282 Investigation of the Epicycloid. 



6. As Bl) is perpendicular to BG, if the radius DK be drawn 

 perpendicular to IB or.GH, the angle BDK will be equal to the an- 

 gle BGH ; and farther, as BD and BG are equal by construction, 

 and likewise as BI or GH equals AD or DK by construction, the 

 two triangles BDK and BGH are identical, and BK equals BH. 



But since BH represents the motion of the point B in the epi- 

 cycloid, its equal BK may be taken as the measure of the same mo- 

 tion, and as the center C of the generating circle has but one motion, 

 viz. the uniform one about the center D, represented by CD, it fol- 

 lows, that the motion of the center C is to that of the point B as 

 CD to BK. Moreover, as DK and CB are both perpendicular to 

 the same hne IB, they are parallel to each other, and as the points 



A, C and D, are in the same right hne, the angles ACB and ADK 

 are equal ; hence the two isosceles triangles ADK and ACB are 

 similar and have the vertex A common, and CD : BK: .'AC : AB. 

 But, as has been shown, CD is to BK as the motion of the center C 

 of the generating circle to that of the point B in the epicycloid ; con- 

 sequently, AC is to AB as the same two motions ; or, the motion of 

 the center C of the generating circle is to that of any point B in its 

 circumference, as the radius of the same circle to the chord drawn from 

 the said point B to the point of contact of the two ciixles at A. 



7. The triangles GBH and DBK being identical, and BG in the 

 one being perpendicular to the corresponding side BD in the other, 

 the side BH is Hkewise perpendicular to the corresponding side BK; 

 and as ABF is a semicircle, BH produced will cut the extremity F 

 of the diameter AF, and as BH is a tangent to the epicycloid at the 

 point B, it follows, that the chord BF in the generating circle is par- 

 allel to the tangent drawn to the 'point B in the epicycloid. 



8. When by the revolution of the generating circle the point B ar- 

 rives at F, the chord AB, which represents the velocity or motion of 



B, will become equal to the diameter AF or double the radius AC. 

 Hence, the motion of the point F is double that of the center C. 



In like manner, when the point B arrives at A, the chord AB will 

 vanish, and consequently, the point A is at rest. 



9. Since the chords of equal arcs have the same ratio to the radii 

 of their respective circles, it follows, (6.) that the relative velocities 

 or motions of the center and generating point of the same circle in 

 describing corresponding portions of the epicycloidal line remains 

 always the same, and consequently, 



