cos. (a+A)+cos. (ds)+cos. (a+a,+s)+cos. (a+a,—s): transposing cos. (z+2,) and subtracting each term from 1 we have

Or in tabular versines (see the author's Trigonometry,

p. 30.)

.. tab. ver. x = tab. ver. (z + z) + tab. ver. (d + ▲)

+ tab. ver. (d — ▲) + tab. ver. (a + a, + ▲)

+ tab. ver. (a + a,

The auxiliary angle A is found in the Nautical Tables of Inman, Riddle, Norie, and others.

The student will be able to determine the relative value of the two methods, by working an example by each.

EXAMPLE.

Required the true distance of the moon from the sun,

35 47 24

And apparent distance of centres

First method. By the common rules of trigonometry.

1. To find angle z, in triangle s, z M,.

... True distance 35° 59' 14".

Second method. The true distance found by versines, the auxiliary angle (taken from the table) being 60° 25′ 16′′.

Sum

Apparent distance

Aux. angle (A)

In practice it is not necessary to take from the table of versines more than the last five figures, rejecting also all but these last five in the sum, since the true distance will be always either in the same column with the apparent distance or the adjacent one. Thus, taking the preceding example, it may be worked thus:

Hence this rule for clearing the distance by means of an auxiliary angle.

Rule XLIX.

To clear the lunar distance.

1. Under the sun's or star's true zenith distance put the moon's true zenith distance; take the sum which mark vers.

2. Under the apparent distance of the two centres put the auxiliary angle A; take their sum and difference, against both, which mark vers.

3. Under the sun's or star's apparent altitude put the moon's apparent altitude and take their sum; under which put the auxiliary angle A; take the sum and difference, against both which mark vers.

4. Add together the five last figures of the versines of the quantities marked vers., rejecting all but the last five in the result, which look for in the column of versines under the apparent distance, or under the adjacent one: take out the arc corresponding thereto, which will be the true distance required. See example, above.

EXAMPLES.

(186.) The apparent altitude of the moon = 50° 54′ 38′′

and the apparent distance of the two centres = 88 49 58

(188.) The sun's apparent altitude is 17° 39′ 31", the moon's apparent altitude 24° 13′ 45", the moon's zenith distance 64° 56′ 45′′, the sun's zenith distance 72° 23′ 22′′, the auxiliary angle A 60° 12′ 33′′, and the apparent distance 111° 20′ 45′′; required the true distance.

Ans., 110° 56' 0".

(189.) The sun's apparent altitude is 54° 47' 4", the moon's apparent altitude 21° 20′ 1′′ the moon's zenith distance 67° 51' 5", the sun's zenith distance 35° 13′ 32", the auxiliary angle A 60° 10' 44", and the apparent distance 71° 16′ 44′′; required the true distance.

Ans., 70° 38′ 5′′.

(190.) The sun's apparent altitude is 12° 19' 30", the moon's apparent altitude 20° 40′ 18′′, the moon's zenith distance 68° 28′ 19′′, the sun's zenith distance 77° 44′ 42′′, the auxiliary angle A. 60° 10′ 53′′, and the apparent distance 124° 44′ 32′′; required the true distance.

Ans., 124° 19' 11".

(191.) The sun's apparent altitude is 57° 53′ 52", the moon's apparent altitude 35° 3′ 2′′, the moon's zenith

distance 54° 11′ 56′′, the sun's zenith distance 32° 6′ 40′′, the auxiliary angle A 60° 17′ 54′′, and the apparent distance 65° 34′ 42′′; required the true distance.

Ans., 64° 58′ 10′′.

(192.) The sun's apparent altitude is 15° 43′ 48′′, the moon's apparent altitude 16° 5' 5", the moon's zenith distance 73° 1' 32", the sun's zenith distance 74° 19′ 28′′, the auxiliary angle A 60° 8′ 36′′, and the apparent distance 119° 44′ 31′′; required the true distance.

Ans., 119° 19' 51". (193.) The apparent altitude of a star is 20° 13′ 26′′, the moon's apparent altitude 31° 17′ 22′′, the star's zenith distance 69° 49′ 11′′, the moon's zenith distance 57° 57′ 44′′, the auxiliary angle A 60° 15′ 21′′, and the apparent distance 72° 42′ 16′′; required the true distance.

Ans., 72° 33′ 4′′.

(194.) The apparent altitude of a star is 29° 59′ 16′′, the moon's apparent altitude 32° 30′ 10′′, the star's zenith distance 60° 2' 24, the moon's zenith distance 56° 41′ 33′′, the auxiliary angle A 60° 17′ 23′′, and the apparent distance 58° 44′ 19′′, required the true distance.

Ans., 58° 30′ 21′′.

Rule L.

To find the longitude by lunar observations.

Objects observed, sun and moon.

Altitudes taken. Ship

mean time determined from sun's altitude.

1. Get a Greenwich date.

2. Take from the Nautical Almanac and correct for Greenwich date the following quantities:

Sun's declination and semidiameter.

Equation of time (noting whether it is to be added to or subtracted from the ship apparent time).

Moon's semidiameter and horizontal parallax.