OF ARTS AND SCIENCES: JANUARY 14, 1873. 489 



definition is apprehended in consequence of the familiarity of the 

 phenomena of motion* 



The most important objections which have been made to the 

 " Method of Fluxions," as developed by Newton and his followers, 

 are those directed against the methods employed in deducing the flux- 

 ions of the different functions. These are usually geometrical methods, 

 often indirect and wanting in generality, even when founded upon well- 

 known and satisfactorily demonstrated properties. The algebraic 

 methods, also, which are employed, are frequently dependent upon an 

 objectionable use of infinite series. 



While a constant rate is easily measured by the increment received 

 in a unit of time, a difficulty is encountered when an attempt is made 

 to employ increments in the measurement of a variable rate. This 

 difficulty probably gave rise to the common method, in which a com- 

 parison of rates is effected by the conception of simultaneous infini- 

 tesimal increments ; to these, while divested of magnitude, ratios are 

 ascribed which are really the ratios of the rates of quantities simulta- 

 neously varying. 



The method of limits is another device for obtaining the values of 

 the same ratios. 



This last expedient, having been adopted by Maclaurin (perhaps the 

 ablest writer on Fluxions), the impression has become prevalent that 

 recourse to it affords the only satisfactory method of treating the sub- 

 ject of rates. 



The following is an attempt to supply a direct method of proving 

 the elementary theorems of the Differential Calculus, which is inde- 

 pendent of all consideration of limits, of infinitesimals, and of alge- 

 braic series. 



Definitions and Notation. 



When a quantity varies uniformly, the constant numerical measure 

 of its rate is the increment received in the unit of time. When, how- 

 ever, the variation is not uniform, we would define the numerical 

 measure of the rate at any instant as the increment which would be 

 received in a unit of time, if the rate remained uniform from and after 

 the given instant. 



This definition corresponds with the usage of mechanics, in accord- 



* See Art. 42, p. 72, Traits d&nentaire de la Th€orie des Fonctions et du Calcul 

 Infinitesimal. Par Cournot. Paris, 1841. 



vol. vin. 62 



