The Complete Mathematical and General Navigation Tables

ZA a, to the sine of the horizontal parallactic angle V Ia; but, as this angle is equal to its opposite angle A I C, it is therefore equal to the true value of the moon's horizontal parallax, or the angle which the earth's semidiameter, A C, subtends at the moon.

Note. In the above method of finding the moon's horizontal parallax, it is indispensably necessary that the latitude of the place of observation be very strictly established; and that the meridional altitude be determined with the most rigid degree of exactness by means of a sextant and an artificial horizon; both of which must be perfectly free from errors; or the value of the errors, if any, correctly known. If these precautions be attended to, the moon's horizontal parallax may be very accurately obtained; for, since the moon at the instant of her transit over the meridian has no parallax in right ascension, but all in declination; and since the circle of declination, at that moment, is in the plane of the meridian which cuts the horizon at right angles, the parallax can only affect the meridional zenith distance in a vertical manner, by making it more than the truth; and therefore the difference between the apparent and the true zenith distances must be the correct parallactic angle.

The most proper time for determining the lunar parallax, is when the moon's declination is at its maximum; because, then, that luminary, like the sun at the solstices, seems to make a momentary pause before returning towards the equinoctial. At that moment, and no other, her parallel of declination is duly bisected by the meridian of the place of observation; and therefore the agency of her parallax must depress her apparent place in the plane of the observer's meridian in a true vertical manner.

Example.

November 6th, 1834, in latitude 51:30:49 North, and nearly under the meridian of Greenwich, the moon passed the meridian at 4:49:36: mean time; at which moment her correct declination was 24:28:37" South, and her true semidiameter 15.36"; required the moon's parallactic angle, and her horizontal parallax?

The altitude of the moon's lower limb, taken by means of a good sextant and an artificial horizon, at the moment of transit on the given day, was 25:46:20"; the half of which, or 12:53:10% = the correct

observed altitude of the moon's lower limb. Let this be reduced to the apparent central altitude in the following manner :

Observed altitude of moon's lower limb =
Moon's semidiameter at time of transit
Augmentation of semidiameter, Table IV. =.

Refraction corresponding to ditto in Table VIII.=.

Apparent altitude of the moon's centre

12:53:10%

+15.36

+ 0. 3

Which, therefore, is the value of the moon's parallax in altitude, or the true measure of her meridional parallactic angle. This being known, the horizontal parallax is to be deduced therefrom in conformity with the fourth section of the rule.-As thus :

As moon's apparent zenith distance 76:55:12", Co-secant = 0.011416

Hence, the moon's horizontal parallax is 57:15"; which differs but one second from that given in the Nautical Almanac.

Remarks.-Were the sun not so very remote from the earth, its horizontal parallax could be determined with as much ease as that of the moon; but since the sun is, at a mean rate, about 400 times farther from us than the moon is, its horizontal parallax, or the difference between its true and apparent places in the heavens as seen by two observers at the same instant, the one on the surface and the other at the centre of the earth, becomes so exceedingly acute as not to be easily determined with any degree of accuracy.

The only correct method of finding the sun's horizontal parallax is by means of the transits of Venus over the face of the sun; these, however, are but very rarely seen; for they can only happen when Venus is between the earth and the sun, and when the earth is in the same right line with one of her nodes: and she cannot be in this favourable position with respect to the earth, but after the long intervals of 105, 235, or 243 years. The last transit was in the year 1769; and there cannot be another till the year

1874. However, from a careful comparison of a great number of observations of the transits which took place in the years 1761 and 1769, it has been found that the mean horizontal parallax of the sun is 858. There is every reason to believe that this is the correct value of the solar parallactic angle reduced to the horizon; because the observations were made in so many different parts of the world, and by men so highly renowned in the annals of science that no reasonable doubts can be entertained of its correctness. The transits were observed at London, at Hackney, and at Liskeard in Cornwall; at Paris, at Stockholm, and at Hernosand in Sweden; at Tobolsk in Siberia; at the Cape of Good Hope; at Madras, and at Calcutta in the East Indies; and yet, in all those places, so widely distant from each other, there was such a remarkable coincidence in the results of the observations as to justify astronomers in coming to the conclusion that the mean value of the sun's horizontal parallax is not more than 8′′58; which is the measure of the angle that the earth's semidiameter of 3958 miles, subtends at the sun.

Now, the moon's horizontal parallax being duly established, as stated in page 33, her absolute distance from the earth may be readily determined; for, in the right angled triangle C A I (diagram page 30), the earth's semidiameter A C is given = 3958. 75 miles, and the opposite angle AIC the moon's horizontal parallax 57'15", to find the side CI = the moon's distance from the centre of the earth. Hence, by plane trigonometry:

As moon's horizontal parallax 57:15", sine comp. arith.
Is to the earth's semidiameter-3958. 75, logarithm
So is radius

90 degrees-sine

To the moon's true distance = 237726 miles logarithm = 5.3760766 But since the moon's horizontal parallax varies from about 53:55" to 61.25%, the moon's distance from the earth, found as above, will decrease or increase according as her horizontal parallax may be greater or less than 57.15%; and hence it is that the distance thus determined is 1034 miles greater than that which is established in page 9.

Note. Because the distances of the sun and planets are in the inverse ratio of the sines of their horizontal parallaxes; therefore, since the solar and lunar parallactic angles are given, and that the moon's distance from the earth is known, the sun's distance from the earth may be readily computed in the following manner, viz. :-

14.3816667

8.2214815

As the sun's horizontal parallax=8′′58 sine comp. arith..
Is to the moon's horizontal parallax =57:15% sine
So is the moon's distance from the earth 237726. 5 miles log.= 5.3760766

To the sun's distance from the earth-95328955 miles log.-7.9792248

If the sun's horizontal parallax be assumed at 865, its distance from the earth will be 94546196 miles: and thus the small difference of 0:07 in the horizontal parallax will produce a difference of 782759 miles in the distance.

TABLE XVII.

Equation of Second Difference.

Since the moon's longitude and latitude require to be strictly determined on various astronomical occasions, 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, 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 and longitude, as deduced from the preceding Table, or obtained by even proportion, will be seen in the solution of the following

PROBLEM.

To reduce the Moon's Latitude and Longitude, as given in the Nautical Almanac, to any given Time under a known Meridian.

Rule.

Turn the longitude into time (by Table I.), and apply it to the mean 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 and latitudes, 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 at the noon or midnight preceding the Greenwich time; but to be applied by addition or subtraction to her latitude, according as it may be increasing or decreasing.

Example.

Required the moon's correct longitude and latitude, August 2nd, 1824, at 310 mean time, in longitude 60:30: west of the meridian of Greenwich?