362 Mr. J. Cockle's Analysis of' the Theory of Equatiotis, 



tions which occur in Sturm's process, that we can look with any 

 degi-ee of hope to obtaining the means of rendering Sturm's criterion 

 of practical usefulness. 



Some very interesting discussions connected with this subject 

 may be consulted with advantage in Professor Young's " Researches 

 on the Imaginary Roots of Numerical Equations" (1844). Those 

 founded on taking away the terms of the direct and reciprocal 

 equations by transformation are peculiarly elegant ; and the entire 

 treatise bears witness to the justice with which the author's high 

 scientific character is accorded to him by mathematicians in general. 

 Perhaps I ought to add that a very faithful translation of Sturm's 

 Memoire was published by Mr. W. H. Spiller, in 4to, 1837. 



6. On the subject of the limits within which the roots of 

 equations lie I shall make a few comments further on. 



7. Among those who have devised processes for determining 

 the numerical values of the roots of equations, the first place 

 must be given entirely and without reservation to Horner. 

 The extension of the approximative principle involved in the 

 extraction of the square root to thegeneral solution of equations 

 was, it is true, arrived at by Vieta ; but it is only by combining 

 that principle with an easy process that we can derive much 

 practical benefit from it. Practically speaking, the whole 

 method of approximation is due to Horner, for his process 

 alone renders it a working instrument. One splendid result 

 of his labours in this field was his invention of synthetic di- 

 vision, which constitutes a radical, material, and important 

 improvement of one of the elementary processes of algebra, 

 and presents the curious spectacle of a simplification of such 

 a kind overlooked for centuries. For appropriate and simple 

 investigations of synthetic division science is indebted to you, 

 as she is for your zealous and unintermitting enforcement of 

 the views and rights of its illustrious inventor [g). 



(g). I am by no means sure that my mode of investigating the 

 synthetic division is more simple than Mr. Horner's, in any other 

 sense than that it is more elementary, and on that account better 

 adapted to the purposes of instruction. Of the two modes which 

 I have given, that first offered (Hutton's Course by Dr. Gregory, 

 11th ed. 1835) is found by experience to be the more easily com- 

 prehended by young students : whilst as an investigation, in the 

 stricter sense of the term, that printed in the Mathematician (vol. i. 

 p. 74) seems to me the preferable one. A different method (but 

 which led to nearly the same praxis in reference to the single 

 purpose of transforming equations, though it was inapplicable to 

 many other important purposes which the sj'nthetic division can 

 be employed to effect) was proposed a little after the synthetic divi- 

 sion was made known by Mr. Peter Nicholson. This method has 

 been repeatedly given in Nicholson and Rovvbottom's Algebra, which 



