POL 



POL 



radius, A F, describe a circle, which will 

 circumsci-ibe the polygon. Next, to cir- 

 cumscribe a polygon, divide 360 by the 

 number of sides required, to find eF d; 

 which set off from the centre, F, and 

 draw the line, de, on which construct the 

 polygon as in the following 1 problem. 2. 

 On a given line to describe any given 

 regular polygon : find the angle of the 

 polygon in the table, and in E set off an 

 angle equal thereto ; then drawing E A 

 =E D through the points E, A, D, de- 

 scribe a circle, and in this applying the 

 given right line as often as you can, the 

 polygon will be described. 3. To find 

 the sum of all the angles in any given 

 regular polygon: multiply the number of 

 sides by 180 ; from the product sub- 

 tract 360, and the remainder is the sum 

 required : thus, in a pentagon, 180 X 5 

 =a 900, and 900 360 = 540, the sum 

 of all the angles in a pentagon. 4. To 

 find the area of a regular polygon : mul- 

 tiply one side of the polygon by half the 

 number of sides ; and then multiply this 

 product by a perpendicular, let fall from 

 the centre of the circumscribing circle, 

 and the product will be the area re- 

 quired : thus, if A B (the side of a pen- 

 tagon) =54 x 24 = 135, and 135 X 29 

 (the perpendicular) = 3915 = the area 

 required. 5. To find the area of an ir- 

 regular polygon, let it be resolved into 

 triangles, and the sum of the areas of 

 these will be the area of the polygon. 



POLYGONAL numbers, are so called, 

 because the units whereof they consist 

 may be disposed in such a manner as to 

 represent several regular polygons. 



The side of a polygonal number is the 

 n umber of terms of the arithmetical pro- 

 gression that compose it ; and the num- 

 ber of angles is that which shows how 

 many angles that figure has, whence the 

 polygonal number takes its name. 



" To find any polygonal number pro- 

 posed," having given its side, n, and an- 

 gles, a. The polygonal number being 

 evidently the sum of the arithmetical 

 progression, whose number of terms is 

 n, and common difference a 2, and the 

 sum of an arithmetical progression being 

 equal to half the product of the ex- 

 tremes, by the number of terms, the ex- 

 tremes being 1, and 1 -f d. 

 n 1 = 1 4. a 2 . n I ; therefore, 

 that number, or this sum, Avilj be 



nV/ n . d 2 n 1 . a 2 n . a 4 

 _ or _ , 



where d\s the common difference of the 



arithmetical^ that form the polygonal 

 number, and is always 2 less than the 

 number of angles, a. 



Hence, for the several sorts of poly- 

 gons, any particular number, whose side 

 is n, will be found from either of these 

 two formulae, by using for d, its values 1, 

 2, j, 4, &c. ; which gives these following 1 

 formulae for the polygonal number in 

 each sort, viz. the 



&c. 



POLYGONUM, in botany, a genus of 

 the Octandria Trigynia class and order. 

 Natural order of Haloracex. Polygonese, 

 Jussieu. Essential character; calyx none; 

 corolla five-parted, calycine ; seed one, 

 angular. There are thirty-six species. 



POLYGYNIA, among botanists, de- 

 notes an order or subdivision of a class of 

 plants, comprehending such plants of that 

 class as have a great number of pistils, or 

 female organs of generation. 



POLYHEDRON, in geometry, denotes 

 a body or solid comprehended under 

 many sides, or planes. A gnomonic po- 

 lyhedron is a stone with several faces, 

 whereon are described various kinds of 

 dials. 



POLYHEDRON, polyscope, in optics, is a 

 multiplying glass or lens, consisting of 

 several plane surfaces disposed into s 

 convex form. 



POLYMN1A, in botany, a genus of the 

 Syngenesia Polygamia Necessaria class 

 and order. Natural order of Composite 

 Oppositifolias. Corymbiferse, Jussieu. Es- 

 sential character : calyx exterior, four or 

 five-leaved ; interior ten-leaved ; the leaf- 

 lets concave ; down none ; reoeptable 

 chaffy. There are five species. 



POLYNEMUS, the poli/neme, in natu- 

 ral history, a genus of fishes of the order 

 Abdominales. Generic character: head 

 compressed, covered wMi scales ; snout 

 very obtuse ;snd prominent; gill-mem- 

 brane, five or seven-rayed; separate fila- 

 ments near ihe base of the pectoral fins. 

 Shaw enumerates ten species; Gmelin 

 onlv four. 



