Mathematicians From France

"This temperature of space is not the same in different regions of the universe; but it does not vary in the regions... [of] planetary bodies... [T]he planets of our system... equally participate in the common temperature... augmented for each... by the rays of the sun, according to the distance of the planet from... [it]. ...The intensity and distribution of heat on the surface of these bodies results from the distance from the sun, the inclination of the axes of rotation to the orbit, and the state of the surface..."

"We conclude... that there exists a physical cause always present which modifies the temperature at the surface of the earth, and gives this planet a fundamental heat, which is... independent of the action of the sun and that internal heat preserved... It is to be attributed to the radiation from all the bodies in the universe, whose light and heat can reach us... rays which penetrate every part of the planetary regions... [A]ny point of space whatever which contains these bodies acquires a fixed temperature."

"We... now consider the second cause of terrestrial heat, which... resides in the planetary spaces. ...[A]scertain what would be the thermometrical state of the terrestrial mass, if it received only the heat of the sun. To facilitate... first leave the atmosphere out of the account. ...[I]f the earth and all the bodies of the solar system, were placed in space deprived of all heat ...The polar regions would be subject to intense cold and the decrease of temperature from... equator to... poles would be incomparably more rapid and extended. In this hypothesis of the absolute cold of space, all the effects of heat... at the surface of the earth, should be attributed to... the sun. The least variance in... [its] distance... from the earth, would occasion... considerable changes in temperature. The interruption of day and night would produce effects sudden... [B]odies, would be exposed... at commencement of night, to a cold of infinite intensity. Animals and vegetables could not resist... the sudden and powerful change... produced at the rising of the sun."

"We shall describe... the principal results of the prolonged action of the solar rays upon the terrestrial globe. ...[T]he state of the mass has varied continually in proportion to the heat received. This variable... internal temperature... has approached... nearer to a final state... subject to no change. Then each point of the solid sphere has acquired, and preserves... a fixed temperature, which depends only on the situation of the point... The final state of the mass, the heat of which has penetrated all... parts, can... be compared to... a vessel which receives by openings at the top, liquid from some constant source, and permits exactly an equal quantity to escape by orifices. Thus the solar heat has accumulated in the interior of the globe and is... continually renewed."

"This distinction of luminous and non-luminous heat, explains the elevation of temperature caused by transparent bodies."

"The interposition of the air very much modifies the effects of the heat upon the surface of the globe. The solar rays traversing the atmospheric strata, which are condensed by their own weight [at decreasing altitudes], heat them very unequally; those which are rarest are likewise coldest, because they... absorb a smaller part of the rays. The heat of the sun... in the form of light, possesses the property of penetrating transparent solids or liquids, and loses this property... when by... terrestrial bodies, it is turned into heat radiating without light."

"Liquids are very poor conductors of heat; but they have, like aeriform media, the property of carrying it rapidly in certain directions. This is the same property which, combining with, combining with the centrifugal force, displaces and mingles all parts of the atmosphere... [and] ocean, and maintains in them, regular and immense currents."

"The earth would have only the same temperature as the heavens, were it not for two causes... One is the internal heat... possessed at its formation... only dissipated through the surface; the other is the continued action of the solar rays... which produce at the surface, the diversities of climate."

"The solar system is situated in a region of the universe, every point of which has a common and constant temperature, determined by the rays of light and heat which proceed from the surrounding stars. This low temperature is a little below that of the polar regions of the earth."

"The heat of the earth is derived from three sources... 1. ...[S]olar rays; the unequal distribution of which causes diversities of climate. 2. ...[T]he common temperature of the planetary spaces; being exposed to the radiation from the innumerable stars which surround the solar system. 3. The earth preserves in its interior that primitive heat which it had at the time of the first formation of the planets. ...We will show ...the principle features of these phenomena."

"The question of terrestrial temperature, one of the most remarkable and difficult in natural philosophy... I have... condensed in a single essay... the results of this theory. The analytical details... I have already published. I was specially desirous of presenting... a complete view of the phenomena and the mathematical relations... between them."

"Profound study of nature is the most fertile source of mathematical discoveries."

"If we consider further the manifold relations of this mathematical theory to civil uses and the technical arts, we shall recognize completely the extent of its applications. It is evident that it includes an entire series of distinct phenomena, and that the study of it cannot be omitted without losing a notable part of the science of nature. The principles of the theory are derived, as are those of rational mechanics, from a very small number of primary facts, the causes of which are not considered by geometers, but which they admit as the results of common observations confirmed by all experiment."

"Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy. Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics."

"The analytical equations, unknown to the ancient geometers, which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and from obscurities, that is to say more worthy to express the invariable relations of natural things. Considered from this point of view, mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. Its chief attribute is clearness; it has no marks to express confused notions. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them."

"Letter, quoted by Vladimir Dobrushkin, "Biography of Joseph Fourier" & Elena Presitini, The Evolution of Applied Harmonic Analysis (2004) p. 42."

"I am sorry not to have known the mathematician who first made use of this method because I would have cited him. Regarding the researches of d'Alembert and Euler could one not add that if they knew this expansion they made but a very imperfect use of it. They were both persuaded that an arbitrary and discontinuous function could never be resolved in series of this kind, and it does not seem that anyone had developed a constant in cosines of multiple arcs, the first problem which I had to solve in the theory of heat."

"I regarded these events as the customary disturbances of a state in which a new usurper tends to pluck the sceptre from his predecessor. ...As the natural ideas of equality were developed it was possible to conceive the sublime hope of establishing among us a free government exempt from kings and priests and to free from this double yoke the long usurped soil of Europe. I readily became enamored of this cause... the greatest and most beautiful which any nation has ever undertaken. ...You will judge whether it is I or my adversaries who are terrorists and persecutors. ...I accuse them of having violated ...all the rules of natural justice, of being ignorant and evil, of profaning the words of humanity and justice in invoking them, just as tyranny was organized in the name of liberty. Finally, of having given themselves up to a boundless revolutionary fury which ought to cover then with disgrace and scorn."

"Fermat had recourse to the principle of the economy of nature. Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law. Fermat, however, not only knew (through Descartes) the law of refraction, but he also invented a procedure—equivalent to the differential calculus—for maximizing and minimizing a function of a single variable. ...Fermat applied his method ...and discovered, to his delight, that the result led to precisely the law which Descartes had enunciated. But although the law is the same, it will be noted that the hypothesis contradicts that of Descartes. Fermat assumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite."

"Perhaps nowhere does one find a better example of the value of historical knowledge for mathematicians than in the case of Fermat, for it is safe to say that, had he not been intimately acquainted with the geometry of Apollonius and Viéte, he would not have invented analytic geometry."

"Descartes' method of finding tangents and normals... was not a happy inspiration. It was quickly superseded by that of Fermat as amplified by Newton. Fermat's method amounts to obtaining a tangent as the limiting position of a secant, precisely as is done in the calculus today. ...Fermat's method of tangents is the basis of the claim that he anticipated Newton in the invention of the differential calculus."

"Fermat's Last Theorem is to the effect that no integral values of x, y, z can be found to satisfy the equation xn+yn=zn if n is an integer greater than 2. ...It is possible that Fermat made some... erroneous supposition, though it is perhaps more probable that he discovered a rigorous demonstration. At any rate he asserts definitely that he had a valid proof—demonstratio mirabilis sane—and the fact that no theorem on the subject which he stated he had proved has been subsequently shown to be false must weigh strongly in his favour; the more so because in making the one incorrect statement in his writings (namely, that about binary powers) he added that he could not obtain a satisfactory demonstration of it. … it took more than a century before some of the simpler results which Fermat had enunciated were proved, and thus it is not surprising that a proof of the theorem which he succeeded in establishing only towards the close of his life should involve great difficulties. ...I venture however to add my private suspicion that continued fractions played a not unimportant part in his researches, and as strengthening this conjecture I may note that some of his more recondite results—such as the theorem that a prime of the form 4n+1 is expressible as the sum of two squares— may be established with comparative ease by properties of such fractions."

"The result of my work has been the most extraordinary, the most unforeseen, and the happiest, that ever was; for, after having performed all the equations, multiplications, antitheses, and other operations of my method, and having finally finished the problem, I have found that my principle gives exactly and precisely the same proportion for the s which Monsieur Descartes has established."

"There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square."

"Et cette proposition est généralement vraie en toutes progressions et en tous nombres premiers; de quoi je vous envoierois la démonstration, si je n'appréhendois d'être trop long."

"Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet."

"Fermat is... honored with the invention of the differential calculus on account of his method of maxima and minima and of tangents, which, of the prior processes, is in reality the nearest to the algorithm of Leibniz; one could with equal justice, attribute to him the invention of the integral calculus; his treatise De æquationum localium transmutatione, etc., gives indeed the method of integration by parts as well as rules of integration, except the general powers of variables, their sines and powers thereof. However, it must be remarked that one does not find in his writings a single word on the main point, the relation between the two branches of the infinitesimal calculus."

"I had a hint of this method [of fluxions] from Fermat's way of drawing tangents, and by applying it to abstract equations, directly and invertedly, I made it general."

"This great geometrician expresses by the character E the increment of the abscissa; and considering only the first power of this increment, he determines exactly as we do by differential calculus the subtangents of the curves, their points of inflection, the maxima and minima of their ordinates, and in general those of rational functions. We see likewise by his beautiful solution of the problem of the refraction of light inserted in the Collection of the Letters of Descartes that he knows how to extend his methods to irrational functions in freeing them from irrationalities by the elevation of the roots to powers. Fermat should be regarded, then, as the true discoverer of Differential Calculus. Newton has since rendered this calculus more analytical in his Method of Fluxions, and simplified and generalized the processes by his beautiful theorem of the binomial. Finally, about the same time Leibnitz has enriched differential calculus by a notation which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing the general results of calculus, that of giving the first approximate values of the differences and of the sums of the quantities; this notation is adapted of itself to the calculus of partial differentials."

"One may regard Fermat as the first inventor of the new calculus. In his method De maximis et minimis he equates the quantity of which one seeks the maximum or the minimum to the expression of the same quantity in which the unknown is increased by the indeterminate quantity. In this equation he causes the radicals and fractions, if any such there be, to disappear and after having crossed out the terms common to the two numbers, he divides all others by the indeterminate quantity which occurs in them as a factor; then he takes this quantity zero and he has an equation which serves to determine the unknown sought. ...It is easy to see at first glance that the rule of the differential calculus which consists in equating to zero the differential of the expression of which one seeks a maximum or a minimum, obtained by letting the unknown of that expression vary, gives the same result, because it is the same fundamentally and the terms one neglects as infinitely small in the differential calculus are those which are suppressed as zeroes in the procedure of Fermat. His method of tangents depends on the same principle. In the equation involving the abscissa and ordinate which he calls the specific property of the curve, he augments or diminishes the abscissa by an indeterminate quantity and he regards the new ordinate as belonging both to the curve and to the tangent; this furnishes him with an equation which he treats as that for a case of a maximum or a minimum. ...Here again one sees the analogy of the method of Fermat with that of the differential calculus; for, the indeterminate quantity by which one augments the abscissa x corresponds to its differential dx, and the quantity ye/t, which is the corresponding augmentation [Footnote: Fermat lets e be the increment of x, and t the subtangent for the point x,y on the curve.] of y, corresponds to the differential dy. It is also remarkable that in the paper which contains the discovery of the differential calculus, printed in the Leipsic Acts of the month of October, 1684, under the title Nova methodus pro maximis et minimis etc., Leibnitz calls dy a line which is to the arbitrary increment dx as the ordinate y is to the subtangent; this brings his analysis and that of Fermat nearer together. One sees therefore that the latter has opened the quarry by an idea that is very original, but somewhat obscure, which consists in introducing in the equation an indeterminate which should be zero by the nature of the question, but which is not made to vanish until after the entire equation has been divided by that same quantity. This idea has become the germ of new calculi which have caused geometry and mechanics to make such progress, but one may say that it has brought also the obscurity of the principles of these calculi. And now that one has a quite clear idea of these principles, one sees that the indeterminate quantity which Fermat added to the unknown simply serves to form the derived function which must be zero in the case of a maximum or minimum, and which serves in general to determine the position of tangents of curves. But the geometers contemporary with Fermat did not seize the spirit of this new kind of calculus; they did not regard it but a special artifice, applicable simply to certain cases and subject to many difficulties, ...moreover, this invention which appeared a little before the Géométrie of Descartes remained sterile during nearly forty years. ...Finally Barrow contrived to substitute for the quantities which were supposed to be zero according to Fermat quantities that were real but infinitely small, and he published in 1674 his method of tangents, which is nothing but a construction of the method of Fermat by means of the infinitely small triangle, formed by the increments of the abscissa e, the ordinate ey/t, and of the infinitely small arc of the curve regarded as a polygon. This contributed to the creation of the system of infinitesimals and of the differential calculus."

"Fermat knew that under reflection light takes the path requiring least time and, convinced that nature does indeed act simply and economically, affirmed in letters of 1657 and 1662 his Principle of Least Time, which states that light always takes the path requiring least time. He had doubted the correctness of the law of refraction of light but when he found in 1661 that he could deduce it from his Principle, he not only resolved his doubts about the law but felt all the more certain that his Principle was correct. ...Huygens, who had at first objected to Fermat's Principle, showed that it does hold for the propagation of light in media with variable indices of refraction. Even Newton's first law of motion, which states that the straight line or shortest distance is the natural motion of a body, showed nature's desire to economize. These examples suggested that there might be a more general principle. The search for such a principle was undertaken by Maupertuis."

"Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits."

"J.M. Child... has made a searching study of Barrow and has arrived at startling conclusions on the historical question relating to the first invention of the calculus. He places his conclusions in italics in the first sentence as follows Isaac Barrow was the first inventor of the Infinitesimal Calculus... Before entering upon an examination of the evidence brought forth by Child it may be of interest to review a similar claim set up for another man as inventor of the calculus... Fermat was declared to be the first inventor of the calculus by Lagrange, Laplace, and apparently also by P. Tannery, than whom no more distinguished mathematical triumvirate can easily be found. ...Dinostratus and Barrow were clever men, but it seems to us that they did not create what by common agreement of mathematicians has been designated by the term differential and integral calculus. Two processes yielding equivalent results are not necessarily the same. It appears to us that what can be said of Barrow is that he worked out a set of geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus. But to say that Barrow invented a differential and integral calculus is to do violence to the habit of mathematical thought and expression of over two centuries. The invention rightly belongs to Newton and Leibniz."

"Since Fermat introduced the conception of infinitely small differences between consecutive values of a function and arrived at the principle for finding the maxima and minima, it was maintained by Lagrange, Laplace, and Fourier, that Fermat may be regarded as the first inventor of the differential calculus. This point is not well taken, as will be seen from the words of Poisson, himself a Frenchman, who rightly says that the differential calculus "consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems.""

"This is another important dispute in the history of how we think about being wrong: whether error represents an obstacle in the path toward truth, or the path itself. The former idea is a conventional one. The latter... emerged during the Scientific Revolution and continued to evolve throughout the Enlightenment. But it didn't really reach its zenith until the early nineteenth century, when... Pierre Simon Laplace refined the distribution of errors, illustrated by the now-familiar bell curve. ...Laplace used the bell curve to determine the precise orbit of the planets. ...By using the normal distribution to graph... individually imperfect data points, Laplace was able to generate a far more precise picture of the galaxy. ...aggregate enough flawed data, and you get a glimpse of the truth."

"Laplace created a number of new mathematical methods that were subsequently expanded into branches of mathematics, but he never cared for mathematics except as it helped him to study nature."

"Laplace made many important discoveries in mathematical physics... Indeed, he was interested in anything that helped to interpret nature. He worked on hydrodynamics, the wave propagation of sound, and the tides. In the field of chemistry, his work on the liquid state of matter is classic. His studies of the tension in the surface layer of water, which accounts for the rise of liquids inside a capillary tube, and of the cohesive forces in liquids, are fundamental. Laplace and Lavoisier designed an ice calorimeter (1784) to measure heat and measured the specific heat of numerous substances; heat, to them, was still a special kind of matter. Most of Laplace's life was, however, devoted to celestial mechanics."

"Whenever I meet in La Place with the words "Thus it plainly appears," I am sure that hours, and perhaps days, of hard study will alone enable me to discover how it plainly appears."

"It is to the influence of the opinion of those whom the multitude [the populous] judges best informed, and to whom it has been accustomed to give its confidence in regard to the most important matters of life, that the propagation of those errors is due, which in times of ignorance, have covered the face of the earth. Magic and astrology offer us two great examples. These errors... having for a basis only universal credence, have maintained themselves during a very long time; but at last the progress of science has destroyed them in the minds of enlightened men, whose opinion consequently has caused them to disappear... through the power of imitation and habit which had so generally spread them... This power, the richest resource of the moral world, establishes and conserves in a whole nation ideas entirely contrary to those... elsewhere... What indulgence ought we not then to have for opinions different from ours, when this difference often depends only upon the various points of view where circumstances have placed us! Let us enlighten those whom we judge insufficiently instructed; but first let us examine critically our own opinions, and weigh with impartiality, their respective probabilities."

"The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought."

"Let us recall that formerly, and at no remote epoch... all the unusual phenomena were regarded as so many signs of celestial wrath."

"All these efforts in the search for truth tend to lead it [the human mind] back continually to the vast intelligence... but from which it will always remain infinitely removed. This tendency peculiar to the human race is that which renders it superior... and their progress in this respect distinguishes nations and ages and constitutes their true glory."

"Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it—an intelligence sufficiently vast to submit these data to analysis—it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes. The human mind offers, in the perfection which it has been able to give to astronomy, a feeble idea of this intelligence. Its discoveries in mechanics and geometry, added to that of universal gravity, have enabled it to comprehend in the same analytical expressions the past and future states of the system of the world."

"Imaginary causes have gradually receded with the widening bounds of knowledge and disappear entirely before sound philosophy, which sees in them only the expression of our ignorance of the true causes."

"The most important questions of life... are indeed for the most part only problems of probability. Strictly speaking it may even be said that nearly all our knowledge is problematical; and in the small number of things which we are able to know with certainty, even in the mathematical sciences themselves, the principal means for ascertaining truth—induction and analogy—are based on probabilities."

"On demandait à Laplace quel était selon lui le plus grand mathématicien de l'Allemagne. C'est Pfaff, répondit-il. - Je croyais, reprit l'interlocuteur, que Gauss lui était supérieur. - Mais, s'écria Laplace, vous me demandez quel est le plus grand mathématicien de l'Allemagne, et Gauss est le plus grand mathématicien de l'Europe."

"It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit.But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity."

"Il est facile de voir que..."

"Nature laughs at the difficulties of integration."

"Lisez Euler, lisez Euler, c'est notre maître à tous."