Two thousand years before Galileo, Greek astronomers argued that the heavenly bodies revolve around the sun. Their reasoning involved sophisticated mathematics and sound physical considerations.
Opinionated History of Mathematics
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Is the earth the center of the universe? Or does it orbit around the sun? The Greeks had some interesting ideas about this. For example, the earth is basically a big rock, right? Well, have you ever seen a rock just hovering in empty space without being supported by anything? Of course not. That would be crazy. On the other hand, think of a rock in a sling. Like a slingshot, the ancient weapon. Like what David used against Goliath, for example. So the rock is at the end of a string and you are whirling it about in a circle. In this way you can in fact keep a heavy rock suspended in the air indefinitely. Without there being any ground or support on which it is resting.
This is a powerful argument in favour of the hypothesis that the earth revolves around the sun. Of course this theory requires that the sun, on the other hand, is in fact hovering in empty space. So aren’t we back to square one, the same problem we started with? No, not at all. Because the sun is made of fire, isn’t it? Fire has no problem levitating. It’s a weightless substance, as everybody knows. Just light a match and see how the flame rises completely unencumbered by gravity and feel how no weight was added to the match even though the flame has considerable volume.
So on the basis of this we can conclude that the earth moves around the sun. Because that agrees much better with everyday physical experience than the outlandish hypothesis that a massive chunk of rock is just sitting there in nothingness without falling.
As you may know, this is not the “official” Greek position, so to speak. It’s not Aristotle’s opinion, and it’s not Ptolemy’s opinion. Those are the main authorities that have come down to us on these issues. Aristotle the philosopher, Ptolemy the astronomer. They are usually taken to express the party line, as it were, of “the Greeks.” These people did indeed put the earth at the center of the universe.
But I want to do some revisionist history here and speak for the underdogs of Greek cosmology. Those guys had some good ideas. Unfortunately their works are lost. We have only some scraps to go by. But the indications we have are very intriguing.
In fact, this idea that I just described, that it’s unrealistic for a heavy rock to simply levitate in empty space, is mentioned by Ptolemy. He brings it up only in order to dismiss it, of course. But more interesting than Ptolemy’s counterarguments is what he is implicitly telling us about his opponents. We must reverse-engineer his text: What does Ptolemy’s dismissals of alternative views tell us about what those views must have been?
For example, Ptolemy tries to argue that, if the earth moved, any loose objects would be thrown off. Just like objects placed on a sleeping animal would immediately fall off as soon as it woke up and started running about. But from the way Ptolemy presents this argument it is clear that he by no means thinks this is self-evident and will be taken for granted by his readers. On the contrary, he addresses counterarguments to this view that show that his opponents, whoever they were, had a well articulated theory based on something like inertia or relativity of motion quite similar to how we today would explain why a moving earth does not throw things off.
It is interesting also that Ptolemy acknowledges the multitude of his opponents. To “all those who” believe in the earth’s motion, I reply as follows, he says. “All those”! Isn’t that an interesting phrase? Evidently there were lots of people who believed the earth moved. That’s what Ptolemy’s own words are saying. What a pity that we don’t have any books by “all those” rebels anymore.
Here’s another important conclusion we can draw from the above example: these forgotten astronomers clearly believed that the heavens should be explained in terms of everyday physics. The earth is like a rock in a sling, for example. That way of thinking was largely rejected in the Aristotelian tradition, where the heavens are seen as a profoundly different kind of thing altogether than what we encounter here on earth. The heavens are some kind of quasi-divine realm, made of some sublime fifth element, and so on. That’s the basic Aristotelian story. But the mathematicians did not care much for that fairytale.
Actually even the Aristotelian tradition made more concessions to heavenly physics than some people like to admit, in my opinion. The Aristotelian story is that the planets are enclosed in enormous crystalline spherical shells that fill up the entire universe like the layers of an onion. The planet itself is just a little speck stuck inside this translucent spherical shell like tiny imperfection in hand-blown glass.
Why did the Aristotelians feel the need for this fiction? Well, they didn’t believe vacuum could exist, so that’s one reason to fill the heavens with some material or other. But I think a more compelling reason was the physics of perpetual motion. Think about it: What kinds of sustained circular motions are you familiar with from everyday experience? Have you ever seen a rock just go in a circle over and over again spontaneously, without being guided by any other material objects? Of course not.
You have, however, seen many sustained circular motions where the object moving circularly is materially connected to the midpoint, such as a rock in a sling or a wheel. Or a millstone disc that has been set in rotational motion and then keeps going by inertia even after you have stopped applying force. A sphere behaves the same way. In fact, it can be a hollow sphere. Like a basketball spinning on your finger. You set in in motion and it keeps going, seemingly forever if it wasn’t for outside resistance slowing it down.
The Aristotelian “onion” universe fits with that physical intuition. The translucent layers of the onion spin on their axis forever just like a basketball. That is why the planets, which are embedded in the shells, go around and around. So it’s a model of how planetary motions work based on a mechanism that is familiar to everyone from everyday experience.
So that’s some physics even in the conservative Aristotelian view. Let’s see how far the more mathematically creative Greeks pushed the idea of a physical analysis of the heavens.
First of all it is quite evident that the spherical shape of the earth can be explained as a consequence of gravitational forces. In fact, Archimedes proves as a theorem in his hydrostatics that a spherical shape is the result or equilibrium outcome of basic gravitational assumptions. This idea, that gravitational forces are the cause of the spherical shape of the earth, is explicitly stated in ancient sources.
But of course other heavenly bodies are round too, such as the moon for example. So that very naturally suggests that they have their own gravity just like the earth. This conclusion too is explicitly spelled out in ancient sources. Here’s Plutarch: “The downward tendency of falling bodies is evidence not of the earth’s centrality but of the affinity and cohesion to earth of those bodies which when thrust away fall back again. … The way in which things here [fall] upon the earth suggests how in all probability things [on the moon] fall … upon the moon and remain there.”
Now, from this way of thinking, it is a short step to the idea that the heavenly bodies pull not only on nearby objects but also on each other. This is again explicit in ancient sources. This is why Seneca, for example, says that “if ever [these bodies] stop, they will fall upon one another.” That is correct, of course. The planets would “fall upon one another” if it wasn’t for their orbital speed.
This point of view explains the motions of the planets in terms of physical forces. It’s not that the planets have circularity of motion as an inherent attribute imbedded in their essence, as Aristotle would have it. Rather, circularity is a secondary effect, the result of the interaction of two primary forces: a tangential force from motion and a radial force from gravity. There are clear indications that ancient astronomers worked out such a theory, including a mathematical treatment.
Here is Vitruvius for example: “the sun’s powerful force attracts to itself the planets by means of rays projected in the shape of triangles; as if braking their forward movement or holding them back, the sun does not allow them to go forth but [forces them] to return to it.” Pliny says the same thing: planets are “prevented by a triangular solar ray from following a straight path.”
All this talk of triangles, in both of these authors, certainly suggests an underlying mathematical treatment. Indeed, the Greeks knew very well the parallelogram law for the composition of forces or displacements, and in fact explicitly used this to explain circular motion as the net result of a tangential and a radial motion.
Isn’t it beautiful how coherently all of that fits together and how naturally we were led from one idea to the other? Just like the water of the oceans naturally seeks a spherical shape, so the spherical shape of the earth has been formed by the same forces. And just as gravity explains why the earth is round, so it must explain why other planets are round. Hence they have gravity. But just as they attract nearby objects, so they attract each other. So the heavens have a perpetual tendency to lump itself up, except this tendency is counterbalanced by the tendency of speeding objects to shoot off in a straight line.
This physical view of the heavens clearly had much to commend it. And clearly the Greeks saw this, even though the original works are lost and we are left with only the kinds of scraps I quoted from the Roman era.
Let’s take a closer look at one of the earlier lost works in particular. Aristarchus. He was a quality mathematician. We know for a fact that Aristarchus wrote a treatise advocating the motion of the earth about the sun. Archimedes, who was a contemporary of Aristarchus, mentions this work, basically with tacit approval.
Archimedes brings this up in connection with a discussion of the size of the universe. Aristarchus’s theory implies that the universe must be very big. This is because of parallax, which means the following. If you move from one side of room to another, your view of everything on the walls will change. The wall you are approaching will appear to “grow,” so to speak, while the wall behind you will shrink and occupy a smaller part of your field of vision. This is what is called parallax. If the earth moves in an enormous circle around the sun, we should be at one moment close to some particular constellation of stars, and then half a year later much further away from them. Hence we should see them sometimes big and “up close,” and sometimes shrunk into a small area, like a faraway wall at the end of a long corridor. But no such effect is observed. The only way to reconcile this with the motion of the earth is to stipulate that the stars are so far away that the diameter of the earth’s orbit around the sun is insignificant by comparison. So the universe must be very big indeed for this theory to work.
From the remark about this in Archimedes we learn quite a lot, even though Aristarchus’s treatise is lost. We learn that Aristarchus’s theory was worked out in some detail. It was a serious scientific proposal. It grappled with nontrivial observational and theoretical implications in a substantive way. And it evidently did so quite convincingly. For why else would Archimedes take the theory seriously?
Many people refuse to believe this. For example, there was a paper on Aristarchus in the January 2018 issue of the Archive for History of Exact Sciences. According to this paper, “pre-Copernican heliocentrisms (that of Aristarchus, for example) have all the disadvantages and none of the advantages of Copernican heliocentrism,” because they postulated only that the earth revolves around the sun, not, as has commonly been assumed, that all the other planets do so as well. This supposedly “explains why Copernicus’s heliocentrism was accepted …, while pre-Copernican heliocentrism” was not.
This is completely wrong, in my opinion. And for an obvious reason. Namely: Why would Aristarchus have affirmed and written a treatise on heliocentrism if it had nothing but disadvantages? What possible reason could he have had done for doing so? None, in fact. Yet this is exactly what this recent article proposes.
It is a fact that Aristarchus asserted the physical reality of his hypothesis. And it is a fact that he recognised the parallax argument against it. Even the recent article I cited admits this. So why, then, would Aristarchus write a treatise proposing this bold hypothesis, discuss a major argument against it (namely the parallax argument) and no arguments in favour of it, and then nevertheless conclude that the hypothesis is true? And, furthermore, why would Archimedes, perhaps the greatest mathematician of all time, cite this treatise with tacit approval as a viable description of physical reality? None of that makes any sense.
The only reasonable explanation is that Aristarchus did in fact recognise an advantage of placing the sun in the center. And the obvious guess for what this was is that he saw the same advantages as Copernicus did. Including the more natural explanations of the retrograde motion of the outer planets and the bounded deviation from the sun of the inner planets that I have discussed before.
Now, Aristarchus’s treatise on heliocentrism is lost, as I said. However, another astronomical treatise by Aristarchus has survived. In this work Aristarchus calculates the relative distances and sizes of the sun, the earth, and the moon. This treatise shows that Aristarchus was at any rate a highly competent mathematician. But I think it shows much more than that. I think it feeds directly into his heliocentrism.
An important argument for heliocentrism is this: Smaller bodies orbit bigger ones. Not the other way around. This conforms with everyday experience. For instance, take a lead ball and a ping-pong ball, and tie them together with a string. If you flick the ping-pong ball it will start spinning around the lead ball. But if you flick the lead ball it will roll straight ahead without any regard for the ping-pong ball, which will simply be dragged along behind it. So the lighter object adapts its motion to the heaver one but not conversely. The planets are not tied to the sun with a string but the point generalises. You can observe the same principle with a big and a small magnet for example: the little one is moved by the bigger, not the other way around.
Kepler used this argument in the 17th century. Here is how he put it: “Just as Saturn, Jupiter, Mars, Venus, and Mercury are all smaller bodies than the solar body around which they revolve; so the moon is smaller than the Earth … [and] so the four [moons] of Jupiter are smaller than the body of Jupiter itself, around which they revolve. But if the sun moves, the sun which is the greatest … will revolve around the Earth which is smaller. Therefore it is more believable that the Earth, a small body, should revolve around the great body of the sun.”
The moons of Jupiter were not known in antiquity but other than that this is a basic idea that fits very well with their extensive attention to the physics of the heavens.
Surely this must have occurred to Aristarchus. Or are we supposed to believe that Aristarchus calculated the sizes of heavenly bodies just for kicks in one treatise and did not see any connection with the heliocentrism he advanced in another treatise even though the obvious connection was right under his nose? What is the probability that he suffered from such schizophrenia? Virtually zero, in my opinion.
In fact there are certain aspects Aristarchus’s treatise on sizes and distances that make much more sense when you read it this way. On its own it is a weird treatise. On the one hand it calculates the sizes and distances of the sun, moon, and earth in a mathematically sophisticated manner. Very detailed, technical stuff, including the completely pointless complication that the sun does not quite illuminate half the moon but only maybe 49.9% of its surface or something like that. This is “pure mathematical pedantry,” as Neugebauer calls it. It makes the geometrical calculations ten times more intricate while having only the most minuscule and completely insignificant impact on the final results.
On the other hand, the observational data that Aristarchus uses for his calculations are extremely crude. He says that the angular distance between the sun and the moon at half moon is 87 degrees. A pretty terrible value. The real value is more like 89.9 degrees. Because of this his results are way off. For instance, his calculated distance to the sun is off by a factor of 20 or so.
So what’s going on? Why do such intricate mathematics with such worthless data? Did he just care about the mathematical ideas and not about the actual numbers? I think it would be a mistake to jump to that simplistic conclusion, even though many people have done so.
In fact, it is easy to see how Aristarchus had a purpose in underestimating the angle. His purpose with the treatise, I propose, is to support his heliocentric cosmology based on the principle that smaller bodies orbit bigger ones.
This hypothesis fits very well with the structure of Aristarchus’s treatise. The treatise has 18 propositions. Proposition 16 says that the sun has a volume about 300 times greater than the earth, and Proposition 18, the very last proposition, says that the earth has a volume about 20 times greater than the moon. These are exactly the propositions you need to explain which body should orbit which. And that is exactly where Aristarchus chooses to end his treatise.
Many commentators have been puzzled by why Aristarchus ends in that strange place. In particular, many have been baffled by why he does not give distances and sizes in terms of earth radii. This seems like the natural and obvious thing to do, and doing so would have been easily within his reach. Many modern commentators add the small extra steps along the same lines needed to fill this obvious “gap.”
Except it’s not a gap at all and there is no need to be puzzled by Aristarchus’s choices. If we accept my hypothesis, everything he does makes perfect sense all of a sudden. He carries his calculations precisely as far as he needs for this purpose, and no further.
And my hypothesis also explains why he chose such a poor value for the angular measurement. He has every reason to purposefully use a value that is much too small. Underestimating this angle means that the size of the sun will underestimated. And his goal is to show that the sun is much bigger than the earth. So he has shown that even if we grossly underestimate the angle, the sun is still much bigger than the earth. So he has considered the worst case scenario for his desired conclusion, and he still comes out on top. That just makes his case all the stronger, of course.
Clearly my interpretation requires that Aristarchus knew that 87 degrees was an underestimate. The standard view in the literature is that he could not have known this. Aristarchus’s numerical data are “nothing but arithmetically convenient parameters, chosen without consideration for observational facts”. That’s a quote from Neugebauer. And he continues:
“It is obvious that [Aristarchus’s] fundamental idea … is totally impracticable. … 87 degrees is a purely fictitious number. … The actual value … must … [have] elude[d] direct determination by methods available to ancient observers.”
Neugebauer tries to prove this as follows. You are trying to measure the angle between the sun and the moon at half moon. But to do that you need to pinpoint the moment of half moon, which can only be done with an accuracy of maybe half a day. But in half a day the moon has moved six degrees, and therefore radically changed the angle you are trying to measure. So your observational value is going to have a margin of error of 6 degrees, which is enormous and makes the whole thing completely pointless.
That’s Neugebauer’s opinion. But I’m not convinced that it’s as hopeless as all that. One way to work around the problem would be to use not one single observation, but the average of many observations. There is little evidence that the Greeks ever made use of averaging that way, but the idea is simple enough.
I did a bit of statistics to see if this would be viable. Let’s assume that our angular measurements are normally distributed about the true value. Neugebauer says that one would be lucky to get the moment of half moon correct to the day. So we can tell it’s today rather than yesterday, but we can’t tell at what exact hour the moon is exactly half full. Let us roughly translate this into statistical terms by saying that the observations have a standard deviation of 12 hours, or six degrees.
Now, an astronomer active for, let’s say, two decades would have occasion to observe about 500 half moons. So say he makes 500 angular measurements and then average them. This would produce an estimate of the true value with a 95% confidence interval of plus/minus about half a degree. A margin of error of half a degree is easily enough to support my interpretation that 87 is a conscious underestimate.
Naturally, Aristarchus would not have reasoned in such terms exactly, but it is not necessary to know any statistical theory to get an intuitive sense of the order of magnitude of the error in such an estimate. As you keep adding observations, and keep averaging them, you will see the average stabilising over time, of course. So certainly it will become clear after a while that, whatever the true value is, it must be greater than 87 degrees at any rate.
It is certainly extremely speculative to imagine that Aristarchus might have had something like this in mind. But in any case my argument shows that it cannot be ruled out as out of the question that Aristarchus could in principle have had solid empirical evidence that his value of 87 degrees was certainly an underestimate.
So, in conclusion: Aristarchus was a good mathematician. He proposed a heliocentric theory that was taken very seriously by Archimedes. There was a long tradition in Greek thought of trying to account for the motions of the planets in terms of everyday physics. This is naturally connected to heliocentrism because of the natural idea that smaller bodies orbit bigger ones. Aristarchus in fact wrote a major treatise devoted specifically to comparing the sizes of the sun and the earth, and the earth and the moon. Several otherwise peculiar aspects of the treatise fit like a glove the idea that it was written precisely to lend credibility to heliocentrism.
On the whole, non-Ptolemaic Greek astronomy was fascinating. It was full of interesting and correct ideas. Nowadays we are stuck with Ptolemy as the canonical source for Greek astronomy. But Ptolemy lived hundreds of years after the golden age of Greek science. It is likely that he was not the pinnacle of Greek astronomy, but rather a regressive later author who perhaps took astronomy backwards more than anything else. Let us keep that in mind as we turn to Galileo’s Dialogue, in which Ptolemy is the designated punching-bag and symbol of stale received wisdom.