What Is New and What Is Old in Viète's analysis restituta and algebra nova, and Where Do They Come From? Some Reflections on the Relations between Algebra and Analysis before Viète (2007)
François Viète considered most of his mathematical treatises to be part of a body of texts he entitled Opus restitutæ mathematicæ analyseos seu algebra nova. Despite this title and the fact that the term algebra has often been used to designate what is customarily regarded as Viète's main contribution to mathematics, such a term is not part of his vocabulary. How should we understand this term, in the context of the title of his Opus, where new algebra is identified with restored analysis? To answer this question, I suggest distinguishing between two kinds of problematic analysis: the former is that described by Pappus at the beginning of the 7th book of his Mathematical Collection, which I will call intra-configurational; the latter is the one Viète applied, which I will call trans-configurational. In order to apply the latter kind of analysis, Viète relies on his new formalism. I argue, however, that the use of this formalism is not a necessary condition for applying it. I also argue that the same kind of analysis was largely applied before Viète for solving geometrical problems, by relying on geometrical inferences of a special sort which I call non-positional, since they do not depend on a diagram. As an example of a similar systematic application of trans-configurational analysis, I consider al-Khayy--am's Treatise of Algebra and Al-muq--a bala. Finally, I suggest that Viète, when speaking of algebra in the title of his Opus, refers to the system of techniques underlying trans-configurational analysis, that is, to the art of transforming the conditions of certain purely quantitative problems, using either an appropriate formalism relative to the operations of addition, subtraction, multiplication, division, root extraction and solving polynomial equations applied to indeterminate numbers, or appropriate geometrical, non-positional inferences. Keywords : Algebra, analysis, Viète, Al-Khayy--a m, Pappus
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