Such are the principal general considerations relating to the _Calculus of Direct Functions_. We have now to pass to the philosophical examination of the _Calculus of Indirect Functions_, the much superior importance and extent of which claim a fuller development. CHAPTER III. TRANSCENDENTAL ANALYSIS: DIFFERENT MODES OF VIEWING IT. We determined, in the second chapter, the philosophical character of the transcendental analysis, in whatever manner it may be conceived, considering only the general nature of its actual destination as a part of mathematical science. This analysis has been presented by geometers under several points of view, really distinct, although necessarily equivalent, and leading always to identical results. They may be reduced to three principal ones; those of LEIBNITZ, of NEWTON, and of LAGRANGE, of which all the others are only secondary modifications. In the present state of science, each of these three general conceptions offers essential advantages which pertain to it exclusively, without our having yet succeeded in constructing a single method uniting all these different characteristic qualities. This combination will probably be hereafter effected by some method founded upon the conception of Lagrange when that important philosophical labour shall have been accomplished, the study of the other conceptions will have only a historic interest; but, until then, the science must be considered as in only a provisional state, which requires the simultaneous consideration of all the various modes of viewing this calculus. Illogical as may appear this multiplicity of conceptions of one identical subject, still, without them all, we could form but a very insufficient idea of this analysis, whether in itself, or more especially in relation to its applications. This want of system in the most important part of mathematical analysis will not appear strange if we consider, on the one hand, its great extent and its superior difficulty, and, on the other, its recent formation. ITS EARLY HISTORY.