# From Euclid to modern geometry: Do the angles of a triangle really add up to 180˚? (28 Feb 2012)

>>Mark Ronan: Well thank

you very much Patricia. That’s very kind of you. The last time I gave this

talk, I’m going to have to move fairly fast

because we want to leave a little

time for questions. But a journalist;

well a journalist in the audience said he learned

more about ancient history than he did about mathematics,

because I’m going to start with Alexander the Great. And I don’t know whether

you know about him, I mean. When I went to school we never

learned that sort of history. But a remarkable guy who was

very keen on scholarly work. I mean he had Aristotle

as his tutor and when he was 20 years old

his father was assassinated; his father the King. And he had to put down

revolts and things, and yet within 2 years he

had got together an army of different Greek states

and crossed Asia Minor; over into Asia Minor,

to beat the Persians. Remember the Persians had

earlier sort of attacked Greece. The battle of Thermopylae

is a famous one which will be 2,500 years

anniversary in 2021. And that’s the place where 300

Spartans died to the last man, to defend Europe from the

invading armies of the Persians. The Persians had

a fantastic empire and Alexander conquered

the whole empire. Now you’ll see why I’m

saying this in a minute. See first of all he was;

he became King of Macedon. But very soon he became King

of Egypt because Egypt was part of the Persian Empire. And even before he conquered

the whole Persian Empire, he took Egypt and there are

many interesting stories involved there. But eventually a

couple years later, he took the whole Persian Empire and became the Great

King of Persia. And he died a young man. I mean, I think what

was his age? 32 when he died. After his death, his generals

divided the empire between them and the person who took over

Egypt was his general Ptolemy. Later known as Ptolemy Soter. The reason he is known as

Ptolemy Soter; he’s also known as Ptolemy the 1st, because

there were lots of Ptolemys; 1, 2, 3, 4, 5, 6, you know. The Rosetta Stone is Ptolemy

the 5th and those who have been to the opera of Giulio Cesare

will meet Ptolemy the 13th. You know, there are

lots of his Ptolemys. And eventually the dynasty

collapsed with Cleopatra, but that’s another story. Ptolemy Soter was called

Savior, Ptolemy Savior, because he saved the island

of Rhodes from an invasion by the King of Cypress and

they gave him the title Savior. And they built a huge monument

there called the Colossus of Rhodes, which exists no more. One of the great 7

Wonders of the World even after it had collapsed,

which happened fairly shortly after it was built actually. It was still one of the

great Wonders of the World. Ptolemy; we think

it’s Ptolemy the 1st, who founded the great Library of

Alexandria where Euclid worked. We don’t know; we

know about Ptolemy. We know about Alexander. We know about all these sort of famous generals

and kings and so on. We know absolutely

nothing about Euclid. I mean we don’t know

where he was born. We don’t know when he was born. We don’t know the

circumstances of his death. We don’t know when he died. There’s a famous quote here that

I quote; when the King asked him if he could you know, give

a sort of neat explanation of his geometry, he said there

are no royal roads to geometry. And you know, the idea of royal

roads existed at that time. In Persia there were royal

roads that were only available to the King and his

entourage, so on. We have no idea who

this guy was. He may have been

quite a young man and he may simply

have disappeared or maybe he didn’t exist at all. I think he probably did exist. Anyway, he wrote this

thing called Elements, Euclid’s Elements,

and they were; let us move on to those;

composed in 13 books. He; this is really

extraordinary stuff because it’s the greatest

textbook of all time. What he did was he sort

of layed out everything. He layed out definitions. He layed out postulates, 5 important axioms

for plane geometry. And it was written using;

if you’ve every done math; any sort of serious mathematics

and people always used to do this in school

at one time. Doesn’t happen anymore;

they all used to. They used to do Euclidian

geometry. And they would prove Lemmas

and Theorems, you know? Wonderful; wonderful for

the brain and development of our brains to do that. I’m very sad it doesn’t;

isn’t done as much as it is; as it was, today. But what were these

5 postulates? We’ll come to that in a minute but the Elements

were transmitted; well they got translated

into Arabic. Here’s how they came to Europe. They got translated into Arabic. A chap called Adelard

of Bath, went to Spain, got hold of a copy and then

translated it into Latin. And this Latin translation;

I’ve mentioned Roger Bacon who was one of the great minds of this country in

the 13th century. Not to be confused

with Francis Bacon. The first translation from Greek

directly into Latin was in 1505 and what I’ve pictured here is

the English edition in 1570, made by Henry Billingsley, who

became Lord Mayor of London. Died in 1606. It contains; it’s difficult

to read this writing actually. But it says that there

is a preference by M-I-D. M-I-D is John D., Dr. John D.,

who was pretty much I think, in charge of the Elizabethan

Secret Service and decoding of messages, which of course led

to the execution of Mary Queen of Scots, as many of

you who have studied that period of history

will know. And the English National

Opera; they were kind enough to mention the fact that

I’m interested in opera; are putting on an opera

called Dr. D. pretty soon. So I’m very much

looking forward to that. It’s a new opera. Now, let us go to

the five postulates because that’s really

what this talk is about. You see I’ve got the

first four pretty simple. There’s a line between

any 2 points. You can extend a line

in both directions. If you’ve got 2 points you

can use one as the center of a circle and the other

one to measure the radius, and then you can draw a circle with that center

and that radius. All right angles are equal. Right angle means 90 degrees in our usual terminology

acquired from the Babylonians. But just call them right angles. You know you take

a straight line. You put a perpendicular there. This angle is the same as that

angle and they’re 90 degrees. But then there’s

the Fifth axiom. You can read it here and you’ll

probably read it and say, I’m not quite sure

I’ve got that, so why don’t I give

you a picture. That’s the picture. So these lines L and M, are sort

of so to speak not parallel. You see the sum of these

2 angles, alpha and beta, is less than 180 degrees. And in that circumstance

the axiom is that these 2 lines

L and M will meet. That’s an axiom. And he; when Euclid wrote his

Elements, he tried to prove as much as he could

without using that axiom. But eventually he had to;

you know he went so far with this theorem, so

eventually he needed this axiom. And many people since then

wanted to get rid of this axiom. They felt it wasn’t

really necessary. I’m only going to mention

a few of these people. We’ll go first of all to

a chap called Al-Haytham. He’s sometimes pronounced

Al-Haytham, also known as Al-Basri

because he came from Basra. Died in Cairo. Famous work on optics

which again, you know, Roger Bacon quoted

Al-Haytham’s work on optics. It was brilliant stuff. At one time, and this is

complete digression, you know. But at one time, people used

to believe that the eye sort of sent out rays that; oh

it’s all very complicated, what they used to think. But this guy had the right idea

that it was just light coming in straight lines and

entering our eyes. You think, well that’s

pretty obvious but you know, it hadn’t been obvious

to an awful lot of people and he did marvelous work

on refraction and so on. Now he also worked on

the parallel postulate. And tried to prove it. But here’s another

famous man who said, well Al-Haytham just got this

wrong, and that was Omar Khyyam. Omar Khyyam is often

remembered as a poet. The Rubaiyat of Omar Khayyam. But in fact he was even better as a mathematician

and an astronomer. And he criticized

Al-Haytham’s proof. I’ll tell you what

Al-Haytham’s proof was. I’ll just do it. He took a straight line and he

took a little perpendicular line of a certain length and he

moved that perpendicular line, keeping it perpendicular to

the straight line, like that. And the point at the top of

that would draw another line. And this would obviously

be you know, so to speak, parallel to the first

line and you just put. But Omar Khayyam said

this is nonsense. This really is nonsense. And it is nonsense actually. The idea of having

points move is okay. I remember my son was doing a

plane geometry course in school. He explained a proof to me

where points moved along here and another point moved

along there, and so on and his teachers [inaudible]. But he insisted it was right. I examined this proof

in some detail and said if you just rephrase it, and he

did; went back to his teacher and his teacher said,

that’s good. Which shows he had

a good teacher. Some teachers wouldn’t do that. But, excellent teacher. Anyway, to jump forward

to the 17th century; well actually 18th century too. An Italian mathematician

called Girolamo Saccheri, thought he had proved

Euclid’s Fifth axiom. In other words, he

thought he could prove it from the other 4

more basic axioms. And he split into 3

cases where triangles; the angles of a triangle

added to less or more or exactly 180 degrees. And then he had to

eliminate the first 2 cases and he published this in

a book which came out; if you read the date on here. I think there is a date on

it somewhere, isn’t there? Yes, right down there

at the bottom, you see? In roman numerals 1733. That was the year he died. Bit of a shame really,

because it was wrong. But he; never the less these

people were very clever people. You know, even they

got it wrong. So I’m going to give

you a false proof; I’ll tell you it’s

false before we start. You know, you can try and figure out what’s wrong

with this if you can. First of all, the first step is

to show that any 3 points lie on a straight line or a circle. So let me just run

through that very quickly. I’m just taking 2 of the points

then I’m drawing a circle through those 2 points and

the red dot is the center of the circle. Okay? Now we want

to make the circle; we can make the circle

smaller, we can make it bigger. Here’s a case of

making it smaller. And here is a case

of making it bigger. It just needs to get

a little bit bigger and it will hit the third point. In this animation actually, it does hit the third

point curiously enough. When I actually do it in the

talk it doesn’t, but never mind. I think it’s rather

good actually. It doesn’t quite get there. All right. So if you’re holding onto your

seats, we’re going to really go into this proof that 2 lines

are going to intersect. Now let me just remind you

of Euclid’s Fifth axiom. You’ve got these lines L

and M. The angles between; the angles that they make with

the third line add up to less than 180 and the idea

is to prove that L and M must intersect, okay? So I’m going to do that. And do it in front of your

eyes and then we’ll move on to some more history. So if you get fed up with this

mathematics, it’s okay, really. We’ll move forward, don’t worry. Right. So let’s take

these lines L and M and what I’m drawing here

is 2 lines; a line cutting L at right angles and a line

cutting M at right angles, yeah? And since L and M don’t have a

third line right angles to both of them, these 2 line segments

must intersect somewhere, yeah? So you just let them intersect

at a point Q. And you take P to be the point opposite

Q, across the line L, and R is the point opposite

Q, across the Line M. And that length of QR is

not necessarily the same as the length of QP. I know it looks like

that in this picture, but it doesn’t have to be. That’s completely

irrelevant to the proof. Okay. Now, let us move on. We’re trying to get

a circle that goes through those 3 points

P,Q and R. First of all let’s think

of it this way. You take a circle through P

and Q, which has its center of course necessarily on the

line L. You take a circle through Q and R, which has

its center on the line M. And you make these

circles bigger until eventually you will get a

circle that’s bigger than either of them and it will go through

P, Q and R. So the center; as you make these circles

bigger, the center will move; the center of this

one, of the top one, will move down on the

line L. The center of the bottom one will

move along the line M. And when you’ve got

a single circle, it goes through the 3 points

P, Q and R. The center of the bottom one and the center

of the top one will coincide because you’ll have

the same circle. Now in this picture

they don’t coincide because I never quite get that

far, because I’m you know. But in principle you’ve got a

circle; you will get a circle through P, Q and R and

its center will lie on L and it will also lie

on M. And therefore L and M have a point in common. Proof. Proof of Euclid’s

Fifth axiom. Now you can wonder

what’s wrong with that and I’ll ask you

at the end, okay? But we’ll move on, because

a great deal more work went into this question and one

particular guy who was a very, very, very smart guy,

Johann Heinrich Lambert. He proved; anybody who has

done circles, you know the area of a circle is pi r

squared and so on. He showed that pi cannot be a;>>Okay are we back again? Yes. Sorry about that. I must have touched the button. He proved that pi can never

be written as a fraction. Many of you who went to school,

particularly in the old days when you know, we used to

deal with fractions a lot because we didn’t

have calculators; used 22/7 as a good

approximation for pi. There are better approximations. 355/113. You try

it on a calculator. Pretty good. Okay, there’s a way of

getting these approximations, we won’t go into that. Anyway, Lambert; now I’ve said

here he showed that the area of a triangle in the

hyperbolic plane; he imagined that you had a plane that didn’t satisfy

Euclid’s Fifth axiom. And then he showed that the

angles of the triangle would add up to less than 180 degrees. We’ll come back to that later. So like Moses, he

saw the Promised Land but never entered it. He died in 1777. And then, I just want to mention

another chap, Schweikart, who was a friend of Gauss. Now we’ll talk about Gauss in

a minute because he was, well, arguably the greatest

mathematician since Newton, at any rate. And one of the 3 greatest

of all time, as I often say. Actually somebody once said

to me; this is a funny story. This is in America and this chap

was a lawyer and he said to me, tell me who are the 3 greatest

mathematicians of all time? You know, question

out of the blue. I said, well, I don’t know. I suppose, Archimedes,

Newton and Gauss. He said, why do you guys

always give the same answer to that question? And, so. Anyway. I think somebody wrote a book and he had quite

fairly strong opinions and that was his opinion. But Archimedes and Newton were

fantastic, they really were. Fantastic. But so was Gauss. Anyway, he wrote a letter to

Gauss saying this being assumed, or whatever, you know. He made some assumptions

about the existence of a plane that wasn’t Euclidian. The sum of the angles in a triangle is less

than 180 degrees. It becomes less as the area of the triangle becomes

greater and etcetera. Let me just talk

about Gauss now. There’s a nice, lovely

picture of him. And there is the statue

to Gauss in Braunschweig. I’ve been there. I’ve seen the statue. And I’ve been around the back of

the statue and if you go in back of the statue you find

there’s a 17 point star because when Gauss

was 19, he proved; well, he didn’t just prove. He showed how you could

construct a 17-gon, a regular 17-gon. A regular polygon with 17 sides. You’ve maybe; in school you

might have constructed a regular triangle. You can construct a regular

pentagon with 5 sides. You can’t construct a

regular heptagon with 7 sides. Let’s talk about prime numbers

there because; but anyway, but you can do a regular 17-gon. The reason is that 17

is a power of 2, plus 1. It’s a prime number;

power of 2 plus 1. There aren’t many of those. But anyway, I digress. This man who one,

he’s just amazed at what Gauss did in his life. But any rate, he discovered

the hyperbolic plane. But he didn’t publish

it because he knew that it wouldn’t really

be accepted, you know, and they’d be in an

awful lot of trouble, see, and everything else. So he intended that these papers of his should be

published after his death. So who did discover

the hyperbolic plane? Well let’s go to

Farkas Bolyai who was; this is a Hungarian

mathematician, Bolyai. He was a friend of Gauss’. They went to University

together. And he was, unfortunately,

he was very poorly paid and he wrote and published

dramas while he worked on his mathematical

masterpieces. He tried to dissuade his son

from ever going into the study of Euclid’s Fifth

axiom and he said, detest it as lewd intercourse. It can deprive you of all your

leisure, your health, your rest and the whole happiness

of your life. But Janos Bolyai persisted

in working on the Fifth; Farkas Bolyai; there are

lots of lovely quotes. I mean he was; this is a man

who wrote plays and so on. I mean he was perfectly

capable of writing and writing very elegantly. And I’m not sure I have

the particular quote here but he said, I’ve been

on this journey before and I’ve always come back with

a broken mast and torn sail. You know, so don’t go there. But Janos Bolyai;

this picture of him is on a Romanian postage stamp. We don’t have a picture

of Bolyai; Janos Bolyai. I know it seems incredible

but we don’t. We have a picture of Evariste

Galois, who died much younger, because a friend of his

sketched his portrait in school. But nobody ever gave; this is

perhaps an invented picture. He wasn’t Romanian,

he was Hungarian, but the town he was born

in is now part of Romania. And he said, I am resolved to

publish a work on parallels as soon as I can put it

in order, complete it and the opportunity arise. I have not yet made

good the discovery but the path I’ve followed

etcetera, etcetera. And his father said, well this

is brilliant stuff actually. We must publish it immediately. And since his father

was publishing a book, he could easily slot his

son’s work into an appendix of 24 pages at the

end of the book. Which he did, and one

historian of science; I can’t remember his name. Very famous historian

of science. Anyway, said he considered this as the 24 greatest

pages every written. It was, you know, it was

really incredible stuff and he never published again. Can you believe it? You see it’s really; it’s really

sad what happened with this. I mean you’d expect that this

man did this incredible work, he would be offered chairs

here and there and everywhere. His father sent the thing to

Gauss but Gauss was a slightly; well, he was not an

outgoing, enthusiastic, what a brilliant thing

this is, kind of chap. What he wrote back

was, and this was read to Janos, who took it badly. If I commenced by saying that I

am unable to praise this work, you would certainly be

surprised for a moment. This he wrote back

to Farkas Bolyai. Indeed the whole contents

of the work; the path taken by your son; the result

to which he’s led, coincide almost entirely

with my meditations which have occupied

my mind partly for the last 30 or 35 years. And so on and so forth. So Janos just felt; you

know, I did all this stuff and it’s already been done? I mean he was really despondent

and he just; he never recovered. Very, very sad. But you see all this

was in the air. I mentioned Schweikart, I

mentioned Lambert and so on. This was sort of in the air and

Nikolai Lobachevsky, a Russian, also discovered this hyperbolic

geometry independently. And he had no better fortune. I mean he became head

of his university and they got rid of him. I mean it is quite incredible. It’s almost as if there was

a curse on this you know? It’s like the assassination

of Kennedy’s. It’s a little cursed thing. You can’t win. You just sort of mustn’t

do anything here at all. Anyway, but I don’t

believe in curses, you know. I was in Egypt recently and

they tell you the big story of how Howard Carter

discovered Tutankhamun’s tomb and all the workmen

who had worked on it all disappeared and so on. I don’t really buy this. Anyway, let me go on, go

back to Saccheri’s Triangles. I remember once and I’m

saying this to people who may not know these

things, mathematics students. I didn’t know this stuff

when I was a student. Not even as a graduate student. A very famous mathematician

came to Chicago and gave a talk on hyperbolic plane

and said that the area of a triangle was proportional

to 180 degrees minus the sum of the angles of the triangle. I was absolutely mind boggled. I thought it was

mind boggling, yeah. Absolutely incredible. I mean even if you send

those corners of the triangle out to infinity, you’ll still

only get the triangle being a certain size. That’s why we tend to

draw them with curved, with concave lines like that. Okay. So what’s the difference

between the Euclidian plane and the hyperbolic plane? Well in the Euclidian

plane, if you have a ; I’ve sort of condensed

the whole Euclidian plane into a disc here. And I’ve taken a point on

the boundary of the disc; in other words a

point out at infinity. And I’ve taken all the

lines that go to that point at infinity so if

that’s the North Pole, then these are all the

lines that are going north. The direction is north and

they all end up at the sort of North Pole at infinity. And at the other end

they end up so to speak at the South Pole at infinity. Hyperbolic plane’s

quite different. There are lines emanating from

the same point and they end up; these are straight lines. I know they look curved on this. Straight lines and they

end up all over the place. Anywhere around the boundary. This is called the Poincare Disc

Model of the hyperbolic plane and I’m just sort of really

mentioning this perhaps for mathematic students. It’s a very interesting

model of the hyperbolic plane where the lines are;

oh, that’s funny. It didn’t; yeah. The lines, if they go

through the center, the straight lines look

like straight lines. If they don’t go

through the center, they look like curved lines

but they are curved lines which are arcs of circles that

meet the boundary at 90 degrees. Here are a few examples. This is the hyper; this is

a wonderful representation of the hyperbolic plane. Whenever lines cross, the angle at which they cross is

always exactly the same as the angle they should cross

at in the hyperbolic plane. It’s a very useful, very useful

thing, this Poincare Disc Model. But I must finish now. There’s another model called

the Upper Half Plane Model. I think I’ll just

skip that actually. What goes on in the

hyperbolic plane? The reason the lines don’t meet

is here are the 2 lines L and M, crossing a third line at

angles alpha and beta, which add up to less than 180. But you see they sort

of drift off and go off to infinity completely

different ways. So what was wrong with my proof? Anybody know? Yeah, go ahead.>>[Inaudible response]>>Mark: Yeah. Actually, well done. Congratulations. I think you’ve hit the

nail on the head there. It is exactly that point

that any three lines lie in a straight line or circle. And what I did was gave you a

very sort of hand wavy proof. Well you see make

the circle bigger and bigger and bigger, yeah? And there’s a limit you see. What happens if I take the model of the hyperbolic plane

given us by Poincare. The Poincare Disc Model. Here are the 3 points in blue. There’s a line going through

2 of them and you can; I haven’t got all

the circles on that because the picture

will just get too messy. I’ve got the biggest

circle you can get that goes through those 2 points, yeah? The 2 lower points there. The biggest circle goes

right out to infinity. The center is out in

infinity believe it or not. And the circle so to speak

goes through infinity as well. That’s the biggest, and it

never hits that third point. So that was a hand wavy proof but you know people

did this kind of thing in geometry at one time. And they would get to

some conclusion like that and then they would use that to

sort of leapfrog onto showing that the Fifth axiom was a

consequence of the other four. And the leapfrogging was fine. What was wrong was their,

their starting position that they managed to get to. But it was just nonsense. So I’m going to; I’m going

to stop there actually because I’ve got through

my talk with enough time for questions I think. So we have about 10

minutes for questions if anybody would like to.>>Right. Do we have

any questions? Yes, we have one.>>[Inaudible]>>Mark: Yes.>>[Inaudible]>>Mark: Yes.>>[Inaudible]. You said it’s only coincidence that that point is

in the middle. But it seems to me how can

it avoid being in the middle.>>Mark: Oh, well let’s just

run back to that very quickly. And we’ll see what you mean. You mean, you mean

this thing, yeah?>>Yes.>>Mark: That picture. Q; well, what I said

was that the length of the line PQ is not

necessarily the same as the length of the line QR. You’ve got one line. You’ve got another line. They will hit, they will meet. Okay? There’s no question. I mean that point Q; if

that point Q could be; if you shifted; if you

shifted this line over slightly to the right, then Q would be

closer to the line L than it is to the line M. But when the

chap who drew these pictures for me drew it, he did actually

make them I think pretty much the same length. But that’s just, you know, yeah. But I was very grateful to

him for drawing the picture because I can’t draw these

pictures on the computer so. Yeah?>>Yeah, I came in slightly late so I might have missed

something, but;>>Mark: Yeah. You missed Alexander the Great. That’s what you missed.>>All right, yeah.>>Mark: Yeah. I was almost going to go back

to that and get a question on it, but go ahead, yeah.>>Nothing to do with

Alexander the Great. I sort of get the feeling

that somehow this is sort of cheesing a bit because;>>Mark: Yeah.>>Surely when Euclid was doing

his geometry he would have assumed that you were talking

about what we call a plane which is a flat plane.>>Mark: Exactly.>>Is there nothing in his

assumptions that sort of relate to that flatness or

is it just sort of;>>Yes, it’s the Fifth axiom. It really is the Fifth axiom.>>Right.>>And it’s that Fifth

axiom which he, I mean, I think Euclid was

a brilliant guy. I mean he just you know got

it down to what he needed, nothing more than he needed. The first 4 axioms are

also things that you know, you’ve got to start somewhere. And the Fifth axiom, you

know, he really needed it. So that’s what makes it fact. And there’s nothing wrong

with the Fifth axiom. You know, it’s very

useful for architects and all sorts of people. And of course now days I don’t

know if you keep in touch with what goes on in physics. I mean I try to, but you know

they talk about dark matter and expanding universes

and all of that stuff. What would happen if you made

a triangle in the universe with the points millions

of light years apart? Would the angles add up

to about 180 degrees? Or wouldn’t they? And the latest thing I read,

probably in the Economist or something, so you

know, [inaudible]. But it said that

actually they would. That’s the latest. So I don’t know. We don’t know, really,

for the universe. But of course on the

earth it’s not true because the earth is curved. But in the ideal flat plane

which Euclid was dealing with, yeah, it’s absolutely true. And yes, sorry. Does that answer your question? Yeah. Okay.>>Any more questions?>>Mark: We should

go back to Alexander.>>[Inaudible]>>Mark: Alexander the Great. Yes?>>So was this; was

the hyperbolic plane; was that the time, was that

the moment that people realized that you could have a

geometry that wasn’t;>>Mark: Yes.>>What we think

of as kind of our;>>Mark: Yes.>>Kind of normal

spatial geometry. So they didn’t have the idea

before they had the example of the;>>Mark: That’s right. That’s a very good point. You needed the example. I mean otherwise, the

whole thing is a bit sort of airy fairy. If this happens then that

happens and the angles of a triangle will be

less than 180 degrees and so on, but can this happen? Because if it can’t,

then forget it. So you needed an

example; this example. It had to be constructed

and that was the great work that Janos Bolyai did. Of course once that’s been done; once that the ice

has been broken, then people can start discussing

other sorts of geometry where things aren’t flat, yeah? They can do not just

2 dimensional geometry but 3 dimensional geometry and

a man called Bernhard Riemann in Germany devised sort of

multi dimensional geometry where you could have curvature that would be intrinsic

to the geometry. It wouldn’t be curved in

some bigger flat space. It would be intrinsic

in your geometry. And that of course would be

vitally important to Einstein when he did his Theory

of Relativity. Sorry, I’ve moved away from here

and I’m not supposed to, I know. When he did his Theory

of Relativity because in spatial

relativity, space-time is flat and there’s a sort of glitch

in spatial; not the right time to go into it but you know, you have to take

account of acceleration. You have to take

account of gravity. And once he did that, he

needed to curve space-time and it was almost Riemannian

geometry that got used; these ideas that

got used by Einstein in the general Theory

of Relativity. So this was absolutely vital

for the development not just of mathematics, but of science. Of physics.>>Thank you.>>Mark: Amazing.>>Any more questions? Yes, we have one over here.>>Mark: Yeah?>>At what point in this version

was replaced by the new one? [Inaudible]>>Mark: Which version?>>This version of Euclid’s.>>Mark: No, we still have

Euclidian geometry, okay?>>Yeah.>>Mark: We have

Euclidian geometry. It’s very, very useful. Students in school

learn it and if you go to a good school they prove

theorems and so on and so forth. Hyperbolic geometry people

don’t tend to learn about. It’s just a bit more

difficult to deal with. But it is extremely

useful in mathematics. And operating in the hyperbolic

plane has a completely different symmetry from operating

in the Euclidian plane. And it gets used in the

study of symmetry in a way that I won’t go into here but;>>Yeah.>>Mark: Both of

them are useful.>>I understand that but

usually the Fifth axiom is; has got a new version that there

is only one parallel outside;>>Mark: Well that’s Euclidian

geometry where the plane is flat and these lines don’t meet and you’ve just got a

unique parallel line. There’s another way

of stating that. It’s called Playfair’s axiom. If you’ve got a line,

it’s Playfair’s axiom. But it’s equivalent to

Euclid’s Fifth axiom.>>Yes.>>Mark: If you’ve got a line

and you’ve got a point not on that line, there is a unique

line through that other point that doesn’t meet this line.>>Yeah, that was my question.>>Mark: Yeah, yeah.>>And these 2 that prove

to be equivalent and that;>>Mark: Ah, and

that was Playfair. I guess Playfair proved that this was equivalent

to [inaudible]. There are other ways, I mean. That business that I said about

3 points lie on a straight line or a circle; if you took that

as an axiom, it’s equivalent to Euclid’s Fifth axiom. You can prove one; you know, you

can go both ways with the proof.>>Well, thank you very much. I think we haven’t any

more time for questions.>>Mark: Oh wait a minute. We’ve got one more here. Go on, yeah.>>If it’s very, very quick

and a very quick answer.>>[Inaudible question]>>Yes, I don’t know. Don’t ask me about that. That’s a very tricky one. I would have thought the

universe sort of curves around on itself

and just expands like a balloon expands

and I agree. But it could just be

sort of generally flat and expanding, so I don’t know. I think we don’t actually know

the answer to that question.>>Well, a nice end

to the lecture. A bit of uncertainty which

inhabits mathematics.>>Mark: Yes.>>And thank you very much Mark. [applause]>>Mark: Thank you.