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

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, may be worked thus:

it

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

and the auxiliary angle A

=60 24 12 Ans., 88 24 17

54° 29′ 33′′

5.25 59

83 48 29

(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

(182.) Aug. 20, 1845 at 0h 30m A.M., mean time nearly, in latitude 50° 20′ N. and longitude by account 142° 0′ E. when a chronometer showed 2h 41m 12, the observed altitude of the star a Aquila (Altair) was 36° 59′ 50′′, west of the meridian, the index correction + 6′ 30′′ and height of eye above the sea 20 feet; required the longitude. On Aug. 1 the chronometer was slow on Greenwich mean time, 17m 459.0 and its daily rate was 4s-3 losing.

Ans., 142° 14′ 15′′ E.

(183.) Sept. 10, 1844, at 4h 21m A.M., mean time nearly, in latitude 40° 36' N., and longitude by account 73° E., when a chronometer showed 11h 21m 56s the observed altitude of ß Geminorum (Pollux) was 39° 0′ 10′′ east of meridian, the index correction - 4' 10" and height of eye above the sea 20 feet, required the longitude. On Aug. 20, the chronometer was slow on Greenwich mean time 3m 198.9, and its daily rate was 9.3 gaining.

Ans., 72° 45′ 45′′ E.

(184.) January 16, 1845, at 8h Om P.M., mean time nearly, in latitude 49° 56′ 50′′ N., and longitude by account 94° 30′ W., when a chronometer showed 2h 24m 30s, the

observed altitude of a Leonis (Regulus) was 8h 4' 20", Eof meridian, the index correction - 4′ 20′′, and height of eye above the sea 25 feet, required the longitude. On January 1, the chronometer was fast on Greenwich mean time 5m 30s-5 and its daily rate was 5s-5 losing.

Ans., 94° 24′ 45′′ W. (185.) January 20, 1846, at 8h 30m P.M., mean time nearly, in latitude 50° 48′ N., and longitude by account 7° 10′ W., when a chronometer showed 8h 32m 50s the observed altitude of & Leonis was 28° 0′ 10′′ east of the meridian, the index correction 6' 20", and the height of eye above the sea 20 feet; required the longitude. On January 2, the chronometer was fast on Greenwich mean time 30m 30s and its mean daily rate was 15s-5 losing.

-

Ans., 7° 18' E.

Longitude by lunar observation.

The time at the ship is obtained by the same kind of observation as that for finding the longitude by chronometer. The time at Greenwich is found by calculating the true distance of the moon from the sun or some other heavenly body, and comparing it with the distance of the moon from the same heavenly body as recorded in the Nautical Almanac for some given time at Greenwich.

To find the true distance.

The true distance is found by clearing the observed distance of the effects of parallax and refraction, by the following or some other similar methods.