Physicists From France

"Methods of drawing tangents were invented by Roberval and Fermat... Descartes gave a third method. Of all the problems which he solved by his geometry, none gave him as great pleasure as his mode of constructing tangents. It is profound but operose, and, on that account, inferior to Fermat's. His solution rests on the method of Indeterminate Coefficients, of which he bears the honour of invention. Indeterminate coefficients were employed by him also in solving bi-quadratic equations."

"It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs sometimes used algebra in connection with geometry."

"In the Greek geometry the idea of motion was wanting but with Descartes it became a very fruitful conception. ...This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree."

"It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs some times used algebra in connection with geometry. The new step that Descartes did take was the introduction into geometry of an analytical method based on the notion of variables and constants, which enabled him to represent curves by algebraic equations. In the Greek geometry, the idea of motion was wanting, but with Descartes it became a very fruitful conception. By him a point on a plane was determined in position by its distances from two fixed right lines or axes. These distances varied with every change of position in the point. This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree."

"In mechanics Descartes can hardly be said to have advanced beyond Galileo. ...His statement of the first and second laws of motion was an improvement in form, but his third law is false in substance. The motions of bodies in their direct impact was imperfectly understood by Galileo, erroneously given by Descartes, and first correctly stated by Wren, Wallis, and Huygens."

"The first important example solved by Descartes in his geometry is the "problem of Pappus"... Of this celebrated problem the Greeks solved only the special case... By Descartes it was solved completely, and it afforded an excellent example of the use which can be made of his analytical method in the study of loci. Another solution was given later by Newton in the Principia."

"Descartes' geometry was called "analytical geometry," partly because unlike the synthetic geometry of the ancients it is actually analytical in the sense that the word is used in logic; and partly because the practice had then already arisen, of designating by the term analysis the calculus [i.e., symbolic calculation or computation] with general quantities."

"His Geometry is not easy reading. An edition appeared subsequently with notes by his friend De Beaune, which were intended to remove the difficulties."

"His philosophy has long since been superseded by other systems, but the analytical geometry of Descartes will remain a valuable possession forever."

"Among the earliest thinkers of the seventeenth and eighteenth centuries, who employed their mental powers toward the destruction of old ideas and the up-building of new ones, ranks Rene Descartes. Though he professed orthodoxy in faith all his life, yet in science he was a profound sceptic. He found that the world's brightest thinkers had been long exercised in metaphysics, yet they had discovered nothing certain; nay, they had even flatly contradicted each other. ...The certainty of the conclusions in geometry and arithmetic brought out in his mind the contrast between the true and false ways of seeking the truth. He thereupon attempted to apply mathematical reasoning to all sciences."

"He dedicated his Book of Principles to his most Illustrious Disciple, Elizabeth, Princess Palatine of the Rhine... The Princess had been Educated in the Knowledge of abundance of Languages, and in whatsoever Learning is comprised under the name of Litterae humaniores, or Politiores; but the elevation of, and profoundness of her genius and natural parts, would not suffer her to dwell long upon these Arts, by which the greatest Wits of her Sex, who are satisfied with desiring to seem somebody, are commonly limited. She desir'd to proceed to those parts of Learning, that the strongest Application of Men had advanced, and accomplish'd her self with, and became a great proficient in Philosophy and Mathematicks; till such time as seeing the Essays of Monsieur Des Cartes his Philosophy, she conceived such high esteem and affection for his Doctrine, that she look'd upon all she had learn'd till that time as good as nothing; and so put her self under his Tuition for to raise a new Structure upon his Principles. Thereupon she sends to him, to come and see her, that she might drink in the true Phi∣osophy at the Fountain Head; and the great desire to do her Service nearer, was one of the reasons that drew him to Leiden & to Eindegeest. Never did Master more happily improve the docibility, aptness, penetration, and withal the solidity of a Scholar's Mind. Having accustomed her insensibly to the profound Meditation of the grand Mysteries of Nature, and sufficiently exercising of her in the most abstracted Questions of Geometry, and the most sublime ones of Metaphysicks. There was no longer any thing abstruse or mysterious to her; and he ingeniously confesseth and owneth, that he had not yet met with any besides her (he excepted Regius in another place) that ever arrived at a perfect understanding of the Works he had published till that time. By this Testimony that he bore to the extraordinary Capacity of the Princess, he intended to distinguish her from those who were not able to apprehend his Metaphysicks, altho' they might have some insight into Geometry; and from those that were not able to understand his Geometry, altho' they might be pretty well vers'd in Metaphysical Truths. She continued to Philosophise with him Viva voce, till a certain Accident obliged her to absent herself from the Presence of the Queen of Bohemia her Mother, and to quit her abode in Holland for Germany; then she changed her Acquaintance into an Intelligence by Letter, which she kept afoot with him, by the Ministery of the Princesses her Sisters."

"[He] came back to Paris towards the middle of October [1644]. At his Arrival, An Edition of his Principles of philosophy... and the Latine Translation of his Essays [he found] finished, and the Copies came out of Holland. The Treatise of Principles did not come out, neither did that Piece he called his World, nor his Course of Philosophy, both of which were suppress'd. He had a mind to divide them into other Parts: The First of which contains the Principles of Humane Knowledge, which one may call the first Philosophy or Metaphysicks: wherein it hath very much relation and connexion with his Meditations. The Second contains what is most general in Philosophy, and the Explanation of the first Laws of Nature, and of the principles of natural things, the Proprieties of Bodies, Space, and Motion, &c.The Third contains a particular Explanation, of the System of the World, and more especially of what we mean by the Heavens and Celestial Bodies.The Fourth contains whatsoever belongs to the Earth. That which is most remarkable in this Work, is, That the Author after having first of all established the distinction and difference he puts between the Soul and the Body, when he hath laid down, for the Principles of corporeal things, bigness, figure and local motion; all which are things in themselves so clear and intelligible, that they are granted and received by every one whatsoever; he hath found out a way to explain all Nature in a manner, and to give a reason of the most wonderful Effects, without altering the Principles; yea, and without being inconsistent with himself in any thing whatsoever. Yet... he [had] not the presumption for all that to believe he had hit upon the explication of all natural things, especially such that do not fall under our senses, in the same manner as they really and truly are in themselves. He should do something indeed, if he could but come the nearest that it was possible to likelihood or verisimilitude, to which others before him could never reach; and if he could bring the matter about, that, whatsoever he had written should exactly agree with all the Phenomena's of Nature, this he judged sufficient for the use of Life, the profit and benefit of which seems to be the main and only end one ought to propose to himself in Mechanicks, Physick, or Medicine; and in all Arts that may be brought to perfection by the help of Physick or natural Philosophy. But of all things he hath explained, there is not one of them that doth not seem at least morally certain in respect of the profit of life, notwithstanding they may be uncertain in respect of the absolute Power of God. Nay, there are several of them that are absolutely, or more than morally certain; such as are Mathematical Demonstrations, and those evident ratiocinations he hath framed concerning the existence of material things. Nevertheless, he was indued with that Modesty, as no where to assume the authority of positively deciding, or ever to assert any thing for undeniable. Altho' what he intended to offer, under the Name of Principles of Philosophy, was brought to that Conclusion, that one could not lawfully nor reasonably require more for the perfecting his design; yet did it give some cause to his Friends, to hope to see the Explication of all other things, which made people say, That his Physick was not compleat. He promised himself likewise to explain after the same manner, the nature of other more particular Bodies, that belong to the Terrestrial Globe; as, Minerals, Plants, Animals, and Man in particular; After which, he proposed to himself (according as God should please to lengthen out his days) to treat with the same exactness of all Physick or Medicine, of Mechanicks, and of the whole Doctrine of Morality or Ethicks; whereby to present the World with an entire Body of Philosophy."

"Man occupies a special place in the Cartesian scheme. He alone is endowed with mind. Descartes believed that animals did not possess one, that they were simply extremely complicated automatons. Other thinkers have rejected this point of view and proposed to endow all matter in the universe—living or inanimate—with consciousness. This "panpsychism" has been promoted by, among others, Teilhard de Chardin and, more recently by the British-American physicist Freeman Dyson, who holds that mind is present in every particle of matter."

"After Bruno's death, during the first half of the seventeenth century, Descartes seemed about to take the leadership of human thought... in promoting an evolution doctrine as regards the mechanical formation of the solar system... but his constant dread of persecution, both from Catholics and Protestants, led him steadily to veil his thoughts and even to suppress them. The execution of Bruno had occurred in his childhood, and in the midst of his career he had watched the Galileo struggle in all its stages. He had seen his own works condemned by university after university under the direction of theologians and placed upon the Roman Index. ...Since Roger Bacon, perhaps, no great thinker had been so completely abased and thwarted by theological oppression."

"The first thing that we know about ourselves is our imperfection. This is what Descartes meant when he said: 'I know God before I know myself.' The only mark of God in us is that we feel that we are not God."

""Catatau" by Paulo Leminski captures that impulse as he imagines the French philosopher René Descartes driven mad by Brazil, a country where everything gets mixed and defies categorization."

"The truth is sum, ergo cogito — I am, therefore I think, although not everything that is thinks. Is not consciousness of thinking above all consciousness of being? Is pure thought possible, without consciousness of self, without personality? Can there exist pure knowledge without feeling, without that species of materiality which feelings lends to it? Do we not perhaps feel thought, and do we not feel ourselves in the act of knowing and willing? Could not the man in the stove [Descartes] have said: "I feel, therefore I am"? or "I will, therefore I am"? And to feel oneself, is it not perhaps to feel oneself imperishable?"

"Having laid down his rules of method, Descartes proceeded to deviate from them. We expect a demonstration of the truth after the principles have already been found, such as been prescribed by Plato. But we get a different way of accounting for the facts: "the way of hypothesis" proscribed by Plato. Unable to proceed further deductively, he resorted to the invention of and choice among different hypothesis, the choice being determined by crucial experiments. Thus he gave up the certainty of the a priori method in favor of the conjectural a posteriori. This meant that in his practice, the synthesis preceded analysis, for it preceded that form of it known as inductive testing."

"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."

"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."

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

"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."

"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."

"Despite Newton's belated appreciation of Euclid's 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 Newton's interest in mathematics, an interest that bordered on obsession."

"[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..."

"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...""

"Descartes's so-called dualism is often taken to represent a fundamental revolution in ideas and the starting point of modern philosophy. ...but in substance his work is... better understood as an attempt to conserve the old truths in the face of new threats. His dualism was in essence an armistice... between the established religion and the emerging science of his time. ...isolating the mind from the physical world... ensured that many of the central doctrines of orthodoxy—immortality of the soul, the freedom of will, and, in general, the "special" status of humankind—were rendered immune to any possible contravention by the scientific investigation of the physical world. ... For men such as Descartes, Malebranche, and Leibniz, solving the mind-body problem was vital to preserving the theological and political order inherited from the Middle Ages... For Spinoza, it was a means of destroying that same order and discovering a new foundation for human worth."

"Modesty was not a condition from which Descartes suffered."

"Descartes may have made a lot of mistakes, but he was right about this: you cannot doubt the existence of your own consciousness. That's the first feature of consciousness, it's real and irreducible. You cannot get rid of it by showing that it's an illusion in a way that you can with other standard illusions."

"Descartes is rightly regarded as the father of modern philosophy primarily and generally because he helped the faculty of reason to stand on its own feet by teaching men to use their brains in place whereof the Bible, on the one hand, and Aristotle, on the other, had previously served."

"In the theory of the state of the seventeenth century, the monarch is identified with God and has in the state a position exactly analogous to that attributed to God in the Cartesian system of the world."

"Descartes subscribed to the doctrine of instantaneous propagation, but with him something new emerged: for his was the first uncompromisingly mechanical theory that asserted the instantaneous propagation of light in a material medium... Indeed, mechanical analogies had been used to explain optical phenomena long before Descartes, but the Cartesian theory was the first clearly to assert that light itself was nothing but a mechanical property of the luminous object and of the transmitting medium. It is for this reason that we may regard Descartes' theory of light as legitimate starting point of modern physical optics."

"I would inquire of reasonable persons whether this principle: Matter is naturally wholly incapable of thought, and this other: I think, therefore I am, are in fact the same in the mind of Descartes, and in that of St. Augustine, who said the same thing twelve hundred years before. ...I am far from affirming that Descartes is not the real author of it, even if he may have learned it only in reading this distinguished saint; for I know how much difference there is between writing a word by chance without making a longer and more extended reflection on it, and perceiving in this word an admirable series of conclusions, which prove the distinction between material and spiritual natures, and making of it a firm and sustained principle of a complete metaphysical system, as Descartes has pretended to do. ...it is on this supposition that I say that this expression is as different in his writings from the saying in others who have said it by chance, as in a man full of life and strength, from a corpse."

"Descartes ... is distinguished from Bacon in respect of the thoroughness of his education in the Scholastic philosophy and in the profound impression that geometrical demonstration had upon his mind, and the effect of these differences in education and inspiration is to make his formulation of the technique of inquiry more precise and in consequence more critical. His mind is oriented towards the project of an infallible and universal method or research, but since the method he propounds is modelled on that of geometry, its limitation when applied, not to possibilities but to things, is easily apparent. Descartes is more thorough than Bacon in doing his scepticism for himself and, in the end, he recognizes it to be an error to suppose that the method can ever be the sole means of inquiry. The sovereignty of technique turns out to be a dream and not a reality. Nevertheless, the lesson his successors believed themselves to have learned from Descartes was the sovereignty of technique and not his doubtfulness about the possibility of an infallible method."

""Do you know who first explained the true origin of the rainbow?" I asked. "It was Descartes," he said. After a moment he looked me in the eye. "And what do you think was the salient feature of the rainbow that inspired Descartes' mathematical analysis?" he asked. "Well, the rainbow is actually a section of a cone that appears as an arc of the colors of the spectrum when drops of water are illuminated by sunlight behind the observer." "And?" "I suppose his inspiration was the realization that the problem could be analyzed by considering a single drop, and the geometry of the situation." "You're overlooking a key feature of the phenomenon," he said. "Okay, I give up. What would you say inspired his theory?" "I would say his inspiration was that he thought rainbows were beautiful." I looked at him sheepishly. He looked at me. "How's your work coming?" he asked. I shrugged. "It's not really coming." I wished I was like Constantine. It all came so easily to him. "Let me ask you something. Think back to when you were a kid. For you, that isn't going too far back. When you were a kid, did you love science? Was it your passion?" I nodded. "As long as I can remember." "Me, too," he said. "Remember, it's supposed to be fun." And he walked on."

"As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition."

"The first and typical example of the application of mathematics to the indirect investigation of truth, is within the limits of the pure science itself; the application of algebra to geometry, the introduction of which, far more than any of his metaphysical speculations, has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences. Its rationale is simple. It is grounded on the general truth, that the position of every point, the direction of every line, and consequently the shape and magnitude of every enclosed space, may be fixed by the length of perpendiculars thrown down upon two straight lines, or (when the third dimension of space is taken into account) upon three plane surfaces, meeting one another at right angles in the same point. A consequence or rather a part of this general truth is that, curve lines and surfaces may be determined by their equations."

"Descartes... fell back on his original confusion of matter with space—space being, according to him, the only form of substance, and all existing things but affections of space. This error... forms one of the ultimate foundations of the system of Spinoza."

"The primary property of matter was indeed distinctly announced by Descartes in what he calls the "First Law of Nature": "That every individual thing, so far as in it lies, perseveres in the same state, whether of motion or of rest.""

"By... confounding the properties of matter with those of space he arrives at the logical conclusion, that if the matter within a vessel could be entirely removed the space within the vessel would no longer exist. In fact he assumes that all space must be always full of matter."

"The one book that turned out to be perhaps the most influential in guiding Newton's mathematical and scientific thought was none other than Descartes' La Géométrie. Newton read it in 1664 and re-read it several times until "by degrees he made himself master of the whole." ...Not only did analytic geometry pave the way for Newton's founding of calculus... but Newton's inner scientific spirit was truly set ablaze."

"While Descartes' theory of vortices was spectacularly wrong (as Newton ruthlessly pointed out later), it was still interesting, being the first serious attempt to formulate a theory of the universe as a whole based upon the same laws that apply on the Earth's surface. In other words, to Descartes there was no difference between terrestrial and celestial phenomena—the Earth was part of a universe that obeyed uniform physical laws."

"Functions are the bread and butter of modern scientists, statisticians, and economists. Once many repeated... experiments and observations produce the same functional interrelationships, those may acquire the... status of laws of nature—mathematical descriptions... Descartes' ideas... opened the door for a systematic mathematization of everything—the very essence of the notion that God is a mathematician. ...[B]y establishing the equivalence of two perspectives of mathematics (algebraic and geometric) previously considered disjoint, Descartes expanded the horizons of mathematics and paved the way to the modern era of analysis, which allows [us] to comfortably cross from one mathematical discipline to another."

""There is one basis of science," says Descartes, "one test and rule of truth, namely, that whatever is clearly and distinctly conceived is true." A profound psychological mistake. It is true only of formal logic, wherein the mind never quits the sphere of its first assumptions to pass out into the sphere of real existences; no sooner does the mind pass from the internal order to the external order, than the necessity of verifying the strict correspondence between the two becomes absolute. The Ideal Test must be supplemented by the Real Test, to suit the new conditions of the problem."

"The rationalism of René Descartes had a liberating effect on women because it assumed that the mind, not the body, was the instrument for sensation and knowledge and that men and women had the same potential for understanding. Cartesianism denied that formal education was the road to higher insight; anyone could think and reason logically. The effect of these ideas was not only to inspire a number of women, such as Mary Astell, Lady Damaris Masham, Marie de Gournay, to enter philosophical discourse with the outstanding male thinkers of their time, but it also helped them to create a new form for such a discourse through personal correspondence."

"As for Descartes, this is... not the place to praise a man whose genius is elevated beyond all praise. He certainly began the true and right way through the ideas, and that which leads so far; but since he had aimed at his own excessive applause, he seems to have broken off the thread of his investigation and to have been content with metaphysical meditations and geometrical studies by which he could draw attention to himself. For the rest, he set out to discover the nature of bodies for the purposes of medicine, rightly indeed, if he had completed the task of ordering the ideas of the mind, for a greater light than can well be imagined would have arisen from these very experiments. His failure to apply his mind to this problem can be explained by no other cause than that he did not adequately think through the full reason and force of the thing. For had he seen a method of setting up a reasonable philosophy with the same unanswerable clarity as arithmetic, he would hardly have used any way other than this to establish a sect of followers, a thing which he so earnestly wanted. For by applying this method of philosophizing, a school would from its very beginning, and by the very nature of things, assert its supremacy in the realm of reason in a geometrical manner and could never perish nor be shaken until the sciences themselves die through the rise of a new barbarism among mankind."

"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."

"As long as algebra and geometry travelled separate paths their advance was slow and their applications limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection. It is to Descartes that we owe the application of algebra to geometry,—an application which has furnished the key to the greatest discoveries in all branches of mathematics."

"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."

"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.""