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co-f. CB rad. (:: co-f. BD: co-f. CD) :: co-f AB: co-f. AC (by Corol. to Theor. 2.) whence, by multiplying means and extremes, we have co-f. S. CB x co-f. AC × T. CD

AB × radius

rad.

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co-f. AC x co-f. BC. But (by Theor. 1.) radius :

co-f. CT. AC: T. CD =

co-f. Cx T. AC

rad.

(by Corol. 1. Prop. 1.) which laft

co-f. C x S. AC

co-f. AC

co-f. AB x rad. —

being fubftituted for its equal, we shall have,
S. CA x S. CB x co-f. C

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co-f. AC x co-f. BC; from whence, if each term be multiplied by radius, the truth of the propofition will appear manifeft.

There is another way of demonftrating this propofition, from the orthographic projection of the fphere; but that is a fubject which neither room, nor inclination, will permit me to treat of

here.

PROP. XXVII.

If AE be the fum, and AF the difference, of the two fides of a spherical triangle ABC, and V be put to denote the verfed fine of the vertical angle, and R R* x J. AF - AB

the radius; then will V =

S. AC × S. BC

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fum of the two former of the three last terms is

co-f. AF x R (by Cor. 1. fore it will be co-f. AB x R S. AC x S. BC x V

R

to Prop. 2.); thereco-f. AF x R

and

confequently V =

Which is the firft

Rx co-f. AF-co-f. AB

S. ACS. BC

cafe. Again, because S. AC x S. BC is R x co-f. AF-co-f. AE (by Corol. 3. to Prop. 2.) we 2Rx cof. AF-co-f. AB

fhall, alfo, have V =

Which is the fecond cafe.

co-f. AF co-f. AE
Moreover, fince R ×
AB+ AF
X S.

co-f. AF co-f. AB is = S.

AB-AF

2

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2

-(by the fame) it follows that V is likewife

2R x S. AB+ AF x S. AB

S. AC x S. BC

AF.

2. E. D.

COROL

COROLLARY I.

Hence, because Rx V is the fquare of the fine of C (by Prop. 1.) it follows that fq. S. C= Rx S. AB+ AF x S. AB-AF

S. AC x S. BC

From whence we have the following theorem, for folving the 11th cafe of oblique triangles, where the three fides are given, to find an angle.

As the rectangle of the fines of the two fides, including the propofed angle, is to the rectangle under the fines of half the bafe plus half the difference of the fides, and half the base minus half the difference of the fides; fo is the fquare of radius, to the fquare of the fine of half the required angle.

COROLLARY 2.

Moreover,because Vis=

R'xco-f.AF-co-LAB S. ACX S. BC we fhall have R: S. AC x S. BC:: V: co-f. AF

co-f. AB; which gives the following theorem, for finding a fide, when the oppofite angle, and the other two fides, are given.

As the fquare of radius, is to the rectangle of the fines of the two fides including the given angle; fo is the verfed fine of that angle, to the difference of the co-fines (or verfed fines) of the difference of those fides, and the fide required.

COROL

COROLLARY 3.

Laftly, because V=

2R x co-f. AF-co-f AB, co-f. AF-co-f. AE

we fhall, by transforming the equation, and putting W for (2R — V) the verfed fine of BCE (the fupplement of the vertical angle) have cof. 2R x co-f. ABW x co-f. AF

AE=

co-fine AF =

V

, and the

2R x co-f. AB-V x co-f. AE

W

From whence the fides themselves may be determined, when their fum, or difference, is given, with the bafe and vertical angle.

The END.

LONDON: Printed by LUKE HANSARD,

N° 6, Great Turnstile, Lincoln's-Inn-Fields.

1799.

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