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Aristotle's wheel paradox

Aristotle's wheel paradox is a paradox or problem appearing in the pseudo-Aristotelian Greek work Mechanica. 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, while the smaller, inner one has the same center and is rigidly affixed to the larger. Assuming the larger circle rolls without slipping 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.

History of the paradox
In antiquity In antiquity, the wheel problem was described in the Greek work Mechanica, traditionally attributed to Aristotle, but widely believed to have been written by a later member of his school. It also appears in the Mechanica of Hero of Alexandria. In the Aristotelian version it appears as "Problem 24", where the description of the wheel is given as follows: For let there be a larger circle a smaller , and at the centre of both; let be the line which the greater unrolls on its own, and that which the smaller unrolls on its own, equal to . When I move the smaller circle, I move the same centre, that is ; let the larger be attached to it. When becomes perpendicular to , at the same time becomes perpendicular to , so that it will always have completed an equal distance, namely for the circumference , and for . 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 comes to , the circumference will be , 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, will be perpendicular and at right angles simultaneously with , the latter to , the former to . So that, when the one will have completed a line equal to , and the other to , and becomes again perpendicular to , and to , so that they will be as in the beginning at and . 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. taking issue with the presumption of its analysis in terms of motion. 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. He then imagines what would happen to the limit as the number of faces on a polygon becomes very large, and finds that the little space that is "jumped" by the inner polygon becomes smaller and smaller. He writes: 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. == Analysis and solutions ==
Analysis and solutions
File:aristotle_wheel_paradox.svg|thumb|250px|link=|CSS animation of Aristotle's wheel paradox. The wheel comprises two concentric circles: the outer one has twice the radius of the inner one and rolls on the lower track. Both circles and tracks are marked with segments of equal length. The inner circle is observed to slip with respect to its track. [ (Animation)] The paradox is that the smaller inner circle moves , the circumference of the larger outer circle with radius , rather than its own circumference. If the inner circle were rolled separately, it would move , its own circumference with radius . The inner circle is not separate but rigidly connected to the larger. First solution If the smaller circle depends on the larger one (Case I), the larger circle's motion forces the smaller to traverse the larger’s circumference. If the larger circle depends on the smaller one (Case II), then the smaller circle's motion forces the larger circle to traverse the smaller circle’s circumference. This is the simplest solution. Second solution This solution considers the transition from the starting to ending positions. Let be a point on the bigger circle and be a point on the smaller circle, both on the same radius. For convenience, assume they are both directly below the center, analogous to both hands of a clock pointing towards six. Both and travel in a cycloid path as they roll together one revolution. While each travels horizontally from start to end, 's cycloid path is shorter and more efficient than 's. travels farther above and farther below the center's path – the only straight one – than does . If and were anywhere else on their respective circles, the curved paths would be the same length. Summarizing, the smaller circle moves horizontally because any point on the smaller circle travels a shorter, and thus more direct, path than any point on the larger circle. ==See also==
Source: Wikipedia ↗
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