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values, tables, sine, chords, alexandria and angle

TRIGONOMETRY, History of the Ele ments of. Among the ancients trigonometry was simply an adjunct to astronomy, and it so remained until comparatively recent times. A slight trace of its application to mensuration is found in the famous papyrus of Ahmes (see ALGEBRA, HISTORY OF THE ELEMENTS OF), where a quotient called seqt is mentioned. In the case of the pyramids the seqt seems to have been the cosine of the angle of slope of the edge, or in some cases the tangent of the angle of slope of the face. Among the Greeks frequent reference to trigonometry is found among the writings of the astronomers. Hypsicles (c. 190 a.c.) used the Babylonian division of the circumfer ence into 360 degrees and from this time the sexagesimal fraction became common in as tronomy. Hipparchus (c. 150 s.c.) was the first to compute a table of chords, the ancients gen erally using the chord instead of the half-chord or sine. Hero of Alexandria (see HERO OF ALEXANDRIA) gave rules which are the equiv alent of certain modern formulas, and in par 2a ticular computed the values of cot — for all n values of n from 3 to 12 inclusive. Menelaus of Alexandria (c. 100 A.D.) carried the study of Spherics to a considerable prominence, his celebrated Regula sex quantitatum relating to the transversal of the sides of a spherical tri angle, and he wrote six books on the calcula tion of chords. It is, however, to Claude Ptol emy (q.v.), c. 125 A.D. that is due the introduc tion of a formal spherical trigonometry into astronomy. The Almaqest made the sexa gesimal fraction more widely known and Ptol emy calculated the chords of arcs to a half degree.

The Hindu astronomers used the half chord instead of the chord which the Greeks usually (but not always) employed. They thus used the sine, and they added the versed sine and the cosine, computing tables for these ratios. They also knew the relation, sin' x cos' x=1.

The Arabs made the greatest advance in trigonometry of any peoples before the Renais sance. Al Battani, or Albategnius as the Latin

writers called him, c. 900 A.D., brought into greater prominence the use of the sine, and computed a table of values of sin x/cos x and its reciprocal, thus practically using the tangent and cotangent. The present names for the vari ous functions are mostly modern. The name sinus seems first to have been used by Gherardo of Cremona, c. 1150, although often attributed to Plato of Tivoli (also c. 1150) in his trans lation of Al Battani. Among the western Arabs, Jabir ibn Allah, often known as Geber, was prominent, his trigonometry covering both the plane and the spherical parts.

In Christian Europe the science is first seri ously considered in the work of Regiomontanus (q.v.), the tamous pupil of Peuerbach (q.v). The latter had done some excellent work in trigonometry, but he died before he could write his projected treatise, and Regiomontanus car ried out his plans. The result was a work which influenced subsequent textbooks much as Euclid's 'Elements) influenced plane geometry. The principal formulas of plane and spher ical trigonometry are set forth and the element ary science became crystallized. Subsequent advances have been chiefly in the nomenclature, the symbolism and the computation of tables, particularly of logarithmic tables. Among the most prominent computers of the values of the functions and of logarithms should be men tioned Rhmticus (1514-76), Pitiscus (1561 1613), Biirgi (1552-1632), Napier (1550-1617), Briggs (1560-1630) and \71acq, whose tables ap peared in 1628.

Bibliography.— Braunmiihl, A von, 'Ge schichte der Trigonometric' (Leipzig 1900, 1904); Cantor, M., 'Geschichte der Mathe matik' (Leipzig, 2d ed. 1894) ; Gow, J., tory of Greek Mathematics' (Cambridge 1884); Cajori, F., 'History of Mathematics' (New York 1900).