The Dissemination of Cartesian Ideas. An Analysis of Practice with Diagrams in Van Schooten’s Reading of Descartes

This paper presents an analysis of the mathematical practice with diagrams that van Schooten develops when editing and commenting on the second edition of the Latin translation of Descartes' Geometry from 1637. This edition (1659–1661), designed to disseminate Cartesian mathematics, was read by Newton, Leibniz, and most of the mathematicians of the time. Its publication in two volumes includes numerous Commentaries with clarifications and elaborations of van Schooten on crucial matters of algebraic geometry that Descartes did not develop in the original text. It also includes a body of Cartesian works by some of van Schooten's students. We argue that van Schooten placed in the work of Descartes a central theme for the dissemination of Geometry, the algebraic representation of curves. We show that diagrams as a semiotic mode of representation and communication were essential for this purpose because, in addition to ensuring the more general algebraic solution of geometric problems, they allowed the passage to the thematization of algebraic curves. We will point out the pedagogical interest of some historical episodes in which this double characteristic of diagrammatic practice is expressed in Cartesian geometry.