Page:A History of Mathematics (1893).djvu/95

 Diophantus also solved determinate equations of the second degree. We are ignorant of his method, for he nowhere goes through with the whole process of solution, but merely states the result. Thus, "$$\scriptstyle{84x^2+7x=7}$$, whence x is found $$\scriptstyle{=\frac{1}{4}}$$." Notice he gives only one root. His failure to observe that a quadratic equation has two roots, even when both roots are positive, rather surprises us. It must be remembered, however, that this same inability to perceive more than one out of the several solutions to which a problem may point is common to all Greek mathematicians. Another point to be observed is that he never accepts as an answer a quantity which is negative or irrational.

Diophantus devotes only the first book of his Arithmetica to the solution of determinate equations. The remaining books extant treat mainly of indeterminate quadratic equations of the form $$\scriptstyle{Ax^2+Bx+C=y^2}$$, or of two simultaneous equations of the same form. He considers several but not all the possible cases which may arise in these equations. The opinion of Nesselmann on the method of Diophantus, as stated by Gow, is as follows: "(1) Indeterminate equations of the second degree are treated completely only when the quadratic or the absolute term is wanting: his solution of the equations $$\scriptstyle{Ax^2+C=y^2}$$ and $$\scriptstyle{Ax^2+Bx+C=y^2}$$ is in many respects cramped. (2) For the 'double equation' of the second degree he has a definite rule only when the quadratic term is wanting in both expressions: even then his solution is not general. More complicated expressions occur only under specially favourable circumstances." Thus, he solves $$\scriptstyle{Bx+C^2=y^2}$$, $$\scriptstyle{B_1x+C_1^2=y_1^2}$$.

The extraordinary ability of Diophantus lies rather in another direction, namely, in his wonderful ingenuity to reduce all sorts of equations to particular forms which he knows how to solve. Very great is the variety of problems considered. The 130 problems found in the great work of Diophantus