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René Descartes

René Descartes (March 31, 1596 – February 11, 1650) was a highly influential French philosopher, mathematician, physicist and writer. He is known for his influential arguments for substance dualism, where mind and body are considered to have distinct essences, one being characterized by thought, the other by spatial extension. He has been dubbed the "Father of Modern Philosophy" and the "Father of Modern Mathematics." He is also known as Cartesius.

 * See also
 * Discourse on the Method (1637)
 * La Géométrie (1637)
 * Meditations on First Philosophy (1641)
 * Principles of Philosophy (1644)

Quotations

 * No doubt you know that Galileo had been convicted not long ago by the Inquisition, and that his opinion on the movement of the Earth had been condemned as heresy. Now I will tell you that all things I explain in my treatise, among which is also that same opinion about the movement of the Earth, all depend on one another, and are based upon certain evident truths. Nevertheless, I will not for the world stand up against the authority of the Church. ...I have the desire to live in peace and to continue on the road on which I have started.
 * Letter to Marin Mersenne (end of Feb., 1634) as quoted by, Pendulum: Leon Foucault and the Triumph of Science (2003)

 * What I have given in the second book on the nature and properties of curved lines, and the method of examining them, is, it seems to me, as far beyond the treatment in the ordinary geometry, as the rhetoric of Cicero is beyond the a, b, c of children.
 * Letter to Marin Mersenne (1637) as quoted by D. E. Smith & M. L. Latham Tr. The Geometry of René Descartes (1925)

 * M. Desargues puts me under obligations on account of the pains that it has pleased him to have in me, in that he shows that he is sorry that I do not wish to study more in geometry, but I have resolved to quit only abstract geometry, that is to say, the consideration of questions which serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the explanation of the phenomena of nature... You know that all my physics is nothing else than geometry.
 * Letter to Marin Mersenne (July 27, 1638) as quoted by Florian Cajori, A History of Mathematics (1893) letter dated in The Philosophical Writings of Descartes Vol. 3, The Correspondence (1991) ed. John Cottingham, Robert Stoothoff, Dugald Murdoch

 * Mr. Clerselier has written me that you are expecting from him my Meditations... in order to present them to the queen of the land. ...If I had only been as wise as they say the savages persuaded themselves that the monkeys were, I never would have become known as a maker of books: Since it is said that they imagined that the monkeys could indeed speak, if they wanted to, but that they chose not to so lest they be forced to work. And since I had not the same prudence to abstain from writing, I now have neither as much liesure nor as much peace as I would have had if I had kept quiet. But since the mistake has already been made, and since I am now known by an infinity of people at the academy, who look askance at my writings and scour them for means of harming me, I do have great hope of being known to persons of great merit, whose power and virtue could protect me.
 * Letter to (Nov. 1, 1646) as quoted by, Descartes' Secret Notebook (2005) citing René Descartes: Correspondance avec Elizabeth et autres lettres (1989) ed., Jean-Marie and M. Beysaade, pp. 245-246.

 * Me tenant comme je suis, un pied dans un pays et l'autre en un autre, je trouve ma condition très heureuse, en ce qu'elle est libre.
 * Staying as I am, one foot in one country and the other in another, I find my condition very happy, in that it is free.
 * Letter to Elisabeth of Bohemia, Princess Palatine (Paris, June/July 1648)

 * So blind is the curiosity by which mortals are possessed, that they often conduct their minds along unexplored routes, having no reason to hope for success, but merely being willing to risk the experiment of finding whether the truth they seek lies there.
 * Rules for the Direction of the Mind: IV

 * The entire method consists in the order and arrangement of the things to which the mind's eye must turn so that we can discover some truth.
 * Rules for the Direction of the Mind: X.379
 * As quoted in

 * No more useful inquiry can be proposed than that which seeks to determine the nature and the scope of human knowledge. ... This investigation should be undertaken once at least in his life by anyone who has the slightest regard for truth, since in pursuing it the true instruments of knowledge and the whole method of inquiry come to light. But nothing seems to me more futile than the conduct of those who boldly dispute about the secrets of nature ... without yet having ever asked even whether human reason is adequate to the solution of these problems.
 * Rules for the Direction of the Mind in Key Philosophical Writings (1997), pp. 29-30

 * Mais apud me omnia fiunt Mathematicè in Natura
 * Loosely translated: With me, everything turns into mathematics.
 * More closely translated as: but in my opinion, all things in nature occur mathematically.
 * Note: "Mais" is French for "but" and the "but in my opinion" comes from the context of the original conversation. apud me omnia fiunt Mathematicè in Natura is in latin.
 * Sometimes the Latin version is incorrectly quoted as Omnia apud me mathematica fiunt.
 * Sources: Correspondence with Mersenne note for line 7 (1640), page 36, Die Wiener Zeit page 532 (2008); StackExchange Math Q/A Where did Descartes write...

Le Discours de la Méthode (1637)

 * Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences

 * Le bon sens est la chose du monde la mieux partagée : car chacun pense en être si bien pourvu, que ceux même qui sont les plus difficiles à contenter en toute autre chose, n'ont point coutume d'en désirer plus qu'ils en ont.
 * Good sense is, of all things among men, the most equally distributed; for every one thinks himself so abundantly provided with it, that those even who are the most difficult to satisfy in everything else, do not usually desire a larger measure of this quality than they already possess.

 * Je pense, donc je suis.
 * I think, therefore I am.

La Géométrie (1637)

 * Translated as The Geometry of René_Descartes (1925) by David E. Smith and Marcia L. Lantham, unless otherwise noted.

But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by anyone at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise.
 * Thus, all unknown quantities can be expressed in terms of a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth.
 * First Book

 * I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and them classify them in order, is by recognizing the fact that all points of those curves which we may call "geometric," that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by a single equation. If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class, which contains only the circle, the parabola, the hyperbola, and the ellipse; but when the equation contains one or more terms of the third or fourth degree in one or both of the two unknown quantities (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class; and if the equation contains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely.
 * Second Book

Principles of Philosophy (1644)

 * In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things.
 * Original Latin: Veritatem inquirenti, semel in vita de omnibus, quantum fieri potest, esse dubitandum
 * Variant translation: If you would be a real seeker after truth, you must at least once in your life doubt, as far as possible, all things.
 * Variant translation: If you would be a real seeker after truth, you must at least once in your life doubt, as far as possible, all things.

A-B

 * Analytical geometry was invented by Descartes and the first exposition of it was given in 1637: that exposition was both difficult and obscure, and to most of his contemporaries, to whom the method was new, it must have been incomprehensible. Wallis made the method intelligible to all mathematicians.
 * W. W. Rouse Ball, A History of the Study of Mathematics at Cambridge (1889) p. 44.

 * René Descartes is more widely known as a philosopher than as a mathematician, although his philosophy has been controverted while his mathematics has not. ...In accordance with the ideals of his age, when experimental science was first seriously challenging arrogant speculation, Descartes set a greater store by his philosophy than his mathematics. But he fully appreciated the power of his new method in geometry.
 * Eric Temple Bell, The Development of Mathematics (1940)

Descartes also saw that his method applies equally as well to surfaces... but he did not develop this. With the extension to surfaces, there was no reason why geometry should stop with equations in three variables; and the generalization to systems of equations in any finite number of variables was readily made in the nineteenth century. Finally, in the twentieth century, the farthest extension possible in this direction led to spaces of a non-denumerable infinity of dimensions. ...The path from Descartes to the creators of higher space is straight and clear; the remarkable thing is that it was not traveled earlier than it was.
 * The implications of Descartes' analytic reformulation of geometry are obvious. Not only did the new method make possible a systematic investigation of known curves, but, what is of infinitely greater significance, it potentially created a whole universe of geometric forms beyond conception by the synthetic method.
 * Eric Temple Bell, The Development of Mathematics (1940)

The principle of undetermined coefficients was also stated by Descartes. A second outstanding addition to algebra was the famous rule of signs... the first universally applicable criterion for the nature of the roots of an algebraic equation. ...it admirably represents Descartes' flair for generality which made him the mathematician that he was.
 * Descartes devised the notation x, x2, x3, x4,... for powers, and made the final break with the Greek tradition of admitting only the first, second, and third powers ('lengths,' 'areas,' and 'volumes') in geometry. After Descartes, geometers freely used powers higher than the third without a qualm, recognizing that representability as figures in Euclidean space for all of the terms in an equation is irrelevant to the geometrical interpretation of the analysis.
 * Eric Temple Bell, The Development of Mathematics (1940)

 * CARTESIAN, adj. Relating to Descartes, a famous philosopher, author of the celebrated dictum, Cogito ergo sum -- whereby he was pleased to suppose he demonstrated the reality of human existence. The dictum might be improved, however, thus: Cogito cogito ergo cogito sum -- "I think that I think, therefore I think that I am;" as close an approach to certainty as any philosopher has yet made.
 * Ambrose Bierce, The Cynic's Dictionary (1906); republished as The Devil's Dictionary (1911).

 * Descartes maintained his confidence in the instantaneity of light. ...Yet in his derivation of the law of refraction, Descartes reasoned that light traveled faster in a dense medium than in one less dense. He seems to have had no qualms about comparing infinite magnitudes!
 * Carl B. Boyer, The Rainbow: From Myth to Mathematics (1959)

 * Descartes was liable to be misled by too easy an acceptance of data that had been handed down by scholastic writers. ...two grand Aristotelian principles helped to condition the form of the universe as he reconstructed it—first, the view that a vacuum is impossible, and secondly, the view that objects could only influence one another if they actually touched—there could be no such thing as attraction, no such thing as . ...Descartes insisted that every fraction of space should be fully occupied all the time by continuous matter... infinitely divisible. The particles were... packed so tightly that one of them could not move without communicating the commotion to the rest. The matter formed whirlpools in the skies, and it was because the planets were caught each in its own whirlpool that they were carried around... all similarly caught in a larger whirlpool, which had the sun as its centre... Gravity itself was the result of these whirlpools of invisible matter which had the effect of sucking things down towards their centre. ...In the time of Newton the system of Descartes and the theory of vortices or whirlpools proved to be vulnerable to both mathematical and experimental attack.
 * Herbert Butterfield, The Origins of Modern Science (1949)

 * [B]elievers in the plenum... the Cartesian idea of a space entirely filled with matter—contributed... to what was to become the Newtonian synthesis. Descartes himself achieved the modern formulation of the law of ... working it out by a natural deduction from his theory of the conservation of momentum, his theory that the amount of motion in the universe always remains the same. It was he rather than Galileo who fully grasped this principle of inertia and formulated it in all its clarity. ...The modern doctrine of had been put forward by Descartes and was quickly gaining acceptance, though people like Borelli... still seemed to think that they had to provide a force actually pushing the planets along...

 * Herbert Butterfield, The Origins of Modern Science (1949)

C-E

 * The great invention... Descartes gave to the world, the analytical diagram, ...gives at a glance a graphical picture of the law governing a phenomenon, or of the correlation which exists between dependent events, or of the changes which a situation undergoes in the course of time. ...the invention of Descartes not only created the important discipline of analytic geometry, but it gave Newton, Leibnitz, Euler, and the Bernoullis that weapon for the lack of which Archimedes and later Fermat had to leave inarticulate their profound and far-reaching thoughts.
 * Tobias Dantzig, Number: The Language of Science (1930).

 * Descartes implicitly assumed a complete correspondence existed between the real numbers and the points of a fixed axis. ...tacitly, because it seemed so natural as to go without saying, he accepted it as axiomatic that between the points of a plane and the aggregate of all pairs of real numbers there can be established a perfect correspondence. Thus the Dedekind-Cantor axiom, extended to two dimensions, was tacitly incorporated in a discipline which was created two hundred years before Dedekind and Cantor saw the day. This discipline became the [tool and] proving-grounds for all achievements of the following two centuries: the calculus, the theory of functions, mechanics, and physics. Nowhere did this discipline, analytic geometry, strike any contradictions; and such was its power to suggest new problems and forecast the results that wherever applied it would soon become the indispensable tool of investigation.
 * Tobias Dantzig, Number: The Language of Science (1930).

 * The famous problems of antiquity [ doubling the cube, angle trisection, and squaring the circle ] ... were now disposed of by Descartes in a matter-of-fact statement that any problem which leads to an equation of the first degree is capable of a geometric solution by straight-edge only; that a straight-edge [and] compass construction is equivalent to the solution of a quadratic equation; but that if a problem leads to an irreducible equation of degree higher than the second, its geometrical solution is not possible by means of a ruler and compass only.
 * Tobias Dantzig, Number: The Language of Science (1930).

 * René Descartes played his part in the world as a man in a mask. The phrase is his own. It implied no conscious duplicity but a certain apartness rendering him incapable of taking his share unreservedly in the game of life. His attitude was that of an unimpassioned spectator. He looked on and learned while others struggled for the stakes. A born recluse, he remained solitary even in the throng of social intercourse. ...The transport felt by Bacon at the glorious vision of a future in which the harvest he had sown would be reaped was not shared by Descartes. He indeed held himself to be the sower and the reaper in one. Calmly, and with settled conviction, he claimed to have virtually expounded all the phenomena of nature. He had crossed the frontier of the new world of knowledge; those who desired to follow him needed only to purge their eyes with the euphrasy of methodic doubt in order to obtain those crystal-clear intuitions from which, by sure reasoning, universal science could be deduced. ...He was a mathematician of first rate originality and power. ...Unfortunately, however, he was misled by false analogies of thought; he sought to universalise what was by its nature special and restricted; and thus pursued the flitting vision of a deceptive unity in knowledge. As the upshot, he gave to his philosophy a pseudo-mathematical character, and sacrificed the solid achievement of much by aiming at the illusory certainty of everything.
 * The : Or Critical Journal, Volume 204 (July, 1906) No. 417, VII. "A Representative Philosopher," pp. 157-158.

 * This momentous finding of nonlocality has, in common with that of Einstein concerning time, the additional feature that it disproves the validity, not of a "view of the World", but of a deeply ingrained concept. And this brings me to the second reason. It is that this disproof of a deeply ingrained concept pointed in fact in a direction quite consonant with my own line of thought. In a way, it brings us back to Descartes, for, as we all know, Descartes was the first scientist who dared to question our common views, including even all the notions that had always seemed so primitive and obvious that thinkers, scientists and so on never hesitated in making use of them. He found out that, at the start, he could doubt of everything but his own thinking, and in this, according to Hegel, he was a hero. Unfortunately he then constructed a grand metaphysical argument that led him to the view that, after all, since God is not a liar, the "obvious" realistic concepts must apply. He thereby founded mechanicism, which is the theory that, apart from thought, everything has to be described by the exclusive means of familiar concepts. We know, of course, how deficient such a view is. But I think Descartes' really significant contribution on these matters is not mechanicism. It is what I just said. In other words, it is the realization that a sharp distinction has to be made between rationality on the one hand and the use of seemingly obvious concepts on the other hand. And that therefore, if you are a rational person you cannot demand that science should be based exclusively on seemingly obvious concepts without first logically justifying this demand. This Descartes tried to do but, since his argument based on God not being a liar is now considered as not convincing enough, we are not bound to his conclusion. Indeed, I consider that we are not even bound to the idea that physics should be expressed in an ontological language, which was more or less Einstein's view. The older Einstein seems to have considered that Reality, and even physical objects in the plural, can be described as they really are, if not by familiar concepts, at least by unfamiliar ones, such as those borrowed from mathematics. I do not think this is necessarily true. I consider that mere predictive rules, such as the Born rule in quantum mechanics, count as fully fledged explanations, or, more precisely, as constituting, when all taken together, a first step in an explanation, the second step being the philosophical idea that these predictive rules dimly reflect some existing, largely hidden structures of Mind-Independent Reality. Of course, these speculations of mine go much further than nonlocality. But you understand that they receive some support from it.
 * Bernard d'Espagnat, "My Interaction with John Bell", in Quantum [Un]speakables (2002) edited by R.A. Bertimann, A. Zeilinger

F-H

 * To say that madness is dazzlement is to say that the madman sees the day, the same day that rational men see, as both live in the same light, but that when looking at that very light, nothing else and nothing in it, he sees it as nothing but emptiness, night and nothingness. Darkness for him is another way of seeing the day. Which means that in looking at the night and the nothingness of the night, he does not see at all. And that in the belief that he sees, he allows the fantasies of his imagination and the people of his nights to come to him as realities. For that reason, delirium and dazzlement exist in a relation that is the essence of madness, just as truth and clarity, in their fundamental relation, are constitutive of classical reason. In that sense, the Cartesian progression of doubt is clearly the great exorcism of madness. Descartes closes his eyes and ears the better to see the true light of the essential day, thereby ensuring that he will not suffer the dazzlement of the mad, who open their eyes and only see night, and not seeing at all, believe that they see things when they imagine them. In the uniform clarity of his closed senses, Descartes has broken with all possible fascination, and if he sees, he knows he really sees what he is seeing. Whereas in the madman's gaze, drunk on the light that is night, images rise up and multiply, beyond any possible self-criticism, since the madman sees them, but irremediably separated from being, since the madman sees nothing. Unreason is to reason as dazzlement is to daylight.
 * Michel Foucault, History of Madness (1961), Part Two: 2. The Transcendence of Delirium

 * Descartes, I think, broke with this when he said, "To accede to truth, it suffices that I be any subject that can see what is evident." Evidence is substituted for ascesis at the point where the relationship to the self intersects the relationship to others and the world. The relationship to the self no longer needs to be ascetic to get into relation to the truth. It suffices that the relationship to the self reveals to me the obvious truth of what I see for me to apprehend the truth definitively. Thus, I can be immoral and know the truth. I believe this is an idea that, more or less explicitly, was rejected by all previous culture. Before Descartes, one could not be impure, immoral, and know the truth. With Descartes, direct evidence is enough. After Descartes, we have a nonascetic subject of knowledge. This change makes possible the institutionalization of modern science.
 * Michel Foucault, "On the Genealogy of Ethics", interview in 1983, published in Michel Foucault, Ethics: Subjectivity and Truth (1997), edited by Paul Rabinow, translated by Robert Hurley and others

 * Newton's proof of the law of refraction is based on an erroneous notion that light travels faster in glass than in air, the same error that Descartes had made. This error stems from the fact that both of them thought that light was corpuscular in nature.
 * John Freely, Before Galileo, The Birth of Modern Science in Medieval Europe (2012)

(1) That the problem should be imagined as done. (2) That then lines, whether known or unknown, which appear necessary in its solution, should be named. (3) And finally, that their relation to each other should be sought, and expressed by means of an equation or equations. Descartes strongly advocates this analytic treatment of Geometry as giving greater clearness and more continuity of argument, qualities lacking in the work of the Ancients, who probably did not understand where such reasoning would carry them, and whose isolated proofs were necessarily wanting in connection and generality.
 * The Geometry is divided into three books. In the first book Descartes briefly explains his method. He says that every geometric problem may be reduced to a problem of straight lines; and he points out that, in order to find these lines, nothing more advanced is required than the five fundamental operations of Arithmetic, viz.: Addition, Subtraction, Multiplication, Division, and Root Extraction. He advises:—
 * Elizabeth Sanderson Haldane, Descartes: His Life and Times (1905) Ch. XVII. Descartes' Geometry

 * It should be remembered that it was Descartes who systematised our Mathematical notation. He used the letters at the end of the alphabet as variables, and those at the beginning as constants, and he brought into general use our present system of indices. He also introduced the method of indeterminate co-efficients for the solution of equations.
 * Elizabeth Sanderson Haldane, Descartes: His Life and Times (1905) Ch. XVII. Descartes' Geometry

 * There seems to me to exist a sort of rationalism which, by not recognizing these limits of the powers of individual reason, in fact tends to make human reason a less effective instrument than it could be. ... This sort of rationalism is a comparatively new phenomenon, though its roots go back to ancient Greek philosophy. Its modern influence, however, begins only in the sixteenth and seventeenth century and particularly with the formulation of its main tenets by the French philosopher, René Descartes.
 * Friedrich Hayek, "Kinds of Rationalism", The Economic Studies Quarterly (1965)

If one follows the great difficulty which even eminent scientists like Einstein had in understanding and accepting the Copenhagen interpretation... one can trace the roots... to the Cartesian partition....it will take a long time for it [this partition] to be replaced by a really different attitude toward the problem of reality.'''
 * [I]n the Copenhagen interpretation of quantum theory we can indeed proceed without mentioning ourselves as individuals, but we cannot disregard the fact that natural science is formed by men. '''Natural science does not simply describe and explain nature; it is part of the interplay between nature and ourselves; it describes nature as exposed to our nature of questioning. This was a possibility of which Descartes could not have thought, but it makes a sharp separation between the world and the I impossible.
 * Werner Heisenberg, Physics and Philosophy: The Revolution in Modern Science (1958).

 * Descartes was an eminent mathematician, and it would seem that the bent of his mind led him to overestimate the value of deductive reasoning from general principles, as much as Bacon had underestimated it.
 * Thomas Henry Huxley, The Advance of Science in the Last Half-Century (1889)

I-K

 * There is no doubt of the enormous influence Descartes has exercised from his own day to ours. But his relation to modern philosophy is not that of father to son, nor of architect to palace, nor of planner to city. Rather, in the history of philosophy his position is like that of the waist of an hourglass. As the sand in the upper chamber of such a glass reaches its lower chamber only through the slender passage between the two, so too ideas that had their origin in the Middle Ages have reached the modern world through a narrow filter: the compressing genius of Descartes.
 * Anthony Kenny, The Rise of Modern Philosophy (2006), Chap. 2 : Descartes to Berkeley

 * Aristotle remarks in his Poetics that poetry is superior to history, because history presents only what has occurred, poetry what could and ought to have occurred, poetry has possibility at its disposal. Possibility, poetic and intellectual, is superior to actuality; the esthetic and the intellectual are disinterested. But there is only one interest, the interest in existing; disinterestedness is the expression for indifference to actuality. The indifference is forgotten in the Cartesian Cogito-ergo sum, which disturbs the disinterestedness of the intellectual and offends speculative thought, as if something else should follow from it. I think, ergo I think; whether I am or it is (in the sense of actuality, where I means a single existing human being and it means a single definite something) is infinitely unimportant. That what I am thinking is in the sense of thinking does not, of course, need any demonstration, nor does it need to be demonstrated by any conclusion, since it is indeed demonstrated. But as soon as I begin to want to make my thinking teleological in relation to something else, interest enters the game. As soon as it is there, the ethical is present and exempts me from further trouble with demonstrating my existence, and since it obliges me to exist, it prevents me from making an ethically deceptive and metaphysically unclear flourish of a conclusion.
 * Søren Kierkegaard Concluding Unscientific Postscript, Hong P. 318-319

 * Thus was the Nixon Administration first exposed to the maddening diplomatic style of the North Vietnamese. It would have been impossible to find two societies less intended by fate to understand each other than the Vietnamese and the American. On the one side, Vietnamese history and Communist ideology combined to produce almost morbid suspicion and ferocious self-righteousness. This was compounded by a legacy of Cartesian logic from French colonialism that produced an infuriatingly doctrinaire technique of advocacy.
 * Henry A. Kissinger, The White House Years

 * The essential difference between Descartes and Vieta is not in the least that Descartes unites "arithmetic" and "geometry" into a single science while Vieta retains their separation. ...both have in mind a universal science: Descartes' "mathesis universalis" corresponds completely to Vieta's "zetetic," by means of which is realized, with the aid of "logistica speciosa," the "new" and "pure" algebra, interpreted as a general "analytic art." But whereas Vieta sees the most important part of analytic in "rhetoric" or "exegetic" in which the numerical computations and the geometric constructions indeed represent two different possibilities of application (so that the traditional conception of geometry is preserved), Descartes begins by understanding geometric "figures" as structures whose "being" is determined solely by their symbolic character. The truth is that Descartes does not, as is often thoughtlessly said, identify "arithmetic" and "geometry"—rather he identifies "algebra" understood as symbolic logistic with geometry interpreted by him for the first time as a symbolic science.
 * Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968) p. 206.

 * As a symbol of the power of absolutism, Versailles has no equal. It also expresses, in the most monumental terms of its age, the rationalistic creed—based on scientific advances, such as the physics of Sir Isaac Newton (1642–1727) and the mathematical philosophy of René Descartes (1596–1650)—that all knowledge must be systematic and all science must be the consequence of the intellect imposed on matter. The whole spectacular design of Versailles proudly proclaims the mastery of human intelligence (and the mastery of Louis XIV) over the disorderliness of nature.
 * Fred S. Kleiner, Gardner's Art through the Ages: A Global History (2009)

 * 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.
 * Morris Kline, Mathematics for the Nonmathematician (1967) pp. 255-256.

 * René Descartes famously argued that a dog howling pitifully when hit by a carriage does not feel pain. The dog is simply a broken machine, devoid of the res cogitans or cognitive substance that is the hallmark of people. For those who argue that Descartes didn't truly believe that dogs and other animals had no feelings, I present the fact that he, like other natural philosophers of his age, performed vivisection on rabbits and dogs. That's live coronary surgery without anything to dull the agonizing pain. As much as I admire Descartes as a revolutionary thinker, I find this difficult to stomach.
 * Christof Koch: The Feeling of Life Itself: Why Consciousness is Widespread but Can't be Computed (2019), p. 26

 * 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."
 * Christof Koch: The Feeling of Life Itself: Why Consciousness is Widespread but Can't be Computed (2019), p. 180

 * 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.
 * Alexandre Kojève, Introduction to the Reading of Hegel: Lectures on the Phenomenology of Spirit, assembled by Raymond Queneau, edited by Allan Bloom, translated by James H. Nichols, Jr. (1969), p. 36

L-M

 * 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.
 * Joseph Louis Lagrange, Lecture V. On the Employment of Curves in the Solution of Problems, Lecons élémentaires sur les mathematiques (1795) Lectures on Elementary Mathematics (1898) as quoted in 2nd ed. (1901) p. 127, Tr. Thomas J. McCormack.

 * 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.
 * Gottfried Wilhelm Leibniz, Letter to Nicolas Malebranche (Jan. 23, 1679) as quoted in Philosophical Papers and Letters (1956) ed., Leroy E. Loemaker

 * 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.
 * Gottfried Wilhelm Leibniz, "On the General Characteristic," (ca. 1679) as quoted in Philosophical Papers and Letters (1956) ed., Leroy E. Loemaker

 * 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.
 * Gerda Lerner The Creation of Feminist Consciousness (1993)

 * "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.
 * George Henry Lewes, Aristotle: a Chapter from the History of Science (1864)

 * 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.
 * Mario Livio, Is God a Mathematician? (2009)

 * 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.
 * Mario Livio, Is God a Mathematician? (2009)

 * 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.
 * Mario Livio, Is God a Mathematician? (2009)

 * 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.
 * James Clerk Maxwell, Matter and Motion (1876)

 * 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."
 * James Clerk Maxwell, Matter and Motion (1876)

 * 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.
 * James Clerk Maxwell, Matter and Motion (1876)

 * 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.
 * John Stuart Mill, An Examination of Sir William Hamilton's Philosophy (1865) as quoted in 5th ed. (1878) pp. 617-618.

N-P

 * 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.
 * Isaac Newton, Opticks (1704)

 * "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.
 * Leonard Mlodinow, Feynman's Rainbow: A Search for Beauty in Physics and in Life (2003)

 * 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.
 * Michael Oakeshott, "Rationalism in Politics" (1947), published in Rationalism in Politics and other essays (1962)

 * 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.
 * Note: see St. Augustine, Civitate Dei, 1. XI, c. xxvi
 * Blaise Pascal, The Art of Persuasion

S

 * 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.
 * A. I. Sabra, Theories of Light, from Descartes to Newton (1981)

 * 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.
 * Carl Schmitt, Political Theology

 * 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.
 * Arthur Schopenhauer, "Sketch for a history of the doctrine of the ideal and the real," Parerga and Paralipomena, E. Payne, trans. (1974) Vol. 1, p. 3

 * 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.
 * John Searle, "Our shared condition — consciousness", TED.com (May, 2013)

 * Modesty was not a condition from which Descartes suffered.
 * Russell Shorto, Descartes' Bones (2008), ISBN 978-0-307-27566-0, p. 15

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.
 * 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. ...
 * Matthew Stewart, The Courtier and the Heritic (2006)

 * 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..."
 * , Mathematics and Its History (1989)

 * [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...
 * , Mathematics and Its History (1989, 2002) 2nd edition, p. 115.

But Descartes had ignited Newton's interest in mathematics, an interest that bordered on obsession.
 * 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.
 * Mitch Stokes, Isaac Newton (2010)

 * 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.
 * , The History of Mathematics in Europe (1925)

T-Z

 * 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.
 * ,, The Architecture of Matter (1962)

 * It was left up to Newton to compute the detailed implications of the vortex-theory... and the result demolished the foundations of Descartes' cosmology.
 * Stephen Toulmin, June Goodfield, The Architecture of Matter (1962)

 * 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 Architecture of Matter (1962)

 * 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?
 * Miguel de Unamuno, The Tragic Sense of Life (1913) Ch.2

 * "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.
 * Cecilia Vicuña The Oxford Book of Latin American Poetry (2009)

 * 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.
 * Simone Weil, Lectures on Philosophy (1959), as translated by Hugh Price (Cambridge: 1978), p. 90

 * 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.
 * Andrew Dickson White, A History of the Warfare of Science with Theology in Christendom Ch.1, p. 57 (1896)

 * 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.
 * Trinh Xuan Thuan, Chaos and Harmony (2001)

The Life of Monsieur Des Cartes (1693)

 * Source, as translated by S. R., from, La Vie De Monsieur Des-Cartes (1691)

 * [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.
 * BOOK VII. From 1644 till 1650, pp. 197-200.

 * 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.
 * BOOK VII. From 1644 till 1650, pp. 200-202.

A History of Mathematics (1893, 1919)

 * Florian Cajori, source of 1919, 2nd edition, revised and enlarged

 * 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.

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

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

 * 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.

 * 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.

 * 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.

 * 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.

A History of Elementary Mathematics (1898, 1901)

 * Florian Cajori, Copyright 1898. Set up and electrotyped January, 1894; Reprinted March, 1895; October, 1897; November, 1901; source of 1901 reprint.

 * 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 sometimes used algebra in connection with geometry.

 * 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.

Misattributed

 * An optimist may see a light where there is none, but why must the pessimist always run to blow it out?
 * Michel de Saint-Pierre, as quoted in Cryptograms and Spygrams (1981) by Norma Gleason, p. 106; attributed to Descartes in The Athlete's Way : Training Your Mind and Body to Experience the Joy of Exercise (2008) by Christopher Bergland, p. 271.

 * Doubt is the origin of wisdom and Latin: Dubium sapientiae initium. This has been attributed to Descartes, including here previously, but no original attribution has been found. Descartes Meditationes de prima philosophia has been cited as the source of Dubium sapientiae initium, but this quote is not found in this work.
 * The Latin version may be found in a footnote of the 1880 edition of "Publilii Syri mimi Sententiae".

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Archimedes

Archimedes of Syracuse (c. 287 BC – c. 212 BC) was a Greek mathematician, philosopher, scientist and engineer.

Quotes

 * εὕρηκα [heúrēka]
 * I have found it! or I have got it!, commonly quoted as Eureka!
 * What he exclaimed as he ran naked from his bath, realizing that by measuring the displacement of water an object produced, compared to its weight, he could measure its density (and thus determine the proportion of gold that was used in making a king's crown); as quoted by Vitruvius Pollio in De Architectura, ix.215;

 * δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω. [Dôs moi pâ stô, kaì tàn gân kinásō.]
 * Give me the place to stand, and I shall move the earth.
 * Said to be his assertion in demonstrating the principle of the lever; as quoted by Pappus of Alexandria, Synagoge, Book VIII, c. AD 340; also found in Chiliades (12th century) by John Tzetzes, II.130. This and "Give me a place to stand, and I shall move the world" are the most commonly quoted translations.
 * Variant translations:
 * Give me a place to stand and with a lever I will move the whole world.
 * This variant derives from an earlier source than Pappus: The Library of History of Diodorus Siculus, Fragments of Book XXVI, as translated by F. R. Walton, in Loeb Classical Library (1957) Vol. XI. In Doric Greek this may have originally been Πᾷ βῶ, καὶ χαριστίωνι τὰν γᾶν κινήσω πᾶσαν [Pā bō, kai kharistiōni tan gān kinēsō [variant kinasō] pāsan].
 * Give me a lever and a place to stand and I will move the earth.
 * Give me a fulcrum, and I shall move the world.
 * Give me a firm spot on which to stand, and I shall move the earth.

 * Noli turbare circulos meos. or Noli tangere circulos meos.
 * Do not disturb my circles!
 * Original form: "noli … istum disturbare" ("Do not … disturb that (sand)") — Valerius Maximus, Memorable Doings and Sayings, Book VIII.7.ext.7 (See Chris Rorres (Courant Institute of Mathematical Sciences) – "Death of Archimedesː Sources"). This quote survives only in its Latin version or translation. In modern era, it was paraphrased as Noli turbare circulos meos and then translated to Katharevousa Greek as "μὴ μου τοὺς κύκλους τάραττε".
 * Reportedly his last words, said to a Roman soldier who, despite being given orders not to, killed Archimedes during the conquest of Syracuse; as quoted in World Literature: An Anthology of Human Experience (1947) by Arthur Christy, p. 655.

On Spirals (225 B.C.)

 * As translated by T. L. Heath, The Works of Archimedes (1897) unless otherwise indicated.

 * How many theorems in geometry which have seemed at first impracticable are in time successfully worked out!

 * Those who claim to discover everything but produce no proofs of the same may be confuted as having actually pretended to discover the impossible.

On the Equilibrium of Planes

 * or The Centres of Gravity of Planes
 * as translated by T. L. Heath, The Works of Archimedes (1897) unless otherwise indicated.

 * Equal weights at equal distances are in equilibrium and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.
 * Book 1, Postulate 1.

 * If two equal weights have not the same centre of gravity, the centre of gravity of both taken together is at the middle point of the line joining their centres of gravity.
 * Book 1, Proposition 4.

 * Two magnitudes whether commensurable or incommensurable, balance at distances reciprocally proportional to the magnitudes.
 * Book 1, Propositions 6 & 7, The Law of the Lever.

 * The centre of gravity of any parallelogram lies on the straight line joining the middle points of opposite sides.
 * Book 1, Proposition 9.

 * The centre of gravity of a parallelogram is the point of intersection of its diagonals.
 * Book 1, Proposition 10.

 * In any triangle the centre of gravity lies on the straight line joining any angle to the middle point of the opposite side.
 * Book 1, Proposition 13.

 * It follows at once from the last proposition that the centre of gravity of any triangle is at the intersection of the lines drawn from any two angles to the middle points of the opposite sides respectively.
 * Book 1, Proposition 14.

The Method of Mechanical Theorems

 * As quoteed in The Method of Archimedes, recently discovered by Heiberg: a supplement to the Works of Archimedes (1912) Ed. T. L. Heath unless otherwise indicated.

 * I thought fit to... explain in detail in the same book the peculiarity of a certain method, by which it will be possible... to investigate some of the problems in mathematics by means of mechanics. This procedure is... no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards... But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.

 * I am persuaded that it [The Method of Mechanical Theorems] will be of no little service to mathematics; for I apprehend that some, either of my contemporaries or of my successors, will, by means of the method when once established, be able to discover other theorems in addition, which have not yet occurred to me.

Any segment of a section of a right angled cone (i.e., a parabola) is four-thirds of the triangle which has the same base and equal height, and after this I will give each of the other theorems investigated by the same method. Then at the end of the book I will give the geometrical [proofs of the propositions]...
 * First then I will set out the very first theorem which became known to me by means of mechanics, namely that

 * The centre of gravity of any cylinder is the point of bisection of the axis.
 * Proposition presumed from previous work.

 * The centre of gravity of any cone is [the point which divides its axis so that] the portion [adjacent to the vertex is] triple [of the portion adjacent to the base].
 * Proposition presumed from previous work.

 * Any segment of a right-angled conoid (i.e., a paraboloid of revolution) cut off by a plane at right angles to the axis is 1½ times the cone which has the same base and the same axis as the segment
 * Proposition 4.

 * The centre of gravity of any hemisphere [is on the straight line which] is its axis, and divides the said straight line in such a way that the portion of it adjacent to the surface of the hemisphere has to the remaining portion the ratio which 5 has to 3.
 * Proposition 6.

Quotes about Archimedes
sorted chronologically

 * Archimedes said, “Give to me a fulcrum on which to plant my lever, and I will move the world.” And I say, give to woman the ballot, the political fulcrum, on which to plant her moral lever, and she will lift the world into a nobler purer atmosphere.
 * Susan B. Anthony from here 1870s

 * Shall we not make an end to this fighting against this geometrical Briareus who uses our ships like cups to ladle water from the sea, drives off our sambuca ignominiously with cudgel-blows, and by the multitude of missiles that he hurls at us all at once outdoes the hundred-handed giants of mythology?
 * Marcus Claudius Marcellus, during the as quoted by, A History of Greek Mathematics (1921) Vol. 2, p. 17, citing Plutarch, Marcellus.

 * When... the Romans assaulted the walls in two places at once, fear and consternation stupefied the Syracusans.... But when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons... that came down with incredible noise and violence... they knocked down those upon whom they fell in heaps, breaking all their ranks and files. ...huge poles thrust out from the walls, over the ships, sunk some by the great weights... from on high... others they lifted up into the air by an iron hand or beak like a crane's... and... plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks... under the walls, with great destruction of the soldiers... aboard them. A ship was frequently lifted up to a great height in the air... and was rolled to and fro... until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall. At the engine [called Sambuca] that Marcellus brought upon the bridge of ships... while it was as yet approaching the wall, there was discharged a... rock of ten talents [600-700 lb. total] weight, then a second and a third, which, striking upon it with immense force and a noise like thunder, broke all its foundation to pieces... and completely dislodged it from the bridge. So Marcellus... drew off his ships to a safer distance, and sounded a retreat... They then took a resolution of coming up under the walls... in the night; thinking that as Archimedes used ropes stretched at length in playing his engines, the soldiers would now be under the shot, and the darts would... fly over their heads... But he... had... framed... engines accommodated to any distance, and shorter weapons; and... with engines of a shorter range, unexpected blows were inflicted on the assailants. Thus... instantly a shower of darts and other missile weapons was again cast upon them. And when stones came tumbling down... upon their heads, and... the whole wall shot out arrows at them, they retired. ...as they were going off, arrows and darts of a longer range inflicted a great slaughter among them, and their ships were driven one against another; while they themselves were not able to retaliate... For Archimedes had provided and fixed most of his engines immediately under the wall; whence the Romans, seeing that indefinite mischief overwhelmed them from no visible means, began to think they were fighting with the gods.
 * Plutarch (75 AD) describing the  in "Marcellus,"  Tr. John Dryden.

 * Abstract enquiries into the most puzzling problems did not arise in the brain of Archimedes as a spontaneous and hitherto untouched subject, but rather as a reflection of prior enquiries in the same direction and by men separated from his days by as long a period — and far longer — than the one which separates you from the great Syracusian.
 * Koot Hoomi, as quoted in The Mahatma Letters to A. P. Sinnett  ~1880 and 1884, p. 2,  (1923).

 * Archimedes originally solved the problem of finding the solid content of a sphere before that of finding its surface, and he inferred the result of the latter problem from that of the former. ...another illustration of the fact that the order of propositions in the treatises of the Greek geometers as finally elaborated does not necessarily follow the order of discovery.
 * Thomas Little Heath, The Method of Archimedes, recently discovered by Heiberg: a supplement to the Works of Archimedes (1897) footnote, p. 21.

 * Some of the later Greeks, such as Archimedes, had just views on the elementary phenomena of and optics. Indeed, Archimedes, who combined a genius for mathematics with physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics.
 * Alfred North Whitehead, An Introduction to Mathematics (1911) p. 37.

Now, this upward force can easily be obtained by weighing the body as it hangs in the water [Fig. 3]...But $$\frac{w}{W}$$ ...is the same for any lump of metal of the same material: it is now called the ... Archimedes had only to take a lump of indisputably pure gold and find its specific gravity by the same process. ...[N]ot only is it the first precise example of '''the application of mathematical ideas to physics, but also... a perfect and simple example of what must be the method and spirit of the science for all time. The discovery of the theory of specific gravity marks a genius of the first rank.'''
 * In these days an infinite number of chemical tests would be available. But then Archimedes had to think... afresh. The solution flashed upon him as he lay in his bath. He jumped up and ran through the streets to the palace, shouting Eureka! Eureka! (I have found it! ...) This day... ought to be celebrated as the birthday of mathematical physics; the science came of age when Newton sat in his orchard. Archimedes... had made a great discovery. He saw that a body when immersed in water is pressed upwards by the surrounding water with a resultant force equal to the weight of the water it displaces. ...Hence if $$W$$ lb. be the [known] weight of the crown, as weighed in air, and $$w$$ lb. be the [unknown] weight of the water which it displaces when completely immersed, $$W - w$$ [from which (knowing $$W$$) the weight w of the equal volume of water can be derived,] would be the extra upward force necessary to sustain the crown as it hung in the water. [Alternatively, the weight of water, equaling the volume of the crown, and overflowing a tub, could be weighed directly.]
 * Alfred North Whitehead, An Introduction to Mathematics (1911) pp. 38-40.

 * The treatises are, without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
 * Thomas Little Heath, A History of Greek Mathematics II (1931).

 * There is here, as in all great Greek mathematical masterpieces, no hint as to the kind of analysis by which the results were first arrived at; for it is clear that they were not discovered by the steps which led up to them in the finished treatise. If the geometrical treatises had stood alone, Archimedes might seem, as Wallis said, "as it were of set purpose to have covered up the traces of his investigations, as if he has grudged posterity the secret of his method of inquiry, while he wished to extort from them assent to his results."
 * Thomas Little Heath, A Manual of Greek Mathematics (1931) p. 281.

 * Modern mathematics was born with Archimedes and died with him for all of two thousand years. It came to life again with Descartes and Newton.
 * Eric Temple Bell, The Development of Mathematics (1940)

 * To conceive of a parabolic segment or of a triangle as the sum of infinitely many line segments, is closely akin to the idea of Leibniz, who thought of the integral $$\int y~dx$$ as the sum of infinitely many terms $$y~dx$$. But, in contrast to Leibniz, Archimedes is fully aware that this conception is... incorrect and that the derivation should be supplemented by a rigorous proof.
 * , Ontwakende wetenschap (1950) translated as Science Awakening (1954) by.

 * The estimations, which occur in the summing of infinite series and in limiting operations, the "epsilontics", as the calculation with an arbitrarily small ε is sometimes called, were for Archimedes an open book. In this respect, his thinking is entirely modern.
 * , Ontwakende wetenschap (1950) translated as Science Awakening (1954) by.

 * In Euclidean geometry the infinitely small was rejected and in the classical treatises of Archimedes we have the finest example of mathematical rigour in antiquity. Notwithstanding, in the discovery method we find him manipulating line and surface indivisibles skilfully, imaginatively and non-rigorously
 * Margaret E. Baron, The Origins of the Infinitesimal Calculus (1969)

 * Using his masterful understanding of mechanics, equilibrium, and the principles of the lever, he weighed in his mind solids or figures whose volumes or areas he was attempting to find against ones he already knew. After determining in this way the answer...he found it much easier to prove geometrically... Consequently The Method starts with a number of statements on centers of gravity and only then proceeds to the geometrical propositions and their proofs. ...[He] essentially introduced the concept of a thought experiment into rigorous research. ...[He] freed mathematics from the somewhat artificial chains that Euclid and Plato had put on it. ...He did not hesitate to explore and exploit the connections between the abstract mathematical objects (the Platonic forms) and physical reality (actual solids and flat objects) to advance his mathematics.
 * Mario Livio, Is God a Mathematician? (2009)

 * Archimedes was the earliest thinker to develop the apparatus of an infinite series with a finite limit ...starting on the conceptual path toward calculus. Of the giants on whose shoulders Isaac Newton would eventually perch, Archimedes was the first.
 * Alex Bellos, The Grapes of Math (2014)

 * Archimedes was a brilliant inventor and a mathematician. He says to the people around him, "Don't just live in the lap of the gods.  Don't be dominated by Mother Nature.  You, as a man, can take control of your own destiny."
 * A&E Television Networks, LLC., "Ancient Einsteins," Ancient Impossible (July 27, 2014).

 * According to legend, nothing could get between him [Archimedes] and his work, and sometimes he would even forget to eat. Ideas would come to him at any moment, and he would scribble them on any available surface.  Famously, he was in the bath when he discovered the laws of buoyancy, leading him to run naked through the streets shouting "Eureka!"  …  Eureka means "I have found it," and it could be argued that Archimedes found out more than anyone else before or since.
 * A&E Television Networks, LLC., "Ancient Einsteins," Ancient Impossible (July 27, 2014).

 * Tragically for all of us, he [Archimedes] was cut down by a Roman soldier because he refused to stop working. …  If Archimedes hadn't been killed before his time, what could have he achieved?  The industrial revolution could have happened two thousand years earlier.  He might have kick-started the modern age.
 * A&E Television Networks, LLC., "Ancient Einsteins," Ancient Impossible (July 27, 2014).

title: Archimedes


text:
Talk:The Time Machine

No source is given! H.G: Wells? -- Mic 08:40 27 Jul 2003 (UTC)

 * H. G. Wells assumed and added. Nanobug 16:23 27 Jul 2003 (UTC)

title: Talk:The Time Machine