# FIGURAL

, the same as Figurate numbers; which see.

FIGURATE *Numbers,* such as do or may represent
some geometrical figure, in relation to which they are
always considered; as triangular, pentagonal, pyramidal,
&c, numbers.

Figurate numbers are distinguished into orders, according to their place in the scale of their generation, being all produced one from another, viz, by adding continually the terms of any one, the successive sums are the terms of the next order, beginning from the first order which is that of equal units 1, 1, 1, 1, &c; then the 2d order consists of the successive sums of those of the 1st order, forming the arithmetical progression 1, 2, 3, 4, &c; those of the 3d order are the successive sums of those of the 2d, and are the triangular numbers 1, 3, 6, 10, 15, &c; those of the 4th order are the successive sums of those of the 3d, and are the pyramidal numbers 1, 4, 10, 20, 35, &c; and so on, as below:

Order. | Name. | Numbers. | |||||

1. | Equals. | 1, | 1, | 1, | 1, | 1, | &c. |

2. | Arithmeticals, | 1, | 2, | 3, | 4, | 5, | &c. |

3. | Triangulars, | 1, | 3, | 6, | 10, | 15, | &c. |

4. | Pyramidals, | 1, | 4, | 10, | 20, | 35, | &c. |

5. | 2d Pyramidals, | 1, | 5, | 15, | 35, | 70, | &c. |

6. | 3d Pyramidals, | 1, | 6, | 21, | 56, | 126, | &c. |

7. | 4th Pyramidals, | 1, | 7, | 28, | 84, | 210, | &c. |

The above are all considered as different sorts of triangular numbers, being formed from an arithmetical progression whose common difference is 1. But if that common difference be 2, the successive sums will be the series of square numbers: if it be 3, the series will be pentagonal numbers, or pentagons; if it be 4, the series will be hexagonal numbers, or hexagons; and so on. Thus: |

Arithmeticals. | 1st Sums, or
Polygons. | 2d Sums, or
2d Polygons. | ||||||||||

1, | 2, | 3, | 4, | Tri. | 1, | 3, | 6, | 10 | 1, | 4, | 10, | 20 |

1, | 3, | 5, | 7, | Sqrs. | 1, | 4, | 9, | 16 | 1, | 5, | 14, | 30 |

1, | 4, | 7, | 10, | Pent. | 1, | 5, | 12, | 22 | 1, | 6, | 18, | 40 |

1, | 5, | 9, | 13, | Hex. | 1, | 6, | 15, | 28 | 1, | 7, | 22, | 50 |

&c. |

And the reason of the names triangles, squares, pentagons,
hexagons, &c, is, that those numbers may be
placed in the form of these regular figures or polygons,
as here below:
*Triangles.*
*Squares.*
*Pentagons.*
*Hexagons.*

But the Figurate numbers of any order may also be
found without computing those of the preceding orders;
which is done by taking the successive products of
as many of the terms of the arithmeticals 1, 2, 3, 4,
5, &c, in their natural order, as there are units in the
number which denominates the order of Figurates required,
and dividing those products always by the first
product: thus, the triangular numbers are found by
dividing the products 1 X 2, 2 X 3, 3 X 4, 4 X 5,
&c, each by the 1st pr. 1 X 2; the first pyramids by
dividing the products 1 X 2 X 3, 2 X 3 X 4, 3 X 4 X 5,
&c, by the first 1 X 2 X 3. And, in general, the figurate
numbers of any order *n,* are found by substituting
successively 1, 2, 3, 4, 5, &c, instead of *x* in this general
expression ; where the
factors in the numerator and denominator are supposed
to be multiplied together, and to be continued till the
number in each be less by 1 than that which expresses the
order of the Figurates required. See Maclaurin's Fluxions,
art. 351, in the notes; also Simpson's Algebra,
pa. 213; or Malcolm's Arithmetic, pa. 396, where
the subject of Figurates is treated in a very extensive
and perspicuous manner.