A Brief History of Infinity

Now we roll the combined wheel along so that it moves onto its next side. As the big wheel turns it pivots on the corner and moves along the track by the length of one side. But what has happened to the smaller wheel? Not only has the big wheel moved on by that distance, so has the small one. It has to: they're fixed together. Yet in turning 1/6 of a rotation, the small one should only have rolled along the track by the length of its own side - a much smaller distance, marked in red on the diagram. To achieve the extra movement, the smaller wheel was lifted entirely off the track.

Now here's the clever bit. Galileo imagined increasing the number of sides. The more sides, the more sets of small movements along the rail and small jumps you get as the wheels rotate. Finally, let's imagine, were it possible, that we take that number of sides to infinity. We end up with circular wheels.

Again we roll the two wheels, joined together, along their respective rails. Again they both travel the same distance - in this case a quarter of the circumference of the big wheel. But now something strange has happened. The rim of the big wheel has rolled out a quarter of its circumference on its track. The rim of the smaller wheel has only rolled out its smaller quarter circumference, but the small wheel still has to travel the same distance as the bigger one, without ever leaving the track. There were no jumps, or at least so it seems.

What Galileo imagined had happened was that as the smaller wheel turns there are an infinite number of infinitesimally small gaps, which add up to make the difference between the circumference of the wheel and the distance it moves. Infinity has come into play in a physical device to make the seemingly impossible happen.

After letting this percolate through his brain in the background, Galileo's traditionalist and frankly rather dim character, Simplicio, has a complaint. What Galileo seems to be saying (or technically Salviati, the character that represents Galileo's voice in the book) is that there are an infinite number of points in one circular wheel and an infinite number of points in the other. But somehow, though each had the same infinity of points, one added up to a greater distance than the other. One infinity was both the same as the other and larger.