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.
No. LXV.—MULTIPLICATION NO VEXATION
This diagram shows an ancient and curious method of multiplication, which will be novel to most of our readers.
In this instance 534 is multiplied by 342. Draw a square of nine cells with diagonals, fill the three top cells, as is shown, by multiplying the 5 by the 3, the 4 and the 2. Then multiply in similar way the 3 and the 4 by these same figures. Turn the square round so that the diagonals are upright, and add. Of course, placing the numbers thus is the same practically as carrying them by our ordinary rule.