Answers 301while Lincoln's head travels a considerable distance, ought to be at once ob-
vious to everybody. It should be quite convincing that the part of the wheel
that is for the time being at the top moves faster through space than the part
at the bottom.- ANOTHER WHEEL PARADOX
I have already shown that, if you mark a spot on the circumference of a
bicycle wheel, that spot, when the wheel is progressing, will describe in space
a curve known as a common cycloid. If, however, you mark the edge of the
flange of a locomotive or railroad-car wheel, the spot will describe a trochoid
curve, terminating in nodes or loops, as shown in the diagram. I have shown
a wheel, with flanges below the railway line, in three positions-the start, a
half-revolution, and a complete revolution. The spot marked Al, has gone to
A2 and A 3. As the wheel is supposed to move from left to right, trace with
your pencil the curve in that direction. You will then find that at the lower
part of the loop you are actually going from right to left.
The fact is that "at any given moment" certain points at the bottom of the
loop must be moving in the opposite direction to the train. As there is an in-finite number of such points on the flange's circumference, there must be an
infinite number of these loops being described while the train is in motion.
In fact, certain points on the flanges are always moving in a direction opposite
to that in which the train is going.- A MECHANICAL PARADOX
The machine shown in our illustration on page 302 consists of two pieces of
thin wood, B, C, made into a frame by being joined at the corners. This frame,
by means of the handle, n, may be turned round an axle, a, which pierces the
frame and is fixed in a stationary board or table, A, and carries within the frame