Mathematical game: Aristotle’s wheels

In ” The Mechanics », a text attributed to Aristotle rediscovered in the Renaissance, we find an interesting paradox. Here it is. Two concentric circles of different radii, one of which draws the other along, travel along the same rectilinear path. Both complete a full turn at the same time.

The paradox consists in concluding from this the equality of the circumferences, which is manifestly false.

Question:

But where is the error?

Responnse :

From a mechanical point of view, the question is simple: if the large wheel rolls without slipping, the small one would slip… if it were in contact with the ground. Galileo responded to this paradox in the spirit of the nascent infinitesimal calculus. It approximates the circle by a regular polygon and, to begin with, by a hexagon. He rotates the set of two hexagons in order to pass from one side of the large one to the next. The surprise comes from the series of short sides, when they follow each other horizontally, because there are holes there.

When the set of two hexagons turn, the sides of the large one follow one another, but not those of the small one. Between two consecutive, there is a gap. When the number of sides of the polygons increases, they get closer to the two circles and the holes decrease in size. We obtain two segments of the same length, but the second, the one corresponding to the small circle, is filled with holes.