POl 
POL 
polygon in the table, and in E set off an an- 
gle equal thereto ; then drawing E A = E D 
through the points, E, A, D, describe a cir- 
cle, 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 
tlie product subtract 360“, and the remain- 
der is the sum required: thus, in a penta- 
gon, 180 X 5 = 900, and 900 — 360 = 540, 
the sum of all the angles in a pentagon. 
4. To find the area of a regular polygon : 
multiply 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 required ; thus, if 
A B (the side of a pentagon) =: 54 X 2^ = 
135, and 135 X 29 (the perpendicular) = 
3915 = the area required. 5. To find the 
area of an irregular polygon, let it be re- 
solved into triangles, and the sum of the 
areas of these will be the area of the poly- 
gon. 
POLYGONAL numbers, are so called, 
because the units whereof they consist may 
be disposed in such a manner as to repre- 
sent several regular polygons. 
The side of a polygonal number is the 
number of terms of the arithmetical pro- 
gression that compose it ; and the number 
of angles is that which shows how many an- 
gles that figure has, whence the polygonal 
number takes its name. 
“ To find any polygonal number pro- 
posed having given its side, n, and angles, 
fl. 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 arith- 
metical progression being equal to half the 
product of the extremes, by the number of 
terms, the extremes being 1, and 1 -j- d . 
n — l=l-|-a — 2.n — 1; therefore, 
that number, or this sum, will be 
d- — n .d — 2 v? .a — 2 — n . a — 4 
where d is the common difference of the 
aritbmeticals that form the polygonal num- 
ber, and is always 2 less than the number 
of angles, a. 
Hence, for the several sorts of polygons, 
any particular number, whose side is n, will 
be found from either of these two formulas, 
by using for d, its values 1 , 2, 3, 4, &c. ; 
which gives these following formulae for 
the polygonal number in each sort, viz, 
the 
Trian.'O'iilai’ 
M- -|-n 
2 ’ 
Square 
11 
IS 
o 
1 
Pentagonal 
3m“ — n 
2 ’ 
4 — 2 m 
2 ' 
Heptagonal 
5 — 3 m 
“ ■ 2 ’ 
&c. 
POLYGONUM, in botany, a genus of 
the Octandria Trigynia class and order. 
Natural order Of Haloraceae. Polygonem, 
Jussieu. Essential character : calyx none; 
corolla five-parted, calycine ; seed one, an- 
gular. 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 polyhedron 
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 seve- 
ral plane surfaces disposed into a convex 
form. 
POLYMNIA, in botany, a genus of the 
Syngenesia Polygamia Necessaria class and 
order. Natural order of Compositae Oppo- 
sitifolim. Corymbiferaa, Jussieu. Essential 
character : calyx exterior, four or fiVe-leav- 
ed ; interior ten-leaved ; the leaflets con- 
cave ; down none ; receptacle chaffy. There 
are five species. 
POLYNEMUS, the polyneme, in natu- 
ral history, a genus of fishes of the order 
Abdominales. Generic character : head 
compressed, covered with scales; snout 
very obtuse and prominent ; gill-membrane, 
five or seven-rayed ; separate filaments 
near the base of the pectoral fins. Shaw 
enumerates ten species ; Gmelin only four. 
The P. paradiseus, or Paradise polyne- 
me, or Mango-fish, inhabits the Indian and 
American seas, and is thirteen inches long, 
elegantly shaped, and with thoracic fila- 
ments frequently far larger than the body ; 
its colour is yellow. At Calcutta it is in 
the highest estimation for the table. 
P. plebeius, or the grey polyneme, 
abounds on the Malabar coast, and has five 
filaments on each side, but all rather short. 
It is sometimes four 'feet long, and is in 
