In Our Time - Euclid's Elements

Episode Date: April 28, 2016

Melvyn Bragg and guests discuss Euclid's Elements, a mathematical text book attributed to Euclid and in use from its appearance in Alexandria, Egypt around 300 BC until modern times, dealing with geometry and number theory. It has been described as the most influential text book ever written. Einstein had a copy as a child, which he treasured, later saying "If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker."With Marcus du Sautoy Professor of Mathematics and Simonyi Professor for the Public Understanding of Science at the University of OxfordSerafina Cuomo Reader in Roman History at Birkbeck University of LondonAnd June Barrow-Green Professor of the History of Mathematics at the Open UniversityProducer: Simon Tillotson.

Thank you for downloading this episode of In Our Time, for more details about in our time, and for our terms of use, please go to BBC.co.ukkah.uket's slash Radio 4. I hope you enjoy the program. Hello, around 300 BC in Alexandria, one of the most important works in mathematics appeared. We know it as Euclid's elements, and it became so influential that it's deeply ingrained in everything mathematicians to do even now. Some of the ideas are a revelation. One of my guests today likens its proof on prime numbers to the discovery of the

atom. Bertrand Russell wrote of Euclid's elements, At the age of 11, I not imagined there was anything so delicious in the world. In the elements for the first time, there were logical arguments for why some mathematics would always work in any situation. It became the core of the teaching and understanding of the subject for over 2,000 years, right down to the school days of many of our listeners. With me to discuss Euclis elements are Marcus Chisotoy, professor of mathematics, and Simony Professor for the Public Understanding of Science at the University of Oxford, Serafina Cuomo,

reader in Roman history at Birkbeck, University of London, and June Barrow Green, Professor of the History of Mathematics at the Open University. Marcus Tiusotoy, when the elements emerged, how well established was mathematics in 300 BC? I think that mathematics have been being done since 2000 BC. You see Egyptians, Babylonians,

doing a sort of mathematics, but it's what we see emerging here is a very, very different sort of mathematics. So they were doing very practical things. They were solving particular equations, finding particular examples of Pythagoras' theorem. But here we see a sea change. And I think for me, this is the beginning of mathematics

as an analytic deductive subject, not just by example, but as you said in your introduction, proving that things will always work, that finding about universal truths. And it's the discovery of the power of proof and deduction from a list of axioms that sees this change. Now I would say that the elements is probably the

culmination of several centuries of work by Greek mathematicians. So this isn't a sudden explosive text that appears on the scene. This is actually sort of bringing together this idea that mathematics actually is something that you can do which proves things with 100% certainty and applies in every single situation. I believe in the Cretion and ancestry and the relevance of the past and so. Nevertheless, at this time in Greece again and again, there's a switch. Yes.

And it is most extraordinary. Yes. I mean, we do keep going back to the Greeks, but it is an extraordinary thing. There was this switch. Now, can you define the switch? Yes, I think for me, why did mathematics appear in Babylon and ancient Egypt? It's because they were starting to create city states. They were measuring the land.

They were building. And so their mathematics was very geared up to very practical needs. but at this time you see political institutions beginning to appear in Greece. And I think this is the change. You have the power of rhetoric, the ability to use language to persuade an audience. I mean the logic of rhetoric. The logic and the pathos. I think actually this is part of something that people may not unappreciated.

Mathematics actually works on the emotions. You're telling a story. Euclid's elements is full of storytelling. It's taking people from things, a place where they kind of understand and are happy, to a new place. And this for me is the transformative moment. I think this is worthy of being compared with the Iliad and the Odyssey.

This is journeys that are being made but in the mind of the reader and the listener to anyone reading these proofs. But you haven't precisely answered the question I ask and perhaps it is unanswerable. Why at that time, among a very few people in a very small place, did the thing change?

Did logic come in? application of logic to law to politics to matter? I think you've identified it. This is when political institutions are appearing. It's not just about building and creating cities. There's a change in society where now we want laws and we want things that can be a way

of thinking which can now be applied to creating a new sort of state. And for me, I think this is why the sort of ideas of mathematics begin to appear it's not just about a practical building. It's about creating political institutions and the way

of logical thinking. It's also, look, get me, you know about this, I don't, okay? But it also seems to me to be some kind of collective determination on the part of people to go for universality. We're not solving this problem. We're solving this condition for all time. Well, I think that's the remarkable change that is happening. And I think, you know, you have, it's extraordinary that when you read the elements, it feels almost like a modern text. And that testament to the fact that what was being done is as true 2,000 years ago today as it

was 2,000 years ago. And this universality is one of the most exciting things about the appearance of the elements on the scene. Serafina Cuomo, who wrote the elements? We call them Euclid elements? There you go. We call them Euclid elements because most of the

manuscripts start with the title, Nuclid's Elements. But the truth is We know very little about Euclid. We know about episodes in his life from later commentators. And some of them have funny anecdotes about Euclid and the king of Alexandria and the rest of Egypt at the time, Ptolemy the first. But historically, Euclid does not give us many clues about his life and his personality. Some other Greek mathematicians, for instance, Archimedes of Syracuse,

start their mathematical treatises with an introduction or a letter to another mathematician, and they tell us more about what they were doing and the process of writing the text, but we don't have anything like that for Euclid. So it remains quite an obscure figure. Obscured to the point where there are various theories about him, that he was a man called Euclid, that he was ahead of a group of people, and he was the chief chap, so they called it Euclid. or that it didn't exist at all and it was later made up as Euclid for reasons, which I can't comprehend.

But anyway, which one of those do you follow? I think there was a person called Euclid. At the same time, what we now have as Euclid's elements has been worked upon by more people than just Euclid. How do you know that? Because the manuscript history, which is very complex and splits into it. different traditions. There's an Arabic tradition. There's a Greek tradition. There's a Latin tradition which is based more on the Arabic side of the transmission than on the Greek one. There are subtle differences between different versions of the text that indicate that there's be more than

one layer of producing the results that today we have as Euclitz elements. Plus there's a very there's a kind of smoking gun. A fourth century mathematician called the Theon of Alexandria, the father of Ipecia, among other things, produced an edition of the elements.

In another book, he also commented on Ptolemy's Almagest, he says there's this passage and I added that to Euclitz elements when I edited the text. If he hadn't told us, we would have never known,

that that particular passage in what people took to be the original text of Euclis elements was actually Theon's Addition. So if that is the case, that could be the case for a lot of other passages. So it's all being added to. How is it transmitted through the next 2,200 years? What would the main, it's a long way to get over, and we've got 43 minutes, but even so, what would the main boosters along the way? So it's a very long, it's a very big question, but let's just focus on a few points. Three will do.

Two will do. Okay. So first of all, there was a transmission of the content of Euclitz elements, if not necessarily the name at the time. Because we have a significant number of papyri. And we have a few pieces of pottery even from Egypt with material that corresponds to Euclitz elements written. on them, they don't start saying Euclid, they don't say the elements, but there seems to be a connection between what we have as Euclid's elements and these pieces of writing from Egypt quite early on.

Another, what I would put as the second big event is, I think, the Arabic translations. Between what we call the 8th and the 9th century CE, in Baghdad, there was an enormous translation movement, which was a sponsor, they encouraged and patronized by the Caliphs at the time, especially the very famous Arunal Rashid, to the point where they would send people to Byzantium, Constantinople, to get manuscripts, Greek manuscripts back, to have them translated. And in this context, Greek manuscripts of Euclid's elements that are now lost to us were translated into Arabic and worked upon and retranslated and transmitted and copied all the way to the West. Thank you very much. June, Barry Green.

The elements are known for their premises, for the premise, axioms and postulates. Can you describe to us what those terms mean in the elements? Well, the elements begins with, well, really two, there are two parts to the beginning, the fundamental parts, are the definitions. You have to start with, you have to know what you're talking about, writing about. So in book one, in fact there are 13 books of the elements and in book one is prefaced with these fundamentals, the definitions and there are 23 definitions at the beginning of book one and they are things as basic as a point is that which has no part.

A line is a breadfulness length. And you went too fast a minute. A line is a what? A breadfulness length. A line between two points, you draw a line and that's called a line. Yes, but it's breathless. It doesn't have any width. Breathless.

Oh, I see. Whittless. I'm not quite sure. So you have these definitions, and he defines triangles, different types of triangles, different types of angles, and so on. Why does you call these axiomatic? Well, these are the definitions. I'm coming on to the axioms.

All right, sorry. So we have to have some definitions in order to be able to have some axioms. So, and then there are definitions carry on through the books. But in book one, as well as definitions, we have the axioms. And there are two types of axioms. the geometric axioms which are generally known as the postulates and there are five postulates and the axioms

I mean these are self-evident truths actually the word comes from the Greek which means sort of take for granted they're things we can just we can assume and so the first one of those is that it's possible to draw a straight line between two points you have to be able to know that you can do these things

and it's possible to extend a line continuously in either direction. It's the second one. It's possible to describe a circle with any centre and any radius. And then fourthly, it's true that all right angles are equal. A right angle is an angle of 90 degrees.

And those four postulates were completely accepted. Then there was the fifth postulate. You were talking about axioms a minute ago, now we're on to postulates. Postulates, yes. Postulates are the geometrists. I'm confusing myself more than you're right. Sorry, the postulates are the geometric axioms.

So you can call them axioms, but these particular ones are known as the postulates. The thing is so fine, I want you to go on on another one. The wonderful thing about this is a blindingly obvious when he says so. But the thing about him saying so at the beginning of what is a very, very complicated work of geometry is that he's using the basic building blocks. And building blocks has to do with it because the engineering component of mathematics at that time is very profound. Yes, absolutely.

And without the definitions and the postulates, you can't go anywhere. So having established the first four postulates, then there's the fifth postulate, which is commonly known as the parallel postulate. And this was problematic. A lot of mathematicians thought the fifth postulate could be proved from the previous four. And I'm sure we're going to get into a discussion about that. But just to tell you what that is, today it's often used in the form where a Scottish 18th century natural philosopher and mathematician John Playfield, used it, well, stated it in his edition of the elements, so it's known as Playfair's postulate. And that's, if you have a plane and you have a line on the plane and you have a point on the

plane that is not on that line, then you can draw at most one line parallel to your original line. So that's a uniqueness of parallels, if you like. So you have a line and you have a point that are separate and you draw a line through that point. There's only one line that will be parallel to it. And this was to cause a lot of problems later on. We then also have some common notions and these are the sort of logical axioms if you like, and they're to do with magnitude. And there are things to do with if you have equals and you subtract equal parts from equals, then the remainders are equals. Or if you have equals and you add equal parts to them, the holes are equals. So once Euclite,

set that up, then he's ready to go. And he can start actually constructing things and he can start proving things. Well, thank you very much for that. I mean, so we know where we are, and I use the word building box, Marcus, and I think that was a, I hope it was useful. Can you develop the idea of building?

Because our listeners will think, well, yeah, a straight line, a straight line, and you draw a circle from a point in the middle, yeah, what's the excitement? Absolutely. This is the magic of the elements, because you start with things which are so blindingly, obviously. is what you've got two points you can draw a line how are you going to get anything interesting out of that um

and so these are these are the foundations i would say and now you use kind of the the ideas of logic and deduction to say well what follows from that and and out of this emerges 13 chapters of the elements where more and more interesting things are discovered so for example uh people have heard of pythagoras's theorem so this is about right angled triangles but why if you have a right angled triangle is the where on the larger side have an area that is the sum of the squares that you can put on the two smaller sides. Now, he uses these rules of deduction from these very basic axioms to show you why that must always be true for every single right-angled triangle. And I would say they're building blocks. Each of these new statements is a building blocks for the next move.

So I would call mathematics like an intellectual pyramid. And these axioms are the foundations of that pyramid. and everything we do and move on is an extra layer on that. And in some ways, that's why, unwittingly, we all start with Euclid. So at school, even though we may not say it's Euclid, actually all the things that we learned, the things that inspired me to become a mathematician, there were two proofs. One proof that there are infinitely many primes, one that the square root of two is an irrational number can't be written as a fraction.

Those are the things which lit up, you know, wow, can you prove that? And that's all in the elements, and it starts from these incredibly basic, things that we've heard these axioms, which seem blindingly obvious, and from there you get these amazing stories emerging. Very, very brief, in Marcus, can we just pick up a stitch from what we said at the side, that this is going when the city-state is burgeoning, when things are being built, and things are being built, not in any haphazard way, and let's put up a hut, and the things are being built according to form, according to space, according to, we're coming to Golden

Meals. So this interrelation with engineering, city building, wealthful leisure, and so-and-so, It's coming into the larger equation. No, I think actually not. I think here we're seeing this as discovering universal truths for their own sake. There's nothing particularly useful about prime numbers that there are infinitely many of them to this. Just a second. Okay, there is there are construction. You've gone way down there.

Prime numbers. Prime numbers is the great mystery later on. But what we've been talking about so far, lines, radiuses. Fair enough. Look, builders need to make pentagons, for example. And you see a lot of pentagons in sort of medieval time, and they're using the constructions to make a pentagon that are in Euclid's elements. But Euclid had a student who said to him, what's the point of all this? How am I going to profit from this?

And Euclid said to his servant, give that student a coin. Now you have profited from this, and let's move on. So actually, I think there's a change. This is mathematics for its own sake. It is useful, but I think that in the elements you're seeing the joy of understanding. and proving universal truths, not because they're going to be useful, but because we're enjoying mathematics for its own sake. Seraphina, what impact did the elements have on the Greek and Roman world that we can draw into this?

They have a very wide influence because the manuscripts are very numerous. So we know that Euclid was read. And more importantly, we know that Euclid's elements were seen as a live, enough piece of mathematics, that people felt almost compelled to not just leave it alone, but add to it, take away from it, insert bits where they didn't understand the text. So the impact it had on the Greek and Roman world is testified to buy the number of manuscripts that still exist. And also by the smaller forms of texts that I was referring to earlier, There are enough papyri from Egypt, from ancient Egypt,

with bits of mathematics that corresponds to Euclitz elements to make us think that people were reading it at the time, studying it, trying to work out the proofs. My favourite piece of evidence about the transmission of the elements at the time is five pieces of a broken pot, which have been found in southern Egypt. It's a very remote location, it's not even papyrus, it's like a very casual medium for writing.

And yet someone around possibly the second of first century BCE found this broken pieces of pot and wrote parts of Euclitz book 13 on them. Euclitz book 13 is the culmination of the elements. It's not basic geometry, quite advanced geometry. And yet there was someone in remote Egypt, just finding the time to scribble on them, maybe because, as Marcus was saying, doing that geometry gave them some kind of satisfaction

or joy. We know that the Romans read some version of Euclite elements because the treatises that the land surveyors used for their measurement contain the Latin versions of the definitions and some of the basic propositions about geometry. June and June Barry Green, the thing that, if I'm right from reading of the

Net-Satria, the technology that he had for all this was a ruler on a compass. Yes. In fact, it was what we'd say a straight-edge and a compass because it was an unmarked ruler. And so the whole of the elements, the only tools you can use

are straight-edge and compasses. Who had developed compasses? Gosh, I don't know. No, it doesn't matter. It's a relevant question. Please go on. So, I suppose builders. Yeah.

I'm taking it back to engineering. I know you don't like that. Don't like getting my hands dirty. So, and so one of the things about the fact that you could only use ruler and compasses, it led mathematicians to think about

what, as well as the proofs and the propositions and the constructions in the elements, what else could you prove using only ruler and compasses? And this led in particular to what are now known as the three classical problems, which, and I'll just tell you what they are, there's squaring the circle. If you have a circle, can you just using ruler and compasses construct a square of identical area? There's duplicating the cube. You have a cube, just using ruler and compasses, can you construct a cube which has double the volume and trisecting the angle?

You have an angle and can you try to set the angle just using ruler and compasses? And the Greeks found that they couldn't do any of these things. They could, but in trying to do them, they developed all sorts of other techniques and they proved to be very fertile. And in fact, I think they knew they couldn't do it, but they couldn't prove that they couldn't do it and they wanted to prove that they couldn't do it. And it wasn't actually until the 19th century that these things were proven

that it wasn't possible to do them with ruler encompasses, which is really quite remarkable. It is. What happened in the 19th century that enabled this to happen? Well, let's just take the squaring of the circle. Yeah, that will do. It was the fact that you needed to be able to construct the number that we now have known as pi,

which is the ratio of the circumference of a circle to its radius. And if you can construct pie, then you can construct the, you can square the circle. And it was only proven in the 19th century that pi is the number which we now call transcendental, which means it has a decimal expansion which goes on infinitely long. And I don't know how many million places

it's been calculated to now. Marcus probably knows. But it was, and they needed to be able to prove that in order to be able to prove that you couldn't do the construction. Marcus, let's come to the thing that interests you more than anything in the world next to a particular subject which you're not going to mention on this programme. I'm not going to talk about the Arsenal.

Oh, okay. And that's the proof on prime numbers. Yes. How did they get at that and why is it so massively significant in your estimation? I think it's interesting to, give a little bit more context of the elements that it's divided into three parts. One is about plain geometry. The middle is about number theory and a particular about prime numbers. And the last

section is about sort of three-dimensional geometry. In this section about numbers, we see what I regard as the most wonderful proof, perhaps the first great proof of mathematics, that the primes, these indivisible numbers like seven and 17, that there are infinitely many of them. And I think here, so if you'll indulge me, I'd love to give listeners an idea. of how you do a proof. So it's a beautiful logical argument, a finite number of steps which we're going to be able to touch infinity. And it goes to the heart of what the elements is about, which is start proving some things already and building on those. So Euclid builds on the fact that every number is either prime or divisible by primes. You can pull it apart. They're the atoms of

mathematics. Now he says, okay, suppose you think there are finitely many of these indivisible prime numbers. He says, okay, take that list, multiply those numbers together, and now here's Euclid's act of genius. He adds one to that number. Why? And then because he says, okay, look at this new number. Which primes divide this new number? I've proved for you already that every number is either prime itself or divisible by primes. Now look at your list. If you try any of your primes and try and divide it, you always get this remainder one. Adding that one, that act of genius has made this number always indivisible by any of the primes on this list. So this new number is either a new prime number or has to be divisible by some

primes that are missing from your list. And so you might say, well, I'll just add those to the list. But Eucl then says, well, I'll just play the same game. So by that little logical argument, he's shown the primes that they go on forever. That's a beautiful thing. The finite mind can conceive of the infinite. And also, it turns out to be one of the underpinnings on the modern world. Absolutely. I mean, the fact that a number can be pulled apart as primes is actually cracking a code on the internet these days, but Euclid didn't know that.

Serafina, how distinctive, let's just go back a bit now. How radical would his work have appeared to mathematicians over the next few hundred years who looked into them? To be honest, I don't think it was that radical. because one thing that was emerging even from what Marcus was saying is that the Euclid is almost the end point of a development as well as the beginning of something else. Before Euclid established basic geometry and arithmetic on undemonstrated premises and then building on that using the axiomatic or didactic method, a lot of philosophers and mathematicians had been discussing how that could be done. So Aristotle, for instance, in the posterior analytics, describes a method of demonstration from undemonstrable premises

that many people have linked to Euclitz elements. So by the time Euclitz produces the elements, there's a lot that's new, but there's a lot that's already known. Building on that, we have people, mathematicians, taking their work in different directions. Back to what June was saying, for instance, they start exploring in greater depth, problems that are not contained uniquely its elements, but using similar methodology. So you have Archimedes square in the circle. You've got various mathematicians all the way down to Papus of Alexandria,

in the cube and dissecting the angle. What you also have, and this has to be said, is different types of mathematics to Euclitz mathematics that run almost parallel to it. So there are non-Euclidean treatises that contain a lot of definitions that are interested in common notions. But then when it comes to a approach in geometrical problems,

take a root that's more similar to what we find in Egyptian and Mesopotameter. mathematics. June, Joe Baragreen, there are many, many, many additions. The number has been compared with profit to the number of editions of the Bible, and it's the

great rival to the Bible as you were. Anyway, which editions had, excuse me, which editions had most influence along the way? Well, the first, you get the first printed edition comes in a 1482. There's a so very soon after the beginning of printing.

But the area that I know about, which is about additions of Euclid in English, and they emerge in the 16th century. And the first one is a book by a textbook writer. The first mathematical textbook writer, we would say, is Robert Record. And he doesn't produce exactly an addition of Euclid, but what he does is he wants to make Euclid available to the common man. So he presents Euclid in a way that's understandable. And this is also new since in the vernacular,

because of course learning at this time was in Latin and so on. But after record, which is 1553 or something, 1551 or something like that, we get the Billingsley Euclid. And this is an incredibly important text. This was published in 1570. It's a folio edition, a magnificent book. I mean, only accessible really to somebody who was pretty well healed, lavishly illustrated.

even in the final books on solid geometry, we've got pop-up diagrams. So it's wonderful. I mean, you look at them and you've got these fold-up. Oh, I want one of those. You open it up, and they're folded bits of paper, and you can make your own pyramids and things. It's absolutely fabulous.

You've just solved Marcus's Christmas. Well, I think he's going to have to have rather a generous benefactor, actually. But as well as the addition itself, what Billingsley, who was a wealthy merchant, he becomes Lord Mayor of London. He invites John D, the well-known astrologer, mathematician, alchemist, works in the service of Queen Elizabeth. He asks John D to write a preface for him.

And D writes this preface, the mathematical preface. And in it, he gives a sort of taxonomy of the mathematical arts and sciences. And what he's doing is he's providing a vindication for producing this work in English. And he's showing the application of mathematics. And he says, basically applied mathematics stems from geometry and from arithmetic. And he gives this great catalogue of different areas where mathematics is used, architecture, music, geography, navigation, and so on. And saying, well, this is, you know, it's important. It's sort of underpinning what we do.

And this mathematical preface has been, was then republished subsequently. So this, I think this text for me is, you know, it really launches Euclid in the English-speaking world. Thank you very much. Marcus, June promised us that we come back to the fifth postulate, so here we go. Yes. Why is it problematic? The fifth postulate says that if you've got a line and a point off that line, that you can always draw a line through that point, which is parallel to the first.

I've drawn a line across the hour. Yes, now draw a point off that line. I just put a dot above it. Yes. The fifth postulate says you can always draw a line through that point which is parallel to the first line. I do that, yeah. Yeah, and there's a unique line which will be parallel.

You mean, I couldn't put another dot and draw another line. If you alter the line a little bit, it will meet the first line. So there's one unique line which, if you extend it, will never meet the first line. And that's what we mean by parallel. Now, as June already pointed out, many people thought, well, maybe we can deduce this from the other axioms. But what transpired was that not every job, geometry actually satisfies this. And in the 19th century, you get three mathematicians,

Gauss in Germany, Boliath in Hungary and Lobbachevsky in Russia, who discover new geometries that don't seem to satisfy this axiom. Now you might think, oh my gosh, Euclid is now going to collapse because that axiom is not true. What it just illustrates is actually mathematics and geometry is much richer than the geometry Euclid was looking at. Everything that Euclid proved from his fifth postulate is still true. For example, that triangle, always have angles adding up to 180. But in these new geometries, we find geometries, for example, the geometry of the surface of the earth,

straight lines are now lines of longitude or great arcs that we can move around. If I take a line of longitude and take a point off that line of longitude, I can't draw any lines which are now curved. These are curved geometries. I can't draw a line through that point which won't meet that line of longitude. So it means now there are no parallel lines in this geometry. on the surface of a sphere. And then there are other geometries

called hyperbolic geometries which were discovered, which actually describe the geometry of the physical space that we live in, the universe, where there are many parallel lines through any point. Now, it's very important

that this doesn't invalidate anything that Euclid is. It just expands our horizons. So everything that Euclid proved from those axioms is true, but now we have new geometries, which are called non-Euclidean geometries, where we have a triangle,

whose angles add up to more than 180 on the surface of a sphere or less than 180 in these negatively curved hyperbolic spaces. Seraphina, was this sort of discussion about the parallel lines, the fifth postage, was this going on in classical times? Was this a constant nag until the 19th century? It was going on in classical times. I don't know if it was a nag, because I think at the time, as I said,

people were looking at Euclid's elements not as something untouchable but there's something live alive to engage with and to change if necessary. So before Euclid there were concerns about the parallels. Again Aristotle has a mention of it.

So some historians think that Euclid established the parallels postulate as a postulate because he realized that it was problematic enough that he wouldn't be able to demonstrate it. The formulation in the element is actually quite different from playfers formulation.

So we are talking about slightly different things. But I think a very important contribution to the debate was made in the fifth century C.E. By someone called Proclos, who wrote a commentary on the first book of Euclitz Elements, and stops at the parallel postulate and says, a lot of people have had problems with it. He mentions that Ptolemy, the author of the Almagest, tried to prove it, but Proclos didn't think it succeeded, and the proof is no longer extant.

And Proclus says the problem is, Euclid says if you prolong those lines, they will meet eventually, and that means they're not parallel. Proclus says, how do we know? Because it could be that we keep prolonging them and we just don't know. Maybe they're like the asymptoties in a type of conic. June, Jean-Barrie Green, has there anything more we want to say? about the non-Euclidean geometry? Marcus Band began to develop that. Yes, I think I'd like to pick up from Serafina and say that it became increasingly a problem

from the ancient times, and that in the 18th century was really where, prior to what Marcus was talking about, there was an Italian, Sarkeri, and he was the one who sort of identified this issue to do with the fact that you can replace the postulate with two other types. You know, you can have these triangles, as Marcus has said, that either you have triangles at a 180-degree angle sum, Euclidean geometry, or you can have triangles that are

less than 180-degree some or more than. And that was, you know, so it's, I think what I find really interesting is that it's, that it's a constant sore. but what happens when Boliere and Lovaceki and Gauz get into the game, it doesn't go anywhere for a bit. Because apart from that, Gauss himself... Are they finding things which cannot be proved? Well, they're finding things...

Which is contradictory to everything you can stand. Well, I think the big problem is that the non-Euclidean geometry that Boliere and Nobacekie came up with, it's non-uclidean geometry in three-dimensional space. And Marcus has talked about this non-uclidean geometry on the surface of a sphere. And that's on a surface. It's not three-dimensional geometry.

And the three-dimensional geometry that Boliai and Obacchese came up with involves trigonometric formulae and things, and they weren't the best expositors. And it took, actually, for... It took a few... Well, I think it was in 1860s when an Italian mathematician called Beltrami

was able to kind of make a model which showed really what they were doing. And then suddenly it sort of... takes off and people understand that, you know, this geometry really does exist. Marcus, having a time briefly, because this is in the end, I want this bit to be brief if we're going to do it at all. Have we got time to do what the platonic solids were? Yes. And how he discovered that there were only five of them, and he's right.

I think this is the culmination for me of Euclis elements. It's the final sort of chapter which proves. This is the first example of a classification theorem, I think. What does it mean, platonic? Platonic solid are the dice that we use in games. So we're used to the cube, but if anyone has played Dungeons and Dragons, you'll know that there are some other sorts of dice,

like a little tetrahedron, which is actually the first sort of dice, used in a game dating back to 2,500 BC, a little made out of four equilateral triangles put together. And then there's the octahedron, the dodecahedron made out of 12 pentagons, and also the icosahedron made out of 20 equilateral triangles. But maybe there's another way to put together

a symmetrical shape to make a sixth dice that we can use. Euclid's elements proves that there's no way to put together a sixth shape. And I think this is the power of proof. You see before Euclid's elements, new discoveries, things being created, but to be able to use mathematics to show that something can't be made, I think it's the power of mathematics. And for me, I'm involved in my daily life as a research mathematician doing the same sort of thing. I'm trying to classify what possible symmetrical shapes can exist beyond our three-dimensional

world. So for me, that's, you know, the beginning of my whole research project is trying to continue what Euclid did, but understand symmetry in higher dimensions. Serafina, there were certain things which were not in the elements. He didn't have algebra, algorithms, zero and negative numbers seems to have come from India and not, can you just ruminate on that for a little? Okay, my first nomination would be to say that it didn't have them

because they didn't exist. I know, I'm not criticising you. I'm just saying it wasn't there. People would be wondering, they'll be scratching their heads. They'll be saying to each other, where's algebra they'll be saying? So you're going to tell us.

There was an algorithm though. It must really, Euclid's algorithm is the first ever algorithm. What we call an algorithm. Right, yeah. What we call algebra, some people thought they could see in book two of Euclid.

I think I've made a bad question. No, you made a very question. controversial questions. Because historians of mathematics are at each other's throats over whether book two of Euclitz elements can be called algebra or not. But sorry,

I'm June, you and Susan. Well, I was going to say that in the beginning of the 18th century, there was the first edition of the elements which was done in an algebraic form. So the proofs are actually much, much shorter because you've got this symbolic

language you can use. Everything in the Euclis elements is really about geometry and links. So even prime numbers are talked about as lengths of lines. Yes. Well, Seraphina wants to look, folks, it's legacy time. What do you think is left behind, Seraphina? What's left behind is the pleasure of reading Euclid. I just enjoying the satisfaction of understanding how he got to the end.

I was rereading the first proposition of the first book in preparation for the programme. And I felt so happy when I understood every single step because he had established it on something that came before. And I felt as if my mind was doing, as they call, a specific kind of gymnastics. June. Well, I think it's the fact that, you know, he was able to build on relatively few, clearly defined ideas, a line and a circle, this magnificent corpus of results, which, and the results aren't necessarily intuitive.

I mean, the Pythagorean theorem, the classification theorem. and so on. And actually, they're general, and yet they're applicable in specific situations. And I think this is kind of what mathematics is about. This is what gives mathematics, it's credibility, it's authority. And briefly and finally, Marcus, up to you. I think as Emmanuel Kant summed it up, if you want to know what mathematics is, just look at Euclid's elements. It's interesting at cross-transference to philosophy, isn't it?

Because Newton's doing it, Kant is using it and so on. Well, Bertrand Russell loved it as well. Einstein. I mean, it's really, you know, it's the seed of many people's passions. This is another program, folks, and I hope we can do it. Thank you very much, Marky de Sertoy, Serafina Cuomo, and June Barrow Green. Next week, I'll be talking about Thomas Hardest novel, Tessa, the Derbervilles, published in 1891, a great controversy.

Thanks for listening. And the In Our Time podcast gets some extra time now with a few minutes of bonus material from Melvin and his guests. Why didn't you disagree on the program? Because I... I could see, I could feel you going. You're saved, Marcus. You're saved, you're just going to be savage by Serafina. Seraphina, I love it to see you.

Thank you. Nice to see you, Serafina. Well, what did we miss out then? Well, I think I would have liked to have expanded a little more on additions of Euclid beyond Billingsley. Oh yeah, there's some popular editions. Yeah, the whole kind of the absolutely extraordinary Euclid industry that went on in the 19th century.

And so, for example, there's a textbook writer called Isaac Todd Hunter and his edition of Euclid went through over half a million copies were printed. I mean, it's phenomenal. I didn't get that in, below. And there were lots of other editors.

So there were just, I mean, you mentioned the fact that it was, you know, the second most sort of popular book really after the Bible. And it's just phenomenal what happens in the 19th century. So I think I would have liked to have to have just got a bat in. Yeah, I think what came out of the discovery of those non-Euclidean geometries was a little bit of a crisis because it made people think, oh, have we actually proved that this is not going to lead to contradiction somehow?

And that led to Russell on Whitehead, trying to sort of, in a way, redo Euclids. And they tried in their, what was it called? Prinkipia, yeah, yeah, yeah, yeah. To show that all of the truths of mathematics could be deduced from a certain set of axioms. And they were trying to produce. What are the complete set of axioms

from which we can prove all truths of mathematics? And this actually led then to the discovery that that's not possible, that there are truths of mathematics which will never be susceptible to this method of proof from axioms. And that's girdles and completeness theorem. So I would have a link from Euclid through to the program that we did about Girdle

because there is a beautiful connection there. I think also we didn't spend long enough on this idea of algebra. Because that's really fascinating. Yeah. That actually Euclid proves that thing about prime numbers. I would use algebra to do it. But he has no lines. He does it all in terms of lines.

You take three lines which are incommensurable, then you can build another line which can't be divided by those three lines. So it's all very geometric. And you know, you need the kind of more linguistic view that the Arabs brought, that brought a language that could articulate this in a much more, you know, without sort of pictures. And I think the other thing on the non-Euclidean geometry, actually, that I would have liked to have got in was the fact that after Beltrami and then Puancares comes along and he says, well, actually, it's a matter of convention which geometry we choose.

And we choose Euclidean geometry because it's most convenient to us. I tell you the thing that we should have talked about a little bit more is the rationality of the square root of two. I mean, this is a discovery of a new sort of number, which wasn't just a fraction, a ratio of two numbers. And this is, for me, was one of the other proofs that inspired me to become a mathematician, reading that you can prove that this length cannot be written as a fraction.

And you can construct it. Yeah. And it's very, it's the most sort of complicated bit in the elements because he hasn't got, or they haven't got algebra, to be able to articulate. We can do it very slickly, but... Yeah, yeah, but it's almost an impedimentary. wit of you could, but an amazing discovery.

Here's Simon with amazing announcement. Well, something much simpler anyway. Tea or coffee? There are many more science and discussion programs from Radio 4 to download for free. Find these on the website at BBC.co.com.uk slash radio 4.