be the Extremes Conjunct. Now by Theorem 28, we
have
CSC: R::CA:t BC.
Then by Alternation we have cs C:t CA:: RtBC.
But (as in Part 1, was fhewn) R: tBC:: ctBC: R;
Wherefore 'twill be
CSC:tCA: ct BC: R;
R:tCF::t CA:s GH.

That is, as

Cafe 2. Let the Complement of the Hypothenufe

2. E. D.

B C,

B, and C,

be the Middle Part; then will the Complements of

the Angles

be the Extremes Conjunct.
Then in the Triangle
(by Theorem 27.) it is
Whence by Alternations
But it is, as
Therefore it will be

DC F,

SCF:R::t DF:tC;
CF:t DF:: R:t C;
R:tC::ct C: R;
SCF:1DF:: ctC: R.
2. E. D.

AB,

Cafe 3. Let the Middle Part be and then the Complement of the Angles B, and AC, shall be the Extremes Conjunct.

Then (by Theorem 27.) it is sAB:R::tCA: tB; Whence by Alternation, sAB:tCA::(R:t B::)

And confequently

ctB: R; R:ctB::tCA:s AB. 2, E. D.

THEOREM XLI.

As Radius, to the Co-fine of one Extreme Difjunct, the Co-fine of the other Extreme Disjunct, the Sine of the Middle Part.

Demonftration.

Cafe 1. Let the Complement of the Angle C, be the Middle Part; then shall

AB,

and

and the Complement of the Angle

B,

be the Extremes Disjunct, In the Triangle D C F Then by (Theorem 25.) we have, csCsD:: csDF: R.

are the Extremes Disjunct; but csBA:cs BC:: R

by Theorem 26; whence, R:cs AC::cs BA:s C F

Cafe 3. Let the Middle Part be

BC, AB, AC,

:cs AC,

=cs BC.

2. E. D.

AB;

BC,

C.

then the Complement of the Hypothenufe and the Complement of the Angle

are the Extremes Disjunct. In the Triangle GHD, we have (by Theorem 25.) csD:s G::csGH: R ; But it is CSDS BA, and s Gcs CF; Therefore it will be R:cs GH::esC F:s BA

2. E. D.

VOL. II.

CHAP.

i

CHAP. VIII.

Of the Cafes, Analogies, and Methods of folving Right-angled Spherical Triangles.

I

fhall here purfue the fame new Method, of Stating the true Nature and Number of the Cafes of Right-angled Spherical Triangles, as I proposed, in Part 1, of Right-angled Plain Triangles; and as I thereby reduced the uncertain Number of Cafes, of Right-angled plain Triangles, to one certain Number, Six; fo a like Effect wil be produced here; for I fhall make it appear, that notwithstanding all Authors, that I have feen on this Subject, make fixteen Cafes of Right-angled Spherical Triangles, there be in reality no more than ten different Cafes; in order to which I fhall again obferve,

That a Cafe of refolving any Triangle, is the having juft fo many of its Parts given or known, as is fufficient for finding and discovering those which are unknown; and fo the Quantity of every Side and Angle in the faid Triangle, may be readily known and understood. And of fuch Cafes there can be no more than ten, in a Right-angled Spherical Triangle; for there are but five Parts unknown, viz. The Hypothenufe, the Angle at Base, the Angle at Perpendicular, and the two Legs; any two of which being given, with the Right-angle, the Reft may be known. But the Combinations of two Quantities is five are but

in

ten,