« What is new and what is old in Viète's analysis restituita and algebra nova, and where do they come from? Some reflections on the relations between algebra and analysis before Viète »

Marco Panza (1)

1 REHSEIS (UMR_7596) - Recherches Epistémologiques et Historiques sur les Sciences Exactes et les Institutions Scientifiques

François Viète considered most of his mathematical treatises to be part of a body of texts which he entitled "Opus restitutae Mathematicae Analyseos Seu Algebrâ novâ". Despite this title and the fact that the term "algebra" has been often used to designate what is customarily regarded as Viète's main contribution to mathematics, such a term is not part of his technical language. 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 this 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 of a special sort of geometrical inferences which I call "non positional", since they do not depend on any diagram. As an example of a similar systematic application of trans configurational analysis, I consider al-Khayyam's "Treatise of Algebra and Al-muqabala". Finally, I suggest that, by speaking of algebra, in the title of his "Opus", Viète was referring 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.

Mots clés: Algebra; Analysis; Viète; al-Khayyam

Domaines: Histoire, Philosophie et Sociologie des sciences

Marco Panza. « What is new and what is old in Viète's analysis restituita and algebra nova, and where do they come from? Some reflections on the relations between algebra and analysis before Viète ». 2006. ⟨halshs-00116725v1⟩