Figurate Numbers. The nature of figurate numbers will be understood from the following table:
| 1, | 2, | 3, | 4, | 5, | 6, | 7, | &c. | |
|---|---|---|---|---|---|---|---|---|
| I. | 1, | 3, | 6, | 10, | 15, | 21, | 28, | &c. |
| II. | 1, | 4, | 10, | 20, | 35, | 56, | 84, | &c. |
| III. | 1, | 5, | 15, | 35, | 70, | 126, | 210, | &c. |
| &c. | &c. |
The natural numbers are here taken as the basis, and the first order of figurate numbers is formed from the series by successive additions; thus, the fifth number of the first order is the sum of the first five natural numbers. The second order is then formed from the first in the same way; and so on. If instead of the series of natural numbers whose difference is 1, we take the series whose differences are 2, 3, 4, &c., we may form as many different sets of figurate numbers. The name figurate is derived from the circumstance that the simpler of them may be represented by arrangements of equally distant points, forming geometrical figures. The numbers belonging to the first orders receive the general name of polygonal, and the special names of triangular, square, pentagonal, &c., according as the difference of the basis is 1, 2, 3, &c. Those of the second orders are called pyramidal numbers, and according to the difference of the basis are triagonally, quadrangularly, or pentagonally pyramidal. The polygonal numbers may be represented by points on a surface; the pyramidal by piles of balls.