146 On Cog or Toothed Wheels. 
has arrived at it, this wheel will virtually be 
divided into an infinite number of teeth, or 
at least into a number greater than that of the 
particles of matter, contained in a circular 
line of the wheel's circumference. For sup- 
pose the surface of a similar, but longer cy- 
linder, stripped from it and stretched on the 
plane ABCE (fig. 4,) where the former ob- 
lique line will become the hypothenuse BC, 
of the right angled triangle CAB, and will 
represent. all the teeth of the given wheel, ac- 
cording to the sketch EG at the bottom of 
the diagram. Here the lines AB and CE, are 
equal to the circumference of the base of the 
cylinder, and AC and BE to its lengths, and 
if between A and B, there exist a number, 
m, of particles of matter, and between A and 
C a number, n, the whole surface ABCE 
will contain mn particles, or the product of 
m and n; and the line BC, will contain a 
number = /m*+n?, froma well known theo- 
rem ; whence it appears that the line BC is 
necessarily longer than AB, and hence con- 
tains more particles of matter.* 
* It need hardly be observed, that whatever is true of 
the whole triangle CAB (fig 4,) is true of every similar 
part of it, be it ever so small: and in fact, when the hy- 
pothenuse BC, is folded again round the cylinder, from 
which we have supposed it stripped, the acting part will 
