Considering ss' to be the base of an isosceles spherical triangle, of which one of the equal sides is (PS + Ps')*=81° 43' 30", and the vertical angle equal to 3 or 45°, let the perpendicular PM be drawn, then we have in the triangle PMS

450

right angled at м, Ps=81° 43′ 30′′, and p= =22° 30';

given to find SM= ss' as follows.

2

1. To find ss' from the triangle PMS.

II. To find Pss' from the triangle ess'.

This angle is acute like its opposite side, (see Art. 128.)

Which we may, without sensible error, where the base is so small.

↑ The proportion employed here is that of Art. 38; в is understood as the first term of the proportion.

IV. To find the two unknown angles of the triangle zsp.

Upon the same principles may the latitude be determined from the altitudes of two fixed stars, taken at the same time; in this case s, s', in the preceding figure, will represent the two stars: Ps, Ps', their known polar distances, and the angle SPS', the difference of their right ascensions; the same quantities are therefore given as in the case of the sun, but, as in the case of two stars Ps, Ps', may differ very considerably; ss' cannot be considered as the base of an isosceles triangle, but must be computed from the other two sides and their included angle. In the Nautical Almanac for 1825, Dr. Brinkley has computed for 1822, and tabulated, the distances SS' for certain pairs of stars, conveniently situated for observation, and has annexed the change of distance corresponding to 10 years. The same table shows also the difference of right ascension for each pair of stars, with the change in 10 years; so that by help of this table the computation for finding the latitude from the simultaneous altitudes of two fixed stars becomes considerably abridged.

For other methods of determining the latitude, the student may consult " Mackay on the Longitude,” Vol. 1., and Captain Kater's Nautical Astronomy, in the Ency. Metropolitana, &c.

On finding the Longitude by the Lunar Observations.

113. There are several astronomical methods of determining the longitude of a place which cannot be accurately employed at sea, on account of the great difficulty of managing a telescope on shipboard; we shall not, therefore, enter here into any explanation of these methods, but shall confine ourselves to the lunar method of determining the longitude, which is justly regarded as the principal problem in Nautical Astronomy. Before entering upon the solution of this problem, it will be necessary to make a few introductory remarks.

The determination of the longitude of a place always requires the solution of these to problems, viz.: 1st, to determine the time at the place at any instant; and, 2d, to determine the time at the first meridian, or that from which the

longitude is estimated, at the same instant; for the difference of the times converted into degrees, at the rate of 15° to an hour, will obviously give the longitude.

When the latitude of the place is known, (and it may be found by the methods already explained,) the time may be computed from the altitude of any celestial object whose declination is known; for the coaltitude, codeclination, and colatitude, will be three sides of a spherical triangle given to find the hour angle, comprised between the codeclination and the colatitude. But to find the time at Greenwich requires the aid of additional data, besides those furnished by observations made at the place. The Greenwich time may, indeed, be obtained at once, independently of any observations at the place, by means of a chronometer, carefully regulated to Greenwich time, provided it be subject to no irregularities after having been once properly adjusted. A ship furnished with such a timepiece always carries the Greenwich time with her*, and the longitude then becomes reduced to the problem of finding the time at the place. Chronometers are now brought to such a state of perfection that very great dependence can be placed on them, and they are accordingly always taken out on long voyages for the purpose of showing the Greenwich time, and are thus of great use to the mariner. Still, however, as the most perfect contrivance of human art is subject to accident, and the more delicate the machine the more liable is it to disarrangement, from causes which we may not be able to control, it becomes highly desirable, in so important a matter as finding the place of a ship at sea, to be possessed of methods altogether beyond the influence of terrestrial vicissitudes, and such methods the celestial motions alone can supply.

The angular motion of the moon in her orbit is more rapid than that of any other celestial body, and sufficiently great to render the portion of its path passed over in so short a

* As chronometers show mean time, the equation of time must be applied to obtain the apparent time at Greenwich.

time as two or three seconds, a measurable quantity even with a small portable instrument (the sextant.)*

It is obvious, therefore, that if the distance of the moon's centre from any celestial body, in or near her path, be computed for any Greenwich time, and this distance be found the same as that given by actual observation at any place, then the difference between the time of observing the phenomenon and the time at Greenwich, when it was predicted to happen, will give the longitude of the place of observation. Now in the Nautical Almanac the distances of the moon from the sun, and from several of the fixed stars near her path, are given for every three hours of apparent Greenwich time, and for several years to come; and the Greenwich time, corresponding to any intermediate distance, is obtainable by simple proportion with all requisite accuracy; so that by means of the Nautical Almanac we may always determine the time at Greenwich when any distance observed at sea was taken.

The distances inserted in the Nautical Almanac are the true angular distances between the centres of the bodies, the observer being considered as at the centre of the earth, and to the true distance therefore every observed distance must be reduced; it is this reduction which constitutes the trigonometrical difficulties of the problem; and it consists in clearing the lunar distance from the effects of parallax and refraction; how to do this it is now our business to explain:

* The sextant is constructed upon the same principles as the quadrant. It is usually of brass, is made to hold in the hand, whereas the quadrant is suspended at the centre. It measures 120°, having an arc equal to the circumference, from which unlike the quadrant it takes its name. The angular distance of two heavenly bodies apart is obtained by making the reflected image of one coincide with the other as seen directly.