parallax, at that noon or midnight, according as it may be increasing or decreasing.

Note. Since the difference of the moon's longitude in 12 hours will always exceed the limits of the Table; if, therefore, an aliquot part, as or of such difference be taken, and the correction resulting therefrom be multiplied by 2 or 3, as the case may be, the required correction will be obtained.

Example,

Required the moon's semidiameter, horizontal parallax, longitude, and latitude, January 10th, 1836, at 2:40:10: mean time, under a meridian which is 70: 10" 45" west of Greenwich.

Greenwich time past noon of the given day 7:20:53:

Moon's correct semidiameter. 15:44", as required.

* See the Article relating to the moon's horizontal parallax, between pages 29 and 35,

To find the Moon's Longitude :

Difference in 12 hours 6:42:55" +3=2:14:18" Prop. log. 0.1272 7:20:53 Prop. log. 1.3891

Greenwich time

Constant logarithm

One third of the correction = 1:22:14 Prop. log. .

Moon's approximate long. 188:36:19", as required.

Remarks.-1. When much accuracy is required, the proportional part, or correction of the moon's longitude and latitude, found as above, must be corrected by the equation of second difference contained in Table XVII, as explained between pages 35 and 37.—And, in all cases, the moon's semidiameter must be increased by the augmentation given in Table IV., as explained between pages 8 and 11.

2. The above problem may be very correctly solved by means of Table XVI. (See the explanation thereof in page 27.)

3. The moon's semidiameter, horizontal parallax, &c., are given in the Nautical Almanac to tenths of a second; but, since the nearest second in those elements will be sufficiently near the tenth for the common purposes of navigation; if, therefore, the decimal be 5, or under, reject it; but if it be more than 5, increase the seconds of the semidiameter, horizontal parallax, &c., by unity or 1; as in the above example.

PROBLEM XVI.

To Reduce the Moon's Right Ascension and Declination, as given in the Nautical Almanac, to any other Meridian, and to any given Time under that Meridian.

RULE.

To the mean time at ship or place, always reckoned from the preceding noon, add the longitude in time if it be west, but subtract it if east, the sum, or difference, will be the mean time at Greenwich.

Enter the Nautical Almanac, between pages V. and XII. of the month, and take out, under the given day, the right ascensions and declinations answering to the hours which are next less and next greater than the hour in the Greenwich time; and find the difference of each: -Then,

To the proportional logarithm of the difference thus found; add the proportional logarithm of the minutes and seconds in the Greenwich time, and the constant logarithm 9.5229;* the sum, abating 10 in the index, will be the proportional logarithm of a correction, which is always to be added to the right ascension at the next less hour; but, to be applied by addition or subtraction to the declination, at that hour, according as it may be increasing or decreasing.

Example.

Required the moon's right ascension, and declination, on the 1st day of July, 1836, at 9:30:36: mean time; the longitude being 70:14:30 cast?

To find the Right Ascension:

Moon's right ascension at 4 hours = 21:28 21: at 5 hours = 21.30.48

Ditto

Comp. of 60")

9.5229

To find the Declination

Moon's declination at 4 hours 20:19:30" south. at 5 hours = 20. 8. 0 ditto

Ditto

Note. The moon's right ascension is given in the Nautical Almanac to the hundredths of a second, and her declination to tenths of a second: however, for the common purposes of navigation, the nearest second in either of those elements will always be sufficiently near the truth; and, therefore, if the decimals in the right ascension be 50, or under, reject them; but if they be more than 50, increase the seconds of R. A. by unity or 1.-And, in like manner, should the decimal in the declination be 5, or under, reject it; but if it be more than 5, increase the seconds in this element by unity or 1; as in the above example.

Remarks. In cases of great nicety, or when the moon's right ascension is required to hundredths of a second; let the following method be adopted, and the value of that element will be obtained to an extreme degree of exactness.

Rule-Multiply the decimals or hundredths of a second by 6, and they will be converted into thirds; observing to cut off the right-hand figure in the product. Then, to the proportional logarithm of the difference of right ascension, in minutes, seconds, and thirds, esteemed as hours, minutes, and seconds, add the proportional logarithm of the minutes and seconds in the Greenwich time, and the constant logarithm 9.5229; the sum, abating 10 in the index, will be the proportional logarithm of a correction in hours, minutes, and seconds, which are to be considered as minutes, seconds, and thirds :- annex a cipher to the thirds, then divide by 6, and they will be reduced to decimals, or to hundredths of a second. In illustration of this, we will compute the correction of the moon's right ascension at the Greenwich time in the last example.

Annex a cipher to the 54; divide by 6, and the result is 90: hence, the true correction of the moon's right ascension is= 2 1:90

To Reduce the Geocentric Right Ascension and Declination of a Planet, as given in the Nautical Almanac, to any given Time under a known Meridian.

RULE.

Let the given mean time at ship be always reckoned from the preceding noon; to which, apply the longitude, in time, (reduced by Problem I., page 341,) by addition if it be west, but, by subtraction if east: the sum, or difference will be the corresponding mean time at Greenwich; subject to the conditions in Problem III., page 342.

Take, from between pages 279 and 302, or from between pages 323 and 346 of the Nautical Almanac, the given planet's geocentric right ascension and declination for the noons immediately preceding and following the Greenwich time, and find their difference; then,

To the proportional logarithm of this difference, add the proportional logarithm of the Greenwich time, (reckoning the hours as minutes, and the minutes as seconds,) and the constant logarithm 9.1249;* the sum, abating 10 in the index, will be the proportional logarithm of a correction; which, being applied by addition or sub

*The arith. comp. of the prop. log. of 24 hours esteemed as minutes.