Music Alright, so good afternoon. I'm Norman Walburger here at the University of New South Wales. And this is a course in the history of mathematics.
So we're going to be looking at a number of different episodes in the history of math from the earliest times up to the end of the 19th century. There will be 12 lectures altogether and the 12 lectures are going to follow very closely the textbook of the subject. The textbook is this weighty tome. It's a very nice book by John Stilwell.
It's called Mathematics and Its History. It's an undergraduate text in mathematics, third edition by Springer. It's available in the bookshop. It will also be on open reserve at the library. What we're going to do is we're going to be following this book rather closely, but we're not going to do all of it because it's a rather thick book.
We're going to do roughly the first half. Roughly. So the outline of the course is as follows.
We're going to have 12 lectures. The first lecture, this is today, we'll talk about Pythagoras'theorem. That's chapter 1 of the book. Second lecture, chapter 2, Greek Geometry.
then Greek number theory infinity in Greek mathematics then number theory in Asia polynomial equations analytic geometry projective geometry calculus infinite series mechanics and non-Euclidean geometry will be our final chapter. So the book is distinguished, I think, amongst other books on history of mathematics by its emphasis on the mathematics. Stillwell is interested in presenting the mathematical ideas, and that's... what I'm interested in doing too. So although there will be historical tidbits here and there, we're mostly interested in understanding and learning some mathematics as it was developed over the centuries.
So this is a nice way of studying mathematics and it's quite different from what you learn in your courses. If you take a course in linear algebra or a course in calculus, you just get sort of the modern condensed version. of a subject after it's been honed by hundreds of people over centuries. You get quite a different view of things when you look at the historical record and ask yourself well what were people thinking, why were they asking certain questions, what is the sort of flow of the subject across time. So we're going to be talking a lot about mathematics, and I want you to learn some mathematics.
So I'm going to ask you to learn quite a lot of mathematics, in fact. And the mathematics that we're going to be learning is quite interesting mathematics. And it's good for you in the sense that it's a lot easier than most math courses because we're not going to focus too much on proofs.
I want to sort of outline some really basic fundamental mathematical ideas and theorems, but we're not going to go into too much detail as to how you prove them and how you set them up formally. I want you nevertheless to understand what the theorems say and to be able to say draw a picture or explain it to somebody else. So the mathematical background that I'm asking of you is relatively minimal, but you will have to do some work.
You will have to try to understand some of the mathematics. Alright, so our first lecture... is a very good place to start with Pythagoras'theorem, which is probably the most important theorem in mathematics. Pythagoras'theorem. And that's a good place to start.
It's also perhaps the first really major theorem in mathematics. So let's talk about Pythagoras'theorem. That's only one of the several topics that I'm going to be talking about today. We're also going to talk about irrationalities and Pythagorean triples or rational parameterization of the circle.
So Pythagoras lived around 600 years. before Christ, and he was one of the first and most influential developers of Greek mathematics. Let's have a quick time frame just to position ourselves historically. So here is a brief history of the world.
Very brief history of the world. Okay, so at some point maybe we'll put year 0 there. And maybe not exactly a linear scale, year 1000 there, year 2000 here. And maybe here we had 1000 BC and 2000...
Okay, so what happened in the world for the last 4,000, 5,000 years? Well, there was a lot of political bickering and armies going back and forth and so on, but that's not really important for the whole big picture, right? as politics.
The really important developments had to do with ideas and the corresponding technological developments that those ideas gave rise to. So this is going to be, let's just position ourselves very quickly, sort of where were the major civilizations at what point. So early on, around 2000 BC, agrarian societies started to develop large-scale agriculture, started to develop in the Mesopotamia, Egyptian regions.
So we had ancient Mesopotamia, or sometimes just called the Babylonians. And sort of roughly concurrent with them, the Egyptians. And in the east, we had Indus Valley Indians and Chinese early empires also. The Greeks don't really, so this is around 2000 BC. What does the Greeks have, like 40,000 or something?
So let's just start with these. So of most of the interest, I suppose, in the story is the Babylonians and the Egyptians because they influenced the Greek period a lot. The Greek period is... as you can tell from our list of topics of fundamental importance, because it was really the ancient Greeks that set up mathematics the way we know now.
And that happened, the Greek period is roughly sort of like this. Sort of roughly a thousand year span from about 6-700 BC to maybe 300 AD. So this was the...
The golden age of Greek mathematics and a lot of what we are studying in mathematics, especially in the history of mathematics, has to do with Greek mathematics or developments that came from the Greek mathematicians. And prominent amongst the Greek mathematicians were some very famous names. Well, Pythagoras was one, but probably more famous than Pythagoras was Euclid and Archimedes.
I would say these are the two towering figures in ancient Greek mathematics. But there are many other important Greek mathematicians too. Okay, then we move on.
There was the Greek civilization died down, and essentially sort of a period of Arabic Middle Eastern influence took over. So we have sort of Arabic development starting from the Middle East but ultimately emerging into Europe and so the Arabs are a bridge from the Greek mathematics to the Europeans. And the Europeans start perhaps roughly around 1400s, 1400-1500.
And then there's a kind of explosion of activity in these years. So there's a lot of activity as the Europeans started to digest the Greek ideas, thanks to the Arabs that basically brought and preserved the Greek ideas. to Europe via Spain and others and then there was this great flowering of mathematics with many many important names but if we had to name the most important names I would say that they were Newton and Euler.
If I had to name the top two I would say they're the most important sort of scientific figures of the modern era. There are also others, Gauss prominently, and many others too. And what happens here is that the mathematics becomes quite rich, and by the time we get to the 20th century, it becomes very complicated, and we're going to cut off at the year 1900 roughly, so we're going to go up to there, up to the beginning of the 20th century, and not much further. Okay, so there's some other things to say.
There was a bit of Chinese work. There was certainly Indian influence. The Indian influence was also around this time here, and so I'll put that there. The Indian influence and the Arabic influence, I should maybe say, they were combined in some ways, at least and certainly in terms of our number system.
But there was certainly a very important also Indian strand in there as well. Okay, that's very roughly, very simplistically, a rough idea of sort of the time scale. And now we want to, in the next few lectures, we're going to go back in time to years before Christ, around 700, 600 BC. And in this sort of thousand year period, which is roughly called the Greek period, we're going to talk about various developments.
Now I should say when one talks about Greek mathematics, one is not really talking about just mathematicians from Greece. One should understand that this is a description of a rather broad area. The whole eastern Mediterranean. was basically Greek culture, Greek language, and some of the main figures were not actually living in Greece at all. For example, Euclid was mostly in Alexandria.
the center of learning in Egypt which was very important for many hundreds of years. There was a big library there. And Archimedes did, well spent a lot of time in Syracuse which is in present-day Sicily, an island off of Greece, off of Italy. And a lot of these mathematicians also traveled rather widely, so it's really a Greek sort of language, culture, or tradition. And there's various portions of it, there's sort of an early and a middle and a late period.
Alright, so that's sort of our rough setup. Now let's talk about Pythagoras'theorem. So Pythagoras'theorem was almost certainly known at some level to the ancient Babylonians.
There's certainly knowledge or indication that the Egyptians knew about it because they had this famous rope example where you have a rope with four lengths there and three lengths there. five lengths there. So this famous triangle, a 3-4-5 triangle, is a right triangle. To make a triangle with sides 3, 4, and 5, then that's a 90 degrees. It's exactly 90 degrees.
And the important relationship here is that this area the area of the square built on that side plus the area of the square built on that side is the area of the square built on the hypotenuse. So these areas, well that one's nine, that one's sixteen, and that one's twenty-five. So these days we usually, the modern formulation, The modern formulation is to say, all right, we have a right triangle with sides lengths d1, d2, d3, then d1 squared plus d2 squared equals d3 squared. But that's not actually the way the ancient Greek...
Greeks thought about the theorem. The ancient Greeks thought about the theorem not in terms of lengths, but rather in terms of areas, rather this way. So the Greeks used area as a fundamental concept.
not length. And so the formulation of the theorem in Euclid, Euclid's Elements, which is the most important book in the history of mathematics. Euclid's Elements, that's written around 300 BC.
So Proposition 47 and Proposition 48 both deal with Pythagoras'Theorem, and they say that A triangle is right angled precisely when the sum of the squares on the sum of squares of No, okay. I should say, a triangle is right angled precisely when the sum of the area of the square, sorry, pardon me, sum of the areas of the squares, areas of the squares on the shorter sides equals the area of the square. On the longer side. All the hypotenuse.
Okay, so it was a theorem actually about areas and not about lengths. So the picture is... It's not good.
This area, let's call it Q1. This area, Q2. This area, Q3. Then we have a right angle precisely when Q1 plus Q2 equals Q3.
So it's important to realize this is different from the modern formulation because it does not require a notion of length. It requires only a notion of area and area is a more fundamental and more simple notion than the notion of length. To define lengths we need square roots.
That's a consequence of Pythagoras'Theorem. But to define areas, in a lot of situations, you don't need square roots. Let me give you an example of that. So suppose you had, suppose we're working in a more modern...
framework. The Greeks did not think this way, but let's work in a grid plane. A piece of graph paper. So we're really introducing coordinates here. We can think of maybe a coordinate system 0, 1, 2, 3, 0, 1, 2, 3, and so on.
X, Y coordinates. So this kind of coordinate system only arose in the European era, basically due to the work of Descartes. Sometimes call this a Cartesian coordinate system because of Descartes. Okay, so suppose you have a little length.
Let me... Do a rather simple minded one. Let's just talk about this, that little segment there. Okay, over two and up one.
Now. What's the length of AB? The length of AB is, well we can use Pythagoras'theorem exactly to define that, because right here is a little right triangle, and it has side length 2 and side length 1 there.
And so the length is going to be the square root of 2 squared plus 1 squared, which is the square root of 5. And calculating the square root of 5 is not an easy thing to do. Even if you use a calculator, you can still only get it approximately. What about the square? Is that simpler? Yes, that's a lot simpler.
So here's a square built on this side. How can you build a square from this segment? We're going to go over 2 and over 1. You're going to go sort of perpendicularly.
Then if you go over 1 and down 2, that's going to be perpendicular. Do you agree? That look perpendicular?
You go over 2 and then up 1, or you go down 2 and over 1, that's perpendicular. So if I keep going like this, then that's a square. Okay, could I calculate the area of that square?
Well, yes we could because, for example, we could do it in a number of ways. One way would be to look at this larger square. Okay, what is that?
Well, that's 3 times 3, so the area of our square, let's call it ABCD to give it some names, the area of the square ABCD, that's the area of the big square, which is 9, 3 times 3, minus the area of these four... Four triangles. Yeah? So I'll say minus four times the area of the triangle.
Can we figure out what the area of the triangle is? Yeah, that's pretty easy because, say, we can put this triangle and this triangle together. We get a rectangle, sides two and one.
So this area plus this area is going to be 2. And this area here of this little triangle and this little triangle also add up to 2. So this is going to be 9 minus the sum of the black triangles is going to be 2 times 1. So altogether, 5. So we can calculate the area of the square on that segment purely by elementary counting, without any sophistication at all. But the square root, the length, is a much more complicated notion. We're going to talk a little bit more about that.
That's also something that was of great interest to the Pythagoreans. The ancient Greeks loved numbers like five. They thought the integers were super really good, especially the positive integers.
Numbers like this worried the ancient Greeks a lot. They didn't have calculators. So if they wanted to say, well, what was this, they'd have to do some...
of calculation and even after the calculation they wouldn't have the square root of 5 exactly they would only have some approximation to it. They knew that this was some kind of hard object, it's some kind of infinite decimal. Nowadays we know that. It's an irrational number.
And the Greeks did not like irrational numbers. Huh? Yeah, that's I suppose, I'm not sure if they called it irrational, what their word was, but it's probably something like that.
But the numbers 1, 2, 3, 4, 5, the natural numbers...... These were number... numbers that made the Greeks happy.
They thought these are very good. We want to explain the world in terms of natural numbers. And so that's why Pythagoras'theorem, the way Euclid stated it, can be phrased in terms of natural numbers. So that's a good thing. Alright, since it's the most important theorem, we need to give a proof.
I've said we're not going to do a lot of proofs in this course, but we will do occasional proof. So let's give a proof of Pythagoras'theorem, probably one that I hope you've seen. At some point, if not, it's a lovely thing.
We want to prove that this is true. And the idea that a statement should be proven was one of the great things. great innovations in Greek thought. The Babylonians probably knew that this was true, but as far as we know, it never occurred to them to ask the question, why is it true? It was just something that was true, like the fact that the sun goes up in the morning and goes down at night.
The Greeks were really the first to ask the question, why is this happening? Can we explain what's going on? Can we deduce this theorem from things that are simpler? So here's a proof that possibly Pythagoras knew. We don't actually know how Pythagoras proved the theorem.
Euclid has a rather elaborate proof, but that was probably one that he created himself. So this proof here... It's a kind of an aha proof. You just sort of look at it, hopefully, and see.
So we're going to take a square, and we're interested in some right triangle. So that's supposed to be a square, and that's supposed to be a square inside the original square. Remember, it doesn't look too square, but that's supposed to be a square inside another square. I'll make that just a little bit. Okay, all right the triangle that we're interested in is this one right here.
There's a right triangle. That triangle there is the one that we're interested in and there's actually four copies of it in this picture. One at each corner. Alright, so this triangle here, imagine that that's the basic one, that's the same one as the one that we're talking about over here. Okay, so we're interested in proving why this, the square on this side plus the square on this side is equal to the square on this side.
And we can see that what's inside here, the square inside, is exactly the area Q3, because it's a square on the hypotenuse. It's a square and it's a square whose side is the hypotenuse of the triangle we're starting with. So that's Q3.
Okay, now what we're going to do is we're going to shift things around. We're going to take these four triangles inside this big square. And we're going to move them. We're going to take this one and we're going to slide it up.
Just left room for it to slide up there. We're going to take this one and we're going to slide it down. We're going to take this one and we're going to slide it across.
So what happens? Well, the one that was up here is still up there. This one has slid up and joined it, so now it's sitting right there.
That was the hatched one that we started with. What's happened to this one? Well, it's slid all the way down there, and it's now right in that corner there. And this one here, as we've slid it across, it's now touching this one right there.
So this one has just moved up, slid up in that direction. This one is just slid across, that one slid down. The original square is supposed to be the same size.
And now we see that we have two smaller squares. whose areas are q1 and q2. Since we've just moved things around, we haven't distorted or changed the areas, the area inside here has got to be the area combined of these two.
So we can deduce that Q3 equals Q1 plus Q2. There are lots of other proofs over the centuries. People have created all kinds of proofs.
There's probably dozens of them. But that's probably one of the most elegant and straightforward. Okay, so one of the reasons why Pythagoras'theorem is so important is because of this formula here that we now use to define the area, rather the length, in Cartesian coordinates.
So in modern... Cartesian coordinates, Pythagoras'theorem is used to define length or distance. Same thing.
And how does that go? Well, in ordinary x and y coordinates, if we have two points, say A1 with coordinates x1, y1, and A2 with coordinates x2, y2, then... If we're interested in the length of the distance between them, what we do is we think about a little right triangle parallel to the coordinate axes.
So if we call this distance d, then D is defined by Pythagoras'theorem. Since D squared, the area of the square, will be the area of the square on this side, which is the difference between the x coordinates squared, so x2 minus x1 squared, plus the area on this side, which is the difference of the y coordinates squared. And now we have to define d, we need to take a square root. So the actual distance itself is defined to be the square root of x2 minus x1 squared plus y2 minus y1 squared. History of maths.
Yes, welcome. So the formula, the modern formula for distance between two points relies on Pythagoras'theorem. Now at this point I'm obliged to correct something that Stilwell says which I don't think is really correct. And it's a point that's a little bit of a source of confusion in other places too. There's this idea that this definition somehow...
reduces Pythagoras'theorem from a theorem to a definition. In other words, there's an idea that somehow we've built Pythagoras'theorem in automatically so that it's no longer a theorem. We've just sort of created length in such a way that Pythagoras'theorem is true. Okay, that's not really correct. So Pythagoras'theorem, even in this Cartesian system, it is still a theorem.
It is not automatically true. It's automatically true if you have a triangle whose sides, which has two sides, parallel to the coordinate axes. If you have a triangle, which is a right triangle, whose sides are not parallel to the coordinate axes, then yes, you can define this distance using this formula, you can define this distance using the formula, you can define this distance using the formula, but you still need to prove that this distance squared plus this distance squared equals this distance squared. So it still requires a proof even in Cartesian coordinates. Legend has it that Pythagoras was very thrilled to have discovered this theorem, and that he sacrificed an ox, as one did back in those days, when something good happened.
So he sacrificed an ox to the gods to thank them for letting him discover this theorem. We don't know if it's true. So Pythagoras was a very mystical figure.
He was perhaps the second important figure in the story of Greek mathematics. The first was Thales. Thales was sort of the father of Greek mathematics.
And Pythagoras was very likely Thales'student. And Pythagoras became very popular and influential because he founded a school. So he founded a school, I think, also in Sicily.
Went to Sicily and founded a school. He had sort of disciples. So there was a Pythagorean school.
And these people... try to understand the world. They thought we're going to try to understand the world using this new way of thinking, this idea that we can deduce things, and that we can use numbers to explain things.
So they had a theory of harmonics. If you take a vibrating string, like a guitar string, and you double it, or you half it, then you go up an octave. Do, do, do, do. If you have it, you go up an octave.
They also discovered that if you have a perfect fifth, was another, given by another proportion, can't remember off-hand what it was, A perfect fourth is another proportion. Proportions between whole numbers had direct connection with musical notes. So they had a kind of a theory of music based on the ratios of whole numbers.
And they had this idea that number could be used to describe almost everything in the natural world. And when they say numbers, we're talking about ordinary natural numbers, and numbers that you could build from them, ratios like this, or maybe what we would call fractions. Okay, so their idea was that the universe, the world, should be explainable only in terms of these kind of numbers. But unfortunately, they were kind of undone by a famous conundrum. The introduction of irrationalities that we talked about earlier.
And it was actually Pythagoras'theorem itself which was the key problem that created the dilemma. And the dilemma is a very simple one. It has to do with just a unit square. There's a square.
And the diagonal of that square, which these days we all know is, well, we call it root 2. Whatever the number d is, by Pythagoras'theorem, it's got to have the property that d squared equals 1 squared plus 1 squared, which is 2. So from their point of view, well, what number squared gives you 2? Well, where is D? So if that's 1 and that's sort of 2, then D should be bigger than 1 but less than 2. And it's not too hard to see that it's got to be sort of somewhere in this range, somewhere between 1 and 1.5. 1.5 being 3 halves.
And at first, they probably just assumed that d could be expressed as a fraction. Maybe it's a big fraction. But that there should be two natural numbers so that you could write the square root of 2 as some fraction over some other fraction. Maybe d equals 351 over 700 and, well it's bigger than 1, so maybe 1300 over 9, okay, maybe 1351 over 973. Maybe that's what d is. Well it's not, but they kind of hoped that it would be.
And then at some point one of them realized that this was not going to work. That there was no way that D was going to be a fraction. And the argument is an important one.
It's a very historically important argument, which we want to understand. So the second sort of important topic of today's lecture, the irrationality of the square root of 2. So theorem, root 2 is not a rational number. There's no way of writing it as some kind of ratio of natural numbers. So let's see why this is true. Okay, it's a famous argument.
So proof. Suppose that root 2 actually is. A fraction, a over b, where a and b are natural numbers. And we can assume that they have no common factor, with no common factor. Okay, so for example, if it was 24 over 8, that's a fraction, but we can simplify because it has a factor of 8 in both top and bottom.
So we can rewrite that as 3 over 1. So that's what we're going to assume. We're going to assume that A and B have no common factor. So what do we know about this number a over b?
Well, we know that a over b squared has to equal 2. We're going to try to get a contradiction. And so we'll look at this equation and we'll first rewrite it as a squared equals 2 b squared. Remember a and b are natural numbers. Okay, what can we conclude from this?
Well, a is a natural number. 2b squared is an even natural number. Because it's 2 times something. So the right-hand side is an even number.
What does that tell us about a? Well, if a was odd, then a squared would also be odd. Because the square of an odd number is an odd number.
So there's no way that could happen. So it must be that a is also even. We can conclude from this. Since the right-hand side is even, a itself must be even. What does it mean to be even?
It means that a can be written as 2 times another number, another natural number. Let's say c. Right? Every even number is 2 times another number.
Like 14 is 2 times 7. 22 is 2 times 11. Every even number is 2 times another number. And so we'll let that other number be c. Alright, now we're going to plug that back in here.
Now we're going to get 2c all squared equals 2b squared. And when we expand this out and divide by 2, we get 2c squared equals b squared. Because there was 4 and a 2 and the 2s cancel, so we have 2 on this side. This looks a lot like what we just had. Now we see that the left-hand side is even, so b must be even too, because if it was odd, then odd squared is odd.
So what can we conclude? We now know that A is even and B is also even. That contradicts our original assumption that they had no common factor. We assumed to begin with that they had no common factor, so 2 cannot be a common factor. This contradicts the assumption.
And so our original setup is impossible. So therefore, root 2 equals a over b is impossible. And root 2, whatever it is, is some new kind of number.
Whatever it is. But the Greeks would have said, well, what does it mean? Maybe it doesn't really exist.
Does root 2 exist? Well of course it's easy enough to write down the symbol. Say, ah, there, now it exists. But if you actually want a number whose square is 2 rather than just a symbol, then it's a little bit more problematic. If you actually get your calculator, it'll spit out 1.414 and blah blah blah, and it'll keep spitting out more and more digits depending on how much time and memory you give it.
And if you let it, it'll go on for trillions and trillions of light years, just spitting out more and more digits. And even after trillions of light years, it still hasn't really stopped. So what exactly is root 2 is in fact more of a conundrum than people generally imagine. And the Greeks were very aware of this. This caused the Greeks an enormous amount of headache.
In fact, the Pythagoreans, when they first discovered this fact, they vowed to keep it a secret. It was an embarrassment. that the diagonal of a simple square could not be made commensurable with the side.
They vowed to keep it a secret, but one of them leaked the secret, according to the legend. And then they were on a trip on a boat, and that person, one of us came out that he had given away the secret, they threw him overboard, fed him to the sharks. That's a famous story, the Pythagoreans. Suggesting just how serious this dilemma was.
Now this dilemma still has a lot of repercussions in modern mathematics. So this is still not entirely unresolved. Still not unresolved.
Most mathematicians pretend that it is, but in fact it's still a little bit problematic. Alright, we'll take a break and then we'll come back and talk about the third interesting development from Pythagoras'theorem, which is how do we describe the points on a circle and Pythagorean triples.