404 Reviews, and Notices respecting New Books. 
The problem, on the authority of very careful researches into the re- 
lation that must subsist amongst the roots themselves in the com- 
position of the coefficients, and the degree of the subsidiary equations 
to which the algebraical expression of those relations conducts us, is 
now known tobe incapable of solution by a general formula. If this 
be established satisfactorily (and to our own minds it is so), the in- 
quiry is ended in this direction; and the only ground to hope for a so- 
lution is in the discovery of some process which shall evolve the several 
roots by one continuous series of operations, figure after figure, till 
either the whole of them areassigned ; or, when the roots are irrational, 
till so many figures shall be assigned as are necessary for the purpose 
had in view in the problem in which the equation originated *. This 
was the method followed by Newton, whose sagacity led him to see the 
hopelessness of a general formula of solution, if not its essential im- 
possibility,—one instance amongst many of his extraordinary pre- 
science of the history of science in after ages, Nor was this done after 
a casual view of the subject, but after careful investigations ofits charac- 
ter, as is evident from the researches which he made respecting the re- 
lations between the roots and the coefficients of a literal equation—re- 
searches, to the results of which, much as they have been since pur- 
sued, the labours of his successors have made comparatively unimpor- 
tant additions. His method of approximation, however, with which we 
are now more immediately concerned, was characteristic of his great 
mind, and remained till our own time, except under peculiar circum- 
stances, not only the briefest, but the best that had been proposed. 
Still it had its difficulties and imperfections, even after the initial figure 
of a root had been found, and these were fully exposed by Lagrange 
in the 5th note to his Traité de Resolution des Equations as far back 
as 1798; and though they have been in some degree removed by 
Mr. Horner (in the Annals of Philosophy,) and Baron Fourier (in his 
Analyse des Equations Determinées,) the method is on many accounts 
incumbered with difficulties that are of a serious practical nature, and 
essentially inherent in the principle of the process. 
A method of approximation, very elegant in theory, and though not 
rapid in execution, yet free from some of the defectsincident to New- 
ton’s method, was given by the celebrated analyst just referred to, 
Lagrange, by which the root was exhibited in a continued fraction. 
This, too, besides its practical tediousness, had other inconveniences, 
several of which, by the labours of Mr. Horner, published in the An- 
nals of Philosophy and the Journal of the Royal Institution, were al- 
most entirely removed. Still the tediousness which is essential to its 
first principle of operation is such as to render it useless in practice, 
except where some very important object arises to justify the employ- 
ment of the great length of time which its practice requires. 
It is to Mr. Horner that we owe a simple, rapid, easy, and complete 
method of continuous approximation, disencumbered of all extraneous 
* The investigation is here referred to cases in which the equation is re- 
duced toa rational form, as indeed are all the general conclusions which are 
deduced respecting equations. A valuable dissertation on irrational or surd 
equations is given by Mr, Horner in the present volume of this Journal, p. 43. 
