In Our Time - Renaissance Maths

Episode Date: June 2, 2005

Melvyn Bragg and guests discuss Renaissance Mathematics. As with so many areas of European thought, mathematics in the Renaissance was a question of recovering and, if you were very lucky, improving upon Greek ideas. The geometry of Euclid, Appollonius and Ptolemy ruled the day. Yet within two hundred years, European mathematics went from being an art that would unmask the eternal shapes of geometry to a science that could track the manifold movements and changes of the real world. The Arabic tradition of Algebra was also assimilated. In its course it changed the way people understood numbers, movement, time, even nature itself and culminated in the calculus of Isaac Newton and Gottfried Leibniz. But how did this profound change come about? What were the ideas that drove it and is this the period in which mathematics became truly modern?With Robert Kaplan, co-founder of the Maths Circle at Harvard University; Jim Bennett, Director of the Museum of Science and Fellow of Linacre College, University of Oxford; Jackie Stedall, Research Fellow in the History of Mathematics, The Queen's College, Oxford.

This BBC podcast is supported by ads outside the UK. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk, forward slash radio four. I hope you enjoy the programme. Hello, as with so many areas of European thought, mathematics in the Renaissance was a question of recovering, and, if you are very lucky, improving upon Greek ideas.

The geometry of Euclid, Apollonius and Ptolemy ruled the day, Yet within 200 years, European mathematics went from being an art that would unmask the eternal shapes of geometry to a signs that could track the manifold movements and changes of the real world. The Indian and Arabic tradition of algebra was assimilated, and both Newton and Leibniz developed the calculus, the maths by which we can still put men on the moon. But how did this profound change come about? What were the ideas that drove it?

And is this the period in which mathematics became truly modern? With me to discuss the new mathematics is Jackie Steddle, Research Fellow in the History of Mathematics at the Queen's College, Oxford. Professor Robert Kaplan, co-founder of the Maths Circle at Harvard University, and Jim Bennett, director of the Museum of Science and fellow of Linneker College at the University of Oxford. Robert Kaplan, the foundation stone for the maths of the Renaissance period was Greek geometry. Can you define geometry and explain why the Greeks considered it to be so important? Here were these forms, these eternal shapes that lay behind changing appearances,

and the Greeks came to grips with these over a long period which becomes solidified, rigorized, made sense of at last by Euclid. One has not only a sense of what these shapes are and how they behave, but a basis for this understanding in logic, a rigorous foundation where the ideas, the basic acceptable ideas, the axioms, those things which, how could one help but accept, are developed through deductive logic into consequences you wouldn't have dreamed of. What a marvelous feat. You talk about eternal shapes being discovered, Robert, and eternal shapes is to do the idea, the platonic idea of there being eternal, let's use the word verities out there, which you have partial knowledge and partially glimpse and so on.

Can you just tell us where they found, you said over a long period, which suggests they were struggling to find these eternal shapes. Where did they, and why did they think they were eternal? I think that one struggles oneself, one finds this in one's own biography. And can you give me some specific examples of the shapes? Yes, let's take a triangle. You look at a triangle, you say, well, what you see is what you get, three lines joined at the corners.

But then there's a point that comes with those triangles, with each triangle, a circumcenter, a point equally far from the three vertices. Well, isn't that strange? It wasn't there when you looked before. All of a sudden, you find it. There's another point which is such that the median, which is the center of gravity of the triangle, where did that come from? How do you know it's there? You can, by a series of small deductive steps, not only see that it's there, but prove to the greatest doubter in the world that it is there. When Shelley says the one remains the many change in past life like a don't of many kinds of many kinds. colored glass, stains the white radiance of eternity, the Greeks found these shapes. Now, did they find them out there? Did they find them within their own minds? Are the two the same? Are these

invented or discovered? That's the great deep mystery of mathematics. Why did they give geometry so much weight, Robert? Why did I think it was so important? Was that because Euclid was so brilliant that they had to take notice of it? Euclid, in a way, comes at the end of the story. I think there's been a long of their seeing that shape. After all, 90% of our brain is devoted to the visual. That shape is something you can come to grips with much more readily than what we'll talk about later, algebra, the abstract dealings with numbers. And these shapes that you see caught in the material world around you, a triangular brace, say. you say, well, but the triangle isn't that brace itself, it's a form behind it.

The brace is imperfect, the triangle is perfect. Let's just very general, and I do apologize for this. I mean, because we were talking about the Renaissance, so that was an excellent, wonderful prologue, but we bump along with Ptolemy and Euclid partially understood through the early Middle Ages. That is to say, only bits of their work were translated and known. in, let us call it, the Western European world. You know, things are going on elsewhere.

I'll come to that later. What happened to geometry in the Renaissance? How important was it when it began to be, when the papers, the theorems, began to be rediscovered? Wonderful feeling of exhilaration in the West that you now had a science which needn't be taken on authority or on faith, but on a rigorous, axiomatic basis.

so that not only was there this science of shapes, and the shapes became manifest in the material world, but that you had a way of thinking that went beyond anything controversial politically, psychologically, these were truths that had to be accepted. So the idea of proof was very important, and something that had proof after ages of faith, that's one of the things that perhaps.

Crucial. Yeah. Jim Bennett, The geometry literally means, as I've discovered, recently, the measurement of the earth. So, were they literally measuring the earth? How practical was the culture of geometry in the Renaissance? That's right. Now I come with a different story from Roberts.

Robert has been talking about the transcendental elements in geometry. And you rightly draw attention to the fact that the Greeks told a story about geometria, literally, measurement of the earth. So their story about the origins of geometry, which is encapsulated. in that word, is that the Egyptians needed to have a method of setting out the boundaries in the land that was inundated annually by the Nile. So it was very important socially that these were re-established in a way that was true and dependable. It was virtuous in the Renaissance sense and also practical. And that's a story that gets repeated endlessly in the Renaissance. That's always a trope that mathematicians reiterate.

So according to that account then, mathematics perhaps may be transcendental, but the focus there is on its practical nature. It's for land surveying, and its origins are in land survey. So right at the very beginning we have what continues still and what was very, very important in the Renaissance, what could be called pure mathematics and applied mathematics, and they're running in harness. And the applied mathematics is what I'm trying to get at now

with geometry in the Renaissance. Exactly. And it's in the Renaissance where this practical geometry comes to the fore, I think. I mean, historians of mathematics are often disappointed with the kind of mathematics that Renaissance geometers do because it seems so practical, it seems so focused on applications. The notion of pure and applied, I don't think, really comes into play. That's a later distinction. But certainly in the Renaissance, that idea of geometry as a practical,

a mathematical art based on a mathematical science is used in land survey but it's expanded into all sorts of other areas. Can you give us some examples as to where geometry is practically effective? Well, it's practically affected in the most unlikely areas such as fighting wars.

In range finding, for instance, finding the distance of the target, elevating the gun so that it hits the target, making calculations. They may not sound geometrical, but they use geometrical instruments to decide on the charge for the particular weight of shot and so on.

That seems a very practical business, but the geometer is the person who comes along and says to the prince or whoever, I'm your geometer, I'm the chap who can deliver this technique to you and help you vanquish your enemies. But architecture, navigation, cartography, all those practical areas

where mathematics can be applied is the mission of Renaissance geometers. So the genius was in the application, particularly the development of instruments, which were used by a lot of architects and painters and inventors and so on. Yeah, instruments are crucial to me. Okay, I work in the museum, so instruments are my life. But if you look at the record of Renaissance mathematics, we mustn't forget that there are lots of these instruments.

Our museums are full of them. For instance. Well, theodalites, sectors, a sector is a calculating device, which we would think of as based on similar triangles, astrolabes, quadrants, so therefore for astronomy, for surveying, for navigation, you navigate your ships using these instruments. And the instrument is a kind of embodiment of what geometry was.

And it seems to have been part of the accepted and necessary sort of job description of a geometer that if you have a new technique, then the way you communicate that, the way you enable people to use it, is by designing an instrument, and that was expected of you. Jackie Settle, can you tell us about what I, perhaps mistakenly, at this stage in the development of mathematics, called pure maths, because as well as the practical side, which has been very vividly described, there were people who just practiced the art, let's say, of mathematics for intellectual satisfaction,

and that was going on. Can you just describe that, please? Yes. This is a mathematics that is neither practical. nor transcendental. It's mathematics for its own sake, and people have always done mathematics simply because they enjoyed it. It's an intellectual exercise and a pleasure to do. And in the Renaissance, people were very interested in Greek geometry. There was a huge rediscovery of Greek geometry, a new discovery of texts. And this was far beyond anything that European mathematicians had known up until then, and so they were very excited about this. They were very profound results that they found in Euclid and Apollonius,

and they wanted to take them further. Why did these new texts come from? I mean, we know in general they steamed in from Santana, but in this particular case, where are these new texts on mathematics, big de geometry, coming from? Byzantium, as you say, also from Sicily and from Spain, through translations from Arabic into Latin. But then in the Renaissance, the original Greek texts were often rediscovered as well,

so they had the original Greek manuscripts alongside the Latin translations. Can we take one example of Euclid, who Robert reminded us, came towards the end of this, but was, most of you would help agree, of enormous importance and significance. When he was translated, the first translating to English was about halfway through the 16th century, 1572. And I presume he was being translated into other vernaculars about that time. So what impact does that have on these people who are, are enjoying the game of mathematics? Well, this was the first translation into English

and probably one of the first translations into any vernacular language. The language of scholarship up till then had been Latin. And there were several Latin translations of Euclid, but this was the first English translation. So that marks an important step. People were beginning to be confident in their own languages that mathematics could be written in their own languages.

It wasn't the first mathematical textbook in English, but it was one of the first. Can you just say a little more, I'm sure listeners will be intrigued, that the idea of people at court, mathematics being one of their pursuits in terms of intellectual pursuits for the sake of it, is something that's rather lost now.

It's now part of a system, a university system, or an industrial system or something like that. It's a very nice idea, and it would be nice to give it a bit more air. It was certainly true that mathematics was part of any educated person, and it was one of the subjects anyone would have studied, along with many other things. And people were what we would now call polymaths. They were interested in many subjects.

And there are beautiful, lavish editions of Euclid and Apollonius, published in the 16th century, which are obviously library editions. They're not for scholars to use, therefore, as you say, princes or dukes to have in their library. We might call these amateur scholars or gentle scholars. Did anything come of that out of that area? It's a heck of a question. I'm often sorry of it, but to turn it. Did something come of that which added to the store-stroke development of mathematics?

Probably not in terms of taking mathematics further, no. I don't think so. But still the idea of it being a way to approach the natural world was kept to flame, wasn't it? Yes, yes, certainly. And actually those people were often patrons of working mathematicians, so the developments might have come at a lower level in society from people who are working.

I think just as one has this interesting dialogue between the pure and the applied in mathematics, between the invented and the discovered, there's this continual back and forth which is enriching and enhancing the other. Practical problems beget mathematical solutions. Mathematical ideas beget practical devices. One has this, well, in the words of Eugene Wigner, the 1950s Nobel laureate in physics, he said that there's an understanding. unreasonable effectiveness of mathematics. Why is it that the world can be explained mathematically? I don't think we understand this. I think we're a long way from understanding it.

But there it is. We look at the world. We see in its imperfect behavior mathematical shapes. We look at mathematics and we see explanations, ways of being able to predict how the world behaves. This is wonderful. And this has helped enormously because we're moving to algebra now, for those of you who want to hang on tight.

It came, as I understand it, I need a hand here, as I understand it from India, originated, maybe started, let's start it in India in the 6th century, then it moves into the Arab world, then it comes into Europe at the beginning of the 13th century, a merchant from pizza called Fibonacci,

publishes a book called Libra Abakai, Book of the Abakas, and Arabic Algebra arrives. Now then, can you take it? it from there. We have to give the Greeks a little credit. There were algebra beginnings among the Greeks. I didn't think I was going to get away with it.

But here we have in India in the 6th and 7th century is this wonderful development of an idea that you can take numbers, numbers apart from geometry, apart from geometric shapes, and deal with them in and for themselves, that you can let loose in the pack of sheep of numbers

the wolf of a variable, an X, which allows you to conjure up solutions, shapes that you hadn't thought of before, shapes like curves that hadn't come up in Euclidean geometry. This gets carried to the West by Arabic merchants and in other ways, and met with a lot of resistance. Algebra, can you really talk about zero, for example, which is how can you talk about nothing? This is devil's work.

And then through Leonardo of Pisa, Fibonacci and others, algebra develops in the West and becomes a beautiful science of its own with many connections to geometry. Jim, but how much initial impact did the Fibonacci's translation have on mathematics in Europe? Well, it has a huge impact, but I suppose again I'm going to introduce this kind of gruviness about the world and try to bring us back down from the ethereal kind of mathematics. I think it's an entertaining dynamic. I think it's an interference at all.

Because, I mean, Fibonacci wrote about all sorts of things and very much the practical, he wrote about practical geometry, for instance. I already talked about land surveying. And the work on what we think of as algebra also contains a lot of material on, on mercantile calculation and so on. And indeed, that's, as you know, Fibonacci's background, and that's how he got exposed to that kind of mathematical practice.

And if you're talking about the influence particularly, perhaps not the influence on the progression of mathematical ideas, but the influence on the ground in terms of what people were doing, then it's those methodologies for merchants, for calculating all sorts of complicated, work that they needed because there were different currencies, there were all sorts of different measures and so on. So how did this help?

I mean, I think fascinating, the intervenants of trade on cultures, absolutely on language, on numbers, on alphabets is absolutely fascinating again and again and again. And it's happening here. Now, can you give us some specific examples as to how it helped? You said it was really helpful. Well, particularly the kind of numbers that he was introduced. The numerical system was much more tractable as far as merchants were concerned.

Instead of the Roman numerals for me. Exactly. And these methodologies get enshrined into an education system, particularly in Italy, in these Abacus schools, which are very famous in Renaissance Italy. And they're based on Leonardo's or Fibonacci's methodology and on his writing, an aspect of his writing. And they provide a kind, I mean, historians of art, say historians of architecture, think of them as providing a kind of shared mathematical culture in Italy. in the Mercantile class in Italy, which was important to the reception of Renaissance mathematics in painting and in architecture and so on.

To take that on, Jackie Stedle, from what Jim's been saying, I mean, for hundreds of years, as I understand it, writing mathematics had been rather laborious, wrote it down in words and so on. And then algebra came in, but also there was a development, as I understand it, from an Englishman called Thomas Hemp. Harriet, can you explain what he contributed that was important in itself?

And then we can talk about whether it had massive influence or whether it was reinvented at the same time by Descartes later. But what did he do that is useful for this story? Can I just first take up a little bit what the previous speakers have said because I don't quite agree with their story about algebra. I don't think we find traces of it in either Greece or India. It comes through the Arabic culture. The word algebra comes from algebra, which is an Arabic.

word. And it was to do... It means bone setting, doesn't it? I'm questioning from your notes. This isn't anything I knew before a few days ago. And it was to do with solving equations. That was what algebra was. You've all done quadratic equations at school probably. That's exactly the kind of equation they were interested in. And it was a technique of solving equations. And although that was published in Fubonacci's Lieber Abbekechit, I don't think there was any practical use for it at all. Again, it was one of these intellectual exercises. It was something people wanted to do. I can't think of any practical problem that leads to a quadratic equation.

I don't know if you can, Jim. No, I think that's right. I was thinking that Fimnachi covers a lot of ground, and some of the ground he covers is practical, but I agree with you about the non-practicality of those. And the algebra is actually a very short section at the end of his book, after he's dealt with many, many practical problems. So, yes, it was about equation solving,

but until about 1600 people had to write those equations in words. Can you give us an example? Well, if you want to take, say, the quadratic equation, X squared plus 10x equals 39, you have to write that verbally, and it takes a whole paragraph, you have to say, my square plus 10 times the side of my square makes 39, and then you give all the instructions for manipulating it, but it all has to be done in words.

I prefer that myself. Well, it's quite difficult to follow, and I think it must have been difficult at the time. And to untangle. And to untangle. And they made amazing progress, I think, considering that that was how they had to do it. Around 1600, you begin to get notation coming in,

and this was Thomas Harriet, who you were talking about, yes, who invented really what is now modern algebraic notation, so lowercase letters to stand for numbers, writing A, B, next to each other, for instance, for A times B, the square root sign that we use now, the three-dot therefore sign, the inequality signs, which are like arrowheads on their signs, all those he invented.

And he wrote in a purely symbolic way. He goes to the other extreme. There are very few words in his manuscripts at all. It's all symbolic. It was extraordinary mad, wasn't it, really? In all sorts of ways. So I understand he didn't publish anything in his life time. He didn't publish anything.

About 8,000 pages of notes, most of which I think you say are rubbish, but some of which is extremely vital and important. A lot of them are waste and rough working. Probably about a quarter of them are, well, extremely good. mathematics, very interesting mathematics. So just to finish this little paragraph, what did this do? This just made it faster and easier of application, easier to handle. It's more than that. It makes it easier to write. It makes it easier

to read so you can communicate mathematics better. But once you write mathematics symbolically, it also helps you to think further, because you can see things that you didn't see before, and you can take the arguments further. So it's an aid to thinking as well as an aid to reading and writing. Robert Kaplan, although you, if I may use the words, wax lyrical, I don't see why cliche shouldn't have a place in this conversation about algebra. It did at the time meet with opposition by some very stern people. Hobbes and Fermat Day, Hobbs said, what was it? I've got it here.

It's a scab of symbols. And they, what were they against it? Well, in Hobbs's case, here's a man who discovered Euclid in his adulthood. came on a proposition, I think it was the Pythagorean theorem, said this can't possibly be, traced it back, saw it had to be, was overwhelmed with the beauty of it, convinced that geometry was a truth. So why should we give up that kind of beauty for a less well-founded algebra? This, I think, is an important issue. Geometry comes with its axiomatic foundations and its rules of deduction. and algebra grows by fits and starts, by leaps and bounds. One has the beauty of new discoveries. The foundations aren't there that they are for geometry.

It's not until the 19th century that a firm footing is put under algebra. So one has objections that, well, it may be powerful, but is it true? What about the influence here? I'm going to go across to Descartes, who was a very powerful mover in all this, but there is a suggestion that he somehow got wind of or had access to Harriet's notation, and his own notation is very similar to it, and therefore did it come from that?

They could have invented it independently. Just as a bit of, you know, fun, what do you think went on there? It's impossible to know. Some of Harriet's work was published posthumously in 1631. Descartes brought his geometry out in 1637, so he could have seen it. He said that he didn't, but Descartes was never very ready to acknowledge other people.

He said he hadn't read a number of people. And even if he hadn't read them, I think he must have had a very good idea of what was around at the time. He was in contact with mathematicians in Paris. He was well connected. My own feeling is that he would have known. But this notation. Anyway, one way and another, the notation came to Descartes. Now, how important was that in this story that we're telling Robert that he...

Well, the important thing about Descartes is that he performs a wedding, between geometry and algebra. He introduces the coordinate plane on which one can draw graphs of equations. One can turn geometric problems into algebraic ones. Would you like to attempt to describe this in detail on Steam Radio now? Why not?

You have a straight line. You have two straight lines which intersect, and you have theorems about intersecting lines in Euclid. Three intersecting lines make a triangle if they intersect in the right way. To deduce from these intersecting lines, this triangle in geometry, that the perpendicular bisectors of the three sides coincide in one point, the circumcenter, that point from which you can draw a circle around the triangle going through the three vertices.

That's not easy. It involves some very nice insights and then some deductive reasoning to back it up. Turn those straight lines into equations. solve the equation simultaneously, why, it's a piece of cake. And Descartes wants to save us the effort of thinking. I think there'll be objections to this, perhaps. He wants to save us the effort of thinking by giving us this mechanism. Can we just push it one bit further?

When you say turn those straight lines into equations, for those of us slow at the back of the class, can you just say how you do that, and then why it works so much easier? With this coordinate plane, you have a horizontal line called the X axis, which has your inputs, 1, 2, 3, 4, and fractions and so on, negative numbers. You have a vertical axis, the Y axis, again, which is calibrated in numbers. And then you can say, well, if I've got two points here on this plane, each with its address, an X and a Y coordinate, and I draw a line through those two points, that line will have an equation, equals mx plus b, where the m is the slope and the b tells you where that line cuts through

the y axis, all the points on that line will have to obey that equation. If you give an X, you'll get the Y out of it, and vice versa. And you can manipulate mechanically those equations to give you places of intersection. So, Jim, we're talking about towards the end of 16th century, early 17th century, that algebra can do things geometry, geometry, can't do. It doesn't work as didong, didong as that. We know that algebra, I think as you said, algebra was not a big stone in the small pond, it was a small stone in a big pond when it arrived. We have to keep

remembering. Geometry ruled. Very much so. But nevertheless, there it is, lurking up algebra, doing things that Robert explained to us all. And so what, in general terms, were people getting hold of it for? What were they using it for? and saying, oh, we've got this as well as geometry, oh, it's better than geometry, for what? Yes, it's tricky, isn't it?

Because as you say, there are conventions of what's acceptable and what we have always accepted and what we trust. It's this business of virtue that I mentioned, I think, at the start, where people trust geometry for all the reasons that Robert has explained. I think one answer to the question comes in thinking about the other contribution of Descartes, and he's not alone here, but he's part of a very strong 17th century trend where we're saying that mathematics really describes the natural world

and really explains the natural world. Now, I know that we said that loosely a little earlier, and I wasn't entirely happy with that because I think in earlier mathematics, it isn't at all clear that mathematics gives you any explanation of nature. It's true that there are these apparently transcendent truths that it encapsulates in some way, but it doesn't seem to tell you about the material world,

or at least that's a disputed notion, certainly very, very troubled and disputed. But in this thought it did that, didn't he? I mean, he announced very emphatically that he thought. It's just a little bit that tipped into my mind. Yes, that's true. I mean, that's part of the debate that takes place in the late 16th century

over the status of mathematics in relation to nature. It isn't, as you say, do dong, di dong. It doesn't just go through because someone says so. But Descartes has this very powerful statement of the mathematical world and the material world being essentially the same thing. So explanations of the material world can be couched in geometry,

or there's as you say, this other thing called algebra, which seems to be more powerful in allowing us to think. Now, there are all sorts of phenomena in the material world that aren't tractable, that don't work easily in geometry, because they flow, they move, they continuously change, in thinking about Copernicus, paths of the planets, for instance, are continuously changing, projectiles are continuously changing the flow of water and so on. Now, to deal with those kinds of phenomena in mathematical terms, as Descartes says, we must,

then it looks as though we're going to need something different from John McRae, something which is more powerful in dealing with motion and change. And it looks as though it may be that the potentiality of this algebra stuff is going to help us in getting to grips with the natural world in all its changing variety. Would you like to take that on, Jackie? Yes. I'm not sure that algebra had that potential

or was seen to have that potential at the time in the early 17th century. And when Descartes told you that you could solve geometric problems by solving equations, for instance, he then told you that you had to go back from the equations and find a geometric construction for your solution. and he was still interested in geometry as the basis of things. The idea of... The geometry was the great given, though, wasn't it?

I mean, I was trying to find an analogy out of my own, rather sort of plumber experience of knowledge than I have. And it seems to me that geometry was to mathematics, what Latin was to language. You had to go back to a rigorous, scyves, solid groups. But it was the power language. Yes, and it was that, you were magnetized by it. Yes.

Is there something in that? Yes. I mean, it was the foundation of all 16th century and early 17th century mathematics. And what a lot of early 17th century mathematicians thought they were doing was rediscovering and reconstructing, sometimes correcting, filling in the gaps in Greek geometry. They weren't really trying to go beyond that because they didn't think they could go beyond it. They were trying to perfect it. In doing so, of course, they moved forward and started to have new ideas.

Yes, we had what we, Paul, Lucy, here, the new maths. And Robert Kaplan, towards the end of the 17th century, this is, as it were, dramatized. by a dispute between Newton and Leibniz about the calculus, who first developed the calculus. The dispute gives it a dramatic presence. But the fact of the calculus is what's important. So why was it so important to claim it, apart from wanting to be first past the post?

What was so important about calculus? And obviously, algebra pushed through that very hard. Yes, calculus allows us to talk about change. The Greeks thought of looking through the veil of change at the permanent forms. Calculus now looks at the veil, at the waving, changing shapes of things. How does it do it? By talking about the very small, talking about not a straight line, but a curved line as if it were made up of very short straight lines,

or as if one could approximate it by lines just touching it, tangent lines, touching it at a point. Well, perhaps at two points that are very close to one another and then letting those two points drift together. This is a remarkable idea. There's a dispute about priority, but a more important dispute about how one is actually imagining the world to be fundamentally. Is it made up of tiny, indivisible particles?

Leibnizabeth's idea. Moanads. Windilless, he called it. Windleless monads. By the way, he says that each of us is a windilless monad. we can't see out of this unity that we are. And Newton, who sees change as fundamental fluence are the basic, well, one can't say elements

because there are no unbroken particles for him. Change is fundamental. These fluents and the ways they change, fluxians. So these two different views, wonderfully enough, develop concurrently in contradiction to one another, evolve and by the dispute between Newton and Leibniz and their followers, evolve into the calculus that we know with its notation, do more to Leibniz now than Newton,

its ideas due to both of them, that allow us to talk about how things change. But it's fascinating, isn't that when Newton writes about this, explaining the principles of calculus in Principia Mathematica, he feels obliged to or he wants to, he chooses to use the language of geometry. So the whole that geometry had is very much, the Latin analogy isn't too terrible.

Milton is still employed by Cromwell because he writes in Latin and you can communicate with everybody across Europe in Latin. It's still there. Were they still giving primacy to geometry? Is that the simple answer to that? I suppose that is the simple answer. You want to be listened to, you want to be believed,

you want to be, you need to achieve credit and for your ideas. And it's clear that to introduce something as fundamentally and innovative as the Newtonian system of things, the whole system of the world. And on the same time, at the same time to introduce that

through a whole new vehicle of mathematical inference, such as the calculus, wouldn't really have been very sensible and credible for a mathematician. So, So there's partly the unwieldiness of doing those two things at once. But I think more important than that is just the virtue that is still vested in geometry.

I doubt if it would even have occurred to Newton to have done otherwise. For us looking back, it maybe looks a bit calculated. Like he sat down one night and thought, well, is it going to be fluxions or is it going to be geometry? I don't think it was anything like that decision. I don't think there was a decision for Newton. Can you tell us, Jack, can you just tell us how? the calculus was received. Did people see its possibilities?

Roberts hinted at them, if you could give us more of the possibilities now. And how it was received around the time, not the next day, but around in the next few years, kind of thing. In the next few years, yes, it was certainly received. I mean, Newton didn't publish his calculus to begin with, and that was part of the problem. Leibniz did, and so Leibniz's calculus was taken up,

and his notation was taken up. And people realized that the calculus could be used to describe the real world, look at problems of change and motion. That wasn't what the calculus was invented for. The way the calculus was arrived at was nothing to do with that, but those were the consequences of it. So it was invented for what?

You suggested it was invented for a particular purpose or purposes? Again, out of a pure mathematical question to do with looking at areas contained by curves and at tangents to curves. So, for instance, if you have an ellipse, you might want to know the area enclosed in it. You might want to know how to draw a tangent to it. And these were problems that people were concerned with, again arising out of Greek study of curves.

And the amazing thing about the calculus is that it tells you that the area question is related to the tangent question. They're inverse problems. So this is an extraordinary mathematical discovery. Nothing to do at the time with ideas of the physical world or change in the world, but it's applicable to those things. Sorry, sir. And this is where I have problems with this. the story. That's to say my, the general approach that I've been plugging that

a lot of mathematics comes out of practical issues and so on, because I'm at a loss to know what I can say about the origins of calculus in that respect. Perhaps all I could say is that mathematicians as people requiring careers and

university positions and so on, in the 17th century, want to move beyond the image that I've been projecting onto them. They want, I mean, Wallace, John Wallace, the professor of geometry in Oxford in the later 17th century, says himself that very famously that when he first got into mathematics, it seemed to be the business of land surveyors to go back to geometry and so on. And carpenters and people like that.

And look how we've moved on is what he's saying. Look, we are now different people. We mathematicians are now up there with the classicists and rhetoricians and all the other departments of proper universities. life. And I think that's part of what's driving this. And that's my practical spin on this, that mathematicians

want to be in a subject that's recognisably, or some of them anyway, that's recognisably cerebral and academic. And certainly those kinds of methods, those kind of detached methods, methods for which I'm

not able to give a practical rationale are part of their program of improvement. Just one second. Before, we should resolve the Leibniz-Nuton thing just quickly, but I'd rather move it to talk about. Newton seems to have written it first, but he didn't publish because it had a bad review for the thing he published for. He went into a sulking, he wouldn't publish for a very long time. Leibniz arrived at it and published first, but that seems to be the sort of deal, is it about? Yes, but I wonder if this dispute over priority is as important.

I was trying to get rid of it by the, by coarsely summarizing it in a rather offhand way, which is meant I've had to explain the course summary, and we've now lost a paragraph. So how did calculus become, as it were, I said at the beginning of the program, the dawn of the new mathematics. Can you just briefly tell us why calculus drove through to become the mathematics that...

Well, to pick up the point that Jim made earlier, once talking about the practical world, the real world with ballistics, with particles, with objects moving in parabolas, for example. One wants to know if they go up here, where are they going to come down? How can a rocket, which travels in a parabola, intersect the moon, which travels in an ellipse? If one can do that with the algebraic expressions of the two curves, wonderful. But one's talking about curves that change their slopes as you move along them.

That's where calculus comes in. And this is the great triumph of calculus, that it's the science, the mathematics, the art, if you will, of change. But I think that there's an important shift in paradigm that we haven't talked about, which lies behind these changes from geometry to algebra, that the style of explanation is changing through the Renaissance to our time from causal to formal. Newton, for example, gives up asking, what is gravity? How does it work? To describing formally what it is. bodies attract one another inversely as the square of the distances between them. And that becomes an explanation.

An explanation one will rest with instead of pushing the inquiry further into geometrical issues that embody gravity. You were the one, Jackie, who said that it was algebra was a small stone in a big pond. By the 19th century, as Robert said earlier, the algebra was being underpinned, and it was now a big stone in a pond. Can you just tell us about that underpinning? Well, by the beginning of the 18th century, really, algebra had become the natural language of mathematics. Newton's Principia, written in its very traditional geometric style, was probably the last book of that kind.

Everyone else was beginning to use algebra as their language. And so, yes, geometry really died away until the 19th century when it was revived in different ways. I'd like to bring up something that the eminent master's, mathematician Michael Attea said that algebra is the deal with the devil. The devil says to Faust, I will give you what you want if you give me your soul. Attia says that this is the deal that the devil offers to mathematicians. He says, I'll give you this powerful machine. It will answer any question you like. All you need do is give me your soul. Give up geometry. Give up meaning. Give up asking what and why. And you will be able to solve all equations. This is where we are now. Thank you. Perfect ending. Thank you very much. Thank you all for that. Next week we'll be talking about the Sclimberus Club with Alexander Pope and John and Swift. They founded it, eminent patrons, all that sort of thing. Thank you for listening.

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