ooo4 I .ooo33 .00026 .00020 252 00232 002 1 3 00 1 94 00177 00160 001 44 00129 001 1 5 00102 00091 00079 00068 ooo58 00049 ooo4o 00032 00025 00019 0.0001 5 000 1 4 .00010 . 00007 .00004 .00002 .00000 00010 00006 oooo3 0000 1 00000 00089 00077 00066 ooo56 00047 00039 ooo3i 00024 00018 (joo 1 3 00009 00006 0000 3 oooo'i 00000 30" 01480 oi43o oi38i oi333 01286 01240 01194 o 1 1 5o 01 106 oio63 01021 00980 00940 C0900 00862 00824 00787 00751 007 1 6 00682 00648 00616 00584' oo553 oo523 00494 00466 00438 oo4i2 oo386 oo36i 00337 oo3i3 00291 00269 00249 002 29 00210 00191 00 1 74 001 57 00142 00127 001 1 3 00099 40" 50" 01472 01422 01373 o 1 3 2 5 0127S 01232 01187 01 142 o 1 099 oio56 oioi4 00973 00933 00894 00855 00S18 00781. 00745 007 1 o 006-6 00643 006 1 o 00579 oo548 oo5i8^ 01464 oi4i4 01 365 oi3i7 01271 01224 01 179 01 135 o 1 09 1 01049 01007 00966 00926 00887 00849 0081 1 00775 00739 00704 00670 00637 oo6o5 00574 00543 oo5i3 00489 oo46 1 00434 00407 oo382 oo357 oo333 oo3 1 o 00484 oo456 00429 oo4o3 oo377 oo353 00329 oo3o6 0028700284 0026600262 00245 00242 00225 00222 00207 00203 0018800185 00171 001 55 00 1 39 00 1 24 00 1 1 o 00097 00087 00075 0006 5 ooo55 ooo46] 000 3 7 ooo3o 00023 00017 000 1 3 00008 0000 5 oooo3 00001 00000 ooo85 00074 ooo63 ooo53 o' )o44 ooo36 00029 00022 00017 0001 2 00008 0000 5 00002 0000 1 00000 00168 001 52 00 1 37 00122 00 [ 08 00095 00083 00072 00061 000 5 2 00043 ooo35 00028 0002 1 00016 3001 1 00007 00004 00002 00(J0 oofioo M. 0" 28597 28648 28697 28746 28793 10" 28840 28S86 2S931 28975 290 1 9 29061 29103 29143 29183 29222 29260 29298 29334 28606 28656 28705 28754 28801 28848 28894 28939 28983 29026 29068 291 10 29150 2919c 29229 20" 28614 28664 28713 28762 28809 28856 28901 28946 28990 29033 29267 29304 29340 29370129375 29404 29410 2943829444 29471 29477 29503 29509 29535 29540 29565 29595 29623 29651 29678 29704 29730 29754 29778 29801 29823 29844 29864 29884 29903 59920 29937 29954 29969 29983 29997 , 3oo I o .3o02 2 ,3oo33 .3oo43 . 3oo53 3oo62 30070 3oo77 3oo83 3ooi 30093 30096 30099 3o I o I 29570 29599 29628 29656 29683J29687 29709 29713 29734:29738 29075 291 16 29157 29196 29235 29273 29310 29346 29381 2941 6 29449 29482 29514 29545 29575 29604 29633 29660 29758 29782 29805 29848 29868 29887 29906 29923 29940 29956 29971 29986 29999 3ooi2 3oo24 29762 29786 29808 29830 I985T 29S71 •'9890 29909 29926 29943 29959 29974 29988 3 0001 3ooi4 3oo26 3oo35j3oo37 3oo45j3oo47 3oo54 3oo56 3oo63i3oo64 30071 30078 3oo84 30072 3oo79 3oo85 30089130090 3oo93j3oo94 30097 30097 3oioo 3oioo 3oio2i3oi02 3oio3|3oio3i3oio3 30" 40" 28623 2863i 28673 2S681 28722 28730 2877028778 2881728825 28863 28871 2890928916 2895328961 28997 29004 29040J 29047 9082 29089 29123 29163 29203 29241 29279 29316 29352 29387 2 9421 29455 29487 29519 29550 29580 29130 29170 29209 29248 29285 29322 29358 29393 29427 29609 29637 29665 29691 2 9717 29742 29766 29790 2981 208; 29854 29874 29893 29912 29929 29946 2996 29976 29990 3ooo4 2946. 29493 29524 29555 29585 29614 19L 28639 28689 8738 878e 28833 28879 28924 28968 29012 29054 29096 29137 29177 29216 29254 29292 29328 29364 29399 29433 29466 29498 29529 29560 29590 29619 29642129647 29669 29674 29696 29700 29721 29726 29746 29750 977029774 29793 29816 29837 29858 >.99i5 !9932 29797 29819 29841 2^ 29881 29900 29918 29935 29951 29966 29981 3ooi6 3ooi8 30028 30029 3oo38|3oo4o 3oo48i3oo5o 3oo57 3oo59 3oo66 30067 3oo73i3oo74 3oo8o 3oo8 29948 29964 29979 : 29993129995 3ooo6i3ooo£ 30020 3oo3i 3oo42 3oo5i 3oo6o 3oo68 30075 30082 3oo86|3oo87 30091 30092 30095130095 30096 30098I30098 3oo86 30090 3o 1 00 3oi02 3oio3 3oioi 3oi02 3oio3 30099 3oioi 30102 3oio3 20 Page 154] TABLE XXIII. To find the Latitude by two Altitudes of the Sun. LOG. RISING OR VERSED SINE Hour. 4o 4 1 4a 43 44 "45 4G 4i 48 il 5o 5i 52 53 _5_4 55" 56 57 58 59 0" Inf.Neg 9 . 97860 o. o . 58o66 0.93284 I . [ . 18271 I .37053 I. 53488 I .66877 1.78474 1.88703 1.97554 2 . 2.061 3 1 2 . [ 3687 2. 2 06 38 2.27073 2.33o63 2.38667 2.43930 2.48893 2.5:^586 2.58o39 2.62274 2.663i2 2.701 70 2.73863 2.774o5 2.80809 2.84o83 •2.87238 2 .90282 2.93223 2 .96067 2 .98820 3. 3. 01 488 3.04077 3.06590 3.09032 3 . 1 1 406 3.13718 3 . 15969 10" 20" 42230 r i25o 65oi9 97980 21S17 4o5oi 55868 68920 80265 90297 02,436 22848 71455 02435 25224 99289 07437 14885 2 1 744 28100 34023 39567 44777 49693 54344 58759 62960 66967 ^0796 74464 59474 63641 67617 71418 75o6o 779S2 81 363 84617 87753 90779 93703 96532 99270 01925 o45oi 3.18162 3 . 2()3oi 3.223S9 3.24427J24762 3.264:8 26745 07001 09432 1 1 796 1 4097 i6338 l8522 2o653 22732 3.28363 3 .3 9 .»236o )492 2 0741 1 19831 12184 14475 16706 8881 2ioo3 2 3., 73 25095 27071 28683:29002 3o579 30891 3243 3425f 36028 37770 39477 4 1 152 42794 4440 5 459S6 47539 49064 5o562 52o35 32739 34549 "632 1 38o57 39759 41427 43o64 44670 46247 47795 493 1 5 50809 52278 37654 33079 77448 06673 28502 45^ 6o44o 72S69 83739 93399 02091 09991 17202 23915 3oi20 35910 41339 46447 51271 5 5841 6m82 643 16 68262 72o36 75652 62642 42 23o 83o54 1071 3 1 660 79124 82461 85675 88773 91765 94656 97454 00164 02792 05342 07819 10227 12570 i485o 17072 48524 62639 74778 85426 94909 o3456 1 1 240 i8382 24980 3i 1 12 50" S2024 5o5o9 88019 4575 34708 5io4i 64784 76646 7080 96394 1 Hour. o48o5 12472 19517 26(j33 32093 37758 43075 4S6S5 52821 36839 42211 47270 52o5o 5658o|573i3 6o8S5 67582 6498765652 68903 69538 72649 76241 79689 S3oo5 86199 89279 92254 19238 2i35i 234 1 4 25428 27396 29320 3 1202 33o44 34847 366 1 3 38343 4oo39 41702 43334 44935 46507 48o5r, 49566 5io56 52520 95129 97912 00608 0322 05760 08225 10622 12954 l5225 1743; 19^94 21699 23753 25759 i-j-jic 29637 3i5i2 333 35i44 36903 38628 4o3i9 4i97fi 36o3 45199 46766 483o5 49816 5i3oi 52761 73258 76825 8025T 83546 86720 S9782 92739 i3 0" 3.53243 3 . 5.4670 3.56074 3.57455 3.58814 3 . 60 1 5 2 3.61469 3.62766 3.64043 3.653o2 95599 9836 01049 o365i 06 1 76 08629 1 ioi5 i3337 i559' 9949 22tj4^ 2409c 26089 28042 9952 31820 33649 35439 37193 38912 40597 4225(1 i387i 45462 47024 48558 5oo6fi 5 1 547 53oo2 23 _2_4 2 5" 26 27 28 ='9 3o 3i 32 33 34 3.66542 3.67765 3.68969 3.70158 3.71329 3.72485 3.73625 3.74750 3.75860 3.76955 3 .78037 3.79105 3.80159 3.81201 3.82230 3.83246 3.&4250 3.85242 3.86223 3.87192 3.88i5( 3 .89097 3 . 90034 3 . 90( 3.91876 10" 53482 54905 563o6 57683 59038 60373 61686 62980 64254 655io 66747 67967 69169 70354 7i523 72676 738 1 3 74936 76043 7 7 '37 78216 79282 80334 81373 82400 20" 53721 55i4o 56537 57910 59262 60593 61903 63 194 64465 65717 66952 68168 69367 7o55o 71716 72S67 74ooi 75l2I 76227 77318 7S395 79458 8o5o8 8 1 545 82570 30" 53959 55375 56767 58i37 59486 t)o8 1 3 62120 63407 64675 65924 67156 68369 69566 70745 71909 834:4 83582 8441684582 6 86385 87352 ,92782 ,9367 9456 95443 963 97170 98021 98862 99696 .o53o4 , 06074 .06838 .07095 .08344 S83o9 S9254 90189 91114 92028 85570 86547 S7513 92933 9382 94712 95588 06455 973 1 3 98162 9900 99834 00657 01473 )228o o3o8o o387 o4656 ^5433 06202 )6965 )77lo 08468 09087 09823 io552 1 1 275 11992 88467 894 II 90344 91267 92179 93082 93975 94859 95733 96599 97455 98302 99141 99972 00794 70057 74189 75307 76409 77498 78573 79634 80682 8.717 82739 83749 84748 85734 40" J4197 556o8 56997 58363 50" 54434 5584 1 57226 58589 59708 59930 6io32i6i 25i 62336!6255i o362o!6383 64885i65o94 66i3ij66337 67359^67562 68570J68770 69763169961 094071 1 35 72101 72293 73247 73436 7437674563 5491 75676 659276774 77678 177858 78750J78928 '09I79985 8o855!8io28 81 888 j8 2059 32908183077 83917:84083 849i385o78 85897 86060 86709j8(.)87o|87o3i 87672i87S32!8799i 8S625 88783188940 89567 89723189879 90498 91-653 '90807 91420 91 572 91724 92331 924S2 92632 09210 09945 10673 1 1395 I i5i5 121 1 1 12229 o 1 608 02414 o32I2 o4oo3 04786 ("^67 o633o 1709 T 07845 08592 09333 1 0067 0794 93232 94123 95oo5 95878 96742, 97597 98443 99280 00109 0093 01743 02547 03344 o4i34 049 1 6 05690 06457 07217 93381I93530 94271 J94418 95i52 95297 9602396167 96885197028 97738 98583 99419 00247 01066 97880 98723 99557 oo384 01202I 01877 02681 03477 04265 o5o45 o58i8 <>6584 07343 07970I08095 08716108840 09456109578 101 88| io3 10 io9i5!i io35 ii634 1 1754 11873! 1 2348] 12466 125841 02814 (j36o8 04395 o5i75 05946 067 1 1 07469 08220 8q64 09701 1 043 1 1 1 155 TABLE XXIII. Li-a.-eiss To find the Latitude by two Altitudes of tiie Sun. LOG. RISING OR VERSED SINE. 2 lloLliS. 0" 23 24 25 26 27 28 ^9 3o 3i 32 33 34 35 36 37 38 4o 4i 42 43 J_4_ 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 ^"9 12702 i34o6 i4io4 14797 1 5483 ,i6i63 i6S3S 17507 18171 18829 .1948 .20129 ,20771 , 2 1 409 , 2Jo4l ,22668 ,23290 2390 24520 25128 25731 2633o 26924 27514 28099 .28681 .29257 .29S30 .30398 .30963 .3.523 .32079 .32631 .33i8o .33724 .34265 .34802 .35335 .35865 .36391 10" 12820 i3523 1 42 20 14911 15597 16276 1695 17618 18281 18938 1959. 20236 20878 2i5i4 22 1 46 22772 23393 24010 24622 25229 83i 26429 27023 276) 2 28197 28777 29353 9925 30493 3io56 3i6i6 32171 32723 33271 338i5 34355 34891 35424 35953 36478 .36913 .37432 .37948 . 3846n .38968 .39473 .39975 .40474 . 40969 .41461 37000 3751S 38o33 38545 39052 39557 4oo58 4o556 4io5i 4i543 42o3 1 42 5 16' ^2998 :.0" 1 2938 i3ri4o i4336 i5o26 i57io 16389 17062 17729 18391 19047 1 9698 2o344 20984 21620 2225o 22S76 23496 241 12 24723 2533o 25931 26529 27121 27710 38294 288^3 29449 3o020 3o587 3 1 1 5o 31709 32264 328i5 33362 33905 34444 34980 355i2 36o4i 36565 37087 37604 38119 38629 39137 39641 40142 40639 4ii33 41624 4195" ,42435 ,42918 ,43398143477 ,4387443953 74434844426 .448! 8144896 .45286J45363'4544i .45750J45S27 45905 . 46 -M 2 146289 4^365 421 12 42597 43078 43557 44o32 445o5 44974 _3t> 73o55 i3756 i445 i5i4o i5824 i65oi 17173 17840 i85oc 19156 1 9806 2045 1 21091 21725 22355 22980 23599 24214 24825 2543o 2603 1 26628 27220 27807 28391 28969 29544 3oi i5 3o68i 3i243 3i8oi 32356 32906 33453 33995 34534 35069 35601 3612S 36653 37173 37690 38204 38714 39221 39725 4o2 25 40722 4i2i5 41706 42193 42677 43i58 43636 44i 44583 45n52 455 18 45982 .-16442 40" I 50 ' i3i72 13S72 14567 i5255 15937 76674 17285 1 7950 18610 19265 19914 20558 21197 2i83i 22459 23f783 28702 243 16 24926 2553i 26i3i 26727 27318 27905 28487 13289 13989 14682 15369 i6o5o 16726 17396 18060 18719 19373 20022 20665 2 1 3o3 2 1036 22064 23'i"87 238o5 244 1 8 25027 2 563 1 26231 26826 27416 28002 28584 29066 29639 30209 30775' 3i337 37894 32448 32997 33543 3408 5 34623 35i58 35689 36216 3674( '37^ 37776 .3S289 38799 393o5 29161 29735 3o3o4 30869 3i43o 37^7 32540 33089 33634 341^5 34713 352.47 35777 363o4 36827 39808 4o3o8 4080. 41297 4r7_82 42274 42758 43238 43716 44190 44667 45i3o 45596 46o5 465 1 8 37346 37S62 38374 38884 39389 39892 40391 4c)887 4 1 379 4r868 2355 42838 433 18 43795 44269 3 Hours. 44740 4520S 45673 461 35 46595 U" 4. i:jOU7i 47127 47580 48o3i 48479 48924 49366 49806 50243 50677 . 5i 109 .5i539 . 5 1 966 .52390 .52812 753717 . 53648 .54u63 .54475 .54885 .55293 .55698 .56101 . 565oi .56900 ,57296 .57690 ,58082 ,58471 ,,58859 59244 59627 60008 6o388 60765 6 1 1 39 JO'' 46747 47203 47656 48 1 06 48553 48998 49440 49879 5o3i6 507 5o 577s7 5i6io 52o37 52461 52882 533oi 53718 54 1 32 54544 549_53 5536o 55765 56 168 56568 56966 57862 57755 58 1 47 58536 58923 59308 59691 60072 6o45o :^U'' 4662 3 47278 47731 48180 48627 49071 4951 3 49952 5o388 50822 5i253 5i68i 52107 5253i 52952 5T3"77 53787 54201 546 I 2 55o2i 55428 55832 56235 56635 57032 57428 57821 58212 58601 58988 59372 59755 60 1 35 6o5i3 60827 608 9(. 61202 I ,6i5i 2 61 574 .6188361945 .62252 .62619 62680 ,62984631745 ,6334763407 .63708 .64068 .64425 .64780 .65 1 34 61264 6 1 636 62006 6231362375 62741 63io5 63468 4.65486 4.65836 4.66184 ,6653o .66875 .672 17 ,67558 .67S97 6376863828 6412764187 64484,64544 64839I64898 65193:65251 65544'656o3 65S94J65952 66242 66299 (76588:66645 6693266989 67274|6733i 67615^67672 67954,68010 682^68347 6S627'68682 68235 68571 6890516896069016 69-.'3-,69>92 69348 69568169623,69678 '.iU' 40" 50" 46899146975:47051 47354J4743o475o5 478o() 47881 47956 48255 4833ol484o4 4S701 4877648850 491454921949293 495864966049733 5ooj5 50098150170 5o46 50894 57374 51753 52178 52601 53o22 53440 5385(i 54269 54680 55089 55496 55900 563oi 56701 57098 5o533 50966 5'7396 51824 52249 52672 53092 sTFio 53925 54338 54749 55i57 53563 55967 56368 56767 57^64 57559 57951 58342 58730 59116 59500 59882 57493 57886 58277 586t,5 59052 5.9436 598 1 8 60198160261 6o576|6o639 60952161015 67r26!{77388 6169S61760 62068J62129 62436I6249- 62802 62S63 6TiW> 635*8 63888 64246 646o3 5o6o 5io38 51467 51895 52319 52742 53162 53579 53994 54407 54817 55225 55630 56o34 ,56435 56834 57230 57625 58017 584o7 58794 59180 59564 59945 6o324 60701 61077 6i45o 61822 62 191 62558 63923 632 2.6 63287 63588 63648 6394864008 6.'f3o6l64365 64662164721 649576501665075 6531016536965427 656616571965777 6601066068 6635766415 66702 6676( 67046167103 67388J67445 67728:67785 68066I68123 66126 66472 668 1 7 67 T 60 67502 67841 68179 685 1 5 684o3T)8459 68738|68794'68849 69071169127 69182 69403 69458 69513 :69733i69788l69842 ''^'^ei5G] TABLli XXllI. To find the Latitude by two Altitudes of the Sun. LOG. RISING OR VERSED SINE. 4 Hours. 5 Hours. Al. i3 i4 i5 i6 17 18 19 20 21 22 23 24 25 26 27 23 29 3o 3i 32 33 34 35 36 37 58 _39_ 4o 41 42 43 44 45 46 47 48 _49. 5o 5i 52 53 54 55 56 57 58 59 0^^ 10" 20 :6^^ ,70224 .70550 .70874 .71197 ,7i5iS ,71837 ,72155 ,72471 ,72785 71571 71890 72208 72523 72838 76196 76492 76787 77081 77373 064 77954 78242 78529 7 8814 79098 : 79381 1 79662 . . 79942 .8017580221 69952 70279 70604 70928 7i25u 73i5i 73462 73772 74oS94 99091 99087 99083 99079 99075 i5i55 i5i84 l5212 i524i 15270 15299 98845 98841 98836 9S832 98827 98823 ioi35 10164 10.192 10221 I025o 10279 99485 99482 99479 99476 99473 99470 1 1869 11898 1 1927 1 1 956 1 1985 I20l4 99293 99290 99286 99283 99279 99276 1 36oo 13629 1.3658 i36S7 13716 13744 99071 99067 99063 99059 99055 9905 1 15327 1 5356 1 5385 i54i4 1 5442 1 547 1 98818 98814 98809 98Sf.5 98800 98796 1 7o5o 17078 17107 17.36 17164 17193 98536 98531 98526 98521 98516 9851 1 io3o8 io337 10 366 10395 10424 10453 99467 99464 99461 99458 99455 99452 12045 I 207 I 12100 12129 i2i53 12187 99272 99269 99265 99262 99258 99255 13773 t38o2 1 383 1 i385o i.3t889 13917- 99047 99043 99039 99035 9903 1 99027 i55oo 15529 15557 1 5586 i56i5 1 5643 98791 98787 98782 98778 98773 98769 N.sine. 17222 17250 17279 17308 17336 17365 98506 98501 98,^96 98491 984S6 98481 N. COS. N. sine. N. COS. N. sine. N. COS. N.sine. N. COS. N. COS. N. sine. 84° 83° 82° 8P 80° Page 162^ TABLE XXIV. Of Natural Sines. Prop, parts 28 o M 10° 11° 12° 13° 14° 60 Prop. p.%rta 6 6 N. sine. N. COS. lV. sine. N. COS. N. sine N. COS. N. sinc.|N. COS. N. sine. N. COS. I73b5 98481 190S1 98168 20791 97S15 22495 97487 24192 97o3o o 1 17393 98476 19109 98ib7 20820 97809 22528 97480 24220 97028 59 6 I 2 17422 98471 19138 98152 20848 97808 22552 97424 24249 97015 b8 6 I 3 17451 98466 19167 98 1 46 20877 97797 22580 97417 24277 97008 57 6 2 4 17479 98461 19195 98140 20905 97791 22608 974 1 1 24805 97001 b6 6 2 5 17508 98455 19224 98185 20988 97784 22687 97404 24333 96994 bb 6 3 3 (i 7 17W7 98450 19252 98129 20962 97778 22665 97898 24862 96987 54 53 5 5 17565 98445 19281 98124 20990 97772 22698 97891 24890 96980 4 8 17594 98440 19809 98118 21019 97766 22722 97384 24418 96978 b2 5 4 9 17623 98435 19338 981 12 21047 97760 22750 97378 24446 96966 bi 5 5 10 1 765 1 98430 19366 98107 2 1 0-6 97754 22778 97871 24474 96959 bo 5 b II 17680 98425 19895 98101 21 104 9774s 22807 97865 245o3 96952 49 5 6 d 12 17708 98420 19423 98096 21182 97742 22885 97858 24581 96945 48 47" 5 5 17737 98414 19452 98090 21161 97735 22863 97351 24559 96987 7 i4 17766 98409 1 948 1 98084 21189 97729 22892 9734b 24587 96980 4b 5 7 lb 17794 98404 19509 98079 21218 97728 22920 97888 246 1 5 96928 4 b 5 7 lb 17823 98399 19538 98073 21246 97717 22948 9733 1 24644 96916 44 4 8 17 17852 98394 19566 98067 21275 97711 22977 97825 24672 96909 4d 4 8 9 18 "19" 17880 983.89- 19595 98061 2i3o8 97705 23oo5 97818 97811 24700 96902 42 4 1 4 17909 98383 19623 98056 2i33i 97698 28088 24728 96894 9 20 17937 98378 19652 9S050 21860 97692 28062 97804 24756 96887 40 4 lO 21 17966 98373 19680 9S044 21888 976S6 28090 97298 24784 96880 39 4 lO 22 17995 98368 19709 98089 2l4l7 97680 28118 97291 24818 96878 38 4 1 1 23 18023 98862 19787 98088 21445 97678 28146 97284 24841 96866 37 4 II 12 24 T5" i8o52 98357 1 9766 98027 21474 97667 23175 97278 97271 24869 9685s 36 35 4 4 180S1 9S352 19794 98021 2l5o2 97661 28208 24897 96851 12 2b 18109 98347 19828 9S016 2i53o 97655 28281 97264 24925 96844 M 3 l3 27 i8i38 9834: 19851 9S010 21559 97648 28260 97257 24954 96887 3d 3 i3 28 18166 98336 19880 98004 21587 97642 28288 97251 24982 96829 3-2 3 1 4 29 18195 98331 19908 97998 2I6I6 97686 28816 97244 25oio 96822 3i 3 i4 f4 3o 17 18224 98325 19987 97992 21644 97680 28345 97237 25o88 96815 3o "29 3 3 18252 98320 19965 97987 21672 97628 28878 97280 25o66 96S07 lb 32 1 828 1 983 1 5 19994 97981 21701 97617 28401 97228 25094 96800 28 3 lb :i6 i83o9 98310 20022 97975 21729 9761 1 28429 97217 25l22 96798 27 3 1 5 M i8338 98004 2O05l 97969 21758 97604 23458 97210 25i5i 96786 2b 3 i6 3b 18367 98299 20079 97963 21786 97598 23486 97208 25179 96778 2b 3 17 17 3b 37 18395 98294 20108 97958 21814 97592 235i4 97196 25207 96771 24 23 2 2 18424 98288 2oi36 97952 21843 97585 23542 97189 25235 96764 18 38 18452 98283 2oi65 97946 21871 97379 28571 97182 25268 96756 2 2 2 18 39 1 848 1 98277 20198 97940 21899 97573 23599 97176 25291 96749 21 2 19 4o 18509 98272 20222 97934 21928 97566 28627 97169 25820 96742 20 2 19 4i i8538 98267 20250 97928 21956 97560 28656 97162 25848 96784 19 2 20 20 42 43 18567 98261 20279 97922 21985 97bb3 23684 97155 25376 96727 18 17 2 2 18595 98256 20807 97916 22018 97547 23712 97148 25404 96719 21 44 18624 98250 20886 97910 2204l 97541 28740 97141 25482 967 1 2 lb 2 21 4b i8652 98245 20864 97905 22070 97534 28769 97134 25460 96705 lb 2 21 4b 18681 98240 20893 97899 22098 97528 28797 97127 25488 96697 i4 22 47 18710 98234 20421 97893 22126 97521 23825 97120 255i6 96690 i3 2 2 l3 48 49" 18738 98229 2o45o 97887 22l55 97b lb 28858 97113 25545 96682 12 1 1 18767 98223 20478 97881 221S8 97508 28882 97106 25578 96675 23 bo 18795 98218 20D07 97875 22212 97502 28910 97100 256oi 96667 10 24 bi 18824 98212 2o535 97869 22240 97496 28988 97093 25629 96660 9 24 b2 iS852 98207 2o563 97868 22268 97489 28966 97086 25657 96653 8 25 bi 18S81 98201 20592 97857 22297 97483 28995 97079 25685 96645 7 2D 26 b4 55 18910 98196 20620 97851 22825 97476 24028 97072 25718 96688 b T -^ 18938 98190 20649 97845 22353 97470 24o5i 97065 25741 96630 26 bb 18967 98185 20677 97889 22882 97463 24079 97o58 25769 96628 4 27 ^7 18995 98179 20706 97883 22410 97457 24108 9705 1 25798 96615 3 27 b8 19024 98174 20784 97827 22438 9745o 24 1 36 97044 25826 96608 2 28 b9 19052 98 1 68 20763 97S21 22467 97444 24164 97087 25854 96600 I 1 28 ()0 1 908 1 98163 20791 97815 22495 %-437 24192 97080 2588?. 96598 N. COS. N. sine. N. COS. ^. sine. N. COS. \. sine. N. COS. N. sine. N. COS. N. sine. 79° 78° 77° 7C° 75» TABLE XXIV. [Page 163 Of Natural Sines. Pttip pans 27 o M 15° 10° 17° 18° 19° 60 Prop. parU 9 9 N. sine . N. fos 96593 N. sine . N. cos N. sine . N. COS. N. sine N. cos N. sine . N. cos 25882 27564 96126 29237 9563o 30902 95106 32557 94552 o I 25910 96585 27592 961 18 29265 95622 30929 95097 32,584 94542 5q 9 I 2 2b938 96578 27620 96110 29293 956 1 3 30957 95088 32612 94533 58 9 I 3 2 59bb 96570 27648 96102 29321 95605 30985 9D079 32639 94523 57 9 2 4 25994 96562 2767b 96094 29348 95596 3l012 95070 32667 945 14 5n 8 2 b 2b022 96555 27704 96086 29376 95588 3io4o 95061 32694 945o4 55 8 3 6 7 26o5o 96347 27731 96078 29404 95579 3i.ob8 95o52 32722 94495 54 53 8 8 2bo79 96540 27759 96070 29432 95571 31095 95043 32749 94485 4 8 26107 96532 27787 96062 29460 95562 3ii23 95o33 32777 94476 52 8 4 9 26135 96524 27815 96054 294S7 95554 3ii5i 95024 32804 94466 5i 8 b lO 26163 96517 27843 9604b 29515 95545 31178 95oi5 32832 94457 5o 8 6 u 2bi9i 96509 27871 96037 29543 95536 3 1 206 95qo6 32859 94447 49 ~6" 12 71 26219 96502 27899 96029 29571 95528 95519 3i233 31261 94997 94988 32887 94438 48 47 7 20247 96494 27927 96021 29599 32914 94428 b i4 26275 96486 279:0 96013 2962b 95511 31289 94979 32942 94418 46 7 7 lb 2b3o3 96479 279S3 96005 29654 95502 3i3ib 94970 32969 94409 45 7 7 lb 2633i 96471 280 1 1 9^997 29b82 95493 3 1 344 94961 32997 94399 AA 7 8 17 26359 96463 28039 959S9 29710 95485 3i372 94952 33o24 94390 43 6 8 9 ■18 '9 26387 96456 28067 95981 29737 95476 3 1399 31427 94943 94933 33o5i 94380 42 4i 6 6 2b4i5 9644s 28095 95972 29765 95467 33079 94370 9 20 2b443 96440 28123 95964 29793 95459 3 1454 94924 33io6 94361 4o 6 9 21 26471 96433 28i5o 95956 29821 95450 3i482 94915 33 1 34 94351 39 6 lO 22 265oo 96425 2817S 95948 29849 95441 3i5io 94906 33i6i 94342 38 6 lO 23 26528 96417 28206 95940 29876 95433 3i537 94S97 33189 94332 37 6 1 1 II 24 25 26556 96410 28234 95931 29904 95424 3 1 565 31593 94888 94878 33216 94322 36 35 5 5 26584 96402 28262 95923 29932 95415 33244 943 1 3 12 2b 26612 96394 28290 95915 29960 95407 31620 94869 33271 943o3 M 5 12 27 26640 96386 283i8 95907 29987 95398 3 1 648 94860 33298 94293 33 5 iJ 28 2666S 96379 28346 95898 3ooi5 95389 31675 9485 1 33326 94284 32 5 1 3 29 26696 96371 28374 95890 3oo43 953S0 3 1 703 94842 33353 94274 3i 5 t4 3o 3i 26724 96363 28402 95882 30071 95372 3 1730 94S32 3338 1 94264 3o 29 5 4 26752 96355 28429 95874 30098 95363 3i758 94S23 33408 94254 i4 32 26780 96347 284 D7 95865 3oi26 95354 3178b o48i4 ZM^6 94245 28 4 ID 33 26S08 96340 28485 95857 3oi54 95345 3i8i3 94So5 33463 94235 27 4 l5 34 26836 96332 285 1 3 9'iS49 3oi82 95337 3i84i 94795 33490 94225 26 4 lb 35 26S64 96324 28541 95841 30209 9532S 3 1868 94786 335i8 94215 25 4 lb I? 2Ci 37 26892 96316 28569 95832 3o237 95319 31896 94777 33545 94206 24 23 4 3 26920 96308 28597 95824 3o265 95310 31923 94768 33573 94196 17 38 26948 96301 28625 95816 30292 95301 3i95i 94758 336oo 94186 22 3 i8 39 26976 96293 2S652 95807 3o32o 95293 31979 94749 33627 94176 21 3 i8 40 27004 96285 286S0 95799 3o348 95284 32006 94740 33655 94167 20 3 i8 4i 27032 96277 28708 95791 30376 95275 32o34 94730 33682 94157 19 3 19 42 43 27060 96269 2S736 95782 3o4f)3 95266 32061 32089 94721 33710 94147 18 17 3 3 270S8 96261 28764 9^774 3o43i 95257 94712 33737 94137 20 U 27116 96253 28792 95766 30459 9524s 32116 94702 33764 94127 i6 2 20 45 27144 96246 28820 95757 3o48b 95240 32144 94693 33792 94118 1 5 2 21 46 27172 96238 28847 95749 3o5i4 9523i 32171 94604 33819 94108 i4 2 21 47 27200 96230 28S75 95740 3o542 95222 32199 946-4 33846 94098 i3 2 2 2 22 48 49 27228 96222 28903 95732 3o570 95213 32227 94665 94656 33874 33901 94088 12 1 1 2 2 27256 96214 28931 95724 3o597 95204 32254 94078 2J 5o 27284 96206 28959 95715 30625 95195 32282 94646 33929 94068 10 2 23 5i 27312 96198 28987 95707 3o653 95186 3^309 94637 3395b 940 5 8 9 23 52 27340 96190 29015 95698 3o68o 95177 32337 94627 33983 94049 8 24 53 27368 96182 29042 95690 30708 95168 32364 94618 34011 94039 7 24 25 _5£ 55 27396 96174 29070 95681 3o736 95159 32392 94609 34o38 94029 6 — 27424 96166 2909S 95673 3(^763 95i5o 32419 94599 34o65 94019 2b 56 27452 96158 29126 95664 30791 95142 32447 94590 34093 94009 4 2b 57 27480 96150 29154 95656 3o8 1 9 95i33 32474 94580 34120 5,3999 3 2b 58 27508 96142 29182 95647 3o84(i 95124 32502 94571 34147 93989 2 27 ■jq 27536 96134 29209 95639 30874 95ii5 02529 94561 34175 93979 I ^ bo 27564 96126 29237 9563o 30902 95106 32557 94552 34202 93969 N. COS. V. sine. N. COS. N..sine. \. co^.N.sino. N. COS. V. sine. V. COS. N. sine. 7-i " 78^ 72° 7P 1 70° Page 104] TABLE XXIV. Of Natural Sines. Prop, paru 27 o M 20° 21° 22° 23° 24° 60 Prop. p:ins 11 II N. sine N. COS. N. sine N. COS. N. sine N. COS N. sine N. COS. i\. sine N. COS. 34202 93969 35837 93358 37461 92718 39073 92o5o 40674 91355 o I 34229 93959 35864 93348 37488 92707 39100 92089 40700 91343 59 II 1 2 34257 93949 35891 93337 375x5 92697 39127 92028 40727 9i33i 58 tl I 3 34284 93939 35918 93327 37542 92686 39153 92016 40753 91819 57 10 ■2 4 343 1 1 93929 35945 93316 37569 92675 89180 92005 40780 91807 56 [O 2 b 34339 93919 35973 93306 37595 92664 39207 91994 40806 91295 55 It; 3 b 7 34366 93909 36ooo 93295 37622 93653 39234 919S2 4o833 91283 54 53" 10 10 34393 93899 36027 93285 37649 92642 39260 91971 40860 91272 4 8 34421 93S89 36o54 93274 37676 92631 39387 91959 40886 9 1 260 53 10 4 9 34448 9J879 36o8r 93264 37703 92620 39814 91948 40918 91248 5 1 9 5 10 34475 93869 36 1 08 93353 37730 92609 39341 91936 40939 91286 5o 9 5 II 345o3 93359 36i35 93343 37757 92598 39367 91925 40966 91224 49 9 5 6 12 34530 93849 36162 36190 93232 37784 92587 39894 91914 40992 91212 48 47 9 34557 93839 93222 37811 92576 8942 1 91902 41019 91200 14 34584 93829 36217 93211 37838 92565 39448 91S91 41045 91188 46 8 7 lb 346 1 2 93819 36244 93201 37865 92554 39474 91879 41072 91176 45 8 7 lb 34639 93809 36271 93190 37S92 92543 39501 9.868 41098 91164 44 8 8 17 34666 93799 36298 93180 37919 92532 39528 91855 4ii35 91 152 43 8 8 9 18 34694 93789 3632 5 93169 37946 92521 39555 39581 91845 91833 4ii5i 91140 42 4i 8 8 3472 1 93779 36352 93 1 59 37973 92510 41178 91128 9 20 34748 93769 36379 93i48 37999 92499 39608 91822 4i2o4 91 116 4o 7 9 21 34775 93759 364o6 93.37 38036 92488 39635 91810 4i23i 91104 39 7 lO 22 348o3 93748 36434 93127 38o53 92477 89661 91799 41257 91092 38 7 lO 23 3483o 93738 3646 1 93t 16 38o8o 92466 39688 917S7 41284 91080 37 7 II II 24 25 34857 34884 93728 36438 93106 38107 92455 39715 91775 4i3io 91068 36 "3c 7_ d 93718 365 1 5 93095 38 1 34 92444 39741 91764 4i337 91056 12 2b 34912 93708 36542 93084 38i6i 92432 39768 91752 4i363 91044 34 b 12 27 34939 93698 36569 93074 38 188 92421 39795 91741 41890 91082 33 () l3 28 34966 93688 36596 93o63 382i5 92410 39822 91729 4i4i6 9102c 32 b i3 29 34993 93677 36623 93o52 38241 92399 39S48 91718 41443 9100B 3i 6 i4 i4 3o 17 35o2i 35o48 93667 36650 93042 38268 92388 39875 91706 41469 41496 90996 90984 3^ 29 6 5 93«7 36677 93o3i 38295 92377 89902 91694 14 32 35075 93647 36704 93020 38322 92866 39928 91683 4 1 52 2 1 90972 28 5 lb 33 35i02 93637 36731 93010 38349 92355 39955 91671 4; 1 54 J 9J980 415.75 90948 37 5 i5 34 35i3o 93626 36758 92999 38376 92343 89982 91660 26 5 i6 3b 35 1 57 93616 36785 929SS 384o3 92332 40008 91648 4i6o'j 1 90986 25 5 i6 17 3b T7" 35i84 93606 368 1 2 92978 92967 38430 92821 4oo35 9i636 41628 90924 24 23 4 4 352II 93596 36839 38456 92810 40062 91625 4i655 90911 17 38 35239 93585 36867 92956 38483 92299 40088 91618 4s68i 90899 22 4 18 39 35266 93575 36894 92945 3S5io 92287 4oii5 91601 41707 90887 21 4 18 4K 35293 93565 3692 1 92935 38537 92276 4oi4i 91590 41734 90875 20 4 18 41 35320 93555 36948 92934 38564 92265 40168 91 578 41760 90868 '9 3 J2. 19 42 43 35347 93544 36975 92913 38591 92254 40195 40221 9:566 91555 41787 9085 1 18 17 3 3 35375 93534 37002 92902 38617 92243 4i8i3 90889 20 ^4 35402 93524 37029 92893 38644 92281 40248 01 543 4 1 840 90S 2 6 16 3 20 43 35429 935i4 37o56 92881 38671 92220 40275 91 53 1 41866 90814 i5 3 21 46 35456 935o3 37083 92870 38698 92209 4o3o I 91519 41892 90802 i4 3 21 47 35484 93493 371 10 93859 38725 92198 40828 9i5o8 41919 90790 .3 2 22 22 48 49' 355 II 93483 37137 37164 92849 92838 38752 92 1 86 4o355 91496 41945 90778 12 11 2 2 35538 93472 38778 92175 4o38i 91484 41972 90766 23 bf) 35565 93462 37191 92827 3SSo5 92164 4o4o8 91472 41998 90753 10 2 23 5i yynp 93402 37218 92816 38832 92152 40434 91461 42024 90741 9 2 23 12 35619 93441 37245 93805 38859 92141 4o46i 91449 42o5i 90729 8 24 bi 35647 93431 37273 92794 38886 92i3o 4o488 91437 42077 90717 7 24 25 b4 55 35674 93420 37299 92784 38oi2 92119 4o5i4 4o54i 91425 42104 90704 6 5 35701 93410 37326 92773 38939 92107 91414 42180 90692 25 5f) 35728 93400 37353 92762 38966 92096 40567 91402 421 56 90680 4 2b 1"" 35755 93389 37380 92751 38993 92085 40594 9 1 390 42183 90668 3 26 5K 35782 93379 37407 92740 39020 92073 40621 91378 42209 90655 2 27 ,.!■; 358io 93368 37434 92729 39046 92062 40647 91866 42235 90643 I 27 bo 35837 93358 37461 92718 39073 92o5o 40674 N. COS. 91355 N. sine. 42262 9063 1 ; N. COS. V. sine. N. COS. N. sine. N. COS. \. sine. M. cos. N. sine. 69° 68° G7° 66° 65° TABLE XXIV. [rage les Of Natural Sines. Prop, parts 26 o o I 1 2 2 3 3 3 i 4 5 5 6 6 7 7 7 8 8 9 9 lO 10 10 1 1 II 12 12 l3 i3 71 i4 i4 i5 i5 i6 i6 i6 17 17 i8 i8 19 19 20 20 20 21 21 22 22 23 23 53 24 24 25 25 26 26 M I 2 3 4 5 6 7 8 9 :o u 12 7T 14 i5 16 17 18 '9 20 21 22 23 M_ 25 26 27 28 =9 3o TT 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 4i 48 49 5o 5i 52 53 54 55 56 57 58 60 25= 26° 27° 28° 29° 60 59 S8 57 56 55 54 53 52 5i 5o 49 48 47 46 45 44 43 42 4i 4o 39 38 37 36 35 34 33 32 3i 3o 29 28 27 26 25 24 23 22 21 20 19 18 "17" 16 i5 i4 i3 12 1 1 10 9 8 7 6 5 4 3 2 I Prop. 14 ~T4 i4 i4 i3 i3 i3 i3 12 12 12 12 11 1 1 11 II II 10 10 10 10 9 9 9 9 8 ~8' 8 8 7 7 __7_ 7 7 6 6 6 6 5 5 5 5 4 4 4" 4 4 3 3 3 ~T 2 2 2 ■k 1 I I I N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. N. sine N. COS. 42262 42288 423i5 42341 42367 42394 42420 91163 1 906 1 8 90606 90594 90582 90569 90557 43837 43863 438S9 43916 43942 43968 43994 89879 89867 89854 89841 89828 89S16 89803 45399 45425 4545i 45477 455o3 45529 45554 89101 89087 89074 89061 89048 89035 89021 46947 46973 46999 47024 47o5o 47076 47101 47127 47153 47178 47204 47229 47255 88295 88281 88267 88254 88240 88226 88213 48481 485o6 48532 48557 48583 ^8608 48634 87462 87448 87434 87420 87406 87391 87377 42446 42473 42^99 42525 42552 4257S 90545 90532 90520 9u5o7 90495 90483 44020 44o46 44072 44098 44124 .44i5i 89790 89777 89764 89752 S9739 89726 45580 45606 45632 45658 45684 45710 89008 88995 88981 8896S 88955 88942 88199 88i85 88172 88 1 58 88144 88i3o 48659 48684 48710 48735 48761 48786 87363 87349 87335 87321 87306 87292 42604 4263 1 42657 42683 42709 42736 90470 90458 90446 90433 90421 9040S 44177 44203 44229 44255 44281 44307 89713 89700 89687 89674 89662 89649 45736 45762 45787 458 1 3 45839 45863 88928 88915 88902 88888 88875 88852 47281 473o6 47332 47358 47383 47409 47434 4746)0 47486 475ii 47537 47562 47588 47614 47639 47665 47690 47716 88117 88io3 88089 88075 88062 S804S 88o34 88020 S8006 -87993 87979 87965 48811 48837 48862 48888 48913 48938 87278 87264 87250 87235 87221 87207 42762 4278S 42815 42841 42S67 42894 90396 90383 90371 90358 90346 90334 44333 44359 44385 444 1 1 444'^! 44464 89636 89633 89610 89597 89584 89571 45891 45917 45942 45968 45994 46020 88848 88835 88822 88808 88795 88782 48964 48989 49014 49040 49065 49090 49116 49141 49166 49192 49217 49242 87193 87178 87164 87150 87136 87121 87107 87093 87079 87064 87050 87036 42920 42946 42972 42999 43o25 43o5i 90321 90309 90296 90284 90271 90259 44490 445 16 44542 44568 44594 44620 89558 89545 89532 89519. 89506 89493 46o46 46072 46097 46123 46149 46175 88768 88755 88741 88728 88715 88701 87951 87937 87923 87909 87896 87882 43077 43io4 43i3o 43 1 56 43182 43209 90246 90233 90221 90208 90196 90163 44646 44672 44698 44724 4475o 44776 89480 89467 89454 89441 89428 89415 46201 46226 46252 46278 463o4 4633o 88688 88674 8S661 88647 88634 88620 47741 47767 47793 47818 47844 47869 47895 47920 47946 47971 47997 48022 87868 87854 87840 87826 87812 87798 49268 49293 49318 49344 49369 49394 87021 87007 86993 86978 86964 86949 43235 43261 43287 433 1 3 43340 43366 90171 901 58 90146 90133 90120 9010S 44802 44828 44854 44880 44906 44932 44958 44984 45oio 45o36 45062 45o88 89402 89389 89376 89363 89350 89337 46355 4638 1 46407 46433 46458 4<^4H 88607 88593 8858o 88566 88553 88539 87784 87770 87756 87743 87729 87715 49419 49445 49470 49495 49521 49546 49571 49596 49622 49647 49672 49697 86935 86921 86906 86892 86878 80863 86849 86834 86820 868o5 86791 86777 43392 43418 43445 43471 43497 43523 90095 90082 90070 90057 90045 90032 89324 8931 1 89298 89285 89272 89259 465 10 46536 4656 1 46587 4661 3 46639 88526 885 1 2 88499 88485 88472 88458 48048 48073 48099 48124 48i5o 48175 87701 87687 87673 87659 87645 87631 43549 43575 43602 43628 43654 436So 90019 90007 89994 89981. 89968 89956 45ii4 45i4o 45i66 45192 45218 45243 89245 89232 89219 89206 89193 89180 46664 46690 46716 46742 46767 46793 88445 8843 1 88417 8S4o4 88390 88377 48201 48226 48252 48277 483o3 48328 87617 87603 87589 87575 87561 87546 49723 49748 49773 49798 49824 49849 86762 86748 86733 8O719 86704 86690 86675 86661 86646 86632 86617 8660; 43706 43733 43759 43785 438 1 1 43837 89943 89930 89918 89905 89892 89879 45269 45295 45321 45347 45373 45399 89167 89153 89140 89127 89114 89101 46819 4(iU4 46870 46896 46921 46947 88363 88349 88336 88322 883o8 88295 48354 48379 484o5 4843o 48456 48481 87532 87518 87504 87490 87476 87462 49874 49899 49924 49950 49975 5 0000 N. COS. .\. sine. N. COS. \. sine. N. co^. N. sine. N. COS. N. s'iie. N. COS. N. sine. 04° 1 6.3= 1 G2° 61° 60° 1 Page 166] TABLE XXIV. Of Natural Sines. Prop, par'.s 16 16" Prop, pans 25 o M o 30° 31° 32° 33° 34° 60 N. sine. N. fos. N. sine. N. COS. N. sine. N. COS. N. sine. N. COS. N. sine. N. COS. 5oooo 866o3 5i5o4 85717 52992 848o5 54464 83867 55919 82904 I 50025 86588 5i529 85702 53017 84789 54488 8385 1 55943 82887 59 16 I 2 5oo5o 86573 5i554 85687 53o4i 84774 545 1 3 83835 55968 82871 58 i5 I 3 50076 86559 bib79 85672 53o66 84759 54537 838i9 55992 82855 57 i5 2 4 5oioi 86544 5i6o4 85657 53091 84743 54561 838o4 56oi6 82839 56 i5 2 b 50126 8653o 51628 85642 53ii5 84728 54586 83788 56o4o 82822 55 i5 3 3 6 7 ,5oi5i 865 1 5 5i653 85627 53i4o 84712 54610 54635 83772 56o64 82806 54 53' i4 50176 865oi 51678 856i2 53 1 64 840 n 8468 1 83756 56oS8 82790 3 8 5020I 86486 5 1 703 85597 53189 54659 83740 56ii2 82773 52 1 4 4 9 50227 86471 51758 85582 53214 84666 54683 83724 56 1 36 82757 5i- 1 4 4 10 50252 86457 51753 85567 53238 8465o 54708 83708 56 1 60 82741 5() i3 5 II 50277 86442 51778 8555i 53263 84635 54732 83692 56i84 82724 4o i3 5 ' 5 12 l3 5o3o2 86427 5i8o3 85536 53288 84619 54756 83676 56208 82708 48 47 i3 5o327 8G4i3 51828 85521 533 1 2 846o4 54781 83660 56232 82692 6 i4 5o352 86398 5i852 855o6 53337 84588 548o5 83645 56256 82675 46 12 6 lb 5o377 86384 bi877 85491 5336i 84573 54829 83629 56280 82659 45 12' 7 i6 5o4o3 86369 51902 85476 53386 84557 54854 836 1 3 563o5 82643 44 12 7 17 50428 86354 51927 85461 53411 84542 54S78 83597 56329 82626 43 I I 8 8 i8 19 5o453 86340 86325 51952 85446 53435 84526 845 1 1 54902 83581 56353 82610 42 4i I 1 I 1 50478 51977 85431 53460 54927 83565 56377 82593 8 20 5o5o3 863 10 52002 85416 53484 84495 54q5i 83549 564oi 82577 40 I I 9 21 5o528 86295 52026 854oi 53509 84480 54975 83533 56425 82561 39 TO 9 22 5o553 86281 52o5i 853S5 53534 84464 54999 83517 56449 82544 38 10 10 23 50578 86266 52076 S5370 53558 84448 55o24 83501 56473 82528 37 10 10 ID 1± 25 5o6o3 S6251 52IOI 85355 53583 84433 55o48 83485 56497 82511 36 IT 10 9 50628 86237 52126 85340 53607 84417 55072 83469 56521 82495 II 26 5o654 86222 52i5i 85325 53632 84402 55097 83453 56545 82478 34 9 II 27 50679 86207 52175 853io 53656 84386 55i2i 33437 56569 82.^62 33 9 12 28 50704 86192 52200 85294 53681 84370 55i45 83421 56593 82446 32 r3 29 50720 86178 52225 85279 53705 84355 55169 834o5 56617 82429 3i 8 i3 i3 3o 3i 50754 86 1 63 5225o 85264 53730 84339 55194 55218 83389 56641 82413 3o 29 8 ~~8' 50779 86:48 52275 85249 53754 84324 83373 56665 82396 i3 32 5o8o4 86i33 52299 85234 53779 843o8 55242 83356 56689 82380 28 7 i4 33 50829 86119 52324 852i8 538o4 84292 55266 8334o 56713 82363 27 7 i4 34 5o854 86104 52349 852o3 53828 84277 55291 83324 56736 82347 26 7 i5 35 50879 860S9 52374 85i88 53853 84261 553i5 833o8 56760 8233o 25 7 i5 1 5" 36 17 50904 86074 52399 85i73 53877 84245 55339 83292 567S4 568o8 823i4 24 23 6 ~6' 50929 86059 52423 85i57 53902 8423o 55363 88276 82297 i6 38 50954 86045 52448 85i42 53926 84214 55388 83260 56832 82281 22 6 i6 39 50979 86o3o 52473 85i27 53951 84198 55412 83244 56856 82264 21 6 17 4o 5ioo4 860 1 5 5249S 85ii2 53975 84182 55436 83228 56880 82248 20 5 17 4i 51029 86000 52522 85096 54000 84167 55460 83212 56904 8223l 10 5 i8 42 43 5io54 85985 52547 85o8i 54024 84i5i 55484 83195 56928 82214 18 17 5 5 51079 85970 52572 85o66 54049 84i35 55509 83 1 79 56952 82198 i8 44 5iio4 85956 52597 85o5i 54073 84120 55533 83i63 56976 82181 16 4 19 45 51129 85941 52621 85o35 54097 84 104 55557 83i47 57000 82165 i5 4 !9 46 5ii54 85926 52646 85o2o 54122 84088 55581 83i3i 57024 b2i48 i4 4 20 47 5x179 85911 52671 85oo5 54 1 46 84072 556o5 83n5 57047 82132 i3 3 20 20 48 49 5i2o4 85896 52696 849S9 54171 84o57 5563b 83098 57071 8211b 82098 12 1 1 3 51229 8588i 52720 84974 54195 84o4i 55654 83o82 57095 21 bo bi254 85866 52745 84959 54220 84025 55678 83o66 57119 82082 10 3 21 5i 51279 8585i 52770 84943 54244 84009 55702 83o5o 57143 82065 9 2 22 52 5i3o4 85836 52794 84928 54269 83994 55726 83o34 57167 82048 8 2 2 2 53 5i329 85821 52819 84913 54293 83978 55750 83oi7 57191 82032 -7 2 23 2 3 _54_ 55 5i354 858o6 52844 52869 84897 84882 54317 83962 55775 83ooi 82985 57215 82015 6 5 2 1 5i379 85792 54342 83946 55799 57238 81999 23 56 5i4o4 8b777 52S93 84866 54366 83900 55823 82969 57262 81982 4 1 24 57 51429 85762 52918 8485 1 54391 8391 5 55847 82953 57286 81965 3 I 24 58 5i454 85747 52943 84836 544 1 5 83899 5587T 82936 5^310 81949 2 I 25 5q 5i479 85732 52967 84820 54440 83SS3 55895 82920 57334 81932 I 25 6o 5i5o4 85717 52992 848o5 54464 83867 55919 82904 57358 81915 IT N. co.s.|N. sine. N. COS. N. sine. N. COS. N. sine. N. COS. N. sine. N. COS. N. sine. 59° 58° 57° 50° 55° TABLE XXIV. [i'age 167 Of Natural Sines. Prop. 23 o M 35° 36° 37° 38° 39° 60 Prop. Dartj 18 :8 N.siiie. N. COS. N. sine. i\. COS. N. sine. N. COS. N. sine. N. COS. N. sine.|N. cos. 57358 81915 58779 80902 60182 79864 61666 78801 62932 777 1 5 c r 57381 81899 588o2 8088 5 60206 79846 616S9 7S783 62955 77696 59 18 I 2 574o5 81882 58826 80867 60228 79829 61612 78765 62977 77678 58 17 r 3 5742Q 8 1 865 58849 8o85o 60261 798 1 1 6i635 78747 63ooo 77660 57 17 2 4 57453 81848 58873 8o833 60274 79793 61668 78729 63022 77641 56 17 2 5 57477 8i832 58896 80816 60298 79776 61681 7871 1 63o45 77623 56 17 2 "3" 6 7 57501 8i8i5 58920 80799 6o32i 79768 61704 78694 63o68 77606 54 53 16 "16" 57524 8.7Q8 58943 80782 60344 79741 61726 78676 63090 77686 3 8 57548 81782 58967 80765 6o367 79723 61749 78668 63ii3 77668 52 16 3 q 57572 81765 58990 80748 60390 79706 61772 78640 63i35 77660 bi i5 4 10 57396 81743 59014 80730 6o4i4 79688 61796 78622 63i58 7753. bo i5 4 n 57619 81731 59037 80713 60437 79671 61818 78604 63 1 80 77bi3 49 i5 b 5 12 7T 57643 81714 59061 80696 60460 79663 6i84i 78686 632o3 77494 48 47 i4 i4 57667 81698 59084 80679 6o483 79635 61864 78568 63226 77476 5 14 57691 816S1 59108 80662 60606 79618 61887 78660 63248 77458 46 i4 6 rj 57715 81664 59131 80644 60629 79600 61909 78532 63271 77439 4b i4 6 16 57738 81647 59154 80627 6o553 79583 61932 78614 63293 77421 44 i3 7 17 57762 8i63i 59.78 80610 6o5v6 79665 61955 78496 633 16 77402 43 i3 7 7 18 19 57786 81614 59201 80693 60699 79547 61978 62001 78478 78460 63338 63361 77384 77366 42 4i i3 12 57810 81597 59225 80676 60622 79630 « 20 57S33 8i58o 59248 8o568 60645 79612 62024 78442 63383 77347 4o 12 8 21 57857 8 1 563 59272 8o54i 60668 79494 62046 78424 634o6 77329 39 12 8 22 57881 8 1 546 59295 80624 60691 79477 62069 78405 63428 77310 38 II 9 23 57904 8i53o 59318 80607 60714 79469 62092 7S387 6345 1 77292 37 1 1 _9 10 24 25 57928 8i5i3 59342 80489 60738 79441 62116 78369 63473 77273 36 35 II II 57952 81496 59365 S0472 60761 79424 62 1 38 78351 63496 77255 10 26 57976 81479 59389 80455 60784 79406 62160 78333 635i8 77236 34 10 ID 27 57999 81462 59412 8o438 60807 79388 62183 783 1 5 6354o 77218 .U 10 II 28 58o23 81445 59436 80420 608 3o 79371 62206 78297 63563 77199 32 10 II 29 58o47 81428 59459 8o4o3 6o853 79353 62229 78279 63585 77181 3i 9 12 12 3o 3i 58070 8i4i2 59482 8o386 60876 79335 62261 78261 636o8 77162 3o 9 58094 81395 59606 8o368 60899 79318 62274 78243 6363o 77144 12 32 58ii8 81378 bQb29 8o35i 60922 79300 62297 78225 63653 77126 28 8 l3 33 58i4i 8i36i 59552 80334 60945 79282 62320 78206 63675 77107 27 8 l3 ■M 58i65 8 1 344 59576 8o3 1 6 60968 79264 62342 78188 63698 77088 26 8 i3 35 58189 81327 59599 80299 60991 79247 62365 78170 63720 77070 2b 8 i4 i4 36 37 582 1 2 8i3io 59622 80282 6ioi5 6io38 79229 62388 78162 63742 77o5i 24 23 7 7 582 36 81293 59646 So 2 64 792 II 6241 1 78134 63766 77033 lb 38 58260 81276 59669 8o2.'[7 61061 79193 62433 78116 63787 77014 22 7 lb 39 58283 81259 59693 8o23o 61084 79176 62466 78098 638 10 76996 21 6 lb 4o 583.17 81242 59716 80212 61 107 79168 62479 78079 63832 76977 20 6 i6 4i 58330 81225 59739 80195 6ii3o 79140 62602 78061 63854 76969 19 6 lb 76" 42 T3" 58354 81208 59763 80178 6ii63 79122 62624 78043 53877 76940 76921 18 17 5 5 58378 81191 59786 80160 61176 79106 62647 78026 63899 17 44 584oi 81174 59809 80143 61 199 79087 62670 78007 63922 76903 lb 5 17 4 b 58425 81157 59832 80126 61222 79069 62692 77988 63944 76884 lb ■ b i8 40 58449 8ii4o 59856 80108 61245 79061 62616 77970 63966 76866 i4 4 i8 47 58472 81123 59879 80091 61268 79033 62638 77952 63989 76847 i3 4 i8 19 49 58496 585 19 81106 59902 80073 61291 79016 62660 77934 64o 1 1 76828 12 1 1 4 3 81089 59926 800 56 6i3i4 78998 62683 77916 64o33 76810 19 5o 58543 81072 59949 8oo38 61337 78980 62706 77S97 64o56 76791 10 3 20 bi 58567 8io55 59972 80021 6i36o 78962 62728 77879 64078 76772 9 3 70 52 58590 8io3S 59995 8ooo3 6i383 78944 62761 77861 64100 76754 8 2 uo 53 586 1 4 81021 60019 79986 6i4o6 78926 62774 77S43 64123 76735 7 2 21 21 54 "55" 58637 5866 1 81004 80987 60042 79968 61429 78908 62796 77824 64 1 45 76717 6 5 2 2 60065 79961 6i45i 78S91 62819 77806 64167 76698 21 bb 58684 80970 60089 79934 61474 788-73 62842 77788 64190 76679 4 1 22 b7 58708 80953 60112 79916 61497 78855 ()2864 77769 64212 76661 3 I 22 58 58731 80936 601 35 79899 61620 78837 62887 77761 64234 76642 2 I 2i b9 58755 80919 601 58 79S81 61643 78819 62909 77733 64256 76623 I 23 bo 58779 80902 60182 79864 61666 78801 62932 77716 64279 76604 IT N. COS. N. sine. N. COS. N. sine. N. COS. N. sine. X. COS. JV. sine. N. COS. N. sine. 54= 5; B° 52° 51° 50° Pagei&s] TABLE XXIV. Of Natural Sines. Prop, pana 22 o M 40° 41° 42° 43° 1 44° 60 Prop. part5 19 19 N. sine. N. COS. N. sine. V. COS. N. sine. N. COS. N. sine. N. cos. N. sine. N. cos. 64279 76604 656o6 75471 66913 743 14 6O200 73i35 69466 71934 o I 643oi 76586 65628 75452 66935 74295 68221 73ii6 69487 71914 59 19 I 2 64323 76567 6565o 75433 66956 74276 68242 73096 69508 71894 58 18 I 3 64346 76548 65672 75414 66978 74256 68264 73076 69529 71873 57 18 I 4 64368 7653o 65694 7b39b 66999 74237 6S285 73o56 69549 71853 55 18 2 5 64390 765 1 1 65716 75375 67021 74217 683o6 73o36 69570 71833 55 17 2 ~3 b 7 644 1 2 76492 65738 75356 67043 74198 68327 73016 69591 7i8i3 54 53 17 17 64435 76473 65759 75337 67064 74178 68349 72996 69612 71792 3 8 64457 76455 65781 753i8 67086 74159 68370 72976 69533 71772 52 lb 6 9 64479 76436 658o3 75299 67107 74139 68391 72957 69654 71752 Dl lb 4 10 6/:)5oi 76417 65825 75280 67129 74 1 20 68412 72937 69675 71732 bo lb 4 II 64524 76398 65847 75261 671 5i 74100 68434 72917 69696 71711 49 lb 4 5 1 2 64546 64568 7.6380 76361 65869 75241 67172 74080 68455 72897 72877 69717 71691 48 47 lb i5 65891 75222 67194 74061 68476 6973,7 71671 1j i4 64590 76342 65913 75203 67215 74o4i 68497 72857 69758 7i65o 46 lb b lb 646 1 2 76323 65935 75 1 84 67237 74022 685i8 72837 69779 7i63o 45 14 b lb 64635 763o4 65956 75i65 67258 74002 68539 7.2817 69800 71610 44 i4 b 17 64657 762S6 65978 75i46 67280 73^83 68 56 1 72797 69821 71590 43 i4 7 7 18 64679 76267 66000 75126 67301 73953 68582 72777 69842 71569 42 '4V i3 i3 64701 76248 66022 75107 67323 73944 686o3 72757 69862 71549 7 2(J 64723 76229 66044 75088 67344 73924 68624 72737 69883 71529 4o- i3 a 21 64746 76210 66066 75069 67366 73904 68545 72717 69904 7i5o8 39 12 8 22 64768 76192 66088 75o5o 67387 73885 68666 72697 69925 71488 38 12 8 23 64790 76173 66109 75o3o 67409 73865 68688 72677 69946 71468 37 12 9 9 24 25 648x2 76154 G6i3i 7301 1 67430 73846 68709 72657 69966 71447 3b I5" 11 II 64834 76135 66 1 53 74992 67452 73826 68730 72637 69987 71427 lO 2b 64856 761 16 66175 74973 67473 73806 68751 72617 70008 71407 34 II lO 27 64878 7O0Q7 66197 74953 G7495 73787 68772 72597 70029 71386 33 10 lO 28 64901 76078 66218 74934 67516 73767 68793 72577 70049 7 1 366 32 10 II 29 64923 76059 66240 74915 67538 73747 68814 72557 70070 71345 3i 10 f I 1 1 Jo yr 64945 76041 66262 74896 67559 675S0 73728 68835 72537 70091 71825 3o 29 10 9 64967 76022 66284 74876 73708 68857 72517 70112 7i3o5 12 32 649S9 76003 663o6 74857 67602 73688 68878 72497 70132 71284 28 9 12 6:i 65oii 75984 66327 74838 67623 73669 68899 72477 7013J 71264 27 9 12 M 65o33 75965 66349 74818 67645 73649 68920 72457 70174 71243 2b 8 iJ 3d 65o55 75946 66371 74799 67666 73629 68941 72437 70195 71223 2D 8 i4 65o77 75927 66393 664 1 4 74780 74760 67688 736'io 68962 72417 70215 71203 24 23 8 7 65 1 00 75908 67709 73590 68983 72397 70235 71182 '4 38 65l22 75889 66436 74741 67730 73570 69004 72377 70257 71162 22 7 i-l 39 65 1 44 75870 66458 74722 67752 73551 69025 72357 70277 71141 21 7 i5 4<' 65 1 66 7585i 66480 74703 67773 7353i 69046 72337 70298 71121 20 b 1 5 4>. 65iS8 75832 665oi 74683 67795 73511 69067 72317 7o3i9 71100 19 b lb 42 43 65210 758i3 66523 74664 67816 67837 73491 73472 69088 72297 70339 71080 18 17 b 5 65232 75794 66545 74644 69109 72277 7o36o 71059 lb 44 65254 75775 66566 74625 67859 73452 69130 72257 7o38i 71039 lb b 17 43 65276 75756 66588 74606 67880 73432 691 5 1 72235 70401 71019 lb b 17 4b 65298 75738 66610 74586 67901 73413 69172 72216 70422 70998 i4 4 17 47 65320 75719 66632 74567 67923 73393 69193 72196 70443 70978 i3 4 i8 i8 48 49 65342 75700 75680 66653 66675 74548 74528 67944 73373 69214 72176 70453 70957 12 11 4 3 65364 67965 73353 69235 72 1 56 70484 70907 i8 bo 65386 75661 66697 74509 67987 73333 69255 72 1 36 7o5o5 70916 10 3 19 bi 654o8 75642 66718 74489 68008 733i4 69277 ■72116 7o525 70896 9 3 '9 32 65430 75623 66740 74470 68029 73294 69298 72095 70546 70875 8 3 19 bi 65452 756o4 66762 744 5 1 68o5i 73274 69319 72075 70557 70855 7 2 20 20 !>4 55 65474 65496 75585 75566 667S3 7443 1 68072 73254 69340 72055 70587 70834 b "5 2 2 668o5 74412 68093 73234 69361 72035 70608 70813 21 bb 655i8 7iM7 668 2 7 74392 6Sii5 732 1 5 693S2 72015 70628 70793 4 I 21 b7 6554o 75528 66848 74373 68 1 36 73195 69403 71995 70649 70772 3 I 21 b8 65562 75509 66870 74353 68157 73175 69424 71974 70670 70752 2 I 22 b9 65584 75490 G6891 74334 68179 73i55- 69445 71954 70690 70731 I 22 b(j 656o6 75471 66913 743 1 4 68200 73i35 69466 1 7 1 934 7071 1 707 II N. cos.|N. sine. i\. ros. N. sine. N. COS. N. sine. N. ros.(N. sine. .\. cos. N. sine. 4 9° 48° 47° 4G° 45° [Page ]C9 TABLE XXV. Of Logarithmic Sines, Tangents, and Secants to every Point and Quarter Point of the Compass. Points. Sine. Co-sine. Tangent. Co-tang. Secant. Co-sccant. o Lif. neg. 10.00000 Lif. neg. Infinite. 10.00000 Infinite. 8 oi 8.69080 9-99948 8.69132 ii.3o868 IO.O0052 II .30920 7^ o h 8.99130 9-99790 8.99340 11.00660 10.00210 II .00870 7 h o.i 9.16652 : 9.99527 9.17125 10.82875 10.00473 10.83348 7 i I 9.29024 9.99157 9.29866 10.70134 10.00843 ic. 70976 7 I i 9.38557 9.98679 9.39879 10.601 2 1 IO.Ol32I 10.61443 Gil I 4 9.46282 9.98088 9.48194 io.5i8o6 IO.OI9I2 10.53718 ' 6 4 1 I % 9.52749 9-97384 9.55365 10. 44635 10.02616 10.47251 6 i 2 9.5S284 9.96562 9.61722 10.38278 I0.03438 10.41716 6 2 i 9.63099 9.95616 9.67483 io.325i7 10.04384 10.36901 5 -I 2 h 9.67339 9.94543 9.72796 10.27204 10.05457 10.82661 5 h 2 5 9.71105 9.93335 9.77770 I0.2223o io.o6665 10.28895 U 3 9-74474 9.91985 9.82489 IO.I75II io.o8oi5 10.25526 5 3 i 9.77503 9.90483 9.87020 10.12980 10.09517 10.22497 4i 3 h 9.80236 9. 888 1 9 9.91417 10. 08583 10. 11 181 10.19764 4 h 3i 9.8270S 9.86979 9.95729 10.04271 10. l302I 10.17292 4 i 4 9.84949 9-84949 10.00000 10.00000 io.i5o5i io.i5o5i 4 Co-sine. Sine. Co-tang. Tangent. Co-secant. Secant. Points. TABLE XXVL Logarithms of Numbers. No. 1 100. Log. 0.00000 2.00000. No. Log;. No. Log. No. Log. No. j Log. No. Log. 1 O . 00000 21 1.32222 4i 1. 61278 61 1.78533 81 I . 90849 2 o.3<)io3 22 I .34242 42 1.62325 62 1.79239 82 1.91381 3 0.47712 23 I. 36173 43 1.63347 63 1.79934 83 I .91908 4 0.6020G 24 i.38o2i 44 I. 64345 64 1.80618 84 I .92428 5 0.69897 P.5 1.39794 45 I .65321 6 5 1.81291 85 1.92942 6 0.77815 26 I. 41497 46 I .66276 6 ^ I. 81954 86 I . 93450 7 o.845io 27 i.43i36 4i I .67210 67 1.82607 87 I .93952 8 0.90309 28 I. 44716 48 I. 68124 68 I.8325I 88 1.94448 9 0.95424 29 I .46240 49 I .69020 69 1.83885 89 1.94939 lO I .00000 3o I. 47712 5o I .69897 7 i.845io 90 1.95424 II 1.04139 3i I .49136 • 5i 1.70757 7 I I.85I26 91 I .95904 12 I .07918 32 i.5o5i5 52 I .71600 72 I. 85733 92 I .96370 1 3 I . II 394 33 i.5fc85i 53 I .72428 73 I. 86332 93 1.96848 • i4 I . i4Gi3 M i.53i48 54 I .73239 74 I .86923 94 1 . 973 1 3 i5 1 . 1 7609 35 1.54407 55 I . 74o36 .75 i.875o( 5 95 1.97772 i6 I .2o4l2 36 I.55630 56 I. 74819 76 1.8808 96 1.98227 17 I .2 3o45 37 1.56820 57 1.75587 77 I .88649 97 1.98677 i8 I .25527 38 1.57978 58 1.76343 78 1 .89209 98 1.99/23 -9 1.27875 39 1 .59106 59 1.77085 79 1.89763 99 I . 99564 20 I .3oio3 40 I .60206 60 1. 77815 80 I .90309 100 2.00000. 2 2 Page 170] TABLE XXVI. Logarithms of Numbers. No. 100- -1600. Loff. 00000- -20412. No. I02 io3 io4_ io5 io6 107 io8 :i3 [2.3 [_24_ 125 [26 [27 [28 t3o [3i l32 i33 1 34 l3:3 [36 i37 1 38 i39_ i4o i4i I 42 13 i44 1 45' '(7 [5i [55 [53 [_54 [55" 1 56 [57 58 No. GOOOO oo432 00860 01284 01703 021 19 0253i 02q38 03342 03743 o4i39 04532 04922 o53u8 05690 06070 06446 068 1 9 071 88 07555 079(8 S279 8636 08991 9342 9691 oo37 o38o 079.1 [059 1394 1727 2o57 2385 2710 3o33 3354 3672 3988 43oi 46i3 4922 5229 5534 5836 (■ii37 6435 6732 7026 7319 7609 7898 8(84 8469 8752 9rr3"3 93 I 2 9590 9866 2()l4o 1 00043 00475 00903 01326 01745 02160 02572 02979 o3j83 03782 04179 04571 04961 o5346 05729 06108 o6483 o6856 07225 07591 07954 oS3i4 0S672 09026 09377 09726 0072 04 1 5 0755 1093 1428 1760 20Q0 2418 2743 3(j66 3386 3704 4019 4333 4644 4953 5259 5564 5866 6167 6465 6761 70 56 7348 7638 7926 82(3 8498 8780 9061 9340 9618 9893 20167 00087 oo5i8 00945 01 368 01787 02202 cfc6i2 o3oi9 03423 03822 04218 o46io 04999 o53S5 05767 06145 o652i 06893 07262 07628 07990 o835o 08707 0906 1 09412 09760 0106 0449 0789 1 1 26 i46[ 1793 2123 2450 2775 3098 34i8 3735 4(i5i 4364 4675 49S3 5290 5594 5897 6197 6495 6791 7085 7377 7667 7955 8241 8526 S80S 9089 9368 9645 9921 20194 ooi3o oo56i 00988 oi4io 01828 02243 02653 o3o6o o3463 o3862 04258 o465o o5o38 o5423 o58o5 o6i83 06558 06930 07298 07664 08027 08386 08743 09096 09447 09795 oi4o o483 0823 1 160 1494 1826 2 1 56 2483 2808 3i3o 345o 3767 4082 4395 4706 5oi4 5320 5625 5927 6227 6524 6820 7114 7406 7696 8270 8554 8S37 939^ 9673 9948 20222 00173 00604 oio3o 01452 01870 02284 02694 o3ioo o35o3 03902 04297 04689 o5o77 o546r o5843 06221 06595 07335 07700 oSo63 08422 08778 09132 09482 09830 0175 o5i7 0857 1 193 1528 i860 2189 25i6 2840 8162 3481 3799 4ii4 4426 4737 5o45 535i 5655 5957 6256 6554 685o 7143 7435 7725 801 3 8298 8583 8865 00217 00647 01072 01494 01Q12 02325 02735 o3i4i 03543 03941 04336 04727 o5ii5 o55oo o588i 06258 06633 07004 07372 07737 08099 08458 08814 09167 09517 9S64 0209 o55i 1227 ij5i 1893 2252 2548 2872 3194 35x3 383o 4i45 4457 7754 8o4i 8327 861 1 8S93 9173 945 1 9728 20003 20776 6 00260 00689 oiiiS oi536 01953 02366 02776 o3i8i 03583 03981 04376 04766 o5i54 05538 05918 06296 06670 07041 07408 07773 081 35 08493 08849 09202 09552 10243 10585 [0924 [1 261 i63i6 1 66 [3 16909 17202 7492 [7782 18070 [8355 18639 3921 oo3o3 00732 01157 01578 01995 02407 02816 03222 o3623 o4o21 044 1 5 o48o5 05192 05576 05956 06333 06707 07078 07445 07809 08171 08529 08884 19237 '9587 09934 0278 0619 0958 ;i294 [1628 [1959 [2287 [26 1 3 [2937 i3258 13577 1 3S93 14208 14520 14829 [5i37 [5442 [5746 16047 1 6346 16643 16938 [7231 [7522 1 78 II 18099 1 8384 18667 18949 9229 9507 9783 2oo58 2o33o oo346 00775 01199 01620 o2o36 02449 02857 03262 03663 o4o6o 04454 04844 o523i o56i4 05994 06371 06744 071 15 07482 07846 08207 08565 08920 09272 09621 09968 03l2 o653 10992 ;i327 1661 [1992 [2320 [2646 2969 16376 16673 16967 17260 17551 17840 I8I27 i84i2 18696 19257 19535 9 00889 00817 01242 01662 02078 02490 02898 o33o2 03703 o4ioo 04493 04883 05269 o5652 o6o32 06408 06781 071 5 1 07518 07882 08243 08600 08955 09307 09656 [1025 i36i 1694 [2024 [2352 [2678 i3oo[ [3322 1 364o [8956 142-0 [4582 i489[ [5198 [55o3 j8u6 16107 16406 16702 16997 [7289 (7580 201 12 2o385 43 I 4 2 9 i3 4 17 5 22 6 26 7 3o 8 M 9 39 41 40 I 4 4 2 8 8 3 12 12 4 16 16 5 21 20 6 25 24 7 29 28 8 33 33 9 J7 36 39 38 1 4 A 2 8 8 3 12 1 1 4 16 i5 5 20 If; 6 23 23 7 27 27 8 3i 3o 9 35 34 37 3i; I 4 4 2 7 7 3 II II 4 i5 i4 5 19 18 () 22 22 7 26 a5 8 3o 20 9 33 32 35 34 I 4 3 2 7 7 3 II 10 4 i4 i4 5 18 17 6 21 20 7 25 24 8 28 27 9 32 3i 33 1 3 2 7 i 10 4 i3 5 17 6 20 7 2J 8 26 9. 3o 32 ~3 6 10 i3 16 19 22 26 19 TABLE XXVI. Logarithms of Numbers. [Page 171 No. 1600- -2200. hocr. 20412- -34242. :6o r6i 162 r63 i64_ i65 166 167 168 K59_ [70 171 [72 [73 [75 .76 177 178 '79 180 181 182 [83 '4 i82_ 190 191 [92 93 _9l .95 196 ■97 198 199 200 201 202 2o3 204 205 206 207 208 209 210 211 212 2l3 2l4 2l5 216 217 218 219 204l2 2o6S3 20952 2I2I9 21484 21748 2201 1 22272 2253l 227S9 23o45 233oo 23553 23So5 24u55 243o4 2455i 24797 25o42 25285 3(jio3 3o3:>o 3o535 3o75o 30963 20439 20710 20978 21245 2l5l I 21775 22037 22298 22557 22814 23070 23325 23578 2383o 24080 24329 24576 24822 25o66 253io 2555i 25792 2603 1 26269 265o5 26741 26975 27207 27439 27669 27898 28126 28353 28578 28803 29026 29248 29469 29G88 29907 3oi25 3o34i 3o557 30771 30984 20466 20737 2I005 21272 2x537 2 1 80 1 2 2o63 22824 22583 22840 23096 2335o 236o3 23855 24 1 o5 24353 2460 1 24846 25091 25334 25575 258i6 26055 26293 26529 26764 26998 27231 27462 27692 27921 28149 28375 28601 28825 29048 29270 29491 29710 29929 3or46 3o363 30578 30792 3 1 oo() 3i2i8 31429 3 1 639 3 1 848 32o56 32263 32469 32675 32879 33082 20493 20763 2Io32 21299 2 1 564 21027 22089 2235o 22608 22866 28121 23376 28629 23880 24i3o 24378 24625 24871 25i i5 25358 256oo 2584o 26079 263i6 26553 267S8 27021 27254 27485 27715 27944 28171 28398 28623 28847 29070 29292 29513 29732 29951 3oi68 3o384 3o6oo 3o8i4 3 [027 33284 33486 33686 33885 34084 31239 3i45o 3 1 660 31869 32077 32 2.84 32490 32695 32899 33io2 333o4 335o6 33706 33905 34 1 04 23i47 23401 23654 23905 24 1 55 244o3 2465o 24S95 25139 25382 25624 2 5864 26102 26340 26576 26S 1 1 27045 27277 27508 27738 27967 28194 28421 28646 28870 29002 29314 29535 29754 29973 3o 1 90 3o4o6 3062 1 3o835 3io48 3 1 260 3i47i 3i68i 81890 32098 323o5 325ro 32715 32919 33 1 22 33325 33526 33726 33925 3412,4 2o548 20817 2io85 2l352 21617 24428 24674 24920 25i64 25406 25648 2 5888 26126 26364 26600 26834 27068 27800 27531 27761 27989 28217 28443 28668 28892 291 1 5 29886 29557 29776 29994 32825 3253i 82786 32940 33i43 33345 33546 33746 33945 34143 20576 20844 21112 21378 21643 21906 22167 22427 22686 22943 28198 23452 28704 28955 24204 24452 24699 24944 25i88 25481 26672 26912 26160 26887 26623 26858 27091 27828 27664 2 7784 28012 28240 28466 28691 28914 29187 29358 29679 29798 3ooi6 3o233 80449 3o664 80878 3 1 09 1 3i3o2 3i6i3 81723 81931 82189 82846 82662 32766 82960 83i63 333$5 33666 33766 33965 34i63 6 20602 20871 21130 2i4o5 21669 21982 22194 22453 22712 28228 23477 28729 28980 24229 24477 24724 24969 26212 26465 28086 28262 28488 28718 28937 29169 29380 29601 29820 3oo38 80255 80471 3o6S6 80899 3X112 3x323 3x534 3x744 3x962 82160 32366 82672 82777 32980 33x83 33386 33586 33786 33986 34 1 83 20629 21 166 2i43x 21696 21968 22220 22479 22737 28249 28602 28764 24006 24264 24502 24748 24998 26237 26479 26720 26969 26198 26435 26670 26906 27188 27370 27600 27880 28068 28286 286x1 28735 28969 29X8X 29408 29628 29842 80060 30276 80492 80707 30920 8ii33 3x345 3i555 3x766 3x978 32i8r 32 387 82693 82797 3 3 00 1 332o3 834o5 336n6 338o6 34006 34208 7 I 8 9 2o656 20926 2X X92 2x468 2x722 21986 22246 22606 22768 28019 28274 23528 28779 24080 24279 24527 24773 260x8 2526X 26603 26744 26988 262 2 X 26458 26694 26928 27161 27898 27623 27862 28081 28307 28533 28-58 289SX 29208 29426 29645 29868 3()o8 1 80298 3o6i4 30728 30942 3x164 3 1 366 8x676 8x786 81994 3220X 32408 326x3 32818 33o2i 33224 33426 33626 33826 34026 34223 9 31 30 I 3 3 2 6 6 3 9 9 4 12 12 5 16 i5 6 19 18 7 22 21 8 26 24 9 28 27 29 28 I 3 3 2 6 G 3 9 8 4 12 II 5 i5 14 6 17 17 7 20 20 8 20 22 9 26 26 27 2G I 3 3 2 5 5 3 8 8 4 11 10 c i4 i3 6 16 16 7 19 18 8 22 21 9 24 23 25 24 I 3 2 2 5 5 3 8 7 4 10 10 5 i3 xa 6 i5 i4 7 18 17 8 20 19 9 23 22 2:3 22 12 i4 i3 71 x6i i5 8 i8'i8 21 I 2 2 4 3 6 4 8 5 X X 6 i3 7 16 8 17 9 19 Page 172] TABLE XXVI. Logarithms of Numbers. No. 2200 2800. Log. 34242 44710. No. 1 1 2 3 4 5 6 7 8 34400 34596 34792 34986 35 1 80 9 34420 34616 34811 35oo5 35199 220 221 222 223 224 225 226 227 228 229 23o 23l 232 233 234 34242 34439 34635 3483o 35o25 34262 34459 34655 34850 35o44 34282 34479 34674 34869 35o64 34301 34498 34694 34889 35o83 34321 34518 34718 34908 35io2 35295 35488 35679 35870 86059 34341 34537 34733 34928 35l22 353i5 35507 35698 35889 86078 34361 34557 34753 34947 j5i4i 84380 34577 34772 34967 35i6o 35353 35545 35736 35927 36ii6 I 2 3 4 5 6 7 8 9. 2 4 6 8 35218 354 II 356o3 35793 35984 35238 35430 35622 358i3 36oo3 35257 35449 35641 35832 36o2i 35276 35468 3566o 3585i 36o4o 35334 35526 35717 3.'59o8 36097 35372 35564 35755 35946 36i35 35393 35563 35774 35965 36 1 54 ic 12 i4 16 18 36 1 73 3636i 36549 36736 36922 36192 3638o 36568 36754 36940 36211 86899 36586 86778 86959 86229 364 1 8 366o5 86791 36977 86248 36436 86624 368 10 36996 86267 36455 36642 86829 87014 86286 36474 3666 1 36847 87083 363o5 36493 36680 36866 87051 36324 365 1 1 86698 36884 87070 36342 3653o 86717 86903 3-u88 19 I 2 3 4 5 6 7 8 9 2 4 6 8 235 236 237 238 239 37107 37291 37475 37658 37840 37125 37310 37493 37676 37858 37144 87828 37511 37694 37876 38o57 88288 384 1 7 88596 88775 37162 87846 37530 877x2 37894 87181 37365 87548 37781 87912 88098 38274 38453 38632 388io 87199 87883 37566 87749 37981 38ii2 38292 38471 38650 38828 87218 37401 37585 87767 37949 87236 87420 87608 37785 87967 87254 87488 37621 87808 87985 38 1 66 38346 38525 88708 3888 1 89058 89235 89410 89585 89759 87278 37457 87689 87822 38(H,3 38 184 38364 38543 38721 38899 89076 89252 89428 89602 39777 39950 40128 40295 40466 4o637 10 1 1 1 3 i5 17 240 241 242 243 244 38o2i 38202 38382 38 56 1 38-39 38917 39094 39270 39445 39620 88089 38220 38399 38578 38757 88075 38256 38435 386i4 88792 38 1 3o 383 10 88489 38668 38846 38i48 38328 385o7 38686 38863 18 I 2 3 4 5 6 7 8 9_ 1 2 4 5 245 246 247 248 249 25o 25l 252 253 254 38934 89111 39287 39463 89687 88952 39129 39805 89480 39655 88970 89146 39322 89498 39672 38987 89164 89840 39515 89690 89005 89182 89358 89533 89707 89028 39199 89875 89550 89724 39041 892 1 7 39893 395es 89742 7 9 II i3 14 16 7 39794 39967 4oi4o 4o3r2 4o483 39811 39985 4oi57 40829 4o5oo 89829 40002 40175' 4o346 4o5i8 40688 4(>858 41027 4i 196 4 1 363 39846 40019 40192 4o364 4o535 40705 40875 4io44 4l2I2 4i38o 89868 40087 40209 4o38i 4o552 89881 4oo54 40226 40898 40569 89898 40071 40243 4o4i5 4o586 89915 400S8 40261 4o432 4o6o3 89933 40106 40278 40449 40620 1 2 3 4 5 6 7 8 9 2 3 5 255 256 257 258 259 4o654 40824 40993 41162 4i33o 40671- 4o84i 4ioio 41179 4 1 347 40722 40892 4io6i 41229 41897 40739 40909 41078 41246 4i4i4 40756 40926 41095 41268 4i43o 40773 40943 4i II I 41280 4 1 447 40790 40960 41128 41296 4 1 464 40807 40976 4ii45 4i3i3 4i4Si 7 9 10 12 i4 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 41497 4 1 664 4i83o 41996 42160 4i5i4 41681 41847 42012 42177 4 1 53 1 41697 4 1 863 42029 42193 4 1 547 41714 41880 42045 42210 4 1 564 41781 41896 42062 42226 4i58i 41747 41918 42078 42243 41597 41764 41929 42095 42259 4i6i4 41780 41946 42111 42275 4i63i 41797 41968 42127 42292 41647 4i8i4 41979 42144 42808 i5 1 I 2 3 4 5 6 7 8 9_ 1 G 2 3 5 6 8 10 II 1 3 42325 42488 4265 1 42813 42975 42341 42504 42667 42880 42991 43i52 433 1 3 43473 43632 43791 42357 42521 42846 43oo8 42374 42537 42700 42862 48024 42890 42553 42716 42878 43o4o 42406 42570 42732 42894 43o56 42428 42586 42749 42911 48072 42439 42602 42765 42927 43o8S 43249 43409 43569 48727 43886 42455 42619 42781 42943 43 104 42472 42635 42797 42959 43i2o 43 1 36 43297 43457 436i6 43775 43169 43329 43489 43648 43807 43i85 43345 435o5 43664 48828 43201 43361 43521 4368o 43838 48217 43377 43537 43696 43854 43233 43398 43553 43712 43870 43265 43425 43584 43743 48902 43281 43441 43600 4.3759 43917 _i4 5 1 2 3 5 6 7 8 9 2 3 275 276 277 278 279 43933 44091 44248 44404 44560 43949 44107 44264 44420 44576 1 43965 44l22 44279 44436 44592 43981 44 1 38 44295 44451 44607 43996 44 1 54 443 1 1 44467 44628 44012 44170 44826 44483 44638 44028 44i85 44342 44498 44654 44044 44201 44358 445i4 44669 44059 44217 44373 44529 44680 44075 44282 44389 44545 44700. 5 6 8 9 11 No. 3 4 5 G 7 8 i) 12 i4 TABLE XXVI. [Pagans Logarithms of Numbers. No. 2300 ^3400. Log. 4471G 53148. No. 1 2 3 1 4 1 5 44778 1 44793 6 7 8 9 280 281 2S2 283 2S4 286 287 288 289 290 291 292 293 294 295 296 297 29S 299 3(jo 3oi 3f)2 3o3 3o4 3o5 3o6 307 3o8 309 3 10 3ri 3l2 3i3 3i4 3i5 3(6 3,7 3!8 3,9 320 3?i 322 323 324 32 5 326 327 328 329 "337." 33i 332 333 334 335 336 337 338 33y No. 447 1 6 44871 45o25 45179 45332 44731 448S6 45o4o 45194 45347 44747 44902 45o56 45209 45362 44762 44917 45071 45225 45378 44809 44963 45117 45271 45423 44S24 44979 45 1 33 45286 45439 44840 44994 45i48 453oi 45454 44865 46010 45i63 453i7 45469 1(3 44932 45o86 45240 45393 44948 45 102 45255 45408 4556i 45712 45864 4601 5 461 65 I 2 3 4 5 6 7 8 9 2 3 5 6 45484 45637 45788 45939 46090 45500 45652 458o3 45954 46io5 455i5 45667 458i8 45969 46 1 20 45530 45682 45834 45984 46i35 45545 45897 45849 46000 46 .5o 45576 45728 45879 46o3o 46180 4633o 46479 46627 46776 46923 45591 45743 45894 46045 46196 46345 46494 46642 46790 46938 45606 46768 46909 46060 46210 45621 45773 46924 46076 46226 8 10 II i3 i4 46240 46389 46538 466S7 46835 46982 47129 47276 47422 47567 46255 464o4 46553 46702 468 5o 46997 47144 47290 47436 47582 46270 46419 46568 46716 46864 46285 46434 46583 46731 46879 463oo 46449 46598 46746 46894 463.5 46464 4661 3 46761 46909 46359 46609 4(>657 46806 46953 46374 46523 46672 46820 46967 1 I 2 3 4 5 6 7 8 9 5 2 3 c e 8 9 1 1 47012 47159 473o5 4745 1 47596 47026 47173 47319 47465 4761 1 47041 47188 47334 47480 47625 47o56 47202 47349 47494 47640 47()70 4^.17 47363 47609 47654 47086 47232 47378 47524 47669 47100 47246 47392 47538 47583 471 14 47261 47407 47563 47698 47712 47857 4S001 48.44 48287 47727 47871 4801 5 48i59 483o2 4774 1 47885 48029 48.73 483 1 6 47756 47900 .48044 48187 4833o 47770 47914 48o58 48202 48344 47784 47929 48073 48216 48359 47799 47943 48087 4823o 48373 47813 47958 48.01 48244 48387 47828 47972 48116 48269 484oi 48544 48686 48827 48968 49108 47842 47986 48i3o 48273 48416 48658 48700 4884 1 48982 49122 12 14 14 4843o 48572 48714 48855 4899G 48444 48586 48728 48869 49010 48458 48601 48742 48883 49024 48473 486 1 5 48756 4SS97 49038 48487 48629 48770 4891 1 49052 485oi 4S643 48785 48926 49066 485.5 48657 4S799 409110 49080 48530 48671 488 1 3 48954 49094 I 2 3 4 5 6 7 8 9_ 1 3 4 6 491 36 49276 4v4i5 49554 49693 49i5o 49290 49429 495(38 49707 49164 49304 49443 49582 49721 49178 49318 49457 49596 49734 49192 49332 49471 49610 49748 49206 49346 49485 49624 497G2 49220 49360 49499 4963s 49776 49234 49374 49613 49661 49790 49248 49388 49627 49666 49803 49262 49402 49541 49679 49817 49966 60092 60229 6o365 5o5oi 7 8 10 II i3 49831 49969 5>ii.,6 5oa43 5o379 49845 49982 5ol20 5o256 50393 49859 49996 5oi33 50270 5o4o6 49872 5ooio 5oi47 50284 5o42o 49886 50024 5oi6i 50297 5o433 49900 5oo37 50174 5o3ii 5o447 49914 5t)o5i 5oi88 5o325 5o46i 49927 60066 6(;202 6o338 5o474 5o6io 60745 60880 5ioi4 5ii48 61282 5i4i5 61648 5i6So 5i8i2 61943 62076 62206 52336 62466 49941 60079 6021 5 5o352 5o488 IS 1 2 3 4 5 6 7 8 9 I 3 4 5 7 8 9 10 12 5.,5;5 5(i(i5i 50786 "^0920 5io55 5i 18S 5i322 5 1455 5 1 587 51720 5i85i 5.9S3 5pii4 5-2 244 52375 5o529 5o664 50799 50934 5 1 068 5 1 202 5i335 5 1 468 5 1 60 1 51733 5.865 5 1 996 52127 52257 52388 5o542 50678 5o8i3 50947 5 1 08 1 5o556 50691 50826 50961 51095 5o56y 5070D 5oS4o 50974 5iio8 5o583 50718 5o853 50987 5 1 1 2 1 50696 607.32. 60866 5iooi 5ii36 61268 5i4o2 616.34 61667 61799 60623 60769 60893 51028 61162 5o637 60772 60907 6k)4i 61176 5i3o8 6i44i 5.674 6 1 706 5 1 838 5i2i5 5.34s 5i48i 5i6.4 51746 51228 5 1 362 51495 51627 51759 51242 5i375 5i5o8 5 1640 51772 5i255 5i388 5i52i 5 1 654 51786 61295 61428 61661 61693 51826 51878 52009 52i4o 52270 52401 51891 52022 52 153 52284 52414 5 1 904 52f)35 52.66 52297 52427 51917 52048 52179 523 10 52440 61930 62061 62192 52323 62453 61967 52o«8 62218 52349 62479 61970 62101 62231 52362 62492 1 I 2 3 4 5 6 7 8 9 2 I 2 4 5 52 5o4 52634 52763 52S92 53o2o 525i7 52647 52776 52905 53o33 5253o 52660 52789 52917 53<>46 52543 52673 52802 52930 53u58 52 556 52(xS6 528.5 52943 53071 52569 62699 52827 5^956 53o84 62682 62711 62840 62969 53097 62696 62724 52853 62982 53iio 62608 52737 62866 62994 53l22 62621 62760 62879 53007 53i35 5 7 8 10 II 1 2 3 4 5 6 7 1 8 9 Page 174] TABLE XXVI. Logarithms of Numbers. Nn Q-ioo Mon 0. Log. 53148- 60206. No. 34o 34 1 342 343 M4 345 346 347 348 349 .350 35. 352 353 354 1 53i6i 53288 534 1 5 53542 53668 2 3 4 5 6 7 8 9 53i48 53275 534o3 53529 53656 53173 53301 53428 53555 53681 53 186 533i4 53441 53567 53694 53199 53326 53453 5358o 53706 53212 53339 53466 53593 53719 53224 53352 63479 536o5 53782 63; 87 53364 53491 536i8 63744 53260 53377 535o4 6363 1 53767 53263 53390 68617 63643 68769 13 I 2 3 4 5 6 7 8 9 I 3 4 5 53782 53908 54o33 54 1 58 54283 53794 53920 54045 54170 54295 53807 53933 54o58 54i83 54307 53820 53945 54070 54195 54320 53832 53958 54o83 54208 54332 53845 53970 54095 54220 54345 53857 53983 64108 64233 64357 53870 53995 54120 54245 54370 63882 54008 64i33 64268 54382 53896 64020 54i46 54270 54394 7 8 9 10 12 54407 ■14531 54654 54777 54900 54419 54543 54667 54790 54913 54432 54555 54679 54802 54925 5UM 54568 54691 548 1 4 54937 54456 54580 54-704 54827 54949 54469 54593 54716 54889 54962 5440 7. 5460L 54728 5485 1 54974 54494 54617 54741 54864 54986 645o6 6463o 64768 54876 54998 64618 54642 54765 54888 66011 355 356 357 358 359 36o 36 1 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 38o 38 1 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 1 399 55023 55i45 55267 55388 55509 55o35 55i57 55279 55400 55522 55642 55763 55883 56oo3 56l22 55o47 55169 55291 554 1 3 55534 55q6o 55f?2 553o3 ,55425 55546 55072 55194 553i5 55437 55558 55o84 55206 55328 55449 55570 66096 66218 66340 55461 66682 55708 66823 66943 66062 66182 55io8 55280 55362 55473 55694 55716 66835 55955 66074 56194 663 1 2 5643 1 56549 66667 66786 66121 55242 55364 55485 556o6 65i33 55265 55376 66497 66618 1 I 2 3 4 6 6 7 8 _9_ 2 I i 55630 55751 55871 55991 56iio 55654 55775 55895 56oi5 56 1 34 55666 55787 55907 56027 56i46 56265 56384 565o2, 56620 56788 55678 55799 55919 56o38 56i58 55691 558ii 55981 56o5o 56170 66727 55847 55967 66086 66206 66324 56443 66661 66679 66797 66739 66869 55979 66098 56217 56336 56466 66673 66691 568o8 A 5 6 7 8 56229 56348 56467 56585 56703 56241 5636o 56478 56597 56714 56253 56372 56490 566o8 56726 56844 56961 57078 57194 57810 56277 56896 565i4 56632 56750 56289 56407 56526 56644 56761 663oi 66419 66538 56656 66778 ic II 56820 56937 57054 571 71 57287 56832 56949 57066 57183 57299 56855 56972 57089 57206 57822 56867 56984 57101 57217 57334 56879 56996 57118 57229 57345 66891 57008 67124 57241 67357 67473 67688 67703 67818 67933 56902 67019 67186 67262 67868 66914 67081 57148 67264 67380 66926 67043 67169 67276 67892 11 574o3 57519 57634 57749 57864 574 1 5 57530 57646 57761 57875 57426 57542 57657 57772 57887 57438 57553 57669 57784 57898 57449 57565 57680 57795 57910 57461 57576 57692 57807 57921 57484 67600 67715 67880 67944. 57496 67611 67726 67841 67955 67607 67623 67738 67862 67967 I 2 3 4 5 6 7 8 9^ I 2 3 4 6 57978 58092 58206 58320 58433 57990 58 104 58218 5833 1 58444 58557 58670 58782 58894 59006 58ooi 58ii5 58229 58343 58456 58ot3 58127 58240 58354 58467 58024 58i38 58252 58365 58478 58o35 58i49 58263 58377 58490 58o47 68161 68274 68388 685oi 68o58 68172 582S6 68899 68612 68070 58 1 84 68297 58410 58524 68081 68195 68809 6842^ 58536 I 9 10 58546 58659 58771 58883 58995 58569 5868: 58794 58906 59017 58580 58692 588o5 58917 59028 58591 58704 588 1 6 58928 59040 58602 58715 58827 58989 59051 586i4 68726 58838 68960 69062 58626 68787 58850 68961 69078 58636 68749 58861 68978 69084 58647 68760 68872 68984 69096 59106 59218 59329 59439 59550 59118 59229 59340 59450 69561 59671 59780 59890 59999 60108 59129 59240 59351 59461 59572 59140 59251 59362 59472 59583 59i5i 59262 59878 59483 59594 59162 69278 69884 59494 69606 69178 69284 69895 69606 69616 59184 69296 69406 69617 69627 69196 69806 69417 59628 69688 69207 69818 69428 69539 69649 1( I 2 3 4 5 6 7 8 _9. ) I 1 59660 59770 59879 599S8 60097 59682 59791 59901 60010 60 1 1 9 59693 59802 59912 60021 6oi3o 59704 59813 59928 6oo32 6oi4i 69715 69824 59934 60043 60162 69726 69835 59945 6oo54 6oi63 59787 69846 69966 60066 60178 59748 69867 69966 60076 60184 69769 69868 69977 60086 60196 6 4 5 6 7 8 9 1 No. 1 2 3 4 5 6 7 8 9 1 ■■ — ' ■■ TABLE XXVI. [Page 175 Logarithms of Numbers. 1 No. 4000 4600. hog. 60206 G6276. No. 1 2 3 4 5 6 7 8 9 4oo 4oi 4o2 4o3 4o4 60206 6o3i4 60423 6o53i 6o633 60217 6o32 5 60433 6o54i 60649 60228 6o336 60444 6o552 60660 60239 60347 60455 6o563 60670 60249 6o358 60466 60574 60681 60260 60869 60477 6o584 60692 60271 60879 60487 60595 60708 60282 60890 60498 60606 60713 60298 60401 60509 60617 60724 60881 60938 61045 6ii5i 61257 60804 604 1 2 6o52o 60627 60735 ] I 2 3 4 5 6 7 8 9 1] I 2 3 4 4o5 4o6 407 408 409 4io 4ii 4l2 4i3 4i4 60746 6o853 60959 61066 61172 60756 6o863 60970 61077 6ii83 60767 60874 60981 61087 61 194 60778 6o885 60991 61098 61204 60788 60895 61002 61109 6i2i5 60799 60906 6ioi3 61119 61225 60810 60917 61028 6ii3o 6i236 60821 60927 6io34 6ii4o 61247 60842 60949 6io55 61162 61268 6 7 8 9 ID 61278 61 384 61490 61595 61700 61289 61395 6i5oo 61606 61711 6i3oo 6i4o5 6i5ii 61616 61721 6i3io 6i4i6 6i52i 61627 61731 61821 61426 6i532 61687 61742 6i33i 61437 61542 61648 61752 61342 61448 6i553 6i658 61768 6i352 6j458 6 1 563 61669 61773 6 1 863 61469 61574 61679 61784 61874 61479 6 1 584 61690 61794 4i5 4i6 417 4i8 419 6i8o5 61909 62014 62118 62221 6i8i5 61920 62024 62128 62232 61S26 61930 62034 62i38 62242 6i836 61941 62045 62149 62252 61847 6 1 95 1 62055 62159 6226J 61857 61962 62066 62170 62278 61868 61972 62076 62180 62284 61878 61982 620S6 62190 62294 61888 61993 62097 62201 62804 61899 6200,3 62107 62211 62815 420 421 422 423 424 425 426 427 428 429 43o 43t 432 433 434 435 436 437 438 439 440 44 1 442 443 444 445 446 447 448 449 45o 45i 452 453 454 62325 62428 6253i 62634 62737 62335 62439 62542 62644 62747 62346 62449 62552 62655 62757 62356 62459 62562 62665 62767 62366 62469 62572 62675 62778 62877 62480 62583 62685 62788 62887 62490 62593 62696 62798 62897 62500 62608 62706 62808 62408 625ii 62613 62716 62818 62418 62521 62624 62726 62829 62839 62941 63o43 63 1 44 63246 62849 62951 63o53 63 1 55 63256 62S59 62961 63o63 63 1 65 63266 62870 62972 63073 63 1 75 63276 62880 62982 63o83 63i85 68286 62S90 62992 68094 68195 68296 62900 68002 63io4 632o5 633o6 62910 63oi2 63ii4 632i5 68817 62921 68022 68124 63225 68827 63428 63528 68629 68729 68829 62981 63o33 63 1 34 68286 63337 1 I 2 3 4 5 6 7 8 9 u 1 2 3 4 63347 63448 63548 63649 63749 63357 63458 63558 63659 63759 63367 63468 63568 63669 63769 63377 63478 63579 63679 68779 63387 63488 63589 63689 63789 63397 63498 63599 68699 68799 63407 63 5o8 68609 68709 63809 63417 685:8 636 19 63719 63819 63438 63538 68689 68789 63839 5 6 7 8 9 63849 63949 64o48 64 1 47 64246 63859 63959 64o58 64i57 64256 63869 63969 64068 64167 64266 64365 64464 64562 64660 64758 63879 68979 64078 64177 64276 64375 64473 64572 64670 64768 68889 68988 64088 64187 64286 68899 68998 64098 64197 64296 68909 64008 64 1 08 64207 643o6 68919 64018 64ii8 64217 643 1 6 68929 64028 64128 64227 64326 68939 64o38 64187 64287 64335 I- 64345 64444 .64542 6464o 64738 64355 64454 64552 6465o 64748 64385 64483 64582 64680 64777 64395 64493 64591 64689 64787 644o4 645o3 64601 64699 64797 644 1 4 645 1 3 646 II C4709 64807 64424 64523 64621 64719 64816 64484 64532 6463 1 64729 648 2 6 J' T 64836 64933 65o3i 65i28 65225 64846 64943 65o4o 65i37 65234 64856 64953 65o5o 65i47 65244 64865 64963 65o6o 65 1 57 65254 64875 64972 65070 65i67 65263 64885 64982 65o79 65176 65273 64S95 64992 65089 65 1 86 65283 64904 65oo2 65o99 65196 65292 64914 65oii 65 1 08 65205 65302 64924 65o2i 65ii8 652 1 5 653 1 2 65321 654 1 8 655i4 656io 65706 6533i 65427 65523 65619 65715 65341 65437 65533 65629 65725 6535o 65447 65543 65639 65734 65360 65456 65552 65648 65744 65369 65466 65562 65658 6:,;^3 65379 65475 65571 65667 65763 65389 65485 65581 65677 63772 65398 65495 65591 65686 65782 654o8 655o4 656oo 65696 65792 9 2 : -J 3 3 4\4 455 456 457 458 459 658oi 65896 65992 66087 66181 658 11 65906 66001 66096 66191 65820 65916 66011 66106 66200 65830 65925 66020 66ii5 66210 65839 65935 66o3o 66124 66219 65849 65944 66089 66134 66229 65858 65954 66049 66143 66288 65868 65968 66o58 661 53 66247 65877 65973 66068 66162 66257 65887 65982 66077 66172 66266 5, 6 7 8 9 3 5 6 7 8 No. 1 2 3 4 5 1 6 7 8 9 k Page 1761 TABLE XXVI. Logarithms of Numbers. No. 4000 5200. Log. C6276 71600. No. 1 2 3 4 5 6 66332 66427 66621 66614 66708 7 66342 66436 66530 66624 66717 8 9 46o 46 1 462 463 464 465 466 467 468 469 470 471 472 473 474 475' 476 477 47S 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 600 5oi 502 5o3 5o4 5o6 5o6 607 5oS 609 610 5ii 612 5i3 5i4 5i5 5x6 617 5i8 619 No. 66276 66370 66454 66668 66662 66286 66380 66474 66667 66661 66296 66389 66483 66577 66671 663o4 66398 66492 66586 66680 663 1 4 66408 66602 66696 66689 66323 66417 66611 66606 66699 6636 1 66445 66539 66633 66727 66361 66455 66649 66642 66736 1 I 2 3 4 5 6 7 8 9 I 2 3 4 66745 66839 66932 67026 67117 66755 66848 66941 67034 67127 66764 66867 66960 67043 67136 66773 66S67 66960 67062 67145 66783 66876 66969 67062 67164 66792 66886 66978 67071 67164 66801 66894 669S7 67080 67173 66811 66904 66997 67089 67182 66820 66913 67006 67099 67191 66829 66922 67016 67108 67201 5 6 7 8 7 67210 67302 67394 674S6 6767S 67219 67311 67403 67495 67687 67228 67321 67413 67604 67696 67237 67330 67422 67614 67606 67247 67339 67431 67623 67614 67266 67348 67440 67532 67624 67265 67367 67449 67641 67633 67274 67367 67469 67660 67642 67284 67376 67468 67660 67661 67293 67335 67477 67669 67660 67669 67761 67862 67943 68034 68124 68216 683o5 68396 68485 67679 67770 67861 67962 68043 68 1 33 68224 683i4 684o4 68494 676S8 67779 67870 67961 68062 67697 67788 67879 67970 68061 67706 67797 67888 67979 68070 67715 67806 67897 67988 68079 67724 67816 67906 67997 68088 67733 67826 67916 68006 68097 67742 6-834 67926 68016 68106 67762 67843 67934 68024 68116 68142 68233 68323 684 1 3 68602 68161 68242 68332 68422 68611 68160 68261 68341 68431 68620 68169 68260 68350 68440 68629 68178 68269 68359 68449 68538 68187 68278 68368 68458 68647 68196 6S287 68377 68467 68666 68205 68296 68386 68476 68565 68674 68664 68763 68842 68931 68583 68673 68762 6885 1 68940 68692 68681 68771 68860 68949 6S601 68690 68780 6S869 68968 68610 68699 68789 68878 68966 68619 68708 68797 68886 68976 68628 6S717 68806 68896 68984 68637 68726 688 1 5 68904 68993 68646 68735 68824 68913 69002 68665 68744 68833 68922 69011 I 2 3 4 5 6 7 8 9 I 2 3 4 69020 69108 69197 69285 69373 69028 69117 69206 69294 69381 69037 69126 69214 69302 69390 69046 69135 69223 69311 69399 69066 69144 69232 69320 69408 69064 69162 69241 69329 69417 69073 69161 69249 69338 69426 69082 69170 69268 69346 69434 69090 69179 69267 69355 69443 69099 691S8 69276 69364 69452 69539 69627 69714 69801 69888 69976 70062 70148 70234 70321 5 5 6 7 f 69461 69548 69636 69723 69810 69469 69667 69644 69732 69819 69478 69666 69663 69740 69827 69487 69674 69662 69749 69836 69496 69683 69671 69768 69846 69604 69692 69679 69767 69854 69613 69601 69688 69776 69S62 69622 69609 69697 69784 69871 69631 69618 69706 69793 69S80 69966 70063 70140 70226 7o3i-2 69897 69984 70070 70167 70243 69906 69992 70079 70166 70262 69914 70001 70088 70174 70260 69923 70010 70096 70183 70269 69932 70018 70106 70191 70278 7o364 70449 70636 7062 1 70706 69940 70027 701 14 70200 70286 69949 7oo36 70122 70209 70296 69968 70044 70i3i 70217 7o3o3 70389 70476 70661 70646 70731 70329 704 1 5 70601 70686 70672 7o338 70424 70609 70695 70680 70346 70432 70618 70603 70689 7o365 70441 70626 70612 70697 70372 70458 70644 70629 70714 7o3Si 70467 70662 7o638 70723 70398 70484 70669 70666 70740 70826 70910 70995 71079 71164 71248 7i332 71416 71600 71584 70406 70492 70678 70663 70749 70834 70919 7ioo3 71088 71172 70767 70842 70927 71012 71096 70766 70861 70935 71020 71106 70774 70869 70944 71029 71113 70783 70868 70962 71037 71122 70791 70876 70961 71046 7ii3o 70800 70885 70969 71064 71139 70808 70893 70978 71063 71 147 7i23i 7i3i5 71399 71483 71667 70817 70902 70986 71071 71166 I 2 3 4 5 6 7 8 9 I 2 2 3 71 181 71265 71349 71433 71617 71189 71273 71357 71441 71626 71 198 71282 7 1 366 7i45o 71533 71206 71290 71374 71458 71642 71214 71299 71383 71466 71660 71223 7 1 307 71391 71475 71669 71240 71324 71408 71492 71676 71267 7i34i 71425 71608 71692 4 5 6 6 ■7 1 (> 3 4 5 i G 7 8 9 - I TABLE XXVI. [Page 177 Logarithms of Numbers. No c;ooo SsiOC ). Log. 71600 76343. No. 520 521 522 523 524 525 526 527 528 529 1 2 3 4 5 G 7 8 9 71600 71684 71767 7i85o 7193^ 71609 71692 71775 71858 71941 71617 71700 71784 71867 71950 71625 71709 71792 71875 7.958 71634 71717 71800 71883 71966 71642 71725 71809 71892 71975 7i65o 71734 71817 71900 71988 71659 71742 71825 71908 71991 71667 7.750 7.884 7.9.7 7.999 7.675 7.759 71842 7.925 72008 72016 72099 72181 72263 72346 72024 72107 72189 72272 72354 72082 72115 72198 72280 72862 72041 72123 72206 7 2 288 72870 72049 72182 72214 72296 72878 72057 72140 72222 72804 72887 72066 72148 72280 72818 72895 72074 72 1 56 72289 72821 72403 72082 72.65 72247 72829 724.1 72090 72178 72255 72887 72419 53o 53i 532 533 534 72428 72509 72591 73673 72754 7^835 72916 72997 73078 73159 72436 72518 72599 72681 72762 72843 72925 73006 78086 73167 72444 72526 72607 72689 72770 72452 72534 72616 72697 72779 72460 72542 72624 72705 72787 72868 72949 78080 78111 78191 78272 73852 78482 73512 73592 72469 72550 72682 72718 72795 72876 72957 78088 78119 78199 78280 78860 73440 73520 78600 72477 72558 -2640 72722 72808 72485 72567 72648 72780 72811 72493 72575 72656 72788 72819 725ui 72583 72665 72746 72827 535 536 537 538 L)39 54o 54 1 542 543 544 545 546 547 548 549 55o 55i 552 553 554 72852 72988 78014 78094 78175 78255 78886 78416 78496 73576 72860 72941 78022 78102 78188 72884 72965 78046 78127 78207 72892 72978 78054 78.35 78215 72900 72981 78062 78143 78328 72908 72989 78070 78.5. 7828. 73239 73320 73400 73480 73560 73247 73328 73408 73488 73568 78268 73344 73424 73504 73584 78288 78868 73448 73528 78608 78296 78876 73456 73536 78616 78804 73384 78464 73544 78624 78812 78892 78472 78552 78682 7364o 73719 73799 73878 73957 73648 73727 73807 73886 73965 73656 78785 738 1 5 73894 78973 73664 73743 78828 78902 78981 78672 78751 73830 78910 78989 78679 78759 78888 78918 73997 78687 78767 78846 78926 74oo5 78695 73775 78854 78988 740 1 3 78708 78788 78S62 7894. 74020 78711 787^ 78870 78949 74028 74o36 74u5 74194 74273 7435i 74o44 74i23 74202 74280 74359 74o52 74i3i 74210 74288 74367 74060 74189 74218 74296 74874 74068 74 1 47 74225 74304 74382 74076 741 55 74233 74312 74390 74084 74162 74241 74820 74398 74092 74.70 74249 74337 74406 74'->99 74178 74257 74335 744.4 74107 74.86 74265 74348 74421 555 556 557 558 559 74429 74507 74586 74663 74741 74437 745)5 74593 74671 74749 74445 74523 74601 74679 74757 74458 74581 74609 74687 74764 74461 74539 74617 74695 74772 74468 74547 74624 74702 74780 74476 74554 74682 74710 74788 74484 74562 74640 74718 74796 74492 74570 74648 74726 74S08 745oo 74578 74656 74788 748 1 1 56o 56 1 562 563 564 74819 74896 74974 75o5i 75128 74827 74904 74981 75o59 75i36 74834 74912 74989 75o66 75i43 74842 74920 74997 75074 75i5i 7485o 74927 75oo5 75082 75i59 74858 74935 75012 75089 75166 74865 74943 75020 75097 75.74 74873 74950 75028 75io5 75.82 7488. 74958 75o35 75ii3 75.89 74889 74966 75o48 75 . 20 75.97 565 566 567 568 569 75205 75282 75358 75435 755ii 752i3 75289 75366 75442 75519 75220 75297 75874 75450 75526 75228 753o5 75881 75458 75534 75286 75312 75389 75465 75542 75248 75820 75397 75473 75549 7525. 75328 75404 7548. 75557 75259 75385 754.2 75488 75565 75366 75343 75420 75496 75572 75274 75351 75427 75504 75580 75656 75782 75808 75884 75959 570 571 572 573 574 575 576 577 578 579 No. 75587 75664 73740 758 1 5 75891 75595 75671 75747 75823 75899 756o3 7567? 75755 7583 1 75906 75610 75686 75762 75888 75914 75618 75694 75770 75846 75921 75626 75702 75778 75853 75929 75633 75709 75785 75861 75987 75641 757.7 75798 75868 75944 75648 75724 75800 75876 75952 75967 76042 761 18 76193 76268 75974 76o5o 7-6125 76200 "6275 75982 76057 76133 76208 76288 75989 76065 76140 76215 76290 75997 76072 76148 76228 76298 76005 76080 76155 76280 76805 76012 76087 76.68 76288 76818 76020 76095 76.70 76^45 76820 76027 76 1 o3 76178 76253 76828 76085 761 10 76185 76260 76335 1 1 2 3 4 5 1 6 7 8 9 3 4 4 5 6 9'6 23 P^s«i78] TABLE XXVI. Logarithms of Numbers. No. 5800 6400. Log r yrpi-ii R 0G18. No. 58o 58 1 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 6o3 604 6o5 606 607 608 609 610 611 612 6i3 6i4 1 2 3 4 5 6 7 8 9 76343 76418 76492 76567 76641 76716 767-vo 76864 76938 77012 76350 76425 76^00 76574 76049 76723 76797 76871 76945 77019 76358 76433 76507 76582 76656 76365 76440 765 1 5 76589 76664 76738 76812 76886 76960 77034 76373 76448 76522 76597 76671 76880 76455 76530 76604 76678 76388 76462 76537 76612 76686 76395 76470 76545 76619 76693 76403 76477 76662 76626 76701 76410 76485 76669 76634 76708 ( I 2 3 4 5 6 7 8 9 4 ■3 76730 76805 76879 76953 77026 76745 76819 76893 76967 77041 76753 76827 76901 76975 77048 76760 76834 76908 76982 77o56 76768 76842 76916 76989 77063 76775 76849 76923 76997 77070 76782 76866 76930 77004 77078 4 5 6 6 7 77085 77159 77232 773o5 77379 77093 77166 77240 773 1 3 77386 77100 77173 77247 77320 77393 77107 77181 77254 77327 77401 77115 77188 77262 77335 77408 77122 77195 77269 77342 774 1 5 77129 77203 77276 77349 77422 77187 77210 77283 77357 7743o 77144 77217 77291 77364 77437 77i5i 77226 77298 77371 77444 77452 77525 77597 77670 77743 77459 77532 77605 77677 77750 77466 77539 77612 77685 77757 77474 77546 77619 77692 77764 77481 77554 77627 77699 77772 77488 77561 77634 77706 77779 77495 77568 77641 77714 77786 77608 77676 77648 77721 77793 77610 77583 77666 77728 77801 77617 77690 77663 77735 77808 77815 77887 77960 78032 78104 77S22 77895 77967 78039 781 II 77830 77902 77974 78046 78118 77837 77909 77981 78053 78125 77844 77916 77988 78061 78132 77851 77924 77996 78068 78140 77859 77931 78003 78075 78147 77866 77938 78010 78082 78164 77873 77945 78017 78089 78161 77880 77962 78025 78097 78168 7S176 70247 78319 78390 78462 78183 78254 78326 78398 78469 78190 78262 78333 78405 78476 78197 78269 78340 78412 78483 78204 78276 78347 78419 78490 7821 1 78283 78355 78426 78497 78219 78290 7S362 78433 78504 78226 78297 78369 78440 78612 78233 78306 78376 78447 78619 78240 78312 78383 78455 78626 7 I 2 3 4 5 6 7 8 9 I 1 2 3 78533 7S604 78675 7S746 78817 78540 78611 78682 78753 78824 78547 78618 786S9 78760 7883i 7S554 78625 78696 78767 78838 78561 78633 78704 78774 78845 78669 78640 7871 1 78781 78852 78576 78647 78718 787S9 78859 78683 78664 78726 78796 78866 78690 78661 78732 78803 78873 78697 78668 78739 78810 7S880 4 4 5 6 6 6i5 616 617 6r8 619 78888 78958 79029 79099 79169 78895 78965 79o36 79106 79176 78902 78972 79043 79113 79183 78909 78979 79o5o 79120 79190 78916 78986 79057 79127 79^97 78923 78993 79064 79134 79204 78930 79000 79071 79141 792 1 1 78937 79007 79078 79148 79218 78944 79014 79086 79166 79226 78961 79021 79092 79162 79232 620 621 622 623 624 79239 79309 79379 79449 79518 79246 79316 79386 79456 79525 79S95 79664 79734 79803 •79872 79941 80010 80079 80147 80216 79253 79323 79393 79463 79532 79260 79330 79400 79470 79539 79267 79337 79407 79477 79546 79274 79344 79414 79484 79553 79281 79351 79421 79491 79660 79288 79358 79428 79498 79667 79295 79366 79436 79606 79574 79302 79872 79442 79611 79681 625 626 627 628 629 63o 63i 632 633 634 635 636 637 638 639 No. 79588 79657 79727 79796 79865 79934 8ooo3 80072 8oi4o 80209 79602 79671 79741 79810 79879 79609 79678 79748 79817 79886 79955 80024 80092 80161 80229 79616 79685 79754 79824 79893 79623 79692 79761 79831 79900 79630 79699 79768 79837 79906 79637 79706 79775 79844 79913 79644 79713 79782 79861 79920 79660 79720 79789 79868 79927 79996 8oo65 8oi34 80202 80271 79948 80017 8oo85 801 54 80223 79962 8oo3o 80099 80168 8o236 79969 80037 80106 80175 80243 79975 80044 8oii3 80182 80260 79982 8oo5i 80120 80188 80267 79989 8oo5S 80127 80195 80264 ( I 2 3 4 5 6 7 8 9 5 I 2 1 80277 8o346 804 1 4 80482 8o55o 80.284 8o353 80421 80489 80557 80291 80359 80428 80496 8o564 80298 8o366 80434 8o5o2 80570 8o3o5 80373 8o44i 8o5o9 80577 8o3i2 8o38o 80448 8o5i6 8o584 5 8o3i8 80387 80455 8o523 80691 80326 80393 80462 8o53o 80698 8o332 8o4oo 80468 8o536 80604 80339 80407 80475 80543 806 II 3 4 4 5 5 1 2 3 4 6 7 8 9 TABLE XXVI. [vagem Logarithms of Numbers. No. G400 7000. Log. B0618 84510. No. 64o 64 1 642 643 644 1 2 3 4 5 6 7 8 9 80618 80686 80754 80821 80889 80625 80693 80760 80828 80895 8o632 80699 80767 8o835 80902 80688 80706 80774 80841 80909 80645 807 1 3 80781 80848 80916 8o652 80720 80787 8o855 80922 80659 80726 80794 80862 80929 8o665 80788 80801 80868 80986 80672 80740 80808 80875 80943 80679 80747 80814 80S82 80949 4 I 2 3 4 5 6 7 8 9 I I a 3 645 646 647 648 649 80956 81023 81090 8ii58 81224 80963 8io3o 81097 81164 8i23i 80969 81037 81104 81171 8i238 80976 81043 81111 81178 81245 80988 8io5o 81117 81184 8i25i 810D7 81124 81191 81258 80996 81064 8ii3i 81198 81265 81008 81070 81187 81204 81271 81010 81077 81144 81211 81278 81017 81084 8ii5i 81218 81285 4 4 5 6 6 65o 65 1 652 653 654 81291 8i358 81425 8 1 491 8i558 81298 8i365 8i43i 81498 81 564 8i3o5 81871 81 438 8i5o5 81571 8i3ii 81878 81445 8i5ii 81578 8i3i8 8i3S5 8i45i 8i5i8 8 1 584 8i325 81891 8i458 8i525 81591 81881 81398 8i465 8i53i 81598 81888 8i4o5 81471 8i58S 8 1 6o4 81845 8i4ii 81478 81544 81611 8i35i 8i4i8 81 485 8i55i 81617 655 656 657 658 659 81624 81690 81757 81823 81889 8i63i 81697 81763 81829 81895 81637 81704 81770 8i836 81902 81644 81710 81776 81842 81908 8i65i 81717 81788 81849 8191S 81657 81728 81790 8 1 856 81921 81664 81730 81796 81862 81928 81671 81787 81808 81869 81985 81677 81748 81809 81875 81941 81684 81750 81816 81882 81948 66o 66 1 662 663 664 81954 82020 82086 82i5i 82217 81961 82027 82092 821 58 82223 81968 82033 82099 82164 82230 81974 82040 82105 82171 82286 81981 82046 82112 82178 82243 82808 82878 82489 82504 82569 81987 82053 821 19 82184 82249 81994 82060 82125 82191 82256 82000 82066 82182 82197 82263 82007 82078 82i38 82204 82269 82014 82079 82145 82210 82276 665 666 667 668 669 82282 82347 824i3 82478 82543 82289 82354 82419 82484 82549 82295 82360 82426 82491 82556 82802 82867 82432 82497 82562 82815 82880 82445 82510 82575 82821 82887 82452 82517 82582 82828 82898 82458 82528 82588 8,2884 82400 82465 82580 82595 82841 82406 82471 82586 82601 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 82607 82672 82737 82802 82866 82614 82679 82743 82808 82872 82620 82685 82750 82814 82879 82627 82692 82756 82821 828S5 82950 88014 83078 83 1 42 88206 82638 82698 82763 82827 82892 82640 82705 82769 82884 82898 82646 82711 82776 82840 82905 82653 82718 82782 82847 82911 82659 82724 82789 82853 82918 82666 82780 82795 82860 82924 82930 82995 83o59 83 1 23 83187 82937 83ooi 83o65 83 1 29 83193 82943 83oo8 83o72 83i36 83200 82956 88020 83o85 88149 832i3 82968 88027 88091 88i55 88219 82969 88o83 88097 83i6i 88225 82975 83o4o 83 1 04 88168 88282 82982 83o46 83iio 88174 88288 82988 88o52 88117 83i8i 88245 8325i 833 1 5 83378 83442 835o6 83257 83321 83385 83448 83512 83264 83327 83391 834D5 835i8 88270 88884 88898 88461 83525 88276 83340 834o4 83467 8353i 88288 83347 83410 88474 83537 88289 83853 83417 83480 83544 88296 88859 88428 83487 83550 88802 83366 88429 83498 83556 833o8 88872 83436 88499 83563 685 686 687 688 689 83569 83632 83696 83:59 83822 83575 83639 83702 83765 83828 83582 83645 83708 83771 83835 83588 8365 1 88715 88778 88841 83594 83658 83721 83784 83847 88601 88664 88727 88790 83858 88607 88670 83734 83797 83S6o 88618 88677 88740 88808 83866 88620 83683 88746 88809 88872 83626 88689 83753 83Si6 83879 I 2 3 4 5 6 7 8 9 I I a 2 690 691 692 693 694 83885 88948 84oii 84073 84 1 36 83891 83954 84017 84080 84 1 42 88897 88960 84028 84o86 84 1 48 88904 88967 84029 84092 84 1 55 88910 83978 84o36 84098 84i6i 88916 83979 84042 84io5 84167 88928 88985 84o48 84111 84173 84236 84298 8436i 84423 84485 88929 88992 84o55 84117 84 180 88935 8899S 84061 84123 84186 88942 84oo4 84067 84i3o 84192 3 4 4 5 5 695 656 697 698 699 No. 84198 84261 84323 84386 84448 842o5 84267 8433o 84392 84454 84211 84273 84336 84398 8446o 84217 84280 84842 844o4 84466 84228 84286 84848 844 10 84473 84280 84292 84354 84417 84479 84242 843o5 84867 84429 84491 84248 848 11 84373 84435 84497 84255 843 1 7 84379 84442 845o4 1 2 1 3 4 5 6 7 8 9 Page 180] TABLE A Logaritlmiri of AV'i. iN umbers. Nn 70nf) 7(11)0 I nir PiVAO CQnQI No. 1 2 3 4 5 6 ! 7 8 9 700 701 702 708 704 7o5 706 707 708 710 711 712 7i3 714 '7! 5 716 717 7.8 719 720 721 722 728 724 725 726 727 728 729 780 73 1 782 788 734 735 786 737 788 789 74o 74 1 742 743 744 845io 84572 84634 84696 84757 84819 84SSo 8^<942 85o65 845i6 84578 84640 84702 84768 84522 84584 84646 84708 84770 84528 84590 84652 84714 84776 84535 84597 84658 84720 84782 84541 84608 84665 84726 84788 84547 84609 84671 84733 84794 84856 84917 84979 85o4o 85ioi 84558 846 1 5 84677 84739 84800 84559 8462 X 84688 84745 84807 84566 84628 84689 8475 1 848 1 3 7 I 2 3 4 5 6 7 8 9 I 2 3 84825 84887 84948 85009 85o7i 84881 84898 84954 85oi6 85o77 84887 84899 84960 85o22 85o88 84844 84905 84967 85o28 85089 84850 8491 1 84973 85o84 85095 84862 84924 84985 85o46 85io7 84868 84980 84991 85o52 85ii4 85i75 85236 85297 85858 854i8 84S74 84986 84997 85o58 85i2o 85i8i 85242 853o8 85364 85425 4 4 5 6 6 85i26 85187 85248 85309 85370 85i82 85198 85254 858i5 85376 85i88 85 1 99 85260 85821 85382 85i44 85205 85266 85827 85888 85i5o 852II 85272 85388 S5394 85i56 85217 85278 85389 854oo 85i63 85224 85285 85345 854o6 85169 8523o 85291 85352 85412 8543 1 85491 85552 856i2 85673 85487 85497 85558 856i8 85679 85443 855o8 85564 85625 85685 85449 855o9 85570 8 568 1 85691 85455 855i6 85576 85637 85697 8 546 1 85522 85582 85648 85708 85467 85528 85588 85649 85709 85473 85534 85594 856d5 857i5 85479 85540 85600 8566i 85721 85485 85546 856o6 85667 85727 85788 85794 85854 85914 85974 85789 85800 85860 85920 85980 85745 858o6 85866 85926 85986 8575i 858i2 85872 85982 85992 85757 858i8 85878 85988 85998 85768 85824 85884 85944 86004 85769 8583o 85890 85950 86010 85775 85836 85896 85956 86016 85781 85842 85902 85962 86022 85788 85848 85908 85968 86028 86084 86094 86i58 86213 86278 86o4o 86100 86159 86219 86279 86o46 86106 86i65 86225 86285 86o52 86112 86171 86231 86291 86o58 86118 86177 86287 86297 86856 864 1 5 86475 86584 86598 86064 86124 86188 86243 86808 86070 86i3o 86189 86249 86808 86076 86186 86195 86255 86814 86874 86483 86498 86552 8661 1 86082 86i4i 86201 86261 86820 S6088 86147 86207 86267 86826 r I 2 3 4 5 6 7 8 9 1 1 I 2 2 86332 86892 8645 1 865 10 86570 86838 86898 86457 865 1 6 86576 86344 86404 86468 86522 8658i 8664 1 86700 86759 86817 86876 8685o 864 10 86469 86528 86587 86862 86421 8648 1 86540 86599 86368 86427 864S7 86546 866o5 86880 86439 86499 86558 86617 86386 86445 865o4,- 86564 86628 86682 86741 86S00 86859 86917 3 4 4 5 5 86629 86688 86747 86806 86864 86685 86694 86753 86812 86870 86646 86705 86764 86823 86882 86652 8671 1 86770 86829 86888 86658 86717 86776 86885 S6894 86664 86728 86782 8684 1 86900 86670 86729 86788 86847 86906 86676 86735 86794 86858 8691 1 86928 86982 87040 87099 87.57 86929 86988 87046 87105 87168 86935 86994 87052 87111 87169 86941 86999 87058 87116 87175 86947 87005 87064 87122 87181 86953 8701 1 87070 87128 87186 86958 87017 87075 87184 87192 86964 87028 87081 87140 87198 86970 87029 87087 87146 87204 86976 87035 87098 87151 87210 745 746 747 748 749 75o 75i 752 753 754 87216 87274 87882 87890 87448 87221 87280 87888 87896 87454 87227 87286 87844 87402 87460 87233 87291 87849 87408 87466 87289 87297 87355 87418 87471 87529 87587 87645 87708 87760 87245 87808 87861 87419 87477 87535 87598 87651 87708 87766 87251 87809 87867 87425 87488 87256 87815 87878 87431 87489 87262 87820 8787-9 87437 87495 87268 87826 87884 87442 87500 87506 87564 87622 87679 87787 87512 87570 87628 87685 87743 87518 87576 87688 87691 87749 87528 87581 87689 87697 87754 87541 87599 87656 87714 87772 87547 87604 87662 S7720 87777 87552 87610 87668 87726 877S8 87558 87616 87674 87731 87789 I I 2 3 4 5 6 7 8 9 ) X I 2 2 755 756 757 758 759 87795 878^2 87910 8^967 8S024 87800 87858 87915 87973 88o3o 87S06 87864 87921 87978 88o36 87812 87869 87027 87984 8804 1 87818 87875 87988 87990 88047 87828 87881 87988 87996 88o53 87829 87887 87944 88001 88o5S 87885 87802 87950 88007 88064 87841 87898 87955 88018 88070 87846 87904 87961 88018 88076 3 3 4 4 5 No. 1 2 3 4 5 6 7 8 9 TABLE XXVI. [P^g« '81 Logarithms of Numbers. i\o. 7(Ji)0— 8200. Log. 88081 91381. No. 1 2 3 4 5 G 7 8 9 760 88081 88087 88093 88098 88104 88110 88116 88i2i 815127 88i33 761 88 I 38 88144 88i5o 881 56 88161 88167 88173 88178 88184 88190 762 88195 88201 8S207 88213 88218 88224 88230 88235 88241 8S247 763 88252 88258 88264 88270 88275 88281 88287 88292 88298 88355 88412 8S3o4 764 883091 883 1 5 88321 88326 88332 88338 88343 88349 8836o 7()5 88366 8S372 88377 88383 88389 88395 88400 884o6 88417 766 88423 S8429 88434 88440 88446 8845i 88457 88463 88468 88474 767 88480 884-85 88491 88497 885o2 8S5o8 885 1 3 88519 88525 88530 768 88536 88542 88547 88553 88559 88564 88570 885-6 8858i 88587 769 88093 8S598 8S604 88610 886i5 88621 88627 88632 88638 88643 770 88649 88655 88660 88666 88672 8S677 88683 886S9 88694 88700 771 8S705 8871 1 88717 88722 88728 88734 88739 88745 88750 88756 772 88762 88767 88773 88779 88784 88790 88795 88801 88807 88812 773 88818 8S824 88829 88835 88840 88846 88852 88857 88863 88868 774 88874 88880 88885 88891 88897 88902 88908 88913 88919 88925 775 88930 88936 88941 88947 88953 88958 88964 88969 88975 88981 776 88986 88992 88997 89003 89009 89014 89020 8qo25 89031 89087 777 89042 89048 89053 89059 89064 89070 89076 89081 89087 89092 77S 89098 89104 89109 89115 89120 89126 89131 89137 89143 89148 779 89154 89159 89165 89170 89176 89182 89,87 89193 89248 89198 89204 89260 780 89209 89215 89221 89226 89232 89237 89243 89254 781 89265 89271 89276 89282 89287 89293 89298 89304 89310 89315 782 89321 89326 89332 89337 89343 89348 89354 89360 89365 89371 783 89376 89382 893S7 89393 89398 89404 89409 89415 80421 89426 7S4 89432 89437 89443 89448 89454 89459. 89515 . 89465 89470 89476 89531 89481 785 89487 89492 89498 89504 89509 89520 89526 89537 786 89542 89548 89553 89559 89564 89570 89575 89581 89586 89592 787 89597 89603 89609 89614 89620 89625 89631 89636 89642 89647 788 89653 89658 89664 89669 89675 89680 89686 89691 89697 89702 789 790 89708 89713 89719 89724 89730 8973:) 89741 89746 89752 89757 89812 89763 89768 89774 89779 89785 89790 89796 89801 89807 791 89S18 89823 89829 89834 89840 89845 S9851 89856 89862 89867 792 89873 89878 89883 89889 89894 89900 89905 89911 89916 89922 79:^ 89927 89933 89938 89944 89949 89955 89960 89966 89971 89977 794 795 89982 89988 89993 89998 90004 90009 90015 90020 90026 90081 90037 90042 90048 90053 90059 90064 90069 90075 90080 90086 796 90091 90097 90102 90108 90113 90J19 90124 90129 90135 90140 797 90146 901 5i 90157 90162 90168 90173 90179 90184 90189 90195 798 90200 90206 902 II 90217 90222 90227 90233 90238 90244 90249 799 90255 90260 90266 90271 90276 90282 90287 90293 90298 90352 90804 800 90309 903 1 4 90320 90325 9033 1 903 36 90342 90347 908 58 801 9o363 90369 90374 90380 9o385 90390 90396 90401 90407 904 1 2 802 90417 90423 90428 90434 90439 90445 90450 90455 9046 1 90466 8o3 90472 90477 90482 90488 90493 90499 9o5o4 90509 905 1 5 90520 804 8o5 90526 9053 1 90536 90542 90547 90553 90558 9o563 90569 90574 9o58o 9o585 90590 90596 90601 90607 90612 90617 90623 90628 806 90634 90639 90644 90650 90655 90660 90666 90671 90677 90682 807 90687 90693 90698 90703 90709 90714 90720 90725 90780 90786 808 90741 90747 90752 90757 90763 90768 90773 90779 90784 90789 809 810 90795 90800 90806 908 II 90816 90S 2 2 90827 90832 90838 90843 90849 90854 90859 90865 90870 90875 9088 1 90886 90891 90897 8n 90902 90907 90913 90918 90924 90929 90934 90940 90945 90950 812 90956 90961 90966 90972 9«977 90982 90988 90993 90998 91004 8i3 91009 91014 91020 91025 9io3o 9io36 91041 9 1 o46 91052 91057 8i4 91062 91068 91073 91078 91084 91089 91094 91100 91105 91110 8i5 911 16 91121 91126 91132 91137 91142 91148 91153 91 158 ' 91 164 1 816 91 169 91 174 91 180 91185 91190 91 196 91201 91206 91212 91217 bl7 91222 91228 91233 91238 91243 91249 91254 91259 91265 91270 818 91275 91281 91286 91291 91297 9i3o2 91307 9i3i2 9i3i8 91828 819 9132S 91334 91339 91344 9i35o 91355 9i36o 91365 91371 91876 No. b 1 2 3 4 5 6 7 8 9 2 1 8 2 4 2 5 3 6 3 7 4 8 4 9 5 Page 1 82] TABLE XXVi. Logarithms of Numbers. No. '^"On 880( ). Log. 91 381- 1448. No. 1 2 3 4 5 6 7 8 9 820 821 822 823 824 9i38i 91434 91487 91540 91593 91387 91440 91492 91545 91598 91392 91445 91498 9i55i 91603 91397 91450 9i5o3 91556 91609 9i4o3 91455 9i5o8 9i56i 91614 91408 91461 9i5i4 91 566 91619 91413 91466 91519 91572 91624 91418 91471 91524 9x577 9i63o 9x424 91477 91529 9x582 91635 9x429 91482 9x535 91587 9x640 ( 1 2 3 4 5 6 7 8 9 I 1 2 2 825 826 827 828 829 91645 91698 91751 91803 91855 9i65i 91703 91756 91808 91861 9i656 91709 91761 91814 91866 91661 91714 91766 91819 91871 91666 91719 91772 91824 91876 91672 91724 91777 91829 91882 91677 91730 91782 91834 91887 91939 91991 92044 92096 92148 91682 91735 91787 91840 91892 91944 91997 92049 9210X 92x53 91687 91740 9x793 91845 91897 91693 91745 91798 9x85o 91903 3 4 4 5 5 83o 83i 832 833 834 91908 91960 92012 92065 92117 9' 9'^ 91965 92018 92070 92122 91918 91971 92023 92075 92127 91924 91976 92028 92080 92132 91929 91981 92033 920S5 92137 91934 91986 92o38 92091 92143 91950 92002 92054 92106 92x58 91955 92007 92059 92 1 II 92x63 835 836 837 838 839 84o 84 1 842 843 844 92169 92221 92273 92324 92376 92174 92226 92278 92330 92381 92179 92231 92283 92335 92387 92184 92236 92288 92340 92392 92443 92495 92547 92598 92650 92189 92241 92293 92345 92397 92195 92247 92298 92350 92402 92200 92252 92304 92355 92407 92205 92257 92309 9236x 92412 922x0 92262 923x4 92366 92418 92215 92267 92319 92371 92423 92428 92480 92531 92583 92634 92433 92485 92536 92588 92639 92438 92490 92542 92593 92645 92449 92500 92552 92603 92655 92454 92 5o5 92557 92609 92660 92459 92511 92562 92614 92665 92464 92516 92567 92619 92670 92469 9252X 92572 92624 92675 92474 92526 92578 92629 92681 845 846 847 848 849 85o 85i 852 853 854 855 856 857 858 859 92686 92737 92788 92840 92891 92691 92742 92793 92845 92896 92696 92747 92799 92850 92901 92701 92752 92804 92855 92906 92957 93008 93059 93x10 93161 92706 92758 92809 92860 92911 92711 92763 92814 92865 92916 92716 92768 92819 92870 92921 92722 9*773 92824 92875 92927 92727 92778 92829 9288 X 92932 92732 92783 92834 92886 92937 X 2 3 4 5 6 7 8 9 X 2 9294^ 92993 93044 93095 93 1 46 92947 92998 93049 93100 93i5i 92932 93oo3 93o54 93io5 93 1 56 92962 93oi3 93064 931 1 5 93166 92967 93018 93069 93120 93171 92973 93024 93075 93x25 93176 92978 93029 93080 931 3 1 93181 92983 93o34 93o85 93x36 93x86 92988 93039 93090 93i4i 93192 3 3 4 4 5 93197 93247 93298 93349 93399 93202 93252 933o3 93354 93404 93207 93258 93308 93359 93409 93212 93263 933i3 93364 93414 93217 93268 93318 93369 93420 93222 93273 93323 93374 93425 93227 93278 93328 93379 93430 93232 93283 93334 93384 93435 93237 93288 93339 93389 93440 93242 93293 93344 93394 93445 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 93450 935f)o 93551 93601 9365i 93455 935o5 93556 93606 93656 93460 93510 93561 9361 1 93661 93465 935i5 93566 93616 93666 93470 93520 93571 93621 93671 93475 93526 93576 93626 93676 93480 93531 93581 9363 1 93682 93435 93536 93586 93636 93687 93490 9354X 93591 9364 X 93692 93495 93546 93596 93646 93697 93-02 93752 93802 93852 93902 93707 93757 93807 93857 93907 93712 93762 93812 93862 93912 93717 93767 93817 93867 93917 93722 93772 93822 93872 93922 93727 93777 93827 93S77 93927- 93732 93782 93832 93882 93932 93737 93787 93837 93887 93937 93742 93792 93842 93892 93942 93747 93797 93847 93897 93947 93952 94002 94o52 94101 94i5i 93957 94007 94057 94 1 06 94 1 56 93962 94012 94062 941 1 1 94161 93967 94017 94067 941 16 94166 93972 94022 94072 941 2 1 94171 93977 94027 94077 94 1 26 94176 93982 94o32 94082 94i3i 94181 93987 94037 94086 94 1 36 94186 93992 94042 94091 94i4i 9/' 1 91 93997 94047 94096 94x46 94196 i I 2 3 4 5 6 7 8 9 1 I I 875 876 877 878 879 No. 94201 94250 94300 94349 94399 94206 94255 943o5 94354 944o4 1 942 1 1 94260 94310 94359 94409 94216 94265 943 1 5 94364 94414 94221 94270 94320 94369 94419 94226 94275 94325 94374 94424 94231 94280 94330 94379 94429 6 94236 94285 94335 94384 94433 y4240 94290 94340 94389 94438 94245 94295 94345 94394 94443 2 2 3 3 4 2 3 4 5 7 8 9 TABLE XXVI. [P'^ge 183 Logarithms of Numbers. • Nn Q^no CfAQ 0. Log. 9444.8 97313. No. 880 88 r 882 883 884 885 886 887 883 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 9i3 914 915 916 917 918 919 920 921 922 923 924 926 926 927 928 929 980 932 933 934 935 936 937 938 939 No. 1 2 3 4 5 6 7 8 94488 94537 94586 94635 94685 9 94448 94498 94547 94596 94646 94453 945o3 94662 94601 94660 94468 94507 94557 94606 94666 94463 94612 94662 94611 94660 94468 94517 94567 94616 94665 94473 94622 94571 94621 94670 94478 94627 94676 94626 94676 94483 94632 94581 9463o 94680 94493 94542 94591 94640 94689 t 1 2 3 4 5 6 7 8 9 ) I I 2 •> 94694 94743 94792 94841 94890 94699 94748 94797 94846 94895 94704 94753 94802 9485: 94900 94709 94768 94807 94866 94906 94714 94763 94812 94861 94910 94719 94768 94817 94866 94916 94724 94773 94822 94871 94919 94968 96017 96066 96114 96163 94729 94778 94827 94S76 94924 94973 96022 96071 96119 96168 94734 94783 94832 94880 94929 94733 94787 94836 94885 94934 3 3 4 4 5 94939 94988 96036 96085 96134 94944 94993 96041 96090 96139 94949 94998 96046 96096 96143 94954 96002 96061 96100 96148 94969 96007 96066 96106 96163 94963 96012 96061 96109 96168 94978 96027 96076 96124 96173 94983 96032 96080 96129 96177 96182 9623 [ 96279 95328 95376 95424 95472 96621 96669 96617 96187 96236 96284 95332 96381 96192 96240 96289 95337 95386 96197 96245 96294 95342 95390 96202 96260 96299 95347 96396 96207 96266 963o3 96352 96400 96211 96260 96308 95357 96406 96216 96266 963i3 96361 96410 96221 96270 95318 96366 96416 96226 96274 96323 95371 95419 95429 95477 95526 96674 96622 95434 95482 95530 96678 96626 96439 95487 95535 96683 9563 1 96444 96492 96640 96688 95636 95448 95497 95546 96693 96641 95453 96601 96660 96698 96646 96458 96606 96554 96602 96660 96463 96611 95569 96607 95666 96468 96616 96664 96612 96660 96666 96713 96761 96809 96866 96670 96718 96766 96813 96861 96674 96722 96770 96818 96866 96679 96727 96776 96823 96871 96684 96732 96780 96828 96876 96689 96737 96786 96832 96880 96694 96742 96789 96837 96886 96698 96746 96794 96842 96890 96703 96761 96799 96847 96896 96708 96766 96804 96862 96899 96904 96962 96999 96047 96096 96909 96967 96004 96062 96099 96914 95961 96009 96067 96104 96918 96966 96014 96061 96109 96923 96971 96019 96066 96114 96928 96976 96023 96071 961 18 96933 96980 96028 96076 96123 96933 96986 96033 96080 96128 96942 96990 96033 96086 96133 96947 96996 96042 96090 96137 96142 96190 96237 96284 96332 96379" 96426 96473 96620 96667 96147 96194 96242 96289 96336 96384 96431 96478 96626 96672 96162 96199 96246 96294 96341 96166 96204 96261 96298 96346 96161 96209 96266 96303 96350 96166 96213 96261 96308 96365 96171 96218 96266 963 1 3 96360 96176 96223 96270 96317 96365 96180 96227 96275 96322 96369 96186 96232 96280 96327 96374 96388 0643 6 96483 96530 96677 96393 96440 96487 96534 96681 96628 96676 96722 96769 96816 96398 96445 96492 96539 96686 96402 96460 96497 96644 96691 96407 96454 96601 96648 96696 96412 96459 96606 96663 96600 96417 96464 96611 96668 96606 96421 96468 96616 96662 96609 96614 9666 r 96708 96765 96802 96619 96666 96713 96769 96806 96624 96670 96717 96764 96811 96633 96680 96727 96774 96820 96638 96686 96731 96778 96825 96642 96689 96736 96783 96830 96876 96923 96970 97016 97063 96647 96694 96741 96788 96834 96662 96699 96745 96792 96839 96666 96703 96760 96797 96844 I 2 3 4 5 5 7 8 9 i I I 7 96848 96896 96942 96988 97035 96S53 96900 96946 96993 97039 96868 96904 96961 96997 97044 97090 97137 97183 97230 97276 96862 96909 96966 97002 97049 96867 96914 96960 97007 97063 96S72 96918 96966 97011 97068 96881 96928 96974 97021 97067 96886 96932 96979 97025 97072 9fj890 96937 96984 97o3o 97077 2 2 3 3 4 9708 1 97128 97174 97220 97267 97086 97132 97179 97225 97271 97096 97142 97188 97234 97280 97100 97146 97192 97239 97286 97104 97161 97197 97243 97290 97109 97155 97202 97248 97294 971 14 97160 97206 97253 97299 971 18 97165 97211 97267 973o4 97123 97169 97216 97262 97308 1 2 3 4 5 G 7 8 9 ^^sem TABLE XXVI. Logarithms of Numbers. IVf, Oinri lOflOO T.n"' Q7"^ir?- J996. No. 940 941 942 943 944 940 946 947 948 949 950 95 1 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 97 1 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 No. 1 2 3 4 5 6 7 8 9 973 1 3 C7359 974o5 974 5 1 97497 97317 97364 97410 97456 97502 97322 97368 974 1 4 97460 97506 97327 97373 97419 97465 975 1 1 97331 97377 97424 97470 97516 97336 97382 97428 97474 975- 97340 97387 97433 97479 97525 97345 97391 97437 97483 97529 97350 97396 97442 97488 97534 97354 97400 97447 97493 97539 5 1 2 3 4 5 6 7 8 9 I I 2 2 97543 97589 976,35 97681 97727 97548 97594 97640 97685 9773: 97552 97598 97644 97690 97736 97557 97603 97649 97695 97740 97562 97607 97653 97699 97745 97566 ■ 97612 97658 97704 97749 97571 97617 97663 97708 97754 97575 97621 97667 97713 97759 97580 97626 97672 97717 97763 97585 97630 97676 97722 97768 3 3 4 4 5 97772 97818 97864 97909 97955 97777 97823 9786^1 97914 97959 97782 97827 97873 97918 97964 97786 97832 97877 97923 97968 97791 97836 97882 97928 97973 97795 97841 978S6 97932 97978 97800 97845 97891 97937 97982 97804 97850 97896 97941 97987 97809 97855 97900 97946 97991 97813 97859 97903 97950 97996 98000 98046 98091 98137 98182 98005 98050 98096 98141 98186 98009 98055 98100 98146 98191 98014 9S059 98105 98150 98195 98')i9 98 064 98109 98155 98200 98023 98068 98114 98 1 59 98204 98028 98073 98118 98164 98209 98032 9S078 98123 98168 98214 98037 98082 98127 98173 98218 98041 98087 98132 98177 98223 98227 98272 983:8 98363 9840S 98453 98498 9S543 98588 98632 98232 98277 98322 98367 98412 98457 98502 98547 98592 98637 98236 9S2S1 9S327 98372 98417 98241 98286 9833 1 98376 98421 98245 98290 98336 98381 98426 98250 98295 98340 98385 98430 98254 98299 98345 98390 98435 98259 98304 98349 98394 98439 98263 9830S 98354 98399 98444 98268 983 1 3 98353 98403 98448 98462 98507 98552 98597 98641 98466 98511 98556 98601 98646 98471 98516 98561 98605 98650 98475 98520 98565 98610 98655 98480 98525 98570 98614 98659 984S4 98529 98574 986:9 98664 98489 98534 98579 98623 98668 98493 98538 98583 9S628 9S673 98677 9S722 98767 9S811 98856 986S2 98726 98771 98816 98860 9S686 98731 9S776 98820 98865 98691 98735 98780 98825 98869 98695 98740 98784 98829 98874 98700 98744 98789 98834 98878 98704 98749 98793 98838 98883 98709 98753 98798 98843 98887 987:3 98758 98802 98847 98892 98717 98762 98807 98851 98896 98900 9S945 989S9 99034 99078 98905 98949 98994 990 3 8 99083 98909 98954 98998 99043 99087 98914 98958 99003 99047 99092 98918 9S963 99007 99052 99096 98923 98967 99012 99056 99100 98927 98972 99016 9906 1 99105 9S932 98976 99021 99065 99:09 98936 98981 99025 99069 991 14 99 1 58 99202 99247 99291 99335 98941 98985 99029 99074 99118 99123 99167 99211 99f5 99300 99127 991 7 1 992 ID 99260 99304 99i3i 99176 99220 99264 99308 99 1 36 99180 99224 99269 993 1 3 99140 99185 99229 99273 99317 99145 99189 99233 99277 99322 99149 99193 99238 99282 99326 99154 99T98 99242 99286 99330 99162 99207 99251 99295 99339 99344 99388 99432 99476 99520 99348 99392 99436 99480 99524 99352 99396 99441 99484 99528 99357 99401 99445 99489 99533 99361 9940 5 99449 99493 99537 99366 99410 99454 99498 99542 99370 994:4 99458 99502 99546 99374 994:9 99463 99506 99550 99379 99423 99467 995 1 1 99555 99383 99427 99471 995:5 99559 L I 2 3 4 5 6 7 8 _9_ I I 99554 99607 99651 99695 99739 99568 99612 99656 99699 99743 99572 99616 99660 99704 99747 99577 99621 99664 99708 99752 99581 99625 99669 99712 99756 99585 99629 99673 99717 99760 99590 99634 99677 99721 99765 99594 99638 99682 99726 99769 99599 99642 99686 99730 99774 99603 99647 99691 99734 99778 •1 1 3 3 4 99782 99S26 99870 99913 99957 99787 99830 99874 99917 99961 99791 99835 99878 99922 99965 99795 99839 99883 99926 99970 99S00 99843 99887 99930 99974 99804 99848 99891 99935 99978 99808 99852 99896 99939 99983 99813 99856 99900 99944 99987 99817 99861 99904 99948 99991 99822 99865 99909 99952 99996 1 2 3 4 6 7 8 9 TABLE XXVIL [Page 185 Log. Sines, Tangents, and Secants. M o V 179° Hour A.nr Hour P.M. Sine. Diir.i Cosecnul. Tang'cnt. Diff.r Cotangent Secant. Cosine. M 60 12 00 000 Inf. Neg Iiifiniie. Inf. Nog. Iiifinitc. 10.00000 10.00000 I II 59 52 8 6.46373 3oio3 13.53627 6.46373 3oio3 13.53627 00000 00000 59 2 59 44 16 7647C 17609 23524 76476 17609 23524 00000 00000 58 3 59 36 24 9408 5 12494 05915 94o85 1 2494 05915 00000 00000 57 4 ^5 59 28 32 7.06579 9691 12.93421 7.06579 9691 12.93421 00000 00000 56 55 II 59 20 4o 7.1627c 7918 12.83730 7.16270 7918 12.83730 1 . 00000 10.00000 6 59 12 48 24188 6694 75812 24188 6694 75812 00000 00000 54 7 59 4 56 30882 58oo 69118 30882 58oo 69118 00000 00000 53 8 58 56 I 4 36682 5ii5 633 18 36682 5ii5 633 1 8 00000 00000 52 _9 lO 58 48 I 12 41797 4576 58203 41797 4576 58203 OOOOCi 00000 5i 5o II 58 4» I 20 7.46373 4i39 12.53627 7.46373 4i39 12.53627 1 . 00000 10.00000 1 1 58 32 I 28 5o5i2 3779 494SS 5o5i2 3779 49488 00000 00000 49 12 58 24 I 36 54291 3476 45709 54291 3476 45709 00000 00000 48 i3 58 16 I 44 57767 3218 42233 57767 3219 42233 00000 00000 47 i4 i5 58 8 I 52 60985 2997 39015 60986 2996 39014 00000 00000 4b 45 II 58 020 7.639S2 2802 1 2 . 36o 1 8 7.63982 2803 12.36018 10.00000 ic .00000 i6 57 52 2 8 66784 2633 33216 66785 2633 332 1 5 00000 00000 44 17 57 44 2 16 69417 2483 3o583 69418 2482 3o582 0000 i| 9.99999 4i i8 57 36 2 24 7 1 900 2348 28 1 00 71900 2348 28 [ 00 00001 99999 42 !9 so 57 28 2 32 74248 2227 25752 74248 2228 25752 00001 99999 4i 4o II 57 20 2 4o 7.76475 21 19 12.23525 7.76476 2119 12.23524 10.00001 9.99999 21 57 12 2 48 7S594 2021 2 1 4o6 78595 2020 2i4o5 00001 99999 39 22 57 4 2 56 So6i5 1930 19385 806 1 5 1931 19385 0000 1 99999 38 23 56 56 3 4 82545 1 848 17455 82546 1 848 17454 0000 1 99999. 37 24 25 56 48 3 12 84393 1773 1 5607 84394 1773 i56o6 0000 1 99999 3b 35 II 56 4o 3 2() 7.861G6 1704 12. 1 3834 7.86167 1704 I2.I3833 10.00001 9.99999 26 56 32 3 28 87870 1639 I2l3o 87871 1639 12129 0000 1 99999 M 27 56 24 3 36 89509 1579 1 049 1 89510 1579 10490 0000 1 99999 ^.i 2S 56 16 3 44 91088 i524 08912 91089 i524 0891 1 0000 1 99999 32 29 3o 56 8 3 53 92612 1472 07388 92613 1473 07887 00002 99998 3i 3^ II 56 4 " 7.940.84 1424 12.05916 7.94086 1424 12.05914 10.00002 9.99998 3i 55 52 4 8 95508 1 379 04492 95510 1879 04490 00002 99998 29 32 55 44 4 16 96S87 1 336 o3i i3 96889 1 336 o3iii 00002 99998 28 33 55 36 4 24 98223 1297 01777 98225 1297 01775 00002 99998 27 34 35 55 28 4 32 99520 1259 00480 99522 1259 00478 00002 99998 26 25 II 55 20 4 4o 8.00779 1223 II .99221 8.00781 1223 II .99219 10.00002 9.99998 36 55 12 4 48 02002 I 1 90 97998 02004 I 190 97996 00002 99998 24 ^7 55 4 4 56 03192 ii58 96808 o3i94 1 1 59 96806 oooo3 99997 23 38 54 56 5 4 043 5o 1128 9565o 04353 I I 28 95647 oooo3 99997 22 39 4o 54 48 5 12 05478 1 100 94522 o548i I 100 94519 oooo3 99997 21 20 II 54 4o 5 20 8.06578 1072 11.93422 8.o658i 1072 II. 93419 io.oooo3 9.99997 4i 54 32 5 28 07650 1046 92350 07653 1047 92347 oooo3 99997 19 42 54 24 5 36 0S696 1022 9 1 3o4 08700 1022 9i3oo oooo3 99997 18 4i 54 16 5 44 09718 999 90282 09722 998 90278 oooo3 99997 17 44 45 54 8 5 5?. 10717 976 89283 10720 976 955 89280 ii.883o4 ooco4 99996 lb 75 II 54 060 3.11693 954 1 1.88 307 8. 1 1696 10.00004 9.99996 40 53 52 6 8 1 2647 934 87353 i265i 934 87349 00004 99996 i4 47 53 44 6 16 i358i 914 86419 i3585 9.5 864 1 5 00004 99996 i3 48 53 36 6 24 14495 89G 855o5 i45oo 895 855oo 00004 99996 12 49 5o 53 28 6 32 15391 877 84609 II .83732 15395 878 846o5 00004 99996 1 1 10 II 53 20 6 4o 8.16268 860 8.16273 860 11.83727 io.oooo5 9.99995 5i 53 12 6 48 17128 843 82872 17133 843 82867 oooo5 99995 9 52 53 4 6 56 1 797 1 827 82029 17976 828 82024 oooo5 99995 8 53 52 56 7 4 18798 812 81202 18804 812 81196 oooo5 99995 7 54 55 52 48 7 12 1 96 10 797 80390 19616 797 8o384 oooo5 99995 6 5 II 52 4o 7 20 8.20407 782 11.79593 8.2o4i3 782 11.79587 1 . 00006 9.99994 56 52 32 7 28 21189 769 78811 21 195 769 78805 00006 99994 4 57 52 ■j4 7 36 21958 755 78042 21964 750 78036 00006 99994 3 58 52 16 7 44 22713 743 77287 22720 742 77280 00006 99994 2 59 52 8 7 52 23456 73o 76544 23462 73o 76538 00006 99994 I 60 52 8 24186 717 758 1 4 24192 718 75808 00007 99993 M Hour P.M. Hour A.M. Cosine. Diff.l' Secant. | Cotangent Diff.r Tangent. Cosecant. Sine. 90° 80^ 24 Pa ;e 18G] TABLE XXVIL 1° Log. Sines, Tang( mts, and Secants. 178° M o Hour A.M.] Hour P.M. Sine. Diir.l'j Cosecant. 11.75814 Tangent. iDiir.l'l Jotangent Secant. Cosine. M II 52 0; 080 8.24186 717 8.24192 718 11.75808 10.00007 ?• 99993 I 5i 521 8 8 24903 706 75097 24910 706 75090 00007 99993 59 2 5i 44 8 16 25609 695 74391 256i6 696 74384 00007 99993 58 3 5i 36 8 24 263o4 684 73696 263 12 684 73688 00007 99993 37 4 ~5 5i 28 8 3^ 26988 673 73012 26996 678 78004 00008 99992 56 55 It 5 1 20 8 40 8.27661 663 II .72339 8 . 27669 663 II .72331 1 . 00008 Q. 99992 6 5r 12 8 48 28324 653 71676 28332 654 71668 OOG08 99992 1 54 7 5i 4 8 56 28977 644 71023 28986 643 71014 00008 99992 53 8 5o 56 9 4 29621 bM 70379 29629 634 70871 00008 99992 52 _9 lO 5t) 48 9 12 30255 624 69745 3o263 625 69787 000C9 99991 5i 5o 1 1 5o 4o 9 20 8.30879 616 II .69121 8.30S80 617 II .691 12 10.000099.99991 1 1 1 5o 32 9 28 31495 608 685o5 3i5o5 607 68495 00009 99991 49 12 5o 24 9 36 32io3 599 67897 321 12 599 67888 00010 99990 48 i3 5o 16 9 44 32702 590 67298 8271 1 59c 67289 000 10 99990 47 i4 i5 5o 8 9 52 10 33292 583 575 66708 333o2 584 66698 00010 99990 46 45 II 5o 8.33875 II .66125 8.33886 575 1 1 . 66 1 1 4 10.00010 9.99990 i6 49 52 10 8 34450 568 65550 34461 568 65539 0001 1 99989 44 17 49 44 10 16 35oi8 56o 64982 35029 56 1 64971 0001 1 99989 43 i8 49 38 10 24 35578 553 64422 35590 553 644 '0 000 1 1 99989 42 1 20 49 28 10 32 36i3i 547 539 68869 11 .63322 36i43 546 63857 0001 1 99989 4i 4o II 49 20 10 4o 8.36678 8.36689 54o II .633ii 10.00012 9.99988 21 49 12 10 48 37217 533 62783 87229 533 62771 00012 99988 39 22 49 4 10 56 37750 526 62250 37762 527 62288 00012 99988 38 23 48 56 II 4 38276 520 61724 38289 520 61711 000 1 3 99987 37 24 25 48 48 II 12 38796 5i4 61204 38809 5i4 61191 000 1 3 99987 3ti" 35 :i 48 4o 1 1 20 8.39310 5o8 1 1 .60690 8.39323 509 II .60677 io.oooi3 9.99987 26 48 32 II 28 39818 5o2 60182 39882 502 60168 000 1 4 99986 34 27 48 24 II 36 4o320 496 59680 40.334 49^ 59666 00014 99986 33 28 48 16 II 44 40816 491 59184 4o83o 491 59170 000 1 4 999S6 32 29 3o 43 8 II 52 4i3o7 485 58693 4i32i 486 480 58679 000 1 5 io.oooi5 99985 3i 3^ (1 48 12 8.41792 480 1 1 .58208 8.41807 11.58193 9.999S5 3i 47 52 12 8 42272 474 57728 4:2287 475 57713 000 1 5 99985 29 32 47 44 12 16 42746 470 57?54 42762 470 57288 00016 99984 28 33 47 36 12 24 43216 464 56784 43282 464 56768 00016 99984 27 34 35 47 28 12 32 4368o 459 56320 43696 46o 563o4 00016 999S4 26 II 47 20 12 45 36 24 24 72597 237 2-j4o3 72659 237 27341 00062 99938 57 4 "5 35 28 24 32 72834 235 27166 72896 236 27104 00062 99958 56 55 1 1 35 20 24 4o 8.73069 234 II .26931 8.73i32 234 11.26868 10.00063 9.99937 ti 35 12 24 48 733o3 232 26697 73366 234 26634 00064 99936 54 ■ 7 35 4 24 56 73535 232 26465 73600 232 26400 00064 99936 53 i " 34 56 25 4 73767 23o 26233 73832 23l 26168 ooo65 99935 52 9 !0 34 48 25 12 73997 8.74226 229 26003 74o63 229 25937 00066 99934 5i 56 1 1 34 4o 2 5 20 228 II .25774 8.74292 229 II .25708 10.00066 9.99934 1 I 34 32 25 28 74454 226 25546 7452 1 227 25479 00067 99933 49 12 34 24 25 36 74680 226 25320 74748 226 25252 0006S 99932 48 i3 34 16 25 44 74906 224 25094 74974 225 25026 00068 99932 47 i4 i5 34 8 25 52 75i3o 223 24870 75199 224 24801 00069 99931 46 45 II 34 26 8.75353 222 1 1 . 24647 8.75423 222 11.24577 10.000709.99930 iti 33 52 26 8 75575 220 24425 75645 222 24355 00071 99929 44 17 33 44 26 16 75795 220 24205 75867 220 24i33 00071 99929 43 i8 33 36 26 24 76015 219 23985 76087 219 23913 00072 99928 42 19 20 33 28 26 32 76234 217 23766 76306 2iy 23694 00073 99927 4i 40 II 33 20 26 4o 8.76451 216 II .23549 8.76525 217 11.23475 10.00074 9.99926 21 33 12 26 48 76667 216 23333 76742 216 23258 00074 99926 39 22 33 4 26 56 768S3 2l4 23l 17 76958 2l5 2 3o42 00075 99925 38 2j 32 56 27 4 77097 2l3 22900 77173 2l4 22827 00076 99924 37 24 25 32 48 27 12 27 20 773 10 212 22690 7738-j 2l3 22613 00077 99923 36 35 1 1 32 40 8.77522 21 I 1 1 .22478 8 . 77600 211 1 1 .22400 10.00077 9.99923 2fa 32 32 27 28 77733 210 22267 77811 211 22189 00078 99922 34 27 32 24 27 36 77943 209 22057 78022 210 21978 00079 99921 33 28 32 16 27 44 78152 208 2184s 78232 209 21768 00080 99920 32 '9 3o 32 8 27 52 28 78360 208 21640 78441 208 21559 00080 99920 3i 3o II 32 8.78568 206 II .21432 8.78649 206 II .2i35i 10.00081 9.99919 3 1 3i 52 28 8 78774 2o5 21226 78855 206 21145 00082 99918 29 32 3 1 44 28 16 78979 2o4 2I02I 79061 205 20939 oooS3 99917 28 33 3 1 36 28 24 79183 203 20817 79266 204 20734 ooo83 99917 27 35 3i 28 28 32 79386 8.79588 202 201 20614 II .20412 79470 203 2o53o 00084 99916 36 25 II 3 1 20 28 40 8.79673 202 II .20327 1O.O0O85 9.99915 36 3i 12 28 48 79789 201 202 I I 79875 201 20I2D 00086 99914 24 37 3i 4 28 56 79990 199 20010 80076 201 19924 00087 99913 23 38 3o 56 29 4 80189 199 198 I I 80277 199 19723 00087 999x3 22 39 4o 3o 48 29 12 8o388 197 I 96 I 2 80476 198 19524 00088 99912 21 20 n 3o 4o 29 20 8.8o585 197 II .19415 8.80674 198 II .19326 10.00089 9.99911 4i 3o 32 29 28 80782 196 1 92 1 8 80872 196 I9128 00090 99910 19 42 3o 24 29 36 80978 195 19022 81068 196 18932 00091 99909 18 43 3o 16 29 44 81173 194 18827 81264 195 18736 00091 99909 "7 44 45 3o 8 29 52 3o 8 1 367 193 i8533 81459 194 i854i 00092 999f)S 16 75 11 3o 8.8i56o 192 II .18440 8.8i653 193 11.18347 10.00093 9.99907 4b 29 52 3o 8 81752 192 18248 81846 192 i8i54 00094 99906 i4 47 29 44 3o 16 81944 190 i8o56 82o38 192 17962 00095 99905 i3 48 29 36 3o 24 82134 190 17866 82230 190 17770 00096 99904 12 49 5o 29 28 3o 32 82324 8.825i3 189 188 17676 82420 190 17580 00(^96 99904 1 1 10 II 29 20 3o 4o 1 1. 17487 8.82610 189 II .17390 10.00097 9.99903 3[ 29 12 3o 48 82701 187 17299 82799 188 17201 00098 99902 9 52 29 4 3o 56 82888 187 17112 82987 188 17013 00099 99901 8 53 28 56 3i 4 83075 186 16925 83i75 186 16825 00 1 00 99900 7 54 55 28 48 3i 12 83261 8.83446 i85 184 16739 1 1. 16554 8336i 186 16639 OOIOI 99899 9.99898 6 ~5 II 28 4o 3i 20 8.83547 i85 1 1. 16453 10.00102 56 28 32 3i 28 8363o ]83 16370 83732 1 84 16268 00102 99898 4 57 28 24 3i 36 838 1 3 1 83 16187 83916 1 84 16084 ooio3 99S97 3 58 28 16 3i 44 83996 !8l 1 600 4 84 100 182 15900 00104 99S96 2 59 28 8 3i 52 84177 181 i5823 84282 182 15718 ooio5 99S95 I 6o M 28 32 84358 Cosine. 181 DiffT' 1 5642 Secant. 84464 182 15535 00106 99894 M Hour P.M. Hour A.M. Cotangent Diff.l' Tangent. Cosecant. ,, ., Sine. 93° 86" TABLE XXVIL [Page 189 Log. Sines, Tangents, and Secants. 4= '•* 175" 31 Hour a.:m Flour P.M. Sine. Difi-. 1 Cosecant. Tang'ciit. 8.8446/ Diff. 1 Cotanj^cnt Secant. Cosine. M II 28 ( 32 8.84358 181 n . 1 5642 182 1 1. 1 5536 10.00106 9.99894 I 27 5i 32 8 84535 179 1 5461 8464G 180 1 5354 00107 99893 5q 2 27 4 32 16 84718 179 15282 ' 8482G 180 i5i74 00108 99892 58 3 27 3c 32 24 84897 178 i5io3 85oo6 179 14994 00109 99891 57 4 5 27 7b 32 32 85()75 177 14925 85i85 178 i48i5 00109 99891 56 55 I I 27 2t 32 4o 8.85252 •77 1 1. 14748 8.85363 177 11 .14637 10.001 1( 9.99890 6 27 I? 32 48 85429 17b 14571 85540 177 1 4460 001 II 1 99''^89 54 7 27 4 32 56 856o5 175 14395 85717 176 14283 001 12 1 998S8 53 8 26 5( 33 4 85780 175 14220 85393 176 14107 001 1 3 1 99887 52 _9 lO 26 48 33 12 85955 173 i4o45 86069 174 1 3931 001 14 998S6 5i 5^ II 26 4< 33 20 8.86128 173 II . 13872 8.86243 174 li .13757 lo.ooiiS 9.99885 1 1 26 3? 33 28 86301 173 13699 86417 174 13583 00116 99884 49 12 26 24 33 36 86474 171 13526 86591 172 13409 001 17 998S3 48 i3 26 i() 33 44 86645 171 i3355 86763 172 i3237 001 ife 99882 47 i4 i5 26 8 33 52 86816 171 i3i84 86935 171 i3o65 00119 99881 46 45 I I 26 c 34 8.86987 169 II .i3oi3 8.87106 171 1 1. 1 2894 10. 0012c 9.99880 [6 2 5 52 34 8 87156 169 12844 87277 170 12723 00121 99S79 44 17 2 5 44 34 16 87325 169 12675 87447 169 12553 00121 99879 43 iB 2 5 3f) 34 2/( 87494 ib7 i2 5o6 87616 169 12384 00122 99878 42 19 20 95 28 34 32 87661 168 166 12339 11 .12171 87785 168 1 22 1 5 00123 99877 4i 4o II 2D 2u 34 4o 8.87829 8.87953 167 1 1 . 1 2047 10.00124 9.99876 21 25 12 34 48 87995 166 1 2 00 5 88120 167 1 1 880 OOI25 99875 39 22 25 4 34 56 88161 165 1 1 839 88287 166 11713 00126 99874 38 23 24 56 35 4 88326 164 11674 88453 1 65 1 1 547 00127 99873 37 24 25 24 48 35 12 88490 164 ii5io 88618 i65 ii382 00128 99872 36 35 II 24 4o 35 2u 8.8S654 i63 11.11 346 8.88783 i65 II .11217 10.00129 9.99871 26 24 32 35 28 888 1 7 1 63 iii83 88948 i63 I1052 ooi3o 99870 34 27 24 24 35 36 88980 162 11020 891 1 1 1 63 10889 ooi3i 99869 33 28 24 16 35 44 89142 162 io858 89274 ib3 10726 OOl32 99868 32 29 3o 24 8 35 52 89304 160 1 0696 89437 ibi io563 00 1 33 99867 3i 3I; II 24 36 8.89464 161 II .io536 8.8959S 162 II. 10402 10.00134 9 . 99866 3i 23 52 36 8 ■89625 159 10375 89760 160 10240 001 35 99865 29 32 23 44 36 16 89784 1 59 10216 89920 160 1 0080 00 1 36 99864 26 33 2 3 36 36 24 89943 159 10057 90080 160 09920 00137 99863 27 34 35 23 28 36 32 90102 1 58 0989S 90240 .59 09760 00 1 38 99862 26 25 I I 23 20 36 40 8.9026(j 1 57 1 1 . 09740 8.90391^ 1 58 1 1 . 0960 1 10.00139 9.99861 3b 23 12 36 48 90417 1 57 09383 90557 1 58 09443 ooi4o 99860 24 37 23 4 36 56 90574 i5b 09426 90715 i57 09285 ooi4i 99859 23 38 22 5(j 37 4 90730 i55 09270 90872 i57 09128 00142 99858 22 39 4u 22 48 37 12 90885 i55 091 1 5 91029 1 5b 08971 00143 99857 21 20 It 22 4o 37 20 8.91 o4o i55 1 1 .08960 8.91 i85 i55 II .oS8i5 10.00144 9.99856 4r 22 32 37 28 91 195 1 54 oS8o5 91340 i55 0S660 00145 99855 iq 42 22 24 37 36 91349 i5J o865i 91495 1 55 o85o5 00 1 46 99854 18 43 22 lb 37 44 91502 i53 08498 91650 133 o835o 00147 99853 17 44 45 22 8 37 52 91655 l52 08345 91S03 1 54 0S197 00148 99S52 iG 75 U 22 38 8.91807 l52 11 .0S193 8.91957 i53 1 1 .08043 10.00149 9.99851 4b 2[ 52 38 8 91959 i5i o£o4i 921 10 l52 07890 ooi5o 99850 1 4 47 2 1 44 38 16 921 10 i5i 07890 92262 l52 07738 OOl52 99848 i3 48 21 36 38 24 92261 i5o 07739 92414 i5i 07586 001 53 99847 12 49 5o 21 28 38 32 92411 i5o 07589 92565 i5i 07435 001 54 99846 1 1 10 11 21 2U 38 4o 8.92561 1 49 11.07439 8.92716 i5o II .07284 io.ooi55 9.99845 bi 21 12 38 48 92710 149 07290 92866 i5o 07134 00 1 56 99844 9 52 21 4 38 56 92859 1 48 07141 93016 149 06984 00157 99843 8 53 20 56 39 4 93007 147 06993 93i65 1 48 o6835 001 58 99842 7 54 55 20 48 39 12 93 1 54 147 06846 933i3 .49 06687 00159 99841 6 5 II 20 4o c 39 20 8.93301 i47 1 1 . 06699 8.93462 147 1 1. 06538 10.00160 9.99840 56 20 32 39 28 9344s 1 46 o6552 93609 '47 06391 00161 99839 4 57 20 24 39 36 93594 i4b o64o6 9I756 147 06244 00162 99838 3 58 20 16 39 44 93740 i45 06260 9 '903 1 46 06097 00163 99837 2 59- 20 8 39 52 93885 i45 o6ii5 94049 1 4b 05951 00164 99S36 1 60 20 40 94o3o 144 0^970 94195 145 o58o5 00166 99S34 M Hour P.M. [lour A.M. Cosine. DiiT.l' Secant. Cotangent Diff. 1' Tangent. | Cosecant. h'iiic. w 85" p ge 1901 TABLE XXVII. S' Log . Sines, Tangents, and Secants. G'. 5= A A B B C C 174° HourA.M. Hour P.M. Sine, Diif. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. M 60 II 20 00 4o 8 . 94o3o II .05970 8.94195 ii.o58o5 10. 00166 9.99834 I 19 52 4o 8 94174 2 05826 94340 2 o5o6o 00167 99833 59 2 19 44 4o 16 94317 4 05683 94485 4 o55i5 00168 99832 58 J 19 36 4o 24 94461 7 05539 9463o 7 05370 00169 9983 1 57 4 5 19 28 4o 32 94603 9 II 05397 94773 9 05227 00170 99830 56 55 II 19 20 4o 4o 8.94746 ii.o5254 8.94917 II ii.o5o83 10.00171 9.99829 b 19 12 4o 48 94887 i3 o5ii3 95060 i3 04940 00172 99828 54 7 19/ 4o 56 95029 i5 04971 95202 i5 04798 00173 99827 53 8 18 56 4i 4 95170 18 o483o 95344 18 04656 00175 99825 52 _9 10 18 48 4i 12 95310 20 04690 95486 20 o45i4 00176 99824 5i 5o II 18 4o 4i 20 8.95450 32 ii.o455o 8.95627 22 11 .04373 10.00177 9.99823 II 18 32 4i 28 95589 24 044 1 1 95767 24 04233 00178 99822 49 12 ifl ■^i 4i 36 95728 2b 04272 95908 27 04092 00179 99821 48 Ij 18 lb 4i 44 95867 29 o4i33 96047 29 03953 00180 • 99820 47 i4 i5 18 8 4i 52 96005 3i 03995 96187 3i o38i3 00181 99819 46 45 II 18 42 8.96143 33 1 1. 03857 8.96325 33 II .03675 10.00x83 9.99817 i6 17 62 42 8 96280 35 03720 96464 35 o3536 00184 99816 44 17 17 44 42 16 96417 37 03583 96602 38 03398 00 1 85 99815 43 i8 IT 36 42 24 96553 39 03447 96739 40 o326i 00186 99814 42 £9 20 17 28 42 32 96689 42 o33ii 96877 42 o3i23 00187 99813 4i 40 II 17 20 42 4o 8.96825 44 II .o3i75 8.97013 44 I" 02987 10.00188 9.99812 21 17 12 42 48 96960 4b o3o4o 97i5o 4b 0285o 00190 99810 39 2 2 17 4 42 56 97095 48 02905 97285 49 02715 00191 99809 38 2J 16 56 43 4 97229 bo 02771 97421 5i 02579 00 1 92 9980S 37 24 25 16 48 43 12 97363 53 02637 97556 53 02444 00193 99807 36 35 II 16 4o 43 20 8.97496 55 II .025o4 8.97691 55 II .02309 10.00194 9.99806 2b 16 32 43 28 97629 57 02371 97825 58 02175 00196 99804 34 27 16 24 43 36 97762 59 02238 97959 bo 0204 1 00197 99803 33 28 16 16 43 44 97894 bi 02106 98092 b2 01908 00198 99802 32 29 3o 16 8 43 52 98026 b4 01974 98225 b4 "66 01775 00199 99801 3i 3^ II 16 44 8.9815"^ 66 II. 01843 8.98358 II .01642 lo. 00200 9.99800 Ji i5 52 44 8 98288 68 OI7I2 98490 69 oi5io 00202 99798 29 J2 i5 44 44 16 98419 70 oi58i 98622 71 01378 002o3 99797 28 JJ i5 36 44 14 98549 72 oi45i 98753 73 01247 00204 99796 27 J4 35 i5 28 44 32 98679 75 Ol32I 98884 75 01116 002o5 99795 26 25 II 1 5 20 44 4o 8.98808 77 II .01192 8.99015 77 II .00985 10.00207 9.99793 db i5 12 44 48 98937 79 oio63 99145 80 oo855 00208 99792 24 ^7 i5 4 44 56 99066 81 00934 99275 82 00725 00209 99791 23 J8 i4 56 45 4 99194 83 00806 9940 5 84 00595 00210 99790 22 39 40 i4 48 45 12 99322 8b 00678 99534 8b 00466 00212 99788 21 20 II i4 4o 45 20 8.99450 88 II .oo55o 8.99662 89 II .oo338 I0.002l3 9.99787 4i i4 32 45 28 99377 90 00423 99791 9' 00209 00214 99786 19 42 1 4 24 45 36 99704 92 00296 99919 93 00081 002l5 99785 18 43 i4 16 45 44 99S30 94 00170 9.00046 95 10.99954 00217 99783 17 44 45 i4 8 45 52 99956 9b 00044 00174 97 99826 00218 99782 16 l5 II i4 46 9.00082 99 10.99918 9.oo3oi 100 10.99699 10.00219 9.99781 4b i3 52 46 8 00207 lOI 99793 00427 102 99573 00220 99780 i4 47 i3 44 46 16 oo332 io3 99668 oo553 io4 99447 00222 99778 i3 48 i3 36 46 24 oo456 io5 99544 00679 106 99321 00223 99777 12 49 5o i3 28 46 32 oo58i 107 99419 oo8o5 108 99795 00224 99776 II 10 II i3 20 46 40 9 . 00704 no 10.99296 9.00930 III 10.99070 10.00225 9.99775 5i i3 12 46 48 00S28 112 99172 oio55 ii3 98945 00227 99773 9 52 i3 4 46 56 00951 ii4 99049 01179 ii5 98821 00228 99772 8 5J 12 56 47 4 01074 lib 98926 oi3o3 1 1 7 98697 00229 99771 7 t>4 55 12 48 47 12 01 196 118 98804 01427 120 122 98573 0023 1 99769 6 5 II 12 4o 47 20 9.oi3i8 121 10.98682 9.oi55o 10.98450 I0.00232 I 9.99768 5b 12 32 47 28 oi44o 123 98560 01673 124 98327 00233 I 99767 4 'i- 12 24 47 36 oi56i 125 98439 01796 126 98204 0O235 I 99765 3 58 12 16 47 44 01682 127 98318 01918 128 98082 00 2 36 I 99764 2 59 12 8 4i 52 oi8o3 129 98197 02040 i3i 97960 00237 I 99763 I 00 12 48 01923, l32 98077 02162 i33 97838 00239 I 99761 M M Hour P.M. HourA.M. Cosino. ' Diff. Secant. Cotangent Difl-. Tangent. Cosecant. Diff.l Sine. 95° Seconds of time 1» 23 3' 4. 5' 6' 7, Prop, parts of cols. < B (c 16 17 33 33 49 5o 66 66 I 82 83 I 99 100 I ii5 116 I TABLE XXVII. ■ [Fiigc 191 S' Log. Sines, Tan gents, and Secants. G'. G° A A B B C C ITS-^ 31 o Hour A.ai. II 12 Hour p. .M. 48 Sine. 9.01923 Diir. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Co.siue. M (k) 10.98077 9.02162 10.97838 10.00239 9.99761 I II 52 48 8 o3o43 2 97957 02283 2 97717 00240 997D0 J9 2 II 44 48 16 02i63 4 97837 02404 4 97596 00241 99759 58 3 1 1 36 48 24 o;.283 6 97717 02525 6 97475 00243 99757 57 4 5 II 28 48 32 02402 7 97598 02645 8 97355 00244 99756 56 55 II II 20 48 4o 9.02520 9 1 . 97480 9.02766 9 10.97234 10.00245 9.99755 6 II 12 48 48 02639 1 1 97361 02885 1 1 97115 00247 99750 54 7 !I 4 48 56 02757 i3 97243 o3oo5 i3 96995 00248 99-52 53 8 10 56 49 4 02874 i5 97126 o3i24 i5 96876 00249 99751 52 _9 10 10 48 49 12 02992 17 97008 03242 17 96758 0025l 99749 5i 5^ 11 10 4« 49 20 9.03109 19 10.96891 9.03361 19 10.96639 10.00252 9.99748 1 1 10 32 49 28 03226 20 96774 o3479 21 96521 00253 99747 ^9 12 10 24 49 36 03342 22 96658 03597 23 96403 00255 99745 48 i3 10 16 49 44 o3458 24 96542 03714 24 96286 00256 99744 47 i4 i5 10 8 49 52 03574 26 96426 o3832 26 96168 00258 99742 46 45 1 1 10 5o 9.03690 28 10.96310 9.03948 28 10.96052 10.00259 9.99741 i6 9 52 5o 8 o38o5 3o 96195 o4o65 3o 95935 00260 99740 44 17 9 44 5o 16 03920 3i 96080 o4i8i 32 95819 00262 99738 43 i8 9 36 5o 24 o4o34 33 95966 04297 34 95703 00263 99737 42 !9 20 9 28 5o 32 o4i49 35 9585i o44i3 36 95587 00264 99736 41 4o II 9 20 5o 40 9.04262 37 10.95738 9.04528 38 10.95472 10.00266 9.99734 21 Q 12 5o 48 04376 3o 95624 04643 39 95357 00267 99733 39 22 9 4 5o 56 04490 4i 95510 04758 41 95242 00269 99731 3b 23 8 56 5r 4 o46o3 43 95397 04873 43 95127 00270 99730 37 24 25 8 48 5i 12 047 1 5 44 95285 04987 45 950 1 3 00272 99728 3t) 35 II 8 4o 5i 20 9.04828 46 10.95172 9.o5ioi 47 10.94899 10.00273 9.99727 26 8 32 5i 28 04940 48 95060 o52i4 49 94786 00274 99726 34 27 8 24 5i 36 o5o52 5o 94948 05328 01 94672 00276 99724 6i 28 8 16 5i 44 o5i64 5? 94836 o544i 53 94559 00277 99723 32 29 3o 8 8 5i 52 05275 54 94725 05553 54 94447 00279 99721 3i 3^ II 8 52 9.05386 56 1 . 946 1 4 9.05666 56 10.94334 10.00280 9.99720 3i 7 52 52 8 05497 57 945o3 05778 58i 94222 00282 99718 29 32 7 44 52 16 o56o7 59 94393 05890 60 941 10 00283 997 '7 2h 33 7 36 52 24 05717 61 94283 06002 62 93998 00284 99716 27 3i 35 7 28 52 32 05827 63 94173 061 13 64 93887 00286 997 '4 2b II 7 20 52 4o 9.05937 65 10.94063 9.06224 66 10.93776 10.00287 9.99713 36 7 12 52 48 o6o46 67 93954 o6335 68 93665 00289 99711 24 37 7 4 52 56 06 1 55 69 93845 06445 69 93555 00290 99710 23 38 6 56 53 4 06264 70 93736 06556 71 93444 00292 99708 22 39 4o 6 48 53 12 06372 72 93628 06666 73 93334 00293 99707 21 20 II 6 4o 53 20 9.06481 74 10.93519 9.06775 75 10.93225 10.00295 9.99705 4i 6 32 53 28 06589 76 934 1 1 06885 77 931 15 00296 99704 '9 42 6 24 53 36 06696 78 93304 06994 79 93006 00298 99702 18 43 6 16 53 44 06804 8c 93196 07103 81 92897 00299 99;o> 17 44 45 6 8 53 52 0691 1 81 93089 07211 83 92789 oo3oi 99699 lb II 60 54 9.07018 83 10.92982 9.07320 84 1 . 92^)80 io.oo3o2 9.99698 40 5 52 54 8 07124 85 92876 07428 86 92572 oo3o4 99(596 14 47 5 44 54 16 07231 87 92769 07536 88 92464 oo3o5 99695 i3 48 5 36 54 24 07337 89 92663 07643 90 92357 oo3o7 99693 12 49 5o 5 28 54 32 07442 91 92558 07751 92 92249 oo3o8 99692 1 1 10 II 5 20 54 40 9.07548 93 10.92452 9.07858 94 10.92142 io.oo3ic 9.99690 5i 5 12 54 48 07653 94 92347 07964 96 92036 oo3ii 99689 9 52 5 4 54 56 07758 96 92242 08071 98 91929 oo3i3 99687 8 53 4 56 55 4 07863 98 92137 08177 99 91823 oo3i4 99686 7 54 55 4 48 55 12 07968 100 92032 08283 lOI 9'7i7 oo3i6 99684 6 "5 1 1 4 4o 55 20 9.08072 102 10.91928 9.08389 io3 10.9161 1 io.oo3i7 9.99683 56 4 32 55 28 08 1 76 io4 91824 08495 io5 9i5o5 oo3i9 9968 1 4 57 4 24 55 36 08280 loC 91720 08600 107 91400 00320 99680 3 58 4 16 55 44 08383 107 91617 08705 loq 91295 oo32? 99678 2 5q 4 8 55 52 0848G IOC 9i5i4 08S10 II I 91190 oo323 95677 I 60 M 4 56 08589 I II 91411 08914 ii3 91086 oo325 99675 Hourp.M HonrA.Ai Cosine. Diff Secant. Cotangent Diff. Tangent. Cosecant. iDiff Sine .% A A B B C C 83= Seconds of time 1' 2^ 3» 4s 5» G' 83 84 I 7' 97 98 I Prop, parts ot cols. < B (c i4 i4 28 28 42 42 I 56 56 I 69 70 I Page 192] TABLE XXVIL S Log. S] nes, Tangents, and Secants. G'. 7° A A B B C C 172° o H cur A.M. Hour P.M. Sine. Diff. Coserant. Tangent. |Diff. Cotangent Secant. DifT. Cosine. M 63 II 40 56 9.08589 10.91411 9.08914 10.91086 io.oo325 9.99675 I 3 52 56 8 08692 2 9i3o8 09019 2 90981 00326 99674 59 2 3 44 S6 16 08795 3 91205 09120 3 90877 00328 00672 58 J 3 36 56 24 08897 5 91103 09227 5 90773 oo33o c ' 99670 57 4 5 3 28 56 32 08999 6 91001 09330 7 90670 oo33i 99669 56 55 II 3 so 56 4o 9.09101 8 10.90899 9.09434 8 10.90566 10.00333 9.99667 t) 3 12 56 48 09202 10 90798 09537 10 Qo463 00334 99666 54 7 3 4 56 56 09304 II 90696 09640 n 90360 oo336 99664 53 8 2 56 57 4 09405 i3 90595 09742 i3 90258 00337 99663 52 _9 10 2 48 57 12 09506 i4 90494 09845 i5 90155 00339 99661 5i 5o 11 2 40 57 20 9 . 09606 16 10.90394 9.09947 16 1 . 90053 io.oo34i 9.99659 11 2 32 57 28 09707 18 90293 10049 18 89951 00342 j 99658 49 12 2 24 57 36 09807 19 90193 ioi5o 20 89850 oo344 99656 48 iJ 2 16 57 44 09907 21 90093 I0252 21 89748 00345 99655 47 i4 i5 2 8 57 52 10006 22 89994 io353 23 89647 oo347 99653 46 45 II 20 58 9.10106 M 10.89894 9.10454 24 10.89546 10.00349 9.99(151 lb I 52 58 8 I0205 2b 89795 io555 2b 89445 oo35o 99650 44 17 I 44 58 16 io3o4 27 89696 io656 28 89344 oo352 99648 43 i8 I 36 58 24 I0402 29 89598 10756 29 89244 00353 99647 42 19 20 I 28 58 32 io5oi 3o 89499 108 56 3i 89144 00355 99645 4 1 4<. II I 20 58 40 9.10599 32 10.89401 9. 10956 33 10.89044 10.00357 9.99643 21 I 12 58 48 10697 34 89303 iio56 34 88944 oo358 99642 39 22 I 4 58 56 10795 35 89205 iii55 36 88845 oo36o I 99640 38 2j 56 59 4 10893 37 89107 1 1254 37 88746 oo362 99638 37 24 25 48 59 12 10990 38 890 1 ii353 39 88647 oo363 99637 36 35 II 40 59 20 9. II 087 4o 10.88913 9.11452 4i 10.88548 io.oo365 9.99635 2b 32 59 28 1 1 184 42 88816 ii55i 42 88449 00367 99633 34 27 24 59 36 11281 43 88719 1 1 649 44 8835, oo368 99632 33 2b 16 59 44 1,377 45 88623 1 1747 46 88253 00370 99630 32 29 3o 8 59 52 1 1474 46 88526 11845 47 88 1 55 00371 99629 3i 3o II 00 I 9.11570 48 I0.88430 9.11943 49 10.88057 10.00373 9.99627 61 10 59 5? 8 1 1666 5o 88334 1204o 5i 87960 00375 99625 29 62 59 44 16 11761 5i 88239 i2i38 52 87862 00376 99624 28 dJ 59 36 24 11857 53 88143 12235 54 87765 00378 99622 27 34 35 59 28 32 1 1952 54 88o48 12332 55 87668 00380- 99620 26 10 59 20 I 4o 9.12047 56 10.87953 9. 12428 57 10.87572 10.00382 9.99618 3b 59 12 48 12142 58 87858 12525 59 87475 00383 97617 24 ^7 59 ,4 56 12 236 59 87764 I262I 60 87379 oo385 1 996 1 5 33 38 58 56 I 4 i233i 61 87669 I27I7 62 87283 oo387 I 99613 22 39 4o 58 48 I 12 12425 62 87575 I28I3 64 87187 00388 996 1 2 21 20 10 58 4o I I 20 9. 12519 64 10.87481 9. 1 2909 65 10.87091 10.00390 9.99610 4i 58 32 I 28 12612 66 87388 i3oo4 67 86996 00392 99608 19 42 58 24 I 36 12706 67 87294 i3o99 68 86901 00393 99607 18 43 58 16 I 44 12799 69 87201 i3i94 70 86806 00395 99605 '7 44 45 58 8 I 52 12892 70 87108 13289 72 86711 10.86616 00397 99603 lb 75 10 58 I 2 9.12985 72 10.87015 9.13384 73 10.00399 9 . 9960 1 4b 67 52 2 8 13078 74 86922 13478 7^ 86522 oo4oo 99600 i4 47 57 44 2 16 i3i7i 75 86829 13573 77 86427 00402 99598 1 3 48 57 3(i 2 24 1 3263 77 86737 1 3667 78 86333 oo4o4 99596 12 49 5o 57 28 2 32 13355 7a 86645 1 376 1 80 86239 oo4o5 99595 I, 10 10 57 20 I 2 4o 9.13447 80 10.86553' 9-13854 81 10.86146 10.00407 9 99^*93 5i 57 12 2 48 13539 82 8646 1 13948 83 86o52 00409 99591 9 52 57 4 2 55 i363o 83 86370 i4o4i 85 85959 004 11 99589 8 53 56 56 3 4 13722 85 86278 i4i34 86 85866 004 1 2 99588 7 54 55 56 48 3 12 i38i3 87 86187 14327 88 85773 oo4i4 2 99586 6 '~5 ID 56 40 I 3 20 9.13904 88 1 . 86096 9. 14320 90 io.8568o io.oo4i6 2 9.99584 5b 56 32 3 28 13994 90 86006 1 44 1 2 91 85588 004 1 8 2 99582 4 57 56 24 3 36 i4o85 91 85915 l45n4 93 85496 00419 2 9958 1 3 58 56 16 3 44 14175 93 85825 14597 95 854o3 00421 2 99579 2 59 56 8 3 52 14366 95 85734 1 4688 96 853i2 00423 2 99577 I bo M 56 4 14356 96 85644 14780 98 85220 00425 2 99575 M lI(,urp.M. IIOUIA.M. Cosine. Difl-. Secant. Cotangent Dill Tangent. Cosecant. DilT. S-ine. U7° A A B B C c 82" Seconds of time 1- 2' 3' 4. 5« 6» 7. 84 86 I Prop, parts of cols. J B 12 12 24 24 36 37 I 48 I 60 61 I 72 73 I TABLE XXVIL fPage 193 s Log. Sines, Tangents, and Secants. G 8° A A B B c C 171° M Hour A.M. Hourp.M. Sine. Diff. Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. M 6^ 10 56 I 4 9-14356 10. 85644 9.147^^0 I0.85220 10.00425 9.99575 I 55 52 4 8 14445 1 85555 14872 1 85i28 00426 99574 59 2 55 44 4 16 14535 3 85465 14963 3 85o37 00428 99572 58 3 55 36 4 24 14624 4 85376 i5o54 4 84946 oo43o 99570 57 4 5 55 28 4 32 i47i4 6 85286 i5i45 b 84855 00432 99568 56 55 10 55 20 I 4 40 9. i48o3 7 10.85197 9.15236 7 10.84764 10.00434 9.99566 6 55 12 4 48 14891 8 85io9 15327 9 84673 00435 99565 54 7 55 4 4 56 14980 10 85o2o i54i7 10 84583 00437 99563 53 8 54 56 5 4 15069 II 84931 i55o8 12 84492 00439 99561 52 _9 10 54 48 5 12 i5i57 i3 84843 15598 i3 84402 0044 1 99559 5i 5o 10 54 4o I 5 20 9.15245 i4 10.84755 9.15688 i4 10.84312 10.00443 9.99557 II 54 32 5 28 15333 16 84667 15777 16 84223 00444J 09556 49 12 54 24 5 36 1 542 1 17 84579 15867 17 84r33 00446 99554 48 i3 54 16 5 44 i55o8 18 84492 15956 19 84044 00448 99552 47 i4 i5 54 8 5 52 15596 20 844o4 16046 20 83954 oo45o 99550 46 45 10 54 I 6 9.15683 21 10.84317 9.16135 22 10.83865 10.00452 9.99548 i6 53 52 6 8 15770 23 84230 16224 23 83776 00454 99546 44 '7 53 44 6 16 1 5857 24 84i43 i63i2 25 83688 00455 99545 43 i8 53 36 6 24 15944 25 84o56 i64oi 26 83599 00457 99'>43 42 19 20 53 28 6 32 i6o3o 27 28 83970 164S9 27 835ii 00459 99541 4i 4o 10 53 20 I 6 4o 9.16116 10. 83884 9.16577 29 10.83423 10.00461 9.99539 21 53 12 6 48 16203 3o 83797 16665 3o 83335 oo463 99537 39 22 53 4 6 56 162891 3i 83711 16753 32 83247 oo465 99535 38 23 52 56 7 4 16374 32 83626 i684r 33 83 1 59 00467 99533 37 24 25 52 48 7 12 1 6460 34 35 83540 10.83455 16928 35 83072 00468 99532 36 35 10 52 4o I 7 20 9.16545 9.17016 36 10.82984 10.004-0 9.99530 26 52 32 7 28 1 6631 37 83369 17103 37 82897 00472 99528 34 27 52 24 7 36 16716 38 83284 17190 39 82810 0047-' 99526 33 28 52 16 7 44 1 680 1 39 83i99 17277 40 82723 00476 99524 32 29 3o 52 8 7 52 16886 4i 83ii4 17363 42 82637 00478 99522 3i 3o 10 52 X 8 9.16970 42 io.83o3o 9.17450 43 10.82550 1 0.00480 9.99520 3i 5i 52 8 8 17055 44 82945 17536 45 82464 00482 99518 29 32 5t A4 8 16 17139 45 82861 17622 46 82378 00483 99317 28 33 5i 36 8 24 17223 47 82777 17708 48 82292 0048 5 995 1 5 27 34 35 5 1 28 8 32 17307 48 82693 17794 49 82206 00487 995 1 3 26 25 10 5i 20 I 8 4o 9.17I391 49 10.82609 9. 17880 5o 10.82120 10.00489 9.9951 1 3o 5[ 12 8 48 17474 5i 82526 17965 52 82035 00491 99309 24 37 5i 4 8 56 17558 52 82442 i8o5i 53 81949 00493 99507 23 38 5o 56 9 4 17641 54 82359 i8i36 55 81864 00495 995o5 22 39 5o 48 9 12 17724 55 82276 18221 5b 81779 00497 995o3 21 20 4o 10 5o 4o I 9 20 9.17807 56 10.82193 9.18306 53 10.81694 10.00499 9.99501 41 5o 32 9 28 17890 58 82110 18391 59 81609 oo5or 99499 19 42 5o 24 9 36 17973 59 82027 18475 bi 8i525 oo5o3 99497 18 43 5o 16 9 44 i8o55 61 8.945 i856o 62 8i44o oo5o5 99495 17 44 45 5o 8 9 52 i8i37 62 8 1 863 18644 63 81 356 oo5o6 99494 16 75 10 5o 11 I 10 9. 1S220 63 10.81780 9.18728 65 10.81272 io.oo5o8 9 • 99492 46 49 52 10 8 i83o2 65 81698 18812 66 81188 oo5io 99490 i4 47 49 44 10 16 18383 66 81617 18896 68 81104 ■ 005l2 99488 i3 48 49 36 10 24 18465 68 8i535 18979 69 81021 oo5i4 2 99486 12 49 5o 49 28 10 49 20 10 32 18547 69 81453 19063 71 80937 oo5i6 2 99484 11 10 I 10 40 9.18628 71 10.81372 9.19146 72 10.80854 io.oo5i8 2 9.99482 5i 49 12 10 48 18709 72 81291 19229 74 80771 00520 2 99480 9 52 49 4 10 56 18790 73 81210 19312 75 80688 Oo52 2 2 99478 8 53 48 56 II 4 1 887 1 75 81 129 19395 76 8o6o5 oo524 2 99476 7 .'>4 5"5 48 48 1 1 12 18952 76 81048 19478 7B 8o52 2 00526 2 99474 6 10 48 40 I 11 20 9. 19033 78 10.80967 9. 19561 79 10.80439 10.00528 2 9.99472 56 48 32 ■ II 28 191 r3 79 80887 19643 81 80357 oo53o 2 99470 4 !)7 48 24 II 36 19193 80 80807 19725 82 80275 oo532 2 99468 3 58 48 16 II 44 19273 82 80727 19807 84 80193 oo534 2 99466 9 59 48 8 II 52 19353 83 80647 19889 85 801 11 oo536 2 99464 I 60 M 48 12 19433 85 80567 1 997 1 87 80029 oo538 2 99462 Hour P.M. Hour A.M. Cosine. Diff. Secant. Cotann-entlDlff. Tangent. Cosecant. Diff. Sine. 9t>"^ C 81°' Seconds of time 1- 2' 3" 32 32 I 4- 42 43 I 5' 53 5.^ I 63 65 I 7" 74 76 a Prop parts of cols < B f C II II 21 22 25 Paye 191T TABLE . XXVIL S'. Log. Sines, Tan gents, and Secants. QK 9° A A B B C C 170° M Hour A.M. Hour P.M. Sine. Diff. Cosecant. Tang'cnt. Uiff. Cotangent Secant. Uiff. Cosine. M 60 10 48 1 12 9.19433 10.80567 9.19971 10.80029 io.oo538 9.99462 I 47 52 12 8 19513 I 80487 2oo53 I 799.17 oo54o 99460 59 2 47 44 12 16 19592 3 8o4o8 20 1 34 3 79866 oo542 99458 58 3 Ai 36 12 24 19672 4 80328 20216 4 79784 oo544 99456 57 4 5 47 28 12 32 1 975 1 5 80249 20297 5 79708 oo546 99454 56 55 10 47 20 I 12 4o 9.19830 6 10.80170 9.20378 6 10.79622 io.oo54S 9.99452 6 47 12 12 48 19909 8 80091 20459 8 79541 oo55o 99450 54 7 47 4 12 56 19988 9 80012 20 540 9 79460 oo552 99448 53 8 A^ 56 i3 4 20067 10 79933 20621 10 79879 oo554 99446 52 _9 10 46 48 i3 12 20145 II 79855 20701 12 79299 oo556 99444 5i 5^ 10 46 4 99438 48 i3 46 16 i3 AA 20458 16 79543 21022 17 78978 oo564 99436 47 i4 i5 46 8 i3 52 2o535 18 79465 21102 18 78S98 oo566 99434 46 45 10 4G I i4 9.20613 19 10.79387 9.21182 19 10.78818 10.00568 9.99482 i6 45 52 i4 8 20691 20 79309 2 1 261 21 78789 00571 99429 AA 17 45 A4 i4 16 2076S 21 79232 2i34i 22 78659 00578 99427 Ai i8 45 36 i4 24 20845 23 79155 21420 23 78580 00575 99425 42 £9 20 45 28 i4 32 20922 24 79078 21499 25 78501 00577 99428 Ai 4() 10 45 20 I 1 4 40 9.20999 25 10.79001 9.21578 26 10.78422 10.00579 9.99421 21 45 12 i4 48 21076 26 78924 21657 27 78343 oo5Si 99419 39 22 45 4 i4 56 2ii53 28 78847 21736 28 78264 oo583 99417 38 23 AA 56 i5 4 21229 29 78771 21814 3o 78186 oo585 994 1 5 37 24 25 AA 48 i5 12 2i3o6 3o 78694 21893 3i 78107 00587 994 1 3 36 35 10 AA 40| I i5 20 9.21382 3i 10.78618 9.21971 32 10.78029 10.00589 9.9941 1 26 AA 32 i5 28 2i458 33 78542 22049 34 77951 00591 99409 M 27 44 24 i5 36 2 1 534 34 78466 22127 35 77873 00593 99407 66 28 AA 16 i5 44 21610 3b 78390 222o5 36 77795 00596 99404 32 29 3o 44 8 i5 52 2 1 635 37 783i5 22283 38 77717 00598 99402 3i 3o 10 44 I 16 9.21761 38 10.78289 9.22361 39 10.77689 10.00600 9.99400 3i 43 52 • 16 8 2 1 836 39 78 164 22433 40 77562 00602 99898 29 32 43 44 16 16 21912 4o 7808S 225l6 4i 77484 00604 99896 28 33 43 36 16 24 21987 42 78013 22593 43 77407 00606 99894 27 34 35 43 28 16 32 22062 43 77938 22670 44 77880 00608 99J92 26 25 10 43 20 I 16 4f cois. < Q 1 (c 8 8 16 16 ! 23 24 I 3i ?2 I 39 40 2 47 4o 3 54 57 a "~ TABLE XXVIL [Page 107 SI. Log . Sines, Tan gents, and Secants. at. 12= A A B B C C 167° o Hour A.M. Hour P.M. Sine. Diff. Cosecant. Tangent. Diff. Cotaiij^enl Secant. Difl.. Cosine. M 60 to 24 I 36 9.31788 10.68212 9.32747 10.67253 10.00960 9 . 99040 I 23 52 36 8 3 1847 I 68 1 53 32810 I 67 1 90 00962 99o38 r)9 ? 23 4i 36 16 31907 2 68093 32872 2 67128 00965 99035 58 3 23 36 36 24 31966 3 68034 32933 3 67067 00968 99032 57 4 5 23 28 36 32 32025 4 67975 32995 4 67005 00970 99o3o 06 55 10 23 20 I 36 40 9.320S4 5 10.67916 9.33057 5 10.66943 10.00973 9.99027 6 23 12 36 48 32143 6 67857 33119 6 66881 o7o 2 9.98930 4i 18 32 4i 28 341 56 39 65844 35229 4i 64771 01073 2 98927 19 42 18 54 41 36 34212 4o 65788 35288 42 64712 01076 2 98924 18 43 18 16 4 1 44 34268 4. 65732 35347 4-i 64653 01079 2 98921 17 44 45 ]8 8 4i 52 34324 42 65676 354o5 44 64595 01 08 1 2 98919 lb l5 n 18 1 42 9.34380 43 10.65620 9.35464 45 10.64536 10.01084 2 9.98916 46 17 52 42 8 34436 44 65564 35523 46 64477 01087 2 98913 14 47 17 44 42 j6 34491 45 655o9 35581 47 64419 01090 2 98910 i3 48 17 36 42 24 34547 46 65453 35640 48 64360 01093 2 98907 12 49 5ci 17 28 42 32 34602 9.34658 47 6539S 35698 49 t)0 643o2 1 096 2 ■ 98904 11 10 uj 17 20 I 42 4o 48 10.65342 9.35757 10.64243 10.01099 2 9.98901 5i 17 12 42 48 3471 3 48 65287 358i5 5i 64i85 01 102 2 98898 9 52 17 4 42 56 34769 49 6523i 35873 52 64127 01 io4 2 98896 8 53 16 56 43 4 34824 5o 65176 35931 53 64069 01 107 2 98893 7 54 55 16 48 43 12 34879 5i 65i2i 35989 54 64oi 1 OHIO 3 98890 b ~5 10 16 4o I 43 20 9.3io3/ 52 io.65o66 9.36047 55 10.63953 10.01 1 i3 3 9.98S87 56 16 32 43 28 34989 53 65oii 36io5 56 63895 01 1 16 3 98884 4 57 16 24 43 36 35o4 54 64956 36i63 57 63837 1 1 1 9 3 98881 3 58 16 16 43 44 3509c 55 64901 36221 58 63779 01122 3 98878 2 59 16 8 43 52 35i54 1 56 64846 36279 59 63721 01 125 3 98875 I 60 16 44 35209 1 57 64791 36336 60 63664 01 128 3 98872 c M Hour p.M Hour .\.:m. Cosine. iDiff .Secant. Colans^ent Diff Tansronl. Cosecant. |Diff. Sine. 102= A A B B C Seconds of time !• 2' 3» 4* 5' 6' 7- Prop, parts of cols. { C 7 i4 i5 1 21 22 I 29 3o I 36 37 2 43 45 2 52 3 C 77« Page 198] TABLE XXVIL 5' Log. Sines, Tangents, and Secants. G 13 A A B B C C 166° M o Hour A.M. Hour P.M. Sine. Diff. Cosecant. Tangent. Dlff. Cotangcn-. Secant. Diff. Cosine. M 6^ 10 16 I 4i 9.35209 10.64791 9.36336 io.63f'd4 10. 0x128 9.98872 I i5 52 44 8 35263 I 64737 36394 I 636o6 oxi3i 98869 59 2 i5 44 44 16 353i8 2 64682 36452 2 63548 01x33 98867 58 3 i5 36 44 24 35373 3 64627 365o9 3 63491 oxi36 98864 57 4 5 i5 28 44 32 35427 4 64573 36566 4 63434 01 139 98861 56 55 10 i5 20 I 44 4o 9.35481 4 10.64519 9.36624 5 10.63376 10.01142 9.9885s 6 i5 12 44 48 35536 5 64464 3668 1 6 633 1 9 01145 98S55 54 7 i5 4 44 56 35590 6 644 K) 36738 6 63262 oii48 98852 53 8 i4 56 45 4 35644 7 64356 36795 7 632o5 01X01 98849 52 _9 lO i4 48 45 12 35698 8 643o2 10.64248 36852 8 63 1 48 01x54 98846 5i 5^ 10 I 4 40 I 45 20 9.35752 9 9.36909 9 10.63091 X0.01157 9.98843 II i4 32 45 28 358o6 10 64194 36966 10 63o34 01 160 98S40 49 12 i4 24 45 36 35860 II 64i4o 37023 ir 62977 oii63 98837 48 i3 i4 16 45 44 35914 11 64086 37080 12 62920 01 166 98834 47 i4 i5 i4 8 45 52 35968 12 64o32 37137 i3 62863 01169 98831 9.98828 46 45 10 i4 I 46 9.36022 i3 10.63978 9.37193 i4 10.62807 X0.01172 i6 i3 52 46 8 36075 i4 63925 i5 62750 01x75 98825 44 17 i3 44 46 16 36129 i5 63871 37306 lO 62694 01178 98822 43 i8 i3 36 46 24 36182 16 638 1 8 37553' 17 62637 0X181 988x9 42 12 20 i3 28 46 32 36236 9.36289 17 18" 63764 10.63711 37419 18 62581 on 84 98816 4i 4o 10 I 3 20 I 46 4o 9.37476 IQ 10.62524 10.01187 9.9S813 21 i3 12 46 48 36342 18 63658 37532 IQ 62468 01190 988x0 39 22 i3 4 46 56 36395 19 636o5 37588 20 62412 01193 98807 38 23 12 56 47 4 36449 20 6355i 37644 21 62356 01 196 9S804 37 24 25 12 48 4? 12 365o2 21 63498 37700 22 62300 01199 98S01 36 35 10 12 4o I 47 20 9.36555 22 10.63445 9.37756 23 10.62244 I0.012U2 9.98798 26 12 32 47 28 366o8 23 63302 37812 24 62188 0X205 9S795 34 27 12 24 47 36 36660 24 63340 37868 25 62X32 OI20S 98792 33 28 12 16 47 44 367x3 25 63287 37924 26 62076 012X X 98789 32 29 3o 12 8 4i 52 36766 25 63234 io.63i8i 37980 27 62020 1 2 1 4 987S6 3i 3^ 10 12 I 48 9.36819 9.38o35 28 10.61965 XO.OI217 2 9.98783 3i II 52 48 8 36871 27 63i29 38091 29 61909 01220 2 98780 29 32 II 44 48 16 36924 28 63076 38i47 3o 6x853 0X223 2 98777 28 33 II 36 48 24 36976 29 63024 38202 3i 61798 01226 2 98774 27 34 35 II 28 48 3 a 37028 3o 62972 38257 32 32 61743 0x229 2 98771 26 25 10 II 20 I 48 4o 9.37081 3. 10.62919 9.383x3 10.61687 XO. 01232 2 9.98768 36 II 12 48 48 37133 32 62867 38368 33 6x632 0X235 2 98765 24 37 II 4 48 56 37185 32 62815 38423 34 6x577 OI238 2 98762 23 38 10 56 49 4 37237 ii 62763 38479 35 6x521 0x241 2 98759 22 39 40 10 48 49 12 37289 9.3734. 34 62711 38534 36 6x466 0x244 2 98756 21 20 10 10 40 I 49 2n 35 10.62659 9.38589 37 io.6i4i I 10.0x247 2 9.98753 4i 10 32 49 28 37393 3b 62607 38644 33 6x356 OX25o 2 9S750 19 42 10 24 49 36 37445 37 62555 38699 39 6x3oi 01254 2 98746 x8 43 10 16 49 A4 37497 38 625o3 38754 4o 6x246 0x257 2 98743 17 44 45 10 8 49 52 37549 39 62451 388o8 4i 6x192 01260 2 98740 16 i5 ID 10 I 5o 9.37600 39 10.62400 9.38863 42 10.61137 10. 01263 2 9.98737 46 9 52 5o 8 37652 40 62348 38918 43 6x082 01266 2 98734 i4 47 9 44 5o 16 37703 4i 62297 38972 44 6x028 01269 2 9S731 i3 48 9 36 5o 24 37755 42 62245 39027 45 60973 0x272 2 98728 12 49 5o 9 28 5o 32 37806 43 62194 39082 45 46 609x8 X 0.60864 01275 2 98755 1 1 10 10 9 20 I 5o 4o 9.37858 44 10.62142 9.39136 10.0x278 3 9.9S722 5i 9 12 5o 48 37909 45 62091 39190 47 60810 0x281 3 98719 9 52 9 4 5o 56 3796<:) 4 9.39677 io.6o323 I0.oi3io 9.98690 I 7 52 52 8 384 18 I 6i582 39731 1 60269 oi3i3 98687 59 2 7 44 52 16 33469 2 6i53i 39785 2 6021 5 oi3i6 98684 58 3 7 36 5a 24 385i9 2 61481 3y838 3 60162 oi3i9 9S881 57 4 5 7 28 52 32 3S570 3 6i43o io.6i38u 3989? 3 60108 01822 98678 5b 55 10 7 20 I 52 40 9.38620 4 9.39945 4 io.6oo55 10 01825 9.98675 6 7 12 52 48 38670 5 6i33o 39999 5 60001 01829 9867 1 54 7 7 A 52 56 38721 6 61279 4oo52 6 59948 oi332 98668 53 8 6 56 53 4 38771 7 61229 4o 1 06 7 59894 01 335 98665 52 _9 lO 6 48 53 12 3882 1 7 61179 40159 8 59841 oi338 98662 5i 5^ 10 6 4o I 53 20 9.38871 8 10.61 129 9.40212 9 10.5978S io.oi34i 9.98659 II 6 32 53 28 38921 9 61079 40266 10 59734 01 344 98656 49 12 6 24 53 36 38971 U) 61029 4o3i9 10 59681 01348 9S652 48 i3 6 16 53 44 39021 11 60979 40372 1 1 59628 oi35i 98649 47 i4 i5 6 8 53 52 39071 1 1 60929 40425 12 59575 oi354 10.01357 98646 46 45 10 6 I 54 9.39121 12 10.60879 9.4047^ i3 10.59522 9.98643 i6 5 52 54 8 39170 i3 6oS3o 4o53i i4 59469 oi36o 9S640 44 17 5 44 54 i6 39220 .i4 60780 4o584 i5 59416 01 364 98686 43 i8 5 36 54 24 39270 n 60730 4o636 16 59364 oi367 98633 42 19 20 5 28 54 32 39319 i5 16 60681 40689 17 59311 01870 98630 4i 4o 10 5 20 I 54 4o 9.39369 io.6o63i 9.40742 17 10.59258 10.01373 9.98627 21 5 12 54 48 39418 17 6o582 40795 18 59205 01877 98623 39 22 5 4 54 56 39467 18 6o533 40847 19 59153 oi38o 98620 38 23 4 56 55 4 39517 >9 6048 3 40900 20 59100 oi383 98617 37 24 25 4 48 55 12 39566 20 60434 4095^ 21 59048 01 386 98614 3b 35 10 4 4o I 55 20 9.39615 20 io.6o385 9.4 lOOD 22 10.58995 10.01390 9.98610 26 4 32 55 28 39664 21 6o336 4io57 23 58943 01893 98607 34 27 4 24 55 36 39718 22 602S7 41109 23 5889. 01896 98604 6d 28 4 16 55 44 39762 23 60288 41161 24 58S39 01899 2 9860 1 32 29 3o 4 8 55 52 3981 1 24 60189 4i2i4 20 58786 oi4o3 2 q85j7 3i 3o 10 4 I 56 9.39860 24 io.6oi4o 9.41266 26 10.5S734 io.oi4o6 2 9.98594 3i 3 52 56 8 39900 25 60091 4i3i8 27 58682 01409 2 98591 29 32 3 44 56 16 3995s 26 6oo4a 41370 28 58630 0l4l2 2 98588 28 33 3 36 56 24 40006 27 59994 41422 29 58578 oi4i6 2 98584 27 34 35 3 28 56 32 4oo55 28 59945 4i474 3o 58526 01419 2 98581 2b 15 10 3 20 I 5t) 4o 9.40103 29 10.59897 9.41526 3u 10.58474 10.01422 2 9.9S578 36 3 12 56 48 4oi52 29 59848 4.578 di 5S422 01426 2 98574 24 37 3 4 56 56 40200 3o 59800 41629 32 58371 01429 2 9857. 23 38 2 56 57 4 40249 01 59751 41681 di , 5S3i9 01432 2 98568 22 39 4o 2 48 57 12 40297 32 59703 41733 M 58267 01435 2 98565 21 20 10 2 4'> I 57 20 9.40346 33 10.59654 9.41784 35 10.58216 10.01439 2 9.98561 4i 2 32 57 28 4039^ 33 59606 4i836 30 58 164 01442 2 98558 19 42 2 2.4 57 36 40442 34 59558 41887 3b 58ii3 01445 2 98555 18 43 2 i6 57 44 40490 35 59510 41939 37 58o6i 01449 2 q855i 17 44 45 2 8 57 52 4o538 36 59462 41990 3S 5So 1 01452 2 9S548 lb i5 10 2 I 58 9.4o586 37 10.59414 9.42041 39 10.57959 10.01455 2 9.98545 46 52 58 8 4o634 37 59366 42093 40 57907 01459 3 9854. 14 47 44 58 16 40682 38 59318 42144 4i 57856 01462 3 98538 i3 48 36 58 24 40730 3q 59270 42195 42 57805 01465 3 98535 12 49 5o 28 58 32 40778 9.40S25 4o 59222 42246 43 57754 01469 3 98531 11 10 10 I 20 1 58 4o 4i 10.59175 9.42297 4i 10.57703 10.01472 3 9.98528 5i 12 53 48 40873 42 59127 42348 4 a 57652 01475 3 98525 9 52 4 58 56 4092 1 42 59079 42399 45 57601 01479 3 98521 8 53 56 59 4 40968 43 59032 42450 46 57550 01482 3 98518 7 54 55 48 59 12 4ioi6 44 58984 42501 47 57499 OI485 3 9851 5 6 5 10 4<> I 59 20 9.41 o63 45 10.58937 9.42552 48 10.57448 10.01489 3 9.98511 56 32 59 28 4i 1 1 1 46 58889 42603 49 57397 01492 3 98508 4 57 24 59 36 4ii58 46 58842 42653 bo 57347 01495 3 985o5 3 58 16 59 44 4i2o5 47 58795 42704 5o 57296 01499 3 98501 2 59 8 59 52 4l252 48 58748 42755 5i 57245 01302 3 98498 I 60 M 200 4l3or 49 58700 Secant. 42805 52 57195 1 5o6 3 98494 M Hour f.M. Hour A.M. Cosine. Dili'. Cotangent DitT. Tangent. Cosecant. DifT. Sine. 104" 75" Seconds of time 1' 2' 3' 18 20 I 4» 24 26 2 5' 3i 33 2 6' 37 39 2 7- 43 46 3 Prop, parts of cols. ^ B (c 6 7 12 i3 I Page 200] TABLE XXVII S' Log. Sines, Tangents, and Secants. G . 15° A A B B C C 164° M Hour A. M . Hour p. M ) 2 Sine. iDiff Cosecant Tangent. Diff Cotangent Secant. Diff Cosine. M 60 10 o c 9.4i3oc 10.58700 9.4280^ 10.57195 10.01 5oe 9 . 98494 I 9 59 5: 8 4i34' I 58653 4285e I 57144 01509' 98491 59 2 59 4^ 16 4139/ 2 586o6 42906 2 57094 Ol5l2 98488 58 3 59 3t ) 24 4i44i 2 58559 4295- 2 57043 oi5ie 9S484 57 4 59 28| 32 5 9 69 20| 3 4o 4i48S 9.41535 3 585x2 43007 3 56993 oi5i9 98481 56 55 4 10. 58465 9.4305- 4 10.56943 IO.OI523 9.98477 6 59 12 : 48 4i582 5 584 1 8 43 1 08 5 56892 oi52G 98474 54 7 59 A 1 56 41628 5 58372 43 1 58 6 56842 01529 98471 53 8 58 5e I 4 41675 6 58325 4320& 7 56792 oi533 98467 52 9 58 48 I 12 41722 7 58278 43258 7 56742 oi536 I 98464 5i 5o 10 9 58 4t 2 I 20 9.41768 8 10.58232 9.43308 8 10.56692 io.oi54( I 9.98460 II 58 32 I 28 4i8i5 8 58 1 85 43358 9 56642 01543 I 98457 49 12 58 24 I 36 41861 9 58i39 43408 10 56592 01547 I 98453 48 i3 58 16 I 44 4190& 10 58092 43458 II 56542 oi55o- I 98450 47 i4 58 & I 52 41954 II 58o46 43508 1 1 56492 oi553 98447 46 45 i5 9 58 220 9.42001 II 10.57999 9.43558 12 10.56442 10. 01557 9.98443 16 57 52 2 8 42047 12 57953 43607 i3 56393 oi56o 98440 44 17 57 44 2 16 42093 i3 57907 43657 i4 56343 01 564 98436 43 18 57 36 2 24 42140 14 57860 43707 i5 56293 01567 98433 41 19 57 28 2 32 42186 14 57814 43756 16 56244 01571 98429 4i 4o 20 9 57 20 2 2 4o 9.42232 i5 10.57768 9.43806 16 10.56194 10.01574 9.98426 21 57 12 2 48 42278 16 57722 43855 17 56i45 01578 98422 3q 22 57 4 2 56 42324 17 57676 43905 18 56095 oi58i 98419 38 23 56 56 3 4 42370 17 57630 43954 19 56046 oi585 98415 37 24 56 48 3 12 42416 18 57584 44oo4 20 55996 01 588 98412 36 35 25 9 56 4o 2 3 20 9.42461 19 10.57539 9.44053 20 10.55947 10.01591 9.98409 26 56 32 3 28 42507 20 57493 44102 21 55898 01595 2 98405 M 27 56 24 3 36 42553 21 57447 44i5i 22 55849 01598 2 98402 33 28 56 16 3 44 42599 21 57401 44201 23 55799 01602 2 98398 32 29 56 8 3 52 42644 22 57356 44 2 5o 24 55750 oi6o5 2 98395 3i 3^ 3o 9 56 240 9.42690 23 10.57310 9.44299 25 10.55701 10.01609 2 9.98391 3i 55 52 4 8 42735 24 57265 44348 25 55652 01612 2 98388 29 32 55 44 4 16 42781 24 57219 44397 26 556o3 01616 2 98384 28 33 55 36 4 24 42826 25 57174 44446 27 55554 1 6 1 9 2 98381 27 34 55 28 4 32 42872 26 57128 44495 28 555o5 01623 2 98377 26 35 9 55 20 2 4 4(1 9.42917 27 10.57083 9.44544 29 10.55456 10.01627 2 9.98373 36 55 12 4 48 42962 27 57088 44592 29 55408 oi63o 2 98370 24 37 55 4 4 56 43008 28 56992 44641 3o 55359 01634 2 98366 23 38 54 56 5 4 43o53 29 56947 44690 3t 553io 01637 2 98363 22 39 54 48 5 12 43098 9.43143 3o 56902 44738 32 55262 01641 2 98359 21 20 4o 9 54 40 2 5 20 3o 10.56857 9-44787 33 I0.552I3 10.01644 2 9.98356 4i 54 32 5 28 43 1 88 3i 568 1 2 44836 34 55 164 oi648 2 98352 19 42 54 24 5 36 43233 32 56767 44884 M 55ii6 oi65i 2 98349 18 43 54 16 5 44 43278 33 56722 44933 35 55067 01655 3 98345 17 44 54 8 5 52 43323 M 56677 44981 36 55019 oi658 3 98342 16 i5 45 9 54 260 9.43367 34 10. 56633 9.45029 37 10.54971 10.01662 3 9.98338 46 53 52 6 8 43412 35 56588 45078 38 54922 01666 3 98334 i4 47 53 44 6 16 43457 36 56543 45126 38 54874 01669 3 98331 i3 48 53 36 6 24 435o2 36 56498 45174 39 54826 01673 3 98327 12 49 53 28 6 32 43546 37 56454 45222 4o 54778 1 676 3 98324 II lO 5o 9 53 20 2 6 4o 9.43591 38 10.56409 9.45271 4i 10.54729 10.01680 3 9.98320 5i 53 12 6 48 43635 39 56365 45319 42 54681 01683 3 98817 9 52 53 4 6 56 4365o 39 1 56320 45367 43 54633 01687 3 9831 3 g 53 52 56 7 4 43724 4o 56276 454 1 5 43 54585 01691 3 98309 7 54 52 48 7 12 43769 4i 5623 1 45463 9.45511 44 ■45 54537 01694 3 98306 6 5 55 9 52 40 2 7 20 9.438i3 42 10.56187 10.54489 10.01698 3 9.98302 56 52 32 7 28 43857 43 56 1 43 45559 46 54441 01 70 1 3 98299 4 57 52 24 7 36 43901 A3 56099 456o6 An 54394 01705 3 9829^. 3 58 52 16 7 44 43946 44 56o54 45654 4i 54346 01709 3 98291 2 59 52 8 7 52 43990 45 56oio 45702 48 54298 01712 3 98288 I 60 52 8 44o34 46 55966 45750 49 54250 01716 4 98284 M M Hour P.M. Hour A.M. Cosine. Difi-. Secant. Cotangent Diff. Tangent. | Cosecant. )iff. Sine. 1U5" A A B B C Seconds of time 1» 2» 3' 4. 5' 6» 7' Frop. parts cf cols. \ C 6 6 II 12 I 17 18 I 23 25 2 28 3i 2 34 37 3 40 43 3 C 74° TABLE XXVIL [Page yOl ^ Log . Sines, Tangents, and Secants. G'. 16-^ A A B B c C 163° M o HourA.JI. Hour P.M. Sine. Ditr. Coseraiit. Tangent. Ditr. Colanf^cnt Secant. Ditr. Cosine. M 60 9 62 280 9.44034 10.55966 9.45750 10.5425(1 10.01716 9.98284 I 5i 52 8 8 44078 I 55922 45797 I 54203 01719 98281 59 2 5i 44 8 16 44122 I 55878 45845 2 54 1 55 01723 98277 58 3 5 1 36 8 24 44166 2 55834 45892 2 54108 01727 98273 57 4 5 5i 28 8 32 44210 3 55790 45940 3 54060 01730 98270 56 55 9 5i 20 2 8 4o 9.44253 4 10.55747 9.45987 4 io.54oi3 10.01734 9.98266 6 5i 12 8 48 44297 4 55703 46o35 5 53965 01738 98262 54 7 5r 4 8 56 44341 5 55659 46082 5 53918 01741 98259 53 8 5o 56 9 4 44385 6 556 1 5 46 1 3o b 53870 01745 98255 52 _? 10 5o 48 9 12 44428 6 55572 46177 7 53823 01749 98251 5i 5o 9 5o 4o 2 9 20 9.44472 7 10.55528 9.46224 8 10.53776 10.01752 9.9824s 1 1 5o 32 9 28 445 16 8 55484 46271 9 53729 01756 98244 49 12 5o 24 9 36 44559 9 55441 463 1 9 9 53681 01760 98240 48 i3 5o 16 9 44 44602 9 5539S 46366 10 53634 01763 98237 47 i4 i5 5o 8 9 52 44646 10 55354 464 1 3 II 53587 01767 98233 4b 45 9 5o 2 10 9.44689 II io.553i 1 9.46460 12 10.53540 10.01771 9.98229 1 6 49 52 10 8 44733 II 55267 465o7 12 53493 01774 98226 44 I? 49 44 10 16 44776 12 55224 46554 i3 53446 01778 98223 43 i8 49 36 10 24 44819 i3 55i8i 46601 i4 53399 01782 98218 42 !9 20 49 28 10 32 44862 i4 55i38 46648 i5 53352 01785 98215 4i 4o 9 49 20 2 10 40 9.44905 14 10.55095 9.46694 i5 io.533o6 10.01789 9.9S211 21 49 12 10 48 44948 i5 55o52 46741 lb 53259 01793 98207 39 2 2 49 4 10 56 44992 16 55oo8 46788 17 532 12 01796 98204 38 23 48 56 II 4 45o35 16 54965 46835 18 53i65 01800 98200 37 24 25 48 48 II 12 45077 17 54923 4688 1 19 53 II 9 01804 98196 3b 35 9 48 4o 2 II 20 9.45120 18 10.54880 9.46928 19 10.53072 I 0.0 I 80S 2 9.98192 26 48 32 II 28 45i63 18 54837 46975 20 53025 01811 2 98189 M 27 48 24 II 36 45206 19 5479i 47021 21 52979 oi8i5 2 98185 33 28 48 16 II 44 45249 20 54751 47068 22 5293: 01819 2 98181 32 29 3o 48 8 II 52 45292 21 5470S 10.54666 47114 22 52886 01823 2 98177 3i 3^ 9 48 2 12 9.45334 21 9.47160 23 io.5284i> 10.01826 2 9.98174 3i 47 52 12 8 45377 22 54623 47207 24 52793 oi83o 2 98170 29 32 47 44 12 16 45419 23 54581 47253 25 52747 01834 2 98166 28 33 47 36 12 24 45462 23 54538 47299 2fa 52701 oi838 2 98162 27 34 35 47 28 12 32 455o4 24 54496 47346 26 52654 01841 2 98 1 59 2b 9 47 20 2 12 4o 9.45547 25 10.54453 9.47392 27 10.52608 10.01843 2 9.98155 36 47 12 12 48 45589 26 544 1 1 47438 28 52563 01849 2 98151 24 37 47 4 12 56 45632 26 54368 47484 29 525i6 01 853 2 98147 23 38 46 56 i3 4 45674 27 54326 47530 29 52470 01856 2 98144 22 39 4o 46 48 i3 12 45716 28 54284 47576 3o 52424 1 860 2 98140 21 20 9 46 4o 2 i3 20 9.45758 28 10.54242 9.47622 3i 10.52378 10.01864 2 9.98136 4i 46 32 i3 28 458oi 29 54199 47668 32 52332 01868 3 98132 '9 42 46 24 i3 36 45843 3o 54157 47714 32 52286 01871 3 98129 18 43 46 16 i3 44 45885 3i 54ii5 47760 33 52240 01875 3 98125 17 44 45 46 8 i3 52 45927 3i 54073 47806 34 52194 01879 3 98121 16 75 9 46 2 i4 " 9.45969 32 io.54o3i 9.47852 35 10.52148 10.01883 3 9 98117 46 45 52 i4 8 4601 1 33 53989 47897 36 52io3 01887 3 98113 i4 47 45 44 i4 16 46o53 33 53947 47943 36 52057 01890 3 98 1 1 i3 48 45 36 i4 24 46095 34 53905 47989 37 5201 1 01894 3 98 1 06 12 49 5o 45 28 i4 32 461 36 35 53864 48o35 38 51965 01898 3 98102 1 1 10 9 45 20 2 i4 4o 9.46178 36 10.53822 9.48080 39 10.51920 10.01902 3 9 . 98098 5i 45 12 i4 48 46220 36 53780 48126 39 51874 01906 3 98094 9 52 45 4 i4 56 46262 37 53738 48171 40 51829 01910 3 98090 8 53 44 56 i5 4 463o3 38 53697 48217 41 51783 01913 3 98087 7 54 55 44 48 i5 12 46345 38 53655 48262 42 5 1 738 01917 3 98083 6 5 9 44 4o 2 i5 20 9.46386 io.536i4 9.48307 43 10.51693 10.01921 3 9.98079 56 44 32 i5 28 46428 4o 53572 48353 43 5 1 647 01925 3 98075 4 ^7 44 24 i5 36 46469 4i 5353 1 48398 44 5 1 602 01929 4 9807; 3 58 44 16 i5 44 465 1 1 4i 53489 48443 45 5i557 01933 4 98067 2 59 44 8 i5 52 46552 42 53448 48489 46 5i5ii 01937 4 98063 I 60 M 44 16 46594 43 53406 48534 46 5 1 466 1 940 4 98060 M Hour P.M. Hour A.M. Cosino. Diff. Secant. Cotangent|Dlfl". Tan!:^cnt. Cosecant. Diff. Sine. 106° A A B B C 1* 2" 3' 4. 5" 6' 32 35 3 7' 37 4i 3 Prop, parts oT cols. (■ 5 6 11 12 16 17 1 21 23 2 27 29 2 2G Page 202] TABLE XXVIL S' Log. S nes, Tangents, and Secants. G'. 17 A A B B c C 162° M HOUFA.AI. Hour P.M. Sine. Diff Cosecant. Tangent. Diff. Cotangent io.5i466 Secant. Diff. Cosine. M 60 9 44 c 2 16 9.46594 10.53406 9-48534 10.01940 9 . 98060 I 43 52 16 8 46635 I 53365 48579 I 5i42i 01944 9S056 59 2 43 44 16 16 46676 I 53324 48624 I 51376 01948 9S052 58 3 43 36 16 24 46717 2 53283 48669 2 5i33i 01932 98048 57 4 5 43 28 16 02 46758 3 53242 48714 3 51286 01956 9S044 56 55 9 43 20 2 16 4o 9.46800 3 10.53200 9-48759 4 io.5i24i 10.01960 9.980^0 6 43 12 16 48 4684! 4 53 1 59 48S04 4 51196 01964 98036 54 7 43 4 16 56 46882 5 53ii8 48849 5 5ii5i 01968 9S032 53 b 42 56 17 4 46923 b 53077 4S894 6 5 1 106 01971 98029 52 _9 10 42 48 17 12 20 46964 b 53o36 48939 7 5 106 1 io.5ioi6 01975 9S025 5i 5o 9 42 4o 2 17 9.47005 7 10.52995 9.48984 7 10.01979 9.98021 II 42 32 17 28 47045 7 52955 49029 8 50971 01983 98017 49 12 42 24 J7 36 47086 8 52914 49073 9 50927 019S7 98013 48 i3 42 16 17 44 47127 9 52873 491 18 10 50882 01991 98000 47 i4 i5 42 8 17 D2 47168 9 52832 49163 10 5o837 01995 98005 46 45 9 42 2 18 9.47209 10 10.52791 9.49207 1 1 10.50793 10.01 999 9.98001 i6 4i 52 18 8 47249 II 52751 49252 12 5074s 02003 97997 44 17 4i 44 i» 16 47290 II 52710 49296 12 50704 r20O7 97993 4i i8 41 36 18 24 47330 12 52670 49341 i3 5o659 02011 97989 42 !9 20 4i 28 18 32 47371 i3 52629 49385 i4 5o6i5 io.5o570 020 (4 97986 4i 40 9 4i 20 2 18 40 9.47411 i3 10.52589 9.49430 i5 10.02018 9.97982 21 4i 12 18 48 47452 i4 52548 49474 i5 5o526 02022 97978 39 22 4i 4 18 56 47492 i5 525o8 ■49519 16 5o48i 02026 97974 38 23 40 56 19 4 47533 i5 52467 49563 17 50437 02030 2 97970 37 24 25 4o 48 19 12 47373 16 52427 49607 9.49652 18 18 50393 io.5o348 O2o34 2 97966 9.979(32 36 35 9 4o 4o 2 19 20 9.47613 17 10. 52387 io.o2o38 2 26 4o 32 19 28 47654 17 52346 49696 19 5o3o4 02042 2 97958 M 27 4o 24 '9 36 47694 18 523o6 49740 20 50260 02046 2 97954 36 28 4o 16 19 44 47734 19 5226b 49784 21 5o2i6 02o5o 2 97950 32 29 3o 40 8 19 52 47774 19 52226 49828 21 50172 o2o54 2 979^6 3t 3o 9 40 2 20 9 . 478 1 4 20 10.52186 9.49872 22 10.50128 10.02058 2 9.97942 3i 39 52 20 8 47854 21 52146 49916 23 5oo84 02062 2 97938 29 32 39 44 20 16 47894 21 52106 49960 24 5oo4o 02066 2 97934 28 33 39 36 20 24 47934 22 52066 5ooo4 24 49996 02070 2 97930 27 'I 35 39 28 20 32 47974 23 52026 5oo48 25 49952 02074 2 97926 2b 9 39 20 2 20 4o 9.48014 23 10.5198b 9 . 50092 26 10.49908 10.02078 2 9.97922 '36 39 12 20 48 48o54 24 51940 5oi36 26 49864 02082 2 97918 24 ^7 39 4 20 5b 48094 25 5 1 906 5oi8o 27 49S20 02086 2 979 '4 23 38 38 56 21 4 48 1 33 25 51867 5o233 28 49777 02090 3 97910 22 39 4o 38 48 21 (2 48173 26 51827 50267 29 29 49733 10.49689 02094 3 97900 21 20 9 38 4o 2 21 20 9.48213 27 10.517S7 9.5o3i 1 10.02098 3 9.97902 4i 33 32 21 28 48252 27 5 1 748 5o355 3o 49645 02102 3 97S98 19 4a 33 24 21 36 48292 28 51708 50398 3i 49602 02106 3 97894 18 43 33 16 21 44 48332 29 5 1 668 5o442 32 49558 02II0 3 97890 17 44 45 38 8 21 52 4837i 39 51629 5o485 32 495 1 5 021 14 3 3 97886 9.97S82 lb Is 9 38 u 2 22 9.4841 1 3o io.5i589 9.50529 33 10.49471 10.02118 46 37 52 22 8 4845o 3i 5i55o 5o572 34 49428 02122 3 97878 i4 47 37 44 22 .6 48490 3. 5i5io 50616 35 49384 02126 3 97874 i3 48 37 36 22 24 48529 32 5i47i 5o659 35 49341 02l3o 3 97870 12 49 5o 37 3-8 22 32 48568 33 5i432 5o7o3 36 49297 02 1 34 3 97S66 II 10 9 37 20 2 22 40 9.48607 33 10.51393 9 . 50746 37 10.49254 10.02139 3 9.97861 5i 37 12 22 48 48647 34 5i353 50789 37 49211 02143 3 97857 9 'j2 37 4 22 56 48686 35 5[3i4 5o833 38 49167 02147 3 97853 8 J3 36 56 23 4 48725 35 51275 50876 39 49124 021 5 1 4 9784Q 7 54 55 36 40 23 [2 48764 36 37 5 1 236 50919 40 49081 10.49038 02i55 4 97845 b "5 9 36 4o 2 23 20 9.48803 1 . 5 1 1 97 9.50962 4o 10.02159 4 9.97841 5b 36 32 23 28 48842 37 5ii58 5ioo5' 4i 48995 02i63 4 97837 4 57 36 24 23 36 48881 38 5i 1 19 5io48 42 48952 02167 4 97833 3 58 36 16 23 44 48920 39 5 1 080 51092 43 48908 02171 4 97839 2 59 36 8 23 52 48959 39 5io4i 5ii35 43 48865 02175 4 97825 I 60 36 24 48998 4o 5 1 002 51178 4i 48822 02179 4 9782! M flour P.M. Hour A ..■\i. Cosine. Diir. Secant. Cotangent Din. Tangent. Cosecant. Diff. Sine. 107° 7'^ Seconds of time . 1» 2» 3^ 4s 5^ 6» 7* r IC 5 10 i5 20 25 5o 35 Prop, pnrts of cols 6 11 17 22 28 33 39 I I 2 2 3 3 TABLE XXVIL [Page 203 S' Log. Sines, Tangents, and Secants. Gi. 18 A A B B C C 161° M D Hour A.M. Hour I'.r.i. Slue. 9.48998 DiflT. Ccsecaut. I0.5l002 Tan°;oiit. Diir. Cotangent Secant. Diff. Cosine. 9 36 2 24 u 9.51 178 10.48822 10.02179 9.97S21 I 35 52 24 8 49037 I 50963 5l22I I 48779 02i83 97S17 5q 2 35 44 24 16 49076 I 50924 5 1 264 1 48736 02188 97812 58 3 35 3(1 24 24 491 15 2 5o885 5i3o6i 2 48694 02192 97808 57 4 5 35 28 24 32 49153 3 5o847 5 1 349 3 48651 02196 97804 56 55 9 35 20 2 24 40 9.49192 3 10.50808 9.51 392 3 10.48608 10.02200 9.97800 b 35 12 24 48 49231 4 50769 5i435 4 48565 02204 97796 54 7 35 A 24 56 49269 4 5073 1 51478 5 48522 02208 9779' 53 8 34 56 25 4 49308 5 50692 5i52o b 48480 02212 97788 52 _9 10 34 4« 25 12 49347 9.49385 6 6 5o653 io.5o6i5 5 1 563 b 48437 02216 97784 5i 5^ 9 34 4o 2 25 20 9.51 606 7 10.48394 10.02221 9-97779 97775 II 34 32 25 28 49424 7 50576 5i648 8 48352 02225 49 12 34 24 2 5 36 49462 8 5o538 51691 8 483o9 02229 97771 48 fci 34 16 25 44 49500 8 5o5oo 51734 9 48266 02233 97767 47 i4 i5 54 8 25 52 49539 9 5o46i 51776 10 48224 02237 97763 46 45 9 34 2 26 9.49577 9 io.5o42 3 9.51819 10 10.48181 10.02241 9.97759 lb 33 52 26 8 49615 10 5o385 5i86i 1 1 48139 02246 97754 4i I? 33 44 26 16 49654 II 5o346 51903 12 48097 0225o 97750 43 i8 33 36 26 24 49692 II 5o3o8 51946 i3 48o54 02254 97746 42 £9 20 33 28 26 32 49730 12 50270 51988 i3 48012 02258 97742 4i 4o 9 33 20 2 26 4" 9.49768 i3 10.5o232 9.52o3i ■ 4 10.47969 10.02262 9-97738 21 33 12 26 48 49806 i3 50194 52073 i5 47927 02266 97734 39 32 33 4 26 56 49844 i4 5oi56 52ii5 i5 47885 G2271 2 97729 38 2j 32 56 27. 4 49S82 i4 5oii8 52157 16 47843 02275 2 97725 37 24 25 32 48 27 12 49920 i5 5oo8o 52200 17 47800 02279 2 97721 36 35 9 32 4o 2 27 20 9.49958 16 io.5oo42 9.52242 17 10.47758 10.02 283 2 9.97717 2b 32 32 27 28 49996 lb 5ooo4 52284 18 47716 02287 2 97713 34 27 32 24 27 36 5oo34 17 49966 52326 19 47674 02292 2 97708 33 28 32 16 27 44 50072 18 49928 52368 20 47632 02296 2 977"4 32 29 3o 32 8 27 52 5oiio 18 49890 52410 20 47590 023oo 2 97700 3i 3^ 9 32 2 28 9.50148 19 10.49852 9.52452 21 10.47548 io.023o4 2 9.97696 di 3i 52 28 8 5oi85 20 49S15 52494 22 475o6 02309 2 97691 29 62 3i 44 • 28 16 50223 20 49777 52536 22 47464 023i3 2 97687 28 di 3 1 36 28 74 5o26i 21 49739 52578 23 47422 02317 2 97683 27 35 3i 28 28 32 50298 21 49702 52620 24 473S0 02321 2 97679 26 25 9 3i 20 2 28 4" 9.5o336 22 I . 49664 9.52661 24 10.47339 10.02326 2 9.97674 cib 3i 12 28 48 5o374 23 49626 52703 25 47297 o233o 3 97670 24 ^7 3i 4 28 56 5o4i 1 23 495S9 52745 2b 47255 02334 3 97666 23 ciS 3o 56 29 4 5o449 24 49551 52787 27 472 1 3 02338 3 97662 22 39 4o 3o 48 29 12 5o486 25 25 49514 52829 27 47171 02343 3 97657 ?.I 20 9 3o 4^1 2 29 20 9.5o523 10.49477 9.52S70 28 .o.47i3o 10.02347 3 9.97653 4i 3o 32 29 28 5o56i 26 49439 52912 29 4708S 0235i 3 97649 19 42 3o 24 29 36 5059S 26 49402 52953 29 47047 02355 3 97645 18 43 3o 16 29 44 5o635 27 49365 52995 3o 470o5 0236o 3 97640 17 44 45 3o 8 29 52 50673 28 49327 53o37 3i 46963 02364 3 97636 16 i5 9 3o 2 3o 9.50710 28 10.49290 9.53078 3 1 10.46922 1 0.02 368 3 9-97632 4b 29 52 3o 8 50747 29 49253 53 120 32 46880 02372 3 976.^8 i4 47 29 44 3o 16 50784 3o 49216 53i6i .33 46839 02377 3 97623 i3 48 29 36 3o 24 5o82i 3o 49179 53202 34 46798 0238i 3 97619 12 49 5o 29 28 3o 32 5o858 9.50896 3i 3i 49142 10.49104 53244 -i-i 46756 02385 3 97615 II 10 9 29 20 2 3o 4" 9.53285 10.46715 10.02390 4 9. 97(1 10 5i 29 12 3o 48 50933 32 49067 53327 i6 46673 02394 4 97606 Q 52 29 4 3o 56 50970 33 49o3o 53368 i6 46632 0239S 4 976J2 8 53 28 5(i 3i 4 51007 3 J 48993 53409 37 46591 o24o3 4 97597 7 64 55 28 48 3( 12 5 1043 34 48957 5345o 38 38 4655o io.465oS 02407 4 97'J93 S 9 28 40 2 3i 20 9.510S0 3') 10.48920 9.53492 10.0241 1 4 9.97589 5b 28 32 3i 28 51117 35 48883 53533 39 46467 02416 4 97584 4 !)7 28 24 3i 36 5ii54 36 48846 53574 40 46426 02420 4 97580 3 58 28 16 3i 44 5i 191 37 4S809 536i5 4i 46385 02424 4 97576 T 59 28 8 3i 52 5l227 37 4S773 53656 41 46344 02429 4 97571 T bo M 28 32 c 51264 38 48736 Secant. 53697 42 463o3 02433 4 97567 M Hour P.M. Hour A.M. Cosine. Difr. Cotangeut DifT. Tangent. Cosecant. DilT. Sine. 108° A A B B C Seconds of time . 1' 2' 3^ i4 16 2 4s 19 21 2 5" 24 26 3 0= 28 3i 3 7" 33 37 4 Prop, parts of cols 5 5 I 9 10 I 71" Page 204] TABLE XXVII SI • Log. Sines, Tangents, and Secants. G'. 19 ° A A B B C C 160° M o Hour A. M Hour P.M. Sine. Dlfl- Cosecant. Tangent. Dirt'. Cotangent Secant. [DifT. Cosine. 9.97567 M 60 9 28 c 2 32 9.5126.^ 10.48736 9.53697 io.463o3 10.02433 I 27 52 32 8 5i3oi I 48699 53738 I 46262 02437 97563 5q 2 27 44 32 61 5i33S I 48602 53779 I 46221 02442 97558 58 J 27 3t 32 24 5 1 374 2 4S626 53820 2 461S0, 02446 97554 57 4 5 27 2& 32 32 5i4ii 2 48589 5386i 3 46 1 39 02450 97550 56 55 9 27 2C 2 32 4o 9-5i447 3 10.48553 9.53902 3 10*46098 10.02455 9-97545 b 27 12 32 48 5 1 484 4 485i6 53943 4 46o57 02459 97541 54 7 27 4 32 56 5i52c 4 48480 53984 5 46016 02464 97536 53 8 26 5(j 33 4 5i557 5 48443 54o25 5 45975 02468 97532 52 _9 10 26 48 33 12 51593 5 48407 54o65 6 45935 02472 97528 9.97523 5i 5o 9 26 4o 2 33 20 9.51629 6 10.48371 9.54106 7 10.45S94 10.02477 II 26 32 33 28 5 1 666 7 48334 54i47 7 45853 02481 97519 4q 12 26 24 33 36 51702 7 48298 54187 8 458i3 02485 975 1 5 48 iJ 26 16 33 44 5 1 738 8 48262 54228 9 45772 02490 97510 47 i4 i5 26 8 33 52 51774 8 48226 54269 9 45731 02494 97506 46 45 926 2 34 9.51811 9 10.48189 9.54309 10 10.45691 10.02499 9.97501 lb 25 52 34 8 5 1 847 10 48i53 54350 II 45650 025o3 97497 44 17 25 44 34 16 5i883 10 48117 54390 II 45610 025o8 97492 43 i8 25 36 34 24 51919 II 48081 54431 12 45569 025l2 97488 42 12 20 25 28 34 32 51955 II 48045 54471 i3 45529 025i6 97484 4i 4o 9 25 20 2 34 4o 9.51991 12 10.48009 9.54512 i3 10.45488 10.02521 9-97479 21 25 12 34 48 52027 12 47973 54552 i4 45448 02525 2 97475 3q 22 25 4 34 56 52063 i3 47937 54593 i5 45407 0253o 2 97470 38 2j 24 56 35 4 52099 14 4790 ' 54633 i5 45367 02534 2 97466 37 24 25 24 48 35 12 52x35 14 47865 54673 16 45327 02539 2 97461 36 35 9 24 4o 2 35 20 9.52171! i5 10.47829 9.54714 17 10.45286 10.02543 2 9-9/457 2b 24 32 35 28 52207I 1 5 47793 54754 17 45246 02547 2 97453 34 2? 24 24 35 36 52242 lb 47758 54794 18 45206 02552 2 97448 33 28 24 16 35 44 52278 17 47722 54835 19 45i65 02556 2 97444 32 29 3o 24 8 35 52 523i4 17 47686 54875 19 45i25 o256i 2 97439 3i 3^ 9 24 2 36 9.52350 18 10.47650 9.54915 20 io.45o85 10.02565 2 9-97435 Si 23 52 36 8 52385 18 4-6 1 5 54955 21 45o45 02570 2 97430 29 J2 23 44 36 16 52421 19 47579 54995 21 45oo5 02574 2 97426 28 JJ 23 36 36 24 52456 20 47544 55o35 22 44965 02579 2 97421 27 34 35 23 28 36 32 52492 20 47508 55075 23 44925 02583 3 97417 26 l5 9 23 20 2 36 4o 9.52527 21 10.47473 9.551 i5 23 10. 44885 10.02588 3 9.97412 6b 23 12 36 48 52563 21 47437 55i55 24 44845 02592 3 97408 24 ^7 23 4 36 56 52598 22 47402 55195 25 448o5 02597 3 974o3 23 38 22 56 37 4 52634 23 47366 55235 25 44765 02601 3 97399 22 39 4o 22 48 37 12 52669 23 47331 55275 26 44725 02606 3 97394 21 20 9 22 4o 2 37 20 9.52705 24 10.47295 9.553i5 27 10. 44685 10.02610 3 9.97390 41 22 32 37 28 52740 24 47260 55355 27 44645 02615 3 97385 IQ 42 22 24 37 36 52775 25 47225 55395 28 446o5 02619 3 97381 18 43 22 16 37 44 528 II 2b 47189 55434 29 44566 02624 3 97376 17 44 45 22 8 37 52 52846 2b 47154 55474 29 44526 02628 3 97372 16 i5 9 22 2 38 9.52881 27 10.47119 9.555i4 3o 10.44486 10.02633 3 9.97367 4b 21 52 38 8 52916 27 47084 55554 3i 44446 02637 3 97363 i4 47 21 44 38 16 52951 28 47049 55593 3i 44407 02642 3 97358 i3 4b 21 36 38 24 52986 29 47014 55633 32 44367 02647 4 97353 12 49 5o 21 28 38 32 53o2i 29 46979 55673 33 44327 o265i 4 4 97349 II 10 9 21 20 2 38 4o 9.53o56 3o 10.46944 9.55712 33 10.4428S 10. 02656 9-97344 5i 21 12 38 48 53092 3o 46908 55752 34 44248 02660 4 97340 9 52 21 4 58 56 53126 3i 46874 55791 35 44209 02665 4 97335 ti 53 20 56 39 4 53i6i 32 46839 5583i 35 44169 02669 4 97331 7 64 55 20 48 39 12 53196 32 46804 55870 36 44 i3o 02674 4 97326 6 5 9 20 4o 2 39 20 9.53231 33 10.46769 9.55910 37 10.44090 10.02678 4 9.97322 5b 20 32 39 28 53266 33 46734 55949 37 44o5i 02683 4 97317 4 i>7 20 24 39 36 533oi M 46699 55989 38 44011 02688 4 97312 3 58 20 16 09 44 53336 M 46664 56028 39 43972 02692 4 97308 2 59 20 8 39 52 53370 35 4663o 56067 39 43933 02697 4 973o3 I bo 20 4o 534o5 3b 46595 56107 4o 43893 02701 4 97299 M M Hour p.ir. HourA.M. Cosine. Ditr. Secant. Cotangent Dilf. Tangent. Cosecant. Ditr. Sine. 109= A A B B C 1' 2» 3' 4. 5» 6' 7- Prop, parts of cols. (• 4 5 I 9 10 I i3 i5 2 18 20 3 22 25 3 27 3o 3 3i 35 4 lOP ^~"" TABLE XXVII. [Page 205 S'. Loot . Sines, Tangents, and Secants. Q'. 20= A A B B C C 159° IM IIourA.M. Hour P.M. Sine. Diir. Cosecant. Tangent. DifT. Cotangent Secant. DifT.; Cosine. M 6^ 59 9 20 19 52 19 44 19 36 2 40 9.53405 G 10.46595 9.56107 10.43893 10.02701 9.97299 I 4o 8 53440 I 46560 56i46 I 43854 02706 97294 9 4o 16 53475 I 46525 56i85 I 438i5 02711 97289 58 57 56 55 54 3 40 24 53509 2 46491 56224 2 43776 02715 97285 4 5 19 28 40 32 53544 2 46456 56264 6 3 43736 10.43697 02720 972S0 9 19 20 2 4o 40 9.53578 3 10.46422 9.563o3 10.02724 9.97276 fi 19 12 ly 4 40 48 536i3 3 46387 56342 4 43658 02729 9771 7 40 56 53647 4 46353 5638 1 4 43619 02734 97266 53 52 iS 18 56 4i 4 53682 5 463 1 8 56420 5 4358o 02738 97262 _9 10 18 48 4i 12 53716 5 46284 56459 6 43541 02743 97257 9.97252 5i 5^ 9 18 40 2 4i 20 9.53751 6 10.46240 9.56498 6 10.43502 10.02748 1 1 18 32 4i 28 53785 6 4621^ 56537 7 43463 02752 97241 49 1 L> 18 24 4 1 36 53819 7 46t8i 56576 8 43424 02757 97243 46 i3 18 16 4i 44 53854 7 46 1 46 566 1 5 8 43385 02762 97238 47 i4 i5 18 8 4t 52 53888 8 461 12 56654 9 43346 02766 97234 46 45 44 43 42 lb 2 42 9.53922 8 10.46078 9.56693 IQ 10.43307 10.02771 9.97229 iG 17 52 42 8 53957 9 46043 56732 10 43268 02776 97224 17 17 44 42 16 53991 10 46009 56771 II 43229 02780 97220 i8 17 36 42 24 54025 10 45975 568 10 12 43190 02785 97215 12 20 17 28 42 32 54059 II 45941 56849 12 43i5i 02790 97210 41 4o 39 9 17 20 2 42 4» 9.54093 II 10.45907 9.56887 i3 io.43ii3 10.02794 2 9.97206 21 17 12 42 48 54127 12 45873 56926 iJ 43074 02799 2 97201 22 17 4 42 56 54161 12 45830 56965 i4 43o35 02804 3 97196 38 23 16 56 43 4 54195 i3 458o5 57004 lb 4299^. 02808 2 97192 37 36 35 24 25 16 48 43 12 54229 i4 45771 57042 i5 42958 02813 2 97187 9 16 4o 2 43 20 9.54263 i4 10.45737 9.57081 16 10.42919 10.02818 2 9.97182 26 16 32 43 28 54297 i5 45703 57120 17 42880 02822 2 97178 34 27 16 24 43 36 54331 i5 45669 57 1 58 17 42842 02827 2 97173 33 28 16 16 43 44 54365 16 45635 57197 18 42803 02832 2 97168 32 29 So 16 8 43 52 54399 16 45601 57235 19 42765 02837 2 97163 3i 3^ 9 16 2 44 9.54433 17 10.45567 9.57274 19 10.42726 10.02841 2 9.97159 3i i5 52 44 8 54466 17 45534 57312 20 42688 02846 2 97154 29 32 i5 44 44 16 54500 18 45500 57351 21 42649 o285i 3 97149 28 33 i5 36 44 24 54534 19 45466 57389 21 42611 02855 3 97145 27 34 35 i5 28 44 32 54567 '9 45433 57428 22 42572 02860 3 97140 26 9 1 5 20 2 44 4o 9.54601 20 10.45399 9.57466 22 10.42534 10.02865 3 9.97135 36 i5 12 44 48 54635 20 45365 57504 23 42496 02870 6 97i3o 24 37 i5 4 44 56 54668 21 45332 57543 24 42457 02874 6 97126 23 38 i4 56 45 4 54702 21 45298 57581 24 42419 02879 3 97121 22 39 4o i4 48 45 12 54735 22 45265 57619 25 4238i 02884 3 97116 21 20 9 1 4 4o 2 45 20 9.54769 23 io.4523i 9.57658 26 10.42342 10.028S9 3 9.97111 4i i4 32 45 28 54802 23 45198 57696 26 423o4 02893 3 97107 ly 42 1 4 24 45 36 5483r 24 45i64 57734 27 42266 02898 3 97102 lb 43 i4 16 45 44 54869 24 45i3i 57772 28. 42228 02903 3 97097 17 44 45 14 8 45 52 54903 25 45097 57S10 28 43190 02908 3 97092 16 75 9 i4 2 46 9.549^6 25 10.45064 9.57849 29 io.42i5i 10.02913 4 9.97087 46 i3 52 46 8 54969 26 45o3i 57887 3o 42ii3 02917 4 97083 14 47 1 3 44 46 16 55oo3 26 44997 57925 3o 42075 02922 4 97078 i3 48 i3 36 46 24 55o36 27 44964 57963 3i 42037 02927 4 97073 12 49 5o i3 28 46 32 55069 28 44931 58ooi 3i 41999 02932 4 97068 1 1 10 9 i3 20 2 46 4o 9.55102 28 10.44898 9.58039 32 10.41961 10.02937 4 9.97063 bi i3 12 46 48 55i3C 29 44864 58077 33 41923 02941 4 97059 y 52 i3 4 46 56 55169 29 4483 1 58ii5 33 4i885 02946 4 97054 8 53 12 56 47 4 55202 3o 44798 58i53 34 4 1847 02951 4 97049 7 54 55 12 48 47 12 55235 3o 44765 58191 35 41809 02956 4 97044 ~5 9 12 4o 2 47 20 9.55268 3i 10.44732 9.58229 35 10.4177' 10.02961 4 9.97039 56 12 32 47 28 553oi 32 44699 58267 36 41733 02965 4 97035 4 ^7 12 24 47 36 5533.J 32 44666 583o4 3? 41696 02970 4 97o3o 3 58 12 16 47 44 55367 33 44633 58342 37 4i658 02975 5 97025 2 ^9 12 8 4i 52 55400 33 44600 58380 38 41620 02980 5 97020 I 60 M 12 48 55433 34 44567 584i8 39 4i582 02985 5 97015 M Hour p.M Hour A.M. Cosinn. Diff. Secant. Cotangen Ditr Tangent. Cosecant. jDifT. Sine. 110° A A B B C 1- 2» 3' 4- 5' 6' 25 29 4 7* 3o 34 4 Prop, parts of cols. (■ 4 5 I 8 10 X i3 i4 a 17 19 a 31 24 3 C 09" Page 206] TABLE XXVIL 5 Log. Sines, Tanaents, and Secants. G'. 21° A A B B C C 158° M o Hour A. M Hour P.M. Sine. Diff Cosecant. Tangent. Diff. Cotangent Secant. Diff. Cosine. M 60 9 12 2 48 9.55433 10.44567 9.58418 io.4i582 10.02985 9.97015 I II 52 48 8 55466 I 44534 58455 I 4i545 03990 97010 59 2 II 4^ 48 16 55499 I 445oi 58493 I 4i5o7 02995 97005 58 6 II 36 48 24 55532 2 44468 58531 2 41469 02999 97001 57 4 5 II 28 43 32 55564 2 44436 58569 2 4i43i o3()o4 96996 56 55 9 1 1 20 2 48 4o 9.55597 3 io.444o3 9.58606 3 10.41394 io.o3oo9 9.96991 b II 12 48 48 55630 3 44370 58644 4 4i356 o3oi4 96986 54 7 II 4 48 56 55663 4 44337 5868 1 4 4i3i9 o3oi9 96981 53 8 10 56 49 4 55695 4 443o5 58719 5 41281 o3o24 96976 52 _9 10 10 48 49 12 55728 5 44272 58757 6 41243 o3o29 96971 5i 5o 9 10 4o 2 49 20 9.55761 5 10.44239 9.58794 6 10.41206 io.o3o34 9.96966 II 10 32 49 28 55793 b 44207 58832 7 4116S o3o38 96962 '40 12 in 94 49 36 55826 b 44174 58869 7 4ii3i o3o43 96957 48 iJ 1 u i 'J 49 44 55858 7 44x42 58907 8 41093 o3o48 96952 47 i4 i5 10 8 49 52 55891 9.55923 7 44109 58944 9 4io56 o3o53 96947 46 45 9 1 2 5o 8 10.44077 9.58981 9 10.41019 io.o3o58 9.96942 lb 952 5o 8 55956 9 44044 59019 10 40981 o3o63 96937 44 17 9 44 5o 16 55988 9 44012 59056 10 40944 o3o68 96932 43 i8 9 36 5o 24 56o2i 10 43979 59094 1 1 40906 o3o73 96927 42 19 20 9 28 5o 32 56o53 10 43947 59131 12 40869 03078 2 96922 4i 4o 9 9 20 2 5o 4o 9.56085 1 1 10.43915 9.59168 12 to.4oS32 io.o3oS3 2 9.96917 21 9 12 5o 48 56ii8 II 43882 59205 i3 40795 o3o88 2 96912 3q 22 9 4 5o 56 56i5o 12 43850 59243 i4 40757 03093 2 96907 38 2 J 8 56 5i 4 56182 12 438x8 592S0 i4 40720 o3o97 2 96903 37 24 25 8 4^ 5i 12 562 15 i3 43785 59317 i5 4o683 o3io2 2 96S98 36 35 9 8 40 ■2 5i 20 9.56247 i3 10.43753 9.59354 i5 10.40646 io.o3!07 2 9.96893 2b 8 32 5i 28 56279 i4 43721 59391 16 40609 o3i 12 2 96888 34 27 8 24 5i 36 563 II 1 4 43689 59429 17 4o57i o3ii7 2 96883 33 28 8 16 5i 44 56343 13 43657 59466 17 4o534 o3l22 2 96S78 32 29 3o 8 8 5i 52 56375 lb 43625 10.43592 5q5c3 18 40497 o3i27 2 96873 3i 3^ 980 2 52 . 564o8 16 0.59540 19 !0.4o46o io.o3i32 2 9 . 96S68 Ji 7 52 52 8 56440 17 43560 5g577 IQ 40423 o3i37 3 96863 29 J2 7 44 52 16 56472 17 43528 5o6r4 20 4o386 o3i42 3 96858 28 33 7 36 52 24 565o4 18 43496 5q65i 20 40349 o3i47 3 96853 27 34 35 7 28 52 32 56536 18 43464 5r^S8 21 4o3i2 o3i52 3 96848 26 9 7 20 2 52 40 9.56568 19 10.43432 9.59725 22 10.40275 io.o3i57 3 9.96843 3b 7 12 52 48 56599 '9 43401 59762 22 40238 o3i62 J 96838 24 ^7 7 4 52 56 5663 1 20 43369 59799 23- 40201 .o3 1 67 3 96833 23 38 6 56 53 4 56663 20 43337 59835 23 4oi65 o3i72 3 96828 22 39 4o 6 48 53 12 56695 21 433o5 59872 24 40128 o3i77 3 96823 21 20 9 6 4o 2 53 20 9.56727 21 10.43273 9.59909 25 10.40091 io.o3i82 3 9.96S18 4i 6 32 53 28 56759 22 43241 59946 25 4oo54 o3i87 3 96813 19 42 6 24 53 36 56790 22 43210 59983 26 40017 03192 3 96808 18 43 6 iG 53 44 5682 2 23 . 43178 60019 27 39981 o3i97 4 96803 17 44 45 6 8 53 52 56854 24 43 1 46 Goo56 27 39944 03202 4 96798 16 75 960 2 54 9.56886 24 io.43ii4 9.60093 28 10.39907 10.03207 4 9.96793 4b 5 52 54 8 56917 25 43o83 6oi3o 28 39870 o32I2 4 96788 i4 47 5 44 54 16 56949 25 43o5i 60166 29 39834 o32i7 4 96783 i3 48 5 36 54 24 56980 26 43o2o 6o2o3 3o 39797 03222 4 96778 12 49 5o 5 28 54 32 57012 26 42988 60240 3o 39760 03228 4 9677? 11 10 9 5 20 2 54 40 9.57044 27 10.42956 9.60276 3i 10.39724 10.03233 4 9.96767 5i 5 12 54 48 57075 27 42925 6o3i3 3i 396S7 o323S 4 96762 9 52 5 4 54 56 57107 28 42893 fo349 32 39651 03243 4 96757 « 53 4 56 55 4 57i38 28 42862 6o386 33 39614 o3248 4 96752 7 54 55 4 48 55 12 57169 29 4283i 6i 422 33 39578 o3253 4 96747 b 9 4 40 2 55 20 9.57201 29 10.42709 9.60459 34 10.39541 io.n3253 5 9.96742 5b 4 32 55 28 57232 3o 42768 60/95 J3 39505 o3263 5 96737 4 t)7 4 24 55 36 57264 3o 42736 6o532 35 39468 03268 5 96732 3 58 4 16 55 44 57295 3[ 42705 6o568 36 39432 03273 5 96727 2 59 4 8 55 52 57326 32 42674 6o6c5 36 39395 03278 5 96722 I bo 4 56 57358 32 42642 6064! 37 39359 o3283 5 96717 Hour p. 51. Hour A.m. Cosine, joiff. Secant. Cotangent DilT. Tangent. Cosecant. Diif. Si.:e. ur A A B B ( *_/ Seconds of time V 2' 3» 4» 5» 6' 7' (^ 4 8 12 16 20 24 28 Prop, parts of cols. ^ 5 9 i4 19 23 28 32 f C I 2 2 1 3 4 4 68* TABLi: XXVII. [Page 'J07 S' Log. Sines, Tangents, and Secants. G'. 22 A A B B C C 157° M I 2 3 4 5 6 7 8 _9 10 II 12 i3 i4 i5 1 6 17 i8 12 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 4o 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 60 M Hour A.M. Hour P.M. Sine. DilT. Cosecant. Tangent. DifT. Cotangent Secant. Difl". Cosine. M 6^ 59 58 57 56 55 54 53 52 5i 53 49 48 47 46 45 44 43 42 4i 40 39 38 37 36 35 34 33 32 3i 3o 29 28 27 26 25 24 23 22 21 20 •9 18 17 16 75 i4 i3 12 1 1 10 9 8 7 6 5 4 3 2 I .?. 940 3 52 3 44 3 36 3 28 2 56 56 8 56 16 56 24 56 32 9.57358 57389 57420 57451 57482 I I 2 2 10.42642 4261 1 42580 42'}4<) 425i8 9 . 6064 1 60677 60714 60750 60786 I I 2 2 10.39359 39323 39286 39250 39214 io.o32S3 08289 08294 08299 o33o4 9.96717 96711 96706 96701 96696 9 3 20 3 12 3 4 2 56 2 48 2 56 40 56 48 56 56 57 4 57 12 9.57514 57545 57576 57607 57638 3 3 4 4 5 to. 42486 42455 42424 42393 42362 9.60823 6u859 60895 60931 60967 3 4 4 5 .5 10.39177 39141 39105 39069 89033 10.08809 o33i4 o33i9 08824 o333o 9.96691 96686 9668 1 96676 96670 9 2 4o 2 32 2 24 2 16 2 8 2 57 20 57 28 57 36 57 44 57 52 9.57669 57700 57731 57762 57793 5 6 6 7 7 10.42331 423oo 42269 42238 42207 9.61004 6io4o 61076 61 1 12 6ii48 6 7 7 8 8 10.38996 38960 38924 38888 38852 10. 03335 o334o o3345 o335o 03355 9.96G65 96660 96655 96650 96645 920 I 52 I 44 I 36 I 28 2 58 58 8 53 16 58 24 58 32 9.57824 57855 57885 57916 57947 8 8 9 9 U) 10.42176 42145 42ii5 42084 42053 9.61184 61220 6i256 61292 6i328 9 10 10 1 1 1 1 io.3S8i6 38780 38744 38708 38672 io.o336o 03366 03371 03376 o338i 2 2 9.96640 96634 96629 96624 966 1 9 9 . 966 1 4 96608 96608 96598 96593 9 I 20 I 12 I 4 56 48 2 58 4o 58 48 58 56 59 4 69 12 9.57978 58oo8 58o39 5S070 5Sioi 10 1 1 II 12 12 10.42022 41992 4 1961 41930 41899 9.61864 6i4oo 6 1 436 61472 6i5o8 12 i3 i3 1 4 i4 10. 38636 386oo 38564 38528 38492 io.o33S6 03392 08897 o34o2 08437 2 2 2 2 9 4o 32 24 16 8 2 59 20 59 28 59 36 59 44 59 52 9.58i3i 58162 58192 58223 58253 i3 i3 1 4 1 4 i5 10.41869 4i838 41808 41777 41747 9.61544 61579 6i6i5 6i65i 61687 i5 i5 16 17 17 10.38456 38421 38385 38349 383 1 3 ro.o34i2 o34i8 03423 08428 03433 2 2 2 2 3 9.965S8 96582 96577 96572 96567 900 8 59 52 59 44 59 36 59 28 3 8 16 24 32 9.58284 583 1 4 58345 58375 584o6 i5 16 16 17 17 :o.4i7i6 4 1 686 4i655 41625 41594 9.61722 61758 61794 6i83o 6 1 865 18 18 '9 20 20 10.08278 38242 382*06 38170 38i35 I0.03438 03444 03449 03454 03459 3 3 3 3 9.95562 96556 96551 96546 96541 8 59 20 59 12 59 4 58 56 58 48 3 4o 48 56 1 4 I 12 9.58436 58467 58497 58527 58557 18 18 19 19 20 20 21 21 22 22 1 . 4 1 564 4i533 4i5o3 41473 41443 io.4i4i2 4i382 4i352 4 I 322 41291 9.61901 61936 61972 62008 62043 21 21 22 23 23 1 . 3S099 38o64 38028 37992 37957 io.o3465 03470 03475 o348o o3486 3 3 3 3 3 9.96535 96530 96525 96520 96514 8 58 4o 58 32 58 2.4 58 16 58 8 3 I 20 1 28 I 36 I 44 I 52 9.58588 586 18 58648 5S678 53709 9.62079 62114 62i5o 62185 6-'2:>l 24 24 25 26 26 10.87921 37886 37850 37815 37779 10.03491 03496 o35o2 o35o7 o35i2 3 4 4 4 4 9.96509 96504 96498 96493 9648S 8 58 57 52 57 4^ 57 36 57 28 320 2 8 2 16 2 24 2 32 9.58739 58769 58799 58829 58859 23 23 24 24 25 25 26 26 27 27 I0.4I26I 4i23i 41201 41171 4ii4i io.4ii 1 1 410S1 4io5i 4l02[ 40991 9.62256 62292 62827 62362 62398 27 27 28 29 29 10.37744 37708 37673 37688 37602 io.o35i7 03523 03528 03533 03539 4 4 4 4 4 9.96488 96477 96472 96467 96461 8 57 20 57 12 57 4 56 56 56 48 3 2 4o 2 48 2 56 3 4 3 12 9.58889 58919 58949 58979 59009 9.62433 62468 62504 62539 62574 3o 3o 3i 32 32 10.87567 37532 37496 37461 37426 10.03544 03549 03555 o856o o3565 4 4 5 5 5 9.96456 96451 96445 96440 96435 8 56 4o 56 32 56 24 56 16 56 8 56 3 3 20 3 28 3 36 3 44 3 52 4 9.59039 59069 59098 59128 59158 59188 28 28 29 19 3o 3i 10.40961 40931 40902 40872 40S42 4oSl2 9.62609 62645 62680 62715 62750 62785 33 33 34 35 35 36 10.37391 37355 37820 37285 37250 37215 10.03571 03576 o358i o3587 08592 03597 5 5 5 5 5 5 9.96429 96424 96419 96418 96408 96408 IIoiiri'.iM. Hour A.M. Cosine. DilT. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. 112° or Seconds of time 1' 2' 3» 4» 5* 6' •ys Prop, parts of cols. ^ B 4 4 I 8 9 I IX i3 3 i5 18 3 19 22 3 23 27 4 27 3i Page2)S] TABLE XXVIL 5' Log. Sines, Tangents, and Secants. (?'. 2-4 A A B B C G 156° M o Hour A. M Hour P.M. Sine. Diff. Cosecant. Tang-ent. Ditr. Cotang'ent Secant. Diff. Cosine. M 6^ 8 56 340 9.59188 10.40812 9.62785 10.37215 10.03597 9 . 96403 I 55 52 4 8 59218 40782 62820 I 37180 o36o3 96397 5q 2 55 44 4 16 59247 I 40753 62855 I 37145 o36o8 96392 58 3 55 36 4 24 59277 1 40723 62890 2 371 10 o36i3 96387 57 4 5 55 28 4 32 59307 2 40693 62926 2 37074 o36i9 96381 56' 55 8 55 20 3 4 4o 9.59336 2 10.40664 9.62961 3 10.37039 10.03624 9.963-6 b 55 12 4 48 59366 3 4o634 62996 3 37004 o363o 96370 '^■^ 7 55 4 4 56 59396 3 4o6o4 63o3i 4 36969 03635 96365 53 8 54 56 5 4 59425 4 40575 63o66 5 36934 o364o 96360 52 _? lO 54 48 5 12 59455 9.59484 4 4o545 63ioi 5 36899 03646 96354 5i 5o 8 54 4^ 3 5 20 5 io.4o5i6 9.63i35 6 10.36865 io.o365i 9.96349 II 54 32 5 28 59514 5 40486 63170 6 36830 o3657 96343 4q 12 54 24 5 36 59543 b 40457 63205 7 36795 03662 96338 48 i6 54 lb 5 44 59573 b 40427 63240 7 36760 03667 96333 47 i4 i5 54 8 5 52 59602 7 40398 63275 8 36725 03673 96327 46 45 8 54 3 b c 9.59632 7 io.4o363 9.63310 9 10.36690 10.03678 9.96322 lb 53 52 6 8 59661 8 40339 63345 9 36655 o36S4 96316 44 17 53 44 6 16 59690 8 4o3io 63379 10 3662 1 03689 2 963 1 1 43 i8 53 36 6 24 59720 9 40280 634 1 4 10 36586 03695 2 963o5 42 !9 20 53 28 6 32 59749 9 4o25i 63449 1 1 3655i 03700 2 96300 4i 4o 8 53 20 3 6 40 9.59778 10 10.40222 9.63484 12 io.3b5i6 10 03706 2 9.96294 21 53 12 6 48 59808 10 40192 635i9 12 3648 1 0371 1 2 96289 3q 22 53 4 6 56 59S37 II 4oi63 63553 i3 36447 03716 2 0/^284 38 2j ■ 52 56 7 4 59S66 II 4oi34 63588 i3 364 1 2 03722 2 90278 37 24 25 52 48 7 12 59S95 12 4oio5 63623 i4 36377 03727 2 96273 36 35 8 52 40 3 7 20 9.59924 12 10.40076 9.63657 i4 10.36343 10.03733 2 9.96267 2b 52 32 7 28 59954 i3 40046 63692 i5 363o8 03738 2 96262 34 27 'n 14 7 36 59983 i3 40017 63726 lb 36274 03744 2 96256 33 28 52 iG ' A' 60012 14 3998S 63761 lb 36239 03749 3 96251 32 29 3o 52 8 7 52 Loc^. — 1 9> ? 63796 9~&383o 17^ 17 36204 10.36170 03755 3 96245 3i 3^ 8 52 3 8 u 9.60070 i5 10.39930 10.03760 3 9.96240 3 1 5i 52 8 8 60090 60128 i5 30901 63865 18 36i35 03769 3 96234 2Q J 2 5i 44 8 16 i5 39872 63899 18 36ioi 03771 3 96229 28 33 5 1 36 8 24 60157 lb 39843 63934 19 36o66 03777 3 96223 27 34 35 5r 28 8 32 60186 lb 39814 63968 20 36o32 03782 3 96218 26 25 8 5i 20 3 8 4o 9.60215 17 10.39785 9.64oo3 20 10.35997 10.03788 3 9.96212 3b- 5i 12 8 48 60244 17 39756 64o37 21 35963 03793 3 96207 2.4 ^7 5r 4 8 56 60273 18 39727 64072 21 35928 03799 3 96201 23 38 5o 56 9 4 6o3o2 18 39698 64106 22 35894 o3So4 3 96196 22 39 4o 5o 48 9 12 6o33i 19 39669 64 1 40 22 35S6o o38io 4 96190 21 20 8 5o 40 3 9 20 9.60359 19 10.39641 9.64175 23 10.35825 io.o38i5 4 9. 96 1 85 4i 5o 32 9 28 6o388 20 39612 64209 24 35791 o382i 4 96179 IQ 42 5o 24 9 36 60417 20 39583 64243 24 35757 03826 4 96174 18 43 5o 16 9 A4 6o446 21 39554 64278 25 35722 o3832 4 96168 17 44 45 5o 8 9 52 60474 21 39526 643 1 2 25 35688 o3838 4 96162 16 75 8 5o 3 10 9.6o5o3 22 10.39497 9.64346 26 10.35654 10.03843 4 9.96157 4b 49 52 10 8 6o532 22 39468 6438 1 26 35619 o3849 4 961 5 1 i4 47 49 A4 10 16 6o56i 23 39439 644 1 5 27 35585 03854 4 96146 i3 48 49 36 10 24 60589 23 394 II 64449 28 3555i o386o 4 96140 12 49 5o 49 28 10 32 60618 24 39382 64483 28 35517 03865 4 96135 II 10 8 49 20 3 10 4o 9.60646 24 10.39354 9.64517 29 10.35483 10.03871 5 9.96129 5i 49 12 10 48 60675 25 ' 39325 64552 29 35448 03877 5 96123 9 52 49 4 10 56 00704 25 39296 64586 3o 35414 03882 5 96118 8 53 48 56 II 4 60732 26 39268 64620 3i 3538o o3888 5 961 1 2 7 54 55 48 48 11 12 60761 26 39239 64654 3i 35346 03893 5 96107 6 5 8 48 4o 3 II 20 9.6C789 27 10.39211 9.64688 32 io.353i2 10.03899 5 9.96101 5b 48 32 II 28 60818 27 39182 64722 32 35278 03905 5 96095 4 ^7 48 24 II 36 60846 28 39154 64756 33 35244 03910 5 96090 3 58 48 16 II 44 60875 28 39125 64790 33 35210 03916 5 96084 2 59 48 8 II 52 60903 29 39097 64824 34 35176 o3i^2I 5 96079 I bo M 48 12 60931 29 39069 64858 35 35i42 00927 b 96073 Sine. M Hour P.M. [lour A.M. Cosine. Diir. Secant. Cotangent Diff. Tangent. Cosecant. Diir. 113° A A B B C Seconds of time V 2' 3» 4' 5» 6' 7' Prop, parts of cols. I C 4 4 I 7 9 I II i3 9 i5 17 3 18 11 3 22 26 4 25 3i 5 66^ TABLE XXVIL fl';i!;e 209 S' Log. Sines, Tar gents, and Secant?. G'. 24° A A B B . C C 155° o Hour A. ill. Hour P.M. Sine. Dirt". Cosecaiil. Tangent. Difl'. Cotang-enl Secant. uiir. Cosine;. ,9.96073 65, 5 48 3 12 9.60931, 10.39069 9.64858 io.35i42 10.03927 1 47 52 12 8 60960' 39040 64892 1 35io8 03933 96067 ■^9 2 47 44 12 16 609881 I 39012 64926 I 35074 03938 96062 58 3 47 3fi 12 24 61016. I 38984 64960 2 35o4o 03944 96056 57 4 5 47 ?B 12 32 61045 2 38955 • 64994 2 35oo6 03950 96050 55 8 47 20 3 12 4o 9.61073 2 10.38927 9.65028 3 10.34972 10.03955 9.96045 6 47 12 12 48 61 101 3 38899 65o62 3 34938 03961 9(1039 54 7 47 4 12 56 61 129 3 38871 65096 4 34904 03966 96034 53 8 46 56 i3 4 6m58 4 3S842 65i3o 4 34870 03972 9')i-»28 52 _9 10 46-48 i3 12 61 186 4 388 1 4 65i64 5 34836 03978 9602 2 5i 5o 8 46 4-0 3 1 3 20 9.61214 5 10.3S786 9.65197 6 io.348o3 10.03983 03989 I 9 . 960 1 7 II 46 32 i3 28 61242 5 38758 65231 6 34769 c6<,i; I 49 12 45 24 i3 36 61270 6 38730 65265 7 34735 o3o95 n6oo5 48 i3 46 16 i3 44 61298 f? 38702 65299 7 34701 OiOOO 96000 4-' 1 4 i5 46 8 i3 52 61326 6 38674 65333 8 34667 o4oo6 ' __?^2?4 9.95988 4b 45 8 46 3 14 9.61354 7 10. 38646 9.65366 8 10. 34634 I0.040I2 i6 45 52 i4 8 6i382 7 386 1 8 654oo 9 34600 o4oi8 2 95982 44 17 45 44 i4 16 6i4' 1 8 38589 65434 9 34566 o4o2 3 2 9^977 43 i8 45 36 i4 24 6i438 8 38562 65467 10 34533 04029 2 9597 1 43 19 20 45 28 i4 32 6i.'i66 9.61494 9 38534 655oi 1 1 34499 o4o35 2 95965 4i 4o 8 45 20 3 14 4" 9 io.385o6 9.65535 11 It). 34465 1 . o4o4o 2 9.9596c. 21 45 12 i4 48 6 I 522 10 38478 65568 12 34432 o4o46 2 95954 t2 22 45 4 i4 56 6i55o 10 38450 656o2 12 34398 o4<'52 2 95948 38 2j 44 56 i5 4 61578 II 38422 65636 i3 34364 o4o5S 2 9594? 3- 24 25 44 48 i5 12 61606 1 1 ■ 38394 65669 i3 3433i o4o63 2 95937 30 35 8 44 40 3 i5 20 9.61654 12 10.38366 9.65703 14 10.34297 10.04069 2 9.95931 2b 44 32 i5 28 6i6()2 12 38338 65736 i5 34264 04075 2 95925 M 27 44 24 i5 36 61689 12 383 1 1 65770 i5 34230 o4o8o 3 95920 33 28 44 16 i5 44 61717 i3 3S283 658o3 16 34197 o4o86 3 95914 32 29 3o 44 8 i5 52 61745 9.61773 i3 3S255 65837 16 34 1 63 04092 959018 9.95902 3i 3o 8 44 3 16 10.38227 9.65870 17 1 0.34 1 3o 10.04098 3 3i 43 52 16 8 6i8<», i4 38200 65904 17 34096 o4 1 o3 3 95897 29 32 43 44 16 16 61828 ID 38172 65937 18 34o63 o4 1 09 95891 Ob a 43 36 16 24 6 1 856 i5 38i44 65971 18 34029 o4i i5 3 95885 2-7 M 35 43 28 16 32 6:883 16 38i 17 66004 19 33996 o4i2i 3 95879 2b 8 43 20 3 16 4" 9.6191 1 16 10.38089 9.66o38 20 10.33962 10.04127 3 9.95873 36 43 12 16 48 61939 17 38o6i 66071 20 33959 o4i32 3 95868 24 37 43 4 16 56 61966 17 38o34 66104 21 33S96 04 1 38 4 9586? 23 38 42 56 17 4 61994 18 38oo6 66 1 38 21 33862 o4i44 4 95856 22 39 40 42 48 17 12 6202 1 18 37979 66171 22 33829 o4i5o 4 9585o 21 20 8 42 4o 3 17 20 9.62049 18 10.37951 9.66204 22 10.33796 io.o4i56 4 9-95844 4i 42 32 17 28 62076 19 37924 66238 23 33762 o4i6i 4 95839 '9 42 42 24 17 36 62104 •9 37896 66271 23 33729 04167 4 95833 18 43 42 16 17 44 62i3i 20 37869 663o4 24 33696 04173 4 95827 '7 44 45 . 42 8 17 52 3 18 .. 62159 20 37841 66337 25 33663 o4 1 79 4 95821 lb 73 8 42 9.62186 21 10.. 378 1 4 9.66371 25 10.33629 io.o4iS5 4 g.958i5 J,0 4 1 52 i3 8 62214 21 37786 66404 26 33596 o4 1 90 4 95810 i4 47 4 1 44 18 16 62241 22 37759 66437 26 33563 04196 5 95804 i3 48 4i 36 18 24 62268 22 37732 66470 27 3353o 04202 5 95798 IS 49 5o 4i 28 18 32 3 18 4" 62296 23 37704 665o3 27 33497 04208 D 95792 II 10 - 8 4i 20 9.62323 23 10.37677 9.66537 28 10. 33463 10.04214 5 9.95786 5i 4i 12 18 48 62350 24 37650 66570 28 * 33430 04220 5 95780 9 ■32 4i 4 18 56 62377 24 37623 666o3 29 333(>7 04225 5 95775 53 4o 56 19 4 62405 24 37595 66636 3o 33364 0423 1 5 95769 7 54 55 4o 48 8 4o 4o 19 12 62432 9»63459 25 25 37568 10.37541 66669 3o 3333i 04237 5 95763 b 3 19 20 9.66702 3i 10.33298 10.04243 D 9.95757 5b 4t> 32 19 28 62486 26 37514 66735 3 1 33265 04249 5 95751 4 ■J7 4o 24 19 36 625i3 26 37487 66768 32 33232 04255 5 95745 3 58 4o 16 19 44 62541 27 37459 66801 32 33199 04261 6 95739 2 59 4o 8 19 52 62 568 27 37432 66834 33 33 166 04267 6 95733 1 bo 4o 20 62595 28 374o5 66867 33 33i33 04272 6 95728 .»1 Kour p.;m. Hour A.M. Cnsiii(>. Diff. Secant. Cotann^enl Difl-. Tangent. Cosecant Ditri Sine. 114° A A B B C P 2' li' 4. 5- G' 7* _ f ^ 3 7 10 i4 17 21 24 Prop, parts of cols. ^ 4 8 i3 17 21 25 29 (c I I 2 3 4 4 5 27 Pa ge 210] TABLE XXVIL S'. Loo '. Sines, Tangents, and Secants. G'. 25 M o 5 A A B B c C 154° Hour A.M. Hour P.M. Sine. Diff. Cosecant. Tangent. Ditr. Cotangent Secant. Diff. Cosine. 31 6^ 8 4o 3 20 9.62595 10.37405 9.66867 10. 33 1 33 10.04272 9.95728 I 39 52 20 8 62622 37378 66900 I 33 100 04278 9J722 59 2 39 4i 20 16 62649 I 37351 66933 I 33067 04284 95716 58 3 39 36 20 24 62676 I 37324 66966 2 33o34 04290 93710 57 4 5 39 28 20 32 62703 2 37297 66999 2 33ooi 04296 957C»4 56 55 8 39 20 3 20 4o 9.62730 2 10.37270 9.67032 3 10.3296S io.o43o2 9.95698 b 39 12 20 48 62757 3 37243 67065 3 32935 o43o8 95692 54 7 39 4 20 56 62784 3 37216 67098 4 32902 043 1 4 95686 53 8 •38 56 21 4 62811 4 37189 67131 4 32869 04320 95680 52 _9 lO 38 48 21 12 62838 4 37162 67163 5 32837 04326 95674 5i 5o 8 38 4" 3 21 20 9.62865 4 10.37135 9.67196 5 1 0.32804 10.04332 9.95668 II 38 32 21 28 62892 5 37108 67229 6 J2771 04337 9566? 49 12 38 24 21 36 62918 5 370S2 67262 7 32738 04343 95657 48 iJ 38 16 21 44 62945 6 37055 67295 7 32705 04349 95651 47 i4 i5 38 8 21 52 62972 b 37028 67327 8 32673 10.33640 04355 9564'j 46 45 8 38 3 22 9.62999 7 10.37001 9.67360 8 10. 04361 2 9.95639 i6 37 5-^ 22 8 63o26 7 36974 67393 9 32607 04367 2 95633 44 17 37 44 22 16 63o52 8 36948 67426 9 32574 04373 2 95627 43 i8 37 36 22 24 63o79 8 36921 67458 10 32542 04379 2 95621 42 19 20 37 28 22 32 63 1 06 8 9 36894 10.36867 67491 10 32509 o4385 2 956x5 4i 40 8 37 20 3 22 4o Q.63i33 9.67624 II 10.32476 10.04391 2 9.95609 21 37 12 22 48 63 1 59 9 3684 1 67556 11 32444 04397 2 956o3 39 22 37 4 22 55 63 186 10 368 1 4 67589 12 3241 1 o44o3 2 95597 38 2J 36 56 23 4 632 1 3 10 36787 67622 12 32378 04409 2 95591 37 24 25 36 48 23 12 63239 1 1 36761 67654 i3 32 346 io.323i3 044 1 5 2 95585 36 35 S 36 40 3 23 20 9.63266 1 1 10.36734 9.67687 i4 10.04421 3 9.95579 26 36 32 23 28 63292 1 1 36708 67719 i4 32281 04427 3 95573 34 i^7 36 24 23 36 63319 12 3668 1 67752 i5 32248 04433 3 95567 33 28 36 16 23 44 63345 12 36655 67785 i5 322l5 04439 3 95561 32 29 3o 36 8 23 52 63372 i3 36628 67817 16 32i83 ro.32i5o 04445 3 95555 3i 3^ 8 36 3 24 9.63398 i3 1 . 366o2 9.67850 16 io.o445i 3 9.95549 3i 35 52 24 8 63425 i4 36575 67882 17 32118 04457 3 95543 29 32 35 44 24 16 6345 1 i4 36549 67915 17 3 208 5 04463 3 95537 28 33 35 36 24 24 63478 i5 3652 2 67947 18 32o53 04469 3 95531 27 35 35 28 24 3a 635o4 i5 36496 67980 18 32020 04475 3 95525 26 I5 8 35 20 3 24 40 9.63531 i5 10.36469 9.6S012 19 10.31988 10.04481 4 9.95519 36 35 12 24 48 63557 16 36443 68044 20 31956 04487 4 955i3 24 37 35 4 24 56 63583 16 36417 68077 20 31923 04493 4 95507 23 38 34 56 25 4 636io 17 36390 68109 21 31S91 o45oo 4 95500 22 39 40 34 48 25 12 63636 9.63662 17 18 36364 68142 21 3i858 o45o6 4 95494 21 20 8 34 40 3 25 20 10.36338 9.68174 22 10.31826 io.o45i2 4 9.95488 4i 34 32 25 28 63689 18 363 1 1 68206 22 31794 o45i8 4 95482 19 42 34 24 25 36 637 1 5 19 36285 68239 23 31761 04524 4 95476 18 43 34 16 25 44 63741 '9 36259 68271 23 31729 o453o 4 95470 17 44 45 34 8 2 5 52 63767 19 36233 683o3 24 31697 04536 4 95464 16 75 8 34 3 26 9.63794 20 10.36206 9.68336 24 io.3i664 10.04542 5 9.95458 46 33 52 26 8 63820 20 36 1 80 68368 25 3i632 04548 b 9^452 14 47 33 44 26 16 63846 21 36 1 54 684oo 25 3 1 600 04554 5 95446 i3 48 33 36 26 24 63872 21 36128 68432 26 3 1 568 04560 b 95440 12 49 5o 33 28 26 32 6389S 9.63924 22 22 36io2 68465 27 3i535 04566 b 95434 1 1 10 8 33 20 3 26 4o 10.36076 9.68497 27 io.3i5o3 10.04573 5 9.95427 5i 33 12 26 48 63950 23 36o5o 68529 28 3 1471 04579 5 95421 9 5 J 33 4 26 56 63976 23 36024 6S56i 28 31439 o45S5 b 95ii5 8 53 32 56 27 4 64002 23 35998 68593 29 3 1 407 04591 b 95409 7 54 55 32 48 27 12 64028 24 35972 68626 29 3i374 04597 b 95403 "5 £ 32 4o 3 27 20 9.64054 24 10.35946 9.68658 3o io.3i342 io.o46o3 6 9.95397 56 32 32 27 28 64080 25 35920 68690 3o 3i3io 04609 6 95391 4 57 32 24 27 36 64 1 06 25 35894 68722 3i 31278 04616 6 95384 3 58 32 16 27 44 64 1 32 26 35868 68754 3i 31246 04622 6 95378 2 59 32 8 27 52 64 1 58 26 35842 68786 32 3i2i4 04628 6 95372 I 60 M 32 28 64»84 26 358i6 68818 33 31182 o4634 6 95366 M Hour P.M. Hour A.M. Cosine. Din: Secant. • Cotangent Diir. Tangent. Cosecant. Diir. Sine. U5" C 64' Seconds of time V 2= 3^ 4. 5» 20 24 5 7, "23 28 5 Prop, parts of cols. < B I C 3 4 I 7 8 2 10 12 3 i3 16 3 17 20 4 TABLE XXVn. [Page 211 S". Log . Sines, Tanrrents, and Secants. G'. 26^ > A A B B C C 153° M o Hour A.M. Hour P.M. Sine. Diff. Cosecant.] Tangent. Diff. Cotangent Secant. Diff. Cosine. M 6?, 8 32 3 28 9.64184 io.358i6 9.6S818 io.3ii82 10.04634 9.95366 I 3i 52 28 8 64210 35790 6885o I 3ii5o o464o 95360 59 2 3i 44 28 16 64236 I 35764 68882 1 3iii8 04646 95354 58 3 3i 36 28 24 6426? I 35738 68914 2 3 1 086 o4652 95348 57 4 5 3i 28 28 32 64^88 2 35712 68946 2 3io54 04659 953.^1 56 55 8 3r 20 3 28 4o 9.643i3 2 10.35687 9.68978 3 10.3l022 I0.04665 I 9.95335 6 3i 12 28 48 64339 3 3 566 1 69010 3 30990 04671 I 95329 54 7 3i 4 28 56 64365 3 35635 69042 4 30958 04677 I 95323 53 8 3o 56 29 4 64391 3 35609 69074 4 30926 04683 I 95317 52 _9 10 3o 48 29 12 64417 4 35583 69106 5 30894 04690I I 95310 5i 5o 8 3o 4o 3 29 20 9.64442 4 10.35558 9.69138 5 10.30862 10.04696 9.95304 II 3o 32 29 28 64468 5 35532 69170 b 3o83o 04702 95298 49 12 3o 24 29 36 64494 5 355o6 69202 6 30798 64708 95292 48 j3 3o 16 29 44 64519 5 35481 69234 7 30766 04714 95286 47 i4 i5 3o 8 29 52 64545 6 35455 69266 7 30734 04721 95279 46 45 8 3o 3 3o 9.64571 6 10.35429 9.69298 8 10.30702 "10.04727 2 9.95273 i6 29 52 3o 8 64596 7 35404 69329 8 30671 04733 2 95267 44 17 29 44 3o 16 64622 7 35378 69361 9 3o639 04739 2 95261 43 lb 29 36 3o 24 64647 « 35353 69393 9 30607 04746 2 95254 42 !9 20 29 28 3o 32 64673 8 8 35327 69425 10 3o575 04752 2 95248 4i 4o 8 29 20 3 3o 4o 9.64698 io.353o2 9.69457 1 1 io.3o543 10.04758 2 9.95242 21 29 12 3o 48 64724 9 35276 69488 1 1 3o5i2 04764 2 95236 39 22 29 4 3o 56 64749 9 3525i 69520 12 3o48o 04771 2 95229 38 23 28 56 31 4 64775 10 3^225 69552 12 3o44S 04777 2 95223 37. ?.4 25 28 48 3i 12 64800 10 35200 69584 1 3 3o4i6 04783 3 95217 36 35 8 28 4o 3 3i 20 9.64S26 1 1 10.35174 9 . 696 1 5 i3 io.3o385 10.04789 3 9.95211 2b 28 32 31 28 6485 1 II 35i49 69647 i4 3o353 04796 3 95204 M 27 28 24 3i 36 64877 1 1 35i23 69679 i4 3o32i 04802 3 95198 33 28 28 16 3i 44 64902 12 35098 69710 1 5 30290 04808 3 95192 32 29 So 28 8 3i 52 64927 12 i3 35073 69742 i5 3o258 o48i5 3 95i85 3i 3o 8 28 3 32 9.64953 io.35o47 9.69774 16 10.30226 10.04S21 3 9.95179 3i 27 52 32 8 64978 i3 35o22 69805 lb 3o 1 95 04827 3 95173 29 32 27 44 32 16 65oo3 1 4 34997 69837 17 3oi63 04833 3 96167 28 3i 27 36 32 24 65029 i4 34971 69868 •7 3oi32 o484o 3 96160 27 34 35 27 28 32 32 65o54 i4 34946 69900 18 3oioo 04846 4 961 54 26 25 8 27 20 3 32 40 9.65079 i5 10.34921 9.69932 18 io.3oo68 10.04852 4 9.96148 3t) 27 12 32 48 65io4 i5 34896 69963 '9 3oo37 04859 4 96141 24 ^7 27 4 32 56 65i3o lb 34870 69995 20 3ooo5 04865 4 q6i35 23 38 26 56 33 4 65i55 16 34845 70026 20 29974 04871 4 96129 22 3<; 4o 26 48 33 12 65 1 So 16 34820 10.34795 70o58 21 29942 04878 4 96122 21 20 8 26 4o 3 33 20 9.652o5 17 9 . 700S9 21 10.2991 1 10.04884 4 9.96116 4i 26 32 33 28 65230 17 34770 7012 1 22 29879 04890 4 96110 19 42 26 24 33 36 65255 18 34745 701 52 22 29848 04897 4 96103 18 ii 26 .16 33 44 65281 18 347 '9 70184 23 2^816 04903 5 96097 J7 44 45 26 8 33 52 653o6 •9 34694 70215 23 29785 04910 5 96090 16 i5 8 26 3 34 9.653ii 19 10.34669 9.70247 24 10.29753 10.04916 5 9.96084 40 25 52 34 8 65356 19 34644 70278 24 29722 04922 5 96078 1 4 47 25 44 34 16 6538: 20 34619 7o3o9 25 29691 04929 5 9607 1 1 3 48 25 36 34 24 654o6 20 34594 70341 25 29659 04935 5 9K)frj 12 49 So 25 28 34 32 6543 1 21 34569 70372 26 29628 04941 5 96069 1 1 10 8 25 20 3 34 40 9.65456 21 10.34544 9.70404 26 10.29596 1C.04948 5 9.96062 ?' 25 12 34 48 6548 1 22 34519 70435 27 29565 04954 5 96046 9 .S2 25 4 34 56 655o6 22 34494 70466 27 29534 0496 1 5 96039 8 53 24 56 35 4 6553i 22 34469 70498 28 2^502 04967 6 96033 7 54 5'5 24 48 35 17 65556 23 34444 70529 28 29471 04973 6 96027 6 8 24 4o 3 35 20 9.655SO 23 10.34420 9.70560 29 10.29440 10.04980 6 9.96020 5 50 24 32 35 28 656o5 24 34395 70592 3o 29408 04986 6 96014 4 '•.7 24 24 35 36 6563o 24 34370 70623 3o 29377 04993 6 96007 3 58 24 16 35 44 65655 25 34345 70654 3i 29346 04999 6 96001 2 59 24 8 35 52 6568o 25 34320 70685 3c 29315 o5oo5 6 94996 I bo VI 24 36 657o5 23 34295 70717 32 29283 o5oi2 6 9498S M Hour p. M Hour A.M. Cosine. Diff. Secant. Cotangent Diff Tangent. Cosecant. Diff. Sine. 116° »/ Seconds of time , 1' 2^ 3' 4- 5' 6» 7' (^ 3 6 10 i3 16 19 t2 Prop, parts of cols U 4 8 :2 16 20 24 28 (c I 1 2 3 4 5 Jij ■'• .i:-\-] TABLE XXVII. . S' hog. S ines, Tangents, and Secants. G'. •27 i\ o A A B B G C 152° HourA.M Hour P.M. Sine. Dirt' Cosecant. Tang-en t. Diff. Cotangent Secant. Dift-. Cosine. M 8 24 3 36 9.65705 10.34295 9.70717 10.29283 io.o5oi2 9.94988 I 23 52 36 8 65729 34271 70748 I 29252 o5oi8 94982 '^9 2 23 4^ 36 16 65754 I 34246 70779 1 29221 o5o25 94975 58 3 23 3b 36 24 65779 1 34221 70810 2 29190 o5o3i 94969 57 4 5 23 28 36 32 658o4 2 34196 70841 2 29159 10.29127 o5o38 94962 56 55 8 23 20 3 36 4<) 9.65828 2 10.34172 9.70873 3 io.o5o44 9.94956 b 23 12 36 48 65853 2 34i47 70904 3 29096 o5o5i 94949 54 7 23 4 36 56 65878 3 34122 70935 4 29065 o5o57 94943 53 « 22 56 37 4 65902 3 34098 70966 4 29034 o5o64 94936 52 J? lO 22 48 37 12 65927 4 34073 70997 5 29003 o5o7o 94'93o 5i 5o 8 22 40 3 37 20 9.65952 4 10.34048 9.71028 5 10.28972 io.o5o77 9.94923 II 22 32 37 28. 65976 4 34024 71059 6 28941 o5o83 94917 4q 12 22 24 37 36 66001 5 33999 7 1 090 b 28910 o5o89 94911 48 IJ 22 16 37 44 66025 5 33975 71 121 7 28879 05096 94904 47 i4 25 22 8 37 52 63o5o b 33950 71 153 7 28847 05l02 2 94898 46 45 8 22 3 38 9.6()075 6 10.33025 9.71184 8 10.28816 io.o5i09 2 9.94891 i6 21 52 38 8 66099 b 3J701 71215 8 28785 o5i i5 2 94885 44 17 21 44 33 16 66124 7 33S76" 71246 9 28754 05l22 2 94878 43 i8 21 36 33 24 66 1 48 7 33852 71277 9 28723 o5i29 2 94871 42 19 so 21 28 38 32 66173 8 33827 7i3o8 10 28692 o5i35 2 94865 4i 4o 8 21 20 3 38 4o 9.66197 8 io.338o3 9.71339 10 10.28661 io.o5i42 2 9-94858 21 21 12 38 48 66221 8 33779 71370 1 1 28630 o5i48 2 94852 3o 22 21 4 38 56 66246 9 33754 7i4oi 11 28599 o5i55 2 94845 38 23 20 56 39 4 66270 9 33730 7>43fc 12 28569 o5i6i 3 94839 37 24 25 20 48 39 12 66295 10 33705 71462 12 28538 o5i68 3 94832 36 35 8 20 4o 3 39 20 9.66319 10 10.33681 9.71493 i3 10.28507 io.o5i74 3 9.94826 2b 20 32 39 28 6634^ 1 1 33657 71524 i3 28476 o5i8i 3 94819 M 27 20 24 39 36 66368 II 33632 71555 14 28445 05187 3 9481 3 33 28 20 16 39 44 6639'^ II 336o8 71586 i4 28414 05194 3 94S06 32 29 3o 20 8 39 52 66416 12 33584 71617 i5 28383 05201 3 94799 3 1 37; 8 20 3 4o 9.66441 12 10.33559 9.71648 i5 10.28352 10.05207 3 9.94793 3i 19 52 4o 8 66465 i3 33535 71679 16 28321 o52i4 3 94786 2Q 32 19 44 4o 16 66489 i3 335ii 71709 16 28291 05220 4 94780 28 33 19 36 4o 24 665 1 3 i3 33487 71740 17 28260 05227 4 94773 27 35 19 28 4o 32 66537 i4 33463 7'77' 17 28229 05233 4 947G7 26 25 8 19 20 3 4o 40 9.66562 14 10. 33438 9.71802 iS 10.28198 io.o524o 4 9.94760 36 19 12 4o 48 66586 i5 334 1 4 71833 19 28167 05247 4 94753 24 ^7 19 4 4o 56 66610 i5 33390 71863 19 28i3t o5253 4 94747 23 38 18 56 4i 4 66634 i5 33366 71894 20 28106 05260 4 9474<-i 22 39 4o 18 48 4r- 12 66658 16 33342 71925 20 28075 o5266 4 94734 21 20 8 18 40 3 4i 20 9.66682 16 io.333i8 9.71955 21 10.28045 10.05273 4 9.94727 4i 18 32 4i 28 66706 17 33294 71986 21 28014 o523o 4 94720 •'9 42 18 24 4i 36 66731 17 33269 72017 22 27983 05286 5 94714 18 4S 18 16 4i 44 66755 17 33245 72048 22 27952 05293 5 94707 17 44 45 18 8 4i 52 66779 9.66803 18 18 33221 10.33197 72078 23 27922 o53oo 5 94700 16 75 8 18 3 42 9.72109 23 10.27891 io.o53o6 5 9.94694 4b 17 52 42 8 66827 19 33173 72i4(j 24 27860 o53i3 5 94687 i4 47 17 44 42 16 66851 19 33 14<) 72170 24 27830 o532o 5 94680 1 3 48 17 36 42 24 66875 19 33125 72201 25 27799 05326 5 94674' 12 49 5o 17 28 42 32 66899 20 33ioi 72231 25 27769 05333 5 94667 1 1 10 8 17 20 3 42 4o 9.66922 20 10,330-8 9.72262 26 10.27738 io.o534o 5 9.94660 5i 17 12 42 48 66946 21 33o54 72293 26 27707 05346 6 9^654 9 52 17 4 42 56 66970 21 33o3o 72323 27 27677 05353 6 9I647 8 t>i 16 56 43 4 66994 21 33oo6 72354 27 27646 o536o 6 94640 7 54 55 16 48 43 12 67018 22 32983 :o. 32958 72384 28 27616 o5366 6 94634 6 1> 8 16 4o 3 4^ 20 9.67042 22 9.72415 28 10.27585 10.05373 6 9.94627 5b 16 32 43 28 67<)66 23 3?93i 72445 20 27555 o538o 6 94620 4 57 16 24 43 36 67090 23 32910 72476 29 27524 05386 6 94614 3 58 16 16 43 44 67113 23 32887 725o6 3o 27494 05393 6 91607 2 59 16 8 43 52 67.37 24 32 863 72537 3o 27463 o54oo b 94600 1 bo M 16 44 67161 24 32839 72567 3i 27433 o54o7 7 94593 Hour P.M. HourA.M. Cosine. Diif. Secant. Cotangent uiir. Tangent. Cosecant. Diff. Sine. 117° A 62' 1' 2' 3^ 9 4' 12 5» 6' iS 7» 21 (^ 3 G i5 I*'ro['. parta of cols. N 4 8 12 i5 19 23 27 (c 1 2 2 3 4 5 6 TABLE XXVIL [Page 21:) i" Log . Sines, Tan gents, and Secants. G'. 28 M c A A B B C C 151° Ilo'.ir A.Bi.jHour p.m. Sino. Diir. Cosscaut. 3D.3a839 Tangent. Diir.jCotaiigcnl Secant. DiiT. Cosine. M 6^ £ 16 3 44 9.67161 9.72567 10.27433 io.o54o7 '9.94593 1 i5 52 44 8 67185 32815 72598 I 27402 o54i3 1 94687 r)9 2 1 5 44 44 16 67208 I 32792 72628 I 27372 05420 ] 94680 58 3 1 5 36 44 24 67232 J 32768 72659 2 27341 05427 94673 ^7 4 5 1 5 28 £ 1 5 20 44 32 3 44 4" 67256 2 32744 72689 2 27311 05433 94567 Ob 55 9.67280 2 10.32720 9.72720 3 10.27280 io.o544o 9.94560 6 [5 12 44 48 67303 2 32697 72750 3 27250 05447 94553 64 7 i5 4 44 56 67327 3 32673 72780 4 27220 05454 94546 63 8 i4 56 45 4 67359 3 3265o 72811 4 27189 o546o 94540 62 ID i4 48 45 12 67374 3 32626 72841 5 27159 05467 94533 5i 57. 8 i4 4o 3 45 20 9.6739S 4 10.32602 9.72072 5 10.27128 10.05474 9.94526 1 I i4 32 45 28 6742 1 4 32579 72902 6 27098 o548i 94519 49 12 1 4 24 45 36 67445 5 32555 72932 6 27068 05487 945 1 3 48 i3 i4 16 45 44 67463 5 32532 72963 7 27037 05494 94606 47 r4 i5 i4 8 45 52 67492 5 32 5o8 72993 7 27007 o55oi 2 94499 4b 46 8 i4 3 46 9.67515 6 10.32485 9.73023 8 10.26977 io.o55o8 2 9.94492 i6 i3 52 46 8 67539 6 32461 73o54 8 26946 o55i5 2 94486 44 17 i3 44 46 16 67562 7 32438 73084 9 26916 o552i 2 94479 4^ 18 i3 36 46 24 67586 7 324i4 73ii4 9 26886 05528 2 94472 42 !9 20 i3 28 46 32 67609 9.67633 7 32391 73i44 10 26856 05535 2 94465 4i 40 8 i3 20 3 46 40 8 10.32367 9.73175 10 10.26825 10.05542 2 9-94458 21 i3 12 46 48 67656 8 32344 732o5 II 26795 05549 2 94451 39 22 i3 4 46 56 67680 9 32320 73235 II 26765 05555 3 94445 38 2j 12 56 47 4 67703 9 32297 73265 12 26735 o5562 3 94438 37 24 25 12 48 47 12 67726 9 32274 73295 12 26705 05569 3 9443 1 3b 35 8 12 40 3 47 20 9.67750 10 I0.3225o 9.73326 i3 10.26674 10.05576 3 9.94424 2b 12 32 47 28 G7773 10 32227 73356 i3 26644 o5583 3 94417 M 27 12 24 47 36 67796 10 32204 73386 i4 26614 05590 3 94410 .ii 28 12 16 47 44 6-7820 II 32180 73416 i4 26584 05596 3 94404 32 29 3o 12 8 47 52 67843 9.67866 1 1 32157 73446 i5 26554 o56o3 3 94397 3i 3^ 8 12 3 48 12 io.32i34 9-73476 i5 10.26524 1 . ()56 1 3 9.94390 3i II 62 48 8 67890 12 32110 73507 16 26493 o56i7 4 94383 29 i> 11 44 48 16 67913 12 32087 73537 16 26463 05624 4 94376 28 34 II 36 48 24 67936 i3 32064 73567 17 26433 o563i 4 94369 27 34 35 II 28 48 32 67959 i3 32o4i 73597 17 26403 05638 4 94362 2b 8 1 1 20 3 48 4o 9.67982 14 10.32018 9.73627 18 10.26373 I0.05645 4 9.94355 3b II 12 48 48 68006 i4 31994 73657 18 26343 o565i 4 94349 24 47 11 4 48 56 68029 i4 31971 73687 '9 263 1 3 05658 4 94342 23 38 10 56 49 4 68o52 i5 31948 73717 19 26283 05665 4 94335 22 39 4<. 10 48 49 12 68075 i5 31925 73747 20 26253 10.26223 05672 4 94328 21 20 8 10 4o 3 49 20 9.68098 16 10.31902 9-73777 20 10.05679 5 9.94321 4i 10 32 49 28 68121 16 31879 73807 21 26193 o5686 5 943 1 4 IQ 42 10 24 49 36 68144 16 3i856 73837 21 26163 05693 6 94307 18 43 10 16 49 44 68167 17 3i833 73S67 22 26133 05700 5 94300 '7 44 45 10 8 4g 52 68190 17 3i8io 73897 22 26103 05707 5 94293 16 16 8100 3 5o 9.68213 17 10.31787 9.73927 23 10.26073 10.05714 5 9.94286 4b 9 52 5o 8 6S237 18 31763 73957 23 26043 05721 5 94279 i4 47 9 44 5o 16 68260 18 3 1740 73987 24 26013 06727 5 94273 i3 48 9 36 5o 24 68283 19 31717 74017 24 25983 05734 5 94266 12 49 5o 9 ?8 5o 32 683o5 '9 31695 74047 25 25953 0574 1 6 94269 1 1 III S 9 20 3 5o 40 9.68328 iQ 10.31672 9.74077 25 10.25923 io.o5748 6 9.94262 61 9 12 5o 48 68351 20 . 3i649 74107 26 25893 05755 6 94246 Q 52 9 4 5o 56 68374 20 31626 74 1 37 26 25863 06762 6 94238 8 53 8 56 5i 4 68397 21 3i6o3 74166 27 25834 06769 6 9423 1 7 54 55 8 48 5i 12 68420 21 3i58o 74196 27 258o4 06776 6 94^24 6 ~5 8 8 40 3 5i 20 9.68443 21 ic.3i557 9.74226 28 10.25774 10.05783 6 9.94217 56 8 32 5i 28 68466 22 3i534 74256 28 25744 06790 6 94210 4 5t 5S 8 24 5 1 36 68489 22 3i5ii 74286 29 25714 06797 7 94203 3 8 16 5i 44 685 1 2 22 3 1 488 743i6 29 2 5684 o58o4 7 94196 2 59 8 8 5i 52 68534 23 3 1 466 74345 3o 25655 o58ii 7 ■ 94189 I 8 52 68557 23 3i443 74375 3o 25625 o58i8 94182 Hour p.?,i. Hour a.m. Cosine. DifiT. Secant. Coiangcnt DilT.| Tangent. Cosecant. Diir. Sine. M 118° A A B B C Seconds of time . . . ..^ 1' 2» 3" 4- 5« 6' 7* Prop, parts of cols. I C 3 4 I 6 8 2 9 II 3 12 !5 3 i5 19 4 17 23 5 20 26 6 GV P; Se 2141 TABLE XXVIL S' Log. S nes, Tancrents, and Secants. G'. 29 A A B B C C 150° o Hour A.M. Hour P.M. Sine. DilT. Cosecant. Tangent. 9-74375 Ditr. Cotangent lo. 25625 Secant. Diff. Cosine. M 880 3 52 9.68557 I 0.3 1443 io.o58i8 9.94182 60 I 7 52 52 8 6858o 3 1420 744o5 25595 o5825 o4i75 5o 2 7 44 52 16 6S6o3 I 3.397 74435 I 25565 05832 94168 58 3 7 36 52 24 6S625 1 3 1 375 74465 I 25535 o5839 94161 67 4 5 7 28 52 32 68648 I 3i352 74494 2 255o6 o5846 94164 56 8 7 20 3 52 40 9.68671 2 io.3i329 9.74524 2 10.25476 io.o5P.:3 : 9.94147 55 b 7 12 52 48 68694 2 3 1 3o6 74554 3 25446 o586o 04 1 40 54 7 7 4 52 56 68716 3 1 284 74583 3 25417 05867 I 1 94i33 63 8 6 56 53 4 68739 3 31261 7461 3 4 25387 05874 I q4i26 52 9 10 6 48 53 12 68762 3 3.238 7 -{643 4 25357 10.25327 o588i 94 119 5. 8 6 4o 3 53 20 9.68784 4 1 . 3 1 2 1 b 9.74673 5 10.05888 9.94112 5o II 6 32 53 28 6S807 4 31193 74702 5 25298 05895 94io5 4c) 12 6 24 53 36 68829 4 3117. 74732 6 25268 o5oo2 9409S 48 i3 6 16 53 44 68852 5 3ii48 74762 6 25238 06910 2 94090 4- i4 i5 6 8 53 52 68875 5 3ii25 7479' 7 25209 05917 2 94o83 46 860 3 54 9.68897 6 io.3iio3 9.74821 7 10.25179 10.06924 2 9 . 94076 46 i6 5 52 54 8 68920 b 3 1080 7485 1 8 25i49 06981 2 94069 44 17 5 44 54 16 68942 b 3io58 74880 8 25 120 05938 2- 04062 43 i8 5 36 54 24 68965 7 3io35 74910 9 25090 05945 2 94o55 42 12 20 5 28 54 32 68987 7 3ioi3 74939 9 25o6i 06962 2 94048 4i 8 5 20 3 54 4o 9 . 690 1 7 1 . 30990 9.74969 10 io.25o3i 10.06959 2 9.9404. 4o 21 5 12 54 48 69032 8 30968 74998 10 250O2 06966 3 94o34 39 22 5 4 54 56 69055 8 30945 75028 1 1 24972 06978 3 94027 38 23 4 56 55 4 69077 9 30923 75o58 1 1 24942 06980 3 94020 37 24 25 4 48 55 12 69100 9 30900 75087 12 24913 06988 3 94012 36 8 4 4o 3 55 20 9.69122 9 10.30878 9.75117 12 10. 24883 10.06996 3 9.94006 35 26 4 32 55 28 69144 lu 3o856 75i46 i3 24854 06002 3 93998 34 27 4 24 55 36 69167 JO 3o833 75.76 i3 24824 06009 3 93991 33 28 4 16 55 44 69189 10 3o8ii 752o5 i4 24795 06016 3 93984 32 29 3o 4 8 55 52 69212 9.69234 II II 30788 75235 14 24765 06028 3 93977 3i 8 4 3 56 10.30766 9.75264 i5 10.24736 io.o6o3o 4 9.98970 3o 3i 3 52 56 8 69256 12 30744 75294 i5 24706 06087 4 93963 29 32 3 44 56 16 69279 12 30721 75323 16 24677 06045 4 98955 28 33 3 36 56 24 69301 12 30699 75353 16 24647 06062 4 93948 27 34 35 3 28 56 32 69323 i3 80677 75382 17 24618 06069 4 93941 26 8 3 20 3 56 40 9.69345 i3 io.3o655 9.75411 17 zo. 24589 10.06066 4 9.93934 25 36 3 12 56 48 69368 i3 3u632 7544 i 18 24559 06078 4 93927 24 37 3 4 56 56 69390 i4 3o6 1 75470 18 2453u 06080 4 98920 23 38 2 56 57 4 69412 i4 3o58S 75500 19 245ou 06088 5 98912 22 39 4o 2 48 57 12 69434 9.69456 i5 i5 3o566 io.3o544 75529 19 24471 06095 5 98906 2. 8 2 4o 3 57 an 9.75558 20 10.24442 10.06102 5 9.98898 20 4i 2 32 57 28 69479 i5 3o52i 75588 20 24412 06109 5 93891 19 42 2 24 57 36 69501 16 3o499 75617 21 24383 061 16 5 98884 18 43 2 16 57 44 69523 lb 3o477 75647 21 24353 06124 5 93876 '7 44 45 2 8 57 52 69545 16 3o455 75676 22 24324 oGi3i 5 98869 lb 820 3 58 9.69567 17 jo.3o433 9.75705 22 10.24295 io.o6i38 5 9.93862 i5 46 I 52 58 8 69589 17 3o4ii 75735 23 24265 06145 b 93865 i4 47 f 44 58 16 6961 1 17 3o389 75764 23 24286 061 53 b 93847 18 48 I 36 58 24 69633 18 3o367 75793 24 24207 06160 b 98840 12 49 5o I 28 58 32 69655 18 3o345 75822 24 24178 06167 b 93833 IT 8 1 20 3 58 4o 9.69677 19 ■io.3o323 9-75852 25 io.24i48 10.06174 6 9.93826 10 5i 1 12 58 48 69699 19 3o3o 1 75881 25 241 19 06181 6 93819 9 52 . 4 58 56 69721 19 30279 7D910 26 24090 06189 b 98811 b 53 56 59 4 69743 2(J 3o25^ 75939 26 24061 06196 b 93So4 7 54 55 48 59 12 69765 20 30235 75969 27 24o3i 06208 6 93797 b 8 Ao 3 59 20 9.69787 20 io.3o2i3 9.75998 27 10.24002 10.0621 1 7 9.93789 5 56 32 59 28 69809 21 80191 76027 28 28978 06218 7 98782 4 57 24 59 36 69831 21 80169 76056 28 28944 06226 7 98773 3 58 16 59 44 69853 22 3oi47 76086 29 28914 06282 7 93768 2 5q 8 59 52 69875 22 30125 761 1 5 29 23885 06240 7 93760 I 60 M 4 69897 22 3oio3 76144 29 23856 06247 7 93753 Hour P.M. Hour A.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Ditr. Sin<;. M 119° A A B B ( ^ 1' 2^ 3= 4s .5^ 6' 7" Prop, parts of cols. I c 3 4 6 7 a 8 11 3 i5 4 14 iS i 17 22 5 20 26 6 GO* TABLE XXVIL • [Page 215 S' Leg. S ncs, Tangents, and Secants. G'. 30 3 A A B B C C 149° M Hour A.M. Hour P.M. S'uio, Ditr. Coserant. Tangent. Diff. Colang-cnl Secant. Diir. Cosine. 6^ 800 400 9.69S97 io.3oio3 9.76144 10.23856 10.06247 9.93753 1 7 59 52 8 69919 3oo8i 76173 23827 06254 93746 59 2 59 44 16 69941 I 3oo59 76202 1 23798 06262 93738 58 3 59 36 24 69963 I 3oo37 76231 I 23769 06269 9373. 57 4 5 59 28 32 69984 3ooi6 76261 2 23739 06276 93724 56 55 7 59 20 4 40 9 . 70006 2 10.29994 9.76290 2 10.23710 10.06283 9.937.7 6 59 12 48 70028 2 29972 76319 3 23681 06291 93709 54 7 59 4 56 70o5o 3 29950 76348 3 23652 06298 9370? 53 8 58 56 I 4 70072 3 29928 76377 4 23623 o63o5 936y5 52 _9 lO 58 48 I 12 70093 3 29907 76406 4 23594 10. 23565 o63i3 93687 5i 5c 7 58 40 4 I 20 9.701 i5 4 10.29885 9,76435 5 io.o632o 9.936S0 1 1 58 32 I 28 70137 4 29863 76464 5 23536 06327 93673 49 12 58 24 I 36 70159 4 29841 76493 6 23507 06335 93665 48 i3 58 16 I 44 70180 5 29820 76522 6 23478 06342 2 93658 4- i4 i5 58 8 I 52 70202 5 29798 76551 9.76580 7 7 23449 o635o 2 9365o -16 45 7 58 420 9.70224 5 10.29776 10.23420 10.06357 2 9.93643 i6 57 52 2 8 70245 b 29755 76609 8 23391 06364 2 93636 -n 17 57 44 2 16 70267 b 29733 76639 8 2336i 06372 2 93628 43 i8 57 36 2 24 70288 6 29712 76668 9 23332 06379 2 9362. A-' !9 20 57 28 2 32 7o3io 7 29690 76697 9.76725 _9. 10 233o3 o6386 2 936.4 4. 4o 7 57 20 4 2 40 9.70332 7 10.29668 10.23275 10.06394 2 9.93606 21 57 12 2 48 70353 8 29647 76754 10 23246 06401 3 93599 3g 22 57 4 2 56 70375 8 29625 76783 1 1 23217 06409 3 93591 38 23 56 56 3 4 70396 8 29604 76812 1 1 23i88 o64i6 • 3 93584 37 24 25 56 48 3 12 70418 9 29582 76841 12 23i59 06423 3 93577 36 35 7 56 4o 4 3 20 9.70439 9 10.29561 9.76870 12 io.23i3o 10.06431 3 9.93569 26 56 32 3 28 70461 9 29539 76899 i3 23l01 06438 3 93562 34 27 56 24 3 36 70482 10 29518 76928 i3 23072 06446 3 93554 33 28 56 16 3 44 7o5o4 10 29496 76957 i3 23o43 06453 3 93547 32 29 3o 56 8 3 52 70525 10 29475 76986 9.77015 i4 1 4 23oi4 10.22985 06461 10.06468 4 4 93539 9.93532 3i 3^. 7 56 440 9.70547 II 10.29453 3i 55 52 4 8 7o568 1 1 29432 77044 lb 22956 06475 4 93525 20 32 55 44 4 16 70590 11 39410 77073 i5 22927 o6483 4 93517 28 33 55 36 4 24 7061 1 12 29389 77101 lb 22899 00490 4 9351c 27 34 35 55 38 4 3} 70633 12 29367 77i3o lb 22870 06^8 4 93502 26 I5 7 55 20 4 4 40 9.70654 i3 10.29346 9.77159 17 10.22841 io.o65o5 4 9.93495 3b 55 12 4 48 70675 i3 29325 77188 17 22812 o65i3 4 93487 ■}4 37 55 4 4 56 .70697 i3 29303 77217 18 22783 06520 5 93480 23 3« 54 56 5 4 70718 i4 29282 77246 18 22754 06528 5 93472 2/ 39 4o 54 48 5 12 70739 i4 29261 77274 '9 22726 o6535 10.06543 5 ~5~ . 93465 9.93457 21 20 7 54 4o 4 5 20 9.70761 i4 10.29239 9.77303 •9 10.22697 41 54 32 5 a8 70783 i5 29318 77332 20 22668 o655o 5 93450 19 42 54 24 5 36 70803 i5 29197 7736. 20 22639 06558 5 93442 .8 43 54 16 5 44 70824 i5 29176 7739(1 21 22610 06565 5 93435 17 44 45 54 8 5 J2 70846 9 . 70867 16 29154 774.8 21 33582 06573 5 93427 16 75 7 54 460 16 10.29133 9-77447 22 10.22553 io.o658o 6 9.93420 4b 53 52 6 8 70888 lb 29112 77476 22 22524 06588 6 q34i2 i4 47 53 44 6 16 70909 17 29091 775o5 23 22495 06595 6 q34<>5 i3 4« 53 36 6 24 70931 17 29069 77533 23 22467 o66o3 6 93397 12 49 5o 53 28 6 32 70952 18 29048 77562 24 22438 10.2 2409 06610 10.06618 6 IT 93390 9.93382 1 1 .0 7 53 20 4 6 40 9.70973 18 10.29027 9.7759. 24 5i 53 12 6 48 70994 18 29006 77619 25 22381 06625 6 93375 9 52 53 4 6 56 71015 '9 289S5 77648 25 22352 o6633 6. 93367 8 53 52 56 7 4 7io36 19 28964 77677 26 22333 06640 7 93360 7 54 55 52 48 7 12 7io58 '9 28942 77706 26 22294 06648 7 93352 6 ~5 7 52 4o 4 7 20 9.71079 20 10.28921 9.77734 26 10. 22266 io.o6656 7 9 93344 5b 52 32 7 28 71100 20 28900 777631 27 22237 06663 7 93337 .'i ^7 52 24 7 36 71121 20 28879 77791! 27 22209 0667 1 7 93329 3 M 52 16 7 44 71 142 21 28858 77820 28 22180 06678 7 93322 2 59 52 8 7 52 71163 21 28837 77849 28 22l5l 06686 7 933i4 I bo M 52 8 71184 21 28816 778771 29 22123 06693 7 93307 c M Hour P.M. Hour .\.m fosliip. Diflr. Secant. Cotang-enlDlflr. Taii£:ent. Cosccnnl. DifT. Sine. 120= Seconds of time 1 2 3 4 5 6 7 Prop, parts of cols. <. B (c 3 4 I 5 7 2 8 1 1 3 1 1 i4 4 i3 18 16 22 6 19 25 7 Page2lG] • TABLE XXVII. .5'. Lo£r. Sines, Tangents, and Secant.s. G' 31 3 A A B B C C 148° o Hour A.M. Hour P.M. Sine. Ditr. Cosecant. 10.28816 Tangent. Diff. Cotangent .Secant. Diir Cosine. 60 7 52 480 9.71184 9.77877 10.22123 10.06693 9.93307 I 5i 52 8 8 7[2o5 28795 77906 2209^ 0670 1 93299 5q 2 5 1 44 8 16 71226 I 28774 77935 I 22065 06709 93291 58 3 5r 36 8 24 71247 I 28753 77963 1 22037 067 1 6 93284 57 4 5 5j 28 8 32 71268 1 28732 77992 2 22008 10.21980 06724 93276 56 55 7 5 1 20 4 8 40 9.71289 2 10.2871 1 9.78020 2 10.06731 9.93269 b 5i 12 8 48 7i3io 2 28690 78049 3 21951 06739 95261 54 7 5i 4 8 56 7i33i 2 28669 78077 3 21923 06747 93253 53 8 5o 56 9 4 7i352 3 28648 78106 4 21894 06754 93246 52 _9 10 5o 48 9 12 4 9 20 71373 3 28627 78135 4 2 1 865 06762 93238 5i 5o 7 5o 40 9.71393 3 10.28607 9.78163 5 10.21837 10.06770 9.93230 II 5o 32 9 28 7i4i4 4 28586 78 1 92 5 21808 06777 93223 49 12 5o 24 9 36 71435 4 28565 78220 b 21780 06785 2 932 i5 48 i3 5o 16 9 44 71456 4 28544 78249 6 21751 06793 2 93207 47 i4 i5 5o 8 952 7'477 5 28523 78277 7 21723 06800 2 93200 46 45 7 5o 4 10 9.71498 5 10.2S502 9.78306 7 10.21694 io.o68r)9 2 9.93192 lb 49 52 10 8 7i5i9 5 28481 78334 8 21666 06816 2 93184 44 17 49 44 10 16 71539 6 28461 78363 8 21637 06823 2 93 '77 43 18 49 30 10 24 7i56o 6 28440 78391 9 2 1 609 o683 1 2 93169 42 !9 20 49 28 10 32 7i58i 7 28419 78419 9 2i58i 06839 2 93161 4i 4o 7 49 20 4 10 4o 9.71602 7 10.28398 9.78448 9 io.2i552 10.06846 3 9.93154 21 49 12 10 48 7lb22 7 28378 78476 10 2l524 06854 3 93t46 3q 22 49 4 10 56 71643 8 28357 785o5 10 21495 06862 3 93 1 38 38 aJ 48 56 • II 4 71664 8 28336 78533 11 21467 06S69 3 93i3i 37 24 25 48 48 II 12 71685 8 283i5 78562 II 21438 06877 3 93123 36 35 7 48 4o 4 II 20 9.71705 9 10.28295 9.78590 12 I0.2l4lO 10.06885 3 9.93115 3b 48 32 11 28 71726 9 28274 78618 12 2i382 06892 3 93108 M 27 48 24 II 36 71747 9 28253 78647 i3 2i353 06900 3 93100 33 28 48 16 II 44 71767 10 28233 78675 i3 2i325 06908 4 93092 32 29 .30 48 8 II 52 71788 10 28212 78704 i4 21296 06916 4 93084 3i 3^ 7 48 4 12 9.718U9 10 10.28191 9.78732 i4 10.21268 10.06923 4 9.93077 3i 47 52 12 8 71829 II 28171 78760 i5 2 1 240 06931 4 93069 29 3?. 47 44 12 16 7i85o II 28i5o 78789 i5 2121 1 06939 4 93061 28 3i 4i 36 12 24 71870 1 1 28i3o 78817 lb 21183 06947 4 93o53 27 34 35 47 28 I^ 32 71891 12 28109 78845 lb 2ii55 06954 4 93o46 26 25 7 47 20 4 12 4" 9-71911 12 10.28089 9.78874 17 10.21 126 10.06962 5 9.93038 Jb 47 12 12 48 71932 12 28068 78902 17 21098 06970 5 93o3o 24 J7 47 4 12 56 71952 i3 28048 78930 17 21070 06978 5 93022 23 38 46 56 i3 4 71973 i3 28027 78959 18 2 104 1 06986 5 93oi4 22 39 4o 46 48 i3 12 71994 9.72014 i3 "i4 28006 10.27986 78987 18 2IOl3 10.20985 06993 5 93007 21 20 7 46 40 4 i3 Qo 9.79015 19 1 . 0700 1 5 9.92999 41 46 3? i3 28 72034 i4 27966 79043 19 20957 07009 5 92991 19 42 46 24 i3 36 72055 i4 27945 79072 20 20928 07017 5 929S3 18 43 46 16 i3 44 72075 i5 27925 79100 20 20900 07024 6 92976 17 44 45 46 8 i3 52 72096 i5 27904 79128 21 20872 07032 6 92968 lb 75 7 46 4 i4 9.721 16 i5 10.27884 9.79156 21 10.20844 10.07040 6 9.92960 4b 45 52 i4 8 72137 16 27863 79185 22 208 1 5 07048 6 92952 i4 47 45 44 i4 16 72157 16 27843 79213 22 20787 07056 6 92944 i3 48 45 36 i4 24 72177 16 27823 79241 23 20759 07064 5 92936 12 49 5o 45 28 i4 32 73198 17 27802 10.27782 79269 23 20731 07071 b 92929 II 10 7 45 20 4 i4 4o 9.72218 17 9.79297 24 10.20703 10.07079 6 9.92921 bi 45 12 i4 48 72238 18 27762 79326 24 20D74 07087 7 92913 9 .12 45 4. i4 56 72259 18 27741 79354 25 2064b 07095 7 92905 8 i>3 44 56 i5 4 72279 18 27721 79382 25 20618 07103 7 92897 7 .'14 55 44 48 i5 12 72299 '9 27701 79410 2b 20590 071 1 1 7 92889 6 5 7 44 4o 4 1 5 20 9.72320 '9 10.27680 9.7943s 26 10.20562 10.07119 7 9.92881 5b 44 32 i5 26 _ 72340 19 27660 79466 26 20534 07126 7 92874 4 57 44 24 i5 36 72360 20 27640 79495 27 2o5o5 07134 7 92866 3 58 44 16 i5 44 72381 20 27619 79523 27 20477 07142 7 92858 2 59 44 8 i5 52 72401 20 27599 79551 28 20449 O7i5o 8 92850 I bii M 44 16 72421 21 27579 79579 28 20421 07 1 58 8 92842 M Hour P.M. Hour A.M. Cosine. Difl-. Secant. Cotangent Diff. Tangent. Cosecant. Dift: Sine. 12r A A B B C Seconds of time . 3 4 I 2» 5 7 St 3' 8 II 3 4- 10 i4 4 5' i3 18 5 6' i5 21 6 7. 18 25 1 Prop, parts of cols !■ C 58« TABLE XXVIL -1 [Page 217 .v. Log. Sines, Tangents, and S ecants. G- 33' M o A A B B c C 147° Hour A.M. Hour P.M. Sine. |Difi'. Cosecant. Tang-ent. Diir. Cotangent Secant. Din: Cosine. M 60 7 44 4 16 9. 7242 J 10.27579 9.79579 10.20421 10.07158 9.92842 I 43 52 16 8 7244r 27559 79607 20393 07166 92834 ^9 2 43 44 16 16 72461 I 27539 79635 I 2o365 07174 92826 58 3 43 36 16 24 72482 I 27518 79663 I 20337 07182 92818 !57 4 5 43 28 16 32 72502 1 27498 79691 2 2o3o9 07190 92810 'Jb 55 7 43 20 4 16 4o 9.72522 2 10.27478 9.79719 2 10.20281 10.07197 9.92803 6 43 12 16 48 72542 2 2745? 79747 3 20253 07205 92795 54 - 43 4 16 -56 72562 2 27438 79776 3 20224 07213 92787 53 8 42 56 17 4 72582 3 27418 79804 4 20 1 96 07221 92779 52 _9 10 42 48 17 12 72602 3 27398 79832 4 20168 07229 92771 0.93763 5^ 7 42 4" 4 17 20 9.72622 3 10.27378 9.79860 5 10.20140 10.07237 1 1 42 32 17 28 72643 4 27357 79888 5 201 12 07245 92755 ^9 13 42 24 17 36 72663 4 27337 79916 6 20084 07253 92747 4 b i3 42 16 17 44 72683 4 27317 79944 6 2oo56 07261 2 92739 47 i4 i5 42 8 17 52 72703 5 27297 79972 7 20028 07269 2 92731 40 45 7 42 4 18 9.72723 5 10.27277 9 . 80000 7 1 . 20000 10.07277 2 9.92723 1 6 4 1 52 18 8 72743 6 27257 80028 7 19972 07285 2 92715 44 I- 4i 4i 18 16 72763 6 27237 8oo56 8 19944 07293 2 92707 43 i8 4 1 36 18 24 72783 6 27217 80084 8 1 99 16 07301 2 92699 42 !9 ao 4i 28 18 32 72803 6 27197 801 12 9 1988S 07309 10.07317 3 92691 9.92683 41 4o 7 4i 20 4 18 40 9.72S23 7 10.27177 9.80140 9 1 . 1 9860 21 4i 12 18 48 72843 7 27157 80168 10 19832 07325 3 92675 39 22 4t 4 18 56 72S63 7 27137 80195 10 19805 07333 3 92667 38 23 40 56 19 4 72883 8 27117 80233 1 1 19777 07341 3 92659 37 24 25 4o 48 19 12 72902 8 27098 8025 1 1 1 19749 07349 3 9265 1 3b 35 7 4o 4»' 4 19 20 9.72922 8 10.27078 9.80279 12 10. 19721 ic. 07357 3 9.92643 26 40 32 19 28 72942 9 27058 8o3o7 12 19693 07365 3 92635 34 27 40 24 19 36 72962 9 27038 8o335 i3 19665 07373 4 92627 ii 28 4o 16 19 44 72982 9 27018 8o363 i3 19637 0738 1 4 92619 32 ^9 3o 40 8 19 52 73002 10 26998 80391 i3 1 9609 07389 10.07397 4 9261 1 3i 3^ 7 4o 4 20 9.73«22 10 10.26978 9.80419 14 10. 19581 4 9.92603 3i 39 52 20 8 73o4i 10 26959 80447 14 19553 074o5 4 92595 29 32 39 44 20 16 73i)6 1 II 26939 80474 i5 19526 0741 3 4 92587 2b 3J 39 36 20 24 73o8 1 1 1 26919 8o5o2 i5 19498 07421 4 92579 27 34 35 39 28 20 32 73 101 9.73121 1 1 26899 8o53o lb ■ 19470 07429 .5 92571 2b ^5 7 39 20 4 20 4(1 12 10.26879 9.8o558 16 10.19442 10.07437 5 9.92563 36 39 12 20 48 73i4o 12 26860 8o586 17 19414 07445 5 92555 24 37 39 4 20 56 73 160 12 26840 80614 17 19386 07454 5 92546 23 38 38 56 • 21 4 73180 i3 26820 80642 18 19358 07462 5 92538 22 39 40 38 48 21 12 73200 i3 26800 80669 18 1 9331 07470 5 92530 21 20 7 38 40 4 21 20 9.73219 i3 10.26781 9.80697 19 10. 19303 10.07478 5 9.93522 4i 38 32 21 28 73239 i4 26761 80725 '9 19275 07486 6 92514 '9 42 38 24 21 36 73259 i4 26741 80753 20 19247 07494 6 92506 18 43 38 16 21 44 73278 14 26722 80781 20 19219 07502 6 92498 '7 44 4!^ 38 8 21 52 73298 i5 26702 80808 20 19192 07510 6 92490 lb 75 7 38 4 22 9.73318 i5 10.26682 9.80836 21 10.19164 10.07518 6 9.92482 40 37 52 22 8 73337 i5 26663 S0864 21 19136 07527 6 92473 14 47 37 44 22 16 73357 16 26643 80892 22 19108 07535 6 92465 i3 48 37 3() 22 24 73377 16 26623 80919 22 1 908 1 07543 b 93457 12 19 5ci 37 28 22 32 73396 16 26604 80947 23 19053 07551 7 93449 1 1 10 7 37 2C 4 22 4'-' 9.73416 17 10. 26584 9.80975 23 10.19025 10.07559 7 9.93441 DI 37 12 22 48 73435 17 26565 8ioo3 24 18997 07567 7 93433 9 bi 37 4 22 00 73455 '7 26545 8io3o 24 18970 07575 7 93425 8 53 36 56 23 4 73474 18 26526 8io58 25 18942 07584 7 93416 7 54 55 36 48 23 12 73494 I? 265o6 81086 25 18914 07592 7 93408 ~5 7 36 4o 4 23 2C1 9.73513 18 io.26'i87 9.81113 26 10.18887 10.07600 7 9.92400 d6 36 32 23 28 73533 19 26467 8ii4i ?6 18859 07608 8 92392 4 37 36 24 23 36 73552 19 26448 81169 26 i883i 076 1 6 8 92 384 3 :)y 36 16 23 ^^ 73572 •9 26428 81196 27 18804 07624 8 92376 2 ^9 36 8 23 52 73591 20 26409 81224 27 18776 07633 8 92367 1 0(1 36 24 736 1 1 20 26389 81252 28 1874s 07641 8 . 9' ^^9 Sntc. [Tour P.M. Hour .\.M. Cosino. DilT. Recant. Cotangent DilT. Tangent. Cosecant Diir. 122=^ A A B B C Seconds of time !• 2» 3' 4- 5' •6' 7' (^ 2 5 7 10 12 i5 >7 Prop, ptirts of cols. " 3 7 10 i4 17 21 24 (c I 2 3 4 5 6 7 C 57" '7 28 -ji 3i 36 74700 18 2 53oo 82817 26 17183 08117 8 91883 3 58 28 16 3 1 44 74719 18 26281 82844 27 17166 08126 8 91874 2 ^9 28 6 3i 52 74737 19 26263 82871 27 17129 08 1 34 8 91866 I bo I\I 28 o| 32 74756 19 26244 8280Q 27 17101 08143 8 91867 Uourr.M lour A.M. Cosine. DiflT. Secant. Cotangent Diff. Tangent. Cosecant. DilT. Sine. | 123° 5e» P 2'^ 3' 4' 5' & '*'« r (^ 2 5 7 10 12 i4 17 Prop, parts of cols. ^ -) 7 10 i4 17 21 24 (c I 2 3 4 5 6 1 TABLE XXVIJ ['"='«« -19 Log. Sines, Tangents, and Secants. '^'• 34^ A A B B C C 345° o I 2 3 4 "5 6 8 _9 10 1 1 12 i3 i4 i5 i6 17 i8 11 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 5o 5. 52 53 54 55 56 57 58 60 M Hour A.M Houip.M. Sine. DiflT Cosecant. Tangent. Ditr. Cotang-ent Secant. Diir. Cosine. M 60 59 58 57 56 55 54 53 52 5. 5o t 4i 46 45 44 43 42 4i 4o 39 38 37 36 35 34 33 32 3. 3Z 29 28 27 26 J5 24 23 22 2. 20 19 .8 17 .6 75 i4 i3 12 11 10 9 8 7 6 5 4 3 2 I 7 28 27 52 27 4^' 27 3t) 27 28 4 32 32 8 32 16 32 24 32 32 9-74756 74775 74794 748(2 7483 1 I I I 10.25244 25225 25206 25.88 25.69 9.82899 82926 82953 82980 83oo8 I I 2 10. 17.01 17074 17047 17020 16992 10.08.43 08. 5. 08160 08.68 08.77 9.9.857 9.849 91840 9.832 91823 7 27 20 27 12 27 4 26 56 26 48 7 26 4'-> 26 32 26 24 26 16 26 8 4 32 4u 32 48 32 56 33 4 33 12 9-74850 74868 74887 74906 74924 2 2 2 2 3 io.25.5o 25.32 25ii3 25094 25076 9.83o35 83o62 83089 83.17 83.44 2 3 ■ 3 4 4 10.16965 .6938 169. . 16883 1 6856 10.08185 08.94 0S202 0S2 1 . 082.9 .0. 08228 08237 08245 08254 08262 9.9.8.5 91806 9' 798 9.789 9.78. 4 33 20 33 28 33 36 33 44 33 52 9.74943 74961 74980 74999 75017 3 3 4 4 4 10 25o57 25o39 25020 25oo. 24983 9.83.7. 83198 83225 83252 83280 5 5 5 6 6 10.16829 16802 16775 16748 16720 2 2 2 2 9.9.772 9.763 9.755 91746 91738 7 26 25 52 25 44 2 5 36 25 28 4 34 34 8 34 16 34 24 34 32 9.75036 75o54 75073 75091 75 1 10 5 5 5 6 6 10.24964 24946 24927 24909 24890 9-83307 83334 8336. 83388 834.5 7 7 8 8 9 10. 16693 16666 16639 16612 16585 .0.0827] 08280 08288 08297 o83o5 2 2 2 3 3 9.91729 91720 917.2 9.703 91695 7 25 20 25 12 25 4 24 56 24 48 4 34 4o 34 48 34 56 35 4 35 12 .9.75128 75i47 75i65 75.84 7520-;' 6 6 7 7 7 10.24872 24853 24835 24816 24798 9-83442 83470 83497 83524 83551 9 9 10 10 II 10.16558 i653o i65o3 16476 16449 io'.o83.4 o832 3 o833i o834o 08349 3 3 3 3 3 9.91686 91677 91669 9 1 660 9.65. 7 24 4o 24 32 24 24 24 16 24 8 4 35 20 35 28 35 36 35 44 35 52 9.75221 75239 75258 75276 75294 8 8 8 9 9 9 9 .0 .0 10 10.24779 2476. 24742 24724 24706 9.83578 836o5 83632 83659 83686 I . 12 12 i3 i3 10.16422 16395 i636S i634i i63i4 10.08357 o8366 08375 08383 08392 4 4 4 4 4 9.91643 9.634 91625 916.7 91608 7 24 23 52 23 44 23 36 23 28 4 36 36 8 36 16 36 24 36 32 9.75313 75331 75350 75368 75386 10.24(187 24669 2465o 24632 24614 9.837.3 83740 83768 83795 83822 i4 14 1 4 i5 i5 ID. 16287 16260 16232 16205 16178 10.08401 08409 084.8 08427 08435 4 4 5 5 5 9.9.599 9.591 9.582 9'573 9.565 7 23 20 23 12 23 4 22 56 22 48 4 36 40 36 48 36 56 37 4 37 12 9.75405 75423 75441 75459 75478 1 1 II 1 1 12 12 10.24595 24577 24559 24541 24522 9.83849 83876 83903 83930 83957 16 16 '7 17 18 10. i6i5i 16124 16097 1 6070 16043 10.08444 08453 08462 08470 08479 5 5 5 5 6 9.9. ::T 91547 9.538 9i53o 9.521 7 22 4o 22 32 22 24 22 16 22 8 4 37 an 37 28 37 36 37 44 3j 52 9.75496 755i4 75533 7555! 75569 .2 .3 i3 i3 i3 io.245o4 24486 24467 24449 2443 1 9.83984 84oi 1 84o38 84o65 84092 18 18 '9 ■9 20 1 . 1 60 1 6 15989 15962 15935 1 5908 .0.08488 0S496 oS5o5 o85.4 o8523 6 6 6 6 6 9.915.2 9i5o4 91495 9.486 91477 7 22 21 52 21 44 21 36 21 28 4 38 38 8 38 16 38 24 38 32 9.75587 756o5 75624 75642 75660 9.75678 75696 757.4 75733 75751 14 i4 i4 i5 i5 75 16 16 16 17 io.244i3 24395 24376 24358 24340 9.84 1 19 84.46 84.73 84200 84227 20 21 21 22 22 10..588I i5854 i5827 i58oo 15773 io.c)853i o854o 08549 o8558 08567 7 7 7 7 7 9-91469 9 1 460 9145. 9.442 91433 7 21 20 21 12 21 4 20 56 20 48 4 38 4o 38 48 38 56 39 4 39 12 10.24322 24304 24286 24267 24249 9.84254 84280 84307 84334 8436. 23 23 23 24 24 10.15746 15720 15693 15666 1 5639 10.08575 08584 08593 08602 08611 7 7 8 8 8 9.9.425 9.416 9.407 9.398 9.389 7 20 4" 2(. 32 20 24 20 16 20 8 20 4 39 20 39 28 39 36 39 44 39 52 40 9.75769 75787 758o5 75823 7584. 75859 17 17 17 18 18 18 10.2423. 24213 24.95 24.77 24.59 24.41 9.84388 844 1 5 84442 84469 84496 84523 25 25 26 26 27 27 10. 1 56 12 i5585 1 5558 i553. i55o4 15477 10.08619 08628 08637 08646 08655 08664 8 8 8 8 9 9 9.9.381 91372 91363 9.354 91345 9.336 IIourP.M.|HourA-Bi. Cosine. Diff. Secant. Co(angent|Difl'. Tangent. Cosecant. Diff Sinc- rM° Seconds of time 1' 2= 3' 4' 9 i4 5' II 17 5 6' i4 20 7 7' 16 24 8 Prop, parts of cols. < B ( C 2 3 I 5 7 2 7 10 3 1\ ISP 220] TABLE XXVIL S'. Log ^ Sines, Tangents, and Secants. g: 35 A A B B C C 144° Hoar A.M. Hour P.M. 4 4o Sine. bifl-. Cosecant. Tang-ent. DilT. Colang^onl Secant. Diir. Cosint;. M 60 7 20 9.75859 Io.24i4i 9.84523 10. 1 5477 10.08664 9.91336 1 19 52 4o 8 75877 24123 8455o i545o 08672 91328 5q 2 19 4 i 4o 16 75895 I 24io5 84576 I 15424 08681 91319 58 3 19 36 4o 24 759.3 I 24087 846o3 1 15397 08690 9i3io 57 4 5 19 28 4o 32 4 4o 4o 75931 I 24069 io.24o5i 8463o 2 15370 08699 I 9i3oi 56 55 7 19 20 9.75949 I 9.84657 2 10.15343 10.08708 I 9.91292 19 12 4o 48 75967 2 24o33 84684 3 i53i6 08717 I 91283 54 7 .9 4 40 56 75985 2 340 1 5 84711 3 15289 08726 I 91274 53 « 18 56 4i 4 76003 3 23997 84738 4 15262 08734 I 91266 52 10 iS 4S 4r 13 7602 I 3 [0. 23961 S4764 9.84791 4 4 15236 08743 I 91257 9.91248 5i 5o 7 18 4o 4 4i 20 9.76039 3 10. l520Q 10.08753 2 1 1 18 32 4 1 28 76057 3 23943 84818 5 ■ i5i82 08761 2 91239 49 12 18 24 4 1 36 76075 4 23925 84845 5 i5i55 08770 2 9 1 2 3o 48 i3 18 16 4i 44 ■ 76093 4 23907 84872 b i5i28 08779 2 91221 47 i4 i5 18 8 41 52 761 1 1 4 23889 10.23S71 84899 b i5ioi 08788 2 91212 46 45 7 iS 4 42 9.76129 4 9.84935 7 io.i5o75 10.08797 2 9.91203 i6 17 52 42 8 76146 5 23854 84952 7 i5o48 08806 2 91 194 44 17 17 44 42 iG 76164 5 23836 84979 8 l502I 08815 3 91185 43 iS 17 36 42 24 76182 b 238i8 85oo6 8 14994 08824 3 91176 42 20 17 28 42 32 4 42 4" 76200 6 238oo 85o33 8 14967 08833 3 91 167 4i 40 7 17 20 9.76218 6 10.23782 9.85039 9 10. 14941 10:08842 3 9.91158 21 17 12 42 48 76236 b 23764 85o86 9 I49I4 o885i 3 91149 3q 22 17 4 42 56 76253 b 23747 85ii3 10 14887 08859 3 91141 38 23 16 56 43 4 76271 7 23729 85i4o 10 14860 08868 ^ 91132 37 24 25 16 48 43 12 4 43 20 76289 9.76307 7 2371 1 85 166 1 1 14834 08877 4 91123 36 35 7 16 4u 7 10.23693 9.85193 1 1 10. 14807 10.08886 4 9.91114 3 b 16 32 43 28 76324 8 23676 852 2U 12 147S0 08895 4 91 io5 34 27 16 24 43 36 76342 8 23658 85347 12 14753 08904 4 91096 33 ■>s 16 16 43 44 76360 8 23640 85273 1 3- 14727 0S913 4 91087 32 29 3o 16 8 7 16 43 52 4 44 7637S 9 2 362 2 853oo i3 14700 08922 4 91078 3i 3^ 9.76395 9 io.236c5 9.85327 .3 10.14673 10.08931 5 9.91 069 h 1 5 52 44 8 764 1 3 9 23587 85354 I A 14646 08940 5 9 1 060 29 J 2 1 5 44 44 16 76431 9 -N^t^l^O 85330 i4 14620 08949 5 9io5i 28 3J 1 5 36 44 24 76448 10 23552 85407 1 5 14593 08958 b 91042 27 34 35 i5 28 44 32 76486 10 23534 io.235i6 85434 i5 14566 08967 b 91033 26 25 7 !5 20 4 44 4o 9.76464 10 9.85460 16 10.14540 10.08977 5 9.91023 3o i5 12 44 48 76501 1 1 23499 85487 16 i45i3 089S6 5 91014 2 4 J7 i5 4 44 56 765 I Q 1 1 23i8i 855i4 16 14486 0S995 6 91005 2 3 36 1 4 56 45 4 76537 1 1 23463 85540 17 1 4460 - 09004 6 90996 2 2 4o i4 48 45 12 4 45 20 76554 12 12 23446 10.23428 85567 1 7 14433 09013 6 909S7 21 20 7 1 4 40 9.76572 9.85594 18 1 . 1 44i 16 10.09023 6 9.9097S 4i 1 4 32 45 28 76590 12 • 23410 85620 18 i43So 09031 b 90969 '9 42 i4 24 45 36 76607 12 33393 85647 19 14353 09040 b 9096(.; 18 43 14 16 45 44 76625 i3 23375 85674 19 14326 09049 6 9095 1 '7 45 i4 8 45 52 4 46 76643 i3 23358 10.23340 85700 9".85727 20 20 1 43oo 09058 7 90943 lb 75 7 i4 9 . 76660 i3 10. 14273 10.09067 7 9.90933 4b 1 3 52 46 8 76677 1 4 23323 85754 20 14246 09076 7 90924 14 47 1 3 44 46 16 76695 i4 233o5 85780 21 14220 09085 7 909 1 5 i3 4a 1 3 36 46 24 76712 14 23288 85807 21 14193 09094 7 90906 12 49 5o i3 28 7 i3 20 46 32 4 46 40 76730 i4 23270 85834 22 i4i66 09104 7 90S 96 1 1 10 9.76747 10 10.23253 9.85860 22 10. i4i4o 10.091 i3 8 9.90887 5i i3 12 46 48 76765 i5 23235 85887 23 i4ii3 09122 8 9087S 9 h2 i3 4 46 56 76782 i5 23218 85913 23 14087 ogi3i 8 90S69 8 53 12 56 47 4 76S0O 16 23200 85940 24 i4o6o 09140 8 90860 7 ^1 12 48 47 12 4 47 20 768,7 lb 16 23i83 io.23i65 85967 24 i4o33 09149 8 9085 1 6 7 12 4<> 9.76835 9.85993 24 10. 14007 10.09158 8 9.90843 30 12 32 47 28 76S52 17 23i48 86020 25 13980 09168 8 90833 4 !37 12 24 47 36 76870 17 23i3o 86o46 25 13954 09177 9 90S 2 3 3 i8 12 16 47 44 768S7 17 23 m 3 86073 26 13927 09 1 86 9 90814 2 59 12 8 47 52 76904 17 23096 86100 26 13900 09195 9 90805 1 7l 12 48 76922 18 23078 86126 27 13874 09204 9 90796 Hour F.. 11. Hi)ur.\..M. Cosine. Diir. Secant. Colang-eiit Diir. Tangent. Cosecant. Difl". Sine. 125= A A B B C . 1' 2» 4 7 2 7 10 ■i 4s 9 i3 5 5^ II 17 6 6^ i3 20 7 7' 16 23 8 Prop, parts of cols. i !• 2 3 I C 54« TABLE XXVIL # ■s-' Log. S mes, Tangents, and Secants. C. 3G A A B B C C 143° vr Hour A.M. 7 12 Hour P.M. Sine. Diff. Cosct-nnl. Tang;ent. I)itr.|Cotangent Secant. Diff. Cosine. 60 4 48 9.76922 10.2307S 9.86126 10.13874 10.09204 9,90796 I II D2 48 8 76939 23o6i 86 1 53 1 3847 09213 90787 59 2 1 1 /\A 48 16 76957 I 23o43 86179, 1 1 382 1 09223 90777 58 3 1 1 26 48 24 76974 I 23o26 86206 I 13794 09232 90768 57 4 5 II 28 48 32 76991 I 23009 86u32 9.86259 2 13768 09241 90759 56 55 7 1 1 20 4 48 4o 9.770U9 I 10.22991 2 10. 1 3-41 10.09250 9.90750 6 1 1 12 48 48 77026 2 22974 86285 3 i37i5 09259 9074 1 54 7 II 4 48 56 77043 2 22957 863 12 3 1 3688 09369 90731 53 8 10 5v> 49 4 77061 2 22939 86338 4 1 3662 09278 90722 52 _9 10 10 48 49. '2 77078 3 22922 86365 4 13635 09287 907 1 3 5i 5^ 7 10 4o 4 49 20 9.77095 3 10.22905 9.86392 4 io.i36o8 10.09296 2 9 90704 1 1 10 Sa 49 28 77112 3 22888 864 18 5 1 3582 09306 2 90694 49 15 10 24 49 36 .77i3o 3 22870 86445 5 i3555 09315 2 906H5 48 iJ 10 16 49 ^^ 77147 4 22853 86471 () 13529 0932.4 2 90676 4- i4 i5 10 8 49 52 77164 4 22836 8649^ 6 i35o2 09333 2 90667 46 45 7 10 4 5o 9.77181 4 10.22819 9.86524 7 10. 13476 10.09343 2 9.90657 1 6 952 5o 8 77199 5 22S01 86551 7 13449 09352 2 90648 AA 17 9 44 5o 16 77216 5 22784 86577 7 1342 3 09361 3 90639 43 iS 9 36 5o 24 77233 5 22767 866o3 8 13397 09370 3 90630 42 20 9 28 5o 32 77250 5 22750 10.22732 8663o 8 9 13370 09380 3 90620 4i 4o 7 9 20 4 5o 4o 9.7726S 6 9.86656 (0.13344 10.093S9 3 9 . 906 1 1 21 9 12 5o 48 77285 b 22715 86683 9 i33i7 09398 3 90602 3q 22 9 4 5o 56 77302 6 22698 86709 10 13291 0940S 3 90592 38 23 8 56 5i 4 77319 7 22681 86736 10 1 3264 09417 4 9o583 37 24 25 8 48 5 1 12 77336 7 22664 86762 I 1 i3238 09426 4 90574 36 35 7 8 40 4 5i 20 9.77353 7 10.22647 9.86789 1 1 10. l3211 10.09435 4 9.90565 2(3 8 32 5i 28 ■77370 7 2263o 868 1 5 1 1 i3i85 09445 4 90555 34 27 8 24 5i 36 77387 8 22613 86842 12 i3i58 09454 4 90546 33 28 8 16 5i A^ 774o5 8 22595 86868 12 i3i32 09463 4 90537 32 29 3o 8 8 5i 52 77422 « 22578 86894 i3 i3io6 09473 5 90527 3i 3^ 780 4 ^2 9.77439 9 I0.2256l 9.86921 i3 10. i3o79 10.09483 5 9.90518 Ji 7 52 52 8 77456 9 22544 86947 i4 i3o53 09491 5 90509 29 i> 7 44 52 !6 77473 9 22527 86974 i4 1 3026 09501 5 90499 28 33 7 36 52 24 77490 9 225l0 87000 lb i3ooo 09510 5 90490 27 34 35 7 28 52 32 77507 10 22493 87027 i5 12973 09520 5 90480 26 7 7 20 4 52 40 9.77524 10 10.22476 9.87053 i5 10.12947 10.09529 5 9.90471 3() 7 12 52 48 77541 10 22459 87079 16 1 292 1 09538 6 90462 24 37 7 4 52 56 77558 1 1 22442 87106 16 12894 09548 6 90452 23 38 6 56 53 4 77575 1 1 22425 87132 '7 12868 09557 6 90.443 2 2 39 40 6 48 53 12 77592 1 1 22408 87158 17 12842 09566 6 90434 2 1 20 7 6 4o 4 53 an 9.77609 ' 11 10.22391 9.87185 18 10. 12815 10.09576 6 9.90424 4i 6 32 53 28 77626 12 22374 8721 1 18 12789 09585 6 904 1 5 19 4? 6 24 53 36 77643 12 22357 87238 18 12762 09595 7 90405 18 43 6 16 53 A/i 77660 12 22340 87264 19 12736 09604 7 90396 17 44 45 6 8 53 52 77677 i3 22323 87290 19 12710 09614 7 9o386 9.90377 16 75 760 4 54 9.77694 i3 10.223o6 9.87317 20 10.12683 10.09623 7 40 5 52 54 8 777" i3 22289 87343 20 12657 09632 7 90368 i4 47 5 44 54 16 7772S i3 22272 87369 21 1 263 1 09642 7 9035s . 48 5 36 54 24 77744 14 22256 87396 21 12604 09651 7 90349 12 49 5o 5 28 54 32 77761 14 22239 87422 22 12578 09661 in. 09670 8 90339 1 1 10 7 5 20 4 54 4o 9.77778 i4 10.22222 9.87448 22 10.12552 8 9.90330 5r 5 12 54 48 77795 i5 22205 87475 22 12525 09680 8 90320 9 52 5 4 54 56 77812 i5 22188 87501 2 3 12499 09689 8 9c:>i 1 8 53 4 56 55 4 77829 j5 2217I 87527 2-3 12473 09699 8 9o3o I 7 54 55 4 48 55 12 77846 ij 22l54 87554 24 12446 09708 « 90292 9.90282 b 5 7 4 4o 4 55 20 9.77862 16 10.221 38 9.87580 24 10.12420 10.09718 9 5b 4 32 55 28 77879 16 22121 87606 25 12394 097271 9 90273 4 57 4 24 55 36 77896 16 22104 87633; 25 12367 097371 9 90263 3 58 4 16 55 AA 77913 lb 22087 87659, 26 12341 09746| 9 90254 2 59 4 8 55 52 77930 17 22070 87685, 26 i23i5 09756 9 90244 I bo M 4 56 77946 17 22054 8771 1 26 12209 09765 9 90235 Sine. Hour P.M. Ilor.r A.M. Cosine. Difr. Secant. Cotangent DifT. Tangent. Cosecant. Dili. ■26° 53' Seconds of time 1' 2' 3' 4. 5» 6= 7a Prop, parts of cols. < B 2 3 I 4 7 2 6 10 4 9 i3 5 II 17 6 i3 20 7 i5 23 8 I' ige 9^21 TABLE XXVn. I 6'- Log. S ines, Tangents, and Secants. G'. :J7 Hour A. M A A B B C C 142^ Hour P.M. Sine. Diff Cosecant. Tangent. Diir Cotangent Secant. Diir. Cosine. M 6^ 7 4 c 4 56 9.77946 10 22054 9.87711 10. 12289 10.09765 9.90235 I 3 52 56 8 77963 22007 87738 12262 09775 90225 59 2 3 44 56 16 7798c I 22020 87764 I 12236 09784 90216 58 J 3 36 56 24 77997 1 2 2003 8779a 1 12210 09794 90206 57 4 '5 3 28 56 32 7S013 I 21987 87817I 2 I2i83 09803 90197 56 55 7 3 20 4 56 40 9.7803c I 10.21970 9.87843 2 10.12157 10.09813 9.90187 b 3 12 56 48 78047 2 21953 87869 3 878951 3 12l3l 09822 90178 54 7 3 4 56 56 78063 2 21937 12105 09832 90168 53 8 2 56 57 4 78080 2 21920 87922 3 I207S 09841 90159 5? 9 lO 2 48 57 12 78097 2 21903 87948 4 I2o52 09851 90149 5i 7 2 4o 4 57 20 9.78113 3 10.21887 9.87974 4 10.12026 10. 09861 2 9.90189 II 2 32 57 28 78i3o 3 21870 88000 5 12000 09870 2 90180 4o 12 2 24 57 36 78147 3 21853 88027 3 11973 09880 2 901 20 /iS iJ 2 iT 57 44 78163 4 2,1837 88o53 6 II 947 09889 2 901 1 1 47 i4 i5 2 8 57 52 78180 4 21820 88079 6 II 921 09899 2 90 1 1 46 45 720 4 58 9.78197 4 io.2i8o3 9.88105 7 10. 1 1895 10.09909 2 9 . 9009 1 lb I 5? 58 8 78213 4 21787 88i3i 7 11869 09910 3 90082 44 17 I 44 58 16 78230 5 21770 881 58 7 II842 09928 3 90072 43 iS I 36 58 24 78246 b 21754 88184 8 I18I6 09937 3 90063 42 12 20 I 28 58 32 78263 b 21737 88210 8 1 1790 09947 3 90053 4i 4o 7 I 20 4 58 4o 9.78280 5 10.21720 9.88236 9 10. 1 1764 10.09957 3 9.90043 21 I 12 58 48 78296 b 21704 88262 9 11738 09966 3 90034 39 22 1 4 58 56 783 1 3 b 21687 88289 10 1171 1 09976 4 90024 88 2j 56 59 4 78329 b 2 1 67 1 883 1 5 10 11 685 09986 4 90014 J/ 24 25 48 59 12 78346 7 2 1 654 8834 1 10 1 1659 09995 4 90005 36 35 7 4o 4 59 20 9.78362 7 [0.21638 9. 88367 1 1 10.11633 10. iooo5 4 9.89995 2b 32 59 28 7S379 7 21621 88393 1 1 1 1607 iooi5 4 899S5 84 27 24 59 36 78395 7 2i6o5 88420 12 ii58o 10024 4 89976 33 28 16 5944 7S412 8 2 1 588 88446 12 11554 ioo34 5 89966 3^ 29 3o 8 59 52 78428 8 21572 88472 i3 ii528 10044 5 89956 3i 3(1 700 5 9-78445 8 I0.2I555 9.88498 i3 10. 1 l502 10. ioo5'3 5 9.89947 :ii 6 59 52 8 78461 9 2 1 539 88524 1 4 1 1476 ioo63 b 89987 20 32 59 44 16 7847S 9 21 j22 8855o 14 ii45o 10073 b 89927 28 JJ 59 3b 24 7S494 9 2i5o6 88577 14 ii423 10082 5 89918 27 34 35 59 28 32 785io 9 21490 886o3 i5 II 397 10092 b 89908 26 25 6 59 20 5 4o 9.78527 10 10.21473 9.8S629 i5 10.11371 10. 10102 6 9.89898 3b 59 12 48 78 543 10 21457 88655 16 ii345 10112 6 89888 24 ^7 59 4 56 78560 10 2i44o 88681 16 ii3i9 10121 6 89879 28 38 58 56 I 4 78576 10 21424 88707 17 1 1293 ioi3i 6 89869 22 39 4o 58 48 I 12 78592 1 1 2i4o8 88733 17 1 1267 ioi4i 6 89859 21 20 6 58 4o 5 I 20 9.78609 1 1 10.21391 9.88759 17 10. 11241 io.]oi5i 6 9.89849 41 58 32 I 28 78625 It 21375 8S786 18 11214 10160 7 89840 '9 42 58 24 I 36 78642 12 2i358 88812 18 1 1 188 10170 7 89880 18 43 58 16 I 44 78658 12 21342 88838 19 1 1 162 10180 7 89820 17 44 45 58 8 I 52 7S674 12 2 1 326 88864 >9 1 1 136 10190 7 89S10 9.89801 16 i5 6 58 320 9 . 7S69 I 12 io.2i3o9 9.88890 20 10.111 10 10.10199 7 40 57 52 2 8 78707 i3 21293 88916 20 11084 10209 7 89791 i4 47 57 44 2 16 78723 1 3 21277 8S942 20 11058 10219 8 897S1 i3 48 57 36 2 24 78739 i3 21261 88968 21 II032 10229 8 89771 12 49 5o 57 28 2 32 78756 i3 2 1 244 88994 21 1 1006 10239 8 89761 1 1 6 57 20 5 2 40 9.78772 i4 10.21228 9.89020 22 10.10980 10.10248 8 9.89752 HI '31 57 12 2 48 7878S 14 21212 89046 22 10954 10258 8 89742 9 ■32 57 4 2 56 78805 i4 21 195 89073 23 10927 10268 8 89782 W 56 56 3 4 78821 i5 21179 89099 23 10901 10278 9 89722 7 54 55 56 48 3 12 78837 i5 21 163 89125 24 10875 10288 9 89712 () ~5 6 56 4o 5 3 20 9.78853 i5 10.21 147 9.89151 24 10.10849 10.10298 9 9.89702 5b 56 32 3 28 7S869 ID 21 i3i 89177 24 10823 io3o7 9 89698 4 57 56 24 3 36 78886 lb 2 1 1 1 4 89203 25 10797 io3i7 9 89688 3 58 56 16 3 44 78902 16 2109S 89229 .25 10771 10327 9 89678 2 59 56 8 3 52 78918 16 21082 89255 26 10745 10337 10 89668 I bo 56 4 78934 18 21066 89281 26 1G719 io347 10 89658 M M riourp.M.j [lour A.M. Cosine. DifT. Secant. Cotangent Ditr. Tangent. | Cosecant. Diff. Sine. 127° rj2« 1' 2' 3' 4. 8 18 5 5- 10 i5 6 6= 12 20 7 7° i4 23 8 Prop, parts of cols 2 3 1 4 7 2 6 10 4 TABLE XXVIL [I'n;;e223 S'. Log Sines, Tan gents, and S ccants ''". 38° A A B B c c l4i°| Hour A.M. Hour I'.M. Slue. Diir. Cosecant. Tnngeiil. Diff. Cotangent Secant. Diir. Cosine. M 6^ 6 56 5 4 9.78934 10.21066 9.89281 10. 10719 , ).io347 :9. 89663 I 55 52 4 8 78950 2io5o 89307 10693 io357 ; 89643 D9 3 55 AA 4 16 78967 I 2io33 89333 I 10667 1 0367 89633 bS 3 55 36 4 24 78983 I 21017 89359 I 1 0641 10376 89624 37 4 55 28 4 32 78999 I 21001 89385 2 io6i5 io386 89614 ^6 6^5 5 6 55 20 5 4 4o 9 . 790 1 5 1 10.20985 9.8941 1 2 10.10689 10.10396 9 . 89604 6 55 12 4 48 7903 1 2 20969 89437 3 10563 I o4o6 89694 64 n 55 4 4 56 79047 2 20953 89463 3 10537 io4i6 89604 63 8 54 56 5 4 79063 2 20937 89489 3 1061 1 10426 89674 b2r 9 54 48 6 54 4" 5 12 79079 9.79095 2 tj 2092 1 89515 4 10485 10. 10469 1 04 36 89664 bi 5o to 5 5 20 3 10.20905 9.89541 4 1 . 1 0446 2 9.89664 1 1 54 32 5 28 791 1 1 3 20889 89567 b 10433 10456 2 89644 49 12 54 24 5 36 79128 3 20872 89593 b 10407 io466 2 89634 48 l3 54 16 5 44 79144 3 20856 89619 Jb io38i 10476 2 89624 ■4 / '4 i5 54 8 6 54 5 52 5 6 79 1 60 4 20840 89645 6 io355 J 0486 2 89614 40 46 9.79176 4 10.20824 9.S9671 6 10. 10329 10.10496 3 9.89604 iG 53 52 6 8 79192 4 20808 89697 7 io3o3 io5o6 3 89496 44 17 53 44 6 16 79208 5 20792 89723 7 10277 io5i6 3 89486 4^' 18 53 36 6 24 79224 5 20776 89749 8 10261 10626 3 89476 r) !9 20 53 28 6 32 79240 9.79256 5 20760 89775 8 10226 io636 3 ~3 89466 9.89466 41 4<'. 6 53 20 5 6 40 5 10.20744 9.89801 9 10. 10199 10. 10646 21 53 12 6 48 79272 6 20728 89827 9 10173 10666 4 89445 39 22 53 4 6 56 79288 6 20712 89853 10 10147 10666 4 89435 38 23 52 56 7 4 79304 6 20696 89879 10 10121 10675 4 89426 ^7 24 25 52 48 7 12 79319 6 20681 89906 10 10096 io585 4 89416 35 6 52 4o 5 7 20 9.79335 7 io.2o665 9.89931 II 10. 10069 10. 10696 4 9.89406 2b 52 32 7 28 79351 7 20649 89957 II 10043 10606 4 89396 11 27 52 24 7 36 79367 7 20633 89983 12 10017 1 061 5 b 89386 ii\ 28 52 16 7 44 79383 7 20617 90009 12 09991 10626 b 89376 32 29 3o 52 8 7 52 79^99 8 20601 90035 i3 09966 io636 b 89364 3i 3c 6 52 5 8 9.79415 8 io.2o585 9 . 9006 1 i3 I p. 09939 10.10646 5 9.89354 3i 5 1 52 8 8 79431 8 20569 90086 i3 09914 10666 5 89344 29 32 5 1 44 8 16 79447 8 20553 901 1 2 i4 09888 10666 b 89334 28 33 5i 36 8 24 79463 9 20537 90 1 38 i4 09862 10676 6 89324 27 34 5[ 28 6 5i 20 8 32 79478 9 20522 90164 lb 09836 1 06S6 b 8y3 1 4 26 26 35 5 8 4o 9.79494 9 io.2o5o6 9.90190 i5 10.09810 10. 10696 6 l9.';93o4 30 5r 12 8 48 79510 10 20490 90216 16 09784 10706 ti ' 89594 >A 37 5i 4 8 56 79526 10 20474 90242 16 09768 10716 6 ' 89284 2 3 38 5o 56 9 4 79542 10 20458 90268 16 09732 10726 6 8^274 22 39 40 5o 48 9 12 79558 10 20442 90294 17 09706 10736 7 89264 21 20 6 5o ^0 5 9 20 9.79573 II 10.20427 9.90320 17 10.09680 10. 10746 7 I9.S9264 41 5o 32 9 28 79589 II 2o4l 1 90346 18 09654 10766 7 89244 '9 42 5o 24 9 36 79605 1 1 20395 90371 18 09629 10767 7 89233 18 43 5o 16 9 M 79621 1 1 20379 90397 19 09603 10777 7 89223 '7 44 45 5o 8 9 52 79636 12 2o364 90423 '9 09677 10787 7 89213 lb 75 6 5o 5 10 9.79652 12 10.20348 9.90449 19 10.09661 10. 10797 8 9.89203 40 49 52 10 8 79668 12 20332 90475 20 09626 10S07 8 89193 i4 47 49 44 .0 .6 79684 12 2o3i6 9o5oi 20 09499 1 08 1 7 8 89183 i3 48 49 3(i 10 24 79699 i3 2o3oi 90527 21 09473 10827 8 89173 12 49 5o 5. 49 38 10 32 79715 i3 20285 90553 21 09447 io838 8 89162 1 1 10 6 49 20 5 10 40 9.79731 i3 10.20269 9.90678 22 10.09422 10.1 084s 8 9.89162 49 12 10 48 79746 i4 20254 90604 22 09396 10858 9 89142 9 52 49 4 10 56 79762 i4 20238 9o63o 22 09370 10868 9 89132 8 53 48 56 II 4 79778 i4 20222 90666 23 09344 10878 9 89122 7 54 55 56 57 58 48 48 II 12 79793 i4 20207 90682 23 09318 10888 9 891 12 6 "5 6 48 4o 5 1 1 uo 9.79809 .5 I0.20I9I 9 . 90708 24 10.C9292 10. 10899 9 9.89101 43 32 11 a8 79825 lb 20175 90734 24 O920tj 1 0909 9 89091 4 48 24 II 36 7984(J lb 20160 m? 25 09241 10919 10 89081 ^ 4* 16 11 44 79856 lb 20 1 44 r^K 002 1 5 10929 10 89071 2 60 M 48 8 11 52 79872 i6 20128 908 1 1 26 09189 10940 10 89060 1 48 12 79887 lb 20Il3 90837 26 09 1 63 10960 10 89060 "SX Hour P.M. flour A.M. Cosliio. DiiT. Scrniit. Cntnno^ent Diir. Tangent. Cosecant. DilT. Sine. 128° A A B B C P Os 3' 4' 5» 6^ 7" f^ 2 4 6 8 10 12 i4 Prop parts of cols. < P 3 6 10 i3 16 19 23 fc I 3 4 5 6 8 9 Page 224] TABLE XXVIL 5''. Log. Sines, Tangents, and Secpails. G'. 39° • A A B B C C 140^ M 1 Hour A.M. Ilourr.M. Sine. Dinr. Cosecant. Tangent. Diir. Cotangent .Secant. Diff. Cosine. M 60 5g 6 48 47 52 5 12 12 8 9.79887 79903 •0 10.201 i3 20097 9.90837 90863 10.09163 09137 10.10950 1096',) 9.89050 S9040 2 47 44 12 '.6 79918 I 20082 90889 I 091 1 1 10970 89030 58 J 47 36 12 24 79934 1 20066 90914 1 09086 10980 80020 57 4 5 47 28 6 47 20 12 32 79950 1 20o5o 90940 3 09060 10.09034 1 099 1 I 1 89009 56 55 5 12 40 9.79965 I io.2oo35 9 . 90966 7 10.11 0( ) 1 9.88999 b 47 12 12 48 79981 2 20019 90992 3 09008 1 1 II 88989 54 7 47 4 12 56 79996 2 200011 91018 3 08982 1 1022 8S978 53 • 8 46 56 i3 4 8001 2 2 19988 9 1 04 3 3 08957 Ilo3i 8806S 5^ _9 io 46 48 6 /i& 4o i3. 12 80027 2 19973 91069 _, 0893 1 1 1042 2 ! 88o58 5i r- 5 i3 20 9.80043 3 10. 19937 9.91095 '4 10.08905 10. 1 11)52 2 9 .8E948 2 , 88937 11 46 32 i3 28 8oo58 3 19942 91 121 6 08879 1 io63 40 12 46 24 1 3 36 80074 3 19926 91147 5 08853 11073 2 1 88027 48 IJ AQ 16 i3 AA 800S9 3 .19911 91172 6 0882S iio83 a 88917 88906 9 88896 '17 i4 i5 46 8 i3 52 8oio5 4 19895 91 198 6 08802 1 1094 2 46 45 6 46 5 i4 9.801 20 4 10.19880 9.91224 6 10.08776 1 . 1 1 1 04 3 lb 45 52 r4 8 801 36 4 19864 91250 7 08750 1 1 1 1 4 3 88886 44 17 45 44 i4 16 8oi5i 4 19849 91276 7 08724 III25 3 88S75 43 i8 45 36 i4 24 80166 5 19834 9i3oi 8 08609 iii35 3 88865 42 !9 20 45 28 i4 32 80182 5 19818 91327 8 08673 1 1145 3 88855 4i 4o 6 45 20 5 i4 4o 9.80197 5 10. 19803 9.91353 9 10.0S647 1 . 1 1 1 56 3 9.88844 21 45 12 i4 48 8021 3 5 19787 . 91379 9 08621 1 1 1 66 4 88834 3q 22 45 4 !4 56 80228 b 19772 91404 9 08596 11176 4 8S824 38 2j 44 5b ■ 5 4 80244 () 19756 9i43o 10 08570 1 1 187 4 8881 3 37 24 25 M 48 ID 12 80259 b 19741 91456J 10 08544 io.o85iS 1 1 197 4 888o3 36 35 6 Ai 40 5 i5 2u 9.80274 6 10. 19726 9.91482 11 10.11207 4 9.88790 2b 44 32 i5 28 80290 7 19710 91507 II 08493 11218 5 88782 34 27 A^ 24 i5 36 8o3o5 7 19695 91533 12 08467 1 1228 5 88772 33 28 AA 16 1 5 44 8o32o 7 19680 91559 12 08441 1 1239 5 88761 32 29 3o AA 8 1 5 52 8o336 9.80351 7 ~8" 19664 9i585 12 084 1 5 II 249 5 88751 3i 3«3 6 44 5 16 10. 19649 9.91610 i3 10.08390 10,11259 5 9.88741 Ji 43 52 16 8 8o366 8 19634 9 1 636 i3 08364 1 1 270 5 88730 29 i2 43 44 16 16 8o3S2 8 19618 91662 i4 08338 1 1280 6 88720 28 J J 43 36 16 24 80397 8 1 9603 91 688 i4 o83i2 11291 6 88709 27 M 35 43 28 6 43 20 16 32 8o4 1 2 9 19588 91713 i5 08287 ii3oi 6 88699 26 25 5 16 4<> 9.80428 10. 19572 9.91739 i5 10.08261 !0.1l3l2 6 9.88688 3b 43 12 16 48 80443 9 19557 91765 i5 08235 Il322 6 88678 23 37 43 4 16 56 8o458 9 19542 91791 lb 08209 1 1 332 6 88668 23 3b 42 56 •7 4 80473 H) '.9527 91816 lb 08184 1 1 343 7 88657 22 39 4o 42 48 17 12 80489 10 1951 1 1 . 1 9496 91842 17 081 58 11353 7 88647 9.88636 21 20 6 42 4" 5 17 20 9.8o5o4 10 9.91868 17 io.o8i32 io.ii36-i 7 4i 42 i-j ,7 28 8o5i9 10 1 948 1 91893 18 08107 1 1 374 7 88626 19 42 42 24 17 36 8o534 1 I 19466 91919 18 08081 ii385 7 8861 5 18 4J 42 16 17 44 SoS'o 1 I 19450 91945 18 o8o55 1 1395 7 88605 17 44 45 42 8 17 52 8o565 1 1 19435 91971 19 08029 10.08004 1 14'')6 8 88594 16 i5 6 42 5 18 9.8o58u 12 10. 19420 9.91996 19 to. ii4i6 8 9.88584 4b 4i 52 18 8 80595 12 19405 92022 20 07978 1 1427 8 88573 .4 47 4 1 44 18 16 8u6io 12 19390 92048 20 07952 1 1437 8 88563 ij 48 4 1 36 18 24 80625 12 19375 92073 21 07927 1 1 448 8 88552 12 49 5o 4 1 28 18 32 8064 1 9.80656 l3 19359 92099 21 07901 ii458 9 88542 1 1 11) 6 4i 20 5 18 40 10.19344 9.92125 21 10.07875 10. 1 1469 9 ■9.88531 5i 4i 12 18 48 8067 1 1-! 19329 92 1 5o 22 07850 11479 9 88521 9 h2 4i 4 18 56 806S6 1 3 19314 92176 22 07824 1 1490 9 885 10 8 :)3 4o 56 19 4 80701 1 4 19299 92202 23 07798 ii5oi 9 88499 7 ;>4 55 4o 48 19 12 80716 1 4 19284 92227 23 07773 10.07747 ii5i 1 9 8S4^i9 b 6 4o 4o 5 19 20 9.80731 i4 10.19269 9 92253 24 10. 11 52 2 10 9.88478 5b 4o 32 19 28 80746 14 19254 92279 24 07721 1 1 532 10 88468 4 37 4o 24 19 36 80762 1 5 19238 92304 24 07696 1 1 543 10 88457 3 38 4o 16 19 ^^ 80777 i5 19223 92330 25 07670 ii553 10 8i447 2 59 4o 8 19 52 80793 i5 19208 92356 25 07644 1 1 561 10 . 88436 1 bo 4o 20 80807 i5 19193 92381 26 07619 1 1 575 10 ; 8842 5 Hour P.M. Hour A.M. Cosine. DilT. Secant. Cotang-enl[Difir. Tangent. Cosecant. DiiT. Sine. 129" V. 5(/ Seconds of lime 1' 2» 4 6 3 3^ 6 10 4 4s 8 ,3 5 5- 10 16 12 '9 8 7^1 . 1 Frop. parts of cols. I B » C 2 3 I 13 23 __9_| S'. 40° TABLE XXVII. Log. Sines, Tangents, and Secants. A ° A B B i3 IIourA.M iHourp.M. 23 24 25 26 27 28 2y So 3i 32 33 34 35 36 37 38 39 4o 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 6 4o o 39 52 39 44 39 36 39 28 6 39 M) 39 12 39 4 3H 5(. 38 48 20 o 20 8 20 16 20 24 20 3:> 6 38 4'J 38 32 38 24 38 16 38 8 20 4ii 20 48 20 5() 21 4 21 12 5 21 20 ! I 28 M 36 n 44 52 21 6 38 o\ 37 52 37 44 37 36 37 28 6 37 20 37 J 2 37 4 36 56 36 48 5 22 o 22 8 32 16 2 2 S4 22 32 6 36 40 36 32 36 24 36 16 36 8 6 36 o 35 52 35 44 35 3() 35 28 35 20 35 12 35 4 34 56 34 48 34 4o 34 32 34 26 34 16 34 8 5 22 4" 22 48 22 56 23 4 23 12 5 23 20 23 28 2 3 36 23 44 23 52 5 24 o 24 8 24 16 24 24 24 32 Siiio 9.80811" 80S 2 2 8: .837 808 5 2 80867 80882 80897 809 I 2 80927 80942 Difl: o Cosecant. 9.81254 8 1 269 81284 81 299 8i3i4 24 4" 24 48 24 56 25 4 25 12 34 o 33 52 33 44 33 36 33 2b 33 20 33 12 33 4 32 56 32 4b 32 4o 32 3 32 24 32 16 32 8 32 M Hour p.. ir. Hour A. M 5 25 2fi 25 28 25 36 25 44 25 52 5 26 o 26 8 26 16 26 24 26 32 26 4" 26 48 26 56 27 .4 27 12 10.18968 18953 18939 18924 18909 10.18894 9.81328 81343 81 358 81372 81387 9.81402 81417 8i43i 81446 81461 9.81475 81/190 8i5o5 8i5i9 8i53 27 20 27 28 27 36 27 44 27 52 28 o 9.81549 81 563 81578 81592 81607 9.81622 8 1 636 8i65i 81 665 81680 81694 Cosine 18864 18849 1 883 10.1S820 i88o5 18790 18775 18760 10.18746 18731 18716 18701 18686 Taiiffent. Diir .92381 92407 92433 93458 9248 1 .92510 92535 92 56 1 925S7 9261 2 9.92638 92663 9.689 927 92740 Cotanafcnl 4 5 5 6 6 9.92766' 6 7 7 8 8 10.07619 07593 07567 07542 07516 10.07490 07465 07439 0741 3 07388 10.07362 07337 0731 1 07285 07260 92792 928 1 7 9^843 92868 9.92894 92920 92945 92971 92996 9.93022 93048 93073 93099 93124 10.18672 18657 18642 18628 i86i3 10.18598 i8583 18569 18554 18539 DitT. 10.18525 i85io 18495 i848i 18466 10. i845i 18437 18422 i84o8 18393 10.18378 1 8 364 18349 18335 i832o t83o6 10.07234 07208 07183 07157 07132 o . 07 1 06 07080 07055 07029 07004 10.06978 0695? 06927 06901 06876 Serant. ,11575 ii585 1 1596 1 1 606 1 1617 [Page SW."! G . _C_139° Diir? Cosine jTl 6? 10. 1 1628 1 1 638 1 1649 1 1 660 1 1670 10. 1 1681 1 1692 1 1702 11713 11724 10. 1 1734 1 1745 1 1756 1 1766 1177- 10.11788 1 1 799 1 1 809 1 1820 ii83 io.o6S5o 06825 06799 06773 06748 9.93406 93431 93457 93482 93508 9.93533 93559 93584 93610 93636 9.93661 93687 93712 93738 93763 Secant. 9.93789 93814 93f 93865 93916 10.06722 06697 0667 1 06646 06620 10.06594 06569 06543 o65i8 06492 1 o . 06467 . 0644 1 064 16 06390 o6364 0.1 1842 ii852 1 1 863 1 1874 1 1 885 9.88425 884i5 884o4 88394 88383 9.88372 88362 8835 1 88340 8833o 9 883 1 9 883o8 88298 88287 88276 9.88266 88255 88244 88234 88223 9.882 12 882C1 8819- 8818^ 88169 9.88i58 881 48 88 1 37 88126 8811 5 10. 1 1895 1 1906 11917 1 1928 11939 [o.i 1949 1 1 960 1 197 1 1 1982 11993 I o . 1 2004 120 12025 1 2o36 1 2047 Cotansrent 10.06339 o63i3 06288 06262 06237 23 23 ~^ 24 24 25 2 5 26 Difil".' Tansjent. 10.062 1 1 06186 06 1 60 o6i35 06109 06084 10. 1 2o58 1 2069 1 20S0 1209 12102 9.88105 88094 8808 3 88072 88061 9.8805 1 88<->4o 88039 88018 88007 87996 87985 87975 87964 87953 8 I9. 87942 8 : 87931 87920 87909 8789S 9.S78S7 87877 87866 87855 8784. 9.87833 87822 878 1 1 87800 87789 87778 Cosecant. Dill'. .Sine. 130° A B B vy Seconds of time 1' 2^ 3^ 4^ 7 i3 5 5^ 9 16 6' 1 1 '9 7^ i3 22 I'rop. parts nf cols. < B f C 2 3 ! 4 6 3 6 10 A S'. TABLE XXVII Log. Sines, Tangents, and Secanls. 41 M o A A B B C C 138° Hour A. M Hour P.M. Sine. DiflT Cosecant. Tangent. Diir. Cotangent Secant. Difl- Cosine. 60 6 32 5 a8 9.81 694 I c . 1 83o6 9.9391b 10.06084 10. 12222 9.87778 I 3i 52 28 8 8 1 709 i8?9i 93942 o6o58 12233 87767 5q 2 3i 44 28 16 8i7i>3 18277 93967 1 o6o33 12244 87756 58 3 3 1 3b 28 24 8 i 73fe 1 18262 93993 I 06007 12255 87745 57 4 5 3i 38 28 32 81752 1 18248 ic. 18233 94018 2 05983 72266 87734 5o 55 6 3i 20 5 28 40 9.81767 I 9.94044 2 10.05956 10.12277 9.87723 6 3i 12 28 48 81781 I 18219 94069 3 05931 12288 87712 54 7 3i 4i 28 56 81796 2 18204 94095 3 05905 12299 87701 53 8 3o 56- 29 4 81810 2 18190 94 1 20 3 o588o i23io 8^690 53 _? 10 3o 48 29 12 81825 2 18175 94 1 46 4 o5854 12321 2 87679 5i 5<. 6 3o 4o 5 29 20 9.81839 2 10.18161 9.94171 4 10.05829 10.12332 2 9 . 87668 1 1 3o 32 29 28 8i854 3 i8i46 94 '97 5 o58o3 12343 2 87657 4o 12 3o 24 29 36 81868 3 i8i32 94222 5 05778 12354 2 87646 48 i3 3o 16 29 44 81882 3 18118 94248 b 05753 12365 2 87635 47 i4 i5 3o 8 29 52 81897 3 i8io3 94273 b 05727 12376 3 87624 46 45 6 3o V. 5 3o 9.81911 4 10.18089 9.94299 6 10.05701 10.12387 3 9.87613 i6 29 5? 3o 8 81926 4 18074 94324 7 05676 12399 3 87601 44 17 29 44 3o 16 81940 4 18060 94350 7 o565o I24IO 3 87590 43 i8 29 36 3o 24 81955 4 18045 94375 8 o5625 1 2421 3 87579 42 19 20 29 28 3o 32 81969 5 i8o3i 94401 8 05599 12433 4 87568 4i 4o 6 29 20 5 3o 4o 9.81983 5 10. 18017 9.94426 8 10.05574 10. 13443 4 9.87557 21 29 12 3o 48 8199S 5 18002 94452 9 o554S 12454 4 87546 39 22 29 4 3o 56 82012 5 17988 94477 9 o5523 1 3465 4 87535 38 2j 28 56 3i 4 82026 5 17974 945o3 10 05497 12476 4 87524 37 24 25 28 48 3i 12 82041 6 17959 94528 10 05472 1 2437 4 87513 36 35 6 28 4o 5 3i 20 9.82055 b 10.17945 9.94554 1 1 io.o5446 10. 12499 5 9.87501 26 .28 32 3i 28 82069 6 17931 94579 1 1 05421 I25lO 5 87490 34 27 28 24 3i 36 82084 6 17916 94604 1 1 05396 I252I ^ 87479 33 28 28 16 3i 44 82098 7 17902 9463o 12 05370 12532 5 87468 32 29 28 8 3i 52 82112 7 17^88 94655 12 05345 12543 5 87457 3i 3o 6 28 5 32 9.82126 7 10.17S74 9.94681 i3 10.05319 10. 13554 6 9-87446 27 52 32 8 82141 7 17859 94706 i3 05294 12 566 6 87434 29 32 27 44 32 16 82155 8 17845 94732 i4 05368 12577 b 87423 28 33 27 36 32 24 82169 8 17831 94757 i4 05243 12588 b 87412 27 34 35 27 28 32 32 82184 8 17816 947S3 i4 05217 12599 6 87401 26 l5 6 27 20 5 32 40 9.82198 8 10. 17802 9.94808 i5 10.05193 [o. 12610 7 9.87390 3b 27 12 32 48 82212 9 17788 94834 i5 o5i66 12622 7 87378 24 37 27 4 32 56 82226 9 "7774 94859 16 o5i4i 12633 7 87367 23 38 26 56 33 4 82240 9 17760 94884 16 o5ii6 12644 7 87356 23 39 4o 26 48 33 12 82355 9 17745 94910 17 05090 12655 _7_ 7 87345 21 20 6 26 4o 5 33 20 9.82269 10 10.17731 9.94935 17 io.o5()65 1 . 1 2666 9.87334 4 1 26 32 33 28 82283 10 17717 94961 17 o5o39 12678 8 87322 19 42 26 24 33 36 82297 10 17703 94986 18 o5oi4 12689 8 87311 18 43 26 16 33 44 82311 10 17689 95012 18 04988 12700 8 87300 17 44 45 26 8 33 52 83326 10 17674 95o37 '9 04963 12712 8 87388 lb i5 6 26 5 34 9.82340 II 10.17660 9.95062 19 10.04938 10.12723 8 9.87377 4b 25 52 34 8 82354 II 17646 95088 20 04912 12734 9 87366 14 47 25 44 Z4 16 82368 1 1 17632 95ii3 20 04887 12745 9 87255 1 3 48 25 36 34 24 82382 1 1 17618 95139 20 o486 1 12757 9 87243 13 49 5o 25 28 34 32 82396 13 17604 95164 21 o4836 12768 9 87233 1 1 10 6 25 20 5 34 4o 9.8241Q. 12 10.17590 9 . 95 1 90 21 10.04810 10.12779 9 9.87321 bi 25 12 34 48 82424 12 17576 95215 22 04785 12791 10 87209 9 b2 25 4 34 56 82439 12 17561 95240 22 0476a 12802 10 87198 8 53 24 56 35 4 82453 l3 17547 95266 22 04734 12813 10 87187 7 54 55 24 48 35 12 82467 i3 17533 10.17519 95291 23 04709 12825 10 87175 b ■5 6 24 4<> 5 35 20 9.8248] i3 9.95317 23 I0.04683 10. 12836 10 9.87164 5b 24 3i 35 28 82495 1 3 i75o5 95343 24 04658 12847 10 8-i53 4 57 24 -M 35 36 82509 i4 17491 95368 24 04632 I28f9 II 87141 3 58 14 16 35 44 82523 14 17477 95393 25 04607 12870 II 87130 2 ^9 24 8 35 52 82537 i4 17463 95418 25 04582 12881 1 1 87119 I bo M 24 36 82551 i4 17449 95444 25 04556 12893 11 87107 M Hourp.M. IIourA.M. Cosine. Diff. Secant. Cotangent Diff. Tangent. Cosecant. Diff. Sine. 131' A A B B C Seconds of time V 2' 3^ 4» 5» 6' 7' Prop, parts of eols. 2 3 4 6 3 5 10 4 7 i3 6 9 16 7 1 1 •9 8 12 1 22 1 m : 4S' TABLE XXVII. 40^ A Sines, Tan A gents, and Secants. B B [Page 2a7 G\ C 137° si IIourA.M.]HourP.M. Sine, DilT. Cosecant. Tangent. DifT. Cotangent Secant. DifT. Cosine. 60 6 24 o| 5 36 9.8255r 10.17449 Q. 95444 10.04556 10.12893 9.87107 1 23 52 36 8 82S65 17435 ' 95469 0453 1 12904 87096 59 2 23 44 36 16 82579 17421 95495 I o45o5 12915 87085 58 3 23 36 36 24 82593 I 17407 95520 I o448o 12927 87073 57 4 '5 23 28 36 32 82607 I 17393 95545 2 04455 12938 I 87062 Dbt 55 8 23 20 5 36 4o 9.82621 1 10.17379 9.95571 2 10.04429 10. 12950 J 9.87060 6 23 12 36 48 82635 I 17365 95596 3 o44o4 12961 87039 54 7 23 4 36 56 82649 2 1 735 1 95622 3 04378 12972 87028 53 s 22 56 37 4 82663 2 17337 95647 3 04353 12984 2 87016 52 _9 10 22 48 37.2 82677 2 17323 95672 4 04328 12995 2 87005 5i 57) 6 S2 4o; 5 37 20 9.82691 2 10.17309 9.95698 4 10.04302 10. 1 3007 2 9.86993 1 1 22 32 37 28 82705 3 17295 95723 5 04277 i3oi8 2 86982 49 t2 22 24 37 3(i 82719 3 17281 95748 5 04252 i3o3o 2 86970 48 i3 22 16 37 M 82733 3 17267 95774 5 04226 i3o4i 3 86959 47 i4 i5 22 8 37 52 82747 3 17253 95799 6 04201 i3o53 3 86947 4b 45 6 22 5 38 9.82761 3 10. 17239 0.95825 6 10.04175 io.i3o64 3 9.86936 16 21 62 38 8 82775 4 17225 95850 7 o4i5o 13076 3 86924 44 17 21 44 38 16 82788 4 17212 95875 7 o4i25 1 3087 3 86913 Ai 18 21 36 38 24 82802 4 17198 95901 8 04099 13098 3 86902 42 !9 20 21 28 38 32 82816 9.82830 4 17184 95926 8 04074 i3iio 4 86890 41 40 6 21 20 5 38 40 5 1 . 1 7 1 70 9.95952 8 1 . o4o48 10. l3l2I 4 9.86879 21 21 12 38 48 82844 5 I7i56 95977 9 .i4o23 i3.33 4 86867 39 22 21 4 38 56 82858 5 17142 96002 9 03998 i3i45 4 86855 38 23 20 56 39 4 82872 5 17128 96028 10 03972 i3i56 4 86844 il 24 25 20 48 39 12 82885 6 17115 96053 10 03947 i3i68 5 86832 3b 35 6 20 4o 5 39 20 9.82899 6 10. 17101 9.96078 II 10.03922 'io.i3i79 5 9.86821 lb 20 32 39 28 82913 6 17087 96104 II 03896 13191 5 86809 34 27 20 24 39 ZG 82927 6 17073 96129 II 0^871 l3202 5 86798 ii 28 20 16 39 44 82941 b 17059 96155 12 03845 i32i4 5 86786 32 29 3o 20 8 39 52 82955 7 17045 96180 12 o382o l3225 b 86775 3i 3^ 6 20 5 4o 9.82968 7 10. 17032 9.96205 >3 10.03795 io.i3237 6 9.86763 U 19 52 40 8 82982 7 4 1 70 1 8 96231 i3 03769 13248 b 86752 29 3.' .■9 44 4o 16 82996 7 1 1 7004 96256 i4 03744 13260 b 86740 28 33 19 36 4o 24 83oio 8 1 6990 96281 14 03719 13272 b 86728 27 34 35 19 28 4o 32 83023 8 16977 96307 i4 03693 i3283 7 86717 2b 6 19 20 5 4o 40 9.83o37 8 10. 16963 9.96332 i5 10.03668 10.13295 7 9.86705 3b 19 12 4o 48 83o5i 8 16949 96357 i5 03643 i33o6 7 86694 24 3? 19 4 40 56 83o65 8 16935 96383 16 o36i7 i33i8 7 86682 23 3S 18 56 4i 4 83o7S Q 16922 96408 16 03592 i333o 7 86670 22 39 40 18 48 4i 12 83092 9 16908 98433 16 o3567 i334i 8 86659 21 20 6 18 4o 5 4i 20 9.83 106 9 10. 16894 9.96459 17 10 o354i 10. 13353 8 9. 86647 4i i8 32 4i 28 83 1 20 9 16880 96484 17 o35i6 i3365 8 86635 19 42 18 24 4i 36 83i33 10 16867 96510 18 03490 13376 8 86624 18 43 18 16 4i 44 83.47 10 16853 96535 18 o3465 1 3388 8 86612 17 44 45 18 8 4i 52 83i6i 10 16839 96560 19 o344o 1 3400 8 86600 16 75 6 18 5 42 9.83174 10 10. 16826 9.96586 19 io.o34i4 10. i34ii 9 9.86589 4b 17 52 42 8 83 188 II 16812 96611 19 03389 1 3423 9 86577 i4 47 17 M 42 16 83202 1 1 16798' 96636 20 o3364 13435 9 86565 i3 48 17 36 42 24 832i5 1 1 16785 96662 20 033.38 13446 9 86554 12 49 5c. 17 28 42 32 83229 11 16771 96687 21 o33i3 13458 9 86543 11 10 6 17 20 5 42 40 9.83242 1 1 10.16758 9.96712 21 10.03288 10.13470 10 9.86530 5i 17 12 42 48 83?5G 12 16744 96738 22 03262 1 3483 10 865 18 9 52 17 4 42 56 83270 12 16730 96763 22 03237 13493 10 86507 8 53 16 56 43 4 83283 12 ■16717 96788 22 o32I2 i35o5 10 86495 7 54 55 16 48 43 12 83297 12 16703 96814 9.96839 23 23 o3i86 i35i7 10.13528 10 86483 6 '5 6 16 4o 5 43 20 9-83310 i3 10.16690 io.o3i6i II 9.86472 5b 16 32 43 28 83324 i3 16676 96864 24 o3i36 1 3540 II 86460 4 !)7 16 24 43 36 83338 i3 16662 96890 24 o3iio i3552 II 86448 3 ;>« 16 16 43 M 8335i i3 16649 96915 25 o3o85 1 3 564 1 1 86436 2 ^9 16 8 43 52 83365 i4 1 663 5 96940 25 o3o6o 13575 1 1 8642 5 I bo 16 4i 83378 i4 16622 96966 25 o3o34 i3587 12 864 1 3 Hour P.M. 'Hour A.M. Cosino. DifT. Secant. Cotanjjcnt DifT. Tangent. Cosecant. DifT. Sine. 132° A A B B C Seconds of time 1' 2» 3' 4. 5" 6^ 10 7' 12 f^ 2 3 5 7 9 Prop, parts of cols. r 3 6 10 i3 16 '9 22 (c > 3 4 6 7 9 10 * ■ ■ rage 228' TABLE XXVIL SI Log. S lies, Tangents, and Secants. G'. 43° A A B B C C 136° M o Hour A.M. Hour P.M. Sine. Ditr. Cosecant. Tangent. Diff. Cotangent Secant DiflT. Co.tnie. 0786413 M 60 6 16 5 44 9.83378 10. 16622 9 . 96966 io.o3o34 10.13587 I i5 52 44 8 83392 16608 9699 I 1 o3oo9 13599 86401 5g 2 i5 44 44 16 834o5 16595 97016 1 02984 i36i I 86389 58 6 i5 36 44 24 83419 I i658r 97042 1 02q5S i3623 86377 57 4 5 i5 28 44 32 83432 J 1 6568 97067 2 02933 i3634 86366 56 55 6 i5 20 5 4440 9 83446 1 10.16554 9.97092 2 10.02908 10. 1 3646 9.86354 b i5 12 44 48 83459 I i654i 97118 3 02882 1 3658 86342 54 7 i5 4 44 56 83473 2 16527 97143 3 02857 1 3670 86330 53 8 r4 56 45 4 83486 2 i65i4 97168 3 02832 1 3682 2 863 1 8 52 _9 lO i4 48 45 12 835oo 2 i65oo 97193 4 02807 13694 2 863o6 5i 5^ 6 i4 4o 5 45 20 9.835i3 2 to. 16487 9.97219 4 10.02781 10. 1 3705 2 9.86^95 II i4 32 45 28 83527 2 16473 97244 5 02756 :37i7 2 86283 49 12 i4 24 45 36 83540 3 i646o 97269 5 02731 13729 2 86271 48 iJ i4 16 45 44 83554 3 t6446 97295 5 02705 i374. 3 86259 47 • 4 i5 i4 8 6 i4 45 52 5 46 83567 3 16433 97320 6 02680 13753 3 86247 46 45 9-83581 3 10.16419 9.97345 6 10.02655 10.13765 3 9.86235 lb i3 52 46 8 83594 4 I 64(j6 97371 7 02629 13777 3 86223 4i J7 i3 44 46 16 836o8 4 16392 97396 7 02604 13789 3 862 1 1 43 i8 i3 36 46 24 8362 1 4 16379 97421 8 02579 i38oo 4 86200 42 !9 20 i3 28 46 32 83634 4 1 6366 97447 8 02553 i38i2 4 86188 4i 40 6 i3 20 5 46 4o 9.83648 4 10. [6352 9.97472 8 10.02528 10.13824 4 9.86176 21 i3 12 46 48 8366 1 5 16339 97497 9 095o3 13836 4 86164 39 22 i3 4 46 56 83674 5 16326 97523 9 02477 13848 4 . 86i52 38 2j 1-2 56 47 4 83638 5 i63i2 97548 10 02452 i386o 5 86 1 40 37 25 12 48 47 12 83701 5 16299 97573 10 II 02427 13872 5 86128 36 35 6 12 4o 5 47 20 9.8^715 6 10.16285 9-97598 10.02402 10. 13884 5 9.86116 2b 12 32 47 28 83728 b 16272 97624 II 02376 13S96 5 86104 34 2? 12 24 47 36 83741 6 16259 97649 1 1 o2 35i 1 3908 5 86092 33 28 12 16 47 44 83755 ' 6 16245 97674 12 02326 13920 6 86080 32 29 3o 12 8 47 52 83768 9.S3781 6 7 16232 97700 12 o2 3oo 13932 6 86068 3i 3^ 6 12 5 48 to. 16219 9.97725 i3 10.02275 10.13944 6 9.86056 Ji II 52 48 8 83795 7 16205 97750 i3 0225o 13956 6 86044 29 J2 II 44 48 16 838o8 7 16192 97776 1^ 02224 13968 6 86o32 28 JJ II 36 48 24 83821 7 16179 97801 i4 • 02199 13980 7 86020 27 M 35 II 28 48 32 83834 8 ~8 16166 97826 i4 02174 10.02149 13992 7 86008 2b 6 II 20 5 48 4o 9.83848 io.i6i52 9.97851 i5 io.i4oo4 7 9.85996 Jb 11 12 48 48 83861 8 16139 97877 i5 02123 14016 7 85984 24 ^7 11 4 48 56 83874 8 161 26 97902 16 02098 14028 7 85972 23 J8 10 56 49 4 83887 8 i6n3 97927 16 02073 i4o4'' 8 85960 22 39 4o 10 48 49 12 83901 9.83914 _9 9 1 6099 97953 9.97978 16 17 02047 10.02022 i4o52 8 85948 21 20 6 '0 40 5 49 20 1 . 1 60S6 io.i4o64 8 9.8593b 41 to 32 49 28 83927 9 16073 98003 17 01997 1407G 8 85924 '9 42 10 24 49 36 83940 9 1 6060 98029 18 OI97I 14088 8 85912 18 43 10 16 49 44 83954 10 i6o46 9S054 18 01946 i4ioo 9 85900 17 44 45 10 8 49 52 5 5o 83967 10 i6o33 98079 19 01921 l4lI2 9 85888 lb 73 6 10 9.839S0 10 10. 16020 9.98104 '9 10.01 896 IO.I4I24 9 9.85876 4b 952 5o 8 83993 10 16007 98130 19 01870 i4i36 9 85864 i4 47 9 44 5o 16 84006 10 1 5994 98155 20 01845 i4i49 9 8585 1 iJ 48 9 36 5o 24 84o2() 1 1 15980 98180 20 01820 i4i6i 10 85839 12 49 5o 9 28 5o 32 84o33 1 1 15967 10.15954 98 2 06 21 01794 14173 10 85827 1 1 10 6 9 20 5 5o 40 9.84046 1 1 9.98231 21 10.01769 10. i4iS5 10 9.8^815 5i 9 12 5o 48 84059 1 1 15941 98256 22 01744 14197 10 85So3 9 h2 9 4 5o 56 84072 12 15928 9S281 22 01719 14209 10 85791 8 53 8 56 5i 4 84oS5 12 15915 9S307 22 01693 14221 11 85779 7 54 55 8 48 5i 12 5 5i 20 84098 12 15902 98332 2 3 01668 14234 11 85766 6 "5 6 8 4o 9.841 12 12 IO.I588S 9.98357 23 ;o. 01643 10.14246 11 9.85754 5b 8 3i 5i 28 84i25 12 15875 98383 24 01617 14258 II 85743 4 ^7 8 24 5 1 36 84 1 38 i3 1 5862 98408 24 01592 14270 11 8573o 3 58 8 16 5i 44 84i5i i3 1 5849 98433 24 01567 14282 12 85718 2 59 8 8 5i 52 84i64 i3 15836 9S45S 25 01 542 14294 12 85706 1 bo M 8 52 84177 Din; 1 5S23 Secant. 984S4 25 oi5i6 1 4307 12 85693 Hour P.M. Hour A.M. Cosine. Cotangent Difl-. Tangent. Cosecant. Difl". Sine. M 133' A A B B C y Seconds of tame 1' 2' 3» 4. 5' '8~ 6' 10 7' 12 (^ 2 3 5 7 1 Prop, parts of cols. l"" 3 6 9 i3 16 19 22 1 Ic 2 3 5 6 8 9_ 1 1 C 4(7' J TABLE XXVIL [ rage 2-39 s Log. Sines, Tangents, and Secants, G'. 14 \i o 3 Hour J .M. A A B B C C 135° Hour P.M. Sine. IDiff. Cosecant. IO.I5823 Tangent. Diir. Cotangent Secant. Diff. Cosine. 6 8 5 52 9.84177 9-98484 io.oi5i6 io.i43o7 9.85693 I 7 52 52 8 84190 i58io 98509 01491 14319 8568i 5.? 2 7 44 52 16 84203 15797 98534 I oi466 i433i 85669 58 3 7 36 52 24 84216 1 15784 98560 I oi44o 14343 85657 57 4 5 7 28 52 32 5 52 4" 84229 1 1 5771 98585 9 . 986 1 ( ) 2 2 oi4i5 10.01390 14355 10. 14368 -j S5645 9.85632 56 55 6 7 20 9.8424.) I 10. 1 5758 6 7 12 52 48 84255 I 15745 98635 3 01 365 i438o 80620 54 7 7 4 52 56 84269 2 1 5731 98661 3 01339 14392 856oS 53 8 6 5() 53 4 8428J 2 15718 98(i86 3 1 3 1 4 i44o4 2 8^596 52 _? 10 6 48 53 12 84295 9.84308 2 2 1 5705 9R71. 4 01289 14417 2 85583 5i 5o 6 6 4(. 5 53 20 10. i5()92 9.98737 4 10.01263 10.14429 2 9 85571 11 6 32 53 28 8432 1 2' 15679 98762 5 01 238 i444i 2 85559 4q 12 6 24 53 36 84334 3 1 5656 98787 5 0I2l3 14453 2 85547 48 i3 6 16 53 44 84347 3 1 5653 98S12 5 OII88 1 4466 3 85534 47 i4 i5 6 8 53 52 8436o 9.84373 3 ~3 1 5640 10. 15627 9883s 6 01 162 14478 3 8552? 46 45 6 6 5 54 9.98863 6 10.01 1 37 10.14490 3 9.85510 i6 5 52 54 8 84385 3 i56i5 • 9888S 7 01 I 12 i45o3 3 85497 44 17 5 44 54 16 8439S 4 i56o2 98913 7 I 087 i45i5 4 85485 43 i8 5 36 54 24 844 m 4 15589 98939 8 01061 14527 4 85473 42 !9 20 5 28 54 32 84424 4 15576 98964 8. oio36 14540 4 85460 4i 40 6 5 20 5 54 4" 9-84437 4 10. .15563 9.98989 8 lO.OIOI 1 10.14552 4 9-85448 21 5 12 54 48 84450 5 i555o 990 1 5 9 00985 14564 4 85436 39 22 5 4 54 56 84463 5 15537 99040 9 00965) 14577 5 85423 38 23 4 56 55 4 84476 5 i5524 99065 10 00935 14589 5 85411 37 24 25 4 48 55 12 84489 5 i55i 1 99090 10 0^9 1 1 4601 5 85399 9.85386 36 35 6 4 40 5 55 20 9.84502 5 10. 15498 9.991 16 1 1 10.00884 10. i46i4 5 26 4 32 55 28 845 1 5 6 15485 99141 1 1 00859 14626 5 85374 34 ^7 4 24 . 55 36 84528 6 15472 99166 I' 00834 1 4639 6 85361 33 28 4 16 55 44 84540 6 1 5460 99191 12 00809 i465i 6 85349 32 29 3o 4 8 55 52 84553 6 6 1 5447 99217 12 00783 14663 6 . 85337 3i 3^) 6 4 5 56 9.84566 10.15434 9.99242 i3 10.00758 10. 14676 6 9.85324 3i 3 52 56 8 84579 7 15421 99267 i3 00733 1 4688 6 853i2 29 32 3 4i 56 16 84592 7 i54o8 99293 i3 00707 1 470 1 7 85299 •28 33 3 36 56 24 846o5 7 15395 99318 i4 00682 i47i3 7 85287 27 34 35 3 28 56 32 84618 7 i5382 99343 i4 00657 14726 10. 14738 _7_ 7 80274 9.85262 26 l5 6 3 20 5 56 4" 9;8463o 8 10.15370 9.99368 i5 io.oo632 36 3 12 56 48 84643 8 15357 99394 10 00G06 i475o 7 852 5o 24 37 3 4 56 56 84656 8 15344 99419 16 oo58i 14763 8 85237 23 38 2 56 57 4 84669 8 i533i 99444 16 oo556 14775 8 85225 22 39 4o 2 48 57 .2 5 57 20 84682 9 . 84694 8 i53i8 99469 16 oo53i io.oo5o5 14788 8 85212 21 20 6 2 40 9 io.i53o6 9.99495 "7 1 . 1 4800 8 9.85200 4i 2 32 57 28 84707 9 15293 . 99520 17 oo48o i48i3 8 85i87 '9 42 2 24 57 36 84720 9 15280 99545 18 00455 14825 9 85.75 18 4i 2 16 57 44 84733 9 15267 99570 18 oo43o 14838 9 85 162 17 44 45 2 a 57 52 84745 9.84758 _9 10 i5255 10.15242 99596 '9 oo4o4 i485o 9 85i5o lb 75 6 2 5 58 9.99621 19 10.00379 10. 14863 9 9. 85 1 37 46 52 58 8 8477' 10 15229 99646 19 oo354 14875 85 125 14 47 44 58 16 84784 10 i52i6 99672 20 00328 14888 85 1 12 i3 48 36 56 24 84796 10 1 5 204 99697 20 oo3o3 1 4900 85ioo 12 ^9 5o 28 58 32 84809 1 1 15191 99722 21 00278 14913 S5087 1 1 10 I 20 5 58 40 9.84822 II 10.15178 9-99747 21 10.00253 10. 14926 9.S5074 5i 12 58 48 84835 II i5i65 99773 21 00227 14938 8 5062 9 52 4 58 56 84847 1 1 i5i53 99798 22 00202 14951 80049 8 53 56 59 4 848Go II i5i4o 99823 22 00177 14963 85o37 . 7 54 55 48 59 12 84S73 12 i5i27 99848 23 00l52 14976 iT 85o24 9 . 8 5o 1 2 6 6 4o 5 59 20 9.84885 12 io.i5ii5 9.99874 23 10.00126 I c. 14988 56 32 .59 28 84898 12 l5l02 99899 24 00101 i5ooi 12 8^999 4 ^7 24 59 36 84911 12 i5cC9 99924 24 00076 i5oi4 12 84986 3 58 16 59 44 84923 12 1 5077 99949 24 000 5 1 1 5026 12 84974 2 59 8 59 52 84936 i3 i5o64 99975 25 00025 1 5o39 12 84961 I 60 M 5 84949 i3 i5o5i 10.00000 25 00000 •i5o5i 12 84949 Hour P.M. Houl A -M . Cosine. Di(r. Secant. Cotangent Ditr. Tangent. Cosecant. Ditr. Sine. 134° 45» Seconds of time V 2' 3' 4= 5' 8 16 8 6» to •9 9_ 7' 1 1 22 11 (A Prop, parts of cols. < B ic 2 3 2 3 6 3 5 9 5 c i3 .1 ^^s^ 230] TABLES XXVIII, XXIX. TABLE XXVIII. TABLE XXIX. For reducing the Time of the Moon's passage over the Merid an of Correction of Moon's Greenwich, to the Time of its passage over any other Meridian. altitude for Paral- The numbers taken from this Table are to be added to the Time at lax and Refrac- Greenwich in West Longitude, but subtracted in East. tion. Daily Variation of the Moon's passing the Meridian. Dak. Deg. Corr. Min. Dalt. Deg. 5i Uorr- Mm. 35 Ship's / / / / 1 / / / / / / / / / Ship's 10 5i Lou. 40 42 44 46 48 50 52 54 56 58 60 62 64 66 Loii. II 12 52 52 52 53 35 34 / 1 1 1 1 / / / 1 / / / 1 / i3 52 54 33 o 14 52 55 32 5 I 1 1 I I I I I I I I I I I 5 i5 52 56 32 lO I I I I I I I I 2 2 2 2 2 2 10 16 52 5? 3i i5 2 2 2 2 2 2 2 2 2 2 2 3 3 3 i5 17 52 58 So 20 2 2 2 3 3 3 3 3 3 3 3 3 4 4 20 18 52 59 60 29 28 25 3 3 3 • 3 3 3 4 4 4 4 4 4 4 5 25 19 20 52 3o 3 3 4 4 4 4 4 4 5 5 5 5 5 5 3o 5i 35 T 4 4 4 5 5' 5 5 T 6 6 ~6" 6 6 35 21 5i ~67~ 27 26 4o 4 5 5 5 5 6 6 6 6 6 7 7 7 7 40 22 5i 62 45 5 5 5 6 6 6 6 7 7 7 7 8 8 8 45 23 5i 63 26 5o 6 6 6 6 7 7 7 7 8 8 8 9 9 9 5o 24 5o 64 25 55 6 7 6 7 7 7 7 8 7 8 8 8 8 9 8 9 _9_ 9 _9_ 10 __9_ ID _9_ 10 10 II 10 11 55 25 26 5o 5o 65 66 24 23 60 60 65 7 8 8 8 9 9 9 10 10 10 I I II 12 12 65 27 49 67 22 70 8 8 9 9 9 10 10 10 1 1 II 12 12 12 i3 70 28 49 68 21 75 8 9 9 10 10 10 II II 12 12 12 i3 i3 i4 75 29 49 69 20 80 85 •_9_ 9 _9_ 10 10 10 10 II II II II I2t 12 1 2 12 T3" 12 73" i3 i4 I 3 T4 i4 i5 i4 i5 i5 16 80 85 00 48 70 19 48 71 i8 90 10 10 II 1 1 12 12 i3 i3 i4 i4 i5 i5 16 16 90 32 47 72- 17 95 1 1 II 12 12 i3 l3 i4 14 i5 i5 16 16 17 17 95 33 47 73 17 100 1 1 12 12 i3 i3 i4 i4 i5 16 16 17 17 18 18 100 34 46 74 16 io5 12 12 i3 i3 i4 i5 i5 16 16 '7 17 18 '9 19 io5 35 46 75 i5 36 37 38 39 40 45 45 44 44 43 76 i4 1 10 12 73" 73" T4" i5 i5 16 16 17 18 18 '9 20 20 1 10 ii5 120 i3 i3 i3 i4 i4 i5 i5 i5 i5 16 16 17 17 17 17 18 18 '9 '9 19 19 20 20 21 20 21 21 22 1x5 120 77 78 i3 12 125 i3o 1 4 1 4 i5 i5 i5 16 16 •7 17 17 17 18 18 _L?_ 19 20 '9 '9 19 20 20 21 21 22 22 22 22 23 23 24 125 i3o 79 80 1 1 ID i35 i5 16" 'x'^ 17 Tb 19 20 21 22 22 77 77 25 i35 . 4i 42 42 4i 4o 4o 39 81 82 83 84 85 86 9 1 40 16 16 17 18 '9 19 21 22 23 23 24 25 26 i4o 42 43 44 45 46 8 i45 16 17 18 '.9 '9 20 21 22 *23 23 ^4 55 26 27 i45 7 6 5 4 i5o 17 17 18 '9 20 21 22 22 23 04 25 26 27 27 i5o 1 55 17 -L^_ _L9 20 21 22 32 23 24 25 26 27 28 28 i55 160 18 19 20 20 21 22 T3" Ta 2 5' 76 27 28" T8~ 39 160 47 38 87 3 ifj-) 18 '9 20 21 22 23 24 25 56 27 27 28 29 3o i65 48 38 88 2 170 '9 20 21 22 23 24 25 25 26 27 28 29 3o 3i 170 it 37 89 I 17') 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 175 36 90 iSo 5 21 22 23 H 25 26 27 28 19 3o 3i 32 33 180 40' 42' 44' 40' 48' 50' 52' 54" 56' 58'] GO' 62' 64' 66' TABLE XXX. [Page 231 For finding the Variation of ih of the ftlooii s Iliglit Ascension eS un's Right Ascension, of the Pec lination, of the Equation of Time or in an\' number of minutes of time , the Horary' Motion being given at the top of the page in seconds, and the number of minutes of time in the side-column y — J Also, for finding the Variation of the Moon's Declination in seconds of time ; the motion in one I minute being given at the top, and the numbers in the side-column being taken (or seconds. Horai-y Motion. M ,■ // // // /; // // // // // n // // II // ti // II // // " II // // ' II II II M 1 3 4 5 6 7 8 9 10 11 \2 13 14 15 IG 17 lb V. 20 21 22 23 24 25 26 27 2b 2!i 30 I ~, c I I 2 I I 1 I I I I 1 I I I I 1 I I 2 3 I 1 I 1 I I I I I I I 1 I I I I I I I 2 3 4 I I I I I I I I I I I I 2 2 2 2 2 2 2 2 4 5 I I I I I I I I 2 2 2 2 2 2 2 2 2 2 2 2 3 5 6 Y I Y 1 J I I 1 2 2 2 2 2 2 2 2 2 2 ~3 "3 ~Z ~3 ~3 3 6 •7 I I I J 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 7 8 I I 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 A 4 4 -s. 9 I 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4. 4 4 5 9 to I 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 10 11 I 2 2 2 2 2 ~3 3 1 3 3 3 4 4 4 4 4 5 5 ~5 ~5 ■y (i 1 1 12 r 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 12 i3 2 2 2 2 2 3 3 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 7 i3 i4 2 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 i4 i5 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 _7 7 7 8 i5 16 2 2 2 2 3 3 ^ "3 4 4 4 5 5 5 5 6 ~6 6 ~6 7 7 7 7 8 8 i6' 17 2 2 2 3 3 3 3 4 4 4 5 5 5 5 6 6 6 7 7 7 7 8 8 8 9 17 18 2 2 2 2 3 3 3 4 4 4 5 5 5 5 6 6 6 .^ 7 7 8 8 8 & 9 9 18 19 2 2 2 3 3 3 3 4 4 4 5 5 5 6 G 6 7 7 7 8 8 8 9 9 9 Id '9 20 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 _7 7 7 8 8 8 _? _9 J' i< Id 20 21 2 2 2 y 3 "4 4 4 5 ~5 5 6 ~6 "6 7 7 7 ~8 "8 ~8 9 9 9 Id Id 1 1 21 22 2 2 3 3 3 4 4 4 5 5 6 6 6 7 7 7 8 8 8 9 9 10 Id Id 1 1 I I 22 23 2 2 2 3 3 3 4 4 5 5 5 6 6 7 7 7 8 8 8 9 9 10 10 10 1 1 I 1 12 23 24 2 2 2 3 3 4 4 4 5 5 6 6 6 7 7 8 8 8 9 9 10 H) 10 1 I 1 1 12 12 24 25 2 2 3 3 -^ 4 4 5 5 5 6 6 _7 7 8 8 8 _9 _? Id 10 Id 1 1 1 1 12 12 l3 25 26 2 2 3 3 3 4 4 5 5 6 "6 7 7 7 7 "8 9 9 Id Id 10 1 I 1 1 12 12 l3 i3 26 27 2 2 3 3 4 4 5 5 5 6 6 7 7 8 8 9 9 9 10 I(J 1 1 1 I 12 12 l3 i3 14 27 28 2 2 3 3 4 4 5 5 6 6 7 7 7 8 8 9 9 id Id 1 1 1 1 12 12 l3 i3 14 i4 28 29 2 2 3 3 4 4 5 5 C 6 7 7 8 8 9 9 10 Id 1 1 1 1 12 12 i3 i3 1 4 1/; i5 29 3<. 2 2 3 3 4 4 5 5 6 6 J_ _7 8 8 _? _? 10 to 1 1 I 1 12 12 l3 i3 i4 i4 i5 i5 3c- 17 2 2 y y 4 4 5 1 6 7) 7 7 8 8 9 9 m Id I 1 1 1 12 12 73 i3 14 i4 i5 7g 3i 32 2 2 3 3 4 4 5 5 6 6 7 7 8 9 9 10 iti 1 1 I 1 12 12 i3 i3 i4 i/i i5 i5 16 32 33 2 2 3 3 4 4 5 6 6 •n 7 8 8 9 9 10 10 1 1 12 12 l3 i3 i4 i4 i5 i5 iG 17 33 34 2 2 3 3 4 5 5 G 6 7 7 8 9 9 10 1 1 I ; 12 12 i3 14 i4 1 5 i5 iG iG 17 M 35 2 2 3 4 4 5 5 6 6 _7 8 8 _9 _? 10 1 1 1 1 12 12 i3 i3 14 i5 i5 iG i(i iZ iS 35 3(5 2 2 y 4 4 "5 y ~6 7 7 8 ~8 9 IC) 10 1 1 1 1 12 i3 73 Ta i4 75 16 7g 17 17 18 36 3- 2 2 3 4 4 5 6 6 7 7 8 9 9 10 10 1 1 12 12 i3 ■ 4 14 i5 i5 16 17 '7 18 '9 37 38 2 3 3 4 4 5 6 6 7 8 8 9 10 10 1 1 1 1 12 i3 i3 14 i5 i5 16 16 17 i8 lb '9 38 39 2 3 3 4 5 5 6 7 7 8 8 9 10 10 1 1 12 12 i3 14 14 i5 16 16 17 ]8 18 '9 20 39 4c. 2 3 3 4 5 5 6 J 7 8 _? _9 10 1 1 II 12 i3 i3 14 i5 i5 16 ■7 17 18 1? 19 20 4o 4i 2 3 3 4 5 y 6 7 8 1 9 10 10 1 1 12 12 i3 i4 i4 1 5 76 77 17 18 77 '9 2d 21 4i 42 2 3 4 4 5 6 6 7 8 8 9 10 II 1 1 12 i3 i3 i4 i5 i5 16 17 18 18 ■9 2d 2d 21 42 43 ' 2 3 4 4 5 6 6 7 8 9 9 10 II 1 1 12 i3 i4 i4 1 5 16 16 17 18 '9 '9 2d 21 22 Ai U 2 3 4 4 5 6 7 7 8 9 10 10 II 12 12 i3 14 i5 i5 16 "7 18 18 '9 2d 21 21 22 A4 45 2 2 3 4 5 5 6 ^ 8 8 9 10 II II 12 i3 14 ■ 4 i5 16 17 17 18 19 20 2d 21 22 23 45 46 2 2 3 4 5 ■5 6 7 8 8 9 9 Ui 11 12 12 73 i4 i5 75 76 17 78 78 '9 20 21 21 22 23 46 47 2 2 3 4 5 5 G 7 8 9 10 1 1 12 i3 i3 i4 i5 16 iG 17 18 '9 2d 2d 21 2 2 23 24 47 48 2 2 3 4 5 6 6 7 8 9 Id 10 1 1 12 i3 14 i4 i5 16 17 18 18 '9 20 21 22 22 23 24 48 49 2 2 3 4 5 6 7 7 8 9 10 II 1 1 12 i3 i4 i5 iC 16 17 18 19 2d 20 21 2 2 23 24 25 49 5o 2 3 3 4 5 6 7 8 8 _9 10 II 12 i3 i3 i4 i5 16 iZ 18 18 19 20 2 1 22 23 23 24 25 5o 5 1 2 y 3 4 5 6 7 y 9 9 10 1 1 12 73 M i4 i5 77) 17 18 '9 2d 20 21 2 2 Vi 24 25 7) 5i 52 2 3 3 4 5 6 7 8 9 10 10 II 12 i3 14 i5 16 16 17 18 ■9 20 21 22 23 23 24 25 26 52 53 2 3 4 4 5 6 7 8 9 10 II II 12 i3 i4 i5 16 17 j8 '9 '9 20 21 22 23 24 25 26 27 53 54 2 3 4 5 5 6 7 8 9 10 II 12 i3 14 i4 i5 16 17 18 '9 20 21 22 23 23 24 25 26 27 54 55 2 3 4 5 6 6 7 8 _9 10 II 12 i3 1 4 i5 16 [7 '7 18 19 20 21 22 23 24 25 26 iZ 28 55 56 2 3' 4" T 6 7 7 "8 9 10 77 12 i3 i4 i5 76 ■7 18 ■9 20 21 21 22 23 24 25 26 27 28 56 57 2 3 4 5 6 7 8 9 10 10 1 1 12 i3 i4 i5 16 17 18 ■9 20 21 22 23 24 25 26 27 28 29 57 58 2 3 4 5 6 7 8 9 10 II 12 i3 i4 i5 i5 16 17 18 >9 20 21 22 23 2zi 25 2G 27 28 29 58 |9 2 3 4 5 6 7 8 9 10 II 12 i3 i4 i5 16 17 18 '9 20 21 22 23 24 25 26 n 28 29 3o 59 60 2 3 4 5 6 7 8 9 10 II 12 i3 i4 i5 16 17 18 '9 20 21 22 23 24 25 26 27 28 29130 60 1 Page 232] TABLE XXX. For fiiKling Ihe Variation of the Sun's Right Ascension, of the Declination, of the Equation of Time or of llie Moon's Rifrlil Ascension, in any number of minutes of time, the Horary Motion being given at the top of llie page in seconds, and the number of minutes of time in the side-column ; — Also, for finding the Variation of tiic ftloon's Declination in seconds of time; l!ie motion in one minute being given at the top, and llie numbers in ihc side-colunui being taken for seeomls. Horary Motion. M '/ 1 " 1 // // ;/ // II '/ i // // /' II II II ;/ II /; ;/ // // // If II II II // // II II II M 31 32 33 34 35 36 37 33; 39 4U 41 42 43 44 45 4G 47 48 49 50 51 52 53 54 55 56 57 5b 59 GO I 2 I ] I I I I I I I I I I I I I I I I I I I I I 1 I I I I 2 I 2 I 2 I 2 I 2 1 2 I 2 I 2 3 I 2 I 2 1 2 I 2 I 2 I 2 1 2 I 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 f, 5 5 6 3 3 "3 "3 ~4 "1 "4 ~4 ~~4 ~4 ~4 4 4 "4 5 5 5 5 5 5 5 5 5 "5 6 ~6 "6 1) ~6 ~6 ^ 7 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 r, 5 6 6 6 6 6 6 6 6 7 7 - 7 7 _ 1 8 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 9 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 ■7 7 8 8 8 8 8 8 8 9 9 9 9 9 10 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 _9 _? _? _9 _9 9 10 10 U' 10 10 II 6 ~6 "6 ~6 ~6 7 7 7 7 7 8 8 8 8 "8 "8 9 9 ~9 9 9 10 10 10 10 10 10 1 1 1 1 II II 12 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 10 K) 10 10 10 11 1 1 1 1 1 1 11 12 1 2 12 12 i3 7 7 7 7 8 8 8 8 8 9 9 9 9 10 10 10 10 10 I 1 II 1 1 1 1 1 1 13 12 13 12 i3 i3 i3 i3 i4 7 7 8 8 8 8 9 9 9 9 10 10 10 10 II II 11 1 1 I I 12 12 12 12 i3 i3 i3 i3 i4 i4 14 i4 i5 8 8 8 _9 _9 _9 _9 10 10 10 10 1 1 I T 1 1 1 1 12 12 12 12 i3 i3 i3 1 3 14 i4 14 i4 i5 i5 i5 i5 i6 "8 9 9 9 9 10 ID 10 10 II II 1 1 \ I 12 12 12 73 73 i3 73 i4 ■ 4 i4 i4 i5 i5 i5 i5 76 16 76 I? 9 9 9 10 10 10 10 1 1 11 II 12 13 12 12 i3 i3 i3 i4 i4 14 i4 i5 i5 i5 16 16 16 16 17 '7 17 i8 9 10 10 10 1 1 11 II II 12 12 12 i3 i3 i3 i4 i4 i4 i4 i5 i5 i5 16 16 16 17 '7 17 17 18 18 18 '9 10 10 10 1 1 II 11 12 12 12 i3 i3 1 3 14 i4 i4 i5 i5 i5 16 16 16 16 17 17 17 18 18 18 19 19 19 20 10 1 1 1 1 1 1 12 12 12 t3 i3 i3 i4 \A i4 i5 i5 i5 16 16 16 17 17 17 18 18 iS !9 19 12 20 20 26 21 1 1 1 1 12 1212 73 73 73 74 i4 1 4 i5 i5 75 16 16 16 17 17 18 7s 18 '9 '9 '9 20 20 20 21 21 21 22 1 1 12 12 12 i3 i3 i4 i4 i4 i5 i5 i5 16 16 17 17 17 18 18 18 •9 '9 •9 20 20 21 21 21 22 22 22 23 12 12 i3 i3 i3 i4 i4 i5 1 5 i5 16 16 16 17 17 18 18 18 •9 '9 20 20 20 21 31 21 2 3 23 23 23 23 24 12 i3 i3 ^4 .4 i4 i5 i5 16 16 16 17 17 18 iS iS '9 19 20 20 20 21 2! 22 22 22 23 23 24 24 24 25 i3 i3 1 4 \4 i5 i5 i5 16 16 '7 17 18 18 r8 12 12 20 20 20 21 21 22 22 23 3 3 23 24 24 25 25 25 16 73 i4 i4 i5 75 76 77i 76 17 1-7 18 78 '9 19 20 20 20 21 21 22 22 73 73 73 24 24 25 25 26 76 26 27 i4 14 i5 i5 16 16 '7 17 18 18 18 '9 ■9 20 20 21 21 22 22 23 23 23 24 ■M 25 25 26 26 27 27 27 28 i4 i5 i5 16 16 17 17 18 r8 '9 19 20 20 21 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 28 29 i5 i5 16 16 17 17 18 18 ■9 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 39 29 29 3o 16 16 17 '7 18 18 !9 i? 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 7C, 29 3o 3o 3o "3 1 76 17 17 18 18 '9 '9 20 20 21 21 22 22 23 73 74 24 25 75 76 76 27 27 78 78 29 29 3o 3^ 3i 3i 32 17 17 18 18 '9 19 20 20 21 21 22 22 23 23 24 25 25 26 26 27 2-? 28 28 29 29 3o 3o 3i 3i 32 32 33 17 18 18 '9 '9 20 20 21 21 22 23 23 24 24 25 25 26 26 27 28 28 29 l^ 3o 3o 3i 3i 32 32 33 33 34 18 18 19 '9 20 20 21 22 22 23 23 24 24 25 26 26 27 27 28 28 29 29 3() 3i 3i 32 32 33 33 34 34 35 18 [9 '1 20 20 21 22 3 2 23 23 24 35 25 26 26 27 27 28 29 29 3o _3o 3i 33 32 33 33 34 34 35 35 36 '9 '9 20 20 21 22 2 2 73 73 24 25 25 26 26 27 28 28 29 29 3o 3i 3 1 32 37 33 34 34 35 35 36 36 3? '9 20 20 21 22 23 23 23 24 25 25 26 27 27 28 28 29 3o 3o 3i 3i 32 33 33 34 35 35 36 36 37 37 ; 38 20 20 21 22 22 2? 23 2.1 25 25 26 27 27 28 29 29 3o 3o 3! 32 32 33 34 34^35 35 36 37 37 38 38 39 iO 21 2 1 22 -.3 23 24 20 25 2b 27 27 28 29 29 3o 3i 3i 32 33 63 34 M 35 36 36 37 38 38 39 3<) : 4o 21 21 22 23 23 24 25 2 5 26 27 27 28 29 29 3o 3i 3i 32 33 33 34 35 35 36 37 37 38 39 39 4o 40 4i 21 22 23 23 24 7s 75 76 27 27 28 29 29 3o 37 3i 52 3^ 33 34 35 36 3''> 37 38 38 39 40 40 4 1 4i • 42 22 2 2 23 24 25 2 5 26 27 27 28 29 29 3o 3i 32 32 33 34 34 35 36 36 37 38 39 39 4o 4i 4i 42 42 : 43 22 2 3 24 24 25 26 27 27 28 29 29 3o 3i 32 32 33 M 34 35 36 37 37 38 39 39 4o 4i 42 42 /, '2 1 ^J, .4 44 23 23 24 25 26 26 27 28 29 29 3o 3i 32 32 33 34 34 35 36 37 37 38 39 4<-> 40 4i 42 43 43 44 44 1 45 23 24 25 26 26 27 28 29 29 3o 3i 32 37 33 34 35 35 36 37 38 38 h 4o 4 1 4i 42 43 44 44 45 45 : 46 "24 25 25 26 27 78 28 29 3o 3i 37 37 33 34 35 35 36 37 38 38 39 4o 4i 47 42 43 44 44 45 46 46 47 24 25'26 27 27 28 29130 3i 3 1 32 33 •3/( M 35 36 37 38 38 39 4o 4i 42 42 43 44 45 45 46 47 4- 48 25 26136 27 28 29 3c 3o 3. 32 33 34 34 35 36 37 38 38 39 4o 4i 42 4- 43 44 45 46 46 47 ^8 48 49 20 26 27 28 99 29 3( 3r 33 33 33 34 35 36 37 38 38 39 40 4i 42 42 43 44 45 46 47 47 48 49 49 5o 20 11 28 28 29 3o 3i 33 33 33 34 35 36 37 38 38 39 4o 4i 42 43 43 44 45 46 47 48 48 49 5o 5o 5i V6 n ^ 29 3o 37 37 37 33 34 35 2G 37 37 38 39 4o 4 1 42 43" 43 44 45 46 47 48 48 49 5o 57 5i 52 27 28 29 29 3o 3i 3? 33 34 35 36 36 37 38 39 40 4i 42 47 43 44 45 46 47 48 49 49 5o 5i 52 52 53 27 2P |29 3o 3 1 32 33 34 34 35 36 37 38 39 40 4i 42 42 43 44 45 46 47 48 49 49 5o 5i 52 53 53 54 28 29 3o 3. 32 3? 33 34 35 36 37 38 39 4o 4i 4i 42 43. 44 45 46 47 48 49 5o 5o 5i 52 53 54 54 55 28 29 3o 3i 32 33 34|35 36 37 38 39 39 40 4i 42 43 44 45 46 47 48 49 5o 5o 5t 52 53 54 ^6 55 56 2y 3. 37 37 33 37 35 35 36 37 38 39 4o 4i 42 43 44 45 46 47 48 49 49 5o 57 52 53 54 55 56 56 57 39 3o 3i 32 33 34 35 36 37 ,38 39 4o 41 42 43 44 45 46 47 48 48 49 5o 5i 5? 53 54 55 56 57 57 58 3o 3. 32 33 34 35 \z(i2- 38 39 40 4i 42 43 44 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 58 59 3o 3i 32 33 34 35 ,36 37 38 39 4o 4i 42 4344 45 46 47 48 49 5o 5i 53 53 54 55 56 '^7 58 59 59 60 3i |3: 33 |34 |35 3r 137 38 39 4o 4i 42 4-i 44'45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59I60 60 TABLE XXX. [P^se^^3 For fiiuling ihe Variation of llie Sun's Rigiit Ascension, of the Declination, of the [■^ciiuuion of Time or of tlie iMoon s Rig;ht Ascension, in any number of minutes of time, the lloiary Motion bci.ig given at llie top of tlie page in seconds, and the number of minutes of lime in the side-column, — Also, for finding the Variation of the Jloon's Declination in seconds of time; the motion in one minute being given at the top, and the numbers in the side-column being taken for seconds. Horary Motion. M II // II /( // „ II \i // II ,7 ;/ /; // // /' 1 II II // II // // // // n // II II II II M 61 (i2 (i:3 (i-J 65 m u 6« 6'J 70 71 72 7:3 74 75 76 77 78 7il 80 SJ 82 83 84 85 86 87 8b 8ii i)0 I 2 1 2 2 I 2 I 2 I 2 I 2 I 2 2 1 2 I 2 1 2 I 2 I 2 I 2 I 3 I 3 1 3 3 I 3 3 I 3 I 3 I 3 3 I 3 I 3 I 3 3 I 3 2 3 I 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 4 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 _7 7 7 J _7 8 5 6 6 () 6 6 7 7 7 7 7 7 7 7 7 7 8 ~8 "8 "8 "8 8 8 '8 8 8 9 9 9 9 9 9 6 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 10 10 10 10 U) 10 K i(j 1 1 7 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 12 12 12 8 9 9 9 9 10 10 10 H) 10 10 1 1 1 1 1 1 1 1 1 1 II 1 1 12 12 12 12 12 12 12 i3 i3 i3 i3 i3 i3 14 9 lO 10 lu 1 1 1 1 1 1 1 1 I I 1 1 12 12 12 12 12 12 i3 i3 i3 i3 i3 i3 14 14 i4 i4 i4 :4 i5 i5 i5 i5 lO 1 1 1 1 1 1 12 12 12 12 12 12 73 73 73 73 73 74 i4 i4 i4 14 74 75 i5 i5 i5 i5 16 16 76 76 16 17 1 1 12 12 1 2 i5 i3 i3 i3 \3 i4 1 4 i4 i4 14 i5 i5 1 5 i5 1 5 16 16 16 16 16 17 ■7 17 17 ■7 18 18 18 12 i3 i3 i3 i4 i4 i4 i4 i5 i5 1 5 i5 i5 16 16 16 16 16 17 17 17 17 18 18 18 18 18 '9 '9 '9 '9 20 i3 i4 i4 i4 i5 i5 i5 i5 16 16 16 16 17 17 17 17 18 18 18 18 18 19 19 '9 19 20 20 2p 2U 21 21 21 i4 i5 1 5 16 16 16 16 17 17 17 17 18 18 18 18 12 i? [9 19 2C1 20 20 20 21 21 21 21 2 2 22 22 22 23 i5 7(3 76 '7 '7 17 '7 18^ 78 18 18 19 19 ^ '9 20 20 20 21 21 21 21 22 22 22 22 73 73 73 73 24 ?4 16 I? 17 18 18 18 18 '9 '9 '9 20 20 20 20 21 21 21 22 22 22 22 23 23 23 24 24 24 24 25 25 25 26 17 i8 18 19 '9 '9 20 20 2CI 20 21 21 21 22 22 22 23 23 23 23 24 24 24 25 25 2 5 26 26 26 26 27 27 18 '9 '9 .)(! 2U 20 2 1 21 21 22 22 22 22 23 23 23 24 24 24 25 25 2 5 26 26 26 27 27 27 28 28 28 29 '9 20 20 2 1 2i 21 22 22 2 2 23 23 23 24 24 24 2 5 25 25 26 26 26 27 27 27 28 28 28 29 29 29 3o 3o 2(1 21 21 22 22 22 73 73 73 74 24 75 25 25 26 76 26 27 27 27 78 28 28 29 29 29 37 3o 3o 3i 37 37 21 22 22 23 23 23 24 24 25 25 25 26 26 26 27 27 28 28 28 29 29 29 3o 3o 3i 3i 3i 32 32 32 33 33 22 23 23 24 24 25 25 25 26 26 26 27 27 28 28 28 29 29 3o 3o 3(; 3i 3i 3i 32 32 33 33 33 34 34 35 23 24 24 25 2 5 26 26 26 27 27 28 28 28 29 29 3o 3o 3o 3i 3i 32 32 32 33 33 34 34 34 34 35 36 36 24 25 25 2() 26 27 27 28 28 28 20 29 3o 3o 3o 3i 3i 32 32 33 33 33 34 34 35 35 35 36 36 37 37 38 25 26 76 27 27 28 28 29 ^9 29 to 3o 37 37 32 32 33 33 33 34 34 35 35 36 36 36 37 37 38 38 39 39 7(7 27 27 28 28 29 29 3u 3c) 3i 3i 32 32 32 33 33 34 34 35 35 36 36 36 37 37 38 38 39 39 40 4v 4i 27 28 28 29 29 3(. 3u 3i 3i 32 32 33 33 34 34 35 35 35 36 36 37 37 38 38 39 39 40 4o 4i 4i 42 42 28 29 29 3.. 3o 3i 3i 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 4o 4u 4i 4i 42 42 43 43 44 29 3o 3i 3i 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 4o 40 41 4i 42 42 43 43 44 44 45 45 3o 3i 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 4(> 4o 4 1 4i 42 42 43 43 44 44 45 45 46 47 3i 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 4\ 4i 42 42 43 43 44 44 45 45 46 46 47 47 48 32 33 34 34 35 35 36 36 37 37 38 39 39 4o 4o 4i 4i 4-2 42 43 43 44 45 45 46 46 47 47 48 48 49 5o 33 34 35 35 3() 36 37 37 38 39 39 4(1 4u 4i 4i 42 43 43 44 44 45 45 46 46 47 48 48 49 49 5o 5(j 5i 34 35 36 3(i 37 37 38 39 l2 4<) 4<. ^< 4> 42 43 43 44 44 45 46 46 47 47 48 48 49 5o 5o 5i 5i 52 53 35 36 37 3^ 38 38 39 4" 40 4> 4i 42 43 43 44 44 45 46 4() 47 47 48 49 49 5o 5o 57 57 57 53 53 54 36 37 38 38 39 4" 39 40 A\ 4i 42 43 43 44 44 45 46 46 47 47 48 49 49 5o 5i 5i> 52 52 53 54 54 55 56 37 38 39 39 41 4i 42 42 43 44 44 45 46 46 47 48 48 49 49 5o 5i 5i 52 53 53 54 54 55 56 56 57 38 3y 4" io 4i'A-^ 42 43 44 44 45 46 46 47 47 48 49 49 5<> 5i 5i 52 53 53 54 55 55 56 57 57 58 59 39 40 4i 4i 42 43 43 44 45 45 46 47 47 4>3 49 49 5o 5i 5i 52 53 53 54 55 2^ 56 57 57 58 5y 59 60 4o 4i 42 42 4^3 44 44 45 46 46 47 48 49 49 5o 5i 5i 57 53 53 54 55 55 56 ^7 57 58 59 59 60 61 67 4i 42 43 43144 45 46 4^^ 47 48 48 49 5<) 5o 5i 52 53 53 54 55 55 56 57 57 58 59 60 60 61 62 62 63 42 43 44 44 45 46 47 4i 48 49 49 5o 5i 52 52 53 54 54 55 56 57 57 58 59 59 60 61 62 62 63 64 65 43 M 45 45 146:47 48 48 49 5o 5i 5[ 52 53 54 54 55 56 56 57 58 59 59 60 61 62 62 63 64 65 65 66 44 45 46 4714748 49 5o 5<. 5i 52 53 53 54 55 56 56 57 58 59 59 60 61 62 62 63 64 65 65 66 67 68 45 46 47 i8 4»l49 5 61 61 62 63 63 64 65 65 66 67 67 68 69 _^ 70 J}.. 71 72 J73 73 74 J^ Jl 76 _77 4o it 62 63 64 64 65 66 66 67 68 68 69 "70 70 71 72 72 73 7'' 74 75 76 77 11 78 79 4i 42 64 64 65 66 67 67 68 69 69 70 71 71 72 73 74 74 75 76 76 77 78 78 79 80 81 42 43 65 66 67 67 68 69 7" 70 71 72 72 73 74 75 75 76 77 77 78 79 8n 80 81 82 82 43 44 ()7 67 68 69 70 70 7' 72 73 73 74 75 76 76 77 78 78 79 80 81 81 82 S3 84 84 44 45 68 69 70 71 71 72 73 74 74 Jl 76 77 77 78 79 80 80 81 82 83 83 84 85 86 86 45 46 70 7' 71 72 73 74 74 75 76 77 77 78 79 80 81 81 82 83 "84 84 "85 "86 07 "87 "88 46 47 71 72 73 74 74 75 76 77 78 78 79 80 81 81 82 83 84 85 85 86 87 88 89 89 90 47 ^S> 7874 74 75 76 77 78 78 79 80 81 82 82 83 84 85 86 86 87 88 89 9,1 90 9' 92 48 49 74 75 76 77 78 78 79 80 81 82 82 83 84 85 86 87 87 88 89 90 91 9! 92 93 94 49 5o 76 77 78 78 79 80 81 82 83 83 84 85 86 87 88 88 .h _22 9' 92 93 93 94 _?5 96 5o 5i 77 78 79 8(1 81 8^ 8"2 83 84 85 86 87 T8 88 "8^ 90 9' 92 93 94 94 95 96 97 98 57 52 79 8(1 8, 81 82 63 84 85 86 87 88 83 89 90 9' 92 93 94 94 95 96 97 98 99 lOo 52 53 8(. 81 82 83 64 85 86 8- 87 88 89 90 9' 92 93 94 95 95 96 97 98 99 100 101 102 53 54 82 83 84 85 86 86 87 88 89 90 9' 92 93 94 95 95 06 97 98 99 100 lOI IC? io3 io4 54 55 83 84 85 86 87 88 89 <)U 21 92 93 _94 94 ^^ 96 _9Z _98 _99 100 lOI 102 i(>3 io4 io5 i.j5 55 56 85 86 87 88 89 90 9' 9' 92 93 94 95 96 97 98 99 100 lOI 102 7^ io4 7^ io5 K-i') 107 56 ^7 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 io3 io4 io5 io5 106 107 108 109 57 58 88 89 9" 91 92 93 94 95 96 97 98 99 100 lOI 102 102 io3 io4 io5 106 107 108 109 I IIJ III 58 59 90 9" 9' 92 93 94 95,9697 98 99 too 101 102 io3 io4 io5 106 107 108 109 1 10 1 1 1 1 12 ii3 59 60 2L 21 93 94 £l 96 97 98 99 100 101 I03 io3 io4 io5 106 107 108 109 IJn 1 1 1 1 12 iij ii4 ii5 bo TABLE XXX. [Page 235 For finding' the Variat on of till Sun's Right Ascension, of the Declination, of the Equation of Time or | of the JMoon's Right A jcensioii, in my number af minutes of time, the Horary Motion Iteing given at the top ol' the page in se conds, and the number of minutes of time in ihe side-column; — Also, for finding ilie Variation f ih e I\Ioon' s Declination in seconds of time : the motion in one minute being given at the top, and the numbers in the side-column being taken for seconds. Ilorat-y Motion. M /' ■ •' // 1 '/ /; / // // // II // // /; /; // II // II /' // // II II M liG 117 llgll!) 120 121 l->2 123 12-1 125 126 127 128 129 130 131 132 133 131 135 13G j37 138 I 2 2 2I 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 "^ 2 I 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 2 3 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 3 4 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 4 5 l3 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 10 12 ID II II n II 1 1 1 1 "73 n ~T3 I' i3 n "73 II Ta II "14 n ~A X2 "74 5 "6 13 i3 ^ ~n H 7 i4 i4 i4 i4 i4 i4 i4 i4 i4 i5 i5 i5 i5 i5 i5 i5 i5 16 16 16 16 16 16 7 a i5 16 16 16 16 16 16 16 17 17 17 17 17 17 17 17 18 18 18 18 18 18 18 8 9 n 18 18 18 18 18 18 18 19 19 19 19 19 19 20 20 20 20 20 20 20 21 21 9 10 Jl 20 20 20 20 20 20 21 21 21 21 21 21 22 22 22 22 22 22 23 23 23 23 10 II 21 21 22 22 22 22 22 ~^ "I3 23 "^ "^ ^3 ~M 24 ~A ~A ~A 25 25 "^ ~b ~^ n 12 23 23 24 24 24 24 24 25 25 25 25 25 26 26 26 26 26 27 27 27 27 27 28 12 i3 25 25 26 26 26 26 26 27 27 27 27 28 28 28 28 28 29 29 29 29 29 3o 3o 1 3 i4 27 27 28 28 28 28 28 29 29 29 29 3o 3o 3o 3o 3i 3i 3i 3i 32 32 3-2 32 i4 i5 29 29 3o 3o 3o 3o 3i 3i 3i 31 32 32 32 32 33 33 33 33 34 34 34 34 35 1 5 i6 3i 3i 3i li^ 1i 32 ~33 33 33 33 34 34 Ta 34 ^5 "3l3 '35 35 36 36 36 37 37 76 '7 33 33 33 M 34 M 35 35 35 35 36 36 36 37 37 37 37 38 38 38 39 39 39 17 i8 35 35 35 36 36 36 37 37 37 35 38 38 38 39 39 39 4o 4o 4o 4i 4. 4i 4i 18 '9 37 37 37 38 38 38 39 39 39 4o 4o 4o 41 4i 4i 4i 42 42 42 43 43 43 44 19 20 39 ^ 39 40 4o 40 4i 4i 4i 42 42 42 A3 43 A3 AA AA AA 45 45 45 46 46 20 ^^ 4i 4i 4i 42 42 42 43 43 ^3 AA M AA 45 45 46 46 46 Ai 47 47 48 48 48 21 ■2 2 43 43 43 M ^^ U 45 45 45 46 46 47 47 47 48 48 48 49 49 5o 5o 5o 5i 22 23 M 45 45 46 46 46 47 47 48 48 48 49 49 49 5o 5o 5i 5i 5i 52 52 53 53 23 ^4 46 47 47 48 48 48 49 49 5o 5o 5o 5i 5i 52 52 52 53 53 54 54 54 55 55 24 25 48 49 _49 5o 5o 5o 51 Si 52 52 53 53 53 54 54 55 55 55 56 56 57 _^ 58 9.5 ^ 5o 5i 5i "5^ 52 52 53 53 54 Ta "55 55 55 56 56 57 57 58 58 59 59 59 60 96 27 52 53 53 54 54 54 55 55 56 56 57 57 58 58 59 59 59 60 60 61 6. 62 62 27 28 54 55 55 56 56 56 57 57 58 58 59 5q 60 60 61 61 62 62 63 63 63 64 64 28 29 56 57 57 58 58 58 59 59 60 60 61 61 62 62 63 63 64 64 65 65 66 66 67 29 3o 58 59 59 60 60 61 61 62 62 63 63 64 64 65 65 66 66 67 67 68 68 _69 69 3o 37 60 60 6i 61 ~6i 1>3 "63 64 64 "65 "65 66 66 67 67 "68 "68 69 69 70 70 71 71 3i 32 62 62 63 63 64 65 65 66 66 67 67 68 68 69 69 70 70 71 71 72 73 73 74 32 33 64 64 65 65 66 67 6- 68 68 69 69 70 70 71 72 72 73 73 74 74 75 75 76 33 34 66 66 67 67 68 69 69 70 70 71 71 72 73 73 74 74 75 75 76 77 77 78 78 34 35 6i 68 _69 J69 70 71 71 72 72 73 74 74 Jl Jl 76 _76 77 78 78 79 79 80 81 35 36 70 "70 71 71 72 73 73 '74 74 75 76 76 11 11 78 79 79 80 80 81 82 "8^ ^ 36 37 72 73 73 73 74 75 75 76 76 77 78 78 79 80 80 81 81 82 83 83 84 84 85 37 38 73 74 73 75 76 77 77 78 79 79 80 80 81 82 82 83 84 84 85 86 86 87 87 38 39 75 76 77 77 78 79 79 80 81 81 82 83 83 84 85 85 86 86 87 88 88 89 90 39 4o _77 78 79 79 80 81 81 82 83 83 84 85 85 86 87 _^ 88 _89 _89 90 9' 9' _92 4o 4 1 79 80 81 81 82 "83 83 "84 "85 "85 86 87 87 88 89 90 90 9' 92 92 93 94 94 4i 42 81 82 83 83 84 85 85 86 87 88 88 89 90 90 91 92 92 93 94 95 95 96 97 42 43 83 84 85 85 86 87 87 88 89 90 90 91 92 92 93 94 95 95 96 97 97 98 99 43 ■U 85 86 87 87 88 89 89 90 9' 92 92 93 94 95 95 96 97 98 98 99 1 00 100 lOI AA 45 87 88 i? ^9 _9^ 9' 92 92 93 94 _95 ^ 96 _?z ^« 98 _99 100 lOI 101 102 io3 io4 45 ^G> 89 "9" 9" 9' 92 93 94 94 95 96 97 97 98 99 100 100 101 102 io3 io4 io4 7^ 106 46 4? 9' 92 92 93 94 95 96 96 97 98 99 99 100 lOI 102 io3 io3 io4 io5 106 107 107 108 47 48 93 94 94 9-) 96 97 98 98 99 100 lOI T02 102 io3 104 io5 106 106 107 108 109 1 10 1 10 48 49 95 96 9!? 97 98 99 too 100 lOI 102 io3 104 [o5 io5 106 107 108 109 109 1 10 n 1 I [2 ii3 49 JO _2Z _?^ _9S _99 100 lOI 102 io3 io3 1 04 io5 106 l^Z 108 108 109 no in 112 ii3 n3 ii4 ii5 5o 5i 99 99 100 101 102 io3 io4 7^ io5 106 107 108 109 no in III 112 773 T74 775 116 116 i'7 57 52 lOI lOI 102 io3 io4 io5 106 107 107 108 109 no in 112 n3 114 1x4 n5 116 117 #£! "9 120 52 53 102 io3 io4 io5 106 107 108 109 no no III 112 1x3 ii4 n5 116 1x7 117 118 119 121 122 53 54 io4 io5 106 107 108 109 no in 112 n3 ii3 ii4 ii5 116 117 118 1x9 120 121 122 122 123 124 54 55 106 107 108 109 no n I 112 n3 ii4 ii5 116 116 117 118 112 120 X2I 122 123 124 125 126 127 55 56 7o8 109 no lU 112 773 774 n5 116 117 n8 119 119 120 121 122 7^3 7^ 7^ 126 127 128 129 56 57 no 1 1 1 112 ii3 ii4 n5 116 117 118 119 120 121 122 123 124 124 125 126 127 128 129 i3o i3i 57 18 112 T I 3 ii4 ii5 n6 117 118 119 120 121 122 123 124 125 126 127 128 129 i3o i3i i3. l32 i33 58 *"^9 ii4 ii5 116 1 17 n8 119 120 121 12a 123 124 125 126 127 128 129 i3o i3i l32 i33 r34 i35 1 36 59 no 116 1 17 118 119 120 121 122 123 124 125 126 127 128 129 i3o i3i l32 i33 i34 i35 1 36 i37 1 38 60 P'»g«236i TABLE XXX. For finding the Variation of the Sun's Right Ascension, of the Declination, c^f tlie Equation of Time or of the Moon's Rigiit Ascension, in any number of minutes of time, the Horary ftlolion being ! given at the top of the page in seconds, and the number of miimtes of time in the side-column ; — Also, for finding the Variation of the Moon's Declination in seconds of time ; the motion in one minute being given at the top, and the numbers in the side-column being taken for seconds. Horarn) Motion. M // II // II // ;/ // // II 1 // // II // '/ // ;/ II // /' // // II M 13!! 140 141 142 143 144 145 14G 147 148 140 150 151 15; 153 154 155 15C 157 158 15< 160 I 2 2 2 2 2 2 2 2 2 2 2 3 ~3 "1 ""3 ""3 1 3 ~~3 ~3 I 2 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 r 5 5 5 5 5 5 2 3 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 6 8 8 8 8 8 8 3 4 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 IC IC 10 10 II n 1 1 4 5 12 12 12 12 12 12 12 12 12 12 12 i3 i3 i3 i3 i3 i3 i3 i3 i3 i3 i3 5 6 ~U i4 "T4 i4 "74 "74 ~75 "75 "75 ~i^ "75 13 l5 "75 ~l5 "75 IC 16 16 "76 "76 "]6 6 7 16 16 16 17 f7 17 17 17 17 17 17 18 18 18 18 i£ 18 18 18 18 •9 '9 7 8 '9 19 19 19 19 19 19 19 20 20 20 20 20 20 20 21 21 21 21 21 2] 21 8 9 21 21 21 21 21 22 22 22 22 22 22 23 23 23 20 23 23 23 24 24 24 24 9 lO 23 23 24 24 24 24 j4 ^ 25 25 25 25 25 25 26 26 26 26 26 26 27 27 10 II 25 ~^ 26 26 26 26 27 27 27 27 27 28 28 28 28 'Tb 28 29 29 29 29 29 1 1 12 28 28 28 28 29 29 29 29 29 3o 3f 3o 3( 3o 3r 3i 3i 3i 3i 32 32 32 12 t3 3o 3o 3i 3i 3i 3i 3; 32 32 32 32 33 33 33 33 33 3A 3A 3A 34 34 35 i3 i4 32 33 33 33 33 34 34 34 M 35 35 35 35 35 36 3G 3b 36 3l 0-, 37 37 i4 i5 35 35 35 36 36 36 36 37 37 37 37 38 38 38 38 39 39 39 39 40 40 4o i5 Te 37 ^7 "38 38 ^8 38 "^ 39 39 39 4o 4r ^ ~A'i ~A^ 4i 4i 42 42 42 Ao 43 76 17 39 40 40 4o 4i 4i 4i 4i 42 42 42 A^ 43 43 43 AA AA AA AA 45 45 45 17 i8 42 42 42 43 43 e M AA AA AA 45 45 45 46 46 46 A- Ai Ai 47 48 48 18 19 AA M 45 45 45 46 46 46 Ai Ai 47 48 48 48 48 49 49 49 5o 5o 5o 5i '9 2 CI 46 47 47 47 48 48 48 49 49 49 5o 5o 5o 5i 5i 5i 52 52 52 53 53 53 20 21 49 49 49 5o 5o 5o 5i 5i 5i 52 52 53 53 53 54 54 54 55 55 "55 "56 56 21 22 5i 5i 52 52 52 53 53 54 54 54 55 55 55 56 56 56 57 57 58 58 58 59 22 23 53 54 54 54 55 55 56 56 56 57 57 58 58 58 59 59 59 60 60 6r 61 6. 23 24 56 56 56 57 57 58 58 58 59 59 60 &o 60 61 6. 62 62 62 63 63 64 64 24 25 58 58 59 59 60 60 60 61 61 62 62 63 63 63 64 64 65 65 65 66 66 67 25 ^ 60 61 61 62 l32 62 63 ~63 64 "64 "65 "65 "65 "66 66 67 "67 ^68 68 68 69 72 69 ^ 27 63 63 63 64 64 65 65 66 66 67 67 68 68 68 69 69 70 70 71 71 72 11 28 65 65 66 66 67 67 68 68 69 69 70 70 70 71 71 72 72 73 73 74 74 75 28 19 67 68 68 69 69 70 70 71 7' 72 72 73 73 73 74 74 75 75 76 76 77 77 29 3o 70 70 71 71 72 72 73 73 74 74 Jl 75 76 76 77 77 78 78 79 79 80 80 3o 37 72 72 73 73 74 74 75 75 76 76 77 78 78 79 79 80 80 81 81 82 "81 "83 37 32 74 75 75 76 76 77 77 78 78 79 79 80 81 8i 82 82 83 83 84 84 85 85 32 33 76 77 78 78 79 79 80 80 81 8i 82 83 83 84 84 85 85 86 86 87 87 88 33 34 79 79 80 80 81 82 8? 83 83 84 84 85 86 86 87 87 88 88 89 90 90 9' 34 35 81 82 82 83 83 84 85 85 86 86 87 88 88 _89 89 _90 90 9' 92 92 k 93 35 36 83 ~84 ^ "85 "86 86 87 "88 88 "89 "89 ~9^ ~9^ 91 92 92 93 94 94 95 9"' 96 36 37 86 86 87 88 88 89 89 90 9' 91 92 93 93 94 94 95 96 96 97 97 98 99 3-^ 38 88 89 89 90 9' 9' 92 92 93 94 94 95 96 96 97 98 98 99 99 100 TGI lOI 38 39 9" 9' 92 92 93 94 •94 95 96 96 97 98 98 99 99 100 lOI lOI 102 io3 io3 1 04 39 4o _93 93 94 Jl 95 96 _97 _9Z j8 __99 _99 100 lOI 101 102 io3 io3 104 io5 io5 106 107 4u 4i 95 96 96 97 98 98 99 100 10(1 lOI 102 77^3 io3 io4 'io5 776 i7^ 107 107 108 109 109 47 42 97 98 99 99 100 101 102 102 io3 io4 io4 io5 106 106 107 108 109 109 no in I 1 1 112 42 43 100 100 lOI 102 102 io3 io4 io5 io5 106 107 108 108 109 1 10 no in 1 12 ii3 n3 n4 n5 43 U 102 io3 io3 104 io5 106 106 107 108 109 109 no 1 1 J 1 1 1 112 1 13 iiA ii4 n5 116 117 117 AA 45 104 io5 106 107 1?2 108 109 no no III 112 ii3 ii3 iiA n5 116 116 117 nS n9 119 120 45 46 107 107 ro8 109 110 1 10 1 1 1 1 12 f73 773 774 i75 n6 117 117 778 119 !20 120 121 122 713 A'^ 47 109 no no III 112 n3 ii4 i^A n5 116 117 118 118 119 1 20 121 121 122 123 124 125 125 47 48 II I 112 ii3 ii4 ii4 ii5 116 117 n8 118 119 120 I2T 122 1 22 123 124 125 126 126 127 128 48 49 ii4 ii4 ii5 116 117 118 iiS 119 120 121 122 123 123 124 125 126 127 :27 128 120 i3o t3i 49 5o 116 117 118 118 il? 120 121 122 123 123 124 125 126 127 128 128 129 l3c: i3i I 32 1 33 i33 5o 57 778 119 120 121 122 122 72! 124 125 126 127 7^8 7^8 129 73^ 73T 732 1 33 733 1 34 i35 736 5 1 52 120 121 ;i ,23 124 125 126 127 127 128 129 i3o i3i I 32 1 33 1 33 i34 i35 1 35 1 37 1 38 .39 52 53 123 124 125 126 127 128 120 i3o i3i I 32 i33 1 33 1 34 i35 1 36 1 37 1 38 139 40 i4o i4i 53 54 125 126 127 128 129 i3o i3i i3. I 32 1 33 1 34 i35 1 36 i37 1 38 139 i4o i4o i4i 42 143 1 44 54 55 127 128 129 i3o i3i l32 i33 1 34 i35 1 36 1 37 1 38 38 .39 1 40 i4i 142 143 i44_ 45 1 46 i47 55 56 i3o 717 732 733 <33 734 TT5 i36 737 738 .39 4o "47 42 iA3 1 44 I45 1 46 1 47 47 1 48 149 56 57 l32 i33 1 34 i35 1 36 .37 i38 .39 1 40 •4i 142 43 i43 AA 145 1 46 1 47 1 48 149 5o i5i 52 57 58 1 34 i35 1 36 ■37 1 38 139 i4o] i4i l42 143 144 45 46 Ai 48 149 i5o .5i' l52 53 1 54 55 58 59 i37 1 38 139 i4o i4i 142 143 1 44 i45i46| i47 48 48 Aq i5o ,5i l52 1 53 1 54 55 i56 57 59 60 139 .4u i4i 14-2 i43 i44i45i46t47!i48| 149 5o i5i| 52 1 53 .54 .551 i56i57i58| 1^9 60 So TABLE XXXI. [ Page 237 For finding the Sun's Right Ascension for any given number of hours. J^umher of hours. Horary Vivrialinn. 1 2 3 4 5 6 7 8 9 10 11 12 Hcirary Viiri:vlion. s II It n n II II II II II II // II S 8. So 8.5 17.0 25.5 34.0 42.5 5i.o 59.5 68.0 76.5 85.0 93.5 102.0 8.5o 8.55 8.6 17. 1 25.7 34.2 42.8 5i.3 59.9 68.4 77.0 85.5 94.1 102.6 8.55 8.60 8.6 17.2 25.8 34.4 43.0 5i.6 60.2 68.8 77.4 86.0 94 .61103. 2 8.60 8.65 8.7 17.3 26.0 34.6 43.3 5i.9 60.6 69.2 77-9 86.5 93.2 io3.8 8.65 8.70 8.7 17.4 26.1 34.8 43.5 52.2 60.9 69.6 78.3 87.0 95.7 104.4 8.70 8.75 8.8 17.5 26.3 35.0 43.8 52.5 61.3 70.0 78.8 87.5 96.3 io5.o 8.75 8.80 8.8 17.6 26.4 35.2 44.0 52.8 61.6 70.4 79.2 88.0 96.8 io5.6 8.80 8.85 8.9 17.7 26.6 35.4 44.3 53.1 62.0 70.8 79-7 88.5 97.4|io6.2 8.85 8.00 8.9 17.8 26.7 35.6 44.5 53.4 62.3 71.2 80.1 89.0 97-9 106.8 8.90 8.95 9.0 17.9 26.9 35.8 44.8 53.7 62.7 71.6 80.6 89.5 98.5 107.4 8.95 9.00 9.00 9.0 18.0 27.0 36.0 45.0 54.0 63.0 72.0 81.0 90.0 99.0 108.0 9.05 9.1 18. 1 27.2 36.2 45.3 54.3 63.4 72.4 81.5 90.5 99.6 108.6 9.05 9.10 9.1 18.2 27.3 36.4 45.5 54.6 63.7 72.8 81.9 91 .0 100. 1 1 09 . 2 9.10 9.15 9.2 18.3 27.5 36.6 45.8 54.9 64.1 73.2 82.4 91.5 100.7 1 09 . 8 9. i5 9.20 9.2 18.4 27.6 36.8 46. 55.2 64.4 73.6 82.8 92.0 IO[ .2 1 1 . 4 9.20 9.25 9.3 18.5 27.8 37.0 46.3 55.5 64.8 74.0 83.3 92.5 101.8 1 1 1 . 9.25 9.30 9.3 18.6 27.9 37.2 46.5 55.8 65.1 74.4 83.7 93.0 102.3 III. 6 9.30 9.35 9-4 18.7 28.1 37.4 46.8 56.1 65.5 74.8 84.2 93.5 102.9 112.2 9.35 9.40 9.4 18.8 28.2 37.6 47.0 56.4 65.8 75.2 84.6 94.0 io3.4 112.8 9.40 9-45 9-5 18.9 28.4 37.8 47-3 56.7 66.2 75.6 85.1 94.5 104.0 ii3.4 9-45 9.50 9.5 19.0 28.5|38.o 47.5 57.0 66.5 76.0 85.5 95.0 104.5 .114.0 9. DO 9.55 9.6 19. 1 28.7 38.2 47.8 57.3 66.9 76.4 86.0 95.5 io5.i 114. 6 9.55 9.60 9.6 19.2 28.8 38.4 48.0 57.6 67.2 76.8 86.4 96.0 105.6 Il5.2 9.60 9.65 9-7 19.3 29.0 38.6 48.3 57.9 67.6 77.2 86.9 96.5 106.2 ii5.8 9-65 9.70 9.75 9-7 19.4 29.1 38.8 48.5 58.2 67.9 77.6 87.3 97.0 106.7 1 16.4 9.70 9.75 9.8 19.5 29.3 39.0 48.8 58.5 68.3 78.0 87.8 97.5 107.3 1 17.0 9.80 t,.8 19.6 29.4 39.2 49.0 58.8 68.6 78.4 88.2 98.0 107.8 1 17.6 9.80 9.85 9.9 19.7 29.6 39.4 49-3 59.1 69.0 78.8 88.7 98.5 108.4 118. 2 9.85 9.90 9.9 19.8 29.7 39.6 49-5 59.4 69.3 79.2 89.1 99.0 108.9 118.8 9.90 9.95 10.00 10. 19.9 29.9 39.8 49-8 59.7 69.7 79.6 80.0 89.6 90.0 99.5 100. 109.5 IIO.O 119.4 120.0 9.95 10. 20.0 3o.o 4o.o 5o.o 60.0 70.0 10.00 io.o5 10. 1 20.1 3o.2 4o.2 5o.3 60.3 70.4 80.4 90.5 100.5 II0.6 120.6 10. o5 10.10 lO.I 20.2 3o.3 40.4 5o.5 60.6 70.7 80.8 90.9 lOI .0 III. I 121.2 10.10 io.i5 10.2 20.3 3o.5 40.6 5o.8 60.9 71. 1 81.2 91.4 loi .5 III. 7 121. 8 10. i5 10.20 10.2 20.4 3o.6 4o.8 5i.o 61 .2 71-4 81.6 91.8 102.0 112. 2 122.4 10.20 10.25 10.3 20.5 3o.8 4i .0 5i.3 61.5 71.8 82.0 92.3 102.5 112. a 123. 10.25 10. 3o 10.3 20.6 3o.9 4i .2 5i.5 61.8 72.1 82.4 92.7 io3.o ii3.3 123.6 io.3o 10.35 10.4 20.7 3i.i 41.4 5i 8 62.1 72.5 82.8 93.2 io3.5 113.9 1 24 . 2 10.35 10.40 10.4 20.8 3l.2 4i.6 52 62.4 72.8 83.2 93.6 104.0 114.4 124.8 10.40 10.45 10.5 20.9 3i.4 4i.8 52.3 "52". 5 62.7 73.2 83.6 94.1 104.5 ii5.o 125.4 10.45 io.5o 10.5 21 .0 3i.5 42.0 03. 73.5 84.0 94.5 io5.o ii5.5 126.0 10. 5o 10.55 10.6 21. 1 3i.7 42.2 52.8 63.3 73 <9 84.4 95.0 io5.5 116.1 126.6 10.55 10.60 10.6 21.2 3i.8 42.4 53.0 63.6 74.2 84.8 95.4 106.0 1 16.6 127.2 10.60 10.65 10.7 21.3 32.0 42.6 53.3 63.9 74.6 85.2 95-9 106.5 117.2 127.8 ■10.65 10.70 10.75 10.7 21.4 32.1 42.8 53.5 64.2 74.9 75.3 85.6 86.0 96.0 107.0 117.7 128.4 10.70 10.8 21 .5 32.3 43.0 53.8 64.5 96.8 107.5 118. 3 129.0 10.75 10.80 10.8 21.6 32.4 43.2 54.0 64.8 75.6 86.4 97.2 108.0 118. 8 129.6 10.80 10.85 10.9 21.7 32.6 43.4 54.3 65.1 76.0 86.8 97-7 108.5 119. 4 l30.2 10.85 10.90 10.9 21.8 32.7 43.6 54.5 65.4 76.3 87.2 98.1 109.0 119. i3o.8 10.90 10.95 II .0 21 .9 32.9 43.8 54.8 65.7 76.7 87.6 98.6 109.5 120.5 i3i.4 10.95 11 .00 II .0 22.0 33.0 44.0 55.0 66.0 77.0 88.0 99.0 IIO.O 121 .0 l32.0 11.00 11 .OD II. I 22.1 33.2 44.2 55.3 66.3 77-4 88.4 99.5 no. 5 121 .6 132.6 II .o5 II .10 II .1 22.2 33.3 M.A 55.5 66.6 77-7 88.8 99.9 III .0 122. 1 i33.2 1 1 .10 II. i5 11 .2 22.3 33.5 44.6 55.8 66.9 78.1 89.2 100.4 III .5 122.7 133.8 II. i5 II .20 II. 2 22.4 33.6 44.8 56.0 67.2 78.4 89.6 100.8 112.0 123.2 1 34. 4 II .20 11 .25 II. 3 22.5 33.8 45.0 56.3 67.5 78.8 90.0 I0I.3 112. 5 123.8 i35.o 11.25 1 1 .3o II. 3 22.6 33.9 45.2 56.5 67.8 79.1 90.4 101.7 ii3.o 124.3 i35.6 1 1. So 11.35 II. 4 22.7 34.1 45.4 56.8 68.1 79.5 90.8 102.2 ii3.5 124.9 136.2 11.35 II .40 II. 4 22.8 34.2 45.6 57.0 68.4 79-8 91 .2 102.6 114.0 125.4 i36.8 11.40 11.45 II. 5 22.9 34.4 45.8 57.3 68.7 80.2 91 .6 io3.i 114.5 126.0 137.4 11.45 J'^e* 2381 TABLE XXXI. For finding the Sun's Right Ascension for any given number of hours. JVumher of hours. Horary Variatinn. 13 14 15 16 17 18 19i 20. 21 1 22 23 24 II Horary Variation. s // // II II II ■/ II II // // II s 8.5o no. 5 119. c 127.5 i36.o 144.5 i53.c 161. 5 170.0 178. f 187.0 195.5 204.0 8.5o 8.55 III. 2 119-7 128.3 i36.8 145.4 i53.(; 162.5 171 .0 i79.e 188. 1 196.7 205.2 8.55 8.60 III. 8 120.4 129.0 137.6 146.2 i54.8 i63.4 172.0 180. c 189.2 IQ7.8 206.4 8.60 8.65 112.5 121. 1 129.8 i38.4 147.1 i55.7 164.4 173.0 181.' 190.3 199.0 207.6 8.65 8.70 8.75 ii3.i 121. 8 i3o.b 139.2 147-9 i56.6 i65.3 174.0 182.: 191.4 200.1 208.8 8.70 ii3.'8 122.5 i3i.3 i4o.o 148.8 157.5 166.3 175.0 i83.£ 192.5 201.3 210.0 8.75 8.80 114.4 123.2 l32.0 i4o.8 149.6 i58.4 167.2 176.0 184. £ 193.6 202.4 211 .2 8.80 8.85 lib. I 123.9 l32.b i4i.6 i5o.5 159.3 168.2 177.0 l85.q 194.7 2o3.6 212.4 8.85 8.90 .15.7 124.6 133.5 142.4 i5i.3 160.2 169. 1 178.0 186. q iq5.8 204.7 213.6 8.90 8.95 9...0 116. 4 125. J 134.3 143.2 l52.2 161 .1 170. 1 179.0 188. c 196.9 205.9 214.S 8.95 1 17.0 126.0 i35.o 144.0 i53.o 162.0 171 .0 180.0 189. c 198.0 207.0 216.0 9.00 9.05 117. 7 126.7 i3b.8 144.8 153.9 162.9 172.0 181. 190. 1 199.1 208.2 217.2 9.05 9.10 118.3 127.4 i3b.b 145.6 ib4.7 i63.8 172.9 182.0 191. 1 200.2 209.3 218.4 9.10 9.15 1 19.0 128. 1 137-^ 146.4 i55.6 164.7 173.9 i83.o 192.2 201.3 210.5 219.6 ?.i5 9.20 119. 6 128.8 i38.o 147.2 i56.4 i65.6 166.5 174.8 184.0 193.2 202.4 211. 6 220.8 9.20 9.25 120.3 129.5 i38.8 i48.o 157.3 175.8 i85.o 194.3 2o3.5 212.8 222.0 9.25 9.30 120.9 i3o.2 139.5 148.8 i58.i 167.4 176.7 186.0 195.3 204.6 213.9 223.2 9.30 9.35 121. 6 1 30.9 140.3 149.6 159.0 168.3 177.7 187.0 196.4 205.7 2l5.I 224.4 9.35 9.40 122.2 i3i.6 i4i .0 i5o.4 159.8 169.2 178.6 188.0 197.4 206.8 216.2 225.6 9.40 9-45 122.9 132.3 i4i.8 i5i .2 160.7 170. 1 179.6 189.0 198.5 207.9I2I7.4 226.8 9.45 9.50 123.5 i33.o 142.5 I 52.0 161.5 171 .0 180.5 190.0 199.5 209.0^18.5 228.0 9.50 9.55 124.2 133.7 143.3 i52.8 162.4 171.9 181. 5 191 .0 200.6 210. 1 219.7 229.2 9.55 9.60 124.8 134.4 i44.o i53.6 i63.2 172.8 182.4 192.0 201 .6 211 .2 220.8 23o.4 9.60 9-65 125.5 i35. 1 144.8 154.4 164. 1 173.7 i83.4 193.0 202.7 212.3 222.0 23i.6 9-65 9.70 126. 1 135.8 i4b.5 i55.2 164.9 174.6 184.3 194.0 203.7 2i3.4 223.1 224.3 232.8 234.0 9.70 9.75 9-p 126.8 136.5 146.3 i56.o i65.8 175.5 i85.3 195.0 204.8 214.5 9.80 127.4 137.2 i47-o i56.8 166.6 176.4 186.2 196.0 2o5.S 2i5.6 225.4 235.2 9.80 9.85 128. 1 137.9 147-8 157.6 167.5 177.3 187.2 197.0 206.9 216.7 226.6 236.4 9-85 9.90 128.7 JJ8.6 148. b i58.4 168.3 178.2 188.1 198.0 207.9 217.8 227.7 237.6 9.90 9.95 129.4 139.3 149.3 159.2 169.2 179.1 189. 1 199.0 209.0 218.9 228.9 238.8 9.95 10,00 i3o.o i4o.o ibo.o 160.0 170.0 180.0 190.0 200.0 210.0 220.0 23o.o 240.0 10.00 10. o5 i3o.7 140.7 i5o.8 160.8 170.9 180.9 191. 201 .0 211 .1 221. I 23l.2 241.2 10. o5 10.10 i3i.3 141.4 i5i.5 161. 6 171. 7 18J.8 191. 9 202.0 212. 1 222.2 232.3 242.4 10.10 10. i5 l32.0 142. 1 lb2.3 162.4 172.6 182.7 192.9 2o3.0 2l3.2 223.3 233.5 243.6 10. i5 10.20 t32.6 142.8 ib3.o i63.2 173.4 i83.6 193.8 204.0 214.2 224.4 234.6 244.8 10.20 10.20 i33.3 143.5 i53.8 164.0 174.3 184.5 194.8 205.0 2i5.3 225.5 235.8 246.0 10.25 10. 3o 133.9 144.2 ib4.5 164.8 175. 1 i85.4 195.7 206.0 216.3 226.6 236-9 247.2 10. 3o 10.35 i34.6 144.9 i5b.3 165.6 176.0 186.3 196.7 207.0 217.4 227.7 238.1 248.4 10.35 10.40 135.2 145.6 ib6.o 166.4 176.8 187.2 197.6 208.0 218.4 228.8 239.2 249.6 10.40 10.45 iJb.9 i36.5 146-3 i47-o ib6.8 157.5 167.2 177.7 188.1 198.6 209.0 219.5 229.9 240.4 250.8 10.45 io.5o 168.0 178.5 189.0 199.5 210.0 220.5 23i .0 241.5 252.0 io.5o 10.55 137.2 i47-7 i58.3 168.8 179.4 l89.q 200.5 211 .0 221 .6 232.1 242.7 253.2 10.55 10.60 137.8 148.4 159.0 169.6 180.2 190.8 201 .4 212.0 222.6 233.2 243.8 254.4 10.60 10. 65 i38.5 149- 1 ib9.8 170.4 181. 1 191. 7 202.4 2l3.0 223.7 234.3 245.0 255.6 10.65 10.70 139. 1 149.8 160.5 171 .2 181. 9 192.6 2o3.3 214.0 224.7 235.4 246.1 256.8 10.70 10.75 139.8 1 5o . 5 161. 3 172.0 182.8 193.5 204.3 2l5.0 225.8 236.5 247.3 258.0 10.75 10.80 i4o.4 l5l.2 162.0 172.8 i83.6 194.4 205.2 216.0 226.8 237.6 248.4 259.2 10.80 10.85 i4i .1 i5i .9 162.8- 173.6 184.5 195.3 206.2 217.0 227.9 238.7 249.6 260.4 10. 85 10.90 141.7 ib2.6 i63.b 174.4 185.3 196.2 207.1 218.0 228.9 239. 8| 25o.7 261 .6 10.90 10.95 142.4 i53.3 164.3 175.2 186.2 197.1 208.1 219.0 23o.O 240.9 251.9 262.8 10.95 11.00 11 .00 143.0 1 54.0 i65.o 176.0 1S7.0 198.0 209.0 220.0 23l .0 242.6 253. 264.0 II. o5 143.7 154.7 165.8 176.8 187.9 198.9 210.0 221 .0 232.1 243.1 254.2 265.2 II .o5 II. 10 144.3 ibb.4 166. b 177.6 188.7 199.8 210.9 222.0 233.1 244.2 255.3 266.4 II .10 II. i5 145.0 i56.i 167.3 17B.4 189.6 200.7 211 .0 223. 234.2 245.3 256.5 267.6 II. i5 II .20 145.6 146.3 1 56.8 168.0 179.2 190.4 201 .6 212.8 224.0 235.2 246.4 257.6 268.8 11.20 11.25 11 .25 157.5 168.8 180.0 191 .3 202.5 2l3.8 225.0 236.3 247.5 258.8 270.0 II .3o 146.9 i58.2 169.5 180.8 1 92. 1 2o3.4 214.7 226. 237.3 248.6 259.9 271 .2 II .3o 11.35 147.6 1 58. 9 170.3 181. 6 193.0 204.3 2i5.7 227.0 238.4 249.7 261 .1 272 .4 11.35 II .4o 148.2 59.6 1 7 1 . 182.4 193.8 2o5.2 216.6 228.0 239.4 25o.8 262.2 273.6 II .40 11.45 14S.9 60.3J171.8 i83.2| 194.7 206.1 217.6 229.0 240.5 25 1 .9I263.4I 274.8 11.45 TABLE XXXII. [Pa-e239 Variation of" the Sun's Altitude in one minute from noon. Lat. Declination of a different name from the Latitude. Lat. 0= 1° 9° 3° 40 5° 6° 7° II 8° 9° 10° 11° // n II II II 28.1 II II i4.o II II II o° 22.4 18.7 16.0 12.4 1 1 . 1 10. 1 c'' I 28.1 22.4 18.7 16.0 i4.o 12.4 1 1 .2 10. 1 9.3 I 2 1 28.1 22.4 18.7 16.0 14.0 12.5 1 1 .3 10.2 9.3 8.6 2 3 28.1 22.4 18.7 16.0 i4.o 12. D 11 .2 10.2 9.3 8.6 8.0 3 •4 28.1 22.4 18.7 16.0 i4.o 12.5 12.5 II .2 II .2 10.2 9.3 8.6 8.6 8.0 7-4 4 5 22.4 .8.7 16.0 i4.o 10.2 9.3 8.0 7-4 7.0 5 6 .8.7 16.0 i4-o 12.5 1 1 .2 10.2 9.3 8.6 8.0 7.5 7.0 6.6 6 7 16.0 i4-o 12.4 II .2 10.2 9.3 8.6 8.0 7.5 7.0 6.6 6.2 7 8 i4-o 12.4 11.2 10.2 9.3 8.6 8.0 7-5 , 7.0 6.6 6.2 5.9 8 9 12.4 ti .2 10.2 9.3 8.6 8.6 8.0 7.5 7.0 6.6 6.2 5.9 5.6 9 lO II . I 10. 1 tl 8.0 7-4 7.0 6.6 6.2 5.9 5.6 5.3 10 1 1 10. 1 9.3 8.0 7-4 7.0 6.6 6.2 5.9 5.6 5.3 5.1 1 1 12 9.2 8.5 7-9 7.4 7.0 6.5 6.2 b.9 5.6 5.3 5.0 4.8 12 i3 8.5 7-9 7.4 6.9 6.5 6.2 5.8 5.6 5.3 5.0 4.8 4.6 i3 1 4 7-9 7.3 7.4 6.9 6.5 6.2 5.8 5.5 5.3 5.0 4.8 4.6 4.4 i4 i5 6.9 6.5 6.1 5.8 5.5 5.3 5.0 4.8 4.6 4.4 4.1 i5 1 6 6.8 6.5 6.1 5.8 5.5 5.2 5.0 4.8 4.6 4.4 4.2 4.1 16 17 6.4 6.T 5.8 5.5 5.2 5.0 4.8 4.6 4.4 4.1 4.1 3.9 17 1 8 6.0 5.7 5.5 5.2 f.o 4.8 4.6 4.4 4.2 4.1 3.9 3.8 18 19 5.7 5.4 5.2 4.9 4.7 4.5 4.4 4.1 4.0 3.9 3.8 3.5 '9 20 5.4 5.1 4.9 4.7 4.5 4.3 4.2 4.0 3.9 3.8 3.6 20 21 5.1 4.9 4.7 4.5 4.3 4.2 4.0 3.9 3.7 3.6 3.5 3.4 21 22 4.9 4.7 4.5 4.3 4.1 4.0 3.9 3.7 3.6 3.5 3.4 3.3 22 23 4.6 4.4 4.3 4.1 4.0 3.8 3.7 3.6 3.5 3.4 3.3 3.2 23 24 25 4.4 4.2 4.1 3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1 4.2 4.1 3.9 3.8 3.7 3.5 3.4 3.3 3.2 3.1 3.1 3 .; 26 4.0 3.9 3.8 3.6 3.5 3.4 3.3 3.2 3.1 3.0 3.0 2.9 26 27 3.9 3.7 3.6 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.9 2.8 27 28 3.7 3.6 3.5 3U 3.3 3.2 3.1 3.0 2.9 2.8 2.8 2.7 28 29 3.5 3.4 3.4 3.3 3.2 3.1 3.1 3.0 2.9 2.8 2.8 2.7 2.6 29 3o 3.3 3.2 3.1 3.0 3.0 2.9 2.8 2.7 2.7 2.6 2.5 3o 3i 3.3 3.2 3.1 3.0 2.9 2.9 2.8 2.7 2.6 2.6 2.5 2.5 3i 32 3.1 3.1 3.0 2.9 2.8 2.8 2.7 2.6 2.6 2.5 2.5 2.4 32 33 3.0 2.9 2.9 2.8 2.7 2.7 2.6 2.5 2.5 2.4 2.4 2.3 33 34 2.9 2.8 2.8 2.7 2.6 2.6 2.5 2.5 2.4 2.4 2.3 2.3 34 35 2.8 2.7 2.7 2.6 2.5 2.5 2.4 2.4 2.3 2.3 2.2 2.2 35 ■ 3G 2.7 2.6 2.6 2.5 2.5 2.4 2.4 2.3 2.3 2.2 2.2 2.1 36 37 2.6 2.5 2.5 2.4 2.4 2.3 2.3 2.2 2.2 2.2 2.1 2. I 37 38 2.5 2.5 2.4 2.4 2.3 2.3 2.2 2.2 2.1 2. 1 2. I 2.0 38 39 2.4 2.3 2.4 2.3 2.3 2.2 2.2 2.1 2. 1 2.1 2.0 2.0 2.0 39 4o 2.3 2.2 2.2 2.2 2.1 2.1 2.0 2.0 2.0 1.9 1.9 40 4i 2.3 2.2 2.2 2. I 2. I 2.T 2.0 2.0 1.9 1.9 1.9 1:8 4i 42 2.2 2. I 2. 1 2. I 2.0 2.0 2.0 1.9 1.9 1 .9 1.8 1.8 42 43 2. 1 2.1 2.0 2.0 2.0 1.9 1.9 1.9 1.8 1 .8 1.8 1-7 43 44 2.0 2.0 2.0 1.9 1.9 1.9 1.8 1.8 1.8 1-7 1-7 1-7 44 45 45 2.0 1.9 I .9 1.9 1.8 1.8 1.8 1 -7 1-7 1-7 1-7 1.6 46 1.9 1.9 1 .8 1.8 1.8 1-7 1-7 1-7 1-7 1.6 1.6 I.b 46 47 1.8 1.8 1.8 1-7 I -7 1-7 1-7 i.b 1.6 1.6 1.6 I.b 47 48 1.8 '•7 ' .7 1-7 1-7 1.6 1.6 1.6 1.6 1.6 1.5 1.5 48 49 1.7 '•7 7 1.6 1.6 1.6 1.6 1.5 1.5 1.5 1.5 1.5 49 5o 5o 1.6 1.6 6 1.6 1.6 1.5 1.5 1.5 1.5 1.5 1.4 1.4 52 1.5 1.5 .5 1.5 1.5 1,4 1.4 1.4 1.4 1.4 1.4 1.3 52 54 1.4 1.4 i./\ 1.4 1.4 I 3 1.3 1.3 1.3 1.3 1.3 1.3 54 56 1.3 1.3 1.3 1.3 1.3 1.3 1.2 1 .2 1 .2 1 .2 1 .2 1 .2 56 58 1 .2 1 .2 1 .2 1 .2 1 .2 1 .2 1 .2 I . I 1 .1 1 .1 1 .1 1 . 1 58 60 f . I I . I 1 .1 1 .1 I . I I . I 1 . 1 I . I 1 .0 1 .0 1 .0 1 .0 60 62 1 .0 1 .0 1 .0 1 .0 I'.O 1 .0 1 .0 I.O 1 .0 1 .0 1 .0 0.9 62 64 1 .0 0.9 0.9 0.9 0.9 0.9 c; 0.9 0.9 0.9 0.9 0.9 64 66 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 66 68 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.7 0.7 0.7 68 70 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 70 QP 1' 2" 3° 40 5° 6° 7° 8° 9° 10° 11° r-i=e240] TABLE XXXII. Variation of the Sun's Altitude in one minute from noon. Declination of a different name from the Latitu de. Lat. ~0° 12° 13° 14° 15° 16° 17° 18° 19^ 20° 21° 22° 23° 24° Lnt. // II // II 7.3 6.8 // II II // II II II II 0° 9.2 8.5 7-9 6.4 6.0 5.7 5.4 5.1 4.9 4.6 4 4 I 8.5 7-9 7.4 6.9 6.5 6.1 5.7 5.4 5.1 4.9 4.7 4.4 4 2 1 2 7-9 7.4 6.9 : 6.5 6.1 5.8 5.5 5.2 4.9 4.7 4.5 4.6 4 I 2 3 7-4 6.9 6.5 6.1 5.8 5.5 5.2 4.9 4.7 4.5 4.6 4.1 3 9 3 4 7.0 6.5 6.2 5.8 5.5 5.2 5.2 5.0 5.0 4.8 4.7 4.5 4.3 4.1 4.0 3 8 4 ' 5 5 6.5 6.2 5.8 5.5 4.5 4.3 4.2 4.0 3.8 3 7 6 6.2 5.8 5.5 5.3 5.0 4.8 4.6 4.4 4.2 4.0 3.9 3.7 3.6 6 7 5.q 5.6 5.3 5'.o 4.8 4.6 4.4 4.2 4.0 3.9 3.7 3.6 3.5 7 8 5.6 5.3 5.0 4.8 4.6 4.4 4.2 4.0 3.9 3.7 3.6 3.5 3.4 8 _9_ lO 5.3 5.0, 4.8 4.6 4^4 4.2 4.2 4.1 4.1 3.9 3.8 3.6 3.5 3.4 3.3 _9_ 10 5.0 4.8 4.6 4.4 3.9 3.8 3.6 3.5 3.4 3.3 3.2 T I 4.8 4.6 4.4 4.2 4.1 3.9 3.8 3.6 3.5 3.4 6.6 3.2 3.1 11 12 4.6 4.4 4.3 4.1 3.9 3.8 3.7 3.5 3.4 3.3 3.2 3.1 3.0 12 t3 A.A 4.3 4.1 3.9 3.8 3.7 3.5 3.4 ■6.6 3.2 3.1 3.0 2.9 1 3 i4 i5 4.2 4.1 3.9 3.8 3.7 3.5 ■6.4 3.3 3.2 3.1 3.0 2.9 2.8 i4 i5 4.1 3.9 3.8 3.7 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.8 2.8 i6 3.9 3.8 3.7 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.8 2.8 2.7 16 17 3.8 3.7 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.8 2.8 2.7 2.6 17 i8 3.7 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.9 2.8 2.7 2.6 2.5 18 20 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.9 ?.8 2.7 2.6 2.6 2.6 2.5 -9. 20 ^.A 3.3 3.2 3.1 3.0 2.9 2.9 2.8 2.7 2.6 2.5 2.4 21 3.3 3.2 3.1 3.0 2.9 2.8 2.8 2.7 2.6 2.6 2.5 2.4 2.4 21 2 2 3.2 3.1 3.0 2.9 2.8 2 8 2.7 2.6 2.6 2.5 2.4 2.4 2.3 22 23 ' 3.1 3.0 2.9 2.8 2.8 2.7 2.6 2.6 2.5 2.4 2.4 2.3 2.3 2 3 24 25 3.0 2.9 2.8 2.8 2.7 2.6 2.5 2.5 2.4 2.4 2.3 2.3 2 .2 24 25 2.9 2.8 2.7 2.7 2.6 2.5 2.5 2.4 2.4 2.3 2.3 2.2 2.2 26 2.8 2.7 2.7 2.6 2.5 2.5 2.4 2.4 2.3 2.3 2.2 2.1 2.1 26 27 2.7 2.7 2.6 2.5 2.5 2.4 2.4 2.3 2.2 2.2 2.1 2. I 2.1 27 28 2.6 2.6 2.5 2.5 2.4 2.3 2.3 2.2 2.2 •2.1 2.1 2.1 .0 28 29 3o 2.6 2.5 2.4 2.4 2.3 2.3 2.2 2.2 2. I i . I 2.0 2.0 2.0 29 "3o 2.5 2.4 2.4 2.3 2.3 2.2 2.2 2.1 2.1 3.0 2.0 2.0 1.9 3i 2.4 2.4 2.3 2.3 2.2 2.2 2. I 2.1 2.0 2.0 2.0 1.9 1.9 3i 32 2.3 2.3 2.2 2.2 2.2 2. I 2. I 2.0 2.0 1.9 1-9 1.9 1.8 32 33 2.3 2.2 2.2 2.1 2.1 2.1 2.0 2.0 1.9 1.9 1-9 1.8 1.8 33 34 35 2.2 2.2 2.1 2.1 2.0 2.0 2.0 2.0 1.9 1.9 1.9 1.8 1.8 1.8 35 2.2 2.1 2.1 2.0 2.0 1.9 1.9 1.8 1.8 1.8 1-7 1-7 36 2.1 2.1 2.0 2.0 1.9 1.9 1.9 1 .8 1.8 1.8 1-7 1-7 1-7 36 37 2.0 2.0 2.0 1.9 1.9 1.9 1.8 1.8 1.8 1-7 1-7 1-7 1.6 37 38 2.0 1.9 1.9 1.9 1.8 1.8 1.8 1.8 1-7 I -7 1-7 1.6 1.6 38 39 40 1.9 1.9 1.9 1.8 1.8 1.8 1-7 1-7 1-7 1.6 1.6 1. 6 1.6 39 40 1.9 1.8 1.8 1.8 1-7 1-7 •■7 1-7 1.6 1.6 16 1.6 1.5 4i 1 .8 T.8 1.8 > -7 1-7 1-7 1.6 1.6 1.6 1.6 1.5 1.5 1.5 4i 42 1.8 1-7 1.7 1-7 1-7 1.6 1.6 1.6 1.6 1.5 1.5 1.5 1.5 42 43 1-7 1-7 1-7 1.6 1 .3 1.6 1.6 1.5 1.5 1.5 1.5 1.4 1.4 43 44 45 1-7 1.6 1.6 1.6 1.6 1.5 1.5 1.5 1.5 1.5 J. 5 1.5 1.4 1.4 1.4 44 45 1.6 1.6 1.6 1.5 1.5 1.5 1.4 1.4 1.4 1.4 1.4 46 1.6 1.6 I .0 1.5 1.5 1.5 1.4 1.4 1.4 1.4 1.4 1.3 1.3 46 47 1.5 1.5 1.5 1.5 1.4 1 .4 1.4 1.4 1.4 1.3 1.3 1.3 1.3 47 48 1.5 1.5 1.4 1.4 1.4 1 .4 1.4 1.4 1.3 1.3 1.3 1.3 1.3 48 49 5o 1.4 1.4 1.4 1.4 1.4 1.3 1.3 1.3 1.3 1.3 1.3 1 .2 1 .2 49 "5o 1.4 1.4 1.4 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1 .2 1 .2 1 .2 52 1.3 1.3 1.3 1.3 1.3 1.3 1 .2 1 .2 1 .2 1 .2 1 .2 1 . 1 I . I 52 54 1 .2 1 .2 1.2 1 .2 1 .2 1 .1 1 .2 t.i I . I 1 . 1 1 . 1 I . I 1 . 1 54 56 1 .2 I . I I . I 1 .1 I . I 1 . 1 1 . 1 1 .1 1 . 1 I . I 1 .0 1 .0 1 .0 56 58 1 .1 I . I 1 .1 I . I 1 .0 1 .0 1.0 1 .0 1 .0 1 .0 1 .0 1 .0 1 .0 58 60 1 .0 1 .0 I.O 1 .0 1 .0 1.0 1 .0 0.9 0.9 0.9 0.9 0.9 0.9 62 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0-8 62 64 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 O.cS 0.8 0.8 0.8 0.8 64 66 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.7 0.7 0.7 66 68 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 68 10_ 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.6 _70_ 12° 13° 14° 15° iQP 17° 18° 19° 20° 21° 22° 23° 24° TABLE XXXII. [I age 24] Variation of the Sun's Altitude in one minute from noon. Lat. Declination of thl same name as tk e Latitude. Lat. 0° 1° 2° 3° 40 5° 6° 7° 8° 9" 10° 11° II ir II // II II n II II „ // II 10. 1 0° 28.1 22.4 18.7 16.0 i4-o 12.4 II.I 0° I 28.0 22.4 18.6 16.0 i3.9 12.4 II.I I 2 28.0 22.3 18.6 i5 9 i3.9 12.3 2 3 27.9 22.3 18.5 i5.8 i3.8 3 4 28.1 27.8 22.2 18.5 i5.8 18.4 4 5 22.4 28.0 27.7 22. 1 5 6 18.7 22.4 28.0 27.6 22.0 6 7 16.0 18. b 22.3 27.9 27.4 7 8 i4-o 16.0 18.6 22.3 27.8 8 9 10 12.4 13.9 i5.9 18.5 22.2 18.5 27.7 22.1 27.6 9 ii.i 12.4 i3.9 i5.8 10 II 10. 1 II.I 12.3 i3.8 i5.8 1S.4 22.0 27.4 II 12 9.2 10. 1 II.I 12.3 i3.8 15.7 18.3 21.9 27.3 12 i3 8.5 9.2 10. II. 12.2 i3.7 i5.b 18.2 21.7 27.1 i3 i4 7-9 8.5 9.2 10. 10.9 12.1 i3.6 i5.5 18.0 21 .6 26.9 i4 i5 7.3 7.8 8.4 9> 9-9 10.9 12. 1 i3.5 i5.4 17.9 21.4 26.7 i5 i6 6.8 7.3 7.8 8.4 9.1 9.8 1U.8 12.0 i3.4 i5.3 17.8 21.3 16 17 6.4 6.8 7.2 7.8 8.3 9.0 9.8 10.7 II. 9 i3.3 l5.2 17.6 17 i8 6.0 6.4 6.8 7 -2 7-7 8.3 8.9 9-7 10.6 II. 8 l3.2 i5.o 18 19 5.7 b.o 6.3 6.7 7-2 6.7 7.6 8.2 8.9 9.6 10.6 II. 7 i3.i 19 20 5.4 5.7 6.0 6.3 7.1 7.6 8.1 8.8 9.5 10.5 II. 6 20 21 5.1 5.4 5.6 5.9 6.3 6.6 7.0 7.5 8.1 8.7 9.5 '10.4 21 22 4.9 5.1 5.3 5.6 5.9 6.2 6.6 7.0 7.5 8.0 8.6 9.4 22 23 4.6 4.8 5.0 5.3 5.5 5.8 6.1 6.5 6. 9 7.4 7.9 8.5 23 24 25 .4.4 4.6 4.8 5.0 5.2 5.0 5.5 5.8 6.1 "5.7 6.4 6.8 7.3 7.8 24 4.2 4.4 4.G 4.7 5.2 5.4 6.0 6.4 6.8 7.2 25 26 4.0 4.2 4.3 4.5 4.7 4.9 5.1 5.4 5.7 6.0 6.3 6.7 26 27 3.9 4.0 4.1 4.3 4.5 4.7 4.9 5.1 5.3 5.6 5.9 6.2 27 28 3.7 3.8 4.0 4.1 4.i 4.4 4.6 4.8 5.0 5.3 5.5 5.8 28 29 3.5 3.7 3.8 3.9 4.1 4.2 4.0 4.4 4.2 4.6 4.3 4.7 5.0 5.2 5.5 29 J(l 3.4 3.5 3.6 3.7 3.9 4.5 4.7 4.9 5.1 3o 3i 3.3 3.4 3.5 3.6 3.7 3.8 4.0 4.1 4.3 4.4 4.6 4.8 3i 3:« 3.1 3.2 S.o ■6.4 3.5 3.7 3.8 3.9 4.1 4.2 4.4 4.6 32 33 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.9 4.0 4.2 4.3 33 34 2.9 3.0 3.1 3.2 3.2 3.3 3.4 3.6 3.7 3.8 3.9 4.1 34 3-') 2.8 2.9 3.0 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.9 35 36 2.7 2.8 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 36 37 2.6 2.7 2.7 2.8 2.9 2.9 3.0 3.1 3.2 3.3 3.4 3.5 37 38 2.5 2.6 2.6 2.7 2.8 2.8 2.9 3.0 3.0 3.2 3.2 3.3 38 39 2.4 2.5 2.5 2.6 2.7 2.7 2.8 2.9 2.9 3.0 3.1 3.2 39 40 40 2.3 2.4 2.4 2.5 2.6 2.6 2.7 2.7 2.8 2.9 3.0 3.0 4i 2.3 2.3 2.4 2.4 2.5 2.5 2.6 2.6 2.7 2.8 2.8 2.9 4i 42 2.2 2.2 2.3 2.3 2.4 2.4 2.5 2.5 2.6 2.6 2.7 2.8 42 43 2.1 2.1 2.2 2.2 2.3 2.3 2.4 2.4 2.5 2.5 2.6 2-7 43 44 2.0 2.1 2.1 2.1 2.2 2.2 2.3 2.3 2.4 2.4 2.5 2.5 44 45 2.0 2.0 2.0 2.1 2.1 2.2 2.2 2.2 2.3 2.3 2.4 2.4 45 46 I .9 1.9 2.0 2.0 2.0 2. I 2.1 2.2 2.2 2.2 2.3 2.3 46 47 1 .8 1.9 1.9 1.9 2.0 2.0 2.0 2.1 2.1 2.1 2.2 2 .2 47 4» 1.8 1.8 1.8 1.9 1.9 1.9 2.0 2.0 2.0 2.1 2.1 2.1 48 49 5o 1-7 1-7 1.8 1.8 1.8 1.8 1.9 1.9 1.9 2.0 2.0 2.1 49 5o 1.6 '•7 1-7 1-7 1.8 1.8 1.8 1.8 1.9 1.9 1.9 2.0 52 1.5 1.6 1.6 1.6 1.6 1.6 1-7 1-7 1-7 1.8 1 .8 1.8 52 54 1.4 1.4 1.5 1.5 1.5 1.5 1.5 1.6 1.6 1.6 1.6 1-7 54 56 1.3 1.3 1.4 1.4 1.4 1.4 1.4 1.4 1.5 1.5 1.5 1.5 56 58 1 .2 1.2 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.4 1 .4 1.4 58 60 1 .1 I.I 1 .2 1.2 1.2 1.2 1 .2 1 .2 1.2 1.2 1.3 1.3 60 62 1 .0 I.O I.I 1 .1 I.I 1 .1 1 .1 1 .1 I.I I.I 1 .2 1 .2 62 64 1 .0 1 .0 1 .0 1.0 1.0 1.0 1 .0 1.0 1.0 1 .0 1 .0 I . I 64 66 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 1 .0 66 68 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.9 0.9 68 70 ®-7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.8 70 0° 1° 2° 3° 40 5° G° 7° 8° 9° 10° 11° ■6i Pa;' u242J TABLE XXXJI. Variation of the Sun's Altitude in one minute from noon. Lat. '0° Declination of the same name as the Latitude. Lat. 0° 12= 13° 14° 15° 16° 17° 18° 19° 20° 21° 22° 23° 2-4° // II II II II II II II II It II II It 9.2 8.5 7-? 7.3 6.8 6.4 6.0 5.7 5.4 5.1 4.Q 4.0 4.4 I 10. 1 9.2 8.5 7.8 7.3 6.8 6.4 6.0 5.7 5.4 5.1 4.8 4.6 I 2 11. 1 10. 9.2 8.4 7.8 7.2 6.8 6.3 6.0 5.6 5.3 5.0 4.8 2 3 12.3 II .0 10. 9.1 8.4 7.8 7-2 6.7 6.3 5.9 5.6 5.3 5.0 3 4 5 i3.8 12.2 10.9 9.9 9.1 8.3 7-7 7.2 6.7 6.3 5.9 5.5 5.2 4 5" 15.7 i3.7 12. 1 10.9 9.8 9.0 8.3 7.6 7-1 6.6 6.2 5.8 5.5 6 18.3 i5.6 i3.6 12. 1 10.8 9.8 8.9 8.2 7.6 7.0 6.8 6.1 5.8 6 7 21.9 18.2 i5.5 i3.5 12.0 10.7 9-7 8.9 8.1 7.5 7.0 6.5 6.1 7 8 27.3 21.7, 18.0 i5.4 i3.4 1 1 .9 10.6 9.6 8.8 8.1 7.5 6.9 6.4 8 _9 10 27.1 21 .6 17.9 i5.3 i3.3 II. 8 10.6 9.5 8.7 8.0 7-4 6.8 _9_ 10 26.9 21.4 17.8 l5.2 l3.2 II. 7 10.5 9.5 8.6 7-9 7.3 11 26.7 21.3 17.6 i5.o i3.i II. 6 10.4 9-4 8.5 7.8 11 12 26.5 21 .1 17.5 14.9 i3.o II. 5 10.3 9.3 8.4 12 i3 26.2 20.9 17.3 i4.8 12.8 II. 3 10. I 9.2 i3 i4 i5 26.0 20.7 17. 1 r4.6 12.7 II .2 10. i4 i5 25.7 20.4 16.9 i4.4 12.5 1 1 .1 lb 2b. 5 25.4 20.2 16.7 i4.3 12.4 16 17 21. 1 26.2 25.1 20.0 16.5 i4.i 17 18 17.6 20.9 26.0 24.8 19.7 16.3 18 _L9 2.0 14.9 17.3 20.7 25. '7 24.5 19.5 11, 20 i3.o i4.8 17. 1 20.4 25.4 24.2 21 II. 5 E2.8 i4-6 16.9 20.2 25.1 21 22 10.3 II. 3 12.7 14.4 16.7 20.0 24.8 22 2j 9.3 10. 1 II .2 12.5 i4.3 16.5 19.7 24.5 23 24 25 8.4 9.2 10. II. I 12.4 14. 1 16.3 19.5 24.2 24 25 7-7 8.3 9.0 9.9 10.9 12.2 13.9 16.1 19.2 23.8 26 7-1 7.6 8.2 8.9 0.8 10.8 12. 1 i3.7 15.9 18.9 23.5 26 27 6.6 7.0 7.5 8.1 8.8 9.6 10.6 1 1 .9 i3.5 i5.6 18.6 23.1 27 28 6.2 6.5 7.0 7-4 8.0 8.7 9.5 10.5 II. 7 i3.3 i5.4 18.3 22.7 28 29 3o 5.7 6.1 6.4 6.9 7.3 7-9 8.6 9-4 10.3 II. 5 i3.i i5.i 18.0 29 3o 5.4 5.7 6.0 6.4 6.8 7.2 7.8 8.4 9.2 10. 1 II. 3 12.8 14.9 3i 5.1 5.3 5.6 5.9 6.3 6.7 7.1 7-7 8.3 9.0 10. II. I 12.6 3i 32 4.8 5.0 5.2 5.5 5.8 6.2 6.5 7.0 7.5 8.1 8.9 9.8 10.9 32 33 4.5 4.7 4.9 5.1 5.4 5.7 6.1 6.4 6.9 7-4 8.0 8.7 9.6 33 M 35 •4.3 4.0" A.^ 4.6 4.8 5.1 5.3 5.6 5.9 6.3 6.8 7.3 7.8 8.6 35 4.2 A.A 4.5 4.7 5.0 5.2 5.5 5.8 6.2 6.6 7-1 7.7 36 3.8 4.0 4.1 4.3 4.5 4.7 4.9 5.1 5.4 5.7 6.1 6.5 7.0 36 ■37 3.6 3.8 3.9 4.0 A. 2 4.4 4.6 4.8 5.0 5.3 5.6 6.0 6.4 37 38 3.4 3.6 3.7 3.8 4.0 4.1 4.3 4.5 4.7 4.9 5.2 5.5 5.8 38 39 4o 3.3 3.4 3.5 3.6 3.8 3.6 3.9 3.7 4.0 4.2 4.4 4.6 4.8 5.1 5.4 39 40 3.1 3.2 3.3 3.4 3.8 4.0 4.1 4.3 4.5 4.7 •5.0 4i 3.0 3.1 3.2 ■6.6 3.4 3.5 3.6 3.7 3.9 4.0 4.2 4.4 4.6 4i 4-2 2.9 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.7 3.8 4.0 4.1 4.3 42 43 2.7 2.8 2.9 3.0 3.0 3.1 3.2 3.3 3.5 3.6 3.7 3.9 4.0 43 44 45 2.6 2.7 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.2" 3.5 3.6 3.8 44 45 2.5 2.6 2.6 2.7 2.8 2.8 2.9 3.0 3.1 3.3 3.4 3.5 4(i 2.4 2.4 2.5 2.6 2.6 2.7 2.8 2.8 2.9 3.0 3.1 3.2 3.3 46 47 2.3 2.3 2.4 2.4 2.5 2.6 2.6 2.7 2.8 2.9 2.9 3.0 3.1 47 48 2.2 2.2 2.3 2.3 2.4 2.4 2.5 2.6 2.6 2.7 2.8 2.9 3.0 48 49 5o 2. I 2.1 2.2 2.2 2.1 2.3 2.3 2.4 2.4 2.3 2.5 2.4 2.6 2.4 2.6 2.7 2.8 2.6 i?. 5o 2.0 2.0 2. I 2.2 2.2 2.3 2.5 2.6 52 1.8 1.9 1.9 1.9 2.0 2.0 2. I 2.1 2.1 2.2 2.2 2.3 2.4 52 54 1-7 1-7 1-7 1.8 1.8 1.8 1.9 1.9 1.9 2.0 2.0 2. I 2. 1 54 56 1.5 1.6 1.6 1.6 I .(i 1-7 1-7 1-7 1.8 1.8 1.8 1.9 1.9 56 58 60' 1.4 1.4 1.5 1.5 1 .5 1.5 1.5 1.6 1.6 1.6 1.6 1-7 1-7 58 "60" 1 .3 1.3 1.3 1.3 I .i 1.4 1.4 1.4 r.4 1.5 1.5 1.5 1.5 02 1 .2 1 .2 1.2 1 .3 I .2 1 .2 1.3 1.3 1.3 1.3 1.3 1.3 1.4 62 ()4 1 .1 1 .1 I . I I . I I . I I . I 1 . 1 1 .2 1.2 1 .2 1 .2 1 .2 1 .2 64 66 1 .0 1 .0 1 .0 1 .0 I .0 1 .0 1 .0 1 .0 I.O I.I 1 .1 1 .1 1 . 1 66 08 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 1.0 1 .0 68 70 0.8 12° 0.8 13° 0.8 0.8 J 5° 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.9 70_ _ I 14° 16\ 17° 18° 19° 20° 21° 22° 23° 24° TABLE XXXIII. [Page 243' Tc reduce the numbers of Table XXXII to other given intervals of time | from noon. Time from Noon. s. o 0' ]/ 2' 3' 4' 5' Q 7' 8' 9' 10' 11' 12' 0.0 1 .0 4.0 9.0 16.0 25.0 36. 49.0 64. 81 .0 100.0 121. 144.0 I 0.0 1.0 4.1 9.1 16.1 25.2 36.2 49.2 64.3 81.3 100.3 121. 4 144.4 I 2 0.0 I . I 4.1 9.2 16.3 25.3 ■66.4. 49.5 64.5 81.6 100.7 121 .7 144.8 n 3 0.0 I . I 4.2 9.3 16.4 25.5 36.6 49.7 64.8 81.9 lOI .0 122. 1 145.2 3 4 0.0 I.I 4.3 9-4 16.5 25.7 36.8 49.9 65.1 82.2 101.3 122.5 145.6 4 5 6 0.0 1.2 4.3 9.5 16.7 25.8 37.0 5o.2 65.3 82.5 101 .7 122.9 i46.o 5 6 0.0 1.2 4. A 9.6 16.8 26.0 37.2 5o.4'65.6 82.8 102.0 123.2 146.4 7 0.0 1 .2 4.5 9-7 16.9 26.2 37.4 5o.6'65.9 83.1 102.3 123.6 i46.8 7 8 0.0 1.3 A.ii 9.8 17. 1 26.4 37.6 5o.9 66.1 83.4 102.7 124.0 l47-2 8 9 0.0 1.3 4.6 9.9 17.2 26.5 37.8 5i.i 66.4 83.7 io3.o 124.3 147.6 9 lo 0.0 1.4 4.7 10. 17.4 26.7 38. 5i.4 66.7 84.0 io3.4 124.7 i48.o 10 I 1 12 0.0 1.4 4.8 10. 1 17.5 26.9 38.2 5i.6 67.0 84.3 84.6 io3.7 125. I i48.4 II 12 0.0 1.4 4.8 10.2 17.6 27.0 38.4 5i.8 67.2 104.0 125.4 i48.8 i3 0.0 1.5 4.9 10.3 17.8 27.2 38.6 02.1 67.5 84.9 104.4 125.8 149.2 i3 i4 0.1 1.5 5.0 10.5 17.9 27-4 38.9 52.3 67.8 85.3 104.7 126.2 149.7 i4 i5 0.1 1.6 5.1 10.6 18. 1 27.6 39.1 52.6 68.1 85.6 io5.i 126.6 i5o.i i5 i6 0.1 1.6 5.1 10.7 18.2 27.7 39.3 52.8 68.3 85. Q io5.4 126.9 i5o.5 16 17 i8 0. 1 1.6 5.2 10.8 18.3 18.5 27.9 28.1 39.5 39.7 53.0 53.3 68.6 68.8 86.2 io5.7 127.3 1 50.9 17 ■ 18 0.1 1-7 5.3 10.9 86.5 106. 1 127.7 i5i.3 19 0.1 1-7 5.4 II. 18.6 28.3 39.9 53.5 69.2 86.8 106.4 128. 1 i5i.7 19 20 0.1 1.8 5.4 1 1 . 1 18.8 28.4 4o. I 53.8 69.4 87.1 106.8 128.4 l52.1 20 21 0.1 1.8 5.5 II. 2 .8.9 28.6 40.3 54.0 69.7 87.4 107. 1 128.8 i52.5 21 22 0.1 1.9 5.6 II. 3 19. 1 28.8 40.5 54.3 70.0 87.7 107.5 129.2 152.9 22 > 23 0.1 1.9 5.7 II. 4 19.2 19.4 29.0 40.7 54.5 70.3 70.6 88. 107.8 129.6 i53.3 23 0.2 2.0 5.8 II. 6 29.2 4i -o 54.8 88.4 108.2 i3o.o i53.8 24 2 5 0.2 2.0 5.8 II. 7 iq.5 29.3 4l.2 55.0 70.8 88.7 108.5 i3o.3 i54.2 25 26 0.2 2.1 5.9 11.8 19.7 29.5 4i.4 55.3 71. 1 89.0 108.9 i3o.7 i54.6 26 27 0.2 2.1 6.0 II. 9 19.8 29-7 4i.6 55.5 71.4 89.3 109.2 i3i .1 i55.o 27 ?8 0.2 2.2 6.1 12.0 20.0 f9.9 4i.8 55.8 71-7 89.6 109.6 i3i.5 i55.4 28 29 3.0 0.2 2.2 6.2 12. 1 20. 1 3o.i 42.0 42.2 56.0 72.0 89.9 109.9 i3i .9 i55.8 29 1 0.2 2.2 6.2 12.2 20.2 3o.2 56.2 72.2 90.2 no. 2 l32.2 i56.2 3o 1 3 1 0.3 2.3 6.3 12.4 20.4 3o.4 42.5 56.5 72.5 90.6 no. 6 i32.6 i56.7 3i 1 3; 0.3 2.4 6.4 12.5 20.6 3o.6 42.7 56.8 72.8 90.9 III.O i33.o 157. 1 32 33 0.3 2.4 6.5 12.6 20.7 3o.8 42.9 57.0 73.1 91 .2 III .3 i33.4 157.5 33 31 0.3 2.5 6.6 12.7 20.9 3i .0 43.1 57.3 73.4 91 .5 III .7 i33.8 157.9 34- 35 3(3 0.3 2.5 6.7 12.8 21 .0 3l.2 43.3 57.5 73.7 91.8 92.2 112. i34.2 i58.3 35 36 0.4 2.6 6.8 i3.o 21.2 3i.4 43.6 57.8 74.0 112. 4 i34.6 i58.8 3- 0.4 2.6 6.8 i3.i 21.3 3i.5 43.8 58. 74.3 92.5 1 12.7 134.9 159.2 37 38 0.4 2.7 6.9 l3.2 21.5 3i.7 44.0 58.3 74.5 92.8 ii3.i i35.3 159.6 38 39 4o 0.4 2.7 7.0 i3.3 21.6 3i .9 44.2 58.5 74.8 93.1 ii3.4 135.7 160.0 39 0.4 2.8 7-1 i3.4 21.8 32.1 44.4 58.8 75.1 93.4 ii3.8 i36.i 160.4 40 4i 42 0.5 2.8 7.2 i3.6 21.9 32.3 44.7 59.0 75.4 93.8 114.1 i36.5 160.9 4i 42 0.5 2.9 7.3 i3.7 22.1 32.5 44.9 59.3 75.7 94.1 114.5 1 36. 9 161.3 43 0.5 2.9 7.4 1J.8 22.2 32.7 45.1 59.5 76.0 94.4 114.8 137.3 161 .7 4i 4f 0.5 3.0 7.5 i3.9 22.4 32.9 45.3 59.8 76.3 94.7 Il5.2 137.7 162. 1 44 45 0.6 3.1 7.6 i4.i 22.6 33.1 45.6 60.1 76.6 9D.1 ii5.6 i3S.i 162.6 45 46 0.6 3.1 7-7 l4.2 22.7 33.3 45.8 60.3 76.9 95.4 115.9 i38.S i63.o 46 47 48 0.6 3.2 7-7 U.i 22.9 33.4 46.0 60.6 77.1 77.4 95.7 n6.3 i38.8 i63.4 47 48 {..6 3.2 7.8 i4.4 23.0 33.6 46.2 60.8 96.0 116. 6 139.2 i63.8 49 0.7 3.3 7-9 14.6 23.2 33.8 46.5 61. 1 77-7 96.4 117. 139.6 164.3 49 5o 7 3.4 8.0 14.7 23.4 34.0 46.7 61.4 78.0 96.7 117-4 i4o.o 164.7 5o 5i 7 3.4 8.1 14.8 23.5 34.2 46. Q 61.6 78.3 97.0 117. 7 i4o.4 i65.i 5i 5t 0.8 3.5 8.2 i5.o 23.7 34.4 47-2 61 .9 78.6 97-4 118. 1 i4o.8 i65.6 ir2 53 51 0.8 3.5 8.3 ID. I 23.8 34.6 47.4 62.1 78.9 97-7 98.0 118. 4 l4l.2 166.0 63 54 0.8 3.6 8.4 l5.2 24.0 34.8 47.6 62.4 79.2 118. 8 i4i.6 166.4 55 0.8 3.7 8.5 i5.3 24.2 35.0 47.8 62.7 79.5 98.3 119. 2 142.0 166.8 65 56 0.9 3.7 8.6 i5.5 24.3 35.2 48. T 62.9 79.8 98.7 119. 5 142.4 167.3 6t) 57 0.9 3.8 8.7 1 5. 6 24.5 35.4 48.3 63.2 80.1 99.0 119. 9 142.8 167.7 67 58 9 3.P 8.8 ID. 7 24.7 35.6 48.5 63.5 80.4 99.3 120.3 l13.2 168. 1 68 A9_ I.O 3.9 8.9 .5.9 ■^4.8 35.8 48.8 63.7 80.7 99-7 120.6 i43.6 i68.6 69 0' 1' 2' 3' 4' 5' 1 6' 7' 8' 9' 10 11' Page 244] TABLES XXXIV., XXXV., and XXXVI . TABLE XXXIV. TABLE XXXV. Errors arising from a deviation ot 1' in Angles Obs'd. the parallelism of the surfaces of the cen- tral mirror. Angle of deviation. 10' II 15' 20' // 25 II 30' II 35' 40' 45' // // 50' II 55' II 60' II Obs'd. Anglo Obs. to Obs. to Obs. Fifth D. 10 20 I I I 2 I 2 ~3 I 3 2 4 6 2 5 7 3 6 9 4 8 12 5 9 14 5 II 17 D. ' II 1 II / II / // o 10 2 I 2 So I 20 5 2 4 2 40 I I 3 4 6 8 10 i3 16 19 23 3o 10 I 6 4 5o 2 3 5 7 10 i3 16 30 25 29 36 4o 4o 16 8 7 60 65 2 3 4 4 6 9 10 12 i4 16 18 20 23 77' 28 37 34 45 19 I 9 9 5o 2S ' 2 II II 70 3 5 8 II i5 20 25 Si 37 44 55 6o 65 70 28 S3 39 46 4 5 7 10 12 i4 16 18 i4 17 21 25 75 ~ 3 5 ~8 12 76" 21 27 S3 4i 48 80 85 2 3 4 4 4 6 6 9 10 i3 i4 16 17 18 20 23 26 78 So So 32 " 35" 39 37 4o 44 ~ 48 44 48 53 58 53 58 63 69 75 80 54 I. 4 i.i5 1.27 12 16 21 24 28 32 So 35 95 2 7 8 12 23 85 90 19 23 4i 48 100 2 2 5 ~5~ 8 "9 i3 _i9. 20 2b 33 l6 42 " 46 b2 ^7~ 33 59" 75 82 io5 95 1.43 28 37 56 no ii5 2 3 6 6 10 16 22 2 5 Si 34 40 44 bo 55 62 68 75 83 90 100 2. I 33 44 I. 6 11 yy io5 no 2.23 2.49 39 46 52 I . I 1. 16 1 .29 120 3 7 12 19 27 ^7 48 61 76 91 109 ii5 3.23 54 i.i4 1.44 120 4.o5 1 . 4 1 .3o 2. 3 i3o 2.5l 1 40 4. 6 TAI 3LE XXXVL Corrections of the mea a refract on for various heights of the Thermometer and Barometer. Ht. Th. 20= 24° 23° 32° 3C ° 40° 44° 43° 52° 5( 3° G0° G4° 68° 72° 7G° Baroin- 32.00 31 .GG 31.32 30.99 30.67 30.3G 30.05 29.75 29.4.^ 29 IG 23.8t: ^ 28.G0 28.33 28.0 n 27.80 app. alt. ' Q '+" '+" '+" '+" 4 Ji '+" '+" 1 II / // 1 // / II 1 II 1 / r > II 2 4i 2 18 I 55 I 33 I 12 5i So ID ir 29 4t I 7 I 25 I 4 32 I 3o 2 18 I 58 I 39 I 20 I 2 44 26 9 6 25 4i 58 I i3 I 2 91 44 I I 59 I 42 I 26 I 9 53 38 22 7 22 se 5o I 3 I I 7 I So I So I 43 I 29 I i4 I 46 33 19 6 e 19 Si 43 55 I 5 I 18 2 I So I 18 I 5 53 4o 29 17 6 e 16 75 2" 37 48 5 Si 8 2 So I 20 I 8 57 46 36 25 i5 5 r 2^ 33 43 5 I I 3 I II I I 5i 4i 32 22 i3 4 z; i3 21 So 38 4 3 53 4 58 49 4i 33 26 18 11 4 / 10 l- 24 3i 3 7 43 5 48 4i 35 28 22 i5 9 3 9 lA 20 26 3 I 36 6 4i 35 So 24 18 iS 8 3 7 12 17 22 2 1 Si 7 36 3i 26 21 16 II 7 2 2 7 II i5 19 2 3 27 8 32 27 23 18 i4 10 6 2 3 6 IC i3 17 2 J 24 9 28 24 20 16 iS 9 5 2 2 5 12 i5 I 3 21 10 26 22 18 i5 11 8 5 2 2 5 10 i4 i( 3 19 12 i4 21 18 i5 12 10 7 4 4 3 e 9 II iz f 16 > i4 18 16 i3 II 8 6 3 t 8 10 i: 16 16 i4 II 9 7 5 3 3 5 7 9 ic ) 12 18 i4 12 10 8 6 4 3 S 4 6 c ? II 21 12 ID 9 7 5 4 2 2 4 5 6 i ) 9 24 10 9 7 6 5 3 2 2 3 4 5 7 8 27 9 8 6 5 4 3 2 2 3 4 5 C ^ 7 So 8 7 6 5 4 3 2 3 4 t 6 35 7 6 5 4 3 2 2 3 3 I 5 40 6 5 4 3 2 2 2 2 3 L. 4 45 . 5 4 3 3 2 I I 2 2 S 5o 4 S S 2 2 I I 2 2 S 60 3| 2 2 2 I I I I I 2 70 \ ' I I I I I I I I I 80 I 1 I o! I I 90 ol c TABLE XXXVII. Longitudes and Latitudes of Stars, for Jan. 1830. [Page 245 Names of STARS. •/ Pegasi Algcnih u Andromcdffi Alpheratz >; Pisciuin a AlUKTIS a (.x-ti Menkar t; Pleiadum ilcyone y Tauri t Tauri u Tauri Aldebaran 1^ Orionis Rigcl a- AurigiB Ctqiclla 3 Orionis |-J Tauri £ Orionis L Orionis L Tauri u Orionis Betergiiese 1} Geminorum u Geminorum •/ Geminorum f Geminorum a Canis Majoris Siritis t Geminorum (5 Geminorum a Geminorum Castor ;•? Geminorum Pollux a Canis Minoris Procyon a i: Caneri Jicuhcns u flydrs Ilphard »j Leonis a Leonis Regulus i Leonis Dcnehola •i Virginis '/ Virginis y Virginis a Virginis Spic a a Bootis 'Ircturus a Coronae Bor Jilphacca a 2 Librce Zuhcncsch a Serpcntis •/ LibrfD b Seorpii S Scorpii rt Seorpii ^ Seorpii a Seorpii Antares Opliiunhi a Opliiuchi Ras Alhaguc a Sagiitarii « Lyras Vega 71 Sagitfarii y Aquiia; a Aquilae Athair 1^ Aquiloe a 2 Capricorni [i Capricorni y Capricorni S Capricorni a Aquarii a Pisca Aust Fomalhaut a Cygni Deneb a Pesasi ]\L\re ab Rlag 4.3 2.3 2 3 3 3.4 4.3 2 3.4 3 4.3 3 2.3 2.3 2.3 3.4 3 I .2 3 3 3 4 3 3 3 Longitude. Ann.Var. aft. 1830. s o I It 0. 6.47.09 o . 1 1 . 56 . 26 0.24.26.32 1. 5.17.07 I .11 .56.4i I .27.36.57 2. 3.25. i4 2. 6. 4.55 2. 7.24.45 2.14.27. 7 2.19.28.47 2. 19.59. 17 2.20. 1 1 .58 2.21 . 5.23 2.22.18.26 2.22.24.32 2.26.22.42 3. I. 3.54 3. 2.55.14 3. 6. .-(3. 35 3. 7.33.46 3. II. 44. 55 3.12.36.52 3.16. 8.43 3.17.52.24 3.20.52.06 3.23.27.09 4. II .i5.52 4.24.54.50 4.25.3i .40 4.27.27.53 5.19. i5.54 5.24.44.16 6. 2.27.42 6. 7.48.04 6.21 .28.05 6.21 .5i .55 7. 9.53.32 7.12.42.48 7. 19.41 .08 7.22.45.27 7.28.45.00 8. o. II. 44 8. 0.33.49 8. 0.48.49 8. 7.23.15 8.19. I. 10 8.20. 3.45 9.10. o.3i 9.12.55.38 9. i3. 52.40 9.28.34.08 9.29.22.38 10, o.o3.34 10. 1.28.52 10. i.4o.i4 10.19.24.25 10.21. 9.29 11. 0.58.57 II. 1.27.58 II. 2.59.30 II .21 .07.05 50.09 49.98 5o. 16 50.27 50.27 5o.i8 5o.2I 5o.2o 5o.2I 50.24 50.19 50.2O 5o.20 5o.2o 5o.2o 5o.20 50.19 5o.2o 5o.2o 5o.i8 5o.2o 50.07 50.19 50.2O 5o.23 49.50 50.12 5o.i6 50.02 5o.23 49.94 5o.3o 5o.20 5o.2i 5o.oo 5o.o8 5o.45 5o.5i 5o.20 5o.32 5o.2 2 5o.i8 5o.i 9 5o.i8 5o.20 5o.I2 5o.20 5o.2I 5o.2I 49.89 So.io 5o.o3 50.79 5o.o5 5o. i5 50.17 5o.2i 5o.2I 5o.ii 50.59 49.42 5o.ii Latitude. Ann.Var. aft. 1830. 12.35.43 N 25.41. qN 5.22. 5 N. 9.57.40 N. 12.35.40 S. 4. 2. 7N. 5.44.56 S. 2.35. I S. 5.28.41 S. 3i. 8.39 S. 22.52.17 N 23.34.29 S 5.22.3i N 24.3i.38 S 25.i8.5r S 2.12.55 S 16. 2.59 S 0.54.28 S 0.49.59 S 6.45.36 S 2. 3.00 N. 39.22.26 S. 2. 3.3i S, o. II .5o S. 10. 5. 4N. 6.40.20 N, 15.57.43 S, 5. 5.35 S, 22.23.36 S, 4.5i.2i N 0.27.41 N. 12. 17.10 N. 0.41.32 N. I .22.22 N. 2.48.42 N. 2. 2.22 S. 3o.53.58N. 44.20.42 N. 0.21 .25 N. 25. 31.27 N. 4.24.20 N. 5.27.49 S. 1.57.42 S. 5.27. 4 S. I. 1.52 N. 4.32.45 S. 1.49. 6 S. 35.52.21 N. 3.25.23 S. 61.44.21 N. I. 27.41 N. 31.15.39N. 29. 18.46 N. 26.42.28 N. 6.56.55 N. 4.36.28 N. 2.32.18 S. 2.33.52 S. 10.40. i4 N. 21. 6.42 S. 59.54.55 N. 19.24.45 N. 4-0-I2 4-0.16 -)-o.25 +0.16 — 0.37 +0.43 —0.45 —0.46 —0.33 -0.47 +0., —0.48 +0.48 — 0.48 —0.48 —0.48 — 0.47 —0.47 +0.46 —0.45 -^.45 -0.44 +0.43 +0.26 — 0.41 — o.3i — 0.22 +0.22 +0.22 +o.o3 — 0.02 —0.08 — 0.1 3 +0.17 — 0.24 —0.35 — 0.37 — 0.40 — 0.42 +0.44 +0.44 +0.45 -0.45 +0.42 +0.." —0.48 +0.46 —0.45 —0.45 — 0.39 +0.08 — 0.38 — 0.37 — 0.37 +0.26 +0.25 —0.18 +0.21 0.16 +0.10 Page 246] TABLES XXXVIIL, XXXIX., XL., AND XLL TABLE XXXVIII. TABLE XXXIX. Reduct. of Lat. and Hor. Par for Ellipticity _i_j Aberration of Planets in Longitude. Elong. LTran. Sat. Jup. Mars. Venus. Mercury. Red. 5 Hor. Par Horizontal Par. Elong. Ab. Elong. A ph. Me a. Peri. Lat. of Lat. D — — D 1 D 53' 57' ' " 01' Con. i5 25" 24 27" 26 29" 28 36" 35 S.C. m S.C. 5 46" 46 5i^" 5i w / // II II i5 4i O 0. 0.0 0.0 o.c 0.0 3o 22 24 2b 33 3o M ID M 48 52 2 0.47.9 0.0 0.0 0.0 45 TO 21 23 28 45 19 i5 4i 43 4i 4 1.35.5 O.I 0.1 O.J 60 ID lb 19 23 Gt.El. i4 20 37 34 6 2.22.7 O.I 0.1 0.1 75 10 12 i4 18 45 9 25 29 8 3; 9.2 0.2 0.2 0.2 90 5 6 9 12 3o Gt. El. 18 18 ^9 10 3.54.8 0.3 0.5 0.6 0.8 I.O I .2 0.3 0.5 0.7 0.9 1 .1 1.3 0.4 0.5 0.7 0.9 1 .2 1.4 io5 120 i35 i5o i65 I 5 10 i3 i5 I + 4 8 II i3 3 I 5 9 1 1 7 3 + 2 3 i5 Inf. C. 3 3^ 25 20 i5 10 5 Inf C 7 I + 2 5 6 6 4 4- 4 8 11 t1 + 2 1 3 18 ^9h 12 i4 i6 i8 20 4.39.3 5.22.4 6. 3.9 6.43.7 7.21.5 22 7.57.2 8.30.7 1.5 T,R 1.6 1-7 Op. 180 i5 i3 1 II 1 4 1 i.y 2.2 2fi 9. 1.6 9.29.9 9.55.4 2-0 3 The aberration of the Sun in longitude is always 20". 1 28 2 3 2 5 2.7 3.1 3.8 Tlie apparent place is given in the INautical Almanac, and 3o 2-7 2.9 3.2 3.6 by adding 20" the Sun's true longitude will be obtained. 32 34 10. 18. r 10.37.8 3.0 3.3 TABLE XL. TABLE XLI. 36 10.54.3 3.7 3.Q 4.2 Equat. Equino.xes in Aberration in Long, and Lat. | 38 II . 7.7 4 4 1 4 6 Longitude. 1 _4^ II. 17. 8 4.A 4.7 5.0 Arg. long. = long. — :)f. long. Arg. lat. = Arg. long. — 3 signs. Long. 5 '3 Node. 42 II. 24. 7 4.7 S.I h.S 44 1 1. 28. 2 5.1 5.5 5.9 6.3 6,7 46 II. 28. 4 II .25.1 5 5 5.0 6.3 D D 48 6.2 + + + + + + 5o II. 18. 6 6.7 7.2 b 7 8 b 7 8 52 II. 8.8 6.6 7.1 7 6 o"o 8"^ i5"5 3o 20"o i7"i io"o 3 54 10.55.6 6.0 7 5 8 n 2 o.b 9.6 i5.8 28 2 Jo.o|i7.c 9-^ 2 « 56 10.39.3 7.3 7.8 8,4 4 1.2 10. c lb. I 26 4 - !0.0 ib.fc 8.8 26 58 10. 19.9 7.6 8.9 8,8 6 1 .0 10.5 lb. 4 24 6 9.9 lb. 2 8.1 24 6o 9-57-4 7-9 8.5 9.1 8 2.5 3,T II .0 II. 5 ib.b 16 P 22 8 9.8 i5.fc t5.,1 7.5 6,8 22 62 64 y-/ 9.32.0 9. 3.8 8.3 8.6 8.9 9.2 9.5 9.8 10. 1 9.5 9.9 10.2 12 3.7 12.0 17.0 18 12 9.6 .4.9 6.2 I 8 66 8.32.9 7.59.6 8.8 i4 4.3 12.4 17.2 16 i4 9-4 14.4 5.5 I b 68 9.1 10,5 16 4.9 12. Q 17.4 i4 16 9.2 13.9 4.8 I 4 70 7.23.8 9-4 10.8 18 5.5 6.1 i3.3 i3.7 17.5 17.6 12 18 I 20 I 9.0 8.8 i3.4 4.2 3.5 12 72 74 6.45.9 6. 6.0 9.6 9.8 10. 10.3 10 5 II .0 II 3 12.9 22 6.7 14. 1 17-7 8 22 I 8.5 12.3 2.8 8 76 5.2.4.3 10.7 TI 5 24 7.3 i4.5 17.8 6 24 I 8.3 II. 8 2.1 b 78 4.41.0 10. 1 10.9 II .1 II .7 26 7.8 i4.8 17.9 4 26 I 8.0 II .2 1.4 4 80 3.56.3 10.3 II. 8 28 8.4 8.9 l5.2 i5.5 17.9 2 28 I 3o 1 7-7 7.3 10.6 0.7 2 82 3. 10.4 2.23.7 10.4 10.5 II .2 11 .3 12.0 12. 1 ly.y 84 — — — + + + 86 83 1.36.2 0.48.2 10.5 10.6 11. 3 11. 4 12. 1 12.2 5 + 4 6 D - b 4 3 D + + — 90 0. 0.0 10.6 II. 4 12.2 II 10 9 II 10 9 Table XI-. contains the etjua tion of the equinoxes in lorjjitiide to be applied with its sign to ily bodies. Thus on July lb, 1830, when the longitude of the moo the inean longilndos of all the heave n s ascending node was 5s. !2° 38' tl e equation of tiso equinoxes was ■ — 5". 3. The correction in Table XLI. ( orresponding to the Argument of Longitude being found, and its lo ;a- rithm added to the loj. secant (Ic.s s radius) of tiie star's latitude, will be tlie log. of the star's aberratiot in lowsilude, to be applied witii its s gn to the mean longitude. The logarithm of the correction in Ta .le XLI. corresponding to the Arpu ncnt of Latitude added to the log. sine of the star's latitude will be the aberration of the star in latitu de, to be apj)lied with its sign lo the mean latitude. Example. Re quired the Aberration of a Pegasi, July 16, 1830? long. 3s. 23° 22'. * long. U. 21. 07. Ar-. loi.-.i. 02. 15. Table 4L- 1-10". 7 log. 1.02938. Arg. lat. Is. 2°. 15 Tab. 41.— IG". 9 log. l.tr. 39 * Latitude 10° 25' Sec. 0.025-13. Sii,e d.Si.\ li * Aboir. lo n^^ + 11"; I. L ^g. l.C .>W1. *• Aber. lat. -5" .6. Log. O.TWf/l [ TABLE XLII. [Page 247 Aberration in Right Ascension and Declination. PART I. PART II. PART III. Arg. R. A. = ^ R. A . — Q Long. Arg. R. A. = * R. A. + Q Long. Ar.2il T)ec.=01on4-^Dec. / Ad.Gsia-na Arg. Drc. =: Arg. R. A. + 3 sigug. Arg. Dec. = Arg. R. A. + 3 signs. Ai-.3il Dcc.=3loii— ^Dcc. S if Decl. s. D 0. G. 1. 7 2. 8. D 0. G. 1. 7. 2. 8. D 0. G. 1. 7. 2. 8. - + - + - + + - + - 0" 41 3o - + - + 3". 45 - 4- o 19". 17 I 6". 60 9". 59 3o o".83 0".72 3'-'. 98 i".99 3o I 19 .17 16 .43 9 .3o 29 I .83 .71 .40 29 I 3 .98 3 .4i I .93 29 2 19 .16 16 .26 9 .00 28 2 .83 .70 .39 28 2 3 .98 3 .38 I .87 28 3 19 .i5 16 .08 8 .70 27 3 .83 .69 .38 27 3 3 .98 3 .34 I .81 27 4 19 .i3 1 5 .90 8 .40 26 4 .82 .69 .36 26 4 3 .97 3 .3o I .75 26 6 19 .10 .71 8 .10 25 5 .82 .68 .35 25 5 3 .97 3 .26 I .68 25 6 19 .07 i5 .5i 7 .80 24 6 .82 .67 .34 24 6 3 .96 3 .22 I .62 24 7 19 .o3 i5 .3i 7 -49 23 7 .82 .66 .32 23 7 3 .95 3 .18 I .56 23 « 18 .99 i5 .11 7 .18 22 8 .82 .65 -31 22 8 3 .94 3 .14 I .49 22 9 18 .94 i4 .90 6 .87 21 9 .82 .64 .3o 21 9 3 .93 3 .09 I .43 21 10 18 .88 i4 .69 6 .56 20 10 .81 .63 .28 20 10 3 .92 3 .o5 I .36 20 II 18 .82 i4 .4? 6 .24 19 II .81 .62 .27 19 II 3 .91 3 .00 I .3o 19 12 18 .75 i4 .25 5 .92 18 12 .81 .61 .26 18 12 3 .8q 2 .96 I ..23 18 i3 18 .68 i4 .02 5 .61 17 i3 .81 .60 .24 17 i3 3 .88 2 .91 I .16 IT i4 18 .60 i3 .79 5 .28 16 i4 .80 .59 .23 16 i4 3 .86 2 .86 I .10 16 i5 18 .5? i3 .56 4 .96 i5 i5 .80 .58 .21 i5 i5 3 .85 2 -82 I .o3 1 5 i6 i8 .43 i3 .32 4 .64 i4 16 .79 .57 .20 i4 16 3 .83 2 .77 .96 i4 17 18 .34 r3 .08 4 .3i i3 17 .79 .56 .19 i3 17 3 .81 2 .72 .90 i3 i8 18 .23 12 .83 3 .99 12 18 .79 .55 .17 12 18 3 .79 2 .66 .83 12 19 18 .i3 12 .58 3 .66 II 19 .78 ,54 .16 II 19 3 .76 2 .61 .76 II 20 21 r8 .02 12 .32 3 .33 10 20 .78 .53 .i4 10 20 3 .74 2 .56 .69 10 17 .90 12 .07 3 .00 9 21 .77 .52 .i3 9 21 3 .72 2 .5i .62 22 17 .78 II .80 2 .67 8 22 .77 .5i .12 8 22 3 .69 2 .45 .55 8 23 17 .65 II .54 2 .34 7 23 .76 .5o .10 7 23 3 .66 2 .4o .49 7 24 17 .52 II .27 2 .00 6 24 ,76 ,49 .09 6 24 3 .64 2 .34 .42 6 2b 26 17 .38 II .00 I .67 5 25 .75 .47 .07 5 25 3 .61 2 .28 .35 5 17 .23 10 .72 I .34 4 26 .74 .46 .06 4 26 3 .58 2 .23 .28 4 27 17 .08 10 .u I .00 3 27 .74 .45 .04 3 27 3 .55 2 .17 .21 3 28 16 .93 10 .16 .67 2 28 .73 M .o3 2 28 3 .52 2 .11 .14 2 29 16 .77 9 -87 .33 I 29 .72 .43 .01 I 29 3 .48 2 .o5 .07 I 3o 16 .60 9 -59 .00 3o .72 .4i .00 3o 3 .45 I .99 .00 - + - + - + r» + - 4- - 4- - D - + - + - + D 11. 5. 10. 4. 9. 3. " \ 11. 5.1 10. 4. y. 3. 11. 5. 10. 4. 9. 3. To J?«r/ ^/;e Aberration of a Star in Rioht Ascension. — Find the Equations in Part I. and II. corresponding to the arguments of R. A. at the top of tliose tables, and connect them ac- cording to their signs, and to t!ie log. of this sum or difference add the log. secant (less ra- dius) of the star's declination, the sum will be tlie log. of the aberration in Piiglit Ascension in seconds of a degree, which divided by 15 will be reduced to time, to be applied to the mean R. A. To find the Aberration of a Star in DecVnmlion. — Increase the former arguments of R. A. by 3 signs, and connect together the corresponding equations of Part I. and II. to the log. of which add the log. sine of the star's declination, the sum will be the log. of arc 1st. With the arguments at the top of Part III. find in the Table arcs 2d and 3d. Those three arcs connected with their signs will be the aberration in declination, to be applied to the mean declination. Example. Required the Aberration in R. A. and Dec. of a P(s-asi, July IG, 1830.' By Tal)lc 8. *R. A.=22h. 5G' 13"=lls. 11°. 5'. *Dcc. 14° 18' N. and b; N. A. Glonj. Ss. 23° 22'. >lcR.A. lis. 14° 5'. ©Lon. 3. 23.22. Difr. 7. 20.4.S. Part I.4-12".M.. Dlff. 4- 3s=10s.20°.43' Part l.—U".U. Sum 3. 7.27. Parlll.— 0. 11. 4-12. 03. log. 1.08027 Suin+3 =6 7 27 Par', rt.- -0 82 -15. 66 og. 1.19479 *Dec. 14°18' 860.0.013^7 *Abcr. R. A. + 12". 4. log.l.093L>4 Arc 1st. — 3".07. iiie 9.39270 og. 0..58749 :t: Ah. in Il.A. in time 0" 83. Olong-f-;f:Dec.:=ls. 7° 40' Arc 2d+2. 43. ©long— >}; Dcc.=3s. 9° 01' Arc 3d--0. 62. ^ .\berr. in Declination — 0. 8?*.. PaB«248] TABLE XLIII. Nutation in Right Ascension and Declination to be applied to the mean values. PART I. PART TI. PART III. Arg. R.A.=: >|<: R. A. — Lon. D node. + 6 signs if Dec. is S. Arg. R.A. = * R.A. + Lon. D node, -j- 6 signs if Dec. is S. Equation Equinoxes in R.A. Arg. Dec. = Arg. R. A. -|- 3 signs. Arg. Dec. = Arg. R. A. + 3 signs. Arg. ^ Long. 5 node a- D 0. C. 1. 7. 2. 8. D 0. G. 1. 7. 2. 8. D 0. C.!l. 7. 2. ~8. - + - + - + - + - -f - + - + - + — ± 8". 33 7". 21 4". 16 3o l".22 i".o6 o".6i 3o 0" .0 8". 2 1 4' .2 3o I 8 .33 7 .14 4 .04 29 I I .22 I .o5 .59 29 I .3 8 .4 i4 .3 29 2 8 .32 7 .06 3 .91 28 2 1 .22 I .o3 .57 28 2 .6 8 .7 i4 .5 28 3 8 .32 6 .99 3 .78 27 3 I .22 I .02 .55 27 3 .9 8 .9 i4 .6 27 4 8 .3i 6 91 3 .65 26 4 I .22 r .01 .53 26 4 I .1 9 -2 i4 .7 26 5 8 .3o 6 .82 3 .52 25 5 I .22 I .00 .52 25 5 6 I .4 9 .4 i4 .8 25 6 8 .28 ~6"74 3 .39 24 6 I .21 .99 .5o 24 I -7 9 .6 i5 .0 24 7 8 .27 6 .65 3 .25 23 7 I .21 .97 .48 23 7 2 .0 9 -9 i5 . I 23 8 8 .25 6 .56 3 .12 22 8 I .21 .96 .46 22 8 2 .3 10 .1 i5 .2 22 9 8 .23 6 .47 2 .99 21 9 I .20 .95 .44 21 9 2 .6 10 .3 i5 .3 21 10 8 .20 6 .38 2 .85 20 10 I .20 .93 .42 20 10 2 .8 10 .5 i5 .4 20 11 8 .18 6 .29 2 .71 II I .20 .92 .40 19 II 3 .1 10 .7 i5 .5 19 12 8 .i5 6 .19 2 .57 18 12 1 .19 .91 .38 18 12 3 .4 II .0 i5 .6 18 i3 8 .12 6 .09 2 .44 17 i3 I .19 .89 .36 17 i3 3 .7 II .2 i5 • 7 17 i4 8 .08 5 .99 2 .3o 16 i4 I .18 .88 .34 16 i4 4 .0 II .4 i5 .7I i6i i5 8 .05 5 .89 2 .16 i5 i5 I .18 .86 .32 i5 i5 4 .2 II .6 i5 .8 i5 i6 8 .01 5 .79 2 .02 i4 16 I .17 .85 .3o i4 16 4 .5 II .8 i5 •9 i4 I? 7 -97 5 .68 I .87 i3 17 I .17 .83 .27 i3 17 4 .8 12 .0 16 .0 i3 i8 7 -92 5 .57 I .73 12 18 I .16 .82 .25 12 18 5 .1 12 .2 16 .0 12 19 7 .88 5 .46 I .59 II 19 I -15 .80 .23 II 19 5 .3 12 .4 16 .1 11 20 7 .83 5 .35 I .45 10 20 I .i5 .78 -21 10 20 5 .6 12 .5 16 .1 10 21 7 .78 5 .24 I .3o 9 21 I .14 .77 .19 9 21 5 .9 12 .7 16 .2 9 22 7 -72 5 .i3 I .16 8 22 I .i3 .75 .17 8 22 6 .1 [2 .0 16 .2 8 23 7 -67 5 .01 I .02 7 23 I .12 .73 .i5 7 23 6 .4 i3 .1 16 .3 7 24 7 .61 4 .90 .87 6 24 I .11 .72 .i3 6 24 6 .7 i3 ,3 16 .3 6 25 7 .55 4 .78 .73 5 25 I .11 .70 .11 5 25 6 .9 i3 .4 16 .3 3 26 7 -49 4 .66 .58 ~4 26 I .10 .68 .09 4 'V6 7 .2 i3 .6 16 73" 4 27 7 -42 4 .54 .44 3 27 ]:1 .66 .06 3 27 7 .4 i3 .7 16 .4 3 28 7 .35 4 .41 .29 2 28 .65 .04 2 28 7 -7 i3 .9 16 .4 2 29 7 -29 4 .29 .i5 I 29 I .07 .63 .02 I 29 7 -9 i4 .0 16 .4 I 3o 7 .21 4 .16 .00 3o I .06 .61 .00 3o 8 .2 i4 .2 16 .4 - + - + - 4- D - + - + - + D + - + - + — D n. 5. 10 4. 9. 3. 11. 5. 10. 4. 9. 3. 11. 5. 10. 4. 9. 3. To find the Katatlon of a Star in Riffht Jisccnsion. — Find in Parts I. II. the Eqi ations corresponding to the arguments of R. A. at the top of the tables, connect them ace ording to the signs, and to the log. of the sum or difference add the log tangent of the star's decli- nation, the sum will be the log. of an arc, to which apply the equation of the equinoxes, Part III. corresponding to the long, of the D 's node (page 3, N. A.) the sum or difference will be the Nutation in Right Ascension in seconds of a degree, which divided by 15 will be reduced to seconds of time. To find the Nattition of a Star in Declination. — Increase the arguments of R. A. Parts I. II. by 3 signs, and connect the corresponding equations of those tables, which will be the nutation of declination. Kote. In putting the R. A. of the star equal to 3 signs, tl e nu- tation in declination will be the equation of the obliquity of the ecliptic. Example. Required the Nutation of a Pegasi, in R. A. and Decl. July 16, 1830 ? >tR.A.Tab.8.11s.l4° .5' D Node N. A. 5. 12. 38 Diff. (). 1. 27 Part I.+8"33 Diff.+3s=9s 1°27' Part I.— 0"21 Sum+3s=7. 20. 43PartII.+0" 07 Sum 4. 26.43PartII.+6.97 +15"3niog.l. 18409 Nut. in Dee. -|-0"40 Gs.— 5 Node=Os. 17°. Parti.— *Dec. 14° 18' tang. 9.4(1(130 -7. !'7 Arch 4-3".9 log. 0.59105 Part III. Eq. Arg. 5s. 12° 38' —4. 9 Gs.4- D Node=l Is. 1 3. Part Eq. Obi. Eclif [I.- 1. 17 t. — 9.14 Nutation in Right Ascension.— I. 0= — C'lof t. If the Declination of the Star was South, the argument of Part I. II. of Right A scen- 8:on and Declination must be increased 6 signs. TABLE XLIV. [Page 249 To find the Augmentation of the Moon's Seniidianieter, by the altitude of the Nonagesimal, and the apparent distance of the Moon therefrom. Arz. Slimi of Pre. Eq. l3 i4 i5 PART IV. d 's Horiz. Semi. Diam. 14' 15' 16' 4(1" o.i6 0.32 50" 0" 0.480.4? 0.640. 56 0.80 0.96 1 . 12 1.28 1 .44j [ .6n 0.70 II ). 12 ).24 ).36 ).48 1.61 70 92 08 24 2.40 56 0.98 1 .12 1 .26 1 .41 0.73 0.85 a. 97 1 .09 1.55 1 .69 [.83 1.97 2.11 2.25 10" o.3o 0.4 1 o.5i 0.61 0.71 0.91 1 .01 1.33 r.45 [ .57I1 .32 r .70 1 .42 82I1.52 1 .94,1 .62 20" ,16 0.33 o.4i 0.49 0.57 0.65 0.73 0.82 30" 0.06 0.12 0.18 0.25 0.3 3.90 3. 98 1 .06 i.i4 0.37 0.43 0.49 0.55 0.62 0.68 0.74 0.80 0.86 0.92 r.3i!o.98, 40" 50" 0' 10" 20" 30" 40" 50" 0.04 16 0.21 II 0.02 0.04 0.06 0.08 o. I( II 0.00 0.00 0.00 0.00 0.00 0.25 9 0.33 0.37 0.41 0T45 0.49 0.54 0.58 0.62 0.12 o.i5 17 •9 0.21 0.00 0.00 0.00 0.00 ).00 // 0.02 0.04 0.06 0.08 o. 10 73 o. i5 o. 17 9 0.21 + o.o4 0.08 i3 4- II 0.06 o.i3 o. 19 0.25 0.32 Find in P. I. llie Iwo equations cor- responding to tlie iirgumcnts at the top, and connect them according to their signs. Willi this sum or dill'er- ence, take out the corres|ionding cor- rection P. II. In occLiitations, tlie orrection P. III. ■i to be found with liic C 's Par. in lat. at the top, and lier true lat. at the side, l>ut in solar Eclip- cs, this p. is nolh- ig. Connect these iree parts, and with the sum en- ter the side col- lur.n of P. IV., and lind the (J 's Hori~. Semi. Dia. at the top ; the corresponding cor. applied, with its sign, to the sum of the three first parts, will give the Autr. of the d 's S. D. Thus in Ex. 1, Prob. 5, Appendi.x. The Alt. Nonag. is 2s. 7'^ 59', Dis. Nonag. (D.-f P.) 20° 46', d S. D. by N. A. 16' 27". 7. Hence Arg. P. I. are 2s. 1° 59'+20° 46', thuMs,2s. 28° 45' and Is. 17° 13', to which correspond + 8" 18 + 6".01 =+ 14" 19. This giv'^ in P. II. + x}" 21. P. HI. is 0". The sum of the three parts is + 14". 4, with which and the d S. D. 16' 27". 7. P. IV. is. ncarl3'+0"8; this connected with 14". 4 gives the .\ug. of d "s S. D. 15" 2. as in Prob. VI. Appendi.x. 0.23 0.2 5 0.29 3i o.66|o.33 0.00 o 00 o 00 0.00 0.00 0.00 0.23 25 0.27 =9 3i 0.34 0.25 0.29 0.34 0.38 0.42 0T46 o.5i 0.55 0.59 0.63 0.67 0.38 0.44 0.5 0.57 0.63 0.70 0.76 0.83 0.89 0.95 4- ,v 0.09 17 0.26 0.34 ck4; o.5i 0.60 0.68 0.77 0.85 0.94 1 .02 1 . 1 1 9 1.28 36 + 3? 0.43 0.53 oT64 0.75 0.86 0.96 07 78 28 39 5< 1 .6fi 1. 71 Pas'.' 250] TABLE XLV. Equation of Second Differences to be appl led to the jnean longitude or latitude with a sign contrary to that of the mean of the second differences. App. Time after noon or midnight. Second Difference. V 2' 3' 4' 5' G' 7' 8' 9' 10' 11' 12' h.m. h. m. II // II II // // // II It II II II 0. 12. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.10 11.50 0.4 0.8 1.2 1.6 2.1 2.5 2.9 3.3 3.7 4.1 4.5 4.9 0.20 11.40 0.8 1.6 2.4 3.2 4.1 4.9 5.7 6.5 7.3 8.1 8.9 9.7 0.30 11.30 1.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 12.0 13.2 14.4 0.40 11.20 1.6 3.1 4.7 6.3 7.9 9.4 11.0 12.6 14.2 15.7 17.3 18.9 0.50 11.10 1.9 3.9 5.8 7.8 9.7 11.5 11.6 13.6 15.5 18.3 17.4 20.6 19.4 22.9 21.3 23.3 27.5 1. 11. 2.3 4.6 6.9 9.2 13.7 16.0 25.2 1.10 10.50 2.6 5.3 7.9 10.5 13.2 15.8 18.4 21.1 23.7 26.3 29.0 31.6 1.20 10.40 3.0 5.9 8.9 11.9 14.8 17.8 20.7 23.7 26.7 29.6 32.6 35.6 1.30 10.30 ' 3.3 6.6 9.8 13.1 16.4 19.7 23.0 26.2 29.5 32.8 36.1 S9.4 1.40 10.20 3.6 7.2 10.8 14.4 17.9 21.5 25.1 28.7 32.3 36.9 39.6 43.1 1.50 10.10 3.9 7.8 11.6 15.5 19.4 20.S 23.3 27.2 29.2 31.1 33.3 34.9 37.5 38.8 41.7 42.7 45,8 46.6 oO.O 2. 10. 4.2 8.3 12.5 16.7 25.0 2.10 9..50 4.4 8.9 13.3 17.8 22 "' 26.6 31.1 35.5 39.9 44..4 48.8 63.3 2.20 9.10 4.7 9.4 14.1 18.8 23.5 28.2 32.9 37.6 42.3 47.0 51.7 56.4 2.30 [)..30 4.9 9.9 14.8 19.8 24.7 29.7 34.6 39.6 44.5 49.5 54.4 59.4 2.40 9.20 5.2 10.4 15.6 20.7 25.9 31.1 36.3 41.5 46.7 51.9 57.0 62.2 2.50 9.10 5.4 10.8 11.2 16.2 21.6 22.5 27.1 32.5 37.9 39.4 43.3 45.0 48.7 60.6 - 54.1 59.5 64.9 3. 9. 5.6 16.9 28.1 33.7 56.2 61.9 67.5 3.10 8.50 5.8 11.7 17.5 23.3 29.1 35.0 40.8 46.6 52.4 58.3 64.1 69.9 3.20 8.40 6.0 12.0 18.1 24.1 30.1 36.1 42.1 48.1 54.2 60.2 66.2 72.2 3.30 8.30 6.2 12.4 18.6 24.8 31.0 37.2 43.4 49.6 55.8 62.0 68.2 74.4 3.40 8.20 6.4 12.7 19.1 25.5 31.8 38.2 44.6 60.9 57.3 63.7 70.0 76.4 3.50 8.10 6.5 13.0 19.6 26.1 32.6 33.3 39.1 40.0 45.7 46.7 62.2 53.3 58.7 60.0 65.2 66.7~ 71.7 73.3 78.3 80.0 4. 8. 6.7 13.3 20.0 26.7 4.20 7.40 6.9 1,3.8 20.8 27.7 34.6 41.5 48.4 55.4 62.3 69.2 76.1 83.1 4.40 7.20 7.1 14.3 21.4 28.5 .35.6 42.8 49.9 57.0 64.2 71.3 70.4 85.6 5. 7. 7.3 14.6 21.9 29 2 36.5 43.7 51.0 68.3 65.6 72.9 S0.2 87.6 5.20 G.40 7.4 14.8 22 2 29.6 37.0 44.4 51.9 59.3 66.7 74.1 81.5 88,9 5.40 6.20 7.5 15.0 22.4 29.9 37.4 44.9 52.3 59.8 67.3 74.8 82 2 89.7 6. G. 7.5 15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 75.0 82.5 90.0 App. Tim or m e after noon dnight. Sec ond Difference. 10" 20" 30" 40" 50" 1" II 2" II 3" 4" 5" 6" 7" 8" 9" n..m. h. m. II // II II " // // II II 0.- 12. 0.0 0,0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 o-.o 0.0 0.10 11.50 0.1 0.1 0.2 0.3 0.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.20 11.40 0.1 0.3 0.4 0.5 0.7 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.30 11.30 0.2 0.4 0.6 0.8 1.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.40 11.20 0.3 0.5 0.8 1.0 1.3 0.0 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0..30 11.10 0.3 0.6 1.0 1.3 1.6 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 1. 11. 0.4 0.8 1.1 1.5 1.9 0.3 ( .3 0.3 1.10 10.50 0.4 0.9 1.3 1.8 2 0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 1.20 10.40 0.5 1.0 1,5 2.0 2..5 0.0 0.1 0.1 0,2 0.2 0.3 0.3 f.4 0.4 1.30 10 30 0;5 1.1 1.6 9 «> 2.7 0.1 0.1 0.2 0.2 03 0.3 0.4 0.4 0.5 1.40 10.20 0.6 1.2 1.8 2.4 3.0 0.1 0.1 0.2 0.2 0.3 4 0.4 0.5 0.5 1.50 10.10 0.6 1.3 1.9 2.6 3.2 3.5 0.1 0.1 0.1 0.1 0.2 0.2 0.3 0.3 0.3 0.3 0.4 0.4 0.6 0.6 0.6 0.6 2. 10. 0.7 1 4 2.1 2.8 0.5 0.6 2.10 9.50 0.7 15 •5 9 3.0 3.7 0.1 0.1 0.2 0.3 0.4 0.4 0.5 0.6 0.7 2 20 9.10 0.8 1,6 2,3 3.1 3.9 0.1 0.1 0.2 0.2 0.3 0.4 0.5 0.5 0.6 0.7 2.30 9.30 0.8 1,6 2.5 3.3 4.1 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.7 2.40 9.20 09 1,7 2 6 3.5 4.3 0.1 0.2 0.3 0.3 0.4 0.5 0.6 0.7 0.8 2.50 9.10 0.9 1,8 2.7 2^ 3.6 3.7 4.5 4.7 0.1 0.1 0.2 0.2 0.3 0.4 0.4 0.5 0.5 0.5 0.6 0.7 0.7 0.7 0.8 0.0 3. 9. 0.9 1.9 0.6 3.1C 8.50 1.0 1.9 2,9 3.9 4.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3 20 8.40 1.0 2.0 3.0 4.0 5.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 8 0.9 3. .TO 8..30 1.0 2.1 3.1 4.1 5.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3 40 8.20 1.1 2.1 Q G) 4.2 5.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 3.50 8.10 1.1 2.2 3.3 3.3 4.3 5.4 5.6 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.7 0.7 0.8 0.8 0.9 0.9 1.0 1.0 4. 8. 1.1 4.4 0.6 4.20 7.40 1.2 2.3 3.5 4.6 5.8 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 4.40 7.20 1.2 2.4 3.^ 4.8 5.9 0.1 0.2 0.4 0.5 0.6 0.7 0.8 1.0 1.1 5. 7.0 1.2 2,4 \G 4.9 6.1 0.1 0.2 0.4 0.5 0.6 0.7 0.9 1.0 1.1 fi. 6.0 1.2 2.5 1 3.7 5.0 6.2 0.1 0.2 0.4 1 0.5 0.6 0.7 0.9 1.0 1.1 TABLE XLVI. [Page 251 Table showing tlie variation of the altitude of an object arising from a change of 100 second 3 in the declination. If the change move the body to- wards the elevated pole, apply the correction to the altitude with the signs in 1 the Table ; otherwise, change the signs. LATITUDE LATITUDE "iF Of same name -^5 declination. Of different name from declination. 0- _a_ 70=' 94" 00° 87" 50° 76" 40° 64" 30° 5o" 20° 34" 10° 0° 10° 20° 34" 30° 5o" 40° 64" 50° 76' G0° 87" 70° 94' 17" 0' 17" 10 95 88 78 6!) 5i 35 18 0' 18 35 5i 65 78 88 95 10 ■20 lOO 92 82 68 53 m 18 0' 18 36 53 68 82 92 too 20 30 100 88 74 57 39 20 0' 20 39 57 74 88 too 30 0° 40 100 84 65 45 22 0' 23 45 65 84 100 40 0° 50 100 78 53 27 0' 27 53 78 too 1 50 GO 100 68 35 0' 35 68 too G'J — 70 ~0~ 94 87 77 64 5o too 34 5i 0' 5i 100 34 5o 64 77 87 94 70 — 17 •7 10 9b By 77 bb 60 34 17 — I 18 35 5i 66 78 88 96 10 20 99 9' 81 67 52 35 17 — I 19 37 54 69 83 93 101 20 :J0 1 07 98 87 73 56 38 18 — 2 22 4i 59 76 90 102 30 2" 40 1 1 1 9^ 82 63 42 20 — 2 25 47 68 86 102 40 90 50 116 97 74 5o 24 —3 3o 57 81 io3 50 GO 124 95 64 3o D 40 73 io3 .iO — 70 94 87 77 64 139 5o 92 M 43 —8 59 108 34 5o 64 77 87 94 70 — 17 17 10 94 87 77 64 5o M 16 — I 19 36 52 67 79 89 97 10 20 9» 90 79 66 5i M 16 —3 21 3q 56 71 84 95 io3 20 30 io5 gb 8b 70 54 36 16 —4 24 U 62 78 93 io4 30 4° 40 107 94 78 59 39 17 —6 29 5i 71 90 106 40 40 bO 1 1 1 92 70 45 19 —8 35 62 86 109 50 GO 117 88 56 23 — 12 47 81 112 GO — 70 94 87 77 65 127 DO 81 34 32 —19 70 119 34 5o 65 77 87 94 70 — 17 — 17 10 94 87 76 64 49 33 16 — 2 20 37 53 67 to 90 98 10 20 97 89 78 65 5o ■6i i5 —4 22 40 57 73 86 96 io4 20 30 .o3 94 8J 69 52 M i4 —6 26 46 64 81 95 107 30 0" 40 .o5 92 76 57 36 i4 —9 32 54 74 93 109 40 c° 50 107 88 66 4i i5 —13 40 66 91 ,i3 50 (=0 III 82 5i 17 —18 53 87 119 tiO /O 118 72 22 -29 80 129 70 95 87 77 65 5() 35 18 —0 18 35 5o 03 77 87 95 10 94 86 76 63 49 33 i5 —3 20 38 54 68 81 91 09 10 20 96 88 77 64 49 32 i4 —5 24 4o 59 74 87 98 1<:6 20 30 lOI 9^ 81 67 5o 32 12 —8 28 48 66 83 97 109 :',() ti'^ 40 102 89 73 54 33 II — 12 35 57 78 Q^ 1 13 40 d° ;j>0 io4 84 62 37 1 1 — 17 d^ 70 95 118 50 (.0 io5 77 45 II —24 59 93 125 i;n — /O 9^ 88 78 65 109 5i 62 35 i3 -39 90 t4o 35 Si 65 7r 88 o5 70 0" — 18 — 18 10 9i 86 75 63 48 32 i5 —3 21 38 55 69 82 92 too 10 21) 95 87 76 63 48 3i 12 —6 25 43 60 76 89 too 20 30 100 9T 80 65 49 3o 10 — 10 3o 5o 69 86 too 30 10°!40 too 87 70 5i 3t 8 — 15 38 60 81 100 40 10° •->0 100 81 58 33 6 — 21 48 75 100 50 OO [00 71 39 5 — 3i 66 100 GO -0 too 53 3 -48 100 70 96 89 78 66 5i 35 18 — 18 35 5i 66 78 89 96 10 94 86 76 63 48 32 r4 —4 22 39 56 70 83 94 lOI 10 20 94 86 76 62 47 29 1 1 —8 27 45 62 78 Q[ 102 20 12" 30 99 90 ; 78 64 47 28 8 — 12 33 53 71 88 io3 30 40 108 98 • 84 68 49 28 5 —18 4i 63 85 io4 40 12° oO 112 97 77 54 29 2 —25 53 80 io5 51) GO 120 95 65 33 — r —37 72 107 GO ;u 70° C0° 50° 1 34 40° 91 30° 20° —6 —58 no 20° 30° 40° 50° 60° 70° 70 < -r- 10° 0° 10° LATIT UDE LATITUDE Of same name a s dcrlination. Of different name from drxlination. ' Page 252] TABLE XLVI. Table showing the variation of the altitude of an object arising from a change of 100 seconds in the declination. If this change move the body to- wards the elevated pole, apply the correction to the altitude with the signs in the Table ; otherwise, change the signs. LATITUDE LATITUDE _Q_ 0° Of same name as declination. Of different name from declination. 1 0- 0) 3... 70° 97" 60° 80" 50° 79" 40° 66" 30° 52" 20° 35" 10° 0° 10° 20° 30° 40° 50° C0° 70° 18" 18" 35" 52" 66" 79" 89" 97" 10 q4 86 76 63 48 3i- i4 —4 23 4o 57 72 85 95 io3 10 20 94 86 75 61 46 27 10 —9 28 45 64 80 93 io4 20 30 97 89 77 62 45 26 6 —14 35 55 74 91 106 30 14° 40 io6 96 82 66 46 25 2 — 21 44 67 88 107 4iJ 14° .^0 109 93 73 5o 25 — 2 — 3o 58 85 no "0 60 ii5 89 60 27 —7 —43 79 ii4 i'i) 70 125 82 35 — 16 -69 121 80 90 98 70 — 98 90 80 67 52 36 18 — 18 36 52 67 10 94 86 76 63 48 3i i3 —5 23 4i 58 73 86 97 io4 10 20 94 85 74 61 45 27 9 — 10 3o 48 66 82 95 106 20 30 96 87 75 61 44 25 4 —17 37 58 77 94 109 30 IGo 40 io4 94 80 63 44 22 -24 48 70 92 III 40 16*^ 50 106 90 70 47 21 —6 -34 62 90 ii5 50 fiO I [0 84 54 21 — 14 — 5o 86 121 60 — 70 IT 99 91 117 73 25 —26 —79 l32 70 — 81 68 53 36 18 — 18 36 53 68 81 9' 99 10 95 87 76 63 48 3i i3 —6 24 42 59 74 88 98 106 10 20 93 85 74 60 44 26 8 — 12 3i 5o 68 84 98 109 20 30 95 86 74 59 42 23 2 —19 40 60 79 97 112 30 18° 40 102 92 78 61 4i 20 —3 —27 5i 74 96 116 40 Ib^ 50 io3 87 66 43 17 — 10 -39 67 95 121 oO 60 io5 79 49 i5 — 20 —56 93 128 60 70 108 64 16 —36 -89 i43 36 53 68 82 92 100 70 ~0" — 100 92 82 68 53 36 18 — 18 10 95 87 76 63 48 3i 12 —6 25 43 60 76 89 100 10 20 93 85 74 60 4i 2D 6 — 13 33 52 70 86 [00 20 30 94 85 73 58 40 21 — 21 42 63 82 100 30 20° 40 100 90 76 59 39 17 —6 — 3i 55 78 100 40 20° 50 100 83 63 39 i3 — 15 -43 72 100 oO 60 100 74 4'^ 10 —26 —63 100 60 70 100 56 6 -46 — 100 93 lOl 70 0" 93 83 69 54 37 19 — n 19 37 54 69 83 10 96 88 77 63 48 3o 12 — 7 26 45 62 78 91 102 10 20 93 85 73 59 43 25 5 -i5 35 54 72 88 io3 20 30 94 85 72 57 39 19 — 2 —23 45 66 86 io3 30 22° 40 98 88 74 57 36 i4 —9 —34 58 82 io4 40 22^^ 50 1 10 97 80 60 36 9 —19 -48 77 106 50 60 117 95 68 38 4 —33 —70 107 (;0 — 70 "o" i3i 92 47 —3 —56 — Ill VO q5 84 70 55 37 19 — 19 37 55 70 84 95 io3 10 97 88 77 64 48 3o II —8 27 46 63 79 93 io4 10 20 93 85 73 59 42 24 4 —16 36 56 74 91 io5 20 30 93 84 71 56 38 18 -4 —26 48 69 89 107 30 24° 40 97 86 72 54 34 12 — 12 -37 62 86 109 40 24^ 50 107 93 77 56 32 5 — 23 —53 83 III 50 60 112 91 64 32 — 2 -39 —77 ii5 60 70 123 83 38 — 13 -67 — 122 96 io5 70 — 96 85 72 56 38 '9 — 19 38 56 72 85 10 98 89 78 64 48 3o II —9 28 47 65 81 95 106 10 20 95 85 73 59 4i 23 3 —18 38 58 77 94 108 20 30 93 83 70 54 36 16 —6 —28 5o 72 92 I II 30 2(;° 40 96 85 70 52 32 9 —16 — 4i 66 91 ii4 40 2G'-' 50 6!) io5 92 108 74 86 53 58 28 27 I —8 —28 —46 —58 —84 88 123 117 50 60 70 ii5 75 29 —23 -78 —1 34 70° 70 < 4) Q < 70° 60° .50° 40° 30° j 20° 10° 0° 10° 20° 30° 40° 50° 00° LATITUDE LATITUDE Of same name as declination. Of different name from dec lination. TABLE XLVII. [Page 253 The first correction is always to be taken at the top. The second correction is to be taken at the top if the apparent distance exceed 90°. o 107/ 1°8' 1° 9' 1°10' \°\V 1°12' 1°13' 1°14' 1°15' 1° 16' 60 9.8879 9.8898 9.8917 9.8935 9-8954 9.8973 9.8992 9.9012 9.9031 9.9050 I 8879 8898 8917 8936 8955 8974 8993 9012 903 1 905 1 59 2 8880 8898 8917 8936 8955 8974 8993 9012 9032 905 1 58 3 8880 8899 8918 893b 8955 8974 8993 9013 9032 905 1 J37 4 5 8880 8899 8918 9.8918 8937 9.8987 8956 8975 8994 9013 9032 9052 5b 55 9.8881 9.8899 9.8956 9.8975 9.8994 9.9013 9.9033 9.9052 6 8881 8900 8918 8937 8956 8975 8994 9014 9033 9o52 54 7 8881 8900 8919 8938 8957 8976 8995 9014 9033 9053 53 « 8882 8900 8919 8938 8957 8976 8995 9014 9033 9053 52 _9 10 8882 8901 8919 8938 8957 8976 8995 9 . 8996 9015 9.9015 9034 9053 5i 9.8882 9.8901 9.8920 9.8939 9.8958 9.8977 9.9034 9.9053 II 8883 8901 8920 8939 8958 8977 8996 9015 9034 9054 49 12 8883 8902 8920 8939 8958 8977 8996 9015 9035 9054 48 i3 8883 8902 8921 8940 8958 8978 8997 9016 9035 9054 47 i4 i5 8884 8902 8921 8940 8959 8978 8997 9016 9035 9055 4b 45 9.8884 9.8903 9 . 892 1 9.8940 9.8959 9.8978 9.8997 9.9016 9.9036 9.9055 i6 8884 8903 8922 8940 8959 8978 8998 9017 9o36 9055 44 17 8884 8903 8922 8941 8960 8979 8998 9017 9o36 9o56 4d i8 8885 8903 8922 8941 8960 8979 8998 9017 9037 9o56 42 19 20 8885 8904 8923 8941 8960 8979 8999 9.8999 9018 9037 9o56 4i 4o 9.8885 9 . 8904 9.8923 9.8942 9.8961 9.8980 9 . 90 1 8 9.9037 9.9057 21 8886 8904 8923 8942 8961 8980 8999 9018 9088 9057 39 22 8886 8905 8924 8942 8961 8980 8999 9019 9o38 905-7 38 23 8886 8905 8924 8943 8962 8981 9000 9019 9088 9o58 37 24 25 8887 8905 8924 8943 8962 8981 9000 9019 9039 9o58 3b 35 9.8887 9.8906 9.8924 9.8943 9.8962 9.8981 9 . 9000 9.9020 9.9039 9.9058 2b 8887 8906 8925 8944 8963 8982 9001 9020 9039 9059 34 27 8888 8906 8925 8944 8963 8982 9001 9020 9040 9039 SS 28 8888 8907 8925 8945 8963 8982 9001 9021 9040 9059 32 29 3o 8888 8907 8926 8945 8964 8983 9002 9021 9040 9060 3i 3^ 9.8888 9.8907 9.8926 9.8945 9.8964 9.8983 9.9002 9.9021 9.9041 9 . 9060 31 8889 8908 8926 8945 8964 8983 9002 9022 9041 9060 29 32 8889 8908 8927 8946 8964 8984 9003 9022 9041 9061 33 8889 8908 8927 8946 8965 8984 9003 9022 9042 9061 27 34 35 8890 8908 8927 9.8928 8946 8965 8984 9003 9023 9042 9061 2() I5 9.8890 9.8909 9.8946 9.8965 9.8985 9 . 9004 9.9023 9.9042 9.9062 3b 8890 8909 8928 8947 8966 8985 9004 9023 9042 9062 24 37 8891 8909 8928 8947 8966 8985 9004 9024 9043 9062 23 38 8891 8910 8929 8947 8966 8985 9003 9024 9043 9063 22 39 40 8891 8910 8929 8948 8967 9.8967 8986 9005 9024 9043 9063 21 20 9.8892 9.8910 9.8929 9.8948 9.8986 9.9005 9.9024 9.9044 9.9063 4i 8892 8911 8929 8948 8967 8986 9006 9025 9044 9064 '9 42 8892 8911 8930 8949 8968 8987 9006 9025 9044 9064 18 Ai 8893 891 1 8930 8949 8968 8987 9006 9025 9045 9064 17 44 45 8893 8912 8930 8949 8968 8987 9007 9026 9045 9064 lb i5 9.8893 9.8912 9.8931 9.8950 9 . 8969 9.8988 9.9007 9.9026 9.9045 9.9065 4b 8893 8912 8931 8950 8969 8988 9007 9026 9046 9e>65 14 47 8894 8913 8931 8950 8969 8988 9007 9027 9046 9065 i3 48 8894 8913 8932 8951 8970 8989 9008 9027 9046 9066 12 49 5o 8894 8913 8932 8951 8970 8989 9008 9027 9047 9066 1 1 10 9.8895 9.8913 9.8932 9.8951 9.8970 9.8989 9 . 9008 9.9028 9.9047 9 . 9<:)66 DI 8895 8914 8933 8952 8971 8990 9009 9028 9047 9067 9 b2 8895 8914 8933 8952 8971 8990 9009 9028 9048 9067 b 53 8896 8914 8933 8952 8971 8990 9009 9029 9048 9067 7 54 55 8896 8915 8934 8952 8971 8991 9010 9029 9048 90G8 b "5 9.8896 9.8915 9.8934 9.8953 9.8972 9.8991 9.9010 9.9029 9 . 9049 9 . 9068 Db 8897 8915 8934 8953 8972 8991 9010 9o3o 9049 9068 4 i)7 8897 8916 8935 8953 8972 8992 901 1 9o3o 9049 9069 3 58 8897 8916 8935 8954 8973 8992 9011 9o3o 9o5o 9069 2 59 8898 8916 8935 8954 8973 8992 90 1 1 903 1 9o5o 9069 I bo 8898 8917 8935 8954 8973 8992 9012 903 1 9o5o 9070 8° 52' 8° 51' 8° 50' 8° 49' 8° 48' 8° 47' 8° 46' 8° 45' 8° 44' 8° 43' b ^he second correct ion is to be taken at the bottom it' t ^le apparent dista ace be less than 90°. Page 254] TABLE XLVII. The first correction is always to be lalien at the top. Tlie second correction is to be taken at the top if the apparent distance exceed 90°. II I 2 3 4 5 6 7 8 _9 lO II 12 i3 i4 i5 t6 17 i8 19 20 21 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 60 1°17' i°18' 1° 19' 1° 20' 1°21' 1° 22' 1° 23' I°24' 1° 25' 1° 26' 60 59 58 57 56 55 54 53 52 5i 5o 49 48 47 46 45 4o 42 4i 40 39 38 37 36 35 34 33 32 3i 3o 29 28 27 26 25 24 23 22 21 20 19 18 17 16 i5 1 4 i3 12 II 10 n 6 ■5 4 3 2 I // 9.9070 9070 9070 9071 9071 9.9089 9090 9090 9090 9091 9.9109 9109 9109 9110 9110 9.9128 9129 9129 9129 9i3o 9.9148 9149 9149 9149 91 5o 9.9168 9168 9169 9169 9169 9.9188 9188 9189 9189 9189 9.9208 9209 9209 9209 9210 9.9228 9229 9229 9229 9280 9.9249 9249 9249 9250 9250 9.9071 9072 9072 9072 9073 9.9091 9091 9091 9092 9092 9.9110 9111 9111 9111 9112 9.9130 9i3o 9i3i 9i3i 9i3i 9.9150 9i5o 9i5i 9i5i 9i5i 9.9170 9170 9170 9171 9171 9.9190 9190 9190 9191 9191 9.9210 9210 92 1 1 9211 9211 9.9230 9280 9231 9231 9231 9.9250 9251 9251 9251 9252 9.90-3 9-.; 9074 9074 9074 9.9092 9093 9093 9093 9094 9.9094 9094 9095 9095 9095 9.9112 9112 9113 91x3 9113 9.9132 9i32 9132 9133 9133 9.9152 9i52 0152 9153 9153 9.9171 9172 9172 9172 9173 9.9191 9192 9192 9192 9193 9.9212 9212 9212 9213 9213 9.9282 9282 9282 9233 9233 9.9252 9252 9253 9253 9253 9.9075 9075 9075 9076 9076 9.9114 9114 9114 9115 9ii5 9.9133 9134 9134 9134 9135 9.9153 9154 9154 9154 9155 9.9173 9173 9174 9174 9174 9.9193 9193 9194 9194 9194 9.9218 9214 9214 9214 9215 9.9233 9234 9234 9234 9235 9.9254 9254 9254 9255 9255 9.9076 9076 9077 9077 9077 9 . 9096 9096 9096 9097 9097 9.9115 911D 9116 9116 9117 9.9135 0135 9i36 9 1 36 9 1 36 9.9155 9155 9i56 9i56 9i56 9.9175 9175 9175 9176 9176 9.9195 9195 9195 9196 9196 9.9215 9215 921G 9216 9216 9.9235 9235 9236 9236 9286 9.9255 9256 9256 92 56 9257 9.9078 9078 9078 9079 9079 9.9097 9098 90 98 9098 9099 9.9117 9117 9118 9118 9118 9.9137 9137 9137 9 1 38 9i38 9.9157 9157 9157 9i58 9i58 9.9176 9177 9177 9177 9178 9.9196 9197 9197 9197 9198 9.9217 9217 9217 9218 9218 9.9237 9237 9287 9238 9288 9.9257 9257 9258 9258 925s 9.9079 9080 90S0 9080 9081 9.9099 9099 9100 9100 9100 9.9119 9119 9119 9:20 9120 9.9138 9139 9i3o 9139 9140 9.9158 9159 9159 9159 91D0 9.9178 9178 9179 9179 9179 9.9198 9198 9199 9199 9199 9.9218 9219 9219 9219 9220 9.9288 9239 9239 9289 9240 9.9259 9259 9259 9260 9260 9 . 908 1 9081 90S 2 9082 9082 9.9101 9101 9101 9102 9102 9.9102 9ig3 9103 9103 9104 9.9120 91 2 1 91 2 1 9121 9122 9.9140 9140 9141 9141 9141 9 . 9 1 60 9160 9161 9161 9161 9.9180 9180 9180 9181 9181 9.9200 9200 9200 9201 9201 9.9220 9220 9221 9221 9221 9.9240 9240 9241 9241 9241 9.9260 9261 9261 9261 9262 9.9083 9083 9083 9084 9084 9.9122 9122 9123 9123 9123 9.9124 9124 9124 9125 9125 9.9142 9142 9142 9143 9143 9.9162 9162 9162 9163 9163 9.9181 9182 9182 9182 9183 9.9201 9202 9202 9202 9203 9.9222 9222 9222 9223 9223 9.9242 9242 9243 9243 9243 9.9244 9244 9244 9245 9245 9.9262 9262 9268 9268 9263 9.9264 9264 9265 92G5 9265 9 . 9084 9085 9085 9085 9086 9.9104 9104 9105 9105 9105 9 9143 9144 9144 9144 9145 9.9163 9164 9164 9164 9165 9.9183 9183 9184 9184 9184 9.9208 9203 9204 9204 9205 9.9223 9224 9224 9224 9225 9.9086 9086 9087 90S 7 9087 9.9106 9106 9106 9107 9107 9.9125 9126 9126 9126 9127 9.9145 9145 9146 9146 9146 9.9165 9165 9166 9166 9166 9.9.85 9185 9185 9186 9186 9.9205 9205 9206 9206 9206 9.9225 9225 9226 9226 9226 9.9245 9246 9246 9246 9247 9.9266 9266 9266 9267 9267 9.9088 908S 90SS 90S9 9.09 9089 9.9107 9107 9108 9108 9108 9' 09 9.9127 9127 9128 912S 9128 9128 9.9147 9147 9147 9148 9148 9148 9.9167 9167 9167 9167 9168 916S 9.9186 9187 9187 9187 9188 918S 9.9207 9207 9207 920S 920S 9208 9.9227 9227 9227 9228 9228 9228 9.9247 9247 9248 9248 9248 9249 9.9267 9268 9268 9268 9269 9269 8° 42' 8° 41' 8° 40' 8^39' 8° 38' 8° 37' 8° 30' 8° 35' 8° 34' 8° 33' The s/cimd correction is to be taken at the bottom if tlie app-ire.il distance be less than 90°. | TABLE XLVII. [Page 255 The firs ! correction is always to be taken at tlie top. The second correction is to be taken at the top if the apparent distance exceed 90°. // o P 27' 1 = 28' 1° 29' 1°30' 9-9331 1°31' 1° 32' 9.9372 1° 33' 1° 34' P35' 1°3G' 60 9.9269 9.9289 9.9310 9.9351 9.9393 9.9414 9.9435 9.9456 T 9269 9290 9310 933 1 9352 9372 9393 94i4 9436 9457 59 ?- 9270 9290 93 1 1 931 9352 9373 9394 94x5 9436 9457 58 i 9270 9290 93 1 1 9332 9352 9373 9394 941 5 9436 9457 37 4 5 9270 9.9271 9291 93 1 1 9332 9353 9373 9394 941 5 9437 9458 56 55 9.9291 9.9312 9-9332 9.9353 9.9374 9.9395 9.9416 9.9437 9.9458 6 9271 9291 9312 9333 9353 9374 9395 9416 9437 9459 54 7 9271 9292 9312 9333 9354 9375 9395 9417 9438 9459 53 8 9272 9292 93i3 9333 9354 9375 9396 9417 9438 9459 32 _9 lO 9272 9292 93i3 9334 9354 9375 9396 9417 9.9418 9438 9460 5l 5^ 9.9272 9.9293 9.9313 9.9334 9.9355 9.9376 9.9397 9.9439 9 . 9460 I r 9273 9293 93i4 9334 9355 9376 9397 9418 9439 9460 49 12 9273 9293 93i4 9335 9355 9376 9397 9418 9439 9461 48 i3 9273 9294 93i4 9335 9356 9377 9398 9419 9440 9461 47 1 4 i5 9274 9294 93i5 9335 9356 9377 939« 9419 9440 9461 46 45 9.9274 9.9294 9.9315 9.9336 9.9356 9.9377 9.9398 9.9419 9.9440 9.9462 i6 9274 9295 93i5 9336 93^7 937S 9399 9420 9441 9462 AA ' 7 9275 9295 9316 9336 9357 9378 9399 9420 9441 9462 A'i i8 9275 9296 93i6 9337 9358 9378 9399 9420 9342 9463 42 12 ■iO 927D 9296 9316 9337 9358 9379 9400 9421 9442 9.9442 9463 9.9464 4i 4o 9.9276 9.9296 9.9317 9.9337 9.9358 9.9379 9 . 9400 9.9421 71 9276 9297 9317 9338 9359 9379 9400 9421 9443 9464 39 >r>. 9276 9297 9317 933-8 9359 9380 9401 9422 9443 9464 38 pj 9277 9297 9318 9338 9359 9380 9401 9422 9443 9465 37 p4 ■i5 9277 9298 9318 9339 9360 9380 9401 9422 9444 9465 36 35 9.9277 9.9298 9. 931 8 9.9339 9.9360 9.9381 9.9402 9.9423 9.9444 9 . 9465 3 b 9278 929S 9319 9340 9360 9381 9402 9423 9444 9466 M ■'7 9278 9299 9319 9340 9361 9381 9402 9424 9445 9466 i6 ■'0 9278 9299 9320 9340 9361 9382 94o3 9424 9445 9466 32 !2 9279 9299 9320 9341 9361 9382 94o3 9424 9445 9467 61 3o 9.9279 9.9300 9.9320 9.9341 9.9362 9.9383 9.9404 9.9425 9.9446 9.9467 u 9279 9300 9321 9341 9362 9383 9404 9425 9446 9467 29 •;■> 92S0 9300 9321 9342 9362 9383 9404 9425 9447 9468 28 ) ) 92S0 9301 9321 9342 9363 9384 94o5 9426 9447 9468 27 -'4 9280 9301 9322 9342 9363 9384 94o5 9426 9447 9469 26 9.9281 9.9301 9.9322 9.9343 9.9363 9.9384 9.9405 9.9426 9.9448 9.9469 J(-> 9281 9302 93" 9343 9364 9385 9406 9427 9448 9469 24 9282 9302 9J23 9343 9364 9385 9406 9427 9448 9470 23 J-S 9282 9302 9323 9344 9364 9385 9406 9427 9449 9470 2 2 9282 93o3 9323 9344 9365 9386 9407 9428 9449 9470 21 20 9.9283 9.9303 9.9324 9.9344 9.9365 9.9386 9-9407 9.9428 9.9449 9.9471 II 9283 93o3 9^24 9345 9365 9386 9407 9428 9450 9471 19 (9. 9533 9304 9324 9345 9366 9387 940S 9429 9450 9471 18 4 J 9284 93o4 9325 9345 9366 9387 9408 9429 9450 9472 17 44 i5 9284 9304 9325 9346 9367 9387 9408 9430 945 1 9472 16 i5 9.9284 9.9305 9.9325 9.9346 9.9367 9. 9388 9.9409 9.9430 9.9451 9.9472 -,(J 9285 93o5 9326 9346 9367 9388 9409 9430 945 1 9473 i4 i7 9285 9.0b 9326 9347 9368 9388 9409 943 1 9452 9473 10 !« 92S5 9306 9326 9347 9368 9389 9410 943 1 9452 9473 12 -19 5AA 0571 0599 0627 o655 o683 G711 0740 0768 49 12 0490 o5i7 0:.. 44 0572 0599 0627 o655 o683 07 1 1 0740 0769 48 iJ 0490 o5i7 0545 0572 0600 0628 o655 0684 0712 0740 0769 47 i4 i5 0491 o5i8 0545 0573 0600 0628 o656 0684 0712 0741 0770 46 45 0.0491 o.o5i8 . o546 . 0573 0.0601 0.0628 o.o656 0.0685 0.0713 0.0741 0.0770 i6 0492 o5i9 o546 0573 0601 0629 0657 068 5 0713 0742 0771 M 17 0492 o5i9 o546 0574 0602 0629 0657 0686 0714 0742 0771 43 i8 0493 o52o o547 o574 0602 o63o o658 0686 0714 0743 0772 42 19 20 0493 o52o o547 0575 0602 o63o o658 0686 0716 0743 0772 4i 4o 0.0493 0.o52I 0.0548 0.0575 o.o6o3 o.o63i 0.0659 0.0687 0.0715 5.0744 0.0773 21 0494 052I o548 0576 o6o3 o63i 0659 0687 0716 0744 0773 39 22 0494 o52i 0549 0576 0604 o632 0660 0688 0716 0745 0774 38 2J 0495 0522 0549 0577 o6o4 o632 0660 0688 0717 0745 0774 37 24 25 0495 o522 o55o 0577 o6o5 o633 0661 0689 0717 0746 0774 36 ~35 0.0496 o.o523 o.o55n 0.0578 0.060 5 o.o633 0.0661 0.0689 0.0718 0.0746 0.0775 26 0496 o523 o55i 0578 0606 0634 0662 0690 0718 0747 0775 34 27 0497 o524 o55i 0579 0606 o634 0662 0690 0719 0747 0776 33 28 0497 0624 o552 0579 0607 o634 o663 0691 0719 0748 0776 32 29 3o 0498 o52 5 o552 0579 0607 o635 o663 0691 0720 0748 0777 3 1 3o 0.0498 o.o525 o.o552 o.o58o 0.0608 o.o635 o.o663 0.0692 0.0720 0.0749 0.0777 Ji 0498 0526 o553 o58o 060S o636 0664 0692 0721 0749 077S 29 32 0499 o526 o553 o58i 0609 o636 0664 0693 0721 0760 0778 28 33 0499 0526 o554 o58i 0609 0637 o665 0693 0721 0750 0779 27 34 o5oo 0527 o554 o582 0609 0637 o665 0694 0722 0751 0779 26 25 35 o.oSoo 0.0527 0.0555 o.o582 0.0610 o.o63S 0.0666 0.0694 0.0722 0.0751 0.0780 36 o5oi 0628 o555 o583 0610 o638 0666 0694 0723 0761 0780' 24 37 o5oi o528 o556 o583 061 1 0639 0667 0695 0723 0752 0781 23 38 o5o2 0529 o556 o584 0611 0639 0667 0695 0724 0762 0781 2 2 39 4o o5o2 0529 0557 o584 0612 o64o 0668 0696 0724 0753 0782 2 1 20 o.o5o2 o.o53o 0.0557 o.o585 . 06 1 2 . 0640 0.0668 0.0696 0.0725 0.0753 0.0782 4i o5o3 o53o o557 o585 o6i3 0641 0669 0697 0725 0754 0783 19 42 o5o3 o53i o558 o585 o6i3 064 1 0669 0697 0726 0754 0783 18 43 o5o4 o53i o558 o586 o6i4 064 1 0670 0698 0726 0755 0784 17 44 45 o5o4 o53i 0559 o586 0614 0642 0670 0698 0727 0755 0784 16 i5 o.o5o5 o.o532 0.0559 0.0587 0.061 5 0.0642 0.0670 0.0699 0.0727 0.0756 0.0785 46 o5o5 o532 o56o 0587 o6i5 0643 0671 0699 0728 0766 0785 i4 47 o5o6 o533 o56o o588 061 5 0643 0671 0700 0728 0767 0786 i3 48 o5o6 o533 o56i o588 0616 0644 0672 0700 0729 0757 0786 12 49 5» o5o7 o534 o56i 0589 0616 0644 0672 0701 0729 0758 0787 1 1 10 o.o5()7 o.o534 0.0 50 2 0.0589 . 06 1 7 0.0645 . 0673 0.0701 0.0730 0.0758 0.0787 5i o5o7 o535 o562 0590 0617 0645 0673 0702 0730 0759 0787 9 i)2 o5o8 o535 o562 0590 0618 0646 0674 0702 0730 0759 0788 8 63 o5o8 o536 o563 0591 0618 0646 0674 0703 0731 0760 0788 7 54 55 o5o9 o536 o563 0591 0619 0647 0675 0703 073 r 0760 0789 6 5 o.o5()9 o.o536 0.0 564 0.0591 . 06 1 9 . 0647 0.0675 0.0703 0.0732 0.0761 0.0789 5b o5io o537 o564 0692 0620 0648 0676 0704 0732 0761 0790 4 ^7 o5io o537 o565 0592 0620 o648 0676 0704 0733 0762 0790 3 58 o5ii o538 o565 0693 0621 0648 0677 0705 0733 0762 0791 2 59 o5ii o538 o566 0593 0621 0649 0677 0705 0734 0769 0791 I bo 05l2 0539 o56fi 0594 0621 0649 0678 0706 0734 0763 0792 // 7° 40' 7° 39' 7° 38' 7=37' 7° 30' 7° 35' 7=31' 7° 33' 7° 32' 7^31' 7°3(y } secmid correct on is to be take n at th 3 hottom if the f ipparen t distanc e be Ic. \« t/ia-n i 0°. T.ABLE XLVII. [Page 2Cl H\ie first correction is always to be taken at the top. The second correction is to be taken at tlie toj) if the apparent distance exceed 90°. II 2°3t/ 2^31' 2° 32' 2^33' 2=34' 2° 35' 2°3G' 2° 37' 2° 38' 2° 39' 2° 40' 60 0.079a 0.0821 o.oS5o 0.0880 . 091 )9 0.0939 . 0969 . 0999 0.1 o3o 0. I06I 0. 1091 I 0792 0821 08 5 1 0880 0910 0940 0970 1000 io3o 1061 1092 59 2 0793 0822 o85i 0881 0910 09.40 0970 1000 io3i 1062 1093 5fi 3 0793 0822 08 52 0881 091 1 0941 0971 lOOI io3i 1 062 1093 57 4 5 0794 0823 o852 0882 091 1 0941 0971 lOOI 1032 io63 1094 56 55 0.0794 0.0823 o.oS53 0.0882 0.0912 0.0942 0.0972 0.1002 0.1032 0.1 o63 0. 109.5 6 0795 0824 o853 088 3 0912 0942 0972 1002 io33 1064 1095 54 7 0795 0824 o854 o883 0913 0943 0973 ioo3 io33 1064 1095 53 8 0796 0825 oS54 o883 0913 0943 0973 ioo3 io34 io65 1 096 52 _9. 10 0796 0825 08 5 5 0S84 0914 0.0914 0944 0974 ioo4 1034 io65 1 09() 5i 'So 0.0797 0.0826 0.0855 0.0884 0.0944 0.0974 0.1004 o.io35 . 1 066 0.1097 11 0797 0826 o855 088 5 0915 0945 0975 ioo5 io35 1066 1097 49 12 0798 0827 o856 088 5 0915 0945 0975 ioo5 io36 1067 1098 48 i3 0798 0827 o856 0886 0916 0946 0976 1006 io36 1067 1098 47 i4 i5 0799 0828 0857 0886 0916 0946 0976 1006 io37 1068 1099 46 45 0799 0.0S28 0.0S57 0.0887 0.0917 0.0947 0.0977 0.1007 0.1037 0.1068 0.1099 i6 0800 0829 oS5S 0887 0917 0947 0977 1007 io38 1069 1100 A4 17 0800 0829 oS58 0S88 0918 0948 0978 1008 1039 1069 1 100 43 18 0801 o83o 0859 08S8 0918 0948 0978 1008 1039 1070 IlOl 42 19 20 0801 o83o 0859 0889 0919 0949 0979 1009 I o4(j 1070 1101 4i 4o 0.0801 o.o83i 0.0860 0.0889 0.0919 0.0949 0.0979 . I 009 . 1 o4o 0.1071 0.1102 21 0802 o83i 0860 0S90 0920 0950 0980 lOIO io4i 1071 1102 39 22 0802 o832 086 1 0890 0920 0950 0980 1011 1 04 1 1072 iio3 38 23 o8o3 o832 0861 0891 0921 0951 0981 1011 1042 1072 iio3 37 24 25 o8o3 o833 0862 0891 0921 0951 0981 I0I2 1042 1073 iio4 36 . 0804 0.0833 0.0862 0.0892 0.0922 0.0952 0.0982 0.1012 0.1043 0.1073 0.1104 26 0804 o834 o863 0892 0922 0952 0982 ioi3 1043 1074 HOD 34 27 o8o5 o834 o863 0893 0923 0953 0983 10x3 1044 1074 iio5 33 28 o8o5 o834 0864 0893 0923 0953 0983 ioi4 1044 1075 1106 32 29 3o 0806 o835 0864 o.o865 0894 . 0S94 0924 0954 0984 ioi4 1045 1075 1106 3i "3^ 0.0806 0.0835 0.0924 0.0954 0.0984 o.ioi5 0.1045 0.1076 0.1107 3[ 0807 o836 086 5 0S95 0925 0955 09S5 ioi5 io46 1076 1108 29 32 0807 o836 0S66 0895 0925 0955 0985 1016 io46 1077 1108 28 33 0808 o837 0866 0896 0926 0956 0986 1016 1047 1078 1 109 27 34 35 0808 0837 0867 0896 0926 0956 0986 1017 1047 1078 1109 56 25 0.0809 0.08 38 o.o8t)7 0.0897 0.0927 0.0957 0.0987 0.1017 0.1048 0.1079 . 1 1 1 36 0S09 o838 0868 0897 0927 0957 0987 1018 1048 1079 IIIO 24 37 0810 0839 0868 0898 0928 0958 0988 1018 1049 1080 nil 23 38 0810 0839 0869 0898 0928 0958 0988 1019 1049 1080 nil 22 39 4o 0811 o84o 0S69 0899 0929 0959 0989 1019 io5o 1081 1 I 12 21 20 0.081 1 o.o34o 0.0870 0.0899 0.0929 0.0959 . 0989 0.1020 . 1 o5o . 1 08 1 0.1112 4i 0812 084 1 0870 0901) 09311 096(j 0990 1020 io5i 1082 iii3 19 42 0812 084 1 0871 0900 0930 C)9()(.) 0990 1021 io5i 1082 iii3 18 43 o8i3 0842 0871 0901 0931 096 1 0991 102 1 io52 io83 1114 17 45 o8i3 0842 0872 0901 0931 0961 0991 1022 io52 io83 1 114 16 75" 0.0814 0.0843 0.0872 0.0902 0.0932 . 0962 0.0992 0. 1022 o.io53 0.1084 o.iii5 46 0814 0843 0873 0902 0932 0962 0992 1 023 io53 1084 iii5 i4 47 oSi5 0844 0873 0903 0933 0963 0993 I023 io54 io85 1116 i3 48 o8i5 0844 0874 0903 0933 0963 0993 1024 io54 io85 1116 12 49 5o 0816 0845 0874 0904 0934 0964 0994 1024 io55 1086 1117 11 10 o.c«8i6 0.0845 0.0875 0.0904 0.0934 0.0064 0.0994 0. I025 0. to55 0.1086 o.ii 17 5i (j8i6 o846 0875 0905 0935 0965 0993 1025 io56 1087 1118 9 52 0817 0846 0876 0905 0935 0965 0995 1026 io5u 1087 1118 b 53 0817 0847 0876 '0906 C936 0966 0996 1026 1067 1088 1119 n 54 55 0818 0847 0877 0906 0936 0966 0996 T027 1057 1088 1119 6 0.0818 0.0848 0.0877 0.0907 0.0937 0.0967 0.0997 0.1027 o.io58 0. 1089 0.1120 56 0819 0848 0878 0907 0937 0967 0997 1028 io58 1089 1 1 20 4 57 0819 0849 0878 0908 0938 0968 0998 1028 1059 1090 1121 3 58 0820 0849 0879 0908 0938 0968 0998 1029 1060 1090 1122 2 59 0820 o85o 0879 0909 0939 0969 0999 1029 ic6o 1091 1122 I 60 0821 o85o 0880 0909 0939 0969 0999 io3o ic6i 1 09 1 II23 II 7° 29' 7° 28' 7° 27' 7°2G' 7° 25' 7° 24' 7° 23' 7° 22' 7°21' 7° 20' 7° 19' The ; second correction is to be taken at the bottom if the ipparen t distan ze be le '^s than 90°. P'^esea] TABLE XLVII. The first correction is always to be taken at the top. The second correction is to be taken at the top if the apparent distance exceed 90°. 1/ o I 2 3 4 5 6 7 8 9 10 1 1 12 i3 i4 i5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 00 4j/ 2° 42' 2°43- 2° 44' 2° 45' 2° 46' 2° 47' 2° 48' 2° 49' 2° 50' 2° 51' 60 • 59 58 . 57 . 56 55 54 53 52 5i 5o 49 48 47 46 45 44 43 42 4i 4o 39 38 37 36 35 34 33 32 3i 1^ 29 20 27 26 25 24 23 22 21 20 19 18 17 16 i5 i4 i3 12 11 10 7 6 5 4 3 2 I 0.1123 1123 1124 1124 ri25 O.I154 ii54 ii55 ii56 ii56 0.1186 1 186 11S7 1187 1188 0.11S8 1189 1189 1 190 1 1 90 0. 1217 1218 1218 1219 1219 0.1249 125o 125o 125l 1252 0.1282 1282 1283 1283 1284 o.i3i4 i3i5 i3i5 i3i6 i3i6 0.1347 1 348 i348 1 349 1 349 0. i38o i38i i38i i382 i382 o.i4i3 i4i4 i4i4 i4i5 i4i6 0.1447 1447 i448 1449 1449 0. 1125 1 126 1126 1127 1127 0.1157 ii57 ii58 ii58 1159 0. 1220 1221 1221 1222 1222 0.1252 1253 1253 1254 1254 0.1284 1285 1285 1286 1287 o.i3i7 i3i7 i3i8 i3i9 i3i9 o.i35o i35o i35i i35i i352 o.i383 1 383 i384 1 384 i385 o.i4i6 1417 1417 i4i8 i4i8 o.i45o i45o i45i i45i 1452 0.1128 II 28 1129 1129 ii3o 0.1 159 1160 1 1 60 1161 1161 0.1191 1191 1190 1192 1193 0. 1223 1223 1224 1224 1225 0.1255 1255 1256 1256 1257 0.1287 1288 1288 1289 1289 0. l320 l320 l321 l321 l322 o.i352 i353 1 354 i354 i355 0.1386 i386 1387 1387 1 388 0.1419 1419 1420 1421 1421 0.1452 1453 1454 1454 1455 o.ii3o ii3i ii3i Il32 ir32 0.1162 1 162 ii63 ii63 ii64 0.1193 1 194 1195 1195 1196 0.1225 1226 1226 1227 1227 0.1257 1258 1259 1259 1260 0.1290 1290 1291 1291 1292 0.l322 i323 i323 i324 i325 0.1355 i356 i356 i357 i357 o.i388 1389 1389 1390 1391 0.1422 1422 i423 x423 1424 0.1455 i456 i456 1457 i458 o.ii33 1134 ii34 ii35 ii35 0.1164 ii65 ii65 1166 • 1 167 0.1196 1197 1197 1198 1 198 0.1228 1229 1229 123o 123o 0.1260 1261 1261 1262 1262 0.1292 1293 1294 1294 1295 o.i325 i326 i326 i327 i327 0.-I358 1359 1359 i36o i36o 0.1391 1392 1392 1393 1393 0.1424 i425 1426 1426 1427 0.1458 1459 1459 1 460 i46o o.ii36 ii36 ii37 1137 ii38 0.1167 1168 1168 1 169 1169 0.1199 1199 1200 1200 1201 0.1 23 1 I 23 I 1232 1232 1233 0.1263 1263 1264 1264 1265 0.1295 1296 1296 1297 1297 0.1328 i328 i329 1329 i33o o.i36i i36i i362 1 362 1 363 0.1394 1394 1395 1396 1396 0.1427 1428 1428 1429 1429 0.1461 i46i 1462 1 463 1 463 o.ii38 1139 1139 I i4o ii4o 0.1170 1170 1171 1171 1172 0. 1201 1202 1202 I203 I204 0.1233 1234 1234 1235 1235 . 1 266 1266 1267 1267 1268 0.1298 1298 1299 i3oo i3oo o.i33i i33i i332 i332 i333 o.i363 1 364 1 365 1 365 i366 0.1897 1397 1398 1398 1399 o.i43o i43i i43i i432 i432 o.i464 1 464 1 465 1 465 1 466 35 36 37 38 39 4o 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 59 60 o.it4i ii4i Il42 II42 II43 0.1172 1173 1173 1174 1 174 . I io4 I 205 1205 1206 1206 0. 1236 1237 1237 1238 1238 0.1268 1269 1269 1270 1270 o.i3oi i3oi l3o2 l302 i3o3 O.I333 1 334 i334 i335 i335 0.1 366 i367 1 367 1 368 1 368 0.1399 i4oo i4oi i4oi l402 0.1433 1433 1434 1435 1435 0.1467 1467 1 468 1 468 1469 0.1143 ii44 1145 ii45 ii46 0.1175 1175 1176 1177 1177 0.1207 1207 1208 I20S 1209 0.1239 1239 1240 1240 1241 0.1 27 1 1271 1272 1273 1273 o.r3o3 i3o4 i3o4 i3o5 i3o6 o.i336 i337 i337 1 338 1 338 0.1369 1370 1370 1371 1371 0. l402 i4o3 i4o3 i4o4 i4o4 0.1436 i436 1437 1437 i438 0.1469 1470 1470 1471 1472 0.1146 ii47 1147 ii48 ii48 0.1178 1178 1179 1179 1 180 0.1209 I2IO I2IO 1211 121 I 0.1241 1242 1242 1243 1243 0.1274 1274 1275 1275 1276 . 1 3o6 i3o7 i3o7 i3o8 i3o8 0.1339 1339 1 340 1 340 i34i 0.1372 1372 1373 1373 1374 o.i4o5 i4o6 i4o6 1407 1407 0.1438 1439 i44o i44o i44i 0.1472 1473 1473 1 474 i474 0.1149 1149 ii5o ii5o ii5i 0.1 1 So 1181 1181 II 82 1182 0.1212 I2l3 I2l3 12l4 12l4 . 1 244 1245 1245 1246 1246 0.1276 1277 1277 1278 1278 0.1 309 1 309 i3io i3io i3ii 0.1342 i342 1 343 1 343 1 344 0.1374 1375 1376 1376 1 377 o.i4o8 i4o8 1409 1409 i4io o.i44i 1442 1442 1443 1443 0.1475 1476 1476 i477 i477 0.1 i5i Il52 Il52 ii53 ii53 ii54 o.ii83 ii83 1184 ii84 ii85 1186 0.12l5 I2l5 I2I6 I2I6 I2I7 1217 0.1247 1247 1248 1248 1249 1249 0.1279 1280 1280 1281 1281 1282 o.i3ii l3l2 i3i3 i3i3 i3i4 i3i4 0.1344 1 345 1 345 1 346 1 346 1 347 0.1377 1378 1378 i379 1 379 i38o o.i4ii i4ii l4l2 l4l2 i4i3 i4i3 0.1444 1445 i445 1 446 i446 1 447 0.1478 1478 1 479 1479 i48o i48i 7=18' 7° 17' 7°1G' 7° IS'- 7° 14' 1 7° 13' 7° 12' 7° 11' 7° 10' 7^ 9' 70 g/ // The second correction is to be taken at the bottom if the apparent distance be less than 90°. TABLE XLVII. [Page 263 The first correction is always to be taken at the top. The second correction is to be taken at the to]) if the apparent distance e.Yceed 90°. // o I 2 3 4 5 6 7 8 lO II 12 i3 i4 i5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 ?9 60 2=52' 2=53' 2° 54' 2° 55' 2° 56' 2° 57' 2° 58' 2° 59' 3°0' 3° 1' 3° 2' 60 59 58 57 56 55 54 53 52 5i 5o 49 48 47 46 45 44 43 42 4i 4o 39 38 37 36 35 34 33 32 3i 29 28 27 26 25 24 23 23 21 20 19 18 17 16 75" 14 i3 12 II 10 7 6 5 4 3 2 1 o.i48i i48i 1482 1482 1 483 o.i5i5 i5i5 i5i6 i5i6 i5i7 . 1 549 i55o i55o i55i i55i o.i584 1 584 1 585 i585 1 586 0. 1619 1619 1620 1620 1621 0.1654 i654 i655 i655 i656 0.1689 1690 1690 1691 1692 0.1725 1725 1726 1727 1727 0.1761 1762 1762 1763 1763 0.1797 1798 1798 1799 1800 0.1834 i835 i835 1 836 i836 0.1837 i838 i838 1839 1839 o.i483 1 484 i485 i485 i486 o.i5i8 i5i8 i5i9 i5i9 I Sao o.i552 i552 1 553 1 554 1 554 0.15S7 i587 1 588 1 588 1589 0.1621 1622 1623 1623 1624 . 1 657 i657 1 658 1 658 1659 0.1692 1693 1693 1694 1694 0.1728 1728 1729 1730 1730 0.1764 1765 1765 1766 1766 0. 1800 1801 1802 1802 i8o3 0.1486 1487 1487 i488 14S9 0.l520 l521 I 522 l522 i523 0.1555 i555 1 556 1 556 1 557 0.1589 1590 1591 1591 1592 0.1624 1625 1626 1626 1627 0.1660 1660 1661 1661 1662 0.1695 1696 1696 1697 1697 0.1731 1731 1732 1733 1733 0.1767 1768 1768 1769 1769 o.i8o3 1804 i8o5 i8o5 1806 . 1 840 i84i ]84i 1842 1843 0.1489 1490 1490 1491 1491 0. i523 1 524 i524 i525 i526 o.i558 i558 1559 1559 i56o 0.1592 1593 1593 1594 1595 0.1627 1628 1628 1629 i63o 0.1663 i663 16&4 1664 1 665 0.1698 1699 1699 1700 1700 0.1734 1734 1735 1736 1736 0.1770 1771 1771 1772 1772 0.1806 1807 1808 1808 1809 0.1843 1844 1 844 1845 1 846 0.1492 1493 1493 1494 1494 0.1526 1527 i527 1528 i528 o.i56i i56i 1 562 1 562 1 563 0.1595 1596 1596 1597 1598 0.1598 1599 1599 1600 1600 0.1 63o i63i i63i i632 i633 0.1665 1666 1667 1667 1668 0.1701 1702 1702 1703 1703 0.1737 1737 1738 1739 1739 0.1773 1774 1774 1775 1775 0.1809 1810 1811 1811 1812 0.1846 1847 1847 1848 1849 0.1495 1495 1496 1496 1497 0.1529 i53o i53o i53i i53i O.I563 1 564 i565 1 565 1 566 0.1633 i634 1 634 i635 i635 0.1668 1669 1670 1670 1671 0.1704 1705 1705 1706 1706 0.1740 1740 1741 1742 1742 0.1776 1777 1777 1778 1778 0.1812 i8i3 i8i4 i8i4 i8i5 0.1849 i85o i85o 165: i852 0.1498 1498 1499 1499 i5oo o.i532 i532 i533 1 534 i534 . 1 568 1567 1 587 1 568 1 569 0.1601 1602 1602 i6o3 i6o3 0.1636 1637 i637 i638 1 638 0.1671 1672 1673 1673 1674 0.1674 1675 1676 1676 1677 0.1707 1708 1708 1709 1709 0.1743 1743 1744 1745 1745 0.1779 1780 1780 1781 1781 0.1816 1816 1817 1817 1818 o.iS52 i853 1 854 i854 i855 o.iSoo i5oi l5o2 l502 i5o3 O.I535 i535 1 536 1 536 i537 . 1 569 1570 1570 1571 i57i . 1 6o4 i6o5 i6o5 1606 1606 0.1639 1640 1640 i64i i64i 0.1710 1711 1711 1712 1712 0.1746 1746 1747 1748 1748 0.1782 1783 1783 1784 1785 0.1819 1819 1820 1820 1821 0.1855 1 856 1857 1857 i858 . I 5o3 i5o4 i5o4 i5o5 i5o6 0.1538 1 538 1 539 1539 1 540 0.1572 1573 1573 1 574 1 574 0.1607 1607 1608 1609 1609 0.1642 1643 1643 1 644 1644 0.1.677 1678 1678 1679 1680 0.1713 1714 1714 1715 I7i5 0.1749 1749 1750 1751 1751 0.1785 1786 1786 1787 1788 0.1822 1822 1823 1823 1824 0.1859 1859 i860 i860 1861 . I 5o6 i5o7 1 507 i5o8 i5o8 . 1 540 i54i 1 542 1 542 1 543 0.1575 1576 1576 1 577 1 577 0.1610 1610 1611 1612 1612 0.1645 1645 i646 1647 1647 0.1680 1681 1681 1682 1 683 0. 1716 1717 1717 1718 1718 0.1752 1752 1753 1754 1754 0.1788 1789 1789 1790 1791 0.1825 1825 1826 1827 1827 0. 1862 1862 1 863 1 863 1864 0.1 509 1 5 1 i5io i5ii i5ii 0.1543 1 544 1 544 1 545 1 546 0.1578 1578 1 579 i58o i58o o.i6i3 i6i3 i6i4 i6i4 i6i5 0.1648 1 648 1649 i65o i65o 0.1683 1684 1 684 1 685 1686 0.1719 1719 1720 1721 1721 0.1755 1755 1756 ,757 1757 0.1791 1792 1792 1793 1794 0.1828 1828 1829 i83o i83o 0.1 865 1 865 1866 1867 1867 0.l5l2 l5l2 i5i3 i5i4 i5i4 i5i5 0.1 546 1 547 1 547 1 548 1 548 1 549 o.i58i i58i 1 582 1 582 1 583 1 584 0.1616 1616 16.7 1617 1618 1619 o.i65i i65i i652 3652 1 653 i654 0.1686 1687 1687 1688 16S9 1689 0.1722 1722 1723 1724 1724 1725 0.1758 1759 1759 1760 1760 1761 0.1794 1795 1795 1796 1797 1797 o.i83i i83i i832 i833 1 833 i834 0.1868 1868 1S69 1870 1870 1871 7° 7' 7° & 7° 5' 70 4/ 7° 3' 7° 2' 7° 1' 7° 0' 6^59' 6° 58' 1 6° 57', The second correction is to be taken at the bottom if the apparent distance be less than 90^. Page 264] TABLE XLVII. The first correction is always to be taken at the top. The second correction is to be Uken at the top if the apparent distance exceed 90°. o 3° 3' 30 4/ 3° 5' 3° & 3° 7' 3° 8' 3° 9' 3° 10' 3° 11' 3° 12' 3° 13' 60 0.1871 . 1 9f )8 . 1 946 . 1 984 0.2022 0.2061 . 2099 0.2139 0.2178 0.2218 0.2259 I 1871 1909 1946 i9«4 2023 2061 2100 2139 2179 2219 2260 59 2 1872 1909 1947 1985 2023 2062 2101 2i4o 2180 2220 2260 58 J 1873 1910 1948 1986 2024 2062 2101 2l4l 2180 2220 2261 57 4 5 1873 1911 1948 1986 2025 2o63 2102 2l4l 2181 2221 2262 56 55 0.1874 0. 191 1 . 1 949 0.1987 0.2025 0.2064 o.2io3 0.2142 0.2182 0.2222 0.2262 b 1875 1912 1950 1987 2026 2064 2io3 2143 2182 2223 2263 54 7 1875 1913 1950 1988 2026 2o65 2104 2143 2i83 2223 2264 53 8 1876 1913 1951 1989 2027 2066 2io5 2144 2184 2224 2264 52 9 10 1876 i9l4 1951 1989 2028 2066 2io5 2145 2184 2225 2265 5i 5o 0.1877 0.1914 0.1952 0.1990 0.2028 0.2067 . 2 I 06 0.2145 0.2185 2220 0. 2266 II 1878 1915 1953 1991 2029 2068 2107 2i46 2186 2226 2266 49 12 1878 1916 1953 1991 2o3o 2068 2107 2147 2186 2227 2267 48 iJ i«79 1916 1954 1992 2o3o 2069 2108 2147 2187 2227 2268 47 14 i5 1880 1917 1955 1993 2o3l 2070 2109 2148 2188 0.2188 2228 2268 46 45 0.1880 0.1918 0.1955 0. 1993 O.2o32 0.2070 . 2 I 09 0.2149 0.2229 0.2269 lb 1881 1918 1956 1994 2032 2071 2110 2149 2189 2229 2270 44 17 1881 1919 1956 1994 2033 1072 2III 2l5o 2190 2 23o 2270 46 i8 1882 1919 1957 1995 2o33 •072 2II1 2l5l 2190 223l 2271 42 19 20 i883 1920 1958 1996 2o34 '.073 2112 2l5l 2191 223l 2272 4i 4o 0.1883 0.1921 0.1958 . 1 996 o.2o35 0.2073 0.21l3 0.21 52 0.2192 2232 0.2272 21 1884 1921 1959 1997 2o35 2074 2Il3 2i53 2192 2 233 2273 39 22 1 884 1922 i960 1998 2o36 2075 21l4 2i53 2193 2233 2274 38 2j i885 1923 1 960 •1998 2037 2075 21l5 2i54 2194 2234 2274 37 24 2,5 1886 1923 1961 1999 2o37 2076 2Il5 2i55 2194 2235 2275 36 35 0.1886 0. 1924 0. 1962 0.2000 o.2o38 0.2077 0.21 16 o.2i55 0.2195 0.2235 0.2276 2b 1887 1924 1962 2000 2039 2077 2116 2i56 2196 2236 2277 34 27 1S88 1925 1963 2001 2039 2078 2117 • 2i57 2196 2237 2277 66 28 1 888 1926 1963 2001 204o 2079 2118 2 1 57 2197 2237 2278 32 29 3o 1889 1926 1964 2002 204l 2079 2118 2i58 2198 2238 2279 3i 3o 0. 1889 0.1927 0.1965 . 20o3 0.204 1 0.2080 0.2119 0.2159 0.2198 0.2239 0.2279 Ji 1890 1928 1965 20o3 2042 2081 2120 2159 2199 2239 2280 29 Ja 1891 1928 1966 20o4 2042 2081 2X20 2160 2200 2240 2281 28 J:i 1891 1929 1967 20o5 2043 2082 2121 2161 2200 224l 2281 27 34 35 1892 1929 1967 2oo5 2044 2o83 2122 2161 2201 2241 2282 26 25 0.1893 . 1 930 . 1 968 0.2006 . 2o44 o.2o83 0.2122 0.2162 0.2202 0.2242 0.2283 Jb 1893 1931 1968 2007 2045 2084 2123 2i63 2202 2243 2283 24 37 1894 1 93 1 1969 2007 2046 2o85 2124 2i63 2203 2243 2284 23 38 1894 1932 1970 2008 2o46 2o85 2124 2164 2 204 2244 2285 22 J9 4o 1895 1933 1970 2009 2047 2086 2125 21 65 2204 2245 2285 21 20 0. 1896 0.1933 0.1971 0.2009 0.2048 . 2086 0.2126 o.2i65 0.2205 0.2245 0.2286 41 1896 1934 1972 2010 2048 2087 2126 2166 2206 2246 2287 19 42 1897 1934 1972 2010 2049 2088 2127 2167 2206 2247 2287 18 43 1898 1935 1973 201 I 2o5o 2088 2128 2167 2207 2247 2288 17 44 "45 1898 1936 1974 2012 2o5o 2089 2128 2168 220S 2248 2289 16 i5 0.1899 0.1936 0.1974 0.201 2 0.2o5l 0.2090 0.2129 0.2169 0.2208 0.2249 0.2289 4b 1899 1937 1975 20l3 2o52 2090 2l3o 2169 2209 2249 2290 i4 47 1900 1938 1975 20 1 4 2052 2091 2i3o 2170 2210 225o 2291 i3 48 1 90 1 1938 1976 2014 2o53 2092 2l3l 2170 2210 225l 2291 12 49 5o 1901 1939 1977 20 1 5 2o53 2092 2l32 2171 22II 225 I 2292 II 10 0. 1902 0.1939 0.1977 0.2016 o.2o54 0.2093 0.2l32 0.2172 0.22 12 0.2252 0.2293 5i 1903 1940 1978 2016 2o55 2094 2i33 2172 2212 2253 2294 9 52 1903 194 1 1979 2017 2o55 2094 2i34 2173 22l3 2253 2294 8 53 1904 1941 '979 2017 2o56 2095 2i34 2174 22l4 2 2 54 2295 7 54 55 1904 1942 1980 2018 2o57 2096 2i35 2174 22l4 2255 2296 ~5' 0.1905 0.1943 . 1 98 1 0.2019 0.2057 . 2096 0. 2 1 36 0.2 1 75 1 0.22l5 0.2256 0.2296 bb 1906 1943 1981 2019 2o58 2097 2i36 2176 2216 2256 2297 4 f)7 1906 1944 1982 2020 2059 2098 2 1 37 2176 2216 2257 2298 3 58 1907 1944 1982 2021 2059 2098 2i37 2177 2217 2258 2298 2 59 1908 1945 1983 2021 2060 2099 2i38 2178 2218 2258 2299 I 60 1 The 1908 1946 1984 2022 2061 2099 2139 2178 2218 2259 23oo If (P 5(;' 6° 5-^' G° 54' 6° 53' 6° 52' G° 51' G° 50' 6° 49' 6° 48' 6° 47' 6°4G' second correcti on is to be taken at the Iiottom if the apparent distance be less than 90°. TABLE XLVIL [fas^ 205 Tlie ^rsi correction is always to be taken at tlie top. The second correction is to be taken at the top if the apparent distance exceed 90°. // o 3° 14' 3^5' 3° 16' 3° 17' 3° 18' 3° 19' 3° 20' 3° 21' 3® 22' 3° 23' 3° 24' 60 0.23u() 0.2341 0.2382*0.2424 0.2467 0.25lO 0.2553 0.2596 0.2640 o.2()85 0.2730 I 23oO 2342 2383 2425 2467 25lO 2553 2597 2641 2686 2781 5q 2 23oi 2342 2384 2426 2468 25 11 2554 2598 2642 2687 2732 58 3 200:^ 2343 2384 2426 2469 2612 2555 2599 2643 2687 2732 57 4 5 23o2 2344 2385 2427 2470 25l2 2556 2599 2643 2688 2733 56 55 o.23o3 0.2344 0.2386 0.2428 0.2470 o.25i3 0.2556 . 2600 . 2644 0.2689 0.2734 6 23o4 2345 2387 2429 2471 25i4 2557 2601 2645 2689 2735 54 7 23o4 2346 2387 2429 2472 25i5 2558 2601 2646 2690 2735 58 8 23o5 2346 2388 243o 2472 25i5 2559 2602 2646 2691 2736 52 9 10 23061 2347 2389 243 1 2473 25 16 2559 2603 2647 2692 2737 5i 5o 0.2307 0.234s 0.2389 0.2431 0.2474 0.2517 0.2560 0.2604 0.2648 . 2692 0.2788 II 2307 2348 2390 2432 2475 25i7 256i 2604 2649 2693 2788 49 12 23oS 2349 2391 2433 2475 25i8 256i 2 60 5 2649 2694 2789 48 IJ 2309 23 5o 2391 2433 2476 2519 2562 2606 265o 2695 274(J 47 i4 i5 2309 235o 2392 2434 2477 2520 2563 2607 265 1 2695 2741 46 45 0.23lO o.235i 0.2393 0.2435 0.2477 0.2520 0.2564 . 2607 0.2652 . 269C 0.2741 i6 23ll 2352 2394 2436 2478 2521 2564 2608 2652 2697 2742 A^ 17 23ll 2353 2394 2436 2479 2522 2565 2609 2653 2698 2743 43 18 23l2 2353 2395 2437 2480 2522 2566 2610 2654 2698 2744 42 19 20 23i3 2354 2396 2438 2480 2523 2566 2610 2655 2699 2744 41 40 o.23i3 0.2355 0.2396 0.2438 0.2481 0.2524 0.2567 0.2611 0.2655 0.2700 0.2745 21 23 1 4 2355 2397 2439 2482 2525 2568 2612 2656 2701 2746 3q 22 23i5 2356 239S 2440 2482 2525 2569 2612 2657 2701 2747 38 23 23i5 2357 2398 2441 2483 2526 2569 2613 2657 2702 2747 37 25 23i6 235- 2399 2441 2484 2527 2570 2614 2658 2703 2748 36 35 o.23i7 0.2358 . 2400 0.2442 0.2485 0.2527 0.2571 0.2615 0.2659 0.2704 0.2749 26 23i7 2359 2401 2443 2485 2528 2572 . 261 5 2660 2704 2750 34 27 23[8 2359 2401 2443 2486 2529 2572 2616 2660 2705 2750 Zi 28 23i9 236o 2402 -^-iU 2487 253o 2573 2617 2661 2706 2751 32 29 3o 2320 236i 24o3 244;" 2487 253o 2574 2618 2662 2707 2752 3i ■3^7 0.2320 0.2362 o.24u3 0.2445 0.2488 0.253 1 0.2574 0.2618 . 2663 0.2707 0.2753 3i 2321 2362 24o4 2446 2489 2532 2575 2619 2663 2708 2753 29 3a 2322 2363 24o5 2447 24S9 2533 2576 2620 2664 2709 2754 28 33 2322 23()4 24o5 2448 2490 2533 2577 2621 2665 2710 2755 27 34 35 2323 2364 2406 2448 2491 25?4 2577 2621 2666 2710 2756 26 25 0.2324 0.2365 0.2407 0.2449 . 2492 0.2535 0.2578 0.2622 0.2666 0.271 1 0.2756 36 2394 2366 2408 245o 2492 2535 2579 2623 2667 2712 2757 24 37 2325 2366 2408 245o 2493 2536 258o 2624 2668 2713 2758 23 38 2326 2367 2409 245 1 2494 2537 2 58o 2624 2669 2713 2750 22 39 4o 2326 2368 2410 2452 2494 2538 258i 2625 2669 2714 2760 0.2760 21 20 0.2327 0.2368 0.2410 0.2453 0.2495 0.2538 0.2582 0.2626 0.2670 0.2715 4i 2328 2369 24 1 1 2453 2496 2539 2583 2626 2671 2716 2761 19 42 232S 2370 2412 2454 2497 2540 2583 2627 2672 2716 2762 18 43 2329 2371 2412 2455 2497 2540 2584 2628 2672 2717 2768 17 44 45 233o 2371 24i3 2455 2498 254 1 2585 2629 2673 2718 2768 16 i5 o.233( 0.2372 0.2414 0.2456 . 2499 0.2542 0.2585 0.2629 0.2674 0.2719 0.2764 4b 233 1 2373 24i5 2457 2499 2543 2586 263o 2675 2719 2765 i4 47 2332 2373 24i5 2458 2 5 00 2543 2587 263 1 2675 2790 2766 i3 48 2333 2374 2416 2458 25oi 2544 2588 2682 2676 2721 2766 12 49 5o 9333 2375 2417 2459 25o2 2545 2588 2632 2677 2722 2767 11 10 0.2334 0.2375 0.2417 . 2460 0.2502 0.2545 ^0.2589 0.2633 0.2678 0.2'722 0.2768 5i 2335 2376 2418 2460 2 5o3 2546 2590 2634 2678 2723 2769 9 52 2335 2377 2419 2461 25o4 2547 2591 2635 2679 2724 2769 8 53 2336 2378 2419 2462 2 5o4 2548 2591 2635 2680 2725 2770 7 54 55 2337 2378 2420 2462 25o5 o.25o6 2548 2592 2636 2681 2725 2771 6 5 0.2337 0.2379 0.2421 0.2463 0.2549 0.2593 0.263-' 0.2681 0.2726 0.2772 5b 2338 238o 2422 2464 25o7 255o 2593 2638 2682 2727 2772 4 ^7 2339 238o 2422 2465 2507 255i 2594 2638 2683 2728 2778 3 58 233y 238i 2423 2465 25o8 255i 2595 2639 2684 2729 2774 2 59 2340 2382 2424 2466 2509 2552 2596 2640 2684 2729 2775 I bo 2341 2382 2424 2467 25io 2553 2596 2640 2685 273(< 2775 6° 45' 6° 44' 6° 43' 6° 42' 6° 41' G°40' 6° 39' 6° 38' 6° 37' 6°3G' 6° 35' Th€ sccund correct on is to be taken at the bottom if the apparent distance be less than 90°. 34 Page 266] TABLE XLVII. The_^r5J correction is always to be taken at the to-p. The second correction is to be taken at the top if the apparent distance exceed 90°. // o 3° 25' 3° 26' 3° 27' 3° 28' 3° 29' 3° 30' 3° 31' 3° 32' 3° 33' 3° 34' 3° 35' 60 0.2775 0.2821 0.286S 0.2915 0.2962 o.3oio o.3o59 0.8108 o.3i58 0.3208 0.8259 I 2776 2822 2S69 2916 2963 3oii 3o6o 3109 3x58 8209 8259 59 2 2777 2823 2869 2916 2964 3oi2 3 060 3iio 8x59 3209 8260 58 3 2778 2824 2870 2917 2965 3oi3 3o6i 3iio 3i6o 3210 326X 57 4 5 . 2779 2825 2871 2918 2965 3oi4 8062 3iii 3i6x 8211 8262 0.3263 56 55 0.2779 2780 0.2825 0.2872 0.2919 . 2966 o.3oi4 o.3o63 0.3ll2 0.8162 0.8212 6 2826 2873 2920 2967 3oi5 3o64 3ii3 3i63 32i3 8264 54 7 2781 2827 2873 2920 2968 3oi6 3o65 3ii4 3x63 8214 3265 53 8 2782 2828 2874 2921 2969 3oi7 3o65 3ii4 3i64 8214 8265 52 9 10 2782 2828 2875 2922 296^ ^ 3oi8 3o66 8ii5 3i65 32x5 3266 5i 5o 0.2783 0.2829 0.2876 0.2923 0.2970 o.3oi8 0.3067 o.3ii6 o.3i66 0.8216 0.8267 II 2784 283o 2876 2924 2971 3019 3o68 8117 8167 8217 8268 49 12 2785 283 1 2877 2924 2972 3020 8069 3ii8 3x68 3218 8269 4^ 13 2785 283 1 2878 2925 2973 302I 3069 3119 3x68 8219 8270 47 i4 1 5 2786 2832 2879 2926 2973 3022 3070 8119 8x69 8220 8270 46 45 0.2787 0.2833 0.2880 0.2927 0.2974 0.3o2 2 0.8071 0.3l20 0.8x70 0.8220 0.327X Tfi 2788 2834 2880 2927 2975 3o23 3072 3l2I 8171 8221 8272 4A 17 2788 2835 2881 2928 2976 3o24 3073 3l22 8x72 8222 8278 43 t8 2789 2835 2882 2929 2977 3o25 8078 3i23 8x78 8228 3274 42 19 20 2790 2836 2883 2930 2977 3026 8074 3i24 8x78 3224 8275 4i 4o 0.2791 0.2837 0.2883 0.2931 0.2978 0.8026 0.8075 0.3 1 24 0.8x74 0.8225 0.8276 21 2792 2838 2884 2931 2979 8027 3076 3i25 8x7b 8225 8276 39 22 2792 2793 2838 2885 2932 2980 3028 3077 8126 8x76 8226 8277 88 23 2839 2886 2933 2981 8029 3078 3i27 8177 3227 8278 il 24 25 2794 2840 2887 2934 2981 3o3o 3078 8128 8x78 8228 8279 3b 85 0.2795 0.2841 0.2887 0.2935 0.2982 . 3o3o o.3o79 0.3129 0.8x78 0.3229 0.3280 26 2795 2842 2888 2935 2983 3o3i 3o8o 8129 8179 8280 8281 84 27 2796 2842 2889 2936 2984 3o32 3o8i 3i3o 3x8o 823x 3282 66 28 2797 2843 2890 2937 2985 3o33 3082 3i3i 8x81 8281 , 82S2 32 29 3o 2798 2844 2891 0.2891 2938 2985 3o34 3082 8i32 8x82 0.3x83 3232J 3283 3i 3o . 2798 0.2845 0.2939 0.2986 o.3o34 o.3o83 o.3i33 0.8233 0.8284 3 1 2799 2845 2892 2939 2987 3o35 3o84 3i33 3x83 8284 3285 29 32 2S00 2846 2893 2940 2988 3o36 3o85 3x34 3x84 3235 3286 28 33 2801 2847 2894 2941 2989 3o37 3o86 3x35 3x85 3236 8287 27 34 3-^ 2801 2848 2894 2942 2989 3o38 8087 3x36 3x86 0.8x87 8236 3288 2b 25 0.2802 0.2848 0.2895 0.2942 0.2990 0.8089 0.8087 0.3 1 37 0.8287 0.8288 3fi 2803 2849 2896 2943 2991 3089 3o88 3x38 8x88 8238 8289 24 37 2804 285o 2897 2944 2992 3o4o 8089 3x38 3x88 8289 3290 28 38 2805 285i 2898 2945 2993 3o4) 8090 3x39 3x89 8240 8291 22 39 40 2805 2852 2898 2946 2993 3o42 8091 3i4o 8x90 8241 8292 21 20 0.2806 0.2852 0.2899 0.2946 0.2994 o.3o43 0.8091 o.3x4i 0.8x91 0.8242 0.8298 4t 2807 2853 2900 2947 2995 3o43 8092 3x42 8x92 3242 3294 19 42 2808 2854 2901 2948 2996 3o44 8098 3x43 8x93 8243 8294 18 43 2808 2855 2901 2949 2997 3o45 3094 8143 8x93 8244 829b 17 2809 2855 2902 2950 2997 3o46 8095 0.8096 ^iM 8194 8245 3296 16 i5 0.2810 0.2856 0.2908 0.2950 0.2998 o.3o47 c.3x45 0.8x95 0.8246 0.8297 40 2811 2857 2904 2951 2999 3o47 8096 3x46 8x96 3247 8298 14 47 2811 2858 2905 2952 3ooo 3o48 8097 3x47 8x97 8247 3299 i3 48 2812 2859 2905 2953 3ooi 3o49 8098 3x48 8x98 3248 33oo 12 49 5o 2813 2859 2906 2954 3ooi 3o5o 8099 3x48 8x98 . 3 X 99 8249 o.325o 3 3 00 o.33ox II 10 0.2814 0.2860 0.2907 0.2954 o.3oo2 3o5i 0.3 IOC 0.3 1 49 'ii 28i5 2861 2908 2955 3oo3 3o5? 3ioi 3x5o 8200 825i 33o2 9 'io 2815 2862 2909 2956 3oo4 3o52 3ioi 8x5i 3201 3252 33o3 8 ^■^ 2816 2862 2909 2957 3oo5 3o53 3l02 3i52 8202 3253 33o4 7 54 2817 2863 2910 2958 3oo5 3o54 3io3 3x53 o.3i53 32o3 8253; 33o5 6 5 G.2818I0.2864 0.2911 0.2958 o.3oo6 o.3o55 . 3 1 04 0.8204 0.3254 o.33o6 'ifi 2818 2865 2912 2959 3007 3o56 8io5 3x54 32o4 3255 33u6 4 ^^7 2819 2866 2912 2960 3oo8 3o56 3io5 3x55 32o5 3256 3307 3 58 2820 2866 2913 2961 3009 3o57 3 1 06 3x56 8206 8257 33o8 2 5o 2821 2867 2914 2962 3009 3o58 8107 3x57 8207 3258 3309 I ^ 2821 2868 2915 2962 3oio 3o59 3io8 3x58 8208 8259 33x0 6° 34' G°33' 6° 32' 6° 31' 6° 30' G°29' 6° 28' C°27' (3°2G' 6° 25' G°24' Th e second correction is to be taken at the bottom if the apparent distance be less than 90°. —■'— TABLE XL VII. t^^s^^oy The first correction is always to be taken at the top. The second correction is to be taken at the top if the apparent distance exceed 90°. o 3=36' 3° 37' 3° 38' 3° 39' 3° 40' 3° 41' 3° 42' 3=43' 3° 44' 0.3745 3° 45' 3° 46' 60 o.33io 0.3362 o.34i5 0.3468 0.3522 0.3576 0.3632 0.3688 o.38o2 . 386o I 33ii 3363 34 1 5 3469 3523 3577 3633 3689 3746 38o3 386 1 59 2 33i2 3364 3416 3470 3524 3578 3634 3690 3746 38o4 3862 58 3 33i3 3365 3417 3471 3525 357Q 3635 3691 3747 3So5 3863 i)7 4 5 33i3 3365 3418 3471 3525 358o 3635 0.3636 3692 0.3693 3748 38o6 3864 56 "5"5" o.33i4 0.3366 0.3419 0.3472 0.3526 0.3581 0.8749 0.8807 0.3865 6 33i5 3367 3420 3473 3527 3582 3637 3693 3750 38o8 3866 54 7 33 16 3368 3421 3474 3528 3583 3638 3694 3751 3809 0867 53 8 33i7 3369 3422 3475 3529 3584 3639 3695 3752 38 10 3868 52 9 10 33i8 3370 3423 3476 353o 3585 364o 3696 3753 38ii 8869 01 "5^ 0.3319 0.3371 0.3423 0.3477 o.353i 0.3586 0.364 1 0.3697 0.3754 0.3S12 0.3870 II 3319 3372 3424 3478 3532 3587 3642 3698 3755 38i3 3871 49 12 3320 3372 3425 3479 3533 3587 3643 3699 3756 38i4 3872 48 i3 3321 3373 3426 3480 3534 3588 3644 3700 3757 38i5 8878 47 i4 i5 3322 3374 3427 3480 0.3481 3535 3589 3645 3701 3758 0.8759 38 16 3874 46 45 0.3323 0.3375 0.3428 0.3535 0.3590 0.3646 0.3702 0.8817 0.3875 i6 3324 3376 3429 3482 3536 3591 3647 3703 8760 38i8 8876 44 17 3325 3377 343o 3483 3537 3592 3648 3704 3761 3819 3877 4i i8 3325 3378 343 1 3484 3538 3593 3649 3705 3762 8820 3878 42 19 20 3326 3379 343 1 3485 3539 3594 3649 3706 8768 0.8764 3820 3879 . 388o 4i 4o 0.3327 0.3379 0.3432 0.3486 0.3540 0.3595 o.365o 0.8707 0.3821 21 3328 338o 3433 3487 3541 3596 365 1 3708 3765 8822 388 1 39 22 3329 338i 3434 3488 3542 3597 3652 3709 3766 8828 3882 38 23 333o 3382 3435 3488 3543 3598 3653 3709 3767 3824 3883 87 24 25 333i 3383 3436 3489 3544 3598 3654 8710 8768 3825 3834 36 "35" 0.3332 0.3384 0.3437 . 3490 0.3545 0.3599 0.3655 0.871 1 0.8768 0.8826 0.38S5 26 3332 3385 3438 3491 3545 36oo 3656 8712 8769 8827 3886 34 27 3333 3386 3438 3492 3546 36oi 3657 3713 8770 8828 38S7 6i 28 3334 3386 3439 3493 3547 36o2 3658 3714 8771 3829 3888 62 29 3o 3335 3387 3440 3494 3548 36o3 3659 3715 8772 383o 3889 3i 3o 0.3336 0.3388 0.3441 0.3495 0.3549 . 36o4 o.366o 0.3716 0.8778 0.3831 0.3890 3i 3337 3389 3442 3496 355o 36o5 366 1 3717 3774 3832 8891 29 32 3338 3390 3443 3497 355i 36o6 3662 3718 3775 3833 3892 28 33 3338 3391 3444 3497 3552 3807 3663 3719 3776 3834 3S93 27 34 35 3339 3392 3445 3498 3553 36o8 3663 8720 3777 3835 8894 26 25 0.3340 0.3393 0.3446 0.3499 0.3554 0.3609 . 3664 0.3721 0.8778 0.3S36 . 3898 36 3341 3393 3446 35oo 3555 36io 3665 8722 3779 3837 3896 24 37 3342 3394 3447 35oi 3555 36io 3666 3723 8780 3838 8897 23 38 3343 3395 3448 35o2 3556 36ii 3667 3724 3781 3839 8898 22 39 40 3344 3396 3449 35o3 3557 36i2 3668 3725 8782 3840 8899 21 20 0.3345 o.33q7 0.3450 o.35o4 0.3558 o.36i3 . 3669 0.8726 0.3788 0.3841 0.8900 4i 3345 3398 345i 35o5 3559 36i4 3670 3727 3784 3842 8901 19 42 3346 3399 3452 35o6 356o 36i5 3671 3727 3785 3843 8902 18 Ai 3347 3400 3453 35o6 356i 36i6 3672 3728 8786 3844 8903 17 44 '45 3348 3400 3454 3507 3562 0.3563 36 17 3673 3729 3787 0.3788 3845 8904 0.3905 16 l5 0.3349 0.3401 0.3454 o.35o8 o.36i8 0.36740.3730 0.3846 46 33 5o 3402 3455 3509 3564 3619 3675 ;?73i 3789 3847 89(16 l4 47 335i 34o3 3456 35io 3565 3620 36-6 J732 8790 3848 8907 i3 48 335i 3404 3457 35ii 3565 3621 36-7 3733 8791 3849 8908 12 49 5o 3352 34o5 3458 35i2 3566 3622 3677 3734 8792 385<. 8909 II 10 0.3353 0.3406 0.3459 o.35i3 0.3567 0.3623 0.3678,0.3735 0.8792 o.385i 0.3910 5i 3354 3407 3460 35i4 3568 3623 3679! 3736 3793 3852 391 1 9 52 3355 3408 3461 35i5 3569 3624 368o 3737 3794 3853 8912 8 53 3356 3408 3462 35 1 5 3570 3625 368 1 8788 3795 3854 8913 7 54 55 3357 3409 3463 35i6 0.3517 3571 3626 3682 3739 3796 3855 3914 6 5 0.3358 0.3410 0.3463 0.3572 0.3627 0.3683 0.3740 0.8797 0.3856 0.8915 56 3358 3411 3464 35i8 3573 3628 3684 3741 8798 3856 8916 4 57 3359 3412 3465 3519 3574 3629 3685 3742 8799 3857 8917 3 58 336o 34i3 3466 3520 3575 363o 3686 37.43 38oo 3858 3918 2 59 336i 34i4 3467 3521 3576 363 1 3687 3744 38oi 3859 3919 I 60 336-i 341 5 3468 3522 3576 3632 3688 3745 38o2 386c 3919 6° 23' G°22' 6° 21' 6=20' G°19' 6° 18' 6° 17' 6° 16' 6° 15' 6° 14' 6°i:y II Th e second correct ion is to be taken at the lottom if the apparent distance be less than 4844 4918 4994 5071 5i49 5329 53io 5393 5477 6^ 1' 6° 0' 5° 59' 5=58' 5° 57' 5= ^0 5° 55' j 5° 54' 5° 53' 5° 52' 5° 51' II Th e srcoml correction is to be taken at the hottmn if the apparent distance be less than 90". Page 270] TABLE XLVII. The Jir St correction is always to be taken at the top. The second correction is to be taken at the top if the apparent distance exceed 90°. n 40 Q, 0.5477 4° 10' 0.5563 4° 11' 4° 12' 4° 13' 4° 14' 4° 15' 4° 16' 4° 17' 4° 18' 4° 19' "60 0.565I 0.5740 0.5832 0.5925 0.602X 0.6118 0.6218 0.6320 0.6425 I 5478 5564 5652 5742 5833 5927 6022 6120 6220 6822 6427 59 2 54S0 5566 5654 5743 5835 5928 6024 6x21 6221 6324 6428 56 3 5481 5567 5655 5745 5836 5930 6025 6x23 6223 6325 643o 57 4 5 5483 5569 5657 5746 5838 0.5839 5931 6027 6x25 6225 6827 6432 56 55 0.5484 0.5570 0.5658 0.5748 0.5933 0.6029 0.6126 0.6226 0.6829 0.6434 6 5486 5572 566o 5749 584 1 5935 6o3o 612S 6228 633i 6435 54 7 5487 5573 566 1 5751 5843 5936 6o32 6x3o 6280 6332 6437 53 8 5488 5575 5663 5752 5844 5988 6o33 6i3i 6282 6334 6439 52 9 10 5490 5576 5664 5754 5846 5939 6o35 6i33 6233 6336 644 1 5i 0.5 '191 0.5578 0.5666 0.5755 0.5847 0.594X 0.6037 o.6x35 0.6235 0.6338 0.6443 II *■ 5579 5667 5757 5849 5942 60 38 6x36 6287 6889 6444 49 12 5494 558o 5669 5758 585o 5944 6o4o 6x38 6288 634 1 6446 48 i3 5496 5582 5670 5760 5852 5946 6042 6i4o 6240 6343 6448 47 i4 i5 5497 5583 5671 5761 5853 5947 6043 6x4i 6242 6344 645o 46 45 . 5498 0.5585 0.5673 0.5763 0.5855 0.5949 0.6045 0.6x43 0.6243 0.6346 o.645x i6 55oo 5586 5674 5765 5856 5950 6o46 6x45 6245 6348 6453 44 17 55oi 5588 5676 5766 5858 5952 6o48 6x46 6247 635o 6455 43 18 55o3 5589 5677 5768 586o 5954 6o5o 6x48 6248 635i 6457 42 19 20 55o4 5591 5679 5769 586i 5955 6o5i 6i5o 6250 6353 6459 4r 40 o.55o6 0.5592 0.5680 0.5771 0.5863 0.5957 o.6o53 o.6i5i 0.6252 0.6355 0.6460 21 5507 5594 5682 5772 5864 5958 6o55 6i53 6254 6357 6462 39 22 55oS 5595 5683 5774 5866 5960 6o56 6x55 6255 6358 6464 38 23 55 10 5596 5685 5775 5867 5961 6o58 6x56 6257 636o 6466 37 24 25 55ii 5598 5686 5777 5869 5963 6059 6x58 6259 6362 6467 36 35 o.55i3 0.5599 0.5688 0.5778 0.58700.5965 . 606 1 0.6160 0.62600.6864 0.6469 26 55i4 56oi 5689 5780 5872 5966 6o63 6161 6262 6365 6471 34 27 55i6 56o2 569! 5781 5874 5968 6064 6x63 6264 6867 6473 33 28 55i7 56o4 5692 5783 5875 5969 6066 6i65 6265 6869 6475 32 29 3o 55i8 56o5 5694 5784 5877 597 X 6067 6166 6267 6871 6476 3i o.55co . 5607 0.56950.5786 0.5878 0.5973 0.6069 0.6168 0.6269 0.6872 0.647S 3i 5521 56o8 5697 5787 588o 5974 6071 6169 6271 6374 6480 29 32 5523 56io 5698 5789 588 1 5976 6072 6171 6272 6876 6482 28 33 5524 56x1 5700 5790 5883 5977 ■6074 6x73 6274 6377 6484 27 34 35 5526 56i3 5701 5792 5884 5979 6076 6x74 6276 6379 6485 26 25 0.5527 o.56i4 0.5703I0.5793 0.5886 0.5981 0.6077 0.6176 0.62770.6381 0.6487 36 5528 56x5 5704 5795 5888 5982 6079 6178 6279 6383 6489 24 37 553o 56i7 5706 5796 5889 5984 608 X 6179 6281 6384 6491 23 38 553i 56x8 5707 5798 5891 5985 6082 6181 6282 6386 6492 2 2 39 40 5533 5620 5709 58oo 5892 5987 6084 0.6085 6x83 6284 6388 6494 21 20 0.5534 0.562 1 0.5710 . 58o t . 5894 . 5989 o.6i85 0.6286 0.6890 0.6496 4i 5536 5623 57X2 58o3 5895 5990 - 6087 6186 6288 639X 6498 19 42 5537 5624 57x3 58o4 5897 5992 6089 6x88 6289 6893 65oo x8 43 5538 5626 57x5 58o6 5898 5993 6090 6190 6291 6895 65oi 17 44 45 5540 5627 57x6 5807 5900 5995 6092 6191 6293 6897 65o3 x6 i5 0.5541 0.5629 0.5718 . 5809 0.5902 0.5997 . 6094 0.6193 0.6294 0.6898 o.65o5 46 5543 563o 5719 58x0 5903 5998 6095 6x95 6296 64oo 65o7 14 47 5544 5632 5721 58x2 5905 6000 6097 6x96 6298 64o2 65o9 x3 48 5546 5633 5722 58x3 5906 600 X 600Q 6198 63oo 64o4 65io 12 49 5o 5547 5635 5724 58x5 5908 6oo3 6100 6200 63ox o.63o3 64o6 65x2 XX 10 0.5549 . 5636 0.5725 o.58i6 . 5909 o.6oo5 c.6xo2 0.620X 0.6407 o.65x4 5 1 555o 5637 5727 58x8 591 X 6006 6io3 6208 63o5 6409 65x6 9 52 555i 5639 5728 58x9 59x3 6008 6io5 6205 63o6 64ix 65x8 8 53 5553 564o 5730 5821 5914 6009 6x07 6206 63o8 64x3 65x9 7 54 55 5554 0.5556 5642 573 1 5823 5916 60XX 6x08 6208 63x0 0.63x2 64x4 6521 b 5 0.5643 0.5733 0.5824 0.59x7 o.6oi3 0.61x0 0.6210 0.6416 0.6523 56 5557 5645 5734 5826 59x9 6ox4 6112 62x1 63i3 64i8 6525 4 57 5559^ 5646 5736 5327 5920 6016 6ii3 62x3 63i5 6420 6527 3 58 55601 5648 5737 5829 5922 6017 61x5 62x5 63x7 642 X 6529 2 59 5562 5649 5739 583o 5924 60x9 6x17 6216 63 1 9 6423 653o I 60 5563 565 r 574c 5832 5925 602 X 6118 6218 6320 642 5 6532 II 5° 50' 1 5° 49' 5° 48' 5° 47' 5= 46' 5° 45' 5° 44' 5° 43' 5° 42' 5° 41' 5^40' Th e second correc ion is t be tak sn at th e bottom if the apparen t distan ce be Ic 5S than 30°. TABLE XLVII. [Pas. ^71 The first correction is always to be taken at the top. The second correction ig to be talien at the top if the apparent dista nee exceed 90°. o 4° 20' 4° 21' 4° 22' 4^23' 4° 24' 4° 25' 4° 20' 4° 27' 4° 28' 4° 2[y ^^9 60 0.6532 0.6642 0.6755 0.6871 0.6990 0.7112 0.7238 0.7368 0.7501 I 6534 6644 6757 6873 6992 7114 7240 7370 75o3 7641 59 2 6536 6646 6759 6875 6994 7116 7242 7372 7506 7G44 58 3 6538 6648 6761 6877 6996 7118 7244 7374 7508 7646 57 4 5 6539 665o 6763 6879 6998 7120 7246 7376 75x0 7648 56 55 0.6541 0.665I 0.6764 0.6881 . 7000 0.7122 0.7249 0.7379 o.75i3 0.7651 6 6543 6653 6766 688 2 7002 7124 725x 738 X 75x5 7653 54 7 6545 6655 6768 6884 7004 7127 7253 7383 75x7 7655 53 8 6547 6657 6770 6886 7006 7129 7255 7385 7519 7658 52 _9 10 6548 6659 6772 6888 7008 7i3i 7257 7387 7522 7660 5x 5^ 0.6550 0.6661 0.6774 0.6890 0.7010 0.7x33 0.7259 0.7390 0.7524 0.7663 II 6552 6663 6776 6892 7012 7x35 726X 7392 7526 7665 49 12 6554 6664 677S 6894 7014 7137 7264 7394 7528 7667 48 i3 6556 6666 6780 6896 7016 7139 7266 7396 753i 7670 47 i4 i5 6558 6668 6782 6898 7018 7141 7268 7398 7533 7672 46 45 0.6559 0.6670 0.6784 . 6900 0.7020 0.7143 0.7270 0.740X 0.7535 0.7674 i6 656 1 6672 6785 6902 7022 7145 7272 74o3 7538 7677 AA 17 6563 6674 6787 6904 7024 7147 7274 74o5 7540 7679 43 i8 6565 6676 6789 6906 7026 7149 7276 7407 7542 7681 42 !9 20 6567 6677 6791 6908 7028 7x52 7279 7409 7544 7G84 4i 4o 0.6568 0.6679 0.6793 0.6910 o.7o3o . 7 1 54 0.7281 0.7412 0.7547 0.7686 21 6570 6681 6795 6912 7032 7x56 7283 74x4 7549 7688 39 22 6572 6683 6797 6914 7034 7x58 7285 7416 755x 7691 38 23 6574 6685 6799 6916 7o36 7x60 7287 74i8 7554 7693 37 24 25 6576 6687 6801 6918 7o38 7162 7289 7421 7556 7696 36 35 0.6578 0.6689 o.68o3 0.6920 . 7040 0.7x64 0.729X 0.7423 0.7558 . 7698 26 6579 6691 68o5 6922 7042 7x66 7294 7425 7560 7700 'M 27 658 1 6692 6807 6924 7044 7168 7296 7427 7563 7703 '66 2b 6583 6694 6809 6926 7046 7x70 7298 7429 7565 7705 32 29 3o 6585 6696 "0T6698" 6810 6928 7048 7172 73oo 7432 7567 7707 3i 3o 0.6587 0.6812 0.6930 0.7050 0.7175 0.7302 0.7434 0.7570 0.7710 3i 6589 6700 68x4 6932 7o52 7177 73o4 7436 7572 77x2 29 32 6590 6702 68 1 6 6934 7o55 7179 7307 7438 7574 77x4 28 33 6592 6704 6818 6936 7057 7x8x 7309 7441 7577 7717 27 35 6594 6706 6820 6938 7059 7x83 73 XX 7443 7579 7719 26 25 0.6596 0.6708 0.6822 . 6940 0.7061 0.7x85 o.73i3 0.7445 0.7581 0.7722 36 6598 6709 6824 6942 7063 7x87 73x5 7447 7583 7724 24 3- 6600 6711 6826 6944 7065 7x89 73x7 745o 7586 7726 23 3S 6601 6713 6828 6946 7067 7191 7320 7452 7588 7729 22 09 40 66o3 6715 683o 6948 7069 7x93 7322 7454 7590 773 1 2X 20 o.66o5 0.6717 0.6832 0.6950 0.7071 0.7x96 0.7324 0.7456 0.7593 0.7734 4i 6607 6719 6834 6952 7073 7198 7326 7458 7595 7736 19 42 6609 6721 6836 6954 7075 7200 7328 7461 7597 7738 x8 43 661 r 6723 6838 6956 7077 7202 7330 7463 7600 774 1 17 45 6612 6725 684o 6953 7079 7204 7333 7465 7602 7743 x6 75 0.6614 0.6726 0.6841 0.6960 0.7081 0.7206 0.7335 0.7467 . 7604 0.7745 46 6616 6728 6843 6962 7083 7208 7337 7470 7607 7748 i4 47 6618 6730 6845 6964 7085 7210 7339 7472 7609 775o i3 48 6620 6732 6847 6966 7087 7212 7341 7474 761 1 7753 X2 49 5o 6622 6734 6849 6968 7089 72x5 7344 7476 76r3 775:) _ IX 10 0.6624 0.6736 0.685I 0.6970 0.7091 0.72x7 0.7346 0.7479 . 76 1 6 0.7757 5i 6625 6738 6853 6972 7093 7219 7348 748 X 76x8 7760 9 52 6627 6740 6855 6974 7096 722X 7350 7483 7620 7762 8 53 6629 6742 6857 6976 7098 7223 7352 7485 7623 7765 7 54 55 663 1 6743 6859 6978 7100 7225 7354 7488 7625 7767 6 5 0.6633 0.6745 0.6861 0.6980 0.7102 0.7227 0.7357 . 7490 0.7627 0.7769 56 6635 6747 6863 6982 7104 7229 7359 7492 7630 7772 4 i)7 6637 6749 6865 6984 7106 7232 736i 7494 7632 7774 d 58 6638 6751 6867 6986 7108 7234 7363 7497. 7634 7777 2 59 6640 6753 6869 6988 7110 7236 7365 7499 7637 7779 I 60 6642 6755 6871 6990 7112 7238 5=34' 7368 7D0X 7639 7782 5° 39' 5° 38' 5° 37' S^' 36' 5° 35' 5° 33' 5° 32' 5= 31' 5= 30' T he seconi I correct) on is to 38 taken at the bottom if tl le apparexit distai ice be la s than 90°. Page a7-2] TABLE XLVII. The first correction is always to be taken at the top. The second correction is to be taken at tlie top if the apparent distance exceed 90°. // o 4° 30' 4° 31' 4= 32' 4° 33' 4° 34' 4° 35' 4° 36' 0.8751 4° 37' 0.8935 4° 38' 4° 39' 60 0.7782 0.7929 0.8081 0.8239 o.84o3 0.8573 0.9128 0.9381 I 7784 7931 8084 8242 84o6 8576 8754 8939 9182 9334 5q 2 7786 7934 8086 8244 8409 8579 8757 8942 9135 9887 58 3 77S9 7936 8089 8247 84ii 8582 8760 8945 9i38 9341 57 4 5 7791 7939 0.7941 8091 . 8094 8250 84i4 8585 8763 8948 9142 9344 56 55 0.7794 0.8253 0.8417 0.8588 0.8766 0.8951 0.9145 0.9348 b 7796 7944 8097 8255 8420 8591 8769 8954 9148 985 1 54 7 -^798 7946 8099 8258 8423 8594 8772 8958 9152 9355 53 « 7S01 7949 8102 8261 8425 8597 8775 8961 9155 9358 52 _9 10 7803 795 1 8io4 8263 8428 8599 8778 8964 9i58 9862 5i 5o . 7806 o.79?4 0.8107 0.8266 o.843i 0.8602 0.8781 0.8967 0.9162 0.9865 1 1 7808 7956 8110 8269 8434 86o5 8784 8970 9165 9869 4q 12 7811 7959 8112 8271 8437 8608 8787 8973 9168 9872 48 ]J 7813 7961 8ii5 8274 8439 861 1 8790 8977 9171 9376 47 i4 i5 7815 7964 8117 8277 8442 86i4 8793 8980 9175 9379 46 45 0.7818 . 7966 0.8120 0.8279 0.8445 0.8617 0.8796 0.8983 0.9178 0.9088 lO 7820 7969 8123 8282 8448 8620 8799 8986 9181 9386 44 17 7823 7971 8125 8285 845 1 8623 8802 8989 9185 9890 4i i8 7S25 7974 8128 8288 8453 8626 88o5 . 8992 9188 9898 42 ■9 2U 7S2S 7976 8i3i 8290 8456 8629 8808 8996 9191 0.9195 9897 4i 40 0.7830 0.7979 o.8i33 0.8293 0.8459 0.8632 0.8811 . 8999 . 9400 21 7832 7981 8i36 8296 8462 8635 88i4 9002 9198 9404 39 22 7835 7984 8i38 8298 8465 8637 8817 9005 9201 9407 38 23 7837 7987 8i4i 83oi 8467 864o 8821 9008 9205 941 1 ^7 24 25 7840 0.7842 7989 8144 83o4 8470 8643 8824 9012 9208 94 i 4 36 35 0.7992 0.8146 o.83o7 0.8473 0.8646 0.8827 0.9015 0.9212 . 94 1 8 26 7845 7994 8149 83o9 8476 8649 883o 9018 9275 9421 84 27 7847 7997 8i52 83i2 8479 8652 8833 9021 9218 9425 66 .a8 78 5o 7999 8i54 83i5 8482 8655 8836 9024 9222 9428 32 29 3o 7S52 8002 8i57 83i8 8484 8658 8839 9028 9225 9432 81 3o 0.7855 0.8004 0.8159 0.8320 0.8487 0.8661 0.8842 0.9031 0.9228 0.9435 3i 7857 8007 8162 8323 8490 8664 8845 9034 9282 9439 29 J 2 7859 8009 8i65 8326 8493 8667 8848 9037 9235 9442 28 33 7862 8012 8167 8328 8496 8670 885i 9041 9238 9446 27 34 35 7864 8014 8170 0.8173 833i 8499 8673 8854 9044 9242 0.9245 9449 0.9453 2b l5 0.7867 0.8017 0.8334 o.85o2 0.8676 0.8857 0.9047 36 7869 8020 8175 8337 85o4 8679 8861 90 5o 9249 9456 24 37 7S72 8022 8178 8339 85o7 8682 8864 9053 9252 9460 23 38 7874 8025 8181 8342 85io 8685 8867 9057 9255 9464 22 39 4o 7877 8027 8i83 8345 0.8348 85i3 8688 8870 9060 9259 9467 21 20 0.7879 o.8o3o 0.8186 o.85i6 0.S691 0.8873 . 9063 0.9262 0.947' 4i 7882 8o32 8188 835o 85i9 8694 8876 9066 9266 9474 '9 42 7S84 8o35 8191 8353 8522 8697 8879 9070 9269 9478 lb 43 7887 8087 8194 8356 8524 8700 8882 9073 9272 9481 17 44 45 7889 8o4o 8196 8359 8527 0.8530 8703 0.8706 8885 9076 9276 9485 lb i5 0.7891 0.8043 0.8199 0.836I 8888 0.9079 0.9279 0.948S 46 7894 8045 8202 8364 8533 8709 8892 9083 9283 9492 14 47 7896 8o48 S204 8367 8536 8712 8895 9086 9286 9496 tJ ■18 7899 8o5o 8207 3370 8539 8715 8898 9089 9289 9499 12 49 5o 7901 0.7904 8o53 8210 8372 8542 8718 8901 9092 9293 95o3 11 10 o.8o55 0.8212 0.8375 0.8544 0.8721 0.8904 0.9096 0.9296 0.9506 Dl 7906 8o58 8215 8378 8547 8724 8907 9099 9800 9510 9 5 2 7909 8061 8218 838i 855o 8727 8910 9102 93o3 95i4 8 53 791 1 8o63 8220 8384 8553 8730 8913 9106 9806 9517 / 54 55 7914 8066 0.8068 8223 8386 8556 0.8559 8733 0.8736 8917 9109 9810 9521 b ~5 0.7916 0.8226 0.8389 0.8920 0.9112 . 93 1 3 0.9524 b6 7919 8071 8228 8392 8562 8739 8923 9115 9817 9528 A i)7 7921 8073 823i 8395 8565 8742 8926 9119 9820 9"^ 3 58 7924 8076 8234 8397 8568 8745 8929 9122 9824 9585 2 59 7926 8079 • 8236 8400 8570 8748 8932 9125 9827 9539 I ()o 7929 8()8i 8239 84o3 8573 8751 8935 9128 9381 9542 5= 2ct' .5° 28' 5° 27' 5° 26' 5° 25' 5° 24' 5° 23' 5= 22' 5° 21' 5° 20 The sccom I correct! on is to 1 )e taken at the bottom if tl e appare nt distan ce be Ics s than 90 3_ TABLE XLVII. [I'wgti -273 The first correction is always to be taken at the toj). Tha St cond correction is to be taken at tlie top if the apparent distance exceed 90°. II 40 40' 4° 41' 40 42/ 40 43/ 40 44/ 4° 45' 4° 46' 40 47, 40 48' 40 49/ 0.9542 0.9765 I . 0000 1.0248 I .05l2 1.0792 I . 1091 I .i4i3 I .1761 I .2139 60 I 9546 9769 ooo4 0252 o5i6 0797 1097 1419 1767 2145 59 2 9550 9773 0008 0257 o52i c8oi 1102 1424 1773 2l52 58 3 9553 9777 0012 0261 o525 0806 1107 i43o 1779 2159 57 4 5 9557 9780 0016 0265 o53o 0811 1112 i436 1785 2i65 56 55 0.9561 0.9784 I .0020 I .0270 1.0534 I. 0816 1.1117 i.i44i 1.1791 1 .2172 6 9^64 9788 0024 0274 0539 0821 II23 1447 1797 2178 54 7 9568 9792 0028 0278 0543 0826 1 1 28 1452 i8o3 2i85 53 8 9571 9796 oo32 0282 o548 o83i ii33 1458 1809 2192 52 lO 9575 9800 oo36 0287 o552 o835 ii38 1 464 1816 2198 5i 5o 0.9579 . 981)3 1 . oo4o I .0291 1 .o557 I .0840 1.1143 1.1469 1.1822 1 .2 2o5 u 9582 9S07 0044 O29D o562 0845 1149 1475 1828 2212 49 12 9586 981 1 0049 o3oo o566 o85o ii54 i48i i834 2218 48 i3 9590 9S15 oo53 o3o4 0571 0855 1159 i486 i84o 2225 47 i4 i5 95y3 9819 00 5 7 o3o8 0575 0860 ii64 1492 1 846 2232 46 45 0.9597 0.9823 I .0061 i.o3i3 I .o58o 1.0865 1 .1170 1.1498 1.1852 I .2239 i6 9(5f)i 9827 oo65 o3i7 o585 0870 1175 i5o3 1859 2245 44 17 9()o4 9830 0069 o32i 0589 0875 1 180 1 509 1 865 2252 43 lb 9f)o8 9S34 0073 0326 0594 0880 1 186 i5i5 1871 2259 42 11 20 9012 9838 0077 o33o 0598 0884 1191 l520 1877 2266 I .2272 4i 40 0.9615 0.9842 I .0081 1.0334 I .o6o3 1.0889 1 . 1 1 96 1.1526 1.1883 21 9619 9846 oo85 0339 0608 0894 1201 i532 1889 2279 39 22 9623 9850 0089 0343 0612 0899 1207 i538 1896 2286 38 23 9626 98 54 0093 o347 0617 0904 1212 i543 1902 2293 ^-i7 24 25 963c. 9S58 0098 o352 0621 0909 I .0914 1217 1 549 1908 23oo I .2307 36 35 0.9634 . 986 1 1 .0102 I .o356 I .0626 I .1223 1.1555 1. 1914 26 9638 9865 0106 o36o o63r 0919 1228 i56i 1921 23 1 3 M ^7 9641 9869 01 10 o365 o635 0924 1233 i566 '927 2320 6i 28 9(345 98/3 oii4 0369 0640 0929 1239 1572 1933 2327 32 29 3o 9649 9877 0118 o374 0645 0934 1244 1578 1939 2334 3i 3^ 0.9652 0.9881 1 .0122 1.0378 1 .0649 1 .0939 I.124Q 1 . 1 584 1 . 1 946 I .2341 3i 91)56 9885 0126 o382 o654 0944 1255 1589 1902 2 348 29 32 9660 9889 oi3i o387 0659 0949 1260 159! 1958 2355 28 33 9664 9S93 oi35 0391 o663 0954 1266 1601 1965 2362 27 34 35 9667 9897 0.9901 0139 0395 0668 0959 1271 1607 1971 2368 1.2875 2b . 967 1 I. 0143 I .o4oo I .0673 1 . 0964 1 .1276 i.i6i3 1.1977 36 9675 990D oi47 o4o4 0678 0969 1282 1619 19S4 2382 24 37 9678 9908 oi5i 0409 0682 0974 1287 1624 1990 2,389 23 38 9()82 99' 2 oi56 o4i3 0687 0979 1292 i63o 1996 2396 22 39 40 9686 99.6 0160 o4i8 0692 0984 1298 1 636 3'X)3 24o3 1. 2410 21 20 . 9(590 0.9920 I .0164 I .0422 1 . 0696 I .0989 I .i3o3 i.:642 I . 2009 4i 9693 9924 0168 0426 0701 0994 1 309 1648 2016 2417 19 42 9697 9928 0172 043 1 0706 0999 i3i4 1 654 2022 2424 18 43 9701 9932 0176 0435 071 1 1004 l320 1660 2028 243 1 17 45 9705 9936 oi8r o44o 0715 1009 1 325 i665 2o35 2438 lb i5 . 97C)8 0.9940 1.0185 I .0444 1 .0720 1 . 1 1 5 i.i33i I . 1671 1 . 204 1 I .2445 40 9712 9944 0189 0449 0725 1020 1 336 1677 2048 2453 14 47 9716 9948 0193 0453 0730 1025 i342 i683 2o54 2460 i3 48 9720 9952 0197 0458 0734 io3o 1347 16S9 2061 2467 12 49 5o 9723 9956 0202 0462 0739 io35 i352 1695 2067 2474 II 10 0.9727 0.9960 I .0206 I . 0467 1.0744 I .1040 i.i358 I .1701 1 .2073 I. 2481 5 1 9731 9964 0210 0471 0749 1045 1 363 1707 2080 2488 9 52 9735 9968 02l4 0475 0753 io5o 1369 1713 20S6 2495 8 53 9739 9972 0219 0480 0758 io55 1 374 1719 2093 2502 V 54 55 9742 9976 0223 o484 0763 1061 i38o 1725 2099 25lO ,b 5 0.9-/46 . 9980 I .0227 1 .0489 1 .0768 1 . 1 066 i.i3S6 1 .1731 1 . 2 1 06 1 .2517 56 9750 9984 023l 0493 0773 1071 1391 1737 2Il3 2524 4 57 9754 9988 0235 0498 0777 1076 1397 1743 21 19 253i 3 58 9758 9992 0240 o5o2 0782 1081 l402 1749 2126 2538 2 59 9761 9996 0244 o5o7 0787 10S6 1 408 1755 2l32 2545 I 60 9765 1 . 0000 0248 o5i2 0792 1 09 1 i4i3 1761 2139 2553 5° 10' 5° 18' 5° 17' Tp m 5° 15' 5= 14' 5° 13' 5° 12' 5° 11' 5° 10' T he sicoiu / correct on is to be taken at the ho tlom if t^ le appar« ;nt distar 1C3 be le. -5 llian 90 o_ •x> p ige 274] TABLE XLVII. The fust correction is always to be taken at the top. The second correction is to be taken at tlie top if the apparent distance exceed 90''. { o 4° 50' 4° 51' 4° 52' 4° 53' 4° 54' 4° 55' 1.5563 4° 5G' 4° 57' 4° 58' 4° 59' 60 1.2553 I .3oio 1.3522 I .4102 1.4771 1.6532 1.7782 I .9542 2.2553 I 256o 3o[8 353i 4lI2 4783 5578 655o 7806 9570 2626 5q 2 2567 3026 3540 4l22 4795 5592 6568 783o 9615 2700 58 3 2574 3o34 3549 4i33 4808 5607 6587 7S55 9652 2775 57 4 5 2582 3o43 3558 4r43 4820 5621 1.5636 66o5 7879 9690 2852 56 55 1.2589 1 .3o5i 1.3567 i.4i54 1.4832 I .6624 1.7904 1.9727 2.2931 6 2596 3o59 3576 4 1 64 4844 565i 6642 7929 9765 3oio 54 7 2604 3067 3586 4175 4856 5666 6661 7954 9803 3091 53 8 261 1 3075 3595 4i85 4869 568o 6679 7979 9842 3i74 52 _9 10 2618 3oS3 36o4 4196 4881 5695 6698 8004 9881 3259 5i 5^ 1.2626 I .8091 I .36i3 1 .4206 I .4894 I .5710 1-6717 1 .8000 I .9920 2.3345 II 2633 3 1 00 3623 4217 4906 5725 6736 8o55 9960 3432 49 12 2640 3io8 3632 4228 4918 5740 6755 8081 2.0000 3522 48 i3 2648 3ii6 364 1 4238 4931 5755 6774 8107 oo4o 36i3 47 i4 i5 2655 3i24 365o 4249 4943 5771 6793 8i33 0081 3707 46 45 I . 2663 i.3i33 i.366o I .4260 I .4956 1.5786 I. 6812 I. 8159 2.0122 2.3802 i6 2670 3i4i 3669 4270 4969 58oi 6832 8186 0164 3900 44 17 2678 3i49 3678 4281 4981 58i6 685 1 8212 0206 4000 43 i8 2685 3i58 3688 4292 4994 5832 6871 8239 0248 4l02 42 12 20 2692 I .2700 3i66 1.3174 3697 43o3 5007 5847 6890 8266 0291 4206 4i 4o 1.3707 i.43i4 I .5019 1.5863 I .6910 1 .8293 2.0334 2.43i4 21 2707 3i83 3716 4325 5o32 5S78 6930 8320 0378 4424 39 22 2715 3191 3726 4335 5o45 5894 6950 8348 0422 4536 38 23 2722 3199 3735 4346 5o58 5909 6970 8375 0467 4652 37 24 25 2730 3208 3745 4357 I .4368 5071 5925 1. 5941 6990 84o3 o5i2 4771 ib 35 1.2738 I .3216 1.3754 I . 5o84 1 .7010 I.843I 2.0557 2.4894 26 2745 3225 . 3764 4379 5097 5957 7o3o 8459 o6o3 5019 M 27 2753 3233 3773 4390 5iio 5973 70 5o 8487 0649 5i49 33 28 2760 3242 3783 44oi 5i23 5989 7071 85i6 0696 5283 32 29 3o 2768 325o I .3259 3792 4412 1.4424 5i36 6oo5 7091 8544 0744 5421 3i 3o 1.2775 1.3802 i.5i49 1 .6021 1 .7112 1.8573 2.0792 2.5563 3i 2783 3267 38i2 4435 5162 6087 7i33 8602 0840 5710 29 32 2791 3276 3821 4446 5175 6o53 7i54 8632 0889 5863 28 33 279S 3284 383i 4457 5189 6069 7175 8661 0939 6021 27 34 35 2806 3293 I .33oi 384i 4468 5202 6o85 7196 8691 0989 6i85 2b 25 1. 2814 1.3851 1 .4480 I.52I5 I .6102 J. 7217 I .8721 2. io4o 2.6355 36 2821 33io 386o 4491 6229 6118 7238 8751 1091 6532 24 37 2829 3319 3870 45o2 5242 6i35 7259 8781 ii43 6717 23 38 2837 3327 388o 45i4 5256 6i5i 7281 881 1 1 1 96 6910 22 39 4o 2845 3336 3890 4525 1.4536 5269 1.5283 6168 i.6i85 73o2 8842 1249 7112 21 20 1.2852 1.3345 I .3900 1.7324 1.8873 2 . I 3o3 2.7324 4i 2860 3353 3910 4548 5296 6201 7346 8904 1 358 7547 19 42 2868 3362 3919 4559 53io 6218 7368 8935 i4i3 7782 18 43 2876 3371 3929 4571 5324 6235 7390 8967 1469 8o3o 17 44 45 2883 i.289r 3379 1.3388 3939 4582 1.4594 5337 6252 I .6269 7412 8999 i526 8293 lb 75 1.3949 i.535i 1.7434 I .9031 2.1584 2.8573 46 2899 3397 3959 4606 5365 6286 7456 9063 1642 8873 i4 47 2907 3406 3969 4617 5379 63o3 7479 9096 1701 9195 i3 48 2915 34i5 3979 4629 5393 6320 75oi 9128 1761 9542 12 49 5o 2923 3423 1.3432 3989 464o 5407 6338 7524 9162 1822 9920 3.0334 II 10 I .2981 I . 4ooo I .4652 1. 5421 1 .6355 1.7547 1. 9195 2.1883 5i 2939 3441 4oio 4664 5435 6372 7570 9228 1946 0792 9 52 2946 345o 4020 4676 5449 6390 7593 9262 2009 i3o3 53 2954 3459 4o3o 4688 5463 6407 7616 9296 2073 i883 7 54 55 2962 3468 4o4o 4699 5477 6425 7639 9331 2139 2553 b "5 1.2970 1.3477 I .4o5o 1. 471 1 1. 5491 I .6443 1.7663 1.9365 2.2205 3.3345 56 2978 3486 4o6i 4723 55o6 646o 7686 9400 2272 43i4 4 57 2986 3495 4071 4735 5520 6478 7710 9435 234i 5563 6 58 2994 35o4 4o8i 4747 5534 6496 7734 9471 2410 7324 2 59 3o02 35i3 4091 4759 5549 65i4 7757 9506 2481 4.0334 I 60 3oio 3522 4l02 4771 5563 6532 7782 9542 2553 5° 9' 5° 8' 5° 7' 5° 6' 5° 5' 5° 4' 5° 3' 5° 2' 5° 1' 5° 0' T he scconc I correct on is to je taken at the bo ttom if tl le appare nt distar ice be les s than 90°. TABLE XLVIIl . [Page 275 Third Correction. Apparent Distance 20°. A pp. apparent Altitude of the Sun, Star or Planet. I>'s App. III. 6° «" 10" 12" 16° 20° 24° 28° 32° 36° 42° 50° 58" 66° 74" 82" Alt. o / II / // / ;; / ^r / n / ;/ 1 II 1 II / // / . // 1 II / 11 II 1 II 6 I 38 I 46 ,2 7 2 34 3 43 4 5i 5 59 6 7 I 46 r 4o I 53 2 12 3 I 3 57 4 5o 7 8 I 55 I 36 I 44 I 56 2 35 3 17 4 4 42 « 9 2 8 I 4o I 39 I 45 2 12 2 47 3 23 3 58 9 lO 2 23 I 46 I 36 I 39 I 56 2 24 2 53 3 23 10 II 2 38 I 54 I 38 I 37 I 46 2 8 2 32 2 56 3 16 II 12 2 53 2 3 I 4i I 35 I 4i I 56 2 16 2 35 2 52 12 i3 3 q 2 r3 I 46 r 37 I 37 I 48 2 4 2 19 2 32 i3 i4 3 25 2 23 t 52 I 3q I 34 I 42 I 54 2 5 2 16 14 lb 3 4i 2 34 I 58 I 42 I 33 I 38 I 45 I 54 2 3 2 i3 i5 1(3 3 58 2 45 2 4 I 46 I 32 I 34 I 38 I 46 I 53 I 59 16 I? 4 i5 2 56 2 10 I 5o I 33 I 32 I 34 I 39 I 44 I 46 17 iS 4 32 3 7 2 17 I 54 I 34 I 3o I 3i I 34 I 37 I 40 18 19 4 49 3 18 2 24 I 58 r 35 I 20 I 28 I 29 I 32 I 33 I 34 19 20 5 5 3 28 2 3r 2 2 I 37 t 28 I 3o I 3o I 29 I 26 20 21 5 21 3 39 2 38 2 6 I 39 I 29 r 27 I 27 I 27 I 25 I 21 21 22 5 36 3 49 2 46 2 II I 4o I 29 I 25 I 25 I 24 I 22 I 18 22 2J 5 5i 3 59 2 53 2 16 I 42 I 29 I 25 I 24 I 22 r 20 I i5 23 24 6 5 4 9 3 2 22 I 43 I 3o I 24 I 23 I 21 I 18 I 12 24 25 6 19 4 18 3 7 2 26 I 45 I 3o I 24 I 21 I 19 I 16 I 9 25 26 6 32 4 27 3 i4 2 3i I 47 I 3i I 25 I 21 I 17 r i4 I 7 26 27 6 454 35 3 20 2 35 I 49 I 32 I 25 I 21 I 17 I i3 I 6 27 28 4 42 3 26 2 38 I 5o I 33 I 25 I 21 I 17 I i3 I 4 5o 28 29 4 49 3 32 2 4i I 52 I 33 I 25 I 21 I 17 I i4 I 5 5o 29 3o 3i 3 37 2 45 I 54 I 56 I 34 I 34 r 25 I 25 r 21 I 20 I 18 I 17 I i5 I i5 I 7 I 7 5o 5r 3o 3i 3 42 2 49 32 2 52 I 58 I 34 I 24 I 19 I 17 I i4 I 7 5i 32 33 2 55 X 59 I a3 I 24 I 19 I 16 I i3 I 8 52 33 34 I 5q I 33 I 23 I 18 I i5 I i3 I 8 53 34 35 I 59 I 32 I 22 I 17 I i4 I 12 I 8 53 35 36 I 59 I 3i I 20 I i5 I i3 I II I 7 54 36 ■66 J7 I 59 I So I 19 I i4 I 12 I 10 I 6 54 37 37 38 I 29 I 18 I i3 I II I 9 I 6 55 38 38 39 I 28 I 17 I II I 10 I 5 55 39 39 4o I 27 I i5 I 10 I 9 I 8 I 4 55 39 40 4i 1 26 I i3 I 9 I 8 I 7 I 3 55 39 4i 42 I II I 7 I 7 I 6 I 2 55 4o 42 43 I 10 I 5 I 5 I 5 I 2 55 40 4i 44 I 9 I 3 I 4 I 4 I I 55 40 29 44 46 I 7 I I I I 2 I 54 4i 3o 46 48 56 56 59 58 53 43 3i 48 5o 52 52 55 55 5i 43 33 5o 52 48 5o 5i 49 43 35 24 52 54 44 45 47 47 43 36 25 54 56 4o 44 45 42 35 27 56 58 35 4o 43 40 M 27 58 60 36 4i 39 33 26 21 60 62 33 38 38 32 26 22 62 64 3o 35 37 32 27 22 64 66 32 36 3i 27 23 66 68 29 M 3o 26 23 68 70 27 32 29 26 22 70 72 25 29 28 25 21 72 74 27 27 24 21 7-i 76 25 26 24 20 76 78 23 25 23 20 78 8n 21 24 22 20 80 8'; 23 21 82 84 22 21 84 86 21 20 tb G° 8° 10° 1 12° 1G° 20° 24° 28° 32° 36° 42° 50° 58° 66° 74° 82° Pase 276] TABLE XLVJ XL . Third Correction. ApparenL Distance 24°. 5's App. Jlpparent Altitude of the Sun, Star or Planet. ])'s App. Alt. G" 7° , ' y" y^ lU^ IP 1^2^ 140 IG^ 18" yu" 22° 24° 26° 28° 30° Ait. ( / / /^ 1 II / ?/ / // 1 II 1 II / II / // / // / // / // 1 II / II / // / II 6 I 28 I 3i I 35 I 42 I 52 2 3 2 16 2 46 3 16 } 47 4 19I i 5o 5 20 5 5o 6 206 5o| 6 -7 I 35 I 27 I 3o I 34 I 39 I 46 I 54 2 i5 2 38 3 3 3 29 i 55 4 20 4 46 5 10 5 34 7 8 I 45 I 32 I 26 I 28 I 00 I 3i) I 4i I 58 2 17 2 37 2 58 i 18 3 39 4 I 4 204 3q| 8 9 I 56 I 39 I 3o I 25 I 26 I 29 I 34 I 44 I 59 2 i5 2 3i 2 48 3 6 3 24 3 40 3 56 9 10 2 8 I 48 I 36 I 29 I 25 I 26 I 28 1-35 I 45 I 57 2 i3 2 27 2 43 2 58 3 12 3 26 10 1 1 2 21 I 58 I 43 I 34 I 28 I 24 I 26 I 3o I 36 I 46 I 58 2 1 1 2 24 2 37 2 4q 3 II 12 2 36 2 9 I 52 I 4i I 33 I 27 I 24 I 26 I 3o 1 07 I 47 I 58 2 9 2 20 2 29 2 38 12 r3 2 5i 2 20 2 I I 48 I 38 I 3i I 27 I 24 I 27 I 32 I 4o I 48 I 57 2 6 2 i4 2 22 i3 i4 3 6 2 3i 2' 10 I 55 I 43 I 35 I 3o I 23 I 25 I 28 I 33 I 4o I 48 I 55 2 2 2 10 i4 i5 3 21 2 42 2 20 2 2 I 5o I 3q I 33 I 24 I 23 I 25 I 24 I 34 I 40 I 46 I 52 I 59 i5 i6 3 36 2 54 2 3o 2 9 I 56 I 44 I 36 I 26 I 22 I 23 I 25 I 29 I 33 I 38 I 44 I 5o 16 17 3 5i J 6 2 40 2 17 2 2 I 49 I 39 I 28 I 23 I 21 I 23 I 26 I 29 I 34 I 39 I 43 17 j8 4 6 3 18 2 49 2 25 2 8 I 54 I 43 I 3i I 24 I 20 I 21 I 23 I 26 I 3o I 3A I 37 18 19 4 21 3 3o 2 59 2 33 2 i4 I 59 I 47 I 33 I 25 I 21 I 20 I 22 I 24 I 27 I 3o I 32 19 20 4 3b J 42 ^ 9 2 4i 2 21 2 5 I 52 I 36 I 27 ; a I 19 I 20 I 22 I 24 I 26 I 28 20 21 4 5u 3 54 3 19 2 5o 2 28 2 11 I 56 I 39 I 29 I 23 I 20 I 19 I 20 I 21 I 23 I 25 21 22 5 4 4 6 3 28 2 58 2 3b 2 17 2 I I 42 I 3i I 24 I 20 I 18 I 19 I IQ I 20 I 22 22 23 5 19 4 18 3 38 3 6 2 43 2 23 2 6 I 46 I 33 I 25 I 21 I 18 I 18 I 18 I 18 I 19 23 24 5 33 4 29 3 48 3 i4 2 5i 2 29 2 12 I 5o I 36 I 27 I 22 I 19 I 17 I 17 I 17 I 17 24 25 5 47 4 4i 3 57 3 22 2 58 2 35 2 17 I 53 I 38 I 28 I 23 I 20 I 18 I 16 I 16 I 16 25 26 6 I 4 52 4 6 3 3o 3 4 2 4i 2 22 I 57 I 4i I 3o I 24 I 20 1 18 I 16 I i5 I i5 26 27 6 i4 5 4 4 lb 3 38 3 10 2 47 2 27 2 I 43 I 32 I 25 I 21 I 18 I i5 I i4 I i3 27 28 6 27 5 i5 4 23 3 45 3 16 2 53 2 32 2 4 I 46 I 34 I 27 I 21 I 18 I i5 I i3 I 12 28 29 6 38 5 26 4 32 3 53 3 22 2 58 2 38 2 8 I 49 I 36 I 28 I 22 I 18 I i5 1 i3 I 1 1 29 So 6 5o 5 36 4 41 4 3 28 3 3 2 44 2 12 I 52 I 38 I 29 I 23 I 19 r i5 I i3 I 1 1 3o 3i 7 5 45 4 5o 4 7 3 34 3 8 2 4q 2 16 I 55 I 40 1 3o I 24 I IQ I i5 I i3 I 1 1 3i 32 5 53 4 58 4 i4 3 40 3 i3 2 54 2 19 I 57 I 41 I 3i I 24* I 19 I i5 I i3 I II 32 33 5 5 4 20 3 46 3 18 2 58 2 22 I 59 I 42 I 3i I 24 I 19 I i5 I i3 I II 33 34 4 35 3 5i 3 22 3 I 2 24 2 1 I 43 I 32 I 25 I 20 I i5 I i3 I II .^4 35 3 56 3 26 3 32 26 2 2 I 45 I 33 I 25 I 20 I i5 t i3 I II 35 36 3 3o 3 52 28 2 4 I 46 I 34 I 25 I 20 I i5 I 12 I 10 36 37 3 7 2 3o 2 6 I 47 I 35 I 25 I 20 I i5 I 12 I 10 37 38 2 32 2 7 1 48 I 35 I 25 I 20 I i5 I 12 I 10 38 39 2 34 2 8 I 49 I 35 I 25 I 19 I i5 I 12 I 10 39 4o 2 9 I 5o I 35 I 2b I 19 I i5 I II I 9 4o 4i 2 ID I 5o I 35 I 25 I 19 I i5 I II I 8 4i 42 I 5i I 36 I 2b I 19 I i4 I 10 I 7 42 43 I 52 I 36 I 2b I 18 I i3 I 9 I 6 43 44 I 36 I 2b I 18 I i3 I 8 I 5 44 46 I 36 I 2b I 17 I 12 I 7 I 3 46 48 I 25 I 17 I 10 I 5 I I 48 5o I 17 I 8 I 4 59 5o 52 I 7 I 3 58 52 54 I 2 57 54 56 56 56 58 58 60 60 62 62 64 64 66 66 68 68 70 70 72 72 74 74 76 78 76 78 80 80 82 82 84 84 86 G° 70 S° 9= 10° 11° 12° 14° 16° 18° 20° 22° 86 24° 26° 28° 30° TABLE XLVIII [Page 277 Third Correction Apparent Distance 24°. D's A,.p. All. Apparent Altitude of the Sun, Sta r or Planet. D's App. Alt. 32° 34° 36° 38° 42° 46° 50° 54° 58° 62° 66° 70° 74° 78° 82° 86° o / . ., / II / // / II / / // 1 II 1 / II 1 II / II / n / // / / 6 6 7 7 8 4 58 8 9 4 12 9 10 3 39 3 5i 10 II 3 II 3 21 3 3o II 12 2 48 2 56 3 5 3 12 12 i3 2 3o 2 37 2 44 2 49 i3 i4 ■2 16 2 22 2 27 2 32 i4 i5 2 4 2 9 2 i4 2 18 i5 itj I 54 1 59 2 3 2 6 2 II 16 17 I 46 r 5o I 53 I 56 2 17 18 I 40 I 43 I 45 I 47 I 5r 18 19 I 35 I 37 I 39 I 4i I 43 19 20 I 3o I 32 I 33 I 34 I 36 I 38 20 21 I 26 I 27 I 28 I 29 I 3o I 3i 21 22 I 22 I 23 I 24 I 24 I 25 I 25 22 23 I 20 I 20 I 21 I 21 I 21 I 21 23 24 I 18 I 18 I 19 I 19 I 18 I 17 I i5 24 2I) I 16 I 16 I 17 I 17 I 16 I i4 I II 25 26 I i4 I i4 I i4 I i4 I i3 I II I 8 26 27 I i3 I i3 I 12 I 12 I II I 9 I 6 27 28 I 12 I 12 I 11 I 10 I 9 I 7 I 4 I I 28 29 I II I II I 10 I 9 I 8 I 5 I 2 D9 29 3o I II I 10 I 9 I 8 I 7 I 4 I 67 3o 3i I 10 I 9 I 8 I 8 I 6 I 2 58 55 3i 32 I 9 I 9 I 8 I 7 I 5 I I 57 54 5i 32 66 I 9 I 8 I 7 I 6 I 4 I I 57 53 5o ■66 M I 9 I 7 I 6 I 5 I 3 I 57 53 49 34 3t) I 9 I 7 I 6 I 5 I 2 I 56 52 48 35 36 I 8 I 7 I 6 I 4 I 2 I 56 5i 47 44 36 ^7 I 8 I 6 I 5 I 3 I I 58 55 5i 46 43 37 38 I 8 I 6 I 5 I 3 I 57 54 5o 46 43 38 39 I 8 I 6 I 4 I 2 59 56 52 48 45 42 39 4o I 7 I 5 I 4 I 2 59 55 5i 47 44 4i 39 40 4i 1 6 I 4 I 3 I I 58 54 5o 47 44 4i 38 4i 42 I 5 I 4 I 3 I I 57 54 5o 47 44 4i 38 42 43 I 4 I 3 I 2 I 56 53 5o 47 43 40 37 34 43 44 I 3 I 2 I I 59 56 53 5o 47 43 40 37 34 44 46 I I I 59 58 55 52 49 46 43 40 37 34 32 46 48 59 59 58 57 54 5i 4q 46 43 4o 37 34 32 48 bo 57 57 56 55 53 5o 48 45 43 40 37 34 32 ?.o 5o 52 55 54 53 52 5i 49 47 45 43 40 37 34 32 3o 52 54 54 52 5i 5o 49 47 46 44 42 39 37 34 32 29 27 54 56 53 5i 49 48 47 45 44 43 4i 38 36 34 3i 29 27 56 58 52 49 47 46 45 44 43 42 4o 37 35 33 3i 29 27 26 58 60 47 45 44 43 42 4i 40 38 36 34 32 3o 28 27 26 60 62 43 43 4i 40 39 38 37 35 33 3i 29 28 27 26 62 64 42 39 38 38 37 36 34 32 3o 29 28 27 26 64 66 38 37 37 36 35 33 3i 29 28 27 26 25 66 68 37 35 35 34 34 33 3i 29 28 27 26 25 68 70 34 34 33 33 32 3o 28 27 26 25 25 70 72 33 33 32 32 3i 29 28 26 25 24 25 72 74 32 3i 3i 3o 29 28 26 25 ■^4 74 76 3i 3o 3o 29 28 27 25 24 24 76 78 29 29 29 28 27 25 24 78 80 28 28 28 27 26 25 24 80 82 27 27 26 25 24 82 84 26 26 25 25 24 84 86 26 25 25 86 32= 34° 36° 38° 42° 46° 50° 54° 58° 62= m° 70° 74° 78° 82° 86° Page 278] TABLE XLVIIL Third Correction. Apparent Distance 28°. App. Apparent Mtitude of the Sui , Star or Planet. App. Alt. o 6° 70 / II 10° / // ir 1 II 12^ 14^ Iti" / // 18° 1 II 20^ 1 1 22° / II 24" 26° / // 28° / // ciU° 1 II Alt. / II 6 I 20 I 23 I 27 I 33 I 4o I 49 2 00 2 28 2 56 3 24 i 53 i 21 4 48 3 58 5 i5 5 42 5 q 6 7 I 25 I 20 I 23 I 27 I 32 I 38 I 45 2 5 2 26 2 49 i i3 3 36 i 20 4 43 5 6 7 8 I 32 I 24 I 20 I 22 I 25 I 29 I 35 I 5o 2 7 2 26 2 46 3 4 3 23 3 42 4 I 4 20 8 9 I 4i I 29 I 23 I 20 I 22 I 24 I 28 I 39 I 52 2 7 2 32 2 37 2 53 3 9 3 25 3 41 9 lO i 53 I 37 I 28 I 23 I 20 I 21 I 23 I 3o I 39 I 52 2 5 2 18 2 3i 2 44 2 58 3 II 10 II 2 6 I 46 I 34 I 27 I 23 I 20 I 21 I 24 I 3i I 4i I 52 2 4 2 i5 2 26 2 37 2 48 II 12 2 19 I 56 I 4i r 32 I 26 I 22 I 19 I 21 I 26 I 33 I 42 I 52 2 I 2 ID 2 20 2 3o 12 i3 2 32 1 6 I 49 I 38 I 3o I 25 I 21 I 20 I 23 I 28 I 34 I 42 I 49 I 57 2 6 2 i5 i3 i4 2 46 2 17 I 58 I 44 I M I 28 I 23 I 19 I 21 I 24 I 28 I 34 I 4o I 47 I 55 2 3 i4 lb 3 00 2 28 2, 7 I 5i I 39 I 32 I 25 I 20 I 19 I 21 I 24 1 28 I 33 1 39 I 45 I 52 i5 i6 3 14 2 39 2 j6 I 58 I 45 I 36 I 28 I 21 I 18 I 19 I 21 I 24 I 28 I 33 I 38 I 44 16 17 3 28 2 5i 2 25 2 5 I 5i 1 4i I 32 I 23 I 19 I 18 I 19 I 21 I 24 I 28 I 33 I 38 17 18 3 4i 3 2 2 35 2 i3 I 58 I 46 I 36 I 25 I 20 I 17 I 18 I 19 I 21 I 24 I 28 I 33 18 19 3 55 3 i3 2 45 2 21 2 5 I 52 I 4i I 27 I 21 I 18 I 16 I 17 I 18 I 21 I 24 I 28 19 20 4 9 3 24 2 55 2 29 2 II 1 57 I 46 I 3o I 23 I 18 I 16 I i5 I 16 I 18 I 21 I 24 20 21 4 23 3 35 3 4 2 37 2 17 2 3 I 5i I 33 I 25 I 19 I 16 I i4 I i5 I 16 I 18 I 20 21 22 4 36 3 46 3 i3 2 45 2 24 2 9 I 56 I 36 I 27 I 20 I 16 I i3 I i4 I i5 I 16 I 17 22 2J 4 49 3 57 3 22 2 53 2 3i 2 i4 2 I I 4o I 29 I 22 I 17 I i3 I i3 I i3 I i4 I i5 23 24 5 2 4 8 3 3i 3 2 37 2 20 2 6 I 43 I 3i I 24 I 18 I i4 t 12 I 12 I 12 I i3 24 2b 5 16 4 19 3 40 3 8 2 43 2 26 2 II I 47 I 34 I 26 I 19 I i5 I i3 I II I II I 12 25 26 5 29 4 3o 3 49 3 i5 2 5o 2 32 2 16 I 5i I 36 I 28 I 20 I i5 1 i3 I II I II I II 26 27 542 4 4i 3 58 3 23 2 57 2 38 2 21 I 55 I 39 I 3o I 21 I 16 I i3 I 11 I 10 I 10 27 28 5 55 4 52 4 7 3 3o 3 4 2 44 2 26 I 59 I 42 I 32 I 22 I 17 I 14 I II I 10 I 10 28 29 6 7 5 3 4 16 3 38 3 II 2 5o 2 3i 2 3 I 45 I 34 I 24 I 18 I i4 I 12 I 10 I 10 29 3o 6 19 5 i3 4 25 3 45 3 18 2 55 2 36 2 7 I 47 I 36 I 26 I 19 I i5 I 12 I 10 I 9 3o 3i 6 3i 5 23 4 34 3 52 3 25 3 I 2 4i 2 10 I 5o I 38 I 27 [ 20 I i5 I 12 I 10 I 9 3i 32 6 42 5 32 4 43 3 59 3 3i 3 7 2 46 2 i3 I 53 I 40 I 29 I 21 I 16 I 12 I 10 I 9 32 33 6 53 5 4i 4 5i 4 6 3 37 3 12 2 5i 2 17 I 56 I 42 I 3i I 22 I 16 I 12 I 10 I 9 33 ■M 7 4 5 5o 4 58 4 i3 3 43 3 17 2 55 2 20 I 58 I 44 I 32 I 23 I 17 I 12 I 10 34 35 7 i5 559 5 5 4 20 3 48 3 21 2 59 2 23 2 00 I 46 I 33 I 23 I 17 I i3 I 10 I 8 35 36 6 8 5 II 4 26 3 53 3 25 3 3 2 26 2 3 I 47 I 34 I 24 I 18 I i3 I 10 I 8 36 37 5 17 4 32 3 58 3 29 3 7 2 29 2 5 I 49 I 35 I 25 I 18 I i3 I 10 I 8 37 38 4 38 4 2 3 33 3 10 2 32 2 7 I 5i I 36 I 26 I IQ I i4 I 10 I 8 38 39 4 6 3 37 3 12 2 34 2 9 I 52 I 37 I 27 I 19 I i4 I II I 8 39 40 3 4i 3 i5 2 37 2 II I 53 I 38 I 27 I 20 I i5 I II I 8 40 41 3 17 2 40 2 i3 I 54 I 39 I 28 I 20 I i5 I II I 8 4i 42 2 42 2 i5 I 55 I 40 I 29 I 21 I i5 I ID I 7 42 43 2 44 2 17 I 56 I 4o I 29 I 21 I i5 I 10 I 7 43 44 2 18 I 57 I 4i I 3o I 21 I i5 I 10 I 7 44 46 48 2 19 1 59 2 I 42 I 43 I 3o I 3i I 22 I 22 I i5 I i5 I 10 I 10 I 7 I 6 46 48 5o I 44 I 32 I 23 I i5 I ID I 6 5o 52 I 33 I 24 I i5 I 9 I 5 52 54 I 25 I i5 I 9 I 5 54 56 58 I i5 I 9 I 9 I 4 I 3 56 58 60 I 3 60 62 62 64 64 66 68 66 68 70 70 72 72 74 74 76 76 78 78 bo 80 82 82 84 84 86 86 6° 7° 8° 9° 10^ |ll° 12° ll4° 16° 18° 20° 22° 24° 26° 28° 30° TABLE XLVIU [Page 279 Third Correction. Appar ent Distance 28°. D's A pp. Apparent Altitude of the Sun, Star or Planet. App. All. ;j-2^ 34" 3G^ 38'^ 42" 4G° 50° 54° 58° (32° 66° 70-' 74° 78^ 82° 86^ Alt. o / /,' 1 II 1 II / // / II / // 1 II / // / // II / II / // / / ; /I 1 1. / // b 6 37 1 4 f 7 5 28 5 49 6 8 7 8 4 40 4 57 3 II 8 9 3 58 4 i3 4 26 4 38 9 lO 3 25 3 38 3 5o 4 2 10 II 3 3 12 3 23 3 33 II 12 2 4o 2 5n 2 59 3 7 3 22 12 i3 2 24 2 33 2 4i 2 48 3 1 3 i4 2 II 2 18 2 25 2 3i 2 42 i4 i5 I 59 2 6 2 12 2 17 2 27 i5 i6 I 5o 1 56 2 I 2 6 2 i4 2 21 16 17 I 43 I 48 I 52 I 56 2 3 2 9 17 i8 I 37 I 4i I 4f) I 48 I 54 I 59 18 19 I 3i I 35 I 38 I 4i I 46 I 5o 19 20 I 26 I 29 I 32 I M I 38 I 42 I 45 20 21 I 22 I 25 I 27 I 29 I 32 I 36 I 38 21 22 I 19 I 21 I 23 I 25 I 28 I 3o I 32 22 23 I 17 I 18 I 20 I 22 I 24 I 26 I 27 23 24 I i5 I 16 I 17 I 18 I 20 I 22 I 23 I 24 24 25 I i3 I i4 I i4 I i5 I 16 I 18 I 19 I 19 25 26 I 11 I 12 I 12 I i3 I i3 I i4 I i5 I i5 26 27 I 10 I II r II I II I 11 I II I 12 I 12 27 28 I 10 I 10 I 10 I 10 I 10 I 9 I 9 I 9 I 9 28 29 I 10 I 10 I 10 I 9 I 9 I 8 I 7 I 6 I b 29 3o I 9 I 9 I 9 I 8 I 8 I 7 I 6 I 4 I 3 3o 3i I 8 I 8 I 7 I 7 I 6 I 5 I 4 I 2 I I 3i 32 I 8 I 7 I 6 I 6 I 5 I 4 I 3 I I I 59 32 33 I 7 I 6 I 5 I 5 I 4 I 3 I 2 I 58 56 33 34 I 7 I 5 I 4 I 4 I 3 I 2 I I 59 57 54 ■ 34 35 I 7 I 5 I 4 I 3 r 2 I I I 58 55 53 35 36 I 6 I 5 I 4 I 3 I I I 58 56 54 52 5i 36 37 I 6 I 4 I 3 I 2 I 59 57 55 53 5i 5o 37 38 I 6 I 4 I 3 I I 59 58 56 54 52 5o 49 38 39 I 6 I 4 I 2 I 59 57 55 53 5i 49 47 39 4o I b I 4 I 2 I 58 57 55 52 5o 48 46 44 4o 4i I 6 I 4 I 2 I 58 56 54 5i 49 47 45 AZ 4i 42 I 6 I 4 I 2 59 57 55 53 5o 48 46 44 42 42 43 I 6 I 3 I I 59 57 55 53 5o 48 46 44 42 4i 43 /d I 6 I J I I 59 56 54 52 5o 47 45 43 4i 40 44 46 I 4 I 2 I 58 55 53 5i 49 47 44 42 4o 39 46 48 I 3 I I 59 57 54 52 5o 48 46 43 4i 39 38 37 48 5o I 3 I I 58 56 53 5i 49 47 45 42 40 38 37 36 5o 52 I 2 I 57 55 52 5o 48 46 44 42 4o 38 36 35 34 52 54 I 2 59 56 54 5i t 47 45 43 4i 3q 37 35 34 33 54 56 I I 58 55 53 5o 46 44 42 40 38 36 35 M 33 32 56 58 I 57 54 52 49 47 45 43 4i 3g 37 36 35 34 32 3. 58 60 58 55 53 5i 48 46 44 42 40 3fe 37 36 35 34 32 3i 60 62 56 i)4 52 5o 47 45 43 4i 3q 38 37 36 35 34 32 3i 62 64 62 5o 49 46 A4 42 40 38 37 36 35 34 33 32 3o 64 66 48 48 45 43 4i 39 38 37 36 35 34 33 3i 29 66 68 46 43 4i 4o 38 37 36 35 34 33 32 3o 28 68 70 42 4o 39 38 38 37 36 35 34 6i 3i 28 70 72 4i 39 37 36 35 34 33 32 3o 72 74 39 37 36 35 34 33 32 3o 28 74 76 38 36 35 M M 33 3i 29 27 76 78 3(. M 34 33 32 3o 28 78 80 35 M 33 32 3i 3o 28 80 82 33 32 3i 3o 29 82 84 32 32 3i 3o 2Q 84 86 .3i 3o 29 86 32° 34° 3G° 38° 1 42° 46° 50° 54° 58° 62° 66° 70° 74° 78° 82° 86° P^?e230] TABLE XLVIII Third Correction Apparent Distance 32°. D's App. Apparent Altitude of the Sun, Stai- or Planet. D's App. Alt. 6^ .;u 8^ V' lU" 11^ 12" 14" IG" 18° 20" 22" 2-1° 2G° 28° 30° Ait. o 1 II / II 1 II 1 II / // / // / // / // / II / // / II / /; / II / // / // / II 6 I 18 I 21 I 25 I 3o I 37 I 47 I 59 2 23 1 48 3 i3 3 39 4 5 4 3o 4 55 5 20 5 45 6 7 I 23 I 18 I 21 I 24 I 28 I 'd^ I 42 2 2 18 2 37 2 58 3 20 3 4-2 4 4 4 25 4 46 7 b I 3(. I 22 1 18 I 20 I 22 I 25 I 29 I 42 I 57 2 i4 2 32 2 5o 3 8 3 26 3 44 4 2 8 9 I 38 I 27 I 20 I 18 I 19 I 21 I 23 I 3i I 44 I 58 2 12 2 26 2 4i 2 56 3 II 3 26 9 10 I 47 I 6^ I 23 I 20 I 18 I 19 I 20 I 25 I 34 I 45 I 57 2 9 2 21 2 34 2 4G 2 59 10 II I 57 I 4i I 28 I 23 I 19 I 17 1 18 I 21 I 27 I 36 I 46 I 56 2 6 2 17 2 28 2 39 II 12 2 9 I 5o 1 M I 27 I 22 I 19 I 17 I 19 I 23 I 29 I 37 I 46 I 55 2 4 2 i3 2 23 12 iJ 2 21 I 59 I 4i 1 32 I 26 I 21 I 18 I 17 I 20 I 24 I 3o I 37 I 45 I 53 2 I 2 9 i3 i4 2 34 2 8 l 5o I 38 I 3o I 24 I 20 I 16 I 18 I 21 I 25 I 3o I 36 I 43 I 5i I 58 i4 if) 2 47 2 18 I 59 I 45 1 35 I 28 I 22 I 17 I 16 I 18 I 21 I 25 I 3n I 35 I 42 I 49 i5 i6 2 59 2 28 2 7 I 52 I 4i I 32 I 25 I 19 I i5 I 16 I 18 I 21 I 25 I 29 I 35 I 4i 16 17 3 12 2 38 2 16 I 59 I 47 I 36 I 28 I 21 I 16 I i5 I 16 I 18 I 21 I 25 I 3o I 35 17 i8 3 25 2 48 2 25 2 7 I 52 I 4i I 32 I 23 I 17 I i4 I i5 I 17 I 19 I 22 I 25 I 29 18 19 3 38 2 58 2 34 2 i4 I 58 I 46 I 36 I 25 I 18 I i5 I i4 I i5 I 17 I 19 I 22 I 25 19 20 3 5u 3 9 2 4i 2 21 2 4 I 5i I 4o I 27 I 20 I 16 I i3 I i4 I i5 I 17 I 19 I 21 20 21 4 3 3 ,9 2 52 2 28 2 10 I 56 I 45 I 3o I 22 I 17 I 14 I i3 I 14 I i5 I 16 I iS 21 22 4 i5 i 3o 3 2 35 2 17 2 2 I 5o I 33 I 24 I 18 I i4 I II I 12 I i3 I i4 I 16 22 23 4 28 3 4o 3 9 2 42 2 24 2 7 I 55 I 36 I 26 I IQ I i5 I 12 I II I 12 I i3 I i4 23 24 4 4o 3 5i i 17 2 5o 2 3o 2 i3 I 59 I 39 I 28 I 21 I 16 I 12 I ID I II I 1 1 I 12 24 2b 4 52 4 I 3 26 2 57 2 36 2 iS 2 4 I 42 I 3o I 22 I 17 I i3 1 II I 10 I 10 I 10 25 26 5 4 4 12 3 34 3 5 2 43 2 24 2 8 I 46 I 33 I 24 I 18 I i3 I II I 9 I Q I Q 26 27 5 16 4 22 3 43 3 12 2 5o 2 3o 2 i3 I 5o I 34 I 26 I 19 I i4 I II I 9 I 8 I 8 27 28 5 28 4 33 3 52 3 20 2 57 2 35 2 17 I 53 I 37 I 27 I 20 I i5 I II I 9 I 7 I 8 28 29 5 4i 4 44 4 I 3 28 3 3 2 4i 2 21 I 57 I 4o I 29 I 21 I 16 I 12 I ID I 8 I 7 29 3o 5 53 4 54 4 10 3 35 3 9 2 46 2 26 2 I 43 1 3i I 23 I 17 I l3 1 10 I 8 1 6 3o 3i 6 5 5 4 4 19 3 42 3 i5 2 52 2 3i 2 4 I 46 I 33 I 24 I 18 I i3 I 10 I 8 I 6 3i 32 6 ,7 5 i4 4 27 3 49 3 21 2 57 2 36 2 8 I 4q I 36 I 26 I 19 I i4 I II I 9 I 7 32 33 6 29 5 23 4 35 3 56 3 27 3 2 2 4i 2 12 I 52 I 38 I 27 I 20 I i5 I II I 9 I 7 33 34 6 4o 5 32 4 43 4 3 3 32 3 7 2 46 2 i5 I 55 I 40 I 29 I 21 I 16 I 12 I 9 I 7 34 35 6 5o 5 4o 4 5o 4 9 3 38 3 12 2 5o 2 19 I 58 I 4-3 I 3i I 22 I 17 I l3 I 9 I 7 35 36 6 59 5 48 4 57 4 i5 3 43 3 16 2 54 2 22 2 I I 45 I 32 I 23 I 18 I i3 I 10 I 7 36 37 7 7 5 56 5 4 4 21 3 49 3 21 2 59 2 25 2 4 I 47 I 34 I 24 I 19 I i4 I 10 I 7 37 38 7 13 6 3 5 10 4 29 3 54 3 25 3 3 2 28 2 6 I 49 I 35 I 25 I 19 I i4 I 10 1 7 38 39 7 22 6 10 5 16 4 33 3 59 3 3o 3 7 2 3i 2 8 I 5. I 36 I 26 I 20 I i5 I 10 I 7 39 4o 6 17 5 21 4 38 4 4 3 34 3 II 2 3^ 2 10 I 52 I 38 I 27 I 20 I i5 I 11 I 8 40 4i 5 26 4 43 4 8 3 38 3 i5 2 36 2 i3 I 54 I 39 I 28 I 21 I 16 I 12 I 8 4i 42 4 47 4 12 3 42 3 18 2 39 2 16 I 56 I 4i I 29 I 22 I 16 I 12 I 8 42 43 4 16 3 46 3 21 2 42 2 18 I 58 I 42 I 3o I 22 I 16 I 12 I 8 43 44 3 5o 3 24 2 45 2 20 2 I 4i I 3i I 23 I 17 I 12 I 8 44 46 3 27 2 5o 2 23 2 2 I 45 I 3, I 24 I 17 I 12 I 8 46 48 2 54 2 26 2 4 Li 47 1 34 I 25 I 18 I 12 I 8 48 bo 2 29 2 6 I 49 I 36 I 26 I 19 I i3 I 8 5o b2 2 8 I 5i I 38 I 28 I 19 I i3 I S 52 64 I 53 I 39 I 29 I 20 I i4 I 8 54 b6 I 4o I 3o I 21 I 14 I 8 56 58 I 3o I 21 I i4 I 8 58 60 I 21 I i4 I 8 60 62 I i4 I 8 62 64 I 8 64 66 66 68 68 70 70 72 72 74 74 76 78 76 78 80 80 82 82 84 84 86 86 G° 7° 8° 9° 10° IP 12° 140 1G° 18° 20° 22° 24° 2G° 28° 30° TABLE XLVIII [Page 281 Third Correction. Apparent Distance 32°. D's App. Alt. Apparent JlltiUide of t he Su I, Sta r or Plane D's App. Alt. 32° 34° 36° 38° 42° 46° 50° 54° 58° 62° 66° 70° 743 78^ 82=' 86° o ( // 1 II 1 II / // / // / /( / " / II 1 1 / /( / // / u / // / ;/ , / / II 6 6 10 6 33 6 55 7 i5 6 7 5 7 5 26 5 44 6 2 7 8 4 20 4 37 4 52 5 7 5 35 8 9 3 4i 3 56 4 10 4 24 4 5o 9 10 3 12 3 25 3 38 3 5o 4 12 10 11 2 5i 3 2 3 i3 3 23 3 42 . II 12 2 33 2 43 2 5i 3 00 3 17 3 33 12 i3 2 18 2 26 2 34 2 42 2 56 3 9 i3 i4 2 5 2 12 2 19 2 27 2 39 2 5o 14 i5 I 55 2 2 2 8 2 i4 2 23 2 35 i5 i6 1 47 I 53 I 58 2 3 2 l3 2 22 2 3o 16 17 I 40 I 45 I 5o I 54 2 2 2 II 2 18 17 18 I 34 I 38 I 42 I 46 I 53 2 2 7 18 ip I 29 I 33 I 36 I 39 I 45 I 5i I 57 19 20 I 25 I 28 I 3i I 33 I 38 I 43 I 49 I 54 20 21 I 21 I 24 I 26 I 28 I 32 I 37 I 42 I 46 21 22 I 18 I 20 I 22 I 24 I 27 I 3i I 35 1 39 22 23 I i5 I 17 I 19 I 20 I 23 I 27 I 3o I 34 23 24 I i3 I i4 I 16 I 17 I 20 I 23 I 26 I 29 I 32 24 25 I II I 12 I i3 I i5 I 17 I 19 > 21 I 24 I 26 25 26 I 9 I 10 I II I 12 I i4 I 16 I 17 I 19 I 21 26 27 I 8 I Q I 9 I 10 I 12 I i3 I i4 I 16 I 17 27 28 r 8 I 8 I 8 I 9 I 10 I II 1 12 I i3 I 14 I i5 28 29 I 7 I 7 I 7 I 7 I 8 I 9 I 9 I 10 I II I II 29 3o I 6 I 6 I 6 I 6 I 6 I 7 I 7 I 7 I 8 I 8 3o 3i I 6 I 6 I 6 I 5 I 5 I 5 I 5 I 5 I 5 I 5 3i 32 I 6 I 5 I 5 I 4 I 4 I 4 I 4 I 4 I 3 I 3 I 3 32 33 I 5 I 4 I 4 I 3 I 3 I 2 I 2 [ 2 I I I I I I 33 34 I 5 I 4 I 3 I 2 I 2 I I I I 59 59 59 34 35 I 5 I 3 t 3 I 2 I I I 59 58 57 57 57 35 36 I 5 I 3 I 2 I I I I I 58 57 56 56 55 54 36 37 I 5 I 3 I I I I 59 57 56 55 55 54 53 37 38 I 5 I 3 I I I 59 58 56 55 54 54 53 52 38 39 I 5 I 3 I I 59 58 57 56 54 53 52 5i 5o 39 40 I 5 I 2 I 59 58 56 55 53 52 5i 5o 49 48 40 4i I 5 I 2 I 59 58 56 54 52 5i 5o 49 48 47 4i 42 I 5 I 2 I 59 57 55 53 5i 5o 49 48 47 47 42 43 I 5 I 2 I 58 56 54 52 5i 49 48 47 47 4b 45 43 44 I 5 I 2 I 58 55 53 5i 5o 49 48 47 46 45 44 44 46 r 5 I 2 I 58 55 52 5i 5o 48 47 46 45 44 43 4b 48 I 5 I 2 59 57 55 52 5o 49 47 46 45 44 43 42 4i 48 5o I 5 I 2 59 57 54 5i 49 48 47 46 44 4i 42 4i 40 5o 52 I 4 I I 58 56 53 5i 49 47 46 45 43 42 4i 4o 39 38 52 54 I 4 I I 58 56 53 5o 48 46 45 44 42 4i 4o 39 38 37 54 56 I 4 I I 58 56 52 49 47 45 44 41 4i 4o 39 38 37 36 5b 58 I 4 I I 58 56 52 49 47 45 43 4i 4o 39 38 37 36 35 58 60 I 4 I 57 55 5i 48 46 44 42 40 39 38 37 3b 35 3b bo 62 I 3 59 56 54 5i 48 45 43 4i 39 38 37 36 35 34 34 62 64 I 3 59 56 54 5o 47 45 43 4i 38 38 37 36 35 34 33 64 66 I 3 59 56 54 5o 47 44 42 40 38 37 36 35 M 33 bo 68 59 55 53 48 46 44 42 4o 38 37 36 35 34 33 68 70 55 52 48 45 43 4i 39 37 36 35 M 33 70 72 52 47 44 42 40 38 37 36 35 33 32 72 l4 47 44 42 4o 38 36 35 M 32 74 76 47 43 4i 39 38 36 35 34 32 7b 78 4i 4i 39 37 35 34 33 78 80 43 4i 39 37 35 M 33 80 82 4o 38 36 34 33 82 84 39 38 36 34 33 84 86 37 35 34 86 32° 34° 36° 38° 42° 46° 50° 54° 58° 62° m° 70° 74° 78° 82^ 86° 36 rage 262] TABLE XLVIII Third Correction. Apparent Distance 3G°. 5's A pp. All. Apparent Altitude of the Sun, Star or Planet. 3's App. Alt. 6" 7° 8° 9° 10° 11° 12° 14° 1G° 18° 20° 22° 24° 2G° 28° 30° o 1 II 1 II / » / /; / // 1 II / // 1 II / II 1 II / /; 1 II / // 1 II / /I / // b I 17 I 19 I 22 1 27 I 33 I 42 I 52 2 i3 2 34 2 56 3 19 3 43 4 7 4 3i 4 55 5 18 6 7 I 20 I 17 1 19 1 22 1 26 1 3i 1 37 1 52 2 10 2 28 2 48 3 8 3 27 3 46 4 6 4 25 7 8 I 2b I 20 I 17 1 19 1, 21 I 23 1 27 I 39 1 53 2 8 2 24 2 4o 2 57 3 i4 3 3o 3 46 8 9 I 32 I 24 1 19 I 17 I 18 1 19 I 21 1 29 I 4o I 52 2 5 2 19 2 33 2 47 3 2 3 16 9 lO I 42 I 3o 1 23 1 19 1 lb 1 17 1 18 1 23 1 3i 1 40 1 5i 2 2 2 14 2 27 2 4o 2 52 10 11 I 52 I 37 1 28 1 22 1 18 1 16 I 17 1 19 I 35 1 33 I 42 1 5i 2 1 2 12 2 23 2 33 11 12 2 3 I 4b 1 34 I 26 1 20 1 17 I lb 1 17 I 21 I 27 I 34 1 4i 1 5o 1 59 2 8 2 17 12 iJ 2 i4 I ti6 I 4o 1 3o 1 23 I 19 1 lb 1 i5 I 18 1 23 I 28 1 34 I 4i I 4q I 57 2 5 i3 14 2 25 2 I I 47 I 6'j 1 26 I 21 1 18 I i4 I 16 I 19 I 24 I 29 1 35 I 4i I 4q 1 55 i4 lb 2 3b 2 10 i.b4 1 4i 1 3o 1 25 I 21 1 16 1 i5 1 17 I 21 1 25 I 3o I 35 1 4i I 46 i5 i6 2 48 2 20 2 2 I 47 I 35 I 29 1 24 1 18 I i3 I i5 1 18 1 21 I 25 1 2q I 3A T 3g ifi 17 3 2 3o 2 ID I 53 I 4o 1 33 I 28 I 20 1 i5 I i4 1 16 1 18 I 21 I 24 I 28 I 3,"^ 17 i8 3 12 2 4o 2 18 2 1 46 1 38 I 32 1 22 1 16 I i3 1 i5 1 16 T 18 I 20 1 23 1 27 18 ^9 3 24 2 49 2 27 2 7 I 5i 1 43 I 36 1 25 1 18 I i5 I i4 1 i5 1 t6 I 18 I 20 [ 23 19 20 21 20 21 3 3b 3 46 2 b9 3 9 2 3b 2 43 2 14 2 21 1 57 2 3 I 48 1 53 1 4o I AA 1 28 1 3i I 21 1 23 I 16 I 17 I 12 1 i3 1 i3 1 12 I i4 1 16 I 18 1 20 I i3 I i4 1 16 I 18 22 3 b7 i 18 2 5l 2 28 2 9 1 58 I 48 1 34 1 25 1 18 I i4 1 11 I 12 1 i3 I i4 1 i5 22 2J 4 9 ;i 28 2 59 2 35 2 16 2 3 1 52 I 36 1 26 I 19 I 14 1 11 I 10 1 11 1 12 I i3 2 3 24 4 20 i 37 i 7 2 42 2 22 2 8 I 5b I 39 1 28 1 20 1 i5 I 11 I 9 I 1 10 1 1 1 24 26 4 32 i 47 3 lb 2 49 2 28 2 i3 2 1 42 1 3o 1 22 I i5 I 11 I 9 I 8 1 8 I 9 25 26 4 43 3 56 3 23 2 56 2 34 2 18 2 4 1 45 I 32 I 23 I 16 1 12 I 9 I 7 I 7 26 27 4 bb i 6 3 3i 3 3 2 4o 2 23 2 9 1 48 I 35 1 25 I 17 1 12 'f 9 1 7 1 6 1 6 27 28 b 6 4 lb 3 39 3 10 2 46 2 28 2 i3 1 52 I 38 1 27 I iq 1 i3 I 9 I 7 I 6 I 6 28 29 b 17 4 2b 3 47 3 17 2 52 2 3A 2 18 I 56 I 4o I 2q I 20 1 i3 I 9 1 7 I 6 I 5 29 do b 28 4 M 3 b4 3 24 2 58 2 39 2 23 2 I A3 I 3i I 21 I i4 I 10 1 8 I 6 1 5 3o 3i 5 39 4 43 4 2 3 3i 3 4 2 44 2 28 2 4 I 46 I 33 1 23 1 16 1 11 1 8 I 6 I 5 3i 32 b 49 4 b2 4 10 3 37 3 10 2 49 2 33 2 7 I 49 I 35 1 25 1 17 1 12 I 9 I 7 I 5 32 J3 b b9 b 4 18 3 AA 3 lb 2 54 2 37 2 10 1 5i 1 37 1 27 1 19 1 i4 I 10 I 7 1 5 33 34 6 9 b 8 4 25 3 5o 3 22 2 59 2 4i 2 i3 I 53 . 39 1 29 1 21 I i5 I 11 1 8 1 6 34 3b 6 19 b lb 4 32 3 56 3 28 3 4 2 46 2 16 I 56 I 4i I 3o 1 22 I 16 1 11 1 8 1 6 35 36 6 28 5 24 4 38 4 2 3 33 3 9 2 5o 2 19 I 59 I 43 1 32 I 23 I 17 1 12 I 9 1 6 3b ^7 6 38 b 32 4 4b 4 8 3 39 3 14 2 54 2 22 2 I I 45 1 33 I 24 I 18 I i3 I 9 I 6 37 38 6 47 3 4o 4 b2 4 i4 3 AA 3 18 2 5t; 2 25 2 4 1 47 1 35 I 26 I 19 I i4 I 10 I 6 38 39 6 b7 b 48 4 b9 4 20 3 49 3 23 3 2 2 28 2 6 I 49 1 36 1 27 1 20 1 i4 1 10 I 7 39 40 7 b b bb b 5 4 25 3 54 3 27 3 6 2 3i 2 8 1 5. I 38 1 28 1 21 1 i5 1 11 I 7 4o 4i 7 16 5 4 5 12 4 3i 3 59 3 3i 3 10 2 33 2 1 1 I 53 1 40 1 3o I 22 I i5 1 11 1 8 4i ■ 42 7 25 3 12 b 18 4 36 4 3 3 35 3 i3 2 36 2 i4 I 55 1 42 1 3i I 22 1 16 I 11 1 8 42 Ai 7 33 J 1.9 b 24 4 4i 4 8 3 39 3 17 2 39 2 16 1 57 1 43 1 32 I 23 I 16 1 11 I 8 43 AA b 2b b 3o 4 46 4 12 3 43 3 20 2 42 2 18 I 5q I 45 I 33 I 24 I 17 I 12 I 9 AA 46 b 4i 4 55 4 20 3 5o 3 26 2 47 2 22 a 2 I 47 I 35 1 25 1 18 I i3 I 9 46 48 4 27 3 57 3 32 2 52 2 26 2 5 I 49 I 37 1 27 1 20 I i4 1 10 48 bo 3 38 2 57 2 3o 2 8 1 5i I 39 I 29 1 21 I i5 I 10 5o • b2 3 1 2 33 2 II I 53 I 4i 1 3i 1 22 1 lb I II 52 b4 2 36 2 i3 I 55 I A3 1 32 1 23 1 lb I 11 54 bb 2 i5 I 57 I 59 I AA I 45 I 46 1 33 I 34 1 35 I 24 I 25 I 26 I 17 I 18 I 18 1 11 1 12 1 12 56 58 60 TaUc P. Effect of Sun's Par. A'l i the Nuiiihers above the lines I 3b 1 26 I 18 1 12 62 tu Third Corrt-clinii ; sublnict thri oMiers. I 26 1 19 I 19 I 12 T l3 64 66 D'g Srin'fi Appiireiit Altitude. Arp. AH. 1 i3 68 70 72 5 1 20 31 •10 50 70 HO 90 5 3 5 7 " 10 1 T 2 4 5 74. 20 4 i I 1 3 4 76 30 6 3 1 •2 3 78 40 9 i S 3 •2 1 '2 80 SO 7 5 4 2 1 1 82 60 7 5 4 3 2 1 84 70 6 3 4 3 2 86 80 6 4 11° 12° 14° 1G° 18° 20° 22° 24° 2G° 28° 30-- TABLE XLVllI [Page 283 Third Correction Apparent Distance 36°. D's Alt. Apparent Altitude of the Sun. Star or Planet. li's App. Alt. 32° 34° 36^ 38° 42° 46° 50° 54° 58° 62° 66° 70° 74° 78° 82° 86° o ( // / // 1 II 1 II 1 II / // / // / II / // / /' / // / // / // / II / // / // 6 5 40 6 I 6 22 6 43 7 24 6 7 4 43 5 I 5 195 36 6 II 7 8 4 I 4 16 4 3i 4 46 5 16 5 45 8 9 3 20 3 42 3 55 4 8 4 33 4 58 9 10 3 4 3 16 3 273 38 359 4 20 10 II 2 43 2 54 3 43 i3 3 32 3 5c. II 12 2 27 2 36 2 45 2 53 3 10 3 25 3 40 12 i3 2 i3 2 21 2 29 2 37 2 5i 3 4 3 16 i3 i4 2 2 2 9 2 16 2 23 2 36 2 47 2 57 i4 i5 i6 r 53 I 45 I 59 1 5o 2 5 I 56 2 II 2 I 2 23 2 12 2 33 2 21 2 42 2 29 2 36 i5 16 17 I 38 I 42 I 47 I 53 2 2 2 10 2 17 2 24 '7 i8 I 32 I 36 I 4o I 45 I 53 2 I 2 7 2 i3 18 19 I 27 I 3o I 34 I 38 I 45 I 52 I 58 2 3 19 20 I 23 I 26 I 29 I 33 I 38 I 4A I 49 I 54 I 58 20 21 I 20 I 22 I 25 I 28 I 33 I 38 I 43 I 47 I 5i 21 22 I 17 I 18 I 20 I 23 I 28 I 33 I 37 I 4i I 45 22 23 I i4 I i5 I 17 I 19 1.24 I 28 I 32 I 36 I 39 23 24 I II I 12 I i4 I 16 I 20 I 23 I 27 I 3i I 'M I 37 24 25 26 I 9 I 10 I II I i3 I 16 713 I 19 I 16 I 22 I 18 I 26 I 21 I 29 I 24 I 3i I 26 25 26 I 8 I 8 I 9 I II 27 I 7 I 7 I 8 I 9 I II I i3 I i5 I 17 I 20 I 22 27 28 I 6 I 6 I 7 I 8 I 9 I II I 12 I i4 I 16 I 18 I 20 28 29 I 6 I 6 I 6 I 7 I 8 I 9 I 10 I II I i3 I i4 I 16 29 So I 5 I 5 I 5 I 6 I 7 I 7 I 8 I 9 I 10 I II I i3 3o 3i I 5 I 5 I 5 I 5 I 6 I 6 I 6 I 7 I 8 I 9 I ID 3i 32 I 4 I 4 I 5 I 5 I 5 I 5 I 5 I 5 I 6 I 7 I 8 I 9 32 33 I 4 I 4 I 4 I 4 I 4 I 4 I 4 I 4 I 4 I 5 I 5 I 6 36 34 I 4 I 3 I 3 I 3 I 3 I 3 I 3 I 3 I 3 I 3 I 3 I 3 34 35 I 4 I 3 I 3 I 3 I 2 I I I I I I I I I I I I I I 35 36 I 4 I 3 I 2 I 2 I I I I I I I I I I 36 37 I 4 I 3 I 2 I 1 59 59 59 59 59 59 59 59 58 37 38 I 4 I 3 I 58 58 58 58 58 68 58 58 57 38 39 I 5, I 3 I 58 58 58 58 57 57 57 56 56 39 40 I 5 I 3 I 58 57 57 57 57 56 56 55 54 53 40 4i I 6 I 3 59 57 56 56 56 56 55 54 53 52 52 4i 42 I 6 I 3 59 57 56 55 55 55 54 53 52 5i 5i 42 43 I 6 I 3 59 56 55 54 54 54 53 52 5i 5o 5o 49 43 44 I 6 I 3 59 56 54 53 53 53 52 5i 5o 49 49 48 44 46 I 6 I 3 59 56 54 53 52 5i 5o 49 48 48 47 47 46 48 I 7 I 3 59 56 54 52 5i 49 48 47 46 46 45 45 45 48 5o I 7 I 3 59 56 53 5i 5o 48 47 46 45 45 44 44 44 5o 52 I 7 I 3 59 55 52 5o 49 48 47 46 45 44 43 42 42 52 54 I 7 I 3 59 55 52 5o 48 47 46 45 44 43 42 4i 4i 54 56 I 7 I 3 X 58 55 52 49 48 47 46 45 44 43 42 41 40 5b 58 I 7 I 3 I 58 55 52 49 47 46 45 44 43 42 4i 40 39 58 60 I 7 I 3 I 58 55 5i 48 46 45 44 43 42 4i 40 39 38 bo 62 I 7 I 3 I 58 54 5i 48 46 44 43 42 4i 4o 39 38 62 64 I 7 I 3 r 58 54 5i 48 46 44 43 42 4o 39 38 37 64 66 I 8 I 3 r 57 54 5o 47 45 43 42 4i 39 38 37 bb 68 I 8 I 3 I 57 54 5o 47 45 43 42 40 39 38 37 68 70 I 8 I 3 I 57 53 5g 47 44 42 4i 40 39 38 70 72 I 3 I 57 53 5o 46 43 4i 4o 39 38 72 74 I 57 52 49 46 43 4i 4o 39 38 74 76 57 52 48 45 43 4i 39 38 37 7b 78 5i 48 45 42 40 39 37 78 80 5i 47 M 42 40 39 37 80 82 47 44 4i 40 38 82 84 47 44 4i 39 38 84 86 44 4i 39 86 32° 34° 36° 38° 42° 46° 50° 54° 58° 62° 6G° 70° 74° 78° 82°j86° P«s''234] TABLE XLVIII Third Correction. Apparent Distance 40°. J)'s App. All. Apparent Altitude of the Sun, Star or Planet. A^'l?.- 1 II 7° 8° 9° / // 10° 11° 12° 14° 16° / // 18° / // 20° ' II 22° 1 II 24° 26° 28° / // 30° / II / » / // / // / // 6 T l6 I i8 I 21 I 25 I 3i I 39 I 47 2 5 2 26 2 48 3 10 3 32 3 54 4 16 4 38 A 59 6 7 I iq I i6 I 18 I 21 I 24 I 28 I M I 48 2 4 2 22 2 40 2 58 3 16 3 3A 3 52 4 10 7 8 I 24 I IQ I 16 I 18 I 20 I 22 I 26 I 36 I 5o a 4 2 18 2 33 2 48 3 4 3 20 3 36 8 9 I 3i I 23 I 19 I 16 I 18 I 19 I 21 I 27 I 38 I 49 2 I 2 i3 2 25 2 38 2 52 3 5 9 10 I 4o I 29 I 23 I 19 I ifa I 17 I 18 I 21 I 29 I 38 I 48 I 58 2 9 2 20 2 32 2 A4 10 II I 5o I 36 I 28 I 22 I 18 I l5 I 16 I 18 I 23 I 3i I 39 I 48 I 57 2 7 2 17 2 27 II 12 2 I I M I 34 I 26 I 20 I 17 I i5 I 17 I 20 1 26 I 33 I 40 I 48 I 57 2 5 2 i3 12 i3 2 I I I 52 I 4o I 3o I 23 I 19 T 16 I 16 I 18 I 22 I 28 I 34 I 4i I 48 I 55 2 2 i3 i4 2 21 2 I 46 I M I 26 I 2] I 17 I i5 I 17 I 19 I 23 X 28 I 34 1 40 I 46 I 53 i4 i5 2 3l 2 8 I '5? I 39 I 3o I 23 I 19 I 16 I i5 I 17 I 20 I 23 I 27 I 32 I 38 I 44 i5 i6 2 4i 2 16 I 58 I A^ I 34 I 26 I 21 I 17 I i4 I i5 I 17 I 19 I 22 I 26 I 3i I 37 16 17 2 52 2 24 2 4 I 49 I 38 I 3o I 24 I 19 I i5 I 14 I i5 I 17 I 19 I 22 I 2fa I 3i 17 i8 3 3 2 32 2 1 1 I 54 I 43 1 iA I 28 I 21 I 16 I i3 I i4 I i5 I 17 I 19 I 22 I 26 18 19 3 i4 2 4i 2 18 2 I 48 I 39 I 32 I 23 I 17 I i4 I i3 I i4 I i5 I 17 1 19 I 22 19 20 3 25 2 5o 2 25 2 6 I 53 I Ai I 36 I 25 I 19 I i5 I 12 I 12 I i3 I i5 I 16 I 19 20 21 3 36 2 59 2 32 2 12 I 58 I 47 I 39 I 27 I 20 I 16 I i3 I 11 I 12 I i3 I i4 I 16 21 22 3 47 3 8 2 4o 2 18 2 4 I 52 I 43 I 3o I 22 I 17 I i3 I II I II I 12 I i3 I i4 22 23 3 58 3 17 2 48 2 25 2 10 I 57 I 47 I 33 I 24 I 18 I i4 I 12 I 10 I 10 I II I 12 23 24 4 9 3 26 2 56 2 32 2 i5 2 2 I 5i I 37 I 26 I 19 I i5 I 12 I 9 I 9 I 9 I 10 24 25 4 2(1 3 35 3 4 2 39 2 21 2 7 I 56 -I 4o I 28 I 21 I 16 I i3 I 10 I 8 I 8 I 9 25 26 4 3o 3 U 3 12 2 45 2 27 2 12 2 I 43 I 3o I 22 I 17 I i3 I 10 I 8 I 8 r-i 9 26 27 4 4i 3 53 3 20 2 52 2 33 2 17 2 4 I 47 I 33 I 24 I 18 I i4 I II I 8 I 7 I 8 27 28 4 5i 4 2 3 28 2 59 2 39 2 23 2 8 I 5o I 35 I 25 I 19 I i4 I II I 8 I 7 I 7 28 29 5 I 4 11 3 36 3 6 2 45 2 28 2 12 I 53 I 38 I 27 I 20 I i5 I 12 1 9 I 7 I 7 29 3o 5 12 4 20 3 M 3 i3 2 5o 2 33 2 17 I 56 I 4o I 29 I 21 I i5 I 12 I 9 ' 7 I 6 3o 3i 5 23 4 29 3 52 3 20 2 56 2 38 2 21 2 I 43 I 3o I 22 I 16 I 12 I 9 I 7 I 6 3i 32 5 33 4 38 3 59 3 27 3 I 2 43 2 26 2 3 I 45 I 32 I 23 I 17 I i3 I ID I 7 I 6 32 33 5 43 4 46 4 6 3 33 3 7 2 48 2 3o 2 6 I 47 I 34 I 24 I 18 I i4 I ID I 8 I 6 33 34 5 52 4 54 4 i3 3 39 3 i3 2 53 2 34 2 9 I 49 I 36 I 26 I 19 I i5 I II I 8 I 6 34 35 6 I 5 2 4 20 3 45 3 19 2 58 2 38 2 12 I 5i I 38 I 27 I 20 I i5 I II I 8 I b 35 36 6 lo 5 10 4 26 3 5i 3 24 3 2 2 42 2 i5 I 54 I 4o I 29 I 22 I 16 I 12 I 8 I 6 36 37 6 i8 5 17 4 32 3 57 3 29 3 7 2 46 2 18 I 57 I 42 I 3i I 23 I 17 I 12 I 9 1 7 37 38 6 26 5 24 4 38 4 3 3 33 3 11 2 5o 2 21 2 I 44 I 33 I 25 I 18 I l3 I 9 I 7 38 39 6 34 5 3i 4 44 4 8 3 38 3 i5 2 54 2 24 2 2 I 46 I 35 I 26 I 19 I i4 I 10 I 7 39 4o 6 42 5 38 4 5o 4 i3 3 42 3 .9 2 58 2 27 2 5 I 48 I 37 I 28 I 20 I 14 I 10 I 7 40 4i 6 5o 5 45 4 56 4 19 3 47 3 24 3 2 2 3o 2 8 I 5i I 39 I 29 I 21 I i5 I II I 8 4i 42 6 58 5 52 5 2 4 24 3 5i 3 28 3 6 2 33 2 10 I 53 I 4i I 3o I 22 I 16 I II I 8 42 43 7 7 5 59 5 8 4 29 3 56 3 323 10 2 36 2 i3 I 55 I 43 I 32 I 23 I 17 I 12 I 9 43 U 7 i6 6 6 5 i4 4 34 4 3 36 3 i3 2 39 2 i5 I 57 I AA I 33 I 24 I 18 I i3 I 9 AA 46 7 33 6 21 5 26 4 44 4 9 3 A^ 3 20 2 AA 2 19 2 I I 47 I 35 I 27 I 20 I i4 I 10 46 48 7 5o 6 35 5 38 4 54 4 18 3 5i 3 27 2 49 2 23 2 5 I 5o I 37 I 29 I 22 I i5 I 1 1 48 5o 5 5o 5 3 4 27 3 58 3 33 2 54 2 27;2 8 I 52 I 39 I 3i I 23 I 17 I 12 5o 52 4 36 4 5 3 39 2 59 2 3l 2 II I 54 I 42 I 32 I 24 I 18 I i3 52 54 3 45 3 4 2 35 2 i4 I 56 I A4 I 34 I 26 I 19 I 14 54 56 3 9 2 39 2 43 2 17 2 19 1 58 2 I 46 I 48 I 36 I 37 I 28 I 29 I 20 I 21 I 14 I i5 56 58 Table v. Effect of Sun's Far. 2 21 2 2 I 49 I 38 I 3o I 22 I i5 60 Aiitl the Numbers above the lines 2 4 I 5o I 39 I 3o I 22 I 16 62 the others. I 5i I 40 I 4o I 3i I 3i I 23 I 24 I 16 I 17 64 66 D's App. Ak. Si.n's Appaa-eiU Altitude. 5 10 -iO 3 U 40 30 50 70 80 90 I 3i I 24 I 17 68 t: r. ., .. '■ .' I 24 I 17 70 5 1 2 6 I 17 72 10 1 1 4 6 74 20 ■1 3 I 2 3 4 76 30 6 5 3 5 1 2 3 78 40 8 7 5 4 2 80 50 9 7 3 4 2 1 , 82 60 9 7 5 4 3 2 2 84 86 70 80 8 6 7 5 6 4 3 4 90 G 11° 12° 14° 1G° 18° 20° 22° 24° 26° 28° -30° TABLE XLVIII [Page ith Third Correction. Apparent Distance 40°. ])'s App. Alt. Apparent Altitude of the Sun, Sta r or Planet. D's App, Alt. 32° 34° 3G= 38° 42° 46° 50° 54° 58° 62° 6G° 70° 74° 82° 86° o ( // 1 II 1 II // / II 1 II / // / // / // 1 II 1 It 1 II / / / // / / / // 6 5 19 5 39 5 59 3 19 6 57 7 33 b 7 4 27 4 44 5 I b 18 5 5i 6 20 7 8 3 5i 4 6 4 20 4 34 5 I 5 26 5 5o 8 9 3 20 3 34 3 46 J 58 4 22 4 44 5 5 9 10 2 56 i 8 3 19 J Jo 3 5o 4 9 4 27 10 II 2 37 2 47 2 57 3 6 3 25 3 42 3 58 11 12 2 22 2 3o 2 39 2 48 3 5 3 20 3 33 3 46 12 i3 2 10 2 17 2 25 2 32 2 47 3 I 3 i3 3 25 i3 i4 2 2 6 2 12 2 18 2 32 2 44 2 55 3 4 14 i5 I 5o I 56 2 I 2 7 2 19 2 3o 2 4o 2 48 i5 i6 I 42 I 47 I 52 I 58 2 8 2 18 2 27 2 35 2 42 16 I? I 36 I 4o I 45 I 5o I 59 2 8:2 16 2 23 2 3o 17 i8 I 3i I 34 r 38 1 43 I 5i I 59 2 6 2 12 2 19 18 ^9 I 26 I 29 I 33 I 36 I 44 I 5i)i 58 2 3 2 9 19 20 I 22 I 24 I 27 I 3o I 37 I 44,1 5o I 5b 2 2 5 20 21 I 18 I 20 I 23 I 26 I 32 I 38|i 44 I 49 I 53 I 57 21 22 I i5 I 17 I 19 I 22 I 28 I 33 I 38 I 4'i I 47 I 5o 22 23 I i3 I 14 I 16 I 19 I 24 I 291 33 I 38 I 42 I 45 23 24 r II I 12 I i4 I 16 I 21 I 25Ji 29 I 33 I 37 I 4o I 43 24 25 I 10 I II I 12 I i4 I 18 I 21 I 25 I 29 I 32 I 35 I 37 25 26 I 9 I 10 I II I 12 I i5 I 18 I 21 I 25 I 28 I 3o I 32 26 27 I 8 I 9 I 9 I 10 I i3 I i5 I 18 I 21 I 24 I 26 I 27 27 28 I 7 I 8 I 8 I 9 I 11 I i3 I 16 I 18 I 20 I 22 I 23 I 0.4 28 29 I 7 I 7 I 7 I 8 I 9 I II I i3 I i5 I 16 I 18 I 19 I 20 29 3o 3i I 6 I 6 I 6 I 6 I 6 I 6 I 7 I 7 I 8 I 9 I 1 1 I 12 I i3 I II I i5 I i3 I 16 I i4 I 17 I i5 3o 3i I 7 I 8 I 9 I 10 32 I 6 I 6 I 6 I 6 I 6 I 6 I 7 I 8 I 9 I 10 I II I 12 I i3 32 33 I 5 I 5 I 5 I 5 I 5 I 5 I 6 I 6 I 7 I 8 I 9 I 10 I 10 ii 34 I 5 I 4 I 4 I 4 X 4 I 4 I 5 I 5 I 6 I 7 I 7 I 8 I 8 34 35 I 5 I 4 I 4 I 4 I 4 I 4 I 4 I 4 I 4 I 5 I 5 I 6 I 6 35 36 I 5 I A, I 3 I 3 I 3 I 3 I 3 I 3 I 3 I 4 I 4 I 4 I 4 I 4 36 37 I 5 I 4 I 3 I 2 I 2 I 2 I 2 I I I I I 2 I 2 I 2 I 2 I 2 il 38 I 5 I 4 I 2 I I I I I I I I I I I I I I I I I I 38 3q I 5 I 4 I 2 I I I I I 59 59 59 59 59 59 ^9 39 4o I 5 I 4 I 2 I I I 59 59 58 58 67 S7 57 !57 57 ^7 40 4i I 6 I 4 I 2 I I 5q 58 58 57 57 56 56 56 56 56 56 4i 42 I 6, I 6 I 4 I 2 I 58 57 57 56 56 55 55 55 55 55 55 42 43 I 4 I 2 I 58 57 56 55 55 54 54 54 54 54 54 54 4i 44 I 6 I 4 I 2 I 58 56 55 54 54 53 53 53 53 53 53 53 44 46 I 7 I 4 I 2 I 58 «56 54 53 53 52 52 5i 5i 5i 5i 5i 46 48 I 8 i 5 I 2 I 58 55 53 52 52 5i 5i 5o 49 49 49 49 48 5o I 8 I '^ I 2 I 57 54 52 5i 5i 5o 49 48 48 48 48 48 5o 52 I 9 I 5 I 2 I 57 54 52 5o So 49 48 47 47 46 46 46 52 54 I 9 I 5 I 2 I 57 54 5i 49 49 48 47 46 46 45 45 45 54 56 I 10 I 6 I 3 I 56 53 5i 49 48 47 46 45 45 44 44 44 56 58 t 10 I 6 I 3 I 56 53 5o 48 47 46 45 45 44 43 43 58 60 r 10 I 7 I 4 56 52 5o 48 47 45 44 44 Ai 42 42 60 62 I II I 7 I 4 56 52 5o 48 46 45 44 43 42 42 62 64 I II I 7 I 4 56 52 49 47 45 44 43 42 4i 4i 64 66 I 12 I 7 I 4 56 52 49 47 45 43 42 42 4i 66 68 I 12 I 8 I 4 56 52 49 47 45 43 42 42 4i 68 70 I 12 I 8 I 4 55 5i 48 46 44 43 42 42 70 72 I i3 I 8 I 4 55 5i 48 46 44 43 42 4i 72 74 I i3 I 8 I 4 55 5i 48 46 44 43 42 74 76 I 8 I 4 55 5i 48 46 44 42 4i 7b 78 I 4 55 5i 48 46 43 42 78 80 55 5i 48 46 43 4i 80 82 55 5i 48 46 43 82 84 55 5i 48 46 43 84 86 5i 48 45 8b 32° 34° 36° 38° 42° 46° 50° 54° 58° 62° GG° 70° 740 78° 82° 86° 1 ^^^^^^^^ TABLE XLVIIL Third Correction. Apparent Distance 44° D's App. Alt. Apparent Altitude of the Sun, Star or Planet. D's App. Alt. 6° 7° 8° 9° 10° 11° 12° 14° 16° 18° 20° 22° 24° 26° 28° 30° o / // 1 II / » / II / // 1 II / // / // / // / // / II / II / // / // / /; / // b X It ) I 18 [ 21 I 25 I 3i I 37 I 45 2 3 2 23 2 44 3 5 3 25 3 45 4 5 4 25 4 44 6 7 I 2C ) I 16 I 18 I 20 I 24 1 28 I 33 I 46 2 1 2 17 2 34 2 5i 3 8 3 25 3 42 3 59 7 8 I 2i I 19 I lb I 17 I 19 I 22 I 25 I 35 I 47 2 2 i4 2 29 2 43 2 58 3 12 3 27 8 9 I 3i I 23 I lb I i5 I lb I 18 I 21 I 27 I 36 I 47 I 5a 2 12 2 24 2 36 2 48 3 9 10 lO I 35 ^I 28 I 21 I 17 I i5 I lb I 18 I 22 I 29 I 38 I 48 I 58 2 8 2 18 2 29 2 39 II I 4t I 34 I 25 I 20 I 17 I i5 I 16 I 19 I 24 I 3i I 39 I 47 I 56 2 5 2 i4 2 24 II 12 I 5fc I 4i I 3o I 23 I 19 I lb I i5 I 17 I 20 I 25 I 32 I 38 I 46 I 54 2 2 2 II 12 i3 2 t I 48 I 35 I 27 I 22 I 18 I 16 I i5 I 17 I 21 I 26 I 32 I 38 I 45 I 52 I 5o t3 i4 2 It I 5b i'4i I 3i I 25 I 20 I 17 I i4 I i5 I 18 I 22 I 27 I 32 I 38 I 44 I 49 t4 i5 2 2fc 2 4 I 47 I 6b I 29 I 23 I 19 I i5 I 14 I 16 I 19 I 23 I 27 I 32 I 37 I 42 i5 i6 2 ,..,2 12 I 53 I 4i I 33 I 26 I 21 I 17 I i4 I i5 I 17 I 20 I 23 I 27 I 32 T 36 ifi 17 2 48 2 20 2 I 47 I 37 I 3o I 24 I 19 I i5 I i5 I 16 I 18 I 20 ) 23 I 26 I 3o 17 i8 2 5b 2 28 2 8 I 53 I 42 I 34 I 27 I 20 I 16 I t4 I i5 I 16 I 18 I 20 I 22 T 25 18 19 3 8 2 37 2 i5 I 59 I 47 I 38 I 3o I 22 I 17 I i4 I i4 I i5 I 16 I 17 I 19 I 22 19 20 3 lb 2 45 2 22 3 5 I 52 I 42 I 34 I 25 I 19 r i5 I i3 I i4 I i4 I i5 I 17 I 19 20 21 3 29 2 54 2 3o 2 12 I 57 I 46 I 37 I 27 I 21 I 17 I i4 I 12 1 i3 I i4 I i5 I 17 21 22 3 39 3 2 2 37 2 18 2 3 I 5i I 4i I 3o I 23 I 18 I i4 I II I 12 I i3 I i4 1 16 22 23 3 49 3 II 2 45 2 24 2 8 I 55 I 45 I 33 I 25 I 19 I i5 I 12 I II I 12 I i3 I i4 23 24 4 3 19 2 52 2 3i 2 14 2 I 49 I 36 I 27 I 20 I 16 I 12 I 10 I 10 I 11 I i3 ?4 25 4 10 3 28 2 59 2 37 2 20 2 5 I 53 I 39 I 29 I 21 I 17 I i3 I 10 I 9 I 10 I II 25 26 4 20 3 36 3 6 2 43 2 25 2 10 I 57 I 42 I 3i I 22 I 17 I i3 I 10 I 8 I 9 I 26 27 4 3o 3 45 3 i3 2 49 2 3i 2 i5 2 I I 45 I 32 I 23 I 18 1 i4 I II I 9 I 8 I 8 27 28 4 39 3 53 3 20 2 55 2 36 2 20 2 5 I 47 I 34 I 25 I 19 I i5 I 12 I 9 I 7 I 7 28 29 4 48 4 I 3 27 3 I 2 4i 2 24 2 9 I 49 I 36 I 27 I 20 I i5 I 12 1 9 I 7 I 6 29 So 4 57 4 9 3 34 3 7 2 4b 2 29 2 i4 I 52 I 38 I 28 I 21 I 16 I i3 I 9 I 7 I 6 3o 3i 5 7 4 17 3 4i 3 i3 2 5i 2 34 2 19 I 55 I 40 I 3o I 22 I 17 I i3 I 10 I 8 I 6 3 1 32 5 16 4 25 3 48 3 19 2 56 2 38 2 23 I 58 I 42 I 3i I 23 I 18 I i4 I 10 I 8 I 6 32 33 5 25 4 33 3 54 3 25 3 I 2 43 2 27 2 I I 44 I 33 I 24 I IQ I i5 I II I 9 I 7 33 34 D 34 4 40 4 I 3 3o 3 b 2 47 2 3i 2 4 I 47 I 35 I 26 I 20 I i5 I II I 9 I 7 34 35 5 43 4 48 4 8 3 m 3 II 2 52 2 35 2 7 I 5o I 37 I 27 I 21 I 16 I 12 : 9 I 7 35 36 5 5i 4 55 4 i4 3 42 3 i5 2 56 2 39 2 II I 53 I 3q I 28 I 22 I 17 I i3 I 10 I 7 36 37 6 5 3 4 21 3 47 3 20 3 2 43 2 i5 I 56 I 4i I 3o I 23 I 17 I i3 I 10 I 8 37 38 6 9 5 10 4 27 3 52 3 24 3 4 2 47 2 18 I 58 I 43 I 32 I 24 I 18 I 14 I II I 8 3S 39 6 18 5 18 4 33 3 58 3 29 3 8 2 5i 2 21 2 I I 45 I 33 I 25 I 19 I i4 I II I 8 39 4o 6 27 5 25 4 39 4 3 3 33 3 12 2 54 2 24 2 3 I 46 I 35 I 26 I 20 I i5 I II I 8 40 4i 6 36 5 32 4 45 4 8 3 38 3 16 2 58 2 27 2 6 I 48 I 37 I 27 I 21 I 16 I 12 I 9 4t 42 6 45 5 3q 4 5i 4 i3 3 42 3 20 3 I 2 3o 2 8 I 5o I 39 I 20 I 22 I 16 I 12 I 9 42 43 6 53 5 46 4 57 4 18 3 47 3 24 3 4 2 33 2 10 I 52 I 4o I 3o I 23 I 17 I i3 I 9 43 44 7 5 53 J 5 4 23 3 5i 3 28 3 7 2 35 2 12 I 54 I 42 I 32 I 24 I 18 I i3 I 10 44 46 7 14 6 6 5 i4 4 33 4 3 35 3 i4 2 4o 2 17 I 58 I 4§ I 35 I 26 I 20 I i4 I 10 46 48 7 27 6 i8 5 25 4 43 4 9 3 43 3 21 2 45 2 21 2 2 I 48 I 37 I 28 I 21 I i5 I II 48 bo 7 4o 6 29 5 35 4 52 4 18 3 5o 3 27 2 5o 2 25 2. 6 I 52 I 4o I 3i I 23 r 16 I II 5o 52 7 52 6 40 5 45 5 I 4 26 3 57 3 33 2 55 2 29 2 10 I 56 I 43 I 33 I 25 I 18 I 12 52 54 3 55 5 9 4 34 4 4 3 39 3 2 33 2 i4 I 59 I 46 I 35 I 26 I 19 I i3 54 56 4 42 4 10 3 45 3 5o 3 5 3 ID 3 i4 2 37 2 4i 2 44 2 17 2 20 2 22 2 2 2 4 2 5 I 49 I 5i I 52 I 37 I 39 I 4o I 27 1 29 I 3o I 20 I 21 I 23 I i4 I ID I 16 56 58 60 1 Table P. Effect of Sun's Pa ^. Add the Numbers above tlie lines 2 47 2 24 2 b I 53 I 42 I 3i I 23 I 17 62 lo Tl}ird Correction ; subtract tlie others. 2 26 2 7 9 8 I 54 I 55 I 43 T 44' I 32 I 33 I 24 T 95 I 18 64 66 1 19 Act All '' Sun's Apparent Altitude. I 56 I 45 I 45 I 34 I 35 I 26 68 5 1 20 30 50 6 70 80 90 u t ,, ,, ,, ,, 70 r 1 9 4 5 6 I 3b I 29 I 22 72 in T I ? 4 5 I 3o I 22 74 20 3 3 T "1 2 3 1 I 23 7b 30 5 5 3 2 1 J 2 78 40 7 7 5 4 2 1 3 1 80 so 9 8 7 5 4 3 i I 1 82 60 8 7 =!^ 3 3 2 84 70 8(1 1 8 6 S 3 86 11° 12° 14° 16° 18° 20° 22° 24° 26° 28° 30° TABLE XLVIII [Page 2E7 Third Correction. Apparent Distance 44°. D's Apixircnt Altitude of the Sun, Star or Planet. Add. Alt. Api.. Alt. 32= 34° 36° 38° 42° 46° 50° 54° 58° 62° 66° 70° 74° 78° 82° 86° o 1 II 1 II / II / // 1 n 1 II 1 II / // 1 II / II / // / II / II J II / // / // 6 5 3 5 22 5 4i 5 59 6 36 7 107 4o 6 7 4 i5 4 3i 4 ^7 5 2 5 '66 6 1,6 29 7 8 3 40 3 53 4 6 4 20 4 46 5 ii5 35 5 58 8 9 3 12 3 24 3 35 3 47 4 10 4 3i'4 5i 5 10 9 lO 2 5o 3 ol 3 10 3 20 3 39 3 58;4 17 4 34 10 II 2 33 2 42 2 52 3 3 17 3 33 3 48 4 3 II 12 2 J9 2 27 2 36 2 44 2 59 3 i3 3'26 339 3 5i 12 i3 2 6 2 i3 2 21 1 29 2 43 2 563 9 3 20 3 29 i3 i4 I 55 2 2- 2 Q 2 16 2 29 2 4i 2 53 3 2 3 10 i4 i5 I 47 I 53 I 59 2 5 2 17 2 28,2 38 2 47 2 54 i5 i6 I 4c I 45 I 5o I 56 2 7 2 17*2 26 2 34 2 4i 2 47 16 I? I 34 I 38 I 43 I 48 I 58 2 7215 2 22 2 29 2 3b 17 i8 I 29 I 33 I 37 1 42 I 5i I 59 2 6 2 12 2 18 2 24 18 19 I 25 I 28 I 32 I 36 r 44 I 52 I 59 2 4 2 9 2 i4 19 20 I 22 I 25 I 28 I 3i 1 38 I 46 I 52 I 57 2 1 2 6 2 II 20 21 I IQ I 22 I 25 I 27 I 33 I 4o'i 46 I 5i I 55 I 59 2 2 21 22 I 17 I 19 I 22 I 24 I 29 I 35!i 4o I 45 I 49 I 53 I 55 22 23 I i5 I 17 I IQ I 21 1 25 I 3o:i 35 I 4o I 44 I 47 I 49 23 24 I i4 I i5 I 16 I 18 I 22 I 26' I 3o r 35 I 39 I 42 I 44 I 46 24 25 I 12 I i3 I i4 i 16 1 19 I 22 I 26 I 3o I 34 I 3- I 39 I 4o 25 26 I 10 I II I 12 I i4 I 16 I IQ I 22 I 26 I 3o I 32 I 34 I 35 26 27 I 9 I 10 I II I 12 I i4 I 16 I 19 1.23 I 26 I 28 I 3o I 3i 27 28 I 8 I 9 I 10 I II I 12 I i4 I 17 I 20 I 22 I 24 I 26 I 27 I 28 28 29 I 7 I 8 I 8 I 9 I 10 I 12 I i5 I 17 I 19 I 21 I 22 I 23 I 25 29 3o I 6 I 7 I 7 I 8 I 9 I 10 I 12 I i4 I 16 I 18 I 19 I 20 I 22 3o 3i I 6 I 6 I 6 I 7 I 8 I 8 I 10 I 12 I 14 I i5 1 17 I 18 I 19 3i 32 I 5 I 6 I 6 I 6 I 7 I 7 I 8 I 10 I 12 I i3 I i4 I i5 I 16 I 17 32 33 I 5 1 5 I 5 I 5 I 6 I 6 I 7 I 8 I 9 I 10 I II I 12 I i3 I i4 33 34 I 5 I 4 I 4 I 4 I 5 I 5 I 6 I 6 I 7 I 8 I 9 I 10 I II I 12 M 35 I 5 I 4 I 4 I 4 I 4 I 4 I 5 I 5 I 5 I 6 I 7 I 8 I 9 I 10 35 36 I 5 I 4 I 3 I 3 I 3 I 3 I 4 I 4 I 4 I 5 I 5 I 6 I 7 I 8- I 9 36 37 I 6 I 4 I 3 I 2 I 2 I 2 I 3 I 3 I 3 I 4 I 4 I 5 I 5 I 6 I 7 37 38 I 6 I 4 I 2 I I I I I 2 I 2 I 2 I 3 I 3 I 4 I 4 I 4 I 5 38 39 I 6 I 4 I 2 I I I I I I I I I 2 I 2 I 3 I 3 I 3 I 4 39 4o I 6 I 4 I 2 I I I I I I I 1 I I I I I I I I I I I 2 I 3 40 4i I 5 I 3 I I I I I I I I I I I I I 4r 42 I 7 I 5 I 3 59 59 59 59 59 59 59 59 59 59 59 59 42 43 I 7 I 5 I 3 59 b9 58 58 58 58 58 58 58 58 58 58 43 44 I 7 I 5 I 3 59 58 57 57 57 57 57 57 57 57 57 57 44 46 I 7 I 5 I 3 59 57 56 56 56 56 55 55 55 55 55 55 46 48 I 8 I 6 t 4 I 2 59 57 55 55 55 54 54 54 53 53 53 53 48 5o I 8 I 6 I 4 I 2 59 57 55 54 54 53 53 53 52 52 52 52 5o 52 I 9 I 6, I 4 I 2 59 56 54 53 53 52 52 5i 5i 5i 5o 5i 52 54 I 10 I 7 I 4 I 2 59 56 54 53 52 5i 5i 5o 5o 49 49 54 56 I 10 I 7 I 5 I 2 59 56 54 52 5i 5o 5o 49 49 48 4- 56 58 I II I 8 I 5 I 3 59 56 53 5i 5o 49 49 48 48 47 58 60 I II I 8 I 5 I 3 5.9 56 53 5i 5o 49 48 47 47 46 60 62 I 12 I 9 I 6 I 3 59 56 53 5i 49 48 47 47 46 62 64 I i3 I 9 I 6 I 3 59 56 53 5i 49 48 47 46 45 64 66 I i4 I 10 I 7 I 4 59 56 53 5i 49 48 47 46 66 68 I i5 I II I 7 I 4 59 56 53 5i 49 47 46 45 68 70 I 16 I II ^ 7 I 4 59 55 53 5i 49 47 46 70 72 I 16 I 12 I 8 I 4 59 55 52 5o 48 46 45 72 74 I 16 I 12 r 8 I 4 59 55 52 5o 48 46 74 76 r 17 I 12 I 8 I 5 59 55 52 49 47 46 76 78 I 17 I 12 I 8 I 5 59 55 52 49 47 78 80 I 12 I 8 I 5 59 55 52 49 47 80 82 I 8 I 5 59 55 52 49 82 84 I 5 59 55 52 49 84 86 59 55 52 86 32° 34° 36° 38° 49° 46° 50° 54° 58° 62° GGP 70° 74° 78° 82° 86° ' : — — — - P^'gesssj TABLE XLVIII. Third Correction. Apparent Distance 48°. 5's App. Alt. .Apparent Altitude of the Sun, Star or Planet. \pp. Alt. 6^ 7° 8° 9° 10° 11° 12° 14° 1G° 18° 20° 22° 24° 26° 28° 30° o 1 II > // / » f // / // 1 II / // / // / II / // / // / // 1 II / II / // 1 II 6 r t6 I 17 t 19 I 23 I 29 I 36 t 43 2 I 2 20 I 39: 58: 16 3 35 3 54|4 i3|4 82 1 6 7 I 19 T ?4 I 16 I 17 I 19 I 23 I 28 [ 33 I 46 2 2 16 2 32]; 47 3 2: 18 3 34: 5o 7 8 I 19 I 16 I 17 I 19 I 22 I 26 I 35 I 47 [ 59 ; I 12 : 25 I 39 = 53 3 73 21 8 9 10 T 3n I 23 I 18 I 16 I 17 I 19 [ 21 I 28 I 37 I 47 58: - 9 2 20 5 32 2 44 2 55 9 I 37 I 27 I 21 I 18 I 16 I 17 I 18 I 23 I 3o I 38 47 56 2 b: 16 I 2b: 36 10 II I 45 [ 33 I 25 I 21 I 18 I 16 I 17 I 20 I 25 I 02 [ 39 47 I 55 2 4\ 2 i3 2 22 II 12 T 53 r 3o I 3o I 24 I 21 I 18 I 16 I 19 I 22 I 27 [ 33 4o I 47 54 2 2210 12 1 3 2 2 I 46 I 36 I 28 I 24 I 20 I 18 I 17 I 19 I 23 I 28 34 I 4o 4b [ 53 2 i3 i4 2 II I 54 1 .42 I 33 I 27 I 23 I 20 I 16 I 17 I 20 I 24 29 I 34 39 I 45 5i i4 i5 2 20 2 I I 48 I 37 I 3o I 26 I 22 I 17 I 16 I 18 I 21 [ 24 I 29 [ 33 I 38 [ 43 i5 i6 2 3u 2 Q I 54 I 42 I 34 I 29 I 24 I 18 I 16 I 17 I 18 [ 20 I 24 t 28 I 32 I 37 16 I? 9 40 2 17 2 I 47 I 38 I 32 I 27 I 20 I 17 I 16 I 17 I 18 I 21 [ 25 I 28 I 32 17 iR 2 5o 2 25 2 7 I 52 I 42 I 3b I 3o I 22 I 18 I i5 I lb I 17 I 19 I 22 I 25 I 28 18 19 3 n 2 32 2 i4 I 58 I 46 I 39 I 33 I 24 I 19 I 16 I i5 I lb I 17 I 19 I 22 I 24 19 20 21 3 9 3 18 2 4o 2 48 2 20 2 26 2 3 2 9 I 5i I 56 I 43 I 47 I 36 I 4o I 27 I 3o I 21 I 23 I 17 I 18 I 14 I i5 I i5 I i4 I lb I 17 I 1^ 1 21 20 21 I i5 I 16 I 17 I 19 22 3 on 9 56 2 33 2 i5 2 2 I 52 I 43 I 32 I 24 I 19 I lb I i3 I 14 I i5 I lb I 18 22 23 3 37 3 3 2 4o 2 21 2 7 I 56 I 46 I 35 I 26 I 20 I 16 I i4 I i3 I i4 I i5 I lb 23 0.4 3 46 3 II 2 47 2 26 2 12 2 I 5o I 37 I 27 I 21 I 17 I i4 I 12 I i3 I 13 I i4 24 25 3 56 3 19 2 54 2 32 2 17 2 5 I 54 I 4o I 29 I 22 I 18 I i5 I i3 I 12 I 12 I i3 25 "^fT 4 'i 3 27 3 I 2 38 2 22 2 9 I 58 I 42 I 3i I 24 I 19 I 16 I i3 I II I II I 12 26 27 4 i5 3 34 3 8 2 M 2 27 2 i4 2 2 I 44 I 6^ I 25 I 20 I lb I i3 I II I 10 I II 27 28 4 ?4 3 42 3 i5 2 5o 2 32 2 18 2 b I 47 I 35 I 27 I 21 I 17 I 14 I 12 I 10 I 10 28 3o 4 33 3 5( 3 21 2 56 2 37 2 23 2 10 I 5o I 37 I 28 I 22 I 18 I i5 I 12 I 10 I 9 29 4 42 3 5b 3 28 3 2 2 42 2 27 2 i3 I 53 I 4o I 3o I 23 I 19 I i5 I 12 1 9 3o 3i 4 5i A 6 3 35 3 8 2 47 2 3i 2 17 I 57 I 42 I 32 I 25 I 20 I lb I i3 I II I 9 3i 32 5 i 1 3 3 4?- 3 i4 2 52 2 35 2 20 2 1 44 I 33 I 2b I 21 I lb I i3 I II I 9 32 33 5 9 5 t8 4 21 3 49 3 20 2 57 2 39 2 23 2 3 I 46 I 35 I 27 I 22 I 17 I i4 I 12 I 10 33 34 4 2& 3 55i3 25 3 2 2 44 2 27 a 6 I 49 I 37 I 28 I 23 I 18 I 14 I 12 I 10 34 35 5 27 4 36 4 I 3 3i 3 7 2 48 2 3i 2 9 I 52 I 39 I 3o I 24 I 19 I i5 I 12 I 10 35 36 5 35 4 43 4 8 3 37 3 12 2 52 2 35 2 12 I 54 I 4i I 3i I 25 I 19 I i5 1 12 I 10 36 37 5 U 4 5c 4 14 3 42 3 17 2 57 2 39 2 :6 I 56 I 4'S I 33 I 26 I 20 I lb I i3 I II 37 38 5 52 4 5- 4 20 3 47 3 22 3 I 2 Ai 2 19 I 59 I 4b I 34 I 27 I 21 I 17 I 14 I 11 38 39 40 6 5 4 4 26 3 53 3 26 3 5 2 47 2 22 2 2 I 47 I 35 I 28 I 22 I 17 I 14 I II 39 6 8 5 II 4 32 3 58 3 3o 3 10 2 5i 2 25 2 5 I 49 I 37 I 29 I 23 I 18 I i5 I 12 40 4i 6 16 -TIT 4 38 4 3 3 35 3 14 2 55 2 28 2 7 I 5i I 39 I 3i I 24 I 19 I 16 I i3 4i 4? 6 24 5 2^ 4 M 4 8 3 4o 3 18 2 58 2 3i 2 10 I 53 I 4i I 33 I 2b I 20 I lb I 1 3 42 43 6 32 5 3t 4 5o 4 i3 3 44 3 22 3 2 2 6i 2 12 I 55 I 43 I 34 I 27 I 21 I 17 I i4 43 44 6 39 5 3- 4 55 4 18 3 48 3 26 3 5 2 36 2 i4 I 57 I 45 I 36 I 28 I 22 I 18 I i5 44 46 6 53 5 4r ,5 5 4 28 3 56 3 34 3 12 2 4i 2 18 2 1 I 4^ I 38 I 3o I 23 I 18 I i5 46 48 7 7 6 5 15 4 37 4 4 3 4i 3 18 2 46 2 22 2 5 I 5i I 4o I 3i I 24 I 19 I 16 48 5o 7 21 6 I,' 5 25 4 46 4 12 3 47 3 24 2 5i 2 26 2 8 I 53 I 42 I 33 I 25 I 20 I lb 5o 59 7 34 6 3. i5 34 4 54 4 20 3 53 3 3o 2 56 2 3o 2 12 I 5b I 44 I 35 I 27 t 22 I 17 52 54 7 47 6 3- 5 5 43 5 I 4 27 3 59 3 36 3 I 2 34 2 i5 I 59 I 4b I 37 I 29 I 23 'I 19 54 56 8 6 4( 55 5i 5 8 4 34 4 5 3 4i 3 6 2 38 2 18 2 2 I 49 I 39 I 3i I 25 I 20 56 58 5 59 5~r5 4 4o 4 46 4 II 4 16 3 46 3 5o 3 54 3 10 3 i3 3 i5 2 42 2 45 2 47 2 21 2 23 2 25 2 4 2 6 2 8 I 5i I 53 I 54 I 41 I 42 I 43 I 33 I 33 I 34 I 26 1 21 58 60 62 TabJc P. Effect of Sun's Par. I 27 I 28 I 22 AJ.I tl>e Nuiiibers ;>ljo»e ilie lines to Third Correcliun j subtract 3 17 2 49 2 27 2 10 I 56 I 45 I 35 I 28 1 22 64 the liiTs. 2 5l 2 29 2 3l 2 II 2 12 2 i3 I 57 I 58 I 59 I 4b I 47 I 48 I 3b I 37 I 38 I 29 I 3o I 3o I 23 I 24 I 24 66 68 70 Ayn. Alt. Sun's Apipareiit Allilutle. 5 lU . U 3L' -10 50 60 r 80 90 " , ,. " 2 I 49 I 39 I 3i I 24 72 5 U 3 4 5 I 5o I 4o I 32 1 25 74 10 1 1 y 3 4 3 I 4i I 33 I 25 76 20 :h H 1 V 3 30 fi .1 ^i T 1 I 33 1 2b 78 40 7 6 . 4 3 i T ] I 2b 80 50 9 8 S 4 3 2 2 1 82 60 9 6 .5 4 3 84 70 9 7 6 5 4 8b SO 90 1 8 7 7 6 10° 11° 12° U° 16° 18° 20° 22° 24° 26° 28° 30° TABLE XLVill [I'ajjf -Jti) Third Correction. Apparent Distance 48°. D's A pp. All. Jlpjmrent Altkuilc of tlie Sun, Uti r or Planet. D's AliC- .32° :m° 3G° 38° 42° 46° .^0° 54° 58° (i2° 6(i° 70° 74° 78° 82° 86" o 1 II / II / // 1 II / II ' II 1 II / '/ / // / /; / II / II / II / // / II / // 6 4 5i 5 10 5 28 5 46 6 18 6 49,7 J9 7 47 6 7 4 6 4 21 4 36 4 5i 5 19 5 45!6 II 6 35 7 8 3 34 3 48 4 I 4 14 4 38 5 I 5 22 5 42 6 I 8 9 3 7 3 19 3 3o 3 4i 4 3 4 24 4 43 5 5 17 9 lO 2 47 2 57 3 7 3 17 3 36 3 54 4 II 4 26 4 40 10 II 2 3i 2 4" 2 49 2 57 3 i4 3 3o 3 44 3 57 4 w II 12 2 17 2 25 2 -63 2 4o 2 55 3 9 3 22 3 34 3 45 3 55 12 i3 2 6 2 i3 2 20 2 27 2 40 2 52 3 4 3 i5 3 25 3 32 i3 i4 I 57 2 4 2 10 2 16 2 27 2 38 2 49 2 59 3 8 3 i5 i4 i5 I 49 I 55 2 I 2 b 2 16 2 26 2 35 2 44 2 53 3 i5 i6 I 42 I 47 I 52 I 57 2 7 2 i5 2 23 2 32 2 4() 2 46 2 52 16 17 I 36 1 4i I 45 I 5o I 59 2 6 2 i4 2 22 2 29 2 34 2 4o 17 i8 I 3i I 35 I 39 1 43 I 5i I 59 2 6 2 i3 2 19 2 24 2 29 18 iQ I 27 I 3i I 34 I 38 I 45 I 52 I 58 2 4 2 10 2 i5 2 19 19 20 I 24 I 27 I 3o I 33 I 39 I 45 I 5i I 57 2 2 2 7 2 II 2 i5 20 21 I 22 I 24 I 27 I 29 I 34 I 40 I 45 I 5i I 56 2 2 4 2 7 21 22 I 20 I 22 I 24 I 26 I 3o I 35 I 40 1 45 1 5o . 54 I 57 I 59 22 23 I 18 I 19 I 21 I 23 I 27 I 3i I 36 I 40 I 45 I 49 I 5i I 53 23 24 I 16 I 17 I 19 I 21 I 25 I 28 I 32 I 36 I 40 I 44 I 46 I 48 I 5o 24 25 26 I i4 I 12 I i5 I i3 I 16 I i4 1 18 I 16 I 22 I 19 I 25 I 23 I 29 I 26 I 32 I 29 I 36 I 32 I 39 I 34 I 4i I 36 I 43 1 38 I 45 25 I 4o 26 27 I 11 I 12 I i3 I i4 I 17 I 20 I 23 I 26 I 28 I 3o I 32 I 34 I 36 27 28 I 10 I II I 12 I 1 3 I i5 I 18 I 21) I 23 I 25 I 27 I 28 I 3c) I 32 I 34 28 29 I 9 I 10 I II I 12 I i4 I 16 I 18 1 20 I 22 I 24 I 25 I 27 I 28 1 3o 29 3o I 9 I 10 I 10 I 11 I 12 I i4 I 16 I 18 I 19 I 21 I 22 I 24 I 25 I 26 3o 3i I 9 I 9 I 9 I 10 I II I 12 I i4 I 16 I 17 I 19 I 20 I 21 I 22 I 23 3i 32 I 8 I 8 I 8 I 9 I 10 I II I i3 I i4 I i5 i 17 I 18 I 19 I 19 I 20 I 21 32 33 I 8 I 7 I 7 I 8 I 9 I 10 I II I 12 I i3 I i5 I 16 I 17 I 17 I 17 I 18 33 34 r 8 I 6 I 6 I 7 I 8 I 9 I 10 I ] I I 12 I i3 I i4 I i4 I l5 I i5 I 16 34 35 I 8 I 6 I 5 I 6 I 7 I 8 I 9 I 9 I 10 I II I 12 I 12 I i3 I i3 I 14 35 36 I 8 I 6 I 5 I 5 I 5 I 6 I 7 I 7 I 8 I 9 I 10 I 10 I II I II I ^2 I i3 36 37 I 9 I 7 I 5 I 4 I 4 I 5 I (J I 6 I 7 I 7 I 8 I 8 I 9 1 9 I 10 I II 37 33 I 9 I 7 I 5 I 3 I 3 I 4 I 5 I 5 I 6 I 6 I 7 I 7 I 8 I 8 I 8 I 9 38 39 I 9 I 7 I 5 I 3 I 3 I 3 I 4 I 4 I 5 I 5 I 6 I 6 I 6 I 7 I 7 I 7 39 4o I 9 I 7 I 5 I 3 I 2 I 2 I 3 I 3 I 4 I 4 I 5 I 5 I 5 I 6 I 6 I 6 4o 4 1 I 10 I 8 I 5 I 3 I I I 2 I 2 I 3 I 3 I 4 I 4 I 4 I 5 I 5 I 5 4i 42 I 10 I 8 I 5 I 3 I I I I I 2 I 2 I 2 I 3 I 3 I 3 I 4 I 4 I 4 42 43 I II I 8 I 6 I 4 I I I I I I I I I 2 I 2 I 2 I 3 I 3 I 3 43 44 I 12 I 9 I 6 I 4 I I I I I I I I I I I I I I I I I 44 4) I 12 I 9 I 6 I 4 59 59 59 59 59 59 59 59 59 59 59 46 48 I i3 I 10 f 7 I 4 59 58 58 58 58 57 57 57 57 57 57 48 5o I i3 I 10 I 7 I 5 59 57 57 57 57 56 56 56 56 56 5o 52 I i4 I II I 8 I 5 59 57 56" 56 56 55 55 54 54 54 52 54 I i5 I II I 8 I 6 I 2 59 57 56 55 55 54 54 53 53 54 56 I i5 I II I 8 I 6 I 2 59 57 55 54 54 53 53 52 52 56 58 I 16 I 12 I 9 I 6 I 2 59 57 55 54 53 52 52 5i 58 6<> I 16 I 12 I 9 I 6 I 2 59 57 55 53 52 52 5i 5o 60 62 I 17 I i3 I 10 I 7 I 2 59 57 55 53 52 5i 5i 62 64 I 17 1 1 3 1 10 I 7 I 2 59 57 55 53 52 5i 5o 64 66 I 18 I i4 I 10 I 7 I 3 59 57 54 52 5i 5o 66 68 I 18 I i4 I 10 I 7 I J 59 56 54 52 5i 5o 68 70 I 19 I i5 I II I 8 I 3 59 56 54 52 5i 70 72 I 19 I i5 I II I 8 I 3 59 56 54 52 5o 72 74 I 20 I i5 I 1 1 I 8 I 3 59 56 53 5i 74 76 78 I 20 I 21 I 16 I ifi I 12 I 12 I 8 • 9 I 3 59 56 53 5i 76 78 I 4 59 56 53 80 I 21 I 16 I 12 I 9 I 4 59 56 53 80 83 I 21 I 16 I 12 I 9 I 4 59 56 82 84 I 16 I 12 I 9 I 4 59 56 84 86 I 12 I 9 I 4 59 86 300 .34° 3G° 38° 42° 46° 50° 54° 58° 62° 60" 70° 74° 78° 82° 86° 37 1 I'age290] TABLE XLVIIl 1 Third Correction Apparent Distance 52°. 1 App. Jl pparcnt Altitude of the Sun, Sta r or Planet. D 's App. Alt. d" 70 8^ y^ 10° 11" 12" 14" IG" lb" 20" 22° 24" 2(i" 28" yo° Alt. o 1 II 1 •! / \i / // 1 II / II / ;/ 1 II / II / .'/ / // / // / // 1 II / // 1 It 6 I 18 I 19 t 21 I 24 I 3o I 37 I 44 2 2 17 2 34 2 5i 3 10 3 28 3 4- 4 6 4 24 6 7 I 21 I 18 I 19 I 21 I 24 I 29 I 34 I 46 2 2 i4 2 28 2 42 2 57 3 12 3 27 3 43 7 8 I 25 I 21 I 18 I 19 I 21 I 24 I 27 I 36 I 47 I 58 2 II 2 23 2 36 2 5o 3 3 3 16 8 9 I 3o I 24 I 20 I 18 I 19 I 21 I 23 t 29 I 3- I 47 I 57 2 8 2 19 2 3i 2 42 2 53 9 lO I 37 I 28 I 23 I 20 I 18 I 19 I 21 I 2D I 3o I 38 I 46 I 56 2 6 2 16 2 26 3 36 Id 1 1 I 45 I 34 I 28 I 23 I 20 I 18 I 19 I 22 I 26 I 32 I 39 I 47 I 56 2 4 2 i3 2 22 11 12 r 54 I 4i I 33 I 27 I. 22 I 20 I 18 I 20 I 23 I 27 I 33 I 40 I 47 I 54 2 2 2 10 12 i3 2 2 I 48 I 38 I 3. I 25 I 22 I 19 I 19 I 21 I 24 I 29 I 35 I 4i I 47 I 54 2 I i3 i4 2 II I 55 I 44 I 35 I 28 I 24 I 21 I 18 I 19 I 23 I 26 I 3o I 35 I 4i I 47 I 52 i4 i5 2 19 2 2 I 5o I 39 I 32 I 27 I 23 I 19 I 18 I 20 I 23 I 26 r 3o I 35 I 40 I 44 i5 i6 2 28 2 q I 55 I 44 I 35 I 3o I 25 I 20 I 17 I 18 I 20 I 23 I 26 I 3o I 34 I 38 16 I? 2 37 2 16 2 I 48 I 39 I 33 I 27 I 21 I 18 I 17 I 18 I 20 I 23 I 26 I 3o I 33 17 i8 2 46 2 23 2 6 I 53 I 43 I 36 I 3o I 23 I 19 I 16 I 17 I 18 I 20 I 23 I 26 I 29 18 '9 2 56 2 3o 2 12 I 59 I 48 I 4o I 33 I 25 I 20 I 17 I 16 I 17 I 18 I 20 I 23 I 26 J9 20 3 5 2 37 2 18 2 4 I 52 I 44 I 37 I 27 I 22 I 18 I i5 I 16 I 17 I 18 I 20 I 23 20 21 3 i4 2 44 2 24 2 9 I 57 1 48 I 4o I 29 I 23 I 19 I 16 I 16 I 16 I 17 I 18 I 20 21 22 3 23 2 52 2 3i 2 i5 2 I I 52 I 44 I 32 I 2b I 20 I 16 I i5 I i5 I lb I 17 I 18 22 23 3 32 2 59 2 38 2 20 2 6 I 56 I 47 1 34 I 26 I 21 I 17 I i5 I i4 I i5 I 16 I 17 23 24 3 4i 3 7 2 44 2 26 2 II 2 I 5i 1 37 I 28 I 23 I 18 I i5 I i4 I i4 I i5 I 16 24 2.5 3 5o 3 i4 2 5i 2 3i 2 16 2 4 I 54 I 4o I 3o I 23 I 19 I lb I 14 I i3 I i4 I i5 25 26 3 5q 3 22 2 58 2 37 2 21 2 8 I 58 I 42 I 32 I 25 I 20 I 16 I i4 I i3 I i3 1 i4 26 27 4 8 3 3o 3 5 2 42 2 26 2 12 2 2 I 45 I 33 I 26 I 21 I 17 I i5 I 14 I i3 I i3 27 28 4 17 3 38 3 12 2 48 2 3i 2 16 2 6 I 48 I 35 I 28 I 22 I 18 I i5 I 14 I i3 I i3 28 2Q 4 26 3 45 3 19 2 53 2 36 2 21 2 10 I 5i I 37 I 29 I 23 I 19 I lb 1 14 I i3 I 12 29 3o 4 34 3 53 3 25 2 59 2 4i 2 25 2 i3 I 54 I 39 I 3i I 24 I 19 I lb I i4 I i3 I J2 3o 3 1 4 43 4 3 32 3 5 2 45 2 29 2 17 I 57 I 4i I 32 I 25 1 20 I 17 I i5 I i3 I 12 3i 32 4 52 4 8 3 38 3 10 2 5o 2 34 2 20 I 59 I 43 I 34 I 27 I 21 I 17 I i5 I i3 I 12 32 33 5 4 i5 3 44 3 16 2 55 2 38 2 24 2 2 I 45 I 36 I 29 I 23 I 18 I i5 I i3 I 12 33 34 5 9 4 22 3 5o 3 21 2 59 2 42 2 27 2 5 I 48 I 38 I 3o I 24 I 19 I 16 I i4 I 12 34 35 5 17 4 29 3 56 3 27 3 4 2 46 2 3i 2 7 I 5i I 40 I 32 I 25 I 20 I 17 I i4 I 12 35 36 5 26 4 36 4 2 3 32 3 9 2 5o 2 34 2 10 I 53 I 42 I 33 I 26 I 21 I 17 I i4 I 12 36 37 5 34 4 42 4 8 3 37 3 i4 2 54 2 38 2 i3 I 56 I 44 I 34 I 27 I 22 I 18 1 i5 I l3 37 38 5 42 4 49 4 i3 3 42 3 18 2 58 2 42 2 16 I 58 I 46 I 36 I 28 I 22 I 18 I i5 I l3 38 3q 5 5o 4 56 4 19 3 47 3 23 3 2 2 46 2 19 2 I I 48 I 38 I 3o I 23 I 18 I i5 I i3 39 4o 5 58 5 3 4 24 3 52 3 27 3 6 2 49 2 22 2 3 I 5o I 39 I 3i I 25 I 19 I lb I 14 4o 4i 6 6 5 9 4 3o 3 57 3 32 3 10 2 53 2 25 2 6 I 52 I 4i I 32 I 26 I 20 I 16 I 14 4i 42 6 i4 5 i5 4 35 4 2 3 36 3 i4 2 56 2 28 2 8 I 54 I 42 I 34 I 27 I 21 I 17 I i5 42 43 6 21 5 21 4 4i 4 7 3 40 3 18 3 2 3i 2 1 1 I 56 I 44 I 35 I 28 I 22 I 18 I i5 43 44 6 28 5 27 4 46 4 12 3 44 3 22 3 3 2 34 2 i3 I 58 I 45 I 37 I 29 I 23 I 19 I 16 44 46 6 42 5 39 4 56 4 21 3 52 3 29 3 10 2 39 2 18 2 I I 48 I 39 I 3i I 24 I 20 I 17 46 48 6 55 5 5i 5 6 4 3o 3 5q 3 36 3 16 2 44 2 22 2 5 I 5i I 4i I 33 I 26 I 21 I 18 48 5o 7 8 6 2 5 16 4 38 4 7 3 43 3 23 2 49 2 26 2 8 I 54 I 43 I 35 I 27 I 22 I 19 5o 52 7 21 6 i3 5 25 4 46 4 i5 3 5o 3 29 2 54 2 3o 2 II I 57 I 45 I 3b I 29 I 24 I 20 52 54 7 33 6 23 5 34 4 53 4 22 3 56 3 35 2 59 2 34 2 i4 2 I 48 I 38 I 3i I 25 I 21 54 56 7 44 6 33 5 43 4 59 4 29 4 2 3 4o 3 4 2 38 2 17 2 2 I 5o I 4o I 32 I 20 I 22 56 58 7 £3 6 42 5 5o 5 6 4 35 4 7 3 45 3 8 2 42 2 20 2 5 I 53 I 42 I 33 I 27 I 23 58 6o 8 2 6 4q 5 56 5 12 4 4o 4 12 3 5o 3 12 2 46 2 23 2 7 I 55 I 44 I 35 I 29 I 24 60 62 6 2 5 17 4 45 4 16 3 54 3 i5 2 49 2 26 2 9 I 57 I 46 I 36 I 3o 1 25 62 64 4 5o 4 20 3 58 3 18 2 5i 2 28 2 II I 59 I 48 I 37 I 3i t 26 64 66 4 I 3 20 2 53 2 3o 2 i3 2 I 49 I 39 I 32 I 26 66 68 3 22 2 54 2 32 2 i5 2 I I 5o I 4o I 33 I 27 68 70 2 55 2 33 2 16 2 2 I 5i I 4i I 34 I 28 70 72 2 34 2 17 2 3 I 52 I 42 I 34 I 28 72 74 2 18 2 4 I 53 I 43 I 35 I 29 74 76 2 5 I 54 I 44 I 36 I 29 76 78 I 55 I 44 I 36 I 3o 78 80 I 45 I 37 I 3o 80 89 I 38 I 3o 82 84 I 3i 84 86 10° 11° 12° 14° 1G° 18° 20° 22° 24° 26° 28° 30° 86 6° T 8° 9° TABLE XLVIII. [PasoSQi Third Correction. Apparent Distance 52°. D's A pp. Apparent Jlltitadc of the Sun, Star or Planet. D's App. Alt. 32° 34^^ 36^^ 38^ 42° 4G^ 5U° 54^ 5b° 62" m° 70° 74° 78° 82° 86° Ah. / II 1 II 1 II / // / II / // / II / // / // / /; 1 II / // / // / // / // 1 II 6 4 43 5 I 5 18 5 34 6 6 6 36 7 4 7 29 7 53 6 7 3 59 4 14 4 29 4 43 5 9 5 M 5 58 6 20 6 42 7 8 3 3o 3 4^ 3 55 4 8 4 3o 4 52 5 i3 5 32 5 5o 6 6 8 9 3 4 3 i5 3 26 3 37 3 58 4 17 4 36 4 5i 5 5 5 18 9 lO 2 45 2 54 3 4 3 i4 3 32 3 II 3 48 3 26 4 4 3 4o 4 20 3 54 4 33 4 45 10 II II 2 3o 2 38 2 4i 2 55 4 6 4 16 12 2 17 2 25 1 32 2 4o 2 54 3 7 3 20 3 32 3 43 3 52 4 I 12 i3 2 7 2 i3 2 20 2 26 2 39 2 5i 3 3 3 i4 3 24 3 32 3 38 i3 i4 I 58 2 3 2 9 2 14 2 26 2 37 2 48 2 58 3 7 3 i4 3 20 i4 i5 I 49 I 54 I 59 2 4 2 i5 2 26 2 35 2 44 2 52 2 59 3 5 i5 ]6 I 42 I 47 I 5i I 56 2 7 2 16 2 24 2 32 2 4o 2 46 2 52 2 57 16 17 I 37 I 4i I 45 I 5o 2 2 8 2 i5 2 22 2 29 2 35 2 4o 2 44 17 i8 I 32 I 36 I 40 I 45 I 53 2 2 7 2 i3 2 19 2 25 2 3o 2 33 18 19 I 29 I 32 I 36 I 4o I 47 I 53 2 2 6 2 II 2 16 2 21 2 24 19 20 I 26 I 29 I 32 I 35 I 4i I 47 I 53 I 59 2 4 2 9 2 i3 2 16 2 19 20 21 I 23 I 26 I 28 I 3i I 37 I 42 I 47 I 53 I 58 2 2 2 6 2 9 2 II 21 22 1 21 I 23 I 25 I 28 I ZZ I 37 I 42 I 47 I 52 I 56 I 59 2 2 2 4 22 23 I 19 I 21 I 23 I 25 I 29 I 33 I 38 I 42 I 47 I 5i I 54 I 56 I 58 23 24 I 17 I 19 I 21 I 23 I 26 I 3o I 34 I 38 I 42 I 46 I 49 I 5i I 53 I 55 24 25 I 16 I 17 I 19 I 20 I 23 I 27 I 3o I 34 I 37 I 4i I 44 I 46 I 48 I 49 25 26 I i5 I 16 I 17 I 18 I 21 I 24 I 27 I 3o I 33 I 36 I 39 1 4i I 43 I 44 26 27 I i4 I i5 I 16 I 17 I 19 I 22 I 24 I 27 I 3o I 32 I 35 I 37 I 39 I 4o 27 28 I i3 I i4 I i5 I 16 I 17 I 20 I 22 I 24 I 27 I 29 I 3i I 33 I 35 I 36 I 37 28 29 I 12 I i3 I 14 I i5 I 16 I 18 I 20 I 22 I 24 I 2b I 28 I 3o I 3i I 32 I 33 29 3o 3i I 12 I II I 12 r II I i3 I 12 I i3 1 12 1 14 I i3 I 16 I i5 I 18 I 16 I 20 I 18 I 22 I 24 I 25 I 27 I 28 I 25 I 29 I 26 I 3o I 27 3o 3i I 20 I 22 I 23 I 24 32 I II I II I II I II I 12 I i4 I 13 I 16 I 18 I 20 I 21 I 22 I 23 I 23 I 24 I 25 32 33 I II I 10 I 10 I 10 I II I i3 I i4 I i5 I 17 I 18 I 19 I 20 I 21 I 21 I 22 I 22 33 M I II I 10 I 10 I 10 I II I 12 I i3 I i4 I ]6 I 17 I 17 I 18 I 19 I 19 I 20 I 20 34 3b I 11 I 10 I 10 I 10 I 10 I II I 12 I i3 I 14 I i5 I i5 I 16 I 17 I 17 I 18 I 18 35 36 I II I 10 I 9 I 9 I 9 I 10 I II I 11 I 12 I i3 I i3 I 14 I i5 t i5 I 16 I 16 36 -il I II I 10 I 9 I 9 I 9 I 10 I 10 I II I II I 12 I 12 I i3 I i3 I i4 1 i4 37 3d I II I 10 I 9 I 8 I 8 I 9 I 9 I 9 I 10 I 10 I II III I 1 1 I II I 12 I 12 38 39 I 1 1 I 10 I 9 I 8 I 8 I 8 I 8 I 8 I 9 I 9 I 10 I 10 I 10 I 10 I 10 I 10 39 4o I 12 I 10 I 9 I 8 I 7 I 7 I 7 I 7 I 8 I 8 I 9 I 9 I 9 I 9 I 9 I 9 4o 4i I 12 I II I 9 I 8 I 7 I 7 : 7 I 7 I 7 I 7 I 8 I 8 I 8 I 8 I 8 I 8 4i 42 I i3 I II I 9 I 8 I 6 I 6 I 6 I 6 I 6 I 6 I 7 I 7 I 7 I 7 I 7 t 7 42 43 X i3 I II I 9 I 8 I 5 I 6 I 6 I 6 I 6 I 6 I 6 I 6 I 6 I 6 I 6 I 6 43 44 I i4 I II I 9 E 8 I 6 I 5 I 5 I 5 I 5 I 5 I 5 I 5 I 5 I 5 I 5 I 5 44 46 I i4 I 12 I 10 I 9 I 6 I 4 I 4 I 4 I 4 I 4 I 4 I 3 I 3 I 3 I 3 46 48 I i5 I i3 I II I 9 I 6 I 4 I 3 I 3 I 3 I 2 I 2 I I I I I I I I 48 5o I lb I i4 I II I 9 I 6 I 4 I 2 I 2 I 2 I I I 1 t I I 5o 52 I 17 I i5 I 12 I 9 I 6 I 4 I 2 I I I I I I 59 58 58 52 54 I 18 I i5 I 12 I 9 I 6 I 4 I 2 I I I 59 59 58 57 54 :jo I 18 I i5 I 12 I 10 I 6 I 4 I 2 I 59 58 58 57 56 56 58 Ho I 19 ! 16 I 16 I i3 I t3 I 10 I 6 I 4 T A I 2 I 59 58 58 5-7 57 56 56 55 1 62 04 I 21 I 22 I 17 I 18 I i3 I i4 I 10 I II I 7 I 7 I 4 I 4 I I I I 59 58 57 56 56 55 54 Table P. Effect of Sun 3 Par. Add the Numbers fibcve (he lines to Third Correction ; sutjtract (),S I 22 I 18 I i4 I II I 7 I 7 I 7 T 4 I J 59 58 57 56 55 the Cillers. 68 I 22 r 18 I i^ I II I ^ 54 App. Alt. Sun's Apparent Altitude. i 70 I 23 I 18 I 14 I u I 3 I 58 56 5 10 2e 30'40L 60 ™ 30 90 1 72 I 23 I 19 I i5 I II I 7 I 3 I 57 55 " " ' " " " 74 I 24 I 19 I i5 I II I 7 I 3 I 57 5 1 2 3 4 4 -6 I 24 I 19 I t5 I 12 I 7 I 3 I 56 10 T 1 1 •2 3 i 4 20 3 3 1 1 i '2 3 78 I 24 I 19 I i5 I 12 I 7 I 3, I 30 5 4 3 "2 T n I 80 I 24 I 19 I i5 I 12 I 7 I 3 I 40 7 6 5 4 2 T T 82 1 25 I 20 I 16 I 12 I 7 I 3 50 8 8 6 5 4 3 3 2 2 84 1 25 I 20 I lb I 12 I 7 I 3 60 9 7 6 5 i i 3 80 . 25 I 21 I 16 I 12 I 7 70 8 7 6 5 5 32° .34° 1 36° 38° 42° 46° 50° 54° 58° 62° 66° 70° 80 90 8 7 7 S PageOiK] TABLE XLVm Third Correction Apparent Distance 5G°. App. Apparent Altiiude of the Sun, Star or Planet. 5 's App. Alt. 6" r b" y^ 10" 11" 12" 14" 16" 18" 20" 22° 24" 20" 28" yo° Alt. o II 1 II ? II / // / // 1 II / // / // / // 1 II 1 II / // / // / // / // 1 II 6 I 20 I 22 I 25 I 29 I 35 I 4i I 48 2 2 2 18 2 35 2 52 3 10 3 27 3 45 4 3 A 20 6 7 I 23 I 20 I 22 I 24 I 27 I 32 I 37 I 48 2 I 2 i5 2 29 2 43 2 58 3 12 3 27 3 42 7 8 r 28 1 23 I 20 I 21 I 23 I 26 I 29 I 38 I 48 2 2 12 2 23 2 35 2 48 3 I 3 i4 8 9 I 34 I 27 I 22 I 20 I 21 I 23 I 25 I 3i I 39 I 48 I 58 2 8 2 18 2 29 2 40 2 5o 9 10 I 4<> I 3i I 25 I 22 I 20 I 21 I 22 I 26 I 32 I 39 I 48 I 56 2 5 2 i5 2 24 2 33 10 11 I 47 I 36 I 29 I 25 I 22 I 20 I 21 I 23 I 27 I 33 I 4o I 47 I 55 2 4 2 12 2 20 II 12 I 54 I 42 I 33 I 28 I 24 I 21 I 20 I 21 I 24 I 28 I 34 I 40 I 47 I 55 2 2 2 9 12 i3 2 2 I 48 I 38 I 3i I 26 I 23 I 21 I 20 I 22 I 25 I 3o I 35 I 4i I 47 I 54 2 i3 i4 2 10 I 54 I 43 I 35 I 29 I 25 I 22 I 19 I 20 I 23 I 27 I 3i I 36 I 4i I 47 I 52 i4 i5 2 18 2 I I 48 I 39 I 33 I 28 I 24 I 21 I 19 I 21 I 24 I 27 I 32 I 36 I 4i I 46 i5 i6 2 27 2 8 I 53 I 43 I 36 I 3i I 26 I 22 I 19 I 19 I 21 I 24 I 28 I 32 I 36 I 40 16 17 2 35 2'l5 I 59 I 47 I 4o I u I 29 I 23 I 20 I I'd I 19 I 22 I 25 I 28 I 32 I 35 17 18 2 44 2 22 2 4 I 52 I 43 I 37 I 3i I 25 I 20 I 17 I 18 I 20 I 22 I 25 I 28 I 3i 18 19 2 53 2 29 2 10 I 57 I 47 I 4o I 34 I 26 I 21 I 18 I 17 I 19 I 20 I 23 I 25 I 28 19 20 3 2 2 36 2 16 2 2 I 5i I AA I 37 I 28 I 22 I 19 I 17 I 18 I 19 I 21 I 2j I 25 20 21 3 II 2 M 2 22 2 8 I 55 I 47 I 4o I 3o I 24 I 20 I 18 I 17 I 18 I 19 I 21 I 23 21 22 3 20 1 5i 2 29 2 i3 2 I 5i I 43 I 32 I 25 I 21 I 18 I 16 I 17 I 18 I 19 I 21 22 23 3 2Q 2 58 2 35 2 18 2 5 I 55 I 46 I 35 I 27 I 22 I 19 i 17 I 16 I 17 I 18 I 19 23 24 3 38 3 5 2 42 2 23 2 9 I 59 I 5o I 37 I 29 I 24 I 20 I 17 I 16 I 16 I 17 I 18 24 25 3 47 3 i3 2 49 2 29 2 i4 2 3 I 53 I 39 I 3i I 25 I 21 I 18 I 16 I 16 I 16 I 17 25 26 3 55 3 20 2 55 2 M 2 19 2 7 I 57 I 42 I 33 I 27 I 22 I 19 I 17 I 16 I 16 I 16 26 27 4 4 3 27 3 I 1 39 2 24 2 12 2 I I 45 I 35 I 28 I 23 I 19 I 17 I 16 I 16 I 16 27 28 4 12 3 34 3 8 2 45 2 29 2 16 2 5 I 48 I 37 I 3o I 24 I 20 I 18 I 16 I i5 I 16 28 29 4 21 3 4i 3 i4 2 5o 2 33 2 20 2 8 I 5i I 39 I 3i I 25 I 21 I 18 I 16 I i5 I i5 29 3o 4 29 3 48 3 20 2 55 2 38 2 24 2 12 I 54 I 4i I 33 I 26 I 21 I 18 I 16 I i5 I i5 3o 3 1 4 38 3 55 3 26 3 2 A'^ 2 28 2 16 I 57 I 44 I 34 I 28 I 22 I 18 I 16 I 16 I i5 3i 32 4 46 4 2 3 32 3 6 2 48 2 32 2 19 2 I 46 I 36 I 29 I 23 I 19 I 17 I 16 I i5 32 33 4 54 4 9 3 39 3 II 2 53 2 36 2 23 2 3 I 49 I 38 I 3i I 25 I 20 I 17 I 16 I i5 33 34 5 2 4 16 345 3 16 2 57 2 4o 2 26 2 6 1 5. I 40 1 32 I 26 I 21 I 18 I 16 I i5 34 35 5 10 4 23 3 5i 3 22 3 2 2 44 2 3o 2 9 I 53 I 42 I 34 I 27 I 22 I 18 I 16 I i5 35 36 5 18 4 3o 3 57 3 27 3 6 2 48 2 33 2 12 I 55 I AA I 35 I 28 I 23 I 19 I 17 I 16 36 37 5 26 4 37 4 3 3 32 3 10 2 52 2 37 2 i5 I 58 I 46 I 37 I 29 I 24 I 20 I 18 I 16 37 38 5 33 4 43 4 8 3 37 3 i4 2 56 2 41 2 17 2 I 48 I 38 I 3o I 25 I 21 I 18 I 16 38 39 5 4i 4 5o 4 i4 3 42 3 19 3 2 45 2 20 2 2 I 5o I 39 I 3i I 25 I 21 I 18 I 16 39 4o 5 48 4 56 4 19 •^ 47 3 23 3 4 2 48 2 23 2 4 I 5i I 40 I 32 I 26 I 22 I 19 I 16 4o 4i 5 55 5 2 4 25 3 52 3 28 3 8 2 5i 2 25 2 6 I 53 I 42 I 33 I 27 I 23 I 20 I 17 4i 42 6 2 5 8 .4' 3o 3 57 3 32 3 II 2 54 2 28 2 9 I 55 I 43 I M I 28 I 24 I 20 I 17 42 43 6 Q 5 i4 4 35 4 2 3 36 3 i5 2 58 2 3l 2 12 I 57 I AA I 35 I 29 I 25 I 21 I 17 43 44 6 16 5 20 4 40 4 7 3 40 3 19 3 I 2 34 2 i4 I 59 I 46 I 37 I 3i I 26 I 22 I 18 A^ 46 6 29 5 32 4 5o 4 16 3 48 3 26 3 8 2 4o 2 18 2 2 I 49 I 40 I 33 I 28 I 23 I 19 46 48 6 42 5 43 4 59 4 24 3 56 3 33 3 i4 2 45 2 22 2 6 I 52 I 43 I 36 I 3o I 25 I 20 48 5o 6 54 5 54 5 8 4 32 4 3 3 40 3 19 2 5o 2 26 2 9 I 55 I 45 I 38 I 32 I 26 I 21 5o 52 7 6 6 4 5 17 4 39 4 10 3 46 3 24 2 55 2 3o 2 12 I 58 I 48 I 4o I 33 I 27 I 22 52 54 7 18 6 t4 5 25 4 46 4 16 3 52 3 29 2 59 2 34 2 i5 2 I 5o I 42 I 35 I 29 I 24 54 56 7 29 6 24 5 33 4 53 4 22 3 57 3 34 3 3 2 37 2 19 2 3 I 52 I 43 I 36 I 3o I 25 56 58 7 40 6 33 5 41 5 4 28 4 2 3 39 3 7 2 41 2 22 2 6 I 54 I 45 I 37 I 3i I 26 58 60 7 5o 6 4i 5 48 5 7 4 34 4 7 3 43 3 II 2 44 2 25 2 8 I 56 I 47 I 39 I 32 I 27 60 62 7 58 6 48 5 55 5 i3 4 40 4 12 3 48 3 i5 2 47 2 28 2 II I 58 I 48 I 4o I 33 I 28 62 64 8 6 6 55 6 1 5 19 4 45 4 17 3 52 3 18 2 5o 2 3o 2 i3 2 I 5o I 4i I 34 I 29 64 66 6 7 5 24 4 5o 4 21 3 56 3 20 2 53 2 32 2 i5 2 2 I 5i I 42 I .-5 I 29 66 (38 4 55 4 25 4 3 22 2 55 2 34 2 17 2 4 I 52 I 43 I 36 I 3o 68 70 4 4 3 24 2 57 2 36 2 18 2 5 I 53 I AA I 37 I 3i 70 72 3 26 2 59 2 37 2 19 2 6 I 54 I 45 I 38 I 32 72 •74 3 I 2 38 2 20 2 7 I 55 I 46 I 39 I 32 74 76 2 39 2 21 2 8 I 56 I 47 I 39 I '66 76 78 2 22 2 8 I 57 I 48 I 40 I 33 78 80 2 9 I 58 I 48 I 4o r 34 80 82 I 58 I 48 I 40 I 34 82 84 I 49 I 4i I 34 84 86 I 4i I M 86 G" 7= 8° 9° 10° 11° 12° 14° 1G° 18° 20° 22° 24° 26° 28° 30° \ TABLE XLVm. ^^''=0293 Third Correction. Apparent Distance 56°. Ap;.. Jipparcnt Altitude of the Sun, Star or Planet. D's App. All. :32° \iA^ 3G° cid^ 42" 4t)" 5U" 54" 58" 02" m° 70" 74° 78" 82" 86" Ah. c / n 1 II / II 1 II / /' / II / // 1 II 1 II / // 1 II 1 II / // 1 II / // / '/ C^ 4 37 4 54 5 TO 5 26 5 56 6 25 6 5i 1 i5 7 37 7 58 6 7 3 57 4 n 4 25 4 38 5 3 5 29 5 52 b 12 6 3i 6 48 7 ft 3 26 3 38 3 5i 4 3 4 26 4 47 5 5 5 23 5 40 5 55 6 8 8 9 lO 3 I 3 12 3 23 3 33 3 53 4 12 4 3o 4 46 5 5 i3 5 25 9 •2 43 2 53 3 2 3 II 3 28 3 45 4 I 4 i5 4 27 4 39 4 5o 10 II 2 29 2 37 2 45 2 53 3 9 3 24 3 38 3 So 4 I 4 12 4 21 1 1 12 2 16 2 23 2 3o 2 38 2 52 3 6 3 18 3 28 3 38 3 47 3 56 4 4 12 1 3 2 6 2 12 2 18 2 25 2 37 2 5n 3 I 3 10 3 19 3 28 3 36 3 42 1 3 i4 I 57 2 3 2 8 2 i4 2 2D 2 3(i 2 47 2 56 3 4 3 12 3 19 3 24 i4 i5 I 5o I 55 I 59 2 5 2 l5 2 25 2 35 2 A4 2 5i 2 5b 3 4 3 10 i5 i6 I 44 I 48 I 53 I 58 2 7 2 16 2 25 2 33 2 39 2 45 2 5i 2 57 3 2 16 '7 I 39 I 43 I 48 I 52 2 2 8 2 16 2 24 2 3o 2 35 2 4o 2 45 2 49 17 i8 I 35 I 39 I 4-i I 47 I 54 2 I 2 8 2 i5 2 21 2 26 2 3i 2 35 2 38 18 19 I 3i I 35 I 38 I 42 I 48 I 65 2 I 2 7 2 i3 2 18 2 23 2 27 2 3u '9 20 21 I 28 I 25 I 3i I 27 I 34 1 3o I 37 I 33 I 43 I 38 . 49 I 44 I 55 I 49 2 I 54 2 6 I 59 2 10 2 3 2 i5 2 7 2 19 2 11 2 22 2 i4 2 24 2 iC 20 21 22 I 22 I 24 I 27 I 3o I 34 I 39 I 44 I 48 I 52 I 56 2 2 4 2 6 2 8 22 23 I 20 I 22 I 24 I 27 I 3i I 3d I 40 I 44 I 47 I 5i I 54 I 57 2 2 2 23 24 I 19 I 20 I 22 I 25 I 28 I 32 I 36 I 4o I A6 I 46 I 49 I 52 I 54 I 5C I 58 24 25 I 18 I 19 I 21 1 23 I 26 I 29 I 33 I 36 I 39 I 42 I 44 I 47 I 49 I 5i I 53 25 26 I 17 I 18 I 19 I 21 I 24 I 27 I 3o|i 33 I 35 I 38 I 40 I 42 I 44 I 4( I 48 26 27 I 16 I 17 I 18 I 19 I 22 I 25 I 27 I 3o I 32 I 35 I 37 I 39 I 40 I 42 1 44 27 28 I 16 I lb I 17 I 18 I 20 I 23 I 25 I 27 I 29 I 32 I M I 36 I 37 I 3q I 40 I 4i 28 29 I i5 I i5 I 16 I 17 I 19 I 21 I 23 I 25 I 27 I 29 I 3i I 33 I 34 I 35 I 36 I 37 29 3o I i5 I lb I 16 I lb I 17 I 19 I 21 I 23 I 25 I 27 I 29 I 3o I 3i I 32 I 33 I 34 3o 3i I i4 I i4 I i5 I i5 I 16 I 18 I 19 I 21 I 23 I 25 I 27 I 28 I 29 I 2c; I 3o I 3i 3i 32 I 14 I 14 I i4 I i4 I i5 I 17 I 18 I 19 I 21 I 23 I 25 I 26 I 27 I 2- I 27 I 28 32 33 I 14 I i3 I i3 I i3 I i4 I lb I 17 I 18 I 20 I 21 I 23 I 24 I 25 I 25 I 25 I 26 33 34 I i4 I i3 I i3 I i3 I i4 I i5 I 16 I 17 I 19 I 20 I 21 I 22 I 23 I 23 I 23 I 24 34 35 I 14 I i3 I i3 I i3 I i3 I 14 I i5 I 16 I 17 I 18 I 19 I 20 I 21 I 21 I 21 I 22 35 36 I i4 I i3 1 12 I 12 I 12 I i3 I 14 I i5 I 16 I 16 I 17 I 18 I 19 I ic; I 19 I 20 36 37 I 14 I i3 I 12 I 12 I 12 I 12 I i3 I i4 I i5 I l5 I 16 I 16 I 17 I 17 I 17 I 18 37 38 I 14 I i3 I 12 I II I II I 12 I i3 I i3 I i4 I i4 I i5 I i5 I 16 I iC I 16 I 17 38 39 I i4 I i3 I 12 I II I 1 1 I II I 12 I 12 I i3 I i3 I i3 I i3 I i4 I i4 I i5 I i5 39 4o I 14 I i3 I 12 I II I 10 I 10 I II I II I 12 I 12 I 12 I 12 I 12 I i3 I i3 I i3 4o 4i I i5 I 14 I 12 I II I 10 I ID I 10 I 10 I II I n I II I II I II I 12 I 12 4i 42 I lb I i4 I 12 I II I 9 I 9 I 9J1 9 I 10 I 10 I 10 I 10 I 10 I II I II 42 4i I i5 1 i4 I 12 I II I 9 I Q I 9|i 9 I 9 I 9 I 9 I 9 I 9 I IC I 10 43 44 I lb I 14 I 12 I II I 9 I 8 I 81 8 I 8 I 8 I 8 I 8 I 8 I c; ' 9 44 46 I 17 I i5 I i3 I 12 I 9 I 7 I 71 7 I b 1 b I 6 I 7 I 7 I 7 46 48 1 17 1 i5 I i3 I 12 I 9 I 7 I 61 6 I 5 I 5 I 5 I 5 I 5 I ( 48 5o I 18 I 16 I 14 I 12 I 9 I b I 5i 5 I 4 I 4 I 4 I 4 I 4 5o 52 I 19 I 17 I i5 I l3 I 9 I b r 4i 4 I 3 I 3 I 3 I 3 I 3 52 54 I 20 I 17 I i5 I i3 I 9 I b I 4i 3 I 3 I 2 I 2 I 2 54 56 58 60 1 21 I 22 I 23 I 24 I 18 I 19 I 19 I 16 I 16 I 16 I i4 I i4 I i4 I K) I 10 1 10 I b I 6 I 6 I 6 I 4i 2 I 4|i 2 I 4. 2 I 2 I I I I I I 56 I I I I I I I I Talile P. Effect of Sun's Par 62 I 20 1 17 I i4 I 10 I 4i 2 I I I Adii tlie Numbers above the liiu-s 64 66 I 24 I 25 I 20 I 21 I 17 I 18 I i4 I i5 I 7 I 7 I 4i 2 I 4 1 2 I I I 10 Tliird Correcliori ; subtract the others. 68 70 I 25 I 26 1 21 I 22 t 19 I i5 I 16 .. . 1 5)'^ Sun's Apparent Ahiluile. I 7 I 7 I 4 I 4 I 2 App. All. 5 20 3 D JO 50 6 70 80 - 90 72 74 76 I 27 I 27 I 28 I 23 I 23 I 23 I 19 I 19 I 19 I 16 I 16 I 16 I 7 I 7 I 7 I 4 I 4 I 4 I 2 5 10 20 T 3 1 2 i _i 3 2 1 3 2 4 3 78 I 28 I 23 I 20 I 17 1 7 30 5 i 3 2 I I ( 1 80 I 29 I 24 I 20 I 17 I 7 40 s 5 5 4 3 2 • 1 1 82 I 29 I 24 I 20 I 17 50 3 7 6 5 4 4 3 84 I 29 I 24 I 20 I 17 60 9 3 7 6 5 5 4 86 !_i9 1 24 I 20 r 17 70 80 9 8 7 H S 6 7 6 32° y4" 30" US'" 42° 4(j" 50" 54° 58° 62° 06° 90 8 P^^.s^^J TABLE XLVIII. Third Correction. Apparent Distance 60° D's App. Alt. o 6 7 8 9 10 II 12 i3 i4 i5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 So 3i 32 33 34 35 36 37 38 39 40 4i 42 43 46 48 5o 52 54 56 58 60 62 64 66 68 70 72 74 1^ 78 80 82 84 86 Apparent Mtitudc of the Sun, Star or Planet. D's 4pp. Alt. . 6 7 8 9 10 11 12 i3 i4 i5 16 17 18 19 20 21 22 23 24 25 26 27 28 =9 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 4^ 46 48 5o 52 54 56 58 60 62 66 68 70 72 74 76 78 80 82 ( 84 1 ^1 6° / // I 22 I 24 I 28 I 33 I 4o I 47 1 55 2 3 2 10 2 18 2 26 2 34 2 42 2 5o 2 59 3 7 3 i5 3 24 3 32 3 4} 3 49 3 58 4 6 4 i5 4 23 4 3i 4 39 4 47 4 55 5 3 5 10 5 18 5 25 5 32 5 39 5 46 5 53 6 6 7 6 21 6 34 6 47 6 59 7 II 7 22 7 3i 7 4o 7 48 7 56 8 3 8 10 6° 7° t II I 23 I 22 I 24 I 28 I 33 I 38 I 43 I 49 1 55 2 I 2 7 2 i3 2 20 2 27 2 34 2 4i 2 48 2 55 3 2 3 9 3 16 3 23 3 3o 3 37 3 44 3 5i 3 58 4 5 4 12 4 18 4 24 4 3i 4 38 4 45 4 5i 4 57 5 3 5 i5 5 26 5 37 5 48 5 58 6 8 6 17 6 25 6 32 6 39 6 46 6 53 6 59 7° 8° 1 II I 25 I 23 I 22 I 24 I 27 I 3i I 36 I 4o I 45 I 5o 1 55 2 2 5 2 II 2 17 2 23 2 29 2 35 2 4i 2 47 2 53 2 59 3 5 3 II 3 17 3 23 3 29 3 34 3 40 3 46 3 52 3 58 4 4 4 10 4 i5 4 21 4 26 4 3i 4 36 4 46 4 55 5 4 5 i3 5 22 5 3o 5 37 5 45 5 52 5 58 6 2 6 6 6 10 8° 9° f II I 28 I 25 I 23 I 22 I 24 I 27 I 3o I 34 I 38 I 42 I 46 1 5o t 54 . 59 2 4 2 9 2 i4 2 19 2 24 2 29 2 34 2 39 2 44 2 49 2 54 2 59 3 4 3 9 3 14 3 19 3 24 3 29 3 34 3 39 3 44 3 49 3 53 3 58 4 3 4 12 4 20 4 28 4 36 4 44 4 5i 4 58 5 4 5 10 5 i5 5 20 ^^ 5 27 9° 10° / // I 33 I 28 I 25 I 24 I 23 I 24 I 26 I 29 I 32 I 36 I 39 I 43 I 46 I 5o I 54 1 58 2 2 2 7 2 10 2 i5 2 20 2 25 2 29 2 33 2 38 2 42 2 47 2 52 2 56 3 3 4 3 8 3 12 3 17 3 21 3 26 3 3o 3 35 3 39 3 47 3 54 4 I 4 8 4 i5 4 21 4 27 4 32 4 38 4 43 4 47 4 5i 4 54 4 57 10° 11' 1 II I 4o I 33 I 28 I 25 I 24 713 I 24 I 26 I 28 I 3i I 34 I 37 I 4o I 43 I 46 7"5^ I 53 1 57 2 I 2 4 2 8 2 12 2 16 2 20 2 24 2 28 2 33 2 36 2 4o 2 44 2 48 2 52 2 55 2 59 3 3 I,] 3 i5 3 19 3 26 3 32 3 37 3 43 3 49 3 55 4 I 4 6 4 II 4 i5 4 19 4 23 4 26 4 29 11° 12° 1 II I 4i 1 37 I 3i I 27 I 25 I 24 I 23 I 24 I 25 I 27 1 29 I 3i I 34 I 36 . 39 I 42 I 45 I 48 I 52 I 55 1 59 2 3 2 7 2 II 2 i4 2 18 2 21 2 25 2 28 2 32 2 35 2 39 2 42 2 46 2 49 2 52 2 55 2 58 3 I 3 7 3 i3 3 19 3 25 3 3o 3 35 3 4o 3 45 3 5o 3 55 3 59 14 2 4 4 4 6 4 8 12° 14° / II 2 I I 47 I 39 I 33 I 29 I 26 I 25 I 24 I 23 I 24 I 25 I 26 I 27 1 29 I 3i I 33 I 35 I 37 I 4o I 42 I 45 I 48 I 5i I 53 I 56 1 59 2 2 2 5 2 8 2 II 2 i4 2 17 2 20 2 22 2 25 2 27 2 3o 2 32 2 35 2 4o 2 45 2 5o 2 55 2 59 3 4 3 8 3 12 3 16 3 19 3 22 3 25 3 27 3 28 3 29 3 3o 14° 1(3° 1 II 2 16 I 59 I 48 I 4o I 34 I 3o I 28 I 26 I 25 I 23 I 22 I 22 I 23 I 24 I 25 I 26 I 28 I 3o I 3i I 33 I 35 I 38 I 40 I 42 I 44 1 46 I 48 I 5i I 53 I 55 I 57 1 59 2 2 2 4 2 6 2 8 2 10 2 i3 2 i5 2 19 2 23 2 27 2 3i 2 35 2 38 2 4i 2 44 2 48 2 5i 2 54 2 56 2 58 3 c. 3 2 3 3 3 4 1G° 18° / II 1 33 2 i3 I 59 I 49 I 4i I 36 I 32 I 29 I 27 I 25 r 23 I 22 I 21 I 22 I 22 I 23 I 24 I 25 I 26 I 27 1 29 I 3i I 32 I 34 I 35 I 37 I 38 I 40 I 4i I 43 I 45 I 47 \f, I 53 I 55 I 56 1 58 2 2 4 2 8 2 II 2 i4 2 18 2 21 2 24 2 27 2 29 2 3i 2 33 2 35 2 36 2 38 2 39 2 4i 2 42 2 43 18° 20° / II 1 5u 2 27 2 11 I 5b I 49 I 42 I 37 I 33 I 3o I 27 I 25 I 23 I 22 I 21 I 20 I 21 I 21 I 22 I 23 I 24 I 25 I 26 I 27 I 28 I 29 I 3o I 3i I 33 I 34 I 35 I 37 I 38 I 4o I 42 I 43 I 45 I 46 I 48 I 49 I 52 I 56 1 59 2 2 2 4 2 7 2 10 2 12 2 i4 2 16 2 18 2 19 2 20 2 21 2 22 2 23 2 24 2 25 2 26 20° 22° / II 3 8 2 41 2 23 2 8 I 57 I 49 I 43 I 38 I 34 I 3o I 27 I 25 I 23 I 22 I 21 I 20 I 20 I 20 I 21 I 22 I 22 I 23 I 23 I 24 I 24 I 25 I 26 I 27 I 28 I 29 I 3i I 32 I 33 I 35 I 36 I 37 I 38 I 40 I 4i I 43 I 46 I 48 1 5i I 53 I 56 1 58 2 2 2 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 II 2 12 2 12 2 12 22° 24° / // 3 25. 2 55. 2 35 2 18 2 6 I 57 I 49 I 43 I 38 I 34 I 3u I 28 I 25 I 23 I 22 I 21 I 20 I 20 I 20 I 20 I 20 I 21 I 21 I 21 I 21 I 22 I 22 I 23 I 24 I 25 I 26 I 27 I 28 I 29 I 3o I 3i I 32 I 34 I 35 I 37 I 39 I 4i I 43 I 45 I 47 I 49 I 5() I 52 I 53 I 55 I 56 I 57 I 58 I 59 1 59 2 2 I 2 I 2 2 2 2 24° 2li° 1 II 3 4i 3 9. 2 48 2 29 2 i5 2 5 I 56 I 49 I 43 I 38 I 34 I 3i I 28 I 26 I 24 I 22 I 21 I 20 I 20 I 19 I 19 I 19 I 19 I 19 I 19 I 20 I 20 I 20 I 21 I 22 I 22 I 23 I 24 I 25 I 26 I 27 I 28 1 29 I 3o I 3i I 33 1 35 I 36 I 38 I 40 I 4i I 42 I 44 I 45 1 46 I 47 I 48 I 49 I 5o I 5o I 5i I 5i I 52 I 52 I 52 26° 28° / ri\ 5 58. 3 23 3 0. 239 2 25 > i3 2 3 I 55 I 49 I 43 I 38 I 34 I 3i I 28 I 26 I 24 I 22 I 21 I 20 I 19 I 19 I 19 I 18 I 18 r 18 I 18 I 19 I 19 I 19 I 20 I 20 I 21 I 21 I 22 I 22 I 23 I 24 I 25 I 26 I 27 I 28 ^^9 I 3i I 33 I 34 I 35 I 36 I 37 I 38 I 39 I 40 I 4i I 4i I 42 I 42 I 43 I 43 I 44 I 44 I 44 28° 30° f i5 } 37 5 12 2 5o 2 34 2 21 2 II 2 2 I 54 I 48 I 43 I 38 I 34 I 3i I 28 I 25 I 23 I 22 I 21 I 20 I 19 I 19 I 18 I 18 1 18 I 18 I 18 I 18 I 18 I 18 I 18 I 19 I 19 I 20 I 20 I 20 I 21 I 22 I 22 I 23 I 24 I 25 1 27 I 28 I 29 I 3o I 3i I 32 I 33 1 34 1 34 I 35 I 35 I 36 I 36 I 37 I 37 I 38 I 38 1 38 30° TABLE XLVllI. [Page 295 Third Correction. Apparent Distance 60°. Al)[). All. o Apparent Altilude thn Sun, Star or Plmirf. 5's App All. 3:2° 34° 1 II 1 II 3G° 1 II 38° 42° / // 46° 1 II 50° / II 64° / II 68° 62° 66° 70" /4° 78° 82° 86° / t ; ir 1 /' 1 II 1 II 6 4 32 4 48 5 3 5 19 5 49 6 17 6 44 7 '' 7 28I7 47 8 3 6 7 3 5i 4 5 4 19 4 32 4 58 5 22 5 44 6 4 6 22 6 38 6 53 7 8 3 23 •J 35 3 47 3 59 4 22 4 42 5 I 5 19 5 35 5 5o 6 2 6 i3 8 9 3 3 10 3 20 3 3o 3 49 4 8 4 25 4 41 4 55 5 8 5 19 5 3o 9 lO 2 43 2 5i 3 3 9 3 26 3 42 3 58 4 12 4 24|4 35 4 45 4 54 10 1 1 2 29 2 37 2 44 2 52 3 7 3 21 3 35 3 48 3 59 4 9 4 18 4 26 II 12 2 18 2 25 2 32 2 39 2 52 3 5 3 17 3 29 3 3g 3 48 3 56 4 3 4 8 12 i3 2 8 2 i5 2 21 2 28 2 39 2 5i 3 2 3 12 3 21 3 3o 3 38 3 44 3 48 i3 i4 2 2 6 2 12 2 18 2 28 2 38 2 48 2 57 3 6 3 14 3 21 3 26 3 29 i4 i5 I 53 I 58 2 3 2 8 2 18 2 27 2 36 2 45 2 53 3 3 6 3 11 3 i5 i5 i6 I 47 1 5i I 55 2 2 9 2 18 2 26 2 34 2 4i 2 48 2 53 2 58 3 2 3 6 16 17 I 42 I 45 I 49 I 53 2 1 2 9 2 17 2 24 2 3i 2 37 2 42 2 46 2 5c 2 53 17 i8 I 37 I 4o I 44 I 47 I 54 2 1 2 9 2 16 2 22 2 27 2 32 2 36 2 4o 2 42 18 '9 I 33 1 36 I 39 I 42 t 48 I 55 2 2 2 9 2 i5 2 19 2 24 2 28 2 3i 2 33 19 20 21 I 3o I 27 I 32 I 29 I 35 I 32 I 38 I 35 I 44 I 40 I 5o I 46 I 56 I 5i 2 2 I 56 2 8 2 I 2 12 2 6 2 16 2 10 2 20 2 i3 2 23 2 i5 2 25 2 27 2 19 20 21 2 17 22 I 25 I 27 I 29 I 32 I 37 I 42 I 47 I 5i I 56 2 2 4 2 6 2 8 2 10 2 12 22 23 I 23 I 25 I 27 I 3(. I 34 I 38 I 43 I 47 I 5i I 55 I 59 2 I 2 3 2 4 2 6 2 3 24 I 22 I 23 I 25 I 27 I 3i I 35 I 40 I 44 I 47 I 5i I 54 I 56 I 58 I 59 2 I 2 3 24 25 I 21 I 22 1 23 I 25 I 29 I 32 I 36 I 40 I 43(1 47 I 49 I 5i 1 53 I 54 I 56 I 57 25 26 I 20 I 21 I 22 1 23 I 26 1 29 I 33 I 37 I 40 I 43 I 45 I 47 I 4q I 5o I 5i I 52 26 27 I 19 I 20 I 21 I 22 I 24 I 27 I 3o I 34 I 37 I 40 I 42 I 43 I 45 I 46 I 47 I 48 27 28 I 19 I 19 I 20 I 21 I 23 I 25 I 28 I 01 I 34 I 37 I 89 I 4o I 4i I 42 I 43 I 44 28 29 I 18 I 18 I 19 I 20 I 22 I 23 I 26 I 29 I 3i I 34 I 36 I 37 I 38 I 39 I 4o I 4i 29 Jo I 18 I 18 I 18 I 19 1 20 I 22 I 24 I 27 I 29 I 3i I 33 I 34 I 35 I 36 I 37 1 38 3o 3i I 18 I 18 I 18 I 18 I 19 I 20 I 22 I 25 I 27 I 29 I 3o I 3i I 32 I 33 I 34 I 35 3i ■3 2 I 17 I 17 I 17 I 17 I 18 I 19 I 21 I 23 I 25 I 27 I 28 I 29 I 3( I 3i I 3i I 32 32 33 I 17 I 16 I 16 I 16 I 17 I 18 I 19 I 21 I 23 I 25 I 26 I 27 I 28 I 29 I 29 I 3c 33 34 I 17 I 16 I 16 I 16 I 16 I 17 I 18 I 20 I 22 I 23 I 24 I 25 I 26 I 27 I 27 I 28 34 35 I 17 I 16 I 16 I 16 I 16 I 16 I 17 I 18 I 20 I 21 r 22 I 23 I 24 I 25 I 25 I 26 35 36 I 17 I 16 1 i5 1 16 I 16 I 16 I 16 I 17 I 18 1 19 I 20 I 21 I 22 I 23 I 23 I 24 36 37 I 17 I 16 I ID I i5 I i5 I i5 I i5 I 16 I 17 I 18 I 19 I 20 I 21 I 21 I 22 37 38 I 17 I 16 I i5 I i4 I 14 I 14 I i4 1 i5 I 16 I 17 I 18 I 19 I 20 I 20 I 21 38 39 I 18 I 16 I i5 I i4 I i3 I 13 I i3 I 14 I i5 I 16 I 17 I 17 I 18 I 18 I 19 3q 4<) I 18 I 16 I i5 I 14 I i3 I i3 I i3 I i4 I i4 I i5 I 16 I 16 I 17 I 17 I 17 40 4i I iS I 16 I i5 I i4 I 12 I 12 I 12 I i3 I i3 I 14 I i5 I i5 I 16 I 16 4i 42 I 18 I 16 I i5 I i4 I 12 I 12 I 12 I 12 I 12 I i3 I i4 I i4 I i5 I i5 42 43 I 19 I 17 I 16 I i4 I 12 I II 1 II I II I II I 12 I i3 I i3 I i4 I i4 43 44 I 19 I 17 I 16 I 14 I 12 I II I II I II I II I 1 1 I 12 I 12 I i3 I i3 44 46 I 30 I 18 I 16 I i4 I 12 I If I ID I 10 I 10 1 10 I II I 1 1 I II 46 48 1 21 1 19 I 17 I i5 I 12 I 10 I 9 I 9 1 9 I 9 I 10 I 10 I 10 48 5o I 22 I 19 I 17 I i5 I 12 I 10 I 9 I 8 I 8 I 8 I 8 I 8 5o 52 I 23 I 20 I 17 I i5 I 12 I 10 I 8 I 8 I 8 I 7 I 7 <■ 7 52 54 I 24 I 21 I 18 I 16 I i3 I 10 I 8 I 7 I 7 I 6 I 6 "14 56 58 60 62 I 25 I 26 I 27 I 28 I 22 I 23 I 24 I 24 I 19 I 20 I 21 1 21 I 16 I 17 I 18 I 18 I i3 1 i3 I 14 1 14 I 10 I 10 I 10 I 10 I 8 I 8 I 8 I 8 <■ 7 I 7 I 7 I 6 t 7 I 6 I 6 56 I 6 I 6 I 5 I 5 I 5 'I'uble P. Effect of Sun's Par. AM the Nn.nlwrs al.ove Iho lines 64 I 29 I 25 I 21 I 18 I 14 I 10 I 8 I 6 I 5 to TliirtI Corictlioii ; suLilnicl 66 68 70 I 29 , 2_9 I 3u 1 25 I 21 1 22 I 22 I 18 I 19 I 19 1 14 I i5 I i5 I 8 I 6 the othf-rs. I 25 I 26 I II I II I 8 I 8 I 6 D's S im's Apparent Al'.itude. Anp. Alt. 5 lU 2ul30 -10 50 fi 70 so so 72 I 3() I 26 I 23 I 20 I t5 I 1 1 T 8 74 7<^) I 3i T 3r I 27 I 27 I 23 I 93 I 20 : 20 I i5 r t5 I 1 1 I 1 1 5 10 1 1 -i \ 1 3 3 2 2 ! 3 ~7tr I i5 20 3 30 5 3 2 1 4 3 3 2 7 ' 7 I 32 I 28 I 24 I 20 80 I 3} I 28 I 24 I 21 I i5 40 6 6 5 4 3 3 ' 2 2 82 I 33 I 28 I 24 I 21 50 7 7 6 5 5 4 . 3 84 1 33 I 28 I 2.4 I 21 60 3 8 7 6 6 5 4 86 I 33 I 28 I 24 70 9 8 7 6 6 32^ 34" 36° 38° 42° 46° 50° 1 54° 58° 62° m'' BO 1 '*■ 1 — ... Page 293] TABLE XLVIII Third Correction. Appai-ent Distance 64*^. D's Apparent Altitude of the Sun, Star or Planet. 5's App. Alt. A pp. Alt. 6=- 7° 8° 9° 1U° 11° 12° 14° 16° 18° 20° 22° 24° 20° 28° ao° o / // 1 II. ? // / II / // / II / // ' // / II / II / // / II / // / // / // / // 6 I 26 I 27 I 29 I 32 I 36 I 42 I 49 2 3 2 19 2 35 2 5i 3 8 3 24 3 4o 3 56 4 12 6 7 T 28 I 26 I 27 I 29 I 32 I 35 I 4 I 43 I 44 I 45 I 46 3o 3i I 25 I 25 I 26 I 26 I 27 I 29 I 3i I 33 I 35 I 37 I 39 I 4o I 4i I 42 I 43 3i 32 I 25 I 25 I 25 I 25 I 26 I 28 I 29 I 3i I 33 I 35 I 37 I 38 I 39 I 40 t 4i 32 33 I 25 I 24 I 25 I 25 I 26 I 27 I 28 I 3o I 3i I 33 I 35 I 36 T 37 T 38 33 M I 25 I 24 I 24 I 24 I 25 I 26 I 27 I 29 I 3o I 3i I 33 I 34 I 35 I 36 34 36 I 25 I 24 I 24 I 24 I 24 I 25 I 26 I 28 I 29 I 3o I 3i I 32 I 33 I 34 35 36 I 25 1 24 I 23 I 23 I 23 I 24 I 25 I 27 I 28 I 29 I 3o I 3o 1 3i I 3? 36 J7 I 2D I 24 I 23 I 23 I 23 I 23 I 24 I 26 I 27 I 28 I 29 I 29 I 3o 37 38 I 25 I 24 I 23 I 22 I 22 I 23 I 24 I 25 I 26 I 27 I 28 I 28 I 29 38 39 I 23 I 24 I 23 I 22 1 22 I 23 I 23 I 24 I 25 I 26 I 27 I 27 I 27 39 4o I 26 I 25 I 24 I 23 I 22 I 22 I 23 I 23 I 24 I 25 I 26 I 26 I 26 40 4i I 26 I 25 I 24 I 23 I 21 I 21 1 22 I 22 I 23 I 24 I 25 I 25 4i 42 I 27 I 25 I 24 I 23 I 21 I 21 I 21 I 22 I 23 I 23 I 24 I 24 42 43 I 27 I 251 I 24 I 23 I 21 I 21 I 21 I 21 I 22 I 22 I 23 I 23 43 44 I 28 I 26 I 24 I 23 I 21 I 20 I 20 I 20 I 21 I 21 I 22 I 22 44 46 I 28 I 26! I 25 I 24 I 21 I 19 I 19 I 19 I 20 I 20 I 20 46 48 I 29 I 27J I 25 1 24 I 22 I 19 I 18 I 18 I 19 I 19 I 19 48 bo I 3<) I 28 I 26 I 25 I 22 I 20 I 18 I 18 I 18 I 18 5o 52 I 3i I 29 I 27 I 25 I 22 ! 20 I 18 I 17 I 17 I 17 52 54 I 3a I 29 I 27 I 20 I 23 I 20 I 18 I 17 I 16 54 56 58 I 33 I 34 I 3o I 3i I 28 I 29 I 26 I 27 I 23 1 23 I 20 I 20 I 18 I 18 I 16 I 16 I i5 56 1 60 62 I 35 I 36 1 32 I 33 I 29 I 3o I 27 I 28 I 23 I 23 I 2(.) I 20 I 18 I 18 I 16 Table P. Effect of Sun's Par. Add tlip Niiinltprs al.ove the lines 64 I 37 I 63 I 3f. I 28 I 24 I 10 I 17 10 Tliird Ciirreciinn ; siiLlracI 66 I 38 1 34 I 3i r 28 1 24 I 20 the others. 68 I 38 I 34 I 35 I 3i I 28 I 24 I 24 D's Sun's Apparent Allilmle. 70 I 39 I 32 I 29 App- Alt. 5 u -io 30 •10 5U6 a 70 ?0 ao 72 I 39 I 35 I 32 1 29 I 24 74 I 40 1 36 I 32 I 29 5 1 u u ' - ~ 76 I 40 I 36 I 3? I 29 10 '•i - 20 H 1 'J 1 ' 1 u 78 1 4i I 36 I 32 30 4 4 3 2 2 2 1 1 80 I 4i I 36 40 6 ; s 4 4 3 3 3 82 I 4i SO 7 r 6 6 5 5 4 84 GO 8 i 7 7 6 6 86 70 SO 9 i 8 1 s 7 8 7 w ;{'7 ' 6 17 5 36 5 : 4 35|4 i: 3 37 3 12 2 5i 2 36 2 23 2 12 2 3 I 5f I 5o 78 80 8 ic 57 K )6 15 5 38 5 e 4 3- 4 14 -i 38 3 i3 2 52 2 37 2 24 2 i3 2 4 I 5- I 5i 80 82 6 21 5 40 5 - 74 3c ;4 i6 3 :-9 3 i32 53 2 38 2 24 2 ij 2 4 i 5- 82 84 5 c ?4 4 4 173 4(> 3 i4 2 54 2 38 2 24 2 IZ 2 5 84 86 4 t8I3 4i 3 i5 2 54 2 38 2 24 2 1/ 86 G° 1 7° 8° 9° 10° 11° 12° 14° K)° 18° 20° 22° 24° 26° 28° 30° 1 TABLE XLVIII. f'''^se3oi Third Correction. Apparent Distance 72°. D's Aun. Jijjparent MlUudc of the Sun, Star or Planet. I A 's aYi. 32^ ;J4" 30^ 36" 42° 4'o° 50" 54" 5a° 02" m° 70° 740 78" 82-' b'o-^ All. 1 1 / // 1 II / // / /' 1 II 1 II II 1 I' / II 1 I' 1 II / // / // / // / // 6 4 27 4 4\ 4 56 5 II 5 38 6 3 6 27 6 48 7 8 7 27 7 427 55 8 6 8 16 6 7 3 5i 4 3 4 16 4 28 4 5. 5 12 5 32'5 5 1 6 8 6 23 6 36 6 48 6 58 7 7 7 8 3 25 3 3(i 3 47 3 58 4 18 4 36 4 54|5 II 5 26 5 39 5 5i 6 I 6 9 6 16 6 22 8 9 3 4 3 i4 3 24 3 33 3 5i 4 8 4 23 4 37 4 5o 5 I 5 II 5 20 5 28 5 35 5 4i 9 lO 2 48 2 57 3 6 3 x4 3 29 3 44 3 58 4 10 4 22 4 33 4 42 4 5(> 4 57 5 3 3 7 KJ II 2 35 2 43 2 5i 2 58 3 II 3 25 3 37 3 48 3 59 4 9 4 17 4 24 4 3o 4 35 4 39I I I 12 2 24 2 3i 2 38 2 45 2 57 3 9 3 20 3 3i 3 4. 3 49 3 57 4 3 4 8 4 12 4 i6'4 ?o 12 i3 2 ID 2 21 2 27 2 -6-^ 2 45 2 56 3 6 3 16 3 24 3 32 3 39 3 45 3 49 3 53 3 56 3 59 i3 i4 2 7 2 i3 2 iS 2 24 2 34 2 44 2 54 3 2 3 10 3 18 3 24j3 29 3 33 3 36 3 393 4 1 1 4 i5 * I 2 6 2 1 1 2 16 2 25 2 34 2 4^ 2 5i 2 58 3 5 3 n 3 16 3 20 3 23 3 25 3 27 i5 i6 I 5(j 2 I 2 5 2 9 2 18 2 26 2 33 2 4i 2 48 2 54 2 59 3 4 3 8 3 II 3 i3 3 i5 16 17 I 52 I 56 I 59 2 3 2 1 1 2 19 2 25 2 32 2 39 2 45 2 5() 2 54 2 57 3 3 2 3 4 17 i8 I 48 1 5i I 54 I 58 2 6 2 i3 2 19 2 25 2 3i 2 37 2 42 2 46 2 48 2 5o 2 52 2 54 18 19 1 44 1 47 I 5o I 54 2 1 2 7 2 i3 2 19 2 25 2 3c) 2 35 2 38 2 4" 2 4^ 2 44 2 45 19 20 I 4i I 44 I 47 I 5o I 56 2 2 2 7 2 l3 2 19 2 23 2 28 2 3i 2 33 2 35 2 3b 2 37 20 21 1 39 I 41 1 44 I 46 I 52 I 57 2 2 2 8 2 i3 2 17 2 21 2 24 2 26 2 28 2 29 2 ,'0 21 22 . 37 . 39 1 4i I 43 I 48 I 53 I 58 2 3 2 7 2 II 2 i5 2 18 2 20 2 22 2 23 2 0.4 22 23 I 36 I 37 I 39 I 4i I 45 I 5o I 54 I 59 2 2 2 6 2 10 2 i3 2 i5 2 16 2 17 2 18 23 24 I 35 I 3d I 37 I 39 I 43 I 47 I 5i I 55 I 58 2 2 2 5 2 8 2 10 2 II 2 12 2 i3 24 25 I M r 35 I 36 I 38. I 4i I 44 I 48 I 5i I 54 I 58 2 I 2 3 2 5 2 6 2 8 25 26 I 33 1 M I 35 I 36 I 39 I 42 I 45 I 48 1 5r I 54 1 57 I 59 2 I 2 2 2 4 26 27 I 32 I 33 I 34 I 35 I 37 I 4o I 43 I 45 I 48 I 5i I 54 I 56 I 57 I 58 2 27 28 I 32 I 32 I 33 I 34 I 35 I 38 I 4 1 I 43 I 46 I 48 I 5i I 53 I 54 I 55 I 56 28 29 I 3i I 32 I 32 I 33 I 34 I 36 . 39 I 4i I 44 I 46 r 48 I 5o I 52 I 53 29 3o I 3i I 3i I 32 I 32 I 33 I 35 I 37 I 39 I 42 I 44 1 46 I 47 I 49 I 5o 3o 3i I 3o I 3i I 3i I 3i I 32 I 34 I 36 I 38 I 4<> I 42 I 44 I 45 1 46 I 47 3i 32 I 29 I 3o I 3o I 3o I 3i I 33 I 35 I 36 I 38 I 40 I 42 I 43 I 44 I 45 32 33 I 29 I 29 I 29 I 3o I 3i I 32 I 33 I 34 I 36 I 38 I 40 I 4i I 42 33 34 I 3o I 29 I 29 I 29 I 3o I 3i I 32 I 33 I 34 I 36 1 38 I 09 I 4o 34 35 I 3o I 29 I 29 I 29 I 3o I 3o I 3i I 3s I 33 I 35 I 36 . 37 1 38 35 36 I 3i I 29 1 28 I 28 I 29 I 3o I 3i I 32 I 33 I 34 I 35 I 36 I 36 36 37 I 3i I 3.,. I 28 I 28 I 29 I 29 I 3o I 3i I 32 I 33 I 34 I 35 37 38 I 3i 1 3() I 28 I 27 I 28 I 29 I 3o I 3i I 32 I 33 I 33 I 34 38 39 I 3i I 3o I 29 I 28 I 28 I 28 I 29 I 3o I 3i I 32 I 32 I 32 39 4o I 3i I 3() t 29 I 28 I 27 I 28 I 28 I 29 I 3o I 3o r 3o I 3o 4o 4i I 3i 1 3o I 29 I 28 I 27 I 27 I 27 I 28 I 28 1 29 I 29 4i 42 1 ii I 3i I 29 I 28 I 26 I 26 I 26 I 27 I 27 I 28 I 28 42 43 1 32 I 3. 1 29 . 28 I 26 I 26 1 26 I 26 I 26 I 27 1 27 43 44 t 33 1 3i I 3o I 28 I 26 I 26 r 25 I 25 I 25 I 26 [ 26 44 46 I 34 I 32 I 3o I 29 I 27 I 25 I 25 I 25 I 25 I 25 46 48 I 35 I 32 I 3o I 29 1 27 I 25 I 24 I 24 I 24 I 24 48 5o I 3() I 33 I 3i I 3o I 27 I 25 t.24 I 23 I 23 5o 52 I 37 I 34 I 3i I 3o I 27 I 25 I 23 I 22 I 23 52 54 I 37 I M I 32 I 3i I 28 I 25 I 23 I 22 54 56 58 60 I 38 . 39 1 39 I 35 1 36 I 36 I 33 I 34 I 34 I 3i I 32 I 3a I 28 I 28 I 28 I 25 I 25 I 25 r 23 I 23 I 23 I 22 — 56 ■ruUe p. F.fecl of Sun's Par. 62 I 4(1 I in I 3b I 32 I 28 I 25 Ail.l Die Numbers nbove tlie lines fM I 4i I 38 I 36 t 33 t ?K 1 25 to Third Correction ; siiblnicl 66 68 70 1 A) 1 38 ■ 39 1 39 1 36 I 36 I 36 I 33 I M I 34 I 28 tlie others. I 4J I 43 App- Alt. Suns Arr^irc'it Allilmle. I 29 I U 20 30 iO .50 S 70 ^0 90 72 I 44 1 4<' I 36 I 34 74 76 I 44 I 45 I 40 I 4c. I 36 5 10 20 ■2 3 r 7 3 2 2 1 T ' I 1 78 1 4:j 30 4 4 4 3 3 ■i ! 2 2 80 40 6 6 5 5 4 4 3 82 50 7 7 6 6 5 5 . 84 60 8 8 7 7 6 6 86 70 9 8 3 7 7 80 9 8 8 yy^ 34" 30° 1 38° 42° 4G° 50° 54° 58° 62° (J()" 90 9 1 P^"«302j TABLE XLVIII. Third Correction. Apparent Distance 76°. i I B's Apparent Mtitude of the Sun, Star or Planet. D'i A pp. A pp. All. 6° 7" 8" y^ lU" 11^ 12*^ 14" 16^ 18" 20" 22° 24^ 2G" 2b" yo° All. o / // / // 1 II 1 II 1 II 1 II ? II / // 1 II / ;/ / /; / (/ 1 II / // 1 II ' // 6 I 37 I 39 I 4i I 44 I 48 I 54 2 2 i3 2 27 2 42 2 57 3 i3 3 28 3 4'i 3 58 4 i3 6 7 I 4o I 37 I 38 I 4o I 4'6 I 47 I 5i 2 I 2 12 2 24 2 37 2 5o 3 3 3 i5 3 28 3 4o 7 8 I 44 I 4o I 37 I 38 I 4o I 42 I 45 I 52 2 2 2 12 2 22 2 33 2 4^ 2 54 3 5 3 16 8 9 I 4q 1 43 I 39 I 37 I 38 I 39 I 4i I 46 I 54 2 2 2 II 2 20 2 3o 2 89 2 48 2 58 9 10 II 1 54 2 I 46 I 5o I 4i I 44 I 39 I 4i I 3- I 6^ I 39 I 42 I 4o I 48 I 44 I 55 I 49 2 2 I 55 2 10 2 2 2 18 2 9 2 26 2 16 2 34 2 23 2 43 2 3i 10 II I 39 I 37 I 38 12 2 6 I 55 I 48 I 44 I 4i I 38 I 37 I 38 I 4i I 45 I 5o I 56 2 2 2 8 2 i5 2 21 ]2 i3 2 12 2 I 52 I 47 I 43 I 4o I 38 I 37 I 39 I 42 I 46 I 5i I 56 2 2 2 8 2 i3 i3 i4 2 19 2 6 I 56 I 5o I 45 I 42 I 4o I 37 I 38 I 4o I 43 I 47 I 52 I 57 2 2 2 7 i4 i5 2 26 2 12 2 i I 54 I 48 I 44 I 42 I 38 I 37 I 39 I 4i I 45 I 49 I 53 I 57 2 1 i5 i6 2 J-, i 18 2 6 I 58 I 5i I 4i I 4A I 39 I 37 I 38 I 40 I 43 I 46 I 49 I 53 I 56 10 I? 2 41 2 24 2 II 2 2 I 54 I 49 I 46 I 40 I 38 I 37 I 39 I 41 I 4i I 4b I 49 I 52 n i8 2 49 2 3o 2 17 2 6 I 58 I 52 I 48 I 42 I 39 I 36 I 38 I 39 I 4i I 4i I 4b I 49 iS 19 2 5- 2 36 2 22 2 10 2 2 I 5b I 5o I 43 I 40 I 37 I 37 I 38 I 39 I 4i I 43 I 4ii 19 20 3 5 2 43 2 27 2 i5 2 6 I 58 I 52 I 45 I 4i I 38 I 36 I 37 I 35 . 39 I 4. I 4'd 20 21 3 12 2 49 2 33 2 20 2 10 2 2 I 55 I 47 I 42 I 39 I 37 I 36 I 37 I 38 I 39 I 4i 21 22 3 20 2 56 2 38 2 24 2 14 2 6 I 6b I 49 I 44 I 40 I 38 I ib I 36 I 3- I 38 I 39 22 23 3 28 3 3 2 44 2 29 2 18 2 9 2 I I 5i I 45 I 41 I 38 I 36 I 35 I 36 I 37 I 38 23 24 3 36 3 9 2 49 2 34 2 22 2 12 2 4 I 54 I 47 I 42 I 39 I 37 I 35 I 36 I 36 I 37 24 25 3 44 3 i5 2 54 2 39 2 26 2 16 2 7 I 56 I 49 I 44 I 4o I 07 I ib I 36 I 36 I 37 25 26 3 5 1 3 21 3 2 44 2 3o 2 20 2 II I 59 I 5i I 45 I 4i I 38 I 36 I 35 I 35 I 36 26 27 3 59 3 28 3 5 2 49 2 34 2 23 2 14 2 2 I 53 I 47 I 42 I 39 I 37 I 36 I 35 I 35 27 28 4 6 3 34 3 10 2 54 2 38 2 27 2 17 2 4 I 54 I 48 I Ai I 39 I 37 I 36 I 35 I 35 28 ' 29 4 t3 3 4o 3 i5 2 58 2 42 2 3i 2 21 2 7 I 56 I 49 I 44 I 40 I 38 I 3b I 35 I 34 29 3o 4 20 3 46 3 21 3 3 2 47 2 34 2 24 2 9 I 58 I 5i I 45 I 41 I 39 I 37 I 35 I 34 3o 3t 4 27 3 52 F^ 3 7 2 5i 2 38 2 28 2 12 2 I 52 I 46 I 42 I 39 I 3- I 35 I 34 3i 32 4 34 3 58 3 3i 3 12 2 55 2 42 2 3i 2 i4 2 2 I 54 I 48 I Ai I 4o I 38 I 36 I 35 32 33 4 4i 4 413 37 3 16 2 59 2 45 2 34 2 17 2 4 I 55 I 49 I 44 I 4i I 38 I 36 I 35 33 34 4 48 4 10,3 42 3 20 3 3 2 49 2 37 2 19 2 6 I 57 I 5o I 45 I 42 I 39 I 37 I 35 34 35 4 55 4 16 3 47 3 25 3 7 2 52 2 4i 2 22 2 8 I 59 I 52 I 46 I 42 I 39 I 37 I 35 35 36 5 2 4 22 3 53 3 29 3 II 2 56 2 44 2 24 2 II 2 I I 53 I 47 I 43 I 40 1 38 I 36 36 37 5 9 4 27 3 58 3 34 3 i5 3 2 47 2 27 2 i3 2 3 I 55 I 48 I 44 I 4i I 38 I 36 37 38 5 16 4 33 4 3 3 38 3 19 3 3 2 5o 2 29 2 i5 2 4 I 56 I 49 I 45 I 42 I 39 I 37 38 39 5 23 4 38 4 8 3 43 3 23 3 7 2 53 2 3i 2 17 2 6 I 58 I 5i I 46 I 42 I 39 I 37 39 40 5 3o 4 44 4 i3 347 3 27 3 10 2 56 2 34 2 19 2 8 I 59 I 52 I 47 I Ai I 4o I 38 40 4i 5 37 4 5o 4 18 3 5i 3 3i 3 i4 2 59 2 36 2 22 2 10 2 I 53 I 48 I 44 I 4i I 38 4i 42 5 43 4 55 4 23 3 55 3 34 3 17 3 2 2 39 2 24 2 12 2 I I 54 I 49 I 45 I 42 I 39 42 43 5 49 5 T 4 28 3 59 3 38 3 20 3 5 2 4i 2 26 2 i4 2 3 I 56 I 5o I 4b I 4i I 40 43 AA 5 55 5 6 4 33 4 3 3 4i 3 24 3 8 2 44 2 28 2 i5 2 4 I 57 I 5i I 47 I 4i I 4o 44 46 6 7 5 16 4 42 4 II 3 49 3 3i 3 i4 2 49 2 32 2 18 2 7 I 59 I 53 1 48 I 44 1 4i 46 48 6 19 5 26 4 5i 4 19 3 56 3 37 3 20 2 54 2 35 2 21 2 10 2 2 I 55 I 5o 1 46 I 43 48 5o 6 3o 5 36 4 59 4 27 4 3 3 43 3 25 2 58 2 39 2 25 2 i3 2 4 I 57 I 5i I 47 I 44 5o 52 6 4i 5 46 5 7 4 34 4 10 3 49 3 3o 3 3 2 43 2 28 2 16 2 6 I 59 I 53 I 49 I 45 52 54 6 5i 5 55 5 i5 4 4i 4 17 3 55 3 35 3 7 2 47 2 3i 2 19 2 9 2 1 I 55 t 5o I 46 54 56 7 1 6 4 5 22 4 48 4 23 4 3 4o 3 II 2 5o 2 34 2 22 2 12 2 3 I 5b I 5i I 47 56 58 7 II 6 12 5 29 4 54 4 28 4 5 3 45 3 i5 2 53 2 37 2 25 2 i4 2 5 I 57 I 52 I 48 58 60 7 20 6 20 5 36 5 o!4 33 4 9 3 49 3 19 2 56 2 4o 2 27 2 16 2 6 I 59 I 53 I 49 60 62 7 28 6 27 5 42 5 5 4 37 4 i4 3 53 3 22 2 59 2 43 2 29 2 18 2 8 2 I 54 I 5o 62 64 7 36 6 34 5 48 5 10 4 4i 4 18 3 57 3 25 3 2 2 45 2 3i 2 20 2 10 2 2 I 56 1 5i 64 66 7 43 6 4o 5 54 5 i5 4 45 4 22 4 1 3 28 3 5 2 47 2 33 2 21 2 II 2 3 I 57 I 52 66 68 7 49 6 45 5 59 5 19 4 49 4 26 4 5 3 3i 3 8 2 49 2 35 2 23 2 i3 2 4 I 58 I 53 68 70 7 5d 6 5o 6 3 5 2314 53 4 29 4 8 3 M 3 10 2 5i 2 36 2 24 2 i4 2 5 I 58 I 53 70 72 8 I 6 54 6 7 5 27 4 57 4 32 4 II 3 37 3 19 2 52 2 37 2 25 2 i5 2 6 I 5v I 54 72 74 8 6 6 58 6 10 5 3o 5 4 34 4 i3 3 39 3 i3 2 53 2 38 2 26 2 16 2 7 2 I 54 74 76 8 II 7 2 6 i3 5 33 5 3 4 36 4 i5 3 4. 3 i4 2 54 2 39 2 26 2 16 2 7 2 I I 55 76 78 8 i5 7 6 6 16 5 36 5 5 4 38 4 17 3 4- 3 i5 2 55 2 40 2 27 2 17 2 8 2 I 78 80 8 18 7 9 6 19 5 38 5 7 4 4o 4 19 3 43 3 16 2 56 2 4o 2 28 2 18 2 9 80 8?. 8 20 7 11 6 21 5 40 5 9 4 42 4 20 3 44 3 17 2 67 2 4i 2 28 2 18 82 84 S 22 7 i3 6 23 5 42 D IC 4 43 4 21 3 45 3 18 2 58 2 41 2 28 84 86 6 20 5 44 ■:! II 4 44 4 22 3 45 3 18 2 58 2 42 86 6= 7== 8° 9° 1C° ir 12° 14° 1C° 18° 20°. 22° 24' 26° 28° 30° TABLE XLVIII. iv.^so^o'i Third Correction. Apparent Distance 7G°. App Alt. o 'Apparent JUtilude of the Su7i, Star or Planet. B's App. Alt. 32° 1 II 34° / II 3G° 1 II 38° / // 42° / // 4G° / // 50° / // 54° 1 II 58° / 62° / // 06° 1 II 70° 74° 78° 82° 86° / // 1 II / /; / // 6 4 28 -\ 42 4 57 5 11 5 37 5 2 6 26 5 47 7 6 7 24 7 4o 7 54 8 5 8 10 8 2C 6 7 3 53 4 5 4 17 4 29 4 52 5 i3 5 33 5 52 6 9 6 24 6 37 6 48 6 b7 7 b 7 12 7 8 3 27 3 38 3 49 3 59 4 19 4 38 4 56 5 12 5 26 5 39 5 5i 6 I 6 9 6 16 6 21 6 26 8 9 3 8 3 17 3 26 3 35 3 52 4 8 4 24 4 38 4 5i 5 3 5 i3 5 22 5 29 b 3o 5 4fJ b 44 9 10 2 52 3 3 8 3 16 3 3i 3 46 4 1 4 i4 4 25 4 35 4 44 4 b2 4 59 b 4 b 8 5 1 1 10 II 2 39 2 46 2 53 3 3 14 3 27 3 4" 3 5i 4 2 4 12 4 20 4 27 4 33 4 38 4 42 4 45 11 12 2 28 2 34 2 4i 2 47 3 3 12 3 23 3 34 3 43 3 52 4 o4 6 4 11 4 lb 4 19 4 22 12 i3 2 19 2 25 2 3o 2 36 2 48 2 59 3 9 3 19 3 28 3 36 3 433 48 3 53 3 57 4 c 4 2 i3 i4 2 12 2 17 2 22 2 27 2 38 2 48 2 58 3 6 3 i4 3 22 3 28 3 33 3 37 3 4i 3 43 3 4b i4 i5 2 5 2 10 2 i5 2 19 2 29 2 38 247 2 55 3 3 3 9 3 i5 3 20 3 24 3 27 3 29 3 3i i5 i6 2 2 4 2 9 2 i3 2 21 2 29 2 37 2 45 2 52 2 58 3 4 3 8 3 12 3 i5 3 17 3 19 16 17 I 56 I 59 2 3 2 7 2 i4 2 22 2 29 2 36 2 43 2 49 2 54 2 58 3 2 3 4 3 e 3 8 17 18 I 52 I 55 I 58 2 2 2 9 2 16 2 23 2 3o 2 36 2 42 2 46 2 5o 2 53 2 55 2 5- 2 b8 18 19 I 49 I 5i I 54 I 58 2 4 2 II 2 17 2 24 2 3o 2 35 2 39 2 42 2 4b 2 47 2 4<; 2 5() 19 20 I 46 I 48 I 5i I 54 2 2 6 2 12 2 18 2 24 2 28 2 32 2 3b 2 37 2 39 2 4i 2 42 20 21 I 4^ I 45 I 48 I 5i I 56 2 2 2 7 2 i3 2 18 2 22 2 26 2 29 2 3i 2 33 2 3/ 21 22 I 41 I 43 I 46 I 48 I 53 I 58 2 3 2 8 2 i3 2 17 2 20 2 23 2 25 2 27 2 2!: 22 23 I 4o I 42 I 44 I 46 I 5o I 55 . 5g 2 3 2 8 2 12 2 lb 2 17 2 19 2 21 2 2.. 23 24 I 39 I 4o I 42 I 44 I 48 I 52 I 56 I 5q 2 4 2 7 2 10 2 12 2 14 2 16 2 it 24 25 I 38 .39 I 40 I 42 I 46 I 49 I 53 I 56 2 2 3 2 6 2 8 2 10 2 12 25 26 I 37 I 38 I 39 I 4i I 44 I 47 I 5c> I 53 I 56 I 5q 2 2 2 4 2 6 2 8 26 27 I 36 I 37 I 38 I 40 I 42 I 45 I 48 I 5o I 53 I 56 I 59 2 I 2 3 2 b 27 28 I 36 I 37 I 38 I 39 I 4i I 43 I 46 I 48 I 5o I 53 I 56 I 58 2 2 2 28 29 I 35 I 36 I 37 I 38 I 4o I 42 I 44 1 46 I 48 I 5o I 53 I bb I 57 29 3o I 35 I 35 I 36 I 37 I 38 I 40 I 42 I 44 I 46 I 48 I 5o I 52 I 54 3o 3i I 34 I M I 35 I 36 I 37 I 3q I 40 I 42 I 44 I 4& I 48 I 5o I 52 3i 32 I M I 34 I 34 I 35 I 36 I 38 I 39 I 4i I 43 I 44 I 46 I 48 I bo 32 33 I 34 I 33 I 34 I 35 I 35 I 37 I 38 I 40 I 42 I 43 I 45 I 46 33 34 I 34 I 33 I 33 I 34 I 35 I 36 I 37 I 3q I 4i I 42 I 44 I 4b 34 35 I 34 I 33 I 33 I 33 I 34 I 35 I 36 I 38 I 39 I 40 I 42 I 43 35 36 I 35 I 34 1 33 I 33 I 33 I 34 I 35 I 37 I 38 I 39 I 4o I 4i 36 37 I 35 I 34 I 33 I 32 I 33 I 33 I 34 I 36 I 37 I 38 I 39 37 38 I 35 I 34 I 33 I 32 I 32 I 33 I 34 1 35 I 36 I 37 I 38 38 39 I 36 I 34 I 33 I 32 I 3a I 33 I 33 I 34 I 35 I 36 I 36 39 40 I 36 I 35 I 34 I 33 I 32 I 32 I 33 I 34 I 34 I 35 I 35 40 4i I 37 I 35 I 34 1 33 I 32 I 32 1 32 I 33 I 33 I 34 4i 42 I 37 I 35 I 34 I 33 I 3i I 3i r 32 I 32 I 33 I 33 42 43 I 37 I 35 I 34 I 33 I 3i I 3o I 3i I 3i I 32 I 32 43 44 I 38 I 36 I 34 I 33 I 3i I 3o I 3o I 3i I 3i I 3i 44 46 , 39 I 37 I 35 I 34 I 3i I 29 I 29 I 3o I 3o 46 48 I 4o I 38 I 36 I 34 I 3i I 29 I 29 I 29 I 29 48 So 52 I 4i I 42 I 43 I 38 I 4o . 37 I 37 I 38 I 35 I 35 T 36 I 32 I 32 I 33 I 3o I 3o I 3c, I 29 I 29 I 29 I 28 5o i 1 2y I 29 56 I 44 I 4i r 38 T 36 I 33 T 3o Table P. Effect of Sun's Par. Add Ihe Numbers above llie line? 58 60 62 I 4b I 46 T 46 I 42 I 43 I ^3 139 1 4o t 4n I 37 I 37 I 37 I 38 1 33 I 33 I 33 I 3o I 3o to Third Correction ; subtr.ict tlie others. D's Alt. Sun's Apparent Altitmle. 64 I 47 I 44 I 4i I 33 5 10 20 3 9 -10 50 so ro s 90 66 I 48 I 44 I 4i I 38 68 I 49 . 45 I 4i I 38 5 1 r n 1 70 I 49 I 4b I 4i 15 2 1 u 72 I 49 I 45 20 25 3 : 2 2 9 2 74 I bo SO 35 4 i A "i 3 3 4 4 3 3 3 3 76 40 45 5 5 4 4 5 5 4 78 50 6 1 6 5 80 55 60 8 8 7 ( 7 - 6 6 7 82 65 8 8 a - 7 84 70 8 9 8 9 8 t 8 f 86 SO 9 9 8 32= 34° 36° 38° 42° 46° 50° 54° 58° 62° 66° 90 9 8 P=^g«304] TABLE XLVIII. Third Correction. Apparent Distance 80°. D's App. Alt. o 6 7 8 9 10 II 12 i3 i4 i5 i6 I? i8 19 20 21 22 23 24 25 26 27 28 ^9 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 46 48 5o 52 54 56 58 Go 62 64 66 68 •,'0 72 74 76 "78 80 82 84 86 Apparent Mtitude of the Sun, Star or Planet. | ])'s App. Alt. 6 7 8 9 10 II 12 i3 i4 i5 16 17 18 19 •20 2] 22 23 24 25 26 27 28 ^9 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 46 48 5o 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 0° / // 1 4i I 44 I 48 I 52 1 57 2 3 2 9 2 16 2 23 2 3o 2 37 2 45 2 53 3 3 8 3 16 3 23 3 3i 3 38 3 46 3 53 4 I 4 8 4 i5 4 22 4 29 4 36 4 43 4 5o 4 57 5 4 5 II 5 18 5 25 5 3i 5 38 5 44 5 5i 5 57 ^^ 6 20 6 3i 6 4i 6 5i 7 I 7 I' 7 20 7 28 7 36 7 43 7 49 7 55 8 c 8 5 8 9 8 i3 8 16 8 19 8 22 8 24 r 1 43 1 41 1 43 I 46 I 5o I 54 1 59 2 4 2 10 2 16 2 22 2 28 2 34 2 4i 2 47 2 54 3 3 6 3 12 3 18 3 24 3 3i 3 37 3 43 3 49 3 55 4 I 4 7 4 12 4 18 4 24 4 29 4 35 4 4i 4 47 4 52 4 57 5 3 5 8 5 18 5 28 5 38 5 47 5 56 6 5 6 i4 6 22 6 29 6 35 6 41 6 46 6 5i 6 55 6 59 7 3 7 6 7 9 7 12 7 i4 7 i£ 7° ( // I 46 I 43 I 41 I 43 I 46 I 49 I 53 1 56 2. 2 5 2 10 2 i5 2 21 2 26 2 3i 2 37 2 43 2 47 2 53 2 58 3 4 3 10 3 i5 3 20 3 25 3 3o 3 35 3 4o 3 45 3 5o 3 55 4 4 5 4 10 4 i5 4 20 4 25 4 3o 4 35 4 44 4 53 5 I 5 9 5 17 5 24 5 3i 5 38 5 44 5 5o ^ 55 6 6 5 6 9 6 i3 6 16 6 19 6 21 6 23 6 25 6 27 8° 9° 1 II I 5o I 45 I 42 I 41 I 4^ I 45 I 48 I 5i I 54 1 58 2 2 2 6 2 11 2 i5 2 20 2 14 1 29 2 33 2 38 2 42 2 47 2 52 2 56 3 1 3 5 3 10 3 i4 3 19 3 23 3 28 3 32 3 37 3 42 3 46 3 5o 3 54 3 58 4 2 4 6 4 i4 4 22 4 3o 4 37 4 44 4 5o 4 56 5 2 5 7 5 12 5 17 5 21 5 25 5 29 5 32 5 35 5 37 5 4o 5 42 5 44 5 46 9° 10° 1 II I 54 I 48 I 44 I 42 I 4i I /^i I 45 I 48 I 5o I 53 I 56 1 59 2 3 2 7 2 10 2 i4 2 18 2 22 2 25 2 29 2 33 2 37 2 41 2 46 2 5o 2 54 2 58 3 2 3 6 3 10 IT4 3 19 3 23 3 27 3 3i 3 35 3 38 3 42 3 46 3 53 4 4 6 4 12 4 18 4 24 4 3o 4 35 4 4o 4 44 4 49 4 53 4 57 5 5 3 5 6 5 8 5 10 5 12 5 i3 5 i4 10° 11° 1 II 159 I 5i I 46 I 44^ I 42 I 4i I 43 I 45 I 47 I 49 I 52 I 54 1 57 2 2 3 2 6 2 9 2 i3 2 16 2 19 2 23 2 26 2 3o 2 34 2 38 2 4i 2 45 2 49 2 52 2 56 3 3 3 3 7 3 II 3 i4 3 18 3 21 3 25 3 28 3 35 3 4i 3 47 3 53 3 58 4 3 4 8 4 i3 4 18 4 22 4 26 4 3o 4 33 4 36 4 38 4 4o 4 42 4 44 4 45 4 46 4 47 11° 12° / // 2 4 I 55 I 49 I 46 I 44 I 42 I 4i I 42 I 44 I 46 I 48 I 5o I 52 I 54 iJl 1 59 2 2 2 5 2 8 2 12 2 i5 2 19 2 22 2 26 2 29 2 32 2 35 2 38 2 4i 2 44 2 47 2 5o 2 54 2 58 3 I 3 4 3 7 3 10 3 i3 3 19 3 25 3 3o 3 35 3 39 3 44 3 49 3 54 3 58 4 2 4 6 4 9 4 12 4 i4 4 16 4 18 4 19 4 21 4 22 4 23 4 24 12° 14° 1 II 2 1- 2 5 I 56 I 5i I 47 I 45 I 43 I 42 I 4i I 42 I 43 I 45 I 4i I 48 I 5o I 52 I 54 I 57 1 59 2 I 2 3 2 6 2 8 2 II 2 i4 2 17 2 19 2 22 2 24 2 27 2 29 2 32 2 34 2 36 2 38 2 4i 2 44 2 46 2 48 2 53 r58 3 3 3 7 3 II 3 i5 3 19 3 23 3 27 3 3i 3 34 3 37 3 39 3 4i 3 43 3 44 3 45 3 46 3 4- 3 48 3 49 14° 16° ' II 2 32 2 17 2 6 I 58 I 53 I 49 I 46 I 44 I 43 I 42 I 4i I 42 I 4^ I 44 I 46 I 47 ;^? I 52 I 54 I 55 I 57 1 59 2 I 2 3 2 5 2 7 2 9 2 11 2 i4 2 16 2 18 2 20 2 22 2 24 2 26 2 28 2 3o 2 32 2 36 2 39 2 43 2 47 2 5i 2 54 2 57 3 3 3 3 6 3 9 3 12 3 i4 3 16 3 18 3 19 3 20 3 21 3 22 3 23 3 24 16° 18° / II 2 47 2 29 2 16 2 6 I 59 I 54 I 5o I 47 I 45 I 44 I 43 I 42 I 4i I 42 I 43 I 44 I 45 I 47 I 48 I 49 I 5o I 5i I 53 I 55 I 56 I 58 1 59 2 I 2 2 2 4 2 6 2 8 2 9 2 10 2 12 2 i4 2 16 2 17 2 19 2 23 2 26 2 29 2 32 2 35 2 38 2 4i 2 44 2 47 2 49 2 5i 2 53 2 55 2 57 2 58 2 59 3 3 I 3 2 3 3 1 Il8° 20° 1 II 3 2 2 4i 2 26 2 i5 2 6 I 59 I 54 I 5i I 48 I 46 I 45 I 43 I 42 I 4i I 41 I 42 I 42 I 43 I 44 I 45 I 46 I 47 I 48 \t I 52 I 53 I 54 I 56 I 57 1 58 2 2 I 2 2 2 4 2 5 2 7 2 8 2 10 2 i3 2 i5 2 18 2 21 2 24 2 26 2 29 2 3i 2 33 2 35 2 37 2 39 2 4i 2 42 2 43 2 44 2 45 2 45 2 46 20° 22° / // 3 17 2 54 2 37 2 25 2 i4 2 6 2 I 56 I 52 1 49 I 47 I 45 I 44 1 43 I 42 I 41 I 40 I 4i I 4i I 42 I 43 I 43 I 44 I 45 I 46 I 47 I 48 149 I 5o I 5i I 52 I 53 I 54 I 55 I 57 I 58 1 59 2 1 2 2 2 4 2 7 2 9 2 12 2 i4 2 17 2 19 2 21 2 22 2 24 2 26 2 27 2 29 2 3o 2 3i 2 32 2 32 2 33 22° 24° / // 3 32 3 6 2 48 2 34 2 22 2 i3 2 6 2 I I 57 I 53 I 5o I 48 I 46 I 44 I 43 I 42 I 4i I 40 1 4o I 4o I 4i I 4i I 42 I 43 I 44 I 44 1 45 I 46 I 47 I 47 I 48 I 49 149 I 5o I 5i I 52 X 53 I 55 I 56 1 58 2 2 2 2 4 2 6 2 8 2 10 2 12 2 i3 2 i5 2 16 2 17 2 19 2 20 2 21 2 22 2 22 24° 26° / // 3 47 3 19 2 59 2 46 2 3o 2 20 2 12 2 6 2 2 I 58 I 54 I 5i I 48 I 46 I 45 I 43 I 42 I 41 I 4i I 40 I 40 I 40 I 41 I 4i I 42 I 42 I 43 I 44 I 44 I 44 I 45 I 46 I 46 I 47 I 47 I 48 149 I 5o I 5i I 53 I 55 I 56 I 58 1 59 2 I 2 3 2 5 2 6 2 7 2 8 2 9 2 10 2 II 2 12 2 i3 26° 28° / II 4 2 3 3x 3 10 2 52 2 38 2 27 2 19 2 12 2 7 2 2 I 58 I 54 I 5i I 49 I 47 I 45 I 43 I 42 I 42 I 4i 1 4i I 40 I 40 I 40 I 40 I 4o I 4i I 42 I 42 I 42 I 4'i I 44 I 44 I 45 I 45 I 46 I 40 I 47 I 48 I 49 I 5i I 57 I 54 I 55 I 57 I 58 1 59 2 2 I 2 2 2 3 2 4 2 5 2 5 28° oO° / /; 4 16 3 44 3 so 3 I 2 45 ^4 2 25 2 18 2 12 2 7 2 2 I 58 I 54 I 5i I 49 I 47 I 45 I 44 I 43 I 42 I 42 I 4i I 4o I 39 I 39 139 I 39 I 40 I 4o I 40 I 4i I 42 I 42 I 43 143 I 44 I 44 I 45 I 45 I 46 I 48 I 49 I 5o I 52 I 53 I 54 I 55 I 56 I 56 I 57 I 58 1 59 2 30° TABLE XL VI II. [r.ge305 Third Correction. Apparent Di.stance 80°. ])'s A pp. Apparent Altitude of tlie Sun, Star or Planet. App. All. 32° 34^ 30" 38" 42° 46° 50° 54° 58° (>2° Gij'^ 70° 74° 7 8° 82° 80° Alt. o 1 II / // / // / II / /' 1 II / // / // / // / // 1 II / II / // / // / // / II G 4 3o 4 44 4 58 5 12 5 39 6 4 6 28 n 49 7 8 7 26 7 4i 7 54 8 58 i38 K ?8 24 6 1 3 56 4 8 4 19 4 3o 4 52 5 i4 5 35 5 54 6 II 6 26 6 39 6 5c, 6 597 67 i: 7 lb 7 8 3 3i 3 4i 3 52 4 2 4 23 4 42 4 59 5 i5 5 29 5 42 5 54 6 4 6126 186 2; b 27 8 9 3 II 3 21 3 3o 3 39 3 56 4 12 4 28 4 42 4 54 5 5 5 i5 5 24 5 32 5 38 5 4: 5 46 9 lU 2 54 3 3 3 12 3 20 3 35 3 5o 4 4 4 16 4 28 4 39 4 48 4 56 5 25 75 11 D i4 10 II 2 42 2 49 2 57 3 5 3 19 3 32 3 44 3 56 4 7 4 16 4 24 4 3i 4 36 4 4i4 4J 4 47 II 12 2 32 2 38 2 45 2 52 3 5 3 17 3 28 3 38 3 48 3 57 4 5 4 II 4 i54 194 2: 4 25 12 i3 2 24 2 3o 2 36 2 42 2 53 3 4 3 i4 3 23 3 32 3 40 3 47 3 53 3 574 i4 ^ 4 6 i3 i4 2 18 2 23 2 28 2 33 2 43 2 53 3 2 3 II 3 19 3 26 3 32 3 38 3 42 3 46 3 4f 3 49 i4 i5 2 12 2 16 2 21 2 25 2 34 2 43 2 52 3 3 7 3 i3 3 .9 3 25 3 29 3 32 3 3^ '3 3b i5 If.) 2 6 2 10 2 i4 2 18 2 26 2 34 2 42 2 5o 2 56 3 2 3 8 3 i3 3 173 20 3 2: 3 24 16 17 2 I 2 4 2 8 2 12 2 20 2 27 2 34 2 4i 2 47 2 53 2 58 3 3 3 63 9 3 I 17 18 I 57 2 2 3 2 7 2 i4 2 21 2 28 2 34 2 4'J 2 46 2 5o 2 54 2 573 o3 : 18 19 I 54 I 56 I 59 2 2 2 9 2 16 2 22 2 28 2 34 2 39 2 43 2 47 2 5o 2 52 2 5; 19 20 I 5i I 53 I 5b I 58 2 5 2 II 2 17 2 22 2 28 2 33 2 37 2 4o 2 432 45 2 4t ) 20 21 I 49 I 5i I 53 I 55 2 I 2 7 2 12 2 17 2 22 2 27 2 3i 2 34 2 37 2 38 21 22 I 47 I 49 I 5i I 53 I 58 2 3 2 8 2 i3 2 17 2 21 2 25 2 28 2 3i 2 32 22 23 I 46 I 47 I 49 I 5i I 55 2 2 4 2 9 2 i3 2 17 2 20 2 23 2 26 2 27 23 24 I 45 I 46 I 47 I 49 I 53 I 57 2 I 2 5 2 2 i3 2 16 2 19 2 21 2 22 24 25 I 44 I 45 I 46 I 48 I 5i I 54 I 58 2 1 2 5 2 9 2 12 2 i4 2 16 25 26 I 43 1 44 I 45 I 46 I 49 I 52 I 55 I 58 2 2 2 5 2 8 2 10 2 12 26 27 I 42 I 43 I 44 I 45 I 47 I 5o I 53 I 56 I 59 2 2 2 5 2 7 2 8 27 28 I 4i I 42 I 43 I 44 I 46 I 48 I 5i I 54 I 57 I 59 2 2 2 4 2 5 28 29 I 4o I 4i I 4i I 42 1 44 I 46 I 49 I 52 I 55 I 57 I 59 2 I 29 3o I 39 I 40 I 40 I 4i I 43 I 45 1 48 I 5i I 53 I 55 I 57 I 59 3o 3i I 39 I 40 I 40 I 4i I 42 I 44 I 46 I 49 I 5i I 53 I 55 I 57 3i 32 I 39 1 39 I 39 I 40 I 4i I 4^ I 45 X 47 I 49 I 5i I 53 I 55 32 33 I 39 I 39 I 39 I 40 i4i I 42 I 44 I 46 I 48 I 49 I 5i 36 34 I 39 I 39 I 39 I 4o I 4i I 42 I 43 I 45 I 47 I 48 I 49 M 35 t 39 I 39 I 39 I 39 I 4o I 4. I 42 I 44 I 45 I 46 I 47 35 36 I 4o I 39 I 39 I 39 I 4o I 4i I 42 I 43 I 44 I 45 I 46 36 37 I 4i I 4o , 39 I 38 I 39 I 4o I 4. I 42 I 43 I 44 37 38 I 4i I 40 I 39 I 38 I 39 I 4u I 4i I 42 I 42 I 4'^ 38 39 I 4i I 40 I 39 I 38 I 39 I 39 I 40 I 4i I 4i 1 42 39 40 I 4i I 40 I 39 i 38 I 38 I 38 I 39 I 40 I 4<-> I 4i 4o 4i I 42 I 4i I 4o I 39 I 38 I 38 I 38 I 39 I 39 4i 42 I 42 I 4i I 4o I 39 I 37 I 37 I 37 I 38 I 38 42 43 I 43 I 4i I 4o I 39 I 37 I 37 I 37 I 37 I 38 4S A^ I 43 I 42 I 40 I 39 I 37 I 37 I 36 I 37 I 37 44 46 I 44 I 42 I 4i I 4o I 38 I J7 I 36 I 36 4b 48 I 45 I 43 I 4i I 4o I 38 I 37 I 36 I 36 48 5o 52 I 46 I 47 I 44 I 45 I 42 r 43 I 4i I 4i I 38 I 38 I 36 I 36 I 36 I 35 5o 1 54 I 48 1 46 I 44 r 42 I 38 I 36 5G I 49 I 47 I 44 I 42 I 38 I 36 raWe P. Effect of Sun's Par. Adil Ihe Numbers above Che lines 58 I 5o I 47 I 45 I 42 I 38 to Third Correction ; sublract 60 69 I 5i I 52 I 48 I 49 I 45 I 46 I 43 I 43 I 38 the olhers. D'3 Sun's Apparent Altitude. 64 66 68 I 52 I 53 I 54 I 49 I 49 I 5o I 46 I 46 I 43 Alt. 10 5 10' 30 40 50 60 70 80 1 90 1 I 1 1 10 1 1 1 I I 1 n 70 I 5-5 15 20 2 2 S 3 2 2 11 3 2 2 2 1 1 2 2 1 2 72 25 4 4 3 3 3 3 3 2 nA 30 1 4 14 4 3 3 3 35 5 i> 5 4 4 4 4 70 40 43 6 6 fi 6 5 5 5 5 6 6 5 5 4 78 50 7 7 6 6 6 6 80 55 7 V 7 7 6 60 8 8 7 7 7 89 1 65 S 8 i 8 84 70 75 8 8 9 9 8 8 8 86 80 9 9 8 32^ 34° 36° 38° 42° 4(;° 50° 54° 58° 62° 00° __ '.\9 iX-esou] TABLE XLVIII . 1 Third Correction. Apparent Distance 84°. 1 ! 1 App. Apparent Mtitude of the Sun, Stt ?• or Planet. D's Add. All. 6° 70 8" 9^ 10^ 11^ 12" 14" l(i° 18" 20° 22" 24" 27 i8 2 3 2 6 2 10 2 i3 2 21 2 28 2 34 2 4o 2 46 2 52 2 57 3 I 3 4 3 6 18 19 2 2 3 2 6 2 2 16 2 23 2 29 2 34 2 4o 2 45 2 49 2 53 2 56 2 58 19 20 I 57 2 2 2 2 5 2 12 2 18 2 24 2 29 2 34 2 38 2 42 2 45 2 48 2 5o 20 21 I 54 1.57 I 59 2 2 2 8 2 l3 2 19 2 24 2 29 2 33 2 36 2 39 2 4i 21 22 I 52 I 54 I 56 I 5q 2 4 2 9 2 i4 2 10 2 24 2 28 2 3i 2 34 2 36 22 23 I 5o I 52 I 54 I 56 2 I 2 5 2 50 2 lb 2 19 2 23 2 26 2 29 2 32 2'3 24 I 49 I 5o I 52 I 54 I 58 2 2 2 7 2 II 2 i5 2 19 2 22 2 25 2 28 24 25 I 48 I 49 I bo I 52 I 56 2 2 4 2 8 2 12 2 i5 2 18 2 21 2b 26 I 47 I 48 1 49 I 5i I 54 I 58 2 2 2 5 2 9 2 12 2 i5 2 17 26 27 I 47 I 48 1 49 I 5o I 53 I 56 2 2 3 2 6 2 9 2 12 2 i4 27 28 I 46 r 47 I 48 I 4q I 5i I 54 I 58 2 I 2 3 2 6 2 P 2 II 28 20 I 46 I 47 I 47 I 48 I 5o I 53 I 56 I 59 2 I 2 4 2 6 29 3o I 45 I 46 I 46 I 47 I 49 I 52 I 55 I 57 2 2 2 2 3 3o 3i I 45 I 45 I 46 I 47 I 49 I 5i I 54 I 56 I 58 2 2 I 3i 32 I 45 I 45 I 45 I 46 I 48 I 5o I 52 I 54 I 56 I 58 I 59 32 33 I 45 I 45 I 45 I 46 I 47 I 49 I 5i I 53 I 54 I 56 33 34 I 45 I 44 I 44 I 45 I 46 I 48 I 5o I 52 I 53 I 54 34 35 I 45 I 44 I 44 I 45 I 46 I 47 r 49 I 5o I 5i I 52 3b 36 I 46 I 45 1 44 I 44 I 45 I 46 I 48 I 49 I 5o I 5o 36 ^7 I 46 I 45 I 44 I 44 I 45 I 45 I 47 I 48 I 49 37 38 I 46 I 45 I 44 I 44 I 44 I 45 I 46 I 47 I 48 38 ^9 I 46 I 45 I 44 I 44 I 44 I 44 I 45 I 46 I 47 39 4o 4i I 46 I 47 I 45 I 46 I 4b I 45 I 45 I 45 1 44 I 44 I 44 I 44 I 45 I 44 I 45 I 44 I 46 40 4i 42 I 48 I 47 I 46 I 45 I 43 I 43 I 44 I 44 42 43 > 49 I 48 I 46 I 45 I 43 I 43 I 44 I 44 43 44 I 49 I 48 I 47 I 45 I 43 I 43 I 43 44 46 . 5o I 49 I 47 I 45 I 43 I 43 I 43 46 48 I 5i I 5o I 48 I 46 I 44 I 43 I 42 48 5o 52 I 53 I 54 I 5i I 5i 149 I 49 I 47 I 47 I 44 I 44 1 43 I 42 bo 1 54 I 55 I 52 I 49 I 47 I 44 56 I 56 I 53 I 5o I 48 I 44 Table P. Eject of Sun's Par. 58 60 62 I 56 I 57 I 58 I 53 I 5o I 5i r 5i I 48 I 48 Tu be subtracted from the third Correction. 1 54 I 54 I 55 t)'s Arp. Alt. Suii'b Apparent Altitude. 64 I 59 5 20 3 ) JO 50 6t 70 SO 90 66 68 I 59 3 10 1 1 1 1 1 1 1 1 1 1 I :'. S 5 5 5 b 76 40 6 J B t (i 45 K 3 K 1 h 78 50 55 7 7 / 7 7 7 7 ■ 7 Ho GO 8 i 8 S 82 65 70 8 8 B 8 8 8 84 75 9 9 86 32° 34° 36° 38° 42° 46° 50° 54° 58° 80 90 9 62° 6(j° 1 I'^-^^ioj TABLE XLVm. ' Third Correction. Apparent Distance 92°. App. Alt. o 6 7 8 9 lO 11 12 i3 i4 i5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 =9 3o 3i 32 33 34 35 36 37 38 39 40 4i 42 43 45 46 47 48 5o 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 jipparetit Altitude of the Sun, Star or Planet. 6 7 8 9 10 II 12 i3 i4 ' i5 ; 16 17 iS 19 20 21 22 23 24 25 : 26 27 28 3o "3r: 32 33 34 35 36 4i 42 43 45 46 47 48 5o 52 "5^ 56 58 60 62 64 66 68 70 72 74 76 78 80 82 6° / H 1 59 2 I 2 4 2 8 2 l3 2 19 2 25 2 32 2 39 2 46 2 53 3 3 8 3 16 3 23 3 3i 3 38 3 46 3 53 4 I 4 9 4 17 4 24 4 3i 4 38 4 46 4 53 5 5 7 5 i4 5 21 5 28 5 34 5 4i 5 47 5 54 6 6 7 6 i3 6 19 6 25 6 3i 6 37 6 47 6 57 7 7 7 17 7 27 7 36 7 45 7 53 8 I 8 8 8 i4 8 20 8 25 8 29 8 32 8 34 8 36 6° 7° 1 H 2 I 1 69 2 I 2 4 2 7 2 11 2 16 2 21 2 27 2 32 2 38 2 44 2 5o 2 56 3 2 3 9 3 i5 3 22 3 28 3 34 3 4o 3 46 3 52 3 58 4 4 4 10 4 16 4 22 4 28 4 34 4 4o 4 46 4 52 4 58 5 3 5 i4 5 20 5 25 5 3i 5 36 5 4i 5 46 5 56 6 5 6 i4 6 23 6 3i 6 39 6 46 6 53 6 59 7 5 7 ic 7 i5 7 19 7 23 7 26 7 28 7 3c 7° 8° 1 H 2 3 2 I 1 59 2 I 2 3 2~6 2 10 2 I4 2 18 2 22 2 27 2 32 2 37 2 42 2 48 2 54 2 59 3 4 3 9 3 i5 3 20 3 26 3 3i 3 36 3 4i 3 47 3 52 3 58 4 3 4 8 4 i3 4 18 4 23 4 28 4 33 4 38 4 43 4 48 4 53 4 58 5 2 5 7 5 II 5,9 5 27 5 35 5 43 5 49 5 56 6 2 6 8 6 i3 6 18 6 23 6 28 6 3i 6 34 6 37 6 39 6 4i 8° 9° / // 2 6 2 3 2 1 59 2 I 2 3 2 6 2 9 2 12 2 i5 2 19 2 23 2 27 2 3i 2 36 2 4i 2 45 2 5o 2 54 2 59 3 3 3 8 3 i3 3 18 3 22 3 27 3 32 3 37 3 4i 3 46 3 5o 3 55 4 4 4 4 8 4 12 4 16 4 21 4 25 4 29 4 33 4 37 4 4i 4 48 4 55 5 2 5 9 5 i5 5 21 5 26 5 3i 5 36 5 4i 5 45 5 49 5 53 5 56 5 58 6 9° 10° / // 2 10 2 5 2 2 2 1 59 2 I 2 3 2 5 2 7 2 10 2 i3 2 16 2 19 2 22 2 26 2 3o 2 34 2 38 2 42 2 46 2 5o 2 55 2 59 3 3 3 7 3 12 3 16 3 20 3 24 3 28 3 32 3 36 3 4o 3 M 3 48 3 52 3 55 3 59 4 3 4 7 4 10 4 i3 4 17 4 24 4 3o 4 36 4 42 4 47 4 53 4 58 5 3 5 8 5 T2 5 16 5 19 5 22 5 25 5 27 5 25 10° 11° / ti 2 i5 2 9 2 4 2 2 2 1 59 2 I 2 2 2 4 2 6 2 8 2 11 2 i4 2 16 2 19 2 23 2 26 2 3o 2 34 2 37 2 4i 2 45 2 48 2 52 2 56 3 3 4 3 8 3 II 3 i5 3 18 3 22 3 25 3 29 3 32 3 35 3 39 3 42 3 46 3 49 3 52 3 55 3 59 4 5 4 II 4 16 4 21 4 26 4 3i 4 36 4 4i 4 45 4 49 4 52 4 55 4 57 4 59 5 I IP 12° / // 2 21 2 i3 2 7 2 4 2 2 2 1 59 2 2 2 2 3 2 5 2 7 2 9 2 II 2 i4 2 17 2 20 2 23 2 27 2 3o 2 33 2 36 2 39 2 43 2 46 2 5o 2 53 2 57 3 3 3 3 6 3 12 3 i5 3 18 3 21 3 24 3 27 3 3o 3 33 3 36 3 39 3 42 3 48 3 53 3 58 4 3 4 8 4 i3 4 17 4 21 4 25 4 28 4 3i 4 33 4 35 4 37 4 39 12° 14° / // 2 34 2 22 2 i4 2 9 2 6 2 3 2 I 2 1 59 2 2 I 2 3 2 4 2 6 2 8 2 10 2 12 2 i4 2 16 2 19 2 22 2 24 2 27 2 29 2 32 2 35 2 37 2 4o 2 42 2 45 2 47 2 5o 2 53 2 55 2 58 3 3 2 3 5 3 8 3 II 3 i3 3 16 3 18 3 22 3 26 3 3o 3 34 3 38 3 42 3 46 3 5o 3 53 3 56 3 58 4 4 I 4 2 14° 16° / // 2 48 2 33 2 23 2 16 2 II 2 7 2 4 2 2 2 I 2 1 59 2 2 I 2 2 2 « 2 5 2 6 2 8 2 9 2 II 2 i3 2 i5 2 17 2 19 2 21 2 23 2 25 2 27 2 29 2 3l 2 33 2 36 2 38 2 4o 2 42 2 45 2 47 2 49 2 5i 2 53 2 55 2 57 2 59 3 4 3 8 3 II 3 i4 3 17 3 20 3 23 3 26 3 29 3 32 3 34 3 35 3 36 16° 18° / // 3 3 2 45 2 33 2 24 2 17 2 12 2 8 2 5 2 3 2 I 2 159 1 59 2 2 2 I 2 2 2 3 2 4 2 5 2 7 2 9 2 II 2 12 2 i3 2 i5 2 16 2 18 2 20 2 22 2 24 2 25 2 27 2 29 2 3i 2 33 2 34 2 36 2 38 2 40 2 42 2 44 2 46 2 49 2 53 2 56 2 59 3 2 3 5 3 8 3 10 3 12 3 i4 3 i5 3 16 18° 20° / // 3 18 2 58 2 A^ 2 33 2 24 2 18 2 i3 2 9 2 6 2 4 2 2 2 I 2 2 I 59 1 59 2 2 2 I 2 2 2 4 2 5 2 6 2 7 2 8 2 9 2 II 2 12 2 i4 2 i5 2 17 2 18 2 20 2 21 2 22 2 24 2 25 2 27 2 28 2 3o 2 3i 2 33 2 35 2 37 2 4i 2 44 2 47 2 49 2 52 2 54 2 56 2 57 2 58 2 59 20° 22° / // 3 33 3 II 2 55 2 42 2 32 2 24 2 18 2 i4 2 10 2 7 2 4 2 3 2 2 2 I 2 I 59 I 59 1 59 2 2 2 I 2 2 2 2 2 3 2 4 2 5 2 7 2 8 2 9 2 10 2 II 2 12 2 i4 2 i5 2 16 2 17 2 18 2 20 2 21 2 22 2 24 2 25 2 27 2 29 2 32 2 34 2 37 2 39 2 4i 2 43 2 44 2-45 2 46 22° 24° / (/ 3 48 3 24 3 5 2 5i 2 4o 2 3i 2 24 2 19 2 i4 2 10 2 7 2 5 2 4 2 2 2 I 2 2 I 59 I 59 I 59 1 59 2 2 2 I 2 I 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 II 2 12 2 i3 2 i4 2 i5 2 16 2 18 2 19 2 20 2 22 2 24 2 26 2 29 2 3i 2 32 2 34 2 35 2 37 24° 26° / // 4 3 3 36 3 16 3 I 2 48 2 39 2 3i 2 24 2 19 2 i5 2 II 2 8 2 6 2 4 2 3 2 2 2 I 2 159 I 59 I 59 1 59 r 59 2 2 2 2 I 2 I 2 2 2 3 2 4 2 5 2 6 2 7 2 7 2 8 2 9 2 10 2 II 2 12 2 i3 2 i4 2 i5 2 16 2 18 2 20 2 22 2 24 2 25 2 26 2 27 26° 28° / // 4 18 3 48 3 26 3 10 2 57 2 46 2 37 2 3o 2 24 2 19 2 i5 2 12 2 9 2 7 2 5 2 4 2 2 2 I 2 2 I 59 I 59 I 59 I 59 1 59 2 2 2 I 2 I 2 2 2 3 2 4 2 4 2 5 2 5 2 6 2 7 2 8 2 8 2 9 2 10 2 II 2 12 2 i3 2 i5 2 16 2 18 2 20 2 21 28° 30° / /' 4 33 4 3 37 3 19 3 5 2 54 2 44 2 36 2 29 2 24 2 19 2 i5 2 12 2 10 2 8 2 6 2 4 2 2 2 I 2 2 2 I 59 I 59 I 59 159 I 59 1 09 2 2 2 I 2 I 2 2 2 2 2 3 2 3 2 4 2 4 2 5 2 5 2 6 2 7 2 8 2 9 2 10 2 II 2 12 2 i3 2 i4 30° TABLE XLVIII. [Page 311 Third Correction. Apparent Distance 92°. 1 D's A Dp. Alt. Apparent Altitude . 2 3 4 4 . 10 2 I 3 4 5 .■) bQ 15 3 ) 4 5 5 6 60 20 23 4 5 5 6 ) fi 6 tj 7 61 30 5 6 7 35 6 7 3 62 40 7 8 63 1 45 SO 7 8 i 8 i 64 1 6.5 9 9 i tib 1 i 1 28° 30° 1 32° 34° 36° 38° 40° 42° 44° 46° 48= 1 ^"^'^'^ TABLE XLVIII. Third Correction. Apparent Distance 120°. ])'s App. Alt. o 6 7 8 9 lO II 12 i3 i4 i5 i6 17 i8 19 20 21 22 23 24 25 26 27 28 11 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 5o 5i 52 53 54 55 56 57 58 60 61 62 63 64 65 Ajjparent Mtitude of the Sun, Star or Planet. 5 's 6= / // 3 I 3 3 3 7 3 12 3 j8 3 25 3 33 3 4i 3 49 3 57 4 6 4 i4 4 23 4 32 4 40 4 49 4 58 5 7 5 16 5 25 5 34 5 42 5 5i 6 6 8 6 17 6 25 6 34 43 6 5i 6 59 7 8 7 16 7 24 7 32 7 4o 7 47 7 55 8 3 8 II 8 18 8 25 8 32 8 39 8 45 8 5i 8 57 9 3 9 9 7° 1 II 3 3 3 2 3 4 3 8 3 12 3 17 3 23 3 28 3 34 3 4i 3 48 3 55 4 3 4 10 4 17 4 24 4 3i 4 39 4 46 4 53 5 I 5 8 5 16 5 24 5 3i 5 39 5 46 5 54 6 2 6 9 6 16 6 23 6 3o 6 37 6 44 6 5o 6 56 7 2 7 9 7 i5 7 21 7 27 7 33 7 39 7 45 7 5i 7 57 8 3 8° , // 3 6 3 4 3 3 3 5 3 8 3 12 3 16 3 20 3 25 3'3o 3 36 3 42 3 48 3 54 4 I 4 7 4 i4 4 21 4 27 4 33 4 4o 4 47 4 53 5 5 6 5 12 5 18 5 25 5 3i 5 38 5 44 5 5o 5 56 6 2 6 8 6 i4 6 19 6 25 6 3i 6 36 6 4i 6 46 6 52 6 57 7 2 7 8 7 i3 9° / // 3 II 3 7 3 5 3 4 3 6 3 8 3 II 3 i5 3 19 3 23 3~^ 3 32 3 37 3 42 3 48 3 53 3 58 4 4 4 10 4 i5 4 20 4 25 4 3i 4 37 4 43 4 48 4 54 5 5 6 5 12 5 18 5 23 5 28 5 34 5 39 5 44 5 5o 5 55 6 6 5 6 10 6 i5 6 20 6 25 6 3o 6 34 10° / // 3 17 3 11 3 8 3 6 3 5 3 6 3 8 3 II 3 i4 3 18 3 22 3 25 3 29 3 33 3 38 3 42 3 47 3 52 3 57 4 2 4 7 4 12 4 17 4 22 4 27 4 32 4 37 4 42 4 47 4 52 4 57 5 2 5 7 5 12 5 17 5 22 5 27 5 32 5 37 5 42 5 46 5 5i 5 55 5 59 6 3 10° 11° / ti 3 24 3 16 3 u 3 8 3 6 3 5 3 6 3 8 3 II 3 i4 3 17 3 20 3 23 3 26 3 3o 3 34 3 39 3 4i 3 47 3 5i 3 56 4 I 4 5 4 10 4 i5 4 19 4 23 4 27 4 32 4 37 4 42 4 46 4 5o 4 55 4 59 5 4 5 8 5 i3 5 17 5 22 5 26 5 3o 5 M 5 37 11° 12° 1 II 3 32 3 22 3 i5 3 II 3 8 3 6 3 5 3 6 3 8 3 n 3 i3 3 i5 3 18 3 21 3 24 3 28 3 32 3 36 3 39 3 43 3 47 3 5i 3 55 3 59 4 3 4 7 4 II 4 i5 4 19 4 24 4 28 4 32 4 36 4 4o 4 44 4 48 4 52 4 56 5 5 4 5 8 5 II 5 i4 12° 13° / // 3 39 3 28 3 20 3 i4 3 10 i 8 3 7 3 6 3 7 3 9 3 II 3 12 3 i4 3 17 3 20 3 23 3 26 3 3o 3 33 3 36 3 39 3 43 3 47 3 5o 3 54 3 57 4 I 4 5 4 9 4 12 4'T5 4 19 4 23 4 27 4 3i 4 35 4 39 4 42 4 46 4 49 4 53 4 56 13° 14° / // 3 47 3 34 3 25 3 18 3 i4 3 II 3 9 3 b 3 7 3 8 3 9 3 10 3 12 3 i5 3 17 3 19 3 22 3 25 3 28 3 3i 3 34 3 37 3 4o 3 43 3 46 3 49 3 52 3 56 3 59 4 2 4 5 4 9 4 i3 4 16 4 20 4 24 4 28 4 3i 4 34 4 37 4 40 15° / /^ 3 55 3 40 3 29 3 22 3 17 3 i3 3 n 3 9 3 8 3 7 3 8 3 9 3 11 3 i3 3 i5 3 17 3 19 3 2i 3 23 3 26 3 29 3 32 3 35 337 3 4o 3 43 3 46 3 49 3 52 3 55 3 58 4 2 4 5 4 8 4 II 4 i5 4 18 4 21 4 24 4 27 16° / // 4 4 3 47 3 35 3 26 3 20 3 16 3 i3 3 II 3 9 3 8 3 8 3 9 3 10 3 II 3 i3 3 i5 3 16 3 18 3 20 3 23 3 25 3 28 3 3o 3 33 3 36 3 38 3 4i 3 44 3 47 3 5o 3 53 3 56 3 59 4 I 4 4 4 7 4 10 4 i3 4 16 17° / // 4 12 3 54 3 4o 3 3i 3 24 3 19 3 i5 3 i3 3 II 3 9 3 9 I 9 3 10 3 12 3 i3 3 i4 3 16 3 18 3 20 3 22 3 25 3 27 3 29 3 32 3 34 3 36 3 39 3 42 3 45 3 47 3 5o 3 53 3 55 3 58 4 I 4 3 4 6 18° / // 4 21 4 I 3 46 3 36 3 28 3 22 3 18 3 i5 3 12 3 II 3 10 3 9 I 9 3 10 3 II 3 12 3 i3 3 i4 3 16 3 18 3 20 3 22 3 24 3 26 3 28 3 3o 3 32 3 35 3 37 3 40 3 42 3 45 3 47 3 5() 3 52 3 55 3 57 19° / II 4 3o 4 8 3 52 3 41 3 33 3 26 3 21 3 17 3 14 3 12 3 11 3 10 3 10 3 9 3 10 3 II 3 12 3 i3 3 i5 3 17 3 18 3 20 3 22 3 23 3 25 3 27 3 29 3 32 3 34 3 36 3 38 3 4; 3 43 3 45 3 47 3 49 20° 2 / // ( 4 394 4 i5 4 3 594 3 473 3 38 3 3 3o3 3 243 3 203 3 173 3 i4 3 3 12 3 3 11 3 3 ii3 3 103 3 10 3 3 103 3 11 3 3 123 3 143 3 i53 3 163 3 183 3 203 3 21 3 3 23 3 3 253 3 273 3 293 3 3i3 3 33 3 3 35 3 3 373 3 393 3 4i 3 43 Apt*. 2° Ah. /' 57 6 3o 7 12 s 58 9 47 10 38 II 3i 12 26 i3 22 :4 18 ]5 16 lb i4 17 i3 18 12 19 12 20 11 21 11 22 12 23 i3 24 i4 25 1 5 26 16 27 17 s8 18 29 19 3o 20 3 1 22 32 24 33 26 34 28 35 3o l6" 3i 37 33 38 39 4o 4i 42 43 44 45 46 47 48 49 5o 14° 15° 16° Tahle P. Effect of Sun's Par. To be sub'.racled from Ihe Third Correction. D's App. Alt. 5 10 15 20 25 30 35 40 4S 50 55 Sun's Apparent Akitupe. 5 2 •2 3 4 5 6 6 8 8 9 lu 20 3 2 3 3 3 3 4 4 4 5 4 5 6 5 6 7 6 7 8 7 8 8 8 8 9 40 50J55 4 5 5 5 6 6 7 7 7 70 80 90 6° 7° 8° 9° TABLE XLVIII. t^^so325 Third Correction. Apparent Distance 120°. D's o 6 7 8 9 lO II 12 1 3 i4 i5 i6 I? i8 19 20 21 22 23 24 25 26 27 28 =9 3o 3i 32 33 34 35 36 37 38 39 4o 4i 42 43 44 45 46 47 48 49 5o 5i 52 53 54 55 56 57 58 61 62 63 64 65 Apparent Altitude of the Sun, Star or Planet. D's App. All. 6 7 8 9 10 II 12 i3 i4 i5 16 17 18 19 20 21 22 23 24 25 26 27 28 3o 3i 32 33 34 35 36 37 38 39 4o 4i 42 43 44 45 46 47 48 49 5o "57 52 53 54 55 56 57 58 60 61 62 63 64 65 24° / '/ 5 i5 4 45 4 25 4 9 3 57 3 47 3 39 3 32 3 27 3 23 3 20 3 18 3 16 3 i5 3 i4 3 i3 3 12 3 12 3 12 3 12 3 i3 3 i4 3 i5 3 16 3 17 26° ! II 5 32 5 4 38 4 20 4 7 3 56 3 47 3 39 3 33 3 29 TI5 3 22 3 20 3 18 3 16 3 i5 3 i4 3 i3 3 i3 3 i3 3 i4 3 i4 3 i4 3 i5 3 16 28° / // 5 49 5 i5 4 5i 4 3i 4 17 4 5 3 55 3 46 3 39 3 34 3 3c. 3 27 3 24 3 21 3 19 3 17 3 16 3 i5 3 i5 3 i5 3 i5 3 i5 3 i5 3 i5 3 16 30° / // 6 6 5 3o 5 4 4 43 4 27 4 i4 4 3 3 53 3 46 3 4o 3 36 3 32 3 28 3 25 3 23 3 21 3 19 3 18 3 17 3 17 3 16 3 16 3 16 3 16 3 16 32° / // 6 23 5 45 5 17 4 55 437 4 23 4 II 4 3 52 3 46 3 4i 3 36 3 32 3 29 3 27 3 24 3 22 3 21 3 20 3 19 3 18 3 18 3 18 34° 1 II 6 4i 6 5 3o 5 7 4 47 4 32 4 19 4 8 3 59 3 52 3 46 3 4i 3 37 3 33 3 3i 3 28 3 26 3 24 3 23 3 21 3 20 36° / n 6 58 6 i5 5 43 5 18 4 57 4 42 4 28 4 16 4 6 3 58 3 52 3 47 3 42 3 38 3 35 3 32 3 29 3 27 3 26 38° / II 7 i4 6 29 5 56 5 28 5 7 4 5i 4 36 4 24 4 i3 4 4 3 58 3 52 3 47 3 43 3 39 3 36 3 33 40° / II 7 3o 6 43 6 8 5 38 5 17 4 59 4 44 4 3i 4 20 4 II 4 4 3 58 3 52 3 47 3 43 40° 42" / // 7 46 6 56 6 20 5 49 5 27 5 8 4 52 4 38 4 27 4 18 4 10 4 3 3 57 42° 44° / '/ 8 2 7 8 6 3i 6 537 5 17 5 4 46 4 34 4 24 4 16 44° 46° / // 8 17 7 21 6 42 6 10 5 46 5 25 5 7 4 53 4 40 46° 48° / II 8 3i 7 33 6 53 6 20 5 55 5 33 5 i4 50° / II 8 M 7 45 7 3 6 29 6 3 52° / // 8 57 7 57 7 i3 54° / II 9 9 3 18 3 19 3 21 3 22 3 24 3 26 24° 3 17 3 18 3 19 3 20 26° 3 17 3 i8 28' 30° 32° 34° 36° 38° 48° 50° 52° 54° 1 *^se 326] TABLE XLIX. To find the correction of the apparent distance of the moon from any planet, on ac- count of the parallax of the planet, supposing its horizontal parallax to be 35". This is to be reduced to the actual horizontal parallax by m.eans of Table L. Apparent Distance. Apparent Distance. * D o o V, o o * » Alt. Alt. 2i) 30 40 50 60 70 80 90 100 110 120 Alt. Alt. 20 30 40 50 60 70 80 90 100 no 120 10 10 + 1 — 1 2 2 — 3 — 4 — 5 — 6 — 7 — 9 -10 25 10 +24 +14 + 9 + 5 + 2 — 1 — 3 — 6 — 9 —12 -15 15 — 8 — 7 — 6 — 6 — 7 — 7 — 8 — 9 —10 —12 —14 15 +15 + 8 + 4 + 1 — 2 — 4 — 6 — 9 —12 -15 -19 20 —17 —13 —11 —10 —10 —10 —11 —12 —13 —15 -17 20 + 7 + 3 — 1 — 3 — 5 — 7 — 9 —12 —15 —18 —22 25 —26 —18 —15 —14 —13 —13 —14 —15 —16 -18 —20 25 — 1 — 3 — 5 — 7 -8 —10 —12 —15 —17 —21 -26 30 —34 —24 —20 —17 —16 —16 —17 —17 —19 —21 —23 30 — 9 — 9 — 9 —10 — U —13 —15 —17 —20 —24 —29 35 —29 —24 —21 —19 —19 —19 —20 —21 —24 —26 35 —17 —14 —13 —14 —14 —16 —18 —20 —23 27 —32 40 —34 og —24 —22 22 22 —23 —24 —26 -29 40 —25 —19 —17 —17 —17 —18 —20 ^22 — 25 —29 45 _?1 —27 —25 —24 —24 —25 —26 —29 —32 45 —32 —23 —21 —20 —20 -21 —22 —25 —28 —32 50 —34 — ::o —27 —26 —26 —27 og —31 —34 50 —28 —24 ^22 22 -23 —24 —27 —3(1 55 —33 —30 —28 -28 —"9 —30 —33 55 _30 _07 —25 —"4 —"5 — ''6 -99 3') GO 65 70 75 80 85 90 —34 —31 —33 —34 —30 —32 —33 —34 —34 —30 —31 -32 —33 —3^ —34 34 —30 —32 —33 —34 —34 —32 —33 —34 —34 GO G5 70 75 80 85 10 —29 —32 —27 —29 —31 —32 —26 ^28 —29 —30 —31 32 —27 —28 —30 —31 —31 —32 —28 —30 —31 —32 —30 —32 15 10 + 9 + 5 + 2 — 2 — 3 — 4 — 6 — 8 —10 12 30 10 +30 +19 +12 + 7 + 3 — 3 — 6 —13 —17 15 — 1 — 3 — 4 — 5 — 6 — 7 — 9 —11 -13 —15 15 +22 +13 + 7 + 3 - 3 — 6 — 9 —12 —16 — 20| 20 — 9 — 7 — 7 — 8 — 8 — 9 —10 —12 ^14 —16 —19 20 +14 + 7 -f 3 — 1 — 4 — 6 — 9 —12 — 15 —19 —24 25 —17 —13 —12 —11 —12 —12 —13 —15 —16 -19 22 25 + (i + 2 2 — 4 — 7 — 9 —12 —15 —18 22 -27 30 —26 -19 —16 —15 —15 —15 —16 —17 —19 22 -25 30 2 — 4 — 6 — 8 —10 —12 — M —17 —21 -25 .-30 35 — 34 —24 —20 —18 —18 -18 —19 -20 22 —25 —28 35 —10 — 9 —10 —11 -13 —15 —17 —20 —23 —28 40 —29 —24 22 —20 —20 —21 —22 —24 -27 —31 40 —17 —14 —14 -14 —16 —17 —20 —23 — 2G —30 45 —34 —25 -23 -23 —23 -25 — 26 —30 —31 45 —24 —19 —17 —17 —18 —20 ^22 —25 —28 50 31 27 ''u 05 26 •''7 oq 3.0 nO on .0" —20 —23 —24 —26 —27 —29 —30 55 . .. . -34 —30 —23 oy -T-27 —29 —30 —34 55 07 -24 -23 -24 GO —32 —30 —29 —29 —30 30 GO -30 —26 —25 —25 -20 —28 —30 (i5 34 31 30 31 3'"' 34 G5 -28 -30 —27 —28 —27 —29 —29 —30 70 -33 —32 -32 —33 70 -28 75 80 85 !)0 —34 —33 —33 —34 -33 —33 —3-! —34 75 80 85 no -29 —30 —29 —30 —30 30 —30 —30 :o !0 +16 + 9 + 5 + a — 4 — 6 — 8 —11 —14 35 10 +23 +15 + 9 + 5 + 1 — 3 — 6 —10 —14 —19 15 + B + 4 + 1 — 1 — 3 — 5 — 7 — 9 —11 -14 —17 15 +29 +17 +10 + 5 + 1 — 2 — 5 - 9 —13 — 1: -22 20 -1-2 — 4 — 5 — 7 — 8 —10 —12 -14 —17 —21 20 +21 +12 + 6 + 2 2 — 5 — 8 —12 —16 —20 —25 25 — 9(— 8 — 8 - 9 —10 —11 —13 —15 —17 —20 —24 25 +13 + 6 + 2 — 2 — 5 — 8 —11 —15 —IS 23 20 30 -17 -14 —12 —13 —13 —14 —16 —17 —20 —23 —27 30 + 5 + 1 — 3 — 6 - 8 —11 —14 —17 —21 —26 35 —2,5 —19 —16 —16 —16 —17 —18 —20 22 -25 —30 35 — 3 — 5 — 7 — 9 —11 —14 —17 —20 —24 —29 tu —33 24 —20 —19 —19 —19 —21 —23 —25 —28 —33 40 —10 —10 —11 —12 —14 -16 -19 —22 0(; Ih —29 —24 22 —21 22 —23 -25 07 —31 45 —16 -M —14 —15 —17 —19 22 -25 —29 ;)!) —33 -27 —25 —24 —24 —25 07 —29 —33 50 —23 —18 —17 —18 —19 —21 —24 27 bo —30 —27 —26 — 2i5 27 -29 —31 55 —29 ^22 -20 —20 —21 —23 —26 —29 GO .... —33 —29 —28 -28 -29 —30 -33 GO — 26 —23 22 —23 05 27 ().'. 31 30 oq 30 1 in G5 —29 70 75 , ... -33 —31 -31 3'> —31 3^ —33 70 75 —27 —29 —26 -27 —28 -29 -27 — 2S -28 —29 —28 —29 i 80 85 90 -33 —32 -33 33 -33 SO 85 90 — O i o 2i) 30 1 40 i.)0 tiO 70 80 •SO 100 110 120 20 30 40 50 60 70 80 90 100 110 120 TABLE XLIX. [i-agess? To find ilie correction of the apparent distance of the moon from any planet, on ac- i count of the parallax of the planet, supposing its horizontal parallax to ':)e 35". This is to be reduced to the actual horizontal parallax by means of Table L. Apparent Distance. Apparent Distance. 1 * Alt. Alt. 20 311 40 50 60 70 80 90 100 no 120 * Alt. D Alt 20 30 40 50 60 70 ^0 90 100 110 1?5 1 ' 40 10 +27 +18 +11 + 6 + 2 _ 2 — 6 —10 —15 —20 55 10 +16 +10 + 4 — 1 — 6 —11 —17 —20 15 +21 +13 + 7 + 3 — 1 — 5 - 9 —13 -18 —23 15 +20 +12 + 6 + 1 — 4 — 9 -14 -20 1 20 +27 +16 + 9 + 3 — 1 — 4 — 8 -12 -16 —21 — 27 20 +16 + 8 + 3 2 — 7 —12 -17 I 25 +19 +10 + 4 — 4 — 7 -11 -15 -19 —24 25 +20 +11 + 5 — 1 — 5 —10 —15 -20 30 +11 i? — 4 — 7 —10 —14 -17 -22 o~ 30 + 15 + 7 + 1 — 4 — 8 —13 —17 35 + 4 — 4 — 7 —10 —13 —16 -20 —24 35 +20+10 + 3 — 2 — 7 —11 -15 —20 40 - 3 — 5 — 8 -10 —13 -10 —19 —23 -27 40 +13 + 5 — 1 — 5 — 9 —13 -18 45 -10 -10 —11 —13 -16 -18 —21 -25 45 + 7+ 1 — 4 — 8 —12 —16 -20 50 -IG -14 —15 —16 -18 —20 —23 —27 50 + 1-4 — 8 —11 —14 -18 55 —22 -18 -IS —18 -20 22 — 25 55 -4-7 —11 —13 —17 —20 CO -27 —21 -20 —21 —22 —24 —27 60 - 9J-11 —13 —15 —18 Co —24 0.3 22 —24 -26 65 -14 —14 —15 —17 -20 70 -27 -24 -24 —25 27 70 —17 —16 —17 —19 75 -26 — 25 -26 75 -20 —18 —19 -20 80 -27 -26 —27 80 —19 —19 85 — 27 85 —20 —20 i 90 07 90 1 146 10 +20 +13 + 7 + 3 — 2 — 6 —10 —15 —21 GO 10 +17 +11 + 5 — 1 — 6 —11 —17 j 15 +25 +15 + 9 + 4 — 5 — 9 -14 —19 —25 15 +14 + 7 + 2 — 4 — 9 —15 1 20 +19+11 + 5 + 1 — 4 — 8 —12 —17 —22 20 +17 +10 + 4 2 — 7 —12 —17 25 +2.5 + 141+ 7 + 2 - 3 — 7 —11 —15 -19 — 25 25 ....j+13 + 6 — 5 —10 —15 i 30 +n + 8+3 — 2 — 6 — 9 —13 —17 —22 30 +17 + 9 + 3 — 3 — 8 —13 —17 35 +10 + 3 — 1 — 5 — 9 —12 —16 —20 —25 35 +13 + 5 — 1 — 6 —10 —15 40 + 3 -5 — 8 —11 -15 —19 -23 40 +17 + 8 + 1 -4 — 8 —13 —17 45 — 4 - 6 -9 —11 —14 —17 —21 —25 45 +11 + 3 -2 -7 —11 -15 50 -ID —10!— 12 -14 —17 -19 -23 50 + 5 -1-6 —10 —13 —17 55 -15 -HJ-lo —17 —19 —21 — 25 55 -5-9 —12 —16 CO -20 -17-18 —19 —21 -23 GO - 5 — 8 —11 —14 -17 05 —25 —20—20 —21 —23 —25 65 — 9 -11 —13 —16 70 -23-5« —22 -24 70 -13 -13 -15 -17 75 -25-23 —23 ■ ■25 75 —16 -15 —16 80 -24 —24 80 —17 -16 —17 85 .... —25 -2-1 85 ... 1 —17 90 90 —17 50 10 +23 + 18 +15 +11 + 9 + 5 + 3 2 g jj IQ 23 65 10 +11 + 8 + 5 + 2 1 — 6 — 9 |0 15 — 5 — 9 -14 -19 15 +15 — 4 -15 20 .... +23;+14j + 7 + 2 — 3 — 8 -12 —17 —23 20 +11 + 5 — 1 — 7 -12 25 .... +17+ 9 + 3 — 2 — 6 -10 —15 -20 25 +15 + 7 + 1 — 4 —10 -15 30 +22;+l2i+ 5 — 5 — 9 —13 —17 —23 30 +11 + 4 2 — 7 —12 35 + 15:+ 7 + 1 - 4 — 8 —11 -16 -20 35 +15 + 7 + 1 — 5 —10 —15 40 + 9+2-3 - 7 —10 —14 —18 -23 40 +10 + 3 — 3 — 8 —12 45 + 2-.3-6 -10 —13 -16 —21 45 +15 + 6 - 1 - 6 -10 —15 50 -4- 71-10 -12 -15 -19 —23 50 + 9 + 2-4] - 8 —13 55 - 9 -10;-12 -15 -18 —21 55 + ^i — 2 -7 —11 —15 60 -Hj-u! -15 -17 —19 -23 60 - 1 — 5 - 9 -13 65 -19-17i-17 -19 —21 65 - 5 — 8 -12 -15 70 -23—191-19 —20 —23 70 - 9 —11 -13 "5 -21 -21 1 22 75 -12 —13 -15 ^■0 -23 —22 -23 80 -14 —14 85 —2a 85 —15 —15 90 .... -23 90 •fo 1 30 40 50 60 70 80 90 100 110 120 20 30 40 50 60 7°0 80 90 100 1?0 115 Page 3281 'se 3281 TABLE XLIX. To find the correction of the apparent distance of the moon from any planet, on account of the parallax of the planet, supposing its horizontal parallax to be 35". This is to be reduced to the actual horizontal parallax by means of Table L. Apparent Distance. 75 20 +12 + — G —10 + 9- 8 30 40 50 +12 + 8 + 3 — 4 — 9 —12 + 2 +12 + 8 + 1 - 2 - 5 - 8 -12 + 9 + 5 + 2 — 1 4 — 7| +12 + 8 + 2 — 2 — 5 7 9 —12 CO 70 80 90 + 9 + 5 + 2 — 1 4 — 7 — 9 +12 + 9 + 5 + 2 — 1 — 4 — 7 —10 —12 + + G + 2 — 1 — 4 6 9 + 2 — 1 — 4 7 — 9 —12 + e + 3 — s — 6 — 9 — 1 — 4 — 7 — 9 —12 — 3 — G — 9 100 Apparent Distance. 80 83 Alt. 20 30 40 50 60 70 80 90 + 6 — 1 — G + G + 3 — 1 — 6 + 3 + 6 + 3 — I — 3 — 6 + 3 — 1 + 3 1 — 3 + 3 — 3 + + 3 — 3 + 3 — 3 + 6 + 3 3 6 + 3 — 3 100 TABLE L To reduce the numbers in Table XLIX., so as to correspond to the actual horizontal parallax of the planet. J^om ontal Parallax of the Planet. a>< // // // // II // // /,' /^ // II II II II // // // II // // II // II // // // // // // II ^^, n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 30 35 1 n 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 I 1 I 1 1 1 1 1 1 1 1 1 2 2 2 2 9 3 I 1 1 1 1 1 1. 1 1 1 1 1 2 2 2 o 2 2 o 2 2 2 2 3 3 3 4 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 o 3 3 3 3 y 3 3 3 4 4 h 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 5 5 6 ] 1 1 1 1 2 2 o 2 o 2 3 3 3 3 3 3 4 4 4 4 4 4 .5 5 5 G "fi" 7 1 1 1 o o o 2 3 3 3 3 3 4 4 4 4 4 5 .5 .5 5 5 G G 7 7 8 1 1 2 2 o 3 3 3 3 3 4 4 4 4 5 .5 .5 .5 5 6 G 6 6 7 8 8 9 u 1 2 2 ') o 3 3 3 3 4 4 4 4 .5 5 5 .5 6 6 6 6 7 7 7 8 9 9 10 1 2 2 2 3 3 3 3 4 4 4 5 5 5 5 6 G G 7 7 7 7 8 8 9 10 10 11 2 o 2 3 3 3 3 4 4 4 5 .5 ,5 G 6 fi 7 7 7 y 8 8 8 9 9 11 TT \> 2 2 3 3 3 4 4 4 5 ."j 5 G G 7 7 7 8 8 8 9 9 9 10 10 12 i'> i:i 2 2 3 3 3 4 4 4 5 5 G 6 G 7 7 7 8 8 9 9 9 10 10 10 11 13 13 11 2 2 2 3 3 4 4 4 5 5 G 6 G 7 7 8 8 8 9 9 10 10 10 11 11 12 14 14 1-) o 2 3 3 3 4 4 5 5 G G G 7 7 8 8 9 9 9 10 10 11 11 12 12 13 15 15 16 o 2 3 3 4 4 5 ry 5 G G 7 7 8 8 9 9 10 10 11 11 11 12 12 13 14 16 16 17 2 2 3 3 4 4 5 6 G G 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 15 17 17 18 2 o 3 3 4 4 5 5 G G 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 1,5 18 IS 19 2 2 3 3 4 4 5 5 G 7 7 8 8 9 9 10 10 11 11 12 12 13 14 14 15 1.5 16 19 19 20 2 2 3 3 4 5 ,"> 6 G 7 7 8 9 9 10 10 11 11 12 13 13 14 14 15 15 16 17 20 20 21 2 2 3 4 4 5 5 6 7 7 8 8 9 10 10 11 11 12 13 13 14 14 15 16 16 17 IS 21 f!1 yo y 3 3 4 4 t) G G 7 8 8 9 9 10 11 11 12 13 13 14 14 15 16 16 17 IS 19 22 22 23 y 3 3 4 5 5 G 7 7 8 9 9 JO 11 11 12 12 13 14 M 15 16 IG 17 IS 18 20 2.3 23 21 y 3 3 4 5 b G 7 8 8 9 10 10 11 12 12 13 14 14 15 16 16 17 IS 19 19 21 24 24 2.') y 3 4 4 5 G G 7 8 9 9 10 11 11 12 13 14 14 15 16 16 17 18 19 19 20 21 25 25 2(i 1 2 3 4 4 5 G 7 7 8 9 10 10 11 12 13 13 14 15 16 16 17 18 19 19 20 21 o>i 2(i 26 a 2 3 4 6 5 G 7 8 8 9 10 11 12 12 13 14 15 1.5 16 17 18 19 19 20 21 oo 23 27 27 28 2 2 3 4 5 G 6 7 8 9 10 10 11 12 13 14 14 15 16 17 IS IS 19 20 SI 22 oo 24 28 28 yj 2 2 3 4 5 6 7 7 8 9 10 U 12 12 13 14 15 IG 17 17 18 19 20 21 22 oo 23 25 29 29 31) 2 3 3 4 5 G 7 8 9 9 10 11 12 13 M 1.5 15 16 17 18 19 20 21 21 2-2 23 24 26 30 30 :ii 2 3 4 4 5 6 7 8 9 10 11 12 12 13 14 1.5 IG 17 18 19 19 20 21 22 2;j 24 2,5 27 31 31 32 y 3 4 5 5 6 7 8 9 10 11 12 13 14 1.5 111 IG 17 18 10 20 21 oo 23 24 25 26 27 32 32 33 3 4 5 G 7 8 8 9 10 11 12 13 14 1.5 IG 17 IS 19 20 21 22 23 24 2.5 25 26 28 33 33 34 2 3 4 5 G 7 8 9 10 11 12 13 11 1,5 If. 17 17 18 19 20 21 oo 23 24 25 26 27 29 34 34 35 2 3 4 6 G 7 8 9 10 11 12 13 14 1.5 16 17 18 19 20- 21 22 23 24 25 26 £7 2b 30. 35 35 TABLE LI. TABLE LIL [f age. 329 To change mean solar time into To change sideral time into mean sideral time. solar time. Solar Solar Sidcr.il Sideral Add. Min- Add. Sec- Add. Sideral Hours. Subtract. iMin- Subtract See- Subtract utes. onds. utps. on Is. M. S. s. s. M. S. S. s. 1 9.9 I 0.2 I 0.0 I 9.8 I 0.2 I 0.0 2 19.7 2 0.3 2 0.0 2 19.7 2 0.3 2 0.0 3 29.6 3 0.5 3 0.0 3 29.5 3 0.5 3 0.0 4 39.4 4 0.7 4 0.0 4 39.3 4 0.7 4 0.0 5 49.3 5 0.8 5 0,0 5 49-1 5 0.8 5 0.0 6 59. 1 6 1 .0 6 0.0 6 59.0 6 1 .0 b 0.0 7 I 9.0 7 1 .2 7 0.0 7 I 8.8 7 1 .1 7 0.0 8 I IS.9 8 1.3 8 0.0 8 I 18.6 8 1.3 8 0.0 9 I 28.7 9 1.5 9 0,0 9 I 28.5 9 1.5 9 0.0 10 I 38.6 10 1.6 10 0.0 10 I 38.3 10 1.6 10 0.0 II I 48.4 11 1.8 II 0.0 II I 48.1 1 1 1.8 II 0.0 12 I 58.3 12 2.0 12 0.0 12 I 58.0 12 2.0 12 0.0 i3 2 8.1 i3 2.1 i3 0.0 i3 2 7.8 i3 2.1 i3 0.0 i4 2 18.0 i4 2.3 14 0.0 i4 2 17.6 i4 2.3 i4 0.0 i5 2 27.8 i5 2.5 i5 0.0 i5 2 27.4 i5 2.5 i5 0.0 i6 2 37.7 16 2.6 16 0.0 16 2 37.3 16 2.6 16 0.0 17 2 47-6 17 2.8 17 0.0 17 2 47-1 17 2.8 17 0.0 i8 2 57.4 18 3.0 18 0.0 18 2 56.9 18 2.9 18 0.0 19 3 7.3 19 3.1 19 0.1 '9 3 6.8 19 3.1 19 O.I 20 3 17. 1 20 3.3 20 O.I 20 3 16.6 20 3.3 20 O.I 21 3 27.0 21 3.5 21 0. I 21 3 26.4 21 3.4 21 O.I 22 3 36.8 22 3.6 22 0. I 22 3 36.2 22 3.6 22 O.I 23 3 46.7 23 3.8 23 O.I 23 3 46.1 23 3.8 23 O.I 24 3 56.6 24 3.9 24 O.I 24 3 55.9 24 3.9 24 25 O.I 25 4.1 25 O.I 25 4.1 «0.I 26 4.3 26 0. I 26 4.3 26 O.I 27 4.4 27 O.I 27 4.4 27 0.1 28 4.6 28 O.I 28 4.6 28 O.I 29 4.8 29 O.I 29 4.8 29 O.I 3o 4.9 3o O.I 3o 4.9 3o 3i O.I 3 1 5.1 3i O.I 3i 5.1 O.I 32 5.3 32 O.I 32 5.2 32 O.I 33 5.4 33 O.I 33 5.4 33 O.I 34 5.6 34 O.I 34 5.6 34 O.I 35 5.3 35 0. I 35 3.7 35 O.I 36 5.9 36 O.I 36 5.9 36 O.I 37 6.1 37 O.I 37 6.1 37 O.I 38 6.2 38 O.I 38 6.2 38 O.I 39 6.4 39 O.I 39 6.4 39 O.I 40 6.6 4o 0. I 4o 6.6 40 O.I 4i 6.7 4i O.I 4i 6.7 4i O.I 42 6.9 7-1 42 O.I 42 6.9 42 O.I 43 43 O.I 43 7.0 43 O.I 44 7.2 44 O.I 44 7.2 44 O.I 45 7-4 45 O.I 45 7.4 4b O.I 46 7.6 46 O.I 46 7.5 46 O.I 47 7-7 47 O.I 47 7-7 47 O.I 48 7-9 48 O.I 48 49 7-9 48 O.I 49 8.1 49 O.I 8.0 49 O.I 5o 8.2 5o O.I 5o 8.2 bo 0. I 5i 8.4 5i O.I 5i 8.4 5i O.I 52 8.5 52 O.I 52 8.5 52 O.I 53 8.7 53 0. I 53 8.7 53 O.I 54 8.9 54 O.I 54 8.8 54 O.I 55 9.0 55 0.2 55 9.0 55 0.2 56 9.2 56 0.2 56 9.2 56 0.2 57 9-4 57 0.2 57 9.3 57 0.2- 58 p. 5 58 0.2 5?> 9.5 58 0.2 59 9-7 59 0.2 59 9-Z 59 2 60 9.9 60 0.2 60 9.8 60 0.2 Page 330] TABLE LIII. This table g ives the Variation of the Compass for 1858, very nearly as in the chart of F. J. Evans, master, R. N. T3 West Longitude. T3 o 1 1 ° O 180 170 160 1.50 140 130 120 110 100 90 1 80 70 j 60 .50 40, 30 20 10 I Variation of the Compass. o o 1 1 1 1 boN i5N 20E 25^57 30^33^35^ 19^ 7Tr39TF 53 1^1561^55 TP' 5iTF 45 F 3qTF 3ir 24 TF 6oiV 58 i5 19 24 28 3i 32 oTr3i 45 1^" 5l :5l 48 43 37 3o 24 58 56 U i8 22 26 28 29 2E 22 46 47 45 41 35 29 23 56 54 U i8 21 24 27 27 2bE 17 62 W 42 44 43 40 34 28 22 34 D2 i3 17 20 23 23 25 24 12 27 36 41 41 38 33 27 22 32 5o 1 3 •7 19 22 23 23 22 8TF 22 3i 37 38 36 3i 26 21 5o 48 .3 i6 IP 20 22 22 21 No rth 18 27 33 36 34 3o 25 20 48 46 i3 i6 i8 19 20 21 14 Ame rica. i3 24 3o Si 32 29 24 20 46 44 i3 i5 17 18 19 19 18 i3 21 27 3o 3o 27 23 19 44 42 1 3 ID 17 17 18 18 17 II 18 |25 28 28 26 23 i9_ 19 42 40 i3 i5 16 17 17 17 \l oE 9 16 22 26 27 25 22 40 ;i8 i3 i4 i5 16 , 16 16 I 8 14 20 24 26 23 22 17 38 36 i3 14 14 i5 l5 i5 i5 I 6 12 18 22 24 24 21 I7TF 36 34 i3 14 14 i4 14 14 14 i3E tiE 2 4 II 16 21 23 23 21 34 32 i3 i3 i3 i3 i3 ij i3 12 8 3 3 9 14 19 22 22 20 32 3o 12 i3 12 12 12 12 12 u io£ 8 3 2 8 i3 18 21 22 20 3o 28 12 12 II II II II II II 10 8 4 iW 6 12 16 20 21 20 IF 28 26 12 12 II II II 11 10 10 9 7 4 3 10 i5 19 20 26 24 12 II 10 10 II 10 10 10 9 7 b ■ lE 4 9 i3 19 20 24 22 II 10 10 9 9 9 9 9 9 7 3 2 3 8 12 17 20 Ajri ca. 22 20 11 10 9 9 8 8 8 9 g 7 5 2 2 7 II 16 19 20 i8 11 10 9 8 8 7 8 8 8 7 5 3 iW 6 II 16 19 18 i6 10 9 8 7 7 7 8 8 7 ^ 3 oE 5 10 i5 19 16 14 10 9 8 7 7 b 7 8 7 6 3 I 4 9 14 18 14 12 10 9 8 7 b 6 6 8 8 6 4 2 3 8 14 18 12 10 Id 9 8 7 6 6 6 8 8 7 4 2 3 8 14 18 19 IF 10 8 10 9 8 7 6 5 6 8 8 7 4 2 2 8 i3 18 20 20 11' 8 6 Q 9 8 7 6 5 5 8 8 7 t)E 6E 2 7 i3 18 20 20 6 4 Q 9 8 7 6 5 5 7 8 8 I 7 i3 17 20 21 4 2 IC 9 9 8 7 6 5 5 8 7 8 I 7 i3 17 20 21 2iV Q 9 8 7 6 5 5 8 9 9 6 12 18 21 22 2^' Q 9 8 7 6 5 5 8 9 9 oW 6 12 18 21 23 2,b' 4 Q 9 8 7 5 5 5 8 10 10 ■6r 12 18 22 23 4 6 P 9 8 7 6 5 3 9 10 10 iSo uth 12 18 22 24 6 8 9 9 8 7 6 5 6 9 11 10 II Ame rica Dir 12 II 18 18 22 22 24 24 8 10 Q 9 8 7 6 fy 6 9 II 10 12 10 9 8 7 6 b 6 10 II II 3 11 18 22 23 12 14 10 9 8 7 6 6 6 8 10 II 12 loE 5 II •7 22 23 14 i6 10 9 8 7 7 6 7 8 10 12 12 10 4 II 17 22 2D 16 i8 10 9 8 8 7 7 7 8 II 12 i3 II 4 10 17 22 2D 18 20 10 9 8 8 7 7 7 9 II i3 i3 II 2E 3 10 16 22 26 20 22 10 9 9 8 7 7 7 9 II i3 14 12 2 3 9 16 22 26 22 24 11 9 9 8 7 7 8 9 12 iJ 14 12 3 3 i3 21 26 24 26 u 10 9 8 8 I 8 10 12 Si4 13 i3 3 2 8 i3 21 23 '26 28 12 10 9 8 8 9 10 12 14 13 14 4 2 8 14 21 25 28 3o 12 10 9 ■9 8 8 9 10 i3 i5 16 14 loE 4 I 7 14 20 23 3o 32 12 II 10 9 8 8 9 11 i3 13 16 10 II 3 III' 7 i3 19 24 32 34 i3 II 10 9 9 9 9 11 i3 16 17 i3 II 6 6 i3 19 24 34 36 1 3 12 10 9 9 9 10 II 14 17 IS 16 12 6 oE 6 12 18 24 36 38 14 12 II 10 9 9 10 12 14 17 ii8 16 i3 7 I 5 II 17 23 38 40 14 l3 11 10 10 10 10 12 i5 18 119 •7 i3 8 2 4 II 17 22 40 42 ID i3 12 10 10 10 11 i3 16 19 ,20 18 14 9 2 4 ID 16 22 42 44 i5 i4 12 II 10 10 II i3 !i7 20 20 18 14 9 3 3 9 i3 21 44 46 i6 14 i3 II II II 12 '4 |i7 21 ,21 19 ID 10 4 2 9 14 20 46 48 i6 i5 i3 12 II II i3 i5 !i8 21 22 20 16 II 5 I 8 14 20 48 5o \l i6 14 i3 12 12 i3 16 20 22 ,23 21 16 12 6 iTF 7 i3 19 5o 52 i6 i5 i3 i3 i3 14 17 21 18 ;22 23 ;23 21 17 12 7 6 12 18 62 54 i8 >7 i5 14 i3 i3 i5 24 ,24 22 18 i3 8 lE 5 II 17 D4 56 '9 •7 16 i5 14 14 16 19 23 25 25 23 19 14 9 2 4 10 16 56 58 20 i8 17 16 rb 16 17 21 25 26 26 24 21 'l5 10 4 3 9 ID 38 6o,S' 21^- jgE 18^ l■^E 11 U 1 7 A' 19^^ 23E 26E, 2-] E 21 E'7bE 21^ ibE 11^ 4E 3Tr gW iDir 6o^' Hi ISO'^ 170^ 160"^ ■150'-' 140*^ 130-^ 120'^ 110'^ 100'^.90^ SO'^ 170^^ loO'^ 50^ 40^^ 30" 30*^ 10^ 0^ 1 West Longitude. Page 331] TABLE LIIL This table gives the Variation of the Compass for 1858, very nearly as n the chart of F. J. Evans, master, R. N. ■o East Longitude. " 8 O 1 o 010 010 1 1 1 '1 1 10 20 30 1 40 50 1 60 70 1 80 90 1 100 1 110 120 130 140 150 160 170 180 Variation of the Compass. o O ] o 1 o 1 ° 6n,V 24TrnTriiTF 5TF 3F, bE 10^ 1 5^1 boN 58 24 17 II 5 IF 3 oW 3 10 i5 58 56 23 '7 II 3 4 10 14 56 54 ■ 22 16 iiW 3 4 9 14 54 D2 5o 22 16 11^ 3 4 9 i3 52 21 Uur ope 3 4 10 i3 5o /fi 20 bW iW I^' AE\ As la 3 4 10 i3 48 .-16 20 i5W 6 2 I 4 bW 3 oW 5 10 i3 46 44 19 i5 6 2 oE 3 5 3 5 10 i3 44 42 19 i5 6 2 2A' 3H' 4 3 5 10 i3 42 40 19 i5 6 3 oW 3 4 3 oE 5 10 i3 40 38 17 i5 6 3F 1 3 4 3 5 10 i3 38 36 iTlTiS 7 ill' 2 3 2 I 6 10 i3 36 34 i5 7 2 3 2 I 6 10 i3 34 32 i5tf 7 iF 2 3 2 I 6 10 i3 32 3o llW 7W I oJr I 2 I 2 6 10 12 3o 28 2 oU' I 2 I 2 6 10 12 28 26 2 7E I 2 I 2 6 10 12 26 24 2 oE 2 lE I oF 3 10 12 24 22 Afr ica 5»' 2ir 2 I I 3 10 1 1 22 20 5 \E 2 I oF I oE 3 10 11 20 i8 5 3F 2 2 I 4 10 II 18 i6 5 3 2 2 A' 2 oE I 4 10 10 16 1 4 5 3 oir 2 2 2 oF I 4 10 10 14 12 b^Y 3 2 2 2 2 4 9 10 12 10 3 2 2 2 oE 2 4 7 9 10 10 8 ■20W1SW 4 2 2 I 2 5 8 9 10 8 6 20 19 4 2 2 I 2 9 8 9 9 6 4 21 20 4 2 qE I I 3 5 8 9 9 4 2A- 21 20 4 2 I I 3 5 8 9 9 2 W 22 21 8Tr 5 3 I I 3 5 8 9 9 2.S' 23 22 8 5 3 1 1 3 5 8 9 9 ■iS 4 23 23 9 5 3 2 I I 3 6 8 9 U 4 6 24 23 10 6 4 2 I oE I I 3 6 8 9 9 6 8 lO 24 '24 24 23 u 12 7 7 4 4 2 3 I I oir oE oE oE 4 4 6 6 8 8 9 9 ') 9 8 10 12 25 j25 i3 8 5 3 2 I iTT oF oF 4 7 9 ID 10 12 14 23 26 14 9 6 4 2 2 I I 4 7 9 ID 10 14 i6 23 lb i5 10 7 4 3 2 2 I I 4 7 9 10 10 16 i8 23 27 16 11 7 3 4 3 3 2 I lE 4 A' 7 9 10 10 18 20 26 27 18 i3 9 6 5 4 3 2 I 8 10 10 10 20 22 26 27 '9 14 10 7 6 5 4 3 iF 8 10 II 10 22 24 26 28 20 i5 II Q 8 I 6 3 Av rfr<( na 8 10 II II 24 26 25 28 21 17 12 10 9 I 4 8iv' II 12 II 26 28 23 28 26 TI 22 18 13 II II 10 6 II 12 12 28 3o 23 23 291^127 24 20 16 14 12 II 10 7 3F oE II i3 12 3o 32 24 28 29 2d 23 21 18 i5 14 i3 II 8 4 bE 9A' 12 i3 12 32 34 24 28 29 29 27 23 20 17 16 i5 i3 9 4 oE 6 10 12 i3 i3 34 36 24 27 3o 3o 28 25 20 '9 '7 16 14 10 5 6 10 i3 14 i3 36 38 23 27 3o 3o 29 26 23 21 19 >7 i5 II 6 6 OF I 6 6 10 II i3 i3 14 i5 14 14 38 40 22 27 3o 3i 3o 28 25 22 21 '9 16 12 40 42 22 27 3o 32 3i 29 27 24 23 21 18 i3 7 I 6 II 14 i5 13 42 44 21 26 3o 32 32 3o 28 26 25 23 19 14 8 I 6 II 14 16 i5 44 46 20 26 3o 32 33 32 3o 28 ?7 26 21 16 8 I 6 12 i5 16 16 46 48 20 23 3o 33 24 33 3i 3o 29 28 23 •7 9 2 6 !l2 i5 .'7 16 17 48 5o 19 25 3o 33 34 34 33 32 32 3i 27 19 10 2 6 ii3 16 17 5o 52 18 24 29 33 35 35 35 35 35 34 29 2T 12 3 6 i3 16 18 18 52 54 17 23 20 33 36 36 37 37 38 36 32 24 14F 3 6 14 •7 19 18 54 56 16 22 28 33 36 37 38 39 40 39 34 26 4 6 14 18 20 19 56 58 i5 21 28 33 36 38 40 42 43 I43 38 28 F 5 6 ii5 19 21 20 58 boS i5Trj2iTii27r33ir 0*^ 110" 20"^ 30-" 37 Tf 39 W'ai TF,45 Tr 47 ir 46 Tf;42 W\ 40'-' ,o0>^ 60*^ ;?0" SO" 00" |100"110" 7F bE \5E 140" 150" 7lE 160" 2 2 A' 170" 21A' 180" 6o^' ^ 120*^ 130" Hi 1^ East Lo> roiTt roE. Page 332] TABLE LIV. Latitudes and Longitudes. This table contains the Latitudes and Longitudes of the most remarkable Harbors, Islands, Shoals, Capes, &c., in the world, founded on the latest and most accurate astronomical observa- tions, surveys, and charts. The longitudes are reckoned from the meridian of Greenwich. I. Coast of the United States of America. ENTRANCE of St. Croix River Island of Campo Bello, (N. point',) Wolf Islands, (northern- most,) Quoddy Head light Grand Manan, N. E. head S. W. head Libby Island lirfit, ( trance Machias Bay . . Titmanan light Mount Desert Rock, (light house,) Isle au Haut Castine Fort, Matinicus Island lights . , Wooden Bald Rock Manhegan Island light. . , Penmaquid Point light. . . , Bantum Ledge Seguin Island light Brunswick College Cape Small-Point Cashe's Ledge, (shoalest part 26 feet,) PORTLAND light-house Cape Elizabeth, (W. light) Wood Island light, en- trance Saco River.... Agamenticus Hills, Tri.Pt. Cape Porpoise Bald Head Cape Neddock Nubble... Boon Island light PORTSMOUTH^''i?lh'r'"'( Isle of Shoals, (White Island light,) Portsmouth, Ft. Constitution Great Boar's Head NEWBURYPORT W. It. on Plum Island 42 Ipswicli entrance light... 42 Squam light 42 CAPE ANN (Thatcher's laland) N, light Eastern pomt Cape Ann Harbor light Liglit-house on Baker's Island Beverly Spire SALEM, Tall spire Mnrblehoad Black-top ch'h Nahant Point, N. E. point of Boston Harbor, Hotel Boston light-housef BOSTON, State-House.. Cambridge, Observatory.. Scituate light Plymouth lights, south. .. Rac(^ Point lin-ht Lat. Loner. D. M. 45 00 N 57 57.5 47-5 45 34 32.5 22 58 59 22.5 47-1 5o 44 56 37-4 33.8 27.4 i3.4 2( -4 l3'2 10 07 -3 o3.5 58 o4.5 55.1 48.4 4t.i 39.7 42 38.3 42 34.8 32.2 43.0 3l.2 3o.4 D. M. 02W 55 43 58 45 53 52 08.5 34.5 48.5 5i 46 j5 29 35 45 57 2 4 50.4 5i.5 12.2 II. 8 19.4 4i .2 25-2 34-4 35 28 6 3 4i-5 37.5 42.2 47-4 48.8 45.8 4o.6 70 34.2 70 39.5 25.1 70 19.6 70 46.8 52.4 53.6 5o.5 54.0 53.1 o3.5 07.4 42.6 35.7 14.3 CAPE COD light... Chatham South light.. Monomoy Point light Shoal of George's. Great Shoal, S. E. point N.W. point From Western Shoals To North Shoal.. Third Shoal East Shoal NANTUCKET light, (Great Point,) Sancoty Head Nantucket South Shoal* . . Cape Poge, (Vineyard,)Lt Cutterhunk Island liglit.. Gay-Head light-house. . . . Noman's Land Triang Pt. New Bedford Court-House liirht-house. Seaconnet Point NEWPORT, Spire Rhode Island light-house, (Beaver Tail light,) Providence Baptist Church Point Judith light Block Island light S. E. point.. Watch Hill light-house Little Gull Island light. . . New London light-house. MONTAUK POINT (E. end Long Island) light-h. Falkner's Island light. . . . NEW HAVEN light.... Stratford Point light Old Field Point light Eaton's Point light NEW YORK, City Hall. Sandy Hook light Neversink lights Barnegat l>ight Ho Great Egg Harbor Cape May light Cape Henlopen light-house Egg Island light PHILADELPHIA St. Ho Smith's Island light Cape C^harlcs Cape Henry light Norfolk Old Point Comfort Yorktown Petersburgh RICHMOND, Capitol... WASHINGTON City... BALTIMORE W. Ml.... Annapolis, Md. St. Ho Currituck Inlet CAPE HATTERAS.... Deep soundings off ditto.. Ocracock Inlet LMt. D. M. 42 02. 40.2 33.5 33 44.7 39.6 35.9 46.4 46.5 43 23.4 17.0 04.2 25.2 24.8 20.9 l5.2 38.1 35.5 27 29.2 26.9 49.6 21.6 i3.4 12.3 19.0 Lonar. 41 i4. 41 09, 40 58, 40 57 40 42. 40 27, 40 23 39 46. 39 19 38 55, 38 46 39 10. 39 56. 37 07. 37 07. 36 55. 36 5o. 37 00. 37 i3 37 i4 37 32- 38 53. 39 '7- 38 58, 36 23 35 i5. 35 06 35 06 D. M. 70 o3.3 69 56.6 69 59.3 67 39 67 47-5 67 49.4 67 53 67 48 67 28.2 67 22.2 70 02.4 69 57.6 69 5 1 .4 70 26.7 70 56.7 70 49. 8 70 48.5 70 56.2 70 53.7 71 i3.5 71 18.5 71 23.6 71 24.2 71 28.6 71 34.2 71 34 71 5r .2 72 06.1 72 o5.i 71 51.1 72 38.9 72 53.9 73 o5.9 73 06.8 73 23.4 74 00. I 73 59.8 73 58.8 74 06.0 74 35 74 57.3 75 o4-7 75 08.0 75 08.7 75 52.2 75 57.9 76 00. 2 76 17 76 18. I 76 34 77 24 77 25-8 77 CO. 2 76 36.6 76 29.1 75 55 75 30.9 5 75 58.9 * New South Slinal. 4ii" .""i + Minot's l.eilee Light, S. E. ] »' N.. 0!)° 51' 5 W. L R, iVom Roston Light. TABLE LIV. Latitudes and Longitudes. [Page 333 CAPE LOOKOUT, Lt.. Deep soundings off do.. . , Old Topsail Inlet Beaufort VViliuington Brunswick Siiiithville New Inlet South point. . . CAFE FE All Bald lid Lt Deep soundings off do.. . . GEORGETOWN Clmrcli light-house .'ape Roman ;harleston, Pinckney,) lioht-liouse (Fort North Edisto River BEAUFORT, (S.C.).... Port Royal Entrance, Tybee light SAVAiNNAH Exchange. . St. Catharine's Island, North point Sapello Bar St. Cath, Is.. Doboy Bar Light Light on St. Sunon's Isl- and, S. point Brunswick St. Andrew's Bar Cumberland Island S. point Amelia Island, S. pt Lot. River St. John's Light. St. Augustine light-house Cape Carnaveral Light. Breakers, S. E. point . . I^as Tortulas,orHummocks iliUaborouii-li Island, N. p. S. p. Mount Pelado, or Bald Head Grenville's Inlet Cooper's Hill Sand Hills Neu- Inlet Middle River CAPE FLORIDA light. Carry.sfort Light Key Tavcrnier Old Metacumbe, S. W. pt. Cayo Sombrero Looe Key Samboes Keys, (eastern,). Key West, Light Sand Key, Light (old).. . . Tortu^as Islands and Banlcs, N. E. point N.W. point S. E. point S. W. point Bush Key light Key Vacas Key Axi Cape Sable Cape Romano Boca Grande, entrance Bay ('arlos D. M. 34 37. 34 28 34 4i 34 43 34 i4 34 02 33 54 35 4r. 33 52. 33 35 33 22 33 i3. 33 01. 32 46. 32 4i. 32 33. 32 27 32 o4. 32°OI . 32 o5 3i 4i. 3i 3i 3i 21 J I 07 3 1 06 3o 53 Lona-. 3o 43 3o 3o 3o 20. 29 52. 28 28 28 24 27 35 27 32 27 i4 27 01 26 47 26 42 26 32 26 18 26 08 25 4o 5 i3. 24 59 24 52 24 37 24 34 24 29 24 33. 24 27, 24 4i 24 4o ■j4 33. 24 3i 24 36- 24 42 24 57 25 06 25 5i 26 43 D. M. 76 30.7 76 4o 76 4o 77 58 77 58 78 01 75 28.5 77 59-8 79 '8 79 IO-7 79 22 2 79 54.4 79 52.5 80 10.7 80 4o 80 37.7 80 5o-6 81 o5.2 8i II 81 i3 81 18,6 26 Bi 3i 81 20 81 28 81 26 24.5 20 34 3o 3o 1 1 02 o3 o3 00 00 09-4 12.7 3o 4i 07 24 40 li 52-7 82 23 Tampa Bay, Egmont Kej Anclote Keys St. Mark's light-house . South-west Cape Dog Island light Cape St. George, Light. . . Cape St. Bias St. Joseph's Bay, entrance St. Andrew's Bay, en- trance Main Pass. .... St. Rosa's Bay, entrance PENSACOLA, town.., liffht..., Mobile Point, light. bar, outer MOBILE Barton's Acad'y Massacre Island, W. pt. .. Ship Island, W. Light Chandeleur Islands, N. point, Liglit S. pt. Palos Island Key Breton, N. E. pt MISSISSIPPI River, Pass a rOutre Balize S. E. Pass.. S. Pass .... S. W. Pass. NEW ORLEANS Barataria Bayou la Fourche Timbalier Island, (Tonba- lier,) N. W. point Racoon point Bayou Decartes, entrance Point au For Light Rabbit Island Sabine River, entrance. .. Galveston, entrance Lat. Long. M. D. M. 36N82 45VV 17.5 o4.5 52 46 35 39.6 5 1 o3 24 25 21 i3.8 09 41.4 12 12.9 o3 44 29 i4 08.5 06 59.7 58.5 57.5 17.5 06 o5 o3 10 19.5 29 40.6 20.5 82 54.3 84 10.6 84 22 84 34 84°58.5 85 16 85 23 85 37.7 86 3 1 87 11-5 87 16.9 00.5 01 01 . 22 57.0 5i 88 5i 89 07 89 00 01 .4 57 89 07.4 89 20 90 00 90 10 90 09 90 23 90 57 91 o4 91 20 91 36 93 49 94 45 II. Islands in the West Indies. TRINIDAD Spanish Town, (fort,) Icacque Point Pomt Galiote Point Galera Tobago, N. E. point S. W. point Grenada, Point Salinus, S. W.pt , Grenada Bank, Barbadoes, S. point , Engineers' wharf, - N. point. St. Vincent's, Kingston S. point . . St. Lucia, T^T, point S. point Martinico, S. point Diamond Rock Port Royal . — Macouba Pt. Dominica, Roseau . N. point Lat. D. M. 39 N o4 5o 1 20 1 06 2 00 I 55 3 o3 3 M 3 20 3 12 3 4 3 4 4 4 4 5 18 5 38 09 06 4i 27 26.6 36 55 Loner. D. M. 57 00 56 27 46 49 57 37 38 4r 16 14 57 54 55 02.7 o4.2 09 25 26 Page 334] TABLE LIV. Latitudes and Longitudes. The Saint's Island, W. pt. Mariegalante, S. point . Guadaloupe, S. W. pt. . N. W. pt. . N. E. pt. .. Point Chateau, S. £. pt Deseada, Antigua, E. point Fort James Montserrat,'N. E. point .. Redondo Island Nevis, Charlestown St. Christopher's or St. Kitt's, N. point Basse Terre St. Eustatia, Town Saba Aves or Birds' Island .... Barbuda, N. pt St. Bartholomew, S. pt. . . St. Martin's, Marigot Fort Anguilla, S. W. pt Anguilleta, N. E. pt Prickly Pear Sombrero St. Croix or St. Cruz, ob- servatory •■ S. W. pt. ._ Anegada, S. point of shoal W. point Virgin Gorda, E. pt Tortola, E. point W. point St. John's St. Thomas, Fort Christian Bird Key Serpent Island, E. part. . . Crab Island, E. part Cape St. John, or N. E. pt. PORTO RICO, St. Au- gustine's Battery, west- ern turret Point Bruquen, or N. W. p. Point St. Francisco Cape Roxo, or S. W. point Caxa de los Muertos Point Coamo Cape Mala Pasqua, or S. E. pt Mona Island, E. pt Monito Island Zacheo or Dessecho Isl.. Cape Engano Saona Island, E. pt St. Catherine's Island.., St. Domingo, Light. . . . . La Catalina Cape Beata AltaA'ela Rock , Cape Jaquemel , Island Vacca (a Vache) Point Gravois Cape Tiberon Lat. D. M 5 5iN 5 52 5 57 7 24 7 17-7 7 29 7 4i 5 40 7 47 7 53.5 8 o5 8 10 8 18 8 20 8 38 7 44.5 7 42 8 32 8 44 8 3o 8 27 8 25 8 18 8 21 8 i5 8 19 8 10 8 24 8 29 8 3[ 8 21 7 57 7 5o 7 55 7 59 8 07 3 !I S 24 8 35 8 12 8 18 8 28 8 08 7 39 7 28 8 10 8 o4 8 ot 8 20 Lous'. M 38W 18 44 5i 29 06 45 52 12 25 37.9 5o 42.2 00 i4 39 02 56-9 o3 i3 58 23 27.4 40.7 48 i3 65 52 67 2- 68 20 3o 00 52. Navaza Island Cape Donna Maria Jeremie Caymito Petit Guave Leogano PORT-AU-PRINCE . . . . Isle Gonave, S. E. part . . N. W. part. . Point St. Mark St. Nicola Mole Tortugas, E. point CAPE HAYTI CITY,.. Shoal off Monte Christe . . Monte Christe Grange Point Point Isabella Old Cape Franqois Cape Samana Cape Raphael Morant, E. point KINGSTON Port Royal, Fort Charles . Portland Point Pedro Bluffs Black River Savannah la Mar Cape Negril, S. point N. point.... Montego Bay Falmouth ; . St. Ann's Port Maria Arnatta Bay N. E. point Morant Keys, or Las Panas Pedro Shoals. Portland R., N. E. p. South Key Rock 5 feet above water. . N. pt. Pedro Shoal Formigas Shoal, N. E. p.. S. W. p. Little Cayman, S. W. p. Caymanbrack, E. p Grand Cayman, E. point , Fort George, W. end Swan Islands, E. pt New shoal, (Sandy Key,) . Cape Mayze Port Negra Point, entr. Cumberland Harbor ST. JAGO DE CUBA, Light...... Tarquin's Peak Cape Cruz Manzanillo Key Breton Trinidad River Bay Xagua. River Vigia . . Stone Keys Lat. D. M. 18 24 N 18 37 18 37 18 39 18 24 18 3o 18 33 18 4o 18 57 19 02 19 49-5 20 02 19 46.4 20 0^ 19 54 19 54 19 59 19 4o 19 10.2 19 o4 18 56 58 56 43 52.5 02 12.3 i5 22.5 18 29 18 28 18 27 18 22 18 16 18 09 17 25 17 07 16 57 16 48 17 36 18 35 18 27 19 36 19 44 19 20 19 i4 17 25 5 52 20 1 5 20 06 19 54 19 58 20 02 19 5o 20 20 4 43 22 02 5-- Long. M. 00 VV 23 o3 43 5o 33 16.3 45 i3 47 61 II . 42 34 36 10.5 53 i5. 52 46 5o.5 II 45 5i 08. 24 23 56 4i i5 54 45 20.5 75 59 ,577 28 77 53 5 78 54 75 5o 76 00 06 74 i3 75 16 75 52 76 5i 77 45 77 II 79 22 80 00 80 42 81 1 5' TABLE LIV. Latitudes and Loniiitudes. [Page 335 Los Jardinellos, S. E. point of the Bank Canal del Rosario Isle Fines, E. pt S. W. pt Indian Keys, N. AV. pt. .. Key St. Philip, E. pt Point Piedras Cape Corrientes Cape St. Antonio Saiicho Pedro Shoal Shoal discovered in 1797 . Los Colorados, S. W. pt. . N. E. pt... Hill Guajibon Bay Honda Port Cabanas Mariel HAV^ANA (the More)... Cape Escondido Point Guanos Pan of Matanzas MATANZAS Point Yeacos Stone Key off do Key Cruz del Padre Las Cabezas Nicholas Shoal Key Verde Key Carenero Key Francis, E. p Key William, northern . . . Pt. St. Juan Centre of Key Coco, S. side Bahama channel . . Key Point Paredon, do. . . The Barrel Cayo Confites Cayo or Key Verde Guajara, N. W. pt Point INIaternillos Neuvitas Point de Mulas Tanamo Key iNIoa Point Guarico Baracoa N. pt. Nativity Bank, or E. Reef. Superb Shoal Silver Key, S. E. end .... iN. E. do . N. do Square HandkercJiief N. E. point S. E. point — S. W. point Turk's Island, N. p. Grand Turk Salt Key • — Sand Key Endymion's Rocks. Great Caycos Isl., Swim- mer's Shoal N. E. p., or Shoal St. Philip N. W. part * See Prefaf'f!. Lat. Long. D. M. D. M. 21 3iN 81 17W 21 35 81 5i 21 4o 82 23 21 22 83 00 21 52 83 i3 21 58 83 22 22 01 83 55 21 44 84 33 21 5o 84 59 22 OI 85 02 22 06 85 02 22 09 84 48 23 00 83 08 22 48 83 24 23 01 83 i3 23 02.5 82 59 23 o3 82 47 23 09 82 22 23 08 81 5i 23 08 81 44 23 02 81 46 23 o3 81 4o 23 i3 81 10 23 i4.5 81 07 23 18 80 55 23 16 80 36 23 i4 80 19 23 09 80 i4 22 52 79 49 22 4o 79 '3 22 34 78 45 22 19 78 57 22 29 78 20 22 3o 78 o5 22 25 77 56 22 II 77 4i 2 2 06 77 38 21 55 77 3o 21 4i 77 08 21 36 77 06 2 1 o5 75 3i 20 44-5 75 12.2 20 43 74 47 20 39 74 4i 20 21 74 24 20 12 68 46 20 58 69 00 20 i4 69 32 20 35 69 17 20 55 69 52 , 21 07 70 26 20 49 70 23 20 55 70 56 21 32 71 10 21 20 71 i4 21 II. 5 71 16 21 07 71 19 21 o5 71 32 21 42.5 71 25 21 53 72 22 North Caycos, middle . . Booby Rocks, oft' do. . . . Providence Caycos, N. VV. p West Caycos, S.W. p.. French Key Soutli Point Shoal Great Inagua, or Heneaga, N. E. p S.E.p S. W. p.... N. W.p.... Little Heneaga, E. p W.p.... Hogsties, or Corrolaes . . . Lookout Bank, (Cuidado) Mayaguana, E. reef. N. do S. W. do. . . . E. point French Keys, or Isle Planas Miraporvos, S. Key Castle Island, or S. Key . , Fortune Island, S. W. pt. North Key, Bird Island . , Crooked Island, W. pt. . , Acklin's Island, N. E. pt Atwood's Keys, or Island Saniana, E. p W. p. . . . Rum Key, E. pt Walland's Island, N. E.pt S. W. pt Conccption,or Little Island St. Salvador, or Guanhari, S. E. pt N. pt Little St. Salvador, N. pt. Eleuthera, or Hctera Isl- and, S. pt N. pt Point Palmeto. Harbor Island New Providence light-h., NASSAU E. pt. . W. pt.. Andros Islands, S. pt. .. N. pt. . Berry Islands, Stirrup Key Blackwood's Bush Ijittle Isaac, (eastern). . Great Isaac Bemini Island, southern fresh water Key Gun Key light Cat Key Riding Rocks, Soutli . . Orange Keys, N S Key Guinchos, Key Lobos, beacon, 20 ft. Las Mucaras, Diamond Point South edge of the Bank Brotlicrs' Rocks Lat. Long. D. M. D. M. 21 56N 72 00 W 21 58 72 00 21 5o 72 20 21 37.5 72 3o 21 3o 72 i4 21 02.5 71 5o 21 20 73 00 20 55 73 08 20 55 73 38 21 09 73 40 21 29 72 55 21 29 73 06 21 4o 73 48 21 57 72 55 22 20 72 40 22 32 73 09 22 22 73 II 22 4i 73 27 22 OD 74 3i 22 07 74 20 22 32 74 23 22 49 .*5 74 24 22 48.5 74 23 22 44 73 5i 23 o5 73 37 23 o5 73 48 23 4i 74 46 24 08 74 25 23 55 74 32 23 5o 75 o5 24 09 75 18 24 42 75 43 24 37 75 55 24 37 76 08 25 34 76 43 25 09 76 08 25 3o 76 38 25 05.2 77 21.2 25 02 77 16 25 01 77 35 23 44 77 38 25 )0 78 02 25 25 77 44 25 49 77 53 25 27 78 o3 25 58.5 78 5t.3 26 02 79 c6.3 25 43 79 19 25 34 79 18.4 25 3i 79 17 25 14.5 79 09 24 57 79 08 24 54 79 08.5 22 46 78 08 22 i 7 J 77 33 22 II 77 i4 22 o5 76 22 22 02 75 42 Page 336] TABLE LIV. Latitudes and Longitudes. Key San Domingo .... Key Verde Island Key Sal, (Ragged Island,) Yuma, or Long Island, S.p N. p Exuma, N VV. p THE HOLE IN THE WALL Light on do N. E. point of Abaco Elbow Reef. Man-of- War Key Great Guano Key Little Bahama Bank, N. p Memory Rock Sand Key Wood Key, or Cape Leno, Great Bahama, W. p E.p Dog Keys, N. W. p Water Key Double-headed Shot Key, (elbow,) light Salt Kej cent beach W. side Anguilla, E. p GEORGETOWN Wreck Hill, westernmost land Best latitude to run for Bermuda Lat. D. M. 42 N 02 12 5o 45 42 5i 5i.5 18 34 37.5 42 35 57 49 45 42 4o 04 59 56.4 4i.8 29 22.2 Lon£ D. M. 75 45W 75 10 75 42 74 5o 75 18 76 00 80 10. o 57 52 57.5 04 06 02 02 01 48 5o 17 27-7 24.3 26 37.6 64 5o liJ, East Coast of Jlmerica, from Gulf of Mexico to Cape Horn. Galveston Inlet W. p. Galveston Island . . Rio Brazos Pasa del Cavallo Aranzas Inlet Corpus Christi Braso de Santiago Rio Bravo del Norte River St. Fernando, entr . . Inlets to Laguna Madre.. Bar de la Marme, entrance River St. Ander Bar del Tordo JVIount Commandante Bar de la Trinidad Bar Ciega River Tampico Point de Xeres Cape Rojo Tamiagua City River Tuspan, entrance . . Point Piedras River Cazones Tenestequepe Boca de Lima River Tocoluta, entrance . Mount Gordo River Nauta, entrance . . . River Palina, entrance . . . Point Piedras River de Santa Nos Lat. Lonor. D. M. 29 17N 29 4 29 3 28 24 27 49 27 36 26 6 25 56 25 22 25 2 23 45 52 48 39 34 16 55 45 16 58 46 42 4o 34 27 16 i3 10 00 55 D. M. 94°45'W 95 26 95 33 96 18 97 4 97 16 97 12 97 12 97 32 97 4i 97 58 97 57 97 58 97 57 97 58 98 2 97 45 97 22 97 29 97 18 97 12 97 12 97 9 97 4 97 o 97 01 96 47 96 45 96 35 96 3o Point Delgada Point M. Andrea Point de Bernat River St. John Angel • Xalapa Peak de Orizaba Point de Sampola River St. Carlos River Antigua Point Gorda VERA CRUZ St. John de Ulloa .... Xamapa River Medellin, entrance . Point Anton Lizardo Bar de Alvarado Tlacotalpan Vigia Point Roca Partida Point Morillos Pic de San Martin Point Olapa Point St. John Barilla Bar Guazacoalcos River Tonato River St. Ann River Cupilco Dos Bocas River Chittepeque River Tabasco River St. Peter and Paul . Island Carmen, W. pt. . . . Point Escondido Tavinal Point Morros CAMPECHE Point Desconocida Point Gorda Point Piedras Igil St. Clara Bocas do Silan El Cuyo Island Jolvas, N. p Island Contoy, N. p L^s Areas Islands S. W. Id Bank Obispo centre Triangles Islands N. W. Id New Shoal Bajo Neuva Island centre. Island Arenas Island Bermeja, or N. W. Shoal Sisal Fort Alacranes N. part of Bank oiF this coast N. E. do Isle de Mugeres, or Wo- men's Island Island Cawkun, S.p New River River Bacales Bay Ascension, entrance . Island Cosumel, N. p. . . , S.W. p Lat. D. M. 9 49 9 43 9 40 9 32 9 32 92 9 3o 26 20 i5 12 12. 4 6 4 46 35 8 38 8 43 8 4o 8 3o 8 34 8 21 8 II 8 II Point Tanack 8 20 8 26 8 26 8 24 8 34 8 38 8 38 8 58 9 10 9 45 9 49 20 46 6 9 21 20 21 22 24 21 3o 21 3o 21 36 20 1 3 20 3o.5 20 57.8 20 53 21 5o.2 22 7 22 33 21 10 22 32.3 23 43 23 27 20 42 20 26 5 19 26 20 36 20 10 18 54 Loner. D. M. N 96 26W 96 21 96 21 96 20 96 5o 97 9 96 j6 96 1 5 96 i4 96 4 96 9 596 8 95 58 96 4 95 58 95 45 95 36 95 18 95 II 94 54 95 10 94 5o 94 38 94 35' 94 22 93 59 93 49 93 26 93 6 93 02 92 40 92 32 91 5i i5 90 58 90 43 90 33 90 26 i3 88 56 87 43 87 II 86 52 9i°59. 92 1 3 92 18.9 91 5o 91° 4. 91 25 91 22 90 2 89 43 88 43 86 37 86 42 86 58 87 i5 87 34 3 86 45 87 00 87 42 TABLE LIV. Latitudes and Longitudes [Page 337 x\. Triangle, N. Key Sandy Key, S. p S. pt. Ambergris Key Isl. BALIZE Turneff Reef, N. pt S. pt Englisli Key Half-Moon Key light-house Hat Key Tobaco Key Island Santanilla, or Swan Island Glover's Raef, N. p S.p Renegade Key Sapotilla's Kej's, S. E.p. Rattan Island, E. p W. p Guanaja, or Bonacca Isl- and, S. pt Cape Three Points Omoa Point Sal Triunfo de la Cruz Utilla, N. p Truxillo Cape Delegado, or llondu ras Cape Cameron Cape False Cape Gracios a Dios .... Caxones, W. p S.E. p Lat. D. M. 8 44 N 8 22 7 52 7 29 7 39 Cayman, or Vivonlla. Key St. Thomas Alao-arte Alia, iN. W. p. . S. p Scranilla, N.E. breaker. W. breaker . . Sarrana, N. p S.p Guana Reefs, N. p. S. p. Roncador . Musketeers, centre Providence Island, N. p. . . Ned Thomas's Keys, S. p. Bracman's Bluff* Man-of-War Keys Little Corn Island Great Corn Island Bluefields, entrance \Ac St. Andrew, middle.. E. S. E. Keys S. S. W. Key, or Albu- querque Paxoro Bovo River St. John, S. pt Port Boco Toro Isle Escudo, N.p River Chatrre, entrance .. PORTO BELLO Point Jlanzanillo Point St. Bias Point Moschitos Isle of Pines Cape Tiburon River Suniquilla, entrance Point Carabana 7 19 7 i3 7 10 6 57 23 55 4i 6 00 6 2 4 i4 9 58 33 24 I 20 o 57 9 25 9 i4 9 19 9 34 9 39-i 9 35 9 8 9 I 8 4i 7 55 8 38 Long. D. M. 87 i5W 87 i8 88 I 88 12 87 4 1 87 56 efi 88 2 s 87 34 87 4i 88 4 a 83 5i u 87 40 87 48 88 n 88 i4 86 i5 86 5i 86 00 88 34 88 I , 87 48 45 87 38 >• 87 2 86 2 ?3 86 6 fi, 85 i4 83 21 83 12 83 18 83 8 83 26 81 49 82 27 82 25 79 4' , 79 58 80 16 80 23 80 44 a 80 41 79 46 80 3 81 20 82 21 83 so 82 39 82 58 83 3 82 54 81 43 81 28 81 52 82 48 83 37 82 12 80 57 79 59 79 4o 79 32 "9 J rt 77 58 a 77 5o s s 77 27 76 56 w 76 58 Point Arboletcs Island Fuerte Isle St. Barnard, N. W. p. CARTAGENA Punta de la Galera de Samba West entrance River Mag- dalen St. Martha. Cape Aguja Bank Navio quebrador . . . Hacha Cape La Vela Point Gallinas Mongcs Islands, N. p Cape Chichibacoa Point Espada St. Carlos MARACAYBO Coro Point Cardon Point Macolla Cape St. Roman Island Oruba, N. W. p.... S. E. p Point Aricula Point Zamuro Point Soldado Key Borracho Point Tucatas PORTO CABELLO .... Point St. John Andres. . . . Point Oricaro Point Trinchera LAGUIRA CARACCAS Centinella Island, or White Rock Cape Codera Curacoa Island, N. p. .. S. E. p Little Curacoa Buenayre, N.p S.p Birds' or Aves Island, western eastern Los Roques, W. p S. E. p Orchilla Island, middle. . Blanca Island, middle... E. point Tortuga Island . Seven Brothers, middle. . Margarita, W. p E.p Island Cuagua, or Pearl Island , Friar's Island Island Sola Testigos Island Morro de Unare New Barcelona Island Borracho Cumana Pta. de Araya Morro Chocopata pjscondido, or Hidden Port Cape INIalapasqua Lat. Lon M. 55 N 24 49 26 47 1 5 I i5 I 20 I 26 1 33 2 1 1 2 25 2 28 2 i5 2 4 o 57 39 1 24 1 36 2 4 24 56 26 i4 57 5i 28 3o 34 37 36 3o 5o 36 24 2 59 00 57 5o 47 48 5i 55 4-h 59 59 49 23 6 10 19 28 38 42 4o 42 D. i\I. 76 3oW 76 16 75 56 75 38 75 3o 74 56 74 18 74 16 73 i5 72 59 72 16 71 44 71 3 71 20 71 i3 71 44 71 45 69 5o 70 23 70 22 70 9 70 12 70 I 69 56 68 59 68 4o 68 22 68 21 68 7 67 5o 67 18 67 8 67 Q 67 li 66 i5 66 J 2 69 17 68 49 68 45 68 3 1 67 46 67 32 67 I 66 38 66 i3 64 41 65 18 64 3i 64 3o 63 52 64 18 63 49 63 40 63 i3 65 22 64 48 64 5i 64 16 64 3o 63 54 63 29 63 7 43 Page 338] TABLE LIV. Latitudes and Longitudes. Cape Three Points Point Galera Point Pena, or Salina Dragon's Mouth River Gaurapiche, entrance Point Redondo Mouth of Oronoco River. . Cape Nassau Essequebo River DEMERARA River en- trance Corrobano Point River Berbice, entrance.. Surinam River, entrance. Paramaribo River IMarouri, entrance.. CAYENNE Mouth of Oyapock River . Cape Orange River Cassipour, entrance Cape North Nortliern mouth of River Amazon Southern do. do Cape Magoany Point Tagioca Para Bay Maracuno Caite harbor Cape Gurapi Slioal off do Fi. point of Island of St. Joao Vigia, fell in with by Mr. Du Sylvia, officer of the Brazilian marine, in 1824 or 1S2.J Vigia of Manuel Luis, Westerly Rock Mondrain Itacolomi. . . . Mt. Alegre, (the summit,) Alcantara, (west church,), Rock E. of Isle Medo .... City of San-Luis de Mar anham, (Cathedral,) .... Fort Sant Antonio das Areias, (the flag-stalF.) Fort San Marcos Isle Maranham, (white sand hills, N. part,) Breakers of Coroa Grande, the North one North-west one West one. . , Isle St. Anne, N. E. point. Breakers of Isle St. Anne E. point Morro Alegre Lancoes Grande, E. point River Perguicas, E. point. River Tnloya, entrance . . Pedra de Sal River Tapuyu. entrance.. Mt. Tapuyu, W. summit . Mt. Ticondiba, summit. .. Point de Jericacoara, the highest sand-hill S:ind-hill near the shore Lat. Lons- M. 45 N 43 43 43 12 5o 5o 32 M. 46 W 34 56 5i 43 43 GO 58 4o 58 26 60 loN 5S 12 32 28 33 46 39 36 19 O 32 o 5i 2 9 2 17 2 24 2 3o 2 3i 2 29 2 28 2 25 2 l3 2 17 2 l5 2 26 2 4i 2 4i 2 47 2 5o 2 58 3 II 2 47 2 5o 58 Hi 57 II 55 3 55 00 53 49 52 i3 5 1 26 5i 1 1 5 1 00 5o 6 5o 00 49 45 48 29 47 58 48 29 47 4i 47 6 45 56 45 56 44 5o ^i 17 U i5 i^A 25 ^4 20 44 23 44 19 44 16 44 17 U 16 44 04 43 58 AA 4 M 5 43 38 43 3o 43 i3 43 00 42 27 42 12 4i 42 4o 5o 4o 5 1 40 37 4o 27 4o 39 Mount Memoca Roccas, (dangerous.) Pernambuquinho Morro Melancia Sand-hill of Parati . « Mountains of Ciara, 1st .. 2d summit. 3d do. , . 4th do. .. 5th do. . . Ciara, steeple in the city . Point Macoripe Morro Aracati, summit. .. Point Reteiro Grande Reteiro Pequeno, remarka- ble sand-hill Morro Tib.ao Point de Mel Point du Tubarro (Breaker) das Ureas dela Lavandela. Pt. Calcanliar, summit . . . Point Petetino-a, low CAPE ST. ROQUE .... Fort of Rio Grande Point Negra, mountain . , . Point Pipa, sand mount . . Bahia Fermosa, S. point.. Bahia da Traicao. N. pt. . . Church of St. Theresa.... Fort Cabcdello Paranahyba de Norte Cape Blanco, steep part . . Point de Guya Point das Pedras Village of Pilar Fort, entrance of Rio Ay . . Nossa Senora Farinha. . , , Olindo, west tovi'er Tower de Recife, Periiam- buco Nossa Senhora de Rosario CAPE ST. AUGUSTIN River Ipojuca, entrance . Mount Sellada, S. peak . Islands of St. Alexio .... Fort de Tamandarc San Bento Village of Quinta La Forqviilha, (hill.) Frenchmen's Port Village at the point of River Alagoas Morro Sant Antonio. . . River San Francisco. . Tabayana IMountain, sum- mit Rio Vasa Barris Rio Real, S. point Torre de Garcia de Avila. River Jacuipe Rock of Itapuan Itapuanzinko, the point. . . ST. ANTONIO, N. W tower Point Caxo Pregos, Isle Itaporica Point Aratuba, do.. . Lat. D. J>1. 3 18 S 3 5i 3 2 3 12 3 24 3 58 3 53 3 5o 3 46 3 39 3 43 3 42 4 42 4 36 4 48 4 49 4 55 5 2 4 52 4 55 5 8 5 22 5 28 5 45 5 53 6 i3 6 23 6 4i 6 57 6 58 7 6 7 8 7 26 735 7 36 7 47 7 57 8 I 8 4 8 9 8 21 8 23 8 25 8 36 8 43 9 5 9 16 9 10 9 4o 9 4o 9 22 10 29 10 47 II II II 28 12 32 12 42 .2 58 i3 I i3 i3 8 i3 5 TABLE LIV. Latitudes and Longitudes. [Page 339 Point laburn, Isle Itaporica Blount Conceicao, do Morro Sant Amarro, do. . . Morro de San Paulo Isle Boypeda Isle Quiepi Point of Sluta Villa of Contas Os Itheos, the largest rock Villa de San George dos Itheos Rio Cachoeira, S. point.. Villa of Unha iNIorro de Commandatuba, S. E. summit Village of Commandatuba Village of Belmont Santa Cruz, steeple Porto Seguro, steeple of the Cathedral I:?olated Mount Mount Pascal, summit... Mount Joao de Siam River Cramimuam Columbiana Villa Prado, fort Abrolhos Islands, (the lar- gest island,) Rio de San Mattheo Rio Doce, entrance Serra dos Reis Magos, the S. summit ^Nlorro Alme}'da Mestrc Alvaro, summit. Cape Zubarro " Piton," at the N. of the city of Victoria Nossa Senhora da Penha, church Mount Morena Pacotcs Rocks Point Jicu Isles Piasas.. Isle Calvada Guarapari Morro Bo, (isolated moun- tnin,) Morro de Benevento . . , Serra de Guarapari Mt. de Campos, S. summit Mountains of Furado, highest CAPE ST. THOMAS.. Isle St. Anne, the largest. Pic do Frade de Macahe . . iNIorro San Joao, summit . Cape Buzios, S. point.... Isles Ancora, easternmost. CAPE FRIO, S. point... Cape Negro Isles Maricas,soutliernmost Redondo . RIO JANEIRO, (Sutrar Loaf,) La Gabia Isle Georgi Grego O. Pakagaio, top of Isle Grande Lat. Long- M. 57 S 3 I 22 38 5i 53 i8 47 49 49 59 58 37 37 5o 57 9 i6 20 19 20 21 20 26 20 43 20 AA 20 AA 20 43 20 55 20 5o 21 23 21 5o 22 3 22 25 22 12 22 32 22 46 22 46 23 I 22 57 23 I 23 4 56 59 i5 D. M. 38 36W 38 4i 38 45 38 54 38 57 38 57 38 57 39 00 38 59 39 00 38 59 38 58 39 8 38 56 38 54' 39 2 39 3 39 3i 39 25 39 37 39 9 39 12 39 12 38 42 39 45 39 5 1 40 22 40 2-0 4o 22 4o 17 4o 23 40 2-0 40 19 40 17 4o 22 4o 25 4o 27 40 33 4o 4 1 40 49 ii 8 41 28 41 43 41 00 4> 46 42 9 42 6 41 56 4t 5r 41 59 42 35 42 5 1 43 9 43 9 43 23 4 19 Ilha Grande, Pt. Acaya. . . Point loatinya Pic de Parati, summit .... Isle Couves, largest Isle Victoria Isle Buzios, S. E Isles dos Porcos, S. sand hill Isle St. Sebastian, highest mountain Pt. Pirasonungo Mouton de Trigo Lage de Santos Point Grossa Taypu Isle Queiniada Grande. Isle Queiniada Pequena. Poin Jurea Mount Cardoz Isle Bon Abrigo Ptochcr Castello Rochcr Figo Isle de Mel, S. top Roc Coral Roc Itascolomi ...-..,.. Point Joao Diaz Isles Tamboretes Isles Remedies Point Itapacoroya Isle Arvooredo, top .... Isle St. Catharine, E. pt Pt. P^apa Steeple of Nossa Senhora do Desterro . . . Morro de Sta Marta Porto St. Pedro Cape St. Mary Island Lobos Maldonado harbor Point Piedras , . MONTE VIDEO, Ratls. BUENOS AYRES Cape St. Antonio Cape Lobos Cape Corientes Point de Ncuva St. Helena St. George's Bay, Cape Cordova Cape Blanco Point Desire, ruins Port St Julian, Cape Curi- oso St. Cruz harbor Cape Fair^yeather Cape Virgin, northern point of entrance to INIagellan Straits Cape Espirito Santo, (ex- treme point of do.) Terra del Fuego ; Cape Penas Cape St. Diego Staten Island, Cape St. John, easternmost land near Cape Horn Cape St. Bar- tholomew Lilt. Long. D. M. D. M. 23 i5S AA 29W 23 18 AA 39 23 19 AA 54 23 26 AA 58 23 48 45 i4 23 AA 45 6 23 34 45 10 23 48 45 22 23 58 45 20 23 5i 45 52 24 18 46 17 23 59 46 24 24 I 46 3o 24 28 46 47 24 21 46 54 24 33 47 19 24 59 48 12 25 7 47 58 25 16 48 3 25 22 48 10 25 33 48 26 25 46 48 3o 25 5o 48 33 26 7 48 40 26 21 48 39 26 29 48 42 26 47 48 AA 27 17 48 29 27 26 48 29 27 .23 48 32 27 36 48 4o 28 39 48 5i 32 07 52 9 34 39 54 10 35 01 54 54 34 53 55 00 35 27 57 5 M 53 56 i3 34 36 58 22 36 19 56 46 36 55 56 47 38 06 57 27 42 53 64 8 AA 3i 65 22 45 46 67 21 47 12 65 43 47 45 65 54 49 II 67 37 5o 09 68 19 5i 32 68 55 52 20 68 ai 52 38 68 35 53 5i 67 33 54 4i 65 7 54 43 63 43 54 54 64 45 Page 340] TABLE LIV Latitudes and Longitudes. Staten Island; Cape del Medio, entrance to Le Maire's Straits New Island, E. part Evout's Island Barnevelt Island, E. part. CAPE HORN, S. part of Hermit's Island Lat. 55 59 Long. D. M. D. M. 54 48 S 55 17 55 33 55 48 64 45 W 66 26 66 45 66 45 67 16 rV. West Coast of America, from Cape Horn to Icy Cape. CAPE HORN Isl. Diego Ramires, S. part ^ N.rock Island St. Ildefonso, S. p.. Terra del Fuego False Cape Horn . York Minster, S. E. extremity Cape Gloucester. Cape Pillars, N. ontr. Magellan's Stpaits. Evangelist Island, W. ent. Magellan's Straits Cape Victory West Cliff Cape Cape Three Points Mount Corso, S. pt Isl. Cainpana, N. point. .. Cape Tres Monies Cape Taytao Island Huafo, N. W. part. Point Quilan Point St. Carles Point Quedal Point de la Galera VALDI'VIA, near Fort Coral ..... Point Tirua. Isl. de la Mocha, N.W. part St. Maria Islands, N. p S. p. Lat. Lons. 5qS 27 22 53.5 55 43 52 43 CONCEPTION, cit^ Talcahuano, Fort . . . Topocahno Pt VALPARAISO, Fort Point Rallena i3i Coquimbo 29 Huasco 28 Copiapo 27 Sugar Loaf Islet, summit. 26 Island St. Felix, Eastern . 26 Western 26 24 23 23 Pt. dos Reyes Morro Moreno, summit Mexilones Hill, summit Point Tames Pt. Francisco Pavellon de Pica Point Piedras Pisagua, Point Pichalo . Arica, Head Point de Coles Ilo, rivulet mouth Point Cornejo 16 34 28 6 45 56 58 3? 28 42 37 52 67 16W 68 36 68 37 69 17 68 6 73 02 74 38 75 3 74 55 75 32 75 21 75 32 75 23 75 28 75 8 74 49 73 33 73 52 74 I 73 47 73 29 73 33 74 I 73 35 73 34 73 5 73 10 72 5 71 4i 71 36 71 19 71 19 71 2 70 47 9 47 u 3 70 4o 70 38 70 35 70 23 70 1 5 70 14 70 i4 70 19 70 24 71 26 71 24 72 21 Point Pescadores .... Atico cove St. Juan, needle hummock Mount Carreta Pisco, middle Point Frayle Point Chilca Isl. St. Lorenzo, N. pt. . . LIMA CALLAO Bay, flagstaff. Island Pescador, summit of largest, W. pt Los Hormigas Rocks. Island Pelada Island Don Martin . . Point Santander Rock seen in 1 792 . . . Ferrol Baj^ent. (Blanco Is.) Truxillo, church Malabrigo, (port,) Island Lobos de Amera fishing-cove '. Island Lobos de Tierre . . . Eten Point de Ajugo Point Payta Cape Blanco Point Malpelo GUAYAQUIL, city Island Puna, S. W. p Point St. Helena Island Pelade Point del Callo Island de la Plata, W. p. . . Cape St. Lorenzo .... Manta Cape Pasado Quito Arbol Cape St. Francisco . . Point de la Galera.. . River Esmeraldas,entrance Point Mangles Island Tumaco Point Guascama Island Gorgona, middle River Cajambrie, entrance Island de Malpelo . . . Island de Palmas . Point Chirembira. . . . Cape Corientes Limones Point St. Francisco Solano Point Garachine .... PANAMA Ft. N.E. Bastion Point Mala Puercos Point Island Quibo, N. p. . . Los Ladrones Point Burica Gulfe Dulce, W. p.. . Isl. Cano, ent.Engl. Harbor Cape Herradura Cape Blanco Nicoya, wat'g pi. on E. side Morro Hermoso Point Culebra. Gorda Pt. . . St. John's Harbor Lat. Lonff. M. D. M. 24 S73 20 W i3 73 45 75 i3 76 20 76 16.5 76 35 76 53 77 19 77 6 77 i3 I 47 I 27 I 2 o 38 9 b 8 8 7 43 56 23 18 4 5? 27 77 20 77 5o 77 53 77 43 77 56 78 48 78 39 79 4 79 28 80 ^/\ 80 53 79 54 81 10 8r 10 81 16 80 3o 79 53 80 8 80 48 80 36 80 34 80 57 80 43 80 32 80 20 i5N 79 48 39 80 6 48 79 5i 58 79 39 ■66 79 7 47 78 5o 4o 78 32 59 78 26 19 77 i5 55 Si 36 37 77 7 i3 77 26 34 77 26 3 77 23 49 78 fO 4 78 3o 57 79 3i 24 80 2 i3 80 27 3i 8r 46 52 82 37 83 23 83 29 AA 84 4 37 84 37 32 85 2 38.5 84 36 8 85 3o 32 85 43 i5 85 5o TABLE LIV. Latitudes and Loncritudes. [Page 341 l^oint Dcsolado I.l'Oll Iloalejo . entr Aserailores i'oiiit Cosignina i\-jiiit Candadillo SacaU'Coluca i\ iat llemedios Point (iuatimala I'uerto V'eutosa Aijualco ACAFULCO Cape Coriciites St. Bias Tres Marias St. Joseph Cape St. Lucas Morro ilermosa lledondo Island Port Sau Quentiu ..... Bay Todos Santos Port Diego , Point Conception Monterey ■ Port St. Francisco Cape Mendocino Port Trinidad Cape Blanco, or Orford Cape G regory Cape Foulweather Cape Rond Cape Disappointment • , Cape Flattery Breakers' Point NOOTKA, N. pt Woody Point , Bay St. Louis , Isles de Sartine, or Scott, Cape Scott Cape Caution Cape Hector, or James.. Bay de la Touche Cape Henry , Bay de Clouard , Point North Cape St. Bartolomo Caj)e Omnianey Port Guibert Port Neckar Cape Engano, or Edge coinb Port Gaudaloupe Port de los Remedios . . , Cape Cross Port des Francais , Cape Fairwcather Behring's Bay , Point de la Boussole .... Mount St. Elias , Cape Hinchinbroke .... Cape Elizabeth Barren Isles Point Banks Cape Douglass Cape Whitsunday Cape Grenville Trinity Islands Foggy Island Lat. D. M. 12 22 N 12 26 12 28 12 35 12 53 .3 7 i3 26 i3 35 i3 54 16 8 16 2 16 55 20 26 21 3o 21 28 23 4 22 52 27 4.6 29 49 30 22 3i 49 32 4i 34 27 36 37 37 48 4o 28 4i 3 42 53 ^3 26 44 52 5 43 46 16 7 49 24 49 36 50 6 5o 34 5o 56 5o 48 5i 12 5i 57 52 42 52 53 53 52 54 20 55 12 56 12 56 38 56 43 57 2 57 10 57 24 57 57 58 37 58 55 59 18 59 5o 60 23 60 1 5 59 9 59 00 58 4i ! 58 56 58 i5 57 33 56 36 56 10 Long. D. M. 86 58 W 86 49 87 8 87 20 87 37 87 57 88 32 89 43 90 53 93 o 96 52 99 48 o5 35 05 i3 06 29 09 38 09 52 14 4i i5 10 i5 57 16 43 17 II 20 26 21 5i 22 21 24 20 24 7 24 37 24 21 24 5 23 48 24 I 24 43 26 3o 26 35 27 43 28 i4 28 5o 28 20 27 52 3i 7 32 10 32 27 33 21 33 i5 33 38 34 35 35 00 35 4 35 5o 35 43 35 43 36 24 37 20 37 52 39 00 40 55 40 45 46 16 5i 28 5i 46 52 6 52 5o 5i 46 52 00 53 4o 56 45 Halibut Head Island Ounalashka Island, N. p. Bristol River, entrance . . Round Island Cape Newnham Shoalness Cape Stephens Cape Denbigh Cape Rodney Cape Prince of Wales. . . Cape Mulgrave Cape Lisburne ICY CAPE Lat. D. M. 54 27 N 53 55 58 12 58 29 58 34 60 00 63 33 64 17 64 34 65 45 67 45 69 5 70 29 Long. D. M. 162 3oW 166 12 1 57 33 159 53 161 55 161 52 162 17 161 53 166 37 168 17 i65 12 i65 22 161 42 From the River St. Croix to Cape Ca7iso. Entrance St. Croix River. Macgone's Isl. (entrance of St. John's River) .... Cape Spencer. Cape Chignecto Haute Island Annapolis Gut Breyer's Island light St. Mary's Cape Cape Fourchu Seal Island light CAPE SABLE Sable Island, E. pt W. pt Cape Ftoseway, Shelburne licrhts LIVERPOOL, Coffin's Island lights Lunenburg, Cross Island lights Sambro light-house HALIFAX Obs. D. Yard Sheet Harbor, entrance . . . Sherbroke Wiiite-Head Island Torbay, Berry Head CAPE CANSO, Cran- berry Island light Lat. 45 00 N 67 2W Loner. 45 i3 66 5 45 12 65 55 45 18 64 58 45 i5 65 U 42 65 45 44 16 66 22 44 6 66 II 43 5o 66 7 43 24 65 58.5 43 24 65 36 43 59 59 47 43 57 60 i4 43 38.565 i5.5 44 3 45 19.5 64 36 44 20 64 7 44 26.6 63 33.3 44 39.4 63 35 44 52 62 29 45 8.5 62 45 12 6j id 45 II 61 20 60 55.3 VI. The Gulf of St. Lawrence. Chedabucto Bay Gut of Canso, S. entrance Cape Hinchinbroke Cape Portland LOUISBURGH CAPE BRETON Scatari Island, N. E. pt... Flint Island Spanish Bay, Sidney light Port Dauphin Cape Egmont Cape North Chetican Harbor, entrance Seal Island Cape Mabou Port Hood, entrance Just au Corps Island Lat. 45 29N' 45 3o 45 34 45 49 45 53.5 45 57 46 2 46 12 46 18 46 24 46 53 47 2 46 4o 46 23 46 12 46 00 46 00 Long. 61 00 W 6r i3 60 42 60 5 60 00 59 48.5 59 4i 59 47 60 9 60 3 1 60 22 60 24 60 59 61 i5 61 26 61 34 61 37 Page 342] TABLE LIV Latitudes and Longitudes GUT OF CANSO, N. entrance Cape St. George, N. end . Pictou Island, E. pt Pictou light Cape Tonnentin, S. E. pt. . Richibucto Harbor, entr.. Cape North Cape West Egmont or 'Halifax Bav, Red Head '. Hillsborougla Bay, St. Pe ter's Isl Bear Cape Cape East Richmond Bay , D. M. 45 42 N 45 53 45 49 45 4i.5 46 o5 46 43 4? o3 46 4i 46 26 Cape Esquiminac Miscou Island, (entrance of Chaleur Bay) Cape Despair Bonaventure Island Flat Island Cape Gaspo Cape Rosier Magdalen River Cape Chatte Bio Island, Riv. St. Law. E. pt Anticosta Island, E. pt. . . West pt, S. W. pt S. point. N. point Deadman's Island Entry Island Amherst Isl. S. W. pt.. Magdalen Isles, E. pt. . . Byron Island, E. pt Bird Island St. Paul's Island Lat. 46 07 46 00 46 28 46 34 4? o4 48 01 48 25 48 3o 48 38 48 45 48 5i. 49 i5 49 06 48 25 49 o5 49 52 49 24 49 o4 49 58 47 16 47 17 47 '3 47 37.6 4? 48 47 5r 47 i4 Lons. D. M. 61 29W 61 52 62 33 62 4o 63 5o 64 5o 64 01 64 23 64 08 63 i4 62 29 6r 59 63 44 64 46 64 3i 64 21 64 10 64 II 64 12 64 14.8 65 22 65 48 68 53 61 45 64 35 63 36 62 18 64 12 62 1 5 61 42 62 o4 61 26 61 25 61 10 60 II VIL JVeiofoundland. Limits of the Great Bank of Newfoundland, N. point S. point Outer Bank Cape Norman Green Island Point Ferrole Point Riche Ingorneclioix Bay, Saunders Bay of St. Paul's . . . Bon Bay Cape St. Gregory .. South Head Red Island Cape St. George . . . Cape Anguile Cape Ray Connoire Bay Port Lat. M. i5N 56 00 38.1 24 02.4 42 39 5c 33 22 06 34 28 54 36.9 4o Lour, D. M. 5i loW 5o 00 45 00 55 56.3 56 37 57 o5.6 57 27 57 21 57 5i 58 oc 58 16 58 21 59 16 59 )5 59 27 59 20.2 58 00 Burgeo's Isles Rainea Islands Penguin's Islands Fortune Head Brunet Island, W. H Great Miguelon, Cape M. Langley's Island, Cape L. St. Peter's Island, S. E. pt. Point May - Cape Chapeau Rouge Mortier Rocks Red Island, S. pt Virgin Rocks Point Breem Cape St. Mary Cape Pine CAPE RACE Cape Race, (Virgin) Rocks Cape Ballard Cape Broyle Bay of Bull Cape Spear St. John's Harbor Cape St. Francis Breakheart Point Trinity Harbor Cape Bonavista Funk Island Cape Frecls WadJiam Islands Gander Bay Fago Islands, cape . — Snap Rock Tuolinguet Islands Cape St. John, N. Bill . Horse Islands, E. pt . . . White Bay, entrance . . Hooping Harbor Belle Isle, southern .... Groais Island, N. pt. . . . Hare Bay, entrance. . . . St. Anthony's Cape. St. Lunaire Bay Cape Bauld Belle Isle, northern .... Oroque Harbor Lat. Loner. D. M. D. M. 47 33 N 57 43W 47 32 57 25 47 22 57 01 47 o5 55 5i 47 16 56 00 47 08 56 26 46 48 56 27 46 45 56 10 46 54 56 o4 46 53 55 27 47 02 54 57 47 23 54 i5 47 10 54 II 46 59 54 16 46 5o 54 i3 46 38 53 35 46 39.4 53 04.6 46 26.3 5o 55 46 47 52 59 4? o5 52 52 47 18 52 47 47 3o.5 52 39 47 34 52 4^ 47 48 52 5i 48 09 52 59 48 22 53 24 48 42 53 08 49 45 53 12 49 18 53 3o 49 34 53 55 49 28 54 26 49 4i 54 00 49 55 53 44 49 42 54 44 5o 00 55 3i 5o i3 55 43 5o i3 56 21 5o 37 56 14 5o 49 55 29 5o 58 55 35 5i 16 55 4i 5i 23 55 3i 5i 29 55 29 5i 39.7 55 27.4 52 01 .3 55 19. I 5i o3 55 5o QUEBEC Coudras Isl. N. W. part. . Bay of Rocks Green Island light Point Mille Vache Bersimis Point, S. E. pt. . Manicougan Point Cape Nicholas Cape Montpelles light-h. Egg Island Seven Islands Bay, Store, Point Moisic Lobster Bay Point Manitou St. Jolin's River D. M. 46 49 I 47 24 b 47 57 48 o3.4 48 34 48 54 I 49 06 2 49 i5 9 49 19.7 49 38 3 5o i3 5o II . 4 49 49. 5 Long. D. M. 71 16W 70 28 69 5o 69 28.2 69 II 68 4t-6 68 i5 67 53.2 67 25 67 i3 66 25 66 07.7 67 06 5o 17.7J65 17. 1 5o 18 64 23 TABLE LIV. Latitudes and Longitudes. [Page 343 Mino-an Island Esquimaux Island Clear-water Point, S.W. ex. Appectetet Bay Mount Joli, Natashquan Pt. Cape Whittle Boat Islands St. Mary's Islands, S. pt.. . Hare Harbor Great Mecatina P.,S.E.p. Mistanoque Island Grand Point Forteau Bay Point Red Cliffs Red Bay York Point Cape Charles Battle Island, S. E. pt. . . . Cape St. Lewis Cape Harrison Enchanted Cape Cardinal's Island Button Islands Lat. Long. D. M. D. M. 5o 12.9 (^ K. 5 5o 1 3 63 4 1 5o 12.6 63 28 5o 16.7 63 01 5o 06 61 46 5o 10.7 60 09. 8 5o 17 59 A6 5o i3 59 45 5o 36.5 59 20. I 5o U 58 53 5i i5.8 58 i5. I 5i 25 57 i4 5i 25.6 56 59. 4 5i 33 56 47 5i Ai 56 28. 4 5i 58 55 55. 9 52 i4 55 22 52 16 55 33 52 21 55 4i 54 54 58 o5 56 40 60 55 58 5o 63 00 60 45 64 53 IX. HudsoTi's Bay dnd Straits, and Davis^s Straits. Cape Resolution Saddle-Back Island Upper Savage Island, E.pt. North Bluft" Capes Charles Cape Dorset Cape Pembroke C:ipe Walsingham Cape Digges, W. ex Salisbury Islands, E. pt Mansfield Island, N. part . S. part . Cape Southampton North Sleepers West Sleepers Portland Point Baker's Dozen Belcher's N. point James's Bay, Cape Henrietta Cape Jones. X. IBear Isle. North Cub. . The Twins. Albany Fort Moose Fort Charlton Island YorkF.)rt Cape Churchill Prince of Wales's Fort . Marble Island Cape Dobbes Cape Walsingham Dyer's Cape Sanderson's Hope Cape Bedford Wayirate Island Lat. M. 29 N 1 1 32 34 46 32 37 3o 37 27 23 3i 6 3 8 48 5 Long. D. 64 67 M. 3o\V 43 o 25 74 78 82 77 -8 76 79 80 84 80 81 79 2 79 3() 80 1 5 X. Greenland. Musquito Cove Gothaah, ent. of River Bal Bear Sound Maab Cape Farewell Whale's Island Herjoisness Bontokoe Island, S. E. pt. . Gael Hamkes Bay John Mayen's I., N.E. Cape Lat. D. M. 64 55 N 64 10 63 20 62 5 59 49 62 3o 65 3 73 29 75 00 71 10 Long-. D. M. 52 57W 5i 47 49 10 48 27 43 54 43 i5 29 5o 20 4o 6 5i 7 26 XI. Iceland. Cape Reikiancss Bessesled Mount Suaesell Patrixfiord Straumness North Cape Hola Grim's Island, N. pt. . . . Rikefiord J^ongnose, Cape Encliuisen Island Wreeland do Cape Hecla, Mt Westman's Island, S. pt. Lat. Long. D. 31. D. M. 63 48 N 22 42W 64 6 21 54 64 52 23 54 65 36 24 10 65 4o 24 29 66 28 22 26 65 44 19 44 66 34 18 o4 66 3o 17 35 66 23 i4 3i 64 20 i4 i5 63 55 18 19 63 58 19 4i 63 20 20 23 XII. Spitzhergen. Lm. Lonir. South Cape Fair Foreland 78 53 Amsterdam Island, (Hack liiyt's Head.) J79 46 Smeerenburg Harbor [79 44 Verlegen Hook 180 2 Hope Island, W. pt. 76 20 Bear or Cherry Island. .. .I74 3o I). 31. ID. 3r. 76 32NI17 23 E 10 35 10 57 11 II 16 37 20 5o 20 00 XIII. Ens;h'sh Coast, from London to St.^Mar^'s Light, {Scilly.) LONDON GREENWICH Observ Woolwich Purfleet Gravesend Rochester Sheerness Nore light North Foreland light. . . South Foreland lights . . Deal Castle DOVER Dunseness Hastings lights Lat. 5o M. 3iN II 3o 28 23 27 29 LniiiT. D 31. 6W o 4E '9 O 22 O 32 o 44 48 1 27 I 22 I 24 I 19 o 58 o 36 Page 344] TABLE LIV. ♦ Latitudes and Lonsitudes. Beachy Head light Brighton lij^ht Shoreliam liglits Arundel Selsey Bill Owers lio-ht PORTSMOUTH, town Isle of Wight, Cowes, Castle. . . Bembridffe Ledge or Point, Ft. light Dunnose St. Calli'ne's Pt, It. Needle's light.s. . . . Hurst Ught Poole light St. Albau's Head Weymouth light Portland lights Exmouth Bar Torbay, Berry Head Dartnaouth Start Point Praul's do Bolt Head Eddystone liglit Hand Deeps Ram Head PLYMOUTH, Mt Fowey Deadman's Point Falmouth light Manacles Rocks Black Head LIZARD Point Mount's Bay Penzance light Bundle's Stone, beac Wolf Rock Land's End St. Agnes' light, (Scilly,) . St. Mary's St. Martin's Lat. M. 44 N 5o 5o 53 43 4i 47 5o 46 Lonsr. D. M. o i3E o 8W o i5 o 35 o 48 4o 1 G i8 34 33 56 3 26 27 4 10 4 38 47 XIV. French Voast from Calais to Ushant. CALAIS Cape Griz Nez Ambleteuse BOULOGiNE Etaples Bay, Lornet light . . Montreul La Rochelle Abbeville Grotoy St. Vallery, River Somme Dieppe light St. Valley, River Cau.x. Fecamp light Cape de Caux Ca])e de le Heve lights . HAVRE DE GRACE PARIS Observatory... Mouth of Seine Harfleur Lat. Lon D. M. 5o 58 N 5o 52 5o 48 5o M 5o 33 So 28 5o 19 5o 7 5o i3 5o I r 49 56 49 52 49 46 49 4i 49 3 1 49 29 48 5o . 27 49 3o D. 31. 5t E 35 36 37 39 45 4o 5o 38 38 5 43 Honfleur lights Caen Bayeux Carentan St. Marcouf Island hght . Cape Barfleur licrht CHERBOURGH Pelee Island Cape la Hogue Alderncy Island, N. point. Caskets lights Guernsey, Pier Hd. light ... Sark Island, N. point Jersey Island, Cape Grosness . . . St. Aubin St. Clement's Point Isle de Chausey hghts . . . Coutances Granville, Mole Hd. light . . Avranches Mount St. Michael Pontorson St. Malo, New Mole light . . Cape Frehel light St. Brieu.x, Oath Brehat Island, Centre Tregucir Morlaix light on T. la Lande St. Pol de Leon Isle de Bas light Roche Blanche St. Anthony's lights USHANT, N.E. point, light Lai. M. 25 N I r 16 18 3o 42 38 4o 46 43 27 26 i5 i3 9 52 3 38 33 39 4i 3i 5i 47 38 4i 45 I 40 29 Loner. D. M. o i4E o 21W 43 1 i5 I 9 I 16 I 37 I 36 1 56 2 12 2 23 2 33 2 23 2 16 2 I I 2 00 I 49 I 27 I 36 I 22 I 3i I 32 46 00 i5 53 00 I 58 29 3 XV. .F)-om the JVorth Foreland to Dun- canshy Head. North Foreland Kentish Knock, Ft. light . Long Sand Head Galloper, N. point S.W. point Ft. It. Shipwash, N. point, Ft. It. S. point Gaberd, outer Orfordness lights A:.ldboro' Steei^le Southwold LoestofF lights Yarmouth W^interton Ness lights . . . , Smith's Knowl, S. pt. ... Hasborough Sand, S. p. . N. p. . Sherringham Shoals . . . . , Hasborotigh lights , Cromer lights Lemon and Ower, N. p. . , -^ S.p... Cromer light Dudgeon light • Outer Dowsing ■ Inner Dowsing Lynn Knock Lat. 31. 23 N 40 45 52 45 02 53 58 5 9 20 29 37 43 48 5i 2 3 49 56 Long'. n. M 27 E 39 38 5 56 38 33 59 34 36 4i 46 A4, 4i 14 48 35 20 32 19 58 o 33 o 29 TABLE LIV. Latitudes and Longitudes [Page 345 Spurn lights Flamborough Head light. . Filey Biig Scarborough liglit Robin Hood's I3ay Wliitby light River Tees', Seatou lights. Stockton River Tyne's Mouth lights Coquet Island Staples light Farn liglits Sunderland Point Holy Island, Castle BERWICK, Pier light . . . St. Abb's Head DUiNBAR May Island lights The Bass N. Ber-\vick FDIiNBURGH Fllyness , Fife Ness .St. Andrew's, Pier Head. Mouth of Tay light Bell Rock, off do light... Buddonness liglits Red Head...-. Montrose light Tod Head NEW ABERDEEN.... Newburgh Peter Head, Pier light Buclian Ness light Ratrie Head Kinnaird's Head light . . . Barnff light Fort St. George Inverness' Croinartie, Pt light Tarbet Ness liglit Clythness Noss Head hght Diincansby Head Lat. D. M. 53 35 N 54 7 54 i5 54 17 54 27 54 3o 54 4o 54 34 55 I 55 20 55 39 55 37 55 36 55 40 55 46 Lonnr. D. M. 7E 5 iW 23 20 37 IS 19 25 32 40 39 38 47 59 2 33 2 38 2 4i 3 12 2 M 1 35 2 47 2 38 2 23 2 45 2 29 2 27 2 i4 46 46 49 XVI. The Orkney Islands. Pentland Skerries light . Slromo South Ronaldsha, S. p.. Copinsha Lamb's Head on Stromsa Island North Ronaldsha, N. p.. Mould Head, on Papa Westra Island Nnup Head, on Westra Isl. Marwick Head, on Pomc-na Island Stromness Hoy Head, on Hoy Wells Island Slue Skerry Fair Island 44 ~ Lat. Lonpliant's Mount Queda Prince of Wales's Island, Fort Cornwallis Cape Caran Salangore Ilill and fort. . Palo Callam or Colonjr, sp .; Parcelar Hill Parcelar Point Tanjong Tuan, (Cape Ra- chado,) Tanjong Clin, or Peer Punjab Fisher's Island Malacca fort Water Islands, southern . Mount 3Iora or Moar. . . . Mount Formosa Mount Battoo Ballo Pulo Pisang Pulo Cocob Sincapore Little Hill, or False Johore HiH Lot. D. M. 20 58: 2£ 2 21 00 21 27 22 21 22 26 21 10 20 34 20 7 9 22 8 5l 8 26 52 541 16 29 18 00 16 32 32 6 42 34 27 I 48 I 23 I 10 9 58 8 43 8 28 8 9 7 4o 6 21 6 10 6 G 2 56 2 52 2 42 2 26 2 17 2 i3 2 II 2 4 I 59 I 49 I 39 I 28 I 19 I 17 I 26 IjOns^. D. M. 88 II E 88 25 88 37 88 M 91 48 91 52 92 4 92 20 92 54 93 16 93 44 93 56 94 6 94 23 94 1 5 94 12 94 i3 94 17 94 i3 96 25 96 52 97 35 98 9 98 i4 97 55 97 42 97 49 98 3fi 97 26 97 38 97 57 98 10 97 4o 97 4o 98 18 98 18 100 1 3 100 21 loi 8 101 22 loi 16 101 25 101 32 loi 5o 102 8 102 12 102 i5 102 20 102 4o 102 54 io3 1 1 io3 i3 io3 25 io3 5o io4 4 Lat. D. M. I 23: 4 47 5 i5 Johore Hill Barbucet Hill POINT ROMANIA....! i False Barbucet Hill i Romania Reef j i Eastern Bank, (outer part) Pulo Tingy Blair's Harbor Pulo Varela Palian Road Tingoram Howard's Shoal .... Pulo Brala, or Capas de Mer Pulo Capas de Terra. Tringany Piiver, entrance Great Redang Island.. Pulo Printian Calantan Road Cape Patani Pulo Lozin Pulo Cara Slam River, E. entrance JUTHIA, or SIAM.. Cape Liant Pulo Way Pulo Oby False Pulo Oby Cambodia Point Cambodia River, W. ent. Cape St. James, (E. en- trance Saigon River,) . Cape Trivoane Point Babeck Brittos Bank, N. E. p Cow Island Point Kega Point Viiiay Mui-guio, or Little Cape. Point Lagan Pulo Ceicer de Terre .... Cape Padaran Padaran Bay Cape Varela False Carmaigne Harbor, ent. . Water Islands Tre Island Pyramid Island Nhiatrang Three Kings Rocks Hone Colie Harbor Cape Varela, or Cape Pa- goda Perforated Rock Phuyen Harbor, entrance Coumong Harbor, ent... Pulo Cambir Cape Sanho Quinhone Harbor Buffalo Island Point Nuoc Ngol TamqiL-xi River Pulo Canton Port Qui-quick, ent. .... Cham Callao Ca])e Turon or Tienchu. Callaohanne Island, (N.' entrance Turon,) I iG 11 i3 23 i4 55 12 34 9 58 8 56 8 25 8 35 9 M o 17 o 21 o 3o o 32 o 39 o 4i o 54 4 9 i3 21 35 A4 49 3 16 21 26 37 45 55 59 23 29 33 4 19 4 39 5 23 5 28 5 59 6 8 Lonsr. D. M. 04 6 E 04 TI 04 16 04 16 04 25 04 35 o4 II o3 4o o3 47 o3 18 o3 3 1 o3 4i o3 12 o3 I 02 56 02 4o 02 i4 01 o5 01 59 00 35 00 34 00 o 01 1 1 03 48 04 38 04 54 04 56 06 20 07 4 07 16 07 33 07 48 07 52 08 4 08 19 08 3i 08 40 08 48 09 00 09 4 09 12 09 12 09 19 09 19 f)9 23 09 10 09 25 09 12 09 28 09 23 09 1 4 09 1 3 09 18 09 i4 09 II 09 16 09 7 08 56 09 6 08 5o 08 4o 08 19 108 12 Page 360] TABLE LIV. Latitudes and Longitudes, Cape Chcuvay Hue or Huesso River, W.e Tiger Island Hainan Island and adja- cent Islands, — Yaitchew Bay — Yulenken Bay, Zenby — South Point of Hainan — Galong Bay — Brother's Islands, east- ern ....'. — Luengso}'^ Point, S. p. — Sail Rock — Saddle Island — Point of land — Nankin Island — Tinhosa Island — False Tinhosa — Toongean Mount, pt. . — Hainan Head, N. E. p. — South Taya Island . . . — North Taya Island . . . Nowchou centre Ty-foong-kyoh Island, (Tienpak Harbor,) Ty-Chook-Chow Island . Song-yue Point Mamee-Chovv, or the Twins, near S.W. p. of Hai-ling-shan Ty-oa Point ;2 1 Nampang Island , Mandarin's Cap Mong-Chow Island . , . . Haw-Cheun, S. W. end Passage Island, (near S. W. p. Haw-Cheun,) Wy-Caup Island, (neat S. point St. John's,), . , Lieu-Chew Island Wizard Piocks Ty-katn Island Cou-cock Island S, W. Pt Tyloo Island, S. p Great Ladrone Potoe or Passage Island . . Laft-Samee Peak Typa,.., Macao, city Lantoa or Tyho Island. S. W. p Lintin Island, peak Asses' Ears Great Lema Isl., N.E. p.. Nine Pin Rock Wlmmpoa anchorage. . . . CANTON Lat. M. 21 N 35 10 8 24 8 II 8 10 8 12 8 ir 8 22 8 26 8 35 8 40 8 38 49 35 00 49 59 52 34 43 34 28 39 35 35 34 36 46 5i 5o 52. 56 24 53. 4 16 6 7 Lons- D 31. 07 59 E 07 41 07 22 08 52 09 35 09 34 09 39 09 4i o 00 o 8 O II o 24 O 2[ o 28 34 1 2 57 1 12 I i5 3a 1 i3 I 25 I 4o I 5o 29 32. 35 53-5 i4 39 49 33 33 51 3 3 48 4 02 -5 4 19 4 22 3 22 3 i4 XLT. hkinds and Skoals in the KVDMJV OCEAJ\r, between the meridians of the Cape of Good Hope and Sumatra, inclu- ding those jr. andJV. IV. ofMw Holland. Dutch Bank, Stot Van Capelle, various ( from situations, > to . , Lat. Lon^ D. M. D. M. 4o 00 S 38 5oE 36 00 43 3o very Telaiuaque Shoal, doubt- ful, various sit- ^ from uations, > to , . Brunswick Bank, doubtful French Shoal, doubtful, , Atlanta's Rock, doubtful Wellington Shoal doubtful Prince Edward's Islands southernmost northernmost Kerguellan's Land, or Isle of Desolation, Bligh's Cap, N. p.. , Christmas Harbor, , Port Paliser Cape Digby, or E. p. Cape George, or S. p. Island Solitaire , , , , Cape Louis St. Paul's or Amsterdam Island Amsterdam or St. Paul's Island, Danish Rock, doubtful . , Cloate's Island, (longitude uncertain,) Tryal Rocks Rosemary Island A reof 10 miles N. W. of Rose- mary Island . , , j Abrohlos Shoals. J Christmas Island .10 Cow Isles, Northern , , . . Southern . . , Very near New Hol- land. Lat. D. M. 9S 00 25 8 43 53 53 40 37 52 46 17 7 40 27 28 Clark's Reef, S. E, point Imperieu.se Shoal Dampier's or Scott's Reef, N. W. end N. E. end . Coral Bank Coral Bank, 9 fathoms.. Coral Bank, 7 fathoms .. Cartier's Sandy Island or Bank Red Island, (very near New Holland,) Coral Bank, 10 fathoms or less Hibernia's Shoal Sahul Shoal, S. W. p., 12 fathoms Echo's Soundings, Rock Coral 7 fathoms Bank . . , Fortune Shoal Union Shoal Dutch Bank Otter's Shoal, doubtful,. Princess Augusta's Shoal, doubtful Union Rocks, doubtful,. Swallow Rocks and Breakers, doubtful . , , . Belliquese Shoal, doubtful i5 3o 3i 5o 23 28 35 52 I 32 25 46 28 i3 25 56 35 16 56 33 8 35 25 3 1 U 33 56 33 44 35 23 28 20 98 43 Lonff. 1). M, 21 57 E 23 24 36 19 43 6 52 00 71 43 37 46 38 8 ■ 68 44 69 4 69 37 70 34 70 10 68 5 68 18 77 35 77 36 98 25 112 3o io5 3o 116 3o 116 23 ii3 35 io5 33 97 4 97 i5 119 20 118 56 121 59 122 16 124 29 124 12 124 32 123 56 124 18 124 1 1 123 28 124 i4 126 00 129 35 43 5 4i 12 44 00 36 00 36 16 4t 20 42 10 42 33 TABLE LIV. Latitudes and Longitudes. [Page 361 CAPE ST. MARY..,, Star Reefs, S. end St. Augustine Bay, Sandy Island Cape St. Vincent Mourondava , Cape St. Andrew Boyanna Bay, entrance. Benibatooka Bay, Majunga Point Majunbo Bay, entrance . . Nareenda Bay Sancasse Island, N. pt. . .. Passandava Bay, Nine Pin Island Dalrymple Bay Nos Bell Island, N. pt. . . . Minow Island, N. pt Cape St. Sebastian CAPE AMBER, N.E.pt. Britisli Sound, entrance . . Port Levon, Nosh How Island Cape East, town A ntongil Bay ,Port Choiseul Cape Bellones St. Mary's Island, N. pt.. - S.pt.. Lat. D. M. 25 39 25 24 23 38 21 54 20 18 6 II 5 59 43 12 4o 3i 28 3o 3 12 2 5o 2 26 1 58 2 I A Foul Point. . Tamatave Point Fong Isles fllanooroo Rangazarah Mananibatoo St. Luce Bay, N. Isle FORT DAUPHIN.. St ir Bank Bassas de India Europa Rocks, S. pt. . Sussex Rocks Bazaruto Islands, Cape . . . Barren Islands, western .. English Bank Juan de Nova or St. Chris- topher's Island Coffin Island Chesterfield Shoal Mayntta Island Moliilla Island, E. pt Johanna Island, peak .... Comoro, S. E. pt Portuguese Slioals John Martin's Island, doubtful Rover Shoal Aldabra Islands, N. W. p.. Assumption Island, Hura'k Cosmoledo Island, N. pt. . Marquis of Huntley's Bank St. Peter's Island Natal Island, doubtful Sandy Island St. L-Twrence Island Zanzibar Island, S. p N. p .'Xmirante Island, N. W. p. S. E. p.. 8 10 8 27 9 55 20 58 24 17 24 45 25 I 25 7 25 25 2 2 23 21 3r 21 25 21 3l 18 4i 17 40 17 3 17 3o 16 17 12 54 12 20 12 i5 11 54 12 3o 10 i5 12 22 9 23 9 46 9 38 9 55 9 20 8 26 9 10 Lonn. 6 20 D. M. 45 7' AA 18 43 38 43 20 44 19 44 3i 45 23 4G 20 46 59 47 26 47 35 i5 48 2 48 19 48 39 48 46 )<; 19 49 23 49 53 50 3o 49 52 9 54 50 5 49 5i 49 37 49 28 49 26 48 52 48 33 25 47 i4 47 2 44 16 4o 24 39 36 42 36 35 33 44 3 40 1 5 42 47 43 47 43 55 45 i4 44 o 44 3o 43 33 46 5o 43 5o 46 25 45 5o 46 34 47 36 50 1 5 5o 5o 47 12 48 10 5o 23 39 33 39 21 53 45 54 3o MaheBank, N. W. p S. E.p Seychelle or Mahe Island -W. pt. Praslin Island French Shoal African Islands Alphonso Island Sandy Island or Bank. Isle Bourbon St. Denis jMauritius,or Isle of France, Port Louis Lat. D. M. 3 20 5 3o Diego Rais or Rodrique . St. Branden or Cargados Garajos, N. part of the Bank Low Sandy Island Islet with huts. — Soutii Islet Nazareth Bank, S. VV. p. . N. E. p... Sandy Island Galega, or S. Roquepiz middle Saya de Malha Bank. limits . ::{ 46 Fortune Bank, 10 fath Joim do Nova, N. pt. . . . Providence Island, N". pt. Coetivy Island, N. pt. • . Chagos Archipelago, DietTO Garcia 37 17 58 55 o 12 4 4 3 4 7 7 20 52 20 10 19 4o i3 4i 16 5 16 27 16 47 16 47 i3 4i i5 52 10 25 11 3o 8 18 7 16 o 7 7 6 7 7 7 6 7 Pitt's Bank Centurion's Bank. . . Ganges Bank I 7 22 Owen's Bank | 6 46 Egmont's or Six Isl-| ands. . . .<. 6 " Danger Island 6 Eagle Island 6 Tiiree Brothers 6 Peros, Banhos Islands 5 Saloman's Isl's, S. W. 5 Sandy Islands 5 Speaker's B'k, N.E.pt. 4 Pona Molubque Atoll, S. p. N. VV. p — N. E.p.. Addon Island, middle Suadiva, southern oroup, South Reef South Island .... S. W. Island..., N.W. Island..., N. Island Northern group, S. W. Island . N.W. Island..., N. E. Island Adoumatis Atoll, — S. \V. extremity. , . . — Southernmost Island — Island — N. W. Island -N. E. Island Long. D. M. 54 40 E 56 59 55 3i 55 44 54 42 53 3o 52 43 52 43 55 29 57 3o 63 24 4i 34 33 9N o 28 o 34 o 5r 58 1 5o I 47 1 5i 2 7 2 7 61 i5 59 47 59 4o 59 24 59 3 1 61 i5 34 56 39 62 20 59 58 57 o 5i 8 5i 7 56 22 72 22 71 18 70 57 71 2 70 20 71 24 71 i3 71 18 71 35 71 48 72 10 72 37 72 24 73 6 73 12 73 25 73 35 73 i5 73 12 3 4 73 2 73 8 73 19 73 20 73 33 73 27 73 22 73 38 73 35 73 35 Page 3C2] TABLE LIV. Latitudes and Lonofitudes. Collomandous Atoll, South Island... Long Island. . . . N. W. extremity West entrance of Coll. Channel Molucque Atoll, S. ex Nillandos Atoll Poulisdous Atoll Ari Atoll, N, Is. S. pt Male Atoll, 'or Maldivia S.E. p Gafer Island Todu Island Cordivia Island Maloss Madoll, S. pt Padipolo Atoll, E. p MillaDoue Atolls, E.pt. . Tilla Dou Matis, or Head of the Islands, northern limit Lat. Long- Minicoy,or Malicoy. Seuvelli Islands, Southern Northern Southern ex- treme Reef. Kalpeni Islands, S. p N. p Courutee Island Pittie Sand Bank Underoot Island, E. pt. . Aucutta Island Bingaro Island Tingaro Island Ameni Island Permulpar Island Cardamuin Island, Elicapeni Bank, E. pt.... Kittan Island, S. pt Betrapar Island, N. ex... Cliittae Island Cherbaniano Bank, (not explored,) S. pt, Angrias Bank, N. p. ... Bale of Cotton Rock, (doubtful,) Le Meme's Reefjfdoubtful) Prp])ari3 Islraid, N. p - S. p Great Coco Island, N. p. S.p.. Little Coco Island Landfall Island Great Andaman, Cape Price, N. end S. E. point Port Cornwallis. . . Port Chatham .... Port Campbell.... Rutland Island, S. p Interview Island, N. p.. .. S.p.... North Centinel 21 3o lO 46 4o 36 3o 27 46 26 58 00 25 5i 7 6 8 17 9 56 o 4 o 10 o 3i o 45 o 48 o 5i o 55 55 1 6 I 9 I i4 I i3 I 25 I 35 I 4o 2 i5 6 38 6 18 5 18 I 20 4 56 4 49 4 II 4 2 3 58 3 39 34 3o D. M. D. M. 2 iSNyS 21 E 73 8 73 8 73 21 73 23 72 54 73 M 72 5o 73 42 73 4o 72 58 73 26 72 58 73 38 73 27 I 43, I 56 1 24 3 I 2 47 I 33 72 53 73 3 72 12 72 i5 72 9 73 35 73 35 72 36 72 32 73 42 72 10 72 16 72 18 72 41 72 o 72 ^i 73 56 73 o 72 II 72 42 72 o 71 43 71 43 94 20 93 4o 93 4o 93 21 93 21 93 1 5 93 4 93 4 92 56 South or Little Centinel. Five Islands, S.p Sisters, southern Brothers, northern Little Andaman, N. p. . • . : S. E. p Invisible Bank, N. p S.p Flat Rock Barren Island Narcondam Car Nicobar Batty Maloe Chowry Island Terressa Island, N. p. . . . S. p. ... Katchall, W. end Noncowry Island and har- bor Comorta, N. p Tillangchong Islands, N.p S.p Meroe Island Little Nicobar, N.p S.p Great Nicobar, N.p S.p D. M. II ooN 17 II 10 Lat. II 00 10 53 10 26 II 27 10 56 II 8 12 16 i3 26 9 10 8 A6 8 28 8 22 8 12 7 54 8 00 8 i5 8 33 8 22 7 29 7 26 7 i3 7 8 6 45 Long. D. M. 92 22 E 92 55 92 46 92 4i 92 38 92 40 93 41 93 40 93 34 93 54 92 46 92 5i 93 3 93 17 93 6 93 23 93 46 93 42 93 40 93 40 93 34 93 42 93 34 93 55 93 54 XLII. The. Islands of Sumatra, Java, Billington, Caspar, Banco, loith the adjacent Islands and Straits. Acheen Golden Mountain .... Pedir Point Elephant Mountain.. Tooloo-Samwoi Point Diamond Point Tanjong Bou Batacarang Point . . . . Fourth Point Third Point Second Point First Point Hog Point Flat Point Billimbing Ba)' Bencoonat Lat. Cawroor ' 4 Manna Point Buffalo Point BENCOOLEN, (Fort Marlborough,) Caytone Moco-Moco Indrapour Point Padang Head Priaman Natal Tappanooly, P. Kaeheel. Tappoose Sinkel Point Bulo Samah M. 35 N 22 1 i3 i4 5S o 20 23 4i 00 54 00 54 35 56 33 58 48 34 10 56 40 33 N 44 00 i5 33 Long. D. M. 95 19E 95 45 96 5 96 5o 97 14 97 38 io4 3o io4 5i io5 i5 io5 32 io5 4i 106 3 io5 45 io4 36 io4 27 io3 34 102 49 102 19 102 19 102 14 loi 20 100 48 roo 20 100 10 99 " 98 45 97 57 97 46 97 54 TABLE LIV. Latitudes and Longitudes. [Page 363 I Lat. Troumon Pulo Duas Baccoongung Pulo Munkie Oujong Coomoowung. .. Oujong Cluet Qualali Bahoo Qualali Assehahn Tainpat Tuan Batto Plyeer South Tallapou Pulo Sooroodung Muckie Laboun Iladjie Mungin North Tallapou Soosoo Pulo Kio Qualali Battoo Oujontr Se Mium Cape Fflix Oujong Trlpah Senaligun Analaboo Oryong-Booboon or Ba^ hoo Pulo Rungass, off Rigas Bay Oujong Chellung Rigas Tellow Goolumpung.. Pulo Cass Pulo Riah and Pulo M. centre Barbce Wee Diah Oujong Dahway Pulo Rondo Pulo Way Pulo Brasse Pulo Rajah Cocos Islands . . . Hog Island, N. p. S.p. ( from (to .. Flat Islands. Pulo Assayo Coral Bank C from (to Burgh Rock Shoal, 10 feet Castlcreagh Siioal .... North Pulo Dua Passage Island Bird Island Pulo Lucotta Londise Shoal, (N. N. E. ^ E. from Lucotta, dis- tant 2^ leagues.) Mcnsular Island, N.W. pt. Pulo Dua Pulo Nyas, S. p Pulo Tamong Pulo Panjang Clappes Island, middle.. Pulo Mintaon, or Batao . Pulo Ayer Besar M. 49 N 54 57 55 5? 4 6 8 i6 4 i3 38 38 39 42 Ao 52 55 6 4 5 49 5 42 4 4o 2 59 3 2 2 57 2 21 2 4i 3 3i 2 4 2 i3 2 47 I 57 I 40 1 27 o 36 o 54 o i3 o o 25 1 24 Lono-- D. M. 97 5iE 97 44 97 42 97 39 97 38 97 32 97 3i 97 3o 97 23 97 19 97 18 97 16 97 i4 97 II 97 4 97 3 96 58 96 57 96 56 96 46 96 42 96 3i 96 24 96 18 96 5 95 38 95 4o 95 4o 95 37 95 34 95 3o 95 3o 95 28 95 25 95 i4 95 23 95 6 95 33 95 33 95 58 96 38 96 34 96 42 96 47 96 54 97 26 97 33 97 6 97 39 97 5o 98 7 98 3o 98 20 97 56 98 40 98 3o 98 3o 98 7 100 17 D. M. G. Fortune Se-beero, Island, N.p S. W. p Se-pora, or South Pora, N. W. p -S.p North Poggy Island, N. p, S. p. South Poggy Island, N. p. S. p, Laage or Larg Islands. Rat Island Trieste or Reefs Island Pulo Pisang Little Fortune Island . . Encrano or Deceit Island, N.p — E. point — S. E. point ... — S. point — W. point Java Head First Point Second Point Third Point , Anger Bantam or St. Nicholas Point Bantam BATAVIA obs Carawang Point Sedary Point Point Pamanoekan. • . , Woerden Castle Rock, Princess Charlotte Shoal Indramaye Point Pulo Rackit Bumkin's Island, or outer Shoal Cheribon Mountain . . , Taggal , Rock Samarang flagstaff, anchorage . . . . Mandalique Island Lerang Point Rambang Point Panka or Panco. . . Sour.abaye, fort Cape Sandana Balainbonang Bay, Ft Goonog Ikan ■ E. point Turtle Bay. Tulan or Dirck Vrie's Bay Wine Cooper's Point Noesa Baron Island, S. p. Tangala Islands, largest. Clappe's Island, about. . . Mew Island Peak on Prince's Island . Peak on Crocatoo Island. Peak on Tamarind Island, or Pulo Bessy , Pulo Sebooko Lat. Loner 56 S 47 00 25 32 52 5o 20 3o 5i 3 8 54 i5 22 3o 3i 6 48 6 44 6 36 6 27 6 3 5 53 6 2 6 5 5 6 5 5 6 5 5 47 6 55 6 5o 8 23 8 46 7 48 7 5o 7 25 8 32 8 26 7 I D. M. 98 38 E 99 2 99 33 99 58 99 37 00 1 3 00 1 5 00 4 1 01 3 02 1 5 01 6 04 6 o4 3o 02 25 02 40 02 38 02 20 o5 1 3 f>5 12 o5 21 o5 4o 05 56 06 4 C16 10 06 5o 07 3 07 27 07 49 07 58 07 54 08 20 08 22 08 23 08 26 04 i4 10 27 10 26 10 5i 1 1 27 11 17 12 32 12 45 i4 22 i4 25 i4 33 09 48 08 1 2 06 26 o5 i5 o5 i5 o5 20 Page 3S4] TABLE LIV. Latitudes and Longitudes. Cap Button Tliwart-tlie-way Zutphen Islands, (largest,) N. p South W.M-'her Man-eater's Island Pulo Baby Thousand Islands, N. . . . Pruysen's Droogte Shoal Annuyden Bank North Watcher Three Sisters North Island Two Brothers, northern . Lynn Shoal Shabunder Shoal Brouwer's Shoal Lueepera, S. entrance St. Banca , Nanka Islands Lat. Banca Island, South Point Tanjong Panjong, or Point Lalary .... Monopin Hill Tanjong Goonting . Tanjong Muncooda N. of Banca Tanjong Tuan Songy Leat Bay .... Tanjong Ryah Goonong JVIarass Mount Tanjong Breket. . . . Rocky Point Entrance Point, or S. E p Essex Shoal, or Fairlie Rock Vansittart's Shoals 5 Pulo Leat or Middle Isl.. Alceste Shoal Shoal Water Island South Island North Island General Hewitt's Rock . . Discover}' Rock Pulo Glassa or Gaspar Isl. Tree Island Warren Hastings's Shoal Belvidere's Shoal, N. pt. Vansittart's Shoal Hillsborough Slioal Magdalen's Shoal Severn's Shoal Billiton Island. S. E.p... S. W. point N.p N. W. Island, off Billiton Shoe Island, (formerly Bird Island and White R) Fo.T Shoal Pulo Mancap Shoal, S. p. D. M. 5 59S 5 53 5 57 5 5o 5 4i 5 54 5 48 5 32 5 17 5 i3 5 12 5 M 5 4i 3 i3 2 25 3 8 2 49 2 00 I 43 I 28 I 38 I 5o I 55 1 53 2 36 2 56 27 3 5 2 5i 2 46 3 20 3 00 2 58 2 53 2 54 2 25 2 28 2 23 2 3 I 56 1 4o 3 22 3 i5 2 33 2 3i 3 47 3 3o 3 5 3 22 Lono-. D. M. o5 57 E o5 57 o5 5i 05 47 06 43 06 3o 06 i4 06 35 06 47 06 48 06 3o o5 48 05 49 06 3 06 1 3 05 56 06 i4 06 8 o5 48 06 28 06 4 o5 12 o5 2C 05 53 06 6 06 9 06 i4 05 52 06 5o 06 54 06 52 07 2 07 2 07 8 07 5 07 2 07 1 3 07 1 5 07 1 5 07 i4 06 56 07 4 06 58 06 57 07 00 06 42 06 22 07 o 06 3o 08 10 07 3i 07 53 07 33 08 o 10 6 10 7 10 7 Discovery 'sWesternBank Eastern Bank Reef Osterly's North Shoal. . Cirencester's Sand Bank Shoal .... Montaran Islands, South Eastt n .< . Toekoekemou, (highest island,) Minto Rocks Ontario's Shoal Rendezvous Island,S.W.p Souroutou, W. p Carimata Island Peak, Pulo Papan Pulo Panumbangan. . , Massa Teega Isles ... Greiff's Shoal , The Seven Islands, N. W Pulo Varela or Barallah . Pulo Taya The Calantigas Ilchister Shoal Ungin, Tanjong Eang, S. E. extremity East Domino Island . . Geldrias Bank, same Dogger Bank Rhio Eastern Island, off Pulo Panjang Island Laage, E. pt. . . Three Brothers, south . Pedro Branco Islands off P. Romaine Bintang Island, (the hill,) - N. W. p Johore Shoal Shoal ent. Rhio Straits Sincapore Island, E. p. Pulo Battain, N. E. p.. St. John's Island, S. p. Rocky Reefs Middle Island Coney Island Buffalo Rock Rocks Red Island Tree Island Alligator Island Rocks Little Carimon Great Carimon, S. pk. The Brothers Pulo Cocob Pulo Pisang Water Islands, or Four Brothers, S. p Fisher's Island Bambeck Shoal Pulo Callam or Colong, S.p Two and a half fathoms Bank Round Arroa Blenheim's Shoal Lat. M. 39 S 33 36 19 17 2 54 2 35 2 3i 2 3i 2 i4 2 I 2 44 I 42 I 36 I 28 I 12 o 55 55 1 8 o 5o o 45 o 35 o 26 o 20 O 10 o 48N o 57 46 3i 20 23 2 i3 2 37 2 56 2 54 2 49 3 3 Loner, D. M. 108 43 E 109 10 108 48 108 40 109 o n.8 58 108 52 io8 36 109 5 1 108 39 no 9 108 38 108 5i 109 26 109 12 109 18 108 35 io5 12 104 25 104 55 io3 5i io4 57 io5 o io5 o io4 58 io4 3o io4 49 104 45 io3 44 io4 23 104 16 io4 26 104 16 104 4 104 II 104 00 104 4 io3 5i io3 55 io3 46 io3 4 1 io3 48 io3 45 U.3 38 io3 36 io3 4o io3 36 io3 22 io3 19 io3 2t io3 25 io3 i3 102 20 102 12 loi 4i loi 16 100 OJ TABLE LIV. Latitudes and Longitudes. [Page 365 Long or Great Arroa. . . . Two Brotliers, Pulo Pan- dan Pulo Salanama Pulo Varela Pulo Jarra Sainbilap.g Isl., southern. Dinding island, W. p. . . . Prince of Wales's Island, Fort Cornwallis Pulo Pera Boonting Island, southern Pulo Bonton, (dome,) . Pulo Ladda, S. p Trotto Island, N. p Sangald or Guilder Rock Pulo Tolibon, S. W. .... The Brothers Pulo Rajah, or P. Taya.. Juiikseylon, S. p Lat. M. 52 N 24 99 54 21 99 52 47 99 36 00 loo lo 3 100 3o i6 loo 35 Lonff. D. 100 M. 44 E 21 57 i8 I? 42 39 45 24 98 18 XLIII. Islcmds and Shoals in the CHIMA SEj]. Si Barbc Island Direction Island Pulo Datoo Welstead's Rock St. Esprit Islands, E Green Island St. Julian Island Tanibolan Islands, East or Great Island (Jap Rock Europe Shoal Rocky Island Camel's Hump Saddle Island French White Rock .... Victory Island Acasta Rock White Rock Macedonian Reef South Anambas, limits < Pulo Domar JMiddle or G. Anambas, W. limit North Anambas PuloTingy Ex. IsletoffP.Tingy.... Pulo AOR or Wawoor . . Pulo Pisang or Pambee- lan Pulo Tinioan, S. p. . . . — N. p — Bay on S. W. side . — N. Islet ofl'N.W. side Pulo Varela Pulo Brala, or Capas de Terre Pulo Capas de Terre . . St. Pierre Islands Ledge of Rocks. . Larkin's Reef Lat. Lons- M. 7N i5 7 32 34 4o 54 9 10 16 32 34 39 2 25 2 18 2 4o 2 45 9 27 17 8 29 37 44 54 48 56 16 47 i5 54 53 [o5 4i [06 i5 [o4 II 104 i4 [o4 35 [o4 i3 [o4 1 5 [o4 i5 [o3 47 South Haycock Island . . South Natunas Islands, — South Island, or Sapata — East Island — West Island — North or Flat Island . . Low Island Hutton's Shoal Diana Shoal North Haycock Island. .. Grand or Great Natuna C Island, limits \ Mt. Peaked Island Pyramidal Rocks N. W. Island Coral Reef Coral Reef North Natunas Islands, S.p N.p Rock above water . . Saddle Island Success Slioal Pulo Oby The Brothers, (eastern,). PuloCONDORE Charlotte's Bank Phaeton Bank Royal Bishop's Bank, S.p. Britto's Bank Holland's Bank, S. W. p. N. E. p. Pulo SAPATA Pyramid Rock, or Little Catwick Round Island, or Great Catwick Pulo Ceicer de Mer Minerva's Bank Investigator's Coral Patch Triton's Island or Bank, S. W. part Passoo Keah, (Sandy Isl.) Bombay Merchant's Shoal, E. p. S. p. Discovery Shoal, W. p.. . E.p. .. Jehangire's Coral Bank.. Vulador's Shoal, E. p W. p... Crescent Chain, Money's Island .... Robert's Island .... Battle's Island Drummond's Island Governor Duncan's Island Antelope's Shoal. . . Observation Bank, N.p.. Pyramid Rock Lincoln Island Rocky Island Woody Island Amphitrite Islands, W. p. —1 E. p. North Shoal, W p E. p Lat. Long. D. M. D. M. 2 9N 109 10 E 2 26 109 8 2 42 109 18 2 5o 108 28 3 3 108 54 3 107 45 3 107 57 3 9 107 44 3 i5 107 18 3 40 108 26 4 16 108 11 4 01 108 10 4 7 107 26 4 7 107 5o 4 I 107 5o 3 57 107 47 4 42 107 58 4 5i 108 4 39 107 57 4 3i 107 M 4 23 107 54 8 25 104 54 8 35 106 1 5 8 40 106 42 7 5 107 37 7 107 29 9 4o 108 21 10 32 107 48 10 36 108 32 10 48 iu8 47 10 I 109 2 TO 2 109 00 10 6 108 52 10 32 108 53 , 10 37 110 18 i4 12 112 52 i5 45 III II 16 3 in 45 t6 4 112 38 i5 59 1 12 26 16 11 11 1 32 16 16 I II 46 16 )8 112 35 i6 19 112 7 16 18 112 16 28 III 3o 16 3i III 34 16 33 I II 36 16 29 III 44 16 27 III 4') 16 27 III 35 16 37 III 4i 16 35 112 37 16 4o 1 12 42 16 52 1 12 20 16 5o 112 18 16 59 1 12 12 16 54 112 23 17 5 III 26 17 6 III 32 Page 366] TABLE LIV. Latitudes and Longitudes. \ Macclesfield Bank, C limits ( Scarborough or Mar- C singola Shoal, limits (^ St. Esprit Shoal, (by Lt Ross,) • (by As- seveido,). , Pratas or Prater's Shoal, N. E. p. . . , N. W. p.., • Anchorage Island .... Great Ladrone [The Islands near Canton are given in No. XL. and in No. XLVI.] Pedro Branco Lamock Islands, outer- most Lat. M 17N 21 4 i3 3o (very Andrade Rock, doubtful,) .... Luconias Shoals, Hard Rocks . .' Two Fathom Shoal. Dry Sand Sea-Horse Reef Half-Moon Breakers • Bank . Paraquas, 5 or 6 leagues from Palawan Euphrates Shoal Kirton's Shoals. Louisa's Breakers . . . Mantannane Isles Barton's Shoals Royal Charlotte's Rocks. Sands. Swallow or Investigator's Rocks Viper's Bank Breakers Ardasier's large coral flats and Taps, — W. p. (Walpole, Corn wallis and A.) — N. E. p. (Walpole and A.) — E. p. (Ardasier) . . — S. p. (Pennsylvania and A.) Gloucester Shoal Stag's Shoal Prince of Wales's Bank, limits London Breakers Reef, western — — Reef, eastern .... Breakers Breakers Ganges Breakers. . . . Investigator's Shoal, W.p 9 56 5 24 5 5 4 57 5 35 8 46 10 57 9 5 5 5 6 6 6 6 10 47 7 23 7 3o 8 00 7 56 7 54 7 4o 7 3o 7 5o 8 24 8 5 8 i3 9 36 8 55 8 48 7 33 7 25 9 25 10 3o 5 Lons- D. M. I i3 44 E ii4 59 117 4,i 117 53 ii3 6 ii3 5 116 54 116 42 116 42 116 45 ii3 A^ ii5 8 117 19 111 4 112 3o 112 24 U2 3o 112 28 1 16 3o 117 53 117 28 ii3 3o ii3 i5 ii3 2 ii3 18 116 7 116 i3 ii3 38 ii4 29 ii3 49 ii5 o ii5 25 ii3 12 ii4 24 ii4 47 ii4 34 ii4 14 112 57 no 27 no 34 n2 26 n2 00 112 24 A 14 ii3 n3 3 ii4 10 n5 ID n4 35 Investigator's Shoal, Shoal Shoal , Coral Rocks. Cavallo Marino's Shoal ) Black Rocks Bank White Sand Low Black Island . . Friendship's Shoal .... Hardwicke's Reef* (or Dolphin's) , Breakers * (ditto) . , Royal Captain's Shoal... Bombay's Shoal , Dolphin's Reef* (or Hard wicke's) Breakers * Breakers * (ditto) . , Great Reef, N.p.*.. Long Island * Breakers * First Island* Ledge * ■ Breakers * Breakers * Falmouth's (or Essex) low Island* Bank, or Gossard's Bank, Essex (or Falmouth) low Island* Gossard's Reef (or Mid- djeburgh R.) Small Island Lat. Cornwallis Breakers Sabut Jung low Island Bank ■\ Gaspar Shoals South Sea Castle's Sandy Islands and dangers, limits (by Lieut. Ross) Two Islands An Island (Investigator) An Island, ditto . . . . A Reef Discovery's Reef. . York Breakers, W. (Viper's) Pennsylvania , p-^ V Krpn rpr« _ _ ^ -z ' D. M. 8 10: 9 12 10 ^/i 9 4o 9 42 5 54 8 3i 9 39 5 52 6 00 9 54 10 2 9 4 9 27 9 59 9 45 o 8 o 7 o 17 o 22 o 35 o 4o 4Q 1 10 58 1 25 I 2 8 58 o 42 00 8 52 1 32 I 34 I 36 I 29 I 21 I 27 I 8 o A^ o i5 o 00 o 8 9 55 8 17 8 5o 8 58 9 4 10 00 9 48 9 32 9 49 9 52 10 25 10 52 Long. \). M. n4 5i £ ! 16 32 ri4 U n3 4 n3 i5 \\4 18 1 14 21 n4 58 n5 7 n5 i3 n5 17 n2 34 n2 49 n2 17 n2 12 116 40 1 16 57 n2 17 n 2 3o n2 i5 n2 9 n2 35 n2 3i n2 38 n2 47 n2 47 112 54 112 4o i\A i3 n2 4o in 5 ii3 26 1 14 22 ii4 12 ii3 29 ii3 5i n3 5i ii4 20 1 14 16 wi^i 22 wA 18 1 14 26 11 3 40 n3 5o 1 17 55 ii4 43 ii5 17 ii5 21 ii5 20 n5 12 ii4 4o 6 28 116 47 116 48 116 37 116 55 The longitudes of these places ought probably to be increased. TABLE LIV. Latitudes and Longitudes Page 367 XLIV. Islands and Shoals between Batavia and J\ew Guinea, South of the Celebes. Carimon Java, W. ex. . . . Lubeck or Babian Island, Arrogant's Shoal , Madura Island, N. W. p., N. E. p. , Pondy Island Great Solombo Island, (hill on S. E. p.) Little Solombo Island . . . Arentes Island Little Pulo Laut, (middle) Four Brothers, sunken Islands Urk Island.. . . Kansrelang or Cangayang Island, N. p S.p S. E. Island, or Hast- ings's Island Kalkoon Islands, north- ern, about . Four small islands, middle Great Paternoster Islands W. p S. W. Island . . S. Island Two low Islands E.p Postilion's Islands. N. W. p... ■ Eastern Island S. post Nocsa Sera Islands. Noesa Comba, about Sd. Bank ofF Noesa Comba Caloeoliij or Rotterdam Island . Hen and Chickens, S. p.. Zalinaff, or Saflanaff, or Lacr's Island Coral Bank ofF ditto. S.p .... ditto E. p., ditto W. p Five Fatlioms Bank Tonyn Islands, S. W. Isl. E. Island D. M. 5 5oS 5 49 5 12 6 53 6 53 7 I Siioal Taiiakeka or Tunikik Isl. Brill Shoal, N. p . S. p. Mansfield Shoal Middle Island Salayer Island, N. p. . . . , Cambyna Island, S. p.. . Peak South Island Hegadis Island Bouton Island, S.p Town N. E. point , Calansoese Harbor Lat. 5 33 5 21 5 lo 4 5i 6 54 i5 32 34 36 42 32 45 58 2 i5 52 28 Loner. D. 5 3i 5 54 45 i3 5 42 5 27 4 23 4 55 E. point I 5 i5 M. o 3E 2 48 3 00 2 45 3 58 4 4 4 24 4 25 4 32 5 53 4 5o 5 16 5 17 5 25 6 II 5 46 5 5o 7 00 7 16 7 3o 7 55 8 3o 8 ^6 9 i5 8 56 7 9 7 9 7 10, 7 38 7 54 8 M 8 26 7 58 8 20 8 36 8 5o 5 24 9 9 9 2 9 o 20 17 20 28 20 28 21 57 22 28 22 4o 22 4i 22 48 23 4 23 II 23 i5 Lat. D. M. Token Bessy's Islands, — Wangiwangi,N.VV. Isl. — Pinnunko, S. lim — Velthoens or Koko C Island \ St. Matthew's Islands, (middle) Mamalakjee Island, (N.W. Tonin Island,) Scliiedam Islands, N. W. S. E.. Shoal. Kalatoa Island Alfred's Shoal Jagger's Reef, or Banga- lore Shoal, about another estimate. Angelica's Shoal another estimate . . Rusa Raji or Lusardy Isl. Rusa Linguette or Rosa- galet Island Tiie Three Bastards Bally Island, — Table Point, or S. p. . . — Volcano — N. E.p Bally Straits, S. entrance A shoal near the anchor- age at Balariibuing, bears S. W. h, W. from the flagstaff, distant % mile from shore. Mynder's Rocks Banditti Island, S. E. pt. Lombock Isl., S. p. about Peak, near N. E. p.. N. E. pt Lombock Island, Isles near N. W. p.. Ampannan River, entrance Loboagee or Bally Town ■. Selonda Island Pulo Majo or Mayo, N. p. Flat Island Sandbuy's Four Shoals limits Sumbava Island, S. W. p Timor Yung Island, (off N. W. p ) . . . . Sumbava Bay Tumbora Mountain. Biema Bay, rugged point ditto, rocky point . . . Sapy Bay, anchorage S. E. Point Goonong Apee Isl. Peak. Comodo Island Flores or Mangerj'e Isl- and, S. W. p. about — S. p. about — Lobetobie Volcano. — N. p. Flores Head, Iron Cape 18 S 5 58 5 18 4i I 12 27 12 9 7 40 7 55 7 4o 8 17 8 5 8 i4 8 5o 8 21 8 18 Lens. 8 5o 7 4i 8 5i 8 5o 8 26 8 19 8 i3 8 33 8 42 8 8 8 7 8 7 7 42 7 56 9 2 8 21 8 27 8 i5 8 II 8 8 8 3o 8 42 8 i; 8 22 8 48 9 00 8 35 8 I D. M. 123 33 E 123 56 124 43 124 16 120 1 4 120 28 120 56 121 i3 121 43 121 39 121 13 121 46 121 25 122 18 121 36 122 3 122 4i ii5 2 ii5 27 ii5 43 ii4 40 ii4 22 ii5 29 116 00 116 26 1x6 4^ II 5 59 6 33 6 57 7 34 7 55 8 4i 8 36 93 i4 5 37 54 3o I'age 368] TABLE LIV. Latitudes and Longitudes. Straits of Florcs,S. ent. Id. Sandal Wood Isl., N. p. . BlufFor W. p S. extremity E. end Padewawy or Bar- ing's Bay Savu Island, W. pt New Island, S. pt Lat. Polo Comba or Cambay Lomlilen Island Peak (on N. W. p.) E.p P-intnr Island, N. E. p East Island, Strait of Aloo Middle Island, ditto... Oinbay or Mallao Island, N. W. p E.p RottoorRottelsl., S.W. p. — Booca Bay, on S. side. Timor Island, S. W. p. .. — Copang,FortConcordia — Peak , — N. W. point — Tulycaon Bay — Batto-gady — point nearest Ombay — Dilly, or Diely — E. end Pulo Batto Pulo Cambing or Passage Island, S. p N. p. . . . Wetter Island, E.p Pulo Baby, near S.W. p Goonong apy or Burning Island Dog Island Kisser Island Pulo Jackee, or Noosa Nessing Lettee Island, W. p Roina Island Lucapin-ha}"- or Lucepera Island Turtle Islands, eastern . . Cerowa Island, about. . . . Babber Island, about .... Timor Laut, S. & W. end Arroe Island, S. extr.... D. M. 8 38S 9 i5 9 42 10 22 10 OO 9 4o lO 32 10 49 7 49 8 12 8 i4 8 10 8 20 8 23 8 9 8 i5 11 2 TO 53 10 23 10 9 9 4i 9 24 9 12 8 57 8 39 8 33 8 2T 9 i4 Lons. 8 II 7 46 8 o5 6 35 7 4[ 8 6 8 21 8 i4 7 42 5 40 5 25 6 10 7 25 8 27 9 00 D. M. 122 58 E 19 00 20 35 20 5 1 20 18 21 35 21 10 23 38 23 47 23 35 24 20 24 00 23 55 24 27 25 i5 22 55 23 5 23 3o 23 35 24 1 1 23 55 24 23 24 5o 25 i3 25 34 27 1 3 23 52 25 29 25 43 26 54 126 4o 125 56 127 7 127 i3 127 4o 127 26 127 21 127 38 129 53 i3o 40 i3i 7 1 35 00 XLV. Borneo, Celebes, Luconia, wilh the adjacent Islands and Shoals, as far East as JYew Guinea. Tanjong Sambar, S. W. p Succadana Tanjong Factie Pontiana or Lewa R., ent, Point Mampava Slackoo Road River Sambas, entrance . Lat. I i3 Lons- X M. D M 1 53 S 109 58 E I 16 1 10 I 16 109 35 2N 109 10 17 109 GO 108 58 Tanjong Apee Tanjontr Datoo BORNEO Read Pulo Teega Abai Harbor Keeney Balloo Mountain Tanjonc Sampanmangio, N. p.° Point Unasang Point Kanneeoongan River Passier, entrance . . Ragged Point Shoal Point Point Salatan, S. p Lat. D. M. I 58 N Lonff. Point Layk, S. W. p... macassar: Town, fort Cape Mandhar Cape William Cape Temoel or Samsa S-P N. W. p Cape Donda Cape Rivers Manado, fort , Cape Coffin , Isle Banca, E. pt Kema Village Castican Baj' Goonong Telia River Cape Talabo, E. pt. , . Weywongy Island, about Waxway Island, middle Cambyna Island Peak. . Middle Island Boele-comba Hill Waller's Shoals and C Laurel Rocks, limits ( Noesa Sera Islands Noesa Comba Shoal off Noesa Comba. . Little Pulo Laut Isl., mid. Moresses or Manevessa Island Dwaalder Island Royal George Shoal Two Brothers Great Pulo Laut, N. E. p — N. p — S. Isl. off the S. E.p.. The Three Alike Islands Dry Sand Bank Triangle Islands, middle Little Paternosters, S. p. N. E. p. N. W. p. Pamaroong or Dondrekin Island, S. p Seven Islands Banguey Peak Balanibang Isl., N. Harb. Balabac Island, (hill,). Mangsee Islands* St. Michael's Islands, (Bangcawang,) Toob-Bataha Slioal, S. extr 3 00 5 00 5 45 6 21 6 8 7 5 I I 17 3 I 44 s 2 10 2 33 4 10 5 37 5 9 3 35 2 37 o iN 48 1 20 I 29 1 42 I 43 I 19 o 48 o 28 o 55S 4 3 3 34 5 21 5 40 5 33 4 3o 4 37 5 2 5 i5 5 26 4 5i 4 25 4 12 4 17 4 26 3 23 D. [09 1 10 5 ii5 6 116 116 119 118 116 116 ii6 ii4 119 M. 19E 36 00 35 20 36 46 2 5o 26 33 25 42 25 i3 5 39 37 3 5o o 5i o 32 7 19N 7 16 7 59 7 32 7 48 8 00 iig 23 119 2 1x8 5o 119 37 119 57 120 45 124 5o 125 II 125 12 125 4 125 00 123 o 123 3o 123 i4 121 57 120 28 120 9 117 7 1 17 i5 117 9 117 9 1 17 o ii5 53 ii5 5o 116 5 1 16 i4 1 16 II 1 16 20 116 II 116 37 117 48 117 53 117 M 117 28 117 33 119 4o 117 6 116 58 116 56 117 19 118 ^G 119 5o * See Table on pnse 451 TABLE LIV. Latitudes and Longitudes. [Page 3G9 Palawan, W. end N.p.... Ragged Island Cagayan Soolo Soolo Island, town*. Takoot Paboonoowan Shoal Pangootaran Island .... Belawn Island, E. p. . , . Tapeantana Island, E. p. Taniook Island , Mataha Island, S. p. . . . . Peelas Island, N. p , Ballook Ballook , Basilan Island, E. p Santa Cruz Island Sangboj's or Hare's Lips, Teynga Island, N. ex. . . . Catanduanes Island, S. p. Cape del Espiritu Santo. N. E. p. Saniur Island. St. Bernardino Island . . . . Ticao Island, Port St. Ja- cinto Lat. D. M 8 24N II 3o II i5 7 00 6 3 6 i5 6 i5 6 00 i4 28 32 4i 47 4i 5o 46 Manilla Cavite Entrance Manilla Bay. Point Caponcs Two Sisters Islands. . . Point Boliano Cape Bajador Point Cavnaion Cape Enganno Mauban Cape St. Ildefonso. . . . Samboongan Point Balagonan Suriago Village, near N point Cape St. Augustine, S. E. p South Point Mindanao Negros, S. point Point Sojoton. Cagayancs Islands, middle Panay Island, Point Na- sog, S. p - Aslomau village — Point Potob or N.p... Dry Sand Bank Sombrero Rock White Rock Cuyos Islands, — Quiniluban (Northern Island) W. ex — Grand Cuyo — Southern Island Caravos or Buffalos Betsey's Bank, .5 fatlioms Ylin "Islands, S. p., off S.p. Mindoro Coral Shoal, W. of ditto, about 6 6 6 6 6 6 6 6 6 52 i3 38 12 34 12 46 12 34 i4 36 i4 20 i4 28 i4 52 1 5 5o 16 27 18 42 18 48 18 39 i4 8 1 5 27 6 55 7 5i 9 47 6 4 5 39 7 10 9 5o 9 M 10 25 10 32 n A(J 11 24 10 45 10 28 II 3o 10 52 10 4o 11 53 11 42 12 9 12 II Lone D. M. 17 141 19 37 19 21 18 28 21 32 20 40 22 8 22 8 21 56 21 5o 21 45 21 5o 22 17 22 12 21 3o 21 27 24 2 25 i5 24 14 23 46 20 55 20 3 19 49 19 5i 21 00 21 i4 22 16 21 44 21 46 22 8 22 6 25 25 26 i3 25 18 24 35 22 56 22 24 21 23 22 6 22 6 21 56 21 34 21 i5 21 5 20 47 21 i5 21 i3 21 48 20 57 21 i5 20 57 Apo Bank, S. p -E.p — N.p — S. W. p. Islet — West or Great Islet.. — Discovery Bank Coron Island, Is.off N.E.p, Green Island Haycock Pinnacle Rock N. W. Rock Sail Rock Busvagnon Island, N. p.. Calavite or High Island. . Group of Islands, S.p..,. -N. p... Turret Island North Rock Mindoro Island, S. p Point Dongan or Pandan Point Calavite Luban, N. pt Goat island Babuyan Islands, Lapurip or Daluperi Island Fuga or New Babu- yan Island Caniiguin Island. .. Guinapac Rocks . . . Didicas Rocks Claro (or Old) Ba- buyan - — Calayan Island Bashee Islands, Balintang or Rich niond Isles, N. cue Sabtang Island, S. pt Bashee Island Goat Island . . Batan or Monmouth Island, S. p ditto Mount, N. p. . , Grafton or High Round Island. . . . Bayat or Orange Isl North Bashee, High Isl.. northernmost Isl Lat. !). M. 2 36] 2 4o 2 45 2 4o 2 39 2 4o 1 59 2 3 2 9 2 18 2 23 2 17 2 27 2 II 2 46 3 28 3 52 3 55 Gadd's Reef Cumbrian's Reef, doubt- ful ; probably the same as Gadd's Reef Little Botel Tobago Xima Botel Tobago Xima Vele-rete Rocks Formosa Island, South Cape Gomano Island Lissamatula Isl., S. E. p Xulla Bessey, S. & E. p. . N. E. p.... N. W. p... Xulla Mangola, W. end . Greyhound Straits .... 5 9 i5 9 I 9 o 9 5 9 12 9 37 9 28 9 58 20 17 20 1 4 20 21 20 17 20 28 20 4i 20 47 21 3 21 9 21 43 21 35 56 5 21 42 21 56 I 46 S 1 46 2 28 I 58 I 58 1 43 I 4o I 56 I Lonff. D. M. 20 33 E 20 36 20 3i 20 29 20 28 20 43 20 36 19 49 19 5i 19 54 19 55 19 56 19 56 19 56 20 23 21 22 20 47 20 26 20 8 20 3 21 10 21 20 21 53 22 5 22 12 22 12 21 46 122 12 121 53 122 9 121 48 122 122 121 57 121 53 121 57 121 59 121 4i 121 43 121 4i 121 38 120 49 120 56' 127 27 126 27 126 7 125 48 125 21 124 3o * See Table on oace 451. Page 370 TABLE LIV. Latitudes and Longitudes- Haycock Island, off S. W. p. Xulla Talaybo Skelton's Island, on N.W. p. ditto Middle Island Albion's Island Bouro Island, N. W. p... N. extr — N.E. p Cajeli on Bouro Bay. S. point Amblaw Island Manipa Island, E. pt Bonoa Island, about Ceram Island, Seeal, or S. W. p Kessing, or E. p Waroo Bay Old Lamata or Flat P. Sawa Bay Leeuwarden Island, S. pt. Shoal Goram Island Matlabella Islands AMBOYNA Island, Fort Victoria Noesa Laut Island, E. pt. Banda Island, anchorage . Lookisong or Landscape Island, S. p Pulo Gasses, S. p Kekik Pulo Pisang Horsburg's Rocks Boo Islands Weeda Islands Kanary Islands, Grand K. Ef be Harbor Pulo Popo, S. E.p Battanta Island, Cape Cambo, W. p. Fisher's Island VVaygecooe Island, S. E. p., or Point Pigot Offiik Harbor Boni Road Amsterdam Island Fow or Faux Island Gagy Island, N. pt Geby Island, N. W. end. S3'ang Island E3'e Island -^ Islet E. of PuloMoar Catharine's Islands Canton Packet Shoal 'Ornisbce's Shoal, N. pt. . Ditto soundings, 15 fath.. Yowl or Aiou Islands, — Aiou, the largest Isle . — N. W. Island — N. E. Island — Reef N. part Asia's Islands, S. W. Isle N. E. Island... Gillolo Island, N. end ... — Ossa village — Maba village — Islet near Pulo Moar. . Lat. D. M. I 47 S I 45 I 45 1 53 3 4 3 2 3 49 3 22 3 54 3 49 3 24 3 00 3 33 3 55 3 25 2 53 2 5l 3 20 2 56 CO 3o 4i 42 3i 39 4i 33 23 32 56 56 19 6 24 2N 22 23 35 46 42 38 36 4i o4 4 23 45 53 9 Lon^. D. 31. 124 24 E 24 36 24 28 24 19 25 57 27 10 27 6 26 37 27 10 27 4o 27 56 27 5i 3i 10 3o 45 29 42 29 6 30 48 3o 43 3i A^ 3i 47 28 10 28 49 3o 00 28 4 28 i5 28 35 28 53 28 20 29 20 28 35 29 42 29 52 30 25 3o 23 3i 18 3o 43 3i 5 32 9 29 28 29 53 29 19 29 55 29 5i 28 58 29 II 28 55 30 4 3o 3 3i o 3i 8 3i i5 3i 21 3i 23 28 22 28 58 Gillolo Island, point en- trance Straits Patientia — Cocoa-nut Point,or S.p. Batchian Island, S. E. p.. Amsterdam Island Kayo or Cayo Island, S. p. N.p. Negory Kalam, N. p Wolf Rock Tidore Island, S. extr. .. Mountain N. E. end Ternate Island Tyfore Island Meyo Island Morty or Mortay Island, (N. cape,) Bangay Island, peak Tagalondo Bejaren Island, peak. . . . Siao Island, S. point .... peak Sangir Island, S. end. . . . — Watering place on the W. side — N. end Glatton's Rock Sallibobo or Toulor Isls. Kabruang, S. p. . . . Tulour or Karka- lang, N. p Meangis or Menangus Isl. Serangi Islands, S.p peak on W. Island . N.p Lat. D. M. i3S 5i 48 20 I 7N 28 20 34 40 A& 49 2 M 1 52 2 23 2 6 2 4o 2 43 3 21 3 47 4 28 5 00 5 20 5 3i Loner. D. M. 127 45 E 128 22 128 3 127 53 127 23 127 37 127 9 127 24 127 25 127 34 127 19 [26 12 126 25 128 25 125 24 125 36 125 25 125 35 125 35 125 46 125 44 125 44 126 4 127 o 126 55 127 17 125 35 125 32 125 43 XL VI. The Coast from CAJVTOJVto KAMTSKATKJl, with the adjacent Islands and Shoals. CANTON Mir's Bay Paint Single Island, or Chueng Chow Mendoza's Island Fokoi Point Pedro Branco Point Chelang Point Tongmi , Point Cup-chi A black conical Mount . . Breaker Point Cape of Good Hope Fort Island Lamo or Namolsl., W. p N. pt.... Lamock Island, S. W. Rk. The Brothers, southern.. Chapel Island Amoy .Chin Chew Bay, mid Lamyet Islands, E. Peak. Ting-hae harbor Hiesham Oroup Saddle Id Quesan Isl's Patahecock . . Lat. M 7N 27 24 3i 33 18.5 39 44 52 56 14 25 26 29 II 32 10 28 Lone D. U. 3 i4E 4 3o 7 » 7 i4 7 42 8 i4 8 4 8 47 9 35 9 5o 22°l4 22 i4 * White Doss S. P. 255 57 120° 01 TABLE LIV. Latitudes and Longitudes. [Page 371 Clmsan Island, Tingbae. . Saddle Group Is; Is Chiu-san Island Amherst Eoclis bliaug-TuDg Prom. S. p. — N.p Cape Zeu-ou-Tau Ten-choo-Foo City Tchoo-san Island Keusen Islands, northern Pekin River, anchorage at Peiho Entrance Alceste Island, S. W. extr. Corea Cape Clouard Sanpon Ternai Bay Suffren Bay Cape Lesseps Castries' Bay Vanjuas Point Bay de Langlc B*ay d'Estaing Monneron Island La Dangereuse Rock. Lat. M iN 5o 25 lO 00 25 36 48 38 58 Cape Crillon, (entrance Perouse's Straits,) .... Cape Aniwa Cape Lowenorn Bay MordwinofF Cape Tonyn Point Siniavin Mount Spenberg or Ber nizet Point MulofTsky Cape Alexander Dalryin- pie Ca])e Soinsonoff River Nova, entrance .... Gulf Patience, N. p Robber Island Reef, N. E. p — S. W. p Cape Patience Cape BiUinghausen Mount Tiara Cape RatmanofF Cape Croyere Downs Point Slioal VVurst Point Cape Klokatschef Cape Lowenstern Cape Elizabeth North Bay Cape Maria Espenberg Peak Cape Golowgtscheff Cape Romberg Cape ChavarofF Jonas Island Ochotsk Yamsk Bolcheretsk C. Lopatka, Kamtskatka St. Peter and St. Paul... 45 54 46 2 46 23 46 48 46 5o 4? i6 47 33 47 58 21 48 52 49 i5 49 19 48 36 48 28 52 49 35 50 3 50 48 5 1 00 5i 53 52 3o 52 57 53 40 54 3 54 24 54 16 54 17 54 4 53 32 53 26 53 38 56 25 59 20 60 46 52 54 5i 2 53 o Lonrr. D. M. 22 6E 22 4i 22 21 22 22 22 4i 22 45 21 28 20 4o 21 2 20 43 17 48 25 21 29 45 28 55 36 00 38 44 4i 3o 4i o 42 42 4i 57 4i 27 4i II 42 9 4i 58 43 3o 43 4o 43 i4 43 33 43 00 42 20 42 44 42 5o 43 2 43 2 44 33 44 lo 44 46 44 26 43 37 43 53 43 43 43 i3 43 29 43 18 43 7 43 i3 42 47 42 37 42 18 42 5o 4i 55 4i 45 4i 26 43 16 43 12 54 3o 56 5o 56 46 58 44 Shipunskey-noss, Cape . . Nisjui Kamtskatka Cape Tschulkolskoi East Cape Cape Serdze Kamen.... North Cape Formosa Island, S. cape N. W. point. N. point N. E. point. . Lamay Island Pehoe or Pescadore Isles, Southern limit High Isl., S.W. limit Pachan Island Northern limit Treble I. S.E.p ditto, nine feet reef . Pat-chow or Madjicose- mah Islands, Southernmost Island Bluff Point, W. ext. Great Island Kumi Island Eastern Island, Ty- pin-san Providence Reef. . . Lew Chew Islands, Great Lew ) From Chew, f To ditto, adjacent Isl- and, N. p Western Island .... Hoapinsu Island Ty-ao-yu-su Island Sulphur Islet Island Ousima Lat. D. M. 53 6 N 56 16 64 i3 66 6 67 12 68 56 21 56 35 II 25 18 25 I 1 22 19 23 1 I 23 19 23 32 23 47 23 3i 23 28 Group of seven Islands, limits Pinnacle Islands Ormsbee's Peak A Rock South Island Gotto Island, S. end .... Ashes' Ears Quelpacrt Island, S. p... Kiusiu Island, Cape TschirikofT. . . Cape Danville Cape NagaefF. Mount Schubert . , . Mount Horner, peak CapeTschitschagofF, S- p CapcTschesma,W.p. Cape Kagul, N. p. . MountUnga,voIcano Nangasaky harbor, entrance Cape Nomo, S. p. of Bay Nan Cape Seurote Sanao-sima Island, N. p.. S. p.. Tenegasima Isl., (middle) 24 6 24 17 34 25 24 42 25 6 26 o3 26 53 27 M 26 20 25 47 25 57 27 5 1 24 48 28 16 29 25 30 06 29 52 29 4o 30 45 3 1 3o 32 35 32 3 33 8 3i 4t 3i 9 3o 57 24 3i 42 3i 43 32 44 32 35 32 58 3o 42 3o 24 3o 23 Lons- D. M. 60 4 E 62 00 71 24W 69 4o 71 49 79 57 20 56 E 21 6 21 34 21 56 20 27 19 23 19 16 19 26 19 32 19 39 19 4i 23 52 23 45 23 00 25 29 25 6 27 34 28 25 27 17 23 29 23 40 28 1 4 4i 20 so 21 29 38 30 04 29 52 4o 20 23 46 4o 00 28 44 28 37 26 19 3i 4i 3i 27 3i II 3i 12 30 28 3o 36 3o 2 3o 7 3o i4 29 46 29 42 29 35 3i 00 |i3o So Page 372] TABLE LIV. Laiitudes and Longitudes. Volcano Island Seriphos Island Apollo Island Julie Island St. Claire Island Symplegados Islands, i\. E.p — S. W. p Meac-Sima Isl., S. W. p. ,— N. E. p. Nadeshda Rocks Tsus Island, S. end Cape Fida-Buen gono Lat. N. p. Colnett's Island Dagelet Island, N. E. p.. Niphon Island, SP Cape Noto A Rock Jootsi-Sima ' Jedo Cape Kennis Zach's Mountain . . . Russian's Promonto- ry, S. p N. E. p Town Cape Gamally Peak Tilesius. ..... Cape Greig Cape Sangar, (ent. Straits of Sangar,) .... Osima Island Kosima Island Okosir Island, (middle,) . Jesso Island, Cape Nadeshda, (ent. of Straits of Sangar,) . . Cape Sineko Matzumay Town . . . Cape Oota-Nizawu. Cape KutusofF Cape Rayten Cape Okaraay, S. p. Cape Taka-sima Mount Rumoifsky . Cape Malespina. . . . Cape SchischkofF . . Pallas Mountain . . . Cape Romanzoff, N. p Cape Soya ■ Cape Shaep Peak de Langle, Rios- chery Island Cape Guibert, Reifuns- chery, N. E. p Jeurire Island Janikesseri Island D. ftl. 3o 43 N 3o A'i 3o U 3o 27 30 45 3 1 3o 3i 26 3t 35 3i 49 3 1 .fa 34 6 M 1; 34 40 34 16 37 25 33 25 37 36 37 36 37 5i 35 40 37 10 35 25 39 46 40 00 4o 5o 4o 38 40 40 4i 9 41 16 I 3i 41 21 42 9 Staten Island, S. W. end Cape Vries, (Vries Straits,) Company's Island, Cape Slionten N. end 4i 25 4i 38 4i 32 42 18 42 38 42 57 ^3 II 43 21 f2 5o 43 42 ■i4 20 ^^ 00 45 26 45 3i 45 21 45 II 45 28 ^i\ 28 44 29 44 26 45 26 46 18 45 28 Loner. D. M. i3o 17 '. i3o 44 i3o 24 i3o 1 3 129 54 129 42 129 37 129 40 129 5i 129 33 .29 17 129 3o 129 29 129 56 i3o 56 1 35 47 i37 20 i36 5o i36 56 i4f> 00 i4i 3o i32 20 139 44 i4o 6 139 48 i4o II i4o 8 i4o i4 39 19 139 46 139 3o i4o 9 i33 53 i4o 4 139 46 i4o I i4o 16 i4o i3 i4o 3i i4i 1 1 i4i 18 i4i 37 i4i 54 i4i 34 i4i 5[ 42 12 i4i 12 i4i 4 i4i 17 i4i 22 147 28 149 43 i5o 58 i5i 20 Marikan Island, N. end S. end,(Bousole Sts.) SarytschefF Island Peak. Raakok Island Mussir Island ' Trap Rocks I Charamukatan Isl. Peak I Poromuschir Island, S. p. ' Peak Fuss, (S.W.p.) E. p Lat. D. M 47 10 N 46 46 48 6 48 8 48 16 48 36 49 8 50 o 5o i5 5o 28 Long. D. 31. i53 6E 152 32 1 53 12 i53 i5 i53 i5 i53 44 i54 39 i55 24 i55 10 i56 9 XLVII. NEW HOLLAND and the adjacent Islands and Slioals. Pedro Branco, (Rubrick,) South-west Cape Mew Stone South Cape Eddystone Sidmouth's Rock Tasman's Head D'Entrecasteaux's Chann Adventure Bay Frederick Henry Bay . . . Cape Pillar Oyster Bay St. Patrick's Head Cape Portland Port Dalrymple Circular Head Cape Grim, N. W. prom. West Cape, or Sandy Pt. Macquarie's Harbor Rocky Point Port Davey Lat. Ent. to Banks's Straits.. Furneaux Islands, Barren Isl., S.E. ext. Clarke's Isl., S. ext. N. Sister, near N. p. of Great Island Endeavor Rock Kent's Group Light The Pyramid Waterhouse Island Hunter's Islands, Black Pyramid. W. Albatross Isl., N. W. King's Island, C. Wickam W. ent. Bass's Straits . Cape Albany Otway Port Philip, "pt. Nepean.. Western Port, Phillip I. . Wilson's Promontory, S.ext Ram Head Cape Howe Cape Dromedary, Mt Jervis Bay, N. pt Rod Point Botany Bay, entrance, (Cape Banks,) Port Jackson, entrance.. M. 59 S 37 46 40 5i 46 3o 32 17 58 12 4o 42 4o 4o 4o 26 4o 36 39 38 39 37 39 3o 39 49 40 48 4o 28 4o 22 39 35 39 23 38 52 38 18 38 27 39 8 37 39 37 3o 36 16 35 7 34 29 34 02 33 5o Lonff. D. M. 47 43 E 46 o 46 3i 46 59 47 4 47 " 47 10 47 12 47 32 47 42 48 6 48 2 48 17 47 57 46 48 45 17 44 42 44 35 45 18 45 3r 45 59 48 20 48 3 1 48 01 47 6 47 21 47 16 47 37 44 2 2 44 39 43 53 43 4o 43 34 44 4o 45 18 46 24 49 45 50 o 5o 9 5o 58 5i i3 5i 18 TABLE LIV. Latitudes and Longitudes. f Page 373 Broken Bay Port Stephens, pt Cape Hawke Smoky Cape Solitary Islands .... Cape Byron Point Danger Slioals oti" ditto Cape Morton Shoal, Dry Eocks.. . . Sandy Cape Sli'l, 9 ft. rock (JronpCapricorn, N. AV. Is. Keppel I Barrier Reef, S. extreme Cape Townsend Cape Palmerston Cape Hillsborough Cape Conway Cape Gloucester. ....... Cape Cleveland Cape Sandwich Cape Grafton Cape Flattery Cape York New Year's Island Van Uieman's Cape Red Island, off P. Vulcan Minstrel's Shoal, N.W. pt. Greyhound Shoal Clarke's Reef, north of Rosemary Island Eastern Rosemary Island. N.E.p .' Western ditto, N. p Doubtful Shoal Piddinn-ton's Islands.... Shoal (land of N. Holland in sio-ht from the mast- head) North-west Cape Dirk Hartog's Road, eiit. to Sharks' Bay Houtinan's or Abrohlos Shoals, N. I Rottenest Island Cape Leuwen or S. W. Cape, S. pt Cape Chatham Cn pe Howe King George III, Harbor Point Hood, Ptocks off . . . Termination Island Endeavor, small island. . P'>rt Lincoln Nepean Bay Cape Jaffa C. Nortbuniberland. Eocks off iMt. M. 34 6 4i 1 5 56 i3 56 38 7 3 56 36 i8 II 23 i4 3o 5/i 20 .)4 20 32 20 20 17 26 35 37 36 i5 5o 25 22 Lons- D. M. i5i 20 E l52 14 l52 3i i53 5 i53 18 1 53 38 1 53 3i i53 39 i53 27 i53 3i 1 53 22 i5i 43 i5i 08 I 52 37 i5o 29 1 49 28 1 49 06 i49 i48 25 i46 59 1 4b 19 i45 57 145 i6 142 33 1 33 o3 i3o 18 124 18 119 10 ii4 40 1 15 4o 112 25 ii4 56 ii4 o4 1 12 57 ii3 35 n5 3i ii5 6 116 25 117 38 118 2 119 33 121 58 127 2 i35 55 i37 55 139 4o i4o 4i XL VI II. Islmids, Rocks, and Shoals, in the JVORTH PACIFIC OCE.IjY. I Aleootskia Islands, I westernmost, W.p. I Ounalashka Lat. D. M. Lons- D. M. 52 52 N 173 24 E 53 54 166 32W Bank (G4 fathoms) Rica de Plata or Crespo. . Reef Island Weeks's Reef, 36' N. E and S. W Island Ganges Island Bank of Soundings Island Island Island Island Island Roca de Oro Island, Rica de Oro Island Island Island Calunas Island ditto (another account) Island Patrocinio Island Disa])])ointment Island .. St. Juan Bassiosos Island Island Reef Copper island Tree Island Laskcr's Island Island Island Reef Bishop's Rock North Island Island Grampus Island Sulphur Island Kendnck's Rock Marcus Island Weeks's Island Dextcr's Island Island Reef Jardines Parel or Peru Island . Abregoes Shoal Reef Douglas Reef Lamira Island Island Bishop's Rock Weeks's or Wilson's Isl. Reef Halcyon Island Folcrer's Island Reef Tarquin Island Reef Island Lat. Guy Rock Urracas, about Assumption Island. Almagan Island . . . Bird Island Tinian D. M. Z3 22 N 32 44 32 00 3 1 3o 3i i5 3 1 00 3o 45 3o 5o 3o 00 3o 00 3o 00 3o 00 3o 00 29 54 29 25 29 33 29 3o 29 35 28 55 28 53 28 3o 28 10 27 14 27 3o 25 58 26 6 26 3 20 o3 26 00 26 o3 25 53 25 42 25 3o 25 22 25 i4 25 12 25 10 24 48 24 35 24 18 24 00 23 24 23 3 6 4o 20 42 20 32 o 20 20 3o 20 16 19 II 19 28 19 6 18 22 17 9 17 00 7 36 6 oc 20 3o 20 10 19 4i 18 5 16 I i5 00 Lonsr. D. M. 78 3oE 70 8 47 00 40 00 53 9 47 6 54 25 77 3o 37 00 39 00 4 1 3o 43 00 44 24 57 3 65 55 37 00 43 00 l4 4i 58 00 62 00 76 5o 75 48 40 57 42 48 73 3i 54 36 60 00 3i 48 45 44 73 42 3i 17 3i i3 52 5o 32 00 4i i4 3i 36 46 4o 4 1 20 34 00 53 42 54 00 62 58 62 57 42 28 5i 35 4 1 40 36 43 53 00 36 6 64 1 5 52 5o 36 53 66 55 66 29 63 33 55 i5 56 i3 60 00 69 3o 71 42 45 3o 45 25 45 27 45 54 46 o3 45 37 Page 3741 TABLE LIV. Latitudes and Longitudes. Guam, Umatac Bay. Radack chain of islands viz. : — Aour, circular group of 32 islands, extending 13 miles N. W. and S. E., anchorage , Kaven group, 33 miles N. W. and S. E. — Araksheef Island, (largest island,). . — Southern Island. . . . Chatham, circular group of islands, N. W. and S. E. 24 mile Eregup Chatham Is.circular group of Co islands, E. and W. 30 miles, and 10 miles wide, enclosing a sea 12 miles wide and 2< miles long. — Otdia Island, eastern (anchorage,) Legiep or Hayden group Ailou group, 15 mile long, 5 miles wide, — Krusenstern Capenius Island, (northern,) Isle Du Nouvel An ... KutosofForUdirick group, separated by a channel from a southern group called Souvoroff or Ta gay, extending N. and S. 25 miles, — Channel Group south of KutosofF, Mille Medjuro Arno Bigar, north of KutosofF. Pescadores Isls,* eastern, western. Ralick chain of islands extend nearly N. and S. about one degree west of the Radack chain, viz. : — Ebon group — Noamureck Island . Kuli group Helut group Odia group Namou group Litel Island Tebot Island Quadelon group Oudia-Milai group.... Radogala group Bigini (northern) Johannes tiion's Island St. Andrew's Island . . . Pulo Anna Lat. D. M. i3 17N 19 54 29 27 25 23 8 Loner. D. M. i44 4oE 171 12 170 49 171 II 170 4 5 5o 167 i5 5 3o 6 4o 7 3o 8 i5 9 00 8 55 8 3o 9 20 10 45 II 00 II 20 167 i5 6 55 l32 3o 5 16 l32 i3 5 20 l32 16 4 38 l32 3 170 iG 169 i3 170 00 170 55 169 5o 170 07 167 37 167 22 Pulo Mariere Lord North's Island. . . Ganges Shoal, S. W. p N. E. p. Helen's Shoal Freewill or St. David's Islands, limits Pelew Islands, — Baubelthouap, E. p. . — Northernmost, Kyan gle , — Large Reef, part dry, — Southernmost, Angour Matelotes, N. extr. Southernmost Yap or Hunter's Isl., N. p -S.p Philip Islands . Thirteen Islands, S.W. ex. Haweis's Island Strong's Island Islands Islands Islands Islands Hope's Islands Baring's Islands* Teyoa Island Providence Islands Ditto Brown's Range, Arthur's Island, N.. Parry's Island, S. . . Margaret's Island ....... Lydia's Island Catharine's Island Arrecife's Island Mosquito group, loi and dangerous Peterson's Island .... Chatham Island, E. pt. . . Reef Calvert's Islands, S. extr. Ibbetson's Islands Elmore Islands, S.I. .... Mulgrave's Islands, mid'le Banham's Island,* E. jit.. Cook's Island Hall's Island! Reef Pitt's Island * N, pt Matthew's Island* Simpson's Island Macasgill's Islands St. Bartholomew Cornwallis or Smyth's Isles Wake's Island* Lamira, \V. pt Gaspar Island Gaspar Rico Island Wake's Rocks St. Peter Barbadoes Krusenstern's Rock Necker Islandt French Fritrate's Shoalt. Lat. D. M. D. M. 4 19N i32 28 E 3 3 i3i 4 2 52 i3i 7 3 6 i3i 23 3 i3i 55 49 i34 17 I 2 1 34 3o 7 4i 8 8 8 18 6 53 8 4i 8 19 9 4o 9 25 8 6 7 18 7 3o 5 12 5 28 5 47 II 4o 1 1 19 8 52 9 4 9 i4 9 3i 7 46 8 10 8 54 9 3o 10 00 8 3o 7 i5 6 16 6 01 I 18' 5i 1 00 3 20 2 3 o 3o 6 12 i5 10 16 5o 19 17 20 20 i4 55 i4 42 17 20 II i3 8 54 22 II 23 34 23 45 Long. 1 34 58 1 34 5o i34 4i 1 34 21 1 37 40 i37 33 i38 I i4o 52 143 53 1 46 28 162 58 1 53 24 1 57 42 160 5i 159 12 i65 9 168 26 162 29 160 58 162 i5 162 25 166 i5 i65 58 166 2 161 8 168 23 168 00 166 35 1 170 i5 179 21 171 II 171 8 168 45 1 171 55 169 48 171 57 173 4 179 34 172 57 •73 26 173 53 160 48 1 1 63 53 169 4o 166 32 164 i5 176 20 169 3 172 40 179 ooW 178 00 175 42 164 43 1 65 59 * See Table on page 451. t See Table on page 450 TABLE LIV. Latitudes and Lonaritudes. [Page 375 Lisiansky's Island . . Owhyhee, N. point* E. point . S. point.. • KarakakoaBay Mowee, E. point*. . , S. point. . . . W. point. . , Talioorowa Ranai, S. point Morotoi, W. point . Woaiioo* N. pt Attoi, Whymoa Bay Talioora Oneelieow Oreehoua Bird's Island* Gardner's Island, discov- ered 1820 Maro's Reef, ditto . Gallego Island .... Cliristmas or Noel Island Sidney or Tanning's Isl Island New York Island* Cocoss Islands, or Chat- ham Bay Palmyra Island, S. pt. . Island Barbary Island Reef Clipperton's low Island Manuel Rodriguez .... Island Island Shoal Slioal Cluster of Islands Island Passion Rock Cornwallis Island New Blada Clarion Island Island Shoal . .-. Socora Island Wilson's I St. Benedicto Freshwater Roca Partida Mallon Island Cloud's Island Copper Island Island Shovel Island Massachusetts Island . . Island Henderson Island another account. . Gardner's Reef Polland's Island Allen's Reef Cooper's Island Maro's Reef, W. pt Island Lat. D. M. 26 3 N 20 23 19 34 18 54 19 28 20 4? 20 32 20 54 20 3i 20 43 21 6 2t 43 21 57 21 4o 21 5o 22 2 23 5 25 8 25 3i I 42 I 58 3 52 4 3o 4 42 5 33 5 48 6 4i 8 55 10 00 10 28 10 67 II 33 i3 4 i3 32 1 4 42 16 00 17 00 16 3o 16 54 16 57 18 12 18 21 18 22 18 27 18 48 19 i3 19 18 19 22 19 6 19 20 19 43 20 6 21 22 6 22 28 24 6 24 12 24 26 25 8 24 52 25 00 25 43 25 3i 22 3 1 Long. D. M. 173 4oVV 1 55 54 1 54 54 1 55 49 1 55 56 i56 7 1 56 25 1 56 48 1 56 39 1 57 1 57 24 i58 7 159 42 160 35 160 i5 160 8 161 49 ce (« 168 09 M 170 46 a io4 5 -^ i57 32 i58 22 169 33 160 i3 87 I 162 23 166 2 178 00 179 28 109 19 i53 45 164 00 168 35 170 26 170 23 1 33 00 1 36 00 i63 54 109 9 i6q 36 ii4 5 . ii4 23 i55 i5 CS 170 3o no 56 166 55 I1 a? Hi 110 45 ii5 8 III 52 i65 23 ii4 57 i3i 43 178 3o 112 1 4 176 36 167 4i 128 6 168 9 168 20 167 57 i3i 35 170 46 i3[ A Rock Laysan's Island Liscanskey's Island Neva Island Maro's Reef, N. pt. E. ... Island and Rock Pearl and Kermes group. Clarke's Reef, 60 miles N.W. and S. E Bunker's Hill Island Ocean I., N. i)t A Bank Culpepper's Island Wenman's Island Redondo Rock Abington Island, Mid Albemarle Island, N. pt S. W. Point James I. Harbor Charles Island, S. p Chatham Island, N. E. p Stephen's Bay Lat. 31. 3oN 46 o3 54 24 43 43 I 4o I 23 o i5 o 32 09 00 S 20 45 53 Lor.s- D. 31. 174 3W 171 49 173 41 172 20 170 32 170 54 175 48 175 48 173 20 176 5o 178 25 118 49 92 4 91 53 91 4o 90 48 91 25 91 32 90 56 90 33 89 37 XLIX. Islands, Rocks, and Shoals, in the SOUTH PACIFIC OCEAjY. New Guinea, — Middleburg Island . . . — Cape of Good Hope . . — Flat Point — Cape Valshe — Cape Rodney — King William's Cape. Torres or Endeavor Straits Eastern Fields or Reefs, N. E. end .' — N. W. part Murray's Islands Wamvax or Darnley Isl.. Pandora's Shoals, N. p... Wreck Reef, S. p. . Portlock's Reef .... cnt. Torres Straits . Boot Reef. Indefatigable's ent. ditto. Halfway Island Booby Isle York Island West I Prince of Wales's Is. S. pt. Kangaroo Coral Reef. . . . Providence Islands, Little Providence or Dann-er Island N. W. ext. of Shoal off ditto Louisiade Isles, Cape Deliverance . . Stephen's Island Durour's Island Lat. D. 31. o 20 S o 20 o 46 8 22 10 i5 6 40 59 56 35 55 25 48 54 59 5o 7 37 9 45 10 46 i3 22 Lon39 4 139 9 i38 5i 1 38 49 160 4o 159 5o 1 33 54 159 3o 173 45 i55 55 1G6 3o 139 54 i58 6 1 56 57 i56 5o 161 10 161 2 166 6 159 25 i56 7 i43 o E.p. Flint Islan * Baunian's Islands. Eng George's Is. . . Tiokea, Oura, Isle des Chiens* . . Isle RomanzofF. . . Isles de Krusen- stern, extend- ing N. N. E. )- centre and S. S. W 15 miles __ Chaine du Rurick, N.E. p — E.p — W. p* Dageraad Island Dean, or Prince of J r^ Wales, or Oan- '*^ na Island ^ Island Island Island Elizabeth Island . . . Eunice Island Armstrong's Island Anderson's Island, (or Elizabeth Island, )N.E.p. Ducie's Islandj N.E. pt. . . Island St. Ambrose Island IST. p. ISr. pt. of St. Felix..- Gray s Island Sales y Gomez Easter Island, Peak Island Group of Islands Massafuera Juan Fernandez, S. W. p. E.p.... Lat. NEW SOUTH SHET- LAND. Clarence Island, Floyd's Promontory Cape Bowles Cornwallis Island Seal Islands Cape Valentine Sarah Island Obrien's Islands Bridgeman's Islands ... Cape Melville Sheriff' Cape Ditto, (another ac count,) , Yankee Straits , Ragged Island Ditto, (another ac count,) Ditto, the harbor, (by another person,) . New Plymouth Monroe's Island, Presi- dent's Bay 1 . . , Castle Rock, (W. of Mon roe's Island,) Mount Pisgali Ditto, (another ac- count.) D. M. II 26 II 52 i4 22 14 U i4 5o i4 57 1 5 00 i5 II i5 20 i5 20 i5 45 1 5 o5 i5 17 16 00 17 00 20 00 21 6 21 8 21 21 24 22 24 4o 25 i3 26 20 26 17 26 24 26 28 27 8 28 6 3i 3 33 45 33 49 33 4i 60 57 61 20 61 o4 61 00 61 3 61 22 61 32 62 06 62 00 62 28 62 2 1 62 3o 6?. 4o 62 42 62 55 62 45 62 46 62 5o 63 00 Long. 62 57 D. M. i5i 48W i55 12 1 44 58 (45 20 1 38 47 1 44 35 48 4i 1 46 47 1 46 3o 1 46 56 147 59 147 i4 139 00 1 38 00 167 5o 178 36 178 47 i6i 4 128 19 124 48 i3o 28 80 GO 80 21 92 24 io5 26 109 17 95 12 129 24 80 47 79 6 78 53 54 6 54 8 54 28 55 32 54 4o 55 3o 55 52 56 4o 57 3o 60 28 61 47 60 22 62 10 62 20 63 5 61 37 62 20 62 3o 63 00 63 4o See Table on page 450 TABLE LV. [I^'ige 379 This Table shows the Times of High Water at the Full and Change of the Moon, at the principal Ports and Harbors of the world, and the vertical rise of the Tide, in feet. [Those marked with a star arc corrected establishments.] Abbeville Aberdeen* Aberystwith Acapulco AchiUHead*.... Aden Ayre, Point of . . . . Aix, Isle d' Albau's Head, St.* Algoa Bay Amazon River. . . . Anibleteuse Anicland* Amhviek Point . . . Anioor Strait . . . . Anioy Amsterdam Amsterdam Island Andrew's Bay, St.* Ani^ra Bay Anliolt Island. . . . Ann, Cape* Annapolis* Anticosta Island, W. end Antwerji* Annamooka Areliangel Arklow* Arran Island Arundt-l Astoria* Augustine, St.*. . . Aug'stine's Bay,St. Avranches Babelmandel Strs. Bali.sore Ballingskellings Bay* Baltimore Baltimore* Baliia Banff Bantry Bay* Bardsey Island .. . Barfleur* Barmouth Barnstable Bay . . Batavia Baudsey Clift' Bay of Islands . . . Bayonne Beachy Head . . . . Bear Islantl Beaumaris Bee's Head, St.... Belfast (entrance) Belle lAe Bembridge Point* Bergen Bermuda Island. . Berwick* Bilboa* Blakeney Blanco, Cape . . . . Fi-ai)ce Scotland Wales Mexico Ireland Arabia Isle of Man. . . France England Africa America France Xorth Sea . . Anglesea Asia China Holland Indian Ocean. Scotland Terceira Cattcgat America America, U. S . America , Belgium , Facilic Ocean . . , Russia Ireland , Scotland England Oregon America , Madagascar . . . . France , Red Sea India . . . Ireland Ireland America Brazil Scotland Ireland Wales France Wales England Java England New Zealand . . . . France England Hudson's Bay. . . . Wales England Ireland Bay of Biscay. . . . Isle of Wight . . . . Norway Atlantic Ocean . . England Spain England Africa h. in. lo 3o o 48 7 3o 3 6 4 56 9 45 lo 3o 46 CO CO 00 00 3o 24 00: 6 25 i5 I 00; 28' 2 45J 2 49 10 25 16 42 7 20 5 3o i3 00 1 3o 00 1 5 46 12 23 10 33: I 3o[ 8 28 10 2 8 40 1 5 45 17 4o 17 3o 19 00 2 3o' i5 9 45 [2 20 20 00 32 21 i5 43 9 °°' / II i4 3o' 4 i4 4 i5 i5 20' 9 3o i5 46 6 (Sibyl Blaskets Head) Block Island*... Bojador, Cape. . . Bolt Head Bombay Borkum Island .. , Boston Light* . . Botany Bay .... Boulogne* Boi'deaux Brassa Sound . . . Bray Head* .... Bremen Brest* Bridgewater .... Bridport Brighton* Bristol Broad Haven . . . Burnt Island .... Button's Islands. Cadiz* Caen Caernarvon Calais* Caldy Island Calf of Man ( 'allao Camj)bel! Town . . Canary Isl., Pt. de la Luz Causo, Cape Cantire, Mull of. . Canton II. (ent). . Capricorn, Cape . . Cardiff Cardigan Bar. . . . Carlingford* Carlisle Caimarthen (baskets Catherine's Pt., St. Catness Cayenne Cedar Keys* . . . . Charente R., Ro- chefort Charles, Cape Charleston, S. C* Charlottetown .. , Chatliam Chepstow Cherbourg* Chester Bar Chicht'ster Harbor Cliristmas Sound.. Churchill, Cape. . . Clear, Cape* . . . . Cod, Cape* Condore Pulo. . . . Conway Copeland Island. . Coringa Bay Coquet Island . . . . Ireland America Africa England India Hrf)lland America New Holland France France Shetland Ireland Germany France England England England England Ireland Scotland Hudson's Bay. . . . Spain France Wales Fiance Wales St. Georg. Chan'l. Peru Gulf St. Lawrence Atlantic Ocean. . America Scotland China New Holland . . . Wales Wales Ireland England Wales English Channel. Isle of Wight . . . White Sea South America. . Florida France America America Prince Edw'd'slsl. England England France England England South America. . . Hudson's Bay. . , . Ireland America China Sea Wales , Ireland India Ensrland ft. 3o 12 37 00 55 4o i5 3o! 9 16 35 5 II i5 i5 56 44 00 10 5o'io 5o I 4o 10 57 9 33 14 11 3219 6 00 34 11 17 16 5 4?! 4 4 00 10 i2 5o 10 8 3o' 6 10 3o 5 10 00 8 00 7 6 59'38 7 00 1 4 10 53, i5 12 1020 3o'2Z 45 i5 00 i5 45 5o 4617 7 45 7 t3 10 45 1 00^17 7 3o|38 7 33,17 10 30,26 11 3o 2 3o 7 20 4 00 II 25 3 00 10 i5 ;o 49 9 i5 3 00 ^age380] TABLE LV. This Table shows the Times of High Water at the Full and Change of the Moon, at the principal Ports and Harbors of the world, and the vertical rise of the Tide, in feet. [Those marked with a star are corrected establishments.] Cornwall, Cape . . Cornwallis, Port. . Cork Harbor (en- trance)* .... Corunna Coutance Cowes Croeotoa Island Cromartie* .... Cromer* Crookhaven. . . . Cross Island . . . Cuxhaven Dartmouth David's Head, St. Deadman's Point. . Deal Dee, River Delaware Bay* (breakwater) . . . Demerara Diamond Point. . . Diego, San* Dieppe* Dingle Bay* Donegal* Dover* Douglas Downs Droglieda Drontheim Dublin* Dudgeon Lights. . Dunbar* Duncansby Head . Dundalk Bay . . . . Dundedy Head .. . Dundee Dungaroon Dungeness* Dunkirk* Dunnose Eastern Brace . . . . EastporL* Eddystone Elbe R., red buoy. Embden Exmouth Bar* . . . Exuma Bar Eyder River Eyemouth Harbor. Fair Head , Falmouth , Fayal Road Fear, Cape* . . . , Fecamp Fernandina* . . . . Fernando Po. . . . Ferrol Ferriters Fifeness Filey Finisterre, Cape. SITUATION. England Prince of Wales' Is. Ireland Spain France Isle of -Wight Strait of Sunda . . Scotland England Ireland White Sea Germany England Wales . . England England Scotland America , S. America Malacca Strait . . California France Ireland Ireland England Isle of Man England Ireland Norway Ireland North Sea Scotland Scotland Ireland Ireland Scotland Ireland England France Isle of Wight . . . Bay of Bengal . . Maine, America . English Channel. North Sea Germany England Bahamas Germany Scotland Ireland . England Azores . . America France. . Florida . Africa . . Spain . . . Ireland . Scotland England Spain . . . TIME. R. h. m. ft. 4 3o 22 I 3o 4 37 3 00 6 00 II l5'l2 7 oo[ 3 II 43 1 3 6 43 i6 4 9|io 4 i5 I 00 10 6 i6 i4 6 00 5 3o 1 1 i5 i6 II 00 23 8 00 4 4 45 9 [2 oo 9 9 38 5 1 1 6 27 3 4o 9 b 8 II II 00 i8 II 12 21 II OO i5 10 45 2 i5 II II i3 6 00 2 oo i4 10 i4 10 10 56 i3 4 00 II 2 32 i4 4 3o lO 54 21 II 55 i6 9 i5 9 45 11 i3 i8 5 5o[i8 12 oo 12 00 6 21 7 20 12 OO 2 l5 9 OO 4 57 II 45 7 19 10 44 23 7 53 4 00 3 00 3 3o 2 00 4 20 3 00 Finmark Fishguard Bay . . . Flamborough* . . . Flushing Fly, or Vlie Gat- way Fly, or Vlie Road. Foreland, North. . Foreland, South. . Formby Point. . . . Fox Island Fowey* Francisco, San* . . Fuuchal Gal way Coast*. . , Galloway, Mull of Gambia River (en trance) , Gay Head Georgetown Bar*. Gibraltar Gloucester* Goa Good Hope, Cape* (St. Simon'sBay) Good Hope* (Ta ble Bay) Goree Gatway. . . Granville* Gravelines Gravesend Grizness, Cape . . . Haerlem Hakodadi Halifax Hamburgh Hartland Point . . Hartlepool Harwich Hastings Hatteras, Cape*. . Havre de Grace*. Helena, St Helen's, St Helvoetsluys* . . . Henlopen, Cape* . Henry, Cape Hobarton Hogue, Cape La. . Holy Isl'd Harbor. Hongkong Road. . Honfieur Hoogley R. (ent.). Hull Humber R. (ent.) . Hurst Castle SITUATION. Lapland Wales . . England Holland. Holland . . Holland . . England . England . England . America . England . California Madeira. . Ireland . Scotland Ice Cove Ichaboe Ipswich Isle Dieu Isle of Man, South side Ives, St Africa . . America America Spain . . . America India . . . Africa , , Africa . . . , North Sea. France . . . , France . . . , England . . France . . . , Holland , Japan Nova Scotia . . Germany . . . . , England , England England England America France Atlantic Ocean , Isle of Wight . , Holland America America Tasmania France England China France India England England Entrland Hudson's Bay. Africa England France St. George's Chan. England 7i. m. ft. i5 6 56 II 4 3o 12 I 20 i5 45 3o i5 16 6 i5 35'28 45 i4i5 6 4 7 24 r5 7 37 7 56 2 20 :i 00 1 3o 2 48 2 29 I 3o 5 54 37 00.19 10 17 27 21 00 00 49 29 00 28 6 53|24 4\ 2 3622 3 II 45 2 3o 8 00 7 14 8 00 8 45 2 3o 10 i5 9 3o.23 10 00 II 6 29'2I 5 i5'i8 11 00! 7 10 00 1 oo 6 2 35ji3 3 00 i4 10 20| 4 44!2t TABLE LV. [i^'^ge 831 This Table sho-ws the Times of IIigu Water at the Full and Change of the Moon, at the principal Ports and Harbors of the world, and the vertical rise of the Tide, in feet. [Those marked with a star are corrected establishments.] Jackaon, Port . . . . Janeiro, Rio John's, St John's, St , River* John's, St Jutland Coast. . . . Kedgeree Kenniaie, River. , Kennebec Kentish Knock . . , Key West* Killibecrs King's Channel. . King's Road .... Kinsale* Kinnaird's Head. , Lambaness Lancaster Land's End Leitli Pier Lemon andOwer. Lerwick* Lewis Islands . . . . Lewis, Butt of . . . Limerick Lisbon Liverpool Lizard Loch Swilly Loire River London Londonderry . . . . Long Sand Head . Longsliips Lookout, Cape* . . Loop Head SITUATION. N^ew Holland . . . South America. . N'ew Brunswick. Florida Newfoundland . . Denmark India Ireland America River Thames. . Florida Ireland River Thames. . Bristol Channel Ireland Scotland Shetland England England Scotland North Sea. . . . Shetland . . . . Scotland Scotland Ireland Portugal England England Ireland France England Ireland River Thames. England America Ireland L'Orient 'France Lundy Island . . . .! Bristol Channel . Lyme Regis Lynn Deeps Macao Machias Madeira Madras Malacca Roads . . . Mulo, St Manilla Marblehead Margate Road. . . , Marks, St.* Martin Vas ...... Mary's, St Maulmain May, Cape* Melbourne Milford Haven . . Miramichi Mizzen Head. . . . Monrovia Monterey* Montrose* Morocco Coast . . Mount's Bav*. . . England England China Amei"ica Atlantic Ocean . . . India India .... France , . . Philippine Islands. America River Thames. . . . Florida Atlantic Ocean. . . Scilly Islands . . . . India America Australia England Canada Ireland Africa California Scotland Afiica England /t. m. 8 i5 3 00 [I 24 7 28 7 3o 9 3o I i5 4 3o 7 00 ID 3o 6 00 2 3o II 26[26 5 GO i4 6 3o 3 4o[i5 2 7 8 00 II 3o 3o 10 i5 6 21 6 00 10 00 11 00 12 48 7 34 7 3o 6 5 10 4o 11 3o II 45|i5 1 i3 3 3 45 4 II 2 o 8 19! 6 I 20 3 6 00124 5 00 4 2 6 22 I 4o 2 i5 4 19 Mount Desert. . . Mozambique. . . ., Xamjasaki Xantucket* Xantes River Loire. Xassau N^atal, Port Needles Newcastle New Bedford* . . . Xewburyport* . . . New Haven* . . . . New London* . . . Newport Newport* New York* Nootka Sound . . . . Nore Light North Cape Olonne Oporto (Bar)*. . OrforJness Orkney Islands. Ornis Head . . . . Ortegal, Cape. . Ostend Owers America Africa . . Japan America , France France New Providence. . .Africa Isle of Wight . . . England America America America America Wales America America North America. . River Thames. . . Lapland Padstow * Panama Road. . . . Para Passamaquoddy River Passier Roads. . . . Penmarks Penobscot River. . Pentland Frith... Penzance Pernanibuco Peter Head* Philadelphia* .. . . Phillips Port Pictou Plymouth Sound.. Plymouth* Pouit de Galle- . . . Pol de Leon, St. . . Poole Port Glasgow. . . . Port Hood Port Howe Port Jackson Portland Bill Portland Race . . . Portland* Port Louis Port Louis Porto Pra3-a Port Roseway . . . Port Royal Island . Portsm'th Harbor Portsmouth* Pulo Pinang SITUATION. France Portugal England North Sea Wales Spain Belgium English Channel. England New Grenada . South America. America Borneo France America Scotland England Brazil Scotland America Australia Nova Scotia . . . . England America India France England Scotland Cape Breton . . . . Nova Scotia . . . . Nova Scotia . . . . England England America France Mauritius Cape Verde Isl. , [Nova Scotia . . . , I North America. , England America India TIME. R, \. m. ft. I 10 i3 4 i5 12 4524 45 5 i3 5 20 7 3oi5 00 3 So 2 3o 11 i5 10 00 10 i5 3 00 12 21 6 3o l4 i5 4 56 20 3 23'i8 12 00' 1 1 3o 25 7 3 6 37|i5 19 II 4 i5 I 00 18 9 ° 8 3o 8 00 7 i5 9 i5 1 25 3 II :2 3o :i ? 8 3o 8 i5 ti 36 [I 23 2 i5 Page 382] TABLE LY. This Table shows the Times of High Water at the Fall and Change of the Moon, at the principal Ports and Harbors of the world, and the vertical rise of the Tide, in feet. [Those marked with a star are corrected establishments.] Quebec Queda Roads. Rachlin's Island*. Ram Head Ramsey Ramsgate* Rangoon (entr.). . Rhe Island Rio Janeiro.. . . '. . Robin Food's Bay Rochef -i u Rochelle Rochester Rodrigues Island. Roman, Cape Roseness Rotterdam Rye Harbor Sable, Cape Sable Island Salem* Salvador, St Sandwich Sandwich Bay.. . Sandy Hook*.. . . Savannah (entr.)* Scarborough .... Scaw Scilly Islands* . . Seal Islands SelseaBill* Senegal R. (entr.) Seven Islands. . . Shanghae Shannon R. (entr. Sheerness Sheepscut Shetland Island, (south end) . . . Shields Shoreham Sierra Leone .... Simoda Simon's Bar, St.* Sincapore Skerries Skerries Sky Island Sligo* Slyne* Smalls Somme River. . . Southampton . . . Southwold Canada. India . . Ireland England Isle of Man England India Bay of Biscay . . . South America. . England France France England Indian Ocean. . . . America Orkneys Holland England TIME. Nova Scotia , America America South America . . England Nova Scotia New Jersey America England Denmark Ijlnglish Channel. Bay of Fundy. . . England Africa Lapland China Ireland England America • h. in. 6 38 12 00 ft. 53 45 44 i3 i5 21 00 00 45 6 3i 00 45 00 3o 3o 20 8 3o lo 3o North Sea . England .... England .... Gruinea Japan America .... Asia Wales Scotland .... Scotland .... Ireland Ireland Wales France England .... England 9 i3 3o 00 00 29 7 20 4 Spurn Point . Start Point . . Stockton . . . . Stonehaven. . Stromness* . . Suez Sunbury . . . . Sunderland* . Surinam Swansey Sweetnose . . Sydney Sydney England England England Scotland Orkneys Red Sea North America. England South America. Wales Lapland Cape Breton I. Australia 8 20 I 40 4 12 o .37 10 45 10 3o 3 23 II 7 5 7 9 10 00 11 00 6 00 5 25 4 32 5 5o II II 16 16 Tees River. . . . , Telling, Cape. . , Terceira Texel (entrance of) Texel Road... Thames River (mouth) ... Tynemouth. . . Todhead Torbay Tory Island. . . Tuscar Rock. . Typa Roads . . England Ireland . Azores. . Holland . Holland. Ushant* Valparaiso Yannes Vincent, Cape St. Wardhuys Watchet Waterford Harb.* Weser River (ent.) Western Brace. . . Wexford Harbor. . Weymouth Whitby* Whitehaven Wicklow Winterton Woolwich Wrath, Cape Yang-tse-Kiang (entrance) .... Yarmouth Roads Yarmouth Yorkshire Coast. Youfrhall* Zanzibar. England .... England .... Scotland .... England .... Ireland Ireland River Canton France Chili. . , France . Spain . Lapland Britisli Channel Ireland Germany Bay of Bengal . Ireland England England England Ireland England England Scotland China England .... Isle of Wight England .... Ireland i5i4 00 5 38 5 3 45 i5 6 00 12 32 6 45 7 45 12 00 17 3 20' 1 5 12 45 6 00 i3 6 00 7 00 10 00 3 39 19 9 82 4 3o 2 3o Africa II 1 5 23 29 9 5o 10 3718 3oi5 i5 9 i5 6 II 00 7 4 3o 5 i4 10 4 20 TABLE LVI. [Page 383 The following table contains extracts from the Nautical Almanac for the year 183G, in those parts which are used in this work, to accommodate those who may not have a copy of that Almanac to refer to. Lunar Distances and Proportional Logarithms. 0;iy of the Month. 183G. January G April 1 May 11 June 20 Oct. 30 Aldebaran W- Antares ..E. Sim E. Venus. . ..W. Sun E. Hours. Di.-tances. P. L 66 6i 4o 46 34 3o 58 112 54 59 1 3 2348 3o97 3(>35 3458 49 3 Hours. Distances. P. L 6j 4i 43 2872 59 55 24 2337 45 5 493108 32 28 lb 3019 III 32 59I3459 G Hours. Distances. P. L. 69 l4 38 2864 58 ID 19 2326 43 37 49 33 58 7 no II 49 3i 17 3oo2 3460 9 Hours. 70 47 43 56 24 58 42 10 00 35 28 17 108 5o 4o P. L, 2856 23i7 3127 2985 3460 Day of tlie Month. 12 Hours. 15 Hours. 18 Hours. 21 Hours. S\Iean'riine. 183G. Feb. 12 Auir. 2G Sun E. Mars E. 5 1 28 10 :i4 55 6 Distances. P. L. 2552 12455 49 48 9 255i 1 13 12 49 2467 Distances. P. L. 48 8 7 III 3o 49 Distances. P. L. 255i 2479 46 109 28 49 255i 2492 Moon^s Semi-diameter, Horizontcd Parallax, S^'C. g tin's Latitude. JVoon. II N. 0.43 0.66 0.34 S. O.IO 0.23 0.33 N. 0.86 S. 0.48 0.44 JN. 0.17 o.o5 S. 0.32 N. 0.32 Day of tlie Month. January 6 April 1 Oct. 30 May 11 Feb. 12 13 June 20 Aus. 26 ° 27 June 2G 27 Sept. 26 Nov. 29 Sun's Longi- tude. Lo^'. Radius Vector. 285 16 II 47 217 7 5o 45 322 5l 323 52 89 5 i53 12 i54 10 94 49 95 46 i83 24 247 22 27.6 34.6 36.5 58.8 54.9 33.0 53.5 57.6 55.6 8.9 20.0 26.6 1 1.8 9926712 ,0000753 .9965769 .0046545 ,9945669 .9946561 .0070882 .0042613 .oo4i6i4 .0071787 .0071880 ,0007178 ,9937878 Moon's Semi-di- ameter. 1 5 4-8 i5 58.7 i4 45.6 i5 16.8 16 16.2 16 17.6 1 5 9.6 16 i4-5 16 4.7 16 30.9 16 39.4 i5 32.4 i4 48.5 Miiln. 8.5 3.5 45.7 12.5 17.2 17.3 l5.2 1 0.0 58.9 35.7 4i.8 26.9 4 5i.4 Moon's Horizontal Parallax. 20.3 38.1 10. 1 44 42.5 47-4 38.1 36.3 0.3 55 33.8 58 55.9 54 10.3 55 48.6 59 46.0 59 46.3 55 58.6 59 19.5 58 39.0 36.4|6o 53.8 7.461 16.4 i.7|56 41.4 20.6:54 3 1. 1 Examples I. IX. X p. 232, 241, 242 Ex. H. p. 233. HI. Vin.234,240 Ex. IV. p. 235. I Ex. V. p. 23G. Ex. VI. p. 237. I Ex. VII. p. 238. I Ex. I. p. 172. Ex. II. p. 172. Ex. p. 214. Swi's Right Ascension, ifc. Dav of the Month. Nov. 29 30 20 27 5 G 8 9 16 17 24 25 March 10 11 Oct. 30 May 11 Feb. 12 IMay Jan. Sept. April July THE SUN'S Riirht Ascension. Declination. Pemi-diam h. m. 16 22 16 26 4 l3 4 17 19 I 19 6 II 7 II II I 38 I 42 8 i5 8 19 23 23 23 26 i4 19 3 i3 21 4o i4.56 32.98 2.5l 5.56 55.40 18.78 47-32 23.47 20.06 2.38 5.79 3.18 10.85 51.37 6.64 18.24 5i.5o S. 21 33 21 43 N. 21 II 21 21 S. 22 4i 22 35 N. 5 35 5 i3 10 i4 10 35 19 5o 19 37 S. 3 58 3 34 i3 54 N.17 57 S. i3 54 J9.7 28.7 i4-o 21.2 53.8 10.5 55.7 14.6 7-4 16.0 18.5 27.4 17.9 45.2 18.2 AU 27.3 6 i4.8 6 14.9 5 48. o 5 47-8 6 17.3 17.3 54.5 54.8 56.6 56.4 46.2 46.3 6.7 6.5 8.5 5o.7 i3.o Equation of Time, to be applied * to Mean Time. + 1' -f-io + 3 + 3 — 5 — 5 + 2 + 2 + + — 6 — 6 4-16 + 3 +t4 21.56 59.70 17.61 II. 12 26.1 I 52.93 3i.3o 51.70 17.81 32.o5 8.74 9.57 25.45 9.41 12.78 53.53 33.06 * Tliosewith the sign -(-are to be added to the mean time ; those with the sign — are to be subtracted, to oMain the apparent time. These signs must be changed if we wish to obtain the mean time from the apparent time. Page 384] TABLE LVI. The following table contains extracts from the Nautical Almanac for the year 1836, in those parts which are used in this work, to accommodate those who may not have a copy of that Almanac to refer to. Sun's Declination, Sfc. Day of the Month. jjpp. l^ime. 183G. May 9 10 March 25 July 25 20 Nov. 25 20 April 11 28 THE SUN'S Ri^lit Ascension. h. ni. s. 3 5 2^.3i 3 9 23.16 17' 56.79 8 19 4.19 8 23 0.98 16 5 5.68 16 9 21.29 1 19 53:95 2 23 i5.i6 Dilf.for \ hour. 9-744 9.769 9.0S1 9.866 9.84 1 io.65o 10.681 9.189 9-484 Declination. N. 17 26 27 17 42 i3 I 56 41 19 37 24 19 24 i3 S. 20 5o i4 21 I 4o N. 8 26 o i4 i5 I Diir. for 1 hour. 39.43 38.69 58.82 32.95 33.75 28.58 27.59 54.^0 46.69 Sid. Time of the S. Diam. passing the me- ridian. in. s. 6.64 6.72 4.39 7.16 7.08 9.68 9.78 4.74 5.77 Equation of Time, to be applied to the Apparent Time. — 3 48.73 — 3 51.42 -\- 6 2.10 + 6 9.57 + 6 9-79 — 12 42.11 12 23.12 + o 58.69 2 40.94 D iff. for 1 hour. O.112 0.088 0.774 0.009 o.oi5 0.791 0.823 0.665 0.371 I Ex. p. 220. Ex. V. p. 157. I Ex. I. p. 247. \ Ex. II. p. 247. Ex. p. 250. Ex. p. 248. Moon^s Declination, fyc. Day of llie Month. Mean Time. 1836. April 18 June 26 Sept. 26 Nov. 29 April 23 h. 7 i5 16 7 3 12 i3 THE MOON'S Ri-'iht Ascension. h. ni. s. 3 52 47.56 16 29 28.69 16 32 8.61 I 38 36.i6 I 4o 35.44 9 24 38.95 9 26 39.34 12 32 52.81 12 34 57.22 21 1 3 5 1. 7 23 87 43.2 23 46 3 1. 1 8 47 27.3 9 I 56.2 20 4i 6.1 20 3 1 3o.i o 20 5o.8 o 4 54.5 Diff. declination for 10 minutes. 87.98 86.33 144.82 i44-27 96.00 96.90 159.38 159.58 Ex. p. 171 I Ex. I. p. 172. I Ex. n. p. 172. Ex. III. p. 173. Ex. p. 213. I Ex. p. 248. Moon's Passage ever the Meridian, fyc. Day of the Month. Mean Time. 1836. April 18 19 June 26 27 Sept. 25 Nov. 28 29 March 17 May 23 JMoon's Longitude. 57 4 35.8 69 I 3o.o 240 r 4 [-7 254 59 4i.8 8 5o 46.8 22 12 43.5 124 8 16.4 i36 2 4-2 358 27 2.5 149 27 19.5 63 3 59.5 74 57 30.9 247 28 26.0 262 34 28.9 i5 34 19.6 28 45 53.9 i3o 4 34.6 142 I 16. 1 5 I 33.1 i55 45 12.4 Moon's Latitude. N.o 38 .36.9 1 42 46.2 S. o 58 3.9 2 i4 35.9 2 47 10.5 I 42 N. 5 5 S. 4 N. 5 7.3 16.5 53.0 54.9 19.7 N. 9.5 9.0 0.0 4.6 2 i3 S. I 37 2 5o 2 i5 23.7 I 7 52.9 N.5 i3 15.9 5 7 6.6 S. 3 4i i4.o N.5 4 46.3 Age. JVoon d. 2.5 3.5 12.3 i3.3 i4.5 i5.5 19.4 20.4 0.1 7-9 Meridian Passage. h. m. 1 55.6 2 43.0 9 55.9 10 59.8 12 42.8 i3 28.0 16 33.1 17 18.6 o 21. 1 6 21. 1 |Es. p. 170. I Ex. I. p. 172. \ Ex. II. p. 172. 1 Ex. III. p. 173. Ex. I. p. 121. Ex. II. p. 121. Declinations, Right .Ascensions, and Time of passing the Meridian of Jupiter, Venus, Sf'C. Mean T. 1836. Oct. 22 23 Sept. 16 17 May 2(5 27 Jan. 5 G April 28 29 GEOCENTRIC Noon. h. m. s. 9 11 ."53.47 9 12 2-2.92 8 41 5."). (50 8 45 14.25 7 8 6.,55 7 8 57.01 14 10 11.19 14 10 215.19 415 47.41 47 27.73 » ( /( N.15 47 17. 16 43 18. 14 49 33. 14 44 22. 22 49 18. 22 4-i 0, S. 10 35 2S, 10 36 33. N.23 16 40, 23 15 5S Log.ufd St. from the E.ir-h. 0.7390451 0.7378510 9.7458961 9.7516468 0.7760038 0.7767647 1.0021683 1.0014895 0.749.5943 0.750G908I 4 m. 5.4 2.0 59.5 58.9 51.4 43.3 10. 7.1 20.3 17.0 HELIOCENTRIC If 124 56 35.6 125 1 24.9 26 19 7.0 27 51 ,59.6 112 52 34.8 112 57 29.1 208 31 44.1 203 33 39.2 110 31 53.1 110 39 48.6 N.O 34 45.9 34 51.9 S.2 33 25.3 2 29 38.0 N.O 19 13.6 19 20.2 2 28 28.7 2 28 28.2 16 9.3 16 15.9 0.7936939 0.7237210 9.8.599886 9.8.599080 0.7193880 0.7194179 0.989.5685 0.9895807 0.7185405 0.7185707 ) Example I. p. 174. j Jupiter, i Example II. p. 175. \ Venus. Example I. p. 215. \ Jupiter. Example II. p. 216. ( Saturn. \ Example, p. 249. \ Jupiter. [Page 385 TABLE LVIL «j X. i Latitude. j^ c =; Q IB ^ 0° 5° 10° 15° 20° 25° 30° 35° 40° 45° 50° 55° 60° G5° 70° 75° O 1 / / / ; / 1 / / 1 1 / / / 1 10 IIO .4 .4 .4 .5 .5 .6 .7 .8 1 .0 1.3 1.8 2.9 no IC 20 .4 .4 .5 .6 •7 .8 1 .u 1 .2 1.6 2.6 20 3o .4 .5 .6 •7 •9 I . I 1.5 2.3 ■ 3o 4o .5 .6 .8 1 .0 1.3 4o bo .7 •9 1 .2 5o bo •9 60 10 io5 .3 .3 .3 .3 .4 .4 .5 .G .8 •9 1.2 1.8 3.0 ro5 10 20 .3 .3 .4 .4 .b .6 • 7 • 9 1 .2 t.6 2.7 20 3o .3 .4 .5 .6 •7 .8 1 . 1 1.5 2.4 3o 4o .4 .5 .6 •7 I.O 1.3 4o bo .4 .6 .8 1 .2 5o 6o .6 •9 60 i5 too .2 .2 .2 .3 .3 .4 .4 .b .6 .8 I.I 1.6 2.9 100 i5 20 .2 .2 .3 .3 .4 .5 .b •7 .9 I.I 1.6 2-7 20 Jo .2 .3 .3 .4 .b .6 .8 r . I 1 .5 2.4 3o 4o .2 .3 .4 .6 •7 •9 1.3 2. 1 4o be. .3 .4 .6 .8 1 .2 5o 6o .3 .6 •9 60 i5 95 .1 .1 .2 .2 .3 .3 .4 .5 .6 .8 I.I 1.7 3.0 "95 i5 20 .1 .2 .2 .3 .3 .4 .b .b .8 I.I 1.6 2.8 20 3o .2 .2 .3 .4 .5 .6 .8 I.O 1.5 2.5 3o 4o .2 .3 .4 .b •7 •9 1.3 2.1 40 5o .3 .4 .6 .8 I . I 5o 6o .2 .3 .6 •9 ■ 60 20 90 .0 .0 .1 .1 .1 .2 .2 .3 .4 .6 • 7 I.I 1.6 3.0 9c 20 Jo .0 .1 .1 .2 .2 .J .4 .b- .7 I.O 1.5 2-7 3o 4o .0 .1 .2 .3 .3 .5 .6 •9 1.3 2.2 4o 5o .0 . I .2 .4 .5 .8 I . I 5o (x> .0 .2 .3 .5 •9 Co 70 .0 .2 .6 I.I 70 20 85 . 1* .1* .0 .0 .0 . I . I .2 .3 .3 .5 .7 I.e. 1.6 3.1 85 20 Jo . I* .0 .0 . 1 . I .2 .2 .4 .b • 7 1 .0 1.5 2.7 3o 4o . 1* .0 .0 . I .2 .3 .4 .6 •9 1.3 2.3 4o bo . 1* .0 . I .2 .0 .b •7 1 .1 5u 6o .7* .0 . I .3 .b •9 60 70 .3* .0 .2 .6 I . I 70 20 80 .2* .2* .1* .1* .1* .0 .0 .0 .1 .1 .2 .4 .5 •9 1.5 3.1 «0 20 Jo .2* .2* . I* .0 .0 . I .1 .2 .3 .4 .6 •9 1.5 2.8 3o 4o .2* .2* . I* .0 .1 .2 .3 .4 .6 • 9 1.3 2.4 40 bo .3' .2* . I* . I .2 .3 .5 •7 I . I 5o 6o .4* .2* .0 .1 .3 .b • 9 fio 70 .6» .3" .0 ,.2 .6 1 .2 70 30 75 .3' .3* .2* .2* .2* . I* . 1* . I* . I* .0 .0 I .2 .3 .fi 1 .2 75 20 Jo .3' .3* .2' .2* . I* . 1* .0 . I .1 .2 .4 .6 . 1.5 3.0 3o 4o .4* .3* .2* .1* .1* .0 .1 .2 .4 .5 .8 1.3 2 . 5 4o' bo .4* .3" .2* .1* .0 .1 .3 .5 • 7 r. I 5o bo .6* .4* .2* .1* .1 .3 .5 •9 fio 70 1 .2* .6* .3* .0 .2 .6 1 .2 70 20 70 .4* .4* .3* .3* .3* .3* .2* .2* .2* .2* .2* .2* .2* .2* .2* .2* 70 20 Jo .4* .4* .3* .3* .2* ,2* .1* . I* .0 .0 .1 .2 .6 .8 1.5 3.T 3o 40 .5* .4* .3* .3* .2* .1* .0 . I .2 .3 .5 .8 1.3 2.6 4n bo .6** .5* .3" .2* .2* .0 .1 .3 .4 .7 I . I 5n bo •9* .6* .4* .3* .1* .1 .2 .5 •9 fin 70 1 .2* .6* .3* .1* .2 .6 1 .2 70 — T. ^ 0^ 5^ 10^ 15° 20° 25° 30° 35° 40° 45^ 50° 55° G0° 65° 70° 75° I. S c; Tj~. Latitude. "'c ■^1 49 Pace 386] TABLE LVII. Table LVII. shows nearly the error hi longitude, in miles and tenths of a mile, occasioned by an error of one mile in the latitude. Tiais, when the sun's altitude is 30°, the latitude 30°, and the polar distance 100°, the error is 8 tenths of a mile. The error affects the longitude as follows : — When in west long., ^ A. M. ^ C decreased; ^ when the correction and the time is > < the long, is < i '^ marked * , the found in column ) P. M. ( ( increased ; ) longitude is ' increased. * decreased. When in east long., ^ A. M. ^ T increased; ^ when the correction and tlie .time is > < the long, is .? V is marked *, the found in column ^P. M. ( (decreased;) longitude is C decreased. ' increased. CATALOGUE OF THE TABLES, EXAxMPLES OF THE USES OF THOSE WHICH ARE NOT EXPLAINED IN OTHER PARTS OF THIS WORK. TABLES I. and II. Difference of Latitude and Departure. — The first table contains the difference of latiluilo and departure corresponding to distances not exceeding 300, and for courses to every quarter-point of the compass. Table II. is of the same nature and extent, but for courses consisting of whole degrees. The manner of using these tables is particu- larly explained under the article of Inspection, in the different Problems of Plane, Middle Latitude, and Mercutor's Sailing. TABLE III. Miridional Parts. — An explanation of this table may be found in pages 78 and 79, and the uses of it are shown in all the Problems of Mercator's Sailing. TABLE IV. Tke Sans Declination. — This table is explained in page I5G. TABLE IV. A. This table contains the equation of time for every noon at Greenwich, and is to be reduced to any other hour by means of Table VI. A. Thus, suppose the equa- tion of time was required for May 2, IStiG, sea ticcount at 10 A. M. apparent time, corre- sponding to May Id. 2'2.\\. by the N. A. Table IV. A. gives the equation May ), at noon. snh. 3m. Cs. and daily increase 7s. Find this at the top in Table VI. A. and 22h. at the side, the corresponding correction Gs. increases the equation 3m. Gs. to 3m. 12s. which is the equation at the proposed time. This Gs. would have been sulitractive if the equation had been decreasing, as it is in JNIarch. The equation of time being thus found, sub. 3m. 12s. is to be subtracted from the apparent time 22ii. as in the table to get the mean time 21h. 56ni. 48s. If the turan time 21h. SGm. 48s. had been given to find the apparent, it must be applied differently from the direction in the table, and in this example must therefore be added to 21 h. 5Gm. 48s. to obtain the apparent time 22h. TABLE V. For reducing the Sun's Declination given for JVoon at Greenwich to any other Time under any other Meridian. — The manner of using this and the preceding Table IV. is explained in pages 15G and 157. TABLE VI. The Sun's Right .Ascension. — The Sun's mean right ascension given in this table may be used when a Nautical Almanac cannot be procured, and no great accuracy is required. The table is to be entered at the top with the month, and at the side with the day of the month. TABLE VI. A. is explained in the precepts for the use of Table IV. A. TABLE VII. Amplitudes. — This table is explained in page 159. TABLE VIII. Right .Ascensions and Declinations of the principal fixed Stars. — This table contains the right ascensions and declinations of the principal fixed stars, adapted to the 1st of Janunry, 1830, and the annual variations in right ascension and declination ; by means of which the right ascensions and declinations of any of these stars may be obtained for any time before or after the year 1830, by the rule at the end of the table. ' To illustrate the method of doing this, we shall here give the following examples : — To find the right ascension of a star at any time. EXA.MrLE r. Required tl)e right ascension of Aklcbaran, Janu- ary 1, 1834. , ■^ ' h. m. s, R. A liy the T.-ihle in 1830 4 2611 Variation in 4 years, add M R. A. in January. 1834 4 26 2o EXAMPLE III. Required the rij-ht ascension of Snica, Utay 20, 1836. Ii. m. s. R. A. hy the Table in 1830 13 16 1.5 Variation in 6 years 4§ niunths, add 20 R. A. May 20, 1830 13 16 35 EXAMPLE IL Required the right ascension of Aldeliaran, Janu- ary 1, 1810. h. m. s R. A. by the Table in ia30 4 26 II Variation in 20 years, sulilract 1 9 R. A. on January 1, 1810 4 2.\ 2 EXAMPLE rV. Required the right ascension of Sirius, November C, 1817. , ' h. m. s. R. A. by the Table in 1830 6 37 39 Variation in 13 years, subtract 34 R. A. in January, 1817 6 37 ."i Variation for 10 ukmiiIjs aiftl 6 days, add.. 9 R.. A. November 6, 1817 ,,... 6 37 7 The sun's right ascension for any time may be found accurately by the Nautical Almanac, by taking proportional parts of the daily difference, as will be explained m the precepts of Table XXX. .XXXi. But in cases where no great accuracy is required, the right ascension may be obtained within 2 or 3 minutes, by means of Table Vi. 388 CATALOGUE OF THE TABLES. To find the declination of a star at any time. EXAMPLE I. Required the declination of Aldebaran, January 1, 1834. Declination by tlie Table in 1830 1C° 10' N. Variation in 4 years 32", add nearly. ... 1 Declination in 1834 16' 11' N. EXAMPLE II L Required the declination of the star Spica, May 20, 1836. Declination by the Table in 1830 10° 16' P. Variation in U years 4^ months 2 Declination May 20, 1836 10° IS* S. EXAMPLE II. Required the declination of Aldebaran, January L 1820. Declination by the Table in 1830 1G° 10' N. Variation in 10 years 1' 20", subtract ... 1 Declination January 1, 1810 16° 9' N. EXAMPLE IV. Required the declination of Sirius, November 6, 1807. Declination by the Table in 1830 1G° 29' S. Var. in 22 years 1 month 24 days, is sub. 2 Declination November 6, 1807 16° 27' S. The right ascensions and declinations obtained by the preceding calculations, are the mean values, to which must be applied the corrections for the Nutation and Aberration Tables XLII. XLllI. in cases where great accuracy is required, as is now done in the Nautical Almanac for 100 of the brightest stars for every 10 days in the year; and the numbers in tlie Nautical Almanac are to be preferred. We must neglect the correction Part III., Table XLllL, when the mean equinox is used, as is the case with the improved Nautical Almanac. To find when a star will be on the meridian, ' • Rule. Find the riglit ascension of the sun and star in the preceding Tables VL and VIIL; subtract the sun's right ascension from tlie star's, having previously increased the latter by 24 hours when the sun's right ascension is the greatest; the remainder will be the time of the star's coming to the meridian. If the remainder be greater than 12 hours, the star will come to the meridian after midnight ; but if less than 12 hours, before midnight EXAIMPLE I. At what time will Aldebaran be on the meridian, January 1 i ,, ^ Aldebaran's right ascension 4 26 Add ^4 28 26 Sun's right ascension 18 46 Aldebaran souths in the evening 9 40 EXAMPLE IIL At what time will the star Regukis be on the me- ridian, December 12? jj ^ Resulus's right ascension 9 59 Add ^4 33 .59 Sun's right ascension 17 17 After midniglit 16 42 Subtract ." ^2 In the morning 4 42 EXAMPLE II. At what time will Pollux be on the meridian, March 31 ? , li. m. Pollux's right ascension 7 35 Sun's right ascension 38 Comes to the meridian in the evening 6 57 EXAMPLE IV. Required the time when the star Fomalhaut comes on the meridian, June 1. j^ Fomalhaut's right ascension 22 48 Sun's right ascension 4 36 After midnight 18 12 Subtract J2 In the morning 6 13 To find what star will come upon the meridian at any given time. Rule. Add the time from noon* to the right ascension of the sun, tlie sum (rejecting 24 hours when it exceeds 24) will be the right ascension of the star required to be known ; with which enter the table of the star's right ascension, and find wh.at star's riglit ascension agrees with, or comes the nearest to it, and tJiat will be the star required, if tlie declination of .the star agrees witji the table, which may be ascertained by observing the meridian altitude of the star, the latitude of the place being given. EXAMPLE I. What star will be on the meridian about 10 at night, January 26? ^ ,„_ Sun's right ascension January 2G 21) 34 Given time 10 hours P. M 10 30 34 Subtract .' 94 Nearly answers to Sirius 6 34 EXAMPLE II. What star will be upon tiie meridian 30 minuteil past four in the morning. May 10? h.m. Sun's right ascension May 10 3 8 Given time 16 hours 30 minutes 16 30 Right ascension of mid. heaven 19 38 Answers nearly to Athair in the Eagle. * The time from noon must be reckoned from the preceding noon, so that 4h. A. M irnst be called 16h. CATALOGUE OF THE TABLES. 389 EXAMPLE Hi What star will be on llie meridian at Gli. 6"Jin. P. M AP"'^-' h.m. Sun's rijht ascension April 1 42 Given time 6 53 Right .iscension of the meridian 7 35 Answers nwirly to Pollux. EXAMPLE IV. What star will be on the nieridiar, September 1, at 5h. 37m. P. M.? ,| ^^ Sun's right ascension Sept. 1 10 41 Given time 5 37 Right ascension of the meridian 16 18 Answers nearly to Aniares. In all the preceding examples, the right ascension of the sun ought to iiave been calculated for the moment of llie star's passing the meridian, as will be more fully explained in the precepts of Tables XXX. XXXL TABLE IX. Sciui-diurnul and Semi-nocturnal arches. — This table exhibits half the time that a celestial object continues above the horizon when the latitude and declination are of the same name, or below when they are of a contrary name ; the former time being usually called the semi-diurnal arch, the latter the semi-nocturnal arch ; whence the time of rising and setting may be computed by the following rules : — ^ Tojind the time of the sun's rising and setting, and the length of the dan ^^"'^ night. Rule. Find tlie sun's declination at the top of the table, and the latitude in eitlier side column ; under the former, and opposite the latter, will be the time of the sun's setting if the latitude and declination are of the same name, but the time of rising if of different names. The time of rising, subtracted from 12 hours, will give the time of setting; or the time of setting, subtracted from \2 hours, will give the time of rising. The time of rising, being doubled, will give the length of the night; and the time of setting, being doubled, will give the length of the day. EXAMPLE I. Let it be required to find the time of the sun's rising and setting, with the length of the day and night, in latitude 51° north, the Dth of July, 1837. The sun's declination on the given day is 20" 52' north, or 21" nearly, under which, and against the latitude ol", stand 7h. 53m., the time of the sun's setting on the given day, in lat. 51° noitli, w'hich doubled, gives 15h. 4Gni., the length of the day ; and by subtracting 7h. 53m. from 12h., the remainder, 4h. 7m., is the time of the sun's rising, which doubled gives 6h. 14m. tlie Icngtli of the night. But, when the sun has 21" south declination in this latitude, tlie time of sun-sgtting be- comes 4h. 7m., the time of rising 7h. 53m., the length of the day 8h. 14m., and the length of the night J5h. 4Cm., as was the case nearly on the 2(Jth of November, 1837. EXAMPLE IL Let it be rci]Mireii to find the time of the sun's ris- ing, setting', anil! tlie Iciigtli of the day and night, at Boston, tlie I2ili of July, 1833. Under ^22", which is nearlv the declination on that (lay, and against 42° 23' or 42=' N., the laliliiile of Boston, stands the time of the h. m. sun's setting 7 25 Subtracted rrom I2h. leaves sun-rising 4 35 Suti-setting clu.ililed is the length of day 14 50 Sun-r.siiig douliled is the length of night. ... 9 10 EXAMPLE III. Required the time of the sun's rising and setting, and length of day, in latitude 34" 29' S., iNlay 15lh, 183ti. Under the declination 18° 57' or 19° N. h.m. and against the lat. 34° S. stands the 12 sun's rising 6 54 Time of sun's setting 5 G 2 The length of the day 10 19 And lih.54in. doubled is length of night 13 49 When a gr(?at degree of accuracy is required, proportional parts may be taken for th( minutes of latitude and declination. To find the time of rising and setting of stars lohosc declination docs not exceed 23 '28'. Enter Table IX. and find the star's declination at the top, and the latitude at tlie side ; under the former, and opposite to tlie latter, will be the semi-diurnal arch, when the latitude and declination are both north or both south; but if one be north and the other south, the difference between the Tabular number and 12 hours will be the semi-diurnal arch. Find the time of the star's coming to the meridian according to the precepts of Table VIII., and subtract therefrom tlxe semi-diurnal arch ; the difference will be the time of rising ; or by adding together the semi-diurnal arch, and the time of passing the meridian, the time of setting will be .obtained. , EXAMPLE IV. Required when the star Arcturus rises an December 1, in latitude 51° N. The time of the star's coming to the meridi- an, or siwithiuL', in the morning, is nearly. Then under star's declination 20° nearly, and against latitude 51°, stand Time of star's rising in the morning Added gives the time of the star's setting. .. Star sets 2r> minutes after 5 in the evening . d sets h. m. 9 30 7 47 1 .52 17 2S 12 5 2G EXAMPLE V. What time will the Dog-star Sirius rise and set at Philadelphia, Feb. 1.' Under the declination, which is near- h. m ly 1G° S., and against the latitude, 12 which is nearly 40° N., stand 6 56 Subtracted from ]2h. leaves half the time the star is above the horizon 5 4 The star conies to the meridian in the evening nearly at 9 40 Sum, rejecting 12 hours, is the time of set- ting in the morning 2 44 Dilference is the time of ri.sing in the evening 4 3U 390 CATALOGUE OF THE TABLES. In like manner may the rising and setting of any planet be found when tlie declination does not exceed 23"^ 28', and the time of the passage over the meridian is iinown. Suppose it was required to find the time of Jupiter's rising and setting, August 7, 1836, civil account, in the latitude of 52'--' N. In the Nautical Almanac for 1836, I find that Jupiter passes the meridian, August 6d. 23h. llm.,or August 7d. llh. 11m. A. ]VL, civil account, his declination being 20'-^ 17' N., or nearly 20"^. Under tlie declination 20°, and opposite to the latitude 52^, stand 7h. 51m., wliich is half the time Jupiter is above the horizon ; this subtracted from 12h. leaves half the time that he is below the horizon, 4h. 9m. j subtracting 7h. 51m. from llh. 11m. A. M. leaves 3h. 20m. A. M., August 7, for the time of Jupiter's rising; and added to llh. llin. gives 7h. 2m. P. M., August 7, for the time of Jupiter's setting, nearly. Suppose it was^ required to find the ti^ne of the moon's setting, Maj' 2, 1836, civil account, in the latitude of' 52° N. In the Nautical Almanac, pages iv. v., ■we find that the moon passes the meridian May Id. 12h. 35)n., or May 2d. Oh. 35m. A. M., civil account ; her declination being about 18° S. Under the declination 18°, and opposite lo the latitude 52°, stand 7h. 38m., half the time the moon is below the^horizon. Subtracting this from 12h. we get half the time she is above the iiorizon, 4h. 22m. ; adding this to Oh. 35in. we obtain the time of the moon's setting May 2d. 4J1. 57m., civil account. If we subtract 4h. 22m., from Oh. 35m. -f- 24h., we get the time of rising May Id. 20h. 13m. or May Id. 8h. 13m. P. M. If greater accuracy is required, you must find the time at Greenwich corresponding to this approximate time of her rising and setting; then find the moon's declination, and the right ascensions of the sun and moon for that moment of time. The former subtracted from the latter leaves the corrected time of the moon's passing the meridian. With these data repeat the operation. In this way we may obtain tlie time of rising and setting to any de- gree of accuracy. Instead of taking the difference of the right ascensions of tlie sun and moon, you may take the daily diflerence in the time of her coming to the meridian of Greenwich, and take a proportional part for the longitude of the place of observation (by means of Talile XXVIII.) and another proportional part, for the interval between the hour of passing tlie meridian, and the time of rising or setting.* It may be noted, that tlie numbers of Table IX. were calculated for the moment the sun's centre appears in the true horizon ; allowance ought to be made for the dip, parallax, and refraction, by whicli tlie sun and stars, when near the horizon, appear in general to be ele- vated above half a degree above their true place, and the moon as much below her true place. TABLE X. For Jinding the Distance of any Terrestrial Object at Sea. — The explanation and use of tliis table is given in Problems useful in Navigation, VIII. — XII., pages 95, 96. TABLE X. A. For the planets is similar to Table XIV. for the sun. The parallax is found by entering at the top with the planet's horizontal parallax, and at tlie side with the altitude of the planet ; the corresponding number is the parallax of the planet in altitude. TABLE XI. Tafile of Projwrlional Parts. — The method of using this table is given in the prejiarations necessary for working a lunar observation page 229. TABLE XII. 7////cr;/AV/mc^:o/i.— Explained in page 154. TABliE XIII. Dip of the Horizon.— Explained in page 154. TABLE XIV. Sun's Parallax in Mltitudc. — Explained in page 153. TABLE XV. Augmentation of tlie Moons Semi-diameter. ^The moon's semi-diameter given in tlie Nautical Almanac is the same as would be seen by a spectator su])poscd to be placed at tlie centre of the earth, or nearly the same as would be seen b}' a spectator on the surface of the eartli, when the moon is in the horizon. Now, when the moon is in the zenith of the spectator placed at the surface, her distance from him is less than when at the horizon by a semi-diameter of the earth ; consequently iier apparent semi-diameter must be aug- mented in proportion as the distance is decreased, that is, about one sixtieth part, or 16". At intermediate altitudes between the horizon and zenith, the augmentation is proportional to the sine of the altitude, and the value for every 5° or 10° of altitude is given in Table XV. The augmentation corresponding to the altitude being found in the table, must be added to the semi-diameter taken from the Nautical Alnianac for the time of observation reduced to Greenwich time, as is explained in the preparations necessary for working a lunar observation. TABLE XV 1. Dip ofj/te Sea at Different Distances from the Observer. — Explained in- page 155. TABLE XVII. For finding the Difference hdioccn 60' and the Correction of the JlUitude of a Star' or Planet, fur Parallax and Refraction ; also the corrcsjionding Logarithm.— The first pag(! of tliis table is to be used for a star, or for the planets Jupiter and Saturn, whose parallax is small. In other cases, that page of the table is to be used, wliich contains, at the top, the horizontal parallax of tiie planet, or comes the nearest to it; the tables being cal- culated for every 5" of iiorizontal parallax, from 0" to 35". TABLE XV III. For finding the Difference beticecn the Correction of the Snn's Altitude for Parallax and Refraction and 60', also a Logarithm corresponding thereto. — The manner of tailing the numbers from the two preceding tables, and the uses, to wliich they may be applied, are explained in the preparations necessary for working a lunar observation, page 230. &c. TABLE XIX. For finding a Correction and Logarithm vscd in the First Method of work * In strict iie-ss, llils liisl correctiim, found by the tahle, o'lglit to he deireased in the ratio of 2!li. to 211i. it" creased Uy llie daily diiieieiite of llie lime of tlie moon's jjassing tlie meridian. CATALOGUE OF THE TABLES. a9i tng a Lunar Observation. — The correction found in this table, being subtracted from 5!)' 42" will leave a remainder equal to the correction of the moon's altitude for parallax and re- fraction. It will be unnecessary here to point out the method of taking out tliis correction, as it is fully explained in the first pages of the table. It may not., however, be amiss to observe, that, after constructing the logarithms of this table, it was concluded to subtract therefro)n the greatest correction of the Table C corresponding, in order to render those corrections additive. Thus the logarithm corresponding to the alt. 30"-' and her. par. 54', was found at first to be 2372 ; and for the hor. par. 54' 10' the correction was 2358 ; so that if these numbers had been published, the correction for seconds of parallax would have been subtractive ; but as this would have been inconvenient, it was thought expedient to subtract from each of the numbers thus calculated, the greatest corresponding correction of Table C, which in the preceding example is 12; by this means the above numbers were reduced to 23t30 and 234(J respectively, and the corrections of Table C were rendered additive. In a similar manner the rest of the logarithms of the table were calculated. It is owing to this circumstance tliat the corrections in Table C for 0" of parallax are greater than for any other number. Similar metliods were used in calculating the other numbers of this table, and in arranging the Tables A and B. TABLE XX. Third Correction of the Apparent Distance. — The manner of finding the correction from this table is explained in the first method of correcting the apparent distance of the moon from the sun, page 231 ; and also at the bottom of the table. TABLE XXI. To reduce Longitude into Time, and the contrary. — In the first column of this table are contained degrees and minutes of longitude, in the second the corresponding hours and minutes, or minutes and seconds of time ; the other columns are a continuation of the first and second respectively. The use of this table will evidently appear by a few examples. EXAMPLE I. Required tlie time (.orresiiondin;; to .50' 3!'. Ii. in. s. Opposite 50° in rol. I is 3 •i^ i) 31' 2 1 Soiijriil time 3 2> 4 EX.\.MI'LE II. Required the de^'rees and minutes corresponding to till. 33jn. 203. Opposite Ch. 32m. Os in col. 4 is. 1 20 in cut. 2 is. 98° 20 98 20 TABLE XXII. Proportional Logarithms. — These logarithms arc very useful in finding 'Jie mean time at Greenwich corresponding to the true distance of the moon from the sun or star, as is explained in the examples of working a lunar observation. They may be also used like common logarithms, in working any proportion where the terms are given in degrees, minutes, and seconds ; or in hours, minutes and seconds, as in the example of taking a lunar observation by one observer. The table is extended only to 3° or 3h .; and if iny of the terms of a given proportion exceed 3° or 3h., you may take all the terms one grade lower ; that is, reckon degrees as minutes, minutes as seconds, &c., and work the proportion as before ; observing to write down the answer one grade higher ; that is, 3-ou must esti- mate minutes as degrees, seconds as minutes, &c. Instead of taking all the terms one grade lower, 3'ou may change two of the terms only, viz. one of the middle term? and one of the extreme terms ; thus the 1st and 3d or the 1st and 2d may be taken one grade less, and the fourth term v^fill be given correctly ; but if the fourth term be taken one grade less, you must, after working llie proportion, write it one grade higher, as is evident. To illustrate '.his, we shall give the following examples: — EXA.MPLE II. If the sun's declination i lianKcs Ifi' 13" in 24 hours, liow much will It chiuige in 81). 2m. ? Here the 1st and 3d terms must he taken one grade less. As 21m. Os... Aiith.Comp....Proii. Log. 9.121'J Is to W Id" Prop. Loir. 1.042fi So is 8m. 23 Prop. Log. 1 .3.^04 To 5* 28" ". Prop. Log. 1.5 1 79 EXA.MPLE IV. ir in Ifim. the sun rises 3° 27', how much will it rise in 3m. 10s. ? Here the 2d and 4tli terms must be taken one grade less. As lOm. Os Arith. Comp Prop. Log. 8.94S*f Is to a* 27" Prop. Log. 1.7173 So is 3m. 10s Prop. Log. 1.7547 To 0' 41" Prop. Log. 2.421C Which, taken one grade higher, is 41', the answer required. TABLE XXIII. For finding the Latitude Inj two Altitudes of the Sun. — The manner of using this table is explained in the examples of double altitudes given in pages 165 — 189. TABLE XXIV. JS'atural Sines. — This table contains the natural sine and cosine for every minute of the quadrant to the radius lOOOOD, and is to be entered at the top or bottom with the degrees, and at fhe side niarkcd M. with the minutes: the corresponding numbers EXAMPLE I. If.in I5in. 10s. of lime the sun rises 2° 40', how nuch will It rise in 3iii. 10s. at the same rate? As 15m. lO.s... Arith. Comp.... Prop. Log. 8.02.^ Is to 2° 40' Prop. Log. .05 12 So is 3m. 10s Prop. Log. 1.7.i47 To 33' 24" Pro)). Log. .731.' EXAMPLE III. If in I2li. the moim's hmgitiide vanes 7° 1', what vill it vary in 4!i. 20m. .' Here all the terms must be taken one grade less. As 12m. Os Arith. Comp Prop. Log. 8.82.39 l.-< to T I" Prop. Log. 1.4091 So is 4m. 203 Prop. Log. 1.0185 To 2' 32" 2'" Prop. Log. 1.8515 Whii h, taken one grade higher, is 2° 32' 2", the an- swer reipiired. 392 CATALOGUE OF THE TABLES. will be the natural sine and cosine respectively, observing that If the degrees are found at the top, the name sine, cosine, and M., must also be found at the top, and the contrary If tlie degrees are found at the bottom. Thus 4336G is the natural sine of 25° 42', or the cosine of 64° 18'. . We have given in tnis edition of the present table, in the outer columns of the margin, tables of proportional parts, for the purpose of finding nearly, by inspection, the, proportional part corresponding to any number of seconds in the proposed angle ; tlie seconds being found in the marginal column marked M., and the correction In the adjoining cohnnn. Thus, if we suppose that it were required to find the natural sine corresponding to 25° 42' 19" ; the difference of the sines of 25° 42' and 25° 43' is 26; being the same as at the top of the left- hand column of the table ; and In this column, and opposite to 19", In the column M., is the correction 8. Adding this to the above number 4336G, because tiie numbers are iiicrcusiiin-, we get 43374 for the sine of 25° 42' 19". In like manner, we find the cosine of the same, angle to be 90108 — 4 = 90104, using the riglit-hand columns, and subtracting because the numbers are decreasing ; observing, however, that the number 14 at tlie top of this column varies 1 from the difference between the cosines of 25° 42' and 25° 43', wiiich is only 13 ; so that the table may give in some cases a unit too much, between tlie angles 25° 42' and 25° 43' ; but this is, in general, of but little Importance, and wlien very great accui" V is required, the usual method of proportional parts is to be resorted to, using tlie actuai labular difference. Similar tables of proportional parts are Inserted in this edition of Tables XXVI. XXVII. for the like purpose. TABLE XXV. Logarithmic Sines, Tangents, and Secants to every Point and Quartet- Point of the Compass. — This table is to be used instead of Table XXVII. when tlie course is given In points. The course is to be found In the side colunm, and opposite tlierelo will be tlie log. sine, tangent, &c. ; the names being found at the top whon the course Is less than 4 points, otiierwise at the bottom. TABLE XXVI. Logarithms of Numbers. — The explanation and uses of this table are given in tlie article treating on logarithms in the body of the work, pages 28 — 33. TABLE XXVII. Logarithmic Sines, Tangents, and Secants. — Th}s table is explained in the corresponding article In the body of the work, pages 33 — 35. TABLE XXVIII. Fur reducing the Time of the Moon's Passage over the Meridia.ji of Greemcich, to the Time of her Passage over any other Meridian. — The manner of doing this Is explained In the corresponding part of the body of the work, page 170. TABLE XXIX. Correction of the Moons Mtitude for Parallax and Refraction. — The mean correction of the moon's altitude is given In this table for every degree of altitude from 10° to 90°. The manner of using this table Is explained in pages 172, 173. TABLES XXX. XXXI. For finding the Suns Right Ascension and Declination, the Equation of Time, and the Moon's Right Ascension. — The uses of these tables will be seen by the following examples, the values for apparent noon being taken from the Nautical Almanac, together with the horary motions. EXAMPLE I. Required tlie sun's right ascension in 1836, May Id. 6h. 35in., apparent time, astronomical account, at Greenwich. Here the horary motion by N. A. is 9s.551. h. m. s. R. A. May 1, at noon, by N. A 2 34 39.6 Hor. motion (Sh. X 9s.55l 57.3 For 35m. in Table XXX 5.5 R. A. May Id. 6h. 35m 2 35 42.4 EXAMPLE in. Required the moon's right ascension in 1836, Sept. lOd. 8h. '20m.30s., J«can time, astronomical account, at Greenwich. h. m. s. By N. A. Rt. As. Sept. ind. 9h. is ll' 16 9.96 Sejit. 10d.8h. isll 14 12.90 Horary motion in Rt. Ascen. .. 1 57.06=117".06 Proportional part fur 20ni. 30s. Table XXX 40" Add to R. A. Se|)t. lOd. 8h 11 14 13 nearly. Gives D's Rt. Asc. Sept. lOd. 8h. 20m. 30s 11 14 53 EXAMPLE V. Required the sun's declination in 1836, May Id. Gh. 3om. apparent time, astronomical account, at Green- wich. Here the honn' motion by N. A. is 44".85. Declination May l,at noon, by N. A. 15° 10' 19" N. Hor. motion (ill. X 44".85 4 29 For 35m. in Table XXX 26 15° 15' 14" If the declination had been decreasing, the horary motion would be .tiditractice instead oi additive, as in the above example. EXAMPLE n. Required the equation of time in 1836, July 9d. 8h 20m. apparent time, astronomical account, at Green- wich. Here the horary motion by N. A. is Os.364. ni. s. Equation of time July 9, at noon, by N. A. -\- \ 49.3 Hor. motion 8h. x Os.364 2.9 For 20m. in Table XXX .1 Right Ascension, 1836, July 9d. 8h. 20m. + 4 52.3 EXAMPLE IV. Required the moon's right ascension in 1836, May lid. 17h. 35m. 36s. incan time, astronomical account at Greenwich. ,, ^ ^ By N. A. Rt. As. May lid. 18h. is 55 40.89 May lid. 17h. is 53 48.72 Ilor. motion in Riglit Ascension 1 52.17=:] 12 J'' Proportional part for 35m. 36s. Table XXX., 66" = 106 Add to Right Asc. May lid. 17h. by N. A 53 49 nearly. Gives C 's Rt. Asc. May lid. 17h. 35in. 36s 54 55 EXAMPLE VI. Required the moon's declination in 1836, Sept. lOd. 8h. 20m. 30s. mean time, astronomical account, at Greenwich. Here the motion in declination for 10m. is by N. A 140i'.07. Motion for 20m. is 2 X 140i'.07=2SO".14 Table XXX. with 140" at top, and 30s. at side, in col. M. the correction, divided by 10, is 7 Motion in declina. in 20m. 303. 287". 1 = 4' 47". 1 Sub. from declination Sept. lOd 8h. 9 32 13".3 !■ '.-, dcclinn. Sept lOd. 8h. 20m. 30s. 9° 27' 26".2 N CATALOGUE OF THE TABLES. 393 EXAMPLE VII. R'-n)i"ired tlie moon's declination in 1S36, May Ud. 171i. 3jni. 3os. mean time, astrononiital aico.inl, a( Greenwich. Here llie motion in declination for 10m. is by N. A. 143".02. Motion for 30m. is 14;;'i.(i2 X 3 = 429''. I 5ni. is 143'i.02x 0.5 71 .5 Tab. XXX. 143" at top, and 3Gs. at side in col. M. the corr. divided by 10 is 8 .6 Mocion in declination is 509". 2 Add to declination May Ud. 17h. by i\. A j)'s declination May I Id. 17h. 3jin. Sis Here the correction 8' 29".2 is added, because the declination is increasing. = & 29i'.2 2 19 25 .9 2 27' 55" 1 N. If we wish to find accurately the time that any star comes to tlie meridian, or the time of rising or setting, we must talie the sun's right ascension for noon at Greenwich, from the Nautical Almanac ; then the star's right ascension from Table VIIL, and with these find the appro.ximate time of rising, setting, or coming to the meridian, by the method already given in the precepts for using Tables VIIL and IX. Then calculate the sun's right ascen- sion for this approximate time, and repeat the operation till the assumed and calculated times agree, and we shall have the true time required. To explain this method, we shall give the following examples : — To Jind the time tvhen a star comes to the meridian. EX.\MPLE I. At what time was Aldcharan on the merid a place in the longitude of 70° 50' W., Jan. 2, sea arconnt? Jan. 2, sea account, is Jan. 1, N. A., on which day the sun's R. A. at noon at h. Greenwich was 18 Aldeliaian's R. A 4h. 2Gm. 32s. Add 24 ian of 183G, m. s. 44 19 28 26 32 D.'tTerenre is the appro.vimate time 9 42 13 Now, calculating tlie sun's R. A. for this time in the long, of 70° 50' W. from h. m. s. Greenwich, we find it was 18 4t> 58 Aldebaran's R. A.-|-24h 28 26 32 App. time of coming to the meridian 9 39 34 EXAMPLE IL At what time was Pollux en the meridian of a place in the longitude of 70° 46' VV., March 31, 1836, sea account .-' March 31 , sea account, is March 30, N. A., on which day, at noon, the sun's right h. m. s. ascension was 36 G This, subtracted from R. A. of Pollux 7 35 17 Gives the approximate time of southing... 6 59 II R. A. for this time in long. 70° 46' VV. from Grecnw ich 37 53 Right ascension of Pollux 7 35 17 Diff. is app. time of comingto the meridian. 6 57 24 To Jind the time of rising or setting of a star. Rule. Enter Table IX. with the declination of the star at the top, and the latitude of the place at the side ; tlie corresponding number will be the time of the star's continuance above the horizon, when the latitude and declination are of the same name ; but if they are of dif- ferent names, the tabular number subtracted from 12h., will be tiie timeof continuance above the horizon. Add this time to the star's right ascension, if we wish to find the time of set- ting ; but subtract the former from the latter if we wish the time of rising. From this sum or difference subtract the sun's right ascension* corrected for tlie longitude of the ',)lace; the remaijider will be the approximate time sought.! Enter Table XXXI. with the distance of this approximate time from noon, and the horary variation of the sun's right ascension : the correction corresponding is to be added to the approximate time in the forenoon, but sub- tracted in the afternoon, and we shall have the corrected time of rising and setting. EXAMPLE L At what time did the star Aldebaran set May 24, '336, sea account, in the latitude of 38° 53' N. and the longitude of 77° VV., or oh. 8m. VV.? The star's declination was 1G° 10' N.,and the lati- tude 38° 53' N., corresponding to which in Table IX. is Gh. 54m. Star's right ascension 4 26 Sum 11 20 May 24, sea ace, or May 23 by N. A. at noon, sun's R. A 4h. Im. Hor. var. 10s. Corr. for long. 5h. 8m. VV.. 1 Sum, subtract 4 2 Remains approximate time of setting 7 18 Corr. in Tab. XXXI. for 7h. 20m., sub.... 1 Corrected time of setting, P. M 7 17 EXAMPLE II. At what time did the Dog-Star Sirius rise in tlit; latitude 39°2U'N., and tlie longitude of 76° 50' VV = ,'.h. 7m. 20s. VV., Jan. 2, 1836, sea account.' The star's declination is 16° 29' S., and the latitude is 39° 20' N., corresponding to which in Table IX. is nearly 6h. 56m Which subtracted from 12 Leaves the time of the star's being above the horizon 5 4 Subtract from star's R. A 6 38 Remainder 1 34 Add 24 Sum 25 34 Jan. 2, sea ace. or Jan. 1, by N. A. at noon, sun's R. A 18Ii. 44m. Hor. var. lis Corr. for long. 5h. 7m. 20s. VV. 1 Subtract the sum 18 45 Remains approxim. time of rising 6 49 Corr. in Tab. XXXI. for Gh. 49m., sub.. 1 Corr. time of rising in the afternoon 6 48 icreasing the number from which the subtraction is to be made, by 24 hours, when necessary. ejecting 2! Iioiirs when it exceeds 24 hcurs. If the time of rising or setting be more than 12'h., it wll bo inidniirht : but if less than 12h.. it v^i'l Ke before midnirlit. * Increasi T I^ejectins ^-i iniuis ^vum ii. ca*. ecus i;t iiu'ird. 11 me nine oi fiRer midnight ; but if less than 12h., it v^i'! Ke before midnight. 50 ;j«J4 CATALOGUE OF THE TABLES. TABLE XXXIL Variation of the Sun's Allitvdc in one J\Ihtutefroin JVoon. TABLE XXXllL To reduce the JVumbcrs of Table XXXII. to other given Intervals oj Time from A"oo7i. The method of using the two preceding tables is explained in the examples of finding the latitude by one altitude taken near noon, given in the body of the work, pages 201 — 20X TABLE XXXIV. Errors arising from a Deviation of V in the Surfaces of the Central Mirror. This table shows the error arising in measuring an angle by an instrument of reflection from a deviation of 1' in the parallelism of the surfaces of the central mirror, the line of intersection of those surfaces (produced if necessary) being perpendicular to the plane of the instrument. If the line of intersection be inclined to that plane, the numbers in the table must, in general, be decreased in proportion to the sine of the angle of inclination. The second, third, and fourth columns of the table are calculated upon the supposition that the surface of the horizon mirVor is inclined 80° to the axis of the telescope, or that the angle intercepted between the raj' incident on the horizon glass and the corresponding re- flected ray passing through the telescope is 20^, which is the case in circular instruments of De Boiida's construction, and on this supposition the errors of an instrument in measur- ing different angles may be ascertained by the rules in pages 136 and 143 ; when the inter- cepted angle is greater or less than 2(y-^, which is the case in most sextants and quadrants, the error in any measured angle corresponding to an inclination of the surfaces of 1', may be obtained as follows : — Find in the first column the intercepted angle, and the sum of that angle and the observed distance ; take the corresponding corrections from column 5th, and their difference will be the sought correction. In a circular instrument yon must find in tl.'e side column the sum and the difference of tbe intercepted angle and observed angle, and taiie out the corresponding corrections from column 5th : half their difference will be the sought correction. Having thus found the correction corresponding to 1', you may find the correction for other angles as in pages 1"36 and 143. TABLE XXXV. Correction for a Deviation of the Telescope of an Instrument of Re- flection from the Parallelism to the Pinnc of the Instrument. — The uses of this table are explained in pages 135, and 143. TABLE XXX VI. Correction of the Mean R fraction for Various Heights of the Barome- ter and Thermometer. — The use of this table is explained in page 154. TABLE XXXVII. Latitudes and Longitudes of the Fixed Stars. — This table contains the latitudes and longitudes of the principa:! fixed stars, adapted to the beginning of the year 1830, with the annual variations for precession nud the secular equation, by which the mean values at any time may be obtained, in lilie manner as the right ascensions and decli- nations are from Table VIII. ; by adding the corrrction of longitude after 1830, subtracting before 1830, and applying the correction of latitude with the same sign as in the table after 1830, but with a contrary sign before 1830. EXAMPLE I. Required the longitude and latitude of a Pegasi, July IG, 1828. La!)g. by Table XXXVII Us. 21° 07' 05" | Latitude by Table XXXVII "lO" 21' 45" N. Vajialion 1 year, 54ni., sub 1 13 Variation 1 year, 5.\ni., sub Loig. July 16, 1828 II 21 05 52 | Latitude July 16, 1828 19 24 45 N. ■ EXAMPLE II. Required the longitude and latitude of a Pegasi, July 1, 1832. Long by Table XXXVII lis. 21° 07' 05" Variation 21 years, add 2 5 Long. July 1, 1832 11 21 09 10 Latitude by Table XXXVII 19' 24' 45" N Variation 2.| years, add Latitude July I, 1832 19 24 45 N The latitudes and longitudes, thus obtained, are the mean values. When great accuracy is required, the corrections for the equation of the equinoxes, Table XL. and aberration, Table XLI. must be applied. TABLE XXXVIII. Reduction of Latitude and Horizontal Parallax. — This table con- tains the corrections to be subtracted from the latitude of the place of observation, and from the horizodtal parallax of the moon, given in the Nautical Almanac, in calculating eclipses of the sun or occultations. Thus, if the latitude of the place was 40'-', and the moon's horizontal parallax 57', the correction of latitude would be nearly — 11' 18", and that of parallax — 4".7, so that the reduced latitude would be 39^ 48' 42", "and the reduced parallax 56 55". 3. These values are to be used in occultations ; but in eclipses of the sun, this parallax is to be further decreased by 8". 6 for the sun's parallax. When the latitude is not given exactly in the table, the two nearest numbers must be found, and a proportional part of their difference is to be applied to one of the numbers, as usual. In calculating this table, the ellipticity of the earth was supposed equal to ^otj-, as in the third edition of La Lande's Astronomy, and in Vince's Astronomy. This value differs but little from -^-^■-^ and ^(j"5'(7"5> deduced by La Place from two lunar equations in the third volume of his immortal v/ork. La Micanujne Ciliste. In the second volume of the same work, he calculated the ellipticity to be ^-^g from the lengths of pendulums observed in different lati- tudes : this calculation corrected for a small mistake in the numerical co-efficient of i/ in the CATALOGUE OF TilC TAJ5LES. 3<)5 tenth of his equations A" becomes 3-^5, which does not clitrer very miicli from tlie value assumed in tiiis tal)le. TABLE XXX IX. Merration of the PUincts. — This table contains the aberration of the planets, to be applied to tlie true longitude or latitude, with the same sign as in tiie table. The argument at tb.e side is the elongation of the planet from the sun ; that is, the dilference of their geoc-entric longitudes, or its supplement to 3(J0 \ Thus, on July 1'.', IS'JO, the longi- tude of the sun was 3s. 2G° 38', the geo. long, of Venus 4s. 13-^ 23', their difii-icnce lU^ 45' is the elongation or distance from the inferior conjunction, corresponding to which is the aberration -|- 3" to be aiiplied to tlie true longitude given by the tables to obtain the ajiparent longitude. The aberration of IMercury is given at its greatest, least and mean distances from tlie sun. At the intermediate places, a proportional part of the differences of the nearest tabular numbers must be applied. TABLES XL. and XLl. Equation of the Equinnzes and Merration in Longitude. — Table XL. contains the equation of the equino.xes in longitude common to all the heavenly bodies. The argument is the longitude of the moon's ascending node; the signs of longitude being found at the top or bottom, and the degrees at the side, the corresponding number with its sign is the equation of the equinoxes in longitude. Table XLI. contains the aberration of the stars in longitude and latitude, to be calcu- lated by the rules at the bottom of the tables ; the signs of the argument being found at the top, and the degrees at the side,* taking proportional parts for minutes. The corrections of longitude found in these tables are to be applied, with their signs, to the mean longitude found in Table XXXVII., and the correction of latitude. Table XLI., is to be applied to the mean latitude deduced from Table XXXVII. Thus, on July IG, 1830, by the e.xamples at the bottom of Tables XL. XLI., the equation of the equinoxes was — 5". 3, and the aberration lu longitude -(- 1 1".3 ; tiiese corrections being applied to tlie mean longitude of the star deduced from Table XXXVII., lis. 2F 7' 32'', gives its apparent longitude lis. 21" 7' 38". In a similar manner the aberration in latitude, — 5".G, found at the bottom of Table XLI., applied to the mean latitude, 19° 24' 4.j" N., deduced from Table XXXVII., gives the ap- parent latitude of the star 19" 24' 30" N. TABLIOS XLII. XLIII. Merration and JVutation in R'ght ^Isccnsion and Drdination. — Table XLII. contains the aberration, and Table XLIII. the nutation in right ascension and declination, to be found by the rules at the bottom of the tables, and applied, with their signs, to the mean values deduced from Table VIII. Thus, by Table VIII., the riglit ascen- sion of a Pegasi, July IG, 1830, was 221i. Sfmi. 20s., and its declination 14" 18' N. The aberration of right ascension in time was nearly -|-0s. 8, in declination — 0".8 ; the nuta- tion in right ascension in time — Os.l, in declination -[- 0''.-'>, as appears i»y the examples at the bottom of the tables. The.se corrections being applied to the mean values, give the apparent right ascension 22h. r)Gm. 2!s., and tlie apparent declination 14" 18' N. The equa- tion of tlie ohliquity of the eclijjtic may be calculated by the rule at the bottom of the table. Thus, on July IG, 1830, the equation was — 9". I, which, applied to the mean obliquity 23° 27' 42".0, gives the apparent obliquity 23° 27' 32". 9. TABLE XLIV. j-inirnientntionoftkeMoonsScmi-diamctcr. — This table is divided into four parts, and is useful in finding the augmentation of the moon's semi-diameter by means of the altitude and longitude of the nonagesimal when the moon's altitude is unknown. The precepts for this calculation are given at the bottom of the table, and Ibr further illus- tration anr)ther example is added, in which it is required to find the augmentation at the commencement of the occultation calculated in Problem VII. of the Appendix, when the D's S. D. by the Nautical Almanac was IG' 13". 9, her true latitude I" 5.7 IJ" S., parallax in lat. 40' 2>".G, altitude of the nonagesimal 81" 17' 32", and the moon's apparent distance tVom the nonairesimal 51" 38' 2G", as in Example III. Prob. V. Appendix. In this case the arguments of Part 1. are 81" 17' 32" -f 51" 38' 2G",or nearly 4s. 12° 5G' and Os 29" 39', and the corresponding corrections -)- G". 00, -f- 4". 05, whoso sum is 10". 05. This in Part II. gives -|- 0". 10. In Part III., with the moon's true latitude, 1" 5.5' 11" S., and her par. in lat. 10' 23".(>. t!ie correction is — 0".10. The sum of these three parts is -f- 10". 05, wliich bein2 found at the side of Part IV., and the moon's horizontal S. D. IG' 18". 9 at the top, gives the corresponding correction -(- 0" .40. This connected with the th.-ce former parts -f- 10" .05, gives the sought augmentation 10". 45, or 10". 4, as in the example Prob. VII. Apjjendix. It may be observed that the calculation by Problem IV. will sometimes produce the supple- ment of the altitude of the nonagesimal; but this requires no alteration in the rule, since tJie result is tli" same whether the altitude or its sujijjlement is used. TABLi'i .XLV. Equation of Second Differences. — This table contains tlie equation of the second differences of the moon's motion, or the correction to be made on account of her unequal velocity between the times marked in the Nautical Almanac. The manner of ap- plying this correction is taught in Problems I. II. III. of the Appendi.x. TABLE XLVI. Variation of the Altitude of an Object, arising from a Chatige of 100 Seconds in the Declination. — This table is useful in finding the latitude by double altitudes of the sun, or any other object. It is explained in the precepts for such calcul.iiions, pages 189, 190, 191, &c. The table is to be entered at the top with tlie latitude of the place, and * The ilc!:rv<'.-! 'n tlrs and the following tables are to be foiinfl in the column marked \) on th? same hori- (^ontril linr \v !h the s^^mis. 'I'lms if the signs are at the top of the table, the dcsrrces mn.^t be foiiiiil n ;Iu k-rt r iliiiiiii, olhiTWiSc in the rifcht. 396 CATALOGUE O^ THE TABLES. at the side with the declination and altitude of the body ; the corresponding number is the variation of the altitude, in seconds, for a cliange of 100" in the declination. TABLES XLVn. XLVIIL are used in finding the First, Second and Third Corrections in Lyons Improved Method of ^corking a Lunar Observation. — The first of these tables gives the first and second corrections. The first correction is always taken out with the degrees and minutes marked at the top of the table. The second correction is also taken at the top when the apparent distance exceeds 90°, but at the bottom when the apparent distance is less than 90-'. TABIjLS XLIX. L. are used in finding the Correction for Parallax in Lyons Improved Method of xcorking a Lunar Observation. — The first of these tables gives the correction, sup- posing the parallax to be 35". It is to be entered at the top with the apparent distance, and at the side with the altitudes of the object ; the corresponding number is the correction fo: the horizontal parallax, 35". This is to be found in the side column of Table L., and the horizontal parallax at the top ; the corresponding number is the actual parallax in altitude, which is to be applied, with the same sign as in Table XLIX., to the apparent distance. Thus, if the app. dist. = C0°, *'s alt. =25^^, D's alt. =45°, the correction in Table XLIX. is — 20" ; and if the planet's horizontal parallax be 15", the corresponding correction in Table L. will be — D" ; to be applied as a third correction to the apparent distance. TABLE LI. To change mean solar time into sideral time. TABLE LI I. To change sideral time into mean solar time. TABLE Lill. Gives the variation of the compass very nearly as in the chart of P, Barlow. TABLE LIV. Table of Latitudes and Longitudes. — This table (as observed in the Preface) has been completely revised for this edition, and the latitudes and longitudes of a great number of places are added to those given in some of the former editions of this work. TABLE LV. Tide Table. — The explanation and uses of this table are given in the body of the work, in treating of the manner of computing the times of the tide, page 121, «S:c. TABLE LVI. Extracts from the Nautical Almanac of the numbers used in the exam- ples of lunar observations &c. TABLE LVII. shows, nearly, the error in Longitude in miles and tenths of a mile, occasioned by an error of one mile in the Latitude.- Thus, when the sun's altitude is 30°, the Latitude 30°, and the Polar distance 100°, the error is 8 tenths of a mile. The error affects the Longitude as follows : When in West Long, and J A. M. t ,. t „„„ j , \ decreased. \ When the cnrrection ia mark- ( increased. ) the time is found in Ool. J 1*. M. < o- » j increased. J ed X tlie Longitude ia ) decreased. \ When in East Long, and > A. M. i .. t „_„ i. j increased, > When the correction is mark- \ decreased. ) the tsjuc is found in Col. JP M. ) e jjon^. jS ( decreased, j ed X the Longitude ia ) increased. \ APPENDIX, CONTAINING METHODS OF DETERMINING THE LONGITUDE BY OBSERVATIONS OP ECLIPSES, OCCULTATIONS, &c. The longitude of a place may be determined in a very accurate manner, by observing th'. beginning or end of a solar eclipse, or occultation of a fixed star by the moon, or tlie differ ence between the times that the moon and a known fixed star pass the meridian. These observations, when made on land with a good telescope and well-regulated time-keeper, furnish by far tlie most accurate method of determining the longitude, and when made on board a ship without a telescope, will in general give it, with a greater degree of accuracy than any other method. For this reason we have inserted, in this Appendix, the usual rules of calculating such observations, by means of the Nautical Almanac. The first thing to be taken notice of, is the method of determining the longitude, latitude, &c. of the moon or other object, having regard to the unequal velocity between the times for which these quantities are given in the Nautical Almanac. This calculation is rendered much more simple by making use of the signs -|- and — , and performing addition and subtraction as in the introductory rules of algebra ; and as it is possible that these rules may not be familiar to some readers of this work, we have given an explanation, as far as will be necessary, in the present problems. Quantities witkuut a sign, or with the sign -\- prefixed, are called positive or affirmative, as 7 or -}- 7 ; and those to which the sign — is prefixed, are called negative, as — 7. Mdi- tion of quantities liavrng the same sign, that is, all affirmative or all negative, is performed by adding them as in common arithmetic, and prefixing the common sign. Thus the sum of -f- 4 and -p 3 is -f- 7. The sum of — 4, — 3, arid — 5, is — 12. When the quantities have not the same sign, the positive quantities must be added into one sum, and the negative into another, as ahore ; the difference of these two sums, with the sign of the greater sum prefixed, will be the sum of the proposed quantities. Thus the sum of -f- 14, — 7,-^5, and — 2, is found by adding -\- 14, -\- 5, whose sum is -f- 19 ; and then — 7 and — 2, whose sum is — 9 ; the dif- ference of 19 and 9 is 10, to which must be prefixed the sign of the greater number, 19, which is -f-, so that the sought sum is -|- 10. The ibllowing examples will illustrate these rules: — Add --4 --3 --7 — 2 Add + 4' Iff' + 2 5 Add — 4' 10' — 2 5 Add — 4' 10" + 2 5 Add + 1 — 1 Add + 6' 0" -2 15 -1-4 13 — 3 7 Sum + a 15 Slim — 6 15 Sum — 2 5 Sum + 12 Sum + 4 51 Subtraction is performed by changing the sign of the number to be subtracted from -f- to — , or from — to -{- ; and then adding the numbers by the preceding rule. Thus to sr.btract -\- 3 from -}- 7, the sign of -f- 3 must be changed, and the numbers — 3 and -f- 7 added together as in algebra, which, by the preceding rule, gives -j- 4 ; and if it were required to subtract — 3 from 7, the sign of — 3 must be changed, and -j- 3, -|- 7 added together ; the sum -\- 10 represents the souglit difference. It is not usual to make an actual change of the sign in any proposed question, it being sufficient to suppose the number to be subtracted to have a different sign from tliat prefixed to it, and to perform the operation accordingly. To illus- trate this, the fallowing examples are added : — From + 4' 10" From + 4' 10" From — 4' 10" From — 4' 10" From + 1 From — 1 From + 1 Sub. -1-2 5 Sill). —2 5 Sub. —25 Sub. +2 5 Sub. —1 Sub. —1 Sub. + 1 Rem. +25 Rem. + 6 15 Rem. — 2 5 Rem. — G 15 Rem. + 2 Rem. Rem. From 108 From — 108 From 108 From — 103 From— 201 Sub. 201 Sub. —201 Sub. — 201 Sub. 201 Sub. 108 Rem. — 93 Rem. + 93 Rem. + 309 Rem. — 309 Rem. — 309 Observing that when no sign is annexed to a quantity, the sign -|- is always understood to be prefixed. PROBLEM I. To find the hvgifude, latitude, S,"c. of the moon at any given time at Greemvich, having regard to the wiequdJ vdociti) between the times marked in the J^aulical Almanac ; ilif inJervuls of these limes being 12 hours. 398 TO FIND I'lIE LONGITUDE &c. OF THE SUN, MOON, &*,. RULE. Take from the Nautical Almanac the two longitudes, latitudes, &c. next preceding the iriven time at Greenwich, and the two immediately following it, and set them down in suc- cession below each other, prefixing the sign -|- to the southern latitudes or declinations, and the sign — to the northern. Subtract each of these quantities from tlie following for the first differences, and call the middle term arc A ; subtract each first difference from the fol- lowing for the second differences, and take the half sum or mean of them, wliich call the arc B, noting the signs of the quantities as in algebra. Find the difference between the given time and the second time taken from the Nautical Almanac, wliich call T ; then to its logarithm add the log. of A and the constant logarithm 5.3G452 ; the sum, rejecting 10 in the index, will be the logarithm of the proportional part,* to which prefix the sign of the arc A ; observing to express all these quantities in seconds. Enter Table XLV. with the arc B at tlie top and the time T at the side :t opposite to this will be the correction of second differences, to which prefix a different sign from tliat of the arc B, and plate it under the proportional part found above, and the second quantity taken from tlie Nautical Almanac, and connect these tliree quantities together as in addition in algebra : tiie sum will be the sought longitude, latitude, &c. ; the latitude or declination being south, if it has the sign -(- ; north, if it has the sign — . EXAMPLE I. Required the longitudes and latitudes of the moon, December 12, 1808, at 15h. 48m. 20s. and 17h. Im. 29s. app. time by astronomical computation at Greenwich, which correspond to the immersion and emersion of Spica, calculated in Problem VII. 1808. Dec. D long. N. A. I) s. ° ' " 12 noon. 10 45 20 12 niiiln. 6 17 51 30 13 noun. 6 25 2 54 13 luidn. 7 2 18 59 1st. aiff. 1 ° ( II 7 fi 16 A 7 11 18 7 It) 5 2.1 diff. + 2 40 58 -f5 2 + 2 6 37 + 4 47 -f- 1 29 52 == + 4 51.5 -f 51 18 IMMERSION. T = 3h, A = 7_ + 2 -l-C 17 Constant 5.36452 48m. 29s. = 13709s Log. 4.13701 11 18 =25878 ....Log. 4.41293 IG 52.2= 8212.2.... Log. 3.91446 51 36 Second longitude. 31.9 Table XLV. B = 4' 51".5 f) 20 56.3 D's longitude. D lat:t. S. 1st d;ir. / II — 34 21 A — 36 45 — 38 34 2d diff — 2 24 -1 4.9 B= — 2 06.5 5.30452 4.13701 •22i)o" Log. 3.34341 699.7 Log. 2.84494 r 37 Second latitude. 13.7 Table XLV. 15 = — 2' 6l'.5 -\- 1. 55 11.0 D's latitude south EMERSION. Constant 5.36452 :.51i. lin.29s.= 180893 Log. 4.25742 7 11 18=25878 Log. 4.41293 -f 3 36.0 = 10836 Log. 4.03487 -1-6 17 51 36 Second longitude. — 3.-..9 Table XLV. B. = 4' 54 '.5 6 20 51 36.1 D's longitude. These quantities are made use of in Problem VII, ■ 3G' 45" = — 2205" , 5.36452 4.2.i742 .Log. 3 34341 — 15 23.3 = — 923.3.... Log. 296535 6 + 2 + 37 Second latitude. 15.4 Table XLV. B. = - -f- 1 51 29.1 D's latitude south. EXAMPLE II. Piequired the longitudes and latitudes of the moon, June IG, 1806, at 2h. 49m. bDs.l, ana r)li. 3'lm. ()s.G, app. time, astronomical account at Greenwich, which correspond nearly to the beginning and end of the total eclipse of the sun as observed at Salem. 18J6. June. D long. N. A. 15d midn. 2 14 48 .58 16 noun. 2 22 6 19 16 mid 11. 2 29 27 12 17 ni'on. 3 6 50 47 1st diff. <> I II 7 17 21 A 7 21) 53 7 23 35 — 1 14 6 -}-3 32 — 34 13 4-2 42 + n (i 33 = + 3 7 -f 47 28 D lat. N. A. -1- 39 ,'■.3 -[- 40 46 -i-40 55 2d diff -f53 + 9 D = -I- 31 * This corrcition may also be found by proportion, by saying, As 12 hours ;ire to tin? time T, so is the arc A to the soiiirht proportional part; and "this nictlMd is the shorte.'st when T is an alii;uol part of 12 hoiws. Thus, if T be 3, 6, or 9 hours, the proporlimial part will be ^, i, or ^ of the arc A respectively. This method is made use of in Problem XVII. in interixilatiu!: the distance of the moon and sun. t If t!ie arc B consists of minutes and seccuuls, the correction fir minutes, tens of seconds, and units of :'econds, must be found separately : the sum of these three parts will be the sought correction. Proportional parts for the m'nutes of the time T may be taken in finding the correction of this table, wtiiui necessary. In this rule, part of the correction of the th'rd difference is neglected. Th':s part never exceeds 1 of the third difference, and rarely amounts to a small fracli(ui of a second. TZS TO FIND THE LONGITUDE, &c. OF THE SUN, MOON, &c. 399 BEGINNING AT 2h. 49m. 50s.l = T fleoond lonsitiide \ 7° 20' 5;{i' Prop. p;irt + l( 3 7 Table XLV — 2s. 22° O 19' 1 43 59.8 16.8 D 's longitude 2 23 50 2.0 Second latitude N A 40' 4G" Prop. part... B 31 Table XLV. D's latitude N.., — 0° 34' + 9 13" 37.0 28 END AT 5h. 34m. Gs.G = T. Second longitude 2s. 22° fi' 19" A 7° 20' 53" Prop, part + 3 24 35.3 U 3 7 Table XL,V — 23_2 D's longitude 2 25 30 31.1 Second latitude N —0° 34' 13" A 40' 4(," Prop, part + 18 55.0 B — 31" Tat)le XLV — 3.& ])'s latitude N. — 15 21.8 The proportional. parts of tlie arc A were calculated in this e.^ample by arithmetic with- out logarithms. By observations of the eclipse on that day, it was found that the moon's longitude was too great by fiS".."), and her Kititude too great by 11".4. These corrections are "applied to tlie above longitudes and latitudes, in calculating the eclipse in Problem VL Remark 1. It will not be necessary to take notice of the second differences in calculating the paralla.x or semi-diameter of the moon, or any of the solar elements useful in calculating an eclipse or occultation. In this case, the quantities immediately preceding and following the proposed time at Greenwich, must be taken from the Nautical Almanac ; and their dif- ference will be the arc A ; also the difference between the proposed time and that taken first from the Nautical Ahnanac is to be called the time T. Then, by proportion, as the interval between the times taken from the Nautical Almanac is to the time T, so is the arc A to tiie correction to be applied to the first quantity taken from the Nautical Almanac ; additive if increasing, subtractive if decreasing. This correction may also be found by logarithms as above, using the constant logarithms 5.30452 if tlie interval of the times in the Nautical Almanac is 12 hours, and 5.00349 if the interval is 24 hours. The proportional part of the moon's paralla,\ and semi-diameter may also be found by Table XL, and that of the solar elements by Tables XXX. XXXL, as taught in the explanation of these tables; these calculations being sometimes much facilitated in the new form of the Nautical Alma- nac, by means of the horary motions, which are given for several of the elements. To exemplify this, the rest of tlie quantities requisite in calculating the eclipse and occultation (Problem VI. VII.) are here found. EXAMPLE III. 1808. Dec. 12, midnight Dec. n, noon Difference .\ Pro. part T = 3h. 48m. 29s. Corrrspomlin;^ values.... Pro. part T :=51i. lin. 29s.. Corresponding values 180G. June IG, noon IG, midnight Differences A Pro. partT=2li 49in. SOs.l Corresponding values Pro. part T =5h. 34m. 6s.6 Corresponding values D S. D. Hi' 17" 1.9 18.9 2.5 19.5 1) S. D. IG' 27" IG t IG + 16 D H. P. • 59' 4G" 60 6 20 6.3 50 52.3 8.4 59 54.4 Dec. 12, noon 13, noon Difference A Pro. part T= loll. 48ni.29s. Corresponding values.... Pro. partT=17h. lui.29s. Corresponding values.... EXAMPLE IV. ]) H.P. 60' 21" 30 GO 34 3 + 13 0.7 + 3.1 27.7 GO 21.1 1.4 + 6.0 28.4 GO 27.0 1806. June 16, noon 17, noon Differences A Pro. partT= 21i. 49in..'')0s.l Corresponding values.... Pro. part T=5h. 34m. 6s. 6 Corresponding values.... O long. ©R. A. 8s. 20° 22' 4" 17h. 18m. 4j 4 8 21 23 10 17 22 29 .5 1 1 6 4 25 .1 40 15 o 54 .6 8 21 2 19 17 20 59 .0 43 21 3 • 8 .1 8 21 5 25 17 21 12 .5 long. OR. A. .84° 34' 18" 5h. 36ni . 2i)s 6 85 31 35 5 40 30 .0 57 17 4 9 .4 -f 6 45.4 + 29 .4 84 41 3.4 5 36 .50 .0 + 13 17.5 + 57 .9 84 47 35.5 5 37 18 .5 e sun's semi-diameter by the Nautical Almanac, June 13, 130G, was 15' 4G".3, and 1!), 1806, was 15' 45".9. Hence, at the above time, it was 15' 40". 1. This, in eclipses The semi-diameters thus found must be decreased 2" for inflexion, and augnwnted by the correction Table XLIV. in calculating an eclipse or occultation by Problem XIII. , or in deducing the longitude from observations by Problems VL VII. VIII. or IX. We may however, observe, that some astronomers neglect the correction of 2" for inflexion. The " . . -. ... June of the sun, must be decreased 3.^" for irradiation. Rcviiirk 2. The above rule for calculating the second differences of the lunar motions where the intervals in the Nautical Almanac are 12 hours, may be made use of whe« the intervals are any number of days, as is the case with the elements of the motions of the planets, by taking two longitudes, latitudes, &c. before, and two after, the given time at Greenwich, and thence deducing the arcs A. B, and the longitudes, latitudes, &c., and then making use, instead of T, cf the qunlient of the difference between the given time and that marked in the Nautical Almanac against the second longitude, &c. divided by the num- ber of lialf days in the given interval. Thus, if the interval is 1 day, the divisor is 2; if the interval is 4 days, the ciivisor is 8 ; and if the interval is 5 days, the divisor is 10. In like shall give the following examples : — 400 TO FIND THE HORARY JVlOTlOiN OF THE SUiN', MOON, &.c. EXAMPLE V. Required the right ascension of Venus, 1836, August, 23d. ICh. 40m. mean time, astronomi cal account, at Greenwicli. Times. Right Ascen. h. in. s. iigust 29 7 44 23.51 ^ 23 7 45 24.05 24 7 4(j 32.84 25 7 47 49. C5 1st diff. 00.54 08.79 16.81 8.25 8.02 B=8.13 Second right ascension 7h. 45ni. 24s.05 A = lin.08s.79 Proportional part... 47.77 B= &S.13 Table XLV —.83 Venua's right ascension 71i. 4eni. lOs.99 In this example, the intervals in the Nautical Almanac being 1 day, we must divide the time, ICh. 40m., by 2, to get T = 8h 20m. EXAMPLE VL Required the declination of Mars, 1836, June, 14d. 13h. 30m., mean time, astronomical account, at Greenwich. Times. Declinations. " 1 ■ II June 13 15 14 19.2 14 15 27 56.9 15 15 41 25.0 16 15 54 43.4 1st diff. 13 37.7 = 13 28.1 13 18.4 — 9.6 — 9.7 B=— 9.6 Second declination 15° 27' 5G''.9 N. A = 13'28".l Proportional part.... 7 34.6 B= — 9".6 Table XLV 1 .2 Mars's declination 15° 35' 32".7 N In this example, as in the last, we divide the time, 13h. 30m., by 2, to get T: EXAMPLE VII. :Gh. 45m. Required the logarithm of the distance of Jupiter from the earth, 1836, June, 2d. 8h., mean time, astronomical account, at Greenwicli. Times. June 1 2 3 4 Log. Dist. 0.7803725 0.7810545 0.7817232 0.7823787 1st djff. G820 A = 6687 6555 2d diff. — 133 — 132 ! = — 132 Second distance 0.7810545 A=: 6687 Proportional part 2229 B= — 132 TableXLV 15 Log. distance Jupiter and Earth 0.7812789 In this example, we also divide the time, 8h., by 2, to get T =4h. EXAMPLE VIII. Required the moon's declinatioij, 1836, January, 16d. 9h. 45m. 50s., mean time, astro- nomical account, at Greenwich. Times. Declination S d. h. » / II Jan. 16 8 26 26 58.3 9 28 08.4 •10 29 06.4 11 29 52.3 70.1 A = 58.0 45.9 2d diff. — 12.1 — 12.1 r— 12.1 Second declination 26° 28' 08''.4 S. A = 58«.0 Prop, part Tab. XXX. 44 .3 B= — 12".l TableXLV 1.1 Moon's declination 26° 28' 5.3".8 S. In this example, the time, 45m. 50s., divided by 5, and changing minutes into hours, Slc. gives T = !'h. 10m., which is used in entering Table XLV. with B= — 12". 1, to find the corresponding correction, l".l. We may, however, remark, that the second differences of the right ascensions and declinations of the moon may generally be neglected as insensible, because these quantities are given in the Nautical Almanac, for every hour, and their second differences are quite small. The same is to be observed relative to Die sun's longitude, right ascension, the equation of time, &c. The second difference of the sun's declination may sometimes be 3" or 4", but is, in general, insensible. The second differences of the log. radius vector must be taken, if we wish to obtain the logarithm correct in the seventh deci- mal place. We can always judge of the necessity' of using the second differences, by observ- ing that the greatest error from neglecting them altogether is equal to J B. Thus, in the last example, the greatest error from neo-lecting the consideration of the second differences is J B = I X 12".l = 1".5. PROBLEM II. To find the Iwrarxj motion of the moon in longitude, latiiiide, fyc. at any given time at Greemoicli ; supposing the intervals of the times i?i the JVautical Almanac to be 12 hours. RULE. Tuke from the Nautical Almanac the four longitudes, latitudes, &c., two immediately preceding the given time at Greenwich, and two immediately following. Prefix the sign -j- to the southern latitudes or declinations, and the sign — to the northern. Then find the first and second differences, the arc B, and the time T, as in Problem I. The mean of the two first differences, noticing the signs as in algebra, will be the approxijnate motion in 12 hours. To tlie proportional logarithm of one fourth part of the time T, add the proportionai logarithm of the arc B : the sum will be the proportional logarithm of the correction of the approximate motion, to be applied to it with the same sign as the arc C, and the corrected TO FIND THE HORARY MOTIONS OF THE SUN, iMOON, &c. 401 motion of the moon in 12 hours will be obtained," which, being divided by 12, will give the horary motion. EXAMPLE I. Required the horary motions of the moon in longitude, Dec. 12, 1803,at loh. 4Sm. 298., and 17h. Im. 2'>s., apparent time, at Greenwich. This corresponds to Example I., preceding, in which T is 3h. 48m. 29s., or oh. Im. 299. The two Jiisi differences in longitude are T^ (i' IG", and 7" 11' 18"; their mean, 7"^ 8' 47", is the appro.ximate motion in 12 hours, and the arc B is 4' 54".5. The rest of the calculation is as follows : — At 151». 48in. 29s. T = 3li. 48m. 2Ds. Ardi B 4' 54" .5 I'n>ii. Lo?. l..')G44 \ T 57 7 Prop. Log . 4985 • Corr. -f 1 33 Prop. Log. 2.0529 Approx. million 7 8 47 Motion 12 hours 7 10 20 In 1 hour 35 51.7 In a similar manv two first differences At 17Ii. Im. 293. T = 51i. Im. 29s. 1.5G44 \ T U\. 15m. 223. I'rop. Log . 3781 Corr. +23 Prop. Log . 1.9425 Approx. motion ..7 8 47 Motion 12 liours.. 7 10 50 In 1 hour 35 51.2 In a similar manner, if the horary motion in latitude was required at 12d. 17h. /{3m., the iw'o firsL dff'creaies in latitude arc — 34' 21", and — 3G' 45" ; their mean, — 35' 33", is the approximate motion in 12 hours. The correction found by the above rule with the time T, 5h. 33m., and the arc B = — 2' G''.5, is — 59", whence the true motion in 12 hours is — 3G' 32", which, divided by 12, gives the horary motion — 3' 2". 7. The negative sign — indicates that the north polar distance is decreasing, the positive sign -(- that it is increasing. In the present example, the north polar distance was decreasing, and as the latitude was south, it was also decreasing, as is evident. EXAMPLE II. Required the horary motions of the moon in longitude, June IC, 1806, at 2h. 49m. 50s.l, and .^>h. 34m. (is.G, apparent time, by astronomical computation, at Greenwich. This corresponds In Example II. preceding, in which T is 2h. 49m. 50s. 1, or 5h. 34in Gs.6; the two first differences are'7° 17' 21", and 7° 29' 53", the mean of which, 7" 19' 7' is the approximate motion in 12 hours. The arch B is -j- 3' 7". At 21i. 49m. 50s.l =T. Ar(hB=-t- 3' 7" Prop. Log. 1.7616 ^TinieT= 42 27 Prop. Log. C274 Correction + 44 Prop. Log. 2.3893 Approx. motion 7 19 7 Motion In 12 hours.. 7 19 51 Motion in 1 hour 3G 39.2 At5h. 34m. Cs.G^T. 1.7616 \ T = 111. 23m. 32s. Prop Log. 33,34 Correction + 1 27 Prop. Log. 2.0950 Motion in 12 hours 7 Blotion in 1 hour.. REMARKS. 1. When it is required to find the motion of the moon in longitude or latitude, for any given interval of time, the motion in 12 hours must be found for the middle of that interval; 2. In calculating an nccultation of a star by the moon, the relative horary motion in longi- tude is the same as the horary motion of the moon, because the star is at rest ; but in calcu- lating a solar eclipse, the sun's horary motion must be found from the Nautical Almanao in the maiincr mentioned below, and subtracted from the moon's horary motion in longitude: the remainder will be the horary motion of the inoon from the sun in longitude. Thus, on the Kith of JvuK', 180(1, tiie sun's horary motion was 2' 23".], which, being subtracted fron. the horary motions found in Exaniple II., 3u' 39". 2, and 36' 42" .8, leaves the correspondino horary motions of tiic moon from the sun in longitude 34' 10". 1, and 34' 19". 7. As the sun has no sensible motion in latitude, the relative horary motion of the moon from the sun in latitude, is the same as the true horary motion of the moon in latitui'o. * Tlie motion in 12 hours tlius olitained, which, for distinct ion, will be called tlie arc M, is not porfectly u', curate, sin i- Ihi-lh rd ami hiL'lier orders of diflVrences are neglected ; but the horary motion deduced there- from is abundantly siiiii -.ent for tlie purpose of projecting an eclipse or occultation. " \\'hL-n creater accuracj- is required, the ill ril d.tiVrences maybe taken into accomit in the following manner: — Having found the second ilijf'n-eiici:< as .-ibove directed, subtract the first of them from the second, not'nsr the sijrns sw in algebra, and call the reuiaiwlir the arc 6. ICnter Table .\LV. with this arc at the top, and the time T at the" side, and take out ilie i orrespoiul ng correition, which is to be imreased by one sixth part of tlie arc U, w'JhoiU noting the sgns. To the i|uanti'ty thus found is to be prefi.ved a sign different from that of the arc h, and then it is to be appl ed to the arc M, w th its sign, to obtain the true motion in 12 hours. This, in the above example, llie .-ecnnd diji'crcnccii of long tudeare+ 5' 2" + 4' 47". Si'litrai ting the former from the latter, leaves the third d If rem e or arc 6 = — 1.5". Correspond ng to this and the time T 3h. 48rn 2;!s. in Table XLV.. is I'M), wliii h, in reased by one sixth of i=z2".5, gives the sought correction 4".l or 4", to wlikh must bie prefixed the s gn -|- (bei ause the s gn of 6 is negative), making it + 4". Th s, connei ted w th the are M = -f-7'' 10' 20', g ves the true motion in 12 hours, 7° 10' 24", whence the horary motion is 3.5' 52". In a similar manner, if the th rd d (Terences were noticed in the above example for find.ng the horary motion in latitude, III'; two sccmid diff'errnce.i — 2' 21" and — 1' 49", the arc 4;= 4- 35", the corteclioa of the motion iii taluKirs — 3i;i 32" is — 10" ; making it — 36' 42", or 3' 3".5 per hour. 51 accu 402 TO FIND THE ECLIPTIC CONJUNCTION OF THE SUN, &c. 3. The hor.iry motions of the sun in longitude were formerly given in page iii. of the Nautical Almanac ; but they are discontinued in its new form, so tiiat we mubl now deduce the»horary motion from the daily difference of longitude, by dividing it by ^4. EXAMPLE III. Thus, if it were required to find the sun's horary motion in longitude, in the interval be- tween July 1 and July 2, 1S36, mean time, astronomical account, at Greenwich; we should have the longitude at noon, July 1, 99" 35' 03".0; July 2, 100" 32' 13". 7. Their difference is 57' 10". 7 ; dividing it by 24, we get the sun's horary motion in longitude 2' 22' .9. The same method may be used in finding the horary motions of the planets, neglecting the second difterences ; but if we wish to notice the second diff'erences, we may proceed as in the three preceding examples, making use of the arc3 A, 13, T, found as in Remark 2. Problem I. EXAMPLE IV. Required the -horary motion iS Venus in right ascension, 1836, August 23d. IGh. 40m., mean lime, astronomical account, .' t Greenwich. Here we have, as in Example V. of tlie nreceding prolilein, T:=8li. 20ni. ; aiul the mean of llie two first dillerences, lin. 00s..54, and Ini. 08s.79, is llie apprii.xiiiiate motion, lm.0-ls.G6; also the arch B = + 8s. 13 Prop. Log. 3.124 JT=2h. 5ni I'rtip. Log. 158 , Correction 5s.66 Prop. Log. 3.282 Appro.ximate motion 1 04 .66 Motion of Venus in 24 hours. ... Im. lOs.32 Dividing it by 24, we get 2s. 93, which represents the horary motion of Ve- nus in riglit ascension, corresponding to August 23d. Ibh. 40m. The hnrarij motion of the moon in right ascension or declination is fouml, by inspection, in the JVaullrnl JJhaanac, taking the differences of the two successive numbers in the Kautical Almanac, the one before, the other after, the time for which the horary motion is wanted. EXAMPLE V. Required the horary motion of the moon in right ascension and declination, between the hours of 10 and 11, on the 4th of August, 1830, mean time, astronomical account, at Greenwich. 1833, August 4d. lOh. Moon's right ascension 3h. 07m.20s.l5 Declination 17° 55' 44".2 N. 4 11 3 09 19 .57 18 03 30 .4 The differences !* the horary motioi.^ !n R. A. Im. 59s.42 In deilnation 10' 4C".2 These horary motions correspond very nearly to the middle of the time between lOh. insl Ilh., that is to say, lOh. 30m. PROBLEM IIL To Jlnd (he lime of the ecliptic conjunction or opposition of the moon tvith the sun, ii planet, or a fixed star. The time of the ecliptic conjunction of the sun and moon is the same as the time of nevr moon given for the meridian of Greenwich in page xii. of the month of the Nautical Alma- nac. Thus, in January, 1836, the ecliptic conjunction is on the I7lh day, at 20h. 27m.8, mean time, at Greenwich. The time of the ecliptic opposition of the sun and moon is the same as at the time of full moon given in the same page of the Nautical Almanac. Thus the full moon or ecliptic opposition in May, 1836, was 3dd. 3h. 59ni.7, at Greenwich. The time of the ecliptic conjunction i« easily coiuputed from the geocentric longitudes of the objects ; and we have here inserted the rule, adapted to the calculation of the conjunc- tion of the sun and moon, which, with a slight modification, will answer for any planet, or a fixed star. RULE. Take from the Nautical Almanac the two longitudes of the sun and moon at the noon and midnight* preceding the time of the conjunction, and the two immediately following. Sub- trait the Inngiludes'of the sun from uiose of the moon, noting the signs as in algebra ; the remainders will represt^nt the distances of the sun from t*'.e moon on the ecliptic. Subtract each of these from the following to obtain the first differences, and call the middle term the arch A ; subtract each of these differences from the following for the second differences, and take their half sum or mean for the arc B, noting the signs as in algebra. To the constant logarithm 4.63548, add the arithmetical complement of the log. of the arch A in seconds, and the log. of the second of the above-found distances in seconds ; the • The sun's longitude at midnight is the mean of the longitudes on the preceding and following noona nearly TO FLND THE ECLIPTIC CONJUNCTION OF THE SUN, &c. 403 Kurn, rejecting 10 in the index, will Be the logarithm of the approximate value of T in seconds. With this time T at the side of Table XLV., and the arc B at the top, find the equation of second diflerences, the logarithm of which, added to the two first logarithms used in find- ing T, will, in rejecting 10 in the index, give tlie logarithm of the correction of the approxi- mate time T in seconds, to be applied to it with the same sign as the arc B, and the mean time of the conjunction at Greenwich, counted from the second noon or midnight, taken from the Nautical Almanac, will be obtained. From which the time of conjunction under any other meridian may be easily obtained, by adding to it the longitude in time when cast, or subtracting when irest. Remark 1. When the time of the ecliptic conjunction of the moon and a planet is re- quired, the longitudes of the planet must be found by Problem I. for the noon and midnight immediately preceding, and those immediately following the time of the conjunction, and these are to be used in the above note instead of the sun's longitudes. If the ecliptic con- i' unction of the moon with a fixed star is required, its longitude must be found in Tabh- ^XXVII., and corrected for the ecjuation of the equinoxes and aberration by Tables XL XL!., as shown in the explanation of tliose tables. This longitude is to be used instead of the sun's, in the above rule. Tlie longitude and latitude of tlie star may also be com- puted more accurately, from the right ascension and declination, given in the Nautical Almanac, by tlie method in Problem XIX. of this Appendix, whenever the star used is one of the 100 stars, whose places are given for every 10 days in the Nautical Almanac. Remark 2. By the same rule, tlie time, when the moon is at any distance from the sun. may be found, by increasing the sun's longitudes given in the Nautical Almanac, by the quantity tiie moon is supposed to be distant from the sun, counted according to the order of the signs ; tlicn supposing a fictions sun to move so as to have these increased longitudes at the corresponding times, and finding by the above rule the time of conjunction of the moon with \.\\\s fictions sun, which will be the sought time when the moon is at the proposed distance from the sun. Thus, to find the time of the first, second, or third quarter of the moon, the sun's longitudes must be increased 3, G, or 9 signs respectively (rejecting, as usual, 12 signs when the sun exceeds that quantity). Thus, if the first quarter of the moon which happened in the afternoon, July 21, 183G, was required : The sun's longitudes increased by 3 signs give the longitudes of the fictions sun, July 20d. 12h. ; 21d. Oh.; 21d. 12h., and 22d. Oh. respectively, 208° 11' 10".0; 208^ 3D' 4S".8; 209° OS' 27".7, and 209° 37' 0G".7. The longitudes of the moon corresponding are 200'^ 22' 15".8 ; 207° 03' 18".4 ; 213° 49' 32" .4, and 220° 41' 13" .8. Hence the time of the conjunction of the moon with the fictions sun found by the above rule, was July 21 d. 3h. 5m. at Greenwich, which is the time of the ♦vst quarter required. In a similar manner, by increasing the longitudes of a planet or a star, the time may be found when the moon is at any proposed distance from it. EXAMPLE. Required the mean time of the ecliptic conjunction of the sun and moon in January, 1836 1838, Jan. 17d. Oh 17 IQ 18 18 12 J) long. long (I 284 41 2,1.1—296 28 53.5 292 07 35.4 — 29G 59 2i;.G 299 31 33.0 — 297 29 59.8 30(3 52 13.2 — 298 00 32.5 Distances. » ( // — 11 47 28.4 — 4 51 51.2 2 01 33.2 8 51 40.7 1st difference. o I II 6 55 37.2 A = 6 53 21.4 6 50 07.5 2(1 difference. — 3 I5 = — 2 19 8 lfi.9 44.8 Constant 4.G3.")48 A = G° 53' 24''.4 = 24804".4 Arith. Conip. Log. 5.U0547 2ddis. 4 51 51 .2 = 17511 .2 Log. 4.24332 3049S3. :8Il. 23m. 18s Log. 4.48427 — 30 4.63548 5.60547 .1.0''. 1.23300 Table XLV. Corr. 17".l.. Correction 303 Lo". 1 .47395 T Correction Conjunction 6h. 27m. 4Ss. past midnight, on January 17d. 20h. 27m. 4Ss., mean time at Greenwich ; being the same as in the Nautical Almanac. The time of conjunction under any other meridian, as for example, 30° W., is found by subtracting the longitude 2h. from 20h. 27m. 4Ss., which leaves Ih'h. 27m. 48s. If the longitude had been 30° E., the time of conjunction would have been 22h. 27m. 48s. The usual method of calculating the parallaxes in eclipses of the sun or occultations, is that by using the longitude and latitude of the nonagesimal or ninetieth degree of the ecliptic above the horizon ; or, in other words, the longitude and complement of the latitude of the zenith, relative to the ecliptic. Several methods have been proposed for calculating the altitude and longitude of this point, which are required at eacli of the phases. The following, which is an improvement I have made on that given in La Lande's Astronomy, seems well adapted to the purpose, since several of the logarithms are the same at each of the phases, which much abridges the calculation, and on this account it admits of considerable simplifications, by a table hke that on page 403. The method of making these calculations will first be given at full length, and then in the abridged form, by means of the proposed table. The process of calculating the parallaxes with the right ascensions and declinations, instead of the longi- tudes and latitudes of the bodies, adapted particularly to the new form of the Nautical Almanac, will be given towards the end of 'his Appendix. 404 rO FIND THE ALTITUDE, &c. OF THE NONAGESLMAl.. PROBLEM IV. Gtveii the apparent time at the place of observation, counted from noon to noo7i, according to the manner of astronomers, the snn^s right ascension, and the latitude of the place, reduced on account of the sphej-oidal fgure of the earth, by subtracting the reduction of latitude, Table XXXVIII. ; to fnd the altitude and longitude of the nonagesimal degree of the ecliptic. RULE NOT ABRIDGED. Add G hours to the sum of the sun's right ascension and the apparent time of observation, and call the sum the time T, rejecting 24 hours when it exceeds that quantity. Seek for this time in the column of hours of Table XXVH., supposing that marked A. M. to be increased by 12 hours, as in the astronomical computation. The corresponding log. co- tan o-ent being found, is to be marked in the first and second columns, as in the following examples. If the reduced latitude is north, subtract it from 90-^ ; if south, add it to 90° ; the sum or difference will be the polar distance. Take half of this, and half the obliquity of the ecliptic, and find their difference and sum. Place the log. cosine of the difference in the first column its \oa. sine in the second column ; the log. secant of the sum in the first column, its log. .•osecant in the second column, and its log. tangent in the third. The sum of the logarithms in the first column, rejecting 20 in the index, will be the log. tangent of the arc G ; the sum of these in the second column, rejecting 20 in the index, wilfbe the log. tangent of the arc F ; these arches being less than 90° when the time T is found in the column A. M.. otherwise greater. This rule is general except in places situated within the polar circles. Within the north polar circle, the supplement of F to 300° instead of F, must be taken ; within the south polar circle, the supplement of G to 180° must be taken instead of G ; the other terms remaining unaltered. In all cases, the longitude of the nonagesimal is equal to the sum of the arcs F, G, thus found, and 90° ; rejecting 300"^ when the sum exceeds that quantity. Place in the third column the log. cosine of G, and the log. secant of F; the sum of the three logarithms of this column, rejecting 20 in the index, will be the log. tangent of half the altitude of the nonagesimal. EXAMPLE. Required the altitude and longitude of the nonagesimal at Salem, in the reduced latitude 42° 22' 4" N., June 15, 1806, at 22h. Cm. 18s. 1, apparent ti)ne, or 22h. Cm. 21s.5, mean time, by astronomical computation, when, by the Nautical Almanac, the sun's right ascension waa 5h. 30m. 50s., and the obliquity of the ecliptic 23° 27' 4S". The sum of the apparent time, sun's right ascension, and G hours, rejecting 24 hours, is 9h. 43m. 8s.l =T. The polar distance is 47° 37' 50" ; its half is 23° 48' 58", and the half obliquity 11° 43' 54" ; hence their difference is 12° 5' 4", their sum 35° 32' 52". The rest of the calculation is as follows : — Column 1. Diff. la* 5' 4" Cosine 9.99027 Sum 35 32 52 .Secant 10.089.57 T 91i. 43m.8s.l P. M Cotang 9.4882J G. 159° 42' 0" Tang 9.5G810 F 173 40 31 90 Column 2. Sine 9.32088 Cosecant 10.23554 9.48823 F Tang 9.04468 Column 3. Tangent 9.85403 G. Cosine 9 97215 F. Secant 10.00095 33= 59' 25" Tang. 9.82883 67 58 50 = Alt. nonagesimal. Sum C3 22 31, rejecting 3C0°, is the longitude of the nonagesimal. The two upper logarithms of the first and second columns, and the upper logarithm of the thifd column, vary but little in several centuries ; and as these numbers occur twice in cal- culating a partial eclipse or occultation, and four times in a total or annular eclipse or transit, it will fend considerably to abridge the calculations, to have a table like the following, con- taining their values for various places, for the obliquity 23° 27' 40", with the variations for an increase of 100" in the latitude or obhquity. The logarithms A, B, C, of the table, were calculated in the following manner : — In north latitudes subtract the reduced latitude from 90°, in south latitudes add the reduced latitude to 90°, the sum or difference will be the polar distance : take half of this and half of the obliquity of the ecliptic, 11° 43' 50", and find the sum and difference. Then, ■ ^ , Loo-. A is equal to the log. cosine of the difference added to the log. secant ot the sum, rejecting 20 in the index. Log. C is equal to the log. tangent of the sum. Log. B is equal to the log. tangent of the difference, increasing the index by 10, less the Thus, for Salem, in the reduced latitude 42° 22' 4", the half polar distance is 23° 48' 58" Ihe half obliquity 11° 43' 50", the difference 12° 5' 8", the sum 35° 32' 48". Difference.. 12° 5' 8" Cosine 9.99027 Tangent + 10 = 19.330C.5 Sum 35 32 48 Secant 10.08956 Tangent = C == 9.85403 Sum A 0.07983 Difference B _9£7663 TO FIND THE ALTITUDE, &c. OF THE NONAGESIMAL. 405 In tills way the logarithms may be found for places not included iu the table. The changes for an increase of 100" in the latitude or obliquity, are found by repeating the operation with these increased values, and ascertaining the corresponding changes in the values of A, B, C. These logarithms are given to six places of figures, thougii, in general, five will be quite sufficient, since the latitude and longitude of the nonagesimal are rarely required to a greater degree of accuracy than 10". Table ca culated for the obliquity 23° 27' 40 ". Reduced Var .A. Var. B. Var.C. riaces. Lattu le A. + 100". B. + 00". C. + 100 '. JV.irtl Lai. Oi,l. •l.,.,. 01.1. L;U. Obi. o , II + _ — — Alhanv, 4-> 27 13 0.079''7n 53 97 9.47,5733 293 739 9.85,3323 223 223 Berlin, 50 20 1 21 9S o.O:;h;o8 fl.(i:;2i(;u 49 49 75 9.324 1:!5 9.331054 618 GOO 1099 loao 9.771197 9.773925 240 240 240 210 Caiiiliriilye, K.. .. C.'iiiilir;d{;e, A.. . . A-?. 1-2 (i.()^i)!5;) 97 9.4783S3 288 733 9.8553.55 222 ').)0 nul)liii Obs 53 12 7 o.Oiionyo 48 73 9.301 IGG G70 1155 9.7G3705 2I2 212 ICilinbiirch, 55 4ii 2 0.0551; 18 47 07 9.233401 878 137G 9.741011 249 219 Greenwich Obs... 51 17 2S 0.0;i34nu 49 77 9.34fi39r) 5G2 1038 9.780232 238 2:18 Iliivanna, aT 3 3t 0.120000 (14 148 9.597i;5S 95 51G 1U.003015 210 210 Kinilerliook, 49. 11 37 0.080 u;3 52 9R 9.478455 289 733 9.855411 222 222 Lan aster, 39 51 18 0.084(i48 51 101 9.501042 249 C88 9.874005 219 2T9 Leon [. Obs 3r, If) .52 0.09 lf80 55 112 9.529940 202 G34 9.902005 21G 21 G 51 31 19 17 29 3(1 0.0o34;)li 0.101899 49 58 77 125 9.345714 9.5';i510 564 1.52 1040 577 9.779944 9.940447 238 212 238 212 N'aichez, Oxibrd Obs 51 34 2S 0.0ti2963 51) 77 9.34058G .576 1051 9.777800 239 239 Par s, 48 39 3S 45 51 41 0.0;i6207 0.0S4828 50 53 83 104 9.394413 9.501872 4.52 2-18 918 687 9.802327 9.874738 233 219 233 219 Pli ladelphJa, Rii hmoiid Obs. .. 51 l(i .51 ; 0.0i;3482 49 78 9.346.576 5n2 1038 9.780308 238 2;)8 Kiitland, ^3 24 32 0.0778fifi .52 95 9.4G5330 312 760 9.845548 224 224 42 19 22 52 4 38 0.079832 0.127485 52 GG 98 157 9.476637 9.607G02 291 78 731 500 9.8540 IG 10.027183 211 222 211 .... Plaie Prob. VII.. These logarithms are calculated for tiic obliquity 23*^ 27' 40". The columns marked Lat. represent the variations of A, B, C, for an increase of 100" in the reduced lat. The column Obi. represents the variations of A, B, C, for an increase of 100" in the obliquity of the ecliptic. The signs must be chanrred if the latitude or obliquity is less than 23° 27' 40' , which is used in calculating the table. EXAMPLE. Required the values of A, B, C, for Salem, when the obliquity is 23° 27' 48". Tab'ilar numbers 0.079832 9.476G37 9.854016 Varati(infor + 8 obliquity +8 —.58 -f 18 Sought values A = 0.079810 D = 9.47G579 C = 9.85 1034 jibridged method of calculating the altitude and longitude of the nonagesimal by the preceding table. Add together the sun's right ascension, the apparent time at the place of observation, (counted from noon to noon), and G hours : the sum, rejecting 24 or 48 hours if greater than those quantities, is to be called the time T : this is to be sought for in the column of houra of Tabic XXVII., supposing the column marked A. M. to be increased 12 iiours, as in t!ie astronomical computation.* The corresponding log. cotangent, added to the log. A of the table, gives the log. tangent of the arc G : this added to the log. B of the table, rejecting 10 in the inde.^, will be the log. tangent of the arc F ; these arcs being less than !)()" whei: T is found in the column A. M., otherwise greater. \ [This rule is general, e-xcejit in ])laces situated within the polar circles, which is a case that very rarely occurs. Within tbe vortli polar circle, the supplement of F to 3G0° is to be used instead of F ; within the so7Uli polar circle, the supplement of G to 180° is to be taken instead of G; the other terms remaining unaltered.] Then the longitude of the nonagesimal is equal to the sum of the arcs F, G, and 90°, neglecting as usual 360° when the sum exceeds that quantity. To the tabular log. C, add the log. cosine of the arc G, and the k)g. secant of the arc F : the sum, rejecting 20 in the index, will be the log. tangent of half the altitude of the nonagesimal. t * Thus, if the t'lne T is 5 hours, it must be called 5h. P. M. ; if T is 14 hours, it must be called 2h. A. M. Ill making use of a common table of logarithms, you must turn the lime T into degrees, and make use of the log cotangent of its half. To prevent mistake, it may be prober to remark, that, in finding 'I', we must add the appureid t me, and not the mean t'lne ; for if the mean time be used, we ought to use also the inr.an right ascension; whereas the o/iparcnj right ascension is given in the Nautical Almanac; and this must be added to the apparent time in finding T. t The arcs F, G, are acute, when the time T is found in the column A. M , otherwise obtuse. This i.-< fasily remembered from the circumstan'-e that a is the first letter of acute and A. M. Some writers liave not taken nntii e of the cases of the values of F, G, within tlie polar circles. J Stri'tly speaking, the quant ty thus obtained is the distance between the north pole of the ecliptic and •i« zenith of the place, whith, in southern laftndes, and between the tropii s, is frequently the supplement .3f tititude of the nonagesimal The above form is made use of to simplify the rules for ajiplying iJi 406 TO FIND THE ALTITUDE. &c. OF THE iNONAGESIMAL. EXAMPLE I. Required the altitudes and longitudes of the nonagesimal at Salem, June IG, 1806, at tlte times of the beginning and end of the eclipse, calculated in Problem VI. BEGINNING OF THE ECLIPSE. ti. m. s. 5 36 .50.0 O riglit ascension. 22 6 18.1 j9pparent time. 6 A 0.07984 9 43 8.1 Cotang. 9.4S826 G 159-42' 0" 90 .Tang. 9.56810 Cosine 9.97215 B 9.47653 C 9.85403 P 173 40 31 Tans 9.04468 Secant 10.00265 9.82883 63 22 31 = long. N. 33 59 25 Tang Altitude nonagesimal.. G7 58 50 END OF THE ECLIPSE, h. m. s. 5 37 13.5 O right ascension. 50 34.6 Apparent time. 6 A 0.07984 1 2 27 53.1 Cotang . 8.78470 4° 11' 13" Tang. 8.86454 Cosine 9.93834 90 B 9.47658 C 9.85403 15 23 Tang. 8.34112 Secant 10.00010 ' 26 35=long. N. 35 23 53 Tang. 9.85297 Altitude nonagesimal.. 70 57 46 EXAMPLE II. Required the altitudes and longitudes of the nonagesimal at the times and places men- tioned in the Example of Problem VII. IMMERSION, h. m. s. 17 20 59 O right ascension. 16 57 29 Apparent time. _6 A 0.12748 T 16 18 28 Cotang. 9.8009& G 40-18' 7" Tang. 9.92846. 90 B 9.607CO osine 9.88233 C 10.02718 F 18 57 48 Tang. 9.53606 Secant 10.02423 149 15 55 =long. N. 40 38 46 Tang. 9.93374 Altitude nonagesimal.. 81 17 32 EMERSION, h. m. s. 17 21 12.5 O right ascension. 18 10 29 Apparent time. 6 A 0.12748 T 17 31 41.5 Cotang. 9.94622 G 49° 50' 18" Tang. 10.07370 Cosino 9.80953 90 B 9.60760 C 10.02718 F 25 38 40 Tang. 9.68130 Secant 10.04504 165 28 58 = long. N. 37 17 39 Tang. 9.88175 Altitude nonagesimal.. 74 35 18 In these calculations, it is usuai l^ take the sun's right ascension, and the apparent times, to tenths of a second, and to take proportional parts for the seconds and tenths in finding the logarithms. Thus, in Example I., in finding the log. cotangent of 9h. 43m. 8s. 1, the near- est logarithms are 9.48849, 9.48804, corresponding to the times 9h. 43m. 4s., 9h. 43m. 12s. These logarithms differ 45, the times 8s.; and the difference between 9h. 43m. 4s., and 9h. 43m. 8s. 1, is 4s. 1. Hence, 8s. : 45 . . 4s. 1 : 23, the correction to be subtracted from the first log. 9.48849 (because it is decreasing), tv obtain the sought log. cotangent 9.48826. PR0BLE3I V. Given the altitude and longitude of the nonagesimal ; the longitude, latitude, and hori- zontal parallax of the moon, and the latitude of the place of obsenation ; to find the moon^s parallax in latitude and longitude. RULE BY C03IM0N LOGARITHMS. From the horizontal parallax of the moon, subtract its correction from Table XXXVIII., corresponding to the latitude of the place ; the remainder, in occultations of a fixed star, will be the reduced parallax ; but in solar eclipses, tliis quantity is to be diminished by the sun's horizontal parallax, 8". 6,* to obtain the reduced parallax. To the logarithm of the reduced parallax in seconds, add the log. sine of the altitude of the nonagesimal, and tlie log. secant of the moon's true latitude .;t the sum, rejecting" 20 in the index, will be a constant log. From the moon's true longitude,! increased by 360° if necessary, subtract the longitude of the nonagesimal ; the remainder will be the vioon's distance from the nonngesimal, which, if less than 180°, is to be called the arc D, other- wise its supplement to SCO" is to be called the arc D. To the constant logarithm add the log. sine of D ; tlie sum, rejecting 10 in the index, will be the logarithm of the approzimate parallax in longitude in seconds, which add to the arc D ; then take the log. sine of the sum, and add it to the constant logarithm, rejecting 10 m tlie index, and the logarithm of the corrected parallax will be obtained. This will, in genera], be sufliciently exact; but when great accuracy is required, the operation may be again repeated, by adding the arc D to the collected parallax ; t then to the log. sine of the sum add the constant logarithm, rejecting 10 in the index, and the logarithm of the parallax in longitude P will be obtained. This is parallaxes. It is immaterial uhctlier the altitude of the nonagesimal, or Its supplement, is made use of in Table XLIV. * This is nearly the mean value of the sun's parallax ; but it will be more accurate to use the actual value D's it is given in page 266 of the Nautical Almanac. t Corrected for the errors of the tables, when known. J This sum D -|- cor. par. -s nearly erpial to !)+ P, the apparent distance of the mnon from the nosiagesi- inal to be made use of in Table XLIV., in finding the augmuntat on of the moon's S. U TO FIND THE PARALLAXES OF THE MOON. 407 to be added to the true longitude of the moon when her distance from the nonagesimal is kss than 130^, otherwise subtracted to obtain her apparent longitude. If the true latitude of the moon is south, prefix the sign -}-to it; i£ north, the sign — . Then to the logaritlim of the reduced paralla.t in seconds, add the log. cosine of tlie altitude of th« nonagesimal, and the log. cosine of the moon's apparent latitude;* the sum, rejecting 20 in the inde.x, will be the logarithm of the first part of the parallax in latitude in seconds, to which prefix lh(^ sign -f- wiien the altitude of the nonagesimal is less than iH)^, otherwise the sign — ; this being added to the true latitude of the moon, due regard being paid to the signs, will give her approximate latitude. To the logarithm of the reduced parallax in seconds, add the log. sine of the altitude of the nonagesimal, the log. sine of the moon's approximate latitude, and the log. cosine of the sum of the arcs 13 and .^ P ; the sum, rejecting 30 in the index, will be the logarithm of the second part of the parallax in latitude in seconds, to which prefix the sign — when the arcs D -{- h P, and the approximate polar distance,! are both greater or both less than !)0°, otherwise the sign -\-; this term, being connected with the approximate latitude, will give the apparent latitude of the moon,+ which will be south if -f-j nortii if — . The moon's true latitude subtracted irom her apparent latitude, noticing the signs, will give the parallax in latitude. BY PROPORTIONAL LOGARITHMS. The above rule will answer in calculating by proportional logarithms, with the following alterations. When the log. sine occurs, read log. cosecant; for log. cosine, read log. secant; for log. secant, read log. cosine ; and for log. cosecant, read log. sine. The parallaxes may be calculated to the nearest second by proportional logarithms. When greater accuracy is required, common logarithms must be made use of. To illustrate this rule, the following examples; corresponding to the timesof the beginning and end of the total eclipse of the sun, of June 1(3, 180G, as observed at Salem, are given. The elements necessary fur tiiis purpose have already been calculated in Problems i. and IV. For greater accuracy, the longitudes and latitudes of the moon are corrected for the errors — 58".5 in longitude, and — 11". 4 in latitude, which were found by comparing several observations of the eclipse made at different places. EXA3IPLE I. Given the altitude of the nonagesimal G7° 58' 50", its longitude G3° 22' 31"; the longi- tude of the moon S'i"^ 4;)' 3".5, her latitude 24' 27" .4 N.,her horizontal parallax GO' 24". 1 ; the latitftde of the place of observation 42'-' 33' 30" ; required the parallaxes in longitude and latitude. The correction in Table XXXVIII. corresponding to the latitude 42° 33' 30", and parallax GO' 24". 1, is 5''.G ; this, and the sun's horizontal parallax, 8". 8, subtracted from the moon's hori- zontal parallax, GO' 24''.1, leaves the rcrfi^cefZ parallax GO' 9".7 = 3G09".7. The longitude of the nonagesimal, 03-^22' 31", subtracted from the moon's longitude, 83° 49' 3", leaves the moon's distance from the nonagesimal, 20° 2G' 32'', equal to the arc D, because it is less than 180**. CALCULATION BY COMMON LOGARITHMS. Keduced parallax 3C09' ,\lt!tude iKMiaKesimal C7 58 j)'s true latitude 24 .7 50 27.4 32 29 1 47 19 46.8 3.5 50.3 Log. Sine Sec. Sine Log. Sine Log. Sine Log. 3.55747 9.90710 10.00001 Reduced parallax Altitude nonagesimal D's app. latitude 1 part paral. 1353".3 = ]) 's true latitude ]) 's approx. latitude Reduced parallax Altitude nonagesimal D-f AP 2 part parallax Approx. latitude 5 '3 app. latitude The sun'B parallax it will be more accura 30Q9".7 G7 53 50 = + ^>' 33".3 — 24 27 .4 Log. 3..55747 Cosine 9.57394 Cosine 10.0000* Constant log. D 20 20 3.52453 9.54315 3.00773 9.54970 3.52458 3.07428 9.54980 3.524,58 3.07438 Log. 3.13141 .\ppr. paralla.t 1 ICO" =19 — 1 54 .1 Sine 6.743 D + Appr. parallai 20 46 Constant log. Cor. paralla.x = 1187'' = 19 D + cor. parallax 20 40 Constant log. Par. long. P 1180".8 = 19 D's true longitude 83 49 I) 's app. longitude 84 8 20 3G 25 — 1 .7 — I .54.1 — 1 55 8 ■brmerly used le to use 8". 6, Log. 3.557 Sine 9.967 Cosine 9.971 Log. 0.238 or 1' 55".8 N. i.s above, is 8".8 j as in the rule. * In solar eclipses, the apparent latitude is so small that its log. cos. may be put equal to 10.00000. In occul- Catioiis, you st calculate the first part of the parallax in altitude by approximation, making use of the true latitude instead of the apparent in the above rule, and deducing the approximate value of the first part ol the parallax; this applied to the t^ue latitude will give the approximate apparent latitude, with which the operat.on is to be repeated, and the first part of the parallax will be obtained to a autficient degree of exactness. fThe apparent polar distance is found by adding -j- 90' to the approximate latitude, due regard being had to the signs. To be perfectly accurate, the apparent instead of the approximate latitude ought to be made u.=e of in this part of the calculation, and the logaritlnus of this term ought to be increased by the log. secant les? radius of .| P ; but these corrections are too small to affect the result. In calculating the second part of the parallax in latitude, it will be sufficient to take the logarithm to three or four places of the decimals. J Th s rule g-ves the apparent latitude in all cases ; but it may not be amiss to observe, that, in several late publicat ons, the cases where the moon is between the zenith and the elevated-pole are by mistake neglected. 408 TO FIND THE PARALLAXES OF THE MOON. EXAMPLE II. Given tlie altitude of the nonagesimal 70° 57' 4G", its longitude 05° 20' 30" ; the longi- tude of the moon 85° 29' 32".0, her latitude 15' 10" .4 N., her horizontal j)arallas 00' 27".0 ; tlie latitude of the place of observation 42° 33' 30" ; required the parallaxes in longitude and 'atitude. The correction in Table XXXVHI., corresponding to the latitude 42° 33' 30", and paral- lax 60' 27", is 5".0 ; this, and the sun's horizontal parallax, 8".8, subtracted from the moon's horizontal parallax, 00' 27".0, leaves the reduced parallax 00' 12". 0. Tiie longitude of the nonagesimal, 05° 20' 30", subtracted from the moon's longitude increased by 3()0°, viz. 445° 2'.)' 33", leaves the vioon's distavcc from the nonagesimal 350° 2' 57", the supplement of which to i;00° is 9° 57' 3'', equal to the arch D. Reduced [laialliix Altitude iioiKigcs. J>'s true latitude 70 60' 15 12' 40 10 .G 4 Cciiistaiit log. D 9 57 3 Approv. purallux 9 50 n-f-a|ipr. parallax Constant log. 10 6 53 Corrected parallax 10 D-f-C(ir. parallax Constant log. 10 7 3 Par. long. P 10 00 D's true longitude 8.5 29 32 .G BY PROPORTIONAL LOGARITHMS Prop. Log. 0.4753 Cosecant 10.0214 Cosine lO.OD-aO 0.5000 Cosecant 10.7024 Prop. Log. 1.2024 Cosecant 10.7.551 0.5000 Prop. Log. 1.2554 Cosecant 10.7553 0.51)00 Prop. Log. 1.25.53 Reduced parallax 00' 12".6 Altitude nonages. 70 57 4G ])'s app. latitude Prop. Log. 0.4756 Seiaul 10.4865 Secant 10.0000 1 part par. lat. Sr 19 38 .5 Prop. I-og. 0.9C21 I)'s true latitude — 15 10 .4 D's appro.x. lat. -f 4 2« . 1 Cosecant 12.8361 Reduced jiar. Altitude nonages. D4-5 P 10 2 3 Prop. Log. 0.475G Cosecant 10.0244 Secant 10.0067 2 part par. lat. -\- 4 .4 + 4 28 .1 Prop. Log. 3.3928 Approx. latitude Appare.1t lat. + 4 32 .5 or 4' 32" .5 S. 5 's app. longitude 85 19 32.6 I EXAMPLE III. Required the parallaxes in longitude and latitude at the time of the occultation of Spica December 12, 1808. at the times and places mentioned in the Example of Problem VIL Reduced parallax .50' 50". 9 Alt. nonagesimal 81 17 32 J's true latitude 1 55 11 Constant log. D 50 52 1 Approx. parallax 45 55 D -{■ appr. parallax 51 37 .56 Constant log. Correited parallax 46 25 D -f- cor. par.allax 51 38 26 Constant log. Par. long. P -|- 43 25 > 's true longitude 200 7 56 .3 > 's app. long. 200 54 21 3 Reduced p.arallax .111. nonagesimal D's true latitude 74 1 .59 35 51 53 18 29 .0 1 Coiuitant log. D 35 22 38 App». parallax 33 26 D -|- appr. par. <.\instant Log. 35 56 4 Corrected parallax 33 54 D -\- cnrr. )iar. -Constant log. 35 56 32 Par. long. P 1>'8 true long. + 200 33 51 54 36 1 >'3 iipp. long. 201 25 30 1 Prop. Lo Cosecant" 10.0050 Cosi::e 9.9908 IMMERSION. 0.4782 Cosecant 4830 10.1103 Prop. Log 5933 Cosecant 10.1057 4830 Prop. Log 5887 Cosecant 10.1056 4830 Prop. Log 5836 ]) 's app. latitude* 1 part par. lat. -}- 9' 311.3 D's true lat. -f 1 .55 11 .0 J's approx. lat. -]- 2 4 14 .3 Reduced parallax Alt. nonagesimal D + i P 0.4783 .Secant 10.8199 Secant 10.0003 51 15 13 2 part par. lat. -f 1 20 .3 ])'s approx. lat. -f 2 4 14 .3 D'sapp. lat. -|- 2 5 34 .6 South. 5's par. latitude -}- 10 23 .6 Prop. Log. 1.2984 Cosecant 11.4421 Prop. Log. 0.4789 Cosecant 10.0050 Secant 10.2035 Prop. Log. 2.1288 EMERSION Prop. Log. 0.4780 Cosecant 10.0159 Cosine 9.9998 Cosecant 0.4937 10.2374 Prop. Log 7311 Cosecant 10.2315 4937 Prop. Log 72.52 Cosecant 10.2314 4937 Prop. Log. 7251 0.4780 10 5755 ]) 's approx. latitude Secant 10.0003 1 part par. lat. -f- 15 54".2 Prop. Log 1.0538 5 's true lat. -(- 1 51 29 .1 ]) 's approx. lat. -\- 2 7 23 .3 Co.secant 11.4313 Eeduced parallax Alt. nonagesimal D-i-iP !5 39 35 Propu Log Cosecant Secant 0.4780 10.01.59 10.0902 2 part par. lat. -f- 1 44 .2 Prop. Log 2.0154 ]) 's approx. lat. -|- 2 7 23.3 B 's apjiar. lat. -J- 2 9 7 .5 South. B 's parallax lat. -|- 17 38.4 * The moon's true lat'tude, 1° 5.5' II", must first be used, its log. secant being 10. 00112, which give the Isl part p:irall;>\ 9' '.V, which, added to the true latitude of the moon, gives the 'aj)prox;nuite lat tude nearly a 4' 14", the log. secant of which is 10.0003, as above. The calculation for the emersion is made in asimilaj manner. TO FIND THK LON'GITUDE BY AN ECLIPSE OF THE SUN. 409 Having thus explained the method of calculating the parallaxes of the moon, it now re- mains to give the rules for finding tJie longitude by eclipses and occultations. The main object in tliese calculations is to determine, from tiie observed beginning or end of tlie eclipse or occultation, the precise time of the ecliptic conjunction of the sun, or star and moon, free from the effects of parallax, counted on the meridian of the place of observation, since the difference of the times of conjunction, obtained in this manner at two places, will be their difference of longitude. If the lunar and solar tables were perfectly correct, t!ie longitude might Le determined by taking the difference between the time of conjunction given in the Nautical Almanac, and that deduced from the observations of the eclijise or occultation ; but it is much more accurate to compare the times deduced from observations actually made at the places for v\-hich the difference of longitude is sought. There are two different methods of finding the ecliptic conjunction, according as the latitude of the moon is supposed to be accurately known or not. If the latitude was given correctly by the lunar tables, or was accurately' known by other observations, the ecliptic conjunction, and the longitude of the place, niigiit be determined by each of the phases of the eclipse or occultation, by tlie method given in Problems VIII. and IX. But the moon's latitude not being generally given to a sufiicient degree of accuracy, it is usual to combine together the observations of the begin- ning and end of the eclipse or occultation, or the beginning and end of total darkness in a total eclip.se, or the two internal contacts of an annular eclipse, to ascertain the error of the moon's latitude, by the method given in Problems VI. and VII. In making the calculations in these Problems, it will be necessary to know nearly the longitude of the place, in order to find tiie supposed time at Greenwich, so as to take out the elements from the Nautical Almanac ; and if the longitude deduced from the observation should diller considerably, the operation must be repeated with the longitude obtained by this operation. PROBLEM VI. Given the latitude of the place, and the apparent times of the beginning and end of a solar eclipse, counted from noon to noon, according to the method of astronomers, tofnd the longitude of the place of observation. In the rule for solving this problem, references will be made to figure 12, Plate XIII, in which DSE represents a small arc of the ecliptic; S, the place of the centre of the sun supposed at rest ; F, L, the apparent places of the centre of the moon at the beginning and end of the eclipse respectively 5 FD, SC, and AEL, are perpendicular to UE ; FA parallel to DE, and SB perpendicular to FL. Then it is evident tiiat FD, LE, represent the apparent latitudes of the moon, which fall below DE if south, above if north ; and SF, SL, represent the sums of the corrected semi-diameters of the sun and moon, at the beginning and end of tlie eclipse respectively. RULE.* To tlie apparent times of the beginning and end of the eclipse, add the estimated longitude of the place in time if it is icest, but subtract if east ; the sum or difference will be the sup- posed time at Greenwich, corresponding to which, in the Nautical Almanac, find, by Prob- lem I., tiie moon's semi-diameter, horizontal parallax, longitude and latitude,! and the sun's semi-diameter, longitude, and right ascension ; also the moon's horary motion from the sun, by Problem II. Decrease the sun's semi-diameter 3^" for irradiation, and the remainder will be his corrected semi-diameter. Decrease the moon's semi-diameter 2" for inflexion, if it be thought necessary, and to the remainder add the correction in Table XLIV.;t the sum will be tiie moon's corrected semi-diameter. Find also, in the Nautical Almanac, the ob- liquity of the ecliptic. With these elements, and the apparent time at the place of observation, calculate the alti- tudes and longitudes of the nonagesimal, by Problem IV. ; the parallaxes in longitude and latitude, and the moon's apparent longitudes and latitudes, by Problem V. Take tlie difference between the apjiarent longitudes of the moon at the bcginninT and end of the eclipse, and subtract therefrom the difference of the sun's longitudes at the same time; the remainder will be the relative motion in longitude DE or FA. 'i'lie relative motion in latitude AL is found by taking the difference of the moon's apparent latitudes at the beginning and end of the eclipse, if they are botii north, or both south, but their sum, if one be north, the other south. From the logarithm FA, increasing tlie index by 10, subtract the logaritliiii of AL ; the remainder will be the log. tangent of the an; the same lalculatioiis by jnt^aiisof the right ascensions and dei linations, adaptMijr the rules to the new form of the Nautical Almanac. The same is to be observed relative to the fol- lowin;; Problems, VH. VIU., &c. t Corre: led for the errors of the tables in lonjilnde and latitude, when known. 1 This correition must be found after the altitude and longitude of the nonaijesimal are calrulated. 5 Th's rule is equally true, whether the latitude be of the same or dfTereut names, [f the latitudes are equal, and of the sajiie name, the angle DSB wdl be 90^. If they are equal, but of d fierent nanu^s, the angle DSB may be taken at ute or obtuse, since, in that rase, the ans;le FSB is flO". Striitly speaking, when the points F li fall on dlTerent sdes of the line DE. the angle DSB is greater or Ics-^ than 90°, according as tb 5"2 410 TO FIND THE LONGITUDE BY AN ECLIPSE OF THE SUN. cosecant of the angle of inclination, add the logarithm of the relative motion in longitude FA; the sum, rejecting 10 in trie index, will be the logarithm of the apparent motion of the moon FL on lier relative orbit. Then, in the triangle SFL, the sides bF, SL, represent the sums of tiie corrected semi-diameters of the sun and moon at the beginning and end of the eclipse, and these, with the relative motion FL, are given to find the angle FSB (^by Case VI. Obi. Trig.) Thus, to the log. arith. comp. of FL, add the logarithm of the sum of SF and SL, and the logarithm of their difference; the sum, rejecting ]0 in the index, will be the logaritlim of the difference of the segments FB, BL ; half of which, beino- added to and subtracted from half of FL, will give the two segments FB, BL ; the greater segment being contiguous to the greater side, whether SF or SL. Then, from the logarithm ol' the segment FB, increasing the index by 10, subtract the logarithm of SF; the remainder will be the log. sine of the angle FSB,* which is always less than 90" ; the difference betu'een this and the angle of inclination DSB will be the central angle DSF. To the log. cosine of the central angle, add the logarithm of the sum of the corrected semi- diameters at the begiiining of the eclipse SF, rejecting 10 in the index ; the sum will be the logarithm of SD, tlie ap])arent difference of longitude of the sun and moon at that time. This is to be subtracted from the longitude of the sun at the beginning of the eclipse, if the central angle is less than !)0", but added if greater than 90°; the sum or difference will be the moon's apparent longitude : to this must be added the moon's parallax in longitude, when her distance from the nonagesimal (found as in Problem V., by subtractinjr ihe longitude of the nonagesimal from the moon's longitude, borrowing 3C0° when necessary) is greater than 180'^ ; otherwise the parallax must be subtracted ; the sum or difference will be the moon's true longitude at the beginning of the eclipse. Take the difference in seconds between the sun's and moon's true longitudes at the be- ginning of the eclipse, to the logarithm of which add the arith. comp. logarithm of the moon's horary motion from the sun t in seconds, and the constant logarithm 3.5r<(J30 ; the sum, re- jecting 10 in the index, will be the logarithm of the time from the conjunction in seconds, which is to be added to the observed apparent time of the beginning of the eclipse, when the sun's longitude at that time is greater than the moon's true longitude, otherwise subtracted; the sum -jx difference will be the apparent time of the true ecliptic conjunction of the sun and moon at the place of observation. The difference between this and the time of con- junction at Greenwich, inferred from the Nautical Almanac by Problem 111., will be the longitude of the place of observation. But if corresponding observations have been made at different places, it will be much more accurate to find the times of the conjunction at each place by the above rule; and the difference of these times will be the difference of meridians, if it does not differ much from the supposed difference of longitude. If there is considerable difference, the operation must be repeated, making use of the longitude found by this opera- tion ; and thus, by successive operations, the true longitude may be obtained. The longitude of the place of observation being accurately known, the errors of the lunar tables in longitude and latitude may be easily found. For the difference between the moon'a true longitude deduced by the above method from the observations, and the longitude found from tlie Nautical Almanac, Vv'ill be the error of the tables in longitude. To find the error in latitude, add the log. sine of the central angle DSF to the logarithm of the sum of the corrected seini-diameters at the beginning of the eclipse SF ; the .sum, rejecting lU in the index, will be the logarithm of the moon's apparent latitude FD at that time; which will be south, if the point F falls below D, otherwise north. Take the difference between this and the moon's apjjarent latitude, found by Problem ^ ., if they are both north, or both south ; but their sum, if one be north and the other south; and the error of the tables in latitude will be obtained.}; REM.4.RK. The above rule will answer for deducing the longitude from the observed beginning and end of the internal contacts of a total or annular eclipse. The differences consist in reading FD EL expression -— is greater or less tlian ■:; — ; but, as the divisors SL and SF are nearly equal, they may be neg- lected (as ill the above rule), exrept in a ease wliicli very rarely occurs, namely, wlien the difference of SL, SF, is greater thiiii the dtiereiice of tlie two anpareiil latitudes EL, FD, in wiiixih case the rule in this note EL FD must be made use of; observing that the fractions , ' — , represent the quotients of the moon's ajiparent latitudes divided by the sum of the semi-diameters of the sun and moon. * When SF, SL, al'e eijiial, or tlieir difference is so small that it may be neglected, the log. sine of the an- f[le FSI) may be ;. hy N. A. — Err. 'I'aljle 58". 5 = True long. ]) I'rob. I Jjongitude of the nonagesinial, by I'roh. IV J) '3 true long. — Long, nonagesinial = 5 's dist. from nonagesimal This distance, or its snpiilenient, if greater than 180°, is arch D... Altitude of nonagesimal, I'rob. IV ])'s horizontal parallax, l)y Prob. I — O's hor. par. 8".8* — Correction Table XXX VIII. 3".G Reduced paralla.v D's semi-diameter by N. A. — Inlle.\ion 2" Add correction Table XLIV 5 's corrected semi-diameter O's semi-diameter by N. A. 15' 4C".l — Irradiation 3".5 Snm of the corrected semi-diameters D's horary motion in longitude by I'rob. II. Example II 0's horary motion D's horary motion from the sunf D's parallax in longitude P D 's apparent longitude — Error Table 58".5 by Prob. V 0's buigi tilde by I'rob. I Difference D ';>" aj>p. longitade = D 's app. motion Difference Q)'^ lungiludcs =r ©'s ajjp. motion Difference of motiuns of Q ]) D's true lat. by N. A. Prob. I. — Error Tal)le 11".4 D's app. lat. corr. -for error Table 11". 1 by Prob. V D 's latitude at end — Latitude at beginning IG d. h. m. 15 22 6 4 43 2 49 5 36 42»22' 23 27 83 49 63 22 20 2t) 20 26 67 58 60 60 16 SF = 16 15 .32 36 2 34 19 84 8 84 41 s. 18.1 32 50.1 50.0 4" 48 3.5 31 33 32 50 24.1 14.4 9.7 25.7 15.2 40.9 42.6 2;J.5 39.2 23.1 Ifi.l 46.8 50.3 3.4 — 24 27.4 FD=— 1 55.8 h. m. s. .50 34.6 4 43 32 5 34 6.6 5 37 18.5 85 29 32.6 95 26 36 350 2 57 9 57 3 70 57 46 60 27.0 — 14.4 60 12.6 16 26.4 16.4 16 42.8 15 42.6 32 25.4 36 42.8 2 23.1 34 19.7 10 0.0 85 19 32 6 84 47 35.5 1 10 42.3 6 32.1 64 10.2 — 15 10.4 EL = -|- 4 32.5 Al. = 4- 6 28.3 SL = FA As the apparent latitude at the beginning of the eclipse is north, and at the end south, tho point F corresponding to this example falls above DE, the point L below it. The rest of the calculation is as follows : — 6.41232 Log. 3..5S983 Log. 0.27875 FA64' 10".2=3S.50".2 Log. 13.58.548 3.58548 AL 6 08.3 = 338 .3 Log. 2..58917 Inclination 84' 14'... Tan. 10.99631 Cosecant 10.00220 Apparent mnlion FL 3S69".7 Log. 3.58768 Its arith. comp SF+SL = 64' 48".9 3388".9 Diff. SF.SL 1.9 Diff. segments 1.91 Log. 0.28090 Its half 0.95 Half of FL 1934.85 Sum is great segment 1935 .8 \y\XX. is le.sser segment FB. 1933 .9 Log. 13.28644 •5F 32' 23'i.5 = 1943.5 Log. 3.28853 Angle FSB 84° 19' Sine 9.99786 Inclination 84 14 Diff. is central angle DSF. ~0 SF SD = 32'23I5 = 1943".5 Log. 3.28853 Cosine 10.00000 Log. 3.2(<858 O's longitude 84° 41' 3".4 SU _ 32 23 .5 D's app. longitude 84 8 30 .9 by obs. D's par. longitude — 19 46.8 D's true longitude 83 48 53.1 O's longitude 84 41 3 .4 Const. 3.55630 Difference 31.30".3 = .50 10.3 Log. 3.49555 D 's hor. mot. from O 34' 17".l=2U57".l A.C. 6.6»675 h. m. 1. Time from conj. 13118.1=5478".! Log. 3.73863 App. time obs. 15 22 6 18.1 App. time conj. 15 23 37 36.2 at Palem. Conjunction 16 4 19 at Greenwich. Diff Mejid. 4 41 23.8 Sine 7 16270 Log. 3.288.58 App. lat. FD=:2".8.. Log. 0.45128 * The mean parallax formerly used was 8". 8 : it is now found to be nearly 8".6. t This horary motion increases from 34' 16".l to 34' 19".7, or 3".6, during the eclipse 2h. 41m. 16s,5, which is 1".32 per hour. Now the ecli|)tic conjun( tion, or time of new moon, at Creeiiwich, by the N. A., was 4h. 19m., or rather 4h. 20m. 473., corresponding to 2.3h. 37m. ISs. at Salem, which is Ih. .^Om. .57s. after the beginning of the eclipse; and the increase of the horary motion in half that time is 1", which, added to 34' I6".l, gives the horary motion 34' 17". 1, corresponding in the middle time between the beginning of the eclipse and the conjunction. This is used in calculaling the correct time of conjunction. We mav remark that, in Ilie above calculations, wehave used the apparent times of observation, to confurm to the arrangement of the Nautical .Almanac in 1806 ; but in the present form of the Nautical .-Mmi U8e the viean time. .•\lmanac, it will be convenient 412 TO FIND THE LONGITUDE BY AN OCCULTATION. In finding tlie time of conjunction or new moon, at Greenwich, 4h. 19m., in the Nautical Almanac, the longitude of the moon was supposed to be given correct)}' by the tables. If the calculation be made by Problem 111., after allowing for the error — 5b". 5, the result will be 4h. 20m. 47s., whence the difference of meridians =4h. 43m. 10s. 8, which differs so little from the assumed longitude, 4h. 43m. o2s., that it will not be necessary to repeat the operation. If the eclipse was observed at Greenwich, the time of conjunction ought to be determined thereby, in a similar manner to the above calculations ; or by those of Problem Vlll., if onl}' one of the phases is observed : by this means the errors of the tables will be wiolly avoided. If the eclipse v.'ss not observed at Greenwich, the observations at any other place whose longi- tude is known might be made use of, and thus the difference of meridians accurately obtained. The moon's true longitude, deduced from the above observation, is 83P 48' 53". 1 ; by the Nautical Almanac it is 83° 50' 2".0 ; the difference, — 68". 9, would be the error of the tables by this observation, if the assumed longitude, 4h. 43' 32", and the solar tables, were correct. By repeating the operation with the assumed longitude, 4h. 43m. lOs.8, the error, 68". 9, would be reduced to nearly the estimated value, 58". 5. The eclipse was so nearly central at Salem, that a variation of a minute in the moon's lati- tude would hardly'alter the times or duration of the eclipse ; so that the latitude could not be determined by the above observations to any considerable degree of accuracy. From this cause it happens that the apparent latitude at the beginning of the eclipse is by the above calculation 2". 8, instead of 1' 55".8, as found by allowing the error, 11".4, deduced from other observations made where the eclipse was not so nearly central, and by the limits of the shadow of total darkness. PROBLEM VII. Given the latitude of the place, and the apparent times of the beginning and end of an oc- cultation of a fixed star by the moon, to find the longitude of the place of observation. In the following rule, reference will be made to figure 13, Plate XIII., in which DSE repre- sents a parallel to the ecliptic passing through the place of the star S ; SF, SL, the corrected semi-diamelens of the moon at the beginning and end of the occultation ; DF, EL, the dif- ferences between the apparent latitudes of the moon and the star, when of the same name, or their sums, when of diff'erent names; either of these lines falling ic/oio DE if tlie moon's apparent latitude is more southerly than that of the star, otherwise above. RULE. To the apparent times of the beginning and end of the occultation, add the estimated longi- tude of the place in time if it is west, but subtract if east : the sum or difference will be the supposed time at Greenwich ; corresponding to which, in the Nautical Almanac, find, h~ Problem I., tiie moon's semi-diameter, horizontal parallax, longitude and latitude,* and the sun's right ascension ; also the moon's horary motion by Problem II., and the true longitude and latitude of the fixed star, by Table XXXVIl., corrected for aberration and equation of equinoxes by Tables XL., XLI. This may also be deduced from the right ascension anc. declination of the star, if it be given in the Nautical Almanac, by means of Problem XIX of this Appendix. Find, also, in the Nautical Almanac, the obliquity of the ecliptic. To the moon's semi-diameter, add the correction in Table XLlV.,t and from the sum subtract the inflexion, 2", if it be thought necessary ; the remainder will be her corrected semi-diameter. With these elements and the apparent times of the place of observation, calculate the alti- tudes and longitudes of the nonagesimal, by Problem IV., and the parallaxes in longitude and latitude, and the moon's apparent longitudes and latitudes, by Problem V. Take the difference between the oppureiit longitudes of the moon at the beginning and end of the occultation, which will be the moon's apparent motion in longitude, the logarithm of which, in seconds, being added to the log. cosine of the meant of the apparent latitudes of the moon at the beginning and end of the occultation, rejecting 10 in the index, will be tlie loga- rithm of the motion of the moon on the parallel FA. The relative motion in latitude AL is found by taking the difference of the moon's apparent latitudes at the beginning and end of the eclipse if they are both north or both south ; but their sum if one be north and the other south. From the logarithm of FA; increasing the index by 10, subtract the logarithm of AL ; the remainder will be the log. tangent of the (mgle of inclination DSB ; this angle is to be taken greater than 90° when the difference of the moon's and star's apparent latitudes at the beginning of the occultation FD is greater than at the end EL, otherwise less. § Then to the log. cosecant of the angle of inclination, add the logarithm of the relative motion FA ; the smn, rejecting 10 in the index, will be the logarithm of the apparent motion of tiie moon in her orbit FL. * Correcteil fur tlie errors of tlie tal)les in longitude and latitude, when known. t This correrlidii must be found alter the alt tuile and longitude of the nonagesimal are calculated. X Tlie menu hil lude is half the sum of the two latitudes, if they are of the same name, but their half differ cnc3, if of d tii-reiil names. In solar ellipses, the correi tioii fur the mean latitude of the moon is neglected as too small to he taken notice of, the d. stance FA being taken equal to the difference of longitude DE (fig. 19. Plate Xlll.). , . , . rvr. § This rule IS ei|ually true, whether the points F, L, fall on the same or on different sides ot the line Ut. If DF, EL, areci|u:il, aild the points F, L,, fall on the same side of Ui:, the angle DSB will be 90°. If they are equal, and those points fall on differtMit sides of the line DE, the angle USB may be taken acute or t'btusa in strictness, when the iioints F, L, fall on different sides of DE, the angle DSB is greater or 'ess than ao* Fl) EI. according as Hie niiantilv — is greater or less than — -. *" ' • SF SI. TO FIND THE LONGITUDE BY AN OCCULTATION. 413 Then in tlie triangle SFL, the sides SF, FL (representing the corrected seini-diumeters» of the moon at the iniinersion and emersion), and the relative motion FL, are yiven to find tlie angle FSB (by Case VL Oblique Trig.). Thus: to the log. arith. comp. of FL, add the logarithm of ihe sum of SF and SIj, and the logarithm of their difference : the sum, rejecting 10 in the index, will be the logarithm of the difference of the segments FB, BIj ; half of this, being added to, or subtracted liom the half of FL, will give the two segments FB, BL; the greater segment being contiguous to the greater side, whether SF or SL. Then, from the logarithm of the segment FB, increasing its index by 10, subtract the logarithm of SF; the remainder will be the log. sine of the angle FSB,* wliich is always less than !)()'-'. The dif- ference between this and the angle of inclination DSB, will be the central avglu DSF. To the log. cosine of the central angle add the logarithm of the moon's corrected serai- diameter at the immersion SF, and the log. secant of the star's latitude : the sum, rejecting 20 in the index, will be the logariliini of the apparent difll-rence of longitude of the moon and star at that time. This is to be subtract! d from the true longitude of the star, if the central angle is less than i)0", but added, if greater than 90": 'the sum or difference will be the moon's apparent longitude; to this must be added the moon's parallax in longitude, when her distance from the nonagesimal (found as in Problem V., by subtracting the longi- tude of the nonagesimal from tlie moon's longitude, borrowing 300° when necessary) is rrreatcr than 180", otherwise tlie parallax must be sitUracUd ; the sum or difference will be the moon's trve loiigiludc at the begiiming of the occultalion. Take the diff"ercnce in seconds between the true longitudes of the star and moon at the beginning of the occultation ; to the logarithm of this add the arithmetical comp. log. of the moon's horary motion 1 in seconds, and the constant logarithm 3.55030 : the sum, rejecting 10 in the index, will be the logarithm of the time from the conjunction in seconds, which is to be a.dded to the observed apparent time of the beginning of the occultation, when the star's longitude is greater than the moon's true longitude at that time, otherwise sahlructcd : the sum, or diffi>rence, will be the a])parcnt time of the true ecliptic conjunction of the star and moon at the place of observation ; the difference between this and the time of conjunction, inferred from the Nautical Almanac by Problem III. for the meridian of Greenwich, will be the longitude of the place. If corresponding observations be made at different pl.ices, it will be much more accurate to deduce from them the time of conjunction at each place, and take the diff'erence of those tiuies for the diff'erence of meridians, if it does not diff'er much from the supposed difference of longitude. If there is considerable diff'erence, the operation must be repeated, making use of the longitude found by this operation ; and thus, by successive operations, tlie true longitude may be obtained. The longitude of the place of observation being accurately known, the errors of the lunar tables in latitude and longitude may be easily found. For the difference between the moon a true longitude, deduced from the observations by the above method, and the longitude found from the Nautical Almanac, will be the error of the tables in longitude. To find the error in latitude, proceed thus : To the log. sine of the central angle DSF add the logarithm of the corrected semi-diameter of the moon at the immersion SF ; the sum, rejecting 10 in the in- dex, will be the logarithm of the apparent difference of latitude of the moon and star, which, being added to the true latitude of tlie star, with the sign -f- if the point V falls licUno the line DE, but with the sign — \i' almve, will give the apparent latitude of the moon at tJiat time : the diff'erence between this and the apparent latitude, found by Problem V., will be the error of the tables, alwaj's supposing the sign -\- to be prefixed to southern latitudes, the sign — to northern, and noting the signs as in algebra.}: REMARK. In the two preceding problems, the time of the true conjunction is calculated by means of the triangle SFD; but it will be useful, for the purpose of verification, to go over the calcula- tion by means of the triangle SLE. The process is nearly the same in both methods. The diflTerences consist in finding the angle LSB, by subtracting the logarithm of SL from the logarithm of LB, increasing its index by 10; tl)e remainder will he tlie log. sine of the acute angle LSB, which, being added to the angle of inclination (lound as before), will give the central angle DSL : with this, and the distance SL, corresponding to the I'nd of the eclipse or occultalion, maybe found the apparent difflsrence of longitude between the sun and moon, and moon and star: this is to be added to the longitude of the sun or star at that lime, if the central angle exceed 5)0°, otherwise subtracted : the sum, or difference, will be the ap. parent longitude of the moon corresponding, from v;liich the time of the ecliptic conjunction may be obtained as before. If the central angle exceed 180", the sine and cosine of the excess of that angle above IdO" must be found instead of the sine and cosine of the central angle. The apparent latitude of the moon is found as in the preceding rules, by making use of the central angle DSL, and the value SL, corresponding to the end of the eclipse or. occul- talion ; whence maybe deduced the apparent latitude, and the error of the tables in latitude. It is evident that both these methods ought to give the same results, and thus furnish, a proof of the correctness of the calculations. All these calculations may be made by propor- tional logarithms, by reading in the rule, log. cotangent for log. tangent, log. cosecant for log. sine, &c., as was mentioned at the end of the rule in Problem V., and by using the constant log'. 0.4771, instead of 3.55030. ^ * Wlien SF = SL, tlie anule may be romul as in tlie nnt<; with this marlc in paje 408. I Wlien this varies, it must be taken to rorrcsjiniid tci the ni (Idle time between the immersion and true riinj;iii< tiiin. t Siie nutc with tliis marit in iia^e 41)8. 414 TO FIND THE LONGITUDE BY AN OCCULT ATION. EXAMPLE. Suppose in a place in the latitude of 20° 0' N., longitude ]h. 9ra. Os. east of Greenwich, by estimation, the occultation of Spica by the moon on December 12, 1808, was observed, the immersion at ICh. 57m. 29s., emersion at 18h. lOni. 29s., apparent time, by astronomical computation. Required the longitude of the place of observation. Most of the elements in the following Table are calculated by Problems I., II. and VI. ELEMENTS OF THE OCCULTATION. Apparent times nf observation Estimated lonj^tiide E. from Greenwich Supposed apparent time at Greenwich 0's riglit ascension Lat. ofpla e20°0'— Reduc. Table XXXVIIL 7' 22" Obliq\iity of the ecliptic D's Ions;, hy N. A, — Prnb. I Longitude of the nonagesimal, by Prob. IV J)'s long. — Long, nonagesimal = D's distance from nonagesimal Thii= 'listance or its snpplement to 3G0° is arch D All I 1 ■ of nonagesimal, Prob. IV •D's iiurizontal parallax .' — Ilednction, Table XXXVIII Reduced parallax D's semi-diameter by N. A. — Inflection 2" Add correction, Table XLIV D's corrected semi-diameter D's horary motion in longitude by Prob. II. Example I.f D 's parallax in longitude D's apjiarent longitude Difference of D's apparent longitudes D's true lat. by N. A. Prob. I South D's parallax iii latitude D's apparent latitude south *'s true lat. = lat. Tab. XXXVII. 2' 2' 13".9 S. — Tab. XLL 0".C Dilleren e of D * apparent latitudes Difference of D's apparent latitudes *'s true long. = Loni. Tab. XXXVn.20r 10' 29".3 + Tab. XL. ) 11".5 — 'Jab. XLI.^IO'M ( d. h. m. 12 16 57 1 9 12 l.'i 48 17 20 19° 52' 23 27 200 7 119 15 50 52 D 50 52 81 17 59 59 16 F 16 35 46 200 54 Emersion. 29 29 59.0 38" 39 56.3 55 1 1 32 52.3 1.4 50.9 16.9 10.4 27.3 51.7 25 21.3 1 55 11.0 10 23.6 2 5 34.6 2 2 13.3 FD = 3 21.3 d. h. ni. s. 12 18 10 29 1 9 12 17 1 29 17 21 12.5 o / II 200 51 36.1 165 28 58 35 22 38 D 35 22 33 74 35 18 59 54.4 1.4 .'•.9 53.0 16 17.5 13.3 L 16 30.8 35 54.2 33 54 501 25 30.1 31 8.8 1 51 29.1 17 38.4 2 9 7.5 2 2 13.3 C 54.2 3 32.9 EL = AL = The difference of the apparent latitudes of the moon and star at the beginning of the oc- cultation 3' 21".3, beinn- less than at the end, 0' 54".2, the angle of inclination Is less than 90^. In tills example the moon's latitude is more southerly than the star's^ hence the points F, L, fall below the line DE. Log. 3.27156 Cosine 9.99970 D 31' 8".8 = 1668".8 2 7 21 212.9 1879.6 . Difference apparent Ion D 's mean ai)iiare;:l lat. Distance FA D's difference lat AL=:3 32.9 Inclination 83° 30' Apparent motion FL.. Its Arilh. Comp gp^SL = 32 53.1=1978.1 Difference SF, SL... Difference segments.. Its half. Half FL FB SF FSB Inclination Log. 13.27126 Log. 2.32818 Tang. 10.94308 6.72594 3.29625 0.541'J7 Log. Log. Log. 0.56626 71° 49' 83 30 Log. Log. Sine 9.97775 2.97220 2.99445 Diff. is central angle.. SF Star's latitude. Cosine 9.99091 Log. 2.99445 Sec. 10.00027 Diff. apparent long *'s longitude .... D * 967".5 = 16 7.5 201 10 30.7 Log. 2.98563 D's apparent longitude. D 's par. longitude 200 54 23.2 by observation. — 46 25 D's true longitude 200 7 58.2 Constant 3.55630 Difference true longitude 3752.5 = 1 2 32.5 Log. 3.57432 D 's horary motion 2153.5= 35 53.5 Ar.Co Log. 6.66686 Time 6273 = lh. 44m. 33s. Log. 3.79748 [muiersion 16 57 29 Conjunction Conjiinction Difference of meridians. . Log 3.27126 Cosecant 10. 00280 Log. 3.27406 Sine 9.3064? Log. 2.99445 FD 199".9 = 3' 19".9 Log. 2.3008f *'s latitude 2 2 13.3 D's app. lat. D 's app. lat. 2 2 5 5 33 .2 hy obs. 34 .6 by N. A. Error Table _ 1 .4 in latitude. 18 42 2 at. place of observation. 17 33 at Greenwich. Ih. 9m. 2s. D 's true Ion. 200 7 .58 .2 by obs. D 's true Ion. 200 7 56 .3 by N. A. EiTor Table -f 1 .9 in longitsfe t The inocui's liorary motion varies from 35' 51".7, to 35' 54".2, during the occultation: hence, at the middle time, 17h. 49tu. 4."s., between the immersion, 16h. 57m. 29s., and the conjunction, 18h. 42m. (deduceil from the Nautiial Ahiiaiiai ), the horary motion was 35' 53". 5, as is ens ly fmirul by a cahulation similar to that in lb<> r.xauipU- "f Problem VI. TO FL\D THE LONGlTUfiE BY AN ECLIPSE OF THE SUxN. 415 The (liiTiTcnce of meridians deduced from the observation, Ih. 9in. 23., differs but 2s. from the assuaied quantity, Ih. 9m. Os. If the difference had been considerable, it would liave been necessary to repeat the operation with the difference of meridians thus calculated, and so on till the assumed and calculated longitudes agree. The errors of the tables above found, v/ero deduced upon the supposition that the observations were actually made at the place mentioned in this example, and that the true longitude of the place of observation was Ih. i>m. Os. For it must be observed, that the errors of the tables in longitude cannot be found by an observation of an eclipse or occultation, without knowing, by other observations, the [irecise longitude of the place of observation. This is evident by observing, that, ly re- peating !he operation till the assumed and calculated longitude of the place of observation agree vvilh each other, the lono-itude of the moon, deduced from the calculation, will agree also with the longitude by the tables. The time of conjunction at Greenwich, 17h. 133m. Os., taken from the Nautical Almanac, is liable to a small error from the incorrectness of the tables. To obviate this eiror, it will be necessary to deduce (by the above method, or by Problem IX. when only the beginning or end is observed) the time of conjunction from observations actually made at two places; the difference of these times will be the difference of meridians free from the errors of the tables. PROBLE3I VIII. To find the longUude of a place by an eclipse of the sun, lohcn the beginning or end only is observed ; the apparent time being estimated from noon to noon, according to the method of astronomers ; the latitude of the place being also known. RULE. To the apparent time apply the estimated longitude of the place in time, by adding ificcst, subtracting iC east ; the sum, or difference, will be the supposed time at Greenwich. Cor- responding to this time in the Nautical Almanac, find, by Problem L,the moon's semi-diame- ter, horizontal parallax, longitude, and latitude;* and the sun's semi-diameter, longitude, and right ascension ; also the moon's horary motion from the sun by Problem II. Decrease the sun's semi-diameter 3A" for irradiation. Decrease the moon's semi-diameter 2" for in- Jlexio 11, if^\i be thought necessary, and to the remainder add the correction to Table XLIV.t; the sum will be the moon's corrected semi-diameter. Find also, in the Nautical Almanac, the obliquity of the ecliptic. With these elements, and the apparent time at the place of observation, calculate the alti- tude and longitude of the nonagesimal by Problem IV., and the parallaxes in longitude and latitude, and the moon's apparent latitude by Problem V To the sum of the corrected semi-diameters of the sun ana moon, add and subtract the moon's a])parent latitude, and find the logarithms of the svm and difference in seconds. Half the sum of these two logarithms will be the logarithm | of an arc in seconds, to be added t<< the sun's longitude if the phase is after the apparent conjunction, but subtracted, if before ;§ the sum, or difference, will be the apparent longitude of the moon. To tliis add the moon's parallax in longitude, when the moon's distance from the nonagesimal (found, as in Problem VI., by subtracting the longitude of the nonagesimal from the moon's longitude, borrowing 3G0'-' when necessary), is greater than 180°, otherwise subtracted ; the sum, or difference, will be the trite longitude of the moon. Take Ihe difference in seconds between the true longitudes of the sun and moon, and to its logarithm add the arithmetical complement log. of the moon's horary motion from the sun in seconds, and the constant logarithm 3.55G30 ; the sum, rejecting 10 in the index, will be the logarilhui \ of the correction of the given time, expressed in seconds. This is to be added to the ajjparent time of observation, when the moon's true longitude is less than the sun's, otherwise subtracted; the sum, or difference, will be the time of the true conjunction at the place of ob.ser'vation. The difference between this and the time of conjunction inferred from the Nautical Almanac for the meridian of Greenwich, by Problem III., will be the longitude of the place of observation in time, supposing the lunar and solar tables to be cor- rect; but it is much more accurate to compare actual observations made at different places, by deducing the times of the ecliptic conjunction from each observation; the difference of these times will be the difference of longitude. EXAMPLE. At Salem, in the latitude of 42- 33' 30" N., longitude by estimation 4h. 43m. 32s. W. from Green wich, the beginning of the total eclipse of June, 1800, was observed at 15d. 22h. Gm. 18s. 1 , * Tlie loiiiiitiuli; ami latitude must be corrected for the errors of the tables, when known, by a previous operaliim, or liy other observations. t Th .s corre linn must lie found after the altitude and longitude of the nonagesimal are ralciilated. j These i alculat ons may be made in the same manner by using proportional logaritlims ; the only differ ence cons'sts In using the constant logarithm 0.4771, instead of 3.55G:!0, in finding tlie time of conjunction. $ In general, the beginning of an eclipse or occultation precedes the apparent conjunction, and the end is after the apparent conjunrtion ; but there is a case (which very rarely occurs) where the contrary may take pince ; namely, where the point F or L (Plate XIII. fig. 12, 13) falls between C ami B, which can happen only when thi; lines FD, RL, are nearly equal to SF or SL. In tliis case, it n ay be ascertaintd whether the phase [(recedes or follows the conjunction, by making the calculation as in I'roblem VI. or VII., with the limes of beginnin;: and end, calculated by Problem XIII. ; and, as the central angle is greater or less than 90° the phase will follow or precede the apparent conj:inction, the latitudes given by the tables being suppos<»<» correct. 416 TO FIND THE LONGITUDE Bt AN OCCULTATION apparent time, by astronomical computation. Piequired the longitude of the place from tills observation. The elements must be calculated, as in the Example of Problem VI., for the beginning of the eclipse, except those marked in italics. The rest of the calculation may be made b} proportional logarithms, as follows : — Sum semi-diameter O ]) 32'23'i.5 J) 's apparent latitude 1 55 .8 Sum 34 19 .3 Prop. Log. /I97 Diflerence 30 27 .7 Prop. Log. 0.7715 Sura 1.4912 Half sum Arc 32 20 corresponding to Prop. Log. 7456 O's longitude 84 41 3.4 D's apparent longitude 84 8 43.4 J>'s par. longitude — 19 46.8 , D's true longitude 83 48 56.6 . 0's true longitude 84 41 3.4 Constant Log. 0.47Vi Difference 52 6.8 Prop. Log. 0..5363 D's horary motion from 34 17.1 Arith. Comp. Prop. Log, 9.2798 Time from conjunrtion 111. 31m. 13s Prop. Log. 0.2952 Apparent time observation... 15 22 6 18 Apparent conjunrtion Salem. 15 23 37 31 App. conjunction Greenwich 16 4 19 by Nautical Almanac. Difference of nierid ans 4h. 41ni.21}s. If we suppose the time of conjunction at Greenwich to be 4h. 20m. 473. as calculated in the Example, Problem VI., the difference of meridians would be 4h. 4:'m. IGs., agreeing nearly with the assumed longitude, so thai it will not be necessary to repeat the operation The remarks at the end of that example, respecting the errors of the lunar tables, and tht comnaring of actual observations at different places, are equally applicable to the presen: problem. PROBLEM IX. To find the. longiiiule of a place hy an occultation of a fijced star by the moon, xchen t'u immersion or emersio7i only is observed ; the apparent time being estimated from 7iom to noon, according to the method of astronomers, and the latitude of the place bein^ known. RULE. To the apparent time apply the estimated longitude of the place turned into time, by adding if 7/;c.9^ subtracting if east ; the sum or difference will be tlie supposed time at Green- wich. At this time find In the Nautical Almanac the sun's right ascension, the moon's semi- diameter, horizontal parallax, longitude, and latitude,* by Problem I.; and the moon's horary motion by Problem II. ; also the latitude and longitude of the fixed star by Table XXXVII., and correct it for aberration and equation of equinoxes by Tables XL. XLI. De'crease the moon's semi-diameter 2" for inflexion, if it be thought necessary, and to the remainder add the augmentation from Table XLIV.; 1 the sum will be the corrected semi-diameter. Find also, in the Nautical Almanac, the obliquity of the ecliptic. With these elements, and the apparent time of observation, calculate the altitude and longitude of the nonagesimal by Problem IV., also the parallaxes in longitude and latitude of the moon's apparent latitude by Problem V. Take the difference between the latitude of the star and the apparent latitude of the moon which add to and subtract from the moon's corrected serni-diameter (these quantities being expressed in seconds) ; half the sum of the logarithms of these quantities, increased by the log. secant of the star's latitude, rejecting 10 in the index, will be the logarithm | of an arc in seconds, to be added to the star's longitude if the moon has passed the apparent conjunc- tion, but subtracted if before ;\ the sum, or difference, will be the apparent longitude of the moon. To this add the moon's parallax in longitude when the moon's distance from the nonagesimal (found as in Problem VII., by subtracting the longitude of llie nonagesimal from the moon's longitude, borrowing 3G0" when necessary) is greater than 180'^, otherwise subtract it ; the sum or difference will be the true longitude of the moon. Take the differ- ence in seconds between tlie moon and star's true longitudes, and to its logarithm add the arithmetical comp. log. of t!ie moon's horary motion, and the constant logarithm 3.55030; the sum, rejecting 10 in the index, will be the logarithm t of a correction in seconds to be applied to the given time of observation by adding when the moon's true longitude is less than the star's, otherwise subtracting ; the sum or difference will be the time of the true * Corrected for the errors of the tables in longitude or laftude when known. f This corrertion nrist he found after the altitude and longitude of the nonagesimal are cnlrulatert. \ Proportionnl logar llims may be used instead of coumion logarithms, the C(>nstani logarithm being 0.4771. instead of 3. .55' 3'), and the log. cosine being used instead of log. secant. 5 See note with this mark in page 413. TO CALCULATE AN LCLIPSE OF THE MOO.N. 417 conjunction at the place of observation. The difference between this and tlie time of con- junr.tion inferred from the Nautical Almanac by Problem III., for the meridian of Green- wich, will be the longitude of the place of observation, if the tables are correct; but it is much more accurate to compare the times of conjunction deduced from actual observations at the different places in the manner mentioned at the end of the rule given in Problem VH. EXAMPLE. Sujvpose in a place in the latitude of 20° 0' N., longitude by estimation Ih. !1m. Os. east from Greenwich, the emersion of the star Spica was observed on December 12. 1808, at 18h. 10m. 2ys., apparent time, by astronomical computation. Requited the longitude of tht place of observation. The elements must be calculated as in the example of Problem VIL, for the emersion of Spica. The rest of the calculation, made by common logaritiuns, is as follows - P's semi-diameter ICi 30".8 = 9f)0".3 Uifloreiice apjiareiit l;U. 5 * 5i .2 414 .2 Slim ]40.').0 Loff. 3.14708 Ui/Tereme 57G .6 Lo''. 2.7tiU87 5.908r« its half 2.9.5427 *'s latitude 2° 2' 13". . . .Sec. lO.OOlW Arc 15' 0".0 = 900". 6 Log. 2.9.')4.=i4 *'s longitude 201 10 30.7 5 's apparent loiiiiitiule 201 2.5 31 .3 D's piir. longitude — 33 54 D 's true longitude 200 51 37 .3 Constant 3..5.5r)30 Ditlerence true longitude D ». 18 53.4 = 1133.4 Log. 3.0.5433 ]) 's horary motion 35 54.7 = 2154.7 Arith. Comp. Log. G.filiGlil Time 0h.31in.34s. = 1894 Log. 3.27729 Time of observation 18 10 29 Conj. at place of observation. 18 42 3 by observation. Conjunction at Greenwich... 17 33 by Nautical Almanac Difference of meridians lli. 9in. 3s. The difference of tueridians by calculation, Ih. 9m. 3s., differs but 3s. from the assumed longitude, so that it will not be necessary to repeat the operation. All the remarks made ai the end of the example in Problem VIL arc applicable to this problem. It may also be further observed, that the emersion or immersion which happens on the dark limb of the moon can be observed with much more accuracy than on the enlightened limb ; because the light from this limb prevents the observer from perceiving the star's immersion or emersioi» so iostantaneously as on the dark side of the moon. PROBLEM X. Tt/ calculate an eclipse of the moon. The time of beginning or end of a lunar eclipse at any place may be found by subtractincr or adding the longitude to the times given in the Nautical Almanac for the meridian of Greenwich, according as the longitude is west or east. But as some readers may wish to know the method of deducing these times from the longitudes, latitudes, Ac. of the moon- and sun, given by the Nautical Almanac or by other tables, it was thought proper to iii.sert the rule for these calculations. An eclipse of the moon can only happen at the time of the full moon. If her longitude at that time is not distant from cither nodet of the moon's orbit more than about 12'^, there may be an eclipse. To find whether there will be one, and to calculate the times and phases, proceed as follows : — • RULE. Find the time of full moon at Greenwich by the Nautical Almanac or Problem III., to which add the longitude of the place turned into time, if east ; but suhirucl if wi:s( ; the sum or difll^rence will be the time of the ecliptic opposition at the proposed place. For the time at Greenwich, find, by Problem I., tlie moon's latitude, horizontal par.allax, and semi-diameter (whicii requires no augmentation) ; also the sun's semi-diameter; then, by Problem IL, the horary motion of the moon from the sun in longitude, and the inoon'3 horary motion in latitude. Draw the line ACB (Plate XIII. figure G) ; and, perpendicularly thereto, the line PCR. Select a scale of equal parts to measure the lines of projection, and from it. take C(r, equal to the moon's latitude, and set it on CR from C to G, ahnve the line AB if the latitude of the moon is north, below if south, t Take CO, equal to the horary motion of the moon from the t The long'tnde of the moon's a.?rending node is given in the Nautical Almanac. The long'tude of the other node is f luiiii by adding or subtracting G signs. I The nortlfern lat tudes fnurid by Problem f. have the sign — , the southern +. In the figure the latitude is south If it b;i I been north, tlie point (i nuist have been placed on the continual on of RC above C 53 418 TO CALCULATE AiN ECLIPSE OF THE MOOK. sun in longitude, and set it on the line CB to the right of C, from C to O. Take CP, equal to the moon's horary motion in latitude, as found with its sign by Problem II., and set it on the line CR, from C to P ; aiore the line AB if its sign is — , heloiv* if +. Join OP, which is equal to the horary motion of the moon from the sun, and parallel thereto through G draw the relative orbit of the moon from the sun NGL, on which are to be marked the places of the moon before and after the full, by means of the horarjr motion OP, so that the moment of full moon, or ecliptic opposition at the proposed place, may fall exactly on the point G. This may be done by making the extent OP equal to the transverse distance of tiO, CO, on the line of lines of the sector, then measuring from the same lines the transverse distance corresponding to the minutes and parts of a minute in the time of full moon at the place of observation, and setting it on the line GN from G towards the right to the point x, where the whole hour preceding the full moon is to be marked.! Then the distance OP set from x to the riglit hand on the line LGN reaclies to the hours preceding the full moon, and set to the left hand reaches successively to the following hours. Tliese intervals are to be divided into 60 equal parts, representing minutes, if the size of the scale will ad- mit of it. Add 50" to the moon's horizontal parallax, + and from the sum subtract the sun's semi- diameter; the remainder will be the semi-diameter of the shadow CB, with which 16 40 .7 1.^ 51 .3 37 37 .8 2 24 .8 35 13 .0 — 3 23 .2 46 12 _2 62 52 .9 2D 31 .5 * III other words, the point P will fall above C if the moon i.s approaching to the north pole of the ecliptic, otherwise below : Ihat is, the point P must fall above C if the mo(m's latitude is smith dccrea-sin g or noTtIt iiicrra.iiiiir, otherwse below. When no great accuracy is re(iuired, the horary motion in latitude need not be found by" Problem 11. Instead of which, the angle COP mav be taken equal to 5° 40', in eclipses of Oie moon or sun, "and tlie line OP equal lo CO increased by 9" or 10" :"bul this method will not answer in occultations in whicli the angle COP varies above 5 degrees. t The d stance Gi may also be found by common arithmetic, by saying, As 60 minutes are to the minnteg and seconds in the time of full moon (which in the present e.xample is 39'), so is OP to Gx. After marking the hours on the line LGN, it is usual to divide them successively into halves and quarters of an hour, then into five minutes and one minute. X The semi-d ameter of the shadow is increased by the earth's atmosphere from 20" to 60", accoruing w the estimates of ditierent astronomers. Mayer supposes this correction to be one 60th part of the shadow, varying from 3:" lo 4i;''. Tlie mean of Mayer's correction addfid.to the sun's paralla.\ is nearly equal to SU* assumed as above. TO PROJECT AN ECLIPSE OF THE SUN. 419 < latitude is south. Make CO equal to the horary motion of the moon from the sun in lonori- tude, 35' 13" .0, and CP perpendicular thereto equal to the horary motion in. latitude, — 3' 2S".2, tlie point P being placed above C, because the moon's horary motion in tlie lati- tude lias tiie sign — prefi-ted ; or, in otlier words, the latitude was south decreasing. Join OP, and parallel thereto draw through G the line NGL, and on it let fall the perpendicular CK. Make the distance OP a transverse distance of 60, GO, on the line of lines of the sector, and measure from the same lines the transverse distance 39, 3!) (corresponding to tiie minutes in tlie time of full moon at tlie place -of observation) ; this distance, set on the line GN, to the riglit of G, reaches to the point x, where tlie hour, 17h., preceding the full moon, is to be marked. Take the extent OP, and lay it from ]71i. to the right hand to IGh., and successively to the left to ]8h. lL)h., &c. Subdivide these lines into GO equal parts, represent- ing minutes, if the scale will permit, and the times corresponding to the points E, e, K, A, H, will represent respectively the beginning of the eclipse, 15h. oGm. ; the beginning of total darkness, IGh. 54m.; the middle of the eclipse, 171i.41m. ; tlie end of total darkness, Idh. 28m. ; and the end of the eclipse, IDli. 2Gm. ; which times agree nearl}' with those in the Nautical Almanac, allowing for the difference of meridians 2 hours. CALCULATION BY LOGARITHMS. The phases of the eclipse may also be calculated by logarithms in a very simple manner. Thus, suppose it was required to find the time of the beginning of the eclipse in the above example. In this case, in tiie rirrht-angled triangle OCP, there would be given CO = 21I3".0, and CP =208".2, to find OP = 2123" .2, and the angle OPC == 84° 22'. This angle is equal to RGE, because GE, OP, are parallel, and its supplement gives the angle CGE =95° 38'. Then, in the triangle CGE, there are given the annle CGE = 95° 38', the moon's latitude CG = G44".8, and the line CE (= CD) =3772".9, to find CEG = 90 48', GCE=74° 34', and GE = 3G54".5. Then say. As OP (2123".2) is to 1 hour (3G00s.), so is GE (3G54"5.) to the time (G19Gs. = ), Ih. 43m. IGs., between the beginning of the eclipse and the full moon at the place of observation, 17h. 39m. ; and as the point E falls to the right hand of G, that time must be subtracted from 17h. 39m., to obtain tlie time of the beginning of the eclipse, 15h. 55m. 44s., which agrees nearly with the projection. As these calculations are very simple, it will be unnecessary to take notice of the different cases, or to give the calcula- tions at full length, the whole being sufficiently evident from the figure. 'The middle of the eclipse is found by means of the triangle GKC, similar to OCP, in which the angles and hypotenuse CG are given to find CK, KG. The time of describing KG being added to, or subtracted from the time of full moon at the place of observation, according as the point K falls to the left or right of G, will give the time of the middle of the eclipse. The distance CK, 10' 41".7, subtracted from the radius CD or CT = G2' 52".9, will leave a remainder equal to the eclipsed part FS (= KT), 52' 11".2 ; and the moon's diameter, 33' 21".4, is to FS,52' 11". 2, as 12 digits to the digits eclipsed, ISJ. In making these calculations, common or proportional logarithms may be made use of. PROELEM XL To project an eclipse of the sun for any given place. An eclipse of the sun can happen only at the time of new moon. If the moon s longitude at that time is not distant from either node of the moon's* orbit more than 17J'-', there may be an eclipse. To find whether there will be one, and to calculate the times and phases, proceed by the following RULE. To the time of the new moon, given in the Nautical Almanac (or calculated by Prob. III.), add the longitude- of the proj)osed place, turned into time, if east; but subtract if west ; the sum or difference will be the time of conjunction at the proposed place. Corresponding to the time of new moon at Greenwich, find, by Problem I., the moon's latitude, horizontal paralla.x, and semi-diameter ; also the sun's longitude, semi-diameter, and declination. Then, by Problem II., find the horary motion of the moon in latitude, and the horary mo- tion of the moon from the sun in longitude. Draw the line ACB (Plate XIII. fig. 10), representing the ecliptic, and, perpendicularly thereto, the line PCR. Take a scale of equal parts to measure the lines of the projection ; measure from it an interval equal to the moon's latitude, and apply it on CR from C to G ; above the line ACB if the moon's latitude is north, below if south.] Take CO, equal to the horary motion of the moon from the sun in longitude, and set it on the line CB, to the right hand of C to O ; take CP, equal to the moon's horary motion in latitude, found by Problem II., and set it on the line CR, from C to P ; above\ the hne ACB if the sign is — , below if -(— •'oin OP, which represents the horary motion of the moon from the sun on the * See note with the mark \ in page 415. All the eclipses that can happen in any part of the earth are indicated In the Nautical Ahiianac. t In the fig'ire, the latitude is supposed north. If it had been as much south, the point G would have been as rnucli below C as it is now aliove it. X See note with the mark * in page 416 420 TO PROJECT AN ECLIPSE OF THE SUN. • relative orbit, and parallel to that line draw the relative orbit of the moon, NGL, on which are to be piarked the places of the moon before and after tlie conjunction, by means of tlie horary motion OP, so that the moment of the new moon, or ecliptic conjunction, at the pro- posed place may fall exactly on the point G, as in the figure, where the new moon is at 23h. 35^m. This may be done by taking the extent OP, equal to the transverse distance of GO, 60, on the line of lines of the sector, then measuring from the same lines the transverse dis- tance corresponding to the minutes and parts of a minute of the time of new moon at the place of observation, and setting it on the line GN from G towards the right hand to the point X,* the place of tJie moon at the first whole hour preceding the conjunction (which in the present figure is 23h.) Then the distance OP being taken in the compasses, and set from a; to the right hand, gives successively the hours preceding the new moon, and the same distance set to the left gives tlie following hours, as in the figure, where they are marked in succession 22h., 23h., 24h., Ih. Tliese hours are to be divided into 60 equal parts, repre- senting minutes, the scale being taken sufficiently large for that purpose.! In the present figure, the subdivisions are carried only to five minutes. From tlie moon's horizontal parallax subtract tlie sun's, 8". 6 ; the remainder is to be taken •from the scale of equal parts for tlie radius CB, with which, on the centre C, describe the circle BRA, cutting CR in R. Open the sector till the transverse distance of 00°, 00°, on the line of chords, is equal to the radius CB, and measure from the same lines the trans- verse distance 23° 23' (equal to the obliquity of the ecliptic), which set on the circle ARB on each side of R to T and U. Join TU, cutting CR in Q. On Q as a centre, with the radius QT, describe the circle TVU, on which set off the arc TV equal to the sun's longi- tude. Through V draw the line VP' parallel to CR to cut TU in P', the place of the pole of the earth, t Draw CP', and continue it on either side so as to cut the circle ARB in the point W, situated above AB if tlie latitude of the proposed place is north, hclow if south. In the present figure, the latitude is north. If it had been south, the lower part of the circle ARB ought to have been made use of. Open the sector so as to make the transverse dis- tance 60°, 60°, on the chords, equal to CB, and measure off the transverse distance equal to the chord of the complement of the latitude of the place, which set from W on each side to D and d. With the same opening of the sector measure the cliord of the sun's declination, and set it on the same circle from U on each side to E and F, and from d on each side to e and/. Draw the dotted lines F/, Dd, Ec, cutting CW in /, q, n. Bisect hi in r, and erect the line VI r XVIII, perpendicular to CW, and make r VI and r XVIII, each equal to qD. Open tlie sector to make the transverse distance 90°, 90°, on the sines, equal to qD, and measure off the transverse distance corresponding to 15°, 30°, 45°, 60°, 75° (or 1, 2, 3, 4, 5 hours), which set on each side of the point r, on r VI and r XVIII, to the points marked witli the numbers 15°, 30°, &c. Through these points draw the lines I XI, 11 X, III IX, &c., as in tlie figure, parallel to CW. Open the sector so as to make rn equal to the transverse distance of 90°, 90°, on the sines, and measure the complements of the former degrees as transverse distances on the sines, viz. 75°, 00°, 45°, 30°, 15°, and set them on the above lines I XI, II X, &c. from the points of intersection with the line VI r XVIII, above and below that line. The points I, II, III, &e. obtained in this man- ner, will represent the situation of tlie spectator at the proposed place, at those hours, and a regular curve drawn through these points will represent his path. In marking tlie hours, it niust be observed, that the place of noon will be at the lower point n, if the sun's declination is north ; but at the -upper point I, if tlie declination is south : the hours must be marked from noon towards the left in numerical succession completely round the curve, ending at 24h., according to the method of astronomers. In the present figure, the declination is nortli, * See note witliectivelv equal to 75° 1, (iO° 9, 45° 3, &c., and the sought points, I, II, III, &c. will be obtained. This inethoil may be used when the line rn is too small to be taken from the sector. The same method may be made use of in projecting an occultation, by drawing Ik (fig. 8, Plate XIII.) perpendicular to ri and equal to rn, and joining rk to cut the dotted lines drawn parallel to CP'in the points 1,2, 3, &c. as above. TO rilOJECT AN ECLli'riL: OF THE SUiN. 421 and the point ii tlie place of noon or hours. If it had been south, the point I would have been marked Oh., and the points marked XI, X, &C. would be I, II, &c. respectively. The path touches the circle ARB in two points, representing the points of sun rising and setting, which, in the present figure, are respectively IGh. 2(Jm. and 7h. 34m. These points divide the path into two parts, of which one represents the path by day, the other b}' night, as is evident from the hours marked on the curve. Half hours, or any other intermediate time, may be marked in a similar manner. Thus, for the time 3h. 30m. = 52° 30', set the sine of f/sJ.^" to the radius r VI, from r to h on the line r VI, and erect the perpendicular /u' equal to tiie sine of 37;^° (whicli is the complement of ^2^'^) to the radius rn, and the point i will be the place of the spectator at the proposed time. In this way the Tialves and quarters of hours may be marked on those parts of tiie path wliere necessary. The smaller subdivisions may generally be obtained to a sufficient degree of accuracy by dividing the quarters of hours into equal parts. Take from the scale of equal parts an extent equal to the sum of the semi-diameters of the sun and moon, and, beginning near N, find, by trials, the point p' of the moon's path, and the point Z' of the path of the spectator, marked with the same time and at that distance apart. That time will be the beginning of the eclipse. If no such jjoints can be found, there will be no eclipse at the proposed place. Proceed in the same way towards the point L, and find the points ^^", Z", at the same distance apart; the corresponding time will be the end of the eclipse. Find, by trials, t!ie point p of the moon's path, and the point Z of the path of the spectator, marked with the same times at the nearest distance from each other (which will in general be nearly the middle time between the beginning and end of the eclipse) ; that time will be the middle of the eclipse. On Z as a centre, with a radius equal to the sun's semi-diameter, describe the circle whose diameter is Ss, representing the sun's disc ; and on the centre p, v/ith a radius equal to the moon's semi-diameter, describe the circle whose diameter is Mm, representing the moon's disc. The part of the sun's disc that is cut off by tliis circle will represent the part of the sun that is eclipsed. In the ex- ample of figure 10, the centre,/;, of the moon's disc is so near that of the sun, Z, that the eclipse is nearly central ; and. as the moon's semi-diameter is greater than the sun's, the eclipse must be total. Under similar circumstances, if the moon's semi-diameter had been least, the eclipse could hava been annular. In case of a partial eclipse, the sun's disc will not be wholly covered by the moon, as in figure 11, Plate XIII., where the circles represent- ing the discs of the sun and moon are marked with the same letters as in figure 10, but the objects are placed in a di§'erent situation. In this case, the number of digits eclipsed may be obtained by drawing a line through the centres p, Z, to meet the discs in the points S, M, s, m, and by saying. As the distance Ss (representing the whole disc) is to the obscured point M*, so are 12 digits to the number of digits eclipsed. The beginning and end of total darkness in a total eclipse are found like the beginning and end of the eclipse, except in taking in the compasses the difference between the semi-diameters of the sun and moon, instead of their sum. For the points of the path of the spectator and of the moon's orbit, marked with the same time, and at that distance from each other, will represent the situations and times of the beginning and end of total darkness. The beginning and end of the internal contacts of an annular eclipse are found in the same manner ; the only differ- ence is that, in a total eclipse, the moon's seiiii-diameter is greatest, but in an annular eclipse the least. In observing the beginning of a solar eclipse, it is of some importance for the accuracy of the observation, to know on what part of the sun's limb the eclipse will begin. This is easily found by means of the projection. Thus at the beginning of the eclipse, v;hich cor- responds to the point p' of the moon's path, and the point Z' of the path of the spectator, the first point of contact g may be obtained by drawing about the centre p', with a radius equal to tiie moon's semi-diameter, a circle representing the moon's disc;* about Z' as a centre, witli a radius equal to the sun's semi-diameter, another circle representing the sun's disc, touching the former in the point ff. Draw the line CZ', meeting the sun's'disc in the points a, c, the point c being the most distant from the centre C. Then the circle sr « c, be- ing held between the eye of the observer and the sun. the engraved or marked side of th;' figure towards the eye, and the line c « in a vertical direction with the point c uppermost. vi^ill represent the appearance of the sun as viewed by the naked eye at that time; r will represent the upper part of the sun, a the lower, and g the point of contact. If the eclips.' be observed with an inverting telescope, the contrary will be observed ; that is. t!;e part it must be uppermost, c the lowest, and g, the point of contact, will appear to the left' hand ore a. In a similar manner the appearance of the objects may be obtained at any other part of the eclipse, but it is not necessary except at the beginning of it, where there is nothing tn direct the eye of the observer. • Instead of this circle, the line p' Z' may be drawn cutting the sun's disc in the sought point of contact g. 422 TO PROJECT AiN ECLIPSE OF THE SUN. EXAMPLE. Required the times and phases of the total eclipse of the snn, June 10, 1806, at Salem, Ih the latitude of 42^ 33' 3U" iN., and the longitude of 4h. 43in. 32s. west from Greenwich. By the Nautical Almanac, the time of new moon at Greenwich was June 16d. 4h. lUm., ELEMENTS. Conjunction at Greenwich, June 16..., Salem W. from Green wicli Ecliptic ccmjunction at Salem, June 15, Latitude of Salem P 's horizontal parallax , 0's horizontal parallax , D 's reduced horizontal parallax , I)'s semi-diameter < 0's semi-diameter Sum of semi-diameters Difference of semi-diameters ]) 's horary motion in longitude, Prob. II 0's horary motion D's horary motion from CO D 's horary motion in latitude CP D's latitude by Prob. I CG 0's longitude TV 0's deciinatibn DF h. m. s. 4 19 00 4 43 32 23 35 28 42^33' 30" CO 25.7 8.6 60 17.1 16 28.1 15 46.1 32 14.2 42.0 33 41.9 2 23. 1 34 18.1 3 22.5 19 37 84 44 36 23 22 N. + corresponding to June 15, 23h. 3.5m. 28s., at Salem. At the time at Greenwich, 4h. 19m. tJie elements of the eclipse were, as in the adjoined table, calculated by the above rule. Draw At;B (Plate XIIL fig. 10), and per- pendicular thereto the line CGPJ.. Make CG equal to the moon's latitude, 19' 37" N., taken from a scale of equal parts, the point G being above C because the latitude is north. Make CO equal to the moon's horary motion from the sun, 34' 18". 1, to the right hand of the point C ; and CP equal to the moon's horary motion in latitude -j- 3' 22". 5, the point P being below C because this hora- ry motion has the sign -(- prefixed. Draw NGL parallel to OP. Make OP a transverse distance of GO, GO, on the line of lines of the sector, and measure from the same lines the transverse distance 35.^, 3.3J (corresponding nearly to the minutes in the time of new moon) ; this distance, set on the line GN to the right of G, reaches the point x, where the hour pre ceding the new moon is to be marked, viz. 23h. Take OP m the compasses, and mark it suc- cessively on the line NL from x, or 23h., to the right to 22h., and to the left to 24h. or Oh., Ih.. &c. These are subdivided into five minutes, the scale not admitting smaller divisions. Take the moon's reduced horizontal parallax, GO' 17". 1, from the scale of equal parts, and with that radius describe about the centre C the circle ARB. Set off (by means of the sector) the arcs RT, RU, each equal to 23" 28'. Join TQU, and about that diameter describe the circle TYU. Make the arc TV equal to the sun's longitude, 84° 44' 36' which is done by setting the radius QT as a chord from T to D, and then the arc nV = 24° 44' 36" by means of the sector. Draw P'V. parallel to CR, to meet TU in the point P'. Join CP', and continue it to meet the circle ARB in W. Make (by the sector) the arcs WD, Wrf, equal to the complement of the latitude of the place, 47° 26.^' nearly, the radius being CB. In a similar manner make the arcs DF, DE, df, de, &c., each equal to the sun's declination 23° 22'. Draw the lines FIf, Dqd, Ene, cutting CW in /, q, n. Bisect In in r. Draw the line VI r XVIII parallel to Dqd, and make r VI, r XVIII, each equal to qD. Through the points I, VI, n, XVIII, /, draw the path of the spectator as taught in the above rule, and mark the hour of noon, Oh., at the point n because the sun's declination is north. Mark the following hours in succession to the left, I, II, III, Sec, as in the figure. Take an extent in the compasses equal to the sum of the semi-diarneters of the sun and moon, 32' 14" .2, and, beginning towards N, find, as above directed, the points p'Z' at that distance apart and marked with the same time, 22h. 7m. nearly, which is the time of the be- ginning of the eclipse. Proceed in the same way for the end of the eclipse corresponding to the points p", Z", and to the time Oh. 53m., which is the time of the end of the eclipse. Take the difference of the semi-diameters of the sun and moon, 42", in the compasses, and proceed in the same way to find the beginning and end of total darkness, 23h. 27m., and 23h. 31m. The points corresponding could not be drawn in the figure, as they are so near to p and Z, and the scale small. Find, by trials, the points p, Z, marked with the same time and at the least distance apart; this will be the time of the middle of the eclipse, 23h. 29m. With an extent equal to the moon's semi-diameter, IG' 28". 1, as a radius, describe about/* the circle whose diameter is Mm representing the moon's disc ; and with the sun's semi- diameter, 15' 46' .1 , describe about Z the circle whose diameter is Ss, representing the sun's disc at the middle of the eclipse. The sun's disc being wholly covered by the moon, in- dicates that tlie eclipse was total. Describe, in the same way, about p' and Z' the discs of the sun and moon, at the beginning of the eclipse, touching each other in g. Draw CZ', cut- ting the moon's disc in c and a. Then the arc e g will be the distance of the first point of contact of the sun and moon from the sun's zenith towards the western part of the limb. REMARKS. 1. The correction for the spheroidal form of the earth, the augmentation of the moon's semi-diameter, inflexion and irradiation, are neglected in the above rule, as not sensibly affecting the result of the projection, though these points might be attended to by the follow- ing precepts. 2. From the latitude of the place subtract the correction of latitude of Table XXXVIII., and from the moon's horizontal parallax, decreased by 8".G, subtract the correction of paral- lax in the same table ; the remainders will be the corrected latitude and parallax to be :nade use of in the above rule to correct for the spheroidal form of the earth. TO PROJECT AJN OCCULTATION 423 3. Decrease liio moon's semi-diameter given by tlie Nautical Almanac by 2" for inflexion, if it be thouglit necessary. 4. Decrease the sun's semi-diameter 3^" for irradiation, and from the remainder subtract a correction equal to tlie aujrnientiition (Table XV.) that the moon's semi-diameter would have when at the same altitude as the sun ; the remainder will be the corrected semi-diame- ter of the sun, to be used in tlie above rule in finding all the times and phases of the eclipse. This metljod of decreasing the sun's semi-diameter produces nearly the same result as that by augmenting tlie moon's semi-diameter, horary motion, and horizontal parallax, and taking the sun's semi-diameter as given in the Nautical Almanac. 5. Besides these corrections, there are otliers, depending on the change of the moon's semi-diameter, horizontal parallax, and horary motion during the eclipse; but all these cor- rections are usually neglected in projecting an eclipse or occultation. 6. The altitude of the sun, which is nearly the same as that of the moon during the eclipse, may easily be found by moans of the projection. Tlius, if it were required at the beginning of the eclipse, when the spectator is at Z' : Take the distance CB, and a])ply it as a transverse distance 90"^, 90°, to the sines of the sector; then the distance CZ', ap|)lied in the same manner to those lines, will give the zenith distance of the sun, about '31^, cor- responding to the altitude 59°. The correction (Table XV.) cofresponding to this altitude is 14", which is nearly the correction to be subtracted from the sun's semi-diameter, 15' 42" .6 (corrected for irradiation), to obtain the corrected semi-diameter, 15' 28". G, as taught in §4. fable XV. was calculated for the mean semi-diameter, 15' 37", and tlie correction of the Table, 14", ought to be increased in ratio of the sun's semi-diameter, 15' 46". 1, to 15' 37", when very great accuracy is required. The difference of tlie corrected semi-diameters of the sun and moon, 15' 28" .6 and IG' 2G".l, is 57^", which is to be used instead of 42" in find- ing the beginning and end of total darkness. The duration of the total darkness found by the corrected value 57.y, is 4;'^ minutes, but with the uncorrected value 42", is only 3^ minutes. It was probably owing to the neglect of this correction that some of the Almanacs publislied in this country, for 180G, mentioned tlie duration as 3 minutes. 7. The path of the spectator, I, II, III, IV, &.C., calculated for the proposed latitude 42° 33' 30", may be made to answer for any other latitude by altering the centre of projection and the scale of equal parts. By this means the trouble of repeatedly describing that patli, when tlie eclipse is to be calculated for several places, may be avoided. To do this, ad(i the prop. log. of the reduced parallax to the log. secant of the latitude of the place; the sum, rejecting 10 in the index, will be the prop. log. of an arc A. To this prop. log. add the log-, secant of the sun's declination (or star's in an occultation), and the log. cotangent of the latitude of tiie place ; the sum, rejecting 20 in the index, will be the prop. log. of the arc B. Take the radius r VI (or qD), in tlie compasses, and make it a transverse distance on the line of lines of the sector corresponding to the arc A, and with that openin,'^ of the sector measure the transverse distance corresponding to the arc B, which, set from r towards C on the line rC (continued if necessary), will reacli to the centre of the projection corresponding to the proposed latitude ; the transverse distance corresponding to the redu'ied parallax., measured from the line of lines with the same opening, will be the radius of tlis projection, and the transverse distance corresponding to the horary motion of the moon from tlie sun or star, in an occultation, will be tiie horary distance to be made use of in marking the hours on the lunar orbit LN ; lastly, the latitude of the moon at the conjunction is to be measured as- a transverse distance, and set from tlie new centre of projection on a line drawn througli it parallel to CR, and the point where it reaches will be the new point G, corresponding to the place of the moon at tiie ecliptic conjunction. Through this point tlie line of the moon's path is to be drawn parallel to the line LN of the figure, and the hours are to be marked on it as before. 'Whence tlie times of beginning and end of the eclipse may be found as in the above rule. An example of this method is not given, as it would render the scheme too confused. PROBLEM XIL To project an occultation of a Jixed star by the moon, at any given placr. The method of projecting an occultation is nearly the same as that of an eclipse of the sun ; but to save the trouble of reference, it was thought expedient to give the rule without abridg- ment. RULE. To the time of the ecliptic conjunction of the moon and star, computed from the Nautical Almanac by Problem III., add tiie longitude of the proposed place turned into time, if east, jut subtract if west; the sum or difference will be the time of conjunction at the ])roposed place. Corresponding to the time of conjunction at Greenwich, find, by Problem I., the moon's latitude, horizontal parallax, and semi-diameter ; also the sun's right ascension. Then, by Problem II., find the horary motion of the moon in longitude and latitude, and by Tables VIII. and XXKVIL, the star's right ascension, declination, longitude and latitude.* * In strictness, these quantities oiislit to be corrected for aberration and nnlation, by Tables XXXIX. XLIK., but the correction is so small th.-rt it may always be neu'lected. If the right ascension and declina- tion only are given, the latitude and lonjiitude may be fonnd by Problem XIX., and if the latter are given, the former may be calculated by Problem XX. It will be found most convenient to nse the right ascensions. and declinations which are given In the Nautical Almanac, wlien any of the stars n'arket' iii it are used 124 TO PROJECT AN OCCULTATJON Draw flie line ACB (Plate XIII. fig. 8), representing a parallel of the eclipfic passing through the star, and perpendicular thereto the line CPR. Take a scale of equal parts tc measure the lines of jjrojection, and from it take an interval equal to the diiTerencc of the latitudes" of the moon and star, and apply it to the line CR from C to G,abuvc the line AC13 if the moon's latitude is north of the star's, otherwise bcloio* Take CO equal to the horary motion of the moon in longitude, and set it on the line CB to the right hand of C to O ; take CP equal to the moon's horary motion in latitude, found with its sign by Problem II., and set it on the line CR from C to P, above \ the line ACB if its sign is — , below if -f-. Join OP, which represents the horary motion of the moon in her orbit, and parallel to that hne draw the orbit of the moon, NGL, on which are to be marked the places of the moon before and after the conjunction by means of the horary motion OP, so that the moment of the ecliptic conjunction at the proposed place may fall exactly at the point G, as in the figure wlierc the conjunction is at 18h. 42m. This may be dune by making OP equal to the transverse distance (iO, 60, on the line of lines of the sector, then measuring from the same lines the transverse distance corresponding to the minutes and parts of a minute in the time of the ecliptic conjunction at the place of observation, and setting it on the line Gi\ from G towards the right to the point x, the place of the moon at the first whole hour t preceding the conjunction (which in Wie present figure is 18h.) Tlien tlie distance OP, being taken in the compasses, and set from x to the riglit hand, gives successively the preceding hours, and the same distance set to the left gives the following hours, as in the figure, where they are marked 17h., 18h., 19h., 20h. These hours are to be divided into GO equal parts representing minutes, the scale being taken sufliciently large for that purpose. § In the present figure the subdivisions are carried only to five minutes. Take the moon's horizontal parallax from ihe scale of equal parts for the radius CB, with which, on the centre C, describe the circle BRA, cutting CR in R. Open the sector till the transverse distance C0°, C0°, on the line of chords is equal co the radius CB, and measure from that line the transverse distance 23- 28' (equal to the obliquity of the ecliptic), which set on the circle ARB, on each side of R to T end U. Join TU cutting CR in Q. On Q as a centre, with the radius QT, describe a circle, TYUV, on which set off the arc TYV, equal to the star's longitude. Through V draw the line VP' parallel to CR. Open the sector till the transverse distance C0°, 90'^, on the sines, is equal to the radius CB; then take in the compasses from the same lines an ex- tent equal to the transverse distance corresponding to the complement of the declination of the star, and with one foot in C sweep a small arc to cut the line VP' in P', the place oi* tlie pole of the earth. || Draw CP',and continue it on either side so as to cut the circle ARB in the point W, situated above AB, if the latitude of the proposed place is north, but bel9w if souih. In the proposed figure the latitude is north. (If it had beensouth, the lower part of the circle ARB ought to have been made use of.) Open the sector as before, so as to make the transverse distance of C0°, 00°, on the chords, equal to CB, and take the chord of the complement of the latitude of the place, which set from W on each side to D and d. With the same opening of the sector measure the chord of the star's declination, which set on the circle ARB from the point D on each side, to E and F, and from d on each side to e and/. Draw the dotted lines F/, Dd, Ee, cutting CW in /, q, n. Bisect Z w in r, and erect the line tru perpendicular to CW, and make rt, ru, each equal to ^D. Oj>en the sector to make the transverse distance 90°, 90°, on the sines equal to r t, and on each side of /• mark on the line trji the sines of 15°, 30°, 45°, G0°, 75° (equal to Ih., 2h., 3h., 4h., 5h., respective- ly) to that radius, and mark tlie points with those degrees as in the figure ; through these points draw the dotted lines parallel to In as in the figure. Open the sector so that the radius rl may correspond to the transverse distance 90°, 90°, on the sines, and measure the complemcnls of the former degrees as transverse distances on the sines, viz. 75°, 00°, 45°, 30°, 15°, and set them on the above dotted lines, on each side of the points 15°, 30°, &c., respectively, above and below the line t ru. A rcgulnr curve, ntlun, drawn through the extremities of these dotted lines, will represent the path of the spectator in the given lati- tude. Subtract the sun's right ascension from the star's (increasing the latter by 24 hours when necessary) ; the remainder will be the hour of the star's passing the meridian, IT which is to be marked at the upper point I of the path if the star's declination is south, but at the lower point n if the declination is north. The other hours are to be marked from this point towards the left, by marking successively, at the points where the dotted lines meet the path, the hour of the star's passing the meridian, increased by Ih., 2h., 3h., &e., completely round the curve, observing to reject 24 hours when the sum exceeds 24h. In the present example, the star's declination is south ; consequently the upper point / of the path is taken for the hour of passing the meridian, l!}h. 54m. ; the extremities of the dotted lines to the left being marked successively 20h. 54m., 21h. 54m., 22h. 54m., 23h. 54m., Oh. 54m., &c. * In the figure the point G is placed above ACB, because the moon is in a less southern latitude than the star. This part of the rule may also be thus expressed : — Find the moon's latitude with its sign as in Prob- lem II. Prefix tlie sign + to the star's latitude if nortli, the sign — if south. Add the latitudes, noticing the signs as in algebra, and the distance CG will be obtained. If its sign is — , the point G is to be placed above C, but below C if the sign is -f-. t See note with tlie mark * in page 416. ^ I See note with the mark | in page 416. ^ See note with the mark f in page 418. I'l The distance of the line \VV from the line CR, the situation of t(je point P', and the path of the spectator, may be found as in the note J page 418. H Or r.ilher the horary distance of the sun and star at the time of the ecliptic coiijiinit on "f the mooin , ana star TO PROJECT a:> Ool^ULTATION, 425 The path touches the circle ARB in two points, representing tlio points of rising and setting of the star, whicii, in the present figure, are J41i. I'm., and Ih. 3l'ni. These points divide tlie path into two parts, of which one represents the path wliile tlie star is above tiie horizon, tlie other when below, as is evident from the hours marked on the curve. I'he half hours, or any other intermediate time, may be marked in a similar manner. Thus, for the time 4h. 24in., which is 3h. 30m., or o2° 30', from the time 7h. 54m., marked at the point ?i, set the sine of 52A'-' to the radius rt from r to h on the line it, and erect the perpendicular //i, equal to the sine of 37.;^° (which is the complement of 52.^") , to the radius rn, and the point t will represent the place of tlie spectator at the proposed time. In this way the halves and quarters of hours may be marked on tho.se parts of the path where necessary. The smaller subdivisions may n;enerally be obtained to a sufficient degree of exactness by dividing the (juarters of hours into equal parts. Take from the scale of equal parts an extent equal to the senii-diamcter of the moon, and, beginning at the line NL, towards N, find, by trials, the point// of the moon's path, and the point Z' of tiie path of the spectator, marked with the same time and at that distance apart. That time will be the beginning of the occultation or immersion at the proposed place. Pro- ceed in the same w.ay towards the point L, and find the points p, Z, at the same distance apart ; the corresponding time will be the end of the occultation or emersion. About the points //, /;. as centres, with a radius equal to the moon's semi-diameter, describe the small circles meeting the paths of the spectator in the points Z', Z. These circles will represent the moon's disc ; the points Z', Z, the places of the star, and the line CZ', CZ, the vertical circles |>assing through the star at the times of immersion and emersion respectively. To render this part of tlie scheme more distinct to the eye, it is drawn separately in figure 9, Plate XIII., in which the points C, />',Z', are similarl}' situated to the corresponding points of figure 8, marked with the same letters. Through// draw the line a' p' c' parallel to CZ , to meet tlie moon's disc in a', c'. Then the circle a' Z' c', being held between the eye of the observer and the sun, the engraved or marked side of the figure towards the eye, and the line CZ' (or a' p' c') in a vertical position with the point Z' above C, will represent the ap- pearance of the moon and star as viewed by the naked eye ; c' will represent the upper part of the moon,' a' the lower part, and Z' the point of contact. The contrary will be ob.served if the object be viewed by an inverting telescope. It will generally be conducive to the accuracy of an observation, to estimate in this manner the point of emersion, so as to keep that point of the moon's limb in the field of view of the telescope, and the eye directed to- wards that point of the liuib, so as to perceive the star at the first instant of its appearance. The situation of the point of emersion with respect to the horns q, 0, of the moon may also be made use of for this purpose. The line (i/; 0, connecting the moon's horns, is nearly parallel to the line CR, except very near the new or full moon ; so that in general it will be sufficiently correct to draw tlirough p the line QpO parallel to CR. If greater accuracy is required, the following construction may be made use of Subtract the sun's longitude from the moon's,! make the arc TYU<\ equal to the remainder, and join QX. Set on the same circle the arc T;i equal to the moon's latitude ; below the point T if that latitude is south, aliove if north. Through (i draw the line ^i 5 parallel to TQ to cut QX in t and CR in S. Take the extent QT and set it on the line 6Y above S to .«. Join u i, and parallel thereto through p draw tlie line QpO cutting the moon's disc in the points qO representmg the horns, the figure being viewed as above directed. The enlightened part of the moon is that nearest to the sun j the dark part is the most distant from it. EXAMPLE. Required the limes of immersion and emersion of Spica, December 12, 1808, at a place in the latitude of 20° N., and in the longitude of Ih. 9m. east from Greenwich. By tile first page of the Nautical Almanac for the month of Decem- ber, 1808, the time of the ecliptic conjunction of the moon and Spica (marked ]) a IT^) was December 12, 17h. 33m. at Greenwich, correspond- ing to 18h. 42m, at the proposed place. This time may also be com- puted by means of the longitudes of the objects, as in Problem III. of tl.is Appendix, At the time at Greenwich, 17h, 33in., the elements of the occultation were, as in the adjoined table, calculated by the above rule. Draw ACB, and perpendicular thereto the line CGY. Make CG equal to the difference between the ELEMENTS. Conjunction at Greenwich, Dec. 12, 1808.... Longitude east from do Conjunction at place of oliservation *'s right ascension. Table Vril 0's right ascen. by Nautical Almanac. subtract * passes the meridian Latitude of the place D's Iiorizontal paral. by Nautical Almanac. CB P'ssemi diameter by Nautical Almanac D's horary motion in longitude, Prob. II.... CO D's horary motion in latitude, Prob. II CP #'3 longitude, Table XXX VH TYV P 's latitude by Nautical Almanac *'s latitude, table XXXVII D (Terence of latitudes 5 N. of * CG *'s declination h. m. ,s. 17 33 00 1 09 00 18 42 00 13 15 08 17 21 13 19 53 50 20° 0' 0" 50 5.i.2 16 19.8 35 55.2 — 3 02.7 201 10 31 1 49 ^,3 S. 2 2 13 S. 12 20 N. 10 10 S. f In strictness, the Irng tude and latitude of the moon at tlie time of immersion or emersion ought to be made use of; but it w 11 be su(lic;ontly e.x.-ict to use the star's longitude instead of the moon's (increasing it by 3(30° when less than the sun's longitude), and the moon's latitude at t!)e conjunction tiuaritities of the same order as the moon's parallax are neglected in the value of the arc TYUA. 54 426 TO PROJECT AN OCCULTATIOM. latitudes of the moon and star, 12' 20", taken from a scale of equal parts, tlic point G beine the star's zenith distance at the immersion. In a similar manner it may be found at the emersion at Z, or at any other point. 5. The curve Itnu may be made to answer for any latitude, as in Problem XL, Remark 7. Calculation of an occultation of a planet by the moon. By a similar process the times of immersion and emersion of a planet may be calculated by finding the planet's right ascension and declination, geocentric longitude and latitude, from the Nautical Almanac, and using them instead of the star's; also, by Problem II., the horary motion of the moon from the planet in longitude and latitude, which are to be used instead of tiie horary motion of the moon. In this projection it will not be necessary to take notice of the parallax of the planet, but it may be easily allowed for by taking the radius CB equal to the difference of the horizontal parallaxes of the moon and planet. The apparent diameter of the planet may also be neglected, making the distances pZ. p'Z' , equal to the moon's semi-diameter. When great accuracy is required, the sum of the semi-diame- ters of the moon and planet must be made use of for finding the external contacts, :md their difference for the internal contacts. //'/AXiy E.tcO.Wrm.T'NT. UUil TO CALCULATE THE BEGINNLNG OR END OF AN ECLIPSE. 427 PROBLEM XIIL To calcvlaie the beginning or end of a solar eclipse. RULE. Tliis must be done by approximation, by assuming a time for tiie beginning or end of the ellipse, as, for example, the time obtained by projection by Problem XL, the time of new moon at the place of observation, or an hour before pr after, according as it is the beginning or end of the eclipse that is sought. With this time calculate the elements of the eclipse and the parallaxes, as taught in Uie first part of Problem VIIL The parallaxes applied to the longitude and latitude of the moon by the Nautical Almanac, will give the apparent longitude and latitude. Find the difference of the apparent longitudes of the moon and sun, and'from its proportional logarithm, increasing the index by 10, subtract the proportional logarithm of the moon's apparent latitude ; the remainder will be the log. tangent of an an°gle, whose corresponding log. cosine is to be added to the proportional logarillim of the dilference of longitudes ; the sum, rejecting 10 in the index, will be the proportional loga- rithm of the apparent distance of the centres of the sun and moon, which ought to be equal to the sum of the corrected semi-diameters, if the assumed time was correct. If this is not the case, the operation must be repeated with an assumed time differing a few minutes from the former, and the apparent distance of the centres of the sun and moon must be calculated in this new supposition. Then add together the arithmetical complement of the proportional logarithm of the difference of the apparent distances thus calculated, the proportional loga- rithm of the difference between the first calculated distance afid the sum of the semi-diame- ters, and the proportional logarithm of the interval of time between the two suppositions ; the sum, rejecting 10 in the index, will be the proportional logarithm of the correction to be applied to the first assumed time, which, at the beginning of an eclipse, is to be added to the first assumed time, if the distance be greater than the sum of the semi-diameters, but sub- tracted if less ; and the contrary in calculating the end of an eclipse; the sum or difference will be the approximate time of the beginning or end of the eclipse. If great accuracy is required, the operation may be repeated with this approximate time, combining this result .with one of the former sjuppositions ; and thus the operation may be repeated till the apparent distance of the centres at the assumed time is found to be exactly equal to the sum of the corrected semi-diameters. REiMARK. This rule, with some modification, will answer for calculating the time of an occultation of a fixed star or planet by the moon. In this case, the star's longitude is to be found in Table XXXVII., and corrected for the equation. Table XLL* (or the planet's longi- tude is to be taken from the Nautical Almanac ;) the difference between this and the moon's apparent longitude corresponding to the assumed time being found, its proportional loga- rithm is to be added to the log. secant of the moon's apparent latitude, and the sum is to be used in finding the distance of the centres instead o& the proportional logarithm of the dif- ference of longitude of the sun and moon, with the index increased by 10. The latitude of the star is to be found by Tables XXXVII. and XLL, or the planet's latitude by the Nautical Almanac, and added to the latitude of the moon, if of a different name ; otherwise their difference is to be taken and made use of, instead of the moon's latitude in the above rule. Lastly, instead of the sum of the semi-diameters, the semi-diameter of the moon is to be made use of When very great accuracy is required in calculating an occultation of a planet by the moon, the difference of tlie parallaxes of the moon and planet, decreased by the cor- rection of parallax, Table XXXVllI., is to be made use of as the reduced parallax, in finding the parallaxes in longitude and latitude. When the apparent distance of the centres of the moon and planet is equal to the sum of their semi-diameters, their limbs will just appear to touch each other; and when that distance is equal to the difference of the semi-diameters, the planet will be wholly covered by the moon. EXAMPLE. Required the time of the beginning of the solar eclipse of June, 180G, at Salem, supposing the errors of the moon's longitude and latitude in the Nautical Almanac to be unknown. To abridge the present calculation, suppose the beginning of the eclipse to be June hence the uncorrected values are 84° 9' 48" .8, and 2' 7" .2 N. The difference between this appatent longitude of the moon, and the sun's longitude, 84° 41' 3".4, is 31' 14".6. Difference of longitufle... 31' 14".6; Prop. Log. 10.7605 0.7G05 D 's apparent latitude 2 7.2 Prop. Log. 1.9289 Tang. 8.8316 — Corresponding Cosine 9.9990 Apparent distance O p. ...31' 19". Prop Lo g. .7595 * We must also apply the correction of Table XL., if the longitudes are counted from the onparent '^oinox, as was the case formerlv m the NaiUical Almanacs 428 TO FIND THE APPARENT TIME AT GREENWICH This apparent distance differs 1' 4".5 from the sum of the semi-diameters, 32' 23". 5. It is therefore necessary to make a second supposition, as for example ton minutes later, or at 22h. ICra. 18s. 1 ; with tliis time tlie elements are to be again calculated as in Problem VI., namely, moon's apparent longitude uAcorrected, 84° 14'17".l ; sun's longitude, 84" 41' 27 .2; their difference, 27' 10". 1 ; moon's apparent latitude uncorrected for error of tables, 1' 58".8 N Difference oflongiturte.,. 27' lO'M..., Prop. Log. 10.8212 0.8212 ]) -3 ai)pareiit latitude 158.8 Prop. Log. 1.9586 Tang. 8.8G2G Corresponding Cosine 9.9988 Second apparent distance Q D 27' 14".7 Prop. Log. .8200 First apparent distance O ]) 31 19.0 Difference 4 4 .3..Arith. Comp. Prop. Log. 8.3545 Difference first distance and semi-diameters 1 4.5 Prop. Log. 2.2238 Interval 10 Prop. Log. 1.25.53 .Prop. Log. ].8338 Correction 2 38 First supposed time , lod. 22h. 6m. 18s.l Appro.ximate time 15d. 221i. 3m. 40s.l If the approximate time differ very much from the assumed times, it will be necessary to repeat the operation till the last assumed and calculated times agree. PROBLEM XIV. Given the moon's true longitude tojind the mean time at Greenivich, IvULE. 1. Take from the Nautical Almanac the two longitudes immediately preceding the given longitude and the two following, and find the first and second differences, as in Problem I. Call the middle term of the first differences the arc A, and the half-sum of the second _ differences, (noticing the signs,) the arc B. 2. To the constant logarithm 4.63548 add the arithmetical complement of t!ie logarithm of A, in seconds, and tlie logarithm of the difference in seconds between the given longitude and the second longitude, taken from the Nautical Almanac ; the sum, rejecting 10 in the index, will be the logarithm of the approximate time T in seconds. 3. Enter Table XLV. with the arc B at the top. and this time T at the side, and find the corresponding correction ; to the logarithm of which add the two first logarithms above found ; the sum, rejecting 10 in the index, will be the correction of tlie approximate time, to be applied witli the same sign as the arc B, and the correct mean time, counted on from the second noon or midnight, will be obtained. EXAMPLE. Suppose the moon's longitude, July 12, 1836, v.-as 98° 10' 16".0. Required the corre- sponding mean time at Greenwich. 2d difference. Mean time. | d. h. July 11 12 12 12 12 13 ])'s lonn tudes. o ( II 89 12 57.4 95 07 44.7 101 03 20.9 106 59 59.8 1st difference. » ( It 5 54 4/ .o 1^ 4Q Q A = 5 55 36.2 TZo7 i> Sb Jo.9 n— -1-55.8 Constant Log. 4.63548 A = 2133fi".2 Arith. Comp. Log. 5.67089 Diff. oflong 10951".3 Log. 4.03946 Appro.v. lime.. Gli. OOiii. 33s. = 22173s Log. 4.34583 Correction.... +14 D's longitude 98° 10' ]G".0 July 12d. Oil 95 07 44 .7 Diff. longitude. 3 02 31 .3 = I0951".3 4.63548 • 5.67089 Eq. Tab. XLV. + 7".0 Log. 0.84510 Correction, +14s Log. 1.15147 Wean time.... Cli. 09in. 47s. past noon, July ISd. The same method might be used in finding the time from the moon's right ascension, supposing the Nautical Almanacs to give the right ascensions at noon and midnight only, as was formerly the case ; but as they are now given for, every hour, we may obtain the time much more simply by the following rule : — RULE. Take from the Nautical Almanac the right ascensions of the ngoon which immediately precede and follow the time at Greenwich, of the proposed observation. Take the differ- ence, D, of those two right ascensions, in seconds of time, also the difference, d, in seconds of time, between the given right ascension and that corresponding to the first hour. Then to the constant logarithm 3.55630 add the arithmetical complement of the logarithm of D, and the logarithnrof d; the sum, rejecting 10 in the index, will be the logarithm of a num ber of seconds to be added to the hour first marked in the Nautical Almanac, to obtain the mean time of the observation at Greenwich, nearly. ru FiiND THE LOiNGlTUDE OF A PLACE. 429 EXA3IPLE. Tlie moon's right ascension, July 12, 1S3G, was, by observation, Gh. 36m. 393.35. Required the mean time of observation. Right Ascension. Difference. July ]Oii Observed right ascension, Gli. SCm. 39s.35 ^ _ oi, no Constant Log. 3.55030 •Julyl2d.Ch byN.A. 6 3(3 17.06 ^ = 1^3 04 XViilV Conn. Lo^' 787602 Julyl2d.7h byN.A.6 38 30.70 "— ^-^J-"* Aran, t^onip. i>o„. /.B/oi« Om. 47s. =5873 Log 2.76858 First hour ISd. Gh. — ; Jlean time of observation, July 12d. 6h.9in.47s. PROBLEM XV. Given the distance of the moon from afxed star not marked in the JVaidical Jllmanac, together with the altitudes of the objects, the mean time of observation, and the estimated longitude, to find the longitude of the place of observation. First sulutlan, using the tnlitudcs and longitudes of the moon and star. RULE. To the mean time of observation, by astronomical computation, add the estimated longitude in time if west, or subtract if east ; the sum or ditlefence will be the supposed mean time at Greenwicii,* corresponding to which, find the moon's latitude, by Problem L, also tlie loniritude and latitude of the star, by Table XXXVIL, and correct them for aberra- tion, by Table XLL VV'ith the apparent altitudes and distance of the objects, find the correct distance by the usual rules of working a lunar observation. To the correct distance, add the latitudes of the moon and star, and find the difference. between the hulf-snm and the distance. Then to the log. secants of the latitudes of the moon and star, rejecting 10 in each index, add the log. cosines o[ the half-sum and differ- ence, if the latitudes are of the same name, or the log. sines, if of a contrary name ; half the sum of these four logarithms will be the log. cosine of half the difference of longitude, if the latitudes are of the same name, or its log. sine, if of a different name. The difference of longitude is to be added to the apparent longitude of the star, if the moon is east of the star, otherwise subtracted, (borrowing or rejecting 300° when neces- sary ;) the sum or difference will be the true longitude of the moon ; whence the mean time ai Greenwich may be found, by Problem XIV. The difference between this ajid the mean time at the ship, will be the longitude, which will be ivcst, if the mean time at Greenwich be greater than the mean time at the ship, otherwise cast. REMARK. This method, with a slight modification, can be used in finding the longitude from the observed distance of the moon from a planet, as Jupiter, Venus, Rlars, or Saturn, in cases where they are not marked in the Nautical Almanac. The only difference in the rule, when a planet is used instead of a star, consists in finding from the Nautical Almanac, by Problem I., the geocentric longitude and latitude of the planet, which are to be used instead of the longitude and latitude of the star in the above rule. For the daily variation of the longitude and latitude of a planet is so small, that no error of moment can arise from calculating those quantities for the sxipposcd instead of the true time at Greenwich; and the parallax and semi-diameter of the planet can be allowed for by the methods pointed out in working a lunar observation. The latitudes of the moon and the fixed star or planet, made use of in these observations, ought not to differ very much, on account of the decrease of the relative motion arisint; from this source. If the latitudes are of a different name, their sum, otherwise their difference, ought to be found, and if it does not exceed one third part of the difference of longitude of the two objects, they may in general be made use of. EXA.'\IPLE. Suppose that, on the 7th of January, 1836, sea account, at Ilm. 57s. past midnight, mean time, in the longitude of 127° 30' E., by account, the observed distance of the farthest limb of the moon from the star Aldebaran, was GS° 36' 0", the observed altitude of the star 32° 14', and the observed altitude of the moon's lower limb 34° 43'. Required the true longitude, without using the distances marked in the Nautical Almanac, upon the supposi- tion that tliey are not given in it. This lunar observation has already been computed by the common m.cthods, in page 232, where we have found that the supposed time at Greenwich is Jan. Gd. 3h. 41m. 57s., the moon's semi-diameter 15' 15'', the moon's horizontal parallax 55' 24", the star's apparent altitude 32° 10', the moon's apparent altitude 34° 55', the apparent distance of the centres of * This time may also be obtained from the chronometer, if you have one which is pretty well rrgiilnl< d •o astronomical time 130 TO FIND THE LONGITUDE OF A PLACE the moon and star G3° 20' 45". With these we find the true distance of the -centres of the moon and star, by the usual rules for working a lunar observation, to be 68° 3' 0'', as in page 232. The moon's latitude, deduced from the Nautical Almanac, by Problem L, is 4° 59' 10" N. Tlien the star's longitude and latitude are found as below, by Tables XXX VIL, XLL, making use of the sun's longitude, 235° 17', as given in the Nautical Almanac, these longitudes being counted from the mean equinox ; with these elements the calculation is made in the following- manner : — Table XXXVII *'j^ longitude, Jan. 6, 1836. 'I'aljle XLI Alieiration 67° 29' 47".l *'s latitude. -|- 15 .9 Aberration. ' 28' 39''. S. + 1 .2 *'s apparent longitude 67 30 03 -*'s apparent latitude 5 28 40 .OS. True distance 68° 03' 00" D's latitude 4 59 10 N Secant 0.00104 *'s latitude 5 28 40 S Secant 00199 Sum....: 1 78 30 50 Half-sum 39 15 25 Sine* Difference of lialf-sum and distance 28 47 35 Sine* 9.80126 9.68274 Ualf-dilTerence of longitude. 2)19.48763 33 40 06 Sine* 9.74381 Difference of longitude 67 20 12 *'3 longitude '. • 67 30 03 D's longitude 134 50 15 D's longitude, Jan. Od. Oh 132 54 51 Difference 1 55 24 = 0024" =difrerence. d. h. D's longitude... Jan. 5 12 6 6 12 7 126 41 37.6 132 54 51.1 139 10 47.8 145 29 34.2 1st differences o / II 6 13 13.5 A = 6 15 56.7 6 18 46.4 2d differences, fi -f 2 43.2 • -f 2 49.7 Mean = +2 46.5 Constant Log. 4.63548 A = 6° 15' 56".7 = 2-255Ci'.7 Log. Ar. Co. 5.64672 Dirt'erence, 6924" Log. 3.84036 Approx. time, 3h. 4Im.01s.: Correction... +34 Time 3h. 41m. 3."8. :13261s Log. 4.1-2256 .... 4.63548 .... 5.64672 Log. 1.24551 Equation, XLV. -{- 17". 0. . Correction + 34s Log. 1.52771 Hence time at Greenwich, Jan. 6d. 3h. 41m. 35s. Mean time at the sliip, Jan. 6 12 11 57 Longitude 8h. 30m. 223. = 127' 35' 30" E. from Grrenwich, differing 5' 15" from the calculation in page 232. The computed time at Greenwich, 3h. 41m. 35s., differs from the assumed time, 3h. 41m. 57s., only 22s. ; and, during this interval, the moon's latitude varies so little, that it will not be necessary to repeat the operation on account of this variation ; observing that an error of one minute in the moon's latitude affects the secant of the latitude about 0.00001, and this produces in the difference of the longitude an error of only 2" or 3" in the present e.xample ; and as the latitudes are always small, it will hardly ever be necessary to repeat tlie operation when this method is used. Second solution, using the right ascensions of the moon and star. RULE. To the mean time of observation, by astronomical calculation, add the estimated longi- tude in time if west, or subtract if east ; the sum or difference will be the supposed mean lime at Greenwich. This time may also be taken from the chronometer, if you have one which is pretty well regulated for mean time at Greenwich. With this time, enter the Nautical Almanac, and find from it the right ascension and declination of the star or planet, and the declination of the moon. With the apparent altitudes and distances of the objects, find the correct distance by the usual rules of working a lunar observation. To the correct distance add the declinations of the moon and star, and find the difference between the half-sum and the distance. Then to the log. secants of the declinations of the moon and star, rejecting 10 in each inde.x, add the log. cosines of the half-sum and of the difference, if tlie declinations are of the same name, or the log. sines, if of a contrary name ; half the sum of these four logarithms is to be sought for in tlie column of log. cosines, if the declinations are of the same name, or in the colunm of log. sines, i? of different names; and ♦ l-'se cosine if the latitude.* arfi of the same name. UY A TRANSIT OF THE MOONS LIMB. 431 Uie correspomling- time in the column p. m. is the difference of tiie right ascensions ol" the moon and star. • This difference of right ascension is to bqi added to tlie apparent right ascension of the star, if the moon is east of the star, otherwise subtracted, (borrowing or rejecting 24h. \\'hen. necessary ;) tiic sum or difference will be the true right ascension of the moon's limb. If the moon's true right ascension can be found exactly in the Nautical Almanac, the corresponding hour will be the mean time at Greenwich. If it cannot be found exactly, as will most commonly happen, take out the right ascensions for the hours immediate!}' pr> ceding and following, and note their difference, D, in seconds of time ; take also the diller ence, d. in seconds of time, between the moon's true right ascension and that right ascension marked for the first hour in the Nautical Almanac. Then, to the constant log. 3.55G30, add the arithmetical complement of the logarithm of D, and the logarithm of d ; the sum, rejecting 10 in the index, will be the logarithm of a number of seconds, to be added to the hour first marked in the Nautical Almanac, to obtain the mean time of the observation at Greenwich. The difference between this and the mean tinie at the ship, will be the longi- tude, which will be jccst, if the mean time at Greenwiclf be greater than the mean time ;il the ship, otherwise cast. We may observe, that we can, as in the first solution, use a planet instead of a star. We shall now calculate, by this method, the same example as in the first solution. Jn this case, for the supposed time at Greenwich, January (jd. 3h. 41m. 57s., we find, by means; of the Nautical ."Mmanac, Aldebaran's right ascension 4h. 2Gm. 31s. 3, Aldebaran's declina- tion 1G° 10' 29" N., and the moon's declination 21° 9' 33" N. True distance, as in page 232 68° 03' 00" J'sdetlinaiion 21 09 33 N Secant 0.03032 *'s declination 16 10 29 N Secant 0.01754 Sum 105 23 02 Half-sum 52 41 31 Cosine* 9.78254 Difference of half-sum and distance 15 21 29 Cosine* 9.98420 2) !9.81460 Diflerence of * and D's right ascensions 4h. 48m. 56s.9 Cosine* 9.90730 • 's right ascension 4 26 31 .3 D's riglit ascension 9 15 28 .2 t,,.^_„, . «,, Constant Log. 3 55630 liy N. A. D 's right ascension, Jan. 6d. 3h. 9 13 59 .8 D Terence d = 88s.4. . . . Log. 1.9- 64.1 •' f b Jan. Od. 4li. 9 16 08.1 iJ'ffei'ence U=: 128 .3. .Arith. Conip. Log. 7.8917V 41m. 20s.= 2480s Log. 3.30453 Add 3h. 00 00 'i'ime at Greenwich 3 41 20 Time al the sliip 12 11 57 Longitude 8 30 37 = 127° 30' 15' E. from Greenwich differing 1' 30|' from tlie calculation in page 232. PROBLEIM XVI. Given the intervals of time between the passages of the moon''s bright limb and a fixed star over two different meridians, to find the difference of longitude between the two meridians. This problem includes, also, the case wliere one of the observations is supposed to be made at Greenwich, considering the time of the transit of the moon's bright limb over that meridian, given in the Nautical Almanac, as an actual observation; the error arising from this supposition being very small, on account of the great degree of accuracy of the lunar tables used in the computation of the Nautical Almanac. We may, however, observe that, where good observations can be obtained at both meridians, it is always best to use them in preference to the computed transits in the Nautical Almanac. The principle upon which the longitude is found in this method is similar to that which is used in a common lujiar observation, and depends on the observed motion of the moon ; but, in the present problem, this motion is ascertained by observing the time when the moon's bright limb passes the meridian, instead of measuring the angular distance of the moon from the sun or a star. The variation of the moon's right ascension, corresponding to a change of 15° in the longitude, is given very accurately by the Nautical Almanac for every transit of the moon's limb at Greenwich. This variation is about 2m. in time for Ih. of longitude, and when the difference of the times of transit under different meridians haa been found by observation, it is easy to get, by proportion, the corresponding longitude, as we shall see in the following examples. This method of computing the longitude is very much facilitated by the new table of moon-culminating stars, inserted in pages 410 — 451 of the Nautical Almanac. To show the construction of the table, we shall insert the following extracts from it, contained in page 433 of the Nautical Almanac for 1836. • ■ ♦ Use sine if the declinations are of different names 432 TO FIND THE LONGITUDE OF A PLACE Col. 1. Col. 2. CoL. 3. Col. 4. • Col. 5. Col. 6. Col. 7. ' 183fi. Name. Magnitude. App. R. Ascens. in time. Declination. Var. J'sRiglit Ascension in Ih. of long. .Sid. Time J) 's semi-diameter pass, nierid. h. m. s. » 1 ,_ 5. Sepl. 15 Moon I u. c. (4.6) 15 07 52.16 18 25 S. 140.92 69.80 15 Moon I. /. c. 15 36 34.71 20 50 S. 146.22 71.16 16 Moon I. u. c. (5.6: 16 05 21.71 22 57 S. 151.02 72.52 16 Moon 1. I. c 16 37 12.45 24 42 S. 156.77 73.79 17 Moon I. u. c. (6.7) 17 09 01.65 213 04 S. 161.29 74.88 17 Moon I. /. c. 17 41 3'J.15 27 00 S. 164.75 75.69 16 a Pcorpii. 1 16 19 23.07 26 04 S. 10 r S<'.orpii. 3.4 10 25 42.40 27 52 S. 17 a t^corpii. 1 16 19 23.05 26 04 S. The stars whose right ascensions and declinations arc inserted in this table, are called inoon-culnnnating stars, because they have nearly the same declination as the moon, and do not differ much in right ascension., so that they are conveniently situated for observa- tions of the differences of tiie times of the transit which are required in tliis problem. The first colunni of this table contains the date ; the second, the name of the star or moon. If the bright limb of the moon be the first whicli passes the njeridian, it is marked I. ; but if it be the second limb, it is marked II. The upper culmination of tJie moon is marked u. c. ; the lower culmination, I. c. ; this last being of frequent use in high latitudes. The third column contains the magnitudes of tlie objects ; that of tlie moon being denoted bv her age, expressed in days and tenths of a day. The fourth column contains the apparent right ascension of the moon's bright limb, at the time of the transit over the meridian of Green- wich; and the fiftii column, its declination at that time : the same columns contain also the right ascensions and declinations of the moon-culminating stars at their upper culmination. Tiie sixth column contains the variations in the right ascension of the moon's bright limb during the intervals of her transit over two meridians ; one of these meridians being 7° 30' W. from Greenwich, and the other 7° 30 E. from Greenwich ; so that the distance of these two meridians is 15°, or Ih. in longitude. For convenience of reference, we shall call this variation the arc H, supposing it to be e.xpressed in seconds of time, as in column G. The arcs H, in the sixth column, are deduced from the right ascensions of the moon's bright limb, contained in the fourth column, so that they include the effect produced by the changes of the moon's semi-diameter. The seventh column contains the intervals of the transit of the moon's semi-diameter over the meridian expressed in sideral time ; this time being generally used in making such observations, and for this purpcjee it is usual to note the times of transit by a clock regulated to sideral time. If the intervals are given in mean time, they may be reduced to sideral time by adding the correction in Table LI. correspond- ing to that time. Thus, if the interval is Ch. mean time, the tabular correction in column 1 of that table is r)!!s.], making the interval Ch. Om. 59s. 1, sideral time. If the interval be Gh. 58m. mean time, tiie corrections in Table LI., columns 1,2, are 59s. 1 -f-9s.5= Im. 8s. 6; consequently the interval in sideral time is Gh. 59m. 8s. G. The numbers in columns 4, 5, G, 7. of tlie table of moon-culminating stars, correspond to the meridian of Greenwich, and may be reduced to any other meridian by the usual method of interpolation, as in Problem I., page 3!JG. Thus, from the above extracts from this table, it appears that, at the time of the upper culmination, September IG, l^SG, the right ascen- sion of the moon's bright limb was IGh. OGm. y]s.7L At the following lower culmination, it was IGh. 37m. 12s. 45, whicli may be considered as corresiponding to the upper culmination, September IG, in a place VZ\i. iu longitude wpst from Greenwich; and at the next upper culmination, the right ascension was 17h. G9m. Ols.GS, which may be considered as apjier- (aining to September IG, in a place 24h. west from Greenwich; according to the ancient method of counting tlie longitude, in a westerly direction completely round the globe. In like manner, in east longitude, we have, at the upper culmination at Greenwich, September IG, 1S3G, the right ascension of the moon's bright limb IGh. OGm. 21s. 71, and we may suppose the preceding transit, 15h. 3Gm. 34s.71, to correspond to the longitude 12h. east, and so on. This being premised, we shall now proceed to show how to find, by interpolation, the moon's rifjht ascension at the time of her transit over any meridian in east or west longitude from Greenwich. The process of calculation is very nearly the same as that in Problem L, page 39(), but for convenience we have reduced it to the following form : — RULE. Tu find llie moons right ascension at Ilct transit over any meridian. 1. Take from the fourth column of the table of moon-cuhiiinating stars, the right asce»- siona of the same limb of the moon corresponding to four successive 'culminations,* so that * Near tlie time of full moon, when the limb marked in the table chan'j-s IVoni I. to II., there may he on« or two of lliese quantities not marlted in column 4th of the tahle for the lunh which is wanted in the calculation. In tliis case, the reiiuired(viaMtities can be obtained from the corresponding tabular nuniliers. by BY A TRANSIT OF THE MOONS LIiMB 483 two nwy precede and tico folloio after the time of transit at the proposed place. Put these numbers below each other in their regular order ; then find their first and sl cond differences. Call the middle term of the first differences, the arc A; the mean of the 2nd differences, the arc B ; and if the longitude be west from Greenwich, put T equal to that longitude in time ; but if the longitude be east, put T equal to the difference between It^h. and that longitude. 2. To the constant logarithm 5.36452 add the logarithm of T in seconds of time, and the logarithm of A in seconds of time ; tlie sum, rejecting 10 in the index, will be a. proportional part, which is to be added to the second right ascension taken from the Nautical Almanac 3. Enter Table XLV. with the arc B at the top, and the time T at tlie side ; opposite to this will be the correction of second differences, to which prefi.x a. different sign from that of the arc B, and place it under the second ascension and the proportional part above found. Connect these three quantities together, as in addition in algebra ; the sum will be the sought right ascension of the moon at the time of her transit over the proposed meridian. The same process may be used for interpolating the numbers in columns 5, 0, 7, as we shall see in the following examples : — EXAMPLE 1. ■ Required the right ascension of the moon, September 16, 1836, astronomical account, at the time of the transit over the meridian of a place whose longitude is 3h. 48m. 298. west from Greenwich; also, the value of the arc H, deduced from the numbers in column 6, for the time of this transit. Here we have T=:3h. 48m. 29s., being the same as in Example I., page 3. 6 ; this value being selected in order to show more readily the similarity of the present calculation with that in page 396. 183G. Sept Uight ascension. h. m. s. 15 I. c. J5 36 34.71 16 u. c. 16 06 21.71 16 I. c. 16 37 12.45 17 u. e. 17 09 01.65 1st difference. A =30 31 50.74 49.20 2d difference. 4-63.74 -1-58.46 B = -t-6I.10 Arc. II. Var. U. A. 146.22 151. 62 156.77 161.29 1st difference. s. 5.40 A = 5. 15 4.52 2d difference. — 0.25 — 0.63 Constant Log. 5.36452 T = 3h. 48m.2r.s. = 13709s Log. 4.13701 A= 30 50.74=1850.74 Log. 3.26735 t 9 47 .33 = 587 .33 Log. 2.76838 16 06 21 .71 second right ascension. 6 .62 Table XLV. B = 613.10. 16h. 16ni. 02S.42 R. A. in long, of 3h. 48ni. 29s. W. 5.36452 4 13701 A= 5s,15 Log. 0.71181 4- 1 .63 Log. 0.31334 4-151 .62 second value of H. -f- .05 Table XLV. B = — Os.4!. H = 153s.30 corresponding to long. 31i. 43ni.293. VV Hence it appears, that, on September 16, 1836, astronomical account, m a place 8h. dSm. 20s. west from Greenwich, the right ascension of the moon's bright limb at tiie time of passing the meridian, was IGh. 16m. 02s. 42, and that the arc H, corresponding to that meridian, was 153s 30. This arc H represents the variation of the moon's right ascension between the times of tlie transit of her bright limb over the two meridians whose longitudes are T — "JOm. and T -|- 30m., corresponding respectively to 3h. 18m. 29s. west, and 41i. 18m. 29s. west, from Greenwich. In the preceding example, t])e longitude of the place is given, to find the moon's right ascension at the time of the passage of her bright limb over the meridian of that place; but we may suppose tliat right ascension to be given, to find, by an inverse process, the longitude of the place of observation, or the time T. The solution of this problem is very similar to that of Problem XIV., page 426, changing longitude into rigid ascension, &.C.; and it may be expressed as in the following rule : — RULE. To find the longitude of any place from the moon's right ascc7isio?i at her transit over the meridian of that place. 1 Take from column 4 of the table of moon-culminating stars, in the Nautical Almanac, the four right ascensions of tlie bright limb of the moon, as in the above example ; and then compute, as in that example, the values of the arcs A, B, in seconds of time. 2. To the constant logarithm 4.63548 add the arithmetical complement of the logarithm of the arc A in seconds of time, and the logarithm of the difi'ercnce in seconds of time between the given right ascension and the second right ascension taken from the Nautical 434 TO FIND THE LONGITUDE OF A PLACE Almanac , the sum, rejecting 10 in the index, will be the logarithm of the approximate time T in seconds. 3. Enter Table XLV., with the arc B at the top, and the time T at the side, and find the corresponding correction ; to the logarithm of which add the two first logarithms above found ; the sum, rejecting 10 in the index, will be the correction of the approximate time, to be applied \vith the same sign as the arc B, and the correct value of T will be obtained, which will express the longitude of the place of observation, if it be west from Greenwich; but if the longitude be east, we must subtract this value of T from 12h. to obtain the true longitudo in time east from Greenwich. EXAIMPLE IL Suppose that, in a place in west longitude, on the IGth of September, 1S36, the moon's bright limb passed the meridian in 3m. 20s.6.5, sideral time, before the star Antares. Re- (^uired the longitude of the place of observation. In the Nautical Almanac, column 4, the star Antares or o Scorpii's right ascension, Sept. IG, 183J5, was ICh. 19m. 23s.07 Subtract the observed difference of the transits in sideral time 3 20 .65 The remainder is the right ascension of the moon's bright limb at the transit IG IG 02.42 The next less right ascension in column 4 of the N. A., corresponds to Sept. IG, u. c. IG OG 21 .7] Difference of these right ascensions is 580s.71 = 9m. 40s.71 The four right ascensions to be taken from the Nautical Almanac, are those corresponding to September 15, Z. c, September 16, u. c, September 1(5,1. c, and September 17, u. c, being the same as those in the preceding example, where we have found A = 30m. 50s.74 = 1850s. 74, B == 4" 61s. 10. The rest of the calculation is as follows : — Equation Table XLV. 6s.58 . Correction 2m. 33s.6 = 153s.6. 4.63548 6.732G5 ..Log. 0.81823 ..Log. 2.18636 Constant Log. 4.63548 A=1850s.74 Arith. Comp. Log. 6.732G5 Diff. of right ascension 580s.71 Log. 2.76396 Approx.T = 3h. 45m. 54s.7 = 13554s.7..Log. 4.13209 Correction = 2 33 .6 T^Sti. 46m. 283.3^ the longitude of the place of observation. This longitude agrees, within a fraction of a second, with the value of the longitude assumed in Example 1.; observing that the computed right ascension in Example L is ICh. ICm. 02s. 42, being the same as that which is supposed to be observed in the present example. When the difference of meridians is small, we may compute their difference from the observed difference of the times of the moon's transit, by means of the arc H, deduced froiiJ column 6 of the table of moon-culminating stars, by the following rule : — RULE. To compute the dijfcrcncc of meridians hy means of the arc H. 1. To the constant logarithm 3.55630 add the arithmetical complement of the logarithm of the arc H, and the logarithm of the difference of the times of the moon's transit over the two meridians in sideral time ; the sum, rejecting 10 in the index, will be the logarithm of the difference of meridians expressed in seconds of time EXAMPLE in. Suppose that, in a place west from Greenwich, Sept. 16, 1836, the moon's bright limb passed the meridian in 20m. 02s. 30, sideral time, after the star Antares. Required the longitude. It appears by column 4 of the table of moon-culminating stars, that, on September 16, the right ascension of Antares was, 16h. 19m. 23s.07. Adding this to 20m. 02s.30, we get 161i 3!tm. 25s. 37 for the right ascension of the moon's bright limb at the time of its transit over the meridian of the place of observation. Subtracting from this the time of transit at Greenwich, IGh. 37m. 12s. 45, taken from column 4 of the table of moon-culminating stars, we get 2m. 12s. 02= 132s.n2, for the diflerence of the times of the transits, to be used in the above rule. Moreover, the arc H, corresponding to the time of the transit at Green- wich, is, by column 6 of the table, H = 156s. 77. Then we have, Constant Log. 3..55fi30 Arc H = 15GS.77 Arith. Comp. Log. 7.80473 Difference of times of transit 132s.92 Log. 2.12359 Difference of longitude 50m. 52s.3 = 3052s.3 Log. 3.48462 In strictness, the value of H, here used, ought to be increased a little; for, by column 6 of the preceding table, its value for Greenwich is 151s.62, and for a place in the longitude of 12h. west, is 156s.77. The difference between these two values of H, is 5s. 15, which repre- oents its increment corresponding to a change of 12h. in the longitude, being at the rate of Os.429 for a change of Ih. in the longitude ; and at this rate the increment for the longitude, 50m. 52s.3, will be Os.364, which will be increased to Os.38, if we notice the correction of BY A TRANSIT INSTRUMENT. 435 second differences depending on the arc B, and compute the arc H as in Example I. Hence tlie value of the arc H, corresponding to the meridian of the place of observation, is 156s.77-{-0s.38=157s.l5. If we take the mean of the values of H at Greenwich, 156s.77, and at the place of observation, 157s. 15, it becomes H == 15Cs.96, and with this we may repeat the above calculation, and obtain a corrected result. Constant Log. 3.556^0 Corrected arc H = 15Cs.96 Arith. Comp. Log. 7.80421 Dilierence of times of transit 132S.92 Log. 2.12359 Correct difTerence of long tude 50m. 48s.6 = 30 18s.C Log. 3.43410 In general, the longitudes of places where such observations are made, are known, within a few seconds, so that it will be easy to find at once the value of the arc H, corre- sponding to the estimated meridian which falls midway between tlie meridians of the two places of observation ; the meridian of Greenwich being used as one of these places, when the times of transit given by the Nautical Almanac are used as if they were actual obser- vations. We shall give the following example of this method : — EXAMPLE IV. In a place whose longitude was known to be 3h. 3Sm. 29s. W. from Greenwich, it wa3 found by observation, on September 16, 1836, that the moon's bright limb passed the meridi- an 3ra. 46s.2, sideral time, before the transit of the star Aldebaran ; and in another place, estimated to be 20m. in longitude west from the first place, or in 3h. 53m. 2!)s. W., the observed difference of the transits was 2m. 55s. 0. Required the difference of longitude which results from this observation. The mean of these two longitudes is 3h. 48m. 29s., and we have found in Example I., that the arc H, corresponding to this meridian on that day, was 153s. 30. Moreover, the difference of the two times of transit, 3m. 46s.2, and 2m. 55s.O, is 51s.2 ; then we have, as in the last example, Constant Log. 3.55630 Arc H = 153S.30 Arith. Comp. Log. 7.81446 Difference of times of transit 51s.2 Log. 1.70927 Difference of longitude 20m. 02s.3 = 1202s.3 Log. 3.08003 Add longitude of the first place 3h. 38 29 .0 Gives the longitude of the second place 3h. 58m. 31s.3 W., as it is deduced from tliis observation. PROBLEM XVIL Given the longitudes of the sun and moon, and the moorCs latitude, tojind their distance. RULE. Find the difference of the two longitudes, and to its log. cosine add the log. cosine of the moon's latitude ; the sum, rejecting 10 in the index, will be the log. cosine of the sought distance, which will be of the same affection* as the difference of longitude. EXAJMPLE. July 20th, 1836, at noon, mean time at Greenwich, by the Nautical Almanac, the sun's longitude was 117° 42' 31", the moon's longitude 193° 46' 05", and the latitude 2° 47' 16" N. Required tlieir distaijce. 0's longitude 117° 42* 31' D's longitude 193 46 05 Difference of longitudes 76 03 34 Cosine 9.38186 D's latitude 2 47 16 Cosine 9.99949 Distance 76 04 35 Cosine 9.38135, as in the Nautical Almanac This is calculated by another method in Example III. of Problem XVIII. In this rule, the sun's latitude is neglected, being only a fraction of a second. Tlie distances being calculated from noon and midnight by this (or by the following) problem, they may be interpolated for every three hours, by Problem I. The following example will serve for an illustration : — EXAMPLE. Given the distances of the sun and moon, in July, 1836, 19d. 12h., 20d. Oh., 20d. 12h. and 21d. Oh., respectively 70° 02' 35", 76° 04' 35", 82° 11' 29", and 88° 23' 32". Required tlie distances, July 20d. at 3h., 6h., and 9h. * Two arcs are said to be of the same affection when they are both crreater than 90', or both lest than 90°, bul .<■ different affection when the one is greater and the other less than 90°. 436 TO FIND THE DISTANCE OF THE MOON AND A STAR d. h. 1836, July 19 12 20 20 12 21 ■•<^c 70 02 35 76 04 35 11 29 23 32 82 1st difference. 6 02 00 A = 6 06 54 6 12 03 2d difference. + 4 54 4-5 09 B= + 5 01 Second longitude Proportional part. Table XLV Distances + 76 04 35 JA= 1 31 43.5 T = 3h. corr. — 28.2 At3h.=:77 35 50 At6h. » ( 1/ + 76 04 35 ^ A = 3 03 27 T = 6h. corr. — 38 At 6h. = 79 07 24 At9h. + 76 04 35 3 A = 4 35 10.5 :9h. corr. —28.2 These distances agree with the Nautical Ahnanac. PROBLEM XVIII. Given the longitudes and latitudes of the moon and a star, tojlnd their distance. RULE. To the log. secant of the difference of longitude of tlie moon and star, add the log. tangent of the greatest latitude ; the sum, rejecting 10 in the index, will be the log. tangent of an arc A, of the same affection as the difference of longitude. Take the sum of the arc A, and the least latitude, if the latitudes are of a different name, but their dfference if of the same name, and call this sum or difference the arc B. Then add together the log. secant of the difference of longitude, the log. secant of the greatest latitude, the log. cosine of the arc A, and the log. secant of the arc B ; the sum, rejecting 30 in the index, will be the log. secant of the distance of the moon and star, which will be of the same affection as B. EXAMPLE I. Required the distance of ♦he moon from the star a Pegasi, at noon, mean time at Greenwich, July 9d. 1836, when, by the Nautical Almanac, the moon's longitude, counted from the mean equinox, was 59° 40' 32", and her latitude 0° 59' 15" N. ; the longitude of the star, computed t as in Problem XIX., being 351° 12' 29", and its latitude 19° 24' 29" N. J 's longitude 59° 40' 3=2" *'s longitude 351 12 29 Difference of longitudes 68 23 03 Secant.. 10.43530 10.43530 Greatest latitude 19 24 29N Tansent 9.54C93 Secant 10.02541 Arc A 43 49 41... Least latitude 59 15 N. Difference X is arc B. . , Tangent 9.98223 Cosine 9.85819 42 50 26 Secant 10.13475 Distance* ]) 69° 24' 00" Secant 10.45365 This distance agrees with the calculated value given in page 146 of the Nautical Almanac. We may observe, that the log. secant of the distance is also equal to the sum of the log. cosecant of the greatest latitude, the log. sine of the arc A, and the log. secant of the arc B, rejecting 20 in the sum of the indices ; but the above rule is in general most convenient, on account of the smallness of the greatest latitude, except when the difference of longitude is nearly equal to 90°. We may use the same method for finding the distance of the moon from the sun, star, or a planet, when tlieir right ascensions and declinations are given, instead of their longitudes and latitudes. The rule is the same as that we have given above, changing longilvde into right ascension, and latitude into declination. To exemplify this, we shall compute the same example by this second method. EXAMPLE II. Required the distance of the moon from the star a Pegasi, at noon, mean time at Green- wich, July 9d. 1836, when, by the Nautical Almanac, the moon's right ascension was 57° 15' 01" from the mean equinox, the moon's declination 21° 3' 55" N. ; tlie star's riglil ascension from the same equinox 344° 9' 20", and the star's declination 14° 19' 32" N. P's richt ascension.... 57°15'01'' *'sriglit ascension.... 344 09 20 Difference 73 05 41 Pecant.. 10.53642 10.53649 Greatest declination .. 91 03 55N Tangent 9..58566 Secant 10.03004 .Tangent 10.12208 ....Cosine 9.779D8 Arc A 52 56 .56... Least decl ination 14 19 32 N. Difference § is arc B . . . '38 37 24 . Secant 10.10720 Distance * ■ BC cosine S -f- cosine (S — A) -|- cosec. B -|- cosec. C — 20 (C ) Co ■ le P^ BC cosine (S — B) -{- cosine (S — C) + cosec. B -f- cosec. C — 20 ^ .; su , - The above include all the cases of Oblique Trigonometry. The 2d and 4th Cases may be solved in a ditferent manner by the following theorems, which, on some occasions, may be found very useful. Thus, both the angles in Case II. may be found by the following theorems : — (7.) Sine h (AC -f BC) : sine ^ (BC M AC) : : cot. h C : tang, i (A — B). (S.) Cosine ^ (AC -f- BC) : cosine i (BC M AC) : : cot. ^ C : tang. ^ (A -f B). i (A — B) is less than 90°, and i (A + B) is of the same affection as ^ (AC + BC) The sum and difference of the terms ^ (A — B) and J (A -|- B) will give A and B. Both the sides in Case IV. may be found thus : — (9.) Sine i (A + C) : sine ^ (A M C) : : tang, i AC : tang. ^ (BC 02 AB). (10.) Cosine i (A + C) : cosine =^ (A CQ C) : : tang. ^ AC : tang. ^ (BC -f- AB). 4 (BCM AB) is less than 90°, and ^ (BC4- AB) is of the same affection as h (A4- C) Then the sum and difference of ^ (BC ^ AB) and i (BC + AB) give AB and BC. The improved rule for solving the cases of Obli/^ue Spheric Trigonometry by the oirculai parts, may be easily deduced from those given by Lord Napier. For if we put M for the middle part, A for the adjacent part, and B for the opposite part of the triangle APC, (fig. 3, 4, 5, 14, Plate XIII.,) m, a, h, for the corresponding parts of the triano-le APB, and P for the perpendicular AP ; then if P is an adjacent part, the rules of Napier will „ sine M sine to , sine M sine m give tang, r = , and tang. r = ; hence = ; consequently, tang. A tang, a tang. A tang, a snie HI : tang. A : : sine m : tang. a. If P is an opposite part, the same rule ... . „ sine M , ^ sine to , sine M sine to will give COS. r = , and cos. r = > hence = j consequentlv. cos. B COS. h cos. B cos. h ' ' sine M . cos. B : : sine m : cos. &, which are the twfl ruks to be demonstrated. 56 442 TO FIND THE LONGITUDE OF A PLACE. PROBLEM XXI. To find the longitude of a place by an eclipse of ike sun, ivlien the beginning or end is observed ; the apparent time being estimated from noon to noon, according to the method of astronomers ; the latitude of the place being also known. RULE. 1. With the longitude by account, find tlie corresponding Greenwich incan time of the observation. For tliis time, take out from the Nautical Ahiianac the sun's right ascension, declination, and semi-diameter, the horizontal parallaxes of the sun and moon, and the moon's declination roughly lu the minute. 2. Reduce the latitude, and the moon's horizontal parallax, by subtracting the corrections found in Table XXXVIII. ; and subtract from the moon's corrected horizontal parallax the sun's horizontal parallax, and the remainder is the relative parallax. 3. To the proportional logarithm of the relative parallax add the log. secant of the reduced latitude, the constant logarithm 1.1761, and the log. cosecant of double the observed time from noon-; (this double time being regarded as P. M. in using Table XXVII., unless it exceeds twelve hours, in which case the excess above twelve hours is to be regarded as A. M. ;) the sum, rejecting 20 in the index, is (S). 4. To the sum (S) add the log. cosine of the moon's declination, and the constant log. 0.3010 ; the sum, rejecting 10 in the index, is the proportional log. of an arc in time, which, subtracted from the observed time from noon, gives the corrected time from noon. 5. With the corrected time, the reduced latitude, and the sun's declination, calculate by Rule, page 247, the sun's true altitude. 6. To the log. secant of the sun's true altitude add the log. sine of double the corrected time from noon, (this double time being regarded as P. M. or as A. M., in the same way as before,) and the log. cosine of the reduced latitude; the sum, rejecting 20 in the index, is the log. sine of the parallactic angle. 7. To the proportional log. of the relative parallax add the log. secant of the parallactic angle, and the log. secant of the sun's true altitude , the sum, rejecting 20 in the index, is the proportional log. of the correction for declination. This correction is of the same name with the latitude, when the observed time from noon is less than six hours, and of the dif- ferent name when this time is greater than six hours. Correct the sun's declination by adding to it the correction for declination if of the same name, and subtracting if of the different name. 8. To the sum (S) add the log. cosine of the sun's corrected declination ; the sum, rejecting 10 in the index, is the proportional log. of an arc in time, which is the correction for right ascension, and is additive if the time is afternoon, but subtractive if the time is forenoon. Correct the sun's right ascension by adding the correction for right ascension when additive, and subtracting it when subtractive. 9. Multiply the nearest numl)er of minutes in the moon's horizontal parallax by the nearest number of minutes in the sun's semi-diameter, and multiply this product by the factor in the annexed table corresponding to the sun's true altitude ; the product, divided by 100, is an arc expressed in seconds, which, subtracted from the sun's semi-diameter, gives the sun's corrected semi-diameter. 10. To the proportional logarithm of tiie moon's horizontal parallax (not corrected) add the constant logarithm 0.5G46 ; the sura is the proportional logarithm of the moon's semi-diameter. 11. When the observation is the beginning or ending of an eclipse, the distance of the centres of the sun and moon is found by adding the sun's corrected semi-diameter to the moon's semi-diameter. But when the obser- vation is that of the beginning or ending of total darkness in a total eclipse, or tliat of the formation or of the breaking up of the ring in an annular eclipse, the distance of the centres of the sun and moon is found by taking the difference between the sun's corrected semi-diameter and the moon's semi-diameter. 12. Assume, from inspection of the Nautical Almanac, a convenient time when the moon's right ascension differs but little from the sun's corrected right ascension, and for this time take out Jiew right ascensions, and neto declinations of the sun and moon, and their horary motions in riglit ascension and declination by Problems I. and II. 13. From the hourly motion of the moon in right ascension subtract that of the sun; the remainder is the relative motion in right ascension. The difference between the hourly motion of the moon in declination and that of the aun Is the relative motion in declination. Correct the sun's new right ascension, by adding the correction for right ascension when it is additive, and subtracting when it is subtractive. Correct the sun's ncio declination, by adding the correction for declination when it is of the same, and subtracting it when it is of the different name. 14. Subtract the logarithm of the difference between the sun's new corrected right ascension and the moon's right ascension from the logarithm of the relative motion in right ascension, and call the remainder R. 15. To the remainder R add the constant log. 0.4771 ; the sum is the proportional loga- -., r • »• i I ( added to ) the assumed time when the sun's C greater rithm of an arc m tune, to be < , , . j /- > * j • u* • • •? i„-. ' ( subtracted from 3 neio corrected right ascension is ^ less than the moon's right ascension, to get the nezc corrected time. 16. To the remainder R add the uroportional logarithm of the relative motion in declina Sun's true al- Factor. titude. 0" 0.01 10 0.31 20 0.61 30 0.89 40 1.15 50 1.37 GO 1.5-1 70 1.G7 80 1.75 90 1.77 BY AN ECLIPSE OF THE SUN. 443 tion; the sum is the proportional logarithm of a correction of the moon's decimation Whether this correction is additive or subtractive is thus determined : — Find three numbers -fo"°- = - ,^ increasing, ^^,^,^^,„^^^,^^.j^; If the moon's declination is (^ decreasi Ul 1 I If the moon's motion in declination is < ? > than the sun's, the second number is < o ? Tf.,, . t A ■ ui • • ( ereater ) than the moon's right ascen- C ] ) If the sun s new corrected right ascension ,s ^ ^^^ ^ ^.^^^ ^1^^ ^,,j,. ^ ^ -^^j^^^ ^3 J 2 5 If the sum of these numbers is J °^^'^ I the correction is ? gui^t/aTtive. > The result gives the moon's neio corrected declination. 17. To the logarithm of the relative motion in right ascension add the log. cosine of tho moon's new declination, (not corrected,) and call the sum (S/). 18. To the sum (S,) add the proportional logarithm of the relative motion in declination, and the constant logarithm 7.1427 ; the result is the logarithm cotangent of the ^rsf orbitical inclination, which is c' ? when the sun's motion in declination is < f'''^'^ ^^ i than the moon's. S. 5 ( less 5 19. To tlie proportional logarithm of the difference between the sun's nac declination corrected and the moon's, add the logarithm secant of the first orbitical inclination, and from the sum deduct the prop, logarithm of the distance between the centres of the sun and moon ; the remainder is tlie log. secant of the second orbitical inclination, which has the name S. ) , ., , .■ ■ C immersion, IV > wlien tlie observation is an < ■ ' JN. 3 ^ emersion. This inclination is greater than 90° when the sun's new corrected declination is greater than tlie moon's; otherwise less than 90°. 20. Jldd together tlie two orbitical inclinations if of the same name, and suhtract them if of different names ; and call the result the relative inclination, which must have the same name as the greater of the two orbitical inclinations. To the log. cosecant of the relative inclination add the sum (S,), the proportional log. of the distance of tlie centres of the sun and moon, and the constant log. 7.G19S; the sum, rejecting 20 in the index, is the prop. log. of an arc in time, to be applied to the new corrected time to get the mean time at Greenwich ; it must be suMrlcted } '^'^^'^ ^^'^ ''•^^^^'^^ inclination is | ^; 21. By applying to the Greenwich mean time the equation of time taken from page II. of the Nautical Almanac, we shall have the apparent time at Greenwich ; tlie difference between it and the apparent time of observation will show the longitude of the place from Greenwich. EXAMPLE. Suppose, at a place in the latitude 42° 31' 13'' N., and estimated longitude 4h. 43m. 38s.6 the end of a solar eclipse was observed, November 30, 1834, at 4h. 5in. 47s.5 apparent time Bequired the longitude. ELEJIENTS OF THE ECLIPSE. Apparent time of observation November Estimated longitude W. Apparent time at Greenwich Equation of time stiUract Mean time at Greenwich 0's rij^ht ascension O's declination S. 0's .semi-diameter 0's horizontal parallax D 's horizontal paralla.x ]) 's declination Latitude of the place— Corr. Table XXX\l\l. = rcdiiced latitude D' horizontal parallax — Corr. Table XXX Vin.=: CO' •20".14— 5".54 ) is 5 's corrected horizontal parallar j B 's corrected horizontal paralla,x — ©'s hor. par. = relative parallax Elements for Nov. 30d. 8h. D 's nciB right ascension D 's iieiB declination S. O's new right ascension 0's neto declination 5 's horary motion in right ascension D's horary motion in declination O's horary motion in right ascension 0's horary motion in declination J) 's horary motion in right ascension — 0's horaiy motion in right ) ascension :=retative motion in right ascension ( D 's horary motion in declination — 0's horary motion in declination | = relative motion in declination \ 0's new right ascension -|- corr. for right ascen. = co7-r. new right as. . O's 7ie«) declination — corr. declination = corr. new declination 30d. 4h. 5 m . 47s. 5 4 43 38 .6 30 8 49 26 .1 11 2 .4 30 8 38 23 .7 Ifi 2.5 53 .05 21° 41' 52".6 16 14 .8 8 .7 no 20 .14 21 7 C .8 42 19 47 .4 CO 14 .6 60 5 .9 16h. 29m . 13S.35 21" 01' 24".6 ]6h. 2.'im .4HS.I5 ,21° 41' 39" 9 2m . 333. 79 & 50" U lOs.79 2-l".05 8 35 .25 23 5C' 33 .3 7".9 444 TO FIND THE LONGITUDE. Relative parallax 60 5'-.9 Prop. Log. 0.47G4 Reduced latitude 42° 19' 47".4 Secant 10.1312 Constant Log. 1.1761 Double obs'd time fr. noon 8h. 1 Im. 353. Cosec. 10.0563 Sum (S) 1.8400 D's declination 21° 7' C".8 Cosine 9.9698 Constant Log. 0.3010 Prop. Log. correction lm.23s.7 2.1108 Observed time fr. noon 41). 5 47 .5 Corrected time from noon 4h.4ni.23s.8 Log.ris. 4.71324 Reduced latitude 42° 19' 47".4 Cosine 9.86881 0's declination 21 4152.6 Cosine 9.96809 Nat. number 35493 4.55014 Nat. cosine 64 01 40 43794 Nat. sine 0's true altitade 08301 = 4° 45' 41" O's true altitude 4°'45'41" Secant 10.00150 Double corrected time fr. noon 8h. 8m. 47s. 6 ) „ „ ,„_„ Sine (P.M.) j 9-9^223 Reduced latitude 42° 19' 47 ".4 Cosine 9.8G881 Parallactic angle sine 40° 30' 00" 9.81254 Rel. parallax CO' 5".9 Prop. Log. 0.4764 Parallactic angle 40° 30' Secant 10.1190 0's true altitude 4° 45' 41' Secant 10.0015 Correction 0's declination P. L. ■ 45* 32" N. 0.5969 O's declination 21° 41 52.6 S 0'3 CORRECTED DECLINATION. . .20 56 20.6 S Sum (S) 1.8400 0's corrected declination 20° 56' 20".6..Cosine 9.9703 Correction ©'s right ascen. P. L. 2m. 47s. 15 1.8103 O's right ascension 16h. 25m. 53s.05 O's CORRECTED BIGHT ASCEN. 16 28 40 .2 60 X 16 X 0.143 ■ = diminution O's semi-diam. 1".4 O's semi-diameter 16' 14". 8 100 O's CORRECTED SEMI-DIAMETER 10 13 .4 D 's horizontal parallax GO' 20". 14... Prop. Log. 4747 Constant Log. 5646 ]) 's semi-diameter Prop. Log. 16' 26".5 1 .0393 O's corrected semi-diameter 16 13 .4 Distance OF THE CENTRES OF 02° 2 33 .5N. Sum..{S) , Const. Log »'s corr. dec. 22 29 .55 .4 N Cosine Correction .' Oh. 2m.37s.93 P. L. *'s right ascension.... 8 33 59 .03 *'s right ascen. corr... 8 36 36 .96 ]) 's right ascen 8 37 5.03 Difference 28 .07 ELEMENTS OF OCCULT.'iTION, Apparent time of observation,... April Estimated longitude W. .Apparent time, at Greenwich Equation of time subtract Mean timeal Greenwich April O's right ascension O's right ascension -|-appar. time of) obs. = right ascension meridian j D 's horizontal parallax Correction Table XXXVIII 5 's corrected horizontal parallax D's declination N. *'3 right ascension *'s hour-angle *'s hour-angle, in degrees, &c W. *'s declination N. Lat. place — corr. Table XXXVIII ) =: reduced latitude j By the Nautical Almanac we find the moon's right ascension to be near- est to the star's corrected right as- cension on 20th April, at 16 hours, fcir which time we get the D 's right ascension D's horary motion in right ascension.. D's declination N. D's horarj' motion in declination. ..S. 20 10 45 35.9 4 44 20 15 29 35.9 1 11.4 20 15 28 24.5 1 52 54.3 12 38 30.2 56' 14".5 5 .] 56 9 .4 22° 42 06 .1 8h.33 Manna. (N. W. pt.)... -^ lOfoo, (N. W. pt.) 20 25 20 42 20 28 17 22 i5 48 18 i3 17 3i i4 32 i4 i3 i4 II Lous'- D. M. i36 2i\V 187 o4 i38 48 i4i 18 i4i 3o i44 39 i44 58 i45 10 145 39 1 46 o4 i46 25 145 10 145 20 i46 5i 1 47 59 i48 i5 1 48 42 i48 i3 i46 32 1 45 49 t44 10 1 44 o3 i44 16 44 59 i43 3i 143 o4 143 18 i4i 3o 142 i5 142 10 149 34 168 07 169 29 169 36 Pago-Pago har., Island of Tutuila Harbor of Ap' i, Itjland of Upolu. ...... . . . . Harbor of Mataalu, Isl- and of Savaii Hoornlsland, (N. W.pt.) Uea, or Wallis Island . . . Jarvis Island, (centre) . . Penrhyn's Island, (N.pt.) Wostock, or Stavers Isl- land, (centre) Flint Island, (centre) .... Phoenix Group, — Birnie's Isl'd, (centre) — Enderbury's Island, (centre) — Hull's Island, (west end) Gardner's Isl'd, (centre) McKean's Isl'd, (centre) Union Group, — Otafu, or Duke of York Island, (N. W.pt.)... — Nukunono, or Duke of Clarence Isl'd, (N. pt.) — Fakaafo, or Bowditch Island, (village) .... Swain's Island, (centre) Roaul, or Sunday Island, (centre) Honolulu har., Oahu Isl Laliaina har., Maui Isl Waiakea har., Hawaii, . . New York, or Washing ton Island, (west end) Necker Island, (centre) . French Frigate School, or Basse de Fregate Franqaise Maro Reef. Smith Island, (centre) , Lat. D. M i4 18 S 3 49 i3 28 i4 i5 i3 24 o 22 8 55 10 o5 11 26 3 35 4 3o 4 38 3 35 8 36 9 o5 9 24 II 10 29 12 21 19N 20 5o 19 44 4 4i 23 35 23 45 N.pt. 25 19 S.pt. 16 48 Taputeouca, or Drum- mond's Island, (Sands- pit at Utivoa) Nanouti, or Sydenham Island, (N.pt.) Nanouki, or Hendervill Island, (S. pt.) Kuria, or Woodle Island, (South pt.) •. . Apamama, or Hopper Isl- and, (N. pt.) Maiana, or Hall's Island, (N.pt.) Tarawa, or Knox Island, (S. W. island) Apia, or Charlotte Island, (entrance) I i4S o 3o o 08N o 17 3o 1 02 I 22 I 48 Long. D. M. 70 38 W 71 4i 72 iS 78 02 76 09 59 5i 58 07 52 16 5i 48 71 39 71 i4 72 20 l4 4i l4 17 72 24 71 38 71 06 70 53 78 i5 57 52 56 4i 55 o3 60 1 3 64.43 65 59 East end 70 32 East end 69 46 74 53 E l4 20 73 4i 73 26 73 54 73 04 73 01 73 02 TABLE LIV. (continued.) Latitudes and Longitudes. [Page 451. DCaraki, or Matthew's Isl and, (N. pt.) Makin, or Pitt's Island, (S.pt.) Funafuti, or Ellice Isl and, (N. W. island) . . . Nukufetau, or Depeys ter's Island, (N.isl.)... Oditupu, or Tracy Isl'nd, (centre) Hudson's Island, (N. pt.) Speiden's Isl'nd, (centre) St. Augustine, (centre). Walpole Island, (centre) Elizabeth Reef, (N. E. pt.) Mathews' Rock MacquarJe's Isl'd, (S. pt.) Lord Auckland Group, (Sarah's Bosom) . . . Ovolau Island, (observa- ^ tory) Lecumba point, (Sandal- wood Bay) Muthuata har. (cemetery on Island) Unda pt. (east entrance Vanu\a Levu) Rewa Roads, (Nukalau Island) LmI. D. M. 2 o3N 3 02 8 26S 7 56 7 28 Long. 35 Chesterfield Group — N. W. pt. Long Island Kenn Reef — Observatory, sand cay S. E. pt. of reef. Frederick Reef — South Sand Islet . . . . North Sand Islet Saumarez Reef — S. W. sand cay S. E. elbow of reef. . . Lihon Reef — N. E. point S. \V. point Percy Group — Mid-Island, "W. bay . N. W. bay, S. islet.. Pine Peak Barrier Reef — Inside, No. i prong . . " No. 4 prong . . Outside, No. i prong. " No. 3 prong. 22 27 29 34 22 27 54 AA 5o 34 17 4i 16 52 16 26 16 08 18 10 19 52 21 16 21 i5 21 02 20 57 21 5i 21 55 17 21 17 39 21 4o 21 45 21 3i 22 09 21 29 20 o5 21 00 M •.16 E 72 46 i4 28 U 23 29 06 07 24 72 10 59 49 79 66 27 53 35 04 55W 32 E 19 5o 17 5o 21 5o 19 52 12 5i 10 5o 55 52 19 Tokanova pt. (S. E. pt Vanua Levu) Direction Island, or Ne- niena Awakalo, or Round I... Malolo, (Avo Town) Vomo Island Kie Island Ongea Island Oneata Island Nanuku Island Turtle Island, (N. pt.) Pescadores Island, (cast island) Korsakoff Island, (west island) Benham's Island, (south end) Hunter's Island, (centre) Bearing's Island, (centre) McKenzie's Island, (cen- tre) Wake's Island, (centre). Antique Roads, (Island of Panay) Caldera Roads, (Island of Mindanao) Soung Roads, (Island of Sooloo) Manghee Islands, (Bala- bac Straits) Lat. Long. D. M. 16 46 S 17 07 16 4i 17 46 17 29 16 4o 19 04 18 24 16 42 19 47 II 23 N II 08 5 47 5 42 5 35 10 08 19 17 10 4o 6 56 6 01 Bodegas Poit . . . San Franci-co . . . ^lonterey .... . . Santa Biu-h.-u-a . . San Pedro Ju:in Diego Quentin Bartolomeo . , Magdalcna Bay ) North Pt. Entr. J Observatory . . . , , Cape St. Lucas . . , 7 3o 38 18 1 37 47. 36 37 M 24. 33 43. 33 26. 32 4i . 3o 21.1 27 39.1 24 32. 24 38.. 22 52.. D. M. 179 56 E 179 0-' 177 43 177 07 177 i4 179 o5 178 3oW 178 32 179 26 178 25 167 37 E 166 22 169 36 169 06 168 26 139 49 166 32 122 122 120 117 123 123 121 "9 1X8 117 117 ii5 ii4 56 19 W 0.7 21 , 5o.8 38.8 i3.S 4i II. 3 56.5 5i.: 112 1.2 112 9 6.3 52. I 452 GREAT CmCLE SAILING. Tlie shortest distance between two points on th.e eartli's surface is on the arc of a great circle. The following rules will enable the navigator to calculate the courses and distance, and to project the great circle track of the voyage, on the chart. By projection, the shortest distance between the two given points will be readily perceived. The greatest difference between the tracks is found in the higher latitudes, and between points in about the same parallels. In crossing the ei^uator, this difference, generally, is not of so much importance. In Case I., we find the courses on the great circle to differ from the course on the rhumb line about two points. This difference, in many cases, is often more ; the knowl- edge ,of which might be of great importance to the mariner, by enabling him to go on his course with a wind which on the rhumb line would be adverse. The course on the rhumb line is always the same ; on the great circle it is continually changing. It would be well to calculate the course two or three times during the day, working out the position of the ship by the usual methods. A vessel sailing on the great circle track, on the same side of the equator, is always in a higher latitude than she would be on the rhumb line ; consequently, in north lat- itude, the great circle line will be north of the rhumb line, and in south latitude, will be south of the rhumb line. This is evident on inspectmg the annexed chart. "When the course found is greater than 90°, its sup- plement will be the course counted from the opposite pole. Thus, in Case V., the course is N. 131° 24' E., or S. 48° 36' E. In calculating the courses PLL' and PL'L, and the distance LL', we have the two sides PL and PL', and their included angle LPL' given, the sides being the co-latitudes of the given places, and the angle P being the difference of longitude. CASE I. Given the latitudes and longitudes of two places on the same side of the equator, to find the courses. By Theorems (8), (7), page 441, we obtaua the following RULE. Make two columns, and write down the following logarithms. The log. cosine ol half th3 difference of the latitudes in Col. 1, and the log. sine in Col. 2, the log. cotangeat of half the difference of longitude in both columns, the log. cosecant of half the sum of the latitudes in Col. 1, and the secant in Col. 2. The sum of the logs, in Col. 1 w^lU give log. tangent of half the sum of the courses, and the sum of the logs, in Col. 2 will give the log. tangent of half the difference of the courses. The sum of these results, rejecting 20 in the index, wUl give the course corresponding to the greatest latitude, and their difference the course corresponding to the least latitude. EXAMPLE I. Required the course from a point in the latitude 40° N. and longitude 70° W. to a place in the latitude 50° N. and 10° W. Half the sum of the lats. = 45°. Col. 1. i diff. lats. 5° COS. I diff. long. 30° cotang. i sum lats. 45° cosec. i sum courses, tang. 10.38741 67° 43' 12 03 Half their difference = 5°. Half the difference of long. = 30°. Col. 2. i diff. lats. 5° sine 8.94030 Same 10.23856 i sum lats. 45° secant 10.15051 9.99834 10.23856 10.15051 i^ diff. courses tang. 9.32937 12° 03' Course N. 79° 46' W., from latitude 50° N. Course N. 55° 40' E., from latitude 40° N. By Morcator's sailing, the course fi-om 50° N. is S. 76° 42' W., making a differenc9 of 23° 32'. When the places have the same latitudes, the sum of the logarithms in Col. 1 will ba the log. tangent of the course from either place. 90° H0° 50° 60° 70" 80*= 90= ;60° 60° 50= 50" 40° 40= SC^ 30' 20° 20= 10= 10= 10= 10= 20= 20= 30« 30= '40= 50° &0° 90= -JK — ==^ 40= 50= 60= 40=" 50° 60" 70° 80° 90 = p 90° 8 o 7C o GO" SO" 40° 30° io" 10° 10 20 30° 40° 5 ° G0° 70" 80° 90° 1 60° 1 " 1 ! 1 -- A y ^,-- ^^ 1 y^' ^'' ^ --""^ -■'' -^ ' 40° '"^ 40° B 30° V^ K - % X M° \ \ 20° 10° 10° X \ \^ \ 10° A V 10° 20° 30° 40° pn° V\ \ !40' 1 ^^„^^ ^ ^^ ^^— ._ _^- fiO° L '^ 50° 60° co° 1 1 1 1 1 ' 0° 80 "■ 70 " 60° 60° 40° 30° 20° 10° I) 1 ° 2 ° 20° 40° J 0° 00° 70° 80° 90° Ij GREAT CIRCLE SAILING. 453 EXAMPLE II. Required tlie course from a point in the latitude of 40° S. and 20° E., to anotiier point in latitude 40° S. and longitude 80° E. The half sum latitudes is 40°, the half difference is 0°, and half the difference of longitude 30°. i difference lats. 0° cosine 10.00000 J^ difference long. 30° cotang. 10.23856 I sum lats. 40° cosec. 10.19193 Course, 69° 38' tang. 10.43049 The courses are S. 69° 38' E., or S. 69° 38' W., and the difference between them and the Mercator course is 20° 22'. CASE n. Given the latitudes and longitudes of the two places, and the courses, to find distance. Theorem (2), page 439, gives the rule. RULE I. Add together the log. sine of the difference of the longitude, the log. cosine of the latitude, the log. cosecant of the course ; the sum, rejecting 20 in the index, ■will be the log. sine of the distance. When the greatest latitude is used, the least course must be taken, and when the least latitude is used, the greatest course must be taken. EXAMPLE I. Let the latitudes and longitudes of the places be as in Example I., Case I., and the courses as therein found, 55° 40' and 79° 46'. Required the distance. Difference of long., 60° sine 9.93753 Greatest latitude, 50° cosine 9.80807 Least course, 55° 40' cosec. 10.08314 Distance, 42° 23' sine 9.82874 60 2543 miles. By Theorem (10), page 441, we obtain RULE II. Add together the log. secant of half the difference of the two courses, the log. cosine of half the sum of the two courses, and the log. cotangent of half the sum of the lat- itudes ; the sum, rejecting 20 in the index, will be the log. tangent of half the distance. EXAMPLE II, Given the parts as in the previous example. Required the distance. h. difference courses, 12° 03' ...secant 10.00968 h. sum courses, 67° 43' cosine 9.57885 h. sum latitudes, 45° cotangent 10.00000 i distance, 21° llil' tangent 9.58853 2_ 42° 23' or 2543 miles. Distance by Mercator's sailing, 2608 Gain on the great circle, 65 Use Rule II. when any doubt exists as to the result, which may be the case when the difference of longitude is about 90°. GREAT CIECLE SAILING. 453 EXAMPLE II. Required the course from a point in the latitude of 40° S. and 20° E., to another point in latitude 40° S. and longitude 80° E. The half sum latitudes is 40°, the half difference is 0°, and half the difference of longitude 30°. i difference lats. 0° cosine 10.00000 i difference long. 30° cotang. 10.23856 I sum lats. 40° cosec. 10.19193 Course, 69° 38' tang. 10.43049 The courses are S. 69° 38' E., or S. 69° 38' W., and the difference between them and the Mercator course is 20° 22'. CASE IL Given tlie latitudes and longitudes of the two places, and the courses, to find distance. Theorem (2), page 439, gives the rule. RULE I. Add together the log. sine of the difference of the longitude, the log. cosine of the latitude, the log. cosecant of the course ; the sum, rejecting 20 in the index, will be the log. sine of the distance. When the greatest latitude is used, the least course must be taken, and when the least latitude is used, the greatest course must be taken. EXAMPLE I. Let the latitudes and longitudes of the places be as in Example I., Case I., and the courses as therein found, 55° 40' and 79° 46'. Required the distance. Difference of long., 60° sine 9.93753 Greatest latitude, 50° cosine 9.80807 Least course, 55° 40' cosec. 10.08314 Distance, 42° 23' sine 9.82874 60 2543 miles. By Theorem (10), page 441, we obtain RULE II. Add together the log. secant of half the difference of the two courses, the log. cosine of half the sum of the two courses, and the log. cotangent of half the sum of the lat- itudes ; the sum, rejecting 20 in the index, will be the log. tar.gent of half the distance. EXAMPLE II. Given the parts as in the previous example. Required the distance. h difference courses, 12° 03' secant 10.00968 I sum courses, 67° 43' cosine 9.57885 I sum latitudes, 45° cotangent 10.00000 i distance, 21° Hi' tangent 9.58853 2_ 42° 23' or 2543 miles. Distance by Mercator's sailing, 2608 Gain on the great circle, 65 Use Rule II. when any doubt exists as to the result, which may be the case when the difference of longitude is about 90°. 454 GREAT CIRCLE SAILING. EXAMPLE III. Required tlie distance between two points, one in 40° S. and 20° E., and the other in 40° S. and 80° E. ; tlie course from eacli point being 69° 38'. Diiference long., 60" sine 9.93753 Latitude, 40° cosine 9.88425 Course, 69° 38' cosec. 10.02804 Distance, 45° 03' '..sine 9.84982 60 2703 Distance by Parallel sailing, 2758 '< " great circle '« 2703 Gain, 55 CASE III. Give7i the latitude and lo7igitude of two places, to find the maximum separation in latitude. In sailing between two places on the same side of the equator, on a great circle, the vessel always keeps in a higher latitude than on the rhumb line. The point on the great circle which is the greatest distance from the rhumb line, measured on the meridian, is the point of maximum separation in latitude. RULE. For the latitude. — Find the course between the places by Mercator's sailing, and take the supplement ; also, the course on the great circle, from the same latitude ; add together the log. cosecant of the Mercator course, the log. sine of the great circle course, and the log. cosine of the latitude of the place of the great circle course ; the sum is the log. cosine of the latitude required. RULE. For the longitude. — Add together the log. secant of half the difference of the given latitude and the latitude just found, the log. sine of half the sum of the latitudes, and the log. tangent of half the sum of the great circle course, and the supplement of the Mercator course ; the sum wUl be the log. cotangent of half the difference of longitude. EXAMPLE I. Required the latitude and longitude of the point of the maximum separation between two points, one in the latitude 40° N. and longitude 70° W., and the other in latitude 50° N. and longitude 10° W. By Case I., Mercator's sailing, we find the course. As merid. diff. of lat., 851 2.92993 Is to radius 10.00000 So is the diff. of long. 3600 .3.55630 To tang, course, 76° 42' 10.62637 Its suppleme7it, 103° 18'. By Case L, Example I., we find the great circle course, from 40° N., to be 55° 40'. To find the latitude. Mercator's course, 76° 42' cosecant 10.01181 Great circle course, 55° 40' sine 9.91686 Latitude, 40° cosine 9.88425 Latitude required, 49° 28' cosine 9.81292 To find the longitude. Lowest latitude, 40° 40° Supplement of Mercator course, ...103° 18 Latitude of max. separation, 49° 28' Great circle course, 55° 40 Sum, 89° 28' Sum, 158° 58 h sum 44° 44' h. sum, 79° 29 Difference, 9° 28 i difi"erence, 4° 44 GEEAT CmCLE SAILING. i diif. of latitudes, 4° 44 secant, i sum of latitudes, 44° 44 sine, i sum of com-ses, 79° 29' tangent, i diff. of longitude, 14° 44' cotangent, 10.58026 2 10.00148 9.84745 10.73133 Diff. of longitude, 29° 28' Longitude left, 70° 455 40° 32' longitude of the required point of max- imum separation. EXAMPLE II. Required the latitude and longitude of the point of maximum separation in latitude, between two places, one in lat. 40° S. and 20° E., and the other in 40° S. and long. 80° E. The courses found in Case I., Example 11., is 69° 38'. The Mercator course is E., or "W., or 90°, and its supplement, therefore, is 90°. Their half sum is 79° 49'. For the latitude. Merc, course, 90° cosec. 10.00000 Great circle co., 69° 38' sine 9.97196 Latitude, 40° cosine 9.88425 Lat. required, 44° 06' cosine 9.85621 For the longitude. In this and similar cases of parallel lat- itudes, the difference of longitude wiU be equal to half the difference of longitude be- tween the given places, which, in this ex- ample, is 30"^, and the long, required, 50.° CASE IV. Given the latitudes and lo7igitudes of the two places, and the great circle courses, to find tJia maximimi latitude and its longitude. "When both courses, counted from the same pole, are less than 90°, then the maximum latitude of the arc will be within the two given points. By Theorem (2), page 439, we get the RULE For the latitude. — Add the cosine of the latitude to the sine of the great circle course, fi-om the same latitude; the sum, rejecting 10 in the index, will be the log. cosine of the maximum latitude. By Theorem (1), page 439, Ave obtain the RULE For the longitude. — To the log. sine of the latitude, add the log. tangent of the cor- responding great circle course ; the sum will be the log. cotangent of the difference of longitude. EXAMPLE I. Given the latitudes and longitudes of two places, as in Example I., Case I. ; viz : 40° N. and 70° W., and 50° N. and 10° W., to find the maximum latitude. The great circle courses are found, in Case I., to be 55° 40' and 79° 46. To find the latitude. Lat. 40° ^. cosine 9.88425 Course, 55° 40' sine 9.91686 Lat. required, 50° 45^' cos. 9.80111 To find the longitude. Latitude, 40° sine 9.80807 Course, 55° 40' tang. 10.16558 Diff. long., 46° 44' cotang. 9.97305 Long, left, 70° Long. req. 23° 16' of the max. lat. "^Tien both places are in the same latitude, the maximum latitude and the point of maximum separation will be the same. EXAMPLE II. Required the maximum latitude between two points ; one in lat. 40° S. and 20° E. and the other in 40° S. and 80° E. 456 GREAT CIRCLE SAILING. In Case I., Example 11., tlie great circle course is 69° 38'. To find the latitude. Latitude, 40° cosine 9.88425 Course, 69° 38' sine 9.97196 Latitude, 44° 06' cosine 9.85621 Being the same results as in Case III., Example II. In this case, the longitude -will be midway between the two given longitudes. CASE V. Given tioo places on the opposite sides of the equator, to find the courses and distance on an arc of the great circle. Find the point of intersection of the great circle with the equator, by the following rule ; then with this point, and the places given, proceed as in Cases I. and II. for the courses and distance. RULE. Add together the sine of the difference between the two latitudes, (not the difference of latitude,) the cosecant of the sum of the latitudes, and the tangent of half the difference of longitude ; the sum, rejecting 20 in the index, will be the tangent of an arc X, which, added to half the difference of longitude, will give the difference of lon- gitude between the greatest latitude and point of intersection. EXAMPLE. Given two points, one in 40° N. and 70° AV., and the other in 30° S. and 10° W. Required the point of intersection of the great circle with the equator, and the courses and distance between the two given places. The difference between 40° and 30° = 10°. The sum is 70°. Half the difference of longitude, 30°. Difference between the lats. Sum of the latitudes. Half difference of long. ArcX, Half diff. of long. Diff. of long, from 40° N. Long, left, 10° sine 9.23967 70° cosec. 10.02701 30° tang. 9.76144 6° 05' tang 30° 9.02812 36° 05' 70° Long, of intersection, 33° 55' "W. Having the latitudes 40° N. and 0°, and the longitudes 70° and 33° 55', we can calculate the courses and distance by rules given in Cases I. and H. To calculate the courses. Half sum lats., 20°, and half difference of lats., 20°. Half difference long., 18° 03 . CoL. 1. 4 diff. lats. 20° cos. 9.97299 I diff. longs. 18° 03' cotang. 10.48694 I sum lats. 20° cosec. 10.46595 i sum courses, 83° 14' tang. 48° 10' 10.92588 Col. 2. i diff. lats. 20° sine 9.53405 Same 10.48694 ^ sum lats. 20° secant 10.02701 i diff. courses, 48° 10' tang. 10.04800 Coxirse N. 35° 04' W. from the equator. 131° 24' 180° or S. 48° 36' E. from latitude 40° N. To find the distance. Difference of long. 36° 05' sine Greatest latitude, 40° cosine Least course, 35° 04' cosec. Distance, 51° 45' sine _60 3105 miles. 9.77009 9.88425 10.24069 9.89503 GREAT CmCLE SAILING. 457 To find the courses and distance fi-om tlic equator, in long. 33° 55' "W., to 30° S. and 10° W. Half sum lats., 15°. Half diff. of lats., 15°. Half diff. long., 11° 57'. To find the couirses. i diff. lats. 15° COS. 9.98494 4 diff. long. 11° 57' cotang. 10.67439 I sum lats. 15° cosec. 10.58700 i diff. lats. 15° sine 9.41300 h. diff. long. 11° 57' cotang. 10.67439 I sum lats. 15° secant 10.01506 h sum courses, 86° 45' . . .tang. 11.24633 1 h. diff- course, 51° 42' tang. 10.10245 51° 42' Course S. 35° 03' E. from the equator. 138° 27' 180° N. 41° 33' W. from 30° S. To find the distance. Difference long. 23° 55' sine 9.60789 Greatest lat. 30° cosine 9.93753 Least course, 35° 03' cosec. 10.24087 Distance, 37° 41' sine 9.78629 60 2261 miles. CASE VI. 7b project the track on a great circle. First, (by Eaper.) \Vlicn the places are on the same side of the equator. — Dra-\v the line connecting the given places, find the position of the point of the maximum separation of latitude, and through this point draw a line parallel to the line connecting the two places. Find the coui-ses on the great circle from the two places, and draAV them on the chart. We can, through these three points, roughly trace the required cuiwe. K the maximum latitude falls on the curve, we shall have a fourth point. EXAMPLE I. Given the latitudes 40° N. and 50° N., and their corresponding longitudes 70° W. and 10° W., to project the great circle track connecting them. By Case I., the courses are found to be N. 55° 40' E., and N. 79° 46' W., and by Case ni., the position of the point of maximum separation of latitude is found to be 49° 28' N. and 40° 32' W. ; and by Case IV., the maximum latitude is in 50° 45^' N., and 23° 16' W. Draw on the chart the line AB, connecting the two points ; from A and B, lay off the courses N. 79° 46' W., and N. 55° 40' E. ; through the point of maximum sep- aration draw a Kne parallel to AB ; through these points, and the point of maximiim latitude, draw the dotted Une, which will be the track required. Second. — "When the places are on opposite sides of the equator. — Find the course at each of the given pomts, and the points of maximum separation of latitude, for both sides of the equator ; find the longitude of the intersection of the great circle with the equator, and the course at that point ; with these five points, construct the track. EXAMPLE II. Given two places, one in the latitude 40° N. and longitude 70° W., and the other in 30° S. and 10° W., to project the track on the great circle, passing through them. The courses are, by Case V., N. 41° 33' W., from 30° S., and S. 48° 36' E., from 40° N. The longitude of the intersection of the great circle on the equator, by Case V., is 33° 55' TV., and the course at the intersection is N. 35° 03' W., and S. 35° 03' E. The maximum separation of latitude north of the equator, is in 25° 33' N. and 53° 31' W. ; and south of the equator, is in 18° 21' S. and 20° 31' W. With these points, the great circle track can be constructed, as in the example pre- ceding. 58 458 GREAT CIRCLE SAILING. EXAMPLE III. Given one place in tlie latitude 40° S. and 20' E., and another in 40° S. and 80° E., to project the track. By Case I., Example II., we find the courses to be S. 69° 38' E., and S. 69° 38' "W. By Case III., Example II., the maximum separation of latitude is in 44° 06' S. and longitude 50° E. By Case IV., Example II., the maximum latitude in this case is the same as the max- imum separation of latitude. With these three points, the track can be easily drawn. The great-circle track, from Cape Clear to the northern portion of the United States, passes so near Cape Race, that mariners, in endeavoring to keep on this track, are often placed in great peril when approaching tlie vicinity of Newfoundland. The following track, from Cape Clear, passing through a point one hundred miles south- east of Cape Race, and thence to Nantucket South Shoal, by Ifercator's sailing, is proposed : Lat. Long. Lat Long. N. W. N. W. 1st. From Cape Clear, in - - - 51° 26' and 9° 29', to 51° 16' and 23° 27'. 2d. From 51° 16' and 23° 27', to 49° 23' and 37° 24'. Sd. From 49° 23' and 37° 24', to 45° 28' and 51° 21'. 4th. From 100' S. E. of Cape Race, 45° 28' and 51° 21', to 41° 04' and 69° 51'. The distance on these four courses, by Mercator's sailing, is - - - - 2532J The distance on the Great Circle, from Cape Clear to a point 100 miles S. East of Cape Race, and thence to Nantucket South Shoal, is - - - - - 2528 Making a saving of only about 4^ The distance on the great circle, direct from Cape Clear to Nantucket South Shoal, is, 2505 Being only 27i miles less than the route proposed. In sailing easterly beyond the Cape of Good Hope, we have (by Example III, page 454) the distance by parallel sailing 2758 mile.s, And by great-circle sailing 2703 " Now, if by Mercator sailing, we lay off the track From 40° S. and 20° E. to 44° 06' S. and 50° E. and thence From 44° 06' S. and 50° E. to 40° S. and 80° E., we shall find the distance 2722 " Only 19 miles more than the great circle. From these examples, it would seem that the advantages in most cases derived from keeping on the great-circle track, are not sufficient to authorize the mariner to run the least risk in pursuing his course ; and that the small saving of distance is not really of any comparative importance. 459 ON THE COMPASS. Tlie British Admiralty have directed that Compasses should be placed at least 4 feet 6 inches apart on board of the ships of war. This is to avoid the disturbance knovpn to exist when two needles are placed near each other. The error from this source has, in some cases, amounted to more than 8°. It is to be hoped that the mercantile interest of the country will adopt this rule. If the steering ap- paratus is sufficiently small one compass is strongly recommended, a standard compass, for reference, being placed on the centre line of the ship. No Iron should be allowed within seven feet, and vertical Iron stancheons, &c., should be at least fourteen feet from the compasses. Binnacles should be made without doors, to prevent improper substances from being placed therein. The common compasses are frequently very imperfectly constructed, and the needles poorly magnetized. Great care in their selection cannot be too strongly recommended ; they should, like the chronometer, be carefully handled, and not subjected to the rough usage they frequently receive. Rules for ascertaining the deviation of the compass caused by the Iron in the ship. 1st. — A good standard compass should be placed on the centre line on the quarter deck, as far as possible from all masses of Iron. It should have such a support as will render bearings and amplitudes easily taken. 2d. — Bearings should be taken only on that part of the ship where the standard compass is placed, or where the observations for deviation were made. 3d. — When the ship is fully ready for sea, with every thing on board, allow her head to come up successively to the thirty-two points of the compass ; then accu- rately observe the hearing of some distant hut well defined object, (the real mag- netic bearing of the same having been ascertained,) and record the same as in Table I. 4th. — The real magnetic bearing may be found by taking the standard compass on shore and placing it on a line with the object observed and that part of the ship where the compass stood, so that they shall be in a line with the observers eye. The difference between this real magnetic bearing and the bearing in col. 2 will give the deviation which is found in col. 3. TABLE I. Real magnetic bearing of the distant object from the ship, N. 80° E. Ship's Head Bearing of by Deviation of Ship's Head Bearing of by Deviation of by the Stand- the Standard the Standard by the Stand- the Standard the Standard ard Compass. Compass. Compass. ard Compass. Compass. Compass. North. N. 81° E. 1° w. South. N. 80° E. Nothing. N. by E. N. 79 E. 1° E. S. by W. N. 81° E. 1° W. N. N. E. N. 78 E. 2° E. s. s. w. N. 82° E. 2° W. N. E. by N. N. 76 E. 4° E. S. W. by S. N. 83° E. 3° w! N. E. N. 75 E. 5° E, s. w. N. 84° E. 4° W. N. E. by E. N. 74 E. 6° E. S. W.byW. N. 85° E. 5° W. E. N. E. N. 73 E. 7° E. W. S. W. N. 80° E. 6° W. E. by N. N. 72 E. 8° E. W. by S. N. 87i E. 7^ W. East, N. 72^ E. 7^ E. West. N. 87° E. 7° W. E. by S. N. 73 E. 7° E. W. by N. N. 801- E. 6} W. E. S. E. N. 74 E. 6^ E. W. N. W. N. 80° E. 6° W. S. E. byE. N. 75 E. 5° E. N. W. by W. N. 85° E. 5° W. S. E. N. 76 E. 4° E. N. W. N. 84° E. 4° W. S. E. by S. N. 77 E. 3° E. N. W. by N. N. 83° E. 3° W. S. S. E. N. 78 E. 2° E. N. N. W. N. 82° E. 2° W. S. by E. N. 79i E. 0^ E. N. by W. N. 81° E. 1° W. The deviation is East when the north end of the needle is drawn to the eastward. 460 or right hand ; and West when the north end of the needle is drawn to the west- ward, or left hand. Example : — When the ship's head is E. by N. the bearing by the standard com- pass was N. 72° E., it follows that the north end of the needle has been attracted 8° to the eastward. Should there be no proper object of sufficient distance visible from the ship, then a second compass must be taken on shore, and the bearing of the two compasses from each other observed at each of the thirty- two points, and the results registered as in Table II. The standard compass should be compared on shore with the second compass, and if any difference is found it should be noted. TABLE II. Ship's Head by the Standard Com- pass. Bearing of the &hore Compass from the Standard Compass. Bearing of the Standard Compass from the 2d Compass on shore with the correction for their difference applied. Deviation Stand ard Ccmpass. Correct Mag- netic Course. N. N. by E. N. N. E. &c. S. 31° W. S. 28i W. S. 27° W. N. 30 E. N. 29^ E. N. 29° E. 1° w. 1° E. 2° E. Nearly N. N. 12° E. N. 24° E. Col. 5th gives the correct magnetic course, and also the points, when the iron causes the least deviation, which generally are the North and South points ; but as this is not always so, especially for steam vessels, we should depend upon obser- vations only. ^ The points once established may be considered permanent, provided every thing remains the same and the compass used in the same place. An azimuth at sea, with the ship's head on the point of no deviation, will give the true variation. The amount of deviation varies with the latitude, and in southern latitudes it be- comes important to form new tables, as the deviation generally changes from West to East and from East to West. The deviations can be examined at sea, by ob- serving azimuths with the ship heading on different points, especially on the point of no deviation. If the results conform to the table they may continue to be used ; if not, then a new table should be made. The standard compass in iron vessels should be raised above the deck much higher than in sailing vessels. In steam vessels with telescopic funnels the deviation is sensibly affected when they are taken down. Observations should be made when up and down. It is recommended that the ship should be directed hy the standard compass, and the binnacle compass should be used by the helmsman only to give the approximate course. Direct reference should be frequently had to the standard compass. E. & G. W. BLUNT. BOOISZS- Nautical Almanac, containing the Moon's right ascension and declination for eyery 5 hours. Bowditch's Navigator, 30th edition. Blunt's Coast Pilot, 18th edition. Erpeditious Measurer, for measuring cargo. Ward's Lunar Tables. Sheet Anchor, 112 quarto plates, with additions by G. W. Blunt. Commercial Digest, by Joseph Blunt, 9th edition. TIDE TABLES for tlie Coast of the United States, by A. D. Bache, Superintendent U. S. Coast Survey. Chart from Cape Cod to Labrador, including the Grand Bank and Gulf of New- foundland, &c. Eastern Coast of the United States, including Nova Scotia, from New- York to Capo Canso. Long Island Sound, on a large scale. Chart from Moutauk Point to Cape Antonio, including Bahama Bank, &c., on a diagonal scale. Chart from New-York to St. Augustine, in three sheets. Bahama Banks, including the Admiralty Surveys up to the present date. Bahama Bank, very large scale, (Pilotage Chart.) Bahama Banks, Island of Cuba and Passages, on a large scale. Florida Reef, on a large scale. North Coast of the Gulf of Mexico, from St. Mark's to New-Orleana. Chart of the Coast of Texas. West Indies to 15*^ North, including Gulf of Mexico, with Admiralty survey! to present date. West Indies to 9° North, including Gulf of Mexico, Spanish Main, Island of Trinidad, &c., two sheets. Chart of Guyana, from recent surveys. Coast of Brazil, three sheets. River Plate. Cape de Verde Islands. NORTH ATLANTIC, new Chart, on a large scale, with a Memoir, PLANS of AZORES, MADEIRA, and TENERIFFE. North Atlantic, with the curves of magnetic variation, and a Memoir. South Atlantic. Do. do. and South Pacific. North Pacific, including China Seas, with plans of Straits Juan de Fuca, &c. Behring's Straits and Sea. Indian and Part of the Pacific Oceans. New Charts of the VINEYARD and NANTUCKET SOUNDS, on a very largo scal«. from actual surveys. New Chart of Sagua La Grande. New Chart of Tapo Cod and Mnf5sachusctts Bays and Coast. New Chart of the Wiiidward Islands, on a largo scale, iJew Chart of Magnetic Variations for the whole world. The subscribers have now published Charts of all the navigable world, from the best authorities, and hope-that American Ship-masters will use American Charts. IDX^^XJDXl^G- Eisrc3-iisrE. They have just completed at their establishment, after a labor of over five years, a Dividing Engine, by which they are enabled to divide Astronomical and Nautical Instruments to a degree of precision -which they vrill guarantee to be equal to the best of foreign make. The subscribers, therefore, ask that American ships may be navigated by American made instruments. Chronometers of the best makers, for sale and to hire. Sextants, Quadrants, &c., of American manufacture. Spy Glasses. Night Glasses, nevr kind. Aneroid Barometers. Compasses, Dent's Improved, and others. Binnacles. Globes, Terrestrial and Celestial, 16 inch. The Terrestrial with Isothermal Lines of Temperature, and Deep Sea Soundings. Massg^, o Patent Logs. Ogden's, Ericsson's, and Massey's Patent Sounding Instruments. Improved Compasses with elastic centres. ABBOTT'S IIOROMETER, a new and simple instrument for working the Longitude either by Lunar Observations or the Chronometer. E. & G. W. BLTJIJT, 179 Water-Street NOTICE TO SHIPMASTERS. Just published, Massachusetts Bay. Office of the Board of Underwriters. New- York, March 20th, 1858. There is reason to believe that disasters to vessels have recently occurred on tiie Southern Coast of the United States, in consequence of the use of old and incorrect. Charts. This Board would earnestly impress upon Shipmasters the great importance of being provided with those that are of recent date and from a reliable source. Blunt's Charts of the Coast of the United States are corrected in conformity with the Government Surveys, and have accurately laid down the position of all the Lights now in use, or in process of construction on our coast, and these Charts should be familiar to every Shipmaster in the trade. ELL WOOD WALTER, Secretary Board of Underwriters Eoctract of a letter from Lieut. John Rodgers, commanding U. S. Ship " Hancock,^' attached to the Surveying Expedition to the China Seas, North Pacific. New-Bedford, January ith, 1852. I had a long discussion on Charts of the extreme North Pacific, Behring's Straits, Sea of Okotsk, &c. All the Whalers say that you are right. COMPASSES. — Attention is invited to the new Compasses constructed at the establishment of the subscribers. It is a fact now well understood, that most of the losses charged to Currents are due to the imperfect construction of Compasses, and to their deviation not being ascertained. Compasses of a superior quality, are manufactured by them, and are constantly on hand. Also, Dent's Patent and other approved Compasses. E. & G. W. BLUNT. Agents for Rogers' Signals.— Office of the Marine Register or American Lloyds. Novtmber^ 18G0. **; H^ ,iW^•■:'^^"■^^^^'! '