Abstract

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 François 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.