g 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.

We had just left Fourier at Paris, submitting to the Academy of Sciences
the analytical memoir of which I have just given a general view. Upon
his retu