He thought it was important to get someone to work just on those folios of the Palimpsest that contained texts not by Archimedes. He wanted to know who kept Archimedes company in this prayer book. I thought this was a good idea. Even if the text of Archimedes was well understood, there was the chance that we could find out more about the other palimpsested texts.
Pat Easterling was her supervisor. Small world. She was helpful in assessing the images, and we will have reason to look at some of her work later. We had to find the right people to image the Palimpsest. This was intimidating. I thought this was a bad idea; it seemed like a lot of work. Mike gently insisted. It would greatly increase the number of imaging procedures that we could perform on the book and it would give the participants the incentive to reduce costs and increase performance in the hope that they would be rewarded with the commission for imaging the entire volume.
This was merely sensible, he said. It sounded like rocket science to me. Then he told me for the first time about a Request for Proposals.
An RFP is quite standard to me now. It is a document in which you outline the problem and ask for a solution. Abigail wrote the RFP. It is one of a number of thorough and brilliant documents that she has written throughout the history of the project.
It started with a goal: to digitally retrieve and preserve for posterity all the writings in the folios of the Archimedes Palimpsest. It mentioned all the constraints: because the manuscript was very fragile, all the handling of the manuscript would be undertaken by Abigail and personnel that she designated. It outlined the phases of work: after the competitive phase, the selected contractor would image the entire manuscript in a disbound state.
The whole proposal ran six pages. In response to the RFP, we received six proposals. Of the six, we submitted three to Mr. B and of the three Mr. B selected two for the competition. One team consisted of Roger Easton, a faculty member at the Chester F. He now works for Boeing in Hawaii. Keith, together with Brian J. Thompson, had achieved fame years earlier by developing and patenting a method—the Knox-Thompson Algorithm— that recovers images from telescopic photographs that have been degraded by the atmosphere.
More recently, Roger and Keith had formed a team together with the late Robert H. Johnston to image degraded texts including a palimpsest in Princeton University Library and several of the Dead Sea Scroll fragments. Some of their images are actually in the catalogue. Roger, Keith, and Bob Johnston were a known quantity and a safe bet. Bill is not an imaging scientist, still less a photographer; he is a physicist.
APL employs nearly three thousand engineers, information technologists, and scientists. It works primarily on development projects funded by federal agencies. Scientists at APL participate in the entire range of data collection and analysis activities of interest to its sponsors, including data from air-, ocean-, and space-borne reconnaissance and imaging platforms. Work on non-defense, non-space projects constitutes a secondary activity of the laboratory. Impressive place; impressive guy.
His proposal was full of ideas that no one else had even considered. I did the talking, and I did the arranging. As Mike put it, I kept an awful lot of plates spinning on their poles. And I was going to have to do it for a long time. By the end of the year, I had talked to the right people and had arranged a lot. I had a plan in place and the key players were on board. I could say what I was doing to anybody who called.
If you had called me up and asked me why any of us wanted to do this work, I would immediately have referred you to Reviel Netz. This conclusion can be reached as follows. The British philosopher A. Later philosophers debated whether it was best to follow Plato or Aristotle. And so, in a real sense, all later Western philosophy is but a footnote to Plato.
The safest general characterization of the European scientific tradition is that it consists of a series of footnotes to Archimedes. By which I mean, roughly the same kind of genealogy that Whitehead meant for Plato applies to Archimedes. This book was published in , by which time Archimedes had been dead for exactly 1, years—a very long time indeed.
Yet throughout it, Galileo is in debt to Archimedes. Essentially, Galileo advances the two sciences of statics how objects behave in rest and dynamics how objects behave in motion. Galileo borrows both of these concepts— explicitly, always expressing his admiration—from Archimedes. No other authority is as frequently quoted or quoted with equal reverence. Galileo essentially started out from where Archimedes left off, proceeding in the same direction as defined by his Greek predecessor.
With Newton, the science of the scientific revolution reached its perfection in a perfectly Archimedean form. Based on pure, elegant first principles and applying pure geometry, Newton deduced the rules governing the universe. All of later science is a consequence of the desire to generalize Newtonian, that is, Archimedean methods.
The mathematics of infinity and the application of mathematical models to the physical world are closely interrelated. This is because physical reality consists of infinitesimal pulses of force acting instantaneously.
This is surprising. We might think that the mathematics of infinity is some kind of flight-of-fancy with no practical application. After all, we might think that there is no infinity to be met with in the ordinary world. Newton, in particular, used the calculus in implicit form to work out how the planets behave. It is also, at its core, the application of Archimedean insights. And so, since Archimedes led more than anyone else to the formation of the calculus and since he was the pioneer of the application of mathematics to the physical world, it turns out that Western science is but a series of footnotes to Archimedes.
Thus, it turns out that Archimedes is the most important scientist who ever lived. There is a special quality to his writings. Again and again, his readers are shocked by the delightful surprise of an unexpected combination. The main reason later scientists were so influenced by him was that he was such a pleasure to read.
Both are equally worthy of our admiration. A major discovery made in made us see, for the first time, how close Archimedes was to modern concepts of infinity. Such was the work on the Palimpsest. I would laboriously pore over a manuscript page or more often over its enhanced image on my laptop screen ; the letters forming into words, into phrases; usually nothing new; occasionally discoveries, sometimes of important historical significance; and then, twice, discoveries that shook the foundations of the history of mathematics.
I never thought that I would ever find myself laboriously poring over manuscript pages. The work of editing major texts from antiquity, based on the transcription of medieval manuscripts, was mostly completed in the nineteenth century.
This is not only because the more interesting authors have already been edited, but also because the intellectual climate today is very different from that in the nineteenth century. Nowadays, people are less interested in the dry details of texts and more interested in the syntheses based on those texts.
A PhD thesis in Classics today is usually some kind of theoretical reflection upon the established texts rather than an addition to the texts themselves. Nor is this necessarily a bad development. Nineteenthcentury scholarship was very impressive, and we owe it a great deal. But it does sometimes make for very boring reading often in Latin, at that , and it is even occasionally naive in its lack of critical and theoretical reflection.
Our understanding of the ancient world was made much richer and more profound by the application of insights from cultural anthropology, for instance, or from general poetics and linguistics. My own PhD thesis, prepared at Cambridge under the supervision of Sir Geoffrey Lloyd, the doyen of Greek science, was very much part of this modern tradition. My first book, The Shaping of Deduction in Greek Mathematics:A Study in Cognitive History, involved specifically the application of insights from cognitive science or the other way around: my hope was that cognitive scientists would find something to learn from what historians had to tell them.
The most important of them all was never translated into English. For Archimedes there existed only T. I, therefore, decided to produce a new translation with a commentary that incorporates my own theoretical angle on Greek mathematics. I was going to do more than just translate Archimedes.
I am one of a number of scholars who have, only recently, begun to pay attention to the visual aspect of science. I mentioned earlier that nineteenth-century scholarship may appear, in some respects, outdated, and here is one respect having to do with the editing work itself. The scholars who edited mathematical texts in the nineteenth century were so interested in the words that they ignored the images.
If you open an edition from that era, the diagrams you find are not based upon what is actually drawn in the original manuscripts. I was shocked to realize this and began to consider whether I should produce, for the first time, an edition of the diagrams.
I researched where those manuscripts were located. It turned out that they were in Paris, Florence,Venice, and Rome. Well, why not?
I decided it was a good idea. This was a very ambitious project and not an altogether likely one. There are some , words of Archimedes to be translated. Difficult , words. Worse, as friends kept pointing out to me— what was I going to do when the text was uncertain? How was I going to decide, given that the most important manuscript was no longer available?
Because, you see, there it was—the Archimedes Palimpsest, the unique source for Floating Bodies, Method, and Stomachion and a crucial piece of evidence for most of the other works—and no one knew where it was.
It had been studied at the beginning of the twentieth century and then it disappeared. Nor did I expect it to resurface, which was my reply to my friends: since the manuscript is likely to remain unavailable, let us just proceed as if it did not exist, otherwise we will never do anything regarding Archimedes. One day I received a letter from her. I mentioned this casually to my colleagues in ancient science, assuming they had known about this all along.
No one did. This letter from Pat Easterling was a bombshell. The news of the imminent sale broke in the community of Archimedes scholars, and the rest is history. Will has already mentioned his own meeting with Pat Easterling and his email to me. As for my reaction to this email, that is, my own wild, childish, embarrassing cries of jubilation; of this I prefer not to speak. Let us speak of Archimedes. Who Was Archimedes? It was a cataclysmic catastrophe of unprecedented proportions, turning the geopolitics of the Mediterranean upside down.
For a moment, it appeared as if Hannibal might conquer Rome. Yet, Rome survived, triumphant, and so powerful at the end of the war that the entire Mediterranean was at its mercy. The independence of Greek states was gone; the civilization that Archimedes represented was humbled.
One of the major turning points of the war came as Syracuse fell. This, the leading Greek city in the western Mediterranean, had made the wrong strategic decision of allying itself with the Carthaginians. In BC, following a long siege, its defenses—set up by Archimedes and undefeated in battle—succumbed to treachery.
We do not know how, but Archimedes died. The above, in point of fact, sums up what we know about Archimedes as a historical figure.
We are lucky to possess several historical documents from antiquity arranged as annals. These documents detail events year by year. The Roman author Livy is a famous example. Their dating system was different from ours, but occasionally such authors provide us with astronomical data, eclipses, in particular.
We can then apply Newtonian physics to calculate the date of those events, and with the results of such calculations, we gain footholds into ancient chronology, constructing the basic equivalencies between ancient dates and modern ones. Without such astronomical data, no chronology could be fixed with any certainty. The siege of Syracuse was a major event etched into ancient memory. It was listed in all the annals, and we know very well when it ended.
The figure of Archimedes, as the chief Syracusan engineer, was of great fascination to his contemporaries, and he appears again and again in ancient accounts. But other pieces of evidence are much less reliable. We know where the date comes from. But what about ? What he has to say about Archimedes comes from a gossipy, fanciful poem.
This is also the main source for the story about Archimedes inventing mirrors that burned enemy ships. Probably Archimedes was quite old so says the reliable Polybius , but nothing more is known. Here is the problem. Archimedes was so famous that legends clung to him.
And now, how are we to separate history from legend? Up until the nineteenth century it was common to accept ancient stories as reality. Since then, skepticism has reigned. Perhaps historians today are too cautious, but we tend to dismiss nearly everything that is said about Archimedes.
I doubt this myself, and let me explain why. Here is the story. Archimedes is lost in thought contemplating the problem of a crown. The crown is supposed to be made of gold, but is it pure? Then Archimedes notices the water splashing out of his bath. Compare this to an equally heavy mass of gold: does it make the same splash? The heavier it is, the smaller the splash it makes. So now you can conclude whether the crown has the specific gravity of gold or not.
The method is sound, but it is based on a trivial observation. To me it appears that Vitruvius or his previous source knew that Archimedes discovered something about bodies immersed in water. This is the pattern with all of the stories dealing with Archimedes, from Vitruvius to Tzetzes. They appear to be urban legends. Some pieces of real evidence can be put together, providing us with the outlines of a fascinating story.
As we will see again and again throughout this book, the pieces of evidence are extremely minuscule and often call for lots of interpretation. This holds for the most important piece of biographical evidence that we have about Archimedes.
You must bear in mind that, until late in the Middle Ages, Greek was written without spaces between the words. This gives rise to the following conjecture. The reason we are allowed to make corrections such as these is that mistakes were certainly made by scribes as they copied their texts; our manuscripts are full of such scribal errors. The text has to be corrected, and the correction offered is so elegant and straightforward that it seems as if it has to be true. On this thin thread hangs the entire family biography of Archimedes, which should give us a sense of how important—and how difficult—the detailed study of manuscripts is.
All of our knowledge of the ancient world is derived from the patient, laborious piecing together of jigsaw puzzles. This is a fact I find very meaningful. Art, as well as craftsmanship in general, were not highly appreciated by ancient aristocrats who generally speaking looked down on anyone dirtying their hands. And, Pheidias is the name of the most famous artist in antiquity—the master sculptor of the Parthenon in the fifth century BC.
The conclusion is quite simple. Otherwise, why give a name that had the lowly associations of craftsmanship? We have not yet exhausted the quarry of names. What about the name Archimedes? This is, in fact, unique—and uniquely appropriate to Archimedes. But it was probably meant to be read from the end to the beginning, as is more often done with Greek names. The grandfather was an artist; the father was a scientist, more specifically an astronomer who turned to the new religion of beauty and order in the cosmos; and the son who created works in which art and science, beauty and order, all worked together in perfect harmony.
Those works are of course the key to understanding Archimedes. The stories may be urban legends, but the works exist for us to read and the surprising thing is that, dry mathematical pieces as they might seem to be, they actually burst with personality. In his pure science, Archimedes keeps splashing out of the bath. Well, this is not what he did. He was wearing a tunic, contemplating diagrams drawn on sand.
We may also be tempted to imagine him as a very earnest man, dedicated entirely to the cause of impersonal truth. This, would be as wrong as the purplish vials. Archimedes was not a modern scientist. The introduction is presented as a letter to a colleague, Dositheus, and Archimedes begins by reminding Dositheus of previous letters. You will recall, says Archimedes, that I put forth a number of mathematical puzzles. I announced various discoveries and asked for other mathematicians to find their own proofs of those discoveries.
Well— notes Archimedes, somewhat triumphantly —no one did! I wish to stress that there is no doubt, on the internal evidence of his own writings, that Archimedes was indeed aware, from early on, that these two claims were false. He is not trying to save face retroactively. So, incidentally, he would most likely have thoroughly relished the future history of his writings. That the effort to read him is so tantalizing, so difficult; is precisely as Archimedes wanted.
There were no universities, no jobs, and no scientific journals. They were more akin to present-day clubs, where like-minded people come together to discuss issues of importance to them usually philosophical rather than scientific.
In Alexandria, the Ptolemaic kings set up a huge library. There were other libraries, as well, but they were not part of research institutions, instead simply marks of enormous wealth and prestige.
So, quite simply, there was no career in science. Nor was there much glory in it. After all, few people could even read science.
The real path to glory was then—as always in the pre-modern world—via poetry. If you wanted to make a name for yourself, to win some kind of eternity, you would write poems—which, after all, was what everyone read starting, in early childhood, with the Iliad and the Odyssey.
How would one become a mathematician? You would have to be exposed to it by chance—say, by your father if he happened to be an astronomer.
And then, you were hooked. This was a rare affliction. I once estimated that in the entire period of ancient mathematics, roughly from BC to AD , there were, perhaps, a thousand active mathematicians—one born every year, on average. I should make clear right now that earlier figures, such as Pythagoras and Thales, were not mathematicians at all.
Mathematics began in the fifth century BC—the age of Pericles and of the Parthenon—but very little is known about the authors. Perhaps the most important was Hippocrates of Chios. This should not be confused with the doctor of the same name, from Cos.
All we know of such authors comes from late quotations and commentaries. Even that does not survive from Eudoxus, later in the century; but Archimedes mentions him twice, with admiration. Apparently Archimedes considered Eudoxus to be his greatest predecessor; but the works of this predecessor are now all lost.
They survive in plenty. And there must have been very few of these. They were private letters—sent out to people in Alexandria who had the contacts to deliver the contents further. Everything depended on this network of individuals.
Archimedes keeps lamenting in his introductions the death of his older friend Conon who was an important astronomer. He was the only one who could understand me! There was no one to write to, no reader good enough.
There would be, in time. He must have known that he was writing for posterity. Many of the works are addressed to Dositheus, of whom very little is known.
We do know one thing, which, yet again, is based on his name alone. It turns out that practically everyone in Alexandria at that time named Dositheus was Jewish. The name, in fact, is simply the Greek version of Matityahu or Matthew. This is very curious: the correspondence between Archimedes and Dositheus is the only one known from antiquity between a Greek and a Jew. It is perhaps telling that the arena for such cross-cultural contact was science. In mathematics, after all, religion and nation do not matter.
This, at least, has not changed. First came a treatise on the Quadrature of the Parabola. Then two separate books on Sphere and Cylinder.
Then a book on Spiral Lines. The one in which the hoax was revealed. And, finally, a book on Conoids and Spheroids. There might have been more, but these are the five books that survive. The five works form a certain unity, as together they constitute the cornerstone of the calculus. However, this is probably not how Archimedes would have thought of them. To him, they were all variations on squaring the circle. That is: time and again, Archimedes takes an object bounded by curved lines and equates it with a much simpler object, preferably bounded by straight lines.
Apparently this task—squaring, or measuring, the circle—was, for Greek mathematicians, the Holy Grail of their science. The very idea of measurement depends on the notion of the straight line. It is not for nothing that we measure with rulers.
To measure is to find a measuring tool and apply it successively to the object being measured. Suppose we want to measure a straight line. For instance, suppose we want to measure your height, which is really saying that we want to measure the straight line from the floor to the top of your head. Then what we do is take a line the length of an inch and apply it successively, well over a sixty times, but probably less than eighty times to measure your height. Since this is very tiresome, we have pre-marked measuring tapes that save us the trouble of actually applying the length successively, but, at the conceptual level, successive application is precisely what takes place.
When measuring area, we do the same successive application, but instead of a straight line, we use a square. This is why floor plans are literally measured by square feet.
Cubes similarly measure volume. Of course, not all objects come pre-packaged in squared or cubed units. Second, every triangle — no matter its shape — is exactly half the rectangle enclosing it, as a consideration of the two symmetries in the figure serves to show. Third, every rectangle can be easily transformed into an equal square, by a proportional reduction and enlargement; reduce the length, and enlarge the width, by exactly the same ratio, so that length and width become equal.
The combination of these three facts means that it is possible to measure any area bounded by straight lines as a sum of squares. It is all this straightforward. Take any object bounded by straight lines.
Take, instead, an object as apparently simple as a baseball—just the most ordinary sphere—and measurement suddenly breaks down. It is impossible to divide the baseball into any finite number of pyramids or triangles. The baseball has an infinitely complex, infinitely smooth surface. Archimedes would measure such objects again and again, pushing the most basic tools of mathematics.
In the Quadrature of the Parabola, Archimedes measured the segment of a parabola: it is four-thirds times the triangle it encloses see fig. A very striking measurement, given that the parabola is a curved line, so this is rather like squaring a circle.
He also, in the same treatise, introduced a certain daring thought experiment: to conceive of a geometrical object as if it were composed of physical slices hung on a balance. The two books on Sphere and Cylinder directly approach the volume of the sphere. It turns out that it is exactly two-thirds the cylinder enclosing it. What is its surface? It turns out it is exactly four times its greatest circle see fig.
This recalcitrant object—the sphere—turns out to obey some very precise rules. In the second book, remarkable tasks are achieved. Instead he invents a new curved object—a complex, counter-intuitive object—and then measures it. The spiral line—invented by Archimedes—turns out to enclose exactly one-third the area enclosed by the circle surrounding it see fig. As for conoids, which are hyperbolas or parabolas turned around so as to enclose space, and spheroids, which are ellipses turned around in similar fashion—these have more complex measurements.
All were obtained, with precision, by Archimedes see figs. This is a major feature of all of his works. Archimedes starts out promising to make some incredible measurement, and you expect him to fudge it somehow, to cut corners. How else can you square the circle? And then he begins to surprise you. He accumulates results of no obvious relevance—some proportions between this and that line, some special constructions of no direct connection to the problem at hand.
In all of them Archimedes was furthering the mathematics of infinity. Imaginary Dialogues In his measurements Archimedes adopts a surprising, circuitous route, which was always his favorite way of approaching things.
Indirect proof is easier to understand, and you have probably engaged in some version of it yourself. You try to convince someone of the truth of your position. Let us say, for instance, that you want to convince your interlocutor that, when you draw a straight line joining two points on the circumference of a circle, all the points on this straight line must fall inside the circle.
Everything you tell him about this line fails to persuade him. And so you resort to indirect proof. You assume the opposite of the truth, as if pretending to agree with your interlocutor.
And now you follow the logic of this situation, until you draw the following conclusion: line DZ is both smaller than DE and bigger than it.
But a line cannot be at the same time both smaller and greater than the same given line. Archimedes did not invent potential infinity, but he made it his own in a series of original applications. What Archimedes does is develop a certain mechanism, capable of indefinite extension, of packing triangles or their like into the curved object.
This is best seen, once again, as an imaginary dialogue between Archimedes and his critic. Let us say that he has packed the curved object in such a way that a certain area has been left out, an area greater than the size of a grain of sand.
This dialogue could go on indefinitely. This is what philosophers refer to as potential infinity. We never go as far as infinity itself in this argument. There is no mention, at any point, of an area which is infinitesimally small, merely of areas that are very, indefinitely small. But we allow ourselves to go on indefinitely.
And this, taken together with indirect proof, allows Archimedes to measure the most incredible objects. Squaring the Parabola Three times in his career Archimedes proved that the parabolic segment—a certain curved object—is exactly four-thirds the triangle it encloses. This was his favorite measurement. Later on we shall see his most spectacular measurement, which transcends geometry itself. This can be made to approach the parabola as closely as we wish. We assume its difference from the parabola is less than a grain of sand FIGURE mathematicians have a hard time unraveling.
It is like an affirmation based on a double double negation. And this is how it works. Since what we are going to prove is that the area of the curve is four-thirds the triangle, how shall we start?
By assuming, of course, that the area of the curve is not four-thirds the triangle! This, after all, is how indirect proof works. Let us assume that the curved area is greater than four-thirds the triangle, by a certain amount: 1.
The curve is greater than four-thirds the triangle, by a certain amount. Let us say, it is greater by a grain of sand. For exactly such occasions, Archimedes has a special mechanism up his sleeve. He fills up the curve with triangles so that the difference between the triangles and the curve is smaller than a grain of sand!
We therefore now have two objects side by side. One is a curve. The curve, with a grain of sand removed, is smaller than the jumble of triangles see fig.
At this point Archimedes leaves aside the results obtained so far. Remember that the jumble of triangles is an object bounded by straight lines. Four-thirds the enclosed triangle is also an object bounded by straight lines, i. It is therefore not a surprise that through the application of geometrical ingenuity one can determine a definite measurement comparing the jumble of triangles and four-thirds the enclosed triangle.
What Archimedes would come up with—applying his geometrical ingenuity—is as follows: 3. The jumble of triangles is smaller than four-thirds the enclosed triangle see fig. Now, recall result 1. The curve, with a grain of sand removed, is equal to fourthirds the triangle. In other words, the jumble of triangles is the greater. It is greater than four-thirds the triangle, which we can put as: 5. This volume provides colour images and transcriptions of three of the texts recovered from it.
Pride of place goes to the treatises of Archimedes, including the only Greek version of Floating Bodies, and the unique copies of Method and Stomachion. This transcription provides many different readings from those made by Heiberg from what he termed Codex C in his edition of the works of Archimedes of Secondly, fragments of two previously unattested speeches by the Athenian orator Hyperides, which are the only Hyperides texts ever to have been found in a codex.
Thirdly, a fragment from an otherwise unknown commentary on Aristotle's Categories. In each case advanced image-processing techniques have been used to create the images, in order to make the text underneath legible. This tenth century manuscript is the unique source for two of Archimedes Treatises, The Method and Stomachion, and it is the unique source for the Greek text of On Floating Bodies. Discovered in by J. Heiberg, it plays a prominent role in his edition of the works of Archimedes, upon which all subsequent work on Archimedes has been based.
The manuscript was in private hands throughout much of the twentieth century, and was sold at auction to a private collector on the 29th October Since that date the manuscript has been the subject of conservation, imaging and scholarship. The version shown here has undergone image processing techniques to suppress the prayerbook writing over the original text. The Manuscripts: Part II. History: 1.
The making of the Euchologion Abigail Quandt; 2. Advanced embedding details, examples, and help! Usage Attribution 3. It originally was a 10th century copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes c.
The manuscript currently belongs to an American private collector It contains The most remarkable of the above works is The Method, of which the palimpsest contains the only known copy. In his other works, Archimedes often proves the equality of two areas or volumes with Eudoxus' method of exhaustion, an ancient Greek counterpart of the modern method of limits.
The work was originally thought to be lost, but was rediscovered in the celebrated Archimedes Palimpsest. Since that date the manuscript has been the subject of conservation, imaging and scholarship, in order to better read the texts.
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