# Aristotle's wheel

Aristotle's wheel is a paradox mentioned in the
ancient Greek text *Mechanica*, whose author is unknown but is suspected
by some to have been Aristotle. The paradox
concerns two concentric circles on a wheel, as shown in the diagram. A one-to-one correspondence exists between points on the larger circle and those on the
smaller circle. Therefore, the wheel should travel the same distance regardless
of whether it is rolled from left to right on the top straight line or on
the bottom one. This seems to imply that the two circumferences of the different
sized circles are equal, which is impossible. How can this apparent contradiction
resolved? The key lies in the (false) assumption that a one-to-one correspondence
of points means that two curves must have the same length. In fact, the
cardinalities of points in a line segment of any length (or even an infinitely
long line or an infinitely large n-dimensional Euclidean space!) are all
the same.