A Discourse on the Theory of Fluxions. 55 
fluxional base with the limit 2va- }.z". An example in quan- 
tities of three dimensions is afforded by the ee. in which © 
the three generating —_— form the limit ; henee the ulti- 
mate ratio is 32x: 1, or 3a22"; x. This 1s also called the 
fluxional ratio. Its latumete use is to illustrate the manner 
in which fluents are ead ae vs shew their me of in- 
crease at any point in their prod 
Leibnitz, in his illustration of “ahii's science, coutetispiteiod 
numerical, otherwise termed discrete quantities. Here t 
arable part, called the measuring unit. But to accommodate 
the genesis to the n ature of variable quantities, he was un- 
to this method the poe is oii produced nl 
the regular aggregation —— oe by by which it suc- 
cessively passes it 
to the given one, Here it hin oth difficult - CORE 
how a quantity can arise into existence by the addition of 
parts that are ee small, and consequently such as w 
cannot arrive at. t the difficult ty will be removed by re- 
curring to the sieieee: method of Sir Isaac Newton, in which 
the principle is exemplified by a body in motion. Should 
the gee metaphysician ask how a fluent can be generated 
by the addition of infinitely small elements, we have only to 
planes before his eyes a body, moving either ‘with an accelera- 
ted, or a retarded motion, and the proposition is illustrated 
by a familiar fact. Ina mathematical view, the co-efficient 
of the fluxion 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 preter 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, ee we must conceive to bea process ana 
to motion, and by contemplating quantities having” n- 
sions Oa ose of length, breadth, and thickness, the 
theory bec more extensive. For, instead of being con- 
general expression x”, extending to any higher The 
supposing z to represent the increment of 2, ihe: anit of the 
