half the fum, of AB and AC: let the radii OD and OC be drawn, and alfo the chord CB, meeting OE in E and OA (produced) in P; draw ES parallel to AO, meeting CH in S, and EF and OK perpendicular to AO, and let the latter meet EC (produced) in I; laftly, draw QDK perpendicular to OD, meeting OA, OC and OI (produced) in Q, Land K.

Because CD BD, it is manifeft that OD is not only perpendicular to the chord BC, but bifects it in E; whence, alfo, EF bifects HG; and therefore CH+BG2EF, and CH-BG 2CS; alfo OG + OH = 2OF, and OGOH= 2HF : but 2EF (CH+BG): 2CS (CH-BG) :: EF: CS: EP: EC (by 14. 4.):: DQ (the tangent of AD): DL (the tangent of DC, by 20. 4.) And 2OF (OG+OH): 2HF (OG-OH): : EI: EC :: DK (the co-tang. of AD): DL (the tang. DC). Q, E. D.

THEOREM VI.

In any fpherical triangle ABC it will be, as the co-tangent of balf the fum of the two fides is to the tangent of half their difference, fo is the co-tangent of balf the bafe to the tangent of the distance (DE) of the perpendicular from the middle of the bafe.

DEMON

-co-fine BC:: co-fine

Since co-fine AC: co-fine BC :: co-fine AD: co-fine BD (by Cor. to Theor. 2.) therefore, by compofition and divifion, co-fine AC+ co-fine BC: co-fine AC AD + co-fine BD: co

fine AD co-fine BD, But (by the preceding
lemma) co-fine AC+co-fine BC co-fine AC-

AC+BC.

AC-BC

co-fine BC co-tang.

: tang.

;

and co-fine AD + co-fine BD: co-fine AD

2

con-fine BD :: co-tang. of AE (AD+BD):
DE (AD-BD);

tang.

2); whence, by equality, co-tang.

AC+BC. AC-BC

2

tang.

:: co-tang. AE: tang.

2

DE.

AC-BC. tang. DE, and it is proved, in p. 13.

2

that the tangents of any two arches are, inversely,
as their, co-tangents; it follows, therefore, that

tang. DE; or, that the tangent of half the bafe, is to the tangent of half the fum of the fides, as the tangent of half the difference of the fides, to the

tangent

{ *

tangent of the distance of the perpendicular from the middle of the bafe.

THEOREM VII.

In any spherical triangle ABC, it will be, as the co-tangent of half the fum of the angles at the bafe, is to the tangent of half their difference, fo is the tangent of half the vertical angle, to the tangent of the angle which the perpendicular CD makes with the line CF bifecting the vertical angle. (See the preceding figure.)

DEMONSTRATION.

It will be (by Corol. to Theor. 3.) co-fine A: cofine B: fine ACD: fine BCD; and therefore, cofine A+ co fine B: co-fine A-co-fine B:: fine ACD+fine BCD: fine ACD-fine BCD. But B-A

(by the lemma) co-tang.

B+A

(co-fine A+co-fine B: co-fine A-co-fine B:: fine ACD+fine BCD : fine ACD-fine BCD) tang. ACF tang. DCF. 2. E. D.

The folution of the cafes of right-angled spherical

triangles.

The oppo-As radius: fine hyp. AC :: fine A fine BC (by the formier part of Theor. 1.)

Given

The hyp IAC and one angle A

fite leg

BC

The hyp.
AC

The oppo-As radius: fine A:: cofite angle fine of AB co-fine of C (by Theor. 3.)

C

9

adjacent

Theor. I.)

angle A

an

(b)

Cafe,

The hyp.
AC

An angle,
fuppofe A

AB

As fin. A : fin. BC: : radius fine AC (by Theor. 1.)

As radius co-fine AB :: co-fine BC: co-fine AC (by Theor. 2.)

¡As fine AB : radius :: tang. BC: tang. A (by Theor. 4.)

As fin. A co fine C :: radius co-fine AB (by Theor.

13.)

As tang. A: co-tang. C:: radius: co-fine AC (by Thear. 5.)

Note, The 10th, 11th, and 12th cafes are ambiguous; fince it cannot be determined, by the data, whether AB, C, and AC, be greater or less than 90 degrees each.