x = ( 1 3 a − z 2 ) ÷ z and substituting, he gets z 6 − b z 3 − 1 27 a 3 = 0 . Putting z 3 = y , he has a quadratic. In the solution of biquadratics, Vieta still remains true to his principle of reduction. This gives him the well-known cubic resolvent. He thus adheres throughout to his favourite principle, and thereby introduces into algebra a uniformity of method which claims our lively admiration. In Vieta's algebra we discover a partial knowledge of the relations existing between the coefficients and the roots of an equation. He shows that if the coefficient of the second term in an equation of the second degree is minus the sum of two numbers whose product is the third term, then the two numbers are roots of the equation. Vieta rejected all except positive roots; hence it was impossible for him to fully perceive the relations in question.
The most epoch-making innovation in algebra due to Vieta is the denoting of general or indefinite quantities by letters of the alphabet. To be sure, Regiomontanus and Stifel in
Germany, and Cardan in Italy, used letters before him, but
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 was written by him "." In numerical equations the unknown quantity was denoted by , its square by , and its cube by . Thus the equation was written . 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 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 arithmetic
of Grammateus, a teacher at the University of Vienna.
His pupil, Christoff Rudolff, the writer of the first text-book
on algebra in the German language (printed in 1525), employs these symbols also. So did Stifel, who brought out a second