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 b cosy sinbsinceosA for the spherical angle. The celebrated astronomer Jabir din 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. Regiomontanus (1436. 1476) wrote a complete plane and spherical trigonometry. Vida ( 1540-1603 ) 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 cosrinc and Flock (15S3) introduced secant and tangent. 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 Byrgins (q.v.) the idea of the
logarithm (q.v.). To Euler much is due for simplifying and classifying the treatment of the whole subject. Lagrange, Legendre, Carnot, Gauss, and others expanded the theory of polygonometry and polyhedrometry. The nine teenth century has contributed the so-called pro jective formulas, and made further generaliza tion of formulas known before. A few leading works on the subject are: Casey, A Treatise on Plane Trigonometry (Dublin, 1888) ; id.. Treatise on Spherical Trigonometry (ib,, 1889) ; Mc Clelland and Preston, Spherical Trigonometry (London, 1890) ; 11elmes, Die Trigonometric (Hanover, 1881) ; Wittstein, Trigonometric (ib., 1887) ; Hobson, Plane Trigonometry (Cambridge, 1891) ; Chauvenet, Plane and Spherical Trig onometry (New York, various editions). For the history of the subject, consult Braumniihl, Beitriige z,nr Orsehichte der Trigonometric (Valle, 1897), and his Forlesungen iibcr Oc sebiehte der Trigonometric (Leipzig. vol. i., 1900; vol. ii., 1903). For reference to valuable tables, see the article on LOGARITHMS.