A Paradox mentioned in the Greek work *Mechanica,* dubiously attributed to Aristotle. Consider the
above diagram depicting a wheel consisting of two concentric Circles of different Diameters (a wheel within a wheel). There is a 1:1 correspondence of points on the large Circle with points on the
small Circle, so 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 different
sized Circles are equal, which is impossible.

The fallacy lies in the assumption that a 1:1 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 Infinite Line, a Plane, a 3-D Space, or an infinite dimensional Euclidean Space) are all the same: (Aleph-1), so the points of any of these can be put in a One-to-One correspondence with those of any other.

**References**

Ballew, D. ``The Wheel of Aristotle.'' *Math. Teacher* **65**, 507-509, 1972.

Costabel, P. ``The Wheel of Aristotle and French Consideration of Galileo's Arguments.'' *Math. Teacher*
**61**, 527-534, 1968.

Drabkin, I. ``Aristotle's Wheel: Notes on the History of the Paradox.'' *Osiris* **9**, 162-198, 1950.

Gardner, M. *Wheels, Life, and other Mathematical Amusements.* New York: W. H. Freeman, pp. 2-4, 1983.

Pappas, T. ``The Wheel of Paradox Aristotle.'' *The Joy of Mathematics.*
San Carlos, CA: Wide World Publ./Tetra, p. 202, 1989.

vos Savant, M. *The World's Most Famous Math Problem.* New York: St. Martin's Press, pp. 48-50, 1993.

© 1996-9

1999-05-25