The Midnight Mathematician · Transcript · · Updated
The Lives Of The Greatest Mathematicians, For Sleep – Transcript
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- The ancient Greek mathematician Uklid wrote a book called the elements that served as the primary way people learned mathematics for over 2,000 years.
- Archimedes measured circles by using a method of exhaustion where he sandwiched a circle between inner and outer straight-sided shapes.
- Brahmagupta treated zero as a real number in 628 and provided clear rules for arithmetic including addition and subtraction.
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Tonight, you don't have to solve anything. You don't have to remember anything. You're just going to drift slowly through some of the most beautiful ideas that human beings have ever had, carried along by the lives of the people who found them.
There's no rush here, no quiz at the end. Just one gentle idea at a time explained softly in the dark until you fall asleep. So let your shoulders drop. Let the day go and let's begin. We start more than 2,300 years ago in a city called Alexandria on the northern edge of Egypt
where the desert meets the sea. And we start with a man we know almost nothing about. His name was Uklid. That's really about all we can say for certain. We don't know when he was born. We don't know when he died. We don't know what he looked like or whether he was kind or married or lonely.
There are no reliable stories about his childhood. He is in a strange way one of the most important people who ever lived and also one of the most invisible.
But here's the thing. He left us a book and that book changed how human beings think. The book was called the elements. And for more than 2,000 years it was the way people learned mathematics.
More editions of it were printed than any book except the Bible. Think about that for a second. A single math textbook written in the ancient world still being taught in classrooms in your grandparents' time. That almost never happens. So, what was in it that lasted so long?
Well, it wasn't really the shapes. It wasn't the triangles and circles themselves. It was something quieter and much deeper. Uklid gave us the idea of proof. The idea that you can be absolutely permanently certain that something is true and that you can show why step by step in a way that no one can argue with. Let me explain what that actually
means because it's gentler than it sounds. Imagine you want to convince someone of something. Normally, you might say, "Well, trust me. I've seen it. It just works." And that's fine for everyday life. But Uklid wanted something stronger than trust. He wanted a way to reach a conclusion that didn't depend on
trusting anyone at all. Not him, not a teacher, not a king, a truth you could check entirely by yourself with nothing but your own careful thinking. So, how do you build something like that? You start with a tiny handful of things so simple, so obvious that no reasonable person could deny them. Uklid began with
just a few of these. that you can draw a straight line between any two points, that you can extend a straight line as far as you like, that you can draw a circle with any center and any size.
Small, plain, humble statements, the kind of thing you'd not along without thinking twice. And then from those few plain beginnings, using only steps that are just as undeniable, you build your way up to something surprising, something you would never have guessed at the start. Each step so small and so obvious that you can't say no to it. And
if you can't say no to any single step, then you can't say no to where they lead. That's a proof, not a trick, not a strong opinion, a chain of tiny yeses that carries you somewhere new. Makes sense so far? Let's actually walk through one, the very first one in his book, and you can picture the whole thing with your eyes closed. Uklid
wanted to build a perfect triangle. Not just any triangle, but an equilateral one, which just means a triangle where all three sides are exactly the same length. The tricky part is the word exactly.
It's easy to sketch a rough triangle that looks about even. It's much harder to build one where you can prove with total certainty that the three sides are equal. So, how do you do it using only the simple tools we just agreed on?
Here's how. Start with a single straight line. Just a short segment lying in front of you with two ends. Let's call one end the left point and the other end the right point. This line is going to be the bottom side of our triangle. Nice and simple. Now, the first move. Put the sharp point of an imaginary compass on
the left end of the line and open it up until the pencil reaches the right end. So the compass is now exactly as wide as your line and you swing it all the way around drawing a full circle. The left point is the center of this circle and the line is its radius, its reach.
Now do the very same thing from the other side. Put the point of the compass on the right end of the line. Open it until the pencil touches the left end, the same width as before, and swing another full circle. Now you have two circles the same size, side by side, each one centered on one end of your line. And because they overlap, they cross each
other. They meet at a point up above the line. Let's call that meeting point the top. Now draw two more straight lines. One from the left point up to the top and one from the right point up to the top. And there it is, a triangle, the bottom line and two lines climbing up to meet at that point above. But hold on.
How do we know it's equilateral? How do we know for certain that all three sides are equal? This is the beautiful part. This is where the proof lives. Think about that first circle, the one centered on the left point. The top point sits right on the edge of that circle. And the right point also sits right on the edge of
that circle. Both of them are on the rim. And here's the quiet truth about any circle. every point on its edge is exactly the same distance from the center.
That's just what a circle is. So the line from the left point to the top and the line from the left point to the right point must be equal. They're both just the radius of the same circle.
They have to match. Now do it again with the second circle, the one centered on the right point. The top point is on its rim and the left point is on its rim. So the line from the right point to the top and the line from the right point to the left point are also equal. Same reason, same radius, same circle.
So now gently put it together. The bottom line equals the left to top line because they're radi of the first circle. and the bottom line equals the right to top line because they're radi of the second circle. And if both of the slanted sides are equal to the bottom line, then they must be equal to each other, too. Two things that each equal
the same third thing must equal one another. All three sides are the same. It is equilateral. And we didn't measure a single one. We didn't trust anyone. We just followed the tiny steps and they left us with no way out but the truth.
That's the whole spirit of Uklid right there in one small shape. You start with almost nothing and you end up certain. [clears throat] Now, what's lovely is that the same way of thinking works on questions that feel much harder.
So, let's do a second one. And this time it's something you probably heard in school and just took on faith. The three angles inside a triangle always add up to the same amount. No matter what the triangle looks like, tall and thin, short and wide, tilted, lopsided, doesn't matter. The three corners always add up to a straight
line, a perfect half turn. You remember your teacher saying that, right? But did anyone ever show you why? Let's see it together. Picture any triangle at all, any three corners you like.
Now, along the very top corner, imagine drawing a straight line that runs perfectly parallel to the bottom side of the triangle. So, the bottom of the triangle and this new top line are like two railway tracks running in the same direction. Never getting closer, never getting farther apart.
Now look at the two slanted sides of the triangle. Each one climbs from the bottom up to that top corner, crossing both of our parallel tracks along the way. And there's a gentle fact about parallel lines that Uklid proves earlier in his book. When a slanted line cuts across two parallel lines, it makes the same angle with each of them.
The angle down at the bottom where a side meets the base shows up again identically at the top where that same side meets our new line. It just gets copied upward. So look what happens up at that top corner. You've got the actual top angle of the triangle sitting in the middle. And on one side of it, the bottom left corner has been copied