A numerical table, of whatever kind, is a series of numbers which possess some common character, and which proceed increasing or decreasing according to some general law. Supposing such a series continually to increase, let us imagine each number in it to be subtracted from that which follows it, and the remainders thus successively obtained to be ranged beside the first, so as to form another table: these numbers are called the _first differences_. If we suppose these likewise to increase continually, we may obtain a third table from them by a like process, subtracting each number from the succeeding one: this series is called the _second differences_. By adopting a like method of proceeding, another series may be obtained, called the _third differences_; and so on. By continuing this process, we shall at length obtain a series of differences, of some order, more or less high, according to the nature of the original table, in which we shall find the same number constantly repeated, to whatever extent the original table may have been continued; so that if the next series of differences had been obtained in the same manner as the preceding ones, every term of it would be 0. In some cases this would continue to whatever extent the original table might be carried; but in all cases a series of differences would be obtained, which would continue constant for a very long succession of terms. As the successive serieses of differences are derived from the original table, and from each other, by _subtraction_, the same succession of series may be reproduced in the other direction by _addition_. But let us suppose that the first number of the original table, and of each of the series of differences, including the last, be given: all the numbers of each of the series may thence be obtained by the mere process of addition. The second term of the original table will be obtained by adding to the first the first term of the first difference series; in like manner, the second term of the first difference series will be obtained by adding to the first term, the first term of the third difference series, and so on. The second terms of all the serieses being thus obtained, the third terms may be obtained by a like process of addition; and so the series may be continued. These observations will perhaps be rendered more clearly intelligible when illustrated by a numerical example. The following is the commencement of a series of the fourth powers of the natural numbers:-- No. Table. 1 1 2 16 3 81 4 256 5 625 6 1296 7 2401 8 4096 9 6561 10 10,000 11 14,641 12 20,736 13 28,561