Polynomials and equations in premodern algebra
Whether writing in Arabic, Latin, Italian, and even in Diophantus, premodern algebraists conceived of polynomials differently than we do today. Where our 3x2 + 4 entails the operations of addition, multiplication, and exponentiation, the medieval equivalent (here from Arabic) “three mals and four dirhams” was conceived as an aggregation of seven objects of two different kinds with no operations present. Premodern equations ideally expressed the equality of two such aggregations, so operations called for in the enunciation of a problem were worked out before the equation was set up.
The Arabic algebraic notation that developed in the Maghreb in the twelfth century was created with the “aggregations” interpretation in mind, as were the various notations in Diophantus and in medieval Italian abacus texts. I will show examples of these from various manuscripts.