And now for something completely different . . .

In the geocentric Ptolemaic model, the planets occupy concentric spheres around the Earth. The order, from nearest-Earth to farthest-from-Earth, is as follows:

- Moon
- Mercury
- Venus
- Sun
- Mars
- Jupiter
- Saturn

This is obviously very similar to the heliocentric Copernican order. In fact the only difference is that the Sun and Moon have switched places. (If the Moon is modeled as orbiting the Sun, its basic orbit, or “deferent” in Ptolemaic terms, would be the same as the Earth’s, with an Ptolemaic-style “epicycle” added to model its motion around the Earth. At any rate, its Copernican location is between Venus and Mars.)

My first thought was that this made perfect sense: the Sun and the Earth swap places, and everything else stays the same. But then I immediately realized that it didn’t actually make any sense at all. The superior planets (Mars, Jupiter, and Saturn) are no problem. But by what logic does the Ptolemaic order Earth-Mercury-Venus-Sun correspond to the Copernican Sun-Mercury-Venus-Earth? To try to figure this out, I made this simplified model of the Sun and the first three (Copernican) planets.

To make the math simpler, the model has circular rather than elliptical orbits, and the orbits are regularly spaced in terms of distance from the Sun.

An arbitrary point on Earth’s orbit, marked E, is selected as a point of reference. In the model, the distance from the Earth to the Sun (S) is always precisely 3 units (a “unit” being the distance indicated by the grid squares, corresponding to one-third of an AU). How far, on average, is Venus from the Earth on this model? Closer than the Sun, or farther? To ask this is to ask the average distance from E of all the (infinitely many) points constituting the circle of Venus’s orbit. Since I lack the mathematical ability to make such a calculation (I never took calculus, which I assume is what is called for), I selected eight easily-calculable and hopefully-representative points on each orbit.

The simplest calculation is to look at two points only: the point on Venus’s orbit which is closest to E (namely, V_{1}), and the one farthest from E (V_{5}). Trivially, the distance from V_{1} to E is 1 unit, and that from V_{5} to E is 5 units — for an average of 3, exactly the same distance as the Sun. Doing the same calculation for Mercury yields the same result. Based on this simplest of calculations, Venus, Mercury, and the Sun are all, on average, equidistant from the Earth.

Although counterintuitive in some ways, that result makes a sort of sense. After all, if you averaged the coordinates of every point on a circle, wouldn’t the result be the coordinates of its center? And doesn’t that mean that the *average* position of each planet is simply the position of the Sun, and that its average distance from the Earth is therefore the same as the Sun’s?

But a little reflection shows that that can’t possibly be right. After all, the orbit of a hypothetical planet 10,000 units from the sun is *also* (in this model) a circle with the Sun at its center, but its average distance from the Earth can’t possibly be 3 units. Nor, in fact, does the two-point calculation suggest that it would be. If “Mars” is added to the model, at a distance of 4 units from the Sun, its average distance from the Earth would be 4 units, not 3. Two-point calculations tell us that for any two planets, A and B, such that A is closer to the Sun, the average distance between A and B is equal to the distance between B and the Sun. Can *that* be right?

Instead of looking at only two points on each orbit, what if we looked at four? Basic math (the Pythagorean theorem) tells us that both V_{3} and V_{7} are √13 units from the Sun, while the corresponding figure for M_{3} and M_{7} is √10 units. Averaging the four points for each planet gives us an average distance of 3.3028 units for Venus and 3.0811 units for Mercury. If we look at all eight of the points marked on each orbit, it yields an average distance of 3.3414 units for Venus and 3.0839 units for Mercury.

In other words, as the number of points included in the sample increases, the average distance-from-Earth figures for Mercury and Venus diverge from one another and from the Sun’s. This suggests that if we were to calculate the average distance-from-Earth of *all* the points on each orbit, we would find that Mercury is (on average) farther from the Earth than the Sun is, and that Venus is farther still. I still can’t understand intuitively *why* that should be the case, but it seems to be what the math indicates.

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Why, you are asking, did I not save myself some trouble and just *look up* the average distance of each planet from the Earth? Actually, I tried that, but all I could find was garbage produced by someone even less mathematically gifted than myself! This page, from the authoritative-sounding site theplanets.org and promisingly titled “Distances Between the Planets of the Solar System,” says that Mercury and Venus average 0.61 and 0.28 AU from the Earth, respectively. The Sun of course averages 1 AU, so these numbers completely contradict mine, saying that Venus is closer than Mercury, and Mercury is closer than the Sun. In fact, it even says *Mars* is closer than the Sun, which is obviously ridiculous!

It turns out that whoever made the site calculated the numbers this way: Earth averages 1 AU from the Sun, and Mercury averages 0.39. Therefore, the average distance between the Earth and Mercury is 1 – 0.39 = 0.61 AU! This is such obvious hogwash that I wonder how someone who thinks that way had the chutzpah to set up an astronomical reference site. Applying the same logic to my simplified model, Mercury is 1 unit from the Sun and the Earth is 3, so Mercury should average 2 units from Earth. Looking at the diagram, you can see that 2 units is the distance between E and M_{1} — and M_{1} represents Mercury’s *closest approach* to Earth, not its average distance.

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Anyway, I’m no closer than before to understanding why the Ptolemaic spheres are arranged as they are. Based on my rough-and-ready math, the “correct” geocentric order of the planets should be

- Moon
- Sun
- Mercury
- Venus
- Mars
- Jupiter
- Saturn

with many of the “spheres” overlapping rather more than not. I’m still trying to figure out what observations led to the actual Ptolemaic model, with its bizarre Mercury-Venus-Sun order.

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**UPDATE:**

After thinking about it a bit more, I now understand the mathematical results I got. In the diagram below, the red arc consists of points that are exactly 3 units away from E (i.e., the same distance as S).

For any given orbit, such as Mercury’s or Venus’s, the portion “north” of the red arc is closer to the Earth than the Sun is, and the portion “south” of the arc is farther. If the planets and Sun were all on average equidistant from the Earth, the red arc would be a horizontal line. Because it is in fact an arc, not a horizontal line, the “northern” part of each orbit is smaller than the “southern” and the difference increases as the orbit increases in distance from the Sun. (If I had been a bit quicker on the uptake, I would have drawn this arc to start with instead of making all those calculations!)

I am now absolutely certain that no planet orbiting the Sun is, on average, closer to the Earth than the Sun itself is, and that the Ptolemaic placing of the Sun in the fourth sphere is simply wrong, even by geocentric standards. The question remains as to why such an extreme error was made and what data it was based on.

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**UPDATE 2:**

Synchronicity alert!

The day after posting this, while in middle of composing a follow-up post also about the “correct” arrangement of the Ptolemaic spheres, I took a break from writing to prepare for an English class I teach. I was using a textbook I had never taught from before, intended to prepare students for a standardized English test administered by the Taiwanese government. For one part of the test, they have to look at a picture, listen to a recorded question and several possible answers, and choose the correct answer based on the picture. The book included this picture as part of a practice exercise for this part of the test.

And here’s the transcript of the audio that goes with the picture.

That’s right. It shows students using calculators to make calculations regarding the orbits of planets, which is precisely what I did in preparing this post. The correct answer for Question 15 is “They are doing calculations.”

Question 14 is poorly designed. Are they learning “physics” or “natural science,” and in any case isn’t the former a subset of the latter? “Ancient history” is clearly not the right answer — but it would be if they were learning geocentric Ptolemaic astronomy!