TRIGONOMETRY. 463 TRIGONOMETRY. The spherical triangle, like the plane triangle, has six elements, the three sides a, b, c, and the angles A, B, C. But the three sides of the spherical triangle are angular as well as linear magnitudes. The triangle is completely deter- mined when any three of its six elements are given, siqce there exist relations between the given and the sought parts by means of which the latter may be found. In the right-angled or quadrantal triangle, however, as in the case of the right-angled plane triangle, only two ele- ments are necessary to determine the remaining parts. Thus, given c, A, in the right-angled triangle, ABC, the remaining parts are given by the formulas sina, = sine, sinA, tanb = tane, cosA, cotB =: cose, tanA. The corresponding formulas when ^any other two parts are given may be obtained by Napier's rules concerning the relations of the five circular parts (q.v.), viz. a, b, complement of A, complement of B, comple- ment of c. In the case of oblique triangles no simple rules have been found, but each case is dependent upon the appropriate formula. Thus in the oblique triangle ABC, given a, b, and A, the formulas for the remaining parts are sinB = sinA sinft ^ , , ,> sin ; (A + B)
- , tanic=
- tanl- (a — b)- - — ^-r-. — -^
sin a ' - ^ "■ ' smi(A — B)' i. 1 ^ J. , / . T.X sin (a-|-6) cot i C = tan A (A — B)- — — ) — -~. ^ ^ ' sin (ri — b) It is evident in spherical trigonometry, as well as in plane, that three elements taken at random may not satisfy the conditions for a triangle, or they may satisfy the conditions for more than one. The treatment of the nmbifiyous cases in spherical trigonometry is quite formidable, since every line intersects every other line in two points and multiplies the eases to be considered. The measurement of spherical polygons may be made to depend upon that of the triangle. For, if, by drawing diagonals, the polygons can be divided into triangles each of which contains three known or obtainable elements, then all the parts of the polygon can be determined. Since the elements of the spherical polygon measure the elements of the polyhedral angle whose vertex is at the centre of the sphere, the fornuilas of spherical trigonometry apply to problems involv- ing the relations of the parts of such figures. E.g., given two face angles and the included dihedral angle of a trihedral angle, the remaining, face and dihedral angle may be determined by the same formulas as ap])ly to the corresponding case of the spherical triangle. By aid of the formu- las of spherical trigonometry the theories of transversals, coaxal circles, poles and polars, may be developed for the figures of the sphere. Spherical trigonometry is eif great importance also in the theory of power circles, stereographic projection, and geodesy. It is also the basis of the chief calculations of astronomy; e.g. the solution of the .so-called astronomical triangle is involved in finding the latitude and longitude of a place, the time of day, the azimuth of a star, and various other data. Some traces of trigonometry exist in the earli- est known writings on mathematics. In the Papy- rus of Ahnies (see Ahmes) a ratio is mentioned called a se<it, and because of its relation to the methods of measuring the pyramids, this ratio seems to correspond to the cosine or the tangent of an angle. But to the Greeks are due the first scientific trigonometric investigations. The sex- agesimal division of the circle was known to the Babylonians, but Hipparchus was the first to complete a table of chords. Heron (q.v.) com- 2ir puted the values of cot"^, for » = 3, 4. . . . 11, 12, and calculated the areas of regular poly- gons. Thirteen books of Ptolemy's Almagest were given to trigonometry and astronomy. The Hindus contributed an important advance by in- troducing the half chord for the whole chord as used in the Greek calculations. They were fa- miliar with the sine and calculated ratios cor- responding to the versine and cosine. The sine, however, first appears in the works of the Arab Al-Battani (q.v.), and to his influence is due the final substitution of the half for the whole chord. Al-Battani knew the theory of the right- angled triangle and gave the relation cos a = cos 6 cose + sinftsinccosA for the spherical tri- angle. The celebrated astronomer Jabir ibn Aflah, or Geber, wrote a work confined chiefly to spherical trigonometry, and rigorous in its proofs, which was translated into Latin by Gerhard of Cremona. Eegiomontanus ( 1430- 1470) wrote a complete plane and spherical trigonometry. Vieta (1,540-1603) made an im- portant advance by the introduction of the idea of the reciprocal spherical triangle. To Napier are due the formulas since called the analogies. Gunter introduced the term cosiine and Finck (1583) introduced secnnf and <a«(7e«Y. Growing out of the desire to construct more accurate tables and to simplify the methods of calculation for astronomical purposes, there was evolved by Napier and Byrgius (q.v.) the idea of the logarithm (q.v.). To Euler much is due
