AND THE METAPHYSICS OF THE FLUXIONAL CALCULUS. 495 



motum superficierum ; anguli per rotationenem laterum ; tempora per fluxum 

 continuum et sic in cseteris. Hae Geneses in rerum natura locum vere hdbent et 

 in motu corporum quotidie cernxmtur^ (Introd. ad Quad. Curv.) 



In a word, Newton's fundamental position is, that the arithmetical concep- 

 tion of quantity is not that with which nature herself presents us, and is not, 

 therefore, universally applicable. On the other hand, every quantity that has 

 objective reality [i.e., is an object of real intuition] is generated by continuous 

 motion, with definite (constant or variable) velocity within definite limits of 

 time. The metaphysical nature of time and motion Newton has nothing to do 

 with. It is enough for him that mathematical time, conceived as an independent 

 variable flowing uniformly, is clearly the true time made known to us in nature 

 (Principia; Schol. to the Defs.), and that the existence of a definite velocity at 

 each point of a motion is in like manner an undoubted physical fact. 



By means of these profound yet simple considerations, Newton is at once able 

 to revolutionise the whole theory of quantity, and to substitute for the relation 

 of unit and sum that of velocity and quantity generated, or, in Newton's own 

 language, of fluxion and fluent. It must be remembered that we have said 

 nothing of space, so that fluent is not limited to extensive quantity, while velocity, 

 or as we should rather say rate, has a correspondingly wide application. Thus, 

 any fluxion may itself be treated as a fluent quantity, and its fluxion sought, the 

 only independent variable being time, which is thus a fluent which has no variable 

 fluxion. 



This conception of time, as the one absolute and independent variable, is 

 undoubtedly one of the most splendid and fruitful in the history of human 

 thought, and well deserves the attention of metaphysicians. Only let it be said 

 that no criticism of Newton's time, which starts from the arithmetical view of 

 quantity, and urges the old objections about infinite divisibility, and so forth, is 

 competent ; for the arithmetical theory is a product of abstract reflection, and so 

 stands on a lower platform than the pure objective notion of Newton. 



There is no difficulty in comprehending the mathematical power which the 

 conception of fluxions at once puts in Newton's hands, if we remember that it 

 is not in any sense an extension of the theory of numbers that he is seeking. It 

 is true that the calculus has revolutionised algebra as well as geometry ; but it 

 has done so by transforming algebra from the abstract science of numbers to a 

 physical science — the science of pure time. In Newton's own mind, however, 

 this conception was probably not explicitly present. What he did see was, that 

 all difficulties in geometry (and to Newton, as to the old geometers, geometrical 

 magnitude is the type and exponent of all magnitude whatsoever, when viewed 

 with respect to its generation) are reducible to the general form : — " Given the 

 fluent as a function of time to determine the fluxion and vice versa.'''' 



The one class of problems that can be thoroughly treated without explicit 



VOL. XXV. PART II. 6 M 



