A Brief History of Infinity

After Galileo's house arrest began in 1634, following the trial over his heretical work on the motion of the Earth around the Sun, Galileo was anything but inactive. It was then that he produced the book that was arguably his greatest work of science, his equivalent of Newton's Principia, named Discorsi e dimostrationi matematiche, intorno a due nuove scienze, or Discourses and Mathematical Demonstrations Concerning Two New Sciences. Galileo had considerable trouble getting this published - the Inquisition made it clear that no work by this heretic would be published in any country where it held sway. When the book was eventually taken up by the great Dutch publisher Elsevier, Galileo expressed his great surprise that it had been published at all, which he claimed had never been his intention. He commented in his introduction:

I was notified by the Elzevirs that they had these works of mine in press and that I ought to decide upon a dedication and send them a reply at once. This sudden and unexpected news led me to think that the eagerness of your Lordship to revive and spread my name by passing these works on to various friends was the real cause their falling into the hands of printers who, because they had already published other works of mine, now wished to honour me with a beautiful and ornate edition of this work.

Hardly "sudden and unexpected news" - Galileo was closely involved in the publication, but at least he was protecting his back.

The book (as did his near-fatal work Dialogue Concerning the Two Chief World Systems on the motion of the Earth) took the form of a conversation between a number of characters, largely over serious matters. But after wondering about what holds matter together (Galileo thought it was tiny pockets of vacuum between the particles of matter), they have a diversion, just for the fun of it, into the nature of infinity.

Galileo brings out a number of points, but two are particularly worthy of note. The first involves the rotation of a pair of wheels.

He starts with wheels with a few sides - for example, they could be hexagons. These are three dimensional shapes - imagine the hexagons are cut out of sheets of marble. The smaller hexagon is stuck to the larger one, and each of them rests on its own horizontal rail.