What More There Is in Early-Modern Algebra than its Literal Formalism

There are two views about early-modern algebra very often endorsed (either explicitly or implicitly). The former is that in early-modern age, algebra and geometry were different branches of mathematics and provided alternative solutions for many problems. The latter is that early-modern algebra essentially resulted from the adoption of a new literal formalism. My present purpose is to question the latter. In doing that, I shall also implicitly undermine the former. I shall do it by considering a single example. This is the example of a classical problem. More in particular, I shall consider and compare different ways of understanding and solving this problem. Under the first understating I shall consider, this problem appears as that of cutting a given segment in extreme and mean ratio. This is what proposition VI.30 of the Elements requires. In section 2, I shall expound and discuss Euclid's solution of this proposition2. Then, in section 3, I shall consider other propositions of the same Elements, and argue that they suggest another, quite different understanding of the same problem. Under this other understanding, this is viewed as the problem that of constructing a segment meeting a certain condition relative to another given segment. One way to express this condition is by stating the first of the three trinomial equations studied in al-Khwarizmi's Algebra, by supposing that this equation is geometrically understood and a particular case of it is considered. In section 4, I shall consider this option, by focusing in particular, on Thabit ibn Qurra's interpretation and solution of this equation.