The hypotenuse is acute, because the legs are alike.

The angle A is acute, being of the same affection with

the opposite leg.

The angle C is acute, being like the opposite leg.

This analogy is like the second.

EXAMPLE. 30' 5", and C

CASE 6.

Given the two angles, namely, A=40° 63° 58' 30"; to find the rest.

*The log. of the 4th term, in this and the third analogy, gives more than half a second beside 55° 8', on account of the fraction of a second, neglected in other parts of the triangle.

The hypotenuse is acute, the angles being of the same

kind.

2. To find the leg AB.

RX cos. C sin. A X cos. AB.

40° 30' 5" ar. co. o'1874433

Sin. A

The leg AB is acute, like the opposite angle:

3. To find the other leg BC.

RX cos. A sin. C X cos. BC.

63° 58' 30" ar. co. o'0464323

Sin. C : R

90

The leg BC is acute, like the opposite angle.

This analogy is like the second.

RECTILATERAL SPHERICAL

TRIGONOMETRY.

If the supplemental triangle to a rectangular spherical triangle be drawn, by Theor. IV. General Properties of Spherical Triangles, one side will be a quadrant, being the supplement of a right angle; and therefore it will be a

quadrantal

Quadrant.

B

quadrantal or rectilateral spherical triangle. And as the sine, cosine and tangent of an arc are the same as the sine, cosine and tangent of its supplement respectively, the equations for the rectangular are all equally applicable to its supplemental rectilateral triangle. Hence a rectilateral spherical triangle ABC may be solved like a rectangular one by the method of five circular parts. The quadrantal side, called the quadrant, or radius, like the right angle, is not considered as separating the adjacent parts. The two angles adjacent to the quadrant, the complement of the other angle, opposite to the quadrant and called the hypotenusal angle, and the complements of the other two sides are to be regarded as the five circular parts.

Thus, in the triangle ABC, AB is the quadrant, or radius, the angle A, the complement of AC, the complement of the hypotenusal angle C, the complement of CB and the angle B are the circular parts.

The ambiguous cases are the same as in rectangular triangles, that is, when one side and its opposite angle are given.

OBLIQUE

SPHERICAL TRIG

ONOMETRY.

THEOREM I.

In a spherical triangle, if the angles at the base be of the same affection, the perpendicular, falling on the base

VOL. II.

Trt

from