JOSEPH FOURIER. 141 



The .subject of algebraic analysis above mentioned, wliicli Fourier had 

 studied with a perseverance so remarkable, is not an exception to this 

 rule. It oifers itself in a great number of ap})lications of calculation to 

 the movements of the heavenly bodies, or to the physics of terres- 

 trial bodies, and in general in the problems which lead to equations of 

 a high degree. As soon as he wishes to <iuit the domain of abstract re- 

 lations, the calculator has occasion to employ the roots of these equa- 

 tions ; thus the art of discovering them by the aid of a uniform method, 

 either exactly or by approximation, did not fail at an early period to 

 excite the attention of geometers. 



An observant eye perceives already some traces of their efforts in the 

 writings of the mathematicians of the Alexandrian school. These traces, 

 it nnist be acJciiowleih/ed, are so slight and so imperfect that Ave should 

 truly be justified in referring the origin of this branch of analysis only 

 to the excellent lal)ors of our countryman Vieta. Descartes, to whom 

 we render v*ry imperfect justice when we content ourselves with saying 

 that he taught us much when he taught us to doubt, occupied his atten- 

 tion also for a short time with this problem, and left upon it the indelible 

 impress of his powerful mind. Hudde gave for a particular but very 

 important case rules to which nothing has since been added, liolle, of 

 the Academy of Sciences, devoted to this one subject his entire life. 

 Among our neighbors on the other side of the channel, Harriot, Xewton, 

 Alaclauriu, 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 afterward the names of Daniel Bernoulli, 

 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 — 



" Lc touips lie fait rieii a I'affaire." 



ISTow, although the processes invented by Lagrange, simple in princi- 

 ple and applicable to every case, have theoretically the merit of leading 

 to the result with certainty, still, on the other hand, they demand cal- 

 culations of a most re[)ulsive 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 Mas 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 ])ositive 

 roots this equation may have. Fourier advanced a step further: he 



