PROPOSITION XXXIX.

In any Spherical Triangle ABC, whofe Legs, containing the Angle B, are BC,AB, and Bafe fubtending the Angle A C; if the Arc AM be taken Difference of the Legs B C—A B; then all the Rectangle under the Sines of the Legs BC, BA, be to the Square of the Radius, as the Rectangle, under the Sine of the Arc AC+AM AC-HM and the Sine of the Arc

is to the Square of the Sine of one half of the Angle B. Vid. Fig. to Prop. 36.

BEcaufe the Rectangle under the Sines of the Legs

A B, BC, is to the Square of Radius, as I L is to the verfed Sine of the Angle B, or as Rx L to R drawn into the verfed Sine of the Angle B (by Props 36. of this.) And fince RxIL=Rectangle, under the AC-AM Sines of the Arcs

AC+AM

2

and

2

by

Prop. 37. of this.) And alfo R drawn into the versed Sine of the Angle B is equal to the Square of the Sine of one half of the Angle B (by Prop. 38. of this.) Therefore the Rectangle under the Sines of the Sides, to the Square of Radius, fhall be as the Rectangle AC-AM, under the Sines of the Arcs AC+AM, and

2

2

is to the Square of the Sine of one half the Angle B. W. W. D.

The

The twelve Cafes of oblique-angled Spherical Triangles are as follow:

3

Given Sought

B, D, and BC.

C.

Make, as

Angles angle R: Cof. BC:: T, B: Cot. BCA In the Ori
(by Prop. 30. of this): Alfo Cof. ginal the
Proportion
B: S, DCA:: Cof. D: S, DCA was thus ;
by 31. of this). Wherefore the Cof. BC:
Sum of the Angles BCA, DCA,

B,BCD, D.

and the

Side BC.

the Perpendicular falls within the Triangle, or the Difference, if it falls without, will be = BCD. Whether the Perpendicular falls within, or without the Triangle, may be known from the Affection of the Angles B and D (by 22. of this); which Admonition ought to be observed in the following Solutions.

R:: T, B Cot. B C A.

was as in

Angles Angle R : Cof. B C : : T, B: Cot. BCA This Pro-
(Prop. 30. of this.) And S, portion in the
BCA: S, DCA :: Cof. B: Col. wa
D(by Prop. 31.) If BCA be the forego-
Hefs than BCD, the Angle D fhall ing. The
Species of
be of the fame Affection with the the Angle
Angle B. If BCA be greater BCA may
than the Angle BCD, then the be known by
Prop. 18.
Angles B and D fhall be of a and 19.
different Affection, by the Con-
verfe of Prop. 22.

The

Sides

BD,

CD, and the

Angle B.

The R: Cof. B :: T, BC : T, BA
Side (by 28. of this.) And Cof. BC:
B D. Cof. BA:: Cof. DC: Cof. CA

(by 32. of this.) The Sum or Dif-
ference of B A and D A, accord.
ing as the Perpendicular falls
within or without the Triangle,
is equal to B D; which cannot
be known, unless the Species of
the Angle D be firft known.

Given Sought

The

Sides

BC,

6 BD,

and the Angle B.

The

Sides

BC,

DC,

and the

Angle
-B.

The

Make, as

The R Cof. B::T, BC: T, BA
Side (by 28. of this.) And Cof. BA :
CD. Cof. B C :: Cos. DA : Cof. DC
(by Prop. 32. of this.) According as
DA is fimilar or diffimilar to CA,
or to the Angle BDC, fo fhall DC
be leffer or greater than a Qua-
drant (by 19, and 20. of this.)
R: Cof. B:: T, BC: T, BA (by
28. of this.) And S, D:T, B::
S, BA: S, DA (by 33. of this.)
The Sum or Difference of BA
and D A is B D.

The

Side
B D.

Angle
D.

Angle

C.

The

BCD, DC.

R: Cof. B : : T, BC : T, Ba (by
Prop. 28. of this.) And S, DA:
S, BA:: T, B.: T, D (by 33. of
this.) According as BD is greater
or leffer than BA, the Angle D
fhall be fimilar or diffimilar to
the Angle B (by 22. of this.)
Cof. BC: R:: Cot. B: T, BCA
(by 30. of this.) And T, DC: T,
BC:: Cof. BCA: Cof. DCA
(by 34. of this.) The Sum or Dif-
ference of the Angles BCA,
D C A, according as the Per-
pendicular falls within or with-
out the Triangle, is equal to the
Angle B C D.

Cof. BC: R:: Cot. B: T,BCA Angles Side (by 30. of this.) Alfo, Cof. DCA Cof. BCA:: T, BC: T, DC (by 34. of this.) If the Angle DCA be fimilar to the Angle B, (that is, if AD be fimilar to CA), then DC fhall be less than

and B, and the

8

Side

B C.

Quadrant. If the Angles DCA and B be diffimilar, then DC fhall be greater than a Quadrant, which follows (from Prap. 18, 19, and 20. of this.

9

Given Sought

The The S, CD: S, B:: S, BC: S, D;
Sides Angle which is ambiguous. The Ana-
D. logy follows from Prop. 35. of
this.

BC, DC, and the

Side
BC.

All the

The As the Rectangle under the Sines Sides Angle of the Legs A B, BC: the Square of Radius: the Rectangle under AC+AM

A B,
BC,

11 CA.

B.

the Sines of the Arcs

2

Prop.

2

36.

All the

The
Angles Side
G, H, GD.
D. Vid.

Fig.

Prop:

14.

Sine of the Angle B. (by Prop. 39.) In the Triangle X N M, the Arc MN is the Complement of the Angle GHD to a Semicircle. XM is the Complement of the Angle G, and X N the Complement of the Angle D: And the Angle X the Complement of the Side GD to a Semicircle. Wherefore, if the Angles be changed into Sides, and the Sides into Angles, the Operation will be the fame as in Cafe 11. of this; fince Arcs, and their Complements to Semicircles, have the fame Sines.

The

The following REMARK, by

ΤΗ

SAMUEL CUNN.

HAT this is true but in a particular Cafe, viz. when two of the Angles of the Triangle are Right ones, and two of the Sides Quadrants, may be thus demonftrated: For, if poffible, let fome Triangle RST, Fig. to Prop. 14th, be fuch, that its Sides RS, ST, TR, be equal to the Measures of GH D, HGD, GDH, the Angles of a Triangle G HD; and, also, that the Measures of RST, STR, TRS, the Angles of the Triangles R S T, be equal to GH; GD, HD, the Sides of the Triangle GHD; and produce M X, M N, two Sides of the fupplemental Triangles to Semicircles, and they will meet fomewhere, fuppofe at E; and there will be conftructed thereby the Triangle N E X, of which X E (the Supplement of XM, which by the 14th Prop. was the Supplement of the Measure of the Angle HG D) is equal to the Measure itself of the fame Angle HGD: And, in like manner, N E, the Supplement of N M, which, by the 14th Prop. was the Supplement of the Measure of the Angle GHD, is equal to the Meafure itfelf of the fame Angle GHD. But the third Side XN is not the Measure of the third Angle GHD, but its Supplement, by the 14th Prop. Moreover, of the Angle EXN (whofe Supplement is N X M), the Measure, by the 14th Prop. is equal to GD; and of the Angle XNE (whofe Supplement is M N X) the Meature, by the 14th Prop. is equal to HD. But of the third NEX (which is equal to N M X) the Measure is not equal to G H, but its Supplement.

Now make N V=R T=BK, the Measure of the Angle G D H, and draw the great Circle E V. And fince RS, by Suppofition, is equal to the Measure of the Angle GHD, which is equal to EN; and fince the Measure of the Angle SRT is, by Suppofition, equal to DH, which is also equal to the Measure of the Angle X NE; the Angle XNE is equal to the Angle R. Then, confequently, by the 4th Prop. the Triangles SRT, EN V, will have the Base ST equal to the Bafe