VIII. The deferred approach to the limit

Abstract

Various problems concerning infinitely many, infinitely small, parts, had been solved before the infinitesimal calculus was invented; for example, Archimedes on the circumference of the circle. The essence of the invention of the calculus appears to be that the passage to the limit was thereby taken at the earliest possible stage, where diverse problems had operations like d/dx in common. Although the infinitesimal calculus has been a splendid success, yet there remain problems in which it is cumbrous or unworkable. When such difficulties are encountered it may be well to return to the manner in which they did things before the calculus was invented, postponing the passage to the limit until after the problem had been solved for a moderate number of moderately small differences. For obtaining the solution of the difference-problem a variety of arithmetical processes are available. This memoir deals with central differences arranged in the simplest possible way, namely, that explained by the writer in the papers cited in the footnote. Advancing differences are ignored, and so are the varieties of central-difference-process in which accuracy is gained by complicating the arithmetic at an early stage.