10 INTRODUCTION. 



SECT. II. OF THli COMPARISON OF VARIABLE 

 QUANTITIES. 



44. Definition. The quantities by which two varia- 

 ble magnitudes are increased or decreased, in the same 

 time, are called their increments or decrements, or their 

 increments positive or negative. 



Scholium. They are sometimes denoted by an accent placed aver 

 the variable quantity ; thus x' and y are the simultaneous incre- 

 ments of X and y. 



45. Definition. The ratio, which is the limit of the 

 ratios of the increments of two connected quantities, as 

 they are taken smaller and smaller, is called the ratio of 

 the velocities of their increase or decrease. 



Scholium. It would be difficult to give any other sufficient defi- 

 nition of velocity than this. If both the quantities vary in the same 

 proportion, the ratio of a;' andj/willbe constant (18), and may be 

 determined without considenng them as evanescent ; but if they 

 vary according to different laws, that ratio must vary, accordingly as 

 the time of comparison is longer or shorter : and since the degree of 

 variation, at any instant of time, does not depend on the change pro- 

 duced at a finite interval before or after that instant, it is necessary, 

 for the comparison of this variation, that the increments should be 

 considered as diminished without limit, and their ultimate ratio de- 

 termined ; and it is indiflerent whether these evanescent increments 

 be taken before, or after the given instant, or whether the mean be- 

 tween both results be employed. 



46. Definition. Any finite quantities, in the ratio of 

 the velocities of increase or decrease of two or more mag- 

 nitudes, are the fluxions of those magnitudes. 



Scholium. They are denoted by placing a point over the variable 



quantity, thus, x,y. And — is always ultimately equal to—. The 



y y 



variable quantity is called a fluent with respect to its fluxion, as x is 

 thcfluentof;if,or a:— ya:. On the continent* the term fluxion is not 



