Resolution of Algebraic Equations . 273 
known law of the binomial theorem, which gives the uncice of 
a regular power. Both depend upon the practicability of a more 
general problem, of which they are but specific cases ; viz. the 
problem “ to give the coefficients of an equation any general de- 
terminate relation” If that were practicable, and it were possible 
to mould them so as to establish a general relation between them, 
(or any required number of them,) it is easy to perceive, that the 
particular relation must be a secondary consideration ; and that, 
wherever the same number of terms are to be acted upon, the 
same means that might make them equal, might give them any 
other proportion at pleasure. 
9. However, of all these methods, and any other of the 
kind, it is to be observed, that the principle is demonstrably a 
false assumption. For, if it be once admitted that the construc- 
tion of equations, and the laws of the successive coefficients 
received ever since Vieta’s time, be true; or that all equations 
are formed invariably in the same manner, from the continual 
multiplication of the simple equations of their roots, which ex- 
perience confirms without any exception ;* it follows, that the 
nature of the roots must infallibly govern that of the equation 
derived from them ; that the same form of equation can only 
be produced by the same forms of roots ; and therefore, before 
all sorts of equations can be made into pure or perfect powers, 
or be given any other general shape, it must be shewn, that all 
quantities are capable of takmg the forms required to produce 
equations of that sort , which will presently be seen to be impos- 
sible. If those who have lost their time and labour in vain 
* Some algebraists, affecting to reject the use of negative quantities, have been 
compelled to dispute the generally received theory of the construction of equations ; 
but they have not been able to suggest any other. 
