In Our Time - Calculus

Episode Date: September 24, 2009

Melvyn Bragg discusses the epic feud between Sir Isaac Newton and Gottfried Leibniz over who invented an astonishingly powerful new mathematical tool - calculus. Both claimed to have conceived it independently, but the argument soon descended into a bitter battle over priority, plagiarism and philosophy. Set against the backdrop of the Hanoverian succession to the English throne and the formation of the Royal Society, the fight pitted England against Europe, geometric notation against algebra. It was fundamental to the grounding of a mathematical system which is one of the keys to the modern world, allowing us to do everything from predicting the pressure building behind a dam to tracking the position of a space shuttle.Melvyn is joined by Simon Schaffer, Professor of History of Science at the University of Cambridge and Fellow of Darwin College; Patricia Fara, Senior Tutor at Clare College, University of Cambridge; and Jackie Stedall, Departmental Lecturer in History of Mathematics at the University of Oxford.

This BBC podcast is supported by ads outside the UK. Every Sunday, we talk about the week's tech news on this week in tech. Hi, this is Leo Leport. inviting you to join me this week with Lisa Schmeiser, Dan Patterson, and Yanko Rekkers. We're going to talk about the new 49 megabyte web page. It's the standard, you know. We'll also talk about Elon Musk. You've got some spleenin to do and the Yassify filter, new from Nvidia.

That's this week on this week in tech. You'll find it at Twitter. or wherever you get your podcasts. 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 program. Hello. Calculus is a mathematical technique created in the 17th century,

which made it possible for the first time in history to measure varying rates of change. This was an astoundingly powerful innovation, which didn't just have a profound impact on mathematics itself, but eventually enable us to do everything, from predicting the pressure building up behind a dam, to tracking the position of a space shuttle. The potential power of calculus was recognised from the start, but the question of who invented it

provoked one of the most bitter and lasting feuds in scientific history. The antagonists were an English natural philosopher, Sir Isaac Newton, and a German philosopher and political advisor Godfried Leibniz. It was a fight set against the backdrop of the Hanoverian succession to the English throne, the formation of the raw society, Spitted England against Europe and geometric notation against algebra and the nature of God.

To discuss calculus and the battle between Leibniz and Newton over who conceived it, I'm joined by Simon Schaffer, Professor of the History of Science at the University of Cambridge, Patricia Farah, senior tutor of Clare College, University of Cambridge, and Jack Ishtadol, Departmental Lecturer in the History of Mathematics at the University of Oxford. Simon Schaffer, can we start with Godfried Leibnitz, and why was he such an important figure? Leibniz is a fascinating figure and I imagine less familiar to a British audience than his great antagonist, Isaac Newton. Both of them have biscuits named after them, Fig Newton's on the one hand and Leibniz, which is a very fashionable form of snack in Germany even today.

Leibniz was born in 1646 in Leipzig, so he's what four years younger than Newton. brilliant young lawyer, diplomat, administrator. Already in his early 20s, he drafted a series of radical and remarkably far-sighted political and legal documents, including the fascinating proposal that the greatest military power in Europe at the time, that of Louis XIV, France, should be persuaded to stop crossing the Rhine and invading the German lands and instead divert all its military energies to Egypt. In order to pursue this program, Leibniz traveled to Paris in the early 1670s and thence to London.

And it was in that period when he's in his 20s and early 30s, that he began his remarkable, rapid, and in a very interesting way, autodidactic project in mathematics and the sciences. And like Newton, he was a classicist. These extraordinary men had such contribution to maths, read classics. There was no such thing as reading mathematics. That's absolutely true.

So when he turned to it, what did he go for and why did he become so very important, so very quickly? Well, let's assume he's very brilliant. That's why I became so important. But what did he go for? What took him to calculus? I think two factors that mattered most to him were in play in the 1670s. On the one hand, we have to remember what the intellectual milieu in which he was working amounted to.

Leibniz was born towards the end of one of the most devastating European wars, what we now call the 30 years war. Almost a third of the German population, it's been estimated, died during that period. So Leibniz, along with many of his contemporaries, dreamt of a new programme which would be rational, which would be harmonic, which would be based on certain knowledge, which would involve a complete overhaul of language of the modes of communication and calculation. and Leibniz's interests in what we now think of as mathematics, after all in that period, range from designing machines, which could do computation. Leibniz is one of the first, and in many ways,

one of the most interesting designers of calculating machines, all the way to an overhaul of the language of mathematics, so that, as Leibniz put it in one of his more pithy remarks, he's not noticed for Pith. One of his more pithy remarks, we will arrange things so that when persons are in dispute, they will simply say to each other, let us calculate.

So that mathematics became, on the one hand, a tool for producing agreement and also a model of what agreement would be like. The fact that this model of what agreement would be like, then flowed into one of the greatest scientific fights in history is an irony to which will return. Patricia Farah, Saracicneed is much better known to our listeners. Nevertheless, let's start from where we are

and have some modelling of him coming into this dispute. Can you start from there and tell us how he got there? Also, interestingly, this intriguing business of these men studying classics, studying Latin, studying Greek, and then turning their minds towards. something which seems to us, as it were, an almost different territory, mathematics. Well, that relationship between classics and maths went on right into the 19th century, and there was a sort of disastrous episode in the Cambridge exam system

in the middle of the 19th century when the mathematics professor suddenly decided he was going to mark all the classicists seriously in the maths exam, and suddenly all the classicists failed their degree, because their maths had been taken seriously. So that went on for many, many, many years afterwards. So you're right. Mathematics and science, as we know, it didn't really exist as a topic that you studied at university.

Newton spent most of his undergraduate years, as far as we can make out, learning about maths, about natural philosophy. On the side, on his own, he had a tutor, didn't he? He had tutors as well. Isaac Barrow was his very famous maths professor, who allegedly the Apocville story runs stood down in order for Isaac Newton to become Lucasian professor

because he recognised that Newton was far more brilliant than he was himself. Like so many of the stories associated with Newton, including the famous apple tree, it's very difficult to know whether that actually happened. Newton put the story about the apple himself. He told it a few years before he died. He told it to four separate people. Let's not be tempted by the apple.

Let's get to the mathematics. So let's go back to Newton, who claimed that he developed the calculus in 1666. By the time that Leibniz first published his own book on calculus in 1684, Newton was nothing like as well known as he is now. He hadn't even published the Principia, his great book on gravity, when Leibniz first published his own work on calculus. So Newton was a fellow of the Royal Society. He was by no means yet president.

He had not yet published his book on natural philosophy. In any case, I think it's very tempting as British people to think that there was an overnight conversion to Newtonian physics, and that certainly wasn't the case. He was slow to be recognised in England, and certainly on the continent in both France and in Germany, two particularly relevant countries in this story. Newtonian natural philosophy wasn't really accepted

and to the middle of the 18th century, and even then it wasn't Newtonian natural philosophy, as Newton himself had set it out. It was a sort of modified version. Now we're going to talk about this one dispute, but it was a time of disputes, perhaps it's still a time of disputes, but leave today out of it.

Then there were a great number of disputes, and Newton was involved in several of them himself. Can you just talk about the root of these disputes? Was it, what, you tell them? me? Well, one of the most famous disputes he was involved in before the Leibniz controversy flared up was with Robert Hook, who remained his rival throughout Hook's life. Hook, when Newton first published some of his work on optics, Hook got very angry and accused Newton of plagiarising his own work. And then, as another example, there was the famous

case when Edmund Halley went to visit Newton and said, could you please, do you, do you, you have any idea what shape the planets would move in, well what orbit the planets would move in if they were governed by a certain arithmetical law? And Newton said, oh, obviously they'd move in an ellipse. I've already proved that. So Hailey said, well, how did you prove it? And so Newton shuffled through all his papers on the desk

and said, well, I can't find the paper just now, but I'll get back to you later. And then you can imagine that he spent the next two or three days desperately sort of working out the mats. It's a real proof of it. Is there a real proof of this? I mean, you talk about the myths around you, the carbuncles of myth around.

There's stories and counter stories. You're here because you're so brilliant academics. We want the real gen. Well, that's the whole point. There isn't any real gen. There's lots of things that are written down. You have to remember that this was an era before the patent system worked effectively.

There was no copyright. There was intellectual copyright hadn't even existed. If you wanted to maintain your priority, the way you had to do it was by writing books, by publishing, by creating a big fuss. And when you get several books coming out and pamphlets coming out, and they all say different things, there is no way of being 100% certain what actually did happen. And in a way, those sort of disputes are what keep historians in business, because people can interpret the existing evidence in all sorts of different ways. and occasionally some new evidence appears.

So we now know much more about the debate than we did in the 18th century. So then we can make new interpretations. But these two men are steering towards this until then unknown, almost the area of mathematics. Or was it unknown? Was there any background to this jackister, though, before we get on to their intervention or invention in this area? Yes, there's always background. It's one of the difficulties about being a historian.

you can always go back a bit further. Yes, there were many results in the early 17th century which both Newton and Leibniz built on, both of them built on their predecessors. And the question really for Newton and which brought about his discovery or invention of calculus were questions about curves.

And curves were something that mathematicians of the time studied intensely because they, again, this goes back to Greek mathematics. They all knew the classics. They knew the classic. mathematicians. And so they were very interested in properties of curves. And two of the questions that people in the 17th century were interested in were how you find areas of curved surfaces, a shape bounded by a curve. And the other question was how you find a tangent to a curve.

And it was these two questions which many people had contributed results to. But they were piecemeal results. But Newton really brought this together in a most remarkable way. And he was still a student when he did this. He was 21, 22 years old, so it was a very remarkable piece of work. But he was in fact the first to do this and to find a general method of approaching these questions. And this is what we now call calculus. Can you give us some idea of that keeping us on board? And what he discovered at the age of 21? Well, one of the things he discovered was this method of finding a tangent. This is the easier part.

This is what we now call differential calculus. This is the easier part of it. How you find the slope of a curve at a given point, how you find its gradient. And this gives you a measure of how rapidly the curve is changing. And this is why it's so useful when you want to do the mathematics of change. You actually need differential calculus. You need to measure the slope of a curve.

Change in anything, the decay of things, the growth of things, finance. The population growth, the spread of swine flu, the movement of the stock market, all these are quantities that are changing, and so you need differential calculus to study them. But that wasn't Newton's problem. He was simply interested in the pure question of how you find a tangent to a curve.

So why was that, can you just go even further and finding a tangent to a curve? A tangent is a straight line, that hits a curve at one point. Now why was that important to this argument? It was simply an intellectual question. How do you do this? This was a puzzle.

Well, it seemed in one way to cause person, like myself, you draw a circle, I've drawn a circle, you whack a thing down, I missed the, I can't go into the circle, but if I had I been jotto, I'd have gone down the side and hit the side of it, so what is the problem now? A circle is easy because you simply draw the radius to the point you want, and then you draw a line at right angles, and that's the

tangent, but if you have a more complicated curve, like parabola, which is the shape of a fountain, or the shape what you get when you throw a ball in the air, for instance, suppose you want the tangent to that curve, which you do, if you want to know how quickly your ball or your drops of water are falling. How do you draw it exactly?

You can approximate it by eye, but to get an exact answer is very difficult. I thought that, and the problem was to know that you're hitting one point on the curve. And which point is one point? Where is the one point? How big is the one point? It's usually the other way around. You choose your one point and then you want to know exactly where you need to draw the line. Okay.

Now, so before we move on, one more thing. So they moved it forward, the two of them before. they moved the thing forward. It had been brooding around. In the background, you've got Descartes, you've got Gregory, but these two moved. And what was the substantiality of the move forward? Well, both of them found general methods of doing these things.

Before they'd been, as I say, piecemeal methods, you could find special methods for special cases. What both Newton and Leibniz managed to do was find general methods that were applicable in a wide range of circumstances. but also recognising that these are what we now call inverse processes, finding the tangent and finding the area, which geometrically don't seem to be related in any way at all,

but they are. They're almost opposites of each other. We now call this tangent process differentiating and finding an area integrating these weren't 17th century terms. But what mathematicians now know what Newton and Leibniz recognized is that if you differentiate and then you integrate, you get back to where you started. And this is an amazing result.

It is an amazing story because they're doing these abstract things, these classical scholars, and within two minutes we know how to measure the acceleration of a car. And the pressure on a dam and the decay of potties. We do. They didn't get that fast. They pointed us very clearly in that direction. I mean, they were going there. I think it's very significant also that although mathematically what they were doing is equivalent

and we now see them as being the same, at the time Newton and Leibniz were approaching what seems to be the same problem. They were approaching it mathematically in slightly different ways. And so part of the nub of this whole debate is were they really doing the same thing? Because in a sense they were and in a sense they weren't. And that's an added complication and a very, very important one to the whole debate. And that's what we're going to talk about in a moment. So we've gone on that.

But one further step before you go into that. Simon Schaffer, has Newton's great discovery of, gravity germane to this? I think it is. As Patricia and the rest of us have been saying, Newton acted at a distance and

attracted many legends to himself. But there's an obvious relationship between the mathematical principles of natural philosophy, as he understood and developed them, and the kinds of problems that press on his

work in analysis, in calculus. The most obvious relationship is that the fundamental question that Newton is going to begin addressing in the middle of the 1680s, when he begins to write his great work on the motion of bodies, is how do we define the motion of a body at an instant? So that that's exactly a tangent problem. Similarly, he wants to know whether there's an effective way mathematically of representing the time, for example, through which a planet has been orbiting. Now, Newton knew from his greatest predecessor, Johannes Kepler, that the planets move so that the radius that links the planet to the sun sweeps out equal areas in equal times.

And he also knew from Kepler that planetary orbits are ellipses. It therefore became vital for Newton to solve what was called the Kepler problem, which is work out a given area contained by a section of an ellipse. Now these are problems of, as we would now see in retrospect, differentiation and integration. As Jackie has said, those are not terms Newton uses. Furthermore, and this would later become very important for the calculus dispute, when one opens the masterwork, the Principia of 1687, there is no calculus there. The methods that Newton actually uses to analyze planetary and terrestrial motion in the Principia

are the methods we see on the face of the page, what he called the method of first and last ratios, which is in fascinating ways equivalent to some of the theorems of the calculus, and indeed Newton wrote some sentences into the printed version of the Principia, hinting tantalizingly about the work that he'd already done on calculus but had not yet published. But the methods of the Principia are not the methods that Leibniz would later seize upon. Okay, it brings us back to what you were saying, Patricia Farah,

about Leibniz, is he basically working on the same thing as Newton, which you started to say, and I wanted to go back to Principia, but now we're back on track with what you were saying. Well, Leibniz was interested in, so if you think of the area under a curve, like the arch of a cathedral, for example, the way that Leibniz approached this problem was to divide it up into... What's it trying to, what's it trying to prove? What is the problem that he's trying to solve?

He's trying to find areas and sums, so he divides an area into infinitesimally thin little slices and then adds them all up together. Whereas what Newton is interested in is rates of change and movement. And so he's looking more at the curve, and Leibniz is interested in adding up these infinitesimally small slices. And Newton sort of fudged the argument rather cleverly and said, this is absolutely ridiculous, infinitesimally small things just don't exist.

And he's sort of glossed over the fact that, in a sense, his own arguments do rely on that. But he managed to conceal that. So Newton is looking, Alibniz is looking at little chunks, little slice, tiny, tiny slices and adding them together. And Newton is more interested in rates of change along the curve and the tangents. Do you like to come in on that, Jackie? Yes, it's true. I mean, Newton had a very, very strong geometric intuition and a strong intuition about movement as well.

That's correct. Both of them fundamentally are using these what you might call infinitesimally thin slices. Newton kind of jumps in at a further level where that's already hidden away, whereas Leibitz is more explicit about it. One of Newton's other great contributions to this debate, and one that I think is overlooked quite a lot, is at the same time as developing his calculus,

he developed a very sophisticated method of finding infinite series, and this is really tied up with his calculus very closely. And this was what Newton was most proud about. Infinite series being... Infinite series being a way of expressing something, mathematically by an infinite sum. So, for instance, the sign of an angle, sign X, which when you first learn about it at school,

you draw lots of right-angle triangles, and you measure the opposite side and the hypotenuse, and you divide the opposite by the hypotenuse, give you the sign. That's fine, but it's quite limited. What Newton found was that you can write sign X equal to X, minus X cubed over six, plus X to the fifth, over 120, plus a whole lot of other terms, infinitely many terms.

This gives you a quite different way of thinking about a sign. It's much, much more powerful. And for Newton, this was his way into the integral calculus. Again, it's not his word. But it's a quite different approach from Leibnitzer's. And I completely agree with Patricia, it's easy for us to look back and say, oh, yes, they were both doing the same thing.

Actually, their approaches are completely different. They look different. Their notation is different. Their conceptual framework is different. Let's talk a bit more about the differences, but if they're so different, why do they have such a blazing row which has gone on for 300 years? We've got to address that too. But on the differences, Simon, at the core difference, you've said there's Newton's focus on time and Leibniz's on changing space. Yes, I think that's right.

So, can you... So, Newton's vocabulary is eloquent here, and it's not the vocabulary that we now use. Newton thinks of time as flux. That's his word. Time for Newton is the absolute measure against which all events in the world take place. Newton is a man obsessed by God and his relationship with God

and in several occasions his identification with God. And God has created a world dominated by temporal change. So time provides the absolute standard against which all events are judged. Call that flux. Now think of quantities varying in time. They flow. Call them fluent.

That's his word. Now ask how fast are these quantities, these fluents, changing. Newton invents a word, it didn't exist before him. Fluxion, which is his word for what we now might call differential. So the entire approach is based on the supposition that we're fascinated by variation in time, by rates of variation in time, and by mathematical techniques which will allow us to capture geometrically, as Jackie has reminded us, how quantities vary in time. Leibniz now. Leibniz is a master of series expansions and of infinite series.

But... Jackie, I must report, is shaking her hand quite vigorously. But what he does with those series expansions is something completely different, which is to try and see if they can be used to solve problems in the geometry of space. So that, for example, one of Leibniz's first priority disputes, one which he lost hands down in a matter of days during his first visit to London, was a claim that a certain series expansion allows you to estimate, with ever greater precision the exact area inside a circle,

in other words, the value of pi. So that for Leibniz, it's not obvious that time is the universal measure of all things. Rather, one's trying to compare how one quantity changes with respect to another quantity, and then to capture, again as Jackie has pointed out to us, how that can be represented in terms of the superposition of areas. So a quick and dirty way of summarising the difference would be a metaphysics of time of temporal change

in the Newtonian worldview and the organisation of spaces and the relationship between spaces, fascinatingly, in the Leibnizian worldview. Jackie, would you like to articulate your head-shaking? I do agree, though, with your summary there. I think Leibniz did you a series, but he wasn't the master that Newton was.

I mean, this was where Newton overtook him by a million miles. I mean, Newton was just way ahead of him in all that. And I think this was one of the causes of the dispute later that Newton knew that he was very far ahead of Magnetts in the other days. Patricia, Patricia, Patricia Farrah, there's also an argument about the language that is being used to write down the workings of natural philosophy. On the whole, Leibniz and Europeans are promoting algebra,

and Newton, on the whole, is resisting and sticking to geometry and coming out of the classics and Greek. Now, can you tell us how that played into the differences of the argument? Newton was always a very, very strong advocate of geometrical methods, and this was another difference between them, is that for Newton, mathematics, including fluxions, had to be rooted in what was actually out there in the world, what you could see and what you could measure.

whereas for Leibniz, all this talk, discussion about infinitesimal small quantities, which don't really exist in some way, is an indication that for him calculus was a tool, a method for manipulating it was symbolic. And this was a sort of difference that remained throughout the 18th century in very, very broad-brushed terms between English mathematicians and continental mathematicians, that roughly speaking through the 18th century, English people followed Newton and stayed with geometry, Whereas on the continent, there was a very, very talented Russian mathematician called Euler,

who developed Leibniz's calculus. He even, big insult, translated Newton's Principia into Leibnizian and Euler's calculus. And so what happened on the continent during the 18th century is Leibniz calculus became further developed by Euler. And then by the end of the 18th century, French mathematicians, were making huge advances by using this very algebraic, what they called analysis, an analytic method of approach, which was repudiated in England. And it wasn't until the early 19th century

when there was a movement amongst Cambridge mathematicians amongst young students to import this French algebraic methods into English physics. So Jackie Stello, before we move into the nub of the dispute itself, I'd just like to go back to the notion of infinitesimal and infinity and these things heaving onto the horizon and trying to deal with that. I'm sorry about that, but I think it would bear a little more explanation. It's a difficult one.

The idea that you could treat what people at the time called infinitely small quantities or infinitely large quantities was really a new idea in the 17th century. It doesn't come into Greek mathematics. This was where European mathematics finally began to break away from Greek mathematics into something quite different. The idea that you could measure things using these very, very tiny quantities

which, as Patricia says, might or might not exist and you don't have to worry too much about it. As an example for the listeners may be, imagine filling a circle with a polygon. And if you give your polygon more and more sides, it becomes closer and closer to the circle. If you imagine it having infinitely many sides, then it becomes the circle. But this is a big step to move from a large number of sides

to somehow saying it has infinitely many sides. It's something quite different. It's something the Greeks would never have done, but which 17th century mathematicians were prepared to do. So where did that lead them? Well, it leads into the calculus. This is what the calculus is founded on, this manipulation of these very small quantities in the limit

as they approach nothing. It's entirely an imaginative in its origins then? Yes, yes, it is. So we're talking about a work of the imagination? We are really, yes, yes. And there were great debates, even in the 17th century and certainly later as to whether these infinitely small quantities are something or nothing, and either way you get a paradox.

And people knew this. Both Leibniz and Newton knew these difficulties and tried in their own ways to overcome them, but at the same time they weren't going to be held back by them because the calculus was so powerful in what it could be. do, that no one was going to give it up because there were a few arguments down at the foundations. So we come to the argument. Bated breath has

been held throughout the contrary. We come to the argument. And there's the argument about priority, there's the argument about plagiarism, and there's an argument about philosophy. Okay, priority. Simon. To oversimplify, this is a period in which the criteria

that you have to satisfy in order to establish that you're first are not at all clear. This is not a world in which you are the first and only begetter if you're the first to publish. That may be true, but you'd have to show that. So the conventions of the fight are being established during the course of the fight. Leibniz was clearly the first to print the kinds of methods that we've been

discussing in the 1670s and 1680s. To such an extent that by the end of the 1680s, when Leibniz had become already probably the single most famous philosopher in Europe employed at the court of Hanover, he'd published these methods and others were learning from arguing with and debating with him in public. So that seemed a game set and match? Only if you already accept that first publication is priority. That was exactly what was in dispute. Not only that, but as we've mentioned in the early 1670s, Leibniz had been in London on one of his magnificently crazy diplomatic missions. He'd met members of the Royal Society and what we might now call maths journalists like the amusing gossipy math journalist John Collins,

who may, or according to some authorities may not, have part. to the young Leibniz documents and summaries of what Newton had already done in the 1660s. So the issue of priority there takes us immediately to the second cause of war, which is plagiarism. Just a second. Had Newton, in your opinion,

scanning across three, had Newton said stuff in his writings, which he'd kept secret because he became, very hurt by any disputation he didn't like to be criticised. So with the hook thing, he'd pulled back, I'm not going to publish, I don't again involved in this. For one psychological and scientific reason and another, but had he written stuff down,

which predated the publication of Leibniz on the calculus? Absolutely. So that's done. The masterpieces that Jackie so well summarized for us, which are produced from the middle of the 16th, 1660s through into the 1670s are such extraordinary, it seems to me, technical and intellectual revolution in mathematics in general and the calculus in particular, that it's right to say that Newton had made major advances before Leibniz, in a certain sense, had started work on mathematics at all. So let's take the plagiarism argument to you, Patricia. What evidence is there for Leibniz, really Leibniz plagiarising Newton?

And then, of course, there are later cries from Leibniz's camp that Newton plagiarised Leibniz. As in your example of Halle are going to see him about the lipses and so on. The evidence depends on how you want to look at it. I mean, one important thing was that in England, everybody seemed to believe Newton almost automatically. Newton claimed that he'd written letters to Leibniz, and Leibniz said, oh, no, that wasn't important.

But as far as the debate panned out over here, everybody believed Newton, and there was an official inquiry at the Royal Society in 1714, where Newton effectively wrote the report that the committee was due to publish. So Newton was very much in control of everything that was happening over here in London. It's also very significant, I think, that Leibniz, in the early 18th century, was desperately trying to get into the Hanoverian court.

I think it's very interesting to compare his fate with Handel. Handler and Leibniz had both been working for the Hanoverian court in Germany. And then when the Hanoverians came over and were on the British throne, Handel was very, very diplomatic and managed to get over here and he's now buried in Westminster Abbey. whereas Leibniz was completely squeezed out and I think a lot of that was due to Newton so conniving against him to keep him away. So Jackie Stollett's at the end of the 17th century that the dispute really takes fire in the beginning of the 18th century as Patricia has indicated.

Can you give us a starting point? We're beginning to run out of times, I've mismanaged this. Anyway, can you give us a starting point so that when did it kick in, the real dispute which did become verbally violent, it did involve courts and so on, so far, which I hope we'll get to. As you say, it didn't really blow up

until the early 1700s, and one of the very remarkable things about this dispute is why it took so long to get going. Leibniz had published in 1684. Why didn't Newton complain then? Why didn't someone else complain? Nothing happened really until the early 1700s when Newton finally did publish.

Leibniz reviewed his book anonymously, as he often did, in his journal, but made rather subtle implications that Newton was really only doing what Leibniz had already done. He didn't openly accuse him of plagiarie, but you could read that message into the subtext. There was a response not from Newton,

but from one of Newton's followers, John Keel, about three years later. All these things take a long time. You know, circulation of journals is slow. People are slow to react. Let us take a long time to get around and so on. So we're now in 1708. Leibniz complains to the Royal Society. that takes another couple of years.

That takes till about 1711. And it's not till 1712 that the Royal Society sets up this committee to look into the dispute. Leibniz appeals to the Royal Society, which he too is a member to judge on this dispute. And as Patricia said, the Royal Society does judge, but essentially it's Newton, writing the report. The other members of the committee don't even sign it.

And at this stage, the dispute really... Newton declares for Newton. Newton. Newton declares for Newton. Totally. And, of course, in one sense, he was right. He was first. If that's all you're looking at,

he... Anyway, this is where the dispute then becomes public. It becomes public property, and everybody can join in. Up till then, it's been rumbling along between Newton and Leibniz. It's one or two followers. Simon.

I think it's also very important to bring out at least two other aspects of why the disputes are so violent, why they're so public. And as Jackie says, why do they happen when they do? On the one hand, there's a great philosophical fight going on. So that John Keel launches the most violent period of the dispute

in the name of a defence of Newton's philosophy against Leibniz. And there the issue is, I think, very closely related to the innermost problems of calculus, since what Kyle is saying is that Newtonian philosophy relies more than anything else on the idea that matter is passive and inert, that it acts instantly between its centres through a force of gravity which has been produced in the world and is still produced in the world by an all-powerful god. Against this, Leibniz, who revivifies and essentially invents the term dynamics to describe his version of the sciences,

argues that no, it's not that God is very powerful, it's that God is really smart, that this is not a world of power but a world of reason, and that the essence of matter is force, and that there will be a science called dynamics, which will describe that force. So two very, very different models of the deity, of creation, of what forces

and how science can describe those forces, And since, which takes us to the other reason why the fight happens when it does, since God provides humans with the ideal model of how to rule, this is always already also a fight about politics. Here are two very different models of good government. So it's no coincidence that the fight begins, more or less at the moment, that it becomes completely obvious that the next king of Britain will be Leibniz's boss. the elector of Hanover, and it comes into it.

And this dispute about the calculus goes into the idea of whether God is an interventionist God or someone who's just set it in motion, and then it goes into politics as well. I mean, one of Leibniz's main criticisms of Newton was that he had, he invented

or imagined a God who was a very, very sloppy watchmaker. Leibniz said if God is infinitely powerful, surely he could have wound up the universe at the beginning so that it would work perfectly. Whereas Newton's God was one who intervened, who affected the universe on a constant basis. And that was one of Leibniz's main critiques that Newton had invented this sloppy watchmaker of a god.

They were both deeply religious men. They had different views of how it works. Newton's philosophy is more like the political structure of Britain, where you have a ruler, who you can make equivalent to the sun, if you like, and independent entities which are subservient, which are attracted by the sun, but also have independence and can act amongst themselves. Whereas Leibniz's rather opaque philosophy of monads is politically more like the German system. Germany at that stage was not one single unified country.

It was lots of individual states with very, very powerful princes, and you've got these groups clustered around the princes. There's a sense in which they're political, and their philosophical systems are very equivalent. Jackie Stelot, does this dispute harm the progress of mathematics? I mean, please respond as you wish to, but then if you could answer my question, we'll just about cover the territory.

Well, I only want to say that I agree with what both Patricia and Simon have said, but it makes it sound like a very high-level philosophical debate. It was also a personal debate. I mean, these men were personally angry. Newton was angry. Leibniz was angry. Both of them told lies. Both of them probably regretted what they had or hadn't done in the past.

I mean, there are two human beings fighting here through their acolytes sometime. But, you know, it gets nasty in the way that human... And the language is nasty. The language is nasty. It's called Newton's people apes, didn't he? It's terrible. It's really bad. But it is very personal as well.

So it isn't just a philosophical debate. So did this hold up the progress of mathematics? Well, because Leibnitz's calculus was so far forward, that was the one that won over really and that we now use. We now use Leibniz's language, Leibniz's notation, and Newton's felt. behind, except in Britain, as Patricia said, where it was used throughout the 18th century. No, it probably didn't hold up mathematics.

These methods were out in the open and people did take them up. It's a pity that Newton didn't publish earlier. It would have changed things earlier, but eventually it did all come out. Well, thank you all very much. Thanks, I'm Shaffa, Patricia Farah, and Jackie Stadol. Next week I'll be discussing the Radical Pharaoh Akanaten and his wife, Nefertiti. Thank you for listening. That's it.

If you've enjoyed this Radio 4 podcast, why not try others, such as Thinking Aloud, where Laurie Taylor discusses the latest social science research. To find out more, visit bbc.co.com.uk forward slash radio four.