René Descartes

"[E]arly analytic geometers—Descartes in particular—did not accept that geometry could be based on numbers or algebra. Perhaps the first to take the idea of arithmetizing geometry seriously was Wallis..."

"Despite Newtons belated appreciation of Euclids geometry, he set it aside as an undergraduate and immediately turned to Descartes Geometrie, a much more difficult text. Newton read a few pages... and immediately got stuck. ...The second time through, he progressed a page or two further before running into more difficulties. Again, he read it from the beginning, this time getting further still. He continued this process until he mastered Descartes text. Had Newton mastered Euclid first, Descartes analytic geometry would have been much easier to understand. Newton later advised others not to make the same mistake. But Descartes had ignited Newtons interest in mathematics, an interest that bordered on obsession."

"Descartes ... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish. ...historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is ... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do."

"Descartes writes in a letter to Plempius in 1638 (page 81 of vol. 3 of his Philosophical Writings, ed. and trans. by Cottingham et al. [Cambridge University Press]): "For this is disproved by an utterly decisive experiment, which I was interested to observe several times before, and which I performed today in the course of writing this letter. First, I opened the chest of a live rabbit and removed the ribs to expose the heart and the trunk of the aorta. I then tied the aorta with a thread a good distance from the heart."

"Starting with "I think," Descartes fixed his attention only on the "think," completely neglecting the "I." Now, this I is essential. For Man, and consequently the Philosopher, is not only Consciousness, but also—and above all—Self-Consciousness. Man is not only a being that thinks—i.e., reveals Being by Logos, by Speech formed of words that have a meaning. He reveals in addition—also by Speech—the being that reveals Being, the being that he himself is, the revealing being that he opposes to the revealed being by giving it the name Ich or Selbst, I or Self."

"He has given us only some beautiful beginnings, without getting to the bottom of things. ...he is still far from the true analysis and the general art of discovery. For I am convinced that his mechanics is full of errors, that his physics goes too fast, that his geometry is too narrow, and that his metaphysics is all these things."

"Algebra had already been applied to geometry by other writers, as we have seen. The wholly new contribution made by Descartes was in importing the idea of motion into geometry. It is said that the idea came to him while lying in bed and watching the movements of a fly crawling near an angle of the room. He saw that its position at any moment could be defined by its perpendicular distance from the ceiling and two adjacent walls. Thus he saw a curve as described by a moving point, the point being the point of intersection of two moving lines which were always parallel to two fixed lines at right angles. As the moving point described the curve, its distances from the two fixed axes would vary in a manner characteristic of the curve, and an equation between these distances could be formed which would express some property of the curve. Algebraical transformations of this equation would then reveal other properties of the curve."

"How you picked your hypotheses (he argued) was of no importance whatever... even if picked at random. ...[A] hypothesis was to be judged by its fruits. ...Descartes analogy between code-breaking and theory-making is excellent."

"It was left up to Newton to compute the detailed implications of the vortex-theory... and the result demolished the foundations of Descartes cosmology."

"His decipherment of Nature might be crude, yet he had the courage to insist that the mechanical sense could be made of the workings of Nature, throughout the realms of physics, chemistry, and even physiology. By reasserting the unity and rationality of Nature, he did as much as any man to put seventeenth-century scientists back on the intellectual road first trodden by the Greeks."

"The mechanical philosophy is a case of being victimized by metaphor. I choose Descartes and Newton as excellent examples of metaphysicians of mechanism malgré eux, that is to say, as unconscious victims of the metaphor of the great machine. Together they have founded a church, more powerful than that founded by Peter and Paul, whose dogmas are now so entrenched that anyone who tries to reallocate the facts is guilty of more than heresy."

"Descartes... clearly understood the power of algebraic methods in geometry. He wanted to withhold this power from his contemporaries, however, particularly... Roberval... La Géométrie was written to boast about his discoveries, not to explain them. There is little systematic development, and proofs are frequently omitted with a sarcastic remark such as, "I shall not stop to explain... because I should deprive you of the pleasure...""