Article ID: CBB825969502

Irrational “Coefficients” in Renaissance Algebra (2017)


From the time of al-Khwārizmī in the ninth century to the beginning of the sixteenth century algebraists did not allow irrational numbers to serve as coefficients. To multiply $\sqrt {18} $ by x, for instance, the result was expressed as the rhetorical equivalent of $\sqrt {18{x^2}} $ . The reason for this practice has to do with the premodern concept of a monomial. The coefficient, or “number,” of a term was thought of as how many of that term are present, and not as the scalar multiple that we work with today. Then, in sixteenth-century Europe, a few algebraists began to allow for irrational coefficients in their notation. Christoff Rudolff (1525) was the first to admit them in special cases, and subsequently they appear more liberally in Cardano (1539), Scheubel (1550), Bombelli (1572), and others, though most algebraists continued to ban them. We survey this development by examining the texts that show irrational coefficients and those that argue against them. We show that the debate took place entirely in the conceptual context of premodern, “cossic” algebra, and persisted in the sixteenth century independent of the development of the new algebra of Viète, Decartes, and Fermat. This was a formal innovation violating prevailing concepts that we propose could only be introduced because of the growing autonomy of notation from rhetorical text.

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Authors & Contributors
García Valverde, José Manuel
Giusti, Enrico
Schütze, Ingo
Martin, Craig
Raimondi, Francesco Paolo
Nenci, Elio
Bollettino di Storia delle Scienze Matematiche
Journal of the Washington Academy of Sciences
Physis: Rivista Internazionale di Storia della Scienza
Asclepio: Archivo Iberoamericano de Historia de la Medicina
Galilæana: Journal of Galilean Studies
Ziran Kexueshi Yanjiu (Studies in the History of Natural Sciences)
Wilhelm Fink Verlag
Leo S. Olschki Editore
European Mathematical Society
Princeton University Press
Harvard University
Scribe Publications
Primary literature (historical sources)
Cardano, Girolamo
Bombelli, Raffaele
Stifel, Michael
Rudolff, Christoff
Viète, François de la Bigotière
Valles, Francisco de
Time Periods
16th century
17th century
15th century
18th century

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