"Book 1 of Géométrie explains how to translate a geometrical problem into an equation ...by making a revolutionary break with the tradition of the theory of proportions to which Galileo adhered. ...Descartes interpreted algebraic operations as closed operations on segments. For instance, if a and b are segments, the product ab is not conceived by Descartes as representing an area, but rather another segment. Prior to the Géométrie the multiplication of two segments... would have been taken as the representation of the area of a rectangle... Descartes' interpretation of algebraic operations was a huge innovation... From Descartes' new viewpoint, ratios are considered as quotients, and proportions as equations. Consequently, the direct multiplication of ratios is allowed... The homogeneity of geometrical dimensions is no longer a constraint for the formation of ratios and proportions... multiplication of time and speed or division of weight by surface's area are all possible. Descartes' approach won great success."
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, "Mathematics and the New Sciences," The Oxford Handbook of the History of Physics (2013) ed. Jed Buchwals, Robert Fox, pp. 238-239.
https://en.wikiquote.org/wiki/La_G%C3%A9om%C3%A9trie
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La Géométrie
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