No. LXIV.—ARITHMETICAL TRIANGLE
The peculiar series of numbers, as arranged in this triangular form, is said to have been perfected by Pascal.
| 1 | |||||||
| 2 | 1 | ||||||
| 3 | 3 | 1 | |||||
| 4 | 6 | 4 | 1 | ||||
| 5 | 10 | 10 | 5 | 1 | |||
| 6 | 15 | 20 | 15 | 6 | 1 | ||
| 7 | 21 | 35 | 35 | 21 | 7 | 1 | |
| 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 |
It has the property of showing, without calculation, how many selections or combinations can be made at a time out of a larger number. Thus to find how many selections of 3 at a time can be made out of 8 we look for the third number on the horizontal row that commences with 8, and find the answer 56.
The series is formed thus: Set down the numbers 1, 2, 3, etc., as far as you please, in a vertical row. To the right of 2 place 1, add them together, and set 3 under the 1. Then add 3 to 3, and set the result below, and so on, always placing the sum of two numbers that are side by side below the one on the right.