Physicists From France

"[I]f all the strata of air of... the atmosphere... preserved their density with their transparency, and lost only the mobility... peculiar to them, this mass of air, thus become solid, on being exposed to the rays of the sun, would produce an effect the same in kind... [as] just described."

"Galileo and Hooke demonstrated that every sound is characterized by a precise number of vibrations per second, but the full understanding of even the simplest... vibrators requires... calculus. ...Leonard Euler, Daniel Bernoulli, Jean le Rond d'Alembert, and Joseph Louis Lagrange all studied the vibrating string.... In 1747 d'Alembert found... the wave equation and in 1753 Bernoulli stated the... decomposition of every motion of a string as a sum of elementary sinusoidal motions. Then at the beginning of the following century Fourier developed "harmonic analysis.""

"The formula and even the method was... used by Euler in... 1777, published... 1798. (Fourier... having failed to refer to earlier works...) ...[A]s with Bernoulli, Fourier had acquired an intimate understanding of the physical meaning of the problem. Over... two years... Fourier repeated all important experiments... carried out in England, France, and Germany and added experiments of his own. ...[S]triking ...confirmations of his new theory ...together with his overcoming ...difficulties advanced by the old masters. Fourier mentioned the motion of fluids... propagation of sounds and...vibrations of elastic bodies, as other applications ...fully aware of having opened up a new era for the solution or partial differential equations... It was the era of linearization that would dominate mathematics for the first half of the nineteenth century and... has remained important... The diffusion equation is a : Linear combinations of solutions are still solutions. It was not the first such equation... in history... but the method opened up enormous... possibilities."

"The novelty of his method... initially perplexed... mathematicians... from Lagrange to Laplace and Poisson. ...[P]ublication ...was ...delayed as many as fifteen years during which he ...defended, explained and extended [his work]."

"He ranks among the most important scientists of the 19the century for his studies in the propagation of heat... and the paternity of the expression ""—effet de serre—is attributed to Fourier."

"He started as a convinced Jacobin... and ended up a cautious liberal."

"It is true that M. Fourier had the opinion that the principal end of mathematics was the public utility and the explanation of natural phenomena; but such a philosopher as he is should have known that the unique end of science is the honor of the human mind, and that from this point of view a question of number is as important as a question of the system of the world."

"He carried on his elaborate investigations on the propagation of heat in solid bodies, published in 1822 in his work entitled La Theorie Analytique de la Chaleur. This work marks an epoch in the history of mathematical physics. "Fourier's series" constitutes its gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807 before the French Academy. The trigonometric series \sum_{n=0}^{n=\infty} (a_n\sin nx+b_n\cos nx) represents the function \phi(x) for every value of x if the coefficients a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}\phi(x) \sin nx\,dx, and b_n be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function."

"Wikipedia recounts research about this much-cited episode."

"The exchange is reported by Victor Hugo (who in turn was citing François Arago) as:"

"[Sire,] je n'ai pas eu besoin de cette hypothèse."

"With respect to the cohesion and of liquids, I have had the good fortune to anticipate Mr. Laplace in his late researches, and I have endeavoured to show, that my assumptions are more universally applicable to the facts, than those which that justly celebrated mathematician has employed."

"Laplace had taken Newton's science and turned it into philosophy. The universe was a piece of machinery, its history was predetermined, there was no room for chance or for free will. The cosmos was indeed an ice-cold clock."

"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."

"L'homme ne poursuit que des chimères."

"Ce que nous connaissons est peu de chose, ce que nous ignorons est immense."

"La dernière chose que nous attendions de vous, Général, est une leçon de géométrie !"

""Les questions les plus importantes de la vie ne sont en effet, pour la plupart, que des problèmes de probabilité." Théorie analytique des probabilités, 1812."

"Said the great and magnanimous Laplace: 'It is India that gave us the ingenious method of expressing all numbers by ten symbols, each 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, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.'"

"On voit, par cet Essai, que la théorie des probabilités n'est, au fond, que le bon sens réduit au calcul; elle fait apprécier avec exactitude ce que les esprits justes sentent par une sorte d'instinct, sans qu'ils puissent souvent s'en rendre compte."

"[Sire,] je n'avais pas besoin de cette hypothèse-là."

"Lagrange, also present (or to whom Napoleon repeated Laplace's reply, in another version), then commented: "Ah ! C’est une belle hypothèse; ça explique beaucoup de choses.""

"Comment, vous faites tout le système du monde, vous donnez les lois de toute la création et dans tout votre livre vous ne parlez pas une seule fois de l'existence de Dieu !"

"The history of science shows that the progress of science has constantly been hampered by the tyrannical influence of certain conceptions that finally came to be considered as dogma. For this reason, it is proper to submit periodically to a very searching examination, principles that we have come to assume without any more discussion."

"It seems a little paradoxical to construct a configuration space with the coordinates of points which do not exist."

"Admettant que la particule possède une vibration interne qui permet de l'assimiler à une petite horloge, je supposais que cette horloge se déplaçait dans son onde de façon que sa vibration interne reste constamment en phase avec celle de l'onde : c'est le postulat de l'accord des phases."

"The actual state of our knowledge is always provisional and … there must be, beyond what is actually known, immense new regions to discover."

"Two seemingly incompatible conceptions can each represent an aspect of the truth … They may serve in turn to represent the facts without ever entering into direct conflict."

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

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

"Analytical geometry was invented by Descartes and the first exposition of it was given in 1637: that exposition was both difficult and obscure, and to most of his contemporaries, to whom the method was new, it must have been incomprehensible. Wallis made the method intelligible to all mathematicians."

"I should like you to consider that these functions (including passion, memory, and imagination) follow from the mere arrangement of the machine’s organs every bit as naturally as the movements of a clock or other automaton follow from the arrangement of its counter-weights and wheels."

"I suppose the body to be nothing but a statue or machine made of earth, which God forms with the explicit intention of making it as much as possible like us."

"Doubt is the origin of wisdom and Latin: Dubium sapientiae initium. This has been attributed to Descartes, including here previously, but no original attribution has been found. Descartes Meditationes de prima philosophia has been cited as the source of Dubium sapientiae initium, but this quote is not found in this work."

"An optimist may see a light where there is none, but why must the pessimist always run to blow it out?"