HEPTAGON

, in Geometry, a figure of seven sides and seven angles.—When those sides and angles are all equal, the Heptagon is said to be regular, otherwise it is irregular.

In a regular Heptagon, the angle Cat the centre is = 51° 3/7, the angle DAB of the polygon is = 128° 4/7, and its half CAB = 64° 2/7. Also the area is = the square of the side or , where t is the tangent of the angle CAB of 64° 2/7 to the radius 1; or t is the root of the equation; ; or ,| where the value of x and y are the roots of the equations .

See my Mensuration, p. 21, 114, and 116, 2d edition.

Heptagon

, in Fortification, a place fortified or strengthened with seven bastions for its defence.

Heptagonal Numbers, are a kind of polygonal numbers in which the difference of the terms of the corresponding arithmetical progression is 5. Thus,

Arithmeticals,	1,	6,	11,	16,	21,	26,	&c.
Heptagonals,	1,	7,	18,	34,	55,	81,	&c.

where the Heptagonals are formed by adding continually the terms of the arithmeticals, above them, whose common difference is 5.

One property, among many others, of these Heptagonal numbers is, that if any one of them be multiplied by 40, and to the product add 9, the sum will be a square number. Thus ; and ; and ; and ; &c. Where it is remarkable that the series of squares so formed is 7², 17², 27², 37², &c, the common difference of whose roots is 10, the double of the common difference of the arithmetical series from which the Heptagonals are formed.—See Polygonals.

HEPTANGULAR Figure, in Geometry, is one that has seven angles; and therefore also seven sides.