Investigation of the Epicycloid. 283 



The lengths of different epicycloidal 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 wath 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 fixed circle, or to 2 AD, tioice the diameter of the 

 generating circle. 



12. The diameter of the internal generating circle 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 arc WCZ, 

 equal to one half of XAY, or equal to one half of the circumfer- 

 ence of the same circle, hence by Art. 9, — The length of an epicyc- 

 loid is to the distance through which the center of its generating 

 circle passes, as tioice the diameter of a circle to one half of its cir- 

 cumference, or as the diameter to one fourth of the circumference, or 

 as I to 0.78539816339744830961575. 



13. If the radius of the fixed circle be supposed extended infi- 

 nitely, as in Art. 10, the center of the generating circle will, in one 

 revolution of the same circle, pass through a distance equal to its 

 circumference, and consequently, by the last Art., — The length of 

 the epicycloid, or more commonly the cycloid, will be equal to four 

 times the diameter of the generating circle. 



14. As the chord DB of the generating circle, as represented 

 in Fig. 3, is equal to the portion of the epicycloid, intercepted between 

 its middle point D, and its generating point B, and as the center of 

 the generating circle, in describing the said epicycloid, passes over a 

 distance WCZ, equal to one half of its circumference, and likewise, 

 as the length of an epicycloid is directly proportional (9.) to the dis- 

 tance described by the center of its generating circle, and moreover 

 as the said chord DB, maintains in all corresponding positions of 



