There exists a close connection between the arithmetical triangle described in the last section, and the series of combinations of letters called the Logical Alphabet. The one is to mathematical science what the other is to logical science. In fact the figurate numbers, or those exhibited in the triangle, are obtained by summing up the logical combinations. Accordingly, just as the total of the numbers in each line of the triangle is twice as great as that for the preceding line (p. 186), so each column of the Alphabet (p. 94) contains twice as many combinations as the preceding one. The like correspondence also exists between the sums of all the lines of figures down to any particular line, and of the combinations down to any particular column. By examining any column of the Logical Alphabet we find that the combinations naturally group themselves according to the figurate numbers. Take the combinations of the letters A, B, C, D; they consist of all the ways in which I can choose four, three, two, one, or none of the four letters, filling up the vacant spaces with negative terms. There is one combination, ABCD, in which all the positive letters are present; there are four combinations in each of which three positive letters are present; six in which two are present; four in which only one is present; and, finally, there is the single case, *abcd*, in which all positive letters are absent. These numbers, 1, 4, 6, 4, 1, are those of the fifth line of the arithmetical triangle, and a like correspondence will be found to exist in each column of the Logical Alphabet. Numerical abstraction, it has been asserted, consists in overlooking the kind of difference, and retaining only a consciousness of its existence (p. 158). While in logic, then, we have to deal with each combination as a separate kind of thing, in arithmetic we distinguish only the classes which depend upon more or less positive terms being present, and the numbers of these classes immediately produce the numbers of the arithmetical triangle. It may here be pointed out that there are two modes in which we can calculate the whole number of combinations of certain things. Either we may take the whole number at once as shown in the Logical Alphabet, in which case the number will be some power of two, or else we may calculate successively, by aid of permutations, the number of combinations of none, one, two, three things, and so on. Hence we arrive at a necessary identity between two series of numbers. In the case of four things we shall have 2 = 1 + 4/1 + (4 . 3)/(1 . 2) + (4 . 3 . 2)/(1 . 2 . 3) + (4 . 3 . 2 . 1)/(1 . 2 . 3 . 4). In a general form of expression we shall have 2 = 1 + *n*/1 + (*n* . (*n* - 1))/(1 . 2) + (*n* (*n* - 1)(*n* - 2))/(1 . 2 . 3) + &c.,