remarkable series of numbers, 1, 3, 6, 10, 15, &c., which have long been called the triangular numbers, because they correspond with the numbers of balls which may be arranged in a triangular form, thus—
The fourth column contains the pyramidal numbers, so called because they correspond to the numbers of equal balls which can be piled in regular triangular pyramids. Their differences are the triangular numbers. The numbers of the fifth column have the pyramidal numbers for their differences, but as there is no regular figure of which they express the contents, they have been arbitrarily called the trianguli-triangular numbers. The succeeding columns have, in a similar manner, been said to contain the trianguli-pyramidal, the pyramidi-pyramidal numbers, and so on.9
From the mode of formation of the table, it follows that the differences of the numbers in each column will be found in the preceding column to the left. Hence the second differences, or the differences of differences, will be in the second column to the left of any given column, the third differences in the third column, and so on. Thus we may say that unity which appears in the first column is the first difference of the numbers in the second column; the second difference of those in the third column; the third difference of those in the fourth, and so on. The triangle is seen to be a complete classification of all numbers according as they have unity for any of their differences.
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 nth line will be 2n–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 21–1; the first two lines give 3 or 22–1; the first three lines 7 or 23–1; the first six lines give 63 or 26–1; or, speaking in general language, the sum of the first n lines is 2n–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 nth line is, as already shown, 2n–1, and the sum of the first n - 1 lines is 2n–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, exclaims10: “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.
Though the properties above explained are highly curious, the greatest value of the triangle arises from the fact that it contains a complete statement of the values of the formula (p. 182), for the numbers of