C

nufe BC is to the radius (or the fine of the right angle at A) as the fine of the fide AC to the fine of the angle B. And, in like manner, the fine of BC is to the fine of the angle A, as the fine of AB to the fine of the angle C; wherefore (11.5.) the fine of the fide AC is to the, fine of the angle B, as the fine of AB to the fine of the angle C.

A

Secondly, Let ABC be an oblique angled triangle, the fine of any of the fides BC, will be to the fine of any of the. other two AC, as the fine of the angle A oppofite to BC, is to the fine of the angle B oppofite to AC. Through the point C, let there be drawn an arch of a great circle CD perpendicular upon AB; and in the right angled triangle BCD, (Prop. 19.) the fine of BC is to the radius, as the fine of

AD

A.

Ꭰ

B

B

CD to the fine of the angle B; and in the triangle ADC, by inverfion, the radius is to the fine of AC as the fine of the angle A to th line of DC: Therefore, ex quo pertur batè, the fine of BC is to the fiffe of AC, as the fine of the angle A to the fine of the angle B. QE. D. AT

PRO P. XXV.

N oblique angled spherical triangles, a perpendicular arch being drawn from any of the angles upon the oppofite fide, the co-fines of the angles at the bafe are proportional to the fines of the fegments of the vertical angle.

Let ABC be a triangle, and the arch CD perpendicular to the base BA; the co-fine of the angle B will be to the co-fine of the angle A, as the fine of the angle BCD to the fine of the angle ACD.

For (Prop. 23.) the co-fine of the angle B is to the fine of the angle BC, as the co-fine of the fide CD is to the radius; and alfo the co-fine of the angle D to the fine of the angle ACD in the fame ratio; therefore by permutation, the cofine of the angle B is to the co-fine of the angle A, as the fine of the angle BCD to the fine of the angle ACD. Q. E. D.

T

C

"

DA

THE fame things remaining, the co-fines of the

THE

fides BC, CA, are proportional to the co-fines of BD, DA, the fegments of the bafe.

For (Prop. 22.) the co-fine of BC is to the co-fine of BD, as the co-fine of DC to the radius, and the co-fine of AG to the co-fine of AD in the fame ratio: Wherefore, by permutation, the co-fines of the fides BC, CA are proportional to the co-fines of the fegments of the bafe BD, DA. Q.E.D.

PROP.

PRO P. XXVII.

'HE fame conftruction remaining, the fines of

procally proportional to the tangents of B and A, the angles at the bafe.

For (Prop. 18.) the fine of BD is to the radius, as the tangent of DC to the tangent of the angle B; and alfo, the radius to the fine of AD, as the tangent of A to the tangent of DC: Therefore, ex æquo perturbate, the fine of BD is to the fine of DA, as the tangent of A to the tangent of B.

THE

HE fame conftruction remaining, the co-fines of the fegments of the vertical angle are reciproprocally proportional to the tangents of the fides.

For (Prop. 20.) the co-fine of the angle BCD, is to the radius, as the tangent of CD is to the tangent of BC; and alfo, (Prop. 21. by inverfion), the radius is to the co-fine of the angle ACD, as the tangent of AC to the tangent of CD: Therefore, ex quo perturbatè, the co-fine of the angle BCD is to the co-fine of the angle ACD, as the tangent of AC is to the tangent of BC, Q. E. D.

PRO P.. XXIX.

from an angle of a spherical triangle there be drawn a perpendicular to the oppofite fide, or bafe, the rectangle contained by the tangents of half the fum, and of half the difference of the feg ments of the bafe is equal to the rectangle contained by the tangents of half the fum, and of half the difference of the two fides of the triangle.

Let ABC be a fpherical triangle, and let the arch CD be drawn from the anglé C at right angles to the bafe AB. Because (Prop.26.), cof. BC: cof. AC:: cof. BD: cof. AD, by compofition and divifion,

cof. AC:: cof.BD + cof. AD: Cor. 3. Pl. Trig.),

cof. AC: cot. (BC+AC):

tan. (BC-AC); and alfo, cof. BD+ cof. AD: cof. BD— cof. AD :: cot. (BD+AD): tan. (BD-AD), therefore cot. 1 (BC+AC): tan. 4 (BC—AC) :: cot. 2 (BD+AD): tan. (BD-AD). And because rectangles of the fame altias their bafes, therefore, tan. (BC+AC) ×

cot. (BC+AC); tan. (BC+AC) x tan. (BC-AC):: 44/444

tan. ±(BD+AD) × cot. 1 (BD+AD) : tan. (BD+AD) × tan. — (BD—AD). But tan. (BC+AC) × cot. ÷ (BC+AC) =R2, (2. Car. def. 9. Pl. Tr.) and also, tan. (BD+AD) ×

cot. (BD+AD) = R2, therefore (9. 5.) tan. (BD+AD) x

2

tan. — (BD—AD) = tan. — (BC+AC) × tan. ÷ (BC—AC). Q.E. D.

COR

COR. 1. Because the fides of equal rectangles are reciprocally proportional, tan. (BD+AD): tan. (BC+AC) :: tan. (BC-AC): tan. (BD-AD).

COR. 2. Since, when the perpendicular CD falls within the triangle, BD+AD AB, the bafe; and when CD falls without the triangle BD-AD AB, therefore in the firft cafe, the proportion in the laft corollary becomes, tan. (AB)= tan. (BC+AC) :: tan. (BC-AC): tan. (BD-AD); and in the fecond cafe, it becomes by inverfion and alternation, tan. (AB): tan. (BC+AC) :: tan. (BC—AC): tan. (BD+AD.

C

C

BD

B

A

SOLUTION