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Note. In these expressions, although the denomimators are negative, the whole fractions under the radical are always positive. The expressions for the tangents and cosines are omitted, to save room, 2. Given A, B, c; to find c. Here, by applying in like manner, the equations (12), (13), to the supplemental triangle, we shall have cot q = cosc tan B.... (23. from which the subsidiary angle 4 may be determined; and thence cos c = ****=2.... (24.) . sin p 3. Given B, C, c. to find A. Find q from equa. (23); then from equa. (24) there results, - _ cos C sin op sin (A - 4) - Tcos IT

from which A is known.

.... (25.)

SECTION IV.

On the Analogies of Napier.

36. These are four simple and elegant formulae discovered by the celebrated inventor of logarithms, of which two serve to determine any two angles of a spherical triangle, by means of the two opposite sides and their included angle; while the other two serve to determine any two sides, by means of the opposite angles and their contained side. Thus, therefore, they together with equa. (1), will serve for the solution of all the cases of oblique spherical triangles. The investigation of these analogies may be given, as below.

If from the first of equa. (2), cos c be exterminated, there will result, after a little reduction,

cos A sin c = cos a sin à — cos c sin a cos ū:

and by a simple permutation of letters, cos B sin c = cos b sin a – cos c sin b cosa: adding these equations together, and reducing, we have sin c (cos A+ cos B) = (1 — cos c) sin (a + b). Now we have from equa. (1) sin a sin b sinc

- sin A sin B sinc Freeing these equations from their denominators, and respectively adding and subtracting them, there results, sin c (sin A + sin B) = sinc §. a + sin b), and sin c (sin A — sin B) = sin c (sin a – sin b). Dividing each of these two equations by the preceding, there will be obtained, sin A + sin B sinc sin a + sin b sin (a + b) sin A — sin B sinc sin a – sin b. cos A+ cosh T 1 — cos c sin (a + b)

Comparing these with the equations in arts. 24, 25, 26, chap. iv. we shall have

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2dly. That § (a + b) and 3 (A + B) are always of the same affection. 3dly. That the difference of two sides is always less than i80°. 4thly. That (a b) and (A – *} have always the same sign; whence it follows, that the greatest angle is opposite to the greatest side, and reciprocally. To these it may be added, 5thly. That the least angle is opposite to the least side, and the mean angle, to the mean side. One or other of these observations will serve to remove the ambiguity in the doubtful cases, where either a, b, and B, or A, B, and b, are given. 38. We may now collect the most commodious theorems, and present in one place all that will be usually required in the solution of oblique angled spherical triangles. sin A — sin B sinc sin a T sin 5 of sinc' I A - sin # (a + b c.) sin #(a + c —b) 2. tan #A = V#: + c a) sin # (a + b + c) sin #(b + c a) sin #(a + b c)

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3. tan B =

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39. As it is, obviously, difficult to retain in recolleetion the necessary rules and formulae in spherical trigonometry, attempts have been made by different mathematicians, to assist the student by contrivances akin to those which occur in repositories of artificial memory.

Napier, to whom this department of science is so much indebted, at the end of his Mirifici Canonis Constructio, obscurely suggested a simple and comprehensive method, characterised by the name of Napier’s Rules for the Circular Parts; which apply, 1st, to right angled spherical triangles; 2dly, by means of the polar triangle, to quadrantal triangles; and, 3dly, by means of a perpendicular from the vertical angle, to oblique angled spherical triangles. These rules were developed much more perspicuously by Gellibrand; and have, since his time, been explained by almost every writer on spherical trigonometry.

40. In a spherical triangle ABC, right angled at A, we have, as was shown in arts. 25–31, of this chapter,

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