An Extension of ordinary Analysis. The transcendental analysis is, then, nothing but a simple though very considerable extension of ordinary analysis. Geometers have long been accustomed to introduce in analytical investigations, in the place of the magnitudes themselves which they wished to study, their different powers, or their logarithms, or their sines, &c., in order to simplify the equations, and even to obtain them more easily. This successive derivation is an artifice of the same nature, only of greater extent, and procuring, in consequence, much more important resources for this common object.
But, although we can readily conceive, à priori, that the auxiliary consideration of these derivatives may facilitate the establishment of equations, it is not easy to explain why this must necessarily follow from this mode of derivation rather than from any other transformation. Such is the weak point of the great idea of Lagrange. The precise advantages of this analysis cannot as yet be grasped in an abstract manner, but only shown by considering separately each principal question, so that the verification is often exceedingly laborious.
Example. Tangents. This manner of conceiving the transcendental analysis may be best illustrated by its application to the most simple of the problems above examined—that of tangents.
Instead of conceiving the tangent as the prolongation of the infinitely small element of the curve, according to the notion of Leibnitz—or as the limit of the secants, according to the ideas of Newton—Lagrange considers it, according to its simple geometrical character, analogous to the definitions of the ancients, to be a right line such that no other right line can pass through the point of contact between it and the curve. Then, to determine its direction, we must seek the general expression of its distance from the curve, measured in any direction whatever—in that of the ordinate, for example—and dispose of the arbitrary constant relating to the inclination of the right line, which will necessarily enter into that expression, in such a way as to diminish that separation as much as possible. Now this distance, being evidently equal to the difference of the two ordinates of the curve and of the right line, which correspond to the same new abscissa x + h, will be represented by the formula
(f'(x) - t)h + qh2 + rh3 + etc.,
in which t designates, as above, the unknown trigonometrical tangent of the angle which the required line makes with the axis of abscissas, and f'(x) the derived function of the ordinate f(x). This being understood, it is easy to see that, by disposing of t so as to make the first term of the preceding formula equal to zero, we will render the interval between the two lines the least possible, so that any other line for which t did not have the value thus determined would necessarily depart farther from the proposed curve. We have, then, for the direction of the tangent sought, the general expression t = f'(x), a result exactly equivalent to those furnished by the Infinitesimal Method and the Method of Limits. We have yet to find f'(x) in each particular curve, which is a mere question of analysis, quite identical with those which are presented, at this stage of the operations, by the other methods.