Gresham College Lectures

The Mathematical Life of Sir Christopher Wren

March 09, 2023 Gresham College
Gresham College Lectures
The Mathematical Life of Sir Christopher Wren
Show Notes Transcript

Christopher Wren, who died 300 years ago this year, is famed as the architect of St Paul’s Cathedral. But he was also Gresham Professor of Astronomy, and one of the founders of a society “for the promotion of Physico-Mathematicall Experimental Learning” which became the Royal Society.

This lecture explores some of Wren’s mathematical work on curves including spirals and ellipses and the mathematics behind his most impressive architectural achievement – the dome of St Paul’s.


A lecture by Sarah Hart recorded on 7 March 2023 at David Game College, London.

The transcript and downloadable versions of the lecture are available from the Gresham College website: https://www.gresham.ac.uk/watch-now/maths-wren

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(air whooshing)- Welcome, everyone, today, I'm going to tell you about Christopher Wren. When we think of Sir Christopher Wren, we remember him as a great architect, but he's also a mathematician. And that's what I want to tell you about today. On the way, I'm going to share with you three mathematical gems that Christopher Wren played with and investigated. So we'll find out the mathematical discovery he made while he was out shopping, we'll find out why he had a battle of wits with a man who didn't exist, and we'll find out what he thought about seashells. We'll finish today by talking about the mathematics behind his greatest architectural achievement, the dome of St. Paul's Cathedral. So Christopher Wren, as I've said, of course, our first thought is the architecture, but actually, this was something he did not do to begin with. He moved into architecture later, but he began as a mathematician, as a scientist. He died 300 years ago this year. 2023 is this year, 1723 he died. And among other things, he was the Gresham professor of astronomy. So obviously good at astronomy, but he was also really a polymath. He investigated everything from meteorology to anatomy. He made progress in mathematics. He was into everything. And it was at Gresham College that he was part of the movement, one of the founder members of a society for the promotion of physico-mathematical learning. And that name, not super catchy, but it would become known as the Royal Society. So this is, you know, one of the most important scientific societies that we have. And that was started at Gresham College. Now, we've said he started out in mathematics and science, then he moved into architecture. I mean, that's a slight simplification of what happened. That step is not such a leap as it might now appear to be, because for Wren, and at the time, architecture was one of the mathematical arts with navigation, with astronomy, with surveying, other practical things. It was part of mathematics. And Christopher Wren was a good mathematician. So it wasn't such a leap, as I say, as it might now appear. And actually, when Christopher Wren was writing about architecture, he put mathematics, in particular, geometry, at the heart of what architectural beauty is all about. So I've got this quote from Wren here where he's talking about what beauty is defined by, the causes of beauty. Says there are two causes. So natural beauty, that comes from geometry. And that's uniformity, symmetry, proportion, those things. That's from geometry, and that's the cause of natural beauty. Then there's customary beauty, you know, which is in the eye of the beholder. So a familiar thing, we may come to love how it looks just because it's lovely to us because we are familiar with them, even though the thing may not be itself lovely. That ragged, old toy that we had and loved as a child is not beautiful, but we love it because it's our thing. But he says, you know, if you are ever in doubt about which of these should take precedence, the error is made if you don't have natural beauty in the forefront. So he said,"This is where the architect's judgment is tried, the true test is natural or geometric beauty." So he's got geometry right at the top there. And he goes further than this. He places mathematics itself kind of at the top of a hierarchy of truth. So he says mathematical demonstrations, proofs, are built on impregnable foundations of geometry and of arithmetic, and they're the only truths that can sink into the mind of man void of uncertainty. And he goes even further. Everything else is participating more or less of truth according as their subjects are more or less capable of mathematical demonstrations. So that's a really strong statement about kind of the importance of mathematics and mathematical truth and geometrical truth being absolutely at the pinnacle. So we can see, yes, he places mathematics very highly in geometry. And actually there's a clue in this picture, in this portrait of Wren, he's of course got the plans of St. Paul's Cathedral there, but he also has mathematical instrument dividers, and there's a book that's on top of it, holding it all there, and that book is "Euclid's Elements." So that's a geometry book. And so there's the clue there in the picture to this importance that Wren placed on mathematics. Now I've said he was a Gresham professor of astronomy. This is Gresham College, as it roughly as it would've been in his time. There was a really exciting and vibrant community of mathematicians and scientists in London, and in Gresham College at that time. It was a great place to be and to study science and mathematics. He, in fact, this is another quotation of Wren, he was very impressed with London and we are here in London now. So I extend to you those congratulations. He says, "I must congratulate this city,"that I find in it's so general a relish of mathematics." And you know, we've got a full room today and I'm pleased that that continues. So well done, London, for relishing mathematics so, it's a good thing. So he was educated in Oxford, became Gresham professor of geometry. He did later get a chair, a very prestigious chair, the Civilian Professorship of Astronomy back in Oxford. But really in this community around Gresham College, and also what later became the Royal Society, he was really at the heart of it. He was a founder member of the Royal Society, he contributed to the meetings, he had papers in the transactions of the Royal Society, and you can also see kind of the esteem of his colleagues. His contributions were really valuable. So just sort of mention a few names, there's Christopher Wren on the left, again, you've got Robert Hooke, who was similar time Gresham professor of geometry, famous for Hooke's law. He's got a spring there showing you that. Isaac Newton next, and it wasn't just in England, he corresponded with mathematicians on the continent. And there's Pascal on the right hand side there with whom he corresponded, among others. And so Newton, I just want to mention this, this here is a little extract from "Principia Mathematica." Now Newton is not someone we necessarily think of as being the first person to credit others with discoveries and be great. But, so he mentions Wren in the "Principia," and this bit that I've quoted there, he's talking about inverse square laws appearing in nature. And he mentions as an example, the motion of the planets. And he says, as Wren, there's Wrennus, it's in, you know, the Latin version, Wren, Hooke and Halley have severally observed, and Wren is first in that list. And so he's really given him clear credit for observations about this. Wren's name is mentioned seven times in the "Principia." So, you know, he's clearly up there speaking and working with the other great mathematicians at the time. So we've got here someone who is clearly involved with these developments. I'm not going to talk about his astronomy really today because that's been addressed in a lecture by my colleague, Gresham professor of astronomy, Katherine Blundell, which you can go and watch online and find online at your leisure. I'm going to talk about the mathematics and I'm going to give you, as I say, three little examples of mathematics that Christopher Wren has done. So let's begin. One of the challenges at this time, at the time of Christopher Wren was making lenses for the exciting new scientific instruments, telescopes, microscopes, and so on. And when you think, what is a lens doing? I mean, the basic idea is you've got a piece of curved, usually glass, a lens, and the ray of light are coming in, and then we would like to focus them to a single point. And how is that focusing happening? Well, it's the rays are coming in and they're passing through the glass, and then when they hit the curved surface, they are refracted and we know we can work out the angle by which their direction is changed using something called Snell's laws, which tell you, depending on what mediums you're passing through and what the angle is, that you are hitting the glass at, what the new angle of the ray will be. Now the easiest kind of lens to make, to grind and at this time, probably the only one that could easily be done was a spherical lens, so the spherical kind of curvature. But the problem with that is it doesn't actually give you this beautiful focusing to a single point. What you get is something more like this. It's gets a little bit blurry at the edges. It's called spherical aberration. It's not the perfect shape. And this, of course is a problem. You want your instruments to give you really precise images. So you would like to know what is actually the right curve to use. Now, just to say, remember this is refraction. If it were reflection, if we were reflecting in a mirror, like if we are thinking of a telescope where you, the rays are coming in and they're reflecting off a reflector and focusing in on a point, we know the answer to that, we would use the parabola for that. But this isn't reflection, it's refraction. So it's a different thing going on. However, conic sections are still relevant. So I'll just remind you what that phrase is. So I mentioned a parabola already, that's one. Conic sections are the curves you get when you slice through a cone at different angles. So if the plane that you are slicing with is has a shallow or is horizontal or has a very shallow angle, you get ellipses. And then if you keep increasing the angle, going the other way, if you keep increasing the angle, at some point you'll hit precisely the same slope as that of the cone, and that's when you get a parabola. And then if you go on past that point, you start to hit both halves of the cone, and that's when you get a hyperbola. So I've just taken that hyperbola shape, it's a bit hard to see, I've turned it on its side. So that's the shape of this hyperbola curve. And it turns out that this is the right or a better at least curved shape for your lens in cross section. So that's the kind of curve we're mountain to aim for with our lens. But course lenses are three-dimensional objects. So just having a cross section is one thing. If you take that hyperbola shape and then you've just rotated about it the vertical axis, you get a three dimensional shape called a hyperboloid, which actually nowadays we most often encounter this shape in the form of a cooling tower. And there's mathematical reasons why this is a good shape for a cooling tower, which I will not go into now, but that's the kind of shape that I'm talking about. And it turns out that yes, this hyperboloid shape is a better solution than spherical, one, full lenses. But the problem is how do you actually make that, how do you make a lens with that particular curvature to it? And so here is where the mind of Christopher Wren comes in, and in particular he's out shopping one day and he has a mathematical inspiration. And so we should all do more shopping. This is what happens, he sees a basket, okay, history doesn't recall whether he bought the basket. But anyway, I've drawn this basket here with my imagination. He sees a basket, a wicker basket, and he notices, so it's got this nice curved shape, but the way it's made is with canes of willow, straight, straight rods or sticks of willow. But the way they're arranged, it gives this curved shape, this curved basket. And he realizes, hang on a minute, I think that means that actually a hyperboloid can be made with straight lights. And I'll show you this sort of happening, take two circles and put sticks or rods or even bits of string or something between them all the way around like this, like I've done. And then what we're going to do is we're going to twist them bit. Now they'll have to get a bit closer together because the sticks are still straight. They're not going to suddenly become curved or stretched. So they'll get a bit closer together when we do this twisting. But see what happens, twist a bit, that's what you get. Twist a bit more and a bit more and a bit more. And look, you get exactly this cooling towel, this hyperboloid shape developing. And if you had infinitely many lines, you'd get the full thing there. So it turns out what Wren realized and then showed properly was that you can make a hyperboloid with straight lines. Now I've got my circles the same size, but if you change the sizes of the circles making different sizes, what you get is kind of the hyperboloid but sliced off at different points, top and bottom. So for instance, that, and then they see it's just a short step to your basket. So just not so the other conic sections don't feel sad, I've made the handle a parabola. I dunno whether it was in the shop that Christopher Wren saw it. But there we are, so that's the basket. And this is not just a theoretical observation, and this is one of the hallmarks of Christopher Wren's work and his attitude to mathematics. It's a practical thing. So of course we use it in architecture, but we, you know, we have practical applications of it. And he doesn't just say, I've had a nice theoretical idea. He does think about how it could be useful. So he realized that this straight line way of making hyperboloids could be then used, this idea could be used because you could then make hyperboloid with a, you start with a cylinder on a lathe, and then you just need to have a straight cutting device at an angle, at a fixed angle, then you rotate the cylinder and it will just produce, then eventually produce a hyperboloid, just with a rotating lathe and one straight edge. So that already is a good thing to have noticed. But he also, then he went further, he came up with a design for a machine to grind hyperbolic lenses. And this is, so this is his paper from his paper in the transactions of the Royal Society and what you've got here, so you'll start with two cylinders at an angle, and then at the back you've got this piece of glass and they're all kind of working against each other to gradually grind the right shape. So the cylinders will work against each other to produce two hyperboloids. And then the cylinder at the back will also simultaneously be working against the piece of glass to grind a hyperbolic lens. So he designed a machine and I do like this diagram. So that's an example of a time where he made a theoretical observation, he did the mathematics, but he also then described and made a machine to show a practical application. So that's gem number one. Next up, a story that began in February, 1658 when several mathematicians in England received a mysterious challenge from France, from one Jean de Montfort, who greatly desired that those distinguished gentlemen, the professors of mathematics and others in England renowned for mathematical skill may condescend to resolve this problem. I'll tell you more about the problem in a minute. But they were issued this challenge by Jean de Montfort. There is no Jean de Montfort, it's a mirage and it's an illusion. There's no one called Jean de Montfort or no mathematician. This was a pseudonym for we don't know who. Now there's speculation, it might have been Pascal, we don't know, we'll never know. But anyway, the challenge was laid down. Can the distinguished gentlemen, professors of mathematics condescend to resolve the problem? So I won't give you the full detail of it, but the basic problem was you are given any ellipse that you know the dimensions of, so you know the length of the like the major axis, you know the length of the minor axis, so you know the ellipse and you are told there's a cord through the ellipse, you are told where it crosses the major axis and at what angle. And the challenge is given that information, can you determine the lengths of those two segments into which the cord is divided, okay? So that's the challenge. And Christopher Wren did condescend to resolve it. So he managed to solve this problem. And so he wrote up his solution, and then what he did was to say, so I've done your problem. Here's a problem for you guys to solve. And he issued a challenge back in return. So let me tell you what this return challenge was, and it has to do with Kepler's laws. So here's Kepler and this is an ellipse, it's illustrating one of Kepler's laws. So these were describing planetary motion and planets move around the sun in elliptical orbits with the sun at one focus, which I've sort of got on the left hand side there. They don't move at a constant speed, their speed changes, but there is something that does remain constant. And this is what one of Kepler's laws tells us. It's the area swept out by the planet in a given amount of time, is constant no matter where it is in its orbit. So the area is constant, but then obviously at different times in the orbit, that will mean the distance it's traveled around the ellipse may not be constant, so the speed will vary. But this thing about the area is a constant. So Kepler observed this, and in order to kind of do the mathematics that lies behind that, there's a problem that he raised, which is okay, if you've got an ellipse, and I'm going to divide it in two, so make it a semi ellipse. The challenge is can you draw a line from your favorite focus to the ellipse that divides that semi ellipse in a specified ratio? So you come to me and say, I want the ratio, I want it to be 50 50 split, and then I've got to be able to draw a line, the relevant line from the focus to the ellipse that divides it in that ratio. I tried for 50 50, I'm not sure I quite managed it, but the yellow might be a bit bigger. But anyway, Kepler had this problem and he realized that you could turn it into a related problem about semi-circles. So it be an equivalent problem, if you could do the semi-circle problem, which I'll show you, then you'd be able to do the ellipse problem. So the semi-circle problem was this time, you've got a semi-circle and you fix a point somewhere on the diameter of that semi-circle, and then you have to do the same thing. Can you find the line from that point on the diameter to the circumference that divides the semicircle in the specified desired ratio? Okay, so that became known as Kepler's Problem and Wren challenged the French mathematicians to solve it. Now Wren had actually solved this problem himself already, and it's quite good to already know if you're going to issue a challenge, it's good to know that you could do it yourself. So he'd solved this problem and he did it using a lovely mathematical curve called a cycloid. So I'm showing you a cycloid now being made. It's the curve that is produced by following the path of a point on the rim of a circle as it rolls along a straight line. So that's it just being produced there. And you can see you get these kind of arches repeating shapes, arches. And that's what a cycloid is. Now cycloids are lovely, the cycloid is a lovely thing. It was studied, it was all the rage really in the 17th century. Galileo started studying cycloids, one of the first to think about them. He gave this curve the name cycloid, he wrote that he'd studied them for 50 years. I can't tell you everything that's marvelous about them right away, but I did give a lecture on them last year. So you can go and find that and watch it if you want to know more. But like everyone was studying cycloids in 17th century. It started with, sort of started with Galileo and then Mersenne of Mersenne Primes fame, Pascal of Pascal's Triangles fame, Newton of being Newton fame, Descartes, Fermat, they were all looking at the cycloid and they all studied it. And the two main questions that were being asked about this were questions that actually tended to be asked about any curve that was being studied. And they were known as the problems of quadrature and rectification. So to our modern language, quadrature means what's the area under the curve? So in our case, obviously the cycloid could continue forever, but we just think, we take one arch of it and care about that. And we're interested in what's the area between that arch and the straight line. So what's the area, you know, in terms of the original circle that generated it? That's quadrature and then rectification, that is what is the length of that curve, of the length of the arch? And those two questions were asked about the cycloid as they were about many other curves. And these are difficult things to do. I mean, you've got to remember at the start of the 17th century, we didn't really even have what you might call modern algebraic notation. The equations of things were not, we, you know, wouldn't have written it down in quite the same way. Calculus hadn't come along. So these are difficult things to do. Galileo couldn't resolve them. To try and find out the area under the cycloid, he actually drew on a piece of sheet metal, cut it out and weighed it to try, and no, and this is fine, it's a valid thing to do, but you know, he wasn't quite able, he got a rough answer but he wasn't quite able to determine it. It took a chap called Gilles de Roberval to work out the quadratic. He was a mathematician who worked this out. He didn't tell anyone for a while. And then he kept his secrets close to his chest, Roberval. But he did work out that the area under the cycloid, if you've got a circle of area pi R squared, making it r is the radius, and it's a beautiful simple formula, the area under each arch is three pi R squared. So it's three times, exactly three times the area of the circle. Lovely, but rectification hadn't been done yet. What's the length of the cycloid? And Christopher Wren solved it. He was the first person to solve it. And here is the expression again, it's really simple. If your circle has a diameter d, then it's circumference is pi d, right? But the length of the arch is four times d. So another lovely whole number, really simple expression. And Christopher Wren was the one who proved that. So that was great. And he used his knowledge of the cycloid, the rectification of the cycloid actually to solve this problem of Kepler. And it's a bit complicated. There is a diagram that sort of tells some of the story. We're not going to go into the details, but he managed to solve it. Essentially what he did, and this is written up in a book by John Wallis, an English mathematician, he wrote a treaties on cycloids in 1659, so just one year after that original challenge. And you can see, I mean, we don't need to be able to understand Latin to see that it looks like it's saying Kepler's problems solved by cycloids. And that's exactly what happens. So Wallis writes up in this book everything that's known about the cycloid, in particular, he writes up Christopher Wren's proof of the rectification, the length of the cycloid and Christopher Wren's solution of the Kepler problem using a cycloid. And essentially what he does, I mean, you can see there's a semi-circle in there, which is divided up by a line from the diameter to the circumference. He relates the areas in question to arcs of a particular circle, of the circumference of a particular circle, and then in turn, those to parts of the length of a cycloid. Or actually it's a sort of stretched cycloid, but it's still within that family. And of course, Wren himself knew he'd found the formula for the length of cycloid. So he was able to then use that to solve Kepler's problem. John Wallis, we'll meet him again in a moment, but he couldn't resist having little jab at the French mathematicians a few years later, in a letter to a friend, 10 years after the original challenge from Jean de Montfort, whoever he was, he couldn't quite resist saying to his friend, no Frenchman has returned any solution to Christopher Wren's problem. Now I don't quite know if that's the case, but anyway, he was, you know, take that, Jean de Montfort, was what he was saying. But of course we're all together in the community of mathematics, of course. So that is Wren's brilliant solution to Kepler's problem and his rectification of the cycloid. Here's a piece of Christopher Wren's mathematics next that is related to spirals. And this spiral shape represents or is reminiscent of many of the shapes that we see in nature. You can think of shells, some of the ways some plants grow in a kind of spiral formation. This is a shape that we do see in nature. And this particular kind of spiral was one that was known to Wren. It's called a logarithmic spiral. I believe the first mention of this kind of spiral in writing is the German artist Durer. Later on, the mathematician Jacob Bernoulli would study it at length and he gave it the nickname spira mirabilis, the miraculous spiral because of its lovely properties. The reason it's called the logarithmic spiral, so what is a spiral? Well, you start at a central point and then you kind of, well, you spiral outwards literally. And in a logarithmic spiral, it's describing how quickly the your distance from the center increases. So that equation at the bottom there, which I've written r equals key to the theta, unhelpfully not telling you what any of those letters are. Now I will tell you, so r is the distance from the center where you are, theta is the angle that you've traveled through, and in this kind of spiral, that's the formula for it, k is just some constant, it's just some fixed number. So you are raising your fixed number to the power of theta at the angle, and that tells you how far out you are. So you're spiraling out. And for this kind of spiral, there's an example of it that I've shown you there, which is involving number two. So here K would be two, and if you can see, but even if you can't, I will tell you that there are points when it hits the axis. So you can see at one point it's at two and then you go full revolution round, now it's distance four, and another full revolution round, it gets to eight, and then it'll be 16, and 32 and so on. So it's powers of two every time you go a full revolution. And the reason it's called logarithmic, so if you know about logarithms, you will notice that if you take logs on both sides, you get that the logarithm of r of the distance is going to be a multiple of the angle theta. So that's why it's called logarithmic now. So these spirals, as I say, you do see things in nature that look quite like it. So just imprint that on your mind for a second. And I will go to the next slide. And you can see this is a cross section through a nautilus shell. It's really, really similar. It looks very like that logarithmic spiral that I've just shown you. And the reason or a reason we think this might be the case is that actually it's the perfect solution to how nature could arrange something that grows constantly through its lifetime and it's growing from a single point. So of course you need to make room for the new growth. So what happens is you have this, you know, it's growing kind of at an angle and the older bits are forced to move outwards, but they're still growing as well. So it's getting bigger as it grows outwards but the new material is still being produced in this spiral formation. And the great advantage of this is that you keep the same basic shape the whole time. It's getting bigger, but it's the same basic shape. So these logarithmic spirals have this self similarity property that when you zoom in or zoom out, it looks the same and that means the animal can, you know, it can be bigger or smaller, but it's still got the same shape body to live in or to, you know, to deal with. It doesn't have to learn new rules of being, whatever. It's just growing but it's keeping the same shape. Now one thing is, okay, nice spiral. This is a cross section though, shells and other things are three-dimensional. So we haven't quite, we need to do something else here. And there's something else again, Christopher Wren thought of it. So brief timeline, John Wallis, we just met him, he was able to perform the rectification, i.e. finding the length of a logarithmic spiral or actually more properly, any portion of it that you desire by a process which he called convolution, which is basically unrolling the spiral until it's a straight line. But of course you have to do that in a very, very careful way to make sure you haven't changed the length by doing whatever mathematical transformation. But he worked out how to do it, sort of unraveling the spiral to make a straight line. And then you can see the length. Now Christopher Wren, who was thinking about seashells and snails shells and things, he realized that you'd need to have a kind of 3D version of this. And so what he managed to do was kind of go up a dimension and to do this convoluting thing, but in reverse. So he found a way to basically roll up a cone to make a three-dimensional version of the logarithmic spiral. So he had the idea of how you could do this to make a sort of three-dimensional logarithmic spiral that would then, he suggested, give you the shapes of shells. And more recently, these ideas have been generalized to something called the power cone construction, which I'll talk more on in a second. But I want to show, I'm so pleased with this slide, it might be the favorite slide I've ever prepared for Gresham because, so what I did was to take that equation for a solid spiral and it has some parameters in it so you can change like how twisty you want it to be just by changing some of the parameters in the equation, in the general equation. And by doing that you can play around and you can try and make things that look like shells. And here look, these things really look like shells and I just sort of, you know, did them on the computer. But you can by changing the parameters, this is coming exactly from Wren's idea about this three-dimensional algorithmic spiral. You can create things that really do look very much like a wide array of different shells. And so this is a, you know, a great idea from Christopher Wren. And as I say, it's really still very relevant. And now, I was looking at a paper from 2021 where they'd made a slightly more general version of the equation, which they call the power cone construction, which can model not just shells but also things like antlers, horns, tusks, teeth, any kind of sharp pointy bits animals may have, even things like rose hips as well in plants. And so I just want to show you two or three things actually from nature. Look right, look at the spirals on those. You've got giant eland on the left. I mean, you can see in the middle you've got a fantastic curly horned ram. And the one on the right, I think this is my favorite one. I mean, it's fabulous. So this is a markhor, it's a kind of goat. Another name for it is, for obvious reasons, the screw horned goat. And look at those corkscrew horns. I mean, they're just fantastic. And you can get those, you can mimic those pretty exactly using these kinds of equations. So this is an example of how Christopher Wren, he was, you know, he was interested in everything as all the best people are. And he had, you know, great ideas and mathematical ideas for how you could model lots of different things, and seashells and snail shells are just one of those. So that is some three little gems of Christopher Wren's mathematics. I want to spend the rest of the time talking about architecture. Now there is lots of mathematics in architecture. I want to spend a little while talking about the mathematics that Christopher Wren used, but not for all of the architecture he ever did. You know, we could have a whole lecture series about the links between mathematics and architecture because it's everywhere, right? You need it for the kind of the engineering aspect, load bearing beams and you know, forces that are acting on the building and being able to construct it so it doesn't fall down, lots of mathematics there. There's also mathematics around the aesthetic side. We already heard how Wren says, you know, geometry is where it's at, that's what beauty is about. It's got to have beautiful proportions and symmetry and you know, these wonderful things that make our buildings look beautiful. That's geometry, that's mathematics. And then of course, while you are doing that, you have to address the practical, what is this building actually for? And can I design it in such a way that it will be, you know, the best feeling as you use this building, you know, this machine for living in. So all of these things have mathematics in them. What I want to talk about specifically is Christopher Wren's most famous building, St. Paul's Cathedral, and even more specifically, the dome of St. Paul's Cathedral. And the mathematics behind that is absolutely, I hope you will soon agree with me, fascinating. So let's just look at the outside of it, the dome. And you may know that the design for St Paul's went through many iterations. And you know, Christopher Wren's getting more and more frustrated 'cause people kept saying, oh, so first of all, the first design, I think I've got the order right, you know, one design was far too ambitious and it was too much and we could never afford it, no, tone it down a bit. So he toned it down a bit and then he was told, well, that's not nearly splendid enough for, you know, the great cathedral of London. You know, you need to make it more splendid but similar, you know, but obviously not too expensive but really splendid. And then the church authorities weighed in saying, well, it's not quite, the shape isn't right for what, you know, the purpose of the building and is it too Catholic or not Catholic enough? All of that, there was all that going on and there were several designs and I think at some point he sort of went, you know what? I'm just going to start building a cathedral and let's see how it goes, I'm paraphrasing little bit. So the thing that we all known recognize instantly about the London skyline is this fabulous dome and the dome that we see is a hemisphere. And really that's for both aesthetic and symbolic reasons. So when you think of the celestial sphere, this hemisphere is kind of representing that universe but also the hemisphere. I mean, it's made of circles and circles because you can go round and round them forever, they symbolize the eternal. So this is a great shape to associate, not just because it's beautiful, it's highly symmetrical, but it's a great shape to associate with, you know, a cathedral where you're worshiping the divine, the eternal. So that was the decision for the dome that we all see, a hemisphere. But there's a slight problem, there's a slight problem with the hemisphere because it's not actually a particularly strong shape and a hemisphere of this size is really going to struggle to support itself even just as it is. But we've got this gigantic lantern on the top, which looks great, but it weighs something like 850 tons. And that added to the weight of the dome itself would absolutely mean this could not support itself as a structure. So we need to have something that is supporting the structure. But you don't want to ruin the look with lots of, you know, buttresses and things on the outside. So you want to put them inside, but then that's going to ruin the look of the inside. You don't be looking up and just seeing load of horrible, you know, things that don't look good. So it's a real problem, what are we going to do? Where are we going to put the supporting structure that allows this hemispherical dome to not, you know, fall over and crush everybody? So the solution was that, okay, we talk about the dome of St. Paul's, I've got news for you, the domes of St. Paul's. The solution was to have an inner dome that would kind of be in between the outer and the inner. Then you could put all the mess in between those two things. So no one would see, so from the inside, it's going to look great, from the outside, it's going to look great. Okay, then what shape should that inner dome be is the next question. So you've got a hemisphere on the outside. What's on the inside? So actually at the time, there was active discussion about the best shape for various things that you could make of masonry. So one of these is an arch. This is such a lovely picture. These are arches in the loft of a building by Gaudi in Spain and they are a beautiful shape and they illustrate the best shape for an arch. But this is being addressed at the Royal Society, what's the best shape? And there's record of meeting in 1671. So Dr. Wren tells a meeting of the society what line it is, which is difficult, sentence three, what line it is which an arch fit to sustain any assigned weight makes? In other words, what's the best shape for an arch? And then they asked Mr. Hooke as well, Robert Hooke, Wren's collaborator, to also weigh in on this. And they promised to write it up and send it to the society. So they're both thinking about the best shape for an arch. They think they've got it worked out. Hooke, well, they did have it worked out and Hooke, he had this habit of writing his answers to things in Latin as an anagram so that he could later on say, I knew that first, look, here's the anagram, right? He did it with Hooke's law and he did it with this. And what he had worked out was that if you're trying to work out the best shape for an arch, there's various forces acting. But he knew something about a seemingly different question. And this picture, this is a not a contemporaneous picture, this was done in 2009 by Rita Greer. So he's holding there a chain and it's just sort of hanging there, and the shape that a chain makes when it's just hanging between two points and you know, under gravity, it's called a catenary from the Latin for chain, and Hooke, who's studying catenaries, he realized that the equation that describes the force acting on a catenary, on a chain, sorry, is exactly the same equation that describes the force acting on an arch. Now the forces are different, with the chain it's gravity and the tension in the chain, with an arch it's gravity and compression. But you get the same equation of each situation. And so here I've got some catenaries and it's the same curve essentially, but you're just sort of stretching it depending on how far apart you're holding the ends of the chain. So these are catenaries, some shallow, some deep. You can just turn those upside down and get the right shape for arches because it's the same equation, and therefore, has the same solution. So Hooke had worked this out and Wren also knew this, they were working together, that the best shape for an arch is this catenary. And so an arch doesn't have to be that shape, but that's the line of thrust. So any arch that's not going to fall down will have that catenary line in it. And so of course that means the most efficient way to do it is just to have a catenary. So the strongest arches are catenary shapes. And so then in the design of St Paul's, the inner dome that you can see there, that's a catenary, has a catenary cross section. And that means like visually it fits in with, it chimes in with the other elements of the internal design. It all looks very nice. So when you are inside the cathedral, it all feels like a harmonious lovely thing. So we've got an inner dome, got an outer dome. The question is what is going on between them? Because we still haven't solved the problem of supporting that gigantic weight of the hemispherical outer dome and the lantern. And Wren and Hooked believed that the strongest shape you could actually have for a dome, which isn't guaranteed to be the same as the answer for an arch because of course an arch is a, you know, essentially two-dimensional thing and a dome is a three-dimensional thing. So we may have different things going on. They thought the best curve to go with is this. So this is the curve Y equals X cubed, not the whole thing. But if you just take half of it, say that bottom bit there and maybe spin it around on the axis, you'll get a kind of long thin dome shape. And they thought this would be the strongest shape. And I want to know why they thought that. Why would you think suddenly out of nowhere, why is a cubic equation come into things? And I to show you that, I want to give you an extremely, like my hands will be waving like anything, approximate rough and ready guide to actually the answer for a catenary and the equation of a catenary. So this what will come out of this, this is not the equation of a catenary, it's an approximation to the equation of a catenary that I hope it'll be clear to you why I've done this. So there's a few symbols in the next couple of minutes. If you don't like them, just let your eyes glaze over for a couple minutes so we can come back after that. So you've been warned, what do they say in the advert? Here comes the science bit, here comes the science bit. So what have we got here? The yellow bit here is your chain that's hanging down and there are various forces acting on it. I am interested in the force acting on the bit that I've got in that rectangle. So just that little bit of the chain there. What are the forces acting on it? So it's just hanging there. So you've got the whole of this left hand half of the chain so that the origin in there I put is the lowest part of the chain, the whole left hand half of it, there's tension pulling and I've called that force F. You've got that bit of the chain that we're interested in is being pulled down due to gravity. So that's downward, its weight is pulling it down. I've called that W. And then at the right hand end, you've got the rest of the chain and that will have a tension that's pulling on our little bit of chain and that's pulling not exactly upwards, but it's at an angle towards sort of the northeast. So if this chain is just at rest, it's come to rest, then everything should be in equilibrium. All the forces should balance out. So what have we got here? Well, we've got the vertical forces at play, got gravity kind of downwards or the force due to gravity pulling you downwards. That's got to be balanced out by the vertical bit of the tension up there, which will be T sin theta. And the tension pulling to the left caused by half of the chain, that has got to be balanced out by the horizontal component of the tension in this direction, which will be T cos theta. So if you're do a few grade level maths, you'll be familiar with this kind of calculation. So we've got these two equations, T sin theta has got to equal W, T cos theta's got to equal F. This is great, Sarah, we dunno what any of those letters are and we dunno any of the values here. We can get rid of one of them because we can divide the top equation by the bottom one and that sin over cos, that'll give you tan, the t's will cancel out, and we get tan theta is W over F. Okay, so far, so good. Now here the hand waving starts in earnest because I'm going to make two approximations. The first one is the force downwards due to gravity. Now force is mass times acceleration. So this will be dependent on proportional to the mass of that bit of chain. And the mass of that bit of chain will depend on its length. We don't know the length'cause we dunno the equation of this curve yet. So we're going to approximate it, we're going to pretend it's a straight line this bit. It's not quite, it's gently curving. We're pretending it's a straight line. And so then the mass would be proportional actually to the horizontal distance you've traveled, which is X. So I'm going to approximate W, I'm going to say it's a proportional to this X horizontal distance. And now F is going to be some constant thing because it's representing half of the chain, which is, you know, we know in theory, we know what that is, it's fixed. And so we can say that W over F is also proportional to x and we'll say that it's some constant a times x. So that's W over F, some constant a times x, approximation number one. Approximation number two, even a bit more dodgy is to say, right, what about this angle theta? Well, I don't know what it is. If I sort of extend this green line dottedly towards the origin, I'm going to make a triangle and the triangle will have width x and height y and approximately, approximately angle theta. It won't be quite, it won't be quite theta, but it's close enough for our approximation. So now I've got the tan theta is y over x. And so putting it all together, y over x is roughly a times x, a is any constant. And so that gives you y is approximately ax squared. And that, if you've ever drawn a curve in class, the first curved line you probably plotted in class was y was x squared, which is a parabola. So approximately a catenary is a bit like a parabola. Now we do know the actual equation for a catenary. It's this, yeah, now we fully understand, it's a bit more complicated and you can change the shallowness by changing this letter B, which is a parameter you can change. That's got some powers in it and things. There's a way of writing that differently as a kind of a thing that just involves x's, powers of x. It's an infinite thing that carries on forever, but it looks like there's the first few terms, something x squared, something x to the four, something x to the six, dot, dot, dot. Now if you just noticed the first term of that, that is looking like that parabola approximation I've got. And if x is small, then if x is like less than one, right? Then x squared is smaller, x to the four is tiny, x to six is even tinier. So actually when X is small, those later terms get very, very tiny and you can almost ignore them. So I'm going to superimpose now just to prove this, I'll superimpose for the correct values of B, the relevant parabola onto these catenary. And you'll see, okay look, the top one you can't even tell'cause it's got a very small bit of the catenary. Next one you start to see it at the edges and the bottom one you can really notice. So the parabola is a really good approximation to the catenary for small-ish angles, for shallow catenaries. And so what's that got to do with anything? Well, Wren and Hooke knew this, they knew the parabola was a good approximation. And you can do a similar but more complicated thing if you go up a dimensions. If you're trying to solve the same sort of situation for a dome, then you know, at some point I had my approximation led me to y over X equals ax. While in the dome situation, we've gone up a direction, the ideal dome as an approximation, at some point in that we get to something like y over X equals approximately ax squared. And so then if you multiply up, y is approximately some constant times x cubed. And that's where the cubic idea came from. Really it's an approximation. And actually they thought, Wren and Hooke thought this was the answer. It's not quite the answer. The real answer wasn't known for a long, long time after then, it's extremely complicated. Here's the first few bits of it. It's a very difficult, complicated expression. But the first term there is that x cubed term. And so the the cube, the cubic, y was x cubic is a really good approximation to the real shape of an ideal, the strongest possible dome which they wanted to support this huge weight. So the dome of St. Paul's actually is two domes. Well, no, how many domes does it take to build a cathedral? Three domes, there are three domes in St. Paul's Cathedral. Here is a picture, this is drawn by Christopher Wren himself. It says at the bottom, Christopher Wren's hand, illustrating the three domes. So, and they called this the cubicle parabolic conoid, conoid catchy, what it is there, and it's might be hard to see at this distance, but you can see there's some numbers down the side perhaps. What those numbers are, he's actually sketching the graph of y equals x cubed. So it goes up in eights so I've drawn that like this, and he's marked off to help him with this, look, I've highlighted it in blue, this is that exact curve y was x cubed. I've just superimposed it and you can see it's a really, really good match'cause that's what he was drawing. And you can actually see where he's marked off, one cubed is one, two cubed is eight, three cubed is 27. You can see the dot at 27, four cubed is 64. And that's the same at the bottom. So he's sketched the x cubed graph in his picture of the not one, not two, but three domes of St. Paul's. So that's really cool to see. And it's what a wonderful mathematical marvel that the domes of St. Paul's are. You've got a catenary, you've got a cubic, you've got a hemisphere. So Christopher Wren then architect, yes. But look how mathematical his work is. And I want to finish by just, this is a justification of how it comes that the Gresham professors in their academic year, we're supposed to give six linked lectures on a theme. And my first lectures this year, three of them were on Matson Money. We did coins and currency, we did game theory, we did how to win the lottery. And those people who attended the how to win the lottery lecture are now sitting here in top hats and fur coats, drinking champagne. So that was at Matson Money. And now that the second half of this year, we have three lectures on unexpected in mathematical lives, Christopher Wren, architect, also mathematician. Next up Florence Nightingale, nurse, also a pioneer of statistics. In June, Alan Turing. Okay, we do know he's a mathematician. What we think of his work mainly in cryptography, cracking the Enigma code in the Second World War. But he also did brilliant work in mathematical biology. So I'll talk about that. So how did I get away with this? So here's my justification, my excuse, I wanted to call these lectures, mathematicians of note because have you spotted the link? They're all on bank notes, so that's my excuse. So Christopher Wren, he got 50 pound note. Florence Nightingale, the old 10 pound note. And on the new 50 pound note, Alan Turing. So we'll finish there and I hope to see you in May for my lecture on Florence Nightingale, thank you.(audience applauding)(audience applauding)- Thank you so much, Professor Hart, for such a fascinating lecture. I've got a couple questions from online before going to audience in the room. First one is from (mumbles). He says, "If they, Wren and Hooke,"knew that the catenary was the best load bearing shape"and they knew its equation,"then why approximate to parabola?"- Parabola, yeah, so they weren't approximating to a parabola, but that was kind of my justification for how they got to the cubic essentially. So Galileo, so some people say that Galileo thought the catenary was a parabola. He didn't think it was a parabola, but he knew that the parabola was close. And so this was a well known thing that the catenary was close to a parabola. And you can get it, you know, you'll notice what I did there, that little slightly dodgy calculation doesn't involve calculus, but if you are pre-calculus, it's very hard to actually derive these things. So a parabola, as we saw is a really good approximation. But it was the fact of it being close to a catenary that made them then think of actually maybe the cubic is the right thing for the dome, yeah.- There's another question referring to another of the slide, I think the slide just before the ideal dome slide.- Yes.- And he asks,"Wouldn't W be closer to the hypothenuse?"- Wouldn't doubly be closer to that? Might have to, oh, there's a W. So the hypothenuse of this triangle. Well, so I know what you're saying. Yes, why wouldn't, I've said it's related to x, well, it's not, it doesn't equal x. So that the length, I think what the question is the length of that bit of the catenary, if we're imagining it, it's a straight line, yes, it would be the hypothenuse of that triangle. But that would be proportional to x. It would be some multiple of x in length. So it's not equal to x, it's a multiple of x. But that's okay 'cause we're only caring about proportions here. Yes, you know, if X were to double, then the hypothenuse would double. You can see it's a, yeah, a proportional relationship, yes.- [Audience Member] You emphasized at the beginning of the lecture, Christopher Wren's emphasis or interest in the applications of mathematics. I just wonder whether you would care to conjecture, if you could imagine Wren meeting, (coughs) excuse me, peers like Birch and Russell, Alfred Whitehead and Godel, would he have any time for what they had to say in the contrast between applications in the purest aspect?- Yeah, so I think, I don't think he'd see it as a contrast. I think he would see all of that as being valuable and important parts of what it is that mathematicians are doing in a way that perhaps has been a bit lost more recently. So, you know, you get Hardy in his mathematician's apology saying like, I'd rather be at the top of Nelson's column and no one can see me or understand what I'm doing, but as long as my important work is, you know, I don't care and I despise application. You know, you get that attitude, which I'm a pure mathematician and I do it 'cause I love it, but I don't despise applications. And I think this is what Wren was like. He was, yeah, he was rectifying the cycloid like there was no tomorrow and he didn't have in mind when he did that because I might be able to use it for something, he was genuinely interested in just finding things out, he had that curious mind. But he didn't then say, oh, I don't care about what it might be used for. He was also interested in that. So he, you know, it was part of what mathematics was to him. The mathematical arts encompass all of these things and I think that's what he what he would've thought, yeah.- Thank you so much Professor Hart. I'm afraid we are now at times, so that's, I think all the time we have for questions, but can everyone give a very warm round of applause for Professor Sarah Hart?(audience applauding)