The paradox of Aristotle's wheel is mentioned in ancient Greek text Mechanica. The problem has hunted some of the greatest mathematicians for centuries. Consider a small wheel, diagrammed as two concentric circles. A one-to-one correspondence exists between points on the larger circle and those on the smaller circle; thats is, for each point in the large circle, there is exactly one point on the small circle and vice versa. Thus, the wheel assembly might be expected to travel the same horizontal distance regardless of whether it is rolled on a rod that touches the smaller wheel or rolled along the bottom wheel that touches the rod. But how can this be? After all, we know that the two circumferences of the circles are different.
Today, mathematicians know that a one-to-one correspondence of points doesn't mean that two curves must have the same length. Georg Cantor (1845-1918) showed that the number, or cardinality, of points on line segment of any length is the sam. He called this Transfinite Number of points the "continuum." For example, all the points in a segment from zero to one can even be put in a one-to-one correspondence with all points of an infinite line. Of course, before the work of cantor, mathematicians had quite a difficult time with this problem. Also note that, from a physical standpoint, if the large wheel did roll along the road, the smaller wheel would skip and be dragged along the lines that touches its surface.
The precise date and authorship of Mechanica may be forever be shrouded in mystery. Although often attributed as the work of Aristotle, many scholars doubt that Mechanica, the oldest-known textbook in engineering, was actually written by Aristotle. Another possible candidate for authorship is Aristotle's student Stratus of Lampsacus (also known as Strato Physicus), who died around 270 B.C.
