344 A Theory of Fluxions. 



the one is less than the length of the other, it is manifest, that 

 the ratio between them of 2 : 1 still exists. For another in- 

 stance, take the proposition, that the areas of similar poly- 

 gons, inscribed in circles, are as the squares of the diameters 

 of those circles. As before observed, when two series of 

 polygons are produced by a continual bisection of the arcs 

 of their circumscribing circles, all the terms will have the 

 same ratio, should the series be continued ever so far. And 

 it is affirmed, that, admitting the series to be infinite, the ra- 

 tio extends through all the infinite number of terms : for if it 

 does not, the ratio stops at some assignable place in the se- 

 ries ; but. there can be no reason given why it should stop at 

 this place ; therefore it follows, that the ultimate ratio, with 

 which the two series vanish, is a thing which can be had, al- 

 though we cannot arrive at the ultimate terms themselves. It 

 hence appears that even the science of the Mathematics has 

 its mysteries : which may well serve to repress that pride, 

 which is apt to arise in those who are conversant about 

 truths, that are clear and demonstrable ; but abstruse, and 

 withdrawn from minds of the ordinary cast. 



As the areas of the polygons are obtained by adding to- 

 gether the small parallelograms, of which the polygons are 

 composed ; it seems to be implied, that, when we arrive at 

 the curvihneal spaces, the ratio supposes the addition of an 

 infinite number of infinitely small parallelograms. No such 

 thing is pretended. Without considering the manner in 

 which the areas of the curves are obtained, it is inferred, that 

 the ratio, which reaches through all the infinite number of 

 terms in the two series of polygons, that are compared, ex- 

 ists also in the curvihneal spaces, which are their limits ; and 

 that the ultimate ratio, which is the ratio formed by the evan- 

 escent terms, is the same with that of the limits themselves, 

 because their difference is less than any assignable quantity. 

 Limits are therefore used instead of evanescent quantities, at 

 which we cannot arrive. Hence the object of exhaustions is 

 not to obtain the last term of an infinite series. That this 

 was Sir Isaac Newton's idea of nascent and evanescent 

 quantities, and limits, is evident from his Scholium to the XI. 

 Lemm a of his Principia. Speaking of the nature of ultimate 

 ratios, he says : " Those ultimate ratios with which quanti- 

 ties vanish, are not truly the ratios of ultimate quantities, but 

 limits, towards which the ratios of quantities decreasing with- 

 out limit do always converge ; and to which they approach 



