Arithmetical, a kind of natural triangle, or triangle of numbers, being a table of certain numbers disposed in form of a triangle. It was so called by Pascal; but he was not the inventor of this table, as some writers have imagined, its properties having been treated of by other authors some centuries before him, as is shewn in Dr Hutton's Mathematical Tracts, vol. i. p. 69. &c.

The form of the triangle is as follows:

``` 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 &c. 1 7 21 &c. 1 8 &c. 1 9 ```

And it is constructed by adding always the last two numbers of the next two preceding columns together, to give the next succeeding column of numbers.

The first vertical column consists of units; the second, a series of the natural numbers 1, 2, 3, 4, 5, &c.; the third, a series of triangular numbers 1, 3, 6, 10, &c.; the fourth, a series of pyramidal numbers, &c. The oblique diagonal rows, descending from left to right, are also the same as the vertical columns. And the numbers taken on the horizontal lines are the co-efficients of the different powers of a binomial. Many other properties and uses of these numbers have been delivered by various authors, as may be seen in the Introduction to Hutton's Mathematical Tables, pages 7, 8, 75, 76, 77, 89, second edition.