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this was an illusion: Aristotle and his followers were confident that, from the correct perspective, all heavenly motion was indeed perfectly uniform and perfectly circular.
    CIRCLES UPON CIRCLES
    This system of nested crystalline spheres was immensely appealing—but anyone who followed the movements of the planets closely came to realize that it was not quite enough; the motion of the planets was too complex. For example, it was still unclear how the circular movement of those spheres could account for retrograde motion. The best guess was that each planet required two such spheres: a large one, to account for the basic eastward motion; and a smaller one, to account for the “loops” that the planet traces out when moving in retrograde. These smaller circles were known as epicycles (from a Greek term meaning “a cycle displaced from the center”).
    The most detailed account of such a system comes from the Greek mathematician and astronomer Claudius Ptolemy (ca. A.D. 90–168). * Ptolemy’s system was intricate and sophisticated, employing geometrical contrivances that today sound unfamiliar to anyone except for historians of astronomy. We will not wade into Ptolemaic astronomy any more than we have to, but it is worth looking at its main elements. As in Aristotle’s system, the Earth lies at the center of the universe. Each planet, as mentioned, has two motions: it moves in a small epicycle, with the center of the epicycle revolving around the Earth in a larger circle called a deferent . The deferent, meanwhile, is not centered precisely on the Earth, but on a nearby point called the eccentric . One more aspect of Ptolemy’s astronomy merits our attention: Ptolemy had imagined not only that the heavenly bodies moved in perfect circles, but that they did so at a constant speed. This was problematic, because, as measured from Earth, the speed would not be constant in the system as described. But the speed would be constant relative to an imaginary point on the “other side” of the eccentric, displaced from the center by the same amount as the Earth. That imaginary point was called an equant .
    If you’re thinking that all of this is frighteningly complex, you’re not alone. In the thirteenth century, the king of León, Alfonso X, commissioned a new set of astronomical tables to be drawn up; the calculations were carried out using the Ptolemaic system, which still reigned supreme in celestial matters. When one of his aides explained the system to him, the king is said to have remarked, “If the Lord Almighty had consulted me before embarking upon the creation, I should have recommended something simpler.”
    Three centuries later, this apparent complexity would trouble the poet John Milton. In Paradise Lost , Adam inquires about the structure of the heavens; the angel Raphael replies that God must surely be laughing at man’s desperate efforts to explain the cosmos:
    â€¦ when they come to model heav’n
    And calculate the stars, how they will wield
    The mighty frame, how build, unbuild, contrive
    To save appearances, how gird the Sphere
    With centric and eccentric scibbled o’er,
    Cycle and epicycle, Orb in Orb …
    (8.79–84)
    Remarkably, as complicated as the Ptolemaic system sounds, it worked: It allowed astronomers (and astrologers) to predict the positions of the planets with reasonable accuracy, allowing them to “save the appearances” of the wandering lights in the night sky. (That phrase, derived from a Greek expression, had long been in common use when Milton borrowed it for use in his poem.) And it worked in spite of a fairly serious glitch. It’s not just that Ptolemy had placed the Earth, rather than the sun, at the center; that, by itself, would not affect the predicted positions, as the two schemes are mathematically equivalent. But his estimates of the sizes of the spheres were all quite far off. They were based on