Infinite SecretsNARRATOR: This is a book that could have changed the history of the world. It contains the revolutionary ideas of a genius who was centuries ahead of his time: Archimedes. The book was lost to the world for more than one thousand years, passing through the hands of scribes, monks, forgers and shady scholars. Yet no one seemed to know the book's true value until it surfaced at auction, selling for $2,000,000. The buyer: a billionaire who wouldn't reveal his identity. But instead of hiding it away, he put the precious manuscript into the hands of those who could unlock its secrets. DR. WILLIAM NOEL (The Walters Art Museum): When the manuscript arrived, you know, shivers ran, ran down my spine. NARRATOR: But interpreting Archimedes' manuscript proved to be more difficult than anyone had imagined. The nearly invisible tracings of his words lay under the writings of a medieval prayer book. As scientists work 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 been hidden for so long, the world today could be a very different place. DR. CHRIS RORRES (University of Pennsylvania): We could have been on Mars today. We could have accomplished all of the things that people are predicting for a century from now. NARRATOR: A story of intrigue, up next on NOVA, Infinite Secrets. Major funding for NOVA is provided by the Park Foundation, dedicated to education and quality television. Science: it's given us the framework to help make wireless communications clear. Sprint is proud to support NOVA. We see an inventor. At Microsoft, your potential inspires us to create software that helps you reach it. Your potential, our passion. Major funding for this program is provided by the National Science Foundation, America's investment in the future. And by the Corporation for Public Broadcasting, and by contributions to your PBS station from viewers like you. Thank you. NARRATOR: He is legendary as the man who made a discovery in the bath, and shouted "Eureka! I have found it!" while running naked through the streets of his town. He is the most famous of the ancient mathematicians and the first to discover the value for Pi—the mathematical equivalent of inventing the wheel. He is Archimedes, one of the most brilliant thinkers of all time. But what we know about Archimedes is about to change. This newly found manuscript reveals that Archimedes' ideas were so advanced they could have altered the course of modern science. And now, the long lost secrets of this genius are about to be rediscovered. DR. 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 really allow him to speak for the first time. NARRATOR: And what he has to say is the culmination of a remarkable journey, one that began two thousand years ago. DR. CHRIS RORRES: We know more about Archimedes than any other Greek mathematician from antiquity, which is not to say we know a lot, because we know practically nothing about the others. NARRATOR: The story starts around 300 B.C., on the island of Sicily, where Archimedes lived in Syracuse, a citystate ruled by Greece. His father sent him to study in the scholarly center of the world, Alexandria, Egypt, where he learned from the writings of Aristotle, Plato and Euclid. Archimedes soon revealed an extraordinary genius for mathematics. PROF. ALEXANDER JONES (University of Toronto): 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: On one particular day, he was trying to solve a problem about a gold wreath that had been given to the King of Syracuse. The King suspected that the goldsmith had slipped in some cheaper, lighter silver. The goldsmith could easily have disguised his deceit by making the wreath bigger so it weighed the same as it would if it were pure gold. PROF. ALEXANDER JONES: Archimedes' insight into how to determine a volume is supposed to have come when he got into a bath and noticed that the more of him that went in, the more water poured out of the edges of the bath tub. And he realized 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 was 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 Syracuse shouting the Greek word for "I've discovered it," "Eureka! Eureka! NARRATOR: What he had discovered was the Principle of Liquid Displacement, and a way to prove that the King's wreath was too big to be pure gold. PROF. 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: Not only did he excel at mathematics, but he was an inventor as well. 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, Chris Rorres, a University of Pennsylvania mathematician, has recreated one of Archimedes' most impressive schemes. DR. CHRIS RORRES: This is a model of the walls of Syracuse, the Greek citystate in Sicily, in which Archimedes lived. He was assigned by his King to be the military adviser and to design the defenses of the city. And his main defenses were these socalled "claws," or iron hands, that lined about a onekilometerlong 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 his arsenal of war machines did not end there. DR. CHRIS RORRES: Although Archimedes didn't invent the catapult, he greatly improved its design. NARRATOR: Until Archimedes, a catapult could only hurl its missile a fixed distance, called its "range." DR. CHRIS RORRES: He made it a variablerange machine. The Romans were in for a surprise since, no matter where their fleet was located, they were bombarded by these catapults. This one here is springloaded: you wind it up, and then you release it, and off the arrow goes. NARRATOR: As his reputation grew, so did the challenges. DR. CHRIS RORRES: The King of Syracuse asked him for some demonstration of his powers. With a combination of pulleys and levers he was able to pull a ship onto the shore all by himself. And the King of Syracuse was so amazed by this that he proclaimed that from now on everything that Archimedes says must be believed. It was on this basis that Archimedes gave his famous boast, "Give me a place to stand, and I will move the Earth." NARRATOR: But Archimedes' true love was mathematics. He devised an ingenious method using straight lines to measure a circle, finding the value for Pi. It's one of the most widely used mathematical values today. PROF. ALEXANDER JONES: Archimedes wants to find a value as close as reasonably possible to what we would call Pi: the ratio of the circumference of a circle—here's our circle—to its diameter—here's its diameter. DR. CHRIS RORRES: And his technique was the following: he began with the circle and then inscribed a triangle in it, like so. Now, the entire perimeter of the triangle is less than the perimeter of the circle, so starting with that fact, here he could easily compute the perimeter of the triangle. And this represented a first approximation, a lower bound, to the circumference of the circle. Next he took this triangle... PROF. ALEXANDER JONES: ...putting a hexagon inside the circle. DR. CHRIS RORRES: Continuing further, he next divided the hexagon, doubled the number of sides to come up with a dodecagon—a 12sided figure—and, and determining its circumference, he has a still better approximation. PROF. ALEXANDER JONES: No need to stop there. We can take each of these and put two sides where one was before. I won't 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. DR. CHRIS RORRES: He continued this way, going from 24 to 48 and finally ending up with 96. PROF. ALEXANDER JONES: On the outside he does the same thing. He starts with a 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. NARRATOR: The circumference lies between the perimeters of the outer and inner polygons, which can be measured because they have straight sides. Now he can estimate how many times the length of the diameter—also a straight line—will fit around the circumference. His answer was 3.14 times, a very good approximation for what is called Pi. DR. CHRIS RORRES: This approximation 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? Ten followed by 62 zeros. Ancient historians reported that Archimedes would become ecstatic as he discovered more and morecomplex mathematical shapes. DR. CHRIS RORRES: Four triangles and four hexagons constitute a truncated tetrahedron. PROF. ALEXANDER JONES: Eight triangles and six squares: a cube octahedron. DR. CHRIS RORRES: Eight triangles and 18 squares constitute a rhombic cube octahedron. PROF. ALEXANDER JONES: Twelve squares, eight hexagons, six octagons: a truncated cube octahedron. DR. CHRIS RORRES: Thirtytwo triangles and six squares constitute a snub cube. PROF. ALEXANDER JONES: Truncated dodecahedron... DR. CHRIS RORRES: ...snub dodecahedron. PROF. ALEXANDER JONES: Truncated icosadodecahedron. DR. CHRIS RORRES: Rhombic cosi dodecahedron. NARRATOR: Archimedes' brilliance caused jealousy among other mathematicians, whom he feared would take credit for his ideas. DR. CHRIS RORRES: There was a small community of mathematicians that Archimedes regularly communicated his ideas with. PROF. ALEXANDER JONES: He would send first the results and then later the proofs. DR. CHRIS RORRES: But Archimedes felt that some of them were taking credit for his discoveries. PROF. ALEXANDER JONES: Because they were claiming that they had found these things first. DR. CHRIS RORRES: So he was in the habit of submitting false theorems. PROF. ALEXANDER JONES: False results, in the hope of tripping them up by having them claim they had proved them first. DR. CHRIS RORRES: So that he could expose them as frauds. 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 Syracuse, instructions were issued to take Archimedes prisoner. PROF. 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, and that was the end of Archimedes. NARRATOR: He was more than 70 years old at the time. Archimedes' death, in 212 B.C., brought a golden age in Greek mathematics to an end. DR. CHRIS RORRES: Archimedes' writings were almost like religious writings: people felt that they were the final word on the matter. For example, nobody tries to improve the Bible today: it's heresy. And back then they had that same attitude about the writings of Archimedes, that these were the definitive texts on geometry. There was no one that could follow him. 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 many of Archimedes' writings did survive, copied by scribes, who passed on his precious mathematics from generation to generation, until, in the 10^{th} century, one final copy of his most important work, called The Method, was made. But interest in mathematics had now died. Archimedes' name was forgotten. Then, one day in the 12^{th} century, a monk ran out of parchment. There were devastating results. PROF. ALEXANDER JONES: The pages were reused to make a prayer book. Each of the sheets that makes a double page in the Archimedes manuscript was 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 clean 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, reused and written over, the manuscript had become what's known as a palimpsest. It began a new life as a book of prayers at the Mar Saba monastery in the Judean desert in the Middle East. DR. WILLIAM NOEL: And there it was used as a prayer book—the Archimedes text completely unread and unknown for many, many centuries. NARRATOR: And so Archimedes' book, The Method, lay hidden in the library tower of the monastery while the rest of the known world moved on. When the Renaissance hit Europe at last, science had advanced far enough for scholars to rediscover and understand Archimedes' surviving works. But no one had the slightest idea that The Method had been lost. Renaissance mathematicians had to grapple with concepts and problems that Archimedes had worked out centuries before. DR. 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 15^{th}, 16^{th} century. NARRATOR: It was hundreds of years before the palimpsest was heard of again, mysteriously turning up in a library in Constantinople. The library catalogue listed no author, but did include several lines from the original manuscript. These caught the eye of Greek expert Johan Ludwig Heiberg who realized they could only come from one source: Archimedes. Determined to find out more, he went to Constantinople in 1906 to take a closer look. DR. 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 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. PROF. 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. DR. 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 a book called The Method. It was just like going inside Archimedes' brain. Here, Archimedes didn't just give the answers, but for the very first time he wrote down his innermost thoughts, revealing how he'd carried out his work. PROF. 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. DR. 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 are 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 dreamed 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. DR. 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 scale. DR. CHRIS RORRES: The final result, after all the arithmetic was done, was that the volume of a sphere is precisely twothirds 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, 1,800 years before it was invented. DR. CHRIS RORRES: Calculus is basically a summation process. Instead of adding just a finite number of things, you have to have some concept of adding up infinitely many things. And it is the integral calculus—the calculus that Archimedes is the father of—that accomplishes this infinite summation. NARRATOR: Building on Archimedes' discoveries, scientists, beginning with Newton and Leibnitz, created the calculus we know today: the mathematics of change, including motion. It's used to calculate everything from the movement of planets to the flight path of a spacecraft. It is a form of mathematics essential for scientists and engineers. Twentyfirst 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 I broke out. Europe and the Middle East were thrown into turmoil, and the palimpsest was lost. 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, England. He decided to take a closer look. 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. Then I suddenly realized 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 traveled 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^{th} century, Heiberg only had a magnifying glass 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 realized 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 tantalizing hightech glimpse of a single page revealed how much more could be gleaned from Archimedes' work if only they knew where the entire manuscript could be found. 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 in London at the auction house Christie's, 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 to have a palimpsest. FELIX DE MAREZ OYENS (Christie's): 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 overexcited. But I did realize immediately that if this thing was authentic that it would be something incredibly exciting. NARRATOR: Amazingly, Felix realized that the owners lived just around the corner, 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 their Parisian family had traveled to Turkey. He was an 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 sell the book. 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 that it would have to be worth about £400,000 to £600,000. Any valuation of something like that is simply a guess, perhaps, if you're lucky, an educated guess. AUCTIONEER: One million, four hundred thousand dollars, one million, eight hundred thousand dollars. NARRATOR: The manuscript sold for far more than even Felix had predicted. An anonymous billionaire paid two million dollars. Word of the sale quickly spread to people like William Noel. As curator of an art museum internationally known for restoring medieval manuscripts, Noel wanted to get his hands on the book. DR. WILLIAM NOEL: I did what I think an awful lot of people did, which was to get in touch with the book dealer who acts on behalf of the anonymous owner of the manuscript. I sent the email to the book dealer, and three days later I got an email 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?" A little alarmed at this stage. And he said, yes, it was in a duffel bag on my table. So I had to sit through a threecourse meal chafing to get back and thinking, "It's on my desk in a duffel bag. I can't believe it." But after lunch, we came back here and opened the duffel bag, and it was an amazing experience. NARRATOR: But it didn't take long for Noel to realize there was a problem. DR. WILLIAM NOEL: I was horrified. I was aghast. 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 burned. 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 bluetack 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 that the Archimedes palimpsest was in. NARRATOR: Will Noel quickly put a team together from all over the world to try to rescue the book: Greek scholars, image experts, and a conservationist. Detailed examination in the conservation lab, by Abigail Quandt, revealed the appalling damage that the book had suffered. ABIGAIL QUANDT (The Walters Art Museum): The manuscript was heavily damaged by mold. 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. NARRATOR: And the thoughts of Archimedes are obscured by other problems, as the team quickly discovered. 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 colors were totally 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: Curiously, 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, in London. JOHN LOWDEN (Courtauld Institute of Art): I was convinced I had seen something very similar before. In 1982, a Gospel book of the 12^{th} century came up for sale, which I investigated because it had miniatures of the four Evangelists. NARRATOR: Lowden found that the miniatures in the 12^{th} 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 Lowden 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 did a test to see how the forger copied these paintings into the manuscript. 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: Quandt did the same experiment with the other three blackandwhite images. She found that each of them was precisely the same size as those in the palimpsest. The forger must simply have traced the images. Much about this 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? DR. 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: Removing the forgeries would compromise the text underneath, so they will remain for now. But the delicate task of cleaning the manuscript and attempting to recover the precious Archimedes text has begun. ABIGAIL QUANDT: On some leaves, the Archimedes text is almost invisible. On some leaves, though, you can see it as a copper brown color. It runs in two columns, perpendicular to the Christian text. I'm in 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, Quandt has to remove it from its binding in the book. This is relatively new binding, probably done in the last 100 years, and removing it has turned out to be far more important than anyone could have realized. 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: In 1906, when Heiberg studied the text, it had been impossible to read the lines in the center of the binding. Only 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. DR. 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 Institute of Technology. ROGER EASTON (Rochester Institute of Technology): We are trying to take advantage of the very slight differences in color 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 at a wide variety of wavelengths. Yeah, that looks good. All right, so, UV. Because the two inks are slightly different colors 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. 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. It's 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 color 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 (Stanford University): 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 is the diagrams. DR. WILLIAM NOEL: When Heiberg transcribed the Archimedes manuscript he was interested in the text. He wasn't interested in the diagrams at all. The diagrams in the Archimedes manuscript have never been transcribed before. They've never been properly studied. NIGEL WILSON: Over the centuries Archimedes' essays weren't copied very often, so the number of intervening copies between the original and our 10^{th} century manuscript may be very small. It might be only four or five. I would think that these 10^{th} 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 Archimedes' thinking and reveal the especially 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. Will Noel rapidly emails the pictures throughout the world to awaiting Greek scholars and mathematicians who 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 hypnotized into this text, until somehow certain traces begin to take shape. NARRATOR: Professors Netz, Wilson and Natalie Tchernetska are meeting to try to 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 (University of Cambridge, England): And then five. NIGEL WILSON: Yeah, there's a row well on, yes, towards the end. NATALIE TCHERNETSKA: Just there is an abbreviation for...probably not the script...Yeah, you have, you have a little hook. NIGEL WILSON: The hook at the bottom is there. Yes, that's good. REVIEL NETZ: Okay, let's make sure I've got the reading. NIGEL WILSON: Yes, right. REVIEL NETZ: Dot. NIGEL WILSON: Yes. REVIEL NETZ: With an omega. NIGEL WILSON: Yeah. NATALIE TCHERNETSKA: Could it be actually epsilon? Look, if you, if you look maybe here, and there are no tail... NIGEL WILSON: Ah, there's no tail. Ah, right. NARRATOR: The painfully slow process of unraveling the text will take years, but already Reviel Netz has made one very important new discovery by close examination of a proof in the Archimedes text. 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 noone 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: A breakthrough because 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 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: It was clear that Archimedes had a made a huge leap beyond ancient mathematics toward a modern understanding of infinity. REVIEL NETZ: Infinity is central to 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. DR. 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: It was Archimedes' work with infinity that ultimately led him to the beginnings of calculus. The new findings reveal that Archimedes was a more sophisticated thinker and closer to modern science than anyone had realized. It's amazing to think that a branch of mathematics so crucial to progress and human advancement 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 and measure a 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? PROF. ALEXANDER JONES: If the book had been available 100 years before the development of calculus, then things would have got going sooner. DR. 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 several hundred years ago. REVIEL NETZ: 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: It's an extraordinary question: if scientists had had access to this document, would mathematics be far more advanced now? Who knows how different the modern world might look, and all because of something written by a man in the 3^{rd} century B.C. On NOVA's Website, examine a single page from Archimedes' manuscript, and learn how scientists brought to light his extraordinary writings. Find it on PBS.org. Educators and other educational institutions can order this or other NOVA programs for $19.95 plus shipping and handling. Call WGBH Boston Video at 18002559424. Next time on NOVA: he was a World War I Ace with 80 kills to his name, but his final dogfight ended in mystery. PBS presents NOVA Who Killed the Red Baron? NOVA is a production of WGBH Boston. Major funding for NOVA is provided by the Park Foundation, dedicated to education and quality television. We see "teacher of the year." We see kids reaching their potential. It's what inspires us to create software that helps you reach yours. Science: it's given us the framework to help make wireless communications clear. Sprint is proud to support NOVA. Major funding for this program is provided by the National Science Foundation, America's investment in the future. And by the Corporation for Public Broadcasting, and by contributions to your PBS station from viewers like you. Thank you. PRODUCTION CREDITSInfinite Secrets Directed and Produced by Edited by Narrated by Associate Producer Consulting Producer Camera Sound Recordists Animation Online Editor Audio Mix Production Manager Production Team Production Coordinators Research Archival Material Special Thanks Writer for BBC Series Producer for BBC Executive Editor for BBC NOVA Series Graphics NOVA Theme Post Production Online Editor Closed Captioning NOVA Administrator Publicity Senior Researcher Production Coordinator Unit Managers Paralegal Legal Counsel Post Production Assistant Associate Producer, Post Production Post Production Supervisor Post Production Editor Post Production Manager Supervising Producers Coordinating Producer Supervising Producer Senior Science Editor Senior Series Producer Managing Director Senior Executive Producer A BBC/WGBH BOSTON CoProduction © BBC MMII Additional program material © 2003 WGBH Educational Foundation All rights reserved Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. 

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