The expressions for tan. (A+B) and tan. (A-B), expand

2

2

ed into a proportion, are called, from their Inventor, Naper's Analogies.

The angles A and B being determined by the above forms, the side c may be determined, as it has been in the foregoing sin. c sin. b Example, from the expression : but it may be sin. C sin. B desirable, as in the corresponding case of rectilinear triangles (see page 78), to determine c immediately without the intervening process of finding the angles A and B : and, in fact, many Problems in Astronomy * require, from the data of two sides and the included angle, solely the determination of the opposite side c.

...cos. c=cos.a.cos. b + sin. a. sin.b.cos. C;

but cos. C=1- ver. sin. C,

(ver. sin. stands for versed sine);

* For instance, in finding the Moon's distance from a Star; in deducing the altitude of a Star from the latitude, declination and hourangle (two Problems useful in determining the Longitude); in deducing, in the case of an occultation, the Moon's distance from a Star; in determining the altitude of the nonagesimal (see Astronomy, p. 364.): in determining the latitude from 2 altitudes and the time between, (see Astronomy, p. 422.), &c.

Y

COS. C=COS. a. cos.b+sin.a. sin. b - sin. a. sin. b. ver sin, C

= cos. (a

6)

sin. a. sin. b. ver. sin. C;

1-cos. c, or, 2 sin.2=ver. sin. (a - b) + sin. a.sin. b. ver. sin. C

2

which in logarithms is 2 log. tan. 0 =

log. sin. a + log. sin. b + log. ver. sin. C-log. ver. sin. (a−b) [p]

then 2 sin.2 = ver. sin. (a - b). sec.20, and

2

log. 2+2 log. sin. =log. ver. sin. (a - b) + 2 log. sec. 0-10

2

Former Example.

c computed independently of A and B by the 2d Method.

Determination of the subsidiary angle & by the form [p]

[9]

This is, perhaps, the most commodious form for computing c; for, when we use it, we need not consider whether the fraction sin. a. sin. b. v. sin. C

ver. sin. (a - b)

is > or < 1, since tan. @ admits of all

degrees of magnitude. It is easy, however, to give another formula of computation, thus:

Third Method of computing c.

COS. c = cos.a.cos. b + sin. a sin. b. cos. C

=

sin. ++M). sin.(+-)

b

2

by the form [c], p. 31; and in logarithms,

b

log. sin. = ={log.sin. log.sin. (++M) M) + log. sin. (+ -)}

2

2

This is the kind of form which Laplace has employed in his Mecanique Celeste, Livre 2, page 227,

M=52 14 23.......... 9.8979455=log.sin, M=log.sin. 52° 14′23′′