538 PHILOSOPHICAL TRANSACTIONS. [ANNO 1700. 



that the number of distinct squares will be only 3. These properties, though with- 

 out any order or connection, and confined merely to particular ranks or numbers of 

 quantities, being general to all possible or imaginable quantities of those classes 

 afford methods general, as to those degrees, but without producing any result 

 really general to equations at large. 



15. Having shown that an indeterminate general equation cannot be resolved by 

 any of the methods whose principle is yet known, because they are all grounded on 

 the assumption of some particularity, either inherent in the roots, or universally 

 communicable to them, which, so far from being general, is seldom found, and 

 absolutely incompatible with many sorts of roots ; that the difficulty is in all cases 

 the same, — the intrusion of superfluous roots and higher radicals ; that a relation of 

 any kind, when known, obviates that difficulty, as far as it extends; and that some 

 orders of quantities have generally a constant and necessary relation, more or less 

 remote, I proceed to examine, more minutely, the application of these observations 

 to the several degrees of equations to which they materially apply. 



Of the Resolution or Reduction of Equations of particular Degrees. 



16. In examining those degrees of equations which submit to be resolved, I shall 

 observe the same order as before ; i. e. first consider the power of obtaining a 

 general formula, or complete resolution; and, if that is not attainable directly, in- 

 quire by what general means the roots can be separately investigated, and what 

 new forms they have taken, or what different functions of them are used in the 

 operation. 



17. If we resume the general indeterminate equation & — px n ~ l -f- qx n ~* — rx"~' 

 &c. = 0, and assign the progressive values 2, 3, 4, &c. to the index n, in the first 

 case it will become the quadratic x 2 — px -}- q = O. Now as this equation has 2 

 roots, in order to obtain a general formula for its resolution, the first step that 

 suggests itself is,- to inquire what is necessary to construct a general representation 

 of 2 quantities in a simple equation. Two quantities are known to be generally 

 expressed by means of their sum and difference ; that half their sum added to half 

 their difference gives the greater, and the same quantities subtracted, the less. The 

 sum being always the co-efficient of the 2d term of the equation, is given in all 

 cases, and here the difference is readily found ; for, the square of the difference of 

 any 2 quantities differs from the square of their sum, by a constant quantity, viz. 4 

 times their product or the co-efficient of the 3d term. If a and b be called the 

 roots of the equation x* — px -f- q = ; then 



p = a + b, andjb 2 = a 2 + lab + b* ; 



q = ab, and — Aq = — 4ab ; 



r the square 



whence a 2 — lab + b* = (a — bf = p* — 4qJ ofthedif- 



L ference. 



The difference itself is therefore y/{p* — 4q). And now, being possessed of the 



parts required to construct a general representation of the 2 quantities, we can at 



