Review of the Principia of Newton, 321 



centre, but never coming nearer to it than a certain distance, 

 or to a circle which is the limit of the spiral motion. 



The philosophical discoveries of Newton, which we have al- 

 ready attempted to review, and the more numerous and pro- 

 found investigations contained in the subsequent portion of 

 the Principia, could not, it is evident, be made or generally 

 demonstrated without a perfect knowledge of that science or 

 art of analysis denominated fluxions, or the differential calcu- 

 lus. This branch of the mathematics is grounded on the re^ 

 lations which subsist between the increments of variable quan- 

 tities considered as evanescent, and the quantities them- 

 selves ; but this principle which constitutes the rationale or 

 metaphysique of the science, of itself would afford no advan- 

 tage unless we were able to determine those relations, and it 

 never could be estimated as a science unless we were in pos- 

 session of general rules, or a general method by which those 

 relations might be calculated in all cases, to which the prin- 

 ciples were applicable, or in the language of our author, 

 " Methodus qum extendit se citra molestum ullum calculum, 

 in terminis surdis ceque ac integris procedens.^ Fluxional 

 principles had before Newton been used by Farmat, Rober- 

 val, Napier, Barrow, Wallis, Mercator, Gregory, and others, 

 for the solution of particular problems of drawing tangents, 

 but in a partial degree, and limited to such quantities as could 

 be expressed by rational functions and of course affording 

 very simple expressions of the ratios of the nascent incre- 

 ments. No rules or system of rules which extend generally 

 to the solution of all kinds of difficult problems were thought 

 of, or if so, were supposed susceptible of discovery, unless the 

 arithmetic of infinites of Wallis may border on systemization, 

 but the grand discovery, the clue which should lay open all 

 the intricate recesses of the Labyrinth, was yet to be made. It 

 was the development of any function of a binomial, whether 

 radical, fractional, or any how involved, so that the second 

 term of that development, which is the nascent increment, 

 fluxion, or differential of the function, may be correctly calcu- 

 lated. That clue was furnished by Newton by his methods 

 of series interwoven with his fluxional calculus, and this long- 

 before any other person had laid claim to this invention ;* he 

 therefore must be considered as the first inventor of a science 



* Vide Lemma 2d of the 2d book of the Principia, and analysis per equationes 

 terminis infinitis, by Stewart. 



Vol. XIII.—No. 2. 16 



