406 Reviews, and Notices respecting New Books. 
tions in 1819, and was immediately pirated by others, yet it has un- 
accountably been neglected amongst mathematicians in general, in 
England. For this we cannot, nor do we pretend to account ; but it 
is unfortunate, less for Mr. Horner’s sake than for the sake of mathe- 
matical science. We hope, however, that the elegant exhibition of 
the principles and the processes of the method which are given in this 
work by Professor Young (together with Mr. Horner’s own illustra- 
tions of the subject in Leybourn’s Mathematical Repository, vol. v.,) 
will have the effect of familiarizing at least the younger and more in- 
quiring English mathematicians with this beautiful system of numeri- 
cal solution. No work could be better calculated to produce such an 
effect, and we doubt whether any (even a minor) improvement can 
be made upon it as it stands in this work, and in Mr. Horner's papers. 
Even amongst those who have felt disposed to do justice to Mr. 
Horner’s labours, there are few, it appears to us, who are fully aware 
of the extent of the applicability of his theorems. He has, indeed, 
only applied them in one particular direction ; and it is, perhaps, too 
much to expect that others will be hasty in making applications of 
them which he has not suggested. The unaccountable neglect with 
which his past labours have been received, may well discourage the 
most ardent and persevering mind. We sincerely trust, however, that 
he has yet many years of activity and usefulness before him, and that 
he may still be able to accomplish some of the purposes, to which his 
previous investigations directly lead. 
As in the extraction of the square and cube roots, so in this gene- 
ral evolution of the roots of an equation, the first figure is to be 
found tentatively ; and as in them, so here, the successive figures are 
determined with greater certainty at every successive step. The divi- 
sional portion also of neither one nor other commences till after the 
first step: or in other words, the initial figure of the root and its sign 
are required as a separate and preparatory step to the operation of 
the method. The difficulty of effecting this first step has always been 
found to be great. Lagrange proposed a method which, though theo- 
ertically perfect, was yet, from the immense labour which it involved, 
utterly incapable of application to practice, except, indeed, in cases in 
which the necessity for its application was partially superseded by other 
methods, namely, in the equations of the first four degrees *. It was 
reserved for Budan to overcome this difficulty in his Nowyelle Méthode, 
published in 1803, with the high approbation of Lagrange himself, 
Most unaccountably, this valuable work lay neglected in France till 
after Navier’s publication of Fourier’s Anal. des Eq. Det. in 1831, and 
in England till it was made known to the readers of our Magazine 
and of Leybourn’s Repository by Mr. Horner. It was soon perceived 
* Common justice requires that the laborious researches of the Abbé de 
Gua (Meém. de l’ Acad., 1741) should not be overlooked in any history of 
this problem. Though only in a very limited degree successful, he yet 
opened the road of inquiry, and deduced several very important results, In 
his expression for the nwmber of conditions necessary to render all or any 
number of the roots of an equation real, he is certainly wrong; but this is 
not the place to discuss the source of his error, or to give the true formula. 
