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 … (original) (raw)
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2000
François Viète considered most of his mathematical treatises to be part of a body of texts he entitled Opus restitutae Mathematicae Analyseos Seu Algebrâ novâ. 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-Khayyām's Treatise of Algebra and Al-muqābala. 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.
François Viète’s revolution in algebra
Abstract Françios Viète (1540–1603) was a geometer in search of better techniques for astronomical calculation. Through his theorem on angular sections he found a use for higher-dimensional geometric magnitudes which allowed him to create an algebra for geometry. We show that unlike traditional numerical algebra, the knowns and unknowns inViète’s logistice speciosa are the relative sizes of non-arithmetized magnitudes in which the “calculations” must respect dimension. Along with this foundational shift Viète adopted a radically new notation based in Greek geometric equalities. His letters stand for values rather than types, and his given values are undetermined. Where previously algebra was founded in polynomials as aggregations, Viète became the first modern algebraist in working with polynomials built from operations, and the notations reflect these conceptions. Viète’s innovations are situated in the context of sixteenth-century practice, and we examine the interpretation of Jacob Klein, the only historian to have conducted a serious inquiry into the ontology of Viète’s “species”.
François Viète and his contribution to mathematics
This paper studies the work of the French mathematician François Viète, known as the "father of modern algebraic' notation". Along with this fundamental change in algebra, Viète adopted a radically new notation based on Greek geometric' equalities. Its letters represent values rather than types, and its given values are undefined. Where algebra had previously relied on polynomials as sets, Viète became the first modern algebraist to work with polynomials generated by operations, and the notations reflect these notions. His work was essential to his successors because it enabled those mathematicians who followed him to develop the mathematics we use today.
Fermat and Descartes in light of premodern algebra and Viète
Handbook of the History and Philosophy of Mathematical Practice, 2021
There were two radically different algebras practiced in Europe in the time of Fermat and Descartes. One was the traditional cossic algebra grounded in arithmetic whose roots stretch back to al-Khwārizmī and Diophantus, and the other was the new geometrical algebra of Franois Viète. Both Fermat and Descartes chose to work in the latter. To understand their choice, and to determine how they understood their algebraic terms, we first outline the conceptual foundations of the two algebras, one that regards a monomial as a premodern number in which the coefficient is a multitude and the power is a kind, and the other that regards a monomial as the product of two magnitudes, one known and the other unknown. We find that cossic algebra could not have been modified to include undetermined coefficients, and thus would have been inadequate for the purposes of Fermat and Descartes. Further, both authors, when solving problems in geometry, worked with precisely the same non-arithmetized algebra as Viète, without a unit measure, respecting dimension, and using magnitudes of arbitrarily high dimension for which no justification is provided.
In 1942, Edgar Zilsel proposed that the sixteenth-seventeenth-century emergence of Modern science was produced neither by the university tradition, nor by the Humanist current of Renaissance culture, nor by craftsmen or other practitioners, but through an interaction between all three groups in which all were indispensable for the outcome. He only included mathematics via its relation to the "quantitative spirit". The present study tries to apply Zilsel's perspective to the emergence of the Modern algebra of Viète and Descartes (etc.), by tracing the reception of algebra within the Latin-Universitarian tradition, the Italian abbacus tradition, and Humanism, and the exchanges between them, from the twelfth through the late sixteenth and early seventeenth century.
Actes D Historia De La Ciencia I De La Tecnica, 2010
The solution of the algebraic equations of third and fourth degree by Italian algebraists in the first half of the 16 th century is considered the beginning of the development of modern mathematics. These results have their roots in Italian vernacular algebra that was taught in Abbacus School and written in a chapter of abbacus treatises since the beginning of the 14 th century. In recent years many of these treatises has been published and studied. Particular attention has been paid to the origins of Italian vernacular algebra that does not seem linked to al-Khwarizmi's and Fibonacci's tradition. In this paper we make a survey of the main 14 th century treatises and give some contribution to the problem of the origins.
In 1942, Edgar Zilsel proposed that the sixteenth-seventeenth-century emergence of Modern science was produced neither by the university tradition, nor by the Humanist current of Renaissance culture, nor by craftsmen or other practitioners, but through an interaction between all three in which all were indispensable for the outcome. He only included mathematics via its relation to the "quantitative spirit". The present study tried to apply Zilsel's perspective to the emergence of the Modern algebra of Viète and Descartes (etc.), by tracing the reception of algebra within the Latin-Universitarian tradition, the Italian abbacus tradition, and Humanism, and the exchanges between them, from the twelfth through the late sixteenth and early seventeenth century.