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 lie fait rien a ? 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 



