On the calculus of finite differences and its application to problems in the doctrine of compound interest and certain annuities

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