Line. First Column. 1 1 Second Column. 2 1 1 Third Column. 3 1 2 1 Fourth Column. 4 1 3 3 1 Fifth Column. 5 1 4 6 4 1 Sixth Column. 6 1 5 10 10 5 1 Seventh Column. 7 1 6 15 20 15 6 1 Eighth Column. 8 1 7 21 35 35 21 7 1 Ninth Column. 9 1 8 28 56 70 56 28 8 1 Tenth Column. 10 1 9 36 84 126 126 84 36 9 1 Eleventh Column. 11 1 10 45 120 210 252 210 120 45 10 1 Twelfth Column. 12 1 11 55 165 330 462 462 330 165 55 11 1 Thirteenth Column. 13 1 12 66 220 495 792 924 792 495 220 66 12 1 Fourteenth Column. 14 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 Fifteenth Column. 15 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 Sixteenth Column. 16 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 Seventeenth Col. 17 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1 Examining these numbers, we find that they are connected by an unlimited series of relations, a few of the more simple of which may be noticed. Each vertical column of numbers exactly corresponds with an oblique series descending from left to right, so that the triangle is perfectly symmetrical in its contents. The first column contains only *units*; the second column contains the *natural numbers*, 1, 2, 3, &c.; the third column contains a 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-- [Illustration] 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.[102] [102] Wallis’s *Algebra*, Discourse of Combinations, &c., p. 109. 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.