INDUCTION.
(2.) In Greek mathematics the method of proving a proposition by resolving it into simpler statements already proved or else as sumed as axiomatic. Pappus (c. 275), in his Synagoge or Collec tion (Bk. VII., "On the Treasury of Analysis"), gives the best extant Greek definition : "Analysis, then, takes that which is sought as if it were admitted and passes from it through its various consequences to something that is admitted as the result of synthesis." The process was probably used by the early Pythagoreans in the 5th century B.C., and was taught in Plato's school. Pappus speaks of it further as "a method which Plato, as they say, communicated to Leodamas, and by which the latter, too, is said to have discovered many things in geometry." In the Renaissance period it came to be considered as the method of solving problems by means of equations, as in analytic geometry (q.v.). At present the word is used in a much more general way, being considered as including the theory of functions of real variables, infinite series, the differential and integral calcu lus, definite and multiple integrals, the calculus of variations, the theory of functions of complex variables, algebraic f unctions, elliptic and modular functions, special functions (Eulerian, Le gendre's, Bessel's, hypergeometric, etc.), metrical and projective properties of quadric surfaces, algebraic curves and surfaces, algebraic configurations in general and infinitesimal and differ ential geometry. These topics are treated in various special articles, including CURVE ; FUNCTIONS; CALCULUS, DIFFERENTIAL AND INTEGRAL ; and DIFFERENTIAL GEOMETRY.
(3.) In chemistry the word analysis was introduced by Robert Boyle to denote the determination of the composition of sub stances. (See CHEMISTRY : Analytical.) a detailed summary of the scope of modern analysis, with references to scientific literature see the Catalogue of Scientific Papers of the Royal Society of London, vol. i. pp. 233-653 (1800–I900) . For later material see the Jahrbuch iiber die Fort schritte der Mathematik (annual summary- of publications) and the Revue Semestrielle.