Science in Context 30(2), 141–172 (2017). C  Cambridge University Press 2017 doi:10.1017/S0269889717000096 Irrational “Coefficients” in Renaissance Algebra Jeffrey A. Oaks University of Indianapolis E-mail: oaks@uindy.edu Argument From the time of al-Khw¯ arizm¯ ı in the ninth century to the beginning of the sixteenth century algebraists did not allow irrational numbers to serve as coefficients. To multiply √ 18 by x, for instance, the result was expressed as the rhetorical equivalent of √ 18x 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 Vi` ete, 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. Premodern algebra, in Greek, Arabic, Latin, and Italian, was a numerical problem- solving technique. To work out a problem by algebra, an unknown number was named in terms of the given names of the powers, the conditions of the problem were applied to set up an equation, and this was then simplified and solved. While this broad approach is comparable to modern algebra, premodern algebraists – and for the time being I mean those who practiced algebra before the sixteenth century – did not solve their problems the same way we do today. Some of the steps they took differ from ours, their phrasing of certain operations makes little sense when interpreted through our symbolic solutions, and even their notations exhibit differences that cannot be attributed to local variation. One example of a difference in practice is that premodern algebraists consistently worked out their operations before setting up equations, where we do not hesitate to https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0269889717000096 Downloaded from https://www.cambridge.org/core. IP address: 99.6.162.195, on 27 Jul 2017 at 02:51:40, subject to the Cambridge Core terms of use, available at