SPHERICAL TRIGONOMETRY, SPHERICAL TRIANGLE, SPHERICS. We shall confine ourselves in the present article to such a collection of the properties of a spherical triangle as may be useful for reference, referring for demonstration to the treatise ou the subject in the Library of Useful Knowledge,' and to that on Geometry ; adding to the former nothing but a shorter mode of obtaining Napier's Analogies.
By a spherical triangle is meant that portion of the sphere which is cut off by three arcs of great circles, each of which cuts the ether two as A B C. It is now usual, however, to consider the spherical triangle as a sort of representative of the solid angle formed at the centre of the sphere by the planes A o B, n o c, c o A, as follows :—The arcs A B, B C, C A, are the measures of the angles A o e, n o c, c o A, and are used for them : the spherical angles D A C, A c D, C D A are by definition the angles made by the planes no A and A 0 C, A o c and con, c o B and B 0 A. The sperical triangle then has six parts corresponding in name to the six parts of a plane triangle ; but a side of it means the angle made by two straight lines of a solid angle, while an angle of it refers to the angle made by two planes of the solid angle.
Throughout this article we shall designate the angles by A, c, the sides opposite to them by a, b, c; the half sum of the sides by s.
And by A', B', a', c', we mean the supplements of A, B, &C., so that A + A' se 180°, a + a' = 180°, &e. No triangle is considered which has either a side or an angle greater than 180°.
Three circles divide the sphere into eight spherical triangles. Of these four are equal and opposite to the other four, with which they agree in every respect but one [SvalaiurnicAL) with which we have nothing hero to do. Of the four which are distinct, if A n o be one, there are three others thus related to it : the first has for its aides a, b', c', and for its angles A, e', c', ; the second has a', b, for sides, and A', B, for angles ; the third has a', b,' c for sides, and A', B', C, for angles. Hence every spherical triangle has another, with one side and its opposite angle remaining unchanged, and all the other parts changed into their supplements.
Again, if the three circles be taken which have A, n, and c for their poles, the intersections of these new circles are themselves the poles of A B, B C, and a A ; and, of the eight new triangles thus formed, each one has all its angles supplemental to the sides of its corresponding triangle in the first set, and all its sides supplemental to the angles. Thus there exists a triangle which has the sides A', B', c', and the angles a', b', c' ; which is called the supplemental triangle of that which a, b, c, for sides, and A, B, c, for angles. Hence if, in any
general formula, sides be changed into supplements of angles, and angles into supplements of sides, the result is also a general formula.
Any two sides of a spherical triangle are together greater than the third, and the sum of the three sides is not so great as Any two angles of a spherical triangle are together less than the third angle increased by and the sum of the three angles is more than two, and less than six, right angles. And the greater side of a spherical triangle is opposite to the greater angle ; and the sum of two sides is greater than, equal to, or less than, according as the sum of the opposite angles is greater than, equal to, or less than, The formula for the solution of a spherical right-angled triangle are six in number. Let c be the right angle, and let c be called the as distinguished from a and b, which are still called sides. [CIRCULAR PARTS.] 1, 2. The cosine of the hypothenuse is equal to the product of the cosines of the sides, and of the cotangents of the angles : cos c = cos a cos b ; cos c = COt A COt D.
3. The eine of a side is the sine of the hypothenuse into the sine of the opposite angle : sin a = sin c sin A ; sin b = sin c sin B.
4. The tangent of a side is the tangent of the hypotlienuse into the cosine of the included angle : tan a = tan c eos B ; tan b = tan c cos A.
5. The tangent of a side is the tangent of its opposite angle into the side of the other side : tan a = tan A sin b; tan b = tau 13 sin a.
C. The cosine of an angle is the cosine of its opposite side into the side of the other angle : cos A = cos a ain ; cos B = cos b sin A.
These formube are sufficient for every case. Name any two out of the five a, b, C, A, B (c being a right angle), and in the preceding six formulae, by repetition ten, will be found those two combined with each of the other three. Thus, having given a side a and its adjacent angle B, we find the other parts from tan a tan b= tan 33 sin a, tan c = cos A= cos a sin B.
These formula, should be committed to memory : the abbreviation, so called, described in CIRCULAR PARTS, is only an expeditious mode of wasting time.
When all the angles are oblique, the principal forrnulm are as follows (in most cases we give only one, those for other sides, &c., being easily supplied) : sin A sin B sin C I. sin a = ein 6 eau - • or the sines of sides are to one another as the sines of their opposite angles.
2. cos c = cos a cos b + sin a sin b cos C.
3. cos 5 = sin s sin 2 sine sin sin / sin (a - a) sin (a - b) 2 'V sin a sin b • 4. tan sin (s - a) sin (a - if •- in a sin (s - sin (s - c)' where x = (s - a) sin (a - b) sin (a - c) sin a).