ABSTRACT
We start from the question: what kind of quantity were the quantities algebra talked about, according to sixteenth-century algebraists? The question springs from Klein’s theses, which we would formulate as follows: Greek numbers were collections of units. Unit was not a number but that by virtue of which anything is called one, as Euclid and Aristotle said. Diophantus introduced abbreviations for the unknown quantities but only Viète’s algebra (together with Stevin’s and Descartes’) produced the shift to symbol. Hence, in early modern Europe, these quantities came to be conceived as secundae intentiones in a shift of Begrifflichkeit (conceptuality) from Greek numbers. Klein also states that this shift in conceptuality depended on the apprehension of the mathematical past by these mathematicians. But what happened in detail in sixteenth-century algebra with respect to this transformation; what appeared before Viète’s species? First, I examine Niccolò Tartaglia’s philosophical discussion about unit, number and some crucial notion Ibn Rushd Tartaglia was familiar with. Tartaglia also ascribes to Savonarola some of his statements, indicating a logical interpretation, leading to species and secundae intentiones. His mathematical production culminating in the General Trattato (1557–1560) develops in great detail many parts of mathematics as listed by Al Farabi’s On the Sciences and Gundisalvi’s Division of sciences, recently transmitted by Savonarola and including algebra. Going back to the main treatise comparable to Tartaglia’s Luca Pacioli’s Summa (1494) and looking at its notion of quantity, we find again Gundisalvi, who has transmitted a few crucial themes relevant to the idea of unit, number, matter and continuous quantity. Finally, a discussion of the solutions for second degree equations introduces sixteenth-century algebraic quantities as a new version of Euclid’s plane and solid numbers, but geometrically constructed from Pacioli to Nunes. Both Pacioli and Tartaglia reconstructed algebra on Euclidean foundations on the philosophical assumption of a primitive doctrine prior to arithmetic and geometry, called practica speculativa, speculative practice.
