Review of the Cambridge Course of Mathematics. 323 



genious and elegant demonstration of the doctrine, that 

 there are two solutions to every equation of the second de- 

 gree. This demonstration, also, contains the germ of the 

 general theory of equations of any degree. 



The binomial theorem, is rigorously demonstrated, and 

 its extensive applications well pointed out. This was es- 

 pecially necessary with respect to a formula, that serves as 

 a foundation for so great a number of important investiga- 

 tions. In the " elements," the demonstration is limited to 

 the case of the theorem, in which the exponent of the bi- 

 nomial is a positive whole number; but in the " Supple- 

 ment," it is extended to the cases, in which the exponent 

 is fractional or negative. In his demonstration, instead of 

 concluding the general expression of the theorem from the 

 observation of some particular powers of a binomial, a 

 method of proceeding which is defective because of the lim- 

 ited number of powers we are able to observe, he investi- 

 gates the law which connects a preceding power with that 

 which succeeds it, and this law thus connecting a series of 

 powers of unlimited extent, makes all the remote results 

 depend upon the tirst, which may be immediately and 

 strictly verified. It is in this way that he has obtained the 

 general expression of the terms of this celebrated formula 

 of Sir I. Newton. 



After employing the binomial formula to explain the ex- 

 traction of the cube and other roots both of numbers and of 

 literal quantities, he treats of " equations with two terms," 

 and gives a specimen of analytical refinement not often to be 

 found in elementary treatises. On "Radical Expressions," 

 there is nothing very noticeable, except his remarks on some 

 peculiar cases which occur in the calculus of these quanti- 

 ties, in vi^hich two or three dark points are illustrated from 

 the theory of equations with two terms. 



The necessary limits within which we are confined, will 

 not permit us to enter into a particular examination of the 

 "general Theory of Equations," and of "Equations exceed- 

 ing the first degree." If justice were done to the subject, 

 the discussion must necessarily be long, and might perhaps 

 be tedious and uninteresting. We know not how a better 

 selection of elementary methods for obtaining universal 

 resolutions of equations, could have been made. The 

 style, also, in which they are presented, corresponds with 



