Thank you all for coming to this this first talk of the evening.
It falls to me to address the mathematical aspects of the things that are in the exhibition, which I hope you get to have a look at afterwards.
So as you look at the exhibition, you see various things relating to geometry over many centuries.
In the coming up to the 19th century where you'll see reference to Ben Freeman with a, um,
what ended up becoming a very much more abstract approach to geometry in the 19th century.
So for example, the, the move to working in more dimensions than three.
So the idea that you can,
you can do geometry in this way that way you don't quite have the same sort of concrete way of visualising things as you do in lower dimensions.
And in fact, this this moved to a more abstract geometry is part of a wider abstraction that happened in mathematics in the 19th century.
And so what I thought I'd do was, was pick up a different aspect of this abstraction.
So not necessarily to to do with the issue of dimension, but it is something that links to several things that are in the exhibition.
So I thought it might be a nice thing to do. So in order to introduce this topic, I have to go back 2000 years to look at Euclid of Alexandria.
Now, some of you may immediately spot that this is not Alexandria.
This is Plato's Academy in Athens. Picture on the on the wall of one of the Raphael rooms in the Vatican, but we'll glossed over that detail.
The reason it's here is that it contains representations of various scholars from antiquity.
And so at the bottom right hand corner here we have Euclid.
Probably some people argue that it's actually Archimedes.
But again, we'll gloss over that detail. And so I suspect Euclid is a name that is known to to most people here.
So around 250 B.C., he collated a collection of of geometrical knowledge, such as it was at the time,
probably gathering together material that was known, possibly adding some new material of his own.
And this is concerned mostly with the very much the sort of day to day geometry,
a sort of plain flat geometry line circles and so on, the kind of thing that he's probably doing on the slate there.
And so this this text in various versions became the basis for the teaching of geometry for 2000 years, essentially.
And it's a very interesting story. And I should perhaps mention that a colleague of mine, Benjamin Walter,
is will shortly be bringing out a book, traces this, uh, this descent because the elements has been.
Sorry, I'm getting ahead of myself. So Euclid's elements,
the text that I'm talking about has been transmitted and translated and corrupted and edited and generally abused down the centuries.
It exists in in umpteen different versions. The one I have on the slide here is the oldest surviving complete version, which dates from 1888.
And it's in the copy. And so this should be in the the exhibition, so you'll be able to have a look at it afterwards.
There are also other versions of Euclid in the exhibition, for example, there is an Arabic one then.
And so I've said that Euclid's elements was the basis for 2000 years of geometrical teaching.
It's also the basis for a lot of the way that we do mathematics in terms of structure, because it begins with basic definitions.
So he defines what he means by a point and by a line and by a circle and so on.
And then he sets up his basic postulates, the things that you can do with these concepts that he's defined.
I'll come to those in a moment. And then once he's got this basic setup established, he then works very systematically, step by step,
building up the geometry through the various books of the elements, building up to very much more complicated geometrical constructions.
And so the page that I have on the slide here, you may just be able to make out the diagram here.
Oops. Which you might recognise as being a representation of Pythagoras theorem.
And so this is the 47th proposition of the first book of Euclid.
So he's worked systematically through a whole series of of earlier results and got to this one at number 47.
And this depends on many of the things that come before.
There's a very logical and rigorous structure to to the book, which is then replicated in other branches of mathematics down the centuries.
And the story that I want to tell very briefly is the story of a certain amount of disquiet that was felt
about one of Euclid's postulates that the postulates were supposed to be basic assumptions about geometry.
And there's one that is somehow less basic, less self-evident than the others.
And it's gets cause some worry for mathematicians down the centuries.
So in order to to access the postulates, I'm going to do it through this, which is also in the exhibition.
This is Henry Billings, these English translation of Euclid, 1570.
This was the first English translation, as he proudly tells us on the the title page there.
And so I've just pulled the postulates out of Billings, these versions.
So we've got a sort of something reasonably authentic to have a look at.
So the first postulate there are five in total. So the first one from any points to any points to draw a right line, meaning straight line.
So if we got two points marked by crosses, we can draw a straight line between them.
Simple as that. Very, very, very basic, very constructive.
He's just telling you what you can do geometrically. The second postulate to produce a right line finite, straight, fourth, continually.
Well, we have to get past some Elizabethan English here.
But basically he says that if you've got the straight line, then you can extend it as far as you like in both directions.
So, again, very straightforward, very constructive. Similarly, the third one upon any centre and at any distance to describe a circle.
So there's a point. There's a distance from the circle as far as distance from the point.
We can draw a circle centred at that point with that distance as its radius.
The fourth postulate is slightly odd compared to the other ones.
It basically says that all right angles are equal, which you would I suppose you would want it to be true.
So this is a right angle and this is also a right angle.
And this is the right angle. This is the right angle. They're all the same no matter where we are.
A right angle means is the same thing. And he's put that in there explicitly, I think,
because a lot of what he goes on to do depends on comparing one angle to another, on comparing an angle to a right angle.
So in a sense, he wants these this to mean the same thing.
Wherever you are in this space that he's he's creating. Otherwise it wouldn't make sense.
So that's why that's in there. And then we get to the fifth postulate, also known as the parallel postulate.
And perhaps without even reading it, alarm bells are starting to go off now because this looks a good deal more involved than the previous ones.
Certainly, there's certainly a lot more going on. So when a right line falling upon two right lines doth make on one and the same selfsame side,
the two inward excuse me are two inward angles, less than two right angles.
Then shall these two right lines being produced at length concur on that part in which are the two angles, less than two right angles.
So I think that needs a bit of unpicking. He's saying if you have two straight lines and a third line falls across these two lines,
and then clearly on the right hand side of the line, there is an angle here and an angle here that are each less than a right angle.
So together they're less than two right angles.
And so on that side of the third line, if we were to extend the first two lines sufficiently far, they would meet at some point.
Okay. And this is Billings, this version of the diagram. So his lines meet at the point date.
And so something that we are sort of invited to think about.
Euclid doesn't quite do this explicitly, but we're invited to think about what would happen if those lines were not angled quite so sharply.
What if they were angled outwards a bit more? Well, the point at which these two lines meet would get further and further over to the right.
Until we get to the case where the first two lines are parallel, in which case the two lines meet metres infinity,
which I suppose is a mathematician's way of saying they never meet. But we say that they metre infinity.
And so this postulate is very different from the other ones.
The other ones are very much about things that you can do in front of you, that you can draw a line, you can draw a circle.
This one is hinting towards things that are going on at infinity.
So this is quite a different thing and. You might ask, what is this a basic postulate?
If you were to write down a list of basic facts about geometry, would you necessarily think of this one?
It's not necessarily that obvious a thing to think about.
And so from very early on, there's a certain amount of disquiet about this postulate.
Even from Euclid, it seems because he doesn't use this until the 29th proposition of book, one of the elements.
He could have used it earlier to simplify certain things, but he seems to have put it off and put it off until he absolutely had to use it.
And this happens at the 29th proposition and then it is used later on.
So it's definitely needed within this geometrical scheme that he's constructing.
But people began to say, Well, do we need it as a postulate?
Does it actually is it actually a theorem? Does it follow from the other postulates?
And so for many centuries, this became the problem that people tried to address.
So a very early or early ish commentator on Euclid was propolis in the fifth century.
I think he attempted to prove the parallel postulate on the basis of the other ones and couldn't do it in the medieval Islamic world.
There was a lot of attention given to this.
So we have al-Hasani, we have Omar Khayyam as all to see, all trying to prove the parallel postulate based on the other ones and failing.
And also in Oxford in the 17th century. So this is John Wallace, who was civilian professor of geometry.
He was obliged to lecture on Euclidean geometry, and he also tried and couldn't get anywhere with this.
And so by the time we get to the 18th century or so, certainly the later 18th century, people are beginning to think,
well, okay, we've not managed to prove the parallel postulate on the basis of the other ones.
So can we actually construct a different kind of geometry where we don't have the parallel postulate all going further?
Can we assume the opposite of the parallel postulate, whatever we mean by that, and still construct a consistent geometry?
And I'm certainly not going to attempt to go into the details of that because I'm likely to come unstuck.
But certainly as the 19th century, it all comes together in sort of the 1820s, 1830s,
where the Hungarian mathematician John Paul II and a Russian Nikolai Ivanovich,
along with Chayefsky, working independently, despite what Tom Lehrer might have you believe.
And they constructed this a consistent non-Euclidean geometry, specifically a geometry in which the parallel postulate is negated.
And these geometries work in a very different way from Euclidean geometries are not going to attempt to, to, to say much about them.
But that's quite a nice representation of one of these types of geometries, which is called the Poincaré disk.
But this particular representation of the Poincaré disk, perhaps recognised by M.C. Escher.
And I suppose just to bring things to a close,
the thing that I would maybe try to point out about this is that there are straight lines in this picture,
but you would not necessarily recognise them as straight lines because in non-Euclidean geometry,
a straight line is not necessarily the same thing as in Euclidean geometry.
That's something that it defines a slightly different shape.
And so this happening in the early 19th century alongside other abstractions that were
happening alongside the realisation that you can move up to more than three dimensions,
we have people rethinking the fundamental notions upon which geometry is based.
So keep thinking what a straight line should be, rethinking how geometry is put together.
And certainly this then feeds into the later work that I mentioned at the beginning by Riemann,
for example, the idea of curved space time, the idea of different,
more exotic types of geometry, which eventually sort of find their, their realisation, if that's the right word in the theory of relativity.
So here in the first half of the 19th century, people this move towards towards a more abstract geometry that frankly,
not a lot of people were interested in at this time. By the time you get to the end of the beginning of 20th century, it's finding a use.
And I think I will stop there.