In Our Time - Archimedes

Episode Date: January 25, 2007

Melvyn Bragg and guests discuss the Greek mathematician Archimedes. Reputed to have shouted “Eureka!” as he leapt from his bath having discovered the principles of floating bodies. Whatever the truth of the myths surrounding the man, he was certainly one of the world’s great mathematicians. The practical application of his work in pulleys and levers created formidable weapons such as catapults and ship tilting systems, allowing his home city in Sicily to defend itself against the Romans. “Give me a place to stand and I will move the earth”, he declared.But despite these triumphs, his true love remained maths for maths sake. Plutarch writes: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life.” His most important breakthroughs came in the field of geometry with his work on the areas and volumes of curved objects.So how did this Greek mathematician in the third century BC arrive at a calculation of Pi? Did he really create a Death Ray to fight off invading ships? And what does a recently discovered manuscript reveal about his methods?With Jackie Stedall, Junior Research Fellow in the History of Mathematics at Queen's College, Oxford; Serafina Cuomo, Reader in the History of Science at Imperial College London; George Phillips, Honorary Reader in Mathematics at St Andrews University

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 4. I hope you enjoy the programme. Hello, today it's Archimedes. The Greek mathematician reputed to have shouted Eureka as he leapt from his bath having discovered the principles of floating bodies.

Whatever the truth of the myths surrounding the man, he was certainly one of the world's greatest mathematicians. The practical application of his work in pulleys and levers created formidable weapons such as catapults and ship-tilting systems, allowing his home city in Sicily to defend itself against the Romans. Give me a place to stand and I will move the earth, he declared. But despite these triumphs, his true love remained for maths for maths's sake. Plutarch wrote,

He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life. His most important breakthrough came in the field of geometry, with his work on the areas and volumes of curved objects. So how did this Greek mathematician in the 3rd century BC arrive at a calculation of pi? Did he really create a death ray to fight off invading ships and what does a recently discovered manuscript reveal about his methods?

Joining me to discuss Archimedes is Jackie Stadol, junior research fellow in the history of mathematics at Queen's College Oxford, George Phillips, honored reader in mathematics at St. Andrews University, and Serafina Cuomo, reader in the history of science at Imperial College London. Jack Estella, what do we know about Archimedes' early life, the dates and so on? Well, we know almost nothing at all.

We know that he lived in the 3rd century BC. We know that he lived in Syracuse, in Sicily, and we know when he died, and really that's about all we know about him. We do know that his father was an astronomer called Phidius. Archimedes tells us that himself in the preface to one of his treatises, so we know a little bit about his father, but nothing else about his family. or his birth or upbringing.

What would we know about that city, the Greek city in Sicily, so near, obviously so near Italy, but it's a Greek city with a Greek culture there. It's a Greek city with a Greek culture. I mean, we talk about the Greeks, but Archimedes actually lived in Syracuse in Sicily, and he was part of a Greek culture that by that time had spread around the Mediterranean. So when we talk about the Greeks, we're talking about Greek speaking or Greek writing people who might live anywhere. around the edge of the Mediterranean,

and there's no evidence that Archimedes ever set foot in Greece itself. He's thought to have visited Alexandria in Egypt and lived in Syracuse, but not Greece. But he would have written in Greek. And his education would have been in Greek?

I don't know. I don't know about that. But we can make a far speculation that being a Greek-speaking person who wrote in Greek, he may well have been educated as something like the Greek Academies which had been in Athens.

Possibly. But we don't know. We don't know. So do you have any idea what influences, though, there might have been on him? I'm not committing it to fact, Jackie, but the sort of the merest speculation. Well, he obviously knew some mathematics. I mean, his father might have educated him to some extent, and then whatever other education he had.

He was a little bit later than Euclid. May have lived even at the same time as Euclid, but we don't know whether he read Euclid himself. He certainly knew some of the propositions that do appear. in Euclid, but whether he read them there or somewhere else, we don't know. So he would have known that kind of Euclidean geometry. That would have been familiar to him. So he was born, as far as an astronomer, we know he lived in a Greek city state in Syracuse.

We know that he died in battle, or he was killed in a battle. Yes, he was killed at the age of 75. Well, the age of 75 comes from much later reports that said he lived until 75, so that's not really. reliable, it may be true, but we were pretty certain that he died in 212 BC during the siege of Syracuse and there are lots of stories about that. See, the thing that's interesting is that

I'm, one of the things, a few things I did know about argument is before I got to grips with this week ago, what now seemed to be a complete encrustation of myths. Yes, yes. The jumping out of the bathroom, running down the street, Karen Eureka, and so and so forth. It's almost as if he had become a

sacred person and you had to have myths to sustain his specialness. I think that's absolutely right. The trouble with these big figures is that they become mythological. And people like Pythagoras or Euclid or Archimedes attract stories around themselves, which have maybe some basis, in fact, but we can't really know. And it also happens that later treatises are attributed to these people

because who else would you attribute it to? You think of a big name and you say he must have written it. So we don't always know that things that are said to be written by Archimedes really were written by him. Well, let's go through this veiled, and let's continue a little with Seraphina Cuomo through this veiled subject of Archimedes who did exist and was born,

and Jackie is going to give us that, that's okay. But it thought he might have gone to study in Alexandria. We're certain, there seems to be records of correspondence with people in Alexandria. They're great museum, university, library combination at the time which is sort of taken over from Athens. Can you tell us about the significance of the contact he had, stroke may have had with Alexandra? Yeah, I think it's almost certain that he went to Alexandria.

And we know for sure that he was in correspondence with people who lived in Alexandria because he tells us in the introductions to some of his works. For instance, one of the people he writes his works for is the chief librarian of the library of Alexandria who was eratosthenes. So there is a direct contact there. Alexandra at the time was the cultural capital of the Mediterranean world. Athens was really in decline at that time. So if he wanted to go somewhere where he would have had the best education

and got in touch with the most famous intellectuals, stroke scientists of the time, Alexander was the place to go. There's been a revision of the idea of Alexandria, and the Egyptians are thought to play in a much bigger part. Egyptian intellectuals, and let's use that word just to keep, and it was previously assumed. Can you tell us about that? Because it's much more mixed place than the great center of Greek learning.

I think the idea of any state or country in the Mediterranean world in antiquity has been monocultural is definitely gone. So I think the recognition that Alexandria was a multicultural capital has finally come. For a long time, people were keen to say there were the Greeks on the one side and there were the Egyptians on the other side and they didn't really mix. But the evidence tells us otherwise. So I think it's fair to say that the knowledge that he got imbibed with in Alexandria was Greco-Egyptian. It was a mixture of the tomb. and people from the two groups, if we want to call them, that mixed because we have evidence that they did.

And what sort of, can you give us one or two instances of what he might have learned in Alexandria or from Alexander, whether he went or it was through letters, whichever? I think, and this is really speculation, that two main areas that he would have been exposed to in Alexandria that may have been of interest to him were mathematics and catapult construction. because we know that Alexander was a huge center for both of them. Euclid, whether it was an older contemporary or a contemporary of Archimedes, is thought to have operated in Alexandria. So it would have been in touch with mathematical knowledge being practiced,

collected, reflected upon. And that's for catapult construction. We know from another third century treatise that the kings of Alexander, Alexandria were particularly keen on that and put money into research. So there would have been one place where engineers from all around the Mediterranean would converge to learn new techniques. So from the beginning you have very practical application of mathematics and inventions generally driven by war and the abstract theories.

Can you what briefly, what sources do we have for his life? Clearly not good enough to convince Jackie that he had anything of life at all, but never mind what sources do we have that the rest of the story? can sort of peg along with. Apart from his own works, we have three historians. The closest to him in time is Polybius, who's a later 3rd century BC, who's a Greek historian.

Then we have Levy, who's a late 1st century BC, who's a Roman historian. And then later on Plutarch, responsible for the image of Archimedes as the genius only interested in pure maths, and this is a Greek from the late 1st century AD. I would say those are the three main ones.

And briefly, would you say they were reliable? Polybius, yes. I like Polybius. He was closer in time to Archimedes. And he's interested in Greek-Roman relationships. That's where Archimedes' history fits in. Livy, yes, why not? Plutarch, I think it was more of a philosopher than a historian.

George Phillips, we mentioned already. He's picking up or being instructed in, perhaps, through Alexandria, or in Alexandria, weapons of war. And he was called on, as we understand it, to help in the defense of his home city of Syracuse when the Romans attacked it. What kind of inventions came up from him in the war context? Well, I'm not sure, again,

whether Archimedes invented all the things that are attributed to him, but there are three described by Plutarch and others. And one is the catapult already. mentioned, which was said to have had an adjustable range. And this was used

chiefly against I understand the Roman naval forces who were besieging the city-state of Syracuse. Is it a catapult as we understand it? Is it a different... How can you describe this cataport?

Because most people will think a Y stick with a bit of elastic and a pouch and there's pebble in it and whack. It's not that sort of... I frankly don't know. Right. Well, that's fair enough.

But I have been imagining, and perhaps Serafina might know more about this. I've been imagining it might be something on the lines of a giant crossbow or something of that kind. But anyway, that was one. Another thing was he had long poles that he levered out over the walls to drop giant boulders on the ships, presumably ships that were too close to use the catapult on. And then there were... Now the third thing is... Seems to me perhaps rather more fanciful.

It is his cranes that leaned out over the city walls towards the invading Roman fleet and grabbed down... And the account goes on to say, picked ships up out of the sea and shook them around. like a terrier shaking a rat until the sailors fell out. And I think this is perhaps getting towards what I used to read in my early days in the Eagle comic.

Now, I should say that he was, as you said in your introduction, he was a master of various tools of the trade like the lever and pulleys. Systems of pulleys, one remembers at school seeing a system of six. poolies where the rope goes around them in a clever way and you pull one end and with only about one sixth of the effort can pull something six times as heavy as the effort applied. Now he was a master of that one understands. Also the lever. He was particularly addicted, one might say, to the lever. And he even used the lever in a theoretical way in his mathematics. The lever remember is it's like a bar

pivoted and a long arm on one side, short arm on the other side apply the pressure of the force on the long arm and with again rather like the pooly thing you get what the physicists call a mechanical advantage with a smaller force

you can move a larger weight and the simplest example I know of this is the child seesaw where we're all familiar those of us who have children and so on of putting the little child on the end of the seesaw and the adult can sit on the other side of the point of balance but nearer the point of balance.

So effectively the weight of the small child is raising a larger weight. So these things he was sturdily acquainted with. There's also a story of him using lenses against invading ships, a kind of death ray, lenses which caught the sun and reflected on the ships and burnt them. Is this fanciful? I rather think so. I rather think so.

But nevertheless, despite all this fancifulness, now we've got the life out the way, there wasn't a great deal to say about it, but such as there was we've done. He left a great deal of mathematical papers for the time. I understand about eight or nine, Jackie. Just to finish the devices, though, can we talk? So there's a lot to talk about in his work.

The works there, he was the man that we think he is in the intellectual development of the West. He was the great mathematician. we think he's just to restore our listeners' faith in this person. And he was people like Galileo and Leibniz and Newton look back to him and so and so forth. So there he still is. Can you tell us about one more invention while we're in the invention before we turn to the mathematics, Jackie Stedle? The Archimedes screw, how useful the device was that, and if you can tell people how it worked.

Yes, the device that's called the Archimedean screw consists of something like a large corkscrew enclosed in a cylindrical casing. and you lower the bottom of it into water, and then by turning the screw, the water starts to rise up the corks screw, rather like the principle of taking a cork out of a wine bottle. The cork rises up the screw as you turn. And it's that principle for drawing water out of a well or a reservoir or a ditch or something and getting it up and out onto your fields.

So that would be the purpose and the use of this. Massively useful for irrigation, especially in military countries. Yes, and the story is that he invented this in Egypt, of course, but I don't know whether that's true. Of course. Disappointed if you did. You've established yourself. Jackie the doubter.

Quite right. Seraphina Quirma, one of the best known stories since discredited on this program is that he jumped out of his bath shouting eureka after it had come to him that he could work out volume in water through floating bodies. Now, can you explain? Let's forget about the eureka and running there. it down the street. It's a nice story, but maybe it's an memoir rather than truth. I don't know

I like to think it's true, but then we're all of our little fantasies. What was he doing there and how original was this discovery? I like to think the story is true as well. What was he doing there? It's worse for you than for me. You come to think that. The story goes that he was there to be a detective for his king, really.

It was interested in the problem because the king of Syracuse had had a crown made for him by an artisan out of gold. When the crown was ready, he thought that the artisan had kept some of the gold and put silver into the mixture. So we wanted to find out whether that was true or not. It is thought, and this comes from a description in Vitruvius so much later, it is thought that Archimedes may have measured the quantity of water displaced by masses of gold and silver respectively and then measured the quantity of water displaced by the crown and detected whether the crown was just gold or a mixture of silver and gold.

But the actual procedure is still a matter for dispute. How important was this discovery, George, and had it been, were there any precedent for it? Did he arrive at this, as it were, through his own thought, or was he picking up on things that were in the air in Alexandria or wherever else? I really don't know what the background was. I really don't know. I mean, the details, as Serfian said, are doubtful. But the interesting thing is how he applied his mind to practice. practical problems, it seems to be. He was really a servant of, in a sense, of the king of the city-state of Syracuse,

King Hiero. And some say he was indeed a relative of his. And the king was very impressed with all Archimedes did. And I think Archimedes plugged this and he made sure the king knew about it. a colleague of mine said it's almost as if he was applying for a research grant it's a dreadful thought

but he was certainly aware of the notion of density I one assumes of course to the idea that he could immerse one theory is that

how he how he solved here his problem was that he immersed the crown in a full vessel of water, perhaps with a tray underneath to catch the water displaced. And so

he could then find the amount of water displaced, therefore the volume of the crown, and knowing the volume and weighing it, he would know the density. And since it would be less, presumably I call it the story, than the density of gold,

then it couldn't be all gold. Jackie Settle, why was this so important? Why is it proved to be so important this work on floating bodies? Well, the principles that Archimedes discovered are still correct principles and still taught in schools today indeed. One of the things, as George described,

is that this principle of displacement of water enables you to calculate the volume of something and therefore it's density. And the other is that you have to know, if you want to float ships, you need to know something about what weight the ship can be, what shape is, It can be what displacement.

It's the displacement of water that the ship makes that gives it the upthrust and allows it to float. So these calculations are the kinds of calculations that he was interested in. But ships had been floating before he made these calculations. Well, people would know from experience which boats floated and which didn't. But if you want to do calculations,

you need the principles that Archimedes was working on. Seraphne. I wanted to mention that Galileo Galilei was inspired. by the episode of the Crown and Archimedes to write a short treatise on hydrodynamics, performing pretty much the same experiment, but in a different way. So what he did with the Crown

was one of the inspirations, if you like, for part of the so-called scientific revolution. Right, 16, 1700 years later. Why are still doing such experiments in school and presumably they still are taught in schools? Can you tell us how we approach the problem of calculating pie and why it was such a tantalising objective. Yes, this was one of the big problems

that Greek mathematicians were interested in the calculation of pi or, to put it another way, finding the area of a circle which involves the calculation. Now here we're talking about, sorry, to interrupt because I really want you to go on, I just want to make a distinction. Here we're talking much more about pure mathematics. Yes, we are. When you're getting catapults that George has talked about

or weighing whether the crown is gold or silver, it's great mathematics, but it's practical. This is just an interesting problem. This is what we would now call pure mathematics. It's a purely mathematical problem. And the problem is, we would say nowadays, to find the area of the circle, and as I say, this involves pi,

and this is a very difficult thing to do. And in fact, finding an exact value of pi, we now know is impossible, so it's not surprising that they had to try so hard to find it and that it was so difficult. Archimedes' approach to this was to take a circle and inscribe it with a polygon, an eight-sided figure, for example. And then by gradually increasing the number of sides, you get closer and closer to the circle.

And so he could calculate, and you can also fit a polygon outside your circle, which gives you an upper bound and, again, keep dividing the number of sides and get closer and closer. And doing this, he found a value of pi that's between three and one-seven, and 3 and 1070 firsts, which is very close. 3 and 1 7th is 22 over 7, which is the approximation that we still use, or still did use before calculators in the 20th century. It's very close, and it's actually the closest approximation

you can get in reasonably small numbers. So he worked out there if you put it, he had a 96-sided... He had a 96-sided polygon. Inside and outside the circle, and could measure these straight lines. You can measure the straight lines very easily. or at least you can calculate the straight lines he wasn't interested in measuring

when he was interested in an exact calculation but he could do that. Sorry George, John, come in. Just to add to the story he began one understands with triangles which of course give very, very crude approximations triangle inside the circle and triangle

touching outside the circle and then he was able he had a formula for calculating the perimeter of a polygon with double the number of sides of the previous one. So he went from 3 to 6 to 12 to 24 to 48, thank you God, to 96 that you mentioned. And so that's how he did it.

And he had terrific control over his arithmetic. He was a wizard of arithmetic. So that as Jackie says, he always kept sure bounds. that his number was not below this or above that. He knew it was always within these bounds. And even beginning with the triangle,

he had to have an accurate number to start with. So he found very good approximations to the square root of three. And remember he's doing all this with a relatively crude method of writing down numbers. He doesn't have our nice decimal number system. He has an alphabetic system for writing numbers.

So alpha is one, beta is two, gamma is three and so on. It's even less useful in a way than the Roman number system. They're actually doing calculations in that system is quite remarkable. So if you had been put to any practical use, would it have been recognised, was it accepted into mathematical law from then on? I would say yes.

But as for the actual applications of that, as you said, there were floating ships before he discovered hydrodynamics. and people did things before it discovered pie. So I think it's a matter more of educated architects maybe wanting to do their things more precisely or educated engineers wanting to do their things more precisely and possibly using this. We have evidence from a commentary on Archimedes by someone who wrote in the 6th century AD so much later, saying that what he did with the circle, was known to be useful for all purposes in life.

And that's as good as it gets in terms of evidence. Can we talk about calculating the area of a circle, Jackie Sutterall? We're still in pure mathematics, aren't we? That drew him to that. So the idea of him, which would describe you, one of the authorities that you don't think a great deal of, is being... Because you have these mathematical papers from him,

so we have the real stuff, the centre stuff. So we must keep firm on that. That has come down to us almost. miraculously through Arab translations and so and so forth. So these are intact and they are there. So tell us about what interested in calculating the area of a circle

and why it was so difficult. Well, why it's so difficult, as I've said, is that we can't find an exact value for pi. What he was trying to do is something that's impossible and we now know that. I'm using the word pi here. Pi is a much later description of this

number. It's the ratio of the diameter of a circle to the the circumference to the diameter, sorry. But what he was able to do was find an equivalent area to the circle. He was able to find a triangle that is exactly equivalent to the circle, and this is a very interesting construction. How do you do that? Well, if you imagine a bicycle wheel, say, with paint on its rim,

and you roll it along the ground for exactly one turn, so you have a line on the ground that's exactly the circumference of the bicycle wheel. and then you are to erect a post at one end of the line, which is the radius of the wheel, so it would be the distance from the bottom of the wheel to the hub. And then you join the top of your post to the other end of the line. I'm drawing this on the table. I hope the listeners can understand it.

The area of that triangle is exactly the same as the area of the circle. Exactly. He's calculated, though? You can't calculate it, but geometrically you can prove that these are the same area. And that's what he did? And that's what he did. and it's a very interesting comparison and it's a very interesting proof that he gives

that these are identical. And this isn't to do with calculation because you can't calculate either of them exactly, but you can prove that they're exactly equal. So if it's not to do with calculation and they didn't have bicycles, so he did it with something else, it is to do with, is it,

I'll be talking about pure thought, abstract thought. Yes, pure abstract thought. And the way he proved that these two areas are equal is that, first of all, you suppose that the circle is a little bit greater than the triangle, and then he proves that that leads to a contradiction, that can't be true. And then you suppose that the circle is a little bit less than the triangle,

and that too leads to a contradiction. This is a double contradiction. The only option left to you is that the circle and the triangle are identical. That seems to be one of his methods, right, doesn't it? He was a master of this method, yes, it's a method of double contradiction, which he used a lot and was extremely adept with it. And it's a very solid method of proof. It's a method that mathematicians still use, proof by contradiction,

So it's not that, and it's not that, so it's bound to be the other. You can't allow contradictions into mathematics, so as soon as you have a contradiction, your hypothesis has to go. Can we talk about George George, can we talk about his findings in volume, the sphere inside the cylinder, for example? Oh, yes, this is what Archimedes seems to have regarded as his greatest achievement. He wished to estimate the volume of a sphere, and how he did it was

he imagined cutting the sphere into two equal bits so we've got two hemispheres take one of the hemispheres and put the flat side down on the table so you've got this little mound this hemisphere and round about the hemisphere

you imagine a cylinder that fits tightly around the hemisphere and has the same height as the hemisphere now he had a good look at that and he had a very very clever argument

and he did this by imagining slicing through this configuration that's to say hemisphere surrounded by a cylinder and he showed that the, of course, what the slice... So he's imagining

he's imagining slicing through the slice would have an inner circle typically part of the hemisphere and the outer circle which is fixed of course the constant height, the constant width of the cylinder. Now, the clever bit was he subtracted the little circle from the big circle, as it were. That gives you what we call an annulus, a ring.

And he demonstrated that the area of that was exactly the same as the area of a slice of a cone. Now, he's not content with having a hemisphere, cylinder. he also imagines, he can't put all these things physically in, of course, but in his imagination, he imagines a cone. Now that would be with the flat side at the top and the point down on the table. And so therefore, he was able to calculate

the area of the, sorry, the volume of the hemisphere by summing up all these slices because it was the difference between the volume of the cylinder and the volume of the cone. Now, the volume of the cone was already known. It was known to be one-third of the area of its base times its height. And so Archimedes deduced that the volume of the hemisphere, it was two-thirds, the volume of what we call the circumscribing cylinder,

his great triumph. Aim solved. Brilliant. I haven't understood that. Very. Right. Saraphena Cuomo, he grappled. He also did some. something as I'm told, very unusual at times,

work with huge numbers considering the universe. Can you explain the sand-reckoner question, which is, I understand it is slightly out of line with most of what he did. It sounds on its own as one of his excrement. The sand-reckoner. Problem, over to you, Serafina. Yes, Jackie was saying right now

that Greek numbers were quite difficult to manipulate. One other limitation of the Greek numerical system was that you couldn't express really big numbers because at some points you ran out of letters of the alphabet. So the Sondre Econer is a kind of display piece that combines a new arithmetical system and astronomy. And it's interesting that it was a display piece because unlike the other treatises, is not addressed to a mathematician, but to the prince of Syracuse, the co-rigent, of hiero, gelon. So it was more of, you know, letter to the prince to show him how good Archimedes was.

The idea is that Archimedes is going to count the number of grains of sand that the whole universe can contain, which is, I think, in modern terms, 10 to the 58 palm. To the 63, I think. To the 60. Thanks, Jackie. So in order to do that, first he has to measure how big the universe is, and it takes an ideal number. It doesn't think this is really how big the universe is.

It takes the biggest number possible. And then he has to fill it up with grains of sand. And in order to do that, he has to devise a whole new numerical system that allows him to express such big numbers. And how does he devise this numerical system, Jackie? Well, he begins with something that he calls a myriad, which in modern terms is, one with eight zeros after it, so 100 million.

No, four zeros for the myriad. Four zeros for the myriad. Okay, so eight zeros is a myriad. I'm so pleased to see that this is going on between the three of you. It makes me feel just a tiny bit smog. Well, when you've counted up to a myriad, then your next unit is a myriad myriad, myriads. Yeah. And then a myriad, myriad, myriads.

So it's similar to what we have now. You might have a thousand and then a thousand thousand, thousand, a thousand, thousand, thousand, but the myriad is four zeros instead of three. But it is entirely imagined, isn't it, because he can't do it with the Greek numbers, and there's nothing for him to play with, and try to fill the universe with grains of sand, apart from sounding like a poem by William Blake, is he makes this a theoretical possibility even,

when in the way he demonstrates it? I'm sorry, what? Does he make it seem even a theoretical possibility, the way he demonstrated in his theory of the sand reckoner? Oh, he demonstrates that he's capable of handling numbers of any magnitude, any magnitude at all because I think he starts off with this myriad times

myriad has got of one giant unit and then he then uses numbers in a sense to describe what happens next because the myriad myriad is like one unit of a new kind of big super number I mean just as a hundred

is a super number compared with one and a thousand and so on but he just keeps piling on what we would they think now is just adding eight zeros at a time in our decimal system. It is often said that the Greeks were scared of infinity, but Archimedes wasn't scared of infinity at all, not of big numbers, not of playing with infinitesimals,

as the parents probably demonstrates. Can we just, before we talk about his legacy, find out if there's any consensus among you about how he died, seraphim, I think I'll give it to you because you, You seem most willing to play in this film. For Libius is my favorite historian here. Unfortunately, it doesn't have a report of his death because the book becomes fragmentary on this point.

So the earliest report is leaving. I think we can all agree that the Roman general who took Syracuse after two years of siege didn't want Archimedes to be killed. It was too valuable a resource. So that's, I guess, consensus. Then how did he die? Probably killed by a soldier, probably unintentionally.

What was he doing when he died? That's a big problem. Because according to some sources, and I think it's Plutarch, Prutarch reports more than one version. He was too engaged in proving one of his geometrical propositions to even care that Syracuse was being taken and that the soldier was about to kill them. him. Hence, the now famous phrase, leave me alone or leave my circles alone, and the soldier being

a Roman soldier just went and killed him. But there is an alternative story, according to which Archimedes was trying to make himself known to the Roman general and transporting all his astronomical and mathematical instruments with him. And he got killed by a soldier, but this time because the soldier mistook the scientific instruments for gold, and so I was trying of getting them. George Phillips, how was he regarded during his lifetime, in the say 100 years that followed? Well, I think that he, my colleagues here, being historians,

which I am really not, will tell us better. but I understand that his reputation went a bit cold in the immediate period after he died, did it not? Yes, I think his work wasn't particularly taken up immediately after his lifetime, as far as we know. But his works were translated first into Arabic. I mean, this is quite a little bit later, not 100 years later, but four or 500 years later. his works were translated and preserved, both in Greek manuscripts and Arabic manuscripts. But it was very, I mean, Cicero is very proud of the fact of finding Archibedes tomb, isn't he? That's true, yes, that time.

Whatever he's called, Governor-General of Syracuse. And this is a great thing that Cicero brings to the world. He has found the tomb, which has been neglected and overgrown. But it proves what an educator indeed, my goodness, man he was. And so we're talking about somebody who resonates with a great thinker in a later, following one quite quickly from that. But the Arabic translations and then their work is transferred from Arabic into Latin

and then we got the Greek the Greek coming from Constantinople. So by the time we're getting into Renaissance, by the time of Galileo, the work is coming through. Do you want to develop that, Serafina? The work is definitely coming through. I think the key point of interest for people in the scientific revolution

was that in modern terms, Archimedes had been able to apply mathematics to physics, and that's what they wanted to do. That's definitely one of the main reasons why Galilei is interested in him. I mentioned a little work on the hydrostatic balance, but the main work by Galilei where he mentions Archimedes and is really influenced by him is the two new sciences, which are application of mathematics to physics. And at some point Galilei says it's justified in doing that, because he is protected under the wings of the great Archimedes.

So he provided both a way of doing it and if you like a justification for doing it. Because being ancient, he also was prestigious. So constituted a prestigious precedent. Yeah. He was also of interest to people working in what we might now call pure mathematics. I mean, his results on areas and volumes were a very great interest to later mathematicians. and one of the problems with Archimedes is that he sometimes gave results without saying how he'd got them

and he'd prove them by contradiction but that doesn't actually tell you how he got those results and mathematicians in the Renaissance in 17th century often criticised Archimedes for not saying how he got his results and they put a huge amount of effort into trying to reconstruct his methods or try to work out other ways of getting to the same results So he was actually the impetus for a lot of new research because he hadn't told everything that they wanted to know. George Phillips recently discovered documents causing great excitement, the Archimedes Palimpsest. Can you tell us why that's so exciting?

Well, this goes back to, well, it goes back as long as you like, but one hinge date is 1906. A professor of classics, Professor Highberg from Copenhagen University, was in Constantinople as it then was. And he was examining a palimpsest. Now, shall I say what I believe a palimcest is? You better be quick, though, because that's fine, yeah.

Okay. Well, it's on vellum, and it's very valuable. And so these things were reused. And when a document in vellum was no longer of interest, rather than toss it out, they would scrape off the existing message and use it again. Now, the document found by Huyberg was a prayer book written in the 13th century, and it had been used for many centuries, waxed.

So why is it important for our comedies studies? It was important because when they recovered this document, they found that seven of Archimedes' treatises were written there, hidden away, through various techniques, imaging techniques, they have now managed, and this is just since 1998, when this document was sold off at auction, and they have recovered. It contained the only copy, the only copy,

of the method, which was a famous. What was that, Jack? What was the method? Well, the method is the treatise in which he describes what George described earlier, how he actually found some of these results. This is what the 17th century mathematicians criticised him for, not giving his method. And in the palimpsest, there is his method. And it's a kind of balancing method that he uses,

which is very close to what later mathematicians started to do and led up to the calculus. Well, we got to the calculus. Thank you very much. Jackie Sodor, Serefina Cuomo, and George Phillips, for getting us so close to Archimedes. And next week it's Genghis Khan, the founder of the Mongol Empire. Thank you for listening.

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