Galileo’s theory of tides

Galileo dismissed the notion that the moon influences the tides as “childish” and “occult.” Instead he argued that tides are a kind of sloshing due to the motion of the earth. This very poor theory is inconsistent with several of his own scientific principles.

Subscribe to

Opinionated History of Mathematics

Or subscribe with your favorite app by using the address below


Transcript

At the end of his famous Dialogue, Galileo lists what he considers to be his three best arguments for proving that the earth moves around the sun. One of these is his argument “from the ebbing and flowing of the ocean tides”. High and low tidal water. Galileo believed the tides were caused by the motion of the earth. This is truly one of his very worst theories. He was so proud of it. But it stinks. And I will tell you why.

First things first. How do the tides work? As we know today, “the ebb and flow of the sea arise from the action of the sun and the moon.” That’s Newton’s accurate statement of the correct explanation for the tides. They are a consequence of gravitational forces, as Newton proved. The moon, and to a lesser extent the sun, pull water towards them, causing our oceans to bulge now in one direction, then the other. This was clearly understood already in Galileo’s time. Kepler explained it perfectly, and many others likewise proposed lunar-attraction theories of the tides. In fact, the lunisolar theory of tides is found already in ancient sources, including the causal role of the sun and moon, and descriptions of the effects in extensive and accurate detail.

Galileo, however, got all of this completely wrong. Why should “the tides of the seas follow the movements of the fireballs in the skies,” as Kepler had put it? Galileo considered the very notion “childish” and “occult,” and declared himself “astonished” that “Kepler, enlightened and acute thinker as he was, … listened and assented to the notion of the Moon’s influence on the water.” Those are Galileo’s words. Continuing in his treatise, he writes: “There are many who refer the tides to the moon, saying that this has a particular dominion over the waters … [and] that the moon, wandering through the sky, attracts and draws up toward itself a heap of water which goes along following it.”

Yes, many indeed believed such things. And they were right. But Galileo would have none of it. This theory is not “one which we can duplicate for ourselves by means of appropriate devices,” he objects. How indeed could we ever “make the water contained in a motionless vessel run to and fro, or rise and fall”? Certainly not by moving about some heavy rock located many miles away. “But if,” says Galileo, “by simply setting the vessel in motion, I can represent for you without any artifice at all precisely those changes which are perceived in the waters of the sea, why should you reject this cause and take refuge in miracles?”

That’s Galileo’s objection to the lunar theory of tides: It’s hocus-pocus. It assumes the existence of mysterious forces that we cannot otherwise observe or test. Proper science should be based on stuff we can do in a laboratory, like shaking a bowl of water.

Actually I think this argument is not a half bad. How is the moon supposed to influence the oceans across thousands of miles of empty space? What reason do we have to believe that such a force exists? It doesn’t fit with any common-sense knowledge. It doesn’t fit with any empirical experience. It’s a wild idea. A kind of mysticism.

But maybe that’s the lesson of the story. Sometimes wild ideas are right. Rational scientific prudence is not all it’s cracked up to be. It’s a good thing we had some Keplers who said “why not?” instead of only Galileos who said “don’t be silly.”

A “childish,” “occult,” theory based on “miracles.” That’s what Galileo thought of the lunar theory of tides. He put it in so many words, literally. And with good reason. But he was wrong. The world is “occult,” it turns out.

Nowadays we’ve been brainwashed into thinking that the moon’s pull on the oceans is just “science” and there’s nothing weird about it. But isn’t it just as occult as it ever was? Isn’t that what science is, actually? A bunch of occult stuff that we’ve gotten so used to that we’ve given it another name?

By the way, if I may editorialise a bit further, don’t we have a lot of mini Galileos running around today? All these self-proclaimed rational disciples of science, who are out to tell you about all your dumb beliefs being nothing but a “childish” faith in “occult” “miracles”? Think of vaccines causing autism, homeopathy, creationism. There’s an army of little Galileo clones waging war on those occult things. Perhaps it is worthwhile to remember, on such occasions, that Galileo was just as sure with his argument about tides in a bucket.

But enough about that. Now, if Galileo didn’t believe any of that stuff, then what did he believe? Let’s turn to Galileo’s own theory of the tides. I like to think of it in terms of a torus, as we say in mathematics. That’s a ring, like a hollow donut, or one of those inflatable rings that children put around their waists when they play in the pool.

To understand Galileo’s theory, picture one of those rings, a torus, filled halfway with water. So it’s lying flat on the ground and the bottom half of it is filled with water. This represents the water encircling the globe of the earth. Now spin it. Spin the torus in place, around its midpoint, like a steering wheel. This represents the rotation of the earth.

For symmetry reasons it is clear that the water will not move. Because there is no reason for the surface of the water to become higher or lower in one place of the tube than another. Every part is rotating equally, everything is symmetrical, and therefore no asymmetrical distribution of water could arise from this process.

But now picture the torus sitting on a merry-go-round. The represents the earth orbiting the sun. And also at the same time it is still spinning around its own midpoint like before. Now there is asymmetry in the configuration. The torus has one part facing inward toward the middle of the merry-go-round, and one part facing outward. As far as the rotation of the torus around its midpoint is concerned, these two parts of the torus are moving in opposite directions. But as far as the rotation of the merry-go-round is concerned they are both going the same way. So one part of the torus spins along with the orbital motion, and one part against the orbital motion. Therefore one part of the water moves faster than the other. It is boosted by the merry-go-round rotation helping it along in the direction it was already going, while the other part of the torus is slowed down; the merry-go-round cancelling its efforts by going in the opposite direction.

So you have fast-moving water and slow-moving water. The fast-moving water will catch up with the slow water and pile up on top of it, creating a high tide. And the space it vacated will not be replenished because the slow water behind it isn’t keeping up. Hence the low tide. Let’s hear Galileo’s own words; here’s how he puts it: “Mixture of the annual and diurnal motions causes the unevenness of motion in the parts of the terrestrial globe. … Upon these two motions being mixed together there results in the parts of the globe this uneven motion, now accelerated and now retarded by the additions and subtractions of the diurnal rotation upon the annual revolution.”

Thus, in fact, “the flow and ebb of the seas endorse the mobility of the earth.” That’s Galileo’s conclusion. This was one of his favourite arguments for the motion of the earth around the sun.

Unfortunately, Galileo’s theory is completely out of touch with even the most rudimentary observational facts about tidal waters. High and low water occur six hours apart. In the lunisolar theory this is explained very naturally as an immediate consequence of its basic principles. Namely as follows. The rotation of the earth takes 24 hours. There’s a wave of high water pointing toward the moon, basically (give or take a bit of lag in the system), and then another high water mark diametrically opposite that that on the other side of the earth. So that’s two highs and two lows in 24 hours, so 6 hours apiece.

Galileo’s theory, on the other hand, implies that high and low water should be twelve hours apart rather than six: “there resides in the primary principle no cause of moving the waters except from one twelve-hour period to another.” Those are Galileo’s own words in his big treatise. So Galileo immediately finds himself on the back foot. He has to somehow talk himself out of this obvious flaw of his theory. To this end he alleges that “the particular events observed [regarding tides] at different times and places are many and varied; these must depend upon diverse concomitant causes,” such as the size, depth and shape of the sea basin, and the internal forces of the water trying to level itself out. The fact that everyone could observe two high and two low tides per day Galileo thus wrote off as purely coincidental. He explicitly says so:

“Six hours … is not a more proper or natural period for these reciprocations than any other interval of time, though perhaps it has been the one most generally observed because it is that of our Mediterranean.”

Galileo even has some fake data to prove his erroneous point: namely that tides twelve hours apart are “daily observed in Lisbon,” he believes, even though that is completely false.

There is a further complication involved in Galileo’s theory, which “caused embarrassment to his more competent readers.” The inclination of the earth’s axis implies that the effects that Galileo describe should be strongest in summer and winter. Unfortunately for Galileo’s theory, the reverse is the case. Actually the tides are most extreme in spring and fall because they receive the maximum effects of the sun’s gravitational pull.

Galileo got himself confused on this point because he was again relying on false data. Galileo—the self-declared enemy of relying on textual authority, who often mocked his opponents for believing things simply because it said so in some book—he was the one in this case who found in some old book the claim that tides are greatest in summer and winter, took this for fact and derived this supposed effect from his own theory.

The mismatch between Galileo’s theory and basic facts is on display in another episode as well: “Galileo … attacked [those who] postulated that an attractive force acted from the Moon on the ocean for failing to realize that water rises and falls only at the extremities and not at the center of the Mediterranean. [But his opponents] can hardly be blamed for failing to detect this [so-called] phenomenon: it only exists as a consequence of Galileo’s own theory.” In other words, Galileo was so biased by his wrongheaded theory that he used its erroneous predictions as so-called “facts” with which to attack his opponents who were actually right.

But all of the above is actually still not yet the worst of it, believe it or not. There is an even more fundamental flaw in Galileo’s theory. Namely that it is inconsistent with the principle of relativity that he himself espoused. Think back to his scenario of the scientist locked in a cabin below deck of a ship that may or may not be moving. Galileo’s conclusion on that occasion was that no physical experiment could detect whether the ship was moving or not. But the torus tidal simulation, if it really worked as Galileo claimed, certainly could detect such a motion. If we put the torus on the floor of the cabin and spun it, one part would be spinning with the direction of motion of the ship, and another part against it. Hence high and low water should arise, by the same logic as in Galileo’s tidal theory. If the ship stood still, on the other hand, no such effect would be observed, of course. So we have a way of determining whether the ship is moving, which is supposed to be impossible. And indeed it is impossible, but if that’s so then Galileo’s theory of the tides cannot possibly work because it is inconsistent with this principle.

This objection against Galileo’s theory was in fact raised immediately already by contemporary readers. Here’s what one contemporary wrote to Galileo in 1633. He is reporting the reflections on Galileo’s book by a group of scholars who studied it:

“They draw attention to a difficulty raised by several members about the proposition you make that the tides are caused by the unevenness of the motion of the different parts of the earth. They admit that these parts move with greater speed when they [go] along [with] the annual motion than when they move in the opposite direction. But this acceleration is only relative to the annual motion; relative to the body of the earth as well as to the water, the parts always move with the same speed. They say, therefore, that it is hard to understand how the parts of the earth, which always move in the same way relative to themselves and the water, can impress varying motions to the water.”

That is to say, picture the earth moving along its orbit and also rotating around its axis at the same time. Hit pause on this animation and mark two diametrically opposite spots on the equator. Then hit play, let it run for a second or two, and then pause it again. Now, compare the new positions of the two marked spots with their original position. One will have moved further than the other. But that’s in a coordinate system that doesn’t move with the earth. That type of inequality of speed is irrelevant.

What is needed to created tides is a different kind of inequality of speed of the water. Namely, a difference in speed relative to the earth itself and to the other water. Tides arise when a fast-moving part of the water catches up with a slow-moving part of the water. But that is to say, these waters are fast and slow in their speed of rotation about the earth. So inequality of speed in a coordinate system centred on the earth. But no inequality of this type arises from the motion of the earth about the sun.

Galileo had no solution to this accurate objection. He’s just wrong. His theory is dumb.

So, to sum up, Galileo small-mindedly rejected the correct theory of the tides, based on the sun and the moon, even though this was widely understood by his contemporaries. He then proposed a completely wrongheaded theory of his own, which is based on elementary errors of physical reasoning that are inconsistent with his own principles. These flaws were readily spotted by his contemporaries. Furthermore, his theory is fundamentally at odds with the most basic phenomena, which he tried to explain away by attributing them to untestable, ad hoc secondary effects. He also adduced several false observational so-called “facts” in support of his theory.

No wonder many have felt that Galileo’s “ill-fated theory of the tides is a skeleton in the cupboard of the scientific revolution,” as one historian puts it. But this is a problem only if one assumes that Galileo was science personified. If we accept instead that Galileo was an exceptionally mediocre mind, who constantly got wrong what mathematically competent people like Kepler got right, then we see that Galileo’s skeletons belong only to himself, not to the scientific revolution. So once again we solve a problem of this type by throwing Galileo under the bus. It’s not that the scientific revolution was flawed. It’s just that Galileo was. If we restrict ourselves to mathematically competent people then we don’t have to deal with this kind of nonsense.

I’m going to use the tides to make the transition from Galileo’s physics, which we have discussed at length, to his astronomy. Galileo, as we saw, wanted to use the tides as a proof that the earth is moving, both rotationally and also orbitally around the sun.

Other people were not fooled by this poor argument. And yet leading mathematicians did have the sense to accept Copernicus’s vision. Why? Why did they become Copernicans? Why did Copernicus, why did Kepler? Certainly not because of Galileo, of course, but what then?

Arguably the most compelling reason was that the Copernican system explains complex phenomena in a simple, unified way. Here are some basics observational facts: The planets sometimes move forwards and sometimes backwards. Also, Mercury and Venus always remain close to the sun. In a geocentric system, with the earth in the center, there is no inherent reason why those things should be like that. Nevertheless these are the most basic and prominent facts of observational astronomy. In the Ptolemaic system, the geocentric system of antiquity, these facts are accounted for by introducing complicated secondary effects and coordinations beyond the basic model of simple circles. So planetary orbits are not just simple circles but combinations of circles in complicated ways that also happen to be coordinated with one another in particular patterns. In the Ptolemaic system there is no particular reason why these complicated constructions should be just so and not otherwise. We have to accept that it just happens to be that way.

So Ptolemy could account for, or accommodate, the phenomena, but he can hardly have been said to have explained them. The basic idea, that planets move in circles around the earth, is on the back foot from the outset. It is inconsistent with the most basic data and is therefore forced to add individual quick-fixes for these phenomena. In the end there are so many layers of patches that it’s like a house that consists more of duct-taped emergency fixes than that the original foundation.

In the Copernican system, it’s the opposite. The phenomena I mentioned are here instead an immediate, natural consequence of the motion of the earth. It becomes obvious and unavoidable that outer planets appear to stop and go backwards as the earth is speeding past them in its quicker orbit. It becomes obvious and unavoidable that Mercury and Venus are never seen far from the sun, since, like the sun, they are always on the “inside” of the earth’s orbit. Ad hoc secondary causes and just-so numerical coincidences in parameter values are no longer needed to accommodate these facts; instead they follow at once from the most basic assumptions of the system. They are built right into the foundation, no duct tape needed.

Galileo makes this point in his Dialogue: “You see, gentlemen, with what ease and simplicity the annual motion … made by the earth … lends itself to supplying reasons for the apparent anomalies which are observed in the movements of the five planets. … It removes them all. … It was Nicholas Copernicus who first clarified for us the reasons for this marvelous effect. … This alone ought to be enough to gain assent for the rest of the [Copernican] doctrine from anyone who is neither stubborn nor unteachable.”

This is all good and well. Galileo is absolutely right. Although of course this point was already a hundred years old and common knowledge by the time Galileo repeated it.

But here now is my fun twist on the story. Compare this story with Galileo’s theory of the tides. In fact, the correct, lunisolar theory of the tides explains the basic phenomena in a simple and natural way as immediate consequences of the first principles of the theory. That’s exactly the same point that we made about the Copernican system. The correct theory of the tides thus has the same kind of credibility as the Copernican system. So, by Galileo’s logic, this “ought to be enough to gain assent.” But in the case of the tides it seems Galileo was the “stubborn and unteachable” one. He insisted on a theory which—like that of Ptolemy—could only account for basic facts by invoking arbitrary and unnatural secondary causes unrelated to the primary principles of the theory.

It’s a sign that your theory has poor foundations if the foundations themselves are good for nothing and all the actual explanatory work is being done by emergency extras duct-taped on later to specifically fix obvious problems with the foundations. Intelligent people realised this, which is why they turned to the sun-centered view of the universe. Galileo paid lip service to the same principle when he wanted to ride on the coattails of their insights. But, if he had been consistent in his application of this principle, he should have used it to reject his foolish theory of the tides.