FIGURATE numbers, are distinguished into orders, according to their place in the scale of' their generation, being all pro duced one from another, viz. by adding continually the terms of any one, the suc cessive 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 succes sive sums of those of the first order, form ing the arithmetical progression 1, 2, 3, 4, &c.; those of the 3d order the successive sums of those of the 2d, and are the trian gular 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. Arithmetical 1, 2, 3, 4, 5, &e.
3. Triangulars 1, 3, 6, 10, 15, &c.
4. Pyramidals.., 1, 4, 10, 20, 35, &c.
5. 2" Pyramidals 1, 5, 15, 35, 70, &c, 6. 3" Pyramidals 1, 6, 21, 56, 126, &c.
7. 4th Pyramidals 1, 7, 28, 84, 210, &c.
The above are all considered as dif ferent sorts of triangular numbers, being formed from an arithmetical progression, whose common difference is 1. But if that common difference is 2, the succes• sive 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 bers, or hexagons, and so on. Thus :
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. The figurate numbers of any order may be found without computing those of the preceding order, 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 divid ing those products always by the first product : thus the triangular numbers are found by dividing the products 1X2 ; 2X3; 3X4, &c. each by the first product 1 x2 : the first pyramids by dividing the products lx2 X 3; 2 X 3 X 4, &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 a in this general expression X X z-F 2 X z+3,&c. , wnere