Galileo thought sunspots were one of the three best arguments for heliocentrism. He was wrong.
Opinionated History of Mathematics
Or subscribe with your favorite app by using the address below
The early days of telescopic astronomy were exhilarating. Listen to this anecdote by Kepler. He is writing in 1610, right after the appearance of Galileo’s first telescope reports. Here’s what Kepler says: “My friend the Baron Wakher von Wachenfels drove up to my door and started shouting excitedly from his carriage: ‘Is it true? Is it really true that he has found stars moving around stars?’ I told him that it was indeed so, and only then did he enter the house.”
Those were the good old days. People shouting in the streets in excitement over scientific discoveries. “Stars moving around stars,” the quote says: that’s moons moving around planets, as we would say today. The moons of Jupiter, in fact. And the “he” who has discovered such things, that’s Galileo.
It’s a big deal, “stars moving around stars,” because it proves that there are several centers of motion in the universe. Not everything revolves around the earth. The earth is not the midpoint of every motion, as the old geocentric vision of the cosmos would have it. Of course in the heliocentric system of Copernicus there are already different centers of motion, because the moon goes around the earth while the planets go around the sun. If you wanted to stick to the old view you could try to argue that it made more sense for everything to have one center, for some metaphysical reason or other. But now that there are moons of Jupiter, that ruins that. Multiple centers of motion are a fact, so this can no longer be used as an argument against Copernicus.
That’s all very important and exciting, but what exactly was Galileo’s role in this? Well, it seems Galileo was indeed the first to observe the moons of Jupiter, but only by the smallest possible margin. His competitor Simon Marius observed them the very next day. In any case one hardly qualifies as the “Father of Modern Science” just by looking.
Nor does Galileo’s account stand out for its scientific excellence. For instance, he tries to “correct” Marius regarding the inclination of the orbits of the moon of Jupiter. Marius found that these orbits sit at an angle to the orbital plane of Jupiter itself. Galileo claimed, no no, they are actually perfectly parallel to Jupiter’s own orbit. But Galileo is wrong, and his opponent is right. As so often happens in such matters. Galileo has no patience for painstaking observations. He oversimplifies, relies on rhetorical points. He is more interested in writing polemics, claiming credit, proving others wrong. He doesn’t have so much time for actual science.
Galileo also tried to do mathematics to the moons of Jupiter, and failed at that too. Amazingly, “Galileo’s first calculations [of the orbital periods of Jupiter’s moons] were geocentric, not heliocentric. … Galileo was treating Jupiter as if it revolved around the Earth, not the Sun. How he ever came to make such an error … is an interesting question,” says one historian. It is interesting, indeed. And we know the answer, don’t we? Galileo couldn’t calculate himself out of a paper bag.
Kepler, meanwhile, of course understood the matter perfectly and realised at once that you need a heliocentric calculation for this to work. The fact that the calculations only work this way is in fact another good argument against the geocentric system.
One Galileo supporter offers a very charitable interpretation of why Galileo didn’t see this: “this throws in doubt the view that by 1611 Galileo was already a Copernican zealot anxious to find every possible argument for the Earth’s motion.” Right. So Galileo didn’t use heliocentric calculations because he was so open-minded, you see. That makes no sense, of course. Galileo was most definitely a “zealot.” A more plausible explanation, in my opinion, is that it was not lack of desire to prove the earth’s motion that made Galileo miss the point, it was lack of ability in mathematical astronomy.
Anyway, let’s change the subject and look at another important novelty discovered when astronomers first turned telescopes to the heavens: sunspots. The sun, the image of brilliance and clarity, turns out to have black spots on it when you study it under the telescope. “Filth on the cheeks of the Sun,” as one contemporary called it. Another was equally disturbed: “Who does not blush that we see the sun occasionally disfigured?”
In Latin there is a saying: adversus solem ne loquitor. Do not speak against the sun. That is, do not argue against what is the clearest and most perfect thing imaginable. How disturbing then that the time had come to in fact argue against the sun. Nothing is sacred as science advances, apparently.
The earliest recorded telescopic observation of sunspots are by Harriot. Before Galileo. Soon many more astronomers across Europe joined in the craze. Of course it is dangerous to stare at the sun, and all the more so through a telescope. Indeed, some people “neglected to observe them, being afraid, … that the image might burn my eye,” as one contemporary put it. But others figured that God had given them two eyes for a reason and “burned” them alternately in the name of science. Thus Harriot “saw it twise or thrise, once with the right ey & [the] other time with the left,” before “the Sonne was to cleare.”
Soon a method was developed for projecting the telescopic image of the sun onto a piece of paper so that no burning of the eyes was needed. That’s convenient enough, but even without this trick there would have been no shortage of martyrs of science willing to pay with their eyes for wisdom.
“Galileo insisted to his dying day that he was the first to have seen” sunspots. But in reality he was “probably preceded by [others, including] the Dutch astronomer Johann Fabricius, who was the first to publish information about them.” To boost his priority case, Galileo later claimed he had seen sunspots already in 1610, rather than in 1611 as documented, but this is almost certainly a lie, as even Galileo’s supporters admit. For that matter, even pre-telescopic astronomers had noticed the phenomenon of sunspots. Already in antiquity there are some allusion to sunspots. It is not impossible to see large sunspots without a telescope.
In any case, the game was now on to explain the nature of the spots. Galileo’s main competitor on this point was Christoph Scheiner, a German astronomer. Scheiner was concerned “to liberate the Sun’s body entirely from the insult of spots,” as he said, for “who would dare call the Sun false?” He found a way of accomplishing this by arguing that the sunspots were “many miniature moons,” rather than blemishes on the sun itself. So, perfection restored.
Galileo on the other hand eagerly embraced sunspots as an opportunity to stick it to his Aristotelian enemies, “for this novelty appears to be the final judgement of their philosophy,” as he said. Thus Galileo placed the spots on the sun itself, arguing that “clouds about the Sun” was the most plausible explanation, for one “would not find anything known to us that resembles them more.” It is true that sunspots are on the surface of the sun—a conclusion, incidentally, which Kepler had already reached before reading Galileo. However, sunspots are not clouds above the surface of the sun, as Galileo believed, but rather dark pits or depressions in the solar surface. “Scheiner … entertained the possibility of this hypothesis, while Galileo resolutely discarded it as unworthy of serious consideration.” Oh well.
Galileo loved claiming new discoveries as his own and using them as ammunition in his philosophical disputes. But he soon lost interest when it came to the detailed work of actual science. Again and again he makes careless errors and jumps to conclusions with premature confidence, while his competitor Scheiner does the meticulous observational work that Galileo had no patience for. For instance, Galileo erroneously claimed—supposedly based on “a great number of most diligent observations of this particular”—that all sunspots had the same orbital period, regardless of latitude. In fact, sunspots near the sun’s equator orbit quicker than those near the poles—a difference of a few days. Galileo was corrected by Scheiner about this.
Galileo also did not miss the opportunity to make some mathematical errors as usual. He tried to compute the perspective aspect of the sunspots’ motion: how does their apparent speed along the sun’s disc vary, given that their actual direction of motion turns more away from us the closer they get to the edge? Galileo’s attempted demonstration covers three pages and contains at least as many errors. Just basic geometry mistakes.
But let’s turn to the most important thing about sunspots. Namely that they can be used as evidence that the earth moves around the sun. In his Dialogue, Galileo considered this one of his three best arguments in favour of Copernicus. Here’s what he says: “The sun has shown itself unwilling to stand alone in evading the confirmation of so important a conclusion [that is, the conclusion that the earth orbits the sun], and instead wants to be the greatest witness of all to this.”
The Copernican argument from sunspots goes as follows. Imagine a standard globe of the earth standing on a table. Its axis is a bit tilted, of course—the north pole is not pointing straight up. Have a seat at one side of the table and face the globe. What do you see? Focus on the equator. What kind of shape is it? If the north pole is facing in your direction, the equator will make a “happy mouth” or U shape. If you move to the opposite side of the table, where you see mostly the southern hemisphere, the equator is instead a sad mouth shape. From the sides, the equator looks like a diagonal line.
Now, suppose the sun had its equator marked on it. And suppose that in the course of a year we would see it as alternately as a happy mouth, straight diagonal, sad mouth, straight diagonal, etc. That would correspond exactly to us moving around the table, looking at a stationary globe from different vantage points. In the same way, if the sun’s equator exhibited those appearances, the most natural explanation would be that the earth is moving around it and we are looking at its equator slightly from above, from the side, from below, etc., just like looking at the globe from different sides of the table.
The sun does not have the equator conveniently marked on its surface, but not far from it. The sun is spinning rather quickly. It makes a full turn in less than a month. As it spins, a point on its surface traces out an equatorial or at least latitude circle. So by tracking the paths of sunspots over the course of a few weeks, we in effect see equatorial and other latitude circles being marked on the surface of the sun.
So the shapes of those paths traced by sunspots show that we are indeed looking at the sun from alternating vantage points. But this does not necessarily mean that we are moving around the sun. The same phenomena could be accounted for from a geostatic or Ptolemaic point of view by saying that the sun is so to speak wobbling. It is showing us different sides of itself in the course of a year, not because we are moving around it but because it is turning different parts of itself in our direction.
You can see the same thing with your globe on the table. Instead of moving around the table and looking at the globe from different sides, you can have a friend tilt the globe, pointing its axis now this way and now that. If you let the axis spin around in a conical motion, this will produce the exact same visual impressions for you as if you had moved around the table.
In order to use the sunspot paths as evidence for Copernicus, then, Galileo needed to dismiss this alternative explanation. He did so by attacking it as physically implausible. To account for the sunspots phenomena from a Ptolemaic point of view, the sun had to orbit the earth, and spin on its own axis, and have its axis wobble in a conical motion. These diverse motions, says Galileo are “so incongruous with each other and yet necessarily all attributable to the single body of the sun.” Surely this is a geometrical fiction that would never happen in an actual physical body.
Actually, such an “incongruous” combination of motions is not only possible but a plain fact. The earth, in fact, has exactly such a combination of motions, as had been known since Copernicus. The earth has a conical wobbling motion which means that the north pole is pointing to a slightly different spot among the stars from year to year. It circles back to its original spot after 26,000 years. This is the explanation for the so-called precession of the equinoxes, an important technical aspect of classical astronomy. So if Galileo’s argument about “incongruous” motions disproves the Ptolemaic explanation of sunspots, it also disproves Copernicus’s correct explanation of the precession of the equinoxes.
Galileo conveniently neglects to bring up this rather obvious problem with his argument. Whether he did so out of ignorance or dishonesty is hard to say, but either way is none too flattering. Any serious mathematical astronomer was well acquainted with the precession of the equinoxes and of course considered it an essential requirement that any serious astronomical system account for this phenomenon. Galileo, though, is not a serious mathematical astronomer. He is a simplistic populariser who simply ignores technical aspects like these. And it is only because of this oversimplification that he is able to maintain his argument against the Ptolemaic interpretation of sunspots.
Or if you like here’s another way we can counter Galileo’s argument. Is it in fact really necessary at all to say that sun’s axis is wobbling in the geocentric explanation of sunspots? Yes and no. It depends on what you mean when you say that one body orbits another. We can picture this with the globe on the table that we considered before. Let’s say you climb onto the table and go sit in the middle of it. Now you grab the globe and make it orbit around you. So you are simulating the hypothesis that the earth is moving around the sun, let’s say. But how exactly do you move the earth? In what way? This will turn out to be more subtle than you might think. You just move the globe around in a big circle, what’s the problem, right? No. Big problem.
Let’s say that you start out with the globe in front of you with its north pole pointing slightly away from you. Remember, the axis is not purely vertical, it’s at an angle. So it’s pointing away from you. So you’re seeing more of Australia than of Siberia or something like that. Now move the globe in a circular orbit around you. Imagine doing this physically. What happens to the axis? Is it still pointing the same way? When you’ve gone halfway around, is the axis pointing toward you now? Or still away from you? Both are quite reasonable ways of conceiving orbital motion.
We can put it this way. Suppose you are not actually touching the globe at all. Instead the globe is standing on a big sheet of paper that covers the entire table. If you want to simulate the orbital motion of the globe around you, how do you move the paper? There are two ways. You can spin the paper around the midpoint, like a roulette wheel. So the midpoint of the paper, where you’re sitting, remains fixed and everything else is spinning around it. Think about what this means for the axis of the globe. If this is how the globe is moved, the axis will keep pointing away from you throughout the entire motion. If the north pole is pointing away or outwards to begin with, it will keep pointing outwards in all the other positions along the orbit as well.
But then there’s a second way you can move the paper. You can slide the entire paper around, without keeping the midpoint fixed. Think of the way you wipe the table with a dishcloth. You put your hand palm down on the table and you make circles with it. You could move the paper that way. Then the globe would go in circles. But if you do it this way, the way the axis of the globe behaves is completely different. If you slide the paper around this way, the axis keeps pointing toward the same end of the room, not the same direction relative to the midpoint of the table as before. So now when you’ve completed half the orbit the north pole is pointing towards you. Unlike before, when it kept pointing away from you.
So those are the two options. What does it mean for a globe to orbit a certain central point? Does it mean that the axis is always pointing the same way with respect to the central point? That’s the roulette wheel case. Or does it mean that the axis is always pointing the same way with respect to the walls of the room? That’s the dishcloth case.
Now, the sunspots. Galileo’s argument that I discussed above assumes that orbital motion is roulette wheel motion and not dishcloth motion. If we put a globe on a roulette wheel and spit it around, it will always show us the same part of itself. So you wouldn’t see that sad-mouth, happy-mouth alteration that happens when you are looking at the equator from above or from below alternately. Instead you would just see for example the sad-mouth equator all the time. But the sunspots show that we do see the equator from different angles, so if we assume that the sun is orbiting the earth, and that orbiting means orbiting like on a roulette wheel, then yes, we must indeed attribute one more motion to the sun; a wobbling of its axis in addition to its orbital motion. This is what Galileo attacks: the multitude of different motions needed. This is what we refuted with the argument about the precession of the equinoxes.
But if we are prepared to say that orbital motion means dishcloth-style motion then Galileo’s objection goes away. In this case the sun will indeed show us happy and sad mouths automatically, without the need for any additional motions. So that makes refuting Galileo even easier.
Now, from a modern point of view, dishcloth motion is a good way to think about orbital motion. The seasons—spring, summer, winter—they are caused by the earth’s axis pointing sometimes toward the sun and sometimes away from the sun. If we were on a roulette wheel the axis would always point for example away and there would always be summer in Australia for example year round. So it makes more sense to think of orbital motion as dishcloth motion. Also from the point of view of Newtonian mechanics, with inertia and stuff, that makes more sense. Gravity doesn’t “turn” things, so to speak.
But in Galileo’s time roulette-wheel motion was the default assumption. Even Copernicus stuck with this, which was a missed opportunity really. It was a kind of relic of older conceptions. Remember the crystalline spheres. Planets are embedded in solid, translucent, spherical shells. The planets each have one, and they fit together like the layers of an onion. The orbital motions of the planets are just a side-effect of the rotations of these entire shells in which they are embedded. So indeed that implies roulette-wheel motion. So before you had Newtonian mechanics this was the most reasonable way of conceptualising the physics of planetary motion.
Anyway, that’s fun to think about but it doesn’t change our point regarding the sunspots. The important point is the one regarding the precession of the equinoxes. This certainly disproves Galileo’s argument. Messing around with the distinction between roulette-wheel and dishcloth motion is more anachronistic and a bit of a tangent that I include for completeness and intrinsic interest. And in any case, bringing that in certainly wouldn’t save Galileo but on the contrary it would completely remove the entire basis for his argument altogether. Either way Galileo loses.
Ok, so we sorted that out. Now let’s get back to history. It is striking that, in the 1610s, when he was first studying sunspots, Galileo completely missed all of this stuff. I don’t mean the point about the precession of the equinoxes, which he never acknowledged at any stage. But even the very idea that sunspots can support heliocentric theory. That entire idea was missed by Galileo for decades. It just didn’t occur to him. And it was his own sloppiness that cost him this discovery. Let’s see how.
As I said, Galileo lacked the disposition to do painstaking scientific research like Scheiner. Instead, with premature hubris, he soon imagined that he had “looked into and demonstrated everything that human reason could attain to” regarding sunspots. Those are his own bombastic words. Many years later he was still convinced that his was the last word on the matter: as one historian observes, “writing … apropos of recent news that Scheiner would soon publish a thick folio volume on sunspots, [Galileo] remarked that any such book would surely be filled with irrelevancies, as there was no more to be said on the subject than he had already published in his Letters on Sunspots.”
But Galileo’s arrogance proved unfounded. When Scheiner’s much better work on sunspots came out, Galileo realised he had to completely reverse his earlier proclamations, even though he had stated those things with such confidence. With unwarranted pomposity, he had claimed to offer “observations and drawings of the solar spots, ones of absolute precision, in their shapes as well as in their daily changes in position, without a hairsbreadth of error.” According to Galileo, the sunspots were “describing lines on the face of the sun”: “they travel across the body of the sun … in parallel lines.” In fact, “I do not judge that the revolution of the spots is oblique to the plane of the ecliptic, in which the earth lies.” In other words, every sunspot path is straight as an arrow, just as the equator of a globe would be from every side if its axis was perfectly vertical.
But Scheiner showed that the sun’s axis in fact does have a inclination of just over 7 degrees and that the paths exhibit exactly those alternating diagonal and U shapes that we discussed before. He published this result in 1630, in the folio Galileo had mocked as bound to be superfluous. It was from this work that Galileo now realised his error.
When Galileo finally realised that inclined sunspot paths spoke in favour of heliocentrism, he immediately threw all his old observations out the window. These were the observations he had called “without a hairsbreadth of error,” if you recall. Galileo had been so proud of those observations for decades, but now they contradicted the point he wanted to make regarding heliocentrism, so he pretended they didn’t exist. Galileo has a very lax relation to empirical data as usual. One minute his observations are “without a hairsbreadth of error” but the next thing you know the Facts According to Galileo have changed radically into something completely different that is in direct contradiction with his own explicit statements just moths before.
Anyway, now that he had made up his mind about which way he was supposed to fudge the data, Galileo rushed the pro-Copernican argument into print without making any new observations. This is clear from the fact that the published argument “displays entire ignorance or complete neglect of the observational data,” his vague descriptions being “utterly wrong” and “almost the exact opposite” of the careful data published by Scheiner. Those are quotes from Stillman Drake, Galileo’s greatest admirer. “Ignorant” and “utterly wrong”—those are very harsh words from your greatest supporter!
But that actually is the most charitable reading of Galileo. Stillman Drake, who is always trying to save Galileo, is driven to call Galileo ignorant in order to avoid an even greater disgrace: namely, that Galileo plagiarised Scheiner’s book and then tried to pass these things off as his own discoveries. So Galileo’s defence lawyer is saving him from the charge of plagiarism by pointing out that Galileo’s account has a thousand errors in it, while Scheiner’s does not. If Galileo was a plagiarist, how come his book stinks and gets all the facts wrong? That’s some “defence,” isn’t it? But there you have it.
Indeed, Galileo did not want to admit his debt to Scheiner, so he pretended that he had come upon this discovery independently. He lied that he had made, as he says, “very careful observations for many, many months, and noting with consummate accuracy the paths of various spots at different times of the year, we found the results to accord exactly with the predictions.” In reality, says one modern scholar, “the evidence is unequivocal: Galileo … must have had a copy of Scheiner’s book in front of him as he wrote this section.” By pretending otherwise, “Galileo has deliberately set out to efface Scheiner from the historical record and to deny his debt to him. It is impossible to find any excuse for this behaviour.”
So let’s stop there with those apt words from David Wootton. But there’s plenty more “inexcusable behaviour” coming up; we haven’t even discussed all of Galileo’s shenanigans with the telescope yet. Until next time.