(B) HISTORY OF ALGEBRA

(xv)
-- Carl Friedrich Gauss (1771-1855)
-- Niels Henrik Abel (1802-29)
-- Budan
-- Joseph Fourier (1768-1830)
-- W G Horner
-- Jacques-Charles-François Sturm (1803-55).

But although the absolute solution of equations of higher orders than the fourth remains amongst the things unaffected, and rather to be hoped for than expected, a very great deal has been done towards preparing the way for approximate, if not for absolute solutions.

Niels Henrik Abel

In the first place, equations of the higher orders, when they assume certain forms, have been shown to be capable of solution. An equation of this kind, to all appearance of a very general and comprehensive form, had been solved by De Moivre in the Philosophical Transactions for 1737. Binomial equations had advanced under the skilful hands of Gauss, who, in his Disquisitiones Arithmeticae, which appeared in 1801, added largely to what had been done by Vandermonde in the classification and solution of such equations; and subsequently, Abel, a mathematician of Norwegian birth, who died too early for science, completed and extended what Gauss had left imperfect. The collected writings of Abel published at Christiania in 1839, contain original and valuable contributions to this and many other branches of mathematics.

But it is not in the solution of equations of certain forms that the greatest advance has been made during the present century. The basis of all methods of solution must evidently be found in the previous separation of the roots, and the efforts of mathematicians have been directed to the discovery of methods of effecting this. The object is not so much to classify the roots into positive and negative, real and imaginary, as to determine the situation and number of the real roots of the equation.

Joseph Fourier

The first writer on the subject whose methods appeared in print is Budan, whose treatise, entitled Nouvelle méthode pour la résolution des équations numériques, appeared in 1807. but there is evidence that Fourier had delivered lectures on the subject prior to the publication of Budan's work, and consequently, without detriment to the claims of Budan, we may admit that the most valuable and original contribution to the science is to be found in Fourier's posthumous work, published by Navier in 1831, entitled Analyse des équations déterminées. The theorem which Fourier gave for the discovery of the position, within narrow limits, of a root of an equation, is one of two theorems, each of which is known by mathematicians as "Fourier's Theorem." The other is a theorem of integration, and occurs in the author's magnificent work Theorie de la Chaleur.

During the interval between the publication of Budan's work and that of Fourier, there appeared a paper in the Philosophical Transactions of the Royal Society for 1819, by W. G. Horner, upon a new method of solving arithmetical equations. From its being somewhat obscurely expressed, the great originality of the memoir did not at once appear. Gradually, however, Mr Horner's method came to be appreciated, and it now ranks as one of the best processes, approaching, in some points, to Fourier's. In the Mémoires des savans étrangers for 1835, appears a memoir, which, if it does not absolutely supersede all that had been previously done in assigning the positions of the real roots of equations, yet in simplicity, completeness, and universality of application, surpasses them all.

The author, M. Sturm, of French extraction, but born at Geneva, has in this memoir linked his name of a theorem which is likely to retain its place amongst the permanent extensions of the domain of analysis as long as the study of algebra shall last. It was presented to the Academy in 1829.





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