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, whe lso 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 XXVII. ;) the remainder will be the proportional logarithm of the moon's parallax in altitude; from which subtracting the moon's refraction, (Table XII.) the 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, m 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 tru 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 D app. alt. 34° 55' app. alt. 32 10 Sum...... 67 05 Difference 2.45 Difference is.. Sum is.. Half-sum.. 33° 33'.. Tang. 9.82161 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 6h, 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 40m at the side, (which is the nearest number to the interval 0h 41m 14',) we get the correction 2a, to be added to the time at Greenwich, 3h 41m 14', (computed in page 232,) because the logarithms are decreasing; hence the corrected time at Greenwich is 3h 41m 16'. EXAMPLE 2. In the example page 237, we find that the two proportional logurithms corresponding, on June 20th, to 9 and 12h, are 2985 and 2969, whose difference is 16. Under this, and opposite the interval 1h 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 1216|20|24|28|32|36|40|44|48|52|56|60|64|68|72 76 80 84 88 92 96 Approxi Correction of the Time at Greenwich for Second Differences. mate Interval. H. M.H. M 3. S. S. s. S. 8. S. S. S. S. 3. S. S. S. S. S. 8. S. 0 03 000 0000 0 102 5001 1122 2 0 202 40 1122233 0 302 30 1 2 2 3 3 4 5 6 6 7 0 0 0 0 0 0 0 0 10 10 11 12 13 mate Interval. S. S. 3. S. S. S. H. M. H.M.) Method of taking a lunar observation by one observer. Three observers are required to make the necessary observations for determining tre 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 computed, 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 altitudes is far more simple, and sufficiently 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 number 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; then, from the two observed altitudes of either of the objects, the sought altitude of that object may be found 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.t of the time elapsed between observing the two altitudes of that object; the remainder will be the prop. log. of the correction, to be applied 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 observed, as in the following table; it is required to find the altitudes at the time of measuring the mean distance. Thus, at the time 2h 4m 30', the mean observed distance of the sun and moon's nearest limbs was 40° 0' 40", the altitude of the moon's lower limb 21° 6', and the altitude of the sun's lower limb 39° 50′; these altitudes must be corrected for dip ana semidiameter as usual. * Table XXII is only calculated as far as 3°, and if the variation of altitude exceed that quantity, 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 respectively In this 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 proportioning he 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 subtracted from the natural cosine of the sum of the declination and latitude, when they are of different 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. 12h 0m 0 5 3 30 49 57 0 N. EXAMPLE I. Required the true altitude of the sun's centre, in latitude 49° 57′ N., and longitude 75° W., July 26, 1836, at 63 56m 30′ in the morning, apparent time, sea account. Apparent time.. Apparent time from noon Declination at that time.. 19 24 15 N 3h 21m 30" 39 20 00 N. 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 3h 21m 30′, apparent time, in the afternoon, sea account? Apparent time from noon Declination at that time 20 53 09 S. Its log. cosine. 9.97048 If the mean time be given, we must deduce 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. |