Changing notions of proportionality in pre-modern mathematics

Abstract

Defined as «a sort of relation in respect of size between two magnitudes of the same kind», a ratio was not a number nor a geometrical magnitude in Euclid's Elements. During the first half of the 18th century, however, ratios were identified with numerical magnitudes for all practical purposes. This paper argues that in order to understand the changing notions of ratio and proportionality in the early modern period two questions are to be answered separately. One concerns the numerical status of the objects compared through a ratio, or term of the ratio and the paper shows that by the turn of the 17th century this difficulty had been overcome. The second question concerns the status of ratios themselves, which was not solved until the 18th century. In studying this development particular attention is paid to the problem of evaluating the influence of the medieval notion of denomination of ratios. Thanks to Pedro Nunez's algebra book it is possible to show that this concept had become fossilized and lost its virtuallity as an «arithmetizing» agent by the mid 16th century. Furthermore, it is argued here that the changes overcoming ratios and proportionality in the 16th century are understandable only with reference to the social background, and particularly to the so-called abbaco books and to 16thcentury algebra.