Take the fun's femi-diameter from page 2 of the month in the Nautical Almanack, from which fubtract the dip of the horizon; the remainder, added to the obfer ed altitude of the lower limb, or the fum fubtracted from the obferved altitude of the upper limb, will give the true altitude of the fun's centre. From the fun's refraction take his parallax in altitude, the remainder will be the correction of the fun's altitude. This correction, fubtracted from the apparent altitude, will give the true altitude of the fun's centre. If a ftar has been obferved, from the obferved altitude fubtract the dip of the horizon, the remainder is the ftar's apparent altitude, from which take the refraction anfwering to that altitude, the remainder is the ftar's true altitude. 3d. To correct the obferved altitude of the moon. Take the moon's femi-diameter and horizontal parallax from page 7 of the month in the Nautical Almanack, for the nearest noon and midnight before and after the reduced time, and find their difference, which multiplied by the number found in Table XVIII, correfponding to the hours and minutes of reduced time, gives a number of feconds, which being added to the moon's femi-diameter at the noon or midnight immediately preceding the reduced time, if it be increafing, but fubtracted therefrom, if decreafing, the fum or difference will be the moon's femi-diameter at the time of obfervation. To the moon's femi-diameter, thus corrected, add the augmentation anfwering to her obferved altitude, the fum will be the moon's true femi-diameter: when the reduced time is any even part of 12 hours, as,, 4, or 4; fuch parts of the difference of the femi-diameter and horizontal parallax may be taken and applied as above, without being at the trouble of working by the numbers in Table XVIII. From the moon's true femi-diameter fubtract the dip of the horizon, the remainder, added to the obferved altitude of the lower limb, or their fum fubtracted from the obferved altitude of the upper limb, gives the apparent altitude of her centre. To obtain the correction of the moon's altitude, proceed as follows: Having taken out the horizontal parallax at the noon and midnight immediately before and after the reduced time, and having found their difference, as before directed, Multiply it by the number found in Table XVIII, correfponding to the hours and minutes of reduced time, gives a num ber of minutes and feconds, which, being added or fubtracted from the horizontal parallax, at the noon or midnight immediately preceding the reduced time, according as it is increafing or decreasing; the fum or difference will be the moon's horizontal parallax at the ducel time. To the prop. log of the moon's horizontal parallax add the log. fecant lefs radius of the moon's apparent altitude, the fum will will be the prop. log. of the moon's parallax in altitude; from which take the retraction, the remainder will be the correction for the moon's altitude. 4th. To correct the obferved distance. To the obferved distance of the fun and moon's nearest limbs, add both their femi-diameters, and the fum will be the apparent. diftance of their centres. To the observed distance of the moon from a star, add the moon's femi-diameter, if her nearest limb was taken, but fubtract it if her fartheft limb was taken, the fum or difference will be the apparent diftance. NOTE. There are 12 pages in each month in the Nautical Almanack. The 'fun's declination is found in page The fun's femi-diameter II. III. The moon's femi-dia. and horizont. parallax VII. The distance of the moon from the fun, &c. VIII. IX. X. XI XII. Having the apparent Altitude of the Objects, and their apparent Distance, to find their true Diflance, by Mr. LYON's Method. ift. Add together the prop. log of the corr tion of the fun or ftar's altitude, the log. co-fine of the fun or ftar's apparent altitude, the log fine of the apparent distance, and the log co-fecant of the moon's apparent altitude; their fum (rejecting 30 in the index) will be the prop. log of the first arch. 2d. Add together the prop log. of the correction of the fun or ftar's altitude, the co-tang, of the fun or ftar's apparent altitude, the log. tang. of the apparent diftance; their fum (rejecting 20 in the index) will be the prop. log of the fecond arch. Take the difference between the firft and fecond arches, which add to the apparent diftance, if Jefs than 90°, and the first arch be greater than the fecond, but if it be lefs fubtract it. But if the dift. be more than 90°, adding both arches to the apparent dift. will give the dift. corrected for the refraction of fun or ftar. 3d. Add together the prop. log. of the correction of the moon's altitude, the log. co-fine of the moon's apparent altitude, the log. fine of the dift. corrected for the fun or ftar's refraction, the log. co-fec, of the fun or ftars true altitude; their fum (rejecting 30 in the index) will be the prop. log of the third arch. 4th. Add together the prop. log. of the correction of the moon's apparent altitude, the log. co-tang, of the moon's apparent alti tude, the log. tang. of the dift. corrected for the fun or ftars refraction; their fum (rejecting 20 in the index) will be the prop. log. of the fourth arch. Take the difference between the third and fourth arches, and subtract it from the diftance corrected for the fun or ftar's refra Rion if less than 9, and the third arch be greater than the fourth; or, add it to the diftance corrected, if the fourth arch be greater than the third; but, if the distance be more than 90°, the fum of both arches must be fubtracted from it; and the fum or difference will be the diftance corrected for the fun or star's refraction, and the principal effect of the moon's parallax. In Table XXVI. look for this laft corrected diftance in the top column, and the correction of the moon's altitude in the left-hand fide column; take out the number of feconds that ftand under the former and oppofite to the latter. Look again in the fame table for the corrected distance in the top column, and the principal effect of the moon's parallax in the lefthand fide column, and take out the number of feconds that ftand under the former and oppofite the latter. The difference between these two numbers muit be added to the corrected distance if lefs than goo, but fubtracted from it if more than 90°; The fum, or difference, will be the true diftance. Having the true Diftance and Time, to determine the Longitude. N the Nautical Almanack, among the diftances of the objects, look for the computed diftance between the moon and the other object obferved on the given day; if it be found there, the time at Greenwich will be at the top of the column, but if it falls between two diftances, as it generally will, take the difference between the diftances that ftand immediately before and after the computed diftance, and alfo the difference between the diftance ftanding before it and the computed diftance. Then take the proportional logarithm of the first difference, which is the difference in three hours, and the proportional logarithm of the fecond difference, which is the difference between the computed diftance and the distance before it. The difference between thefe two logarithms will be the propor tional logarithm of a number of hours, minutes, and feconds, which being added to the time ftanding over the first diftance in the Nautical Almanack, will give the true time at Greenwich. The difference between Greenwich-time and that at the fhip turned into longitude, will be the longitude in, at the time the obfervations were made, which will be eaft if the time at the fhip be greater than that at Greenwich, but if it be leis, the longitude will be weft. Or the proportional part of time may be found by faying; As the firft difference: is to 3 hours :: fo is the fecond difference ; to a proportional part of time, which being added as above directed will give the true time at Greenwich. NOTE. In working the following examples, it will fave fome time, if all the logarithmic fines, tangents, fecants, and proportional logarithms, which fall at the fame opening opening of the book, be taken out at the fame time, both in the first and fecond part of the operation. Thas, the co-fine and co-tsngent of the ftar's apparent altitude, and co-fecant of its altitude, may all be taken out at the fame time, and written down in different parts of the paper (or in a formula) and fo may the co-fine, co-tangent, and co-fecant of the moon's apparent altitude, the fine and tangent of the apparent diftance, and the fine and tangent of the diftance corrected, for the refraction of the fun or far. EXAMPLE I. Suppofe, on the 23d of May, 1809, in longitude 13° 13' weft of Greenwich by account at 6h. 10m. P. M. by a watch well regulated, the diftance of the fun and moon's nearest limbs were obferved to be 104° 38′ 14′′, when the moa's altitude of her lower limb was 43° 20′20′′, the altitude of the fun's lower limb 12° 39′ 28", the eye of the obferver 20 feet above the furface of the fea. Required the true longitude? 4 Diff. in 12 hours 15 41 7's hor. par. at noon 57 33 + 8 Diff. in 12 hours 'sobf.alt.12°39'28's femi-dia.noon 5 41's par. at noon 58 A + + 16 T's femi-dia.n. 54 Do. midnight 57 33 Dift. correc. for C's refrac. 105 13 0 6434 Cor. for's ap.alt.° 40′ 55′′ P. L.. D's ap. alt. 43.32 oCo fine 9 8603 Correc. dift. 165 13 7 Sine 9 9845 O's alt. 12 47 3 Co-fec. 0 6551 Fourth arc 10' 34" P. L. I 2310 Principal effects of the D's par. Dift. correc. for O's refraction 105 13 74 23 31 Dif. זי Suppofe, on the 10th of March, 1809, in longitude 23° east of Greenwich, at 5 h. 36 m. P. M. by a watch well regulated, the diftance of the fun's neareft limb to the fun was 68° 9' 57", when the altitude of the fun's lower limb was 31° 48' 9", the alt. of the moon's lower limb 23° 41′ 7′′, the height of the eye of the ob ferver 18 feet above the fea, the true longitude is required? M. 5. femi dia. at noon 16 I hor. par. noon 58 48 Red. time diff. in 12 hours 3 diff. in 12 hours 11 Obf.alt.of LL 31 48 9 femi-dia. at n. 16 +12 3 femi-dia. Dip. } 16 Augmentation 7 Refrac. 131-23 dip. for 18 feet 4 1 11 X,3388 gives- 4 I hor par. noon 58 46 o H. P. red. time 58 42 PL 4866 app. alt. 24 O 10 Sec. 0393 53 38 PL 5259 - 2 8 femi-dia. 16 7 par, in alt. par. 4 refrac. |