20? 



by substitution of large numbers, we may diminish the results ; 

 but if the absolute term is not great, the smaller the numbers 

 substituted, the Jess the results : and this, with the facility 

 of involving small numbers, would make us to prefer the sub- 

 stitution of the natural numbers near cypher. 



Yvlien the roots are integral, by substituting the terms of 

 the natural series, (which contains all integral numbers), we 

 may be able, by mere substitution, to discover those roots: 

 for, when the root is substituted for the unknown, the result 

 is equal to cypher, since the absolute term of the transformed 

 is equal to cypher when the root is diminished by a quantity 

 equal to itself. When there are fractional roots, we might dimi- 

 nisli the roots by an arithmetical series of fractions, and find 

 results equal to cypher, but the substitution of these, and the 

 ridding the terms of denominators, would be the same as to 

 multiply the roots, and then substitute the natural numbers. 

 Thus also, when all the roots have a common factor, we might 

 substitute multiples of natural numbers, or vise the natural 

 numbers multiplied by that common factor; but since the 

 results of substitution or the absolute terms, which are the pro- 

 ducts of the roots with tl^eir signs changed, would, in the trans- 



Ih 



formed, be all divisible by the n power of the common factor, 

 after this division, the result would be the same as if we had 

 first divided the roots by their common factor, and substituted 

 the natural numbers themselves. 



The 



