altitudes, as in the preceding examples. Moreover, as the ship most probably sails during the interval of the observation, an additional reduction becomes necessary; the first altitude must be reduced to what it would have been if taken at the place where the second was taken this correction will be known if we know the number of minutes or miles which the ship has made, directly towards or directly from the sun, since leaving the place where the first observation was made, To find this, take the angle included between the ship's course and the sun's bearing, at the first observation; and considering this angle as a course, and the distance sailed as the corresponding distance, find by the traverse table, or by the operation of plane sailing, the difference of latitude, which will be the amount of the approach to, or departure from, the This must be added to the first altitude if the angle is less than 90°, because the ship will have approached towards the sun; but it must be subtracted when the angle exceeds 90°. If the angle is 90°, no correction for the ship's change of place will be necessary, since in that case sailing in a perpendicular to the direction of the sun, she maintains always the same distance from him,

sun.

Where great accuracy is aimed at, account should be taken of the ship's change of longitude during the interval of the observations; when converted into time it must be added to the interval of time between the observations when the ship has sailed eastward, and subtracted when she has sailed westward. This correction is very easily applied.

Having thus mentioned the necessary preparative corrections, we shall now give an example of the trigonometrical operation.

EXAMPLES.

Let the two zenith distances corrected be (see last fig.)zs= 73° 54′ 13′′, zs' = 47° 45′ 51′′, the corresponding declinations 8° 18' and 8° 15' north, and the interval of time three hours; to determine the latitude,

Considering ss' to be the base of an isosceles spherical tri

(PS + ps')*=81° or 45°, let the per

angle, of which one of the equal sides is 43' 30", and the vertical angle equal to 3 pendicular PM be drawn, then we have in the triangle PMs

45°

right angled at M, Ps=81° 43′ 30′′, and P= =22° 30';

given to find sм= ss' as follows.

2

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

sin, ss' 44° 30′ 22.8"

sin. Ps' 81 45 0

sin. SPS' 45 0

arith. comp. 0.154290

9.995482

9.849485

9.999257

0

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.

cos. (zs+Ps) 77°48′52′′ ar: comp. 0.675555 ar. comp. sin. 0.009897

*COS. (ZS PS) 7°49′17′′

9.995941 sin

.

9.133811 10.422276

11.093772 tan (z)20°12′32′′ 9.565984

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.