140 On Cog or Toothed Wheels. 
of De la Hire. (about a century ago,) have 
uniformly taught that the trae form of the 
teeth of wheels depends upon the curve called 
an epicycloid, and that. of teeth destined to 
work in a straight rack depends upon the 
simple cycloid. The cycloid is a.curve which 
may be formed by the trace of a nail in the 
circumference of a cart wheel, during, the 
period of one revolution of the wheel, or, from 
the nail’s leaving the ‘ground to its return; 
and the epicycloid is a curve that may be 
formed by the trace of a nail, in the cireum- 
ference of a wheel, which wheel rolls ( with- 
out sliding) along the circumference of ano- 
ther wheel. 
Let AB (fig. 1.) be part of the weionenaors 
rence of a wheel ABF'to which it is designed 
to adapt teeth, so formed as to produce equa- 
ble motion in the wheel C, when that of the 
wheel ABF is also equable. Also, let the 
teeth so formed; act upon the indefinitely 
small pirs, 7, 7, t, let into the plane of the 
wheel C, near its circumference. To give 
the teeth of the wheel ABF a proper form 
(according to the present prevailing, system), 
a style or pencil may be fixed in the circum- 
ference. of a circle D equal to the wheel C, 
and a paper may be placed behind both circles, 
on which by the rolling of the eirele D on 
