3.1416 = ratio of circumference to diameter.
8 = diameter.
Number of teeth = 12)25.1328 = circumference.
2.0944 = arc pitch of wheel.

If now we subtract the chord pitch from the arc pitch, we shall obtain the difference between the arc and the chord pitches of the wheel; here

2.0944 = arc pitch.
2.0706 = chord pitch.
.0238 = difference between the arc and the chord pitch.

Now suppose this cutter to be used upon a wheel having the same pitch, but containing 18 teeth; then we have

2.0944 = arc pitch.
2.0836 = chord pitch.
.0108 = difference between the arc and the chord pitch.

Then

.0238 = differenceon wheelwith12teeth.
.0108 = „„„18„
.0130 = variation between the differences.

And the thickness of the tooth equalling the width of the space, it becomes obvious that the thickness of the cutter at the pitch line being correct for the 12 teeth, is one half of .013 of an inch too thin for the 18 teeth, making the spaces too narrow and the teeth too thick by that amount.

Now let us suppose that a cutter is made correct for a wheel having 96 teeth of 2.0944 arc pitch, and that it be used upon a wheel having 144 teeth. The proportion of the wheels one to the other remains as before (for 96 bears the proportion to 144 as 12 does to 18).

Then we have for the 96 teeth