A Discourse on the Theory of Fluxions. 57 



to explain this part of the theory, they have involved it in a 

 greater obscurity and mystery, than the nature of the sub- 

 ject renders necessary. That the principle of cause and ef- 

 fect in the fluxional calculus, which considers the fluxion as 

 the cause, and the fluent as the effect, does not explain the 

 relation, is manifest from the fact, that the fluxion is not the 

 precise cause. To explain it in a satisfactory manner, it is 

 necessary that we assume neither more, nor less than the en- 

 tire cause for the reasoning proceeds upon the principle, that 

 every effect is proportional to its cause. But the fluxion, 

 which operates at the moment the fluent is completed, is in 

 a great measure diflferent from that, which operated, when 

 the fluent began to be produced. In the constantly varying 

 motion by which the fluent is generated, either some part of 

 the generating cause has gone out of existence, or a conge- 

 ries of new causes has arisen, which did not operate at the 

 commencement. From this consideration it is manifest, that 

 the theory requires some additional principle to be introdu- 

 ced. That it is not embraced in the supposition, that a flux- 

 ion is an elementary part of its fluent, is evident, first, from 

 the consideration that the derivation of the fluent from the 

 fluxion does not depend upon any rules for the summation of 

 a series of elements, as the proposition implies ; secondly, 

 the very supposition of the relation of a part to the whole, 

 makes it necessary that the element should be assigned, and 

 if assigned, a small part, being the difference between the 

 increment and the corresponding fluxion, is lost. If any 

 dissatisfaction should arise on this account, a new assign- 

 ment may be made still nearer to the truth : but yet this is 

 found to make no difference in the final result. This con- 

 sideration is a suflScient evidence, that no use is made of 

 this elementary part, but the relation of fluxion and fluent 

 depends upon other principles. 



In a paper communicated to the Connecticut Academy of 

 Arts and Sciences, and published in Vol. XIV. of the Jour- 

 nal of Science, I attempted to supply a few links in the chain 

 of illustration which I judged to be wanting. It may, per- 

 haps, be thought a fruitless undertaking, to presume to add 

 any thing to the elaborate researches of Newton, Leibnitz, 

 Euler, La Grange, La Place, &c. but when it is considered, 

 that they were urged on by the attractions of a most sublime 

 and beautiful discovery, to make still new advances in the 

 practical part, it need not be thought strange, if they have 



Vol. XVI.— No. U 8 



