Prof. Young on Algebraic Equations. 409 
which result from finding the coefficients of the derived polynomes ; 
as the ratios alone are sought, and these can hardly ever be required 
to extreme exactness, since the general character, not the particular 
values, are generally sought. Atthe same time some experience, and 
the foresight which experience alone can give, is requisite to distin- 
guish when, and under what circumstances, this abbreviation can be 
used with perfect safety. If we take, for instance, the biquadratic 
equation 32 2* + 41 2° — 184 2* — 24x + ] = 0, the application of 
Sturm’s method brings us to products (and in these cases logarithms 
cannot be used with perfect safety, except where the contractions can 
be used) of szrteen figures; and as the coefficients of equations which 
arise out of any inquiry to which algebra may be made subservient, 
and under all conditions of the data, are generally less likely to be so 
simple as the example given above, some subsidiary methods of less- 
ening the actual trouble are yet not only desirable, but necessary. 
Cannot Mr. Horner so apply or modify his principle of “ Synthetic 
Division”’, as to furnish a more direct and easy algorithm for Sturm’s 
Rule? We think we see more than one way by which this may be 
accomplished ; but we leave it in better hands, when we refer over the 
problem into his*. 
The space which we can devote to a review will not allow us to 
give even a general analysis of Professor Young’s treatise. It is suf= 
ficient to say that it contains all that can be interesting to the student 
on the subject of equations, developed with his usual perspicuity and 
elegance ; and that it is brought, in all essential points, to the state of 
Science at the present hour ; and though principally intended for the 
use of students who have only mastered the first principles of algebra, 
and happily adapted to their wants, yet as a syllabus for recalling to 
the minds of the most extensively read mathematicians on the subjects 
of equations, the essentials of what they already know, we are per- 
suaded that it will be of considerable utility. With this conviction 
we take our leave of the work, happy if our favourable notice shall be 
the means of rendering it more extensively known, and that less for 
the sake of Professor Young than of the numerous persons who may 
derive advantage from his labours. 
* We have often wondered that the method of working with the “ de- 
tached coefficients” in algebraic multiplication and division has never been 
introduced into practice, and even into elementary works: and that the 
beautiful contrivance of what its inventor has called “ Synthetic Division” 
(see Leybourn’s Repos., vol. v.,) has not also become a school-boy practice 
ere now. This is also one amongst the many valuable improvements in 
algebra and arithmetic conferred on mathematicians by Mr. Horner. The 
eleventh edition of Hutton’s course, edited by Dr. Gregory, is the only e/e- 
mentary work in which it has yet appeared, 
