Diff. Time 2 51 28 42° 52′ the Long, in, and it is Weft because the Time at Ship is less than at Greenwich. Suppofe that on the 15th of May, 1796, in Long. 1° 30' E. the Eye 21 Feet above the Sea, the following Obfervations were made, and the true Longitude be required? 240 Time at Ship Diff. of Time between Ship and Greenwich 12 41 19 I 215' the Longitude in and it is Weft, the Time at the Ship being lefs than at Greenwich. Suppofe the Dift. of the Sun and Moon's nearest Limbs, with the respective Altitudes of their lower Limbs, were obferved as under on the 13th of April 1796, the Eye being 18 Feet above the Sea, in Lat. 34° 17' N. and Long. by Account 30° 46′ W. of Greenwich, the Watch not previoufly regulated, and the Ship's true Longitude be required? As the Sun,must be then at a fufficient Distance from the Meridian, the Mean of the three Altitudes may be fuppofed preferable to any fingle Alt. taken for finding the Time. Sun's Moon" H. M. S. D. M.D. M. 4 55 26 77 18 21 Sum 14 32 51231 42 66 9242 3 Means 4 5 57 77 14 22 3 80 41 Obf. Dift. of Sun and Moon 77°14′0′′ Sun's Semdi. 15' 58} Moon's ditto 16 265 App. Dift. of Centres Cor. Sun's Alt. 0° 2' 11" Moon's Obf. Alt. Moon's Semidi. 16' 26"-Dip 4′ 3′′: Moon's app. Alt. Moon's Hor. Par. 0°59′ 21′′ P. L. Diff. II 55 1 Dip 4 3 80 41 =0 12 23 8053 23 10,482 1 L. co-f. 9,1998 P. L. of Moon's Par. in Alt. 32 24 Refraction 77 46 24 Cor. of Moon's Alt. 915 P. L. Co-tan. 1,9161 10,3881 Dift. corrected prin. Effect of Par. and Ref. True Dift. 77 38 26 the Correction To find the Time. By the fecond Method thus: 932 S's ap. Alt. 22 14 55 Dec. 9°32' Lat. 34 17 | Co. Lat. 55°43′ 55 43 M. Alt. 65 15 N. S. 90814 L. R. 08892 Tr. Alt. 22 13 N. S. 37811 Co-fec. 0,08288 53003 Log. 4.72428 203.58 16 Co-fec. 0,00604 In Col. of Rifing gives 4h. 38' 10" 4,81320 Sum 101 59 8 Sine Zen. Dift. 67 47 16. ་ H. Ang. 34 46 20 Co-f. 9,91457 Ho Ang. 69 32 404 38 11 True Time. Diff. of Time between Ship and Greenwich True Time at Greenwich 6 50 4 2 11 53 32° 59' 32" the Long. in Weft of Greenwich. In finding the Time the Decl. is proportioned to the reduced Time 6h. 54'. Here I have given one Method of finding the Longitude, illuftrated by a fufficient Number of Examples, all of which are reduced to the Year 1796, in order that the Reader, or Teacher, may have fufficient Time to furnish himself, with a Nautical Almanack for that Year, which is now printed.. But as many would wish to have fome other Method of reducing the Distance, that, by comparing them together, they may not only have the Advantage of proving their Calculations, but also of making Choice of which they prefer to work by. The fecond Method I fhall present the Reader with, is chiefly deduced from that invented by Mr. Witchell, late Master of the Royal Academy at Portsmouth, and as it is fhort, requires but four Places of Figures in the Logarithms, befides the Index; the Preparations in both Methods being exactly the fame. To find the true Distance, obferve this general Rule. Firft. Add the Sun or Star and Moon's apparent Altitudes together, and take half the Sum; fubtract the lefs from the greater, and take half the Difference; then add together The Co-Tangent of half the Sum, The Tangent of half the Difference, and The Co-Tangent of half the apparent Distance. Their Sum, rejecting 20 in the Index, will be the Log. Tangent of an Angle, which call A. Secondly. When the Sun or Star's Altitude is greater than the Moon's, take the Difference between the Angle A and half the apparent Distance, but if lefs take their Sum. Then add together The Co-Tangent of this Sum or Difference, The Co-Tangent of Sun or Star's apparent Altitude, and The proportional Log. of the Correction of the Sun or Star's Their Sum, rejecting 20 in the Index, will be the proportional Logarithm of the first Correction. Thirdly. If the Sum of Angle A and half the apparent Distance was taken in the last Article, take now their Difference; but if their Difference, take now their Sum. Then add together The Co-Tangent of this Sum or Difference, The Co-Tangent of Moon's apparent Altitude, and The proportional Log. of the Correction of the Moon's apparent Altitude. Their Sum, rejecting 20 in the Index, will be the proportional Log. of the fecond Correction. Fourthly. When the Angle A is less than half the apparent Diftance, the firft Correction must be added to, and the second subtracted from the apparent Distance; but, when the Angle A is greateft, their Sum must be added to the apparent Distance, when the Sun or Star's Altitude is less than the Moon's; but when the Moon's Altitude is leaft, their Sum must be fubtracted to give the corrected Distance. Fifthly. In Tab. XX. look for the corrected Dist. in the Top Column, and the Correction of Moon's Alt. in the Left-hand Side Column; take out the Number of Seconds that ftand under the former and oppofite to the latter. Look again in the fame Table for corrected Distance in the Top Column, and the fecond Correction in the Left-hand Side Column; take out the Number of Seconds that ftand under the former and oppofite the latter, the Difference between these two Numbers will be the third Correction, which must be added to the corrected Distance, if lefs than 90°, but fubtracted from it, if more than 90°; the Sum, or Difference, will be the true Distance. To illuftrate this laft Method of reducing the apparent Distance to the true Diftance, I fhall take the apparent Altitudes and Diftances as they ftand in the three first Examples, worked by the former Method. EXAMPLE I. See Page 235. Given the apparent Distance of the Moon's Centre from the Star 63° 35′ 13", the Moon's apparent Altitude 24° 29′ 44′′, and that of the Star Regulus 45° 9' 12", the Moon's Horizontal Parallax 55' 2". Required the true Distance? Given the apparent Altitude of the Sun's Centre 32°, that of the Moon's 24°, and the apparent Distance of their Centres 68° 42' 11", and the Moon's Horizontal Parallax 58' 10". Required the true Distance? |