radius as the co-sine of the hypothenuse BC to the co-sine of the other side BA. Q. E. D.

PROP. XXII. FIG. 14.

IN right angled spherical triangles, the co-sine of either of the sides is to the radius, as the co-sine of the angle opposite to that side is to the sine of the other angle.

The same construction remaining; in the triangle CEF, the sine of the hypothenuse CF is to the radius as the sine of the side EF is to the sine of the angle ECF opposite to it; that is, in the triangle ABC, the co-sine of the side CA is to the radius, as the co-sine of the angle ABC opposite to it, is to the sine of the other angle. Q. E. D.

3 S

Fig. 15.

OF THE CIRCULAR PARTS.

IN any right angled spherical triangle ABC, the complement of the hypothenuse, the complements of the angles, and the two sides are called The circular parts of the triangle, as if it were following each other in a circular order, from whatever part we begin: thus, if we begin at the complement of the hypothenuse, and proceed towards the side BA, the parts following in order will be the complement of the hypothenuse, the complement of the angle B, the side BA, the side AC, (for the right angle at A is not reckoned among the parts), and, lastly, the complement of the angle C. And thus at whatever part we begin, if any three of these five be taken, they either will be all contiguous or adjacent, or one of them will not be contiguous to either of the other two: in the first case, the part which is between the other two is called the Middle part, and the other two are called Adjacent extremes. In the second case, the part which is not contiguous to either of the other two is called the Middle part, and the other two Opposite extremes. For example, if the three parts be the complement of the hypothenuse BC, the complement of the angle B, and the side BA; since these three are contiguous to each other, the complement of the angle B will be the middle part, and the complement of the hypothenuse BC and the side BA will be adjacent extremes but if the complement of the hypothenuse BC, and the sides BA, AC be taken; since the complement of the hypo. thenuse is not adjacent to either of the sides, viz. on account of the complements of the two angles B and C intervening between it and the sides, the complement of the hypothenuse BC will be the middle part, and the sides, BA, AC opposite extremes. The most acute and ingenious Baron Napier, the inventor of Logarithms, contrived the two following rules concerning these parts, by means of which all the cases of right angled spherical triangles are resolved with the greatest ease.

RULE I.

The rectangle contained by the radius and the sine of the middle part, is equal to the rectangle contained by the tangents of the adjacent parts.

RULE II.

The rectangle contained by the radius, and the sine of the middle part is equal to the rectangle contained by the co-sines of the opposite parts.

These rules are demonstrated in the following manner:

First, Let either of the sides, as BA, be the middle part, and Fig. 16. therefore the complement of the angle B, and the side AC will be adjacent extremes. And by Cor. 2. prop. 17. of this, S, BA is to the Co-T, B as T, AC is to the radius, and therefore RXS, BA=Co-T, B×T, AC.

The same side BA being the middle part, the complement of the hypothenuse, and the complement of the angle C, are opposite extremes; and by prop. 18. S, BC is to the radius, as S, BA to S, C; therefore RXS, BA=S, BCXS, C.

Secondly, Let the complement of one of the angles, as B, be the middle part, and the complement of the hypothenuse, and the side BA will be adjacent extremes: and by Cor. prop. 20. Co-S, B is to Co-T, BČ, as T, BA is to the radius, and therefore RxCo-S, B=Co-T, BCXT, BA.

Again, Let the complement of the angle B be the middle part, and the complement of the angle C, and the side AC will be opposite extremes: and by prop. 22. Co-S, AC is to the radius, as Co-S, B is to S, C: and therefore RxCo-S, B⇒Co-S AC×S, C.

Thirdly, Let the complement of the hypothenuse be the middle part, and the complements of the angles B, C, will be adjacent extremes: but by Cor. 2. prop. 19. Co-S, BC is to Co-T, B as Co-T, B to the radius: therefore RxCo-S, BC=Co-T CXCo-T, C.

Again, Let the complement of the hypothenuse be the middle part, and the sides AB, AC will be opposite extremes: but by prop. 21. Co-S, AC is to the radius, Co-S, BC to Co-S, BA; therefore RxCo-S, BC=Co-S, BAXCO-S, AC. Q. E. D.

SOLUTION OF THE SIXTEEN GASES OF RIGHT ANGLED SPHE RICAL TRIANGLES.

GENERAL PROPOSITION.

IN a right angled spherical triangle, of the three sides and three angles, any two being given, besides the right angle, the other three may be found.

In the following Table the solutions are derived from the preceding propositions. It is obvious that the same solutions may be derived from Baron Napier's two rules above demonstrated, which, as they are easily remembered, are commonly used in practice.

..

Case Given |So't

2

3

AC,C B

AC, BC

B, CAC

R: Co-S, AC :: S, C : Co-S, B and B is of the same species with CA, by 22. and 13. Co-S, AC: R:: Co-S, B: S, C: by 22. {S, C : Co-S, B :: R: Co-S, AC : by 22. and AC is of the same species with B. 13. R: Co-S, BA :: Co-S, AC: Co-S, BC. 21. and if both BA, AC be greater or less 4 BA, AC BC than a quadrant, BC will be less than a quadrant. But if they be of different affection, BC will be greater than a quadrant. 14.

Co-S, BA: R:: Co-S, BC: Co-S, AC. 21. 5 BA, BC AC and if BC be greater or less than a quadrant, BA, AC will be of different or the same affection: by 15.

S, BA: R:: T, CA: T, B. 17. and B is

6 BA, AC B of the same affection with AC. 13.

Casel Given So't]

་

7

8

9

10

BA, BAC

R: S, BA :: T, B: T, AC, 17. And AC is of the same affection with B. 13. AC, B BAT, B: R:: T, CA : S, BA. 17.

R: Co-S, C:: T, BC : T, CA. 20. If BC BC, CAC be less or greater than a quadrant, C and B will be of the same or different affecItion. 15. 13.

Co-S, CR:: T, AC: T, BC. 20. And AC, C BC BC is less or greater than a quadrant, according as C and AC or C and B are of the same or different affection. 14. 1.

T, BC: R:: T, CA: Co-S, C. 20. If BC 11 BC, CA C be less or greater than a quadrant, CA and AB, and therefore CA and C, are of the same or different affection. 15.

12

BC, B AC

R: S, BC :: S, B : S, AC. 18. And AC lis of the same affection with B.

13 AC, BBC S, B: S, AC :: R: S, BC : 18. BS, BC: R: :S, AC : S, B : 18. And B is jof the same affection with AC.

14 BC, AC

15

16

B, C

BC, C

[T, C : R : : Co-T, B: Co-S, BC. 19. And BC according as the angles B and C are of different or the same affection, BC will be greater or less than a quadrant. 14.

R: Co-S, BC::T,C: Co-T,B. 19. If BC B be less or greater than a quadrant, C and B will be of the same or different affection. 15.