are such as have three legs or feet, by which any triangle, or three points, may be taken off at once. These are very useful in the construction of maps, globes, &c.

TRIANGULAR Numbers, are a kind of polygonal numbers; being the sums of arithmetical progressions, which have 1 for the common difference of their terms.

Thus, from these arithmeticals 1 2 3 4 5 6, are formed the triangular numbers 1 3 6 10 15 21, or the third column of the arithmetical triangle above-mentioned.

The sum of any number n of the terms of the triangular numbers, 1, 3, 6, 10, &c. is

\[ \frac{n^3}{6} + \frac{n^2}{2} + \frac{n}{3} \text{ or } \frac{n \times (n + 1) \times (n + 2)}{6} \]

which is also equal to the number of shots in a triangular pile of balls, the number of rows, or the number in each side of the base, being n.

The sum of the reciprocals of the triangular series, infinitely continued, is equal to 2; viz.

\[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots = 2 \]

For the rationale and management of these numbers, see Malcolm's Arith. book 5. ch. 2.; and Simpson's Alg. sec. 15.