On Cog or Toothed Wheels. 143 
no more than the small portion de, of the 
curve developed, whereas a second equal step 
of the generating circle cb, will extend the 
curve forward frome to f, a greater distance 
than the former; while a third equal step ab, 
will extend the curve from f to g, a distance 
greater than the last; and the successive in- 
crements of the curve will be still greater, as 
it approaches its summit ; yet all these parts 
correspond to equal advances of the wheel, 
namely, to the equal parts ab, be and cd of 
the base, and to equal ones of rotation of the 
generating circle. Surely then the parts sg, 
gf, of the epicycloidal tooth will be worn out 
sooner than those fe, ed, which are rubbed 
with so much less velocity than the other, 
even though the pressure were the same. But 
the pressure is not the same. For, the line 
ag is the direction in which the pressure of 
the curve acts at the point g, and the line pg 
is the length of the lever-arm on which that 
pressure acts, to turn the generating circle on 
its axis (now supposed to be fixt;) but, as 
the turning force or rotatory efforts of the 
wheels, is by hypothesis uniform, the pres- 
sure at g must be inversely as pq; that is, 
inversely as the cosine of half the angle of 
rotation of the generating circle; hence it 
would be infinite at s, the summit of the 
