MEMOIR ON THE RESOLUTION OF NUMERICAL EQUATIONS.

About the close of the year 1789 Fourier repaired to Paris and read before the Academy of Sciences a memoir on the resolution of numerical equations of all degrees.  This work of his early youth our colleague, so to speak, never lost sight of.  He explained it at Paris to the pupils of the Polytechnic School; he developed it upon the banks of the Nile in presence of the Institute of Egypt; at Grenoble, from the year 1802, it was his favourite subject of conversation with the Professors of the Central School and of the Faculty of Sciences; this finally, contained the elements of the work which Fourier was engaged in seeing through the press when death put an end to his career.

A scientific subject does not occupy so much space in the life of a man of science of the first rank without being important and difficult.  The subject of algebraic analysis above mentioned, which Fourier had studied with a perseverance so remarkable, is not an exception to this rule.  It offers itself in a great number of applications of calculation to the movements of the heavenly bodies, or to the physics of terrestrial bodies, and in general in the problems which lead to equations of 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: