=

Arch fourth 7702 ar-co-log. 1.11340 arch first

log. 8.32284 arch sec.

5° 6' sine, 8.94887 7 co-s. 9.99664

Arch third 12° 13′ sine, 9.32553 arch sev. 8821

log. 8.94551

In the preceding problems for computing the apparent time from an observation of the altitude of a celestial object, the Earth was supposed to be spherical, and the computations were performed agreeable to that hypothesis. But because the figure of the Earth is that of an oblate spheroid, the latitude of a place, as inferred directly from observation, is, therefore, greater than the angle at the Earth's center, contained between the equatoreal radius, and a line joining the center of the Earth and the place of observation. This reduction of latitude, according to Sir Isaac Newton's hypothesis, is contained in Table XXXVI.

As the altitude of a celestial object is observed from the visible horizon, whose pole is the apparent zenith; therefore, in order to make an allowance for the spheroidal figure of the Earth, a correction must be applied to the observed altitude, to refer it to the reduced horizon. Table XXVI. contains this reduction. Since the reduced zenith is always more distant from the elevated pole than the apparent zenith, the above correction is, therefore, additive, when the azimuth reckoned from the meridian is less than 90°, but substractive, when it exceeds that quantity.

In order to illustrate the above, the following example of the computation of the apparent time, from an observation of the Sun's altitude, is inserted.

June 24, 1802, in latitude 57° 8'.9 N. and long. 2° 8' W. about 3h. P. M. the altitude of the Sun's lower limb was 42° 24', and height of the eye 13 feet. Required the apparent time of observation?

Of the Methods of clearing the Apparent Distance between the Moon and the Sun, or a fixed Star, from the Effects of Refraction and Parallax,

SINCE the observed altitude of a celestial object is affected by two physical causes, the refraction and parallax, whose effects are produced in a vertical direction; it is, therefore, obvious, that the observed distance between any two objects will be also affected by these causes. Indeed, with regard to the fixed stars, the parallax vanishes; and, therefore, these objects are affected by refraction only. But in observations of the Moon particularly, the effect of parallax is very sensible, upon account of its proximity to the Earth. Therefore, by reason of the above causes, the true distance between the Moon and any celestial object is, for the most part, considerably different from

that observed.

Let Z (fig. 24) represent the zenith, s the apparent place of the

Sun,

Sun, and m that of the Moon; the arch sm will, therefore, be the apparent distance between these objects. Also ZS, ZM, being vertical circles, passing through the centers of the Sun and Moon, the true and apparent places of these objects will be found therein.

Now, since the refraction is ever greater than the parallax of the Sun at the same altitude, the true place of the Sun will, therefore, be lower than the apparent place, which let be S; and because the Moon's parallax, at any given altitude, is greater than the refraction at that altitude, its true place will, therefore, be higher than the apparent place, which let be M; hence, SM will be the true distance.

The method of reducing the apparent to the true distance, or, in other words, that of clearing the apparent distance from the effects of refraction and parallax, being the most tedious part of the calculus for ascertaining the longitude, when the calculation is performed by the common spherical análogies; many eminent astronomers and mathematicians have, therefore, given compendiums to facilitate the solution of this problem; among which are those by the Chevalier de Borda, the Abbé de la Caille*, Messrs. Delambre, Dunthorne, Elliot, Emerson, Jeaurat, Krafft, De la Lande, Legendre, Lyons, Maskelyne, Robertson, Romme, Witchel, Vince, &c.; but the largest and most elaborate work, that has hitherto appeared for the purpose of correcting the apparent distance, is the Cambridge Tables +. These tables were calculated by Messrs. Lyons, Parkinson, and Williams, under the inspection of Dr.Shepherd, Plumian Professor of Astronomy at Cambridge, by the rule formerly given by Mr. Lyons, in the first edition of the Requisite Tables.

All the methods that have hitherto been given, for the purpose of reducing the apparent to the true distance, depend on one or other of the two following principles; of which, the first appears to be the most simple and accurate.

FIRST GENERAL PRINCIPLE.

With the apparent zenith distances Zm, Zs, (fig. 24) and the apparent distance between the objects ms, compute the vertical angle mZs: with which, and the true zenith distances, ZS, ZM, the true distance Zm, may be found.

SECOND GENERAL PRINCIPLE.

Let Z (fig. 25) be the zenith; Zm, Zs, the apparent zenith distances of the Moon and star, and sm the apparent distance. Let Ss, mn, be the refractions ip altitude of these objects respectively.

Mr. James Ferguson's Parallactic Rotula is constructed on the same principles as the Abbe de la Caille's Chassis de Reduction.

+These tables are adapted to give the correction answering to the distance between the Moon and a star only; therefore, when the true distance between the Sun and the Moon are required, it becomes necessary to apply another correction depending on the Sun's parallax. This correction may be taken from Tables v. and v1. supplemental of the first edition of the Requisite Tables. Mr. Margett's Longitude Tables are deduced from the Cambridge Tables, Join

Join Sn, and draw sa, mb perpendicular thereto; and Sa, nb will be the effects of refraction; which, therefore, being applied to the apparent distance, sm will give Sn, the distance corrected by refraction.

Again, let nM be the parallax of the Moon in altitude; then SM being joined, will be the true distance. From M draw Mc perpendicular to Sn, and cn will be the principal effect of parallax in distance; which, being applied to the distance corrected by refraction, will give the arch Sc. Now, in the right-angled spherical triangle ScM, Sc and Mc being given, SM may be found, or rather the difference between SM and Sc may be computed, which, being applied to Sc, will give the true distance SM.

If the object with which the Moon is compared be the Sun, another correction depending on the parallax of that object is necessary.This correction may be computed on the same principles as the effect of the Moon's parallax.

The first of the following methods is, perhaps, as easy a solution of this problem as has hitherto appeared, especially when the table of natural versed sines is used, whieh accompanies this work. The second is another solution, which will be found extremely easy, by using the table of log. sines, now inserted in the second volume of this edition; and the third and fourth methods are given, merely because they may be performed entirely by natural versed sines. These methods depend on the first general principle. The other methods are deduced from the second general principle.

PROBLEM I.

The Apparent Distance between the Moon and the Sun, or a fixed Star, together with the Altitude of each being given, to find the True Distance.

METHOD I

Of reducing the Apparent to the True Distance.

RULE.

Take the correction of the Moon's altitude from Table IX. to which add the correction* of the Sun's altitude, or the refraction of the star. Now, this sum added to, or subtracted from, the difference of the apparent altitudes, according as the Moon is higher or lower than the Sun or star, will give the difference of the true altitudes.

From the natural versed sine of the observed distance, subtract the natural versed sine of the difference of the apparent altitudes, and to the log. of the remainder add the log. from Table XLII. answering to the Moon's apparent altitude and horizontal parallax, corrected by the number from Table XLIII. or XLIV. according as the distance between

The difference between the refraction and parallax of the Sun in altitude. The refraction is contained in Table vr. and the Sun's parallax in Table vii.

the

the Moon and the Sun, or a fixed star, is observed. Now, the natural number answering to the sum of these two logs. being added to the natural versed sine of the difference of the two altitudes, will give the natural versed sine of the true distance*.

EXAMPLES.
I.

Let the apparent distance between the centers of the Sun and Moon be 81° 23′ 38′′, the apparent altitude of the Sun 27° 43′, the apparent altitude of the Moon 48° 22′, and the Moon's horizontal parallax 58'45". Required the true distance?

Let the apparent distance between the centers of the Sun and Moon be 72° 21' 40", the apparent aititude of the Moon 19° 19, that of the Sun 25° 16', and the Moon's horizontal parallax 56' 32". Required the true central distance?

Let the apparent distance between the centers of the Sun and Moon be 96° 19′25′′, the apparent altitude of the Sun's center 8° 37′, that of the Moon's 5° 30, and the horizontal parallax 56′ 20′′. Required the true distance?"

*Mr. Keith, in his Trigonometry, page 297, says, "Dr. Mackay's first method, page 112 of his Treatise, (first edition), is the simplest I ever met with, where his Tables are used."