300 On the Principles of Motion, fyc. 



ther is assignable by an arithmetical or algebraical calculus.* 

 It is not easy to conceive, why a principle, which has been 

 the most productive source of discovery in the mathematics, 

 particularly in descriptive geometry, and the reasonings and 

 demonstrations of the properties of curve lines, should be 

 exploded from the mathematics and exclusively confined to 

 physics. 



What is objected to Leibnitz's use of infinitesimals will 

 equally apply to the ancient method of exhaustions, and 

 even to the doctrine of limits, which never will be intelligible 

 to those, who confound metaphysical, and mathematical in- 

 finity. In a full discussion of this subject I must refer to 

 Maclaurin's Fluxions, or the works of Benjamin Robins. 

 The other position of Lagrange, that the subject belongs 

 peculiarly and exclusively to analysis, or algebra, is in our 

 view wholly gratuitous, unless it be in reference to the calcu- 

 lus only, or the numerical operations, as indicated by the 

 algebraic expressions, according to which the values of the 

 variable quantities must finally be determined. But this is 

 not peculiar to the science under consideration. Every 

 branch of the mathematics, when brought to practice, must 

 be reduced to the idea of quantity as constituted by aggre- 

 gation, or as composed numerically of some quantity as the 

 standard of measure, or the unit of that species of it, to 

 which it belongs. In geometry, quantities are represented 

 by lines, or some figure of the magnitude itself. In Algebra, 

 the same are expressed by symbols, or letters, which desig- 

 nate the numerical parts of which they are composed. Ac- 

 cording to either mode of designation, the demonstration 

 of the science depends on the same principles, viz. the de- 

 termination of the ratio of increase, or diminution at a point. 

 For this purpose, the ratio of the increments, or decrements 

 of the variables is first considered, and then what that ratio 

 will be at the point of vanishing. We know of no other 

 method by which that ratio can be established ; and to us, as 

 it respects the demonstration, it appears wholly immaterial 

 whether that be expressed geometrically, or algebraically. 

 But according to either mode, it must rest on the doctrine 

 of limits and limiting ratios, as a legitimate subject of math- 

 ematical demonstration. To have recourse to far fetched 

 principles, which themselves are built entirely on fundamen- 

 tal truths, is argumentum in circulo, wholly inconsistent with 



* See Robins' Mathematical Tracts, vol. 2. 



