So also I hope to show for continuous quantities that some problems can be solved by straight lines and circles alone others only by other curved lines, which, however, result from a single motion and can therefore be drawn with new forms of compasses, which are no less exact and geometrical, I think, than the common ones used to draw circles and finally others that can be solved only by curved lines generated by diverse motions not subordinated to one another, which curves are certainly only imaginary (e.g., the rather well-known quadratrix). Yet each problem will be solved according to its own nature, as, for example, in arithmetic some questions are resolved by rational numbers, others only by irrational numbers, and others finally can be imagined but not solved. by which all questions can be resolved that can be proposed for any sort of quantity, either continuous or discrete. 5ĭescartes expressed his second programmatic goal in a letter to Beeckman in 1619 at the time it appeared to him to be unattainable by one man alone. Finally, there have been some most ingenious men who have tried in this century to revive the same for it seems to be nothing other than that art which they call by the barbarous name of “algebra,” if only it could be so disentangled from the multiple numbers and inexplicable figures that overwhelm it that it no longer would lack the clarity and simplicity that we suppose should obtain in a true mathematics. some traces of this true mathematics seem to me to appear still in Pappus and Diophantus. As Descartes wrote in his Rules for the Direction of the Mind ( ca. The first stemmed from a belief, first expressed by Petrus Ramus, 4 that cossist algebra represented a “vulgar” form of the analytical method employed by the great Greek mathematicians. 3ĭuring this decade Descartes sought to realize two programmatic goals. Whatever the early influences on Descartes’s mathematics, it nonetheless followed a relatively independent line of development during the decade preceding the publication of his magnum opus, the Géométrie of 1637. 2 Descartes’s treatise De solidorum elementis, which contains a statement of “Euler’s Theorem” for polyhedra ( V + F = E + 2), was quite likely also a result of their discussions. Descartes’s command of cossist algebra (evident throughout his papers of the early 1620’s) was perhaps strengthened by his acquaintance during the winter of 1619–1620 with Johann Faulhaber, a leading German cossist in Ulm.
Descartes apparently received the stimulus to study these works from Isaac Beeckman his earliest recorded thoughts on mathematics are found in the correspondence with Beeckman that followed their meeting in 1618. 1 Its historical foundations lie in the classical analytical texts of Pappus ( Mathematical Collection) and Diophantus ( Arithmetica) and in the cossist algebra exemplified by the works of Peter Rothe and Christoph Clavius. The mathematics that served as model and touchstone for Descartes’s philosophy was in large part Descartes’s own creation and reflected in turn many of his philosophical tenets. The physics is discussed in two subsections: Optics and Mechanics. In this section, Descartes’s mathematics is discussed separately.
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