Investigation of the Epicyeloid. 283 
The lengths of different eprcycloidal lines, formed by the same gen- 
erating circle, are directly proportional to the distance through which , 
the center of the circle passes in describing the same. 
10. If the radius of the fixed circle be extended infinitely, the 
above properties and relations still obtain, and since, when thus ex- 
tended, the circumference becomes in effect a right line, and as the 
epicycloid becomes what is commonly termed a cycloid, it follows, 
that every proportion or fact deduced in the above investigation, will 
apply to all curves or cycloids formed by the revolution of a circle 
upon a plane. 
11. If the diameter AF of the internal generating circle, (fig. 2,) 
be made equal to AD, the radius of the fixed circle, the point F will 
coincide with the point D, and the tangent HF will, in every position 
of the point B, pass through D the center of the fixed circle; conse- 
quently, the epicycloid will be a right line, equivalent to XY, (fig. 3,) 
the diameter of the fied circle, or to 2AD, twice the diameter of the 
generating circle. 
12. The diameter of the internal generating cirele being supposed 
equal to the radius of the fixed circle, as in the last case, its center 
in describing the epicycloid line XY, passes through the are WCZ, 
equal to one half of XAY, or equal to one half of the circumfer- 
face of the same circle, hence by Art. 9,—The length of an epreye- 
loid 1s to the distance through which the center of its generating 
avele passes, us twice the diameter of a circle to one half of tts cw- 
cumference, or as the diameter to one fourth of the circumference, or 
% 1 to 0.78539816339744830961575. 
13. If the radius of the fixed circle be supposed extended infi- 
lutely, as in Art. 10, the center of the generating circle will, in one 
"evolution of the same circle, pass through a distance equal to its 
“teumference,. and consequently, by the last Art.,—The length of 
. emeycloid, or more commonly the cycloid, will be equal to four 
fies the diameter of the generating circle. 
14. As the chord DB of the generating circle, as represented 
0 Fig. 3, is equal to the portion of the epicycloid, intercepted between 
tS middle point D, and its generating point B, and as the center of 
© generating circle, in describing the said epicycloid, passes over a 
distance WCZ, equal to one half of its circumference, and likewise, 
’8 the length of an epicycloid is directly proportional (9.) to the dis- 
nce described by the center of its generating circle, and moreover 
*S the said chord DB, maintains in all corresponding positions of 
