POLYGON I Lat. polygonum. from Gk.-.702.v PO/ygon (fii, polygon, net. sg. of 7,-o7.i.)wroc, polygonos, having many angles, from ro;ti.c, polys, much. many + yenta, gonia, angle). if the two end-points of a broken line coincide, the figure ob Amazons, and of the Lapith:r with the Centaurs, and the descent of Theseus to Amphitrite in the depths of the sea. In the Stoa were represented the capture of Troy and the council of the Greeks to judge Ajax for his outrage on Cassandra. which was certainly by Polygnotus, also the bat twined is called a polygon, and the broken line its rimuter. The vertices of the angles made by the various segments of the perimeter are called the vertices of the polygon, and the segments themselves the sides of the polygon. The peri meter of a polygon divides the plane into two parts, one finite (the part inclosed), the other infinite. The finite part is called the surface of the polygon, or for brevity simply the polygon. A polygon is said to be convex when no side pro duced cuts the surface of the polygon, concave when a side produced cuts the surface of the polygon, and cross when the perimeter crosses it self. The word polygon, in elementary geometry, is understood to refer to a polygon that is not cross unless the contrary is stated. if all of the sides of a polygon are indefinitely pro duced, the figure is called a general poly gon. If a polygon is both equiangular and equilateral it is said to be regular. A poly gon is called a tri angle, quadrilateral. pentagon, hexagon, hepta gon, octagon, nonagon, decagon. . .dodecagon. .
pentedecagon. . . n-gon, according as it has 3, 4, 5, 6. 7, S, 9, 10, .. .12 . . .15, . . at sides.
According to the principle of continuity (q.v.) polygons may he regarded as positive or as nega tive. E.g. consider the triangle ABC, which in general, regarded as positive. If C moves down to rest on AB, then A, ABC' becomes zero: and as C passes through AB A ABC passes through zero and is considered as having changed its sign and become negative; that is, A AC"B is nega tive. In the case of polygons in general, the law of signs will readily he understood from the annexed figures. In Figs. 1. 2. 3, both the up
per and lower parts of the polygon are considered equals (n — 2) straight angles. The sum of the exterior angles equals a perigon, or 360°. In concave polygons certain exterior angles lie in side of the polygon and are taken as negative ac cording to the principle of continuity. The num ber of diagonals of a simple convex polygon is n(n-3) 9 a being the number of sides. If a polygon of an even number of sides be circum scribed about a circle, the sums of its even and odd sides are equal; and if a polygon of an even number of sides be inscribed in a circle, the sums of its even and odd angles are equal. The in scription and circumscription of regular poly gons depend upon the partition of the perigon. Thus to inscribe an equilateral triangle in a cir cle depends upon trisecting the circumference, hence the perigon at the centre. It was known as early as Euclid's time that the perigon could he divided into 2", 3.2', 5.2", 152° equal angles, and no other partitions were deemed possible by the use of the straight edge and compasses. But in 1796 Gauss found, and published the fact in 1S01, that a perigon could also be divided into 17.2° equal angles; furthermore, that it could be divided into 2m + 1 equal angles if 2m + 1 rep iesents a prime number; and, in general. that it could he divided into a number of equal angles represented by the product of different prime numbers of the form 2°. + 1. Hence it follows that a perigon can be divided into a number of equal angles represented by the product of 2° and one or more different prime numbers of the form 2 m+ 1. It is shown in the theory of numbers that if 2 m± 1 is prime. vi must equal 25; hence the general form for the prime numbers men tioned is + 1. Elementary geometry is thus limited to the inscription and circumscription of the regular polygons mentioned. Consult Klein's Fa mous Problems of Elementary Geometry (American edition, Boston, 1S97).