f 
382 JOSEPH FOURIER. 
on the other side of the channel, Harriot, Newton, Mac- 
laurin, Stirling, Waring, I may say all the illustrious 
geometers which England produced in the last century, 
made it also the subject of their researches. Some years - 
afterwards the names of Daniel Barnoulli, of Euler, and 
of Fontaine came to be added to so many great names. 
Finally, Lagrange in his turn embarked in the same 
career, and at the very commencement of his researches 
he succeeded in substituting for the imperfect, although 
very ingenious, essays of his predecessors, a complete 
method which was free from every objection. From 
that instant the dignity of science was satisfied ; but in 
such a case it would not be permitted to say with the 
poet: 
‘“* Le temps ne fait rien & 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 cer- 
tainty, 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 depriv- 
ing 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 numeri- 
cal 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 
