Answers 299- A WHEEL FALLACY
The inner circle has half the diameter of the whole wheel, and therefore
has half the circumference. If it merely ran along the imaginary line CD it
would require two revolutions: after the first, the point D would be at E. Butthe point B would be at F, instead of at G, which is absurd. The fact is the
inner circle makes only one revolution, but in passing from one position to
the other it progresses partly by its own revolution and partly by carriage on
the wheel. The point A gets to B entirely by its own revolution, but if you
imagine a point at the very center of the wheel (a point has no dimensions
and therefore no circumference), it goes the same distance entirely by what
I have called carriage. The curve described by the passage of the point A to
B is a common cycloid, but the point C in going to D describes a trochoid.
We have seen that if a bicycle wheel makes one complete revolution,
so that the point A touches the ground again at B, the distance AB is the ex-
act length of the circumference, though we cannot, if we are given the length
of the diameter, state it in exact figures. Now that point A travels in the di-
rection of the curved line shown in our illustration. This curve is called, as I
have said, a "common cycloid." Now, if the diameter of the wheel is 28 inches,
we can give the exact length of that curve. This is remarkable-that we can-
not give exactly the length from A to B in a straight line, but can state
exactly the length of the curve. What is that length? I will give the answer at
once. The length of the cycloid is exactly four times that of the diameter.