PHYSICAL AND MATHEMATICAL SCIENCES. 299

The idea common to both, is that of considering allquantities expressed by some combination of algebraicterms which receive variable values, dependent solelyon the value successively given to some one or more ofthe simple quantities involved, and which are supposedto vary according to a simple and uniform manner;the complex quantity, thus dependent on the simplequantity or quantities, is called a function of thatvariable, or those variables, if more than one. Weshall then in future speak of a function of one or morevariables, without risk of being misunderstood. Ac-cording to the particular nature of the algebraic com-bination which constitutes the function, it will be easilyseen that it will vary at very different rates, as com-pared with the variable, which we suppose to increaseuniformly. To take a simple illustration : the square ofa quantity is a function of it: as the quantity is sup-posed to increase uniformly, its square increases in amuch more rapid ratio. Its logarithm is another func-tion, but this increases at a totally different rate. Was itthen possible to devise any general method of comparingthe rate of variation of a function with the uniformvariation of its variable ? The rate of the function'sincrease was changing at every moment: it could never,therefore, be expressed by any constant finite ratio ;but might it not he of such a nature, as to be suscep-tible of a limit ? This was the essential point, andboth the great inventors showed, in their several ways,that the ratio was in all cases bounded, as it were, by alimiting ratio, which they showed how to express interms of the variable. This, in whatever language ornotation it might he couched, or in whatever way de-monstrated or illustrated, w r as the great, essential prin-ciple, for which the world is indebted to the originalunaided independent inventive powers of Newton andLeibnitz .

Neither of them, however, expressed the principleby an explicit reference to the simple notion of a limit:if they had, not only would all the mysticism and ob-