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Co-f. CB : rad. (: : 20-f. BD : 20-f. CD) :: Co-fa 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 XT. CD AB x radius =

+

rad. co-f. AC x 20-. BC. But (by Theor. 1.) radius :

co-f. C x T.AC Co-L.C:: T. AC : T. CD =

rad. Co-. C XS. AC Co-f. AC

(by Corol. 1. Prop. 1.) which last being substituted for its equal, we shall have,

S. CA X S. CB x co-f. C CO-1. AB x rad. =

rad.

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

There is another way of demonstrating this proposition, from the orthographic projection of the sphere; but that is a subject 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 sides of a Spherical triangle ABC, and V be put -to denote the versed fine of the vertical angle, and R the radius ; then will V =

RX Ço-. AF-C0-4: AB

S. AC XS. BC
2R x CO-S. AF -- Co-S. AB

Co-f. AF-60-). AE
2R x S. AB + AF X. AB - JAF.

S. AC X S., BC

It

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It appears from the last
Prop. that co-f. AB x R is
= co-f. AC x co-f. BC +
S. AC x S. BC x co-f. C

R
in which, for co-f. C. let
its equal R-V be fub-
stituted, and then we shall
have co-fine AB x R =-

F
Co-fine AC x co-fine BC

B
+ fine. ACX fine BC - A
S. AC X S. BC XV

: but the
R
fum of the two former of the three last terms is
= co-f. AF XR (by Cor. 1. to Prop. 2.); there -
fore it will be co-f. AB XRC0-1. AF XR --
S. ACX S. BC x V

and consequently V =
R
R? x co-f. AF-CO-f. AB

Which is the first
S. AC X S. BC
cafe. Again, because S. AĆ x S. BC is = IR X
co-f. AF CO-. AE (by Corol: 3. to Prop. 2.) we

2R x co-f. AF-CO-s. AB
İhall, also, have V =

CO-L. AF - Co-f. AE
Which is the second cafe.

Moreover, since RX

AB + AF
CO-T. AF -CO-. AB is = S,

x S.

2 AB-AF

(by the fame) it follows that V is likewise
2R x S. ŽAB + AF x S. ŽAB AF.

S. AC X S, BC
Q. E. D.

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Hence, because IR x V is = the square of the fine of įC (by Prop. 1.) it follows that iq. S. ¿C= R? x S. AB + AF * S. AB - JAF

S. AC x S. BC From whence we have the following theorem, for solving the uth case of oblique triangles, where the three sides are given, to find an angle.

As the rectangle of the lines of the two sides, including the proposed angle, is to the rectangle under the fines of half the base plus half the difference of the sides, and half the base minus balf the difference of the sides; fo is the Square of radius, to the Square of the fine of half the required angle.

COROLLARY 2.

R*XC0-f.AF-CO-LAB Moreover, because Vis=

S. AC X S. BC we shall have R’: S. AC x S. BC::V: Co-f. AF - Co-f. AB; which gives the following theorem, for finding a lide, when the opposite angle, and the other two sides, are given.

As the square of radius, is to the rectangle of the fines of the two sides including the given angle; so is the versed fine of that angle, to the difference of the co-fines for versed fines) of the difference of those sides, and the fide required.

COROL

COROLLARY 3.

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2R x Co-f. AF - CO-CAB,
Lastly, because V=

co-f. AF – co-f. AE
we shall, by transforming the equation, and put-
ting W for (2R – V) the versed sine of BCE
(the supplement of the vertical angle) have co f.
AE =
2R x Co-f, AB - W X co-f. AF

and the
V

2R X Co-f. AB –V x co-f. AE
Co-fine AF =

W
From whence the sides themselves may be de-
termined, when their fum, or difference, is given,
with the base and vertical angle.

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