Ptolemy and Copernicus

I’m teaching my first-year seminar, Space is Big, again this fall. In the first part of the course we look at the Copernican revolution, when people first* figured out that the Earth goes around the Sun.

Before we get to Copernicus, we spend a while looking at what the ancient Greeks thought about the way the planets move, focusing especially on the Ptolemaic system. I made a few little animations to show some connections between the old Earth-centered view and the Copernican one.

There’s nothing original in what I’m about to say, and nothing that the cognoscenti don’t already know, but there are a few animated gifs that show exactly why the ancient system and the Copernican system were observationally equivalent (and why both were equivalent to the less well-known Tychonic system).

In the Ptolemaic system, the Earth was at rest at the center, and the Sun, Moon, and planets all went around it. Here’s a simplified version of the Ptolemaic model, showing just the Earth (blue dot), Sun (orange dot), and Venus (red dot):

In this model, the Sun goes around the Earth in a circle, but Venus has a more complicated motion, involving a deferent (the big red circle) and an epicycle (the little red circle). This is necessary because Venus (like the other planets) goes through periods of retrograde motion during which its apparent path through the sky reverses direction. In this model, retrograde motion occurs when the epicycle carries Venus close to the Earth: at those times, the epicycle is making Venus go around backwards faster than the deferent is making it go forwards, so it reverses direction.

Actually, Ptolemy’s system was somewhat more complicated than this: to get the motions of the planets right in detail, he needed extra epicycles, as well as things called eccentrics and equants. But the circles in this diagram are the most important ones. They’re all you need to get the gross features of the Sun’s and Venus’s apparent motion right. (Similar epicyle-deferent pairs work for all the other planets; I’m just focusing on one planet to keep things uncluttered.)

Here’s a funny thing about this model: Venus’s epicycle goes around the Earth at exactly the same rate as the Sun (once per year, if you must know). That’s Ptolemy’s way of accounting for the fact that Venus has bounded elongation, which is just a fancy way of saying that Venus always appears near the Sun in the sky. (You’ll never see Venus rising in the east when the Sun is setting in the west, for instance.) One thing that Copernicus found unsatisfying about the Ptolemaic system was that there was no good reason for these two motions to be synchronized like this.

Ptolemy put Venus’s orbit inside the Sun’s orbit, as I’ve shown. But there’s no reason he had to. You could make the Sun’s orbit bigger or smaller by as much as you want, and everything an earthbound observer sees would remain exactly the same. In particular, you could shrink the Sun’s orbit until it was exactly the same size as Venus’s deferent:

Physically, Ptolemy wouldn’t have liked this model. He probably believed that the epicycles and deferents were actual physical objects, so Venus’s and the Sun’s shouldn’t cross each other. But, as he would surely have agreed, this model has the exact same appearance as his original model: if you use the two models to predict where the Sun and Venus will appear in the sky on any given date, you’ll get the same answers.

As a matter of fact, the greatest astronomer of the medieval period, Tycho Brahe, advocated this model, in which the Sun goes around the Earth while Venus (and the other planets) go around the Sun.

Now take the above picture and imagine what it would look like from the point of view of someone standing on the surface of the Sun. That person would see both Venus and Earth circling the Sun like this:

This picture is exactly the same as the previous one, but with a change of reference frame: everything is drawn from the point of view of the Sun rather than the Earth. Once again, the two models are observationally equivalent. If you freeze the two pictures at any moment, the relative positions of Earth, Sun, and Venus will be exactly the same. That means that an Earthbound observer in either of these two pictures will see the exact same motions of Venus and the Sun.

The last picture is how Copernicus explains the motions of the Sun and Venus. The key point is that, although the three pictures are conceptually quite different, they’re all  observationally equivalent. They’re exactly equally good at predicting where Venus and the Sun will appear on any given day.

No matter which of the three models you use, you get a pretty good approximate model of the planetary positions. In all three cases, if you want greater accuracy, you have to throw in extra complication (e.g., extra little epicycles). People sometimes say that Copernicus’s model was better because it eliminated epicycles, but that’s not true: he got rid of the great big ones, but he still needed the little ones.

Copernicus’s model was not any more accurate than Ptolemy’s (in fact, they were essentially equivalent), and it still had some (although not as many) of the clunky features like epicycles. Moreover, it required everyone to believe in the very unlikely-sounding proposition that the Earth, which feels awfully stationary, was in fact whizzing around at enormous speeds. Given all that, it’s not too surprising that people didn’t immediately fall in love with the new theory.

*Well, not quite first. There was Aristarchus, who proposed the idea that the Earth goes around the Sun about 1700 years before Copernicus. But it appears that nobody listened to him.

2 Responses to “Ptolemy and Copernicus”

  1. A few minor points which I am sure you, but perhaps not all readers, are aware of.

    First, the models are observationally equivalent as long as the distance between a planet and the Earth isn’t observed.

    Second, Tycho Brahe suggested his hybrid model not because he couldn’t fully break with the past (as some have claimed—actually, this probably wasn’t a worry of Tycho at all: anyone who didn’t marry his long-time live-in girlfriend and mother of his children probably didn’t care much for tradition), but because he didn’t understand wave optics. Stars appear to be about an arc minute in size, but this doesn’t reflect their true angular size, which is orders of magnitude smaller. Thus, Tycho assumed that stars are about 30 times as far away as the Sun. At that distance, he (correctly) concluded that he should be able to measure aberration (even though he didn’t have a telescope), but he didn’t, so based his stationary Earth on (wrongly interpreted) astronomical evidence.

    Third, I would classify Tycho as Renaissance and not Medieval.

    Fourth, those interested in the reception of Copernicus’s ideas might want to check this out: http://thonyc.wordpress.com/2012/09/10/dava-sobel-tries-her-hand-at-historical-fantasy/

    Fifth, in Tycho’s system, planets cannot move on crystalline spheres, as in Ptolemy’s model, since they would pass through each other. (Tycho had shown that comets are at interplanetary distances, which proved that the spheres can’t exist.)

    Question: Is the term “Copernican revolution” meant to be a pun?

  2. Ted Bunn says:

    Thanks for the comments, which I agree with completely!

    I had heard that the word “revolution” meaning “dramatic change” actually derived from the title of Copernicus’s book (in which the word has its literal meaning). But Wikipedia (and the OED) say that the term was used in this sense as far back as 1450, so I guess I heard wrong.

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