ATTEMPTS AT ARITHMETISATION 237 



expressed by " a fraction whose denominator is 

 g(R-Q)/Q^ and whose numerator is obtained by ex- 

 panding and simplifying Ai^/^-(A - N)R/Q. *'But 

 if A + N and A — N stand to B in relations nearer 

 to that of equality than by any given or assigned 

 magnitude of the same Kind, these general expres- 

 sions become R/0 . A^^-QVQ. N-fB^^-QVQ. This I 

 cali the antecedental of the magnitude which has 

 to B such a ratio as has to the ratio of A to B the 

 ratio of R to Q. Now if N the antecedental of A 



be denoted by À or A . . . [and] if Q = i and 



"^AÀ 

 R = 2, 3, 4, 5, etc. , this expression gives 



3A^A 

 B* 



B 



respectively. " For the ''antecedent 



A 2AÀ 



A— he finds the '* antecedental " or 2 A 



B B 



2MÀ 

 H- *^ T. (putting M for A — B). Glenie shows that 



at a point of a curve the antecedentals of the ab- 

 scissa, ordinate and curve, are as the sub-tangent, 

 the ordinate and the tangent, respectively. 



Glenie's calculus involves extremely complicated 

 identities of ratios and examines the antecedents of 

 ratios having given consequents. The style of ex- 

 position is poor. In deriving the antecedentals, 

 Glenie quietly drops out ali the terms in the 

 numerator that involve powers of N higher than 

 the first power. As this calculus plays no part in 

 the later history of fluxions, we shall give only one 

 more quotation ; it relates to the Binomial Theorem 



