ELISHA MITCHELL SCIENTIFIC SOCIETY. 25 



wards a true "iufinitesimal method" can be easily logically deduced (as 

 Duhamel and others have shown) that wdll offer all the advantages and 

 abbreviated processes of the Leibnitz method, with none of its errors of 

 reasoning. 



It is well to remark just here that because j' ^=/{x) can 

 always be represented by a locus, and since its derivative 

 with respect to x represents the slope of the tangent at the 

 point (a-, y\ it will generally be finite. It is only at the 

 points where the tangent is parallel or perpendicular to the 

 axis of X that the derivative is zero or infinity. Hence the 

 ratio of aj' to aa', whose limit is the slope, has generally a 

 finite limit. 



We have studied now, with some thoroughness, the 

 theory of tangents and will next take up, a no less impor- 

 tant subject, the method of rates. When a variable changes 

 so that, in consecutive equal intervals of time, the incre- 

 ments are equal, the change is said to be uniform; other- 

 wise variable. For uniform change, the increment of the 

 function in the unit of time is called the rate. Thus in 



space 



the case of uniform motion, velocity = rate = . 



time 



For a variable change, the rate of the function at any 

 instant is what its increment would become in a unit of 

 time if at that instant the change became uniform. 



In looking for an illustration to show clearly the spirit 

 and method of the calculus, perhaps none is more satisfac- 

 tory to the beginner than the consideration of falling bod- 

 ies in vacuo. If we call the space in feet, described by 

 the falling body in / seconds, s and ^ the acceleration due 

 to gravity, we have the relation between the space and 

 time, as given by numerous experiments, expressed in the 

 following equation, 



s= Yi g i-\ 



