Spherical Trigonometry

From the law of cosines flow the two triplets of formula adapted to logarithmic computation and being respectively of the types (18) tan IA = V sin (s — b)sin (s -- a) : V sin s(s—a), and (19 ) tan la = — cos (S — A)cos S : V cos s and S being the half sums of the sides and of the angles.

Napier's From (17) by help of the Sine Law are found the so-called first set of Napier's analogies, namely (20) cos i (a b) : cos }(a— b).

= cot iC : tan i(A B), (21) sin i(a + b) : sin it(a—b) = cot iC : tan i (A — B); and from these, by use of the polar triangle, this second set: (22) cos 1(A B) : cos i(A — B) = tan c : tan 1(a b).

(23) sin “A B) : sin i(A — B) tan c : tan i(a b).

For ways of resolving the ambiguity incident to the use of the Sine Law, the reader is re ferred to any standard work on Spherical Trigonometry. (See Bibliography).

Plane Trigonometry a Special Case of The ground of the notable resem blances between corresponding plane and spherical formula may be made evident by the following considerations: Suppose a plane p tangent to a sphere of radius r at a point P. If r increase without limit, the one-based zone having P for mid-point will flatten, swelling out toward p so as to include any given finite point Q near at will but not on p however far Q be from P. Plane p is said to be the limit of the sphere surface as r increases limitlessly — a relationship commonly expressed briefly by saying that a plane is a sphere of infinite radius. Accordingly the geometry and the trig onometry on a sphere of radius r ought to de grade respectively into plane geometry and plane trigonometry, on taking r infinitely great. To show that and how, in case of trigonometry, such degeneration actually occurs, consider the spherical Sine Law (s) sin a sin b sin c where a, b, c are the sides (i.e., the central angles they subtend), and a, the corresponding angles, of a triangle on a sphere of radius,. Denoting the lengths of a, b, c by 1, m, n re spectively, a, b, c= - 1 m n , — (radian measure).

r 1r Denoting the lengths of a, b, c by 1, m, n re / !It sin-sin- sin 1 r mr S --•. • 1 sin a msin g n sin r _ r Now it may be proved that the limit of the ratio of the sine to the angle as the angle ap proaches zero is 1. Hence as r increases limit

1 lessly and consequently the angles m n • r r r(numerators being kept constant, or finite at any rate) approach zero, the foregoing Sine Law degrades into sin as in el = — = —, the Sine s sinLaw for plane triangles. Similarly, by use of the Sine and Cosine Series, it may be shown that the Cosine Law for the sphere degenerates for r = m into the Cosine Law for the plane, and that the Tangent Law for the plane is but a special case of the fourth Napierian analogy, the Law of Tangents for the sphere.

Hyperbolic These are asso ciated with the rectangular hyperbola (see GEOMETRY, CARTESIAN and CALCULUS) in a man ner similar to the connection of the trigonomet ric or circular functions with the circle. The hyperbolic functions invented by Lam bert (1768), may be defined as follows (compare with the Eulerian formulae) : nam ing them hyperbolic sine, etc., and denoting them by sinh, etc., their definitions are, sinh a = }(ea = cosh a = Yea tanh a = sinh a:cosh a, etc. Each of the six is ex pressible in terms of each of the others. Thus coshla -sinhla = 1. = 1, etc. These tunctions are most instructively intro duced through the integral calculus. For their geometric interpretation the reader is referred to such works as W. B. Smith's 'Infinitesimal Analysis,' Vol. I, and Greenhill's 'Differential and Integral Calculus.' A given sphere has constant positive curvature (see CAL CULUS). If the radius be infinite, the curvature is zero. The plane is a sphere of constant zero curvature. Suppose the curvature to be con stant and negative. The surface is then called pseudo-sphere. This, too, has its trigonometry. Its formulae are obtainable from those of spher ical trigonometry by replacing the circular func tions by the corresponding hyperbolic functions. See TRIGONOMETRY, HISTORY OF THE ELEMENTS OF.

Chauvenet, 'Treatise on Plane and Spherical Trigonometry' (19th ed., Philadelphia 1881) ; Loney, 'Plane Trigonom etry> (Cambridge 1895) ; Todhunter, 'Plane Trigonometry' (London 1891) ; 'Spherical Trigonometry' (London 1901).