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From: Baich <cowley@ke...>  20091224 08:36:54

A high degree. As soon as he wishes to quit the domain of abstract relations, the calculator has occasion to employ the roots of these equations; thus the art of discovering them by the aid of an 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 must be _acknowledged_, are so slight and so imperfect, that we should truly be justified in referring the origin of this branch of analysis only to the excellent labours of our countryman Vieta. Descartes, to whom we render very imperfect justice when we content ourselves with saying that he taught us much when he taught us to doubt, occupied his attention 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; Rolle, of the Academy of Sciences, devoted to this one subject his entire life. Among our neighbours on the other side of the channel, Harriot, Newton, Maclaurin, 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 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 