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Author Topic: X-rays Reveal Archimedes Secrets  (Read 1134 times)
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« on: August 31, 2006, 01:15:18 AM »

X-rays reveal Archimedes secrets

By Jonathan Fildes
Science and technology reporter, BBC News

A series of hidden texts written by the ancient Greek mathematician Archimedes are being revealed by US scientists.

Until now, the pages have remained obscured by paintings and texts laid down on top of the original writings.

Using a non-destructive technique known as X-ray fluorescence, the researchers are able to peer through these later additions to read the underlying text.

The goatskin parchment records key details of Archimedes' work, considered the foundation of modern mathematics.

The writings include the only Greek version of On Floating Bodies known to exist, and the only surviving ancient copies of The Method of Mechanical Theorems and the Stomachion.

In the treatises, the 3rd Century BC mathematician develops numerical descriptions of the real world.

"Archimedes was like no-one before him," says Will Noel, curator of manuscripts and rare books at the Walters Art Museum in Baltimore, Maryland and director of the imaging project.

"It just doesn't get any better than re-reading the mind of one of the greatest figures of Western civilisation."

'Eighth wonder'

Revealing Archimedes' writings presents a huge challenge to the imaging team.

The original texts were transcribed in the 10th Century by an anonymous scribe on to parchment.

Three centuries later a monk in Jerusalem called Johannes Myronas recycled the manuscript to create a palimpsest.

Palimpsesting involves scraping away the original text so the parchments can be used again. To create a book, the monk cut the pages in half and turned them sideways.

To create a book Myronas also used recycled pages from works by the 4th Century Orator Hyperides and other philosophical texts.

Mr Noel describes the palimpsest as "the eighth wonder of the world".

"You never get three unique palimpsested texts from the ancient world together in one book," he told the BBC News website. "That's just completely unheard of."

The monks filled the recycled pages with Greek Orthodox prayers.

Later, forgers in the 20th Century added gold paintings of religious imagery to try to boost the value of the tome.

The result was the near total obliteration of the original texts apart from faint traces of the ink used by the 10th Century Scribe.

Bright light

Previously the privately-owned palimpsest has been investigated us

The palimpsest contains pages from several bodies of work
ing various optical and digital imaging techniques.

However, much of the text remained hidden behind paint and stains.

The researchers have now turned to a technique known as X-ray fluorescence to tease out the final details of the writings.

The method is used in many branches of science including geology and biology. It has previously been used by other researchers to decode ancient texts.

In August 2005 a team from Cornell University successfully deciphered a series of 2,000-year-old worn down stone inscriptions.

The X-rays are formed in a synchrotron - a particle accelerator that uses electrons travelling at close to the speed of light to generate powerful "synchrotron" light.

The light covers a wide range of the electromagnetic spectrum, including powerful X-rays, a million times more intense than a transmission X-ray used in medical imaging.

"In fluorescence it's like looking at the stars at night whereas in transmission it's like looking during the day," explains Dr Uwe Bergmann of the Stanford Synchrotron Radiation Lab in the US, where the work is being done.

The light enables scientists to look inside matter at the molecular and atomic scale.

Glowing words

The technique is particularly useful for probing the palimpsest because the ink used by the scribe to record Archimedes' work contains iron.

"When the X-rays hit an iron atom it emits a characteristic radiation, it glows," says Dr Bergmann. "When you record the glow you can reconstruct an image of all of the iron in the book."

The glowing words are displayed on a computer screen, giving the researchers the first glimpse of the text in nearly 800 years.

"It's like receiving a fax from the 3rd Century BC," said Mr Noel. "It's the most sensational feeling."

Each page takes 12 hours to reconstruct as the highly focused beam of X-rays, the width of a human hair, sweeps across the page.

The team have until the 7 August this year to scrutinise the palimpsest, before the synchrotron is switched off for maintenance.

During that time they hope to scan between 12 and 14 pages, paying particular attention to the areas covered with the forged paintings.

The public can watch the researchers as they reveal the glowing ancient text during a live webcast at 2300 GMT on 4 August.


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« Reply #1 on: August 31, 2006, 09:57:40 AM »

Archimedes (Greek: Αρχιμήδης ) (c. 287 BC ? 212 BC) was an ancient Greek mathematician, physicist, engineer, astronomer, and philosopher born in the seaport colony of Syracuse, Sicily. He is considered by some historians of mathematics to be one of the greatest mathematicians in antiquity; Carl Friedrich Gauss considered him one of the two greatest ever (the other being Isaac Newton).

Writings by Archimedes

    * On the Equilibrium of Planes (2 volumes)

    This scroll explains the law of the lever and uses it to calculate the areas and centers of gravity of various geometric figures.

    * On Spirals

    In this scroll, Archimedes defines what is now called Archimedes' spiral. This is the first mechanical curve (i.e., traced by a moving point) ever considered by a Greek mathematician.

    * On the Sphere and The Cylinder

    In this scroll Archimedes obtains the result he was most proud of: the relation between the area of a sphere to that of a circumscribed straight cylinder is the same as that of the volume of the sphere to the volume of the cylinder (exactly 2/3).

    * On Conoids and Spheroids

    In this scroll Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.

    * On Floating Bodies (2 volumes)

    In the first part of this scroll, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This was probably an attempt at explaining the observation made by Greek astronomers that the Earth is round. Note that his fluids are not self-gravitating: he assumes the existence of a point towards which all things fall and derives the spherical shape. One is led to wonder what he would have done had he struck upon the idea of universal gravitation.
    In the second part, a veritable tour-de-force, he calculates the equilibrium positions of sections of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, which is reminiscent of the way icebergs float, although Archimedes probably was not thinking of this application.

    * The Quadrature of the Parabola

    In this scroll, Archimedes calculates the area of a segment of a parabola (the figure delimited by a parabola and a secant line not necessarily perpendicular to the axis). The final answer is obtained by triangulating the area and summing the geometric series with ratio 1/4.

    * Stomachion

    This is a Greek puzzle similar to a Tangram. In this scroll, Archimedes calculates the areas of the various pieces. This may be the first reference we have to this game. Recent discoveries indicate that Archimedes was attempting to determine how many ways the strips of paper could be assembled into the shape of a square. This is possibly the first use of combinatorics to solve a problem.

    * Archimedes' Cattle Problem

    Archimedes wrote a letter to the scholars in the Library of Alexandria, who apparently had downplayed the importance of Archimedes' works. In these letters, he dares them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations, some of them quadratic (in the more complicated version). This problem is one of the famous problems solved with the aid of a computer. The solution is a very large number, approximately 7.760271 ? 10206544 (See the external links to the Cattle Problem.)

    * The Sand Reckoner

    In this scroll, Archimedes counts the number of grains of sand fitting inside the universe. This book mentions Aristarchus of Samos' theory of the solar system (concluding that "this is impossible"), contemporary ideas about the size of the Earth and the distance between various celestial bodies. From the introductory letter we also learn that Archimedes' father was an astronomer.

    * The Method

    In this work, which was unknown in the Middle Ages, but the importance of which was realised after its discovery, Archimedes pioneers the use of infinitesimals, showing how breaking up a figure in an infinite number of infinitely small parts could be used to determine its area or volume. Archimedes probably considered these methods not mathematically precise, and he used these methods to find at least some of the areas or volumes he sought, and then used the more traditional method of exhaustion to prove them. Some details can be found at how Archimedes used infinitesimals.

Archimedes Palimpsest
A team of imaging scientists from the Rochester Institute of Technology and Johns Hopkins University has used computer processing of digital images from various spectral bands, including ultraviolet and visible light, to reveal the Archimedes text. Dr. Reviel Netz [2] of Stanford University has been trying to fill in gaps in Heiberg's account with these images.

Four pages that had been painted over with Byzantine-style religious images, which turned out to be 20th-century forgeries intended to increase the value of the prayer book, rendered the underlying text of Archimedes forever illegible, it appeared. Then, in May 2005, highly focused X-rays produced at the Stanford Linear Accelerator Center in Menlo Park, California were used to begin deciphering the parts of the 174-page text that have not yet been revealed. The production of x-ray fluorescence was described by Keith Hodgson, director of SSRL. "Synchrotron light is created when electrons traveling near the speed of light take a curved path around a storage ring?emitting electromagnetic light in X-ray through infrared wavelengths. The resulting light beam has characteristics that make it ideal for revealing the intricate architecture and utility of many kinds of matter?in this case, the previously hidden work of one of the founding fathers of all science."

A page from "On Floating Bodies"

The Archimedes Palimpsest
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.
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« Reply #2 on: August 31, 2006, 12:53:20 PM »

Is the Stomachion puzzle the same thing Mobius tried to do with a two dimensional object?


« Reply #3 on: August 31, 2006, 01:42:37 PM »

Is the Stomachion puzzle the same thing Mobius tried to do with a two dimensional object?

Maybe similar, Doc, though not quite the same. The Mobius Strip can be represented by a tape in a loop and a twist in it, so that if one cuts it lengthways, it is still a loop, and if one cuts its again, it becomes two interlocking loops. When I cut this piece of paper in half, I will still have one piece of paper?


The stomachion is a 14-piece dissection puzzle similar to tangrams. It is described in fragmentary manuscripts attributed to Archimedes Eric Weisstein's World of Biography as noted by Magnus Ausonius (310-395 A.D.). The puzzle is also referred to as the "loculus of Archimedes" (Archimedes' box) or "syntemachion" in Latin texts. The word stomachion has as its root the Greek word sigmatauomicronmualpha^'chiiotaomicronnu, meaning "stomach." Note that Ausonius refers to the figure as the "ostomachion," an apparent corruption of the original Greek.

The puzzle consists of 14 flat pieces of various shapes arranged in the shape of a square, with the vertices of pieces occurring on a 12x12 grid. Two pairs of pieces are duplicated. Like tangrams, the object is to rearrange the pieces to form interesting shapes such as the elephant illustrated above (Andrea).

« Reply #4 on: August 31, 2006, 09:49:55 PM »

Consider this: after the losses of the Alexandra and Constantinople libraries, we have only a minute fraction of what the Classical Greeks knew and we know of their analog computer and these papers of Archimedes. We've missed out on a lot.
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« Reply #5 on: September 01, 2006, 03:28:34 AM »

Very interesting and intriguing piece of Ancient History. Never any end to learning, is there?


« Reply #6 on: September 01, 2006, 10:16:32 AM »

I hope not  Smiley
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« Reply #7 on: December 28, 2006, 08:15:30 AM »

December 26, 2006:

 13th century Text Hides Words of Archimedes

The pages of a medieval prayer text also contain words of ancient Greek engineer Archimedes. It takes high-tech imaging to read between the lines.

     The book cost $2 million at auction, but large sections are unreadable. Some of its 348 pages are torn or missing and others are covered with sprawling purple patches of mildew. Sooty edges and water stains indicate a close escape from a fire.

     "This manuscript is, by far, the worst of any manuscript I've ever seen," said William Noel, curator of manuscripts for the Walters Art Museum in Baltimore, where it now esides. "It's a book that is on its last legs."

     The sheepskin parchment originally contained a 10th century Greek text, which was erased by a 13th century scribe who replaced it with prayers. Seven hundred years later, a forger painted gilded pictures of the Evangelists on top of the faded words.

     Underneath it all, however, is an exceptional treasure ? the oldest surviving copy of works by the ancient Greek mathematician and engineer Archimedes of Syracuse, who lived in the 3rd century BC.

     About 80% of the text had been transcribed and translated in the 1910s after it was rediscovered in an Istanbul monastery, but since then much of it became unreadable again because of deterioration.

     Fully deciphering its mysteries has had to wait for advanced technologies, some of which had never been applied to ancient manuscripts.

     The unusual cast of detectives includes not only the imaging specialists who helped photograph the Dead Sea Scrolls, but also a Stanford University physicist who studies trace metals in spinach with a particle accelerator.

     Together, they have been carrying out one of the most remarkable "salvage jobs" in the history of codicology, the study of ancient manuscripts.

     Archimedes, it turns out, is only one secret of the text. Among the mathematicians of antiquity, Archimedes was one of the greatest and most cunning.

     He was one of the earliest to devise ways to calculate the area beneath curves and was the first to prove that a circle's circumference and diameter are related by the constant pi. He developed the Archimedes Screw to lift water and invented deadly devices, such as the Claw of Archimedes, which was designed to grapple enemy warships.

     Archimedes died in 212 BC, when Syracuse was sacked by the Romans. Legend holds that he was drawing figures in the sand. "Don't disturb my circles," he supposedly told the soldier who killed him.

     Knowledge of Archimedes' work is derived from three books. Codex A, transcribed around the 9th century, contained seven major treatises in Greek. Codex B, created around the same time, had at least one additional work by Archimedes and survived only in Latin translation.

     Codex C has been an enigma. It was originally copied down in 10th century Constantinople, now known as Istanbul. Three centuries later, the manuscript was in Palestine. By then, it was no longer a precious vestige of ancient learning but an obscure text that could be put to better use as a prayer book.

     A scribe began by unbinding the pages. He washed them with citrus juice or milk and sanded them with a pumice stone. He cut the sheets in half, turned them 90 degrees and stitched the new book down the middle.

     The scribe wrote prayers over the blank pages. Codex C had become a "palimpsest" ? a recycled book.    The book eventually was brought back to Constantinople, where it sat until the 1890s, when a Greek scholar wrote down a fragment of erased text that he was able to read.

     That fragment was brought to the attention of Danish philologist Johan Ludvig Heiberg in 1906, then the foremost authority on Archimedes. Armed with a magnifying glass, he translated everything he could read, publishing his work in 1910.

     The palimpsest disappeared amid the chaos of World War I, only resurfacing in 1998, when a French family named Guersan offered it for auction at Christie's in New York. An anonymous book collector paid $2 million and deposited it at the Walters Art Museum for conservation.

     Mold had attacked much of the manuscript, and four forged paintings of the Evangelists made in the 20th century covered some of its most important pages.

     "That was our worst nightmare," said Abigail Quandt, senior conservator of rare books and manuscripts at the Walters Art Museum.

     Roger L. Easton Jr., a 56-year-old imaging specialist at the Rochester Institute of Technology, had just come off his success revealing hidden text in the Dead Sea Scrolls.

     Christie's had commissioned him to make ultraviolet images of the palimpsest for the auction catalog, and now he offered his help to the museum. Easton and his colleagues began their work in 2000. They tinkered with different methods for capturing the image with the ultraviolet light, which makes the parchment glow more whitish.

     They then merged those images with another set taken under a tungsten light, which enhanced the reddish hue of the Archimedes text. The resulting "pseudocolor" image made it easier to distinguish the black prayer book writing from the burnt sienna words of Archimedes.

     Using this painstaking method, Easton and his team took two years to uncover another 15% of the text. They were stymied in penetrating the rest. Two more years passed before Stanford physicist Uwe Bergmann, 43, read a magazine article about the Archimedes palimpsest that mentioned it had originally been written with iron gall ink.

     One of Bergmann's projects at Stanford was investigating the process of photosynthesis in plants by using the synchrotron X-rays to image small clusters of manganese atoms in spinach.

     "Why not find traces of iron in an ancient book?" he asked. Bergmann sent an e-mail to the Walters Art Museum, and the museum agreed to a test.

     Bergmann set up the palimpsest experiment at the Stanford Synchrotron Radiation Laboratory. Spread over an area the size of a football field, the synchrotron is part of the Stanford Linear Accelerator Center, a Department of Energy facility set in the foothills of Menlo Park.

     The synchrotron hurls electrons at near light speed, forcing them to give off X-rays as they veer around bends. That X-ray beam is channeled away into the laboratories.  Bergmann figured the powerful and precise beam could be used to make iron molecules fluoresce, thus allowing him with a sensitive-enough detector to pick up even the faintest traces of ink.

     Bergmann first had to determine the exposure time. Too much time and the powerful synchrotron X-ray could damage the parchment. Then, they adjusted the intensity of the beam, which could be so strong that it blinded the detectors that picked up the glow from the iron gall ink.

     After two years of refining their technique, Bergmann and his colleagues began the laborious process of imaging the palimpsest this summer. Each side of a page, mounted in frame that moved in front of the beam, took 12 hours to record. The machines processed the pages continuously for two weeks.

     Beneath a moldy, torn painting of St. John emerged two layers of writing. On the edge of the first page, they saw a signature dated April 14, 1229: "By the hand of presbyter Ioannes Myronas."

     It was the name of the priest who had erased Archimedes.

     In an office near Memorial Church at Stanford, Reviel Netz flicked off the lights. Netz, a slight 38-year-old with dark hair, leaned close to the screen of his laptop. Bergmann's X-ray work had produced a black-and-white picture of a page from "The Method of Mechanical Theorems," a text found only in the palimpsest.

     One phrase ? "let them be arranged so they balance on point theta" ? had already been translated by Heiberg, although he had had to guess about the word "on," which was unreadable.

     Netz, a professor of classics, looked at the X-ray image and nodded. He smiled.
The actual word was "around."

     "That's not trivial," he said, explaining that the change altered the meaning of Archimedes' calculations involving an object's center of gravity.

     The X-ray image also revealed a section of "The Method" that had been hidden from Heiberg in the fold between pages. It contained part of a discussion on how to calculate the area inside a parabola using a new way of thinking about infinity, Netz said. It appeared to be an early attempt at calculus ? nearly 2,000 years before Isaac Newton and Gottfried Wilhelm Leibniz invented the field.

     The discoveries may seem small, but they are significant in the understanding of ancient mathematics, Netz said.

     One passage he studied several years ago involved the innumerable slices and lines that could be made from a triangular prism similar to a wedge of cheese. Netz said the passage, which was unreadable to Heiberg, showed that Archimedes was grappling with the concept of infinity long before other mathematicians.

     For Netz, a specialist in ancient mathematics and cognitive history, the chance to decipher the palimpsest "is the fulfillment of an incredible dream," he said.

     One of his biggest breakthroughs involves a quirky part of the palimpsest called the "Stomachion," which literally means "Belly-Teaser."

     Stomachions were children's games in which 14 geometrical shapes were rearranged to create new shapes. Heiberg translated fragments of the manuscript but paid little attention to it, thinking it was just a game.

     Netz saw a deeper significance. Archimedes asked a more restricted question in his "Stomachion": How many different ways could you combine the 14 triangles to make a square?

     Netz believes the fragments address an area of mathematics known as combinatorics that scholars have only recently believed interested the Greeks. For all the high-tech efforts, there are still gaps remaining in the Archimedes text, perhaps 2%, Netz guessed.

     Among the jumbled fragments are clues that perhaps the deepest secrets are yet to be found. A century ago, Heiberg copied down two lines that he couldn't identify. They began: "The youngest had been abroad for so long that the sisters wouldn't even know who was who."

     The passage was not Archimedes. In 2002, scholars were able to cross-reference the quote. It came from "Against Timandros," written by a 4th century BC Athenian orator named Hyperides.

     Although Hyperides is little-known now, contemporaries frequently compared him to Demosthenes, an acknowledged master of oratory. No complete versions exist of "Against Timandros," which Hyperides had written as part of a lawsuit over an inheritance, said Judson Herrman, a classicist at Allegheny College in Pennsylvania.

     Further study determined there were 20 pages of Hyperides in the palimpsest, including a previously unknown text called "Against Diondas." The palimpsest, it turns out, took parchment from seven texts, including what are believed to be a commentary on Aristotle's "On the Soul" and a group of biographies of the saints, plus two still unidentified texts.

     The works are even more difficult to discern than the Archimedes because the ink is different and the pages more thoroughly scrubbed.

     "I have been cursing all morning," Herrman said of his work on a few lines of Hyperides.

     The scientists aren't giving up. Easton's team recently began experimenting with precisely tuned light-emitting diodes, or LEDs, to illuminate the text. The team also is using angled light to detect the outlines of letters etched in the parchment by the acid in the ink.

     The team made progress on a few pages, but it may take decades ? or longer ? before technologies are developed that can unveil all the texts. "We'll probably leave something for future scientists to work on," Netz said.


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« Reply #8 on: December 28, 2006, 08:23:11 AM »

     The Claw of Archimedes was an ancient weapon devised by Archimedes to defend the seaward portion of Syracuse's city wall against amphibious assault. Although its exact nature is unclear, the accounts of ancient historians seem to describe it as a sort of crane equipped with a grappling hook that was able to lift attacking ships partly out of the water, then either cause the ship to capsize or suddenly drop it.

     These machines featured prominently during the Second Punic War in 213 BC, when the Roman Republic attacked Syracuse with a fleet of at least 120 Quinqueremes under Marcus Claudius Marcellus. When the Roman fleet approached the city walls under cover of darkness, the machines were deployed, sinking many ships and throwing the attack into confusion. Historians such as Polybius and Livy attributed the Romans' defeat to these machines, together with catapults also devised by Archimedes.

     The plausibility of this invention was tested on the television show Superweapons of the Ancient World by bringing in a group of engineers to try and conceive of, design and implement a design that was realistic given what we know about Archimedes. Within seven days they were able to test their creation. They did indeed succeed in tipping a model of a Roman ship over so that it would sink. While this does not prove the existence of the Claw, it does, at least, demonstrate its possibility.


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« Reply #9 on: December 28, 2006, 08:34:09 AM »

     Hypereides (c. 390-322 BC) was a logographer (orator for the courts) in Ancient Greece. He was one of the ten Attic orators included in the Alexandrian Canon compiled by Aristophanes of Byzantium and Aristarchus of Samothrace in the third century BC.

     William Noel, the curator of manuscripts and rare books at the Walters Art Museum in Baltimore, Maryland and the director of the Archimedes Palimpsest project, called Hyperides "one of the great foundational figures of Greek democracy and the golden age of Athenian democracy, the foundational democracy of all democracy.

Rise to power

     Little is known about his early life except that he was the son of Glaucippus, of the deme of Collytus and that he studied logography under Isocrates. In 360 BC he prosecuted Autocles for treason.[2] During the Social War (358?355 BC) he accused Aristophon, then one of the most influential men at Athens, of malpractices,[3] and impeached Philocrates (343 BC) for high treason. Although Hypereides supported Demosthenes in the struggle against Phillip II of Macedon; that support was withdrawn after the Harpalus affair. After Demosthenes' exile Hypereides became the head of the patriotic party (324 BC).


     After the death of Alexander the Great, Hypereides was one of the chief promoters of war against Macedonian rule. His speeches are believed to have led to the outbreak the Lamian war (323-322 BC) in which Athens, Aetolia, and Thessaly revolted against Macedon rule. After the decisive defeat at Crannon (322 BC) in which Athens and her allies lost their independence, Hypereides and the other orators, were condemned to death by the Athenian supporters of Macedonia.

     Hypereides fled to Aegina only to be captured at the temple of Poseidon. After being put to death his body (according to others) taken to Cleonae and shown to the Macedon general Antipater before being returned to Athens for burial.

Personality and Oration style

     Hypereides was an ardent pursuer of "the beautiful," which in his time generally meant pleasure and luxury. His temper was easy-going and humorous; and hence, though in his development of the periodic sentence he followed Isocrates, the essential tendencies of his style are those of Lysias, whom he surpassed, however, in the richness of his vocabulary and in the variety of his powers. His diction was plain and forcible, though he occasionally indulged in long compound words probably borrowed from the Middle Comedy, with which, and with the everyday life of his time, he was in full sympathy. His composition was simple. He was especially distinguished for subtlety of expression, grace and wit, as well as for tact in approaching his case and pseudo-Longinus [4]

Surviving speeches

     Seventy-seven speeches have been attributed to Hypereides, of which seventy-five were regarded as spurious by his contemporaries. It is said that a manuscript of most of the speeches survived as late as the 15th century in the library of Matthias Corvinus, king of Hungary, but was later destroyed after the capture of Buda by the Turks in the 16th century. Only a few fragments were known until relatively recent times. In 1847 large fragments of his speeches, Against Imosthenes and For Lycophron (incidentally interesting clarifying the order of marriage processions and other details of Athenian life, and the Athenian government of Lemnos) and the sole of the For Euxenippus (c. 330, a locus classicus on state prosecutions), were found in a tomb at Thebes in Egypt. In 1856 a considerable portion of a eulogy for Leosthenes and his comrades who had fallen in the Lamian war. Currently this is the best surviving example of epideictic oratory.

     Towards the end of the ninteenth century further discoveries were made including the conclusion of the speech Against Philippides (dealing with an indictment for the proposal of unconstitutional measure, arising out of the disputes of the Macedonian and anti-Macedonian parties at Athens), and of the whole the Against Athenogenes (a perfumer accused of fraud in the sale his business).

New discoveries

     It was Dr Natalie Tchernetska of Trinity College Cambridge who discovered and identified in 2002 fragments of two speeches of Hyperides that have been considered lost, Against Timandros and Against Diondas. Dr Tchernetska's discovery led to a publication in ZPE 154 (2005), which in turn prompted the establishment of a working group under the auspices of the British Academy, which includes scholars from the UK, Hungary, and USA.

     In 2006 the Archimedes Palimpsest project, based at the Walters Art Museum in Baltimore, Maryland, cooperated with imagers at Stanford University who used powerful X-ray fluorescence imaging to read the final pages of the Palimpsest, which were interpreted, transcribed and translated by a group of scholars in the United States and Europe.

     The new Hyperides revelations include two previously unknown speeches, effectively increasing this renowned orator?s body of work by 20 percent, said Judson Herrman, a 36-year-old professor of classics at Allegheny College in Meadville, Pennsylvania. He is one of a handful of classicists who have written doctoral dissertations on Hyperides.
"It?s a spotlight shining on an important moment in history,? said Mr. Herrman, currently a fellow at the National Humanities Center in Research Triangle Park, North Carolina. Until the new leaves were found in the palimpsest, most scholars believed only fragments of Hyperides survived beyond the Classical period. [...] W. Robert Connor, the president of the Teagle Foundation, which provides education and financial resources for education, called the discovery of new Hyperides text a ?tour de force of the first order.?[5]

Lost speeches

     Among the speeches not yet recovered is the Deliacus [6] in which the presidency of the Delian temple claimed by both Athens and Cos, which was adjudged by the Amphictyonic League to Athens. Also missing is the speech in which he defended the illustrious courtesan Phryne (said to have been his mistress) on a capital charge: according to Plutarch and Athenaeus the speech climaxed with Hyperides stripping off her clothing to reveal her naked breasts; in the face of which the judges found it impossible to condemn her.


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« Reply #10 on: December 28, 2006, 08:53:28 AM »

    Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. In particular, it is concerned with "counting" the objects in those collections (enumerative combinatorics), with deciding when the criteria can be met, with constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), with finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and with finding algebraic structures these objects may have (algebraic combinatorics).

     Combinatorics is as much about problem solving as theory building, though it has developed powerful theoretical methods, especially since the later twentieth century. One of the oldest and most accessible parts of combinatorics is graph theory which is now connected to other areas.

     An example of a combinatorial question is the following: What is the number of possible orderings of a deck of 52 playing cards? That number equals 52! (fifty-two factorial), which is equal to 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000.

      An example of another kind is this problem: Given a certain number n of people, is it possible to assign them to sets so that each person is in at least one set, each pair of people is in exactly one set together, every two sets have exactly one person in common, and no set contains everyone, all but one person, or exactly one person? The answer depends on n. See "Design theory" below.

     The earliest recorded statements of combinatorical rules appear in India. The medical treatise Sushruta Samhita written by Sushruta in the 6th century BC states that 63 combinations can be made out of six different tastes ? bitter, sour, salty, sweet, astringent, and hot ? by taking them one at a time, two at a time, three at a time, etc. In other words, there are 6 single tastes, 15 combinations of two, 20 combinations of three, etc. The Bhagabati Sutra, written by a Jaina mathematician circa 300 BC, contains rules on combinations and permutations corresponding to:

(see link at bottom for these examples)

     Numbers are calculated in the cases where n = 2, 3 and 4. The author then says that one can compute the numbers in the same way for larger n: "In this way, 5, 6, 7, ..., 10, etc. or an enumerable, unenumerable or infinite number of things may be specified. Taking one at a time, two at a time, ... ten at a time, as the number of combinations are formed they must all be worked out." This suggests that the arithmetic can be extended to various infinite numbers. The relation of the number of combinations to the coefficients occurring in the binomial expansion was noted by Pingala in the 3rd century BC in a musical composition. He gave the different combinations of guru and laghu sounds as a meru-prastara (Pascal's triangle) and gave a rule simpler than that of Blaise Pascal, based on the simple formula.

     Varahamihira in the 6th century CE states that "if a quantity of 16 substances is varied in four different ways, the result will be 1820." He found this result using rules related to Pascal's triangle. In the 9th century, Mahavira gave an explicit algorithm for calculating the number of combinations and provided the well-known general formula.

     Jewish philosopher and astronomer Abraham ibn Ezra studied binomial coefficients in the early 12th century. [1][2] He found several recurrence relations and their fundamental symmetry: (see link at bottom)

     Muslim mathematicians later studied combinatorial analysis from at least the 13th century. Ibn Mun'im, in the Maghreb of North Africa in the early 13th century, dealt with combinatory problems. He stated the rule for determining all possible combinations of n colours p times and established, inductively, the resulting arithmetic triangle of the relationship as (see link at bottom).

     He applied similar formulae for permutations with and without repetitions using the Arabic alphabet for illustrative purposes. He also did some work on combinatorial reasoning.

     Recently it was discovered that Archimedes's work "Stomachion", believed to be lost, was one of the first works of combinatorics ever written. A palimpsest discovered a few years ago, contains most of this lost work and involves finding the number of ways that the problem posed in the Stomachion can be solved. This game consists of 14 irregular strips of paper that can be placed in many different ways in order to form a square. The solution to this problem involves very high level combinatronics even for today's standards. [3][4]

     Persian mathematician Al-Farisi, in the late 13th century, introduced ideas concerning factorisation and combinatorial methods. Al-Farisi's approach is based on the unique factorisation of an integer into powers of prime numbers. He states and proves this fundamental theorem of arithmetic. Al-Farisi saw the relation between polygonal numbers and the binomial coefficients and he presented arguments, using an early form of mathematical induction, which showed a relation between triangular numbers, the sums of triangular numbers, the sums of the sums of triangular number, etc., and the combinations of n objects taken k at a time.

     Enumerative combinatorics came to prominence in Europe after counting configurations became essential to elementary probability, starting with the work of Pascal and others from the 17th century. Modern combinatorics began developing in the late 19th century and became a distinguishable field of study in the 20th century, partly through the publication of the systematic enumerative treatise Combinatory Analysis by Percy Alexander MacMahon in 1915 and the work of R.A. Fisher in design of experiments in the 1920s. [verification needed] Two of the most prominent combinatorialists of recent times were the prolific problem-raiser and problem-solver Paul Erdős, who worked mainly on extremal questions, and Gian-Carlo Rota, who helped to formalize the subject beginning in the 1960s, mostly in enumeration and algebraization. The study of how to count objects is sometimes thought of separately as the field of enumeration.

WIKI/ COMBIATORICShttp://en.wikipedia.org/wiki/Combinatorics

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« Reply #11 on: April 26, 2007, 01:22:12 PM »

Scientists used advanced imaging technology on the original parchment (left) to reveal the under-text (right), which was previously hidden

Text reveals more ancient secrets

LONDON. April 26

Experts are "lost for words" to have found that a medieval prayer book has yielded yet another key ancient text buried within its parchment. Works by mathematician Archimedes and the politician Hyperides had already been found buried within the book, known as the Archimedes Palimpsest.

But now advanced imaging technology has revealed a third text - a commentary on the philosopher Aristotle.

Project director William Noel called it a "sensational find".

The prayer book was written in the 13th Century by a scribe called John Myronas.

But instead of using fresh parchment for his work, he employed pages from five existing books.

Dr Noel, curator of manuscripts at the US-based Walters Art Museum and a co-author of a forthcoming book on the Archimedes Palimpsest, said: "It's a rather brutal process, but it means you can reuse parchment if you are short of it.

"You take books off shelves, you scrub off the text, you cut them up and you make a new book."

In 1906 it came to light that one of the books recycled to form the medieval manuscript contained a unique work by Archimedes.

And in 2002, modern imaging technology not only provided a clearer view of this famous mathematician's words, but it also revealed another text - the only known manuscript of Hyperides, an Athenian politician from the 4th Century BC.

"At this point you start thinking striking one palimpsest is gold, and striking two is utterly astonishing. But then something even more extraordinary happened," Dr Noel told the BBC News website.

One of the recycled books was proving extremely difficult to read, explained Roger Easton, a professor of imaging science at Rochester Institute of Technology, US.

"We were using a technique called multispectral imaging," he said.

This digital imaging technique uses photographs taken at different wavelengths to enhance particular characteristics of the imaged area. Subtle adjustments of this method, explained Professor Easton, suddenly enabled these hidden words to be revealed.

"Even though I couldn't read Ancient Greek, just the fact that I could see the words gave me shivers," he said.

Foundations of logic

An international team of experts began to scrutinize the ancient words, explained Reviel Netz, professor of ancient science at Stanford University, US and another co-author of the palimpsest book.

A series of clues, such as spotting a key name in the margin, led the team to its conclusion.

"The philosophical passage in the Archimedes Palimpsest is now definitely identified as a relatively early commentary to Aristotle's Categories," said Professor Netz.

He said that Aristotle's Categories had served as the foundation for the study of logic throughout western history.

Further study has revealed the most likely author of this unique commentary is Alexander of Aphrodisias, Professor Robert Sharples from the University College London told BBC News.

If this is the case, he said, "it gives us part of a commentary previously supposed lost by the most important of those ancient commentators on Aristotle".

A provisional translation of the commentary is currently being undertaken.

It reveals a debate on some aspects of Aristotle's theory of classification, such as: if the term "footed" is used for animals, can it be used to classify anything else, such as a bed?

The passage reads:

For as "foot" is ambiguous when applied to an animal and to a bed, so are "with feet" and "without feet". So by "in species" here [Aristotle] is saying "in formula".

For if it ever happens that the same name indicates the differentiae of genera that are different and not subordinate one to the other, they are at any rate not the same in formula.

Dr Noel said: "There is no more important philosopher in the world than Aristotle. To have early views in the 2nd and 3rd Century AD of Aristotle's Categories is just fantastic."

"We have one book that contains three texts from the ancient world that are absolutely central to our understanding of mathematics, politics and now philosophy," he said.

"I am at a loss for words at what this book has turned out to be. To make these discoveries in the 21st Century is frankly nutty - it is just so exciting."

There will be a live webcast of Dr Noel and Professor Easton presentation of this latest discovery at the Annual General Meeting of the American Philosophical Society on 1415 BST (0915 ET) Thursday 26 April.

Categories (Aristotle)
Categories (Lat. Categoriae, Greek κάτέγόρίά) is a text from Aristotle's Organon that enumerates all the possible kinds of thing which can be the subject or the predicate of a proposition.

The Categories places every object of human apprehension under one of ten categories (known to medieval writers as the praedicamenta). They are intended to enumerate everything which can be expressed without composition or structure, thus anything which can be either the subject or the predicate of a proposition.

The text begins with an explication of what is meant by "synonymous," or univocal words, what is meant by "homonymous," or equivocal words, and what is meant by "paronymous," or denominative words. What we say:

    * Is either simple, without composition or structure, such as "man," "horse," "fights," etc.
    * Has composition and structure, such as "a man fights," "the horse runs," etc.

Next, we distinguish between a subject of predication, namely that of which anything is affirmed or denied, and a subject of inhesion. A thing is said to be inherent in a subject, when, though it is not a part of the subject, it cannot possibly exist without it, e.g., shape in a thing having a shape.

Of all the things that exist,

   1. Some may be predicated of a subject, but are in no subject; as "man" may be predicated of James or John, but is not in any subject.
   2. Some are in a subject, but can be predicated of no subject. Thus my knowledge in grammar is in me as its subject, but it can be predicated of no subject; because it is an individual thing.
   3. Some are both in a subject, and may be predicated of a subject, as science, which is in the mind as its subject, and may be predicated of geometry.
   4. Last, some things can neither be in a subject nor be predicated of any subject. These are individual substances, which cannot be predicated, because they are individuals; and cannot be in a subject, because they are substances.

Then we come to the categories themselves, (1)-(4) above being called by the scholastics the antepraedicamenta. Note, however, that although Aristotle has apparently distinguished between being in a subject, and being predicated truly of a subject, in the Prior Analytics these are treated as synonymous. This has led some to suspect that Aristotle was not the author of the Categories.

The ten categories, or classes, are

   1. Substance. As mentioned above the notion of "substance" is defined as that which can be said to be predicated of nothing nor be said to be within anything. Hence, "this particular man" or "that particular tree" are substances. Later in the text, Aristotle calls these particulars "primary substances," to distinguish them from "secondary substances," which are universals. Hence, "Socrates" is a primary Substance, while "man" is a secondary substance.
   2. Quantity. This is the spatial extension of an object. All medieval discussions about the nature of the continuum, of the infinite and the infinitely divisible, are a long footnote to this text. It is of great importance in the development of mathematical ideas in the medieval and late scholastic period.
   3. Quality This is a determination which characterises the nature of an object.
   4. Relation This is the way in which one object may be related to another.
   5. Place Position in relation to the surrounding environment.
   6. Time Position in relation to the course of events.
   7. Position The relative position of the parts of the object (usually a living object).
   8. State The determination arising from the physical accoutrements of an object; for example, clothed or sleeping.
   9. Action The production of change in some other object.
  10. Affection The reception of change from some other object.

The first six are given a detailed treatment in four chapters, the last four are passed over lightly, as being clear in themselves. Later texts by scholastic philosophers also reflect this disparity of treatment.

After discussing the categories, four ways are given in which things may be considered contrary to one another. Next, the work discusses five senses wherein a thing may be considered prior to another, followed by a short section on simultaneity. Six forms of movement are then defined: generation, destruction, increase, diminution, alteration, and change of place. The work ends with a brief consideration of the word 'have' and its usage.

See also:
Four passages from Athenian Neo-Platonists on complex curves
Proclus, Commentary on Euclid's Elements I (271.1-273.10)
Proclus, Commentary on Euclid's Elements I  (p. 356.6-16)

These are the same quotation from the lost work: Iamblichus, Commentary on Aristotle's Categories:
Simplicius, Commentary on Aristotle's Categories, 192.12-25
Simplicius, Commentary on Aristotle's Physics, 60.6-18

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« Reply #12 on: April 27, 2007, 12:26:35 AM »



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« Reply #13 on: September 13, 2007, 10:51:16 AM »

Super-powerful x-rays could peer beneath the skin of manuscripts

'Super-scope' to see hidden texts

The hidden content in ancient works could be illuminated by a light source 10 billion times brighter than the Sun.

The technique employs Britain's new facility, the Diamond synchrotron, and could be used on works such as the Dead Sea Scrolls or musical scores by Bach.

Intense light beams will enable scientists to uncover the text in scrolls and books without having to open - and potentially damage - them.

The research was presented at the British Association science festival.

Iron gall ink, which is made from oak apples, has been in use from the 12th Century, but causes parchment to deteriorate rendering precious documents unreadable.
Both paper and parchment - thinly stretched skins from cows, sheep or goats - contain collagen, which reacts with iron ink to become gelatine.

When dry, gelatine is very brittle; but as soon as it gets wet, it turns into jelly, destroying some documents if they are disturbed.

Unrolling the scroll

Now, scientists from the University of Cardiff have developed a technique that uses a powerful X-ray source to create a three-dimensional image of an iron-inked document.

The team then applies a computer algorithm to separate the image into the different layers of parchment, in effect using the program to unroll the scroll.

Diamond opens for business

Professor Tim Wess, who led the research, said: "We've folded up a real piece of parchment and then done a process of X-ray tomography on it. We've been able to recover the structure where we can see the words that are written inside the document."

The team now plans to use the Diamond synchrotron's powerful X-ray source to penetrate many layers of parchment.

The synchrotron, which covers the area of five football pitches, generates light beams that can probe matter down to the molecular and atomic scale.

Professor Wess explained: "The letters have got iron in them, so you shine a band of X-rays through, and you end up with an absorption image, rather like your bones would absorb on an X-ray.

"This is something we can take forward with Diamond, to try to unravel the secrets inside documents that we're too scared to try to open, or that are beyond the point of conservation."

Wish list

The National Archives has donated some 18th Century fire-damaged scrolls that have never been unrolled, due to their condition. But the team also has a wish-list of works that they plan to probe.

Professor Wess said: "There are some parts of the Dead Sea Scrolls which have not been unrolled."

Reading books without opening them was a goal of the project, added Professor Wess. The technique works best with rolled parchment. The flat nature, as well as the thickness of books, presents a challenge.

He said: "I know of books which have been damaged by iron gall ink corrosion where the conservators are actually afraid to open the book because of all the letters. You really end up with a stencil rather than the lettering."

Conserving works

Another target of the project is to image documents before they become too damaged, to monitor levels of gelatine.

The team can then advise on the most appropriate conservation methods, depending on the state of the parchment.

Using modern technology to reveal secrets of the past is a rewarding task.

Professor Wess said: "If you can bring together a �260m ($527m) synchrotron, and the cutting edge science from that, the provenance and the depth of history that you can access when you see these things is actually a revelatory moment."

History turns to jelly

By Angela Newton

Impossible to read a book without opening the cover? Research led by Professor Tim Wess, head of the Institute of Vision at Cardiff University, suggests that it is only a matter of time.
By using Britain�s new Diamond Light Synchrotron, an X-ray source 10 billion times brighter than the sun, non-invasive techniques are being developed that could allow the secrets of priceless documents, such as the Dead Sea Scrolls, to be revealed to the world without any damage to the artefacts themselves.
Much of written history has been recorded on conditioned animal skin, which over time degrades from collagen to gelatine, a substance that is brittle when dry and jelly-like when wet.
Researchers have focussed on examining the nanoscopic structure of this collagen substrate, aiming to determine exactly how it degrades. They hope that this will give clues about how the gelatination process can be delayed or prevented, allowing us to better preserve key documents of our cultural heritage such as the Doomsday Book and the Magna Carta.
Iron Gall, the ink used for over two millennia on such records, has been found to play a key role in accelerating the rate of this degradation. However, using this more powerful X-ray scattering technique, it could also be the saving grace of documents feared to be too fragile to disturb.
Iron- and Copper-containing molecules found in the ink can essentially be used as a contrast agent. They absorb high-intensity X-rays with a different frequency to the collagen substrate.
Through depth-profiling and scanning techniques the exact position of ink can be mapped out, from which a virtual 3D image of the document can be constructed.
Documents currently being investigated by the team have been sourced from national archives and include fire-damaged scrolls dating from the eighteenth century, musical manuscripts thought to be written by Beethoven and Mozart, and postage-stamp sized fragments of the Dead Sea Scrolls.
Although some of the 800 scrolls have already been opened throughout their long history, it is hoped that by piecing together the fragments answers may be found to the many questions researchers are still asking about them.

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« Reply #14 on: September 14, 2007, 02:47:19 PM »


I am amazed at this break through in technology.

I can't even begin to imagine the documents that will be read in the future.

Thanks for the post.


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