43<3 
GEOMETRY. 
the circle; and the Tides AB, ah, of the infcribed and 
circiunfcribing polygons will be parallel. 
185. Cor. 3. Tiie chords or tangents qf like arcs in 
difterent circles, are in the Tame proportion as the radii 
of thoTe circles. For if a circle cir-cumfcnbe the poly¬ 
gon abcdef-, then the Tides of the polygon ahcdef, 
ABCDEl', are chords of like arcs in their refpedlive 
circumTcfibin^>; circles. And if a circle be infcribed in 
th.e polygon ABCDEF, the fides AB, ab, &:c. are tan¬ 
gents ot like arcs alfo : and thefe have been fliewn to be 
proportional to their radii SA, Sa. 
1S6. Cor. 4. The perimeters of like polygons (or 
the lum ot their tides) are to one another as the radii of 
their infcribed or circiiinfcribed circles ; for SA : Sa :: 
AB : ab (182); and AB, ab, are like parts of the peri¬ 
meters of their polygons; therefore SA : Sa peri¬ 
meter ABCDEF ; peiimeter aicifry (151). 
t 87. If there be two regular and like polygons applied to the 
fame circle, the one infcribed and 
the other circumfcribcd: then will 
the cirjcwnferenfe of that circle, 
and half the fum of the perimeters 
~qf thofe polygons, approach nearer 
to equality, as the number of fides 
in the polygons increafe. —It is 
evident at tight, that tliccircum- 
_feribing hexagon FGHIKL 
B X IE is lefs than the circumTcribing 
Tquare ABED; and alfo that the infcribed hexagon 
fgfiikl is greater than the infcribed Tquare abed. And 
in bot.h cafes, the dilference between the hexagon and 
tile circie is lefs than the difference between the circle 
and the fqiiare; therefore the polygon, wltether in- 
feribed or circiiinfcribed, differs lefs from the circle, as 
the number of its tides is inareafed. And when the 
number of tides in both is very great, the perimeters of 
the polygons will nearly coincide with the circumference 
of the circle ; for then the difference of the polygonal 
perimeters becomes fo very fmall, that tliey may be 
efteemed as equal. And yet fo long as there is any dif¬ 
ference between thefe polygons, though ever fo fmall, 
the circle is greater than the infcribed, and lefs than the 
circumfcribcd, polygons : therefore half their funis may 
be taken for the circumference of the circle, when the 
'number of thofe Tides is ver^ great. Hence, the cir¬ 
cumferences of circles are in proportion to one another, 
as the radii of thofe circles, or as their diameters. For 
' the perimeters of the infcribed and clrcumfcribing po¬ 
lygons are to one another, as the radii of the circles 
(i86). And thefe peiimeters and circumferences con¬ 
tinually approach to equality. 
189, In a circle A F B, if lines, BA, DA, F A, be drawn 
from the extremities of two equal arcs, 
B D, D F, to meet in that point A 
of the circumference determined by one 
of them, BA, pajjing through the 
centre-, then fall the middle Line AD, 
be a mean proportional between the 
the fum A B -E A F' of the extreme 
lines, and the radius B C c/' that 
circle.' —On D, with the dilhince 
Da, cut AF produced in E. 
Then drawing DE, DF, DB, 
the triangles A D B, E D F, are 
B C A. congruous (102). For ^ E F D 
= (ZFDA-IZFAD (96)=) ZDBA (128). Becaufe 
the arc DFA=DF^-FA. AndZE = Z''AD (104) 
— ^ D A B, by conftrudtion ; andDEziiDA. T. here- 
fore E F A B ; and AFi—AB-)-AF. Draw CD; 
then the triangles A C D, A D E, are Tnnilar. For they 
are ifofceles and equiangular. Therefore A C : A D :: 
AD; (AE=:) AB-f A F. 
190. HencCj whence the radius of the circle is ex- 
pjelled by i, and one of the extreme lines, or chords, 
paffes through the centre; then if the number 2 be 
added to the other extreme chord, the fquare root of 
that lum will’be equal to the length of the mean chord. 
For fince AC: A D ::‘A D : A B + A F (189). There- 
fore AD =;AC XnBXAF (162). Now if AC—i, rlien 
AB =32 ; and AD = 2 4-AF ; becaufe multiplying by 1 
is ufelefs liere. Tliercfore AD 2= \/ 2-i-Ai' . 
As the arcs BD and DFA make a femicircle, tliey 
are called tlie fupplements of one another ; tlierefore if 
tlie arc BD is any part (as, i, Jrr, &c.) of tlie femi- 
circumterence ; theii,is the line D A called the fupple- 
mental chord of that part. 
191. Remark. In the pofthtimous works of the marquis 
de I’Hopital, (page 319, Engliffi edition,) this principle 
is applied to the doftiine of angular febtions; that is, 
to_ the dividing of a given arc into any propofed number 
ot equal parts; or the finding of the chord of any pro- 
poled arc. For if B F was any afl'umed arc, the chord 
ot which had a known ratio to the given radius BC; 
then as BFA is a rigiit-angled triangle (130), tiie fide 
A F— 3/ AB^— B !• " ( 113) will alfo be known. And by 
this theorem the mean chord AD wdll be known; and 
alfoDB (2=3/rt.B^—AD^) the chord of half the arc BF' 
will alfo be known. 
And by bifefting the arc D B in G, and drawing 
AG, G B, the mean chord A G is known (189) ; and 
GB (=:-j/AB*—AG*) is alfo given. And in this 
manner, by a continual bifeCtion, the chord of a very 
fmall arc may be obtained; the pratlice of which is fa¬ 
cilitated by our article 190, deduced from page 330 
of the faid work. 
192. Required the chord of the -^^2 f circumfe¬ 
rence of a circle, the radius of which 
is I ; or, required the fide of a re. 
gular polygon of 3072 fides, in- 
feribed in a circle, the diameter of 
which is 2.—Let ADF be a te- 
micircle, the diameter AFzza, f— 
and centre C. 'Fake the arc 
AD2=^ of tlie feniicircumference, or equal to 60° ; and 
draw DC, DA, DF. Let d reprefent the point where 
the arc is bifedted; d¥ the fupplemental chord to that 
bifedtion: and let the marks d, d", d‘", d", &c. ex- 
prefs tlie bifedted points agreeable to the number of 
bifedlions. 
193. Now fince ^ACD=26o°, 
Therefore ^CAD-fzADC=2(iSoO—6oO=:) 120° (98). 
But ^CADr^Z ADC ( 104); then ^CAD2=(^22:)6oo. 
Therefore DA=(DC=AC=) i. And as the triangle, 
ADF is right-angled at D (130), then DFc:; 
(v/AT —AD [113)2=3/4 —i=)v'3=D732050S075688773 
Therefore the fupplemental chord 
of the arc AD, or of 
— Vs _=1,7320508075688773 
2=3/2 + FD =1,9318516525781366 
—3/2-i-F(^' 2=1,9828897227476208 
Fff" 2=3/2-EFiT'' 2=1,9957178464772070 
Fif’'' =3/2-EF<i"' =i,998929i7495273r3 
Td'' 
Td''^ 
I'' d \/ 2 . i'd'^ 
2 = 3 / 2 -fF 4 '' 
of the femicircum* FD 
3 ference is 
^ of the fame [190) is ¥d' 
-i- Fd" 
4 ¥ 
Cl 
95 
3 54 
T6? 
--■[/2-j-¥d''' =21,9997322758191236 
zfo.f-Vd'' =1,9999330678348022 
= 1,9999832668887013 
:3/’T+Ft7™”=i,9999958167178004 
‘=1,9999989541791767 
Now 
X 
