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But if AD be without the triangle, BAD-CAD=BAC, and there

fore sin (AB+AC): sin (AB-AC): :
cot (BAD+CAD) : tan BAC ; or because

'cot (BAD+CAD) : tan { BAC :: cot | BAC : tan ; (BAD+CAD), sin (AB+AC): sin (AB ---AC) :: cotBAC :: tan 1 (BAD+ÇAD). Wherefore, &c. Q. E. D.

LEMMA.

The sum of the tangents of any two arches, is to the difference of their

tangents, as the sine of the sum of the arches, to the sine of their difference.

+

COS A

cos B

sin (A + B) that is, because

Let A and B be two arches, tan A + tan B : tan A-tan B :: sin (A+B) : sin (A--B). For, by § 6. page 243, sin A Xcos B + cos A Xsin B=sin (A+B),

sin A sin B and therefore dividing all by cos A cos B,

sin A

tan A, tan A+tan B cos A x cos B'

COS A sin (A+B)

In the same manner it is proved that tan A-tan B cos A Xcos B sin (A-B) Therefore tan At tan B : tan A-tan B':

:: sin cos A Xcos B* (A+B): sin (A– B). Q. E. D.

PROP. XXXI.

The sine of half the sum of any two angles of a spherical triangle is to

the sine of half their difference, as the tangent of half the side adjacent to these angles is to the tangent of half the difference of the sides opposite to them; and the cosine of half the sum of the same angles is to the cosine of half their difference, as the tangent of half the side àdjacent to them, to the tangent of half the sum of the sides opposite.

Let C+B=2S, C-B=2D, the base BC=2B, and the difference of the segments of the base, or BD-CD=2X. Then, because (30.) sin (C+B): sin (C-B) : : tan BC : tan} (BD - CD), sin 28 : sin 2D :: tan 'B : tan X. Now, sin 28=sin (S + S)=2 sin S x cos S, (Sect. III. cor. Pl. Tr.). In the same manner, sin 2D= 2 sin DX cos D. Therefore sin S x.cos $: sin D x cos D :: tan B: tan X.

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2

But (29.)

D

B

D с Again, in the spherical triangle ABC it has been proved, that sie C+sin B : sin C-sin B : : sin AB+sin AC : sin AB-sin AC, and sine sin C + sin B=2 sin } (C+B)X cos } (C-B), (Sect. III. 7. Pl. Tr.)=2 sin S Xcos D; and sin C - sin B=2 cos (C+B) Xsin * (C-B) 2 cos $ X sin D. Therefore 2 sin S Xcos D : 2 cos SX sin D :: sin AB + sin AC : sin AB - sin AC. But (3. Pl. Tr., sin AB+sin AC : sin AB -sin AC :: tap } (AB+AC): tan (AB-AC):: tan : tan A, & being equal to 1 (AB + AC) and A to } (AB-AC.). Therefore sin SX cos D: cos SXsin D:: tan : tan A. Since then tan X sin D Xcos D

tan A

cos S X sin D and

by multiplying equals tan B

sin SX cos S tan sin S Xcos D

tan X
by equals,
tan A (sin D): X cos SXcos D

(sin D):
Х
tan B
tan Σ

(sin S)' X cos S X cos D
tan ! (BD-DC) taní (AB+AC) that is,

(sin 8)

tan X tan Σ tan (AB-AC) tan | BC

'tan Atap B' tan X

tan X X tan A and therefore,

tan X

tan A_ (tan A) as also

X
tan B
(tan B) ?

tan B tan Σ (tan B): tan X

tan A (sin D)2 (tan a)(sin D)) But X tan B“tan < (sin S)*

; whence

(tan B)* (sin s) sin D

or sin S : sin D :: tan B : tan 4, that is, sin (C+B) : sin (C-B) :: tan : BC : tan (AB-AC), which is the first part of the proposition.

tan A
cos Sxsin D

tan & sin SX cos D. Again, since tan & sin Sxcos D'or inversely

tan A tan X sin D Xcos D and since

tan B sin Sxcos g; therefore by multiplication, tan _ (cos D) tan A (cos S2

tan X tan Extan A But it was already shewn that

tan B

(tan B)3 tan E_ (tan 2)2

X tan B

tan A (tan B)2
tan X tan (cos

(cos Da)
X
tan Acos S)??

as has just been shewn.
(cos Da (tan
(tan 2)2

cos D tan »

and consequently (cos S) (tan B)?'

tan A ; and

tan B

sin s'

cos S X sin D

tan X

Х

tan B

,wherefore also

tan X

Now,

tan B

Therefore (cos D)

,or cos

cos Stan B

9:cos D :: tan B : tan E, that is cos (C+B): cos (C-B):: tan į BC : tan į (C+B); which is the second part of the proposition. Therefore, &c. Q. E. D.

Cor. 1. By applying this proposition to the triangle supplemental to ABC (11.), and by coksidering, that the sine of half the sum or half the difference of the supplements of two arches, is the same with the sine of half the sum or half the difference of the arches themselves; and that the same is true of the cosines, and of the tangents of half the sum or half the difference of the supplements of two arches; but that the tangent of half the supplement of an arch is the same with the cotangent of half the arch itself; it will follow, that the sine of half the sum of any two sides of a spherical triangle, is to the sine of half their difference as the cotangent of half the angle contained between them, to the tangent of half the difference of the angles opposite to them : and also that the cosine of half the sum of these sides, is to the cosine of half their difference, as the cotangent of half the angle contained between them, to the tangent of half the sum of the angles opposite to them.

Cor. 2. If therefore A, B, C be the three angles of a spherical triangle, a, b, c the sides opposite to them,

I. sin } (A+B): sin } (A-B) :: tan c : tan} (amb).
II. cos Į (A+B): cos (A-B) :: taníc : tan } (a+b),
III. sip 1 (a+b): sin ; (a-b) :: tan įC : tan} (A-B).
IV. cos } (a+b): cos} (0-6):: tan fC : tan* (A+B).

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PROBLEM I.

In a right angled spherical triangle, of the three sides and three angles,

any two being given, besides the right angle, to find the other three.

This problem has sixteen cases, the solutions cf which are contained in the following table, where ABC is any spherical triangle right angled at A.

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TABLE for determining the affections of the Sides and Angles found

by the preceding rules.

1

AC and B of the same affection, (14)
If BC_90°, AB and B of the same affection, otherwise dif-
different,

(Cor. 15.) If BC_90° C and B of the same affection, otherwise different,

(15.)

2

3

4

AB and C are of the same affection,

(14.) If AC and C are of same affection, BC 2900; otherwise BC 790°,

(Cor. 15.) B and AC are of the same affection,

(14.)

5 6

Ambiguous.
Ambiguous.
Ambiguous.

7 8 9

When BC 290°, AB and AC of the same ; otherwise of different affection,

(15.) | 10 AC and B of the same affection,

(14.) 11 When BC 2 90°, AC and C of the same ; otherwise of different affection,

(Cor. 15.) 12

BC 290°, when AB and AC are of the same affection,

(1. Cor. 15.) ; 13 B and AC of the same affection,

(14.) 14 C and AB of the same affection,

(14.) 14 AB and C of the same affection,

(14.) 15 AC and B of the same affection,

15 When B and C are of the same affection, BC_90°, otherwise, BC 7900,

(15.) | 16

The cases marked ambiguous are those in which the thing sought bas two values, and may either be equal to a certain angle, or to the supplement of that angle. Of these there are three, in all of which the things given are a side, and the angle opposite to it; and accordingly, it is easy to shew, that two right angled spherical triangles may always be found, that have a side and the angle opposite to it the same. in both, but of which the remaining sides, and the remaining angle of the one, are the supplements of the remaining sides and the remaining angle of the other, each of each.

Though the affection of the arch or angle found may in all the other cases be determined by the rules in the second of the preceding tables, it is of use to remark, that all these rules except two, may be reduc

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