Galileo committed scores of errors in his physics. These are bad in themselves and also undermine Galileo’s claim to credit for the things he did get right.
Opinionated History of Mathematics
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Nostradamus published a famous book of prophesies in 1555. Some people like to praise him for having predicted the future. Allegedly he foresaw all kinds of things about world history. With a bit of imagination you can see him speaking about Napoleon and Hitler and god knows what else. Of course for every such pseudo-truth he also said a hundred things that turned out dead wrong. But people are less excited about that. It’s easy to be a prophet if you’re allowed a thousand guesses and people only count the few that came true.
Galileo is another Nostradamus. He too threw a thousand guesses out there and hoped that one or two would stick. Like Nostradamus, Galileo’s reputation rests on his admirers having selective amnesia, and remembering only the rare occasions when he got something right.
That’s our thesis for today. So, let’s have a go at Galileo’s catalogue of errors. A bunch of them have to do with the law of fall. Supposedly Galileo’s strong point. But in fact if we judge Galileo’s grasp of the law of fall by the way he applied it, then we must conclude that he did not understand it very well at all. He gets it wrong more often than not when he tried to apply his own law.
All objects fall with the same acceleration. But how fast is that exactly? What is this same universal acceleration that every object shares? The answer is well-known to any student of high school physics: the constant of acceleration, that famous lower case g, it is approximately 9.8 meters per second squared. But Galileo messed this up. He gives values equivalent to less than half of the true value. According to his defenders, and I quote, “clearly, round figures were taken here in order to make the ensuing calculation simple.” In other words, Galileo “used arbitrary data.” That’s again a quote. And that’s what the people trying to *defend* Galileo are saying.
Isn’t the law of fall supposed to be one of Galileo’s greatest discoveries? Why did he use fake data? Why not use real data? It was readily doable. His contemporaries did it. Why not do a little work to get the details right when you are publishing your supposed key results in your mature treatise? Is that really too much to ask? Instead of reporting make-believe evidence with a straight face, as Galileo does.
Competent and serious readers, like Mersenne and Newton, were all in disbelief at Galileo’s inaccurate data. They certainly did not think it was fine to “use arbitrary data” in order to get ”round” numbers for simplicity. Nor did they think it was “clear” that this is what Galileo was doing, contrary to what Galileo’s defenders are forced to argue when they try to excuse his inexcusable behaviour.
Mersenne put it clearly: “I doubt whether Mr Galileo has performed the experiment on free fall on a plane, since … the intervals of time he gives often contradict experiment.” Indeed. Mersenne was a serious and diligent scientist. He did the work to find the correct value, unlike Galileo. As usual, Galileo’s supposedly scientific treatises are popularising polemics and little else. To actual scientists, what he has to say is disappointingly shallow and lacks serious follow-through.
Here’s another error Galileo made with his law of fall: his so-called “Pisan Drop” theory of planetary speeds. The planets orbit the sun at different speeds. Mercury has a small orbit and zips around it quickly. Saturn goes the long way around in a big orbit and it is also moving very slowly. Galileo imagines that these speeds were obtained by the planets falling from some faraway point toward the sun, and then being somehow deflected into their circular orbits at some stage during this fall. That supposedly explains why the planets have the speeds they do.
I’ll read Galileo’s description in the Dialogue. “Suppose all the [planets] to have been created in the same place … descending toward the [sun] until they had acquired those degrees of velocity which originally seemed good to the Divine mind. These velocities being acquired, … suppose that the globes were set in rotation [around the sun], each retaining in its orbit its predetermined velocity. Now, at what altitude and distance from the sun would have been the place where the said globes were first created, and could they all have been created in the same place? To make this investigation, we must take from the most skilful astronomers the sizes of the orbits in which the planets revolve, and likewise the times of their revolutions.” Using this data and “the natural ratio of acceleration of natural motion” (that is, the constant g), one can compute “at what altitude and distance form the center of their revolutions must have been the place from which they departed.” According to Galileo this shows that indeed all the planets were dropped from a single point and their orbital data “agree so closely with those given by the computations that the matter is truly wonderful.”
Galileo omits the details though. Galileo has one of the characters in his dialogue say: “Making these calculations … would be a long and painful task, and perhaps one too difficult for me to understand.” Galileo’s mouthpiece in the dialogue confirms that “the procedure is indeed long and difficult.”
Actually there is nothing “difficult” about it. At least not to mathematically competent people. Mersenne immediately ran the calculations and found that Galileo must have messed his up, because his scheme doesn’t work. There is no such point from which the planets can fall and obtain their respective speeds. Later Newton made the same observation. Galileo’s precious idea is so much nonsense, which evidently must have been based on an elementary mathematical error in calculation.
Here’s another example. Again involving the law of fall. Galileo wished to refute this ancient argument: “The earth does not move, because beasts and men and buildings” would be thrown off. Picture an object placed at the equator of the earth, such as a rock lying on the African savanna. Imagine this little rock being “thrown off” by the earth’s rotation. In other words, the rock takes the speed it has due to the rotation of the earth, and shoots off with this speed in the direction of a tangent line of its motion. The spectacle will be rather underwhelming at first: since the earth is so large, the tangent line is almost parallel to the ground, and since the speed of the rock and of the earth are the same they will keep moving in tandem. So rather than shooting off into the air like a canon ball, the rock will slowly begin to hover above the ground, a few centimeters at a time.
Of course this is not what happens to an actual rock, because gravity is pulling it back down again. The rock stays on the ground since gravity pulls it down faster than it rises due to the tangential motion. How can we compare these two forces quantitatively? Since we know the size and rotational speed of the earth, it is a simple task (suitable for a high school physics test) to calculate how much the rock has risen after, say, one second. This comes out as about 1.7 centimeters. We need to compare this with how far the rock would fall in one second due to gravity. Again, this is a standard high school exercise (equivalent to knowing constant of gravitational acceleration g). The answer is about 4.9 meters. This is why the rock never actually begins to levitate due to being “thrown off”: gravity easily overpowers this slow ascent many times over.
But of course this conclusion depended on the particular size and speed and mass of the earth. We could make the rock fly by spinning the earth fast enough. For example, if we run the above calculations again assuming that the earth rotates 100 times faster, we find that, instead of rising a measly 1.7 centimeters above the ground in one second, the rock now soars to 170 meters in the same time. The fall of 4.9 meters due to gravity doesn’t put much of dent in this, so indeed the rock flies away.
Galileo, alas, gets all of this horribly wrong. Even though we are supposed to celebrate Galileo as the discoverer of the law of fall, it is apparently too much to ask that he work out this very basic application of it. As we noted, Galileo did not offer a serious estimate for the constant of gravitational acceleration g, unlike his contemporaries who were proper scientists. Therefore he did not have the quantitative foundations to carry out the above analysis, which high school students today can do in five minutes.
Worse yet, Galileo maintains that no such analysis is needed in the first place, because he can “prove” that the rock will never be thrown off regardless of the rotational velocity. “There is no danger,” Galileo assures us, “however fast the whirling and however slow the downward motion, that the feather (or even something lighter) will begin to rise up. For the tendency downward always exceeds the speed of projection.” Galileo even offers us “a geometrical demonstration to prove the impossibility of extrusion by terrestrial whirling.” Those are quotes from his big treatise on this.
Galileo’s claim to fame as a “mathematiser of nature” is certainly done no favours by this episode. He doesn’t know how to quantify his own law of fall, and doesn’t understand basic implications of it. His physical intuition is categorically wrong on a qualitative level, and worse than that of the ancients he is trying to refute (whose stance was quite reasonable and would be accurate if the earth was spinning faster). Galileo even offers a completely wrongheaded geometrical “proof” that the ancients’ conception is impossible, even though so-called “Galilean” physics leads to the opposite conclusion in an elementary way.
Galileo’s error in effect amounts to assuming that speed in free fall is proportional to distance rather than time. This is a crucial distinction in “Galileo’s” law of fall, which Galileo and others at times got wrong. By messing up this very point in his mature work, Galileo is undermining his claim to being the rightful father of the correct law of fall.
Galileo also makes another, independent error in this connection when he claims that the earth has the same whirling potential as a wheel with the same rotational period. In other words, that centrifugal force doesn’t depend on radius, only period. “Anyone familiar with simple merry-go-rounds will know that this is false,” as one scholar observes. Is it really just as hard to “hang on” to the speeding carousel whether you are close to the center or right out at the periphery? Alternatively, think of the pottery wheel used when shaping the clay when making ceramic pots. Is a piece of clay equally likely to be thrown off the wheel by its rotation if it sits near the middle as if it is placed near the edge? According to Galileo, yes.
Another example. Galileo tried to compute how long it would take for the moon to fall to the earth, if it was robbed of its orbital speed. “Making the computation exactly,” according to himself, he finds the answer: 3 hours, 22 minutes, and 4 seconds. This is way off the mark because Galileo assumes that his law of fall (that is, constant gravitational acceleration) extends all the way to the moon, which of course it does not. Ironically, Galileo’s purpose with this calculation was to refute the claim of another scholar that the fall would take about six days, which is a much better value: in fact it would take the moon almost five days to fall to the earth. That’s Galileo, the great hero of quantitative science, in action for you: bombastically claiming to refute others with his “exact calculations,” only to make fundamental mistakes and err orders of magnitude worse than his opponents did.
Another example. A rock dropped from the top of a tower falls in a straight line to the foot of the tower. But its path of fall is not actually straight if we take into account the earth’s rotation. Seen from this point of view—that is to say, in a coordinate system that doesn’t move with the rotation of the earth—what kind of path does the rock trace? Galileo answers, erroneously, that it will be a semicircle going from the top of the tower to the center of the earth. Here’s what he says: “If we consider the matter carefully, the body really moves in nothing other than a simple circular motion, just as when it rested on the tower it moved with a simple circular motion. … I understand the whole thing perfectly, and I cannot think that … the falling body follows any other line but one such as this. … I do not believe that there is any other way in which these things can happen. I sincerely wish that all proofs by philosophers had half the probability of this one.”
This is inconsistent with Galileo’s own law of fall. Once again he doesn’t understand basic implications of his own law. Mersenne readily spotted Galileo’s error. And Fermat observed that the path should be a spiral, not a semi-circle. When his embarrassing error was pointed out to him, Galileo replied that “this was said as a jest, as is clearly manifest, since it is called a caprice and a curiosity.” Some defence this is! Far from offering exonerating testimony, Galileo actually openly pleads guilty to the main charge: namely, that his science is a joke.
And if he meant is as “a caprice and a curiosity” then why did he say, in the quote I just read, that he “considered the matter carefully” and “sincerely wished that all proofs by philosophers had half the probability of this one” and so on? He always says cocky stuff like that. Just look back at the errors I already mentioned today. All of them came with bombastic claims where Galileo is editorialising about how remarkably convincing his own arguments are.
Isn’t that convenient? Throw out a bunch of half-baked guesses, and when they turn out right you can claim credit for stating it with such confidence while a more responsible scientist may have been exercising prudent caution. And when the guesses turn out wrong, you can apparently just write it off as a “joke” and pretend that that was what you intended all along, even though you published it with all those extremely assertive phrases right in the middle of your big definitive book on the subject. It’s easy to be “the father of science” if you can count on posterity to play along with this double standard.
So much for the law of fall. Let’s look at some of Galileo’s other physics errors. As we have seen, Aristotelians were often as inclined to experiment as Galileo—a point obscured by Galileo’s pretences to the contrary when it suited his purposes. Elsewhere it suited Galileo better to feign other straw men. As Butterfield has observed, “in one of the dialogues of Galileo, it is Simplicius, the spokesman of the Aristotelians—the butt of the whole piece—who defends the experimental method of Aristotle against what is described as the mathematical method of Galileo.”
Consider for example the question of the resistance of the medium (such as air or water) on a moving object. Aristotle had stated a law regarding how a body moves faster in a rarer medium than in a dense one. Galileo, in an early text, criticises Aristotle for accepting this “for no other reason than experience”; instead one must “employ reasoning at all times rather than examples,” “for we seek the causes of effects, and these are not revealed by experience.” Alas, despite his avowed allegiance to “reasoning,” Galileo’s own law as to how resistance depends on density of the medium is itself “incompatible with classical mechanics” as one study puts it—a polite, scholarly way of saying that it’s wrong.
Employing some more “reasoning” along the same lines, Galileo decided that air resistance doesn’t really increase appreciably with speed: “The impediment received from the air by the same moveable when moved with great speed is not very much more than that with which the air opposes it in slow motion.” A surprising conclusion to modern bicyclists, among others. Yet “experiment gives firm assurance of this,” Galileo promises. Alas, once again “the statement is false, and the experiment adduced in its support is fictitious.” So more fake data, in other words. This is quickly becoming a pattern.
The pendulum is a case that is similar in this regard. “With regard to the period of oscillation of a given pendulum, [Galileo] asserted that the size of the arc [that is, the height of the starting position of the pendulum] did not matter, whereas in fact it does.” Galileo’s allegedly experimental report on pendulums in the Discourse is clearly fabricated—or “exaggerated,” to use the diplomatic term preferred by some scholars. Mersenne did the experiment and rejected Galileo’s claim. Galileo’s friend Guidobaldo del Monte did the same, but when he told Galileo of his error Galileo rejected the experiment and insisted in his claim. Instead of admitting what experiments made by sympathetic and serious scientists showed, Galileo preferred to defend his false theory with what one historian calls “conscious deception.” Perhaps more commonly known as lying.
Another example. The shape of a hanging chain, like a neckless suspended from two points, looks deceptively like a parabola. It is not, but Galileo fell for the ruse. As he says: “Fix two nails in a wall in a horizontal line … From these two nails hang a fine chain … This chain curves in a parabolic shape.” More competent mathematicians proved him wrong: Huygens demonstrated that the shape was not in fact parabolic. Admittedly, his proof is from 1646, which is four years after Galileo’s death. So one may consider Galileo saved by the bell, as it were, on this occasion, since he was proved wrong not by his contemporaries but only by posterity. It is not fair to judge scientists by anachronistic standards. On the other hand, Huygens was only seventeen years old when he proved Galileo wrong. So another way of looking at it is that a prominent claim in Galileo’s supposed masterpiece of physics was debunked by a mere boy less than a decade after its publication.
In any case, Galileo thus ascribed to the catenary the same kind of shape as the trajectory of a projectile. He considered this to be no coincidence but rather due to a physical equivalence of the forces involved in either case. Indeed, Galileo made much of this supposed equivalence and “intended to introduce the chain as an instrument by which gunners could determine how to shoot in order to hit a given target.”
Galileo also tried to test experimentally whether the catenary is indeed parabolic. To this end he drew a parabola on a sheet of paper and tried to fit a hanging chain to it. His note sheets are preserved and still show the needle holes where he nailed the endpoints of his chain. The fit was not perfect, but Galileo did not reject his cherished hypothesis. Instead of questioning his theory, he evidently reasoned that the error was due merely to a secondary practical aspect, namely the links of the chain being too large in relation to the measurements. Therefore he tried it with a longer chain, and found the fit to be better. In this way he evidently convinced himself that he was right after all.
The catenary case thus undermines two of Galileo’s main claims to fame. First it bring his work on projectile motion into disrepute. The composition of vertical and horizontal motions that we are supposed to admire in that case looks less penetrating and perceptive when we consider that Galileo erroneously believed it to be equivalent to the vertical and horizontal force components acting on a catenary. Secondly, Galileo’s reputation as an experimental scientists par excellence is not helped by the fact that his experiments in this case led him to the wrong conclusion, apparently because his love of his pet hypothesis led him to a biased interpretation of the data and a sweeping under the rug of an experimental falsification. So those are two conclusions we can draw from the hanging chain episode.
Another example. The brachistochrone problem. This is the challenge to find the path along which a ball rolls down the quickest from one given point to another. Galileo believed himself to have proved that the optimal curve was a circular arc. “The swiftest movement of all from the terminus to the other is … through the circular arc.” That’s Galileo. Actually the fastest curve is not a circle but rather a cycloid. But this was only proved in the 1690s, using quite sophisticated calculus methods. We cannot blame Galileo for not possessing advanced mathematical tools developed only half a century after his death.
Nevertheless it is one more on his pile of erroneous assertions about various physical curves and problems. We are supposed to celebrate him for being the first to discover the parabolic path of projectile motion, and conveniently forget that at the same time he was wrong on the brachistochrone, wrong on the catenary, wrong on the isochrone, and so on. With all these errors stacking up, one may be forgiven for beginning to wonder whether the one thing he got right was any more than dumb luck. Galileo’s accounts of his correct discoveries may sound very convincing and emphatic, but knowing that he was equally sure of a long list of errors gives us reason to suspect that some of the things he got right are to some extent guesswork propped up with overconfident rhetoric in the hope that readers will mistakenly take his case to be stronger than it is.
That’s it for today. And I have still only dealt with physics. Galileo’s astronomy is a whole other can of worms with a parade of blunders all of its own. But we’ll get to that another time.