270 LIMITS AND FLUXIONS 



without having occasion to introduce any ideas 

 foreign to geometry. Sir Isaac Newton, however, 

 in first delivering the principles of the method, 

 thought proper to employ considerations drawn from 

 the theory of motion. But he appears to have done 

 this chiefly for the purpose of illustration, for he 

 immediately has recourse to the theory of limiting 

 ratios, and it has been the opinion of several mathe- 

 maticians of great eminence (such as Lagrange, 

 Cousin, La Croix, etc. , abroad, and Landen in this 

 country) that the consideration of motion was intro- 

 duced into the method of fluxions at first without 

 necessity, and that succeeding writers on the subject 

 ought to have estabhshed the theory upon principles 

 purely mathematica!, independent of the ideas of 

 time and velocity, both of which seem foreign to 

 investigations relating to abstract quantity." **By 

 the fluxions then of two variable quantities having 

 any assigned relation to each other, we are in the 

 foUowing treatise always to be understood to mean 

 any indefi^tite quantities which have to each other the 

 limiting ratio of their simultaneous inci'cjuents (we 

 . . . mean the ratio of the nuinerical values of the 

 increments, which may always be compared with 

 each other, whether the variable quantities be of 

 the same kind, as both lines, or both surfaces, etc, 

 or of different kinds, as the one a line, and the other 

 a surface). The Newtonian notation is used in the 

 article exclusively." 



