Page:A History of Mathematics (1893).djvu/169

 Vieta extended the idea and first made it an essential part of algebra. The new algebra was called by him logistica speciosa in distinction to the old logistica numerosa. Vieta's formalism differed considerably from that of to-day. The equation $$\scriptstyle{a^3+3a^2b+3ab^2+b^3=(a+b)^3}$$ was written by him "a cubus + b in a quadr. 3 + a in b quadr. 3 + b cubo æqualia $\overline{a + b}$ cubo." In numerical equations the unknown quantity was denoted by N, its square by Q, and its cube by C. Thus the equation $$\scriptstyle{x^3-8x^2+16x=40}$$ was written $$\scriptstyle{1 C - 8 Q + 16 N}$$ æqual. 40. Observe that exponents and our symbol ( = ) for equality were not yet in use; but that Vieta employed the Maltese cross ( + ) as the short-hand symbol for addition, and the ( — ) for subtraction. These two characters had not been in general use before the time of Vieta. "It is very singular," says Hallam, "that discoveries of the greatest convenience, and, apparently, not above the ingenuity of a village schoolmaster, should have been overlooked by men of extraordinary acuteness like Tartaglia, Cardan, and Ferrari; and hardly less so that, by dint of that acuteness, they dispensed with the aid of these contrivances in which we suppose that so much of the utility of algebraic expression consists." Even after improvements in notation were once proposed, it was with extreme slowness that they were admitted into general use. They were made oftener by accident than design, and their authors had little notion of the effect of the change which they were making. The introduction of the + and — symbols seems to be due to the Germans, who, although they did not enrich algebra during the Renaissance with great inventions, as did the Italians, still cultivated it with great zeal. The arithmetic of John Widmann, printed A.D. 1489 in Leipzig, is the earliest book in which the + and — symbols have been found. There are indications leading us to surmise that they were in use first among merchants. They occur again in the