NARRATOR (JULIET STEVENSON): This is the story of a book that could have changed the history of the World. Lost for over a thousand years, it contains a unique record of the world and mind of one of the greatest men ever, a mathematical genius who was centuries ahead of his time: Archimedes.

DR WILLIAM NOEL (The Walters Art Museum): When the manuscript first arrived, you know, shivers ran, ran down my spine. I have never before in my life handled a book that is the only material witness to the mind of someone who died 2,200 years ago.

DR ROGER EASTON (Rochester Institute of Technology): This is a manuscript of incalculable value to the history of science. That feeling, that, that excitement was, was my, my first reaction.

PROF. REVIEL NETZ (Stanford University): I think it's fair to say that western science is a series of footnotes to Archimedes. People trying to come to terms with the problems of Archimedes, people trying to produce works that are as great as Archimedes, that are greater than Archimedes. This is the goal, this is the goal of western mathematics.

NARRATOR: As scientists worked to recover the text from this fragile document, they are discovering that Archimedes was further ahead of his time than they had ever believed. If his secrets had not lain hidden for so long the World today could be very different from what we know. Archimedes' manuscript is one of the most valuable ever found. Sold at auction for $2 million. The buyer refused to reveal his identity, only that he was a billionaire who'd made his money in IT, but research institutes all over the world wanted to work on his precious manuscript.

WILLIAM NOEL: I did what I think an awful lot of people did which was to get in touch with the book dealer who acted on behalf of the anonymous owner of the manuscript. I sent the e-mail to the book dealer and three days later I got an e-mail back. It said, "Dear Mr. Noel, I'm sure that you can borrow the Archimedes manuscript and the owner would be delighted in this idea." The owner visited the museum, together with the book dealer, and they left their equipment and kit on this table, so we went out to lunch and I said to him that it was extremely kind of him to even consider thinking of depositing the Archimedes manuscript with the Walters. He looked at me and he said "I've already deposited it with you," and I said "I'm sorry," you know, I was a little alarmed at this stage and he said "Yes, it, it was, it was on a duffle bag on my table," so I had to sit through a three course meal shaking to get back and thinking sitting on my desk on a duffle bag, I can't believe it and but after lunch we came here and opened the duffle bag and, and it was an amazing experience.

NARRATOR: This book contains unique works of Archimedes, lost for centuries, including the most important mathematics he ever wrote, because unlike other known writings by the Ancient Greek genius, this book does much more than list his mathematical achievements. This book explains how he made his discoveries.

WILLIAM NOEL: The Archimedes manuscript is, to all intents and purposes, the material remains of the thought of the man. I like to think of it as his brain in a box and it's for us to dig into that box and to pull out new thought. I wake up every day knowing that Archimedes is actually dependent upon the team of people that we've gathered together to, to, to really, to really allow him to speak for the first time.

NARRATOR: Retracing the story of how the Archimedes manuscript ended its journey at the museum is a remarkable tale of mystery and intrigue. The story starts back in Sicily in 287BC when Archimedes was born. Much about his life remains shrouded in obscurity. Historians have had to rely on the few surviving records of his work to try and piece together a picture of Archimedes, revealing a man with an extraordinary genius for mathematics.

DR CHRIS RORRES (University of Pennsylvania): In antiquity he stands alone. There is no other mathematician in antiquity, or for that matter in history, that comes close to Archimedes.

NARRATOR: Archimedes has become famous as the man who shouted "Eureka!" in the bath. He was trying to solve a problem with a gold wreath given to the King. The King suspected the goldsmith who had made it had slipped in some cheaper silver. The wreath weighed the right amount, but silver is lighter than gold, so the question was: was it greater in volume than it would have been if it was made with pure gold?

PROF. ALEXANDER JONES (University of Toronto): Archimedes' insight into how to determine a volume is supposed to have come when he got into a bath, noticed that the more of him went in, the more water poured out of the edges of the bath tub, and realised that this was, in fact, giving an exact measure of the volume of him going in and this would apply to the crown too. You could find how big the crown was by immersing it in a vessel of water and seeing how much water is displaced. He's supposed to have been so excited by this discovery that he immediately leaped out of his bath and without throwing any clothes on ran naked through the streets of Siracusa shouting the Greek word "I've discovered it - eureka, eureka!"

NARRATOR: It's probably unlikely that the citizens of Sicily ever saw Archimedes' naked body, but he did go on to reveal the truth behind the King's wreath.

ALEXANDER JONES: When the wreath was immersed in the water then it turned out that in fact its volume was greater than it should have been if it had been pure gold, so the smith was clearly not an honest one and Archimedes had successfully worked out some good detective work.

NARRATOR: During his life Archimedes became famous for invention and many of his ideas are used in machines today, but he was best known, and feared, for his weapons of war. In a garden in Philadelphia an Archimedes enthusiast has re-created one of his hero's most impressive schemes.

CHRIS RORRES: This is a model of the walls of Siracusa, the Greek city state, in Sicily in which Archimedes lived. He was assigned by his King to be the military adviser and to design the defences of the city, and his main defence were these so-called claws, or iron heads, that line about a one kilometre long piece of the wall. The ships would come in close to the wall, then the, then the claw would be swung around and the grappling hook dropped. The ship would be raised a certain amount. Then the grappling hook would be suddenly released. The ship would come smashing into the ground. All of these actions just frightened the Romans to death.

NARRATOR: But it's through his mathematics that the true genius of Archimedes is revealed. He came up with a value for Pi (p), probably the most famous mathematical symbol of all. Vital for calculating the area of a circle it's one of the most basic building blocks of science, the mathematical equivalent of the invention of the wheel.

ALEXANDER JONES: The way he goes about it is to try to squeeze the circle between polygons. You can find the perimeter of polygons because they're straight sides and if he can get polygons that wrap closer and closer to the perimeter of the circle then he will have a closer and closer pair of bounds within which Pi must lie. He begins by putting a hexagon inside the circle.

CHRIS RORRES: Continuing further he next divided the hexagon, doubled the number of sides to come up with a dodecagon, a 12 sided figure and, and determining its circumference he has a still better approximation.

ALEXANDER JONES: No need to stop there. We can take each of these and put two sides where one was before and we'll put them all in because you see it's so close to the circle already that on the drawing it starts to look like the circle. We now have 24 sides.

CHRIS RORRES: He continued this way going from 24 to 48 and finally ending up with 96.

ALEXANDER JONES: On the outside he does the same thing. He starts with the hexagon and for every side he makes two sides by putting more in like that so we now have 12 and so on until again you have 96 sides outside as well as in.

CHRIS RORRES: So in this way he guarantees that the number Pi is trapped between three and ten seventy-firsts and three and one-seventh, an estimate which is accurate to within one part in 2,000, better than one part in 2,000 and indeed this approximation, three in one-seventh is still used by engineers today and is more than good enough for all practical purposes.

NARRATOR: Obsessed by mathematics there was no problem too ambitious for Archimedes. He even tried to calculate the number of grains of sand to fill the universe. The answer: 10, followed by 62 zeros.

ALEXANDER JONES: We're told that Archimedes was often so preoccupied with his mathematical work that sometimes even just to get him to go to bathe was difficult. His, his slaves would have to carry him off forcibly we're told and even in the bath he would spend his time drawing little diagrams with the soapsuds presumably on his body.

NARRATOR: Ancient historians reported that Archimedes would become ecstatic as he discovered more and more complex mathematical shapes.

CHRIS RORRES: Four triangles and four hexagons constitute a truncated tetrahedron.

ALEXANDER JONES: Eight triangles and six squares a cubeoctahedron.

CHRIS RORRES: Eight triangles and eighteen squares constitute a rhombic- cubeoctahedron.

ALEXANDER JONES: Twelve squares, eight hexagons, six octagons - truncated cubeoctahedron.

CHRIS RORRES: Thirty-two triangles and six squares constitute a snub cube.

ALEXANDER JONES Truncated dodecahedron...

CHRIS RORRES: Snub dodecahedron.

ALEXANDER JONES: Truncated icosa...

CHRIS RORRES: Rhombicosi- dodecahedron.

DIRECTOR (V/O): Cut!

NARRATOR: But tragically Archimedes' genius had brought him to the attention of the Romans who were eager to capture him. When they finally managed to invade Siracusa instructions were issued to take Archimedes prisoner.

ALEXANDER JONES: Some soldiers had apparently been delegated to find Archimedes to take him to the Roman General, but a soldier who wasn't given these instructions barged into Archimedes' house, found him entirely preoccupied with doing mathematics, making drawings on a dust board and Archimedes didn't even, hadn't even heard the bustle that was going on and when he turned to the soldier he made some remark to the effect of don't disturb my circles and the soldier killed him with his sword. That was the end of Archimedes.

NARRATOR: Archimedes' death in 212BC brought a golden age in Greek mathematics to an end.

CHRIS RORRES: There was no-one that could follow him in Europe. Greek mathematics then gradually declined and then the Dark Ages, the Age of Faith entered where all interest in mathematics was lost and as a result, nothing really interested us scientifically.

NARRATOR: But Archimedes' writings did survive, copied by scribes who passed on his precious mathematics from generation to generation, until in the 10th-century one final copy of his most important work was made, but interest in mathematics had now died. Archimedes' name was forgotten. When one day in the 12th-century a monk ran out of parchment, with devastating results.

ALEXANDER JONES: The pages were re-used to make a prayer book. Each of the sheets that makes a double page in the Archimedes manuscript was unbound, cut down the binding, turned sideways and then folded to make a new double page in a prayer book. The pages were washed, or scraped plain enough so that it would then be possible to write over them with the religious texts that are now the, the obviously visible part of this manuscript.

NARRATOR: The ancient works of a mathematical genius were systematically consigned to oblivion, washed clean, re-used and written over the manuscript had become what's known as a palimpsest. It had begun a new life as a book of prayers at the Mar Saba monastery in the Judean desert in the Middle East.

WILLIAM NOEL: And there it was used as a prayer book, the Archimedes text completely unread and unknowing for many, many centuries.

NARRATOR: And so Archimedes' secrets lay hidden in the library tower of the monastery while all around them the rest of the known world moved on. In the 15th-century the Renaissance hit Europe. Now at last science had advanced enough for scholars to understand Archimedes' mathematical arguments, but no-one had the slightest clue that some of his greatest ideas had been lost. Renaissance mathematicians had to grapple with concepts and problems that Archimedes had worked out in his bath 1,500 years before.

CHRIS RORRES: If the mathematicians and scientists of the Renaissance had been aware of these discoveries of Archimedes this could have had a tremendous impact on the development of mathematics. This was a crucial period in mathematics, the 15th /16th-century.

NARRATOR: It was hundreds of years before the manuscript was heard of again. No-one knows how, but it turned up in a library in Constantinople. The library catalogue listed several lines from the manuscript. These caught the eye of Greek expert Johan Ludwig Heiberg. He realised at once that the words could only come from one source: Archimedes. Determined to find out more, he had arrived in Constantinople in 1906 to take a closer look.

WILLIAM NOEL: Heiberg must have had his hopes up, but when he saw the manuscript itself he must have been flabbergasted, and he of all people knew the significance of what he was reading. It must have been an extraordinary moment for him.

NARRATOR: Heiberg wasn't allowed to remove the manuscript from the library, so instead he asked a photographer to take pictures of every page and from these photos he attempted to reconstruct Archimedes' work. It was an incredibly difficult task.

ALEXANDER JONES: It is remarkable how much Heiberg was able to get out of this manuscript given the condition of the manuscript. The Archimedes text is really very faint on most of the pages. He had limited time and, so far as we can tell, the only help that he had in reading the manuscript was a magnifying glass.

NARRATOR: Heiberg's discovery revealed ideas that had never been seen before.

WILLIAM NOEL: The discovery of the Archimedes manuscript was, was so significant that it, that it made the front page of the New York Times. This was understood at the time as a major breakthrough in the history of mathematics.

NARRATOR: What Heiberg had stumbled across was like going inside Archimedes' brain. Here Archimedes didn't just give the answers to his calculations, but for the very first time he wrote down his innermost thoughts revealing how he'd carried out his work. It was a book he had called The Method.

ALEXANDER JONES: It's a book about discovery instead of a book about how you get to the result in the first place before you do the proof. This is very rare, in fact there's no whole book that we have from antiquity, aside from The Method, that addresses that kind of question.

CHRIS RORRES: This was a spectacular find for the history of mathematics. It was very much like getting a glimpse into Archimedes' mind. If you were a painter, for example, you would certainly be interested in the finished works of the Masters, but more than that you want to learn the techniques, the methods of the Masters. What paint did they use, how did they outline their subjects? Likewise with mathematicians, they want to know not just what his finished works were, what his finished theorems were, but how did he arrive at them.

NARRATOR: The Method revealed that Archimedes had come up with a radical approach that no mathematician had come close to inventing. In his head he had dreamt up an entirely imaginary set of scales to compare the volumes of curved shapes. He used this to try and work out the volume of a sphere.

CHRIS RORRES: Now prior to Archimedes the volume of a cone and a cylinder was already known and so he tried to use those previously known results to compute the volume of a sphere and so he concocted this very interesting balancing act. He attempted to balance the sphere and the cone on one side with the cylinder on the other.

NARRATOR Using very complex mathematics in his head in which he imagined cutting the shapes into an infinite number of slices, Archimedes was able to work out how to balance the objects on the scales.

CHRIS RORRES: The final result after all the arithmetic was done was that the volume of a sphere is precisely two-thirds of the volume of the cylinder that encloses this sphere. This was a result that he considered so important that he asked that it be inscribed on his tombstone as his most important mathematical discovery.

NARRATOR: Working out volumes using infinite slicing suggested that Archimedes was taking the first step towards a vital branch of mathematics known as calculus 1800 years before it was invented. The modern world couldn't live without calculus. It is a form of mathematics essential for scientists and engineers. 21st century technology depends on it. But back in 1914 as he was poised to uncover the true genius of Archimedes, Heiberg's plan to study the manuscript further in Constantinople was brutally interrupted. World War 1 broke out. Europe and the Middle East was thrown into turmoil and the palimpsest was lost again. No-one had any idea of the secrets that still lay buried within its covers. Scholars had little hope that they would ever see the document again. Then in 1971 Ancient Greek expert Nigel Wilson heard about a single page of a manuscript in a library in Cambridge. He decided to take a closer look.

DR. NIGEL WILSON (University of Oxford): I transcribed a few sentences almost completely. They included some rather rare technical terms and if you go to the Greek lexicon and check where those terms occur you soon find that you're dealing with essays by Archimedes and then I suddenly realised this must be a leaf detached from the famous palimpsest It was a very good moment. I became rather excited.

NARRATOR: But why had a single page, and only a single page, of the Archimedes palimpsest turned up in Cambridge? A clue lay in a collection of papers that were handed over to the university, papers that had belonged to a scholar called Constantine Tischendorf. a man of few scruples.

NIGEL WILSON: Tischendorf travelled a lot in the Near East. When he got to Constantinople he visited the library, he said that at the time it had about 30 manuscripts in it, they weren't of any interest with one exception. He mentions a palimpsest with a mathematical text in it. He doesn't say anymore.

NARRATOR: When Tischendorf discovered the palimpsest he was tempted to examine it in much more detail.

NIGEL WILSON: I don't think there's any alternative to the assumption that he stole this page. He must have waited until the librarian was out of the room and I think he wasn't sufficiently interested in Greek science to know enough to identify the text as Archimedes, but he probably had a hunch that it was something important.

NARRATOR: At the turn of the 20 century Heiberg only had a magnifying glass with which to read the manuscript. Now Nigel Wilson had the advantage of modern technology.

NIGEL WILSON: When I got the leaf to work on most of it was legible, not quite all, but with the ultraviolet lamp the corners, which one couldn't read, became clear. I realised that with an ultraviolet lamp one ought to be able to read most, if not everything, that had remained a mystery to Heiberg.

NARRATOR: This tantalising hi-tech glimpse of a single page revealed how much more could be gleaned from Archimedes' work if only they knew what had happened to it since Heiberg had last held it in his hand. After World War I, Paris and other European cities were flooded with works of art from the Middle East, yet there had been no sign of any manuscript of Archimedes, but in 1991 Felix de Marez Oyens arrived at Christies to discover that the Archimedes palimpsest might have been in Paris all along. At his new office he found a letter from a French family who claimed they had a palimpsest.

FELIX DE MAREZ OYENS (Former Director of the International Book Dept., Christies): They were talking about this amazing palimpsest manuscript of this incredibly important scientific classical text, so you take a little bit of distance and you know you don't immediately get over-excited, but I did realise immediately that if this thing was authentic that it would be something incredibly exciting.

NARRATOR: Intrigued by the letter, Felix wrote back and discovered that the owners lived just around the corner from his in Paris, so he set off to examine the book.

FELIX DE MAREZ OYENS: But it was quite, immediately quite clear that this had to be that manuscript that was in the literature and that had been seen, or studied for the first time, by Heiberg in 1906.

NARRATOR: The owners had an unusual story to tell. In the 1920s a member of the Parisian family had travelled to Turkey. He was a keen amateur collector and somehow had acquired the manuscript in Constantinople. All the time that it had been assumed lost the palimpsest had been lying in the family's Paris apartment, but now they had decided to alert the art world because they wanted to sell. Felix had to decide how much the manuscript was worth.

FELIX DE MAREZ OYENS: Well the valuation of important manuscripts, let alone palimpsests, are terribly difficult in general, but I think I told them it would have to be worth about £400,000-600,000. Any valuation of something like that is simply a guess. Perhaps if you're lucky in it you get an educated guess.

AUCTIONEER: $1,400,000. $1,800,000.

NARRATOR: The manuscript sold for far more than ever Felix had predicted. An anonymous billionaire paid $2 million and so finally the manuscripts arrived at The Walters Art Museum in Baltimore and into the hands of curator Will Noel. He was in for a terrible shock.

WILLIAM NOEL: I was horrified, I was aghast. It's, it's, it's really a disgusting document. It really looks very, very, very ugly. It doesn't look like a great object at all. It looks dreadful, I mean it really looks dreadful. It's been burnt, it's got modern PVA glue on its spine. The Archimedes text that we're trying to recover goes behind that PVA glue. It's got blue tack on it, it's got strips of paper that have been stuck on top of it. It's very hard to describe adequately the poor condition of the Archimedes palimpsest is in.

NARRATOR: Will quickly put a team together from all over the world to try and rescue the book: the Greek experts Reviel, Natalie and Nigel, the imagers Bill, Roger and Keith and finally conservationist Abigail. Detailed examination in the conservation lab revealed the appalling damage that the book has suffered.

ABIGAIL QUANDT (The Walters Art Museum): The manuscript was heavily damaged by mould. This is seen in a lot of these purple spots all over the surface of the leaves. In this area of the parchment there's a very intense purple stain. The parchment is perforated where the fungi have actually gone through and digested the collagen and it means that the Archimedes text is just totally missing in these areas.

WILLIAM NOEL: It's like BSE. What BSE does to the brain mould does to the Archimedes manuscript. I tend to think of the Archimedes manuscript as Archimedes' brain in a box, so this really is an appalling wasting away of a great mind.

NARRATOR: And the team have discovered there is another problem preventing them reading the thoughts of Archimedes. On several pages of the palimpsest there are mysterious illustrations which completely cover over the text.

ABIGAIL QUANDT: When we first saw the manuscript we were very curious about these paintings. They could possibly be Medieval, but the colours were tonally wrong for this period. The other curious thing was that these leaves seem to have been intentionally mutilated, possibly to make them look Medieval by adding additional knife cuts to the edges of the leaves.

NARRATOR: Puzzlingly Heiberg made no mention of these pictures when he studied the manuscript in 1906, so the conservation team showed these illustrations to Byzantine expert John Lowden.

DR JOHN LOWDEN (Courtauld Institute of Art): I was convinced I had seen something very similar before. In 1982 a Gospel book of the 12th-century came up for sale which I investigated because it had miniatures of the four Evangelists.

NARRATOR: John had discovered that the miniatures in the 12th-century Gospel book were forgeries and had been copied from a French book.

JOHN LOWDEN: This is the book which is Ormont's Manuscrits Grecs de le Bibliotheque Nationale in Paris which was published in 1929.

NARRATOR: When John saw the illustrations in the Archimedes manuscript he thought they looked very similar to the forgeries in the Gospel book. He suspected that this same French book had been used to forge the pictures in the palimpsest.

JOHN LOWDEN: And indeed I was able, within a few minutes, to identify Plate 84 of Ormont as the source for the four images of the Evangelists in the Archimedes manuscript. I am convinced that they're forgeries.

NARRATOR: Back in Baltimore, Abigail Quandt has does a test to see how the forger copied their paintings from the book.

ABIGAIL QUANDT: I made this tracing from the painting of John that appeared in the 1929 publication and then if you overlay it with the forgery in the Archimedes palimpsest it lines up almost exactly.

NARRATOR: Abigail did the same experiment with the other three black-and-white images. She found each of them was precisely the same size as those in the palimpsest. The forger must simply have traced the images. Much about this mysterious forger remains a mystery, but John Lowden's investigation has been able to reveal one vital clue.

JOHN LOWDEN: The earliest the forgeries could have been done is 1929 because that's the date of publication of the book.

NARRATOR: But why would someone have spent so long carefully putting forgeries into this manuscript?

WILLIAM NOEL: There's only one reason that there are forgeries put into the Archimedes manuscript and that's straightforwardly money. It increases the manuscript's value. Now this is true whether you know that the manuscript contains the work of Archimedes or not. The reason for that is that it becomes art and it belongs to a completely different clientele. The question is whether these forgeries were done in the knowledge that Archimedes was underneath it. I rather hope not and it's a horrific thought to think that, that, that, that the illuminations were, were painted over Archimedes text in full knowledge of the fact.

NARRATOR: Back in the lab the delicate conservation work on the manuscript has started. Abigail has the vital job of cleaning it to help try and recover the precious Archimedes text.

ABIGAIL QUANDT: On some leaves the Archimedes text is almost invisible. On some leaves though you can see it is a copper brown colour. It runs in two columns perpendicular to the Christian text. I mean the process of removing wax droplets - the wax is there because, of course, the manuscript during Medieval times was read by candlelight. The wax droplets are actually interfering with the success of the imaging.

NARRATOR: To study each page fully Abigail has to remove it from its binding in the book. This is relatively new binding, probably done in the last hundred years and removing it has turned out to be far more important than anyone could have realised.

ABIGAIL QUANDT The way the manuscript is put together there are four folded sheets, one nested inside the other. The Archimedes text goes across the fold, but Heiberg's difficulty was that he couldn't actually see those writings.

NARRATOR: It had been impossible to read the Archimedes text that went right into the centre of the binding. Only now, by taking the book apart, can the writing be seen.

ABIGAIL QUANDT: So we're seeing lines of Archimedes text for the very first time.

NARRATOR: And examining the original photos that Heiberg took of the palimpsest has also shown there was something else that he missed. Some of the most important pages of the manuscript were never photographed at all.

WILLIAM NOEL: People have assumed that Heiberg knew this manuscript extremely well. He didn't know it that well. Now that we've got the manuscript we can fill in large chunks of Heiberg's transcription for the first time. This is really going to surprise the scholarly community. It's much more than we ever thought that we would, we would pull out of the Archimedes palimpsest when we began this project.

NARRATOR: Recovering the words of Archimedes is a huge technical challenge. Tackling the problem are teams from Johns Hopkins University and the Rochester Research Institute.

ROGER EASTON: We are trying to take advantage of the very slight differences in colour of the two inks, of the, the inks from the Archimedes text and the, and the later ink, so we did that by taking images of a wide variety of wavelengths. Yeah, that looks good. Alright so UV. Because the two inks are slightly different colours they reflect slightly differently in these different bands of wavelengths. Now we want to go to flash.

NARRATOR: Their latest approach uses a combination of visible and ultraviolet lights to try and make the Archimedes text as simple to read as possible.

DR KEITH KNOX (Rochester Institute of Technology): Here's a piece of the parchment shown in visible light. The horizontal text is the top text that's obscuring the Archimedes text. The Archimedes text is the vertical lines that you see. They're very hard to see to the naked eye. In red light the Archimedes text really is not very visible at all. As you can see from this image only the top text is showing. On the other hand, if you look at this piece of parchment the two sets of text show most clearly using the blue image from the ultraviolet and if you now take and combine the blue image which shows both texts very well and the red light image and put them together and do a little extra processing you can create a false colour image which is this image.

NARRATOR: The Archimedes text shows up far more clearly than before. It now appears as the red writing, much easier to read.

REVIEL NETZ: I was amazed by the fact that now for the first time I can look at pages that looks, that look hopeless with the naked eye and begin to use them as text from which you just read. We are able to recover the original text of Archimedes where it appears to have been lost. I think we will be able to read everything there for the first time.

NARRATOR: One of the most exciting things to emerge already are diagrams. This is the first time they have ever been examined for Heiberg never copied them when he made his great translation and Nigel Wilson believes the diagrams bring us even closer to the mind of Archimedes.

NIGEL WILSON: Over the centuries Archimedes' essays weren't copied very often, so the number of intervening copies between the original and our 10th century manuscript may be very small. It might be only four or five. I would think that these 10th century drawings reflect very accurately the diagrams that Archimedes himself intended to be an essential part of his treatise.

NARRATOR: The drawings are providing a real insight into the work of Archimedes revealing the specially important role that diagrams played in Greek mathematics.

REVIEL NETZ: Our mathematics is always based on what you write. Only what you write down is part of the proof. In Greek mathematics the proof relies not only on what you write down as part of the text, it's also, it also relies on what you write into the diagram, what you draw in the diagram. You draw and write simultaneously - the two work together in Greek mathematics. If you want to recover the thought of Archimedes you don't need just the text of Archimedes, you want also the diagrams of Archimedes.

NARRATOR: But the diagrams are only the beginning. Image after image of the original writing is now emerging. As Will Noel rapidly e-mails the pictures to Moscow, Cambridge and London the awaiting Greek experts attempt to decipher the text. Trying to piece together the faint red words of Archimedes is an incredibly painstaking task.

REVIEL NETZ: What you do is first of all to stand back from the page and think what could Archimedes in principle have been saying there. Once you have some sort of guesses as to what he could say then you begin to apply this to the text. It will look very, very hard and just to get yourself hypnotised into this text until somehow certain traces begin to take shape.

NARRATOR: Today Reviel, Nigel and Natalie are meeting for the first time to try and decipher the pages of the manuscript together.

NIGEL WILSON: So that means uprights, upright lines.

REVIEL NETZ: And then a row a bit later on.

NATALIE TCHERNETSKA: And then five.

NIGEL: Yeah, there's a row well on yes, towards the end.

NATALIE: Just there is a (TALKING TOGETHER) probably not the script (TALKING TOGETHER) Yeah, you have, you have a little hook (TALKING TOGETHER)

NIGEL: The hook at the bottom is there, yes, that's good.

REVIEL: OK, let's make sure I've got the reading. (GREEK)

NIGEL: Yes, right.

REVIEL: Dot.

NIGEL: Yes.

REVIEL: (GREEK) with an omega.

NIGEL: Yeah.

NATALIE: Could it be actually epsilon (GREEK) Look if you, if you look maybe here and there are no tail...

NIGEL: Ah, there's no tail (TALKING TOGETHER) Ah, right.

NARRATOR: The painfully slow process of unravelling the text will take years, but already Reviel has made one very important new discovery. He examined a proof in Archimedes revolutionary work The Method. In it Archimedes was trying to work out the volume of an unusual shape by dividing it into an infinite number of slices. Archimedes had drawn a diagram of a triangular prism. Inside this he drew a circular wedge. This was the volume that he wanted to calculate. He then drew a second curve inside the wedge. Modern mathematicians already understood that Archimedes had used some very complex ideas to work out that a slice through the wedge equals a slice through the curve times a slice through the prism divided by a slice through the rectangle. But what no-one knew was how Archimedes had added up an infinite number of these slices to work out the volume of the wedge. The frustration was that the lines explaining how he had done this appeared in Heiberg's translation merely as a row of dots. These vital lines were missing, but then, with the help of the very latest images of the palimpsest, Reviel Netz went back to study the manuscript again.

REVIEL NETZ: I was looking at those missing lines in the page. At this point I was stuck until I saw a faint trace just above the line. It didn't appear as if it was part of the writing because it was just above the line, but I began to think perhaps it is something which belongs to the text of Archimedes and that's something which was absolutely new. This was a breakthrough in the history of Greek mathematics.

NARRATOR: Reviel realised that Archimedes had come up with a set of rules for dealing with infinity. He'd worked out a system for calculating the value of each slice and then adding up an infinite number of them.

REVIEL NETZ: Well I was there totally shaken. I was exhilarated and surprised when I saw this argument and I definitely had the sense that without my knowing yet what the argument is, what this argument represents in terms of the mathematical interest of Archimedes which we didn't know about, it represents something very important, something very deep for the history of mathematics.

NARRATOR: What was clear was that Archimedes had made a huge step towards the understanding of infinity.

REVIEL NETZ: Infinity is central to where the history of Western mathematics because the history of Western mathematics was determined by a very Greek problem, the problem to which Archimedes contributed more than anyone else: how to calculate the properties of curved objects. The theorem of the wedge is the first time that we see any Greek mathematician doing something with infinity, actually producing an argument using infinity. That's something which we simply thought could not happen.

NARRATOR: Even today infinity is a concept that mathematicians can struggle to deal with.

CHRIS RORRES: Humans are finite creatures and to talk about infinity in any context, whether it's in a religious context or in a mathematical context, has always caused us problems. Possibly the fact that we can even think about infinity, about the concept, even come up with the concept, involves that we have some kind of a, a passport to God. Now I'm getting very religious here, but whenever you talk about infinity you almost have to confront religious issues. Will we live for infinity, will the Universe last for infinity, where did the Universe come from, is infinity something that exists only in our minds and has no reality in basis?

NARRATOR: The new finding in the wedge theorem reveals not only that Archimedes was confident in dealing with infinity, but also that his use of infinite slices to calculate a volume was far more sophisticated than anyone had realised. In fact his technique is similar to the concept used in modern calculus for the same kind of problem. Archimedes was even closer to modern science than had been believed. It's amazing to think that a branch of mathematics that has been so crucial to our development was first begun by a man who died over 2,000 years ago.

REVIEL NETZ: We always knew that Archimedes was making a step in the direction leading to modern calculus. What we have found right now is that, in a sense, Archimedes was already there. He already did develop a special tool with which you can sum up infinitely many objects in measure of volume.

NARRATOR: But perhaps the most interesting question of all is what might have happened if this document had not been lost for a millennium. Supposing it had been available to those mathematicians of the Renaissance.

ALEXANDER JONES: If the book had been available a hundred years before the development of the calculus then things would have got going sooner.

CHRIS RORRES: It would have, of course, changed mathematics, but mathematics has an influence on all of the sciences. It serves as the foundation, the language of all of the sciences, so it's not just mathematicians who need mathematics, it's all scientists, all physicists, all engineers who need it and you would basically be raising the tide by increasing the knowledge of mathematics so several hundred years ago.

NARRATOR: It's an extraordinary thought that if scientists had had access to this document mathematics might be far more developed now. Who knows as a result how different the modern world might look and all because of something written by a man in the 3rd century BC.

CHRIS RORRES: If we had been aware of the discoveries of Archimedes hundreds of years ago we could have been on Mars today, we could have have developed the computer that is as smart as a human being today, we could have accomplished all of the things that now people are predicting for a century from now.