Mathematicians From France

"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."

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

"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."

"... 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."

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

"The first appears to have been made by Baron Charles Dupin in 1826 to illustrate an address to the Conservatoire des Arts et Metiers in Paris. The map shows the number of persons per male child in school for each department and is the first moral ."

"In the 19th century, the French geometer Charles Pierre Dupin discovered a nonspherical surface with circular lines of curvature. He called it a cyclide in his book, Applications de Geometrie published in 1822. Recently, cyclides have been revived for use as surface patches in computer aided geometric design (CAGD). Other applications of eyelides in CAGD are possible (e.g., variable radius blending) and require a deep understanding of the geometry of the cyclide."

"Charles Dupin's Discourse on the Condition of the Workers (1873) introduced such concepts as time study and balanced ."

"Of the early management pioneers, history has provided us with the best records for four men: Robert Owen, Charles Babbage, Andrew Ure, and Charles Dupin... Ure knew the French engineer and management writer Charles Dupin, and when Dupin visited Great Britain in 1816–1818, Ure escorted him around the Glasgow factories. Dupin commented that many of the managers of these factories were Ure’s own students."

"The successes obtained in the government of the arts, are similar to the successes obtained in the government of men. We may succeed for a time, by fraud, by surprise, by violence: we can succeed permanently only by means directly opposite. It is not alone the courage, the intelligence, the activity of the manufacturer and the merchant which maintain the superiority of the productions and the commerce of their country; it is far more their wisdom, their economy, above all their probity."

"This is what it behoves us to know: as Frenchmen, for the advantage of France; as friends of all humanity, by that just and generous sentiment which makes us feel interest in the dignity, the peace, the independence, the happiness of all nations, on whatever spot of the globe nature may have placed their country."

"If ever, in the British Islands, the useful citizen should lose these virtues, we may be sure, that for England as well as for any other country, notwithstanding the protection of the most formidable navy, notwithstanding the foresight and activity of diplomacy the most extended, and of political science the most profound, the vessels of a degenerate commerce, repulsed from every shore would speedily disappear from those seas whose surface they now cover with the treasures of the universe, bartered for the treasures of the industry of the three kingdoms."

"For 12 years I have had the honor of teaching geometry and mechanics applied to the arts, in favor of the industrial class... on the most important questions to the well-being, education, and morality of the workers, to the progress of national industry, to the development of all means of prosperity that work can produce for the splendor and happiness of our country."

"In the following work, I have endeavoured to exhibit the full extent of the Military and Naval Forces which the government of Great Britain can bring into the field, or launch upon the ocean. I have likewise described the connection of these forces with the government of the country, and also the discipline usually exercised in order to produce a hardihood in battle, invulnerable to fear and unassailable by cowardice. My observations on these subjects were derived from a residence of five years in England; during which time I was constantly employed in visiting and viewing every object and institution worthy of notice relative to the British Army and Navy."

"Amongst the important results of the recent attempts to extend Science to the labouring classes, maybe ranked the elementary treatises published by Baron Dupin. Possessing an extraordinary fund of scientific information, as well as of practical knowledge collected during a period of twenty years, in the workshops and manufacturing establishments of the most enlightened nations of Europe, combined with a singular degree of clearness, elegance, and ingenuity, in mathematical and physical expositions, this distinguished individual might have continued to delight and instruct inquirers of the highest description, by works classical and profound, but without having witnessed the occurrences alluded to, he might never have directed his attention and his efforts to this most interesting object, the improvement of humble and neglected intellect."

"I found myself obliged, through perhaps unique circumstances, to devote myself to my mathematical research, almost without help, advice or even books... Endlessly occupied by a thousand different matters and constrained my state duties, it is the work of an engineer that I herewith present and not the fruit of the meditations of a savant."

"It is to the director of workshops and factories that it is suitable to make, by means of geometry and applied mechanics, a special study of all the ways to economize the efforts of workers... For a man to be a director of others, manual work has only a secondary importance; it is his intellectual ability (force intellectuelle) that must put him in the top position, and it is in instruction such as that of the Conservatory of the Arts and Professions, that he must develop it."

"The total extent to which steam power is applied in Great Britain was estimated by Baron Dupin, 1825, to be equivalent to the power of 320,000 horses in constant action; and since that period it has prodigiously increased, independently of our rapidly extending railways. To this immense command of power our country owes much of its commercial prosperity, besides a vast addition to the comforts and conveniences of life."

"By another way, therefore, I endeavoured to attain the same end; and since it is a property of water that a small quantity of it, converted into steam by the force of heat, has an elastic force like that of the air, but, when cold supervenes, is again resolved into water, so that no trace of the said elastic force remains; I felt confident that machines might be constructed wherein water, by means of no very intense heat, and at small cost, might produce that perfect vacuum which had failed to be obtained by aid of gunpowder. But of the various constructions which can be contrived for this purpose, the following seemed to me to be the most suitable."

"In the machine for a new use of gunpowder, which is described in the 'Acta Eruditorum' for the month of September, 1688, the first desideratum was, that the gunpowder fired in the bottom of the tube AA should fill the whole cavity with flame, so that the air might be entirely expelled from it, and the tube remain a perfect vacuum beneath the piston BB. But there it was mentioned, that the desired effect could not be sufficiently attained... But hitherto such attempts have been in vain; and always, after the flame of the gunpowder is extinguished, about a fifth part of the air remains in the tube AA."

"AA is a tube of uniform diameter throughout, close shut at the bottom; BB is a piston fitted to the tube; DD a handle fixed to the piston; EE an iron rod moveable round an axis in F; G a spring, pressing the cross rod EE, so that the said rod must be forced into the groove H as soon as the piston with the handle has arrived at such a height as that the said groove H appears above the lid II; L is a little hole in the piston, through which the air can escape from the bottom of the tube AA, when first the piston is forced into it. The use of this instrument is as follows: A small quantity of water is poured into the tube AA; to the depth of 3 or 4 lines; then the piston is inserted, and forced down to the bottom, till a portion of the water previously poured in comes through the hole L; then the said hole is closed by the rod MM. Next the lid II, pierced with the apertures requisite for that purpose, is put on, and a moderate fire being applied, the tube AA soon grows warm, (being made of thin metal), and the water within it, being turned into steam, exerts a pressure so powerful as to overcome the weight of the atmosphere and force up the piston BB, till the groove H of the handle DD appears above the lid II, and the rod EE is forced, with some noise, into the said groove by the spring G. Then forthwith the fire is to be removed, and the steam in the thin metal tube is soon resolved into water, and leaves the tube entirely void of air. Next, the rod EE being turned round so far as to come out of the groove H, and allow the handle DD to descend, the piston BB is forthwith pressed down by the whole weight of the atmosphere, and causes the intended movement, which is of an energy great in proportion to the size of the tube. Nor is it to be doubted that the whole weight of the atmosphere exerts its force in tubes so constructed; for I have established by experiment, that a piston, raised to the top of the tube by the force of heat, shortly afterwards descends again to the bottom, and so on alternately for a number of times, so that no suspicion can arise of air pressing beneath. Now my tube, the diameter of which does not exceed 2 ½ inches, yet raises sixty lbs. aloft with the same velocity as the piston is forced down into the tube, and the tube itself scarcely weighs five ounces. I therefore have little doubt but that tubes may be manufactured, the weight of each of which would scarcely amount to 40 lbs., and yet which could raise, at each operation, two thousand lbs. to a height of four feet. ...If any one now will consider the magnitude of the forces to be obtained in this way, and the trifling expense at which a sufficient quantity of fuel can be procured, he will certainly admit that this my method is far preferable to the use of gunpowder above spoken of, especially as in this way a perfect vacuum is obtained, and so the inconveniences above recounted are avoided."

"Turning a small surface of water into vapour by fire, applied to the bottom of the cylinder that contains it; which vapour forces up the plug (or piston) in the cylinder to a considerable height, and which, as the vapour condenses, (as the water cools when taken from the fire,) descends again by air's pressure, and is applied to raise water out of the mine."

"What I say here is not to give room for believing, that Mr. Savery, who has since published this invention at London, is not actually the inventor. I do not doubt that the same thought may have occurred to him, as well as to others, without having learnt it elsewhere."

"In what manner that power can be applied to draw water or ore from mines, to discharge iron bullets to a great distance, to propel ships against the wind, and to a multitude of other similar purposes, it would be too long here to detail; but each individual, according to the particular occasion, must select the construction of machinery appropriate to his purpose."

"Dr. Denys Papin, a native of Blois, a man of great ingenuity, and of considerable acquirements as a philosopher, is considered by his countrymen to be the true inventor of the Steam Engine: a claim strongly contested by some English authors of eminence who have written on the subject,—but on grounds which appear to have been taken from very erroneous and prejudiced statements. It is due to Papin, to state, that no one, whose labours have produced so many important results, has in his writings shewn so little of the vanity and absurd enthusiasm proverbially characteristic of an inventor."

"I have endeavoured to attain this end (viz. the production of a vacuum in the cylinder) in another way. As water has the property of elasticity, when converted into steam by heat, and afterwards of being so completely recondensed by cold, that there does not remain the least appearance of this elasticity, I have thought that it would not be difficult to work machines in which, by means of a moderate heat and at a small cost, water might produce that perfect vacuum which has vainly been sought by means of gunpowder."

"About the year 1761, or 1762, I tried some experiments on the force of steam in a Papin's digester, and formed a species of steam-engine by fixing upon it a syringe, one-third of an inch diameter, with a solid piston, and furnished also with a cock to admit the steam from the digester, or shut it off at pleasure, as well as to open a communication from the inside of the syringe to the open air, by which the steam contained in the syringe might escape..."

"In 1680, Robert Boyle published the Second Part of his Continuation of New Experiments Physico-mechanical, Touching the Spring and Weight of the Air. ...According to Boyle's preface, the experimental work... was mainly done by a remunerated technician... Denis Papin. The air-pump with which the experiments were performed was... of Papin's own design... At least some, and perhaps the greatest part, of the design of the experimental project was also owing to the technician. ...It seems also that the technician was partly, if not mainly, responsible for the composition of the experimental narratives."

"The first machine of Papin was very similar to the gunpowder-engine... of Huyghens. In place of gunpowder, a small quantity of water is placed at the bottom of the cylinder, A; a fire is built beneath it, "the bottom being made of very thin metal," and the steam formed soon raises the piston, B, to the top where a latch, E, engaging a notch in latch engaging the piston rod, H, holds it up until it is desired that it shall drop. The fire being removed, the steam condenses, and a vacuum is formed below the piston, and the latch, E, being disengaged, the piston is driven down by the superincumbent atmosphere and raises the weight which has been, meantime, attached to a rope... passing from the piston rod over pulleys... The machine had a cylinder two and a half inches in diameter, and raised 60 pounds once a minute; and Papin calculated that a machine of a little more than two feet diameter of cylinder and of four feet stroke would raise 8,000 pounds four feet per minute—i.e., that it would yield about one horse-power."

"What better way to establish his name than to tackle the most fundamental philosophical question man could ask: When was the world created? And how? ...Unlike all previous theorists, Buffon chose to ignore the Bible. ...Instead, his inspiration came from Newton. ...he believed that Newton's mechanics would enable man to unravel the mysteries of the universe. If Newton's laws could explain the motion of the moon and planets... why shouldn't they also reveal our history?"

"Buffon, the director of the Jardin du Roi... watched... the sun... Besides him.. stood... his latest experiment: four large wooden frames holding between them an array of over 150 mirrors. Screw threads ensured that each mirror was precisely aligned... Buffon had been inspired to conduct the experiment by the famous story of how... Archimedes defended his hometown from an attack by the Roman fleet... [by] [[w:Archimedes#Heat ray|focus[ing] the sun's rays onto the ships' wooden hulls]]. ...In 1747, at the age of thirty-nine, [Buffon] felt it was time to make his mark... Within two minutes the deal plank began to char, then to smoke. Just as it looked as if it would burst into flames, however, the sun disappeared behind a cloud. ...That summer, crowds flocked... to see the now celebrated scientist set fire to buildings over two hundred feet away."

"Buffon produced in the fifty years from 1749 an Histoire Naturelle... one of the signal products of eighteenth-century science. ...He attempted to see nature as a whole, produced a vast synthesis and sought to give a picture of the history of the earth... [If] Newton had appeared to reduce the inanimate world to a system of law, Buffon... set his mind on a similar achievement, and even a wider one—comprising... biological phenomana and expanding into the realm of history. ...Along with Leibnitz he believed that the earth had once been in an incandescent state... part of the sun, but had broken away after a collision with a comet. He rejected the tradition that this globe was only six thousand years old and made an attempt to set out the periods or stages of its history; a time when mountain ranges were formed...a time when waters entirely covered... the globe... and a time when the continents came to be separated from one another. ...He held something like Leibnitz's idea that every plant and animal was composed of a mass of minute particles, each of which was a pattern of the whole individual; and this enabled him to explain the origin of living creatures without reference to an act of creation. He tried to show that no absolutely definite boundary existed between the animal and vegetable kingdoms. Nature always proceeds by nuances, he said."

"Le style c'est l'homme."

"La génie n'est utre chose qu'une grande aptitude à la patience."

"Never think that God's delays are God's denials. Hold on; hold fast; hold out. Patience is genius."

"[F]rom the earliest periods of time [man] alone has divided the empire of the world between him and Nature. ...[H]e rather enjoys than possesses, and it is by constant and perpetual activity and vigilance that he preserves his advantage, for if those are neglected every thing languishes, changes, and returns to the absolute dominion of Nature. She resumes her power, destroys the operations of man; envelopes with moss and dust his most pompous monuments, and in the progress of time entirely effaces them, leaving man to regret having lost by his negligence what his ancestors had acquired by their industry. Those periods in which man loses his empire, those ages in which every thing valuable perishes, commence with war and are completed by famine and depopulation. Although the strength of man depends solely upon the union of numbers, and his happiness is derived from peace, he is, nevertheless, so regardless of his own comforts as to take up arms and to fight, which are never-failing sources of ruin and misery. Incited by insatiable avarice, or blind ambition, which is still more insatiable, he becomes callous to the feelings of humanity; regardless of his own welfare, his whole thoughts turn upon the destruction of his own species, which he soon accomplishes. The days of blood and carnage over, and the intoxicating fumes of glory dispelled, he beholds, with a melancholy eye, the earth desolated, the arts buried, nations dispersed, an enfeebled people, the ruins of his own happiness, and the loss of his real power."

"Man alone consumes more flesh than all the other animals together devour; he is, then, the greatest destroyer; and this more from custom than necessity. Instead of using with moderation the blessings which are offered him, instead of disposing of them with equity, instead of increasing them in proportion as he destroys, the rich man places all his glory in consuming, in one day, at his table, as much as would be necessary to support many families: he equally abuses both animals and his fellow-creatures, some of whom remain starving and languishing in misery, and labour only to satisfy his immoderate appetite, and more insatiable vanity, and who, by destroying others through wantonness, destroys himself by excess. Nevertheless, man, like some other animals, might live on vegetables."

"To discover something in mathematics is to overcome an inhibition and a tradition. You cannot move forward if you are not subversive."

"Trotskyism gave me … a remarkable education, clearly more advanced and sophisticated than that of most youngsters of my age. But by the extremism and sectarianism of its ideas, and by its stereotyped language, it neutralised me during the occupation. My judgment remains extremely severe on my own actions as well as those of the majority of the Trotskyist party during that period."

"I have always thought that morality in politics was something essential, just like feelings and affinities."

"I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent. And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully. Even when I was the first to answer the teacher's questions, I knew it was because they happened to be questions to which I already knew the answer. But if a new question arose,usually students who weren't as good as I was answered before me. Towards the end of the eleventh grade, I secretly thought of myself as stupid. I worried about this for a long time. Not only did I believe I was stupid, but I couldn't understand the contradiction between this stupidity and my good grades. I never talked about this to anyone, but I always felt convinced that my imposture would someday be revealed: the whole world and myself would finally see that what looked like intelligence was really just an illusion. If this ever happened, apparently no one noticed it, and I’m still just as slow. (...)At the end of the eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn't have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant. Naturally, it's helpful to be quick, like it is to have a good memory. But it's neither necessary nor sufficient for intellectual success."

"Mathematics, politics and butterflies were the three great loves of Laurent Schwartz."

"Schwartz never offered blind allegiance to Left or Right. Above all he was, in his own words, a man who hated to see systems not working properly."

"One day, I had to give a lecture at the Chevalley Seminar, a group theory seminar in Paris. [...] When I got to the room, fifteen or so researchers were there, along with a few students seated in the rear. A couple of minutes before the talk was to start, Serre came in and sat in the second row. I was honored to have him in the audience, but I let him know right off that the presentation might not be very interesting to him. It was intended for a general audience and I was going to be explaining very basic things. [...] At the end of the seminar, Serre came up to me and said—and here I quote verbatim: “You’ll have to explain that to me again, because I didn’t understand anything.” [...] the most troubling aspect was the abruptness, the frankness with which Serre had overplayed his own incomprehension. It takes a lot of nerve to listen closely to a presentation, then go up to the speaker, smile, and tell him that you “didn’t understand anything.” I never would have dared. Why did he do it? I first told myself it must be one of the things you have the right to do when you’re Jean-Pierre Serre. Then I realized that could also work the other way: what if this technique had actually helped him become Jean-Pierre Serre?"

"You see, some mathematicians have clear and far-ranging. "programs". For instance, Grothendieck had such a program for algebraic geometry; now Langlands has one for representation theory, in relation to modular forms and arithmetic. I never had such a program, not even a small size one."

"If Serre was a Mozart, Grothendieck was a Wagner."

"Je pourrais dire, en exagérant à peine, qu’entre le début des années cinquante jusque vers l’année 1966, donc pendant une quinzaine d’année, tout ce que j’ai appris en "géométrie" (dans un sens très large, englobant la géométrie algébrique ou analytique, la topologie et l’arithmétique), je l’ai appris par Serre, quand je ne l’ai pas appris par moi-même dans mon travail mathématique. C’est en 1952 je crois, quand Serre est venu à Nancy (où je suis resté jusqu’en 1953), qu’il a commencé à devenir pour moi un interlocuteur privilégié - et pendant des années, il a été même mon seul interlocuteur pour les thèmes se plaçant en dehors de l’analyse fonctionnelle. - Grothendieck, Récoltes et Semailles."

"It strikes me that mathematical writing is similar to using a language. To be understood you have to follow some grammatical rules. However, in our case, nobody has taken the trouble of writing down the grammar; we get it as a baby does from parents, by imitation of others. Some mathematicians have a good ear; some not (and some prefer the slangy expressions such as 'iff'). That's life."

"He used capital vowels for the unknown quantities and capital consonants for the known, thus being able to express several unknowns and several knowns."