2 cos. (a+a+d)cos.{(a+a'~dicos.Acos.A
cos. a cos. a' cos. (A + A').

or calling the first term within the brackets 2 cos.2F,

−1}cos. (A+A');

cos. D=(2 cos.2 F — 1) cos. (A + A′)= cos. 2 F cos. (A + A') . . (2).

The formulas marked (1) and (2) are both of them convenient for the computation of D; a third formula may be obtained from (1), as follows. Subtract each side of (1) from 1; then since (p. 55,)

1 -cos. D = 2 sin.2 D, 1+ cos. (A + A')=2 cos.2 (A + A'),

or, calling the second term within the brackets sin.20,

We shall solve an example by each of these formulas.

EXAMPLES.

1. Suppose the apparent distance between the centres of the sun and moon to be 83° 57′ 33′′, the apparent altitude of the moon's centre 27° 34′ 5′′, the apparent altitude of the sun's centre 48° 27′ 32′′, the true altitude of the moon's centre 28° 20′ 48′′, and the true altitude of the sun's centre 48° 26′ 49′′; then we have

d = 83° 57′ 33′′, a = 27° 34′ 5′′, a′ = 48° 27' 32"

A = 28° 20′ 48′′, A′ = 48° 26′ 49′′;

and the computation for D, by the first formula, is as follows:

(Reject 40 from index) Ī∙5369260 = log. ·3442921 +

AA 76 47 37

nat. cos. 2284595

True distance 83° 20′ 54′′

nat. cos. 1158326.

By glancing at the formula (1), we see that 30 must be rejected from the sum of the above column of logarithms, so that the logarithmic line resulting from the process is 9.5369260. Now, as in the table of log. sines, log. cosines, &c., the radius is supposed to be 1010, of which the log. is 10, and, in the table of natural sines, cosines, &c., the rad. is 1,

of which the log. is 0; it follows that when we wish to find, by help of a table of the logarithms of numbers, the natural trigonometrical line corresponding to any logarithmic one, we must diminish this latter by 10, and enter the table with the remainder. Hence the sum of the foregoing column of logarithms must be diminished by 40, and the remainder will be truly the logarithm of the natural number represented by the first term in the second member of the equation (1). If this natural number be less than nat. cos. (A+A'), which is to be subtracted from it, the remainder will be negative, in which case D will be obtuse.

is to be subtracted from

In adding up the logarithms to find cos. F, 20 must be rejected from the index; and the logarithm marked that marked +. Moreover, if A+ A' and 2F are both acute or both obtuse, D will be acute, otherwise it will be obtuse.

We shall now exhibit the process by Borda's formula.

An estimate may now be formed of the relative advantages of these three methods, as regards practical facility. We are inclined to prefer the first method, which we believe is new, as fewer references to the tables are requisite, and as moreover there are no arithmetical operations required, besides those which are actually exhibited. The second and third methods seem to offer nearly equal advantages; in the first of these, however, it may be observed that the trigonometrical lines involved are all of one name, viz. cosines, and that the final reference to the tables gives the true distance instead of its half, as in the last method.

Each of the foregoing processses may be shortened by using a subsidiary table, containing the various values of the expression cos. A cos. A' Such a table computed to every degree of the moon's

cos, a cos. a

apparent altitude, and to every 10 seconds of her horizontal parallax, forms Table IX. of the Requisite Tables, published by order of the Commissioners of Longitude. But a more complete table of this kind is given in the second volume of Dr. Mackay's work, on the Longitude. If each number in this table were increased by the constant number 3010300, the table itself would become somewhat simplified, and the process of clearing the distance by our first method would be rendered remarkably short and convenient.

The preceding example is taken from Woodhouse's Astronomy, part II., p. 859, where the day of observation is stated to be June 5, 1793. Now by the Nautical Almanack, for that year, we have

Distance at 15h 83° 6' 1", Also at time of observation D = 83°20′ 55′′ at 18h 84 28 26 at 15h D83 6 1

Increase in 3h 1 22 25, Increase between 151 and time of obs. 0 14 54

.*. 1° 22' 25": 14' 54" : 3h: 32m 33s.

Hence, when the observation was made, the apparent time at Greenwich was 15h 32m 33s.

To find the time at the ship, requires that we know the latitude of the place and the sun's declination. The former, therefore, must have been previously ascertained, and the latter may now be found by means of the apparent Greenwich time just deduced, and the Nautical Almanack. We shall suppose the latitude to be 10° 16′ 40′′ S.; the sun's declination will be 23° 22′ 28′′, and taking the true altitude of the sun = 48° 46′ 49′′, we shall thus have, in order to find the time, three sides of a spherical triangle to find an angle. The computation is as follows.