186 
em a ing circle at P and: FE, fe, the 
= diameters which passthrough ts in which 
it touches the arc DA ; join p> will be tan- 
~ gents to the curve at P and 2. Cor. 2.); and 
Gre the lines PE, pe, which will be normals 
curve; draw Vr to Hy ; also f's pe 
dicular to FP, and e x ‘PE, and 
Pate 
ccLui. 
Fig. 21. 
" 
4 
= 
3 
5 
: 
eT 
Ae 
= 
Zz 
K, we have, by reject- 
are infinitely small in of 
ax. K fk: % Rei f-PraH v—HV=r», 
and PK4K ; and since 
pi—PF=(p tke cP 5 J crakeneres RE 
s eee ers p=rv+sF 
the are V v as coincidin, vith its tangent) the 
sngle rV vis equal tothe angle DH, or to EF P, that 
is to F fs; therefore V v: Ff::rv:Fs; but Vr=Ee, 
and Ev or Ve: Ff::CE (=e): CF (=c4-2.); there- 
fore c:c42a::rv: Fs, and by composition ¢ : 2Qe+ 
Qa::re: re+Fs, that is, c: 2c-4+2a::rv: arc Pp. 
Now rv is the increments of the chord HV, and P p is 
os increment of the are HP ; and it ap- 
pears that the HV and arc HP, which begin 
together, are augmented by increments which have to 
the cndeeak Yeiho of c to 2c+42a; there- 
fore the chord HV, and arc HP themselves, will have 
to each other the same ratio. 
Cor. The are HPA, half the epicycloid, is a fourth 
seperti es , C, 2c-4-2a, and 2a. 
~ Senorium 1. If the epicycloid be interior, then the 
chord of the g circle determined as in the 
will be to the bp roped oy Ash Sager 
ence of the diameters of the fixed and generating cir- 
cles to the radius of the fixed circle. The 
demonstrated in this case exactly 
2. If we suppose the radius of the fixed circle to be 
infinite, its circumference is to be reckoned a strai 4 
line, and the ratio of c to 2 c42 ais that of c to 
or-of 1 to2. The curve is then the common eyelaia 
and the proposition agrees with what has been shewn 
that curve. 
3. It a that any it wee whe 
= may be rectified {chat in; straight line rha auig' be 
found equal to it), when the curve is described by a 
=< the circumference of the generating circle. 
pent meee point is with or without the 
circle, the rectification of the curve is redu- 
cible to of the ellipse ; and therefore cannot be 
efected but by approximation, See Fiuxions, 
Prop. IV. 
The same thi being supposed as in last proposi- 
tion, let PE, sneer P, meet the ci 
ition is 
Do, and draw H iV, : 
ferent ii to EP. ft incor sciar Bd aes.” and 
4 @e indefinitely small, they may be as 
x 
tng ale the bases eral F 
EPIC YCLOID. 
coinciding with en may be t 
Ken a, recline ti rds be equal, be 
¢ En is equal to E to DUY =EFP ; the 
fn LS not ch eis ate 
position or en 
aft epcycarhar to Sef thee 
position, rv: inet p SB es pl ad 
witely little triangle br v, and 
the same altitude ; ta bie Detendete 
therefore the triangle Drv is to the 
as r#, the base of the former, to n e+ maneer’ 
the parallel sides of the latter ; that is asc to S$c-42a. 
But the triangle and trapezoi d are the increments b, 
which the circular space DHVD, and the « ; 
DHPED, are augmented, in 1 
ce of the 
epics DUPED. ot ee eee e; there 
fore, these spaces are continually fasrensett oy : 
ties which have to each other the cobstant falao nt'e.tp 
Sc4+2a; and cdnsequently the Spaces Saeeneer ia mas 
the same ratio. 
Cor. The whole epicycloidal space DHPAD is to 
half the area of the generating circle as $¢-4+2 toc. 
Scuotium. When the radius of the circular base is 
infinitely great, the epicycloid becomes the common 
cycloid ; Y ind the tatio of $042 ¢ to ¢ becomes the ra- 
tio of $c toc, or $ to 1, as was demonstrated in ; 
8. of the Cyctorp. oi sale 
~ Prop. Vv. ide bows 
Ifa thread be fastened ‘at A, one 
epicyc’oid, and applied thre AH 
ares ina he oy 
ed into a strai line, tee extremity sng ne 
another epeyeoid HXZ, smile tthe epicycloid 
i ees accngh the pata EL Pte circle, when 
through the ints Hand P respectively. i 
Ae centre of the cle describe ahs 
and PV ; join HV, and PE ; and make EF to FY 
CE to CF, that is as ¢ to ¢-4-2a, and j in XY 
cause PF is equal to VH, and the are wiraight 
line PX, PFs XP: BK, 2c4-2a (Prop. 2) aad hy gin 
version, PF: FX :: ¢: c+ 2a, that is, 
as EF to FY; henes ‘the triangles PFE, OK PY are simi- 
lar ; the angle FXY is pea fh a pa and 
a circle described on FY as a 
X, and touch the are HR in F. again case 
angles XYF, PEE Te cone the arc XF is similar tothe 
arc PF, and arc XF :arc PF: : chord XF: chord PE. But 
XF: PF :: YF: FE (or by construction,) : ; CF : ie 
: are FH: arc ED; therefore arc XF: are PE: 
are FH : are ED: But the arc PF is equal to the 
ED, because, by the generation of the curve, are 
arc EP, andere AE are EPF ; therefore the : 
XF is equal to the are FH: Hence it follows 
i Ee OL Hick the sto 
XY the 
because, be Cantertition, EF BY Nay CF ; the di 
meters of the HM pope | circles, have 
tio as the diameters’ of. the f circles ; 
the same ra 
tee the epicycloids will be si 
Con. The radius of curvature at ay Wa 3 et 
epicycloid is to the chord of the arc of the 
por between that ae pie bee 
constant ratio of the sum 
Wy bigeye yey OF the tation, 
former, and the diamnster of the later. For a 
aaa i cCLitt. ; 
