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Aristotle's Wheel Paradox

© Copyright 2002, Jim Loy

This is a "paradox" erroneously (probably) attributed to Aristotle, in the ancient book Mechanica. We have a wheel with a smaller wheel attached to it (left side of the diagram). The larger wheel rolls on a flat surface from point A to point B. During this movement, the smaller wheel touched the line CD all the way. The point of contact between the smaller wheel and that line moved the same distance (CD) as did the bottom of the larger wheel. This would seem to imply that the circumference of the smaller wheel is the same as that of the larger wheel, which is impossible.

The logic of that is poor. There are several things wrong with it. For one thing, while the wheel did roll from A to B, the smaller wheel did not roll from C to D; it was dragged along that line. The above "paradox" is often begun by mentioning that there is a one-to-one correspondence between the points on the two wheels, and from that we get a one-to-one correspondence between the points on the two line segments. This is perfectly true, and of course saying that two sets of points (on a line or curve) correspond does not necessarily mean they have the same length.

One can "deduce" from this that motion is impossible. This "paradox" is similar to Zeno's Paradoxes.

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