1911 Encyclopædia Britannica/Polygonal Numbers

POLYGONAL NUMBERS, in mathematics. Suppose we have a number of equal circular counters, then the number of counters which can be placed on a regular polygon so that the tangents to the outer rows form the regular polygon and all the internal counters are in contact with its neighbours, is a “polygonal number” of the order of the polygon. If the polygon be a triangle then it is readily seen that the numbers are 3, 6, 10, 15 . . . and generally ⁠1/2⁠𝑛 (𝑛 + 1); if a square, 4, 9, 16, . . . and generally 𝑛²; if a pentagon, 5, 12, 22 . . . and generally 𝑛(3𝑛−1); if a hexagon, 6, 15, 28, . . . and generally 𝑛(2𝑛−1); and similarly for a polygon of 𝑟 sides, the general expression for the corresponding polygonal number is ⁠1/2⁠𝑛[(𝑛−1) (𝑟−2)+2].

Algebraically, polygonal numbers may be regarded as the sums of consecutive terms of the arithmetical progressions having 1 for the first term and 1, 2, 3, . . . for the common differences. Taking unit common difference we have the series 1; 1+2＝3; 1+2+3 ＝6; 1+2+3+4＝10; or generally 1+2+3 . . . + 𝑛＝⁠1/2⁠𝑛(𝑛+1); these are triangular numbers. With a common difference 2 we have 1; 1+3＝4; 1+3+5＝9; 1+3+5+7＝16; or generally 1+3+5+ . . . + (2𝑛−1)＝𝑛²; and generally for the polygonal number of the 𝑟th order we take the sums of consecutive terms of the series

1, 1+(𝑟−2), 1+2 (𝑟−2), . . . 1+𝑛−1.𝑟−2;

and hence the 𝑛th polygonal number of the 𝑟th order is the sum of 𝑛 terms of this series, i.e.,

1+1+(𝑟−2)+1+2(𝑟−2)+ . . . +(1+𝑛−1.𝑟−2)

 ＝𝑛+⁠1/2⁠𝑛.𝑛−1.𝑟−2.

The series 1, 2, 3, 4, ... or generally 𝑛, are the so-called “linear numbers” (cf. Figurate Numbers).