Since each line is formed by adding the previous line to itself, it is evident that the sum of the numbers in each horizontal line must be double the sum of the numbers in the line next above. Hence we know, without making the additions, that the successive sums must be 1, 2, 4, 8, 16, 32, 64, &c., the same as the numbers of combinations in the Logical Alphabet. Speaking generally, the sum of the numbers in the *n*th line will be 2^{*n* - 1}. Again, if the whole of the numbers down to any line be added together, we shall obtain a number less by unity than some power of 2; thus, the first line gives 1 or 2^{1} - 1; the first two lines give 3 or 2^{2} - 1; the first three lines 7 or 2^{3} - 1; the first six lines give 63 or 2^{6} - 1; or, speaking in general language, the sum of the first *n* lines is 2^{*n*} - 1. It follows that the sum of the numbers in any one line is equal to the sum of those in all the preceding lines increased by a unit. For the sum of the *n*th line is, as already shown, 2^{*n* - 1}, and the sum of the first *n* - 1 lines is 2^{*n* - 1} - 1, or less by a unit. This account of the properties of the figurate numbers does not approach completeness; a considerable, probably an unlimited, number of less simple and obvious relations might be traced out. Pascal, after giving many of the properties, exclaims[103]: “Mais j’en laisse bien plus que je n’en donne; c’est une chose étrange combien il est fertile en propriétés! Chacun peut s’y exercer.” The arithmetical triangle may be considered a natural classification of numbers, exhibiting, in the most complete manner, their evolution and relations in a certain point of view. It is obvious that in an unlimited extension of the triangle, each number, with the single exception of the number *two*, has at least two places. [103] *Œuvres Complètes*, vol. iii. p. 251.