“Le temps ne fait rien a l’affaire.”

Now although the processes invented by Lagrange, simple in principle and applicable to every case, have theoretically the merit of leading to the result with certainty, still, on the other hand, they demand calculations of a most repulsive length.  It remained then to perfect the practical part of the question; it was necessary to devise the means of shortening the route without depriving it in any degree of its certainty.  Such was the principal object of the researches of Fourier, and this he has attained to a great extent.

Descartes had already found, in the order according to which the signs of the different terms of any numerical equation whatever succeed each other, the means of deciding, for example, how many real positive roots this equation may have.  Fourier advanced a step further; he discovered a method for determining what number of the equally positive roots of every equation may be found included between two given quantities.  Here certain calculations become necessary, but they are very simple, and whatever be the precision desired, they lead without any trouble to the solutions sought for.

I doubt whether it were possible to cite a single scientific discovery of any importance which has not excited discussions of priority.  The new method of Fourier for solving numerical equations is in this respect amply comprised within the common law.  We ought, however, to acknowledge that the theorem which serves as the basis of this method, was first published by M. Budan; that according to a rule which the principal Academies of Europe have solemnly sanctioned, and from which the historian of the sciences dares not deviate without falling into arbitrary assumptions and confusion, M. Budan ought to be considered as the inventor.  I will assert with equal assurance that it would be impossible to refuse to Fourier the merit of having attained the same object by his own efforts.  I even regret that, in order to establish rights which nobody has contested, he deemed it necessary to have recourse to the certificates of early pupils of the Polytechnic School, or Professors of the University.  Since our colleague had the modesty to suppose that his simple declaration would not be sufficient, why (and the argument would have had much weight) did he not remark in what respect his demonstration differed from that of his competitor?—­an admirable demonstration, in effect, and one so impregnated with the elements of the question, that a young geometer, M. Sturm, has just employed it to establish the truth of the beautiful theorem by the aid of which he determines not the simple limits, but the exact number of roots of any equation whatever which are comprised between two given quantities.

PART PLAYED BY FOURIER IN OUR REVOLUTION.—­HIS ENTRANCE INTO THE CORPS OF PROFESSORS OF THE NORMAL SCHOOL AND THE POLYTECHNIC SCHOOL.—­EXPEDITION TO EGYPT.