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Published online by Cambridge University Press: 18 August 2016
The calculus of finite differences was created by Taylor, in his celebrated work entitled Methodus Incrementorum, and it consists, essentially, in the consideration of the finite increments which functions receive as a consequence of analogous increments on the part of the corresponding variables. These increments or differences, which take the characteristic Δ to distinguish them from differentials, or infinitely small increments, may be in their turn regarded as new functions, and become the subject of a second similar consideration, and so on; from which results the notion of differences of various successive orders, analogous at least in appearance to the consecutive orders of differentials. Suc h a calculus evidently presents, like the calculus of indirect functions, two general classes of questions:—
1. To determine the successive differences of all the various analytical functions of one or more variables, as the result of a definite manner of increase of the independent variables, which are generally supposed to augment in arithmetical progression.
2. Reciprocally to start from these differences, or, more generally, from any equation established between them, and go back to the primitive functions themselves or to their corresponding relations
page 335 note * The immense increase of money accumulating at compound interest for a long period, is sufficient to astonish the human mind, and to stagger the credibility of persons who may not be conversant with the properties of geometrical progression, ex. gr.:— The amount of a farthing placed out at compound interest at the commencement of the Christian era, and continued to the end of the eighteenth century, would be 144,035 quintillions of pounds; but of the magnitude of this sum, spoken of in the abstract, no just conception can be formed. When, however, by a further calculation we ascertain, that to coin such a quantity of money (were it possible) into sovereigns of the present weight and fineness, we should require 60,308,170 solid globes of gold, each as large as the earth, we are enabled to entertain a more adequate idea of the sum, whose fastness, without having recourse to this adscititious assistance, placed it almost beyond the reach of our limited understandings.