•with Mr. T. S. Davies's Notes on some of the Topics. 365 



and able young analyst, Mr. Weddle, has also published a method 

 of solving numerical equations, as far as the evolution of the figures 

 of the root is concerned, which, like Horner's, is a th'm^ per se. It 

 has, moreover, no one principle or idea in common with Horner's 

 method ; and, of course, there can be nothing in common as to the 

 modes of investigation. Horner's consecutive corrections are addends 

 to the root already evolved : Weddle's are factoi'S of it. Horner 

 uses as many columns as there are units in the dimension of the 

 equation : Weddle only employs the columns that have coefficients 

 in the given equation different from zero. Horner's is the briefer 

 and more manageable where the terms of the equation are nu- 

 merous and the figures of the root required to be many ; Weddle's 

 when the terms are few and the figures of the root not required to 

 be numerous. Weddle's may be benefited as to applicability by 

 such processes as Mr. Jerrard's for diminishing the number of terms 

 of the equation (if, indeed, the actual labour of doing this would 

 not more than counterbalance the opposite advantage) : but such 

 transformations would not subserve Horner's process in the slightest 

 degree. 



Of the valuable contributions of Professor De Morgan to the 

 illustration of Horner's process, and of his earnest enforcement of 

 its merits on all occasions, I ought to express the very high estimate 

 that I form. It would however only be " to gild refined gold" to 

 attempt that expression ; and I need only refer to his articles in the 

 Penny Cyclopaedia and its Supplement, and to his paper in the Com- 

 panion to the Almanac and the recent edition of his Elements of 

 Arithmetic. 



has given a quadratic transformation, of vvliich, if I recollect 

 right, there is no example in the treatises of Dr. Hymers or 

 Mr. Stevenson, and to which I think there is but one ap- 

 proach in that of Murphy — I mean in his remarks on Sir J. 

 W, Lubbock's improvements in Bezout's processes. 1 think 

 that it was Professor Young who first introduced Sturm's 

 theorem into this country; he has introduced considerable 

 improvements into its praxis. Upon this subject I concur 

 with the note at p. v. of the preface to your Hutton. 



The treatise of Mr. Stevenson is a very elegant one. Its 

 author has given what I conceive to be a very proper and 

 prominent position to Cauchy's demonstration of the existence 

 of a root of an algebraic equation, from which the existence 

 of 7J roots may be arrived at without much difficulty {J). 



(_/ ). Of the works enumerated by Mr. Cockle I am not disposed to 

 offer any general opinion : but I may remark, that Mr. Young's is 

 the only one amongst them which even professes to aim at giving 

 the actual practice of solution. Whatever other merits the several 

 works may have (and each has its merits), they are of an algebraical 

 rather than of a numerical character : whilst in Young's, the alge- 



