PROPOSITION II

Given the polar and zenith distances of a heavenly body, and the zenith distance of the pole [= colatitude], to find the time.

The hour, found approximately by the longitude of the vessel and the chronometer, regulated to Greenwich time, is to be corrected by observation.

Let A be the pole, B the position of the heavenly body, and C the zenith of the place; then the hour angle A will be found by (540) or by (541), the first being preferable when A is small.

Fig. 135.

(614)

Example. In the preceding example, after sailing 3 hours due west, the zenith distance of the sun's centre was found to be 47° 43′ 51′3′′. Required the hour angle, A, or the time referred to the meridian of the place arrived at.

Therefore the ship had sailed 20 92 miles.

Scholium I. The time of setting of a heavenly body, or, rather, the hour angle when the body is 90° distant from the zenith, will be better found by (569), since we have

where it is to be remembered that the declination will become minus when measured in a direction opposite to the elevated pole. Ex. When did the sun set to the place in the last example but one? Ans. 6h. 42m. 38s.

19s. for change of dec. Scholium II. The time of the apparent setting of a heav- (616) enly body will be found by substituting for a, the apparent zenith distance of the horizon, which will generally differ from 90°. Scholium III. The hour when twilight ends will be found (617) by putting a = 90° +18° 108, since the sun is 18° below the horizon at that time.

=

Ex. When did twilight end in the above example?

Ans. 7h. 3m. 27s.

Scholium IV. The azimuth, C, or bearing of a heavenly (618) body, when its zenith distance, polar distance, and either the hour angle or the zenith distance of the pole, are known, will be found by (551), or by (540).

Ex. What was the bearing of the sun at setting in the above? Ans. 73° 40′ 14′′.

How might the variation of the magnetic needle be determined? Scholium V. When on shore, the magnetic variation will be determined, with greater accuracy, by ascertaining the greatest east

ern or western elongation of a circumpolar star, i. e., its azimuth when B = 90°. We have (568)

COSC= sinb sin C, or log. sin C = log. sinc-log. sinb. (619) For this purpose the pole star is preferable. Its polar distance on the 1st of January, 1847, was 1° 30′ 22.85", and decreases 19.273" annually.

The time of greatest elongation will be found by taking the difference of the right ascensions of the sun and star from the Nautical Almanac. The elongation will be easily observed by the aid of any altitude and azimuth instrument, carefully levelled, as a theodolite-or, if no such instrument be at hand, by suspending a long plumb line, with its weight swimming in a pail of water to prevent agitation by the wind, and, at a suitable distance south of the line, placing upon a table, a piece of board carrying a sight vane, by the motion of which, east or west, the required point will be obtained when the star appears no longer to depart from the line. Next alline a lighted candle at a considerable distance north-ten or a dozen rods--which is to be blown out and left till morning, when it may be observed with a common compass, or the true meridian may be traced by calculating a triangle, and constructing it with a rod graduated for measuring lengths. In making the above observation it will be necessary to illumine the plumb-line in the direction of the star.

Scholium VI. When the zenith distance, polar distance, and either the hour angle or the azimuth, of a heavenly body are given, the latitude may be found by combining (535) with (383 ... 4); for we obtain

Having determined by the aid of a theodolite, on the same or different evenings, the altitudes and azimuths of the same or different stars, the mean of several computations by (621) will give the latitude of a place on land with a considerable degree of accuracy. The same observations will also make known the hour

The error that is likely to be committed in the hour angle, A, the chronometer not indicating the exact time, may be corrected by trial, from couple of observations, the first made soon after

the meridian passage and the other several hours later. It is to be observed that a change of sign in the error of A will correspond to a change of sign in the error of b, and that A is to be varied till two values of b are found answering to the difference of latitude made by the vessel between the observations.

Scholium VII. THE METHOD OF DOUBLE ALTITUDES has for its object to determine the latitude and time from two zenith distances, either of the same body taken a few hours apart, or of different bodies taken at the same time,

Let CB be the first zenith distance of the sun, and C, the zenith arrived at when the second C2B2 is taken; the number of miles the ship sails gives the number of minutes in the arc CC, and the angle, BCC2 = C, is determined by the course and the azimuth of B at the first observation.

Therefore, in the triangle, BCC, to find c, we have

COSC = cosb cosc,+ sinb sinc, cos C ;

Fig. 1352.

but, as b is small and c differs little from c2, putting c = c, d, and expanding by (317), reserving only the first and second powers of d and b, the reduced equation becomes,

or, better,

d2-2 tanc, d = b2 - 2b tanc2 cos C,

c2- c=d=b cos C,

(622)

(623)

when, as in the present case, the second powers of the small quan

tities, d, b, may be rejected.

There are now three steps in the operation:

(624)

1o. In the triangle, BAB, we have the two sides, AB, AB2, and the included angle A = the time elapsed between the observations, or to the difference of right ascensions, if two stars are observed at the same time, to find the remaining parts;

2o. Therefore the angle, CB,B, becomes known from the three sides of the triangle, CB,B;

3°. And, lastly, in the triangle ABC, we have the two sides, BA, BC, and the included angle, B, to find the colatitude, AC2, and the hour angle, CAB.

Note 1. When the declination changes but little between (625) the observations, the first part of the operation may be shortened by regarding the triangle, BAB, as isosceles, taking instead of AB or AB,, (AB+ AB2) for the equal side and solving by (570).

Note 2. In any of the preceding operations, when a star, (626)

instead of the sun, is employed, the hour angle will be converted into degrees by observing that a sidereal day is equal to 23h. 56m. 4'0906s.; whence it follows that

1 solar hour = 1'002738 sidereal h. = 15° 04107.

Note 3. Before taking out any quantity from the Nautical (627) Almanac, or the Connaissances des Tems, the time of the place where the observation was made must be reduced to the meridian and denomination of time specified in the ephemeris. "Equations" (as these numbers are technically called) for reducing apparent to mean time, will be found in the ephemeris itself.

Example. On the first day of August, 1847, at 10 o'clock 43m. 25s. P. M., apparent time, and in 20° W lon., it is required to find the moon's declination.

11h. 3m. 25s. + 6m. 1 s. =11h. 9m. 26s.

6° 27′ 13'6" + 1′ 40·8′′ = 6° 28' 54'4" N.Dec. Ans.

Example. At the time specified in the example preceding the last, the ship tacks and sails N 43° W, nine and a half miles an hour for two hours, when the zenith distance of the sun's centre is found to be 37° 55' 40". Required the latitude of the place arrived

at and the exact time.

PROPOSITION III.

The Lunar Method for the Longitude consists in find- (628) ing by the ephemeris the time corresponding to the geocentric distance of the moon from a heavenly body. The problem requires for its solution the apparent angle which these bodies make with each other, as well as their apparent and geocentric zenith distances.

Let A, B, C, be the places of the zenith, moon, and some other heavenly body, as the sun, as seen from the earth's centre. On account of parallax

and refraction, to be explained hereafter, the moon

Fig. 136.

B

will appear at B2, below B and in the same vertical, and the second body will be seen in C2, vertically above C; hence BC, will be the apparent distance of the two luminaries,