The alien realm of the minus: Deviatory mathematics in Cardano's writings

This is a companion paper to my preceding one on Harriot's experimentations in the field of the sign-rule of multiplication in algebra. Cardano had earlier attacked the conventional rule in a chapter (to which Harriot refers) of his De Aliza regula liber , published in 1570 as an appendix to the second edition of his Ars magna . He returned to the subject in a brief tract, published nearly a century later in his collected works as Sermo de plus et minus . Only Cardano's valid contention that the standard ‘proof' of the accustomed sign-rule was faulty made impact, in the Euclidean background to which the ‘proof' belonged. His cause was to preserve the purity of the Euclidean tradition. It overshadows the true aim in the De Aliza chapter, but the Sermo reveals it plainly as far from retrograde: he wished to retain all the new techniques of algebra that brought in not only ‘minus' quantities but also their square roots, and to escape the ‘impossible' status of these last. This the rebel sign-rule did for him by making every root of a ‘minus' just another ‘minus'.