Every mathematical table consists of a series of values of a function corresponding to successive values of the variable, which last series of values forms the argument of the table. Any such table may be constructed therefore, when the function to be tabulated—which may be called the characteristic function—is known, by the evaluation of that function in terms of the successive values of the argument. It is only, however, when the table to be constructed is of limited extent that this method of formation would be employed. If the table be extensive, and especially if the characteristic function be complex, this, which may be called the direct method, would become too laborious, each value when formed in this way also requiring separate verification. In these circumstances the Method of Differences becomes available for the end in view. This method dispenses with all reference to the characteristic function beyond what is necessary for the formation of a few values (which I call fundamental or primitive values), at stated intervals; and in applying it, each value being dependent on the preceding, verification is obtained by the periodical coincidence with those fundamental values of the corresponding terms in the series in course of being formed.

A Table of mortality is designed to represent the number of lives which, according to the best deductions from past experience, may be expected to survive at the termination of each successive year of age, supposing these survivors to be derived exclusively from a certain number of persons originally taken either at birth or at a given age. As a practical index of the mathematical law of average mortality such a table may be taken as a trustworthy guide for the future, provided that the number of lives which enter into its formation be sufficiently large and the particulars respecting them be correctly registered; and, what is equally important, that the observations shall extend over a considerable number of years.