POLYGON

POLYGON (Yunanca köken — orijinale bakınız.), according to Euclid, is any plane rectilinear figure having more than four sides or four angles; but, in treating of polygons generally, geometers also include the triangle and the quadrangle. If the sides of the polygon are all equal, it is said to be a regular polygon; otherwise it is irregular. Every regular i)olygon can be circumscribed by a circle, or have a circle inscribed in it; but of irregular polygons, excepting triangles, there is only one case in which a circle will pa through all their angular points; namely, when the polygon has an equal number of sides, one half of which are equal to one another, and the other half also equal to one another, but different from the former, and the equal and unequal sides are placed alternately. It is obvious that all the angles of such a figure are equal. Euclid has shown in the ElementsY^ovj to'm%cnhe a triangle, a square, and a pentagon in a given circle; and as any arc of a circle may be bisected geometrically, and the lialves again bisected continually, it follows that any regular polygon, of which the number of sides is 2" (» being any number whatever), or 3 x 2", or 5 x 2", may be inscribed in a circle by elementary geometry. Until a recent discovery respecting the division of angles was made by Gau , it was supposed that if the number of sides of a polygon was any other prime number than 3 or 5, the figure could not be inscribed in a circle; but this geometer, in his Dis qui sit i ones Arithmeticce, has demonstrated that every polygon, the number of whose sides is a prime number of the form 2n + 1, may be geometrically inscribed in a circle. When w = 1, this form gives 3, or the triangle; when ra = 2, it gives 5, or the pentagon; when « = 3, it gives 9, which is not a prinne; when w = 4, it gives 17, whence a seventeen-sided figure may be inscribed in a circle geometrically. The next prime is found by making w = 8, when the form gives 2.57. In order to investigate the general properties of polygons, it is nece ary to divide them into two cla es, convex and concave; the first comprehending those of /n \ which all the interior '^ '' angles are le than two ■Sy\ right angles (fig. 1.), and the second those which have one orraorere-enter'iV ^'"'S angles, as C (fig. 2.). If we call those the interior angles of the polygon which belong to the interior of the figure (whether le or greater than two right angles), and those exterior angles which are obtained by substracting each interior angle from four right angles, the two following theorems will be true of the polygons of both cla es: — 1. The sum of the interior angles of a polygon is equal to as many times two right angles as there are sides minus two. Thus, let S denote the sum of the interior angles, R a right angle, and n the number of sides of the polygon; then S = 2 (« — 2) R. 2. The sum of the exterior angles of a polygon is equal to as many times two right angles as there are sides plus two. Let S' be the sum of the exterior angles, as above defined; then S' = 2 (» + 2) R. Any polygon may be decomposed into triangles by drawmg straight lines from one of its angular points to each of the opposite angles, and the area of the polygon is the sum of the areas of all the component triangles. But a beautiful theorem was found by LHuilier of Geneva, by means of which, when the sides and angles of a polygon are known, the area is found without decomposmg it into triangles, which, when the number of sides is considerable, leads to laborious calculations. The theorem is this: — The double of the surface of any rectilinear figure is equal to the sum of the rectangles of its sides, taken two and two, excepting one, multiplied by the sine of the sum of the supplen»ents of the interior angles contained between each pair of sides. Thus, in the preceding figure (fig. 1.), let A,B, C,D, Ebe the supplements of the interior angles at those points; then 2 area = A B x B C sin. B + AB X C Dsin.(B + C) -1- A B X D E sin. (B -t- C + D) + B C X C D sin.C + B C X D E sin. (C + D) + C D X D E sin. D. This formula also gives the area of the polygons of the second cla (fig. 2.); only the supplement of the reentering angle C must be taken with the negative sign. (See LHuilier's PolygonomStrief Genfive, 1789.) Polygon, in Fortification, is either exterior or interior. The exteri«>r polygon is the figure formed by lines connecting the angles of the bastions with one another all round the work; the interior polygon by lines connecting the centres of the bastions all round. POLYGONACEiE. (Polygonum, one of the genera.) A natural order of herbaceous, rarely shrubby, apetalous Exogens, inhabiting the whole world; distinguished from most other plants by the cohesion of the scarious stipules into a sheath, technically called an ochrea or 957 POLYOPTRON. boot, and by their triangular fruit. Sorrel on the one hand, and rhubarb on the other, represent the general qualities of this order. While the leaves and young shoots are acid and agreeable, the roots are universally nauseous and purgative. Rumex acelosa contains pure oxalic acid, and many species of Polygonum are used in dyeing. The Rheum or rhubarb, and Rumex or dock, are well-known plants of this order; which is also sometimes remarkably astringent, as in the case of the Coccoloba uvifera, or sea-side grape of the West Indies, an extract of whose bark forms a kind of kino. POLYGONAL NUMBERS, in Arithmetic, are the succe ive sum.s of a series of numbers in arithmetical progre ion. When the common difference of the series in arithmetical progre ion is 1, then the sums of the terms give the triangular numbers; when the common difference of the terms of the arithmetical series is 2, the sums of the terms are the square numbers; when the difference is 3, the sums are the pentagonal numbers; and so on. Thus: — C Common difference = 1; 1, 2, 3, 4, 5, 6, . I Triangular numbers 1, 3, 6, 10, 15, 21, . (■ Common difference = 2; 1,3, 5, 7, 9, 11, . X Square numbers 1,4, 9, 16, 25, 36, . f Common difference = 3; 1,4, 7, 10, 13, 16, . I Pentagonal numbers 1, 5, 12, 22, 35, 51, . and so on. These numbers are called, in general, polygonal, from po e ing this property, that a the same number of points may be ar*^ ranged in the form of that polygonal ^> J5' V--J >' figure to which it belongs. For example. ■<: d: * the pentagonal numbers 5, 12, 22, 35, 51, . may be severally arranged in a V'^ V pentagonal form. Thus, in the annexed I ■ *"*5» figure, 5 points form the pentagon a b c d e; 12 the pentagon afghi, with the former enclosed; 22 the pentagon a k Imn, with the two former enclosed. A very general and remarkable property of polygonal numbers was discovered by Fermat, though it has yet been demonstrated only in respect of the triangular and square numbers. It is this:— Every number whatever is the sum of one, two, or three triangular numbers; the sura of one, two, three, or four squares; the sum of one, two, three, four, or five pentagonal numbers; and so on. See FiGURATE Numbers. [s. 970]