Page:A History of Mathematics (1893).djvu/94

 suggestions of algebraic notation, and of the solution of equations, then his Arithmetica is the earliest treatise on algebra now extant. In this work is introduced the idea of an algebraic equation expressed in algebraic symbols. His treatment is purely analytical and completely divorced from geometrical methods. He is, as far as we know, the first to state that "a negative number multiplied by a negative number gives a positive number." This is applied to the multiplication of differences, such as $$\scriptstyle{(x-1)(x-2)}$$. It must be remarked, however, that Diophantus had no notion whatever of negative numbers standing by themselves. All he knew were differences, such as $$\scriptstyle{(2x-10)}$$, in which $$\scriptstyle{2x}$$ could not be smaller than 10 without leading to an absurdity. He appears to be the first who could perform such operations as $$\scriptstyle{(x-1) \times (x-2)}$$ without reference to geometry. Such identities as $$\scriptstyle{(a+b)^2=a^2+2ab+b^2}$$, which with Euclid appear in the elevated rank of geometric theorems, are with Diophantus the simplest consequences of the algebraic laws of operation. His sign for subtraction was , for equality $$\scriptstyle{\iota}$$. For unknown quantities he had only one symbol, $$\scriptstyle{\varsigma}$$. He had no sign for addition except juxtaposition. Diophantus used but few symbols, and sometimes ignored even these by describing an operation in words when the symbol would have answered just as well.

In the solution of simultaneous equations Diophantus adroitly managed with only one symbol for the unknown quantities and arrived at answers, most commonly, by the method of tentative assumption, which consists in assigning to some of the unknown quantities preliminary values, that satisfy only one or two of the conditions. These values lead to expressions palpably wrong, but which generally suggest some stratagem by which values can be secured satisfying all the conditions of the problem.