Galileo’s errors on projectile motion and inertia

Galileo gets credit he does not deserve for the parabolic nature of projectile motion, the law of inertia, and the “Galilean” principle of relativity. In reality, his treatments of all of these matters were riddled with errors and fundamental misunderstandings.

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Opinionated History of Mathematics

Transcript

Pick up a rock and throw it in front of you. It makes a parabola. The path of its motion is parabolic. That’s Galileo’s great discovery, right? Well, not really. Galileo does claim this but he doesn’t prove it. Even Galileo’s own follower Torricelli acknowledged this. The result is “more desired than proven,” as he says, very diplomatically.

And the reason why Galileo doesn’t prove it is a revealing one. It is due to a basic physical misunderstanding.

The right way to understand the parabolic motion of projectiles like this is to analyse it in terms of two independent components: the inertial motion and the gravitational motion. If we disregard gravity, the rock would keep going along a straight line forever at exactly the same speed. That’s the law of inertia. But gravity pulls it down in accordance with the law of fall. The rock therefore drops below the inertial line by the same distance it would have fallen below its starting point in that amount of time if you had simply let it fall straight down instead of throwing it. A staple fact of elementary physics is that the resulting path composed of these two motions has the shape of a parabola.

Galileo does not understand the law of inertia, and that is why he fails on this point. If the projectile is fired horizontally, like for instance a ball rolling off a table, then Galileo does prove that it makes a parabola. He proves it the right way, they way I just outlined, by composition of inertial and gravitational motion.

But if you throw the rock at some other angle, not horizontally, then Galileo doesn’t dare to give such an analysis. This is because he thinks the law of inertia is maybe not true for such motions. He thinks, if you throw a rock at an upward angle, then maybe the rock won’t have such an inertial disposition to keep going in that direction with that speed. Instead, we thinks maybe the motion is going to slow down gradually, like a ball struggling to roll up a hill or an inclined plane.

Galileo asserts neither this wrong form of inertia nor the right one. He equivocates and never takes a stand, because he isn’t sure. And this is why he cannot give a correct proof of the theorem of parabolic motion. Even though such a proof was very much within his reach. In fact, Cavalieri, who was a better mathematician, had already published this proof, the correct analysis of parabolic motion, before Galileo wrote his book.

So it’s not that this stuff was beyond the reach of the mathematical and scientific methods of the time by any means. On the contrary, it was already explicitly spelled out completely and correctly in a published book that Galileo was aware of. And still Galileo gets it wrong in his famous work. He’s just not a very good physicist.

Ok, so that’s the big picture on parabolic motion. Now I want to go into more detail on these things. First let’s take a step back and look at inertia generally.

Here is Newton’s law of inertia: “Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.” That’s from Newton’s great Principia of 1687. It’s Law 1 of that work. A cornerstone of the whole thing.

In Galileo there’s nothing like that. Even the most ardent Galileo admirers admit this. Here’s Stillman Drake, Galileo’s great defender. Even he, and I quote, “freely grant that Galileo formulated only a restricted law of inertia” and that “he neglected to state explicitly the general inertial principle” that everyone knows today, which was instead correctly “formulated two years after his death by Pierre Gassendi and René Descartes.”

That’s the charitable interpretation. That’s the view of Galileo’s most committed supporters. And it is rather too kind, in my opinion. Trying to attribute to Galileo some kind of “restricted law of inertia” is a dubious business. Stillman Drake tries to do so, and here is what he says: “in my opinion the essential core of the inertial concept lies in the ideas of a body’s indifference to motion or to rest and its continuance in the state it is once given. This idea is, to the best of my knowledge, original with Galileo.”

You could very well argue that that’s not really inertia at all because it doesn’t involve the straightness of the direction of the motion, nor does it explicitly say that the motion keeps going at a perpetual uniform speed. It only focusses on indifference of motion versus rest and preservation of the state of motion.

So that’s “the essential core of the inertial concept” according to Galileo’s defenders. That’s very convenient. Galileo got half the properties of inertia right and half wrong, so his supporters try to spin it and say that the parts he did get right are “the essential core”, you see, and the other stuff is just secondary anyway so it doesn’t really matter that Galileo was wrong about all of that.

Sure enough, if you’re allowed to pick and choose like this which half of inertia you think is important then you can find some evidence for that part is Galileo. For example, Galileo says quite correctly: “No one could say why a thing once set in motion should stop anywhere; for why should it stop here rather than there? So that a thing will either be at rest or must be moved ad infinitum, unless something more powerful get in its way.”

Sure enough, that’s indifference of motion versus rest and preservation of the state of motion, the alleged “core” of the inertial concept. How much credit do you think Galileo deserves for this? For getting half of inertia right? Maybe you think that was the difficult step, the conceptual revolution, and then it was easy for Newton and others to fill in the details by just continuing what Galileo started.

Actually I tricked you. The quote I just read is not from Galileo at all. I lied. The quote is from Aristotle. It’s from Aristotle’s physics, written two thousand years before Galileo. So if you think that’s “the essential core of the inertial concept,” then Aristotle was the pioneering near-Newtonian who conceived it, not Galileo.

This claim is rather isolated in Aristotle and didn’t really form part of a sustained and coherent physical treatment of motion comparable to how we use inertia today. Aristotle as usual is focussed on much more philosophical purposes. So you might say: that’s a one-off quote taken out of context which sounds much more modern than it really is.

Indeed. But then again the same could be said for Aristotle’s so-called law of fall that Galileo refuted with so much fanfare. This too is only mentioned in passing very briefly and plays no systematic role in Aristotle’s thought. Yet Galileo takes great pride in defeating this incidental remark, and his modern fans applaud him greatly for it. So if we want to dismiss Aristotle’s inertia-like statement as insignificant, then, by the same logic, we ought to likewise dismiss all of Galileo’s exertions to refute his law of fall as completely inconsequential as well. If we argue that statements such as those of Aristotle don’t count as scientific principles unless they are systematically applied to explain various natural phenomena, then we would have to conclude that there was no Aristotelian science of mechanics at all. This, of course, would be a disastrous concessions to make for advocates of Galileo’s greatness, since so much of Galileo’s claim to fame is based on contrasting his view with so-called “Aristotelian” science.

So take your pick. Here are the three options:

Option 1. Galileo’s understanding of inertia was very poor.

Option 2. Galileo’s understanding of inertia was pretty good, but so was Aristotle’s.

Option 3. Galileo’s understanding of inertia was pretty good, but not Aristotle’s, because Aristotle’s statements, even though they say pretty much what Galileo says, should be disqualified because they are philosophy rather than science.

I, of course, advocate the first solution: throw Galileo under the bus. He and Aristotle were both stupid. Problem solved.

If you want to preserve Galileo’s reputation you’re in a trickier position. Are you going to admit that Aristotle understood inertia? But then what was Galileo’s contribution, and how could it be revolutionary, if that kind of stuff was already well understood two thousand years before? Or do you want to say: No, Aristotle didn’t really understand this, because his text wasn’t meant as science anyway. Well, then what is the value in Galileo spending hundreds of pages of his most important works arguing against Aristotle?

You tell me how you’re gonna solve these puzzles. Trying to maintain Galileo’s alleged greatness, it just doesn’t add up. You’re left having to bend over backwards with these inconsistent rationalisations.

What about the *rectilinear* character of inertia? The thing keeps going *straight*. Is that in Galileo? The following passage may appear to suggest as much. Quote: “A projectile, rapidly rotated by someone who throws it [like a rock in a sling], upon being separated from him retains an impetus to continue its motion along the straight line touching the circle described by the motion of the projectile at the point of separation. The projectile would continue to move along that line if it were not inclined downward by its own weight. The impressed impetus, I say, is undoubtedly in a straight line.”

That’s Galileo, and it’s straight up rectilinear inertia, right? Done and dusted. No, not so. It’s not inertia, it’s impetus. That’s what Galileo calls it. The projectile has “impetus” to go straight. But what does that mean? What is “impetus”? Is it the same thing as inertia? Will “impetus” run out, for example? Is the motion caused by the “impetus” perpetual and uniform? Galileo doesn’t say, and most likely he didn’t believe so.

In many other sources at the time, “loss of impetus by projectiles was likened to the diminution of sound in a bell after it is struck, or heat in a kettle after it is removed from the fire,” as Drake remarks. This conception is perfectly compatible, to say the least, with what Galileo writes. In fact, Galileo nowhere asserts the eternal conservation of rectilinear motion. On the contrary, he explicitly rejects it: “Straight motion cannot be naturally perpetual.” That’s an exact quote form his major work. “It is impossible that anything should have by nature the principle of moving in a straight line.” Again, a literal quotation right out of Galileo’s main work. It is easy to understand, then, why Galileo’s defenders are so eager to insist on characterising “the essential core of the inertial concept” in a way that does not involve its rectilinear character, since Galileo clearly and explicitly *rejected* rectilinear inertia.

If there’s any inertia in Galileo it is horizontal rather than rectilinear inertia. Here are some quotes from Galileo.

“To some movements [bodies] are indifferent, as are heavy bodies to horizontal motion, to which they have neither inclination or repugnance. And therefore, all external impediments being removed, a heavy body on a spherical surface concentric with the earth will be indifferent to rest or to movement toward any part of the horizon. And it will remain in that state in which it has once been placed; that is, if placed in a state of rest, it will conserve that; and if placed in movement toward the west, for example, it will maintain itself in that movement. Thus a ship, for instance, having once received some impetus through the tranquil sea, would move continually around our globe without ever stopping.”

Another one:

“Motion in a horizontal line which is tilted neither up nor down is circular motion about the center; once acquired, it will continue perpetually with uniform velocity.”

Again, as with the sling and the projectile, one can debate whether this is inertia per se. In Newtonian mechanics too a hockey puck on a spherical ice earth would glide forever in a great circle, even though this is not inertial motion. But this agreement with Newtonian mechanics only holds if the object is prevented from moving downward, as the puck is by the ice, or the ship by the water. Galileo seems to have believed horizontal inertia to hold also for objects travelling freely through the air, which is of course not compatible with Newtonian mechanics. For example, Galileo says:

“I think it very probable that a stone dropped from the top of the tower will move, with a motion composed of the general circular movement and its own straight one.”

Once again it is not entirely clear that this is supposed to represent inertia at all. It is conceivable that, in Galileo’s conception, the circular motion itself is not a force-free, default motion, but rather a motion somehow caused or contaminated by gravity-type forces. Who knows? Galileo just isn’t clear about these kinds of things. Newton and Descartes, like the good mathematicians that they are, state concisely and explicitly what the exact fundamental assumption of their theory of mechanics are. Their laws of inertia are crystal clear and specifically announced to be basic principles upon which the rest of the theory is built. Galileo never comes close to anything of this sort. He uses the casual dialogue format of his books to hide behind ambiguities. One moment he seems to be saying one thing, then soon thereafter something else, like an opportunist who doesn’t have a systematically worked out theory but rather adopts whatever assumptions are most conducive to his goals in any given situation.

Let’s get back to parabolic motion. Some have tried to argue that “if Galileo never stated the law [of inertia] in its general form, it was implicit in his derivation of the parabolic trajectory of a projectile.” That’s a quote from Stillman Drake. It would have been a very good argument if Galileo had treated parabolic trajectories correctly. But he didn’t, so the evidence goes the other way: Galileo’s bungled treatment of parabolic motion is actually yet more proof that he did not understand inertia.

His restriction of inertia to horizontal motion only is clear in his treatment of projectiles. He speaks unequivocally of “the horizontal line which the projectile would continue to follow with uniform motion if its weight did not bend it downward.” But he does *not* make the same claim for projectiles fired in non-horizontal directions. Rather he studiously avoided committing himself on that point because he was afraid it wasn’t true, like we said.

Since he only trusted the horizontal case, Galileo tried to analyse other trajectories in terms of this case. To this end he assumed, without justification, that a parabola traced by an object rolling off a table would also be the parabola of an object fired back up again in the same direction. In other words, “he takes the converse of his proposition without proving or explaining it.” That judgement is in fact a quote from Descartes, a mathematically competent reader who immediately spotted this blatant flaw in Galileo’s book.

Here’s another interesting point that Descartes makes: Galileo “seems to have written [this theory] only to explain the force of cannon shots fired at different elevations.” That is to say, Galileo made no theoretical use of his theory of projectile motion whatsoever. For example, he makes no connection to the motion of planets, the moon, comets; nothing like that. That’s a huge missed opportunity.

Instead Galileo erroneously claimed that his theory would be practically useful for people who were firing cannons. That’s quite naive, as Descartes pointed out. Here’s a quote on this by the historian A. Rupert Hall: “In many passages Galileo remarks that the theory of projectiles is of great importance to gunners. He made little or no distinction between his theory and useful ballistics; he believed—though without experiment—that he had discovered methods sufficiently accurate within the limitations of military weapons to be capable of direct application in the handling of artillery.”

This belief, however, was completely wrong. A contemporary put the matter to experimental test, and reported as follow: “I was astonished that such a well-founded theory responded so poorly in practice. If the authority of Galileo, to which I must be partial, did not support me, I should not fail to have some doubts about the motion of projectiles, and whether it is parabolical or not.” That’s a follower of Galileo writing shortly after his work was published.

Galileo foolishly thought his theory would work without testing it. This is evident for example from the extensive tables that he printed as an appendix to his big book: ballistic range tables based on his theory. These long tables make no sense at all other than as a practical guide for firing cannons. So clearly Galileo thought his theory was practically viable, which it is absolutely not.

Here’s a more theoretical issue related to inertia: the relativity of motion.

When teaching basic astronomy at Padua, Galileo explained to his students that Copernicus was undoubtedly wrong about the earth’s motion. The earth doesn’t move, Galileo explained. Because, if the earth moved, a rock dropped from a tower would strike the ground not at its foot but some distance away, since the earth would have moved during the fall. In support of this claim, “Galileo observed that a rock let go from the top of a mast of a moving ship hits the deck in the stern.” This had indeed been reported as an experimental fact by people who actually carried it out.

Of course this is completely backwards and the opposite of Galileo’s later views that he is famous for. To be sure, these lectures do not necessarily say anything about Galileo’s personal beliefs. In all likelihood he simply taught the party line because it was the easiest way to pay the bills. But at least the episode does show that the simplistic narrative that “the experimental method” forced the transition from ancient to modern physics is certainly wrong. On the contrary, experimental evidence was among the standard arguments for the conservative view well before Galileo got into the game.

In his later works Galileo of course affirms the opposite of what he said in those lectures: the rock will fall the same way relative to the ship regardless of whether the ship is standing still or travelling with a constant velocity. He gives a very vivid and elaborate description of this principle. I’ll quote in it full, it’s a long quote but it’s quite fun:

“Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other.”

Ok, so that’s that famous passage. Galileo’s prose is as embellished with fineries as this little curiosity-cabinet of a laboratory that he envisions. But is it any good of an argument? Insofar as it is, the credit is perhaps due to Copernicus himself, who had already made much the same point a hundred years before. Here are his words:

“When a ship floats over a tranquil sea, all the things outside seem to the voyagers to be moving in a movement which is an image of their own, and they think they themselves and all the things with them are at rest. So it can easily happen in the case of the movement of the Earth that the whole world should be believed to be moving in a circle. Then what would we say about the clouds and the other things floating in the air or falling or rising up, except that not only the Earth is moved in this way but also no small part of the air [is moved along with it]?”

So Galileo’s relativity argument, like so much else he says, is old news. The primary contribution of his version is literary ornamentation. Adding some butterflies and whatnot, while saying nothing new in substance.

Perhaps one could argue that Galileo goes beyond Copernicus’s passage by asserting more definitively that no mechanical experiment of any kind could prove that the ship is moving. Today the so-called “Galilean” principle of relativity says that the phenomena in the cabin cannot be used to distinguish between the ship being at rest or moving with constant velocity in a straight line. But Galileo clearly has another scenario in mind: he sees the ship as travelling along a great circle around the globe. This is the kind of motion he believes cannot be distinguished from rest, in keeping with his misconceived idea of horizontal inertia. This principle of relativity—the actually “Galilean” one—is of course false. In fact it’s even worse than that. Galileo’s purpose with this passage about the ship is to argue, erroneously, that the rotation of the earth cannot be detected by physical experiments, which in fact it can. The Foucault pendulum is a device that can detect this.

So the attribution of the principle of relativity of motion to Galileo in modern textbooks is doubly mistaken. First of all, relativity of motion and the idea of an inertial frame had been noted long before and was invoked by Copernicus to much the same end as Galileo. Moreover, Galileo’s principle is wrong in itself (because it’s about motion in a great circle, not in a straight line), and furthermore his purpose in introducing it is to draw another false conclusion from it (namely that the earth’s motion is undetectable). So errors at every turn as usual with Galileo. And there’s plenty more where that came from.