Geometry

Summary

Traditionally, the science of geometry was taken to be an abstract inquiry into the properties of continuous spatial magnitudes (lines, surfaces, angles, and solids). The geometrical method proceeds by strict deduction from clearly grasped and transparently true first principles, such as Euclid's first axiom (see common notion): “Things which are equal to the same thing are equal to one another” (Euclid 1926[1956], 1:222). This account of geometry distinguishes it from the science of arithmetic on the basis of its object: whereas geometry is the science of continuous quantities, arithmetic is the deductively organized study of discrete quantities (“multitudes” or “numbers” in the traditional parlance). The great geometers of Greek antiquity, Euclid, Archimedes, Apollonius, and Pappus, left behind an immense body of results whose level of technical sophistication was unmatched until the seventeenth century. Much of the work of geometers in Descartes’ day was directed toward the preparation of editions and translations of the work of classical Greek geometers, as well as speculative “restorations” of lost treatises on a variety of geometrical subjects. With the 1637 publication of Descartes’ Geometry, the subject underwent a profound and permanent change.

The particulars of Descartes’ involvement with geometry before the publication of the Geometry are very difficult to assess. His mathematical education at La Flèche was certainly guided by study of the works of the Jesuit mathematician Christopher Clavius, notably his Latin edition of Euclid's Elements with its extensive commentary (first published in 1574 and a standard work in the Jesuit mathematical curriculum) as well as a 1608 Algebra. On his arrival in the Netherlands in 1628, Descartes was in possession of what Isaac Beeckman called an “Algebra” (Journal 3: 94–95) that was evidently an early version of the Geometry. In a 1619 letter to Beeckman, Descartes announced that he had “discovered four remarkable and completely new demonstrations” and declared his intention to produce “a completely new science that will provide a general solution of all possible sorts of problems involving any sort of quantity, whether continuous or discrete, each according to its nature” (AT X 154, 157). It is apparent that even at this early stage in his mathematical career Descartes sought to apply algebraic techniques to study all types of quantity, with the result that the classical division between arithmetic and geometry was undermined.