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, ro, 15 .. and generally Zn (n + I) ; if a square, 4, 9, 16, . . . and generally n'; if a pentagon, 5, 12, 22... and generally n(3n--1); if a hexagon, 6, 15, 28, . . . and generally n(2n— r) ; and similarly for a polygon of r sides, the general expression for the corresponding polygonal number is 2n[(n—I) (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; I+2+3+4= 10; cr generally I+2+3 ,, + n= an(n+r); 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-I.r—2; and hence the nth polygonal number of the rth order is the sum of is terms of this series, i.e., 1+I+(r—2)+I+2(r—2)+ ... +(I+n—I.r—2) =n + ;n.n — I.r -2. The series 1, 2, 3, 4, . . . or generally n, are the so-called, " linear numbers " (cf. FIGURATE NUMBERS).