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Proof-oriented geometry began with Thales. The theorems attributed to him encapsulate two modes of doing mathematics, suggesting that the idea of proof could have come from either of two sources: attention to patterns and relations that emerge from explorative construction and play, or the realisation that “obvious” things can be demonstrated using formal definitions and proof by contradiction.

**Transcript**

How did proofs begin? It’s like a chicken-or-the-egg conundrum. Why would anyone sit down and say to themselves “I’m gonna prove some theorems today” when nobody had ever done such a thing before? How could that idea enter someone’s mind out of the blue like that?

In fact, we kind of know the answer. The Greek tradition tells us who had this lightbulb moment: Thales. Around the year -600 or so. Hundreds of years before we have any direct historical sources for Greek geometry. But we still sort of know what Thales proved, more or less. Later sources tell us about Thales. History is perhaps mixed with legend in those kinds of accounts, but key aspects are likely to be quite reliable. More fact than fiction. Let’s analyse that question, the credibility question, in a bit more depth later, but first let’s take the stories at face value and see how we can relive the creation of deductive geometry as it is conveyed in these Greek histories.

So, here we go: What was the first theorem ever proved? What was the spark that started the wildfire of axiomatic-deductive mathematics? The best guess, based on historical evidence, goes like this. That love-at-first-sight moment, that theorem that opened our eyes to the power of mathematical proof, was: That a diameter cuts a circle in half.

Pretty disappointing, isn’t it? What a lame theorem. It’s barely even a theorem at all. How can you fall in love with geometry by proving something so trivial and obvious?

But don’t despair. It is nice, actually. It’s not about the theorem, it’s about the proof.

Here’s how you prove it. Suppose not. This is going to be a proof by contradiction. Suppose the diameter does not divide the circle into two equal halves. Very well, so we have a line going through the midpoint of a circle, and it’s cut into two pieces. And we suppose that those two pieces are not the same. Take one of the pieces and flip it onto the other. Like you fold an omelet or a crepe. The pieces were not equal, we assumed, so when you flip one on top of the other they don’t match up. So there must be some place where one of the two pieces is sticking out beyond the other. Now, draw a radius in that direction, from the midpoint of the circle to the place on the perimeter where the two halves don’t match up. Then one radius is longer than the other. But this means that the thing wasn’t a circle to start with. A circle is a figure that’s equally far away from the midpoint in all directions. That’s what being a circle means.

So we have proved that two things are incompatible with one another: You can’t be both a circle, and have mis-matched halves. Because if you have mis-matched halves you also have “unequal radii” and that means you’re not a circle.

So a circle must have equal halves. Bam. Theorem. It’s a boring result but a gorgeous proof. Or a suggestive proof. It’s a proof that hints at a new world.

Thales must have felt like a wizard who just discovered he had superpowers. “Woah, you can do that?!” By pure reasoning, by drawing out consequences of a definition, one can prove beyond any shadow of a doubt that certain statements could not possibly be wrong? That’s a thing? That’s something one can do? Wow. Let’s do that to everything! Right?

So that’s how Thales discovered proof. As best as we can guess.

A few other theorems are attributed to Thales as well. I want to bring up one in particular that I think is also a kind of archetype of what mathematics is all about.

The theorem we just saw, about the diameter bisecting the circle, perfectly embodies one prototypical mode of mathematical reasoning. The pure mathematics paradigm, you might call it. Logical consequences of definitions, proofs by contradiction. That kind of thing. Thales’s proof really hits the nail on the head with that whole aesthetic. We’ve been doing the same thing over and over ever since. A modern course in, say, group theory, for example, is just Thales’s proof idea applied five hundred times over, basically.

Now I want to take another one of the results attributed to Thales, and I want to argue that it is emblematic of another mode of mathematical thought. It’s a second road to proof. This second way is based more on play, exploration, discovery, rather than logic and definitions.

The example I want to use to make this point is what is indeed often called simply “Thales’s Theorem.” Which states that any triangle raised on the diameter of a circle has a right angle. So, in other words, picture a circle. Cut it in half with a diameter. Now raise a triangle, using this diameter as one of its sides, and the third vertex of the triangle is on the circle somewhere. So it looks like a kind of tent, sticking up from the diameter. And it could be an asymmetrical tent that is pointed more to one side or the other. No matter how you pitch this tent, as long as the tip of it is any point on the circle, then the angle between the two walls of the tent at that point, at the tip, is going to be a right angle, 90 degrees. That’s Thales’s Theorem.

How might Thales have proved this theorem? We don’t really know that based on historical evidence unfortunately. But let’s consider one hypothesis that makes sense contextually.

We must imagine that Thales would have stumbled upon the proof somehow. We are not trying to explain how someone might think of a proof of this theorem per se. That’s the wrong perspective because it takes for granted that in mathematics one tries to prove things. What we need to explain is where this vision to prove everything in geometry came from in the first place. How could someone have struck upon Thales’s Theorem unintentionally, as it were, and through that accident become aware of the idea of deductive geometry?

Indeed Thales’s Theorem is not terribly interesting or important in itself. If you had this vision of subjecting all of geometry to systematic proofs, why would you start with this theorem, or make this theorem such a center piece, as Thales supposedly did? You wouldn’t.

The interesting thing about Thales’s Theorem is not that is was one of the first results to which mathematicians applied deductive proof. Rather, the interesting thing about it is that it was the occasion for mathematicians to stumble upon the very idea of proof itself, unintentionally.

There’s a story about Thales falling into a well because he got so caught up in astronomical reasoning that he forgot his surroundings. It’s recorded in Plato: “While he was studying the stars and looking upwards, he fell into a pit. Because he was so eager to know the things in the sky, he could not see what was before him at his very feet.”

A legend maybe, but the discovery of Thales’s Theorem must have been a little bit like that too. Discovering mathematical proof must have been like falling into a pit. You are looking in one direction, and boom, suddenly you find yourself having accidentally smashed face first into this completely unrelated new thing that you didn’t know existed.

How could Thales’s Theorem be like that? Among all the world’s theorems, what makes Thales’s Theorem particularly conducive to this kind of fortuitous discovery of proof?

Here’s my hypothesis. In this age of innocence, before anyone knew anything about proof, people still liked shapes. The had ruler and compass. They used these tools for measuring fields and whatnot, but they also liked the aesthetic of it.

They were playing around with ruler and compass. Playing with shapes. After five minutes of playing with a compass you discover how to draw a regular hexagon. Remember? You probably did this as a kid. Draw a circle, and then, without changing the compass opening, run the compass along the circumference. It fits exactly six times. A very pleasing shape.

We know for a fact that people did this before Thales. There are hexagonal tiling patterns in Mesopotamian mosaics from as early as about -700.

Dodecahedra are another one of those things. The dodecahedron is like those twelve-side dice that you use in Dungeons and Dragons and stuff like that. Do-deca-hedron, it’s literally: two-ten-sided. So twelve-sided, in other words. Twelve faces, each of which is a regular pentagon. These things are in the archeological record. People made them of stone and bronze. A couple of dozen of dodecahedra from antiquity have been found, the oldest ones even predating Thales. They were used perhaps for oracular purposes, like tarot cards or something. Or maybe for board games, who knows?

In any case, my point is that people were interested in geometrical designs for various purposes: artistic, cultural, and so on. Not just measuring fields for tax purposes. And they were clearly working with instruments such as ruler and compass to make these things.

It’s easy to arrive at Thales’s Theorem by just playing around with ruler and compass, trying to draw pretty things. Start with a rectangle. Draw its diagonals. Put the needle of a compass where they cross, right in the midpoint of the rectangle. Set the pen of the compass to one of the corners of the rectangle. Now spin it. You get a circle that fits perfectly, snugly, around the rectangle.

But look what emerged. A diagonal of the rectangle becomes a diameter of the circle. And the rectangle pieces sticking out from it are precisely those kind of “tent” triangles that Thales’s Theorem is talking about. This suddenly makes the theorem obvious.

Why is Thales’s Theorem true? Why does any of those “tents” raised on the diameter of a circle have a right angle? It’s because it comes from a rectangle. Any such tent is half a rectangle. This is a powerful shift of perspective. By looking at the triangle this way we reveal hidden relationships, a hidden order in the nature of things. Certain angles must always be right angles by a sort of metaphysical necessity, as it were. Our eyes have been opened, maybe for the first time, to the existence of these kinds of necessities, these kinds of hidden relationships that are out there for the thinking person to uncover.

So the key is this shift of perspective that the triangle is “really” half a rectangle. Suppose instead that we had been stuck in the point of view is that we are staring at a triangle inscribed in a circle. Then the kinds of associations and ideas that suggest themselves to us are not so useful for proving this theorem. From that point of view, if you were looking for a proof, what would you do? Maybe you would for example connect the midpoint of the circle to the tip of the triangle. So now you have two smaller triangles. What are you going to do with those? Something with angle sums and so on? Or maybe you would be tempted to drop the perpendicular instead from the tip of the triangle, and then you can use the Pythagorean Theorem of the two small triangles you get.

These kinds of things are not what we want. Those kinds of approaches quickly become too technical. This was supposed to be the beginnings of geometry, remember. You are not supposed to use a bunch previous results for the proof. It should be a proof from first principles. A proof before all other proofs.

The idea that the triangle is “really” half a rectangle is different. It transforms how we look at the diagram. It changes the emphasis. It changes what we think of as primary. Now the rectangle comes first, and the triangle second, and the circle last. The theorem actually isn’t so much about circles at all, so to speak, from this point of view. The circle is just a kind of secondary artefact.

With this proof we are like artists. We take a step back from the canvas and tilt our heads and have this epiphany. And the epiphany was made possible by the way we had played with these ideas previously. We were just playing around with ruler and compass, we explored triangles and rectangles and circles with an open-minded affection. Epiphanies like Thales’s Theorem emerge from this play. Inspiration comes naturally in that context.

Unlike those other boring proofs I alluded to, that were based on cutting the triangle up and throwing the book at it: angle sums, Pythagorean Theorem, everything we can think of. That’s an uninspired approach, a brute force approach. It lacks that aesthetic inspiration, that epiphany of revealing the true nature of the triangle, and its other half that it was destined to be reunited with.

Geometry could not have started with these kinds of by-the-book proofs, because they only make sense after there is a geometry book to begin with. But geometry could have started with the epiphany type of proof. So that’s a way in which someone like Thales might have arrived at the idea of proof through playing around with ruler and compass.

Perhaps you are familiar with “Lockhart’s Lament”: a great essay on what is wrong with mathematics education. Go read it, it’s available online. It is interesting that Lockhart uses this very example to make his point. He describes how his students discovered Thales’s Theorem basically the way I’m saying that Thales might have done so. He also eloquently captures how this is so much more satisfying than a dry by-the-book proof.

It’s not for nothing that history and education go together on this point. Proof must have started with a compelling aesthetic experience or wow moment. There was no other way at the time. There was no one to force Thales to memorise facts for an exam. Discovery compelled him to value mathematics. If we want to foster intrinsic motivation in our students, it’s a good idea to consider what made people fall in love with these ideas in the first place. First love is always the purest and most innocent. Modern textbooks are like arranged marriages forced upon the students. But history always has the true love story.

Nevertheless, for all this, you might still think that Thales’s Theorem is a bit boring. Something something is always a right angle. So what? Who cares?

As I tried to argue, it was probably not the theorem per se that was impressive to Thales and his contemporaries, but rather the idea that there is such a thing as theorems and proofs at all. There are hidden truths out there that can be uncovered through reasoning. Remarkable.

But in fact even the theorem itself is quite interesting. Let me show you something cool you can do with Thales’s Theorem.

There’s an ancient legend about Queen Dido. Daughter of the king of Tyre, a major city in antiquity. You can still see the ruins of this ancient city in present-day Lebanon. At a certain point Dido had to flee, because of court intrigues. Murders and betrayals and so on. So she grabs a couple of diadems off her nightstand, maybe a chest of gold she put aside for a rainy day, and hastily sails off into the night. With hardly a friend left in the world.

She has to go all the way to present-day Tunisia, thousands of kilometers away, and try to start over somehow, in a manner befitting a royal. Using her treasure chest, she strikes a bargain to buy some land. As much land as she can enclose with the skin of an ox, the story goes. So she cuts the ox hide into thin strips and ties them together, and now what? So now she has this long string, which she can use as a kind of fence to seal off the land she wants.

But what shape to make it? A square, a rectangle, a triangle? No. Dido knows better. Perhaps her royal education included mathematics. Make it round. That’s the best way. The circle has the maximal area among all figures with a given perimeter. Or in this case, since she was by the ocean: a semi-circle, with the shoreline as a natural boundary on the other side.

Let’s prove this. That the semi-circle is the best choice. I’m going to prove this by contradiction: Suppose somebody has fenced in an area that is not semi-circular; then I can show how to make it better: how to move the fence so that the area becomes even bigger, without adding any more fence.

Ok, so you have the shoreline, that’s a straight line. And from one point on the shore, going inland you have this fence which then comes back down and meets the shore again in some other point. So together with the shoreline it closes off a certain area.

Suppose this shape is not a semi-circle. If it was a semi-circle, Thales’s Theorem would apply. And it would tell you that this angle, what I called the tent angle, at any point along the fence would be a right angle. So if the shape is not a semi-circle, there must be some point along the fence where this angle is not a right angle.

I say that making this angle a right angle improves the amount of area covered. You can picture it like this. So you have this shape enclosed by the fence: imagine that you have that cut out of cardboard. And on the perimeter you have some point marked where the tent angle is not a right angel. So on your cardboard you have that triangle drawn: a triangle consisting of the straight shoreline on one side, and the two lines from its endpoints going up to meet at the tent point on the perimeter.

Let’s cut that triangle out of the cardboard. So you’re left with two pieces: whatever bits that were sticking out from the triangle sides. Now move those two pieces so that you make the tent angle a right angle. This means moving the endpoints along the shoreline. As you move the two points on the shoreline, you change the angle at which the two cardboard pieces meet. The two cardboard pieces meet in a single point, the tent point, and that’s like a hinge that can open or close to a bigger or smaller angle. So you slide these things around until that hinge angle becomes 90 degrees.

Note that you didn’t change the perimeter this way. You just moved the same amount of fence around.

But you did increase the area enclosed, in fact. Because if you have two sticks of fixed length, and you want to make the biggest triangle you can with those sticks, the best way is to make the angle between them a right angle. That’s quite clear intuitively. You know that the area of a triangle is base times height over two. So if one of your sticks is the base, then to maximise the area you want to maximise the height, that is to say the perpendicular height going up from the base, which is obviously done by pointing the other stick straight up at right angles.

So what this proves is that, for any fence enclosure that is not a semi-circle, you can make a better one. You can move the fence around and make the area bigger. So the semi-circle is the best solution, and all other ones are less good.

I don’t know if you could visualise all of that. But maybe try reconstructing this argument for yourself later. It really is very intuitive and beautiful.

So what’s the moral of the story then? Mathematically, it is an answer to the “so what?” question regarding Thales’s Theorem. It may have seemed like a boring enough theorem, but here we see it in action in a beautiful and unexpected way, as a key ingredient in this proof about how to enclose land. Who would have seen that coming?

This suggests that mathematics has a kind of snowballing or self-fertilising aspect to it. Thales’s Theorem, what’s the big deal? Just some boring observation about a triangle in a circle. May not seem like much. But one thing leads to another. Once Thales’s Theorem is a thing to you, you start seeing it in other places, unexpected places. Like this problem about area. You wouldn’t think it was related, but the more mathematics you do, the more connections you find.

Pick any theorem, no matter how boring, like Thales’s Theorem, and you can find these amazing things where the boring theorem is actually a key insight that opens entirely new ways of thinking about seemingly unrelated problems. That’s mathematics for you. No wonder it caught on like a bug among the Greeks, once they got the ball rolling. One moment you stumble upon some random result like Thales’s Theorem, and the next thing you know you’re seeing mathematics everywhere.

So that’s the mathematical moral of the story. Now we must go back and say something about the historical side of all this. What do we really know about Thales and his theorems and Queen Dido and all that? How much is history and how much is legend?

If we start with Dido, that story comes to use primarily through Virgil. The Aeneid, the famous epic poem. That was written in Roman times, around the year -20. But it is referring to historical, or supposedly historical, events that took place even before Thales, maybe two centuries before Thales, so -800-ish. We have Virgil’s version, that’s what has come down to us, but he is just stealing an older story. These things would have been around for centuries in Greek culture, in various literary and historical retellings that are now lost.

It is perfectly plausible that there really was such a historical queen, who really did flee her royal home in Tyre, and really did land on the north shores of Africa where she founded this new settlement, which was to become the great city of Carthage. Maybe indeed she even made the city walls semi-circular, who knows? It is perfectly conceivable that she might have wanted to minimise the perimeter for whatever reason, and that she might have known that a semi-circular shape was optimal for this purpose.

But at that time there would not have been any mathematical proofs of this, like the one I sketched above. The proof I outlined is from Jakob Steiner, in the early 19th century. From Greek times we have a different proof of this result. So they were certainly very much aware of the result, that the semi-circle is optimal, if perhaps not the particular proof I suggested.

If the story of Queen Dido says anything about the history of mathematics, it probably illuminates most neither the time when the events took place, around -800, nor the time when the sources we have were written, around year 0. But maybe it says something about the centuries in between, where the story would have been passed on and reworked.

The story was marinated, as it were, in Greek culture. Maybe they were the ones who gave it a mathematical flavour. The shoe fits: The Greeks valued wise, aristocratic, well-educated rulers, who design rational policy for the common good informed by reason and mathematics. Maybe they let these ideals colour the way they retold the story of Queen Dido and her round city.

From this point of view we could also speculate that by the time Virgil comes around and writes the Roman version of the story, this appreciation of mathematics is no longer what it once was. Indeed Virgil doesn’t really spell out the mathematical optimisation aspect of the story. Dido is just a side character altogether. His epic is about Aeneas, who is on a quest that will eventually lead to the founding Rome.

Aeneas is shipwrecked and blown ashore at Carthage, Dido’s round city. Dido falls in love with him, but he does not return her love. He sails away and Dido kills herself because of her broken heart. Morris Kline concludes the story: “And so an ungrateful and unreceptive man with a rigid mind caused the loss of a potential mathematician. This was the first blow to mathematics which the Romans dealt.” Sure enough there’s plenty more where that came from.

One can view this story as symbolic of this transition from the wise philosopher kings (or queens in this case) of the Greek world, who cherished mathematics and used it to improve the world. The transition from that to the heartless Roman, who only think of themselves and couldn’t care less about Thales’s Theorem. In the Greek world math nerds were considered attractive, but somehow these ignorant Romans didn’t think a geometer queen was girlfriend material at all evidently.

Ok, so the story about Dido and the round city and the optimisation proof and all that, it is very interesting in terms of the broader mathematical and cultural points its connects to, but in and of itself its is not directly history per se.

It’s different with Thales. That’s more fact than legend. As best as we can determine, Thales really did prove that diameter bisects a circle, most likely with the proof discussed above.

The sources that we have for this are far from perfect. Primarily Proclus, who was writing in year 450 or so, basically one thousand years after Thales lived. These kinds of late sources are hit and miss. They have no authority in and of themselves. Proclus was nobody. His own understanding of history and mathematics is very poor. A mediocre thinker, a mediocre scholar, living in a mediocre age.

Those are the kinds of sources that we have. Basically as authoritative as a factoid you read on the back of a cereal box or something.

But there is hope. Back in its glory days, Greece was just an outstanding intellectual culture. And some of the stuff about for example Thales can be traced back to then, which makes it highly credible. Aristotle’s student, Eudemus, wrote a history of geometry. It’s no longer with us alas. Ignorant ages neglected it and now it’s gone. But what a work that would have been.

These people knew what they were doing. Later people like Proclus are like some online rando posting half-baked ideas on blogspot or poorly informed comments on Facebook. That’s how credible they are.

But people like Eudemus is a very different story. That is more like a first-rate scholar at a research institution with all the infrastructure one could dream of: libraries, extremely knowledgable and intelligent colleagues with a range of expertises, broad financial and cultural support from the public and from politicians, and so on. Eudemus’s History of Geometry would be a proper “University Press” book, peer-reviewed to the teeth and with a nice dust-jacket blurb by Aristotle.

People like Eudemus were not in the business of passing on random gossip and unchecked factoids because they sound cool. They were proper scholars and intellectuals.

And indeed a lot of the stuff about Thales can be traced back to this lost source. When Proclus says that Thales was the first to prove that a circle is bisected by its diameter, the source of this is Eudemus. Hence it is very credible. This Thales stuff really happened. Actually that part about the diameter bisecting the circle is more certain than the part about Thales’s Theorem. Was Thales’s Theorem really Thales’s? Maybe. But we cannot trace that part specifically back to the best sources. Unlike the diameter bisection one and some other details. But contextually it makes sense.

The stories of Thales and the origin of geometry were evidently well known not only to specialised scholars but to the general Athenian public. Aristophanes the playwright uses the name of Thales as a symbol of geometry a few times in his plays. Just as today one might use the name of Einstein for instance to evoke the image of a scientist. Aristophanes has one speakers in a dialogue say: “The man is a Thales.” Meaning that the person is a geometer. Evidently the theatre-going public in classical Athens could be expected to understand this reference. Every educated person would know about Thales and the origins of geometry.

In fact, public respect for geometry and its history was apparently so great that Aristophanes even has one of his characters lament it as excessive, saying: “Why do we go on admiring old Thales?” What a time to be alive that would have been. When playwrights had to tackle issues such as there being too much respect and interest in mathematics among the general public. “Hey guys, maybe we need to cool it with how much we love geometry.” What a luxury problem. Hardly one that Hollywood blockbusters today have to grapple with.

Anyway, we should maybe not read too much into those isolated quotes. But the general intellectual credibility of this age is important. These very intelligent and serious people recorded in scholarly histories the accounts about Thales founding deductive geometry and proving that a circle is bisected by its diameter. That’s only some two or three hundred years after Thales, and in a direct lineage from him, probably with entire works by Thales still around in libraries and so on.

So there you go. The origins of proof and deductive geometry. We really do know quite a bit about it, and it’s a story worth knowing if you ask me.