A

the theorem may be put under this form, viz.

sin S. sin S - a. cosec b. cosec c

cos2

Whence cos

And by a like process, similar formulæ may be deduced for

and cos

C

But cos b c = cos b. cos c + sin b. sin c, therefore

And by a like process, similar formulæ may be deduced for

We may now, for the convenience of reference, collect and arrange the formula which have been demonstrated in this proposition and the preceding one.

If A, B, C measure the angles of a spherical triangle, and a, b, c be the sides respectively opposite those angles, then if S be pút for a+b+c we have

2

sin

=33

(6)

Formulæ 2, 5, and 6, may be thus expressed logarithmically; the first only of each class is put down, as the others, being perfectly analogous, will present no difficulty,

log sin Slog sin S — a + log cosec b + log cosec c — 20

In any spherical triangle A B C (see figure to Prop. 16.) if CD be a perpendicular drawn from the angle C to the opposite side A B, or A B

Let B C = a, A C = b, B D = m, and A D = n; then (Prop. 17.) cos a cos b:: cos m : cos n, therefore (Theo. 72. Geo.) cos a + cos b: cos a - eos :: cos m + cos n: cos m

cos n. But

because rectangles of equal altitudes are to each other as their bases,

proposition are equal, the second and fourth are also equal, or tan

m + n

Cor. 1. The above equation, converted into an analogy, gives us

Cor. 2. When the perpendicular C D falls within the triangle BD + A D = A B, and when C D falls without the triangle B D – A D AB BC+AC = A B, therefore in the first case we have tan : tan

2

2

Cor 3. By this proposition, an oblique angled spherical triangle, whose sides are given, may be divided into two right angled spherical triangles, in each of which the hypothenuse and a side will be given : and hence the angles of the triangles may be determined without reference to the formulæ in the scholium to the last proposition.

CONSTRUCTION AND USE OF THE MARINER'S SCALE.

THE scale commonly used by mariners is two feet in length, and on one side of it are drawn several scales of equal parts, lines of natural sines, tangents, chords, &c.; and on the other side scales of the logarithmic relations of the numbers representing these lines.

The line marked Rum, or Rhumbs, exhibits the relative lengths of the chords to every point and quarter-point of the mariner's compass, or to every thirty-second part of the quadrant.