A Discourse on the Theory of Fluxions. 55 



fluxionalbase with the limit 'S.xx' ; x'. An example in quan- 

 tities of three dimensions is afforded by the cube, in which 

 the three generating squares form the limit ; hence the ulti- 

 mate ratio is 3xx : 1, or 3x^x' I x\ This is also called the 

 fluxional ratio. Its legitimate use is to illustrate the manner 

 in which fluents are generated, and to shew their rate of in- 

 crease at any point in their production. 



Leibnitz, in his illustration of this science, contemplated 

 numerical, otherwise termed discrete quantities. Here the 

 increase is made by the continued addition of a distinct sep- 

 arable part, called the measuring unit. But to accommodate 

 the genesis to the nature of variable quantities, he was un- 

 der the necessity of considering this elementary part as infi- 

 nitely small, or as some call it an infinitessimal. According 

 to this method the integral is supposed to be produced by 

 the regular aggregation of insensible parts, by which it suc- 

 cessively passes through every assignable magnitude, from o 

 to the given one. Here it may seem difficult to conceive, 

 how a quantity can arise into existence by the addition of 

 parts that are infinitely small, and consequently such as we 

 cannot arrive at. But the difficulty will be removed by re- 

 curring to the clearer method of Sir Isaac Newton, in which 

 the principle is exemplified by a body in motion. Should 

 the subtle metaphysician ask how a fluent can be generated 

 by the addition of infinitely small elements, we have only to 

 place before his eyes a body, moving either with an accelera- 

 ted, or a retarded motion, and the proposition is illustrated 

 by a familiar fact. In a mathematical view, the co-efficient 

 of the ffuxion is the limit, towards which the increment ap- 

 proaches, when it is made to vanish, and is in effect equal 

 to the evanescent quantity, which is supposed to exist at the 

 moment when the fluent is completed ; and the fluent is the 

 limit of the aggregate of all the nascent quantities, which 

 are supposed to arise successively during the time of its gen- 

 eration. The differential calculus illustrates the genesis of 

 variable quantities by the aggregation of infinitely small ele- 

 ments, which we must conceive to be a process analogous 

 to motion, and by contemplating quantities having dimen- 

 sions beyond those of length, breadth, and thickness, the 

 theory becomes more extensive. For, instead of being con- 

 fined to the second and third powers, we may introduce the 

 general expression x"", extending to any higher power. Then 

 supposing z to represent the increment of x, the ratio of the 



