Today, we look at another classical paradox: Aristotle’s wheel. The paradox was introduced in the text *Mechanica*, attributed, not without controversy, to Aristotle. It runs as follows. Consider two circular wheels, fixed rigidly, one within the other. The wheels have the same center, but the radius of the outer wheel is twice that of the inner wheel. Suppose this combined wheel rolls without slipping for exactly one full revolution, and consider the paths traced by the bottoms of the two wheels. These paths are evidently equal in length to the circumferences of the respective circles, yet the two paths are the same length, while the circumference of the outer wheel is twice that of the inner wheel. This would seem, then, to yield a contradiction.

Unlike Zeno’s paradoxes, discussed on Monday, Aristotle’s wheel contains no real mystery today. Some combination of the following two observations should be enough to convince you of this.

- It is physically impossible for the two joined wheels to roll without at least one of them “slipping” relative to the ground. Therefore, there is no reason to think that the paths traced by the bottoms of the wheels are equal in length to the circumferences of the respective wheels.
- Even though the length of one of the paths may not be equal to the circumference of the wheel that creates it, the set consisting of the points on the path and the set consisting of the points on the circumference of the wheel have the same cardinality, so there is no contradiction in there being a one-to-one correspondence between points on the circumference of the wheel and points on the path.

If this paradox can be resolved in such a straightforward manner, you may be wondering why I decided to dedicate an entire post to it. The reason is that Aristotle’s wheel plays a significant role in Galileo’s last published work, the influential *Discourses and Mathematical Demonstrations Relating to Two New Sciences*, and Galileo’s solution to the paradox is both remarkable in its own right and influential in the history of mathematics and physics.

To attack the problem of Aristotle’s wheel, Galileo makes a move that had been common at least since ancient Greece: reasoning about circles by approximating them with regular polygons. To illustrate his ideas, Galileo considers the case in which the two wheels are not circular but rather are regular hexagons. He then considers what happens when this hexagonal wheel “rolls” (or, rather, lurches in six discrete steps) along the ground for one full revolution. The situation is illustrated in the diagram below.

Consider first the outer hexagonal wheel. Initially, the wheel is at rest, with side AB resting on the ground. When the wheel makes its first step in its “roll,” it pivots around point B, and side BC comes to rest on the ground, occupying the segment BQ. After the second step, side CD comes to rest on the segment QX, and so on. Through the course of the wheel’s revolution, the entire segment AS is thus covered successively by sides of the outer wheel. Therefore, the length of the segment AS is equal to the perimeter of the outer wheel.

Now consider the inner hexagonal wheel. Initially, side HI is resting on an initial segment of HT. After the first step of the revolution, though, side IK does *not* come to rest on the segment IO but rather “jumps ahead” and lands on the segment OP. Similarly, after the second step, side KL jumps across the segment PY to land on the segment YZ, and so on. Therefore, the parts of the segment HT that are covered by sides of the inner wheel during the revolution alternate with parts that are skipped over. This explains why AS is the same length as HT (or, rather, HT extended by a bit equal in length to one of the sides of the inner wheel, as shown in the diagram) while the perimeter of the outer wheel is twice that of the inner wheel.

The same situation holds for polygonal wheels of any number of sides. Thus, if, for example, the wheels are regular 100,000-gons, then the lower path will be entirely covered by the sides of the outer wheel in its revolution, while the upper path will be split into 200,000 equal pieces, and these pieces will alternately be covered or skipped over by the sides of the inner wheel in its revolution. Put another way, the path traced by the bottom of the inner wheel will consist of 100,000 pieces, each the length of one of the sides of the wheel, interspersed with 100,000 “voids” of equal length. The path traced by the outer wheel will have no such voids. As a polygonal wheel gets more and more sides, though, it more and more closely approximates a circle (a circle could even be seen as a regular polygon with infinitely many infinitely short sides), so, Galileo argues, a similar situation must hold in the case of circular wheels.

I will let Galileo explain this idea in his own (translated) words:

Let us return to the consideration of the above mentioned polygons whose behavior we already understand. Now in the case of polygons with 100000 sides, the line traversed by the perimeter of the greater, i. e., the line laid down by its 100000 sides one after another, is equal to the line traced out by the 100000 sides of the smaller, provided we include the 100000 vacant spaces interspersed. So in the case of the circles, polygons having an infinitude of sides, the line traversed by the continuously distributed [cantinuamente disposti] infinitude of sides is in the greater circle equal to the line laid down by the infinitude of sides in the smaller circle but with the exception that these latter alternate with empty spaces; and since the sides are not finite in number, but infinite, so also are the intervening empty spaces not finite but infinite. The line traversed by the larger circle consists then of an infinite number of points which completely fill it; while that which is traced by the smaller circle consists of an infinite number of points which leave empty spaces and only partly fill the line. And here I wish you to observe that after dividing and resolving a line into a finite number of parts, that is, into a number which can be counted, it is not possible to arrange them again into a greater length than that which they occupied when they formed a continuum [continuate] and were connected without the interposition of as many empty spaces. But if we consider the line resolved into an infinite number of infinitely small and indivisible parts, we shall be able to conceive the line extended indefinitely by the interposition, not of a finite, but of an infinite number of infinitely small indivisible empty spaces.

In essence, what Galileo is saying is this: the reason that the paths traced by the bottoms of the wheels can be the same length is that the path traced by the inner wheel consists of infinitely many points interspersed with infinitely many infinitely small empty spaces, while the path traced by outer wheel consists only of the points and not the empty spaces!

One should not let the fact that Galileo’s solution is, by modern standards, misguided at best detract from its remarkable inventiveness or take away from the tremendous influence it and related ideas have had on mathematics and science. The ideas in Galileo’s exposition of Aristotle’s wheel are central to the development, also in the *Discourses*, of his celebrated law of free fall, which states that if an object starts moving from rest with uniform acceleration (as does (approximately) an object in free fall), then the distance traveled by the object is proportional to the square of the time during which it is moving. This law, commonplace today to any physics student, was groundbreaking in its time and anticipated Newton’s famous laws of motion. The *Discourses* also appeared at the beginning of a period of renewed interest in the question of the composition of the continuum and a revolution ushered in by the increasing acceptance of the use of infinitesimal quantities in mathematics. Galileo’s work on infinitesimals was extended and refined through the 17th century by mathematicians such as Cavalieri, Torricelli, and Wallis, who paved the way for the development of the infinitesimal calculus at the end of the century by Newton and Leibniz.

Like any radical idea, the mathematical use of infinitesimals was not immediately and universally accepted. In fact, there was strong opposition to the idea, most prominently (though certainly not exclusively) from the Catholic Church and especially the Jesuits, who were seeking to reestablish order and hierarchy in the wake of the chaos brought about by the Reformation. In the 17th century, the use of infinitesimals had not been provided with a rigorous mathematical foundation, and paradoxes frequently arose from their indiscriminate application. This was seen as threatening to the status of mathematics, as exemplified by Euclidean geometry, as an orderly realm of absolute certainty and, in some circles, as a model for theology and indeed for society as a whole. As Amir Alexander asserts in his book *Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World:*

The infinitely small was a simple idea that punctured a great and beautiful dream: that the world is a perfectly rational place, governed by strict mathematical rules … By demonstrating that reality can never be reduced to strict mathematical reasoning, the infinitely small liberated the social and political order from the need for inflexible hierarchies.

I suspect that Alexander may be exaggerating the importance of the controversy over infinitesimals in the social history of Europe, but there is no doubt that the Church came down strongly against them. Many times through the first half of the 17th century, the Jesuit Revisors, who were in charge of determining what could or could not be taught in the many Jesuit colleges throughout the world and therefore, in effect, what ideas would be endorsed or condemned by the Catholic Church, issued rulings against the doctrine of infinitesimals. For example, the following is from such a ruling in 1632:

We consider this proposition to be not only repugnant to the common doctrine of Aristotle, but that it is by itself improbable, and … is disapproved and forbidden in our Society.

Finally, in 1651, the Revisors published an official list of 65 forbidden philosophical theses, including no fewer then four forbidden theses regarding infinitesimals:

25. The succession continuum and the intensity of qualities are composed of sole indivisibles.

26. Inflatable points are given, from which the continuum is composed.

30. Infinity in multitude and magnitude can be enclosed between two unities or two points.

31. Tiny vacuums are interspersed in the continuum, few or many, large or small, depending on its rarity or density.

And now we have made a full revolution back to Aristotle’s wheel, as item 31 is precisely the theory of the continuum that Galileo developed in order to explain the paradox: the path traced by the inner wheel is the same length as the path traced by the outer wheel precisely because the “tiny vacuums” interspersed in the path of the inner wheel are larger than those in the path of the outer wheel.

Eventually, of course, infinitesimals became widely accepted in mathematics and have even been given rigorous foundations that do away with the paradoxes that plagued their early use. Even though Galileo’s solution to Aristotle’s wheel did not last, the ideas it helped usher in transformed mathematics and remain with us to this day.

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