POL                       (85
find the other a Perpendicular drawn from the Centre to
one of the Sides of the Polygon. See TRIANGLE.
Hence alfo every Polygon circumfcribed about a Circle is
bigger than it; and every Polygon. inscribed, lefs than the
Oircle.-The fame likewife appears hence, that the thing
containing is ever greater than the thing contain'd.
And hence again, the Perimeter of every Polygon circum-
fcribed about a Circle, is greater than the Circumference
of that Circle; and the Perimeter of every Polygon in-
fcribed, lefs: whence it follows, that a Circle is equal to
a Right Angle Triangle, whoIe Bafe is the Circumference
of the Circle, and its Height the Radius , fince this Triangle
is iefs than any Polygon circumfcribed, and greater than any
infcribed. See CIRCUMSCRIBING.
Nothing therefore is wanted to the Quadrature of the Circle;
but to find a right' Line equal to the Circumference of a
Circle. See CIRCLE, CIRCUMFERENCE, QUADRATURE,
To find the Area of a Regular Polygon.-Multiply a Side
of the Polygon, as A B, by half the Number of the Sides, e. gr.
the Side of a Hexagon by 3. Again, multiply the Produ&k
by a Perpendicular let fall from the Centre of the circum-
fcribing Circle to the Side A B i the Produ& is the Area re-
quired. See AREA.
Thus, fuppofe A B, 54; and half the Number of Sides z 2;
the Produ&t or Semiperimeter is 135. Suppofing then the
Perpendicular Fg, zg; the Produft of there two, 3915, is the
Area of the Pentagon required.
To find the Area of an irregular Polygon, or Trapezi.um.-
Refolve it into Triangles; find the feveral Area-, of the feve-
ral Triangles, fee TRIANGLIS; the Sum of there is the Area
of the Polygon required. See TRAPEZIUM.
To find the Sum of all the Angles in any Polygon.-Multiply
the Number of Sides by i8o0: From the Produft fubtraa
360; the Remainder is the Sum required.
Thus in a Pentagon, Igo being multiplied by 5 gives goo;
whence fubtraffing 360 there remains 540; the Sum of the
Angles of a Pentagon.
Hence, if the Sum found be divided by the Number of
Sides; the Quotient will be the Angle of a regular Poly-
gon.
Or, the Sum of the Angles is more fpeedily found thus:
Multiply!8o by a Number lefsby two than theNumber of Sides
of the Polygon; the Produ&t is the Quantity of the Angles
required: thus i8o being multiplied by 3, a Number lefs by
z, than that of its Sides; the ProduCt is 540, the Quantity
of Angles as before.
The following Table exhibits the Sums of the Angles in
all rectilinear Figures, from a Triangle to a Dodecagon; and
is of good ufe both for the describing of regular Figures, and
for proving whether or no the Quantity of Angles have been
truly taken with an Inftrument. See R E G U L A, F I-
0 U R E, &C.


Numb. jSum.
Sides. Ang.
111  I 1800
IV    360
V    540
VI   720
VIl  0oo


.Ang. of Numb.
Reg. Fig  Sides
600       Vill
90         IX
108         X
120        Xi
I28 1      XII


Numb
Angl.
xo800
1 260
1440
i620
1800


Aung. of
Reg. Fig
i35
140
144
147 -X3-
IS0


To infcribe is regular Polygon in a Circle.-Divide 3 6o by
the Number of Sides in the Polygon required, to find the
Quantity of the Angle E F D. Set off the Angle at the
Centre, and apply the Chord thereof E D, to the Periphery,
as often as 'twill go.-Thus will the Polygon be inicribed
in the Circle.
The Refolution of this Problem, tho' it be Mechanical;
yet is not to be defpis'd, becaufe both eafy and univerfal.-
Euclid, indeed, gives us the Confiruftion of the Pentagon,
Decagon, and Quindecagon; and other Authors give us
thofe of the Heptagon, Enneagon, and Hendecagon; but
they are far from Geometrical Stri~tnefs.
Renaldinus lays down a Catholic Rule for the describing
of all Polygons, which many other Geometricians have bor-
row'd from him ; but Wagnerus and Wolfius hive both demon-
Itrated the Falfity thereof.
On a Regular Polygon to circumfcribe a Circle: or to circum-
fcribe a regular Potygonupon a aircle.-Bifel two of the Angles
of the given Polygon A and E, by the right lines A F and
: F, concurring in F. And from the Point of Concourfe
wvith the Radius  F defcribe a Circle.
To circumfcribe a Polygon, &c. Divide 360 by the Number
of Sides required, to find c d- which fet off from  the
Centre F. and draw the Line e2; on this ConfiruCt the Pa-
Jygon as in the following Problem:
On a given Line, ED, to defribe any given regular Poly-
won  Find an Angle of the Polygon in the Table; and in E
3An p gA  sua1 thercto, 4xawing E   F _   D. ThWo


the three Points AE D defcribe a Circle. Sed C RCLU. E.
this apply the given right Line as often as it will go.-Thu
will the required Figure be defcribed.
To infcribe or circumfckibe a Regular Polygon, Trigono.
metrically.-Find the Sine of the Arch produced by divi-
ding the Semi-Periphery i8o by the Number of Sides of
the Polygon: the double of this is the Chord of the double
Arch, and therefore the Side A E to be infcribed in the
Circle.-If then the Radius of a Circle wherein, e. gr. a
Pentagon is to be inscribed, be given in any certain Mea;
fure e. gr. 345. the Side of the Pentagon is found in the
fame Meafure by the Rule of Three, Thus as Radius zoDoo
isto 1I76:: fo is 345, to4057.  The Side ofthe Pentagon.
-With the given Radius therefore defcribe a Circle; and
therein fet off the Side of the Polygon as often as 'twill go;
thus will a Polygon be inscribed in the Circle.      X
To fave the trouble of finding the Ratio of the Side of
the Polygon to radius, by the Canon of Sines; we Ihall add
a Table expreffing the Sides of Polygons in fuch Parts
whereof Radius contains aoooooooo. In practice, as many
Figures are cut off from  the Right-Hand, as the Circumll.
Qarices of the Cafe render needlefs.


Numb.
Sides
Ill
IV
V
VI
VJi


Quantity
Side
I7320508
14T4Z1 35
11755705
10000000
' 8677674


Numb.
Sides
.~ _
Vill
Xl
XII


Q antity
Side
- _
7653668
68404OZ
6180339
563465i I
5176380_


To de/cribe a Regular Polygon, on a given right Line, and to
circumfcribe a Circle about a given Polygon, Trigonometricraly._
Taking the Ratio of the Side to the Radius 'out of the Ta-
ble; find the Radius in the fame Meafure wherein the Side
is given. For the Side and Radius being had, a Polygon may
be defcribed by the laft Problem. And if with the Interval
of the Radius, Arches be lTruck from the two Extremes of
the given Line the Point of Interfeftion will be the Centre
of the circumfcribing Circle.
P 0 I Y G 0 N, in Fortification, is the Figure or Perimeter
of a Fortrefs or fortified Place. See FORTIFICATION
Exterior-P o LY G o N is a right Line drawn from  the
Vertex or Point of a Baffion, to the Vertex or Point of the
next adjacent Baftion. See BASTION.
Such is the Line C F, Fab. Fortification, 1g. K
Interior-P 0 L Y G 0 N is a right Line drawn firom the Cen-
ter of one Baftion to the Centre of another, fuch is the
Line G H.
Line of PO L Y G O N S, is a Line on the French Seaors,
containing the homologous Sides of the firft gregularPolygons
infcribed in the faame Circle, i. e. from an Equilateral Tri-
angle to a Dodecagon. Soe SECTOR.
P 0 L Y G 0 N A L Numbers, in Algebra, are the Sums of
Arithmetical Progreilions, beginning from Unity. See SE-
RIES, NUMBER, PROGRESSION, RC.
Polygonal Numbers are divided, with refpeft to the Nuai-
ber of their Terms, into E iangular, which are thofe whofe
difference of Terms is l; quzadrangular orfquare, where 'tis z;
Pentagonal, where 3 i Hexagonal, where 4; Heptagonal,
where 5; Ollagonal, where 6, &c.
They have their Names from the Geometrical Figures in-
to which Points corresponding to their Units, may be dif-
pofed e, gr. three Points correfponding to the three Unint
of a triangular Nnmber may be difpofed into a Triangle a
and fo of the reft. See TRIANGULAR, &C.
The Genefis of the feveral kinds of Polygonal Numbers
from the feveral Arithmetical Progrefflions, may be conceived
from the following Examples.


Arithmetical Progreflion
Triangular Numbers
Arithmetical Progrefflon
Square Numbers
Arithmetical Progreffilon
Pentagonal Numbers
Arithmetical Progrefljon
Hexagonal Numbers


x, 2,  3  42 5, 6     7, 8
I, 3, 6, lo, i5s 21Z 28X 36
1 3, 5) 7w 9X IT- 13i 15
1, 4, 9, i6; 25, 36, 49, 64
Ix 4, 7, Ix0 13, 16, x9, 21
I) 5, Id 2Z2s 35, 51, 70, 93
I, 5X 9, 13, 17, 21, 25, 29
I, 6, 15, z8 45) 66, 9.1 I20'


The Side of a Polygonal Number is the Number of Terms of
the Arithmetical Progrefflon that are fumm'd up to co;li-
tute it: And the Number of Angles is that which fhews how
many Angles that Figure has whence the Polygonal Nuaber
takes its Name.
The Number of Angles, therefore, in Triangular Numbers
is 3. In Tetragonal 4. In Pentagonal A, &c. confequetly
the Number of Angles exceeds the difference of T¢OiS
fumm'd up, by two Units.
To find a Polygonal Number, the Side and Nuxber of i:
A4u4is being given. The Canon is this.lThe fPo0t4