TABLE XVI.

To reduce the Moon's Longitude, Latitude, Semidiameter, and Horizontal Parallax, as given in the Nautical Almanac, to any given Meridian, and to any given Time under that Meridian.

This Table is so arranged that the proportional part corresponding to any given time and change of longitude &c. in 12 hours may be taken out to the most rigid degree of astronomical exactness.

Precepts.

In the general use of this Table it will be advisable to abide by the solar day; and hence to estimate the time from noon to noon, or from 0 to 24 hours, after the manner of astronomers, without paying any attention to either the nautical or the civil division of time at midnight. And to guard against falling into an error, in applying the tabular proportional part to the moon's longitude &c. &c., it will be best to reduce the mean time at ship to the Greenwich mean time, as thus :

Turn the longitude into time (by Table I.), and add it to the given mean time at ship or place, if it be west; but subtract it if east: and the sum or difference will be the corresponding mean time at Greenwich.

Take from pages III. and IV. of the month, in the Nautical Almanac, the moon's longitude, latitude, semidiameter, and horizontal parallax, (or any one of these elements, according to circumstances,) for the noon and midnight immediately preceding and following the Greenwich time, and find their difference; which difference will express the variation of those elements in 12 hours.

Enter the Table with the variation, thus found, at top, and the Greenwich time in the left-hand column: in the angle of meeting will be found the corresponding equation, or proportional part, which is always to be added to the moon's longitude at the preceding noon or midnight, but to be applied by addition, or subtraction, to the moon's latitude, semidiameter, and horizontal parallax, according as they are increasing or decreasing. And, since the Greenwich time and the variation in 12 hours will be very seldom found to correspond exactly; it is the sum, therefore, of the several equations making up those terms, that will, in general, express the required proportional part.

Example.

Required the moon's longitude, and latitude, semidiameter, and horizontal parallax, August 2nd, 1824, at 3:10", mean time, in longitude 60:30: west of the meridian of Greenwich?

To find the Moon's Longitude :

Moon's longitude at noon, August 2nd, 1824, per Nautical

Propor. part to 7:12" and 6:31:59% is 3.55. 11. 24 = +3:55:11?

Moon's latitude at noon, August 2nd, 1824, per Nautical

1:40% 0
0.20. 0

Proportional part to 7:12 and 23:35 is 14. 9. 0 = 14: 9"

Note. In consequence of the unequal motion of the moon in 12 hours, (when her place is to be determined with astronomical precision,) the ркоportional part of the variation of her longitude and latitude, found as above, must be corrected by the equation of second difference contained in. Table XVII.

Moon's horizontal parallax at noon, August 2nd, 1824, per

Nautical Almanac,

decreasing and var. in 12 hours = 23"

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.

Parallax of the Heavenly Bodies.

As the nature of celestial parallax is but little understood by those who have not had an opportunity of studying the elementary parts of astronomy, the following concise and practicable method of finding the lunar horizontal parallax is, therefore, submitted to their consideration; and,

although remarkably familiar, yet, if duly attended to, it cannot fail of giving a correct idea of the cause and effects of celestial parallax in general.

Definition. The parallax of the moon, sun, or planet, is the distance between its true and apparent places in the starry heavens. The true place of any celestial object, referred to the sphere of the fixed stars, is that in which it would appear if seen from the centre of the earth; the apparent place is that in which it appears to an observer on the earth's surface.

Illustration.

B

E

In the annexed diagram, let ABDE be the earth, and C its centre; the quadrant HI KLMN the concave crystalline arch, or the azure sky in which the moon is seen; and the quadrant U V W X Y Z an arc of the sphere of the fixed stars. Let the dotted line Ala be the sensible horizon of an observer on the earth's surface at A: to him the moon at I will appear in the horizon, extended to the starry heavens, at the horizontal point a; but to an observer at the centre C (supposing the earth to be transparent) she will appear above the horizon at V, which is her true place. The angle AIC is called the moon's horizontal parallax; and is equal to the opposite or parallactic angle VI a. To an observer at the centre C, the moon KD will appear at W, her true place; but to an observer at A, she will appear below her true place at b: the difference, or parallactic angle WK b, equal to the angle AK C, is the moon's parallax at the altitude KD; and so on to the zenith Z, where the parallax entirely vanishes: for, it is evident that the ray of light flowing from the moon, when in the zenith at N», must be in the same right line with the points A and C. From this it will appear manifest that the parallax decreases from the horizon to the zenith in proportion to the co-sine of the altitude; and, vice versa, that it increases from the zenith to the horizon in proportion to the sine of the zenith distance of the object.

D

The parallax causes the moon, sun, or planet, to appear nearer to the

horizon than it really is; hence it increases the zenith distances of those objects. The fixed stars have no sensible parallax; because their distance from the earth is so inconceivably great that, though seen from opposite points of the earth's orbit, they always appear under the same angle. Hence the diameter of the earth's orbit, which is upwards of 190 millions of miles in extent, is but as a dimensionless point compared with the immeasurable distance of those refulgent luminaries.

These being premised, we will now proceed to the proposed method of finding the moon's horizontal parallax.

Rule.

1. Reduce the mean time of the moon's transit over the meridian of Greenwich, as given in page IV. of the month in the Nautical Almanac, to the meridian of the place of observation; as thus:

Find the difference of transit between the given day and the day following, if the longitude be west, but the day preceding if east; and it will be the daily retardation of transit. Then say, As the sum of 24 hours and the retardation of transit, is to the retardation; so is the longitude, in time, to a correction: which being applied by addition to the time of transit over the meridian of Greenwich, on the given day, if the longitude be west, but by subtraction if east; the sum, or difference, will be the mean time of transit over the meridian of the place of observation. To this time, let the moon's declination and semidiameter be carefully reduced. Then,

2. If the latitude of the place of observation and the moon's corrected declination be of the same name, take their difference; but if of contrary names, their sum in either case the moon's correct meridional zenith distance will be obtained.

3. Let the meridional altitude of the moon's lower or upper limb be very carefully taken at the moment of transit; to which apply her corrected semidiameter, and the corresponding refraction; the result will be the apparent meridional altitude of the moon's centre; the difference between which and SO degrees will be the moon's apparent meridional zenith distance; this will be always greater than the true meridional zenith distance; the excess will be the parallactic angle corresponding to the apparent meridional zenith distance. The parallactic angle being thus known, the horizontal parallax may be readily computed in the following manner, viz.:

4. Suppose the moon's apparent place in the concave arch of the firmament to be at KD in the preceding diagram, then her apparent zenith distance is Z b, and her true zenith distance Z W; the difference between these, viz., the angle W Kb, is the parallactic angle in altitude. Then, As the sine of the apparent zenith distance Z A b, is to the sine of the parallactic angle W K b, so is the sine of the zenith distance or right angle