J88 



by all its simple divisors, in whicli case no compound divisor 

 can have a simple divisor, for such would be also a simple di- 

 visor of the quantity proposed. If, as in the example 2x — b, 

 Newton had merely considered the quantity in the relation of 

 a divisor, he would not first represent the divisor in the form 

 .r — ^ and thence 2x — b. For, from division alone it would 



not follow, that, if the sub-multiple found should divide the 

 proposed, so should also the multiple. Therefore, Newton, 

 in the examples i/ + j, a — f, &c. must have deduced those 

 expressions from the nature of fractional roots, as they enter 

 the factors of an original equation, where from y + ± = fol- 

 lows 3y + 4=p» The case of a divisor of one dimension, in 

 which the co-efficient of the highest term is unit, has been 

 proved from the nature of equations by JM'Laurin. In the 

 following Chapter, I shall give a demonstration for every 

 case, by distinguishing betv^reen integral, fractional, and surd 

 roots. 



If an equation admits of no rational divisor of one or of 

 two dimensions, the rules which have been usually given are 

 inadequate to discover a binomial or trinomial divisor which 

 might serve for investigating roots. But since equations of 

 higher than two dimensions may often appear as trinomials, 

 of a form which is similar to that of quadratic equations, and 

 thus be capable of a similar resolution, a statement of the 

 rules which would extend to the finding of such divisors may 

 be practically advantageous,- 



Besides 



