Derrière la série de Fourier, d'autres séries analogues sont entrées dans la domaine de l'analyse; elles y sont entrees par la même porte; elles ont été imaginées en vue des applications.
After the Fourier series, other series have entered the domain of anylsis; they entered by the same door; they have been imagined in view of applications.

Deviner avant de démontrer! Ai-je besoin de rappeler que c'est ainsi que se sont faites toutes les découvertes importantes.
Guessing before proving! Need I remind you that it is so that all important discoveries have been made?

En un mot, pour tirer la loi de l'expérience, if faut généraliser; c'est une nécessité qui s'impose à l'observateur le plus circonspect.
In one word, to draw the rule from experience, one must generalize; this is a necessity that imposes itself on the most circumspect observer.

Il faut bien s'arrêter quelque part, et pour que la science soit possible, il faut s'arrêter quand on a trouvé la simplicité.
Analyse data just so far as to obtain simplicity and no further.

L'analyse mathématique, n'est elle donc qu'un vain jeu d'esprit? Elle ne peut pas donner au physicien qu'un langage commode; n'est-ce pa là un médiocre service, dont on aurait pu se passer à la rigueur; et même n'est il pas à craindre que ce langage artificiel ne soit pas un voile interposé entre la réalité at l'oeil du physicien? Loin de là, sans ce langage, la pluspart des anaologies intimes des choses nous seraient demeurées à jamais inconnues; et nous aurions toujours ignoré l'harmonie interne du monde, qui est, nous le verrons, la seule véritable réalité objective.
So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the initimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality.

La pensée n'est qu'un écliar au milieu d'une longue nuit. Mais c'est cet éclair qui est tout.
Thought is only a flash in the middle of a long night. But this flash means everything.

Les faits ne parlent pas. (Facts do not speak)

Les mathématique sont un triple. Elles doivent fournir un instrument pour l'étude de la nature. Mais ce n'est pas tout: elles ont un but philosophique et, j'ose le dire, un but esthétique.
Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.

Longtemps les objets dont s'occupent les mathématiciens étaient our la pluspart mal définis; on croyait les connaître, parce qu'on se les représentatit avec le sens ou l'imagination; mais on n'en avait qu'une image grossière et non une idée précise sure laquelle le raisonment pût avoir prise.
For a long time the objects that mathematicians dealt with were mostly ill-defined; one believed one knew them, but one represented them with the senses and imagination; but one had but a rough picture and not a precise idea on which reasoning could take hold.

Qu'une goutee de vin tombe dans un verre d'eau; quelle que soit la loi du movement interne du liquide, nous verrons bientôt se colorer d'une teinte rose uniforme et à partir de ce moment on aura beau agiter le vase, le vin et l'eau ne partaîtront plus pouvoir se séparer. Tout cela, Maxwell et Boltzmann l'ont expliqué, mais celui qui l'a vu plus nettement, dans un livre trop peu lu parce qu'il est difficile à lire, c'est Gibbs dans ses principes de la Mécanique Statistique.
Let a drop of wine fall into a glass of water; whatever be the law that governs the internal movement of the liquid, we will soon see it tint itself uniformly pink and from th at moment on, however we may agitate the vessel, it appears that the wine and water can separate no more. All this, Maxwell and Boltzmann have explained, but the one who saw it in the cleanest way, in a book that is too little read because it is difficult to read, is Gibbs, in his Principles of Statistical Mechanics.

Quand les physiciens nous demandent la solution d'un problème, ce n'est pas une corvée qu'ils nous impsent, c'est nous au contraire qui leur doivent des remercîments.
When the physicists ask us for the solution of a problem, it is not drudgery that they impose on us, on the contrary, it is us who owe them thanks.

Chance ... must be something more than the name we give to our ignorance.

If we ought not to fear mortal truth, still less should we dread scientific truth. In the first place it can not conflict with ethics? But if science is feared, it is above all because it can give no happiness? Man, then, can not be happy through science but today he can much less be happy without it.

It is by logic that we prove, but by intuition that we discover.

It is often said that experiments should be made without preconceived ideas. That is impossible.

One does not ask whether a scientific theory is true, but only whether it is convenient.

Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.

The aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relation between things.

[Poincaré was] the last man to take practically all mathematics, pure and applied, as his province. ... Few mathematicians have had the breadth of philosophic vision that Poincare had, and none in his superior in the gift of clear exposition.

Poincaré was a vigorous opponent of the theory that all mathematics can be rewritten in terms of the most elementary notions of classical logic; something more than logic, he believed, makes mathematics what it is.