Application of the Fluxional Ratio, &fc. 301 



consequently the rate of increase is momently varying, the fluxion 

 in all its parts is produced by a uniform motion. This may be re- 

 garded as a true definition of fluxions. In algebra the fluxion is ex- 

 pressed by a single terra in the binomial x-{-x' raised to the given 

 power ; while the increment is expressed by all the terms that re- 

 main, after the first is withdrawn. In as much as an abstruse prin- 

 ciple is best elucidated by an example, let us suppose that x-\-x' is 

 the variable root, and this root raised to the fourth power, or a?* + 

 4x^x'+6x^x'^-{-4xx'^-i-x"^ is a function of x+x'. Now with- 

 draw x'^, and the increment is expressed by all the remaining terms 

 4x^x'-{-6x'^x'^-\-4:Xx'^-{-x"^. But the first fluxion is expressed by 

 4x'^x', the second term of the series, not because the following terms 

 6x^x'^ -\-4xx'^-{-x"^ are so exceedingly small as not to deserve no- 

 tice, but because by the definition of a fluxion, just given, they do 

 not enter into its expression. The doctrine of ultimate ratios and 

 limits, applied to fluxions, is only a particular way of arriving at the 

 second term of a binomial series, by which all the following terms 

 are exterminated. The consideration of infinity, however, has 

 spread over the theory a degree of obscurity and mystery, which is 

 altogether unnecessary. For although the method of exhaustions, 

 and of ultimate ratios, and the limits of variable quantities, are very 

 useful in the solution of certain problems ; yet they are not essential 

 to the theory of fluxions. Without any scruples we may assume the 

 second term in the binomial series, as containing all the elements, 

 which are necessary to form a relation between the unknown quanti- 

 ty sought, and a known quantity, by means of which the value of that 

 unknown quantity can be obtained. 



The foregoing remarks regard the determination of the fluxion 

 itself. But the relation of the fluxion to its corresponding fluent is 

 quite a different thing. Since this, in my apprehension, depends on 

 a similar principle with the relation of the sides of two similar trian- 

 gles to each other in trigonometry, it may perhaps receive some elu- 

 cidation from a comparison. A ratio is the quotient of any number 

 or quantity divided by another, and is either that of the antecedent 

 divided by its consequent, or that of the consequent divided by its 

 antecedent. When the ratio between any two quantities is equal to 

 the ratio between any other two quantities, those four quantities are 

 said to be analogous or proportional ; and the two figures, which are 

 compared, are said to be similar. When three of these proportion- 

 al quantities are given ; or when two are given, and one of them is 



Vol. XXIV.~No. 2, 39 



