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.
Transcript
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.
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