But, regarding merely the present constitution of the science, the only auxiliary quantities habitually introduced in the place of the primitive quantities in the _Transcendental Analysis_ are what are called, 1⁰, _infinitely small_ elements, the _differentials_ (of different orders) of those quantities, if we regard this analysis in the manner of LEIBNITZ; or, 2⁰, the _fluxions_, the limits of the ratios of the simultaneous increments of the primitive quantities compared with one another, or, more briefly, the _prime and ultimate ratios_ of these increments, if we adopt the conception of NEWTON; or, 3⁰, the _derivatives_, properly so called, of those quantities, that is, the coefficients of the different terms of their respective increments, according to the conception of LAGRANGE. These three principal methods of viewing our present transcendental analysis, and all the other less distinctly characterized ones which have been successively proposed, are, by their nature, necessarily identical, whether in the calculation or in the application, as will be explained in a general manner in the third chapter. As to their relative value, we shall there see that the conception of Leibnitz has thus far, in practice, an incontestable superiority, but that its logical character is exceedingly vicious; while that the conception of Lagrange, admirable by its simplicity, by its logical perfection, by the philosophical unity which it has established in mathematical analysis (till then separated into two almost entirely independent worlds), presents, as yet, serious inconveniences in the applications, by retarding the progress of the mind. The conception of Newton occupies nearly middle ground in these various relations, being less rapid, but more rational than that of Leibnitz; less philosophical, but more applicable than that of Lagrange. This is not the place to explain the advantages of the introduction of this kind of auxiliary quantities in the place of the primitive magnitudes. The third chapter is devoted to this subject. At present I limit myself to consider this conception in the most general manner, in order to deduce therefrom the fundamental division of the _calculus of functions_ into two systems essentially distinct, whose dependence, for the complete solution of any one mathematical question, is invariably determinate.