300 Application of the Fluxional Ratio, fyc. 



is momently varying to the increment of its base or root, which ib 

 considered as uniform. To render the results of any two flowing 

 quantities analogous, their rates of increase must be similar. A 

 quantity, therefore, generated by a uniform motion, cannot be com- 

 pared with a quantity generated by an accelerated motion ; for it 

 would involve the same absurdity as to say, that a certain given line 

 is three times greater than a certain given cube. Mathematicians 

 were aware of this, but instead of taking the limits according to Sir 

 Isaac Newton's view of the subject, they have taken the increments, 

 for they say by way of objection : " but it is then at its limit, and 

 the ratio is that of the limit, and not of the increment." Under the 

 influence of this view, they have sought for an analogous term by 

 making the increment small. Hence they have been deceived by 

 contemplating a quantity, in which the error is less than the imagin- 

 ation can reach, and have dignified it with the name infinitesimal. 

 Instead of laying hold of the finite quantity, they have sought for the 

 increment, and have attempted to overtake it at the end of an infi- 

 nite series ! — a thing impossible. So far from its being correct that 

 an error is committed by casting away the increment e, it is true, 

 that if we introduce the quantity e at all, an error exists exactly pro- 

 portional to the magnitude of e. The petitio principii, then, lies on 

 the other side, for the ratio is not that of a whole to its part, but that 

 of a whole to the sum of its sources of increase, which consists of 

 the limits made analogous by combining them with the fluxional base. 

 These sources of increase are dependent on the dimensions of the 

 variable function, and in case of the first fluxions are expressed by 

 the second term in the development of the binomial series. That 

 this term is the true fluxion, and not a petitio principii is made evi- 

 dent, as will be shown hereafter, by the coincidence of the second 

 term with the first fluxion, of the third term with the second fluxion, 

 of the fourth term with the third fluxion, and so on, in expanding 

 any power of a binomial quantity. Fluxions, therefore, are the ele- 

 ments, that arise in the development of a binomial quantity. We 

 may, then, describe a fluxion to be an artificial finite quantity, aris- 

 ing from the sources of increase belonging to any power of a varia- 

 ble quantity ; which sources are exhibited, when that quantity is ex- 

 panded in the form of a binomial series. Quantities thus constitu- 

 ted may be analogous, and may admit of the existence of a ratio be- 

 tween them. The fundamental principle of fluxions is, that while 

 the fluent is generated with an accelerated or retarded motion,, and 



