342 A Theory of Fluxions. 



ces are less than any assignable quantities ; then upon the 

 principle of exhaustions the evanescent antecedent becomes 

 in effect equal to the nascent one, likewise the evanescent 

 and nascent consequents become equal, and according to 

 Prin. V., the one may be taken for the other. The equality 

 of these terms forms the connecting link, by which the two 

 infinite series are united, so as to become one series, having 

 the same ratio throughout. To exemplify this in symbols, 

 a J n : : a' t n' : '. a" J n" '. '. , &c in infinitum evanes- 

 cent of a : evanescent of n : ; nascent of A : nascent of N. 

 Nascent of A : nascent of N : : , &c in infinitum .... 



A"' : N'"::A" : N"::A' : N'::A : N. 



Hence a : n '. \ a : n 1 : : a" : n" : : , &c in inf. .... A" : 



N"::A' : N'::A : N. 



It hence appears, that the series, expressed in terms of 

 the curvilineal spaces, is identified with the series, expressed 

 in terms of the rectilineal spaces. And, since any antece- 

 dent and its consequent may be taken at pleasure, to ex- 

 press the ratio, let A ! N be taken, and it will be, the fluent 

 AHN8C : the fluent DFMEC: \a : n\ :A : N: : the fluxion 

 BxzC : the fluxion Eun/C: \yx' : myx: 



a. e. d. 



Here, without any error whatever, a transition is made 

 from the ratio a : n, expressed in terms of the curvilineal 

 spaces, to the ratio A : N, expressed in terms of the rectili- 

 neal spaces. 



It will readily be perceived, that the foregoing proportion 

 illustrates the important analytical fact, that the ratio of two 

 infinite series, or rather, of the incommensurable quantities, 

 which they are designed to express, can be had in finite 

 terms ; although, taken separately, their exact sum or meas- 

 urement, cannot be obtained. 



To represent this in symbols, let the division NBCT=a = 

 B-f B'+B"+B'" 4- , &c, in infinitum, (Fig. 6.) and the divis- 

 ion MECH=»=C+C'+ C"-fC"'+, &c, in infinitum, then, 



B+B'+B 1 +B'"+, &c in inf. = A 



C+C'+C H 4-C , "+,&c .in inf. N' 



Here it will be well to notice, that although the two vari- 

 able quantities, which are compared in the foregoing demon- 

 stration, are, even at first, so taken, that they are always ac- 

 curately in the same ratio ; yet it is unknown, until we ar- 

 rive at what Sir Isaac Newton terms the ultimate ratio, 



