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XLIX. Reviews, and Notices respecting New Booh, 



On the Solution of Numerical Equations: translated byW.H. Stiller, 



from a Memoir of M. C. Sturm, presented to the Royal Academy 



of Sciences of Paris. 4to. J. Souter, London j Bachelier, Paris. 



rj^HE mcmoire, of which the present publication of Mr. Spiller is 

 A a translation, was printed in Paris about three months ago, in the 

 sixth volume of " Memoires presentes par des Savans etrangers." 

 It forms the most valuable addition to our knowledge of the theory 

 of numerical equations that has been made since the time of La- 

 grange ; in fact, it fully accomplishes the great object which, from 

 Descartes till now, has engaged the thoughts of mathematicians 

 in this department of analysis, and which is no other than the com- 

 plete determination of the number and situation of the real roots 

 of every numerical equation, a problem to which the labours of La- 

 grange were, as is well known, especially directed. The success 

 of these labours, however, extended no further than to show that 

 the solution of the problem in question was mathematically possible, 

 although practically nearly impossible, on account of the mass of 

 numerical labour, and the consequent risk of error, involved in the 

 method which he proposed. 



As a means, therefore, of facilitating the actual evolution of the 

 real roots of numerical equations of the higher order, the process 

 of Lagrange was entirely useless. Very recently, however, the re- 

 searches of Budan and Fourier have proved that the rule of signs 

 of Descartes was far more comprehensive than had hitherto been 

 supposed ; and that by employing it in its full extent, it was ca- 

 pable of furnishing much more explicit information respecting the 

 nature of the roots of an equation than had as yet been expected 

 from it. Still these additional advantages did not reach the length of 

 supplying us with the exact number of real roots comprised between 

 any two proposed limits, and yet nothing short of this knowledge 

 would suffice fully to answer the demands of the practical computer. 



It is true that the last-named mathematician, Fourier, in his 

 posthumous work entitled " Analyse des Equations determinees," 

 has pushed his inquiries beyond those of Budan, and has added 

 to the rule of Descartes a process of his own for ascertaining the 

 nature of those numerical intervals which that rule left in doubt; 

 he has therefore the merit of having thus supplied the deficiencies 

 of Descartes' rule; and there can be no question that the combina- 

 tion of this rule as improved by Budan, with the supplementary 

 process of Fourier for ascertaining the precise character of the 

 doubtful intervals, is sufficient to make known the nature and situa- 

 tion of the roots of a numerical equation, without having recourse 

 to the method of the squares of the differences. Still the method of 

 Fourier is slow and tedious, tentative and inelegant; and is not, 

 after all, capable of making known & priori the exact number of the 

 real and imaginary roots, without proceeding to the actual solution 

 of the equation. Now it is a remarkable peculiarity in the theorem 

 of Sturm that it completely solves the problem by a perfectly in- 



