534 PHILOSOPHICAL TRANSACTIONS. [ANNO 17 QQ. 



cients received ever since Vieta's time, be true; or that all equations are formed in- 

 variably in the same manner, from the continual multiplication of the simple equa- 

 tions of their roots, which experience 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 shown, that all 

 quantities are capable of taking the forms required to produce equations of that 

 sort, which will presently be seen to be impossible. If those who have lost their 

 time and labour in vain endeavours to improve these general methods, had, instead 

 of involving themselves in a labyrinth of substitution and process, on the chance 

 of some means of simplification presenting itself, considered before-hand the pro- 

 bability of success, the imperfection of Cardan's rule would never have appeared 

 a paradox, nor the interruption of all further progress by it have given room for 

 surprize. They must have seen, that no equation beyond a quadratic can admit of 

 a real extinction of its intermediate terms. In the general equation x" — px"~ l -f. 

 qx"-* — rx"-* + sx n ~* &c. = O, p being the sum of the roots, and q the sum of 

 their combinations in pairs, by Sir I. Newton's theorem for finding the sums of the 

 powers of the roots, /> 2 — 1q will be the sum of their squares; and therefore, if 

 both p and q vanish, the sum of the squares of the roots must vanish also; which 

 can never happen with real quantities. Besides this, in attempting to destroy many 

 intermediate terms at once, we know by experience, that the equations which be- 

 come incidentally necessary to be solved, rise to a much higher dimension than the 

 given equation ; so that our labour, in this respect, defeats itself. 



10. Nor will these difficulties be avoided, if we abandon the idea of a general 

 resolution, and attempt to work out the roots separately: though the number of 

 co-efficients is always sufficient to afford a distinct equation of each root, and there- 

 fore, by the common principles of indeterminate equations, will clearly determine 

 them all; and would also find them, if the equations afforded by the co-efficients 

 were all of the same degree; but they rise successively, and, from the drawing them 

 together, in order to expunge the several unknown quantities, the index of the 

 reducing equation increases so as to defeat the operation. To show this, let us recur 

 to the general equation before given, x n — px n ~ x -J- qx"'* — rx"~ 3 -f- sx n ~* = O; 

 suppose its n roots to be represented by a, b, c, d, &c. w;-f- then, by the construc- 



* 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. — Orig. 



+ The nature of the roots is not material in this place; whether affirmative or negative, real or ima- 

 ginary, they have just the same operation in forming the co-efficients of the equation. I have however 

 throughout chosen, wherever I could, to give examples capable of being tried by real and affirmative 

 roots; and, for that purpose, have uniformly made the signs of the co-efficients alternately affirmative 

 and negative. — Orig. 



