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