If we had to trace here the systematic history of the successive formation of the transcendental analysis, it would be necessary previously to distinguish carefully from the _calculus of indirect functions_, properly so called, the original idea of the _infinitesimal method_, which can be conceived by itself, independently of any _calculus_. We should see that the first germ of this idea is found in the procedure constantly employed by the Greek geometers, under the name of the _Method of Exhaustions_, as a means of passing from the properties of straight lines to those of curves, and consisting essentially in substituting for the curve the auxiliary consideration of an inscribed or circumscribed polygon, by means of which they rose to the curve itself, taking in a suitable manner the limits of the primitive ratios. Incontestable as is this filiation of ideas, it would be giving it a greatly exaggerated importance to see in this method of exhaustions the real equivalent of our modern methods, as some geometers have done; for the ancients had no logical and general means for the determination of these limits, and this was commonly the greatest difficulty of the question; so that their solutions were not subjected to abstract and invariable rules, the uniform application of which would lead with certainty to the knowledge sought; which is, on the contrary, the principal characteristic of our transcendental analysis. In a word, there still remained the task of generalizing the conceptions used by the ancients, and, more especially, by considering it in a manner purely abstract, of reducing it to a complete system of calculation, which to them was impossible.