COROLLARY I.

Hence, because R x V is the Square of the Sine of C (by Prop. 1.) it follows that Sq. S. + C=

R2 x S. AB+ AF ×S.

2

S. AC × S. BC

AB — AF

2

From whence we have the following Theorem, for folving the 11th Cafe of oblique Triangles, where the three Sides are given, to find an Angle.

As the Rectangle of the Sines of the two Sides, including the propofed Angle, is to the Rectangle under the Sines, of half the Bafe plus half the Difference of the Sides, and half the Bafe minus half the Difference of the Sides; fo is the Square of Radius to the Square of the Sine of half the required Angle.

COROLLARY 2.

Moreover, because V is=

R2xCo-f. AF-Co-f.AB

S. AC x S. BC

we fhall have R2: S. AC x S. BC :: V: Co-f. AF

Co-f. AB; which gives the following Theorem, for finding a Side, when the oppofite Angle, and the other two Sides, are given.

As the Square of Radius, is to the Rectangle of the Sines of the two Sides including the given Angle, fo is the verfed Sine of that Angle, to the Difference of the Co-fines (or verfed Sines) of the Difference of thofe Sides, and the Side required.

Co

COROLLARY 3.

Lastly, because V=

2 Rx Co-f. AF-Co-f. AB Co-f. AF-Co-f. AE

we fhall, by transforming the Equation, and putting W for (2 R — V) the versed Sine of BCE (the Supplement of the vertical Angle) have Co-f. 2 R x Co-f. AB-W x Co-f. AF

AE=

Co-fine AF=

V

and the

2 R x Co-f. ABV x Co-f. AE

W

From whence the Sides themfelves may be determined, when their Sum, or Difference, is given, with the Bafe and vertical Angle.

The EN D.

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