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]
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, 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.
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 combinations of m things out of n, for all possible values of m and n. Out of seven things one may be chosen in seven ways, and seven occurs in the eighth line of the second column. The combinations of two things chosen out of seven are 7 × 6/1 × 2 or 21, which is the third number in the eighth line. The combinations of three things out of seven are 7 × 6 × 5/1 × 2 × 3 or 35, which appears fourth in the eighth line. In a similar manner, in the fifth, sixth, seventh, and eighth columns of the eighth line I find it stated in how many ways I can select combinations of 4, 5, 6, and 7 things out of 7. Proceeding to the ninth line, I find in succession the number of ways in which I can select 1, 2, 3, 4, 5, 6, 7, and 8 things, out of 8 things. In general language, if I wish to know in how many ways m things can be selected in combinations out of n things, I must look in the n + 1th line, and take the m + 1th number, as the answer. In how many ways, for instance, can a subcommittee of five be chosen out of a committee of nine. The answer is 126, and is the sixth number in the tenth line; it will be found equal to 9 . 8 . 7 . 6 . 5/1 . 2 . 3 . 4 . 5, which our formula (p. [182]) gives.
The full utility of the figurate numbers will be more apparent when we reach the subject of probabilities, but I may give an illustration or two in this place. In how many ways can we arrange four pennies as regards head and tail? The question amounts to asking in how many ways we can select 0, 1, 2, 3, or 4 heads, out of 4 heads, and the fifth line of the triangle gives us the complete answer, thus—
| We can select | No | head and 4 tails in 1 way. |
| " | 1 | head and 3 tails in 4 ways. |
| " | 2 | heads and 2 tails in 6 ways. |
| " | 3 | heads and 1 tail in 4 ways. |
| " | 4 | heads and 0 tail in 1 way. |
The total number of different cases is 16, or 24, and when we come to the next chapter, it will be found that these numbers give us the respective probabilities of all throws with four pennies.
I gave in p. [181] a calculation of the number of ways in which eight planets can meet in conjunction; the reader will find all the numbers detailed in the ninth line of the arithmetical triangle. The sum of the whole line is 28 or 256; but we must subtract a unit for the case where no planet appears, and 8 for the 8 cases in which only one planet appears; so that the total number of conjunctions is 28 – 1 – 8 or 247. If an organ has eleven stops we find in the twelfth line the numbers of ways in which we can draw them, 1, 2, 3, or more at a time. Thus there are 462 ways of drawing five stops at once, and as many of drawing six stops. The total number of ways of varying the sound is 2048, including the single case in which no stop at all is drawn.