POL
(850)
POL
and 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 Grcle is bigger than it , and every Polygon infcribed, left than the Circle.— 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, whole Bafe is the Circumference of the Circle, and its Height the Radius ; fince this Triangle is lefs 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, &c.
To find the Ann 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 Product by a Perpendicular let fall from the Centre of the circum- fcribing Circle to the Side A B •, the Produft is the Area re- quired. See Area.
Thus, fuppofe AB, 54; and half the 'Number of Sides 2j; the Produft or Semiperimeter is 135. Suppofing then the Perpendicular F g, 29 , the Produft of thefe two, 3915, is the Area of the Pentagon required.
To find the Area of an irregular Polygon, or Trapezium. — _ Refolve it into Triangles ', find the feveral Areas, of the feve- ral Triangles, fee Triangle-, the Sum of thefe 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 180 : From the Produft fubtraft 360 ; the Remainder is the Sum required.
Thus in a Pentagon, 180 being multiplied by 5 gives 900; whence fubtrafting 360 there remains 540; the Sum of the Angles of a Pentagon.
Hence, if the Sum found be divided by the Number of Sides i the Quotient will be the Angle of a regular Poly- gon.
Or, the Sum of the Angles is more fpeedily found thus: Multiply 1 80 by a Number lefs by two than theNumber of Sides of the Polygon; the Produft is the Quantity of the Angles required: thus 180 being multiplied by 3, a Number lefs by 2, than that of its Sides-, the Produft 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 defcribing 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, Fi- gure, &c.
the three Points AED defame a Circle. See Circle, fa
this apply the given right line as often as it will go Thus
will the required Figure be defcribed.
To inferibe or circumfcribe a Regular Polygon, Trigono- metrically. Find the Sine of the Arch produced by divi- ding the Semi-Periphery 180 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, c. gr. a
Pentagon is to be infcribed, 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 10300 is to 1176:: fo is 3450, to 4057. The Side of the Pentagon. — With the given Radius therefore defcribe a Circle ; and therein fet off the Side of the Polygon as often as 'twif I go : thus will a Polygon be infcribed in the Circle.
To fave the trouble of finding the Ratio of the Side of the Polygon to radius, by the Canon of Sines ; we (hall add a Table exprefling the Sides of Polygons in fuch Parts whereof Radius contains 100000000. In praftice, as many Figures are cut off from the Right-Hand, as the Circum- llances of the Cafe render needlefs.
Numb. Sides.
Sum.
Ang.
180
360
540
720
900
Ang. of Reg. Fig
6o u 90 108 120 128 f
Numb. Sides
VIII
IX
X
XI
XII
Numb Angl.
1080° 1260 1440 1620
1S00
Ang. of Reg. Fig
I3S
140 144
H7 IV
150
Ill IV V VI VII
To inferibe a regular Polygon in a Circle. Divide 360 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 infcribed
in the Circle.
The Refolntion of this Problem, tho' it be Mechanical ;
yet is not to be defpis'd, becaufe both eafy and univerfal
Euclid, indeed, gives us the Conftruftion of the Pentagon, Decagon, and Quindecagon ; and other Authors give us thole of the Heptagon, Enneagon, and Hendecagon ; but they are far from Geometrical Striftnefs.
Renaldims lays down a Catholic Rule for the defcribing of all Polygons, which many other Geometricians have bor- row'd from him ■, but Wagnerus and Wolfius have both demon- ftrated the Falfity thereof.
On a Regular Polygon to circumfcribe a Circle: or to circum- fcribe a regular Polygonupon a Circle. — BifTeft two of the Angles of the given Polygon A and E, by the right lines A F and E F, concurring in F. And from the Point of Concoutfe with the Radius EF defcribe a Circle.
To circumfcribe a Polygon, &c. Divide 360 by the Number of Sides required, to find efd\ which fet off from the .Centre F, and draw the Line erf; on this Conftruft the Po- lygon as in the following Problem :
On a given Line, ED, to defcribe any given regtdar Poly
/on.- Find an Angle of the Polygon in the Table •, and in E
l?t off an Angle equal thereto, drawing E A = E D. Thro'
Numb.
Ouantity
Numb.
Quantity
Sides
Side
Sides
Side
111
17320508
VIII
7653668"
IV
14142135
IX
6840402
V
H755705
X
6180339
VI
1 0000000
XI
5634651
VII
8677674
XII
5176380
_ To defcribe a Regular Polygon, on a given right Line, and to circumfcribe a Circle about a given Polygon, Trigonometric-ally.— 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 ftruck from the two Extremes of the given Line the Point of interleftion will be the Centre of the circumfcribing Circle.
Polygon, in Fortification, is the Figure or Perimeter of a Fortrefs 01 fortified Place. See Fortification.
Exterior-P olygon is a right Line drawn from the Vertex or Point of a B.iftion, to the Vertex or Point of the next adjacent Baftion. See Bastion.
Such is the Line C F, Fab. Fortification, Fig. I.
Interior-? olygon is a right Line drawn from the Cen- ter of one Baftion to the Centre of another, fuch is the Line G H.
Line (/Polygons, is a Line on the French Seftors, containing the homologous Sides of the firft 9Fegulari'oAj;o»j, infcribed in the fame Circle, i.e. from an Equilateral Tri- angle to a Dodecagon. Sse Sector.
Polygonal Numbers, in Algebra, are the Sums of Arithmetical Progreilions, beginning from Unity. See Se- ries, Number, Progression, r>c.
Polygonal Numbers are divided, with refpeft to the Num- ber of their Terms, into T, Lingular, which are thofe whofe difference of Terms is 1 ; quadrangular or/>aar.",where'tis2 - , Pentagonal, where 3 ; Hexagonal, where 4 ; Hcptagonal, where 5 ; Octagonal, where 6, &c.
They have their Names from the Geometrical Figures in- to which Points correfponding to their Units, may be dif- pofed e, gr. three Points correfponding to the three Units of a triangular Number may be difpofed into a Triangle ; and fo of the reft. See Triangular, &c.
The Genefis of the feveral kinds of Polygonal Numbers from the feveral Arithmetical Progreffions, may be conceived from the following Examples.
Arithmetical Progreflion Triangular Numbers Arithmetical Progreflion Square Numbers Arithmetical Progreflion Pentagonal Numbers Arithmetical Progreflion ' Numbers
- i 2> 3, 4. 5> 6, 7, 8
1, 3, 6, 10, 15, 21, 28, 36 '■> 3, 5. 7, 9> 11, I3j 'S 1, 4, 9, 16, 25, 36, 49, 64 I) 4. 7> *°i t3) 16, 19, 21 1, 5, 12, 22, 35, ?i, 70,92. »i 5) 9> I3> '77 2i> 25, 29 1, 6, 15, 28, 45, 66, 9.1, 120
The Side of a Polygonal Number is the Number of Terms of the Arithmetical Progreflion that are fumm'd up to confti- tute it : And the Number of Angles is that which {hews how many Angles that Figure has whence the Polygonal Number takes its Name.
The Number of Angles, therefore, in Triangular Numbers is 3. la Tetragonal 4. In Pentagonal 5, &c. confequently the Number of Angles exceeds the difference of Terms fumm'd up, by two Units.
To find a Polygonal Number, the Side and Number of its
Angles being, given. The Canon is this, The Polygonal
' Number
