Prof. Young on Algebraic Equations. 407 
by the Continental mathematicians that Fourier’s was but a slight mo- 
dification of Budan’s method, and accordingly the French elementary 
writers since that time have invariably given the name of Budan, not 
that of Fourier, to the method. The few English mathematicians who 
have spoken on the subject, following Navier and Fourier, or rather 
Mr. Peacock’s account of the matter, have designated them as ‘‘ Fou- 
rier’s rules.” Professor Young ascribes them rightly to Budan ; and we 
hope that, as his work must of necessity obtain an extensive circulation, 
the mistake will be gradually corrected. We hope it is not too late, 
though we well know how difficult it is to eradicate a familiar epithet, 
however unjust ; as, for instance, in the case of “ Cardan’s Rule” for 
cubics, and “ Mercator’s Projection” of the Sphere, neither of which, 
it is well known, was the invention of the persons whose names they 
bear, whilst the names of their authors, Tartaglia and Wright, are 
almost unknown, except to well-read mathematicians. This rule was 
an immense advance in the progress of actual solution, as it enables 
us to discover the number of roots which lie between any assigned 
limits, a and b, and to determine whether they be real or imaginary. 
The initial (or, if need be, any number of figures,) of the real roots 
may be successively determined ; and hence the methods of actual ap- 
proximation, whether that of Newton, of Lagrange, or of Horner, may 
be immediately commenced, and the determination of it gradually 
and systematically effected*. This method, however, though in com- 
parison of Lagrange’s “ Equation of the Squares of the Differences 
of the Roots” such as to induce any one to rejoice in its discovery, 
and value it as perfect, still a simpler, more direct, and effective me- 
thod has been since discovered by M. Sturm, already alluded to. It 
was read to the French Institute in 1829, before the publication of 
Fourier’s Traité, but was not published in its Mémoires till a few 
months ago. It was, however, printed in Crelle’s Journal fiir die 
reine und angewandte Mathematik, about a year after, and was 
introduced in an abridged form into the works of Lacroix, Bourdon, 
and Lefebre de Fourcy, soon and successively. No allusion to it, 
however, appeared in any English work till the publication of Profes- 
sor Young's treatise. It is the more extraordinary that Mr. Pea- 
cock should have overlooked it when writing his “ Report”, as it was 
so easy of access from so many quarters, The memoir has since been 
accurately and elegantly translated into English, as we noticed in a 
late Number, by Mr. W. H. Spiller. The best and most simple of all 
the abstracts of this important paper that we have seen is that of Mr. 
Young, in the work before us. Mr. Horner has well termed it the 
« gem of the book,”—well, as modestly coming from him ; but still, to 
our thinking, not more a gem than the version of his own methods in 
the same work : and to follow the metaphor, we would add that it is 
here cut, polished, and set in the most tasteful and elegant manner 
of which it seems capable. Even to accomplished mathematicians, to 
whom the subject is new, we strongly recommend the reading of 
Mr. Young’s chapter before taking up the original memoir, as it will 
* Fourier employs the Newtonian, though he has put it in almost the 
worst form, perhaps, of which it is susceptible for actual working. 
