"wiih Mr. T. S. Davies's Notes on some of the Topics. 361 



vol. viii. pp. 408-409. Analysts were so enrajjtured with the theore- 

 tical beauty of Sturm's criterion, that some time elapsed before they 

 thought of testing its practical value — the only special value, indeed, 

 to which the author made the least pretension. So far is it deficient 

 in this respect, that it has not in the hands of any one whose pur- 

 suits lead him to resolve numerical equations, been able to supersede 

 the method of Budan, indirect and sometimes unsatisfactory as this 

 is admitted on all hands to be. Seldom, however, has so small a 

 degree of scientific service been so brilliantly rewarded ; for the 

 French Academy elected him one of their own august body ; and 

 the Roj'al Society awarded him a Royal Medal. Sturm, however, 

 has earned for himself a name of more enduring honour by his other 

 researches, than future times will accord to him for this once-eulo- 

 gised criterion. 



The same remark applies to that of Fourier and Lagrange. 

 I fully agree with the view that you have taken of this subject 

 in the last edition of Mutton's Course, and with your amend- 

 ment of the criterion of De Gua, which forms a very pretty 

 corollary to the rule of signs. The criterion of Budan is well 

 worthy of attention (_/). 



(/). The criterion of Budan in the form in which he himself 

 presented it, is entirely useless ; and I am bold to say that it would 

 not have attracted the least notice but for the form in which Mr. 

 Horner arranged the algorithm and explained the principle of it. 

 Hov.'ever (to use the language of Mr. Horner with respect to the 

 criterion of De Gua), the employment of the method is more eifective 

 than might at first sight appear. For instance, under one aspect, 

 an application of the Hornerian transformation to the reciprocal 

 equation is often effective, as is shown by Mr. Christie, Phil. Mag. 

 vol. xxi. p. 96 : and other methods might be suggested but for the 

 space they would occupy. I believe that those mathematicians who 

 employ themselves on numerical solutions in this country, inva- 

 riably employ Budan's criterion ("either immediately or modified) — ■ 

 availing themselves, however, where possible from the occurrence 

 of zero-coefficients in the transformees , of De Gua's as an auxiliary. 



The drawback upon Sturm's method arises from the uncouth, 

 inartificial and laborious process of finding the common measure of 

 two integer functions of x. The mere ceconomy of space and of 

 writing by a diff'erent arrangement of the work and by the use of 

 detached coefficients, which I first gave in the " Solutions of the 

 Questions in Hutton's Course " (1839), is altogether insufficient to 

 render the process a practicable one, except in very special cases. 

 The same may be said of Professor De Morgan's mode of working, 

 which is a little different from mine (Penny CycL art. Involu- 

 tion). Professor Young, in his "Dissertations" and in his "Theory 

 of Equations," has given a method of taking two steps of the pro- 

 cess at once, which is a material improvement. It is, however, only 

 by the employment of some totally new consideration (I ought, per- 

 haps, to say some new principle) as to the formation of those func« 



