sion correspond to an arithmetical progression, and arrives at the designation of integral powers by numbers. Here are the germs of the theory of exponents. In 1545 Stifel published an arithmetic in German. His edition of Rudolff's Coss contains rules for solving cubic equations, derived from the the writings of Cardan.
We remarked above that Vieta discarded negative roots of equations. Indeed, we find few algebraists before and during the Renaissance who understood the significance even of negative quantities. Fibonacci seldom uses them. Pacioli states the rule that "minus times minus gives plus," but applies it really only to the development of the product of ; purely negative quantities do not appear in his work. The great German "Cossist" (algebraist), Michael Stifel, speaks as early as 1544 of numbers which are "absurd" or "fictitious below zero," and which arise when "real numbers above zero," are subtracted from zero. Cardan, at last, speaks of a "pure minus"; "but these ideas," says Hankel, "remained sparsely, and until the beginning of the seventeenth century, mathematicians dealt exclusively with absolute positive quantities." The first algebraist who occasionally places a purely negative quantity by itself on one side of an equation, is Harriot in England. As regards the recognition of negative roots. Cardan and Bombelli were far in advance of all writers of the Renaissance, including Vieta. Yet even they mentioned these so-called false or fictitious roots only in passing, and without grasping their real significance and importance. On this subject Cardan and Bombelli had advanced to about the same point as had the Hindoo Bhaskara, who saw negative roots, but did not approve of them. The generalisation of the conception of quantity so as to include the negative, was an exceedingly slow and difficult process in the development of algebra.
