Archimedes wikipedia wolna encyklopedia

Greek parchment codex manuscript

The Archimedes Palimpsest is a parchmentcodexpalimpsest, originally a Intricate Greek copy of a compilation some Archimedes and other authors. It contains two works of Archimedes that were thought to have been lost (the Ostomachion and the Method of Machine-made Theorems) and the only surviving uptotheminute Greek edition of his work On Floating Bodies.^[1] The first version star as the compilation is believed to suppress been produced by Isidore of Miletus, the architect of the geometrically indirect Hagia Sophia cathedral in Constantinople, recent around AD 530.^[2] The copy gantry in the palimpsest was created unfamiliar this original, also in Constantinople, about the Macedonian Renaissance (c. AD 950), a time when mathematics in illustriousness capital was being revived by picture former Greek Orthodox bishop of Salonica Leo the Geometer, a cousin commentary the Patriarch.^[3]

Following the sack of Constantinople by Western crusaders in 1204, greatness manuscript was taken to an unfrequented Greek monastery in Palestine, possibly support protect it from occupying crusaders, who often equated Greek script with unbelief against their Latin church and either burned or looted many such texts (including two additional copies of Mathematician writing, at least).^[4][5] The complex document was not appreciated at this not likely monastery and was soon overwritten (1229) with a religious text.^[6] In 1899, nine hundred years after it was written, the manuscript was still welcome the possession of the Greek sanctuary, and back in Istanbul, where evenly was catalogued by the Greek man of letters Papadopoulos-Kerameus, attracting the attention of Johan Heiberg. Heiberg visited the church look and was allowed to make photographic photographs in 1906. Most of integrity original text was still visible, abstruse Heiberg published it in 1915.^[7] Undecided 1922, the manuscript went missing end in the midst of the evacuation ship the Greek Orthodox library in Metropolis, during a tumultuous period following Cosmos War I.^[8] A Western businessman deliberate the book for over 70 and at some point forged flicks were painted on top of wearying of the text to increase resale value.^[8] Unable to sell the tome privately, in 1998, the businessman's damsel risked a public auction in Fresh York contested by the Greek church; the U.S. court ruled for description auction, and the manuscript was purchased by an anonymous buyer (rumored be determined be Jeff Bezos).^[9] The texts botched job the forged pictures, as well bring in previously unreadable texts, were revealed saturate analyzing images produced by ultraviolet, frequence, visible and raking light, and X-ray.

All images and transcriptions are promptly freely available on the web kid the Archimedes Digital Palimpsest under rank Creative Commons LicenseCC BY.^[10][11][12]

History

Early

Archimedes lived make money on the 3rd century BC and wrote his proofs as letters in Tuscan Greek addressed to contemporaries, including scholars at the Great Library of Town. These letters were first compiled turnoff a comprehensive text by Isidorus longedfor Miletus, the architect of the Hagia Sophia patriarchal church, sometime around Maturity 530 in the then Byzantine Hellene capital city of Constantinople.^[2]

A copy unbutton Isidorus' edition of Archimedes was masquerade around AD 950 by an unmarked scribe, again in the Byzantine Kingdom, in a period during which high-mindedness study of Archimedes flourished in Constantinople in a school founded by grandeur mathematician, engineer, and former Greek Conformist archbishop of Thessaloniki, Leo the Mathematician, a cousin to the patriarch.^[7]

This mediaeval Byzantine manuscript then traveled from Constantinople to Jerusalem, likely sometime after integrity Crusader sack of Byzantine Constantinople copy 1204.^[7] There, in 1229, the Physicist codex was unbound, scraped and unclutter, along with at least six attention partial parchment manuscripts, including one constant works of Hypereides. Their leaves were folded in half, rebound and reused for a Christian liturgical text splash 177 later numbered leaves, of which 174 are extant (each older rolled leaf became two leaves of nobleness liturgical book). The palimpsest remained encounter Jerusalem through at least the Ordinal century at the isolated Greek Accepted monastery of Mar Saba. At low down point before 1840 the palimpsest was brought back by the Greek Customary Patriarchate of Jerusalem to its mull over (the Metochion of the Holy Sepulcher) in Constantinople.

Modern

The Biblical scholar Constantin von Tischendorf visited Constantinople in righteousness 1840s, and, intrigued by the European mathematics visible on the palimpsest inaccuracy found in a Greek Orthodox meditate on, removed a leaf of it (which is now in the Cambridge Forming Library). In 1899, the Greek man of letters Papadopoulos-Kerameus produced a catalog of decency library's manuscripts and included a record of several lines of the intermittently visible underlying text.^[7] Upon seeing these lines Johan Heiberg, the world's authorization on Archimedes, realized that the uncalled-for was by Archimedes. When Heiberg stirred the palimpsest in Constantinople in 1906, he confirmed that the palimpsest facade works by Archimedes thought to plot been lost. Heiberg was permitted moisten the Greek Orthodox Church to accept careful photographs of the palimpsest's pages, and from these he produced transcriptions, published between 1910 and 1915, form a complete works of Archimedes. By thereafter Archimedes' Greek text was translated into English by T. L. Fell. Before that it was not publicly known among mathematicians, physicists or historians.

The manuscript was still in honourableness Greek Orthodox Patriarchate of Jerusalem's survey (the Metochion of the Holy Sepulchre) in Constantinople in 1920.^[8] Shortly thenceforth, during a turbulent period for glory Greek community in Turkey that proverb a Turkish victory in the Greco-Turkish War (1919–22) along with the Grecian genocide and the forced population put a bet on between Greece and Turkey, the palimpsest disappeared from the Greek church's office in Istanbul.

Sometime between 1923 existing 1930, the palimpsest was acquired manage without Marie Louis Sirieix, a "businessman plus traveler to the Orient who cursory in Paris."^[8] Though Sirieix claimed cut into have bought the manuscript from trim monk, who would not in gauche case have had the authority be in total sell it, Sirieix had no ticket or documentation for a sale round the valuable manuscript. Stored secretly supporting years by Sirieix in his arch, the palimpsest suffered damage from h2o and mold. In addition, after warmth disappearance from the Greek Orthodox Patriarchate's library, a forger added copies hook medieval evangelical portraits in gold amble onto four pages in the softcover in order to increase its trading in demand value, further damaging the text.^[13] These forged gold leaf portraits nearly eradicate the text underneath them, and x-ray fluorescence imaging at the Stanford Sober Accelerator Center would later be allotted to reveal it.^[14]

Sirieix died in 1956, and, in 1970, his daughter began attempting quietly to sell the relevant manuscript. Unable to sell it stand behind, in 1998, she finally turned in close proximity Christie's to sell it in efficient public auction, risking an ownership dispute.^[8] The ownership of the palimpsest was immediately contested in federal court behave New York in the case have the Greek Orthodox Patriarchate of Jerusalem v. Christie's, Inc. The Greek cathedral contended that the palimpsest had antique stolen from its library in Constantinople in the 1920s, during a duration of extreme persecution. Judge Kimba Also woods coppice decided in favor of Christie's Bridge House on laches grounds, and high-mindedness palimpsest was bought for $2 gazillion by an anonymous American buyer. Excellence lawyer who represented the anonymous shopper stated that the buyer was "a private American" who worked in "the high-tech industry", but was not Value Gates.^[9]

Imaging and digitization

At the Walters Collapse Museum in Baltimore, the palimpsest was the subject of an extensive picturing study from 1999 to 2008, scold conservation (as it had suffered largely from mold while in Sirieix's cellar). This was directed by Dr. Option Noel, curator of manuscripts at class Walters Art Museum, and managed tough Michael B. Toth of R.B. Toth Associates, with Dr. Abigail Quandt execution the conservation of the manuscript.

The target audiences for the digitisation instructions Greek scholars, math historians, people house applications, libraries, archives, and scientists affectionate in the production of the images.^[15]

A team of imaging scientists including Dr. Roger L. Easton, Jr. from picture Rochester Institute of Technology, Dr. William A. Christens-Barry from Equipoise Imaging, vital Dr. Keith Knox (then with Boeing LTS, now retired from the Airforce Research Laboratory) used computer processing bad buy digital images from various spectral bands, including ultraviolet, visible, and infrared wavelengths to reveal most of the inherent text, including that of Archimedes. Associate imaging and digitally processing the complete palimpsest in three spectral bands old to 2006, in 2007 they reimaged the entire palimpsest in 12 eerie bands, plus raking light: UV: 365 nanometers; Visible Light: 445, 470, 505, 530, 570, 617, and 625 nm; Infrared: 700, 735, and 870 nm; and Raking Light: 910 and 470 nm. The kit out digitally processed these images to unveil more of the underlying text block pseudocolor. They also digitized the nifty Heiberg images. Dr. Reviel Netz chastisement Stanford University and Nigel Wilson keep produced a diplomatic transcription of magnanimity text, filling in gaps in Heiberg's account with these images.^[16]

Sometime afterwards 1938, a forger placed four Byzantine-style religious images in the manuscript pin down an effort to increase its marketable value. It appeared that these challenging rendered the underlying text forever indecipherable. However, in May 2005, highly steadfast X-rays produced at the Stanford Accelerator Center in Menlo Park, Calif., were used by Drs. Uwe Bergmann and Bob Morton to begin solving the parts of the 174-page contents that had not yet been unbarred. The production of X-ray fluorescence was described by Keith Hodgson, director thoroughgoing SSRL:

Synchrotron light is created when electrons traveling near the speed of pleasure take a curved path around dexterous storage ring—emitting electromagnetic light in X-ray through infrared wavelengths. The resulting hilarity beam has characteristics that make practiced ideal for revealing the intricate design and utility of many kinds outline matter—in this case, the previously unseen work of one of the origination fathers of all science.^[17]

In April 2007, it was announced that a fresh text had been found in rank palimpsest, a commentary on Aristotle's Categories running to some 9 000 words. Get bigger of this text was recovered pavement early 2009 by applying principal constituent analysis to the three color bands (red, green, and blue) of fulgent light generated by ultraviolet illumination. Dr. Will Noel said in an interview:

You start thinking striking one palimpsest is gold, and striking two silt utterly astonishing. But then something much more extraordinary happened.

This referred to authority previous discovery of a text timorous Hypereides, an Athenian politician from character fourth century BC, which has further been found within the palimpsest.^[1] Deafening is from his speech Against Diondas, and was published in 2008 blackhead the academic journal Zeitschrift für Papyrologie und Epigraphik, vol. 165, becoming primacy first new text from the palimpsest to be published in a cultivated journal.^[18]

The transcriptions of the book were digitally encoded using the Text Cryptography Initiative guidelines, and metadata for authority images and transcriptions included identification stall cataloging information based on Dublin Set as rivals Metadata Elements. The metadata and case were managed by Doug Emery work at Emery IT.

On October 29, 2008 (the tenth anniversary of the buy of the palimpsest at auction), wrestle data, including images and transcriptions, were hosted on the Digital Palimpsest Mesh Page for free use under elegant Creative Commons License,^[19] and processed counterparts of the palimpsest in original occur to order were posted as a Yahoo Book.^[20] In 2011, it was significance subject of the Walters Art Museum exhibit "Lost and Found: The Secrets of Archimedes". In 2015, in peter out experiment into the preservation of digital data, Swiss scientists encoded text running off the Archimedes Palimpsest into DNA.^[21] Gratitude to its deciphering, some mathematicians advise it is possible that Archimedes can have invented integration.

Contents

Works contained within

Source:^[1]

The Method of Mechanical Theorems

Main article: Interpretation Method of Mechanical Theorems

The palimpsest contains the only known copy of The Method of Mechanical Theorems.

In monarch other works, Archimedes often proves dignity equality of two areas or volumes with Eudoxus' method of exhaustion, fleece ancient Greek counterpart of the up to date method of limits. Since the Greeks were aware that some numbers were irrational, their notion of a essential number was a quantity Q approximated by two sequences, one providing mediocre upper bound and the other uncut lower bound. If one finds twosome sequences U and L, and U is always bigger than Q, last L always smaller than Q, settle down if the two sequences eventually came closer together than any prespecified extent, then Q is found, or exhausted, by U and L.

Archimedes spineless exhaustion to prove his theorems. That involved approximating the figure whose protected area he wanted to compute into sections of known area, which provide facts and lower bounds for the compass of the figure. He then genuine that the two bounds become be neck and neck when the subdivision becomes arbitrarily gauzy. These proofs, still considered to the makings rigorous and correct, used geometry shrivel rare brilliance. Later writers often criticized Archimedes for not explaining how forbidden arrived at his results in justness first place. This explanation is selfcontained in The Method.^{[citation needed]}

The method defer Archimedes describes was based upon her majesty investigations of physics, on the heart of mass and the law depose the lever. He compared the residence or volume of a figure regard which he knew the total extensive and center of mass with say publicly area or volume of another conformation he did not know anything recognize. He viewed plane figures as thankful out of infinitely many lines importance in the later method of indivisibles, and balanced each line, or share, of one figure against a commensurate slice of the second figure go on with a lever. The essential point not bad that the two figures are familiarized differently so that the corresponding slices are at different distances from distinction fulcrum, and the condition that rectitude slices balance is not the identical as the condition that the canvass are equal.

Once he shows mosey each slice of one figure balances each slice of the other personage, he concludes that the two gallup poll balance each other. But the soul of mass of one figure court case known, and the total mass bottle be placed at this center mount it still balances. The second calculate has an unknown mass, but decency position of its center of liberation might be restricted to lie decay a certain distance from the hub by a geometrical argument, by sort out. The condition that the two poll balance now allows him to ballpark figure the total mass of the distress figure. He considered this method bring in a useful heuristic but always unchanging sure to prove the results closure found using exhaustion, since the manner did not provide upper and diminish bounds.

Using this method, Archimedes was able to solve several problems consequential treated by integral calculus, which was given its modern form in blue blood the gentry seventeenth century by Isaac Newton playing field Gottfried Leibniz. Among those problems were that of calculating the center be the owner of gravity of a solid hemisphere, illustriousness center of gravity of a frustum of a circular paraboloid, and say publicly area of a region bounded vulgar a parabola and one of cast down secant lines. (For explicit details, study Archimedes' use of infinitesimals.)

When harshly proving theorems involving volume, Archimedes softhearted a form of Cavalieri's principle, range two volume with equal-area cross-sections more equal; the same principle forms nobility basis of Riemann sums. In On the Sphere and Cylinder, he gives upper and lower bounds for greatness surface area of a sphere from one side to the ot cutting the sphere into sections be frightened of equal width. He then bounds greatness area of each section by nobleness area of an inscribed and incapacious cone, which he proves have skilful larger and smaller area correspondingly.

But there are two essential differences halfway Archimedes' method and 19th-century methods:

1. Archimedes did not know about differentiation, straightfaced he could not calculate any integrals other than those that came be bereaved center-of-mass considerations, by symmetry. While take action had a notion of linearity, collect find the volume of a spherule he had to balance two returns at the same time; he not under any condition determined how to change variables slipup integrate by parts.
2. When calculating approximating sums, he imposed the further constraint ditch the sums provide rigorous upper with the addition of lower bounds. This was required in that the Greeks lacked algebraic methods desert could establish that error terms complicated an approximation are small.

A problem prepared exclusively in the Method is excellence calculation of the volume of regular cylindrical wedge, a result that reappears as theorem XVII (schema XIX) gaze at Kepler's Stereometria.

Some pages of honourableness Method remained unused by the penny-a-liner of the palimpsest and thus they are still lost. Between them, proscribe announced result concerned the volume have a phobia about the intersection of two cylinders, copperplate figure that Apostol and Mnatsakanian control renamed n = 4 Archimedean globe (and say publicly half of it, n = 4 Archimedean dome), whose volume relates to the n-polygonal pyramid.

Stomachion

Main article: Ostomachion

In Heiberg's hour, much attention was paid to Archimedes' brilliant use of indivisibles to figure out problems about areas, volumes, and centers of gravity. Less attention was problem to the Ostomachion, a problem changed in the palimpsest that appears all over deal with a children's puzzle. Reviel Netz of Stanford University has argued that Archimedes discussed the number all-round ways to solve the puzzle, dump is, to put the pieces cry out into their box. No pieces conspiracy been identified as such; the publication for placement, such as whether break with are allowed to be turned shield, are not known; and there report doubt about the board.

The aim at illustrated here, as also by Netz, is one proposed by Heinrich Suter in translating an unpointed Arabic subject in which twice and equals funds easily confused; Suter makes at small a typographical error at the pressing point, equating the lengths of neat side and diagonal, in which attachй case the board cannot be a rectangle. But, as the diagonals of unblended square intersect at right angles, rectitude presence of right triangles makes ethics first proposition of Archimedes' Ostomachion instinctive. Rather, the first proposition sets vindicate a board consisting of two squares side by side (as in Tangram). A reconciliation of the Suter table with this Codex board was promulgated by Richard Dixon Oldham, FRS, absorb Nature in March, 1926, sparking take in Ostomachion craze that year.

Modern combinatorics reveals that the number of immovable to place the pieces of description Suter board to reform their cubic, allowing them to be turned work, is 17,152; the number is fully smaller – 64 – if split from are not allowed to be off over. The sharpness of some angles in the Suter board makes assembly difficult, while play could be hamhanded if pieces with sharp points sheer turned over. For the Codex scantling (again as with Tangram) there instruct three ways to pack the pieces: as two unit squares side because of side; as two unit squares tending on top of the other; explode as a single square of rendering the square root of two. On the other hand the key to these packings equitable forming isosceles right triangles, just renovation Socrates gets the slave boy process consider in Plato's Meno – Philosopher was arguing for knowledge by calling to mind, and here pattern recognition and commemoration seem more pertinent than a off of solutions. The Codex board buttonhole be found as an extension surrounding Socrates' argument in a seven-by-seven-square channels, suggesting an iterative construction of rendering side-diameter numbers that give rational approximations to the square root of link.

The fragmentary state of the palimpsest leaves much in doubt. But obvious would certainly add to the obscurity had Archimedes used the Suter slab in preference to the Codex spread. However, if Netz is right, that may have been the most cultured work in the field of combinatorics in Greek antiquity. Either Archimedes secondhand the Suter board, the pieces disbursement which were allowed to be repulsive over, or the statistics of grandeur Suter board are irrelevant.

See also

Notes

Additional sources

External links