Page:The New International Encyclopædia 1st ed. v. 16.djvu/255

POLYGON. 217 POLYGONACE^. 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 jt- A GENERAL QUADRILATERAL. self. The word polygon, in elementary geometry, is understood to refer to a polygon that is not cross xmless 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. . . Ji-gon, according as it has 3, 4, 5, 6. 7, 8, 9, 10, . . .12 . . .15, . . .n sides. According to the principle of continuity (q.v. ) polygons may be regarded as positive or as nega- tive. E.g. consider the triangle ABC, which iS) in general, regarded as positive. If C moves down REGULAR COX- TEX-POLY- GO.N. REGULAR CROSS-POLY- GON. to rest on AB, then / ABC becomes zero; and as C passes through AB l ABC passes through zero and is considered as having changed its sign and become negative: that is, l AC"B is nega- tive. In the case of polygons in general, the law of signs will readily be understood from the annexed figures. In Figs. 1, 2, 3, both the up- per and lower parts of the polygon are considered equals (» — 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 p — -, n 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 be divided into 2°, 3-2, .5-2", lo-'I" 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 ISOl, that a perigon could also be divided into 17'2° equal angles : furthermore, that it could be divided into 2™ + 1 equal angles if 2" + I rep- lesents a prime number; and, in general, that it could be divided into a number of equal angles represented by the product of different prime numbers of the form 2°" + I. 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 "+ I. It is shown in the theory of numbers that if 2 "^ -f I is prime, m must equal 2p : hence the general form for the prime numbers men- tioned is 2-P + 1. Elementary geometry is thus limited to the inscription and circumscription of the regular polygons mentioned. Consult Klein's Famous Proilems of Elementary Geometry (American edition, Boston, 1897). POL'YGONA'CE.a! (Xeo-Lat. nom. pi., from Lat. polygonum, from Gk. TroKuyovov, knot-grass, polygeny, neu. sg. of ttoXi^opoj, polygonos. pro- lific, from iroXiJs, poli/s. much, many -f- yimt, gonos. seed). The Buckwheat Family. A nat- ural order of about 30 genera and 750 species of widely distributed dicotyledonous herbs, a few sliruhs and trees, particularly abundant in the temperate regions of the Northern Hemisphere. The principal genera are Chorizanthe. Eric- gonum. Euniex, Rheum. Polygonum, Fagopyrum, and Coceoloba. The genus Polygonum, which is typical of the order, consists of abut 150 species, mostly weeds. Knot-grass {Polygonum avicu- tare) is one of the most extensively distributed A B Fig. 1. as positive; in Fig. 4, P has reached BC plants of the world: it is an annual of low and the upper part of the polygon has become growth, but very variable, with much branched zero; in Fig. 5, P has passed through BC trailing stems, small lanceolate leaves, and very and the upper part of the figure has passed .small flowers, two or three together, in the axils through zero and become negative. of the leaves. Thunberg says that in Japan a The sum of the interior angles of a polygon blue dye is prepared from the plant. Polygonum