Mathematicians From France

"No one ever squared the circle with so much genius, or, excepting his principal object, with so much success."

"Mathematics and philosophy are cultivated by two different classes of men: some make them an object of pursuit, either in consequence of their situation, or through a desire to render themselves illustrious, by extending their limits; while others pursue them for mere amusement, or by a natural taste which inclines them to that branch of knowledge. It is for the latter class of mathematicians and philosophers that this work is chiefly intended j and yet, at the same time, we entertain a hope that some parts of it will prove interesting to the former. In a word, it may serve to stimulate the ardour of those who begin to study these sciences; and it is for this reason that in most elementary books the authors endeavour to simplify the questions designed for exercising beginners, by proposing them in a less abstract manner than is employed in the pure mathematics, and so as to interest and excite the reader's curiosity. Thus, for example, if it were proposed simply to divide a triangle into three, four, or five equal parts, by lines drawn from a determinate point within it, in this form the problem could be interesting to none but those really possessed of a taste for geometry. But if, instead of proposing it in this abstract manner, we should say: "A father on his death-bed bequeathed to his three sons a triangular field, to be equally divided among them: and as there is a well in the field, which must be common to the three co-heirs, and from which the lines of division must necessarily proceed, how is the field to be divided so as to fulfill the intention of the testator?" This way of stating it will, no doubt, create a desire in most minds to discover the method of solving the problem; and however little taste people may possess for real science, they will be tempted to try iheir ingenuity in finding the answer to such a question at this."

"There is reason, however, to think that the author would have rendered it much more interesting, and have carried it to si higher degree of perfection, had he lived in an age more enlightened and better informed in regard to the mathematics and natural philosophy. Since the death of that mathematician, indeed, the arts and sciences have been so much improved, that what in his time might have been entitled to the character of mediocrity, would not at present be supportable. How many new discoveries in every part of philosophy? How many new phenomena observed, some of which have even given birth to the most fertile branches of the sciences? We shall mention only electricity, an inexhaustible source of profound reflection, and of experiments highly amusing. Chemistry also is a science, the most common and slightest principles of which were quite unknown to Ozanam. In short, we need not hesitate to pronounce that Ozanam's work contains a multitude of subjects treated of with an air of credulity, and so much prolixity, that it appears as if the author, or rather his continuators, had no other object in view than that of multiplying the volumes. To render this work, then, more worthy of the enlightened agt in which we live, it was necessary to make numerous corrections and considerable additions. A task which we have endeavoured to discharge with all diligence"

"It was the business of the Sorbonne doctors to discuss, of the pope to decide, and of a mathematician to go straight to heaven in a perpendicular line."

"Arithmetic and geometry, according to Plato, are the two wings of the mathematician. The object indeed of all mathematical questions, is to determine the ratios of numbers, or of magnitudes ; and it may even be said, to continue the comparison of the ancient philosopher, that arithmetic is the mathematician's right wing; for it is an incontestable truth, that geometrical determinations would, for the most part, present nothing satisfactory to the mind, if the ratios thus determined could not be reduced to numerical ratios. This justifies the common practice, which we shall here follow, of beginning with arithmetic."

"Jacques Ozanam, whose fame is established as an eminent Mathematician, was born at Bouligneux in Brescia, in the year 1640: he was descended from a family of Jewish extraction, but which had long been converts to the Romish faith, and some of whom had held considerable places in the parliaments of Provence. Being a younger son, though of an opulent family, it was thought proper to educate him for the church, that he might be qualified for some small benefices belonging to the family: he accordingly studied divinity four years, but this was purely in obedience to the will of his father, upon whose death he relinquished his theological pursuits, and, following his natural inclinations, devoted himself to the study of the mathematics. Having considerable genius, as well as great industry, he made very great progress, though unassisted by a master, and at the juvenile age of 15 years, he wrote a mathematical treatise."

"There are two general Methods made use of in the Mathematicks, viz. Synthesis and Analysis, which we shall explain, after having acquainted the Reader, that the Method we make use of to resolve a Mathematical Problem, is called Zetetick; and that that Method which determines when, and by what way, and how many different ways a Problem may be resolved, is called Poristick. But in treating of Methods, we will first premise, that in general, a Method is the Art of disposing a Train of Arguments or Consequences in a right Order, either to discover the Truth of a Theorem, which we would find out, or to demonstrate it to others, when found."

"Ozanam possessed a mild and calm disposition, a cheerful and pleasant temper, an inventive genius, and a generosity almost unparalleled. After marriage, his conduct was irreproachable; and, at the same time that he was sincerely pious, he had a great aversion to disputes about theology."

"The sun, as we have already said, is placed in the middle of our system, as a source of light and heat, to illuminate and vivify all the planets subordinate to it. Without his benign influence the earth would be a mere block, which in hardness would surpass marble and the most compact substances with which we are acquainted ; no vegetation, no motion would be possible: in a word, it would be the abode of darkness, inactivity and death. The first rank therefore among inanimate beings cannot be refused to the sun ; and if the error of addressing to a created object that adoration which is due to the Creator atone could admit of excuse, we might be tempted to excuse the homage paid to the sun by the ancient Persians, as is still the ease among the Guebres, their successors, and some savage tribes in America."

"The Usefulness of the Mathematicks in General, and of some Parts of them in Particular, in the common Affairs of Humane Life, has rendered some competent Knowledge of them very necessary to a great Part of Mankind, and very convenient to all the Rest that are any way conversant beyond the Limits of their own particular callings."

"To be perfectly ignorant in all the Terms of them is only tolerable in those, who think their Tongues of as little Use to them, as generally their Understandings are. Those whom Necessity has obliged to get their Bread by Manual Industry, where some Degree of Art is required to go along with it, and who have had some Insight into these Studies, have very often found Advantages from them sufficient to reward the Pains they were at in acquiring them. And whatever may have been imputed (how justly I'm not now to determine) to some other Studies, under the Notion of Insignificancy and Loss of Time ; yet these, I believe, never caused Repentance in any, except it was for their Remissness in the Prosecution of them. And though Plato's Censure, that those who did not understand the 117 Prop. of the 10th Element, ought not to be ranked among Rational Creatures, wax unreasonable and unjust: Yet to give a Man the Character of Universal Learning that is destitute of a competent Knowledge in the Mathematics, is no less so."

"Although the Mathematicks according to its Etymology, signifies only Discipline, yet it merits the Name of Science better than any other, because its Principles are self-evident, and independent on any sensible Experience, and its Propositions demonstrated beyond all possible Doubt or Opposition. Youth were anciently instructed herein before Philosophy, on which Account Aristotle called it the Science of Children. This was taught them not only to raise and excite their Genius, but also as a fit preparative to the Study of Nature; and it was upon this Account that the Divine Plato inscribed on his School... that none wholly ignorant of Geometry should be admitted there."

"By Science is understood a Knowledge acquired by, or founded on clear and self evident Principles, whence it follows that the Mathematicks may truly be stiled such."

"Mathematicks therefore is a Science which teaches or contemplates whatever is capable of Measure or Number as such. When it relates to Number, it is called Arithmetick; but when to measure, as Length, Breadth, Depth, Degrees of Velocity in Motion, Intenseness or Remissness of Sounds, Augmentation or Diminution of Quality, 6tc. it is called Geometry."

"The Essential Parts of the Simple or Pure Mathematicks are Arithmetick and Geometry, which mutually assist one another, and are independent on any other Sciences, except perhaps on Artificial Logick: But doubtless Natural Logick may be sufficient to a Man of Sense. The other parts are chiefly Physical Subjects explained by the Principles of Arithmetics or Geometry."

"Those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression."

"The employment of mathematical symbols is perfectly natural when the relations between magnitudes are under discussion; and even if they are not rigorously necessary, it would hardly be reasonable to reject them, because they are not equally familiar to all readers and because they have sometimes been wrongly used, if they are able to facilitate the exposition of problems, to render it more concise, to open the way to more extended developments, and to avoid the digressions of vague argumentation."

"Anyone who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains."

"In the act of exchange, as in the transmission of power by machinery, there is friction to be overcome, losses which must be borne, and limits which cannot be exceeded."

"In 1823 he took a license degree in mathematics at Sorbonne University. He then became the private secretary of a field marshal who required assistance in writing his memoirs. This position must have left Cournot with considerable time for his own pursuits, for in the course of his ten years in the field marshal's employment he took two doctoral degrees, one in mechanics and one in astronomy. In addition, he published a number of articles and even acquired a degree in law."

"There are many anticipators of marginal analysis. Three major names were Augustin Cournot (1801-1877), J. H. von Thünen (1783-1850), and H. H. Gossen (1810-1858). Cournot's originality and ingenuity can hardly be exaggerated. In 200 small pages, he described and defined the downward-sloping , completely analyzed the maximization of profit under conditions of monopoly, advanced an ingenious explanation of pricing, proved that equilibrium price occurred when equaled , and exactly defined the market from which we call perfect competition and he called "unlimited competition." And the book went unread."

"It took from a hundred to a hundred and fifty or two hundred years for the astronomy of Kepler to become the astronomy of Newton and Laplace, and for the mechanics of Galileo to become the mechanics of d'Alembert and Lagrange. On the other hand, less than a century has elapsed between the publication of Adam Smith’s work and the contributions of Cournot, Gossen, Jevons, and myself."

"Faith makes us live by showing us that life, although it is dependent upon reason, has its well spring and source of power elsewhere, in something supernatural and miraculous. Cournot the mathematician, a man of singularly well-balanced and scientifically equipped mind has said that it is this tendency towards the supernatural and miraculous that gives life, and that when it is lacking, all the speculations of reason lead to nothing but affliction of the spirit. ...And in truth we wish to live."

"So far we have studies how, for each commodity by itself, the law of demand in connection with the conditions of production of that commodity, determines the price of it and regulates the incomes of its producers. We considered as given and invariable the prices of other commodities and the incomes of other producers; but, in reality the economic system is a whole of which the parts are connected and react on each other. An increase in the incomes of the producers of commodity A will affect the demand for commodities Band C, etc., and the incomes of their producers, and, by its reaction will involve a change in the demand for A. It seems, therefore, as if, for a complete and rigorous solution of the problems relative to some parts of the economic system, it were indispensable to take the entire system into consideration. But this would surpass the powers of mathematical analysis and of our practical methods of calculation, even if the values of all the constants could be assigned to them numerically."

"Well, it has been a long time since manna last fallen from heaven. We cannot live alone; we rely on others to produce the stuff of our material and intellectual life, and we have to organize society so that its members will cooperate toward the common good."

"What is needed is courage: it is always so much easier to accept what you are being told than to think for yourself."

"Power is no linger seen as inheriting its legitimacy from some divine authority; it is a mere convention which we adhere to because we are born and educated into it, and because we see others conform to it. Its strength lies in the fact that we believe that others believe in it: power is no more than the illusion of power. The exercise of power is a constant fight to keep up appearances."

"The measurement of time was the first example of a scientific discovery changing the technology."

"The moment when the scientists became engineers was a historical turning point."

"The social world is not driven by natural laws and randomness alone, as the physical world is, but also by human wills."

"Chaos cuts with two edges. We have seen how it is impossible to retrieve past history from current observations. We will now show that it is impossible to predict future states from the current observations."

"Many great failures and many great successes are due to chance and not to human folly or ingenuity."

"If there is a God, he has left no tracks in the laws of physics; or if he has, he has covered them up very well."

"An equilibrium is not always an optimum; it might not even be good. This may be the most important discovery of game theory."

"It is a testimony to the power of education that classical mechanics could operate for so long under a mistaken conception. Teaching and research concentrated on integrable systems, each feeding the other, until in the end we had no longer the tools nor the interest for studying nonintegrable systems."

"In the struggle for life, or in the struggle for power, there is no reason why their victory would make the world better than it was. There is no invisible hand guiding these processes, dealing out victory to the most deserving. Chance is their leader."

"The transition from integrable to non integrable systems is quiet interesting to observe."

"The optimist believes that this is the best of all possible worlds, and the pessimist fears that this might be the case."

"We do not discover mathematical truths; we remember them from our passages through this world outside our own."

"The world is full, at every scale, and every scale ignores the higher and lower ones."

"So every possible reality, once God gives it existence, will reveal its own identity. Its life will unfold slowly, whereas God encompasses it at a single glance."

"Nowadays, however, we are much more aware of the fact that the best proof in the world is worth no more than its premises: every scientific theory is transitory and provisional, in wait for a better one, and accepted only as long as the experimental results conform to its predictions."

"The experimental investigation by which Ampère established the law of the mechanical action between electric currents is one of the most brilliant achievements in science. The whole, theory and experiment, seems as if it had leaped, full grown and full armed, from the brain of the 'Newton of electricity.' It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics."

"Écoute les savants, mais ne les écoute que d'une oreille!... Que l'autre soit toujours prête à recevoir les doux accents de la voix de ton ami céleste!"

"Either one or the other [ analysis or synthesis ] may be direct or indirect. The direct procedure is when the point of departure is known-direct synthesis in the elements of geometry. By combining at random simple truths with each other, more complicated ones are deduced from them. This is the method of discovery, the special method of inventions, contrary to popular opinion."

"There is synthesis when, in combining therein judgments that are made known to us from simpler relations, one deduces judgments from them relative to more complicated relations. There is analysis when from a complicated truth one deduces more simple truths."

"For we may remark generally of our mathematical researches, that these auxiliary quantities, these long and difficult calculations into which we are often drawn, are almost always proofs that we have not in the beginning considered the objects themselves so thoroughly and directly as their nature requires, since all is abridged and simplified, as soon as we place ourselves in a right point of view."

"Quels que soient les progrès des connaissances humaines, il y aura toujours place pour l'ignorance et par suite pour le hasard et la probabilité."

"Concevons qu’on ait dressé un million de singes à frapper au hasard sur les touches d’une machine à écrire et que, sous la surveillance de contremaîtres illettrés, ces singes dactylographes travaillent avec ardeur dix heures par jour avec un million de machines à écrire de types variés. Les contremaîtres illettrés rassembleraient les feuilles noircies et les relieraient en volumes. Et au bout d’un an, ces volumes se trouveraient renfermer la copie exacte des livres de toute nature et de toutes langues conservés dans les plus riches bibliothèques du monde. Telle est la probabilité pour qu’il se produise pendant un instant très court, dans un espace de quelque étendue, un écart notable de ce que la mécanique statistique considère comme la phénomène le plus probable."

"Just as Borel, the pure mathematician interested in probability and statistics, had no counterpart in England so Keynes, the logician-economist, had no counterpart in France."