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the Angles ACD and ABC are equal, may be thus proved: Suppofe the two Sides AB and BD to be continued to a Semicircle at E, then will the faid Angle be equal to its oppofite at B; the Side AC by Suppofition is equal to the Side AE, the Complement of the Side AB to a Semicircle; but equal Sides fubtend equal Angles; therefore the Angle at C is equal to the Angle at B, or at E: which being admitted, retaining the Side AD and Angle at D, we have another Angle oppofite thereto, either C or B, which are equal and common to both Triangles; and fo, if the Side oppofite to the given Angle at D, were fought, a double Answer fhould be given, either the Side AC, or the other Side AB, its Complement to 180°: and the interjacent Side might be CD or BD, and the third Angle the leffer Angle CAD, or the greater BAD. So it is plain, that by only the three Terms given, these Cafes are doubtful without the affection of a fourth.

In all fuch Cafes, where the affection of the Side or Angle required is not known, it will be the best way to delineate the Triangle truly; and by that you may commonly difcern whether it be more or lefs than a Quadrant. How to delineate any Triangle by the three Terms given, fhall be fhewed in the next Chapter.

I fhall next fhew how to vary the fecond Proportions in each of the Ten firft Cafes: Which may be done by the Theorems laid down in Page 108, &c.

The first Proportions being right angled Triangles, the Variations are already fhewn.

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CASE I.

The Proportion is,

As s. BD: s. C:: s. CB: s. D. But it may be,
As fe.c. C: fe.c. BD:: s. CB: s. D

of Theor. 6th.

By the Inverfe

And As s. C: s. BD:: fe.c. CB: fe.c. D By the 5th. As fe.c. BD: fe.c, C:: fe.c. CB: fe.c. D By Theo. the 4th.

CASE II.

The Proportion is,

Ass. C: s. BD: : s. D: s. CB, But it may be,
As fe.c. BD: fe.c. C:: s. D: s. CB By the In-
verfe of Theor. 6th.

And As s. BD: s. C:: fe.c. D: fe.c. CB By the 5th.
As fe.c. C: fe.c. BD:: fe.c. D: fe.c. CB By the 4th.

CASE III.

The fecond Proportion is,

Ass. AB: ct. B:: s. AD: ct. D

Ast. B: fe.c. AB :: s. AD: ct. D By the 7th Theor. As fe.c. AB: t. B:: fe.c. AD: t. D By the 7th Theor.

CASE IV.

The fecond Proportion is,

As cs. ACB: ct. BC: : cs. ACD: ct. DC

As t. BC: fe. ACB:: cs. ACD: ct. DC By the 7th. As fe. ACB: t. BC:: fe. ACD: t. DC By the 7th.

CASE

CASE V.

The fecond Proportion is,

As ct. BC: cs. BCA :: ct. CD: cs. ACD

As fe. BCA: t. BC :: ct. CD: cs. ACD SBy the Inverse As t. BC: fe. BCA:: t. CD: fe. ACD 2 of the 7th The.

CASE VI.

The fecond Proportion is,

As cs. BC: cs. AB:: cs. CD: cs. AD

As fe. AB: fe. BC:: cs. CD: cs. AD By the 6th Theo.
As cs. AB cs. BC :: fe. CD: fe. AD By the 5th.
As fe. BC: fe. AB:: fe. CD: fe. AD By the 4th.

CASE VII.

The fecond Proportion is,

As ct. B: s. AB:: ct. D: s. AD

As fe.c. AB: t. B:: ct, D: s. AD By the 7th Theor: As t. B: fe.c. AB :: t. D: fe.c. AD By the 7th.

CASE VIII.

The fecond Proportion is,

As cs. B: s. ACB:: cs. D: s. ACD

-As fe.c, ACB: fe. B:: cs. D: s. ACD By the 6th. As s. ACB cs. B:: fe. D: fe.c. ACD By the 5th. As fe. B: fe.c. ACB:: fe. D: fe.c. ACD By the 4th.

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The fecond Proportion is,

As cs. AB cs. BC:: cs. AD: cs. CD

As fe. BC: fe. AB:: cs. AD: cs. CD By the 6th Theo.
As cs. BC: cs. AB: fe. AD: fe. CD By the sth.
As fe. AB: fe. BC :: fe. AD: fe. CD By the 4th.

CASE X.

The fecond Proportion is,

As s. ACB cs. ABC :: s. ACD: cs. D

:

As fe. ABC: fe.c. ACB :: s. ACD: cs. D By the 6th. As cs. ABC: s. ACB :: fe.c. ACD: fe. D By the sth. As fe.c. ACB: fe. ABC :: fe,c. ACD: fe. D By the 4th.

Thus have I fhewed all the Variations, and fufficiently explained all the Cafes both in right and oblique angled spherical Triangles: I fhall next fet down a Synopfis of all the Cafes in both.

A

A SYNOPSIS of the Sixteen Cafes of right angled Spherical Triangles, and their Proportions.

Note, That the middle Part in this Synopfis, whether given or fought, is noted with the Old English Letter.

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