The extreme importance and significance of this formula seems to have been first adequately recognised by Pascal, although its discovery is attributed by him to a friend, M. de Ganières.[99] We shall find it perpetually recurring in questions both of combinations and probability, and throughout the formulæ of mathematical analysis traces of its influence may be noticed.
The Arithmetical Triangle.
The Arithmetical Triangle is a name long since given to a series of remarkable numbers connected with the subject we are treating. According to Montucla[100] “this triangle is in the theory of combinations and changes of order, almost what the table of Pythagoras is in ordinary arithmetic, that is to say, it places at once under the eyes the numbers required in a multitude of cases of this theory.” As early as 1544 Stifels had noticed the remarkable properties of these numbers and the mode of their evolution. Briggs, the inventor of the common system of logarithms, was so struck with their importance that he called them the Abacus Panchrestus. Pascal, however, was the first who wrote a distinct treatise on these numbers, and gave them the name by which they are still known. But Pascal did not by any means exhaust the subject, and it remained for James Bernoulli to demonstrate fully the importance of the figurate numbers, as they are also called. In his treatise De Arte Conjectandi, he points out their application in the theory of combinations and probabilities, and remarks of the Arithmetical Triangle, “It not only contains the clue to the mysterious doctrine of combinations, but it is also the ground or foundation of most of the important and abstruse discoveries that have been made in the other branches of the mathematics.”[101]
The numbers of the triangle can be calculated in a very easy manner by successive additions. We commence with unity at the apex; in the next line we place a second unit to the right of this; to obtain the third line of figures we move the previous line one place to the right, and add them to the same figures as they were before removal; we can then repeat the same process ad infinitum. The fourth line of figures, for instance, contains 1, 3, 3, 1; moving them one place and adding as directed we obtain:—
| Fourth line . . . | 1 | 3 | 3 | 1 | |||
| 1 | 3 | 3 | 1 | ||||
| Fifth line . . . . . | 1 | 4 | 6 | 4 | 1 | ||
| 1 | 4 | 6 | 4 | 1 | |||
| Sixth line . . . . . | 1 | 5 | 10 | 10 | 5 | 1 |
Carrying out this simple process through ten more steps we obtain the first seventeen lines of the Arithmetical Triangle as printed on the next page. Theoretically speaking the Triangle must be regarded as infinite in extent, but the numbers increase so rapidly that it soon becomes impracticable to continue the table. The longest table of the numbers which I have found is in Fortia’s “Traité des Progressions” (p. 80), where they are given up to the fortieth line and the ninth column.
THE ARITHMETICAL TRIANGLE.
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—
