Aristotle's wheel paradox

Aristotle's wheel paradox is a paradox appearing in the Greek work Mechanica traditionally attributed to Aristotle.^[1] There are two wheels, one within the other, whose rims take the shape of two circles with different diameters. The wheels roll without slipping for a full revolution. The paths traced by the bottoms of the wheels are straight lines, which are apparently the wheels' circumferences. But the two lines have the same length, so the wheels must have the same circumference, contradicting the assumption that they have different sizes: a paradox.

History of the Paradox

In Antiquity

In antiquity, the wheel problem was described in the Aristotelian Mechanica, as well as in the Mechanica of Hero of Alexandria.^[1] In the former it appears as "Problem 24", where the description of the wheel is given as follows.

For let there be a larger circle ΔZΓ a smaller EHB, and A at the centre of both; let ZI be the line which the greater unrolls on its own, and HK that which the smaller unrolls on its own, equal to ZΛ. When I move the smaller circle, I move the same centre, that is A; let the larger be attached to it. When AB becomes perpendicular to HK, at the same time AΓ becomes perpendicular to ZΛ, so that it will always have completed an equal distance, namely HK for the circumference HB, and ZΛ for ZΓ. If the quarter unrolls an equal distance, it is clear that the whole circle will unroll an equal distance to the whole circle, so that when the line BH comes to K, the circumference ZΓ will be ZΛ, and the whole circle will be unrolled. In the same way, when I move the large circle, fitting the small one to it, their centre being the same, AB will be perpendicular and at right angles simultaneously with AΓ, the latter to ZI, the former to HΘ. So that, when the one will have completed a line equal to HΘ, and the other to ZI, and ZA becomes again perpendicular to ZΛ, and HA to HK, so that they will be as in the beginning at Θ and I.^[2]

The problem is then stated:

Now since there is no stopping of the greater for the smaller so that it [the greater] remains for an interval of time at the same point, and since the smaller does not leap over any point, it is strange that the greater traverses a path equal to that of the smaller, and again that the smaller traverses a path equal to that of the larger. Furthermore, it is remarkable that, though in each case there is only one movement, the center that is moved in one case rolls a great distance and in the other a smaller distance.^[1]

In the Scientific Revolution

The mathematician Gerolamo Cardano discusses the problem of the wheel in his 1570 Opus novum de proportionibus numerorum,^[3] taking issue with the presumption the analysis of the problem in terms of motion.^[1] Mersenne further discussed the wheel in his 1623 Quaestiones Celeberrimae in Genesim,^[4] where he suggests that the problem can be analysed by a process of expansion and contraction of the two circles. But Mersenne remained unsatisfied with his understanding, writing,

Indeed I have never been able to discover, and I do not think any one else has been able to discover whether the smaller circle touches the same point twice, or proceeds by leaps and sliding.^[1]

In his Two New Sciences, Galileo uses the problem of the wheel to argue for a certain kind of atomism. Galileo begins his analysis by considering a pair of concentric hexagons, as opposed to a pair of circles. Imagining this hexagonal wheel "rolling" on a surface, Galileo notices that the inner hexagon "jumps" a little space, with each roll of the outer wheel onto a new face.^[5] He then imagines what would happens in the limit as the number faces on the polygon becomes very large, and finds that the little space that is "jumped" by the inner wheel becomes smaller and smaller, writing:

Therefore a larger polygon having a thousand sides passes over and measures a straight line equal to its perimeter, while at the same time the smaller one passes an approximately equal line, but one interruptedly composed of a thousand little particles equal to its thousand sides with a thousand little void spaces interposed — for we may call these "void" in relation to the thousand linelets touched by the sides of the polygon.^[5]

Since the circle is just the limit in which the number of faces on the polygon becomes infinite, Galileo finds that Aristotle's wheel contains material that is filled with infinitesimal spaces or "voids", and that "the interposed voids are not quantified, but are infinitely many".^[5] This leads Galileo to conclude that a belief in atoms, in the sense that matter is "composed of infinitely many unquantifiable atoms" is sufficient to solve the problem of the wheel.^[5]

In the 19th Century

Bernard Bolzano discussed Aristotle's wheel in The Paradoxes of the Infinite (1851), a book that influenced Georg Cantor and subsequent thinkers about the mathematics of infinity. Bolzano observes that there is a bijection between the points of any two similar arcs, which can be implemented by drawing a radii, remarking that the history of this apparently paradoxical fact goes back to Aristotle.^[1]

Analysis

One way to understand the paradox of the wheel is to reject the assumption that the smaller wheel indeed traces out its circumference, without ensuring that it, too, rolls without slipping on a fixed surface. In fact, it is impossible for both wheels to perform such motion. Physically, if two joined concentric wheels with different radii were rolled along parallel lines then at least one would slip; if a system of cogs were used to prevent slippage then the wheels would jam. A modern approximation of such an experiment is often performed by car drivers who park too close to a curb. The car's outer tire rolls without slipping on the road surface while the inner hubcap both rolls and slips across the curb; the slipping is evidenced by a screeching noise.^[6]

Alternatively, one can reject the assumption that the smaller wheel is independent of the larger wheel. Imagine a tire as the larger wheel, and imagine the smaller wheel as the interior circumference of the tire and not as the rim. The movement of the inner circle is dependent on the larger circle. Thus its movement from any point to another can be calculated by using an inverse of their ratio.

References

Further reading

• Rota Aristotelica, The Archimedes Project, Digital Research Library
• Weisstein, Eric W. "Aristotle's Wheel Paradox". MathWorld.