To find the Moon's Semidiameter : 15:33 Moon's semidiameter at noon, August 2d, 1824, per Nautical Almanac, decreasing, and var. in 12 hours = 6" Proportional part to 72 05 and 6". = 35.30”. Do. to 0.12 and 6 = 0, 6 Proportional part to 7?12" and 6” is 3.36 4" Moon's semidiameter, as required 15:29 To find the Moon's Horizontal Parallax : 57.6 Moon's horizontal parallax at noon, August 2d, 1824, per Nautical Almanac, decreasing, and var. in 12 hours = 23. Proportional part to 7! 05 and 20 = 11:40 Do. to 0.12 and 20 0. 20 Proportional part to 712". and 23. is 13.48 14! Moon's horizontal parallax, as required 56:52! Remarks.-1. It is evident that, in the above operations, the greater part of the figures might have been dispensed with, by taking out two or more of the proportional parts at once ; but since they were merely intended to simplify and render familiar the use of the Table, the whole of the proportional parts have been put down at length. 2. This Table was computed according to the rule of proportion; viz. : As 12 hours are to the variation of the moon's longitude, latitude, right ascension, &c. &c. &c., in that interval, so is any other given portion of time to the corresponding proportional part of such variation. TABLE XVII. Equation of Second Difference. Since the moon's longitude and latitude, and also her right ascension and declination, require to be strictly determined on various astronomical occasions ; particularly the two latter when the apparent time is to be D inferred from the true altitude of that object; and since the reduction of these elements, to a given instant, cannot be performed by even proportion, on account of the great inequalities to which the lunar motions are subject;-a correction, therefore, resulting from these inequalities, must be applied to the proportional part of the moon's longitude or latitude, right ascension or declination, answering to a given period after noon or midnight, as deduced from the preceding Table or otherwise, in order to have it truly accurate. This correction is contained in the present Table, the arguments of which are,—the mean second difference of the moon's place at top; and the apparent or Greenwich time past noon, or midnight, in the left or right-hand column; in the angle of meeting stands the corresponding equation or correction. The Table is divided into two parts: the upper part is adapted to the mean second difference of the moon's place in seconds of a degree, and in which the equations are expressed in seconds and decimal parts of a second ; the lower part is adapted to minutes of mean second difference; the equations being expressed in minutes and seconds, and decimal parts of a second. In using this Table, should the mean second difference of the moon's place exceed its limits, the sum of the equations corresponding to the several terms which make up the mean second difference, in both parts of the Table, is in such case to be taken. The manner of applying the equation of second difference to the proportional part of the moon's motion in latitude, longitude, right ascension, or declination, as deduced from the preceding Table, or obtained by even proportion, will be seen in the solution to the following PROBLEM. To reduce the Moon's Latitude, Longitude, Right Ascension, and Declina tion, as given in the Nautical Almanac, to any given Time under a known Meridian, Rule. Turn the longitude into time, (by Table 1.) and apply it to the apparent time at ship or place by addition in west, or subtraction in east longitude; and the sum, or difference, will be the corresponding time at Greenwich. Take from the Nautical Almanac the two longitudes, latitudes, right ascensions, and declinations immediately preceding and following the Greenwich time, and find the difference between each pair successively; find also the second difference, and let its mean be taken. Find the proportional part of the middle first difference, (the variation of the moon's motion in 12 hours,) by Table XVI., answering to the Greenwich time. With the mean second difference, found as above, and the Greenwich time, enter Table XVII., and take out the corresponding equation. Now, this equation being added to the proportional part of the moon's motion if the first first difference is greater than the third first difference, but subtracted if it be less, the sum or difference will be the correct proportional part of the moon's motion in 12 hours. The correct proportional part, thus found, is always to be added to the moon's longitude and right ascension at the noon or midnight preceding the Greenwich time; but to be applied by addition or subtraction to her latitude and declination, according as they may be increasing or decreasing. Example. Required the moon's correct longitude, latitude, right ascension, and declination, August 2d, 1824, at 3? 10apparent time, in longitude 60;30: west of the meridian of Greenwich? Apparent time at ship or place . .' 3:10. Greenwich time. 7:12: To find the Moon's correct Longitude: First Second Mean Diff. Diff. 2d Diff. Moon's long. Aug. Ist, at midnt. 7:10:38:49 6:37.387 Š 5:39 Do. 2 at midnt. 7. 23.48. 26 }6.31.59 Do. 3 at noon 8. 0. 15. 96. 26. 43 {5.16 }5:274: Propor. part from Table XVI., ans. to 712" and 6:31:59. is 3:55:1124"! Eq. from Tab. XVII., corres. to 7?12" and 5! 0” = 36".0 and 0. 20 = 2 .4 and 0. 71 = 0.9 Eq. of mean second diff. ans. to 7" 12" and 5:271" is 39.3= + 3918" Correct proportional part of the moon’s motion in longitude 3:55:50:42" Moon's longitude at noon, August 2d, 1824. 7:17. 16. 27. 0 Moon's correct longitude, at the given time . 7:21:12:17"42" To find the Moon's correct Latitude : First Second Mean Diff. Diff. 2d Diff. Moon's lat. Aug. Ist, at midnt. 4:27:37" . S.}20:38.? 2:57") "}2:44: Do, 23.35 2 at midnt. 3. 43. 24 2.31 Do. 3. at noon 3. 17. 18 26. 6 Pro. part from Table XVI., ans. to 7! 12" and 23:35". is 0:14! 9". and 0.40 = 4 .8 Eq. of mean sec. diff., ans. to 7:12 and 2:44"is 19 .7 = -19”.7 Correct proportional part of the moon's motion in lat. 0:13:49".3 4. 6.59.0 S. To find the Moon's correct Right Ascension: First Second Mean Diff. Diff. 2d Diff. Moon's R. A. Aug. Ist, at midnt. 216:44:43" } 6:48:53" Do. 2 at noon 223. 33. 36 12:32 Do, 2 12. 8 648!53??2:56". Propor. part from Table XVI., ans. to 7:12" and 6:51:49" is 4: 7 5"24" and 0.30 = 3.6 Eq. of mean sec. diff., ans. to 7" 12" and 2:32" is 18 .2 = 18:12 Correct propor. part of the moon's motion in right ascension 4: 6:47"12"!! Moon's right ascension at noon, August 2d, 1824 223. 33. 36, 0 Moon's correct right ascension at the given time. 227:40:23!12! To find the Moon's correct Declination : First Second Mean Diff. Diff. 2d Diff. 2 17:35" s 1.23. 13 18. 21 23. 26. 12 }1. 5. 22 35"}17:58 Pro. part fr. Tab. XVI., ans. to 7:12" and 1:23:43" is 0:50:13"48" and 2. 0 = 0.14 .4 Eq. of mn sec.diff., ans. to7:12 and 17:58"?is 2. 9 .4=+2' 9"24". Correct prop. part of the moon's motion in declination 0:52:23"12" 20.57. 7. OS. Moon's correct declination at the given time 21:49:30"12" south. Note.--It frequently happens that the three first differences first increase and then decrease, or vice versa, first decrease and then increase ; in this case half the difference of the two second differences is to be esteemed as the mean second difference of the moon's place : as thus, First Second Mean Diff. 2d Diff. 14'26!) Do. 26534:26 } 3.55 19 at midt. 24. 37.52 23. 18 Do. 20 at noon 24. 10. 39 27. 13 Here the two second differences are 14:26", and 23:18! respectively; therefore half their difference, viz., 8:52" + 2 = 4:26. is the mean second difference. Now, if the Greenwich time be 5740% past noon of the 19th, the corresponding equation in Table XVII. will be 33. subtractive, because the first first difference is less than the third first difference ; had it been greater, the equation would be additive. Remark. When the apparent time is to be inferred from the true altitude of the moon's centre, the right ascension and declination of that object ought, in general, to be corrected by the equation of second difference; because an inattention to that correction may produce an error of about 2 minutes in the right ascension, and about 4 minutes in the de. clination ; which, of course, will affect the accuracy of the apparent time. See the author's Treatise on the Sidereal and Planetary Parts of Nautical Astronomy, pages 171 and 172. The equation of second difference, contained in the present Table, was computed by the following Rule. To the constant log. 7.540607 add the log. of the mean second difference reduced to seconds; the log. of the time from noon, and the log of |