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Excerpt from Proceedings of the London Mathematical Society, Vol. 12: From November 1880 to November 1881 In the direct application of Finite Differences, James Stirling rightly lays great stress on the selection of a proper form for interpolation, by means of transformation, if necessary. His words (translated from the Latin) are For interpolation is not to be rashly undertaken; but before beginning the work we must inquire what is the simplest series, upon the interpolation of which that of the proposed series depends. And this preparation is usually absolutely necessary, in order to arrive at neat and definite His own illustration consists in forming a series whose nth term is the product of the nth terms of three other series, and then remarking that it is simpler to interpolate to these three series separately, and to take the product of the results of these separate interpolations, than to interpolate directly in the more complex series. This is evidently meant as an illustration of the wider truth, that such transformation, as is necessary to make the interpolation as small a matter as possible, should always be adopted. A familiar case of this is the interpolation of the logarithms of numbers differing but little from unity, or of the logarithmic sines or tangents of small angles, in which a transformation is always necessary to exactness. It is but an extension of this principle in another direction, that when any definitely known law governs the phenomenon, to which the obsor vations relate, either as a whole, or in part, that law should enter into the formulas of interpolation. Thus, for accurate work, the motion of a projectile subject to gravity is often better treated by interpolating its departure from a parabola, than by direct treatment of its ordinates. This advantage, however, disappears when the departure is excessive, as when the resistance of the medium is very great, or when the trajectory is followed to an extreme length. In these cases a formula must be used which takes account of resistance, and the subject of interpolation or comparison must be the departure from the path thus indicated. This caution is even more necessary to be observed where there is periodicity. When that exists, no interpolation which fails to take notice of it is worth anything if it covers more than a very small fraction of the period. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

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