EXAMPLE IX. [Same as EXAMPLE I., page 232.] Suppose that, on the 7th of January, 1836, sea account, at 11m 57 mean time, past midnight, in the longitude of 127° 30′ E., by account, the observed distance of the farthest limb of the moon from the star Aldebaran, was 68° 36' 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. Sum · 10° 68° 03'01"=True distance, differing 1" from the first method, in page 232. = Difference is longitude in time..... 8 30 41 = 127° 40′ 15′′ E. from Greenwich. 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 XII.) take his parallax in altitude, (Table XIV. ;) the remainder call the correction of the sun'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 the planet's altitude. But 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, (increasing 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 apparent distance of the sun and moon, (planet 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 halfsum and the moon's apparent zenith distance call the first remainder; and the difference between the half-sum and the sun's (planet or star's) apparent zenith distance, call the second remainder. To the constant log. 9.6990 add the cosecant of the half-sum, and the sine of the apparent 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 prop. 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 first correction, to be 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 from the index) will be the prop. log. of the second correction, to be found in Table XXII. Then, to the apparent distance add the correction of the moon's altitude, and the first correction, and subtract the sum of the second correction and the correction of the sun's (planet 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.‡ 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 longitude, as it is very nearly the same as in page 232. EXAMPLE X. [Same as EXAMPLE I., preceding.] Suppose the apparent distance of the centre of the moon from the star Aldebaran was 68° 20′ 45′′, the apparent altitude of the star 32° 10′, the apparent altitude of the * We may also find this correction by means of Table XVII., or Table XVIII.; taking the difference between the tabular number and 60' for the correction; using Table XVIII. for the sun, and Table XVII. for a planet, or a fixed star. + This correction may very easily be found by means of Table XIX., by subtracting the tabular number from 53' 42"; for the remainder will be the correction of the moon's altitude for parallax and refraction. moon's centre 34° 55', and the moon's horizontal parallax 55′ 24′′. Required the true distance of the moon from the star. This method, as well as the first, was invented by the author of this work, who also improved Witchell's method, and reduced considerably the number of cases. These improvements were made in consequence of a suggestion of the late Chief Justice Parsons, (a gentleman eminently distinguished for his mathematical acquirements,) who had somewhat simplified Witchell's process; and it was found, upon examination, that this improvement could be extended farther than he had done it, and that the number of cases, with the manner of applying 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 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 the planet's altitude. But if a star be observed, 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, (increasing the index by 10,) take the cosine of the moon's apparent altitude, (Table XXVIL) the remainder will be the proportional logarithm of the moon's parallax in altitude; from which subtracting the moon's refraction, (Table XII.) thẻ remainder will be the correction of the moon's altitude.† This correction may be found in Table XVII. or XVIII., as is shown in a note to the third method, in page 242. This correction may be found by Table XIX., as is shown in a note to the third method, in page 242. 1. Add together the apparent altitudes of the moon and sun, (planet or star,) and take the half-sum; subtract the least altitude from the greatest, and take the halfdifference; then add together The tangent of the half-sum, The cotangent of the half-difference, The tangent of half the 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 the index) will be the prop. log. of the first correction. Or the refraction (Table XII.) corresponding to the first angle, or its supplement, 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 true distance. EXAMPLE XI. [The same as EXAMPLE I., preceding.] Suppose the apparent distance of the centre of the moon from the star Aldebaran was 68° 20′ 45′′, the apparent altitude of the star 32° 10', the apparent altitude of the moon's centre 34° 55', and the moon's horizontal parallax 55′ 24′′. Required the true distance of the moon from the star. * Every cotangent in Table XXVII. corresponds to two angles, the one greater than 90°, the other Method of correcting for the second differences of the motions of the bodies in computing a lunar observation. In all the preceding calculations, 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 source is generally quite small, and may, in most cases, be neglected, as coming within the limits of the usual errors of such observations. It is, however, very easy to find this correction by means of the following table, which is similar to that in page 484 of the Nautical Almanac for 1836. In using this table, we must find the difference between the two proportional logarithms, corresponding to the distances in the Nautical Almanac, which include the given distance. This difference is to be sought for at the top of the table; and at the side we must find the interval which is calculated in the last part of the process of computing the true longitude, being the time between the hour marked first in the Nautical Almanac, and the mean time of observation at Greenwich. The number of seconds in the table corresponding to these two arguments is to be applied, according to the directions in the 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 logarithms corresponding, on January 6th, to 3 and 6, are 2872, 2864, whose difference is 8; and the interval past 3", computed in page 232, is 0h 41m 143. Entering the table with 8 at the top, and 0 40 at the side, (which is the nearest number to the interval 0h 41m 14',) we get the correction 2, to be added to the time at Greenwich, 3h 41m 14", (computed in page 232,) because the logarithms are lecreasing; hence the corrected time at Greenwich is 3h 41m 16o. EXAMPLE 2. In the example page 237, we find that the two proportional logarithms corresponding, on June 20th, to 9 and 12", are 2985 and 2969, whose difference is 16. Under this, and opposite the interval 1" 55m 28, computed in page 237, (or the nearest tabular number 2h 0m,) we find a correction 4* to be added to the time at Greenwich 10h 55m 28, computed in page 237, making the corrected time at Greenwich 10h 55m 32*. Table, showing the Correction required on account of the Second Differences of the Distances in the Nautical Almanac, in working a Lunar Observation. Find at the top of the table the difference between the proportional logarithm taken from the Nautical Almanac, in working a lunar 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 number 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 subtracted if the proportional logarithms are increasing; the sum or difference will be the corrected time at Greenwich. Difference of the Proportional Logarithms in the Nautical Almanac. Approxi- 48 12 16 20/24 28 32 36 40 44 48 52 56 60 64 68/72 7680 84 88 92 96 Approxi mate Interval. Correction of the Time at Greenwich for Second Differences. mate Interval. H. M.H. M. S.S. S. s. s. S. S. 8. S. 8. | 8. S. S. S. S. S. S. S. 8. S. S. S. S. S. H. MH. M. 0 03 0000 00 00000000 000000000 03 0 223 567 1 2 3 4 67 2 4 5 6 7 8 9 10 11 12 13 911 12 13 14 10 11 12 14 15 9 10 11 12 14 15 9 10 10 11 12 13 14 16 17 18 19 20 13 14 14 15 16 17 21 22 23 24 25 27 0 302 30 0 402 20 0 50 2 10 1 002 00 1 101 50 1 201 40 1 301 30 |