for instance in Clavius’ edition of Euclid’s Elements. But in the case of Viète, precisely because of the generality he sought to associate with his methods, this choice raised an issue when applied to areas multiplied by areas, for example, or to quantities of higher dimensions. The results thus obtained admit of no direct geometric interpretation, and yet Viète’s methods allow them to be handled without the need for such an interpretation. Fortunately, Viète does not seem to have been bothered by this apparent conceptual difficulty, and he just went on to work out the methods as part of his ambitious plan. Indeed, Viète’s general analysis—aimed at extending to quantities of all kinds what in algebra had been thus far applied mainly to numbers, and in which preservation of dimensional homogeneity was still fundamental—provided a convenient framework against which the idea of abstract quantities could gradually evolve, which are neither numbers nor geometric magnitudes, and which represent the two species at the same time. A revealing statement in this regard appears in the text where he presented his method for approximating the value of π. He wrote:2 Arithmetic is absolutely as much a science as geometry [is]. Rational magnitudes are conveniently designated by rational numbers, and irrational [magnitudes] by irrational [numbers]. If someone measures magnitudes with numbers and by his calculation gets them different from what they really are, it is not the reckoning’s fault bu the reckoner’s.
So, Viète’s “new algebra” not only contributed to the intensifying trend towards eliminating the separation between discrete and continuous magnitudes, but was also instrumental in facilitating the soon-to-be-completed fusion between algebra and geometry in the framework of the analytic geometry that Fermat and Descartes would develop about forty years later. Viète had no direct interest in the kind of problems that would lead these two mathematicians to understand the benefits of applying symbolic methods to the study of curves. But whereas for all of his predecessors, Cardano and Bombelli included, algebraic methods required a justification that only geometry was seen to provide, Viète inverted the conceptual hierarchy and laid the priority on algebra. This was a step of momentous consequences. His most important book, In Artem Analyticam Isagoge (“Introduction to the Art of Analysis”), published in 1591, reached wide mathematical audiences all around Europe. It is easy to spot its direct influence in texts that appeared soon thereafter in the British Isles, Italy, Holland and France.
7.2 Stevin and decimal fractions A second contemporary focus of influential activity related to numbers and arithmetical practice is found in the work of Simon Stevin (1548–1620), a Flemish mathematician and engineer. Stevin was a highly ingenious and original thinker, with many intellectual and practical contributions to various fields of knowledge ranging 2
Quoted in (Berggren et al (eds.) 2004, p. 759). STEVIN AND DECIMAL FRACTIONS
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