An introduction to the concept of function within Descartes's algebra of segments

Abstract

In his G\'eom\'etrie (1637) Descartes introduces the algebra of segments. This is a fundamental step in the mathematical treatment of variable quantities before the creation of differential calculus. It is an algebra with symbols but without numbers, in which the covariation between geometric variables, constrained by ruler and compass constructions or with other geometric constructions, can be expressed with symbolic equations. By using algebraic manipulations, it is possible to easily deduce the properties of the corresponding geometric constructions, including those that produce graphs of rational functions. We believe that the study of functions through Descartes's algebra can be didactically effective in teaching and learning the concept of function in secondary school. Firstly, it avoids the reference to real numbers; secondly, the interpretation of formulas as geometric constructions and vice versa facilitates the "transition" from functions understood as processes to functions understood as objects.