Aristotle's wheel paradox is a paradox or problem appearing in the Greek work Mechanica, traditionally attributed to Aristotle. It states as follows: A wheel is depicted in two-dimensional space as two circles. Its larger, outer circle is tangential to a horizontal surface (e.g. a road that it rolls on), while the smaller, inner one has the same center and is rigidly affixed to the larger. (The smaller circle could be the bead of a tire, the rim it is mounted upon, or the axle.) Assuming the larger circle rolls without slipping (or skidding) for one full revolution, the distances moved by both circles' circumferences are the same. The distance travelled by the larger circle is equal to its circumference, but for the smaller it is greater than its circumference, thereby creating a paradox. The paradox is not limited to wheels: other things depicted in two dimensions display the same behavior such as a roll of tape, or a typical round bottle or jar rolled on its side (the smaller circle would be the mouth or neck of the jar or bottle). In an alternative version of the problem, the smaller circle, rather than the larger one, is in contact with the horizontal surface. Examples include a typical train wheel, which has a flange, or a barbell straddling a bench. American educator and philosopher Israel Drabkin called these Case II versions of the paradox, and a similar, but unidentical, analysis applies. 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. 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