Add together the co-secant of arc the first, the co-secant of the least zenith distance, (rejecting the indices,) the sine of the half sum, and the sine of the remainder; half this sum is the logarithmical cosine of half the third arc, double of which gives arc the third. The difference between arc the third and arc the second gives arc the fourth.

IV. Add the sine of the polar distance, the sine of the least zenith distance, and double the sine of half the fourth arc into one sum, and reject 30 from the index. Consider the remainder as a common logarithm, and find the natural number answering to it, to as many places of figures as the tables extend to, if the index be 9; if 8, to one less; if 7, to two less; and if 6, to three less (for this index can never exceed 9, nor fall short of 6). Subtract double this natural number from the natural cosine of the difference between the polar distance and the least zenith distance, the remainder will be the natural sine of the true latitude.

EXAMPLE 111.

In a supposed latitude of 50°.41′ N. at 10h.17.30′′ per watch, the true altitude of the sun's centre was 17°.13', and at 11.17.30" it was 19°.41', the sun's declination on that day was 20° South; required the true latitude independent of the supposed one?

Here 11.17.30"-10b.17.30"-1h the elapsed time, hence the elapsed time=30m. =7°.30'.

* 1. When the supposed latitude, or latitude by account, is less than, or nearly equal to, the declination, and both the same way, the magnetic azimuths of the sun ought to be attended to at the time of taking the altitudes. For when the magnetic azimuth at taking the least altitude is greater than the magnetic azimuth at taking the greatest altitude, counting from the elevated pole; the supplement of arc the second must be added to the supplement of arc the third to obtain arc the fourth. 2. Should the double natural number used near the conclusion of the problem, at any time exceed the natural cosine of the difference between the least zenith distance and polar distance, the latitude will be of a different name with the supposed latitude. 3. Hence it appears that in places near the equator where the azimuths increase and decrease slowly, the above method of solution requires a little more attention. However, should any doubt remain whether the latitude found by the general rule be true or not: find the fourth arc as directed in the first note, and use it instead of that found by the rule; one of the two results will most certainly be the true latitude, and some circumstances will generally occur that will seldom fail to point out which is the proper one.

Note. This example, by the approximating rules, requires a repetition of the process, vide page 18. Requisite Tables, Example 2.

EXAMPLE IV.

In a supposed latitude of 51°.0' North, when the sun's declination was 22°.23′ South, some time past noon the sun's corrected altitude was found by observation to be 14°.46' and 1h.22' afterwards, his altitude was 8°.27′; required the true latitude independent of the supposed one?

Answer. The true latitude is 50°.33'.44′′ N.

EXAMPLE V.

In a supposed latitude of 47°.19′' North, when the sun's declination was 12°.16′ North, some time in the afternoon the true altitude of the sun's centre was observed to be 51°.59', and 2b.50' afterward it was 49°.9'; required the true latitude independent of the supposed one?

Answer. The true latitude is 49°.19.30" North.

EXAMPLE VI.

In a supposed latitude of 30°.0' North, when the sun's declination was 10°.24′ North, at 8b.54' in the forenoon the sun's true altitude was 41°.30′; and at 1.24′ in the afternoon his altitude was 61°.47'; required the true latitude independent of the supposed one?

Answer. The latitude is 31°.33'.26" North.

EXAMPLE VII.

In a supposed latitude of 40°.0′ North, when the sun's declination was 2°46′ South, at 9b.20' in the forenoon the sun's altitude corrected was 33°.11', and at 1.20′ in the afternoon his altitude was 42°.44′; required the true latitude independent of the supposed one?

Answer. The latitude is 40°.50′.10′′ North.

EXAMPLE VIII.

In a supposed latitude of 50°.40' North, when the sun's declination was 20° South, the true altitude of the sun's centre was 19°.41', and one hour afterward his altitude was 17°.13′; required the true latitude independent of the supposed one? Answer. The true latitude is 50° North.

EXAMPLE IX.

In a supposed latitude of 60° North, when the sun was on the equinoctial, or had no declination, the true altitude of his centre was 28°.53', and two hours afterwards his corrected altitude was 20°.42′; required the true latitude independent of the supposed one?

Answer. The true latitude is 59°.59′.32′′.

In this example the polar distance is equal to 90°; and in all such examples, the elapsed time will be arc the first, and arc the second will be 90°.

EXAMPLE X.

In North latitude when the sun's declination was 23°.29' North, at 8h.54' in the forenoon the sun's corrected altitude was 48°.42′, and at 9b.46' the altitude was 55°.48'; required the true latitude? British Palladium 1773, page 72. Answer. The true latitude is 49°.49'.28" North.

EXAMPLE XI.

In a supposed latitude of 6°.50' North, at 7h.30' in the forenoon the true altitude of the sun's centre was 22°.30', and at 10b.36'.40" his altitude was 63°.40', the sun's declination being 22°.48' North; required the true latitude, without using the supposed one?

Here 10h.36'.40′′-7h.30′-3h.6′.40" the elapsed time, hence half the elapsed time=11.33′.20′′=23°.20′.

This latitude being so very different from the latitude by account, it will be necessary to examine it by the notes to the rule.

There can be no error, except in the fourth arc, and this is examined with very little additional trouble, there being only three numbers to take out of the tables:

Arc the second= 80°.30′.37′′ its supplement=99°.29'.23" Arc the third =155°.33'.20" its supplement 24°.26′.40"

One of the two latitudes found above is certainly the true one, the latter, being the nearest to the latitude by account, may be taken as the proper one.

EXAMPLE XII.

In latitude 1°.50' North, by account at 10h.24' in the forenoon the true altitude of the sun's centre was 64°.59′, and at 11. 20' it was 78°.57', the sun's declination being 0°.30' North;