Origins of the Ptolemaic System
We’ve worked backwards from Kepler to Ptolemy. What inspired Ptolemy and his predecessors (Apollonius and Hipparchus) to come up with this scheme in the first place?
Almost none of the astronomical works of Apollonius and Hipparchus have survived. As for Ptolemy, to quote Evans (p.357), “Ptolemy’s style in the Almagest is the style of most scientific writing. It is lean, elegant, and efficient and discloses very little of the original process of discovery.” (Evans refers here to the equant; see below.) Historians have partially filled in the gap with closely-reasoned speculation.
All the planets exhibit retrograde motion at times: they usually travel in one direction, staying close to the ecliptic, but periodically they reverse course for a while. (For example, a typical retrogradation for Mars might last a couple of months, with the retrogradations occurring a little over two years apart.) Here’s an example:
Retrograde Motion of Mars
Given this pattern, it doesn’t take a genius to come up with the deferent-epicycle idea—and Apollonius was a genius.
Eccentric deferents pop up naturally for the orbit of the Sun. The seasons have different lengths: nowadays summer lasts around 93 days and winter only 89. The Sun moves faster around the ecliptic during the winter than during the summer. Imagine a car zipping around a perfectly circular racetrack, and picture yourself standing inside but off-center: you’ll see that the car will appear to move faster when it is closer to you. Hipparchus devised this explanation for the so-called solar or zodiacal anomaly.
For the planets, we have a similar anomaly: the retrogradations (say of Mars) are spaced unevenly along the zodiac (i.e., near the ecliptic). Also the widths of the retrograde loops are different. (Evans-1998 has diagrams illustrating this quite clearly: see p.341, 343, 357 and Evans-1984, available on-line.) Imagine a row of trees, planted around the rim of a circular lawn; the spacing appears uneven to your sight, and the trees appear to have different sizes. Perhaps you are standing off-center? Perhaps the planting really is uneven? Thus:
Tree Analogy
Viewed from Earth, the size of the trees (width of the retrogression loops) looks uneven, also the spacing. Moving to the deferent center makes the trees look equal in size because they are all equally distant. Spacing still looks uneven, because they are not planted evenly (planet does not move uniformly). Moving to the equant makes the spacing appear even (motion is uniform when viewed from equant).
For the planets, epicycles riding on eccentric deferents cannot account for the observed motion accurately and in detail—not if we also insist on uniform motion. Ptolemy probably invented the equant1 to resolve the discrepancies. Pedersen (pp.277–279, 306–307), Evans, and others have offered various ideas for Ptolemy’s innovation.
Without going into gory detail, all the suggestions make use of one fundamental idea. How large the epicycle looks to us at a given time, depends on how close we are to its center at that time. In principle this can tell us the center of the deferent. Of course, we can’t actually see the epicycle, and so we can’t directly observe its size. But the apparent size reflects itself in various effects: for example, the width of the retrograde loops. Having determined the eccentricity of the deferent, we can then tell if the motion of the epicycle’s center is indeed uniform. As it happens, it’s not.
We can’t see the epicycle’s center, any more than the epicycle’s apparent size. But its location betrays itself indirectly, for example by the position of the retrograde loops.
Using the tree analogy: suppose all the trees enjoy the same physical size, but their sizes appear different, likewise their spacing (again, see the figure above). You conclude that you are standing off-center in the lawn. Using the tree sizes, you compute how far off-center; then you use that to compute the actual spacing of the trees. Suppose that comes out non-uniform. You notice though that the tree spacing would appear uniform from a different spot. You conclude that you are standing in the first spot (to account for the apparent sizes), and the gardner was standing in the second spot when he told his assistants where to plant the trees (to account for the non-uniform spacing). You are standing at the center of the deferent; the gardner is standing at the equant2.
Complicating these historical reconstructions, Ptolemy uses different procedures to find the equants of different planets. For the outer planets, he computes the equant circle first, then halves its eccentricity (without explanation) to find the eccentricity of the deferent. For Venus, he computes the eccentricities of the deferent and the equant circle independently, using different techniques.
Ptolemy deserves admiration for his innovation. Modern historians have been ready to grant it. Such was not the reaction in antiquity. Ptolemy had found a loophole around Aristotle’s doctrine of uniform circular motion; to many ancient astronomers, it looked like a form of cheating, although one not easily discarded. The Arab astronomer Ibn al-Shāṭir (c. 1304–1375) found a replacement for the equant: another epicycle! Later, Copernicus used the same technique (maybe independently, maybe not). Most astronomers of his day regarded this as progress. (See Evans-1998, p.422.)
[1] Most historians believe this, but some disagree. We have no solid evidence one way or the other. I will go along with the consensus.
[2] Kepler, and some modern authors, have suggested that the latitudes of the planets (angular distance above or below the ecliptic) played a key role. Latitudes are analogous to the heights of the trees.
Diagonal Argument 