witli Mr. T. S. Davies's Notes 07i some of the Topics, 363 



has gone through eight or perhaps nine editions ; and it has been 

 recently reproduced by Mr. Christie in his Algebra for the use of 

 the Royal Military Academy, 1844. 



Mr. Horner first arrived at the method by considerations founded 

 on the doctrine of recurring series ; and he was so well satisfied 

 Avith the investigation as he first gave it in a paper sent to the 

 Royal Society in 1823, that he never attempted any change. It is 

 easy to see, however, that a common principle runs through his in- 

 vestigation and mine ; though there may be very little in either 

 method to suggest the other. My own, indeed, occurred to me 

 almost momentarily, in the midst of my professional duties, at a time 

 when they included that subject. 



I feel it necessary once for all to say, that as far as regards the 

 subject of numerical equations, I have never had any object in view 

 further than to faciUtate the operations of resolution and mainly to 

 render the remarkable methods of my deceased friend more gene- 

 rally accessible and more facile in practice. As the possessor of 

 Mr. Horner's papers, I feel it to be due alike to my own honour 

 and to Mr. Horner's memory, never to publish anything of my own 

 on the subject. 



Another method of approximation is that by recurring series^ 

 a subject treated by Euler and other able mathematicians, and 

 to which Fourier has contributed considerable extensions. 

 This method is of great importance in the solution of equations 

 with unreal roots. 



It is not my intention to discuss at any length the different 

 methods that have been proposed for the numerical solution 

 of equations; but that of Mr. Weddle appears to be deser- 

 ving of mention, as one of those which is likely to be generally 

 and permanently adopted in those cases for which it is more 

 peculiarly adapted. 



The series l;y which Murphy expresses a root (Theory of 

 Equations, art. 64, pp. 83-84) of an equation is most remark- 

 able, not only in itself and for the process by which it is de- 

 rived, but also for its principle of derivation, which is capable 

 of application in a variety of other cases. 



The principal treatises on the theory of equations in the 

 English language are those of Murphy, Hymers, Young, and 

 Stevenson. 



That of Murphy abounds in traces of the great genius of 

 that lamented analyst. I have just alluded to an interesting- 

 feature of his work : amongst the many others I shall content 

 myself with citing his discussion of the theorems" of Fourier 

 respecting the solution of equations by recurring series. Ad- 

 mitting the originality of the view taken by the latter philo- 

 sopher. Murphy shows that he is incorrect in his details, and 

 supplies the necessary corrections. 



