DI'OPHAN'TINE ANALYSIS. That por tion of the theory of indeterminate problems which seeks rational and commensurable roots of equations involving the squares and Cubes of the unknown quantity. This class of problems was first and chiefly treated by Diophantus, whose name the method bears. Although Dio phantus gave problems involving the indetermi nate linear equations, he was chiefly interested in solving equations of the second degree: e. g. to find two whole numbers. the sum of whose squares is a square: or to find three square numbers in arithmetic progression. The method of Diophantns, being neither concise nor definite, does not admit of a brief exposition. The follow ing are, however. some of its characteristics: (1) His quantitative symbolism was limited to that of one unknown. (2) His rules of operation arc the common axioms of adding to and sub tracting from both members of an equation. (3) Ile showed much adroitness in selecting the unknown; e.g. so as to avoid affected quadratics or complete cubics. He also showed skill in his tentative assumptions, in assigning sepa rately to the unknown preliminary values which one or two only of the necessary condi tions: then. from its failure to satisfy the re maining conditions, he discovered the required number, a method somewhat analogous to the require falsi (q.v.). (5) He used the symbol for the unknown in different senses, which amounted to the same thing as substituting One unknown for a function of another, (6) Ile made some use of limits: e. g. to find a square between 10 and
11, Diophantus proceeds thus: No square lies between tO and 44. nor between and 99, but 169 lies between 160 and I in. These numbers being DI time.. 10 and 11. respectively, the square between 10 and II is (7) He developed the use of synthesis (q.v.). (Si Ile showed in the introduction of arbitrary conditions: e. g. two numbers being sought such that the cube of one is greater by 2 than the square of the ot!.er. Diophantus arbitrarily assumes that the tin hers arm' ,r 1. — I. The strength of this I{ xis not consist in the elegance and deli. niteness of the method, hut in the c1)11,11iii111:11.0 Skill with which Diophantus makes use of the above devices, Diopliantirs's flame is also assoei ated with the following theorem: The sum of the squares of any integers can never be ex pres•ed as the sum of two such squares. Fermat tir-t proved this theorem and added the vorol hiry: It. is impo—ihle that any multiple of a pritne of the form 4a-1, by a number prime to it, can either be a square or the sum of two squares, integral or fraetional.