Galileo's bumbling attempts at determining the area of the cycloid suggests a radical new interpretation of his scientific opus. Archimedes's work on floating bodies is an example of excellent Greek science that has not been sufficiently appreciated.
Opinionated History of Mathematics
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Galileo is the most overrated figure in the history of science. That’s the thesis of Season 1 of this podcast. It’s going to be proper revisionist history fun. If you enjoy scholarly polemics and received wisdom turned on its head, then this is the story for you.
But it’s also more than that. Galileo is at the heart of fundamental questions. What is the relation between science, mathematics, and philosophy? Between ancient and modern thought? What’s the history of our scientific worldview, of scientific method? All this big-picture stuff. Galileo is right in the thick of the action is on all of these issues. So that’s all the more reason to study him.
But let’s start small. Here’s a simple snapshot of Galileo at work.
The cycloid. It’s a famous geometrical curve. The cycloid is the path traced by a point on a rolling circle. So, in other words, say you have a bicycle wheel. And you attach a piece of chalk to the rim of it. Then you roll the wheel along a wall, so that the chalk is drawing on the wall as it’s rolling. That makes a cycloid. It’s a kind of arch shape.
What’s the area of the cycloid? That was a natural question in Galileo’s time. Finding areas of shapes like that is what geometers had been doing for thousands of years. Archimedes for instance found the area of any section of a parabola, and the area of a spiral, and so on. The cycloid was a natural next step. It fit right into this tradition.
Of course nobody cared about the area of the cycloid as such. That’s not the point. Think for example of portraits painted by great artists. Of course the value in such a painting is not that it accurately depicts some prince and whatever pompous costume he was wearing. Obviously the value of this as art is not the subject but the method. How does the artist manipulate composition and subtle detail to achieve a particular impact? How does the artist build on tradition, yet innovate beyond it? Those are the things we admire when we view the works of great artists.
And it’s the same in mathematics. Archimedes, when he found his areas, gave clever geometrical arguments; gorgeous proofs. The point is not that he gave you a “formula” to compute various areas. How often have you needed to know the area of a spiral anyway? Never, of course. The point is not the result. The point is that Archimedes took human thought to a new level. His proofs are beautiful; they are logically flawless. They give you a sense that you are at the pinnacle of what the human mind can achieve. Everybody wanted to see more that kind of thing.
So solving problems like the area of the cycloid then, in this sense, in the Archimedean sense, was the way to prove yourself a worthy geometer. So Galileo tried. And failed. All those brilliant feats of ingenuity that Archimedes and his friends had blessed us with; it just wasn’t happening for Galileo. He just wasn’t any good at it.
In fact he said so himself. Here is a quote from Galileo on Archimedes: “Those who read his works realise only too clearly how inferior are all other minds compared with Archimedes’, and what small hope is left of ever discovering things similar to those he discovered.”
That’s Galileo. And he’s quite right. Except maybe it’snot that *all* other minds are inferior to Archimedes. Although certainly Galileo’s is.
So picture Galileo sitting in his study, racking his brain, staring at the cycloid. The books of Archimedes lie open on his desk. Agh, all that math, it’s giving him a headache. Thinking isn’t really working out for Galileo. It’s not his strong suit.
Desperate, frustrated, he turns to the failed mathematician’s last resort: trial and error. He cuts the cycloid out of thick paper and starts weighing it. It’s much like in modern mathematics classrooms where certain students prefer pushing buttons on a calculator and have some device do the thinking for them instead of trying to figure stuff out for themselves. So also Galileo starts fiddling around with scales and cardboard cut-outs as a substitute for thinking.
It is striking to compare Galileo’s take on this problem with that of his contemporaries. They in fact did solve it. Roberval in France, Torricelli, Galileo’s countryman in Italy, and also Descartes. They all solved it. And not with some middle school cut-and-paste project. They solved it as mathematicians, using reasoning and proof. These are contemporaries of Galileo. If they could do it, why couldn’t he?
Galileo himself says the cycloid area is a “very difficult” problem. “I worked on it fruitlessly,” he says. Compare that with Descartes’s reaction. Descartes, a mathematically competent person. When Descartes receives the problem—it was being passed around in correspondence at this time, as a challenge to various mathematicians—when Descartes receives it, he at once replies: “I do not see why you attribute such importance to something so simple, that anyone who knows even a little geometry could not fail to observe, were he simply to look.” And he back this up by immediately sending his solution, which he composed on the spot.
Now, Descartes, he is not famous for his humility, to put it mildly. So we should not necessarily read too much into those words there. But even so, the contrast with Galileo is very striking. The fact is that a number of mathematicians solved the cycloid problem with relative ease, while Galileo was fumbling with scissors and glue.
And Galileo got it wrong, too. Not only did he fail to find one of those beautiful Archimedean proofs, which was the whole purpose of the exercise in the first place. Not only that, but he also got the result itself wrong. The area of the cycloid is in fact 3 times the area of the generating circle, as these mathematicians showed. Galileo through specifically concluded based on his bumbling experiments that it was not exactly 3 but a bit off from 3. By relying on experiments unchecked by proper mathematics, Galileo got the answer wrong, and not for the first time nor the last.
That just goes to show why mathematicians have little respect for experiment. Christiaan Huygens, another very competent mathematician, once said: “Do not think that I am relying on experiments, because I know they are deceitful.” That was in a different context, nothing to do with the cycloid. But all the same. This is a universal attitude among mathematicians. And for good reason, as we see in the Galileo case. Haphazard trial and error has to be superseded by rigorous demonstration, as able mathematicians have always known.
So the moral of the story is this. In the case of the cycloid, it is undeniable that Galileo turned to empiricism precisely because he lacked the mathematical ability to tackle the problem any other way. It is beyond any shadow of a doubt that if he had had the ability to compose a mathematical proof, like some of his contemporaries did, he would have loved nothing more than to do so. In the case of the cycloid, these are facts.
This leads us to my revisionist thesis. I suggest that the cycloid episode is typical of Galileo’s science altogether. As with the cycloid, so with science. Galileo was bad at mathematics. And it is precisely because he was so bad at mathematics that he was so keen on experiments. He was not a pioneer of scientific method. He was not the father of modern science. He was not a heroic knight defeating dogmas and superstitions with the light of empirical truth. No, he was none of those things. Galileo was, first and foremost, a failed mathematician. This is the key to understanding his role in the history of science, in my opinion.
Galileo’s contribution to the history of thought is to cut off mathematical reasoning at the training-wheels stage; to air in public what true mathematicians considered unworthy scratch work at best. He experiments because he cannot think. He cannot reach insights by reason, so he turns to more simplistic methods, hands-on methods. In physics this blatant shortcoming has been mistaken for methodological innovation. But in the case of the cycloid we see its true colours. We see that it is a sign of failure rather than genius. Galileo’s empiricism is the last resort of a failed mathematician. It is not science being born; it is science being dumbed down.
I will argue for this conclusion in considerable detail in many future episodes. In any case, the cycloid example obviously fits this thesis like a glove. So that’s certainly food for thought at the very least.
It is instructive to compare Galileo with Archimedes, who was a proper mathematician. Consider Archimedes’s work on floating bodies. This is science done right, almost two thousand years before Galileo. Archimedes’s treatise is an outstanding masterpiece of science by the standards of any age. Only the mathematically illiterate could fail to grasp its immense significance. As indeed they have.
Archimedes gives a thorough theory of the floatation behaviour of paraboloids. That’s the shape generated when a parabola is rotated about its axis. Think of a wine glass or a champagne glass for example. The cups of those glasses are kind of paraboloids. Suppose you put a shape like that in water. Sometimes it will float upright, sometimes it will tip over, and so on. Under certain conditions the thing is in equilibrium, in other conditions not so it will wobble one way or the other.
Three parameters determine the state of the system. One is the inclination or tilt; so the angle the axis of the paraboloid makes with the water. Another is density. These are solid paraboloids. The material they are made of is either heavier or lighter than water in some certain ratio. Then there’s the steepness of the paraboloid. So whether it is more round like a wine glass or pointy like a champagne glass.
The equilibrium conditions of floating paraboloids depend on these parameters in highly nontrivial ways. It is a complicated and complex matter to know what’s going to happen if you vary any of these values. Perhaps you are familiar with dynamical systems catastrophe theory stuff from the 20th century? It’s like that. So you have a phase space and the stable states make some complicated surface and so on.
Here’s an example of an application. Consider an iceberg. It’s floating upright, let’s say. But it’s melting. It’s becoming thinner and thinner, it goes from a wine glass to a champagne glass kind of shape. There will be a critical point when upright floating is no longer an equilibrium state. The iceberg topples over. Even though the parameter changed only very slowly, the effect on the floating position was at first nothing for a long time and then a dramatic collapse all of a sudden as some critical threshold was met.
Archimedes does all of this. Well, he doesn’t mention applications like icebergs but he has all the mathematics of the thing. He gives detailed, exact, quantitative predictions of the floatation behaviour of paraboloids. When will it float, when will it tip over. Archimedes tells you all of that.
And he’s right. All of these very specific empirical predictions that he derives, they are spot on correct.
Now, poor Archimedes, he is often misunderstood. So many people who don’t care for mathematics, they hardly even know this work exists. But if they do look at it they say: What’s this? It’s just a bunch of technical geometry about parabolas and stuff. Archimedes says not a word about any experiment, not a word about any empirical data, nothing about testing his theory, nothing science-y like that at all. It’s just a bunch of intricate geometry. You might as well have opened any work on geometry, maybe Euclid or Apollonius or whoever. It looks just like that, like more geometry. It seems to have very little to do with the real world.
Archimedes has a few very basic postulates in the beginning: some common-sense assumptions about how fluids push on submerged objects. That’s the only link to the physical world. After stating these two or three assumptions at the beginning, it’s all-out geometry from there on.
So it has been easy for people to say: Well, let some specialised historian of mathematics read that technical gibberish, I’m a historian of science, I don’t care about that because it says nothing about empirical science. The Greeks may have been excellent geometers but they didn’t really do science, you see. They were speculative thinkers, philosophers. They were great with abstract stuff but they didn’t have the sense to ground their fanciful theories in reality.
That attitude is completely wrong, in my opinion. Ask yourself: What are the odds that Archimedes got his detailed, quantitative theory of floating bodies absolutely spot-on right if he was ignorant of empiricism and experiment and scientific method? Was he just sitting around doings speculative armchair geometry and, whoops, it just happened to come out exactly equal to empirical facts in a range of far from obvious ways? Are we supposed to believe that was just dumb luck? It doesn’t make any sense.
Of course Archimedes knew about the scientific method. Of course Archimedes tested his theory by experiment. That’s obvious. His text doesn’t say that because he was too good of a mathematician to think that kind of kid’s stuff counted for much of anything. He only published the actual theory, not the obvious tests that any fool with half a brain could do for themselves.
Galileo, though, was precisely that fool with that half brain. He spent his whole life spelling out those parts that Archimedes thought were too trivial to mention. People ignorant of Archimedes are readily tricked into thinking that this was somehow profound. But mathematicians know better.
This is why we need the mathematician’s point of view represented in historical scholarship. And that’s what you will get in this podcast. And that will be the basis for my revisionist interpretation of Galileo’s contribution to the history of science.