Mathematicians From France

"We shall have [a complete theory] only when the laws of Physics shall be extended enough, generalized enough, to make known beforehand all the effects of heat acting in a determined manner on any body."

"[T]he production of heat alone is not sufficient to give birth to the impelling power: it is necessary that there should also be cold; without it, the heat would be useless."

"The phenomenon of the production of motion by heat has not been considered from a sufficiently general point of view. We have considered it only in machines... [for which] the phenomenon is... incomplete. It becomes difficult to recognize its principles and study its laws. ...[T]he principle of the production of motion by heat... must be considered independently of any mechanism or... particular agent. It is necessary to establish principles applicable not only to steam-engines but to all imaginable heat-engines, whatever the working substance and whatever the method by which it is operated."

"The maximum of motive power resulting from the employment of steam is also the maximum of motive power realizable by any means whatever.1878"

"The production of motion in steam-engines is always accompanied by... the re-establishing of equilibrium in the caloric; that is, its passage from a body in which the temperature is more or less elevated, to another in which it is lower."

"The production of motive power is then due... not to an actual consumption of caloric, but to its transportation from a warm body to a cold body... to its re-establishment of equilibrium..."

"[W]e have just described the re-establishment of equilibrium in the caloric, its passage from a... heated body to a cooler one."

"What happens... in a steam-engine... ? The caloric developed in the furnace by the effect of the combustion traverses the walls of the boiler, produces steam, and in some way incorporates itself with it. The latter carrying it away, takes it first into the cylinder, where it performs some function, and from thence into the condenser, where it is liquefied by contact with the cold water... [T]he cold water of the condenser takes possession of the caloric... It is heated by the intervention of the steam as if it had been placed directly over the furnace. The steam is here only a means of transporting the caloric."

"Heat is simply motive power, or rather motion which has changed form. It is a movement among the particles of bodies. Wherever there is destruction of motive power there is, at the same time, production of heat in quantity exactly proportional to the quantity of motive power destroyed. Reciprocally, wherever there is destruction of heat, there is production of motive power."

"Life is a short enough passage. I am half the journey. I will complete the remainder as I can."

"... string theory has uncovered beautiful relations between physics and different parts of mathematics, mostly differential geometry, enumerative algebraic geometry and complex analysis. In fact string theory started like this. At the very beginning when and others were starting string theory motivated by the for strong interactions they found solutions to the duality equations for the scattering amplitudes in terms of s and these were nice mathematical functions like generalized s which are very natural in terms of complex analysis. It was then realized, by and others, that these models could be understood geometrically from the propagation of strings. It is a very powerful idea to “test” a given complex space using the space of complex curves inside or of maps from s to that target space. And physicists could use all the arsenal of which is quite powerful. This generated a very interesting group of people that do a kind of “physics motivated” mathematics which rejuvenated some parts of complex geometry. They adopt a rather free attitude towards mathematics, which is original and productive and had a very positive influence. It started as mathematics and had a very positive impact on mathematics up to now."

"I am not really motivated by confidence. Nor by curiosity. What I would say is, it’s more anxiety. I spend much more time being anxious than being confident or being curious. My mind sort of constantly worries. It’s not confidence — okay, I have of course some self-confidence, but it’s not a kind of overreaching confidence, by no means. I knew only one person who had overreaching confidence, that was Michael Atiyah. I really liked him a lot. He could jump to other topics. But I am not like him. I am much more motivated by the fact that when I do not understand something, it makes me suffer. It puts me into a state of misery. I am feeling bad until I understand. That’s exactly the motivating force."

"Connes is one of the revolutionaries of the subject (Riemann hypothesis), a benign Robespierre of mathematics to Bombieri's Louis XVI."

"In some situations however, when you are deeply with your problem, you feel at home anywhere just thinking about your problem. Some of my best work was done in hotels, on the train, and there's no rule. More important when you think is what goes on inside rather outside. [...] The best thoughts can be nearly everywhere."

"When you are into mathematics, you have been so high on the scale of complexity of reasoning that you are living in some kind of altered reality. You think everybody on the street is able to understand complicated reasoning [...]. And you get very frustrated, when you discover that's not the case."

"If paparazzi specialized in mathematical celebrities they'd camp outside the dining hall at the IAS and come away with a new batch of pictures every day."

"There are two fundamental sources of ‘bare’ facts for the mathematician. These are, on the one hand the physical world which is the source of geometry, and on the other hand the arithmetic of numbers which is the source of number theory. Any theory concerning either of these subjects can be tested by performing experiments either in the physical world or with numbers. That is, there are some real things out there to which we can confront our understanding."

"Fields medalists are nothing out of the ordinary at Princeton—you sometimes find yourself seated next to three or four of them at lunch!"

"Boyle entertains the hypothesis of a universal matter, the concept of atoms of different shapes and sizes, and the possibility of existence of substances that might properly be called elements... The atomic theory as originally conceived by Democritus and Epicurus, developed by Lucretius, and resurrected by Gassendi from about 1647 on, was doubtless the source from which Boyle derived his ideas, ...as he cites both Epicurus and Gassendi. Boyle, however... avoids any dogmatic assertion of these hypotheses. It is plain, however, that these atoms or "corpuscles" as he calls them are a constant element of his thought."

"Man lives very well upon flesh, you say, but, if he thinks this food to be natural to him, why does he not use it as it is, as furnished to him by Nature? But, in fact, he shrinks in horror from seizing and rending living or even raw flesh with his teeth, and lights a fire to change its natural and proper condition. … What is clearer than that man is not furnished for hunting, much less for eating, other animals? In one word, we seem to be admirably admonished by Cicero that man was destined for other things than for seizing and cutting the throats of other animals. If you answer that ‘that may be said to be an industry ordered by Nature, by which such weapons are invented,’ then, behold! it is by the very same artificial instrument that men make weapons for mutual slaughter. Do they this at the instigation of Nature? Can a use so noxious be called natural? Faculty is given by Nature, but it is our own fault that we make a perverse use of it."

"Gassendi, le meilleur philosophe des littérateurs et le meilleur littérateur des philosophes... [Gassendi, the greatest philosopher among literati and the greatest literato among philosophers...]"

"The ancient Greek philosopher, Democritus, propounded an hypothesis of the constitution of matter, and gave the name of atoms to the ultimate unalterable parts of which he imagined all bodies to be constructed. In the 17th century, Gassendi revived this hypothesis, and attempted to develope it, while Newton used it with marked success in his reasonings on physical phenomena; but the first who formed a body of doctrine which would embrace all known facts in the constitution of matter, was Roger Joseph Boscovich, of Italy, who published at Vienna, in 1759, a most important and ingenious work, styled Theoria Philosophiæ Naturalis ad unicam legem virium, in Natura existentium redacta. This is one of the most profound contributions ever made to science; filled with curious and important information, and is well worthy of the attentive perusal of the modern student. In more recent days, the theory of Boscovich has received further confirmation and extension in the researches of Dalton, Joule, Thomson, Faraday, Tyndall, and others."

"An interesting theorem bearing his name and typical of projective geometry is as follows:—If two triangles ABC and A'B'C' are so related that lines joining corresponding vertices meet in a point O, then the intersections of corresponding sides will lie in a straight line A"B"C". It remained for Monge, the inventor of descriptive geometry... and others more than a century later to carry this development forward. Desargues's work was indeed practically lost until Poncelet in 1822 proclaimed him the Monge of his century."

"Perceiving that the practitioners of these arts ["...among others, the cutting of stones in architecture, that of sun-dials, that of perspective in particular"] had to burden themselves with the laborious acquisition of many special facts in geometry, he sought to relieve them by developing more general methods and printing notes for distribution among his friends."

"Desargues contented himself with enunciating general principles remarking:—"He who shall wish to disentangle this proposition will easily be able to compose a volume.""

"He met Descartes while employed by Cardinal Richelieu at the , and they with others met regularly in Paris for the discussion of the new Copernican theory and other scientific problems."

"In his chief work Desargues enunciates the propositions:— 1. A straight line can be considered as produced to infinity and then the two opposite extremities are united. 2. Parallel lines are lines meeting at infinity and conversely. 3. A straight line and a circle are two varieties of the same species. On these he bases a general theory of the plane sections of a cone."

"The researches of Kepler and Desargues will serve to remind us that as the geometry of the Greeks was not capable of much further extension, mathematicians were now beginning to seek for new methods of investigation, and were extending the conceptions of geometry. The invention of analytical geometry and of the infinitesimal calculus temporarily diverted attention from pure geometry, but at the beginning of the last century there was a revival of interest in it, and since then it has been a favourite subject of study with many mathematicians."

"Desrgues commences [in the Brouillon project] with a statement of the doctrine of continuity as laid down by Kepler: thus the points at the opposite ends of a straight line are regarded as coincident, parallel lines are treated as meeting at a point at infinity, and parallel planes on a line at infinity, while a straight line may be considered as a circle whose center is at infinity. The theory of involution of six points, with its special cases, is laid down, and the projective property of pencils in involution is established. The theory of polar lines is expounded and its analogue in space suggested. A tangent is defined as the limiting case of a secant, and an asymptote as a tangent at infinity. Desargues shows that the lines which join four points in a plane determine three pairs of lines in involution on any transversal, and from any conic through the four points another pair of lines can be obtained which are in involution with any two of the former. He proves that the points of intersection of the diagonals and the two pairs of opposite sides of any quadrilateral inscribed in a conic are a conjugate triad with respect to the conic, and when one of the three points is at infinity its polar is a diameter; but he fails to explain the case in which the quadrilateral is a parallelogran, although he had formed the conception of a straight line which was wholly at infinity. The book, therefore, may be fairly said to contain the fundamental theorems on involution, homology, poles and polars, and perspective."

"The influence exerted by the lectures of Desargues on Descartes, Pascal and the French geometricians of the seventeenth century was considerable; but the subject of projective geometry soon fell into oblivion, chiefly because the analytical geometry of Descartes was so much more powerful as a method of proof or discovery."

"Hardly less interesting than the new ideas of Descartes and Cavalieri are those of their contemporary Desargues... who made important researches in geometry. But for the still more brilliant geometrical achievements of Descartes, these might have led to the immediate development of projective geometry, the elements of which are contained in Desargues's work."

"... and more recently, Bernard Cache, have argued that Girard Desargues' mathematics provided a model for Leibniz's monad. ...Desargues was a founder of projective geometry, which offers a mathematical model for the intuitive notions of perspective and horizon by studying what remains invariable in projections. Outlining the concept of the "invariant," he gives his name to the "Desargues theorem," focusing on homological triangles. His disciple was the engraver, , author of a Treatise on Projections and Perspective (1665), who later taught linear perspective to stone cutters, carpenters, engravers, manufacturers of instruments and, less successfully, to painters. The perspective that Bosse teaches implicitly introduces the idea of infinity, in that he uses parallel lines with an infinitely extending vanishing point... Moreover, permeated by the knowledge of Desargues, Bosse develops a method for tracing shadows, which was inspired by his master."

"The discovery of a method for correct perspective is usually attributed to... Brunelleschi... The first published method appears in... On Painting [Della Pittura] by Alberti. ...Alberti's veil, used a piece of transparent cloth, stretched on a frame... viewing the scene with one eye... fixed... one could trace the scene directly onto the veil. ...The mathematical setting... is the family of lines ("light rays") through the point (the "eye"), together with the plane... (the "veil"). ...In this setting, the problems of perspective and were not very difficult, but the concepts were... a challenge to traditional geometric thought. Contrary to Euclid, one had... (i) Points at infinity ("vanishing points") where parallels met. (ii) Transformations that changed lengths and angles (projections). The first to construct a mathematical theory incorporating these ideas was Desargues,... although the idea of points at infinity had already been used by Kepler..."

"Girard Desargues... gave some courses of gratuitous lectures in Paris from 1626 to about 1630 which made a great impression upon his contemporaries. Both Descartes and Pascal had a high opionion of his work and abilities, and both made considerable use of the theorems he had enunciated."

"In 1639, nine years after Kepler's death, there appeared in Paris a remarkably original but little-heeded treatise on the conic sections. ...The work was so generally neglected by other mathematicians that it was soon forgotten and all copies of the publication disappeared. ...in 1845 Chasles happened upon a manuscript copy... made by Desargues' pupil, ... and since that time the work has been regarded as one of the classics in the early development of synthetic projective geometry."

"Kepler (and Desargues) regarded the two "ends" of the ["straight"] line as meeting at "infinity" so that the line has the structure of a circle. In fact, Kepler actually thought of a line as a circle with its center at infinity."

"In 1636 Desargues issued a work on perspective; but most of his researches were embodied in his Brouillon project on conics, published in 1639, a copy of which was discovered by Chasles in 1845."

"In general this geometry instead of dealing with definite triangles, polygons, circles, etc., in the Euclidean manner, is based on a consideration of all points of a straight line, of all lines through a common point and of the possible effects of setting up an orderly one-to-one correspondence between them. In particular, Desargues makes a comparative study of the different plane sections of a given cone, deducing from known properties of the circle analogous results for the other conic sections."

"Blaise Pascal... was one of the very few contemporaries who appreciated the worth of Desargues. He says in his Essais pour les coniques, "I wish to acknowledge that I owe the little that I have discovered on this subject to his writings.""

"He gives the theory of involution of six points, but his definition of "involution" is not quite the same as the modern definition, first found in Fermat, but really introduced into geometry by Chasles. On a line take the point A as origin (souche), take also the three pairs of points B and H, C and G, D and F; then, says Desargues, if AB \cdot AH = AC \cdot AG = AD \cdot AF, the six points are in "involution." If a point falls on the origin, then its partner must be at an infinite distance from the origin. If from any point P lines be drawn through the six points, these lines cut any transversal MN in six other points, which are also in involution; that is, involution is a projective relation."

"After his own fashion, Desargues discussed cross ratio; poles and polars; Kepler's principle (1604) of continuity, in which a straight line is closed at infinity and parallels meet there; involutons; assymptotes at tangents at infinity; his famous theorem on triangles in perspective; and some of the projective properties of quadrilaterals inscribed in conics. Descartes greatly admired Desargue's invention, but happily for the future of geometry did not hesitate on that account to advocate for his own."

"Pascal made grateful acknowlegement to Desargues for his skill in projective geometry."

"Desargues also gives the theory of polar lines. What is called "" in elementary works is as follows: If the vertices of two triangles, situated either in space or in a plane, lie on three lines meeting in a point, then their sides meet in three points lying on a line, and conversely. This theorem has been used since by Brianchon, Sturm, Gergonne, and others. Poncelet made it the basis of his beautiful theory of homological figures."

"He who shall wish to disentangle this proposition will easily be able to compose a volume."

"Nous démontrerons aussi la propriété suivante dont le premier inventeur est M. Desargues, Lyonnois, un des grands esprits de ce temps, et des plus versés aux mathématiques, et entre autres aux coniques, dont les écrits sur cette matière, quoiqu'en petit nombre, en ont donné un ample témoignage à ceux qui auront voulu en recevoir l'intelligence. Je veux bien avouer que je dois le peu que j'ai trouvé sur cette matière à ses écrits, et que j'ai tâché d'imiter, autant qu'il m'a été possible, sa méthode..."

"One of the first important steps to be taken in modern times... was due to Desargues. In a work published in 1639 Desargues set forth the foundation of the theory of four harmonic points, not as done today but based on the fact that the product of the distances of two conjugate points from the center is constant. He also treated the theory of poles and polars, although not using these terms."

"The book of Desargues (1639), Brouillon project d'une atteinte aux événemens des recontres du cône avec un plan (Schematic Sketch of What Happens When a Cone Meets a Plane), suffered an extreme case of delayed recognition, being completely lost for 200 years."

"The famous geometer Desargues worked on the lines of Kepler and is now commonly credited with the authorship of some of the ideas of his predecessor. ...the oneness of opposite infinities followed simply and logically from a first principle of Desargues, that every two straight lines, including parallels, have or are to be regarded as having one common point and one only. A writer of his insight must have come to this conclusion, even if the paradox had not been held by Kepler, Briggs, and we know not how many others, before Desargues wrote. ...Desargues must have learned directly or indirectly from the work in which Kepler propounded his new theory of these points, first called by him the Foci (foyers), including the modern doctrine of real points at infinity."

"When no point of a line is at a finite distance, the line itself is at an infinite distance."

"I freely confess that I never had taste for study or research either in physics or geometry except in so far as they could serve as a means of arriving at some sort of knowledge of the proximate causes... for the good and convenience of life, in maintaining health, in the practice of some art,... having observed that a good part of the arts is based on geometry, among others that cutting of stone in architecture, that of sundials, that of perspective in particular."