A Theory of Fluxions. 345 



nearer than by any given difference, but never go beyond, 

 nor in effect attain to, until the quantities are diminished in 

 infinitum." 



I have given the ideas of Sir Isaac Newton concerning 

 some of the properties and relations of infinite quantities in 

 his own words, because I conceived, that a just notion of 

 prime and ultimate ratios, nascent and evanescent quanti- 

 ties, and especially the limits of infinite series, was of great 

 importance in gaining a knowledge of a science, at once the 

 most sublime, beautiful, and subtle, that has ever exercised 

 the ingenuity of man ; and because I conceived, that the 

 doctrine could not be expressed in a manner more clear and 

 perspicuous, than in the words of that illustrious philosopher. 



The limit of x n is nx n_1 , and the limit of a x is N«#, N being 

 the Napieran logarithm of a. In curvilineal figures it is the 



same with the ordinate, and in the circle it is (ax — x 2 ) 2 , in 



c — -i 



the ellipsis it is ~(tx— a 2 ) 2 , in the parabola it is (jps) 2 , and 



in the hyperbola - — These limits are also the same with 



Ax 



the fluxional co-efficients, and bear the same relation to their 

 corresponding fluents, that a line does to a superficies, or a 

 superficies, to a solid. Hence limits are always one dimen- 

 sion less, than the fluents to which they stand related. The 

 fluxional base x' is a fundamental quantity, which supplies 

 the defect in dimensions ; and, by making fluxions to be 

 things of the same kind with fluents, renders it possible, that 

 a proportion between them may exist. This insertion of a 

 fluxional base constitutes the difference between fluxions and 

 indivisibles. Fluxions, then, may be defined to be the ratio 

 of variable quantities. 



Some Mathematicians have expressed themselves con- 

 cerning infinitely great, and infinitely small quantities, as 

 though they were quantities, at which we can arrive ; hence 

 they have treated of them much in the same way, as of finite 

 ones. This is evidently inconsistent with that accuracy and 

 clearness of reasoning, for which mathematical science is 

 justly celebrated. Owing, probably, to this erroneous idea 

 of infinity, they have spoken in a very loose wav of a differ- 

 ential quantity, as though the curvilineal space MECX, (Fig. 

 6.) was intended by it ; considering the part MEK to be so 

 far diminished by repeated subdivisions, as to become of no 



Vol. XIV.— No. 2. 18 



