Interpolations. This great class of problems being then set aside, the most essential of the veritable applications of the analysis of Taylor is, undoubtedly, thus far, the general method of interpolations, so frequently and so usefully employed in the investigation of the empirical laws of natural phenomena. The question consists, as is well known, in intercalating between certain given numbers other intermediate numbers, subjected to the same law which we suppose to exist between the first. We can abundantly verify, in this principal application of the calculus of Taylor, how truly foreign and often inconvenient is the consideration of differences with respect to the questions which depend on that analysis. Indeed, Lagrange has replaced the formulas of interpolation, deduced from the ordinary algorithm of the calculus of finite differences, by much simpler general formulas, which are now almost always preferred, and which have been found directly, without making any use of the notion of differences, which only complicates the question.
Approximate Rectification, &c. A last important class of applications of the calculus of finite differences, which deserves to be distinguished from the preceding, consists in the eminently useful employment made of it in geometry for determining by approximation the length and the area of any curve, and in the same way the cubature of a body of any form whatever. This procedure (which may besides be conceived abstractly as depending on the same analytical investigation as the question of interpolation) frequently offers a valuable supplement to the entirely logical geometrical methods which often lead to integrations, which we do not yet know how to effect, or to calculations of very complicated execution.
Such are the various principal considerations to be noticed with respect to the calculus of finite differences. This examination completes the proposed philosophical outline of abstract Mathematics.
Concrete Mathematics will now be the subject of a similar labour. In it we shall particularly devote ourselves to examining how it has been possible (supposing the general science of the calculus to be perfect), by invariable procedures, to reduce to pure questions of analysis all the problems which can be presented by Geometry and Mechanics, and thus to impress on these two fundamental bases of natural philosophy a degree of precision and especially of unity; in a word, a character of high perfection, which could be communicated to them by such a course alone.
BOOK II.
GEOMETRY.
BOOK II.