III. On the developement of exponential functions; together with several new theorems relating to finite differences

In the year 1772, Lagrange, in a Memoir, published among those of the Berlin Academy, announced those celebrated theorems expressing the connection between simple exponential indices, and those of differentiation and integration. The demonstration of those theorems, although it escaped their illustrious discoverer, has been since accomplished by many analysts, and in a great variety of ways. Laplace set the first example in two Memoirs presented to the Academy of Sciences,* and may be supposed in the course of these researches, to have caught the first hint of the Calcul des Fonctions Generatrices with which they are so intimately connected ; as, after an interval of two years, another demonstration of them, drawn solely from the principles of that calculus appeared, together with the calculus itself, in the memoirs of the Academy. This demonstration, involving, however, the passage from finite to infinite, is therefore (although preferable perhaps in a systematic arrangement, where all is made to flow from one fundamental principle) less elegant ; not on account of any confusion of ideas, or want of evidence ; but, because the ideas of finite and infinite, as such, are extraneous to symbolic language, and, if we would avoid their use, much circumlocution, as well as very unwieldy formulæ must be introduced. Arbogast also, in his work on derivations, has given two most ingenious demonstrations of them, and added greatly to their generality ; and lastly, Dr. Brinkley has made them the subject of a paper in the Transactions of this Society,* to which I shall have occasion again to refer. Considered as insulated truths, unconnected with any other considerable branch of analysis, the method employed by the latter author seems the most simple and elegant which could have been devised. It has however the great inconvenience of not making us acquainted with the bearings and dependencies of these important theorems, which, in this instance, as in many others, are far more valuable than the mere formulæ.