1.
Archimedes Palimpsest
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The Archimedes Palimpsest is a parchment codex palimpsest, which originally was a 10th-century Byzantine Greek copy of an otherwise unknown work of Archimedes of Syracuse and other authors. It was overwritten with a Christian religious text by 13th-century monks, the erasure was incomplete, and Archimedes work is now readable after scientific and scholarly work from 1998 to 2008 on images produced by ultraviolet, infrared, visible and raking light, and X-ray. The Palimpsest is the only copy of Stomachion and The Method of Mechanical Theorems. Archimedes lived in the 3rd century BC and wrote his proofs as letters in Doric Greek addressed to contemporaries, a copy of this text was made around 950 AD, again in the Byzantine Empire, by an anonymous scribe. This medieval Byzantine manuscript then traveled to Jerusalem, likely sometime after the Crusader sack of Constantinople in 1204, there, in 1229, the original Archimedes codex was unbound, scraped and washed, along with at least six other parchment manuscripts, including one with works of Hypereides. The parchment leaves were folded in half and reused for a Christian liturgical text of 177 pages, the palimpsest remained near Jerusalem through at least the 16th century at the isolated Greek Orthodox monastery of Mar Saba. At some point before 1840 the palimpsest was brought back by the Greek Orthodox Patriarchate of Jerusalem to their library in Constantinople, in 1899 the Greek scholar Papadopoulos-Kerameus produced a catalog of the librarys manuscripts and included a transcription of several lines of the partially visible underlying text. Upon seeing these lines Johan Heiberg, the authority on Archimedes. When Heiberg studied the palimpsest in Constantinople in 1906, he confirmed that the palimpsest included works by Archimedes thought to have been lost. Heiberg was permitted to take photographs of the palimpsests pages. Shortly thereafter Archimedes Greek text was translated into English by T. L. Heath, before that it was not widely known among mathematicians, physicists or historians. The manuscript was still in the Greek Orthodox Patriarchate of Jerusalems library in Constantinople in 1920, sometime between 1923 and 1930 the palimpsest was acquired by Marie Louis Sirieix, a businessman and traveler to the Orient who lived in Paris. Though Sirieix claimed to have bought it from a monk, who would not in any case have had the authority to sell it, stored secretly for years in Sirieixs cellar, the palimpsest suffered damage from water and mold. These gold leaf portraits nearly obliterated the text underneath them, Sirieix died in 1956, and in 1970 his daughter began attempting quietly to sell the manuscript. Unable to sell it privately in 1998 she finally turned to Christies to sell it in a public auction, indeed, the ownership of the palimpsest was immediately contested in federal court in New York in the case of the Greek Orthodox Patriarchate of Jerusalem v. Christies, Inc. The plaintiff contended that the palimpsest had been stolen from its library in Constantinople in the 1920s, judge Kimba Wood decided in favor of Christies Auction House on laches grounds, and the palimpsest was bought for $2 million by an anonymous buyer. Simon Finch, who represented the anonymous buyer, stated that the buyer was a private American who worked in the high-tech industry, at the Walters Art Museum in Baltimore, the palimpsest was the subject of an extensive imaging study from 1999 to 2008, and conservation. This was directed by Dr. Will Noel, curator of manuscripts at the Walters Art Museum, Toth of R. B. Toth Associates, with Dr. Abigail Quandt performing the conservation of the manuscript
2.
Archimedes
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Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics and he is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Archimedes was born c.287 BC in the city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years, in The Sand Reckoner, Archimedes gives his fathers name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, a biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt and he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c.212 BC during the Second Punic War, according to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, the soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives an account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable asset and had ordered that he not be harmed. Marcellus called Archimedes a geometrical Briareus, the last words attributed to Archimedes are Do not disturb my circles, a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is given in Latin as Noli turbare circulos meos. The phrase is given in Katharevousa Greek as μὴ μου τοὺς κύκλους τάραττε
3.
Arabic
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Arabic is a Central Semitic language that was first spoken in Iron Age northwestern Arabia and is now the lingua franca of the Arab world. Arabic is also the language of 1.7 billion Muslims. It is one of six languages of the United Nations. The modern written language is derived from the language of the Quran and it is widely taught in schools and universities, and is used to varying degrees in workplaces, government, and the media. The two formal varieties are grouped together as Literary Arabic, which is the language of 26 states. Modern Standard Arabic largely follows the standards of Quranic Arabic. Much of the new vocabulary is used to denote concepts that have arisen in the post-Quranic era, Arabic has influenced many languages around the globe throughout its history. During the Middle Ages, Literary Arabic was a vehicle of culture in Europe, especially in science, mathematics. As a result, many European languages have borrowed many words from it. Many words of Arabic origin are found in ancient languages like Latin. Balkan languages, including Greek, have acquired a significant number of Arabic words through contact with Ottoman Turkish. Arabic has also borrowed words from languages including Greek and Persian in medieval times. Arabic is a Central Semitic language, closely related to the Northwest Semitic languages, the Ancient South Arabian languages, the Semitic languages changed a great deal between Proto-Semitic and the establishment of the Central Semitic languages, particularly in grammar. Innovations of the Central Semitic languages—all maintained in Arabic—include, The conversion of the suffix-conjugated stative formation into a past tense, the conversion of the prefix-conjugated preterite-tense formation into a present tense. The elimination of other prefix-conjugated mood/aspect forms in favor of new moods formed by endings attached to the prefix-conjugation forms, the development of an internal passive. These features are evidence of descent from a hypothetical ancestor. In the southwest, various Central Semitic languages both belonging to and outside of the Ancient South Arabian family were spoken and it is also believed that the ancestors of the Modern South Arabian languages were also spoken in southern Arabia at this time. To the north, in the oases of northern Hijaz, Dadanitic and Taymanitic held some prestige as inscriptional languages, in Najd and parts of western Arabia, a language known to scholars as Thamudic C is attested
4.
Ancient Greek
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Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often divided into the Archaic period, Classical period. It is antedated in the second millennium BC by Mycenaean Greek, the language of the Hellenistic phase is known as Koine. Koine is regarded as a historical stage of its own, although in its earliest form it closely resembled Attic Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects, Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical phases of the language, Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Attic and Ionic, Aeolic, Arcadocypriot, some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are also several historical forms, homeric Greek is a literary form of Archaic Greek used in the epic poems, the Iliad and Odyssey, and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic, the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period and they have the same general outline, but differ in some of the detail. The invasion would not be Dorian unless the invaders had some relationship to the historical Dorians. The invasion is known to have displaced population to the later Attic-Ionic regions, the Greeks of this period believed there were three major divisions of all Greek people—Dorians, Aeolians, and Ionians, each with their own defining and distinctive dialects. Often non-west is called East Greek, Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, thessalian likewise had come under Northwest Greek influence, though to a lesser degree. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, and Northern Peloponnesus Doric. The Lesbian dialect was Aeolic Greek and this dialect slowly replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek, by about the 6th century AD, the Koine had slowly metamorphosized into Medieval Greek
5.
Byzantine Empire
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It survived the fragmentation and fall of the Western Roman Empire in the 5th century AD and continued to exist for an additional thousand years until it fell to the Ottoman Turks in 1453. During most of its existence, the empire was the most powerful economic, cultural, several signal events from the 4th to 6th centuries mark the period of transition during which the Roman Empires Greek East and Latin West divided. Constantine I reorganised the empire, made Constantinople the new capital, under Theodosius I, Christianity became the Empires official state religion and other religious practices were proscribed. Finally, under the reign of Heraclius, the Empires military, the borders of the Empire evolved significantly over its existence, as it went through several cycles of decline and recovery. During the reign of Maurice, the Empires eastern frontier was expanded, in a matter of years the Empire lost its richest provinces, Egypt and Syria, to the Arabs. This battle opened the way for the Turks to settle in Anatolia, the Empire recovered again during the Komnenian restoration, such that by the 12th century Constantinople was the largest and wealthiest European city. Despite the eventual recovery of Constantinople in 1261, the Byzantine Empire remained only one of several small states in the area for the final two centuries of its existence. Its remaining territories were annexed by the Ottomans over the 15th century. The Fall of Constantinople to the Ottoman Empire in 1453 finally ended the Byzantine Empire, the term comes from Byzantium, the name of the city of Constantinople before it became Constantines capital. This older name of the city would rarely be used from this point onward except in historical or poetic contexts. The publication in 1648 of the Byzantine du Louvre, and in 1680 of Du Canges Historia Byzantina further popularised the use of Byzantine among French authors, however, it was not until the mid-19th century that the term came into general use in the Western world. The Byzantine Empire was known to its inhabitants as the Roman Empire, the Empire of the Romans, Romania, the Roman Republic, Graikia, and also as Rhōmais. The inhabitants called themselves Romaioi and Graikoi, and even as late as the 19th century Greeks typically referred to modern Greek as Romaika and Graikika. The authority of the Byzantine emperor as the legitimate Roman emperor was challenged by the coronation of Charlemagne as Imperator Augustus by Pope Leo III in the year 800. No such distinction existed in the Islamic and Slavic worlds, where the Empire was more seen as the continuation of the Roman Empire. In the Islamic world, the Roman Empire was known primarily as Rûm, the Roman army succeeded in conquering many territories covering the entire Mediterranean region and coastal regions in southwestern Europe and north Africa. These territories were home to different cultural groups, both urban populations and rural populations. The West also suffered heavily from the instability of the 3rd century AD
6.
Ausonius
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Decimius Magnus Ausonius was a Roman poet and teacher of rhetoric from Burdigala in Aquitaine, modern Bordeaux, France. For a time he was tutor to the future emperor Gratian and his best-known poems are Mosella, a description of the river Moselle, and Ephemeris, an account of a typical day in his life. His many other verses show his concern for his family, friends, teachers, Decimius Magnus Ausonius was born c. Ausonius was given a strict upbringing by his aunt and grandmother and he received an excellent education at Bordeaux and at Toulouse, where his maternal uncle, Aemilius Magnus Arborius, was a professor. Ausonius did well in grammar and rhetoric, but professed that his progress in Greek was unsatisfactory, when his uncle was summoned to Constantinople to tutor one of the sons of emperor Constantine I, Ausonius accompanied him to the capital. Having completed his studies, he trained for some time as an advocate, in 334 he became a grammaticus at a school of rhetoric in Bordeaux, and afterwards a rhetor or professor. His teaching attracted many pupils, some of whom became eminent in public life and his most famous pupil was the poet Paulinus, who later became a Christian and Bishop of Nola. After thirty years of this work Ausonius was summoned by emperor Valentinian I to teach his son, Gratian, when Valentinian took Gratian on the German campaigns of 368-9, Ausonius accompanied them. In recognition of his services emperor Valentinian bestowed on Ausonius the rank of quaestor, Gratian liked and respected his tutor, and when he himself became emperor in 375 he began bestowing on Ausonius and his family the highest civil honors. In 376 Ausoniuss son, Hesperius, was proconsul of Africa. In 379 Ausonius was awarded the consulate, the highest Roman honor, when Magnus Maximus was overthrown by emperor Theodosius I in 388, Ausonius did not leave his country estates. His estates supposedly included the land now owned by Château Ausone, Ausonius appears to have been a late and perhaps not very enthusiastic convert to Christianity. His grandson, Paulinus of Pella, was also a poet, about 120 epigrams on various topics. A description of the occupations of the day from morning till evening, in various meters, only the beginning and end are preserved. 30 poems of various lengths, mostly in elegiac meter, on deceased relations, composed after his consulate, a continuation of the Parentalia, dealing with the famous teachers of his native Bourdeaux whom he had known. 26 epitaphs of heroes from the Trojan war, translated from Greek Caesares, on the 12 emperors described by Suetonius. 14 pieces, dealing with 17 towns, in hexameters, a kind of puppet play in which the seven wise men appear successively and have their say. 20 pieces are grouped under this title, the most famous of which is the Mosella
7.
Tangram
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The tangram is a dissection puzzle consisting of seven flat shapes, called tans, which are put together to form shapes. The objective of the puzzle is to form a shape using all seven pieces. It is reputed to have invented in China during the Song Dynasty. It became very popular in Europe for a then. It is one of the most popular dissection puzzles in the world, a Chinese psychologist has termed the tangram the earliest psychological test in the world, albeit one made for entertainment rather than for analysis. The origin of the word tangram is unclear, the -gram element is apparently from Greek γράμμα. The tan- element is variously conjectured to be Chinese tan to extend or Cantonese tang Chinese, the tangram had already been around in China for a long time when it was first brought to America by Captain M. Donnaldson, on his ship, Trader, in 1815. When it docked in Canton, the captain was given a pair of Sang-Hsia-kois Tangram books from 1815 and they were then brought with the ship to Philadelphia, where it docked in February 1816. The first Tangram book to be published in America was based on the pair brought by Donnaldson. The puzzle was originally popularized by The Eighth Book Of Tan, a history of Tangram. The book included 700 shapes, some of which are possible to solve, the puzzle eventually reached England, where it became very fashionable. The craze quickly spread to other European countries and this was mostly due to a pair of British Tangram books, The Fashionable Chinese Puzzle, and the accompanying solution book, Key. Soon, tangram sets were being exported in great number from China, made of materials, from glass, to wood. Many of these unusual and exquisite tangram sets made their way to Denmark, danish interest in tangrams skyrocketed around 1818, when two books on the puzzle were published, to much enthusiasm. The first of these was Mandarinen and this was written by a student at Copenhagen University, which was a non-fictional work about the history and popularity of tangrams. The second, Det nye chinesiske Gaadespil, consisted of 339 puzzles copied from The Eighth Book of Tan, Tangrams were first introduced to the German public by industrialist Friedrich Adolf Richter around 1891. The sets were out of stone or false earthenware. More internationally, the First World War saw a resurgence of interest in Tangrams, on the homefront
8.
Lucretius
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Titus Lucretius Carus was a Roman poet and philosopher. His only known work is the philosophical poem De rerum natura about the tenets and philosophy of Epicureanism. Lucretius has been credited with originating the concept of the system which was formalised from 1834 by C. J. Thomsen. Very little is known about Lucretiuss life, the certain fact is that he was either a friend or client of Gaius Memmius. De rerum natura was an influence on the Augustan poets, particularly Virgil. Virtually nothing is known about the life of Lucretius and he was probably a member of the aristocratic gens Lucretia, and his work shows an intimate knowledge of the luxurious lifestyle in Rome. In a letter by Cicero to his brother Quintus in February 54 BC, Cicero said, The poems of Lucretius are as you write, they exhibit many flashes of genius, and yet show great mastership. By this time, both Cicero and his brother had read De rerum natura, and so might have many other Romans, a literary evaluation of Lucretiuss work, however, reveals some repetition and a sudden end to Book 6 during a description of the plague at Athens. The poem appears to have been published without a final revision, if this is true, Lucretius must have been dead by 54 BC. A brief biographical note is found in Aelius Donatuss Life of Virgil, the two consuls of 70 BC, Pompey and Crassus, stood together as consuls again in 55, not 53. There is insufficient basis for a confident assertion of the date of Lucretiuss birth or death in other sources, another yet briefer note is found in the Chronicon of Donatuss pupil, Jerome. Writing four centuries after Lucretiuss death, he enters under the 171st Olympiad the following line, Titus Lucretius the poet is born. The claim that he was driven mad by a potion, although defended by such scholars as Reale and Catan, often is dismissed as the result of historical confusion. In some accounts the administration of the toxic aphrodisiac is attributed to his wife Lucilia, jeromes image of Lucretius as a lovesick, mad poet continued to have significant influence on modern scholarship until quite recently, although it now is accepted that such a report is inaccurate. Similarly, the statement that Cicero emended the work prior to publication is doubtful, the exact date of his birth varies by manuscript, in most it is recorded under 94 BC, but in others under 93 or 96. Lucretius and Jerome wrote for opposing purposes, and whether or not Jerome attempted to disparage Lucretiuss work as the work of a madman is an open question. If 55 BC is Lucretiuss most likely year of death, however and his poem De rerum natura transmits the ideas of Epicureanism, which includes Atomism, and psychology. Lucretius was the first writer to introduce Roman readers to Epicurean philosophy, the poem, written in some 7,400 dactylic hexameters, is divided into six untitled books, and explores Epicurean physics through richly poetic language and metaphors
9.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
10.
Monumenta Germaniae Historica
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The editor from 1826 until 1874 was Georg Heinrich Pertz, in 1875 he was succeeded by Georg Waitz. The MGH was founded in Hanover as a text publication society by the Prussian reformer Heinrich Friedrich Karl Freiherr vom Stein in 1819. The first volume appeared in 1826, the editor from 1826 until 1874 was Georg Heinrich Pertz, who was succeeded by Georg Waitz. Many eminent medievalists from Germany and, eventually, other countries, joined in the project of searching out and comparing manuscripts, the motto chosen, Sanctus amor patriae dat animum is explained as linking Romantic nationalism with professional scholarship. In 1875 the MGH was established as a formal institution with headquarters in Berlin. It was taken over by the state in 1935 and renamed the Reichsinstitut für ältere deutsche Geschichtskunde and it moved into its current premises in the building of the Bavarian State Library in 1967. The project, an effort of historical scholarship, continues in the 21st century. The series falls into five divisions, Antiquitates, Diplomata, Epistolae, Leges and Scriptores. Many subsidiary series have also established, including a series of more compact volumes for school use. Historiography of Germany Wilhelm Levison Knowles, M. D, presidential Address, Great Historical Enterprises III. Transactions of the Royal Historical Society, Great Historical Enterprises, problems in monastic history. 2015 list of publications The MGH homepage Digital MGH homepage Monumenta Germaniae Historica on Archive. org
11.
Fan Chung
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Since 1998 Chung has been the Akamai Professor in Internet Mathematics at the University of California, San Diego. She received her doctorate from the University of Pennsylvania in 1974, after working at Bell Laboratories and Bellcore for nineteen years, she joined the faculty of the University of Pennsylvania as the first female tenured professor in mathematics. She serves on the boards of more than a dozen international journals. Since 2003 she has been the editor-in-chief of Internet Mathematics and she was selected to be a Noether Lecturer in 2009. Chung has two children, the first born during her studies, from her first marriage. Since 1983 she has married to the mathematician Ronald Graham. They were close friends of the mathematician Paul Erdős, and have published papers with him –13 in her case, thus, both have Erdős numbers of 1. She has published more than 200 research papers and three books, Erdős on Graphs, His Legacy of Unsolved Problems, A K Peters, Ltd. 1998, ISBN 1-56881-079-2 Spectral Graph Theory, American Mathematical Society,1997, ISBN 0-8218-0315-8 Complex Graphs and Networks (CBMS Regional Conference Series in Mathematics,107, American Mathematical Society,2006, ISBN 0-8218-3657-9 In 2012, she became a fellow of the American Mathematical Society. Fan Chung was born on October 9,1949 in Kaohsiung, under the influence of her father, an engineer, she started to be interested in mathematics, especially in the area of combinatorics in high school in Kaoshiung. After high school, Chung entered the National Taiwan University to start her career in mathematics formally, while Chung was an undergraduate, she was surrounded by many women mathematicians. They talked about the mathematics and helped each other, seeing more and more women doing well in mathematics also encouraged Chung to study mathematics. After graduating from NTU with a B. S. in mathematics, there she obtained the highest score in the qualifying exam by a wide margin, catching the attention of Herbert Wilf, who would eventually become her doctoral advisor. Wilf suggested Ramsey Theory as a subject Chung could work on, during a single week studying material Chung had come up with new proofs for established results in the field. Prof. Wilf said, My eyes were bulging, I asked her to go to the blackboard and show me. In just one week, from a start, she had a major result in Ramsey theory. I told her she had just done two-thirds of a doctoral dissertation, Chung was awarded a M. S. in 1972 and a Ph. D. two years later. By this time, she was married and had given birth to her first child
12.
Ronald Graham
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He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness. Graham was born in Taft, California, in 1962, he received his Ph. D. in mathematics from the University of California, Berkeley and began working at Bell Labs and later AT&T Labs. He was director of information sciences in AT&T Labs, but retired from AT&T in 1999 after 37 years and his 1977 paper considered a problem in Ramsey theory, and gave a large number as an upper bound for its solution. Graham popularized the concept of the Erdős number, named after the highly prolific Hungarian mathematician Paul Erdős, a scientists Erdős number is the minimum number of coauthored publications away from a publication with Erdős. He co-authored almost 30 papers with Erdős, and was also a good friend, Erdős often stayed with Graham, and allowed him to look after his mathematical papers and even his income. Graham and Erdős visited the young mathematician Jon Folkman when he was hospitalized with brain cancer, between 1993 and 1994 Graham served as the president of the American Mathematical Society. He has published about 320 papers and five books, including Concrete Mathematics with Donald Knuth and he is married to Fan Chung Graham, who is the Akamai Professor in Internet Mathematics at the University of California, San Diego. He has four children, daughters Ché, Laura and Christy, in 2003, Graham won the American Mathematical Societys annual Steele Prize for Lifetime Achievement. The prize was awarded on January 16 that year, at the Joint Mathematics Meetings in Baltimore, in 1999 he was inducted as a Fellow of the Association for Computing Machinery. Graham has won other prizes over the years, he was one of the laureates of the prestigious Pólya Prize the first year it was ever awarded. And the Carl Allendoerfer prize which was established in 1976 for the reasons, however for a different magazine. In 2012 he became a fellow of the American Mathematical Society, with Paul Erdős, Old and new results in combinatorial number theory. L’Enseignement Mathématique,1980 with Fan Chung, Erdős on Graphs, a. K. Peters,1998 with Jaroslav Nešetřil, The mathematics of Paul Erdős. Springer,1997 Rudiments of Ramsey Theory, American Mathematical Society,1981 with Donald E. Knuth & Oren Patashnik, Concrete Mathematics, a foundation for computer science. Addison-Wesley,1989,1994 with Joel H, spencer & Bruce L. Rothschild, Ramsey Theory. Wiley,1980,1990 with Martin Grötschel & László Lovász, Handbook of Combinatorics
13.
On the Equilibrium of Planes
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On the Equilibrium of Planes is a treatise by Archimedes in two volumes. The first book establishes the law of the lever, and locates the centre of gravity of the triangle, according to Pappus of Alexandria, Archimedes work on levers caused him to remark, Give me a place to stand on, and I will move the Earth. The second book, which contains ten propositions, examines the centres of gravity of parabolic segments, book one contains fifteen propositions with seven postulates. In proposition six Archimedes establishes the law of the lever, concluding that Magnitudes are in equilibrium at distances proportional to their weights. In propositions ten and fourteen, respectively, Archimedes locates the centre of gravity of the parallelogram, additionally, in proposition 15, he establishes the centre of gravity of the trapezium. The second book, which contains ten propositions, studies parabolic segments exclusively and it examines these segments by substituting them with rectangles of equal area, an exchange made possible by results obtained in the Quadrature of the Parabola. Archimedes proof of the law of the lever is executed within proposition six and it is for commensurable magnitudes only, and relies upon propositions four, and five, and on postulate one. In postulate one Archimedes states that Equal weights at equal distances are in equilibrium, now, double the length of ED by duplicating the longer arm on the left, and the shorter arm on the right. Note, additionally, that EH carries the common divisor N, a number of times, as does EC. It remains then to prove that A applied at E, and B applied at D, will have their centre of gravity at C. Therefore, as the ratio of LH to HK is not four to three, but eight to six, similarly divide the magnitudes A and B, and align them as per the diagram opposite. Further, the system consists of an even number of equal weights equally distributed. Thus A applied at E, and B applied at D, have their centre of gravity at C, whilst the authenticity of book two is not doubted, a number of researches have highlighted inconsistencies within book ones presentation. Berggren, in particular, questions the validity of one as a whole, highlighting, inter alia. However, Berggren follows Dijksterhuis, in rejecting Machs criticism of proposition six, further, it should be noted that proposition seven is incomplete in its current form, so that book one demonstrates the law of the lever for commensurable magnitudes only
14.
Measurement of a Circle
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Measurement of a Circle is a treatise that consists of three propositions by Archimedes. The treatise is only a fraction of what was a longer work, any circle with a circumference c and a radius r is equal in area with a right triangle with the two legs being c and r. This proposition is proved by the method of exhaustion, proposition two states, The area of a circle is to the square on its diameter as 11 to 14. This proposition could not have been placed by Archimedes, for it relies on the outcome of the third proposition, proposition three states, The ratio of the circumference of any circle to its diameter is greater than 31071 but less than 317. This approximates what we now call the mathematical constant π and he found these bounds on the value of π by inscribing and circumscribing a circle with two similar 96-sided regular polygons. This proposition also contains accurate approximations to the root of 3 and other larger non-perfect square roots, however. He gives the upper and lower bounds to √3 as 1351780 >3 >265153. Discussion of this goes back at least to Thomas Fantet de Lagny, FRS in 1723, but was treated more explicitly by Hieronymus Georg Zeuthen.4,7. Although only one route to the bounds is mentioned, in there are two others, making the bounds almost inescapable however the method is worked. But the bounds can also be produced by a geometrical construction suggested by Archimedes Stomachion in the setting of the regular dodecagon. In this case, the task is to give rational approximations to the tangent of π/12
15.
On Spirals
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On Spirals is a treatise by Archimedes, written around 225 BC. Although Archimedes did not discover the Archimedean spiral, he employed it in book to square the circle. Archimedes begins On Spirals with a message to Dositheus of Pelusium mentioning the death of Conon as a loss to mathematics and he then goes on to summarize the results of On the Sphere and Cylinder and On Conoids and Spheroids. He continues to state his results of On Spirals, the Archimedean spiral was first studied by Conon and was later studied by Archimedes in On Spirals. Archimedes was able to find various tangents to the spiral and he defines the spiral as, The construction as to how Archimedes trisected the angle is as follows, Suppose the angle ABC is to be trisected. Trisect the segment BC and find BD to be one third of BC, draw a circle with center B and radius BD. Suppose the circle with center B intersects the spiral at point E. Angle ABE is one third angle ABC, to square the circle, Archimedes gave the following construction, Let P be the point on the spiral when it has completed one turn. Let the tangent at P cut the line perpendicular to OP at T. OT is the length of the circumference of the circle with radius OP, so the area of the circle with radius OP is equal to the area of the triangle OPT
16.
On the Sphere and Cylinder
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On the Sphere and Cylinder is a work that was published by Archimedes in two volumes c.225 BC. It most notably details how to find the area of a sphere and the volume of the contained ball and the analogous values for a cylinder. The principal formulae derived in On the Sphere and Cylinder are those mentioned above, the area of the sphere, the volume of the contained ball. On the sphere, he showed that the area is four times the area of its great circle. In modern terms, this means that the area is equal to. The result for the volume of the contained ball stated that it is two-thirds the volume of a circumscribed cylinder, later, Roman philosopher Marcus Tullius Cicero discovered the tomb, which had been overgrown by surrounding vegetation. Archimedes used an inscribed half-polygon in a semicircle, then rotated both to create a conglomerate of frustums in a sphere, of which he determined the volume. It seems that this is not the original method Archimedes used to derive this result and his original method likely involved a clever use of levers. 7, pp. 473–476 Lucio Lombardo Radice, La matematica da Pitagora a Newton, Roma, attilio Frajese, Opere di Archimede, Torino, U. T. E. T
17.
On Floating Bodies
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On Floating Bodies is a Greek-language work consisting of two books written by Archimedes of Syracuse, one of the most important mathematicians, physicists, and engineers of antiquity. On Floating Bodies, which is thought to have written around 250 BC, survives only partly in Greek. It is the first known work on hydrostatics, of which Archimedes is recognized as the founder, the purpose of On Floating Bodies was to determine the positions that various solids will assume when floating in a fluid, according to their form and the variation in their specific gravities. It contains the first statement of what is now known as Archimedes principle, Archimedes lived in the Greek city-state of Syracuse, Sicily. He is credited with laying the foundations of hydrostatics, statics, a leading scientist of classical antiquity, Archimedes also developed elaborate systems of pulleys to move large objects with a minimum of effort. The Archimedes screw underpins modern hydroengineering, and his machines of war helped to hold back the armies of Rome in the First Punic War. Archimedes opposed the arguments of Aristotle, pointing out that it was impossible to separate mathematics and nature, the only known copy of On Floating Bodies in Greek comes from the Archimedes Palimpsest. In the first part of the treatise, Archimedes establishes various general principles, such as that a solid denser than a fluid will, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not self-gravitating, since he assumes the existence of a point towards all things fall in order to derive the spherical shape. Further, Proposition 5 of Archimedes treatise On Floating Bodies states that, the second book is a mathematical achievement unmatched in antiquity and rarely equaled since. It is restricted to the case when the base of the paraboloid lies either entirely above or entirely below the fluid surface, Archimedes investigation of paraboloids was probably an idealization of the shapes of ships hulls. Some of his sections float with the base water and the summit above water. Of his works survive, the second of his two books of On Floating Bodies is considered his most mature work, commonly described as a tour de force
18.
The Quadrature of the Parabola
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The Quadrature of the Parabola is a treatise on geometry, written by Archimedes in the 3rd century BC. The statement of the problem used the method of exhaustion, Archimedes may have dissected the area into infinitely many triangles whose areas form a geometric progression. He computes the sum of the geometric series, and proves that this is the area of the parabolic segment. A parabolic segment is the bounded by a parabola and line. To find the area of a segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the chord of the parabola. By Proposition 1, a line from the third vertex drawn parallel to the axis divides the chord into equal segments, the main theorem claims that the area of the parabolic segment is 4/3 that of the inscribed triangle. Archimedes gives two proofs of the main theorem, the first uses abstract mechanics, with Archimedes arguing that the weight of the segment will balance the weight of the triangle when placed on an appropriate lever. The second, more famous proof uses pure geometry, specifically the method of exhaustion, of the twenty-four propositions, the first three are quoted without proof from Euclids Elements of Conics. The main idea of the proof is the dissection of the segment into infinitely many triangles. Each of these triangles is inscribed in its own segment in the same way that the blue triangle is inscribed in the large segment. In propositions eighteen through twenty-one, Archimedes proves that the area of each triangle is one eighth of the area of the blue triangle. Using the method of exhaustion, it follows that the area of the parabolic segment is given by Area = T +2 +4 +8 + ⋯. This simplifies to give Area = T, to complete the proof, Archimedes shows that 1 +14 +116 +164 + ⋯ =43. The formula above is a geometric series—each successive term is one fourth of the previous term, in modern mathematics, that formula is a special case of the sum formula for a geometric series. Archimedes evaluates the sum using a geometric method, illustrated in the adjacent picture. This picture shows a square which has been dissected into an infinity of smaller squares. Each successive purple square has one fourth the area of the previous square, however, the purple squares are congruent to either set of yellow squares, and so cover 1/3 of the area of the unit square
19.
The Method of Mechanical Theorems
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The Method of Mechanical Theorems, also referred to as The Method, is considered one of the major surviving works of the ancient Greek polymath Archimedes. The Method takes the form of a letter from Archimedes to Eratosthenes, the librarian at the Library of Alexandria. The work was thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. Archimedes did not admit the method of indivisibles as part of rigorous mathematics, in these treatises, he proves the same theorems by exhaustion, finding rigorous upper and lower bounds which both converge to the answer required. Nevertheless, the method was what he used to discover the relations for which he later gave rigorous proofs. To explain Archimedes method today, it is convenient to use of a little bit of Cartesian geometry. His idea is to use the law of the lever to determine the areas of figures from the center of mass of other figures. The simplest example in modern language is the area of the parabola. Archimedes uses an elegant method, but in Cartesian language, his method is calculating the integral ∫01 x 2 d x =13. The idea is to balance the parabola with a certain triangle that is made of the same material. The parabola is the region in the x-y plane between the x-axis and y = x2 as x varies from 0 to 1, the triangle is the region in the x-y plane between the x-axis and the line y = x, also as x varies from 0 to 1. Slice the parabola and triangle into vertical slices, one for each value of x, imagine that the x-axis is a lever, with a fulcrum at x =0. The law of the states that two objects on opposite sides of the fulcrum will balance if each has the same torque. Since each pair of balances, moving the whole parabola to x = −1 would balance the whole triangle. This means that if the original uncut parabola is hung by a hook from the point x = −1, the center of mass of a triangle can be easily found by the following method, also due to Archimedes. If a median line is drawn from any one of the vertices of a triangle to the opposite edge E, the triangle will balance on the median, considered as a fulcrum. The reason is if the triangle is divided into infinitesimal line segments parallel to E, each segment has equal length on opposite sides of the median. This argument can be made rigorous by exhaustion by using little rectangles instead of infinitesimal lines
20.
Book of Lemmas
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The Book of Lemmas is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions on circles, the Book of Lemmas was first introduced in Arabic by Thābit ibn Qurra, he attributed the work to Archimedes. In 1661, the Arabic manuscript was translated into Latin by Abraham Ecchellensis, the Latin version was published under the name Liber Assumptorum. T. L. Heath translated Heiburgs Latin work into English in his The Works of Archimedes, another possibility is that the Book of Lemmas may be a collection of propositions by Archimedes later collected by a Greek writer. The Book of Lemmas introduces several new geometrical figures, Archimedes first introduced the arbelos in proposition four of his book, The figure is used in propositions four through eight. In propositions five, Archimedes introduces the Archimedes twin circles, and in eight, he makes use what would be the Pappus chain. Archimedes first introduced the salinon in proposition fourteen of his book, Archimedes proved that the salinon, if two circles touch at A, and if CD, EF be parallel diameters in them, ADF is a straight line. Let AB be the diameter of a semicircle, and let the tangents to it at B, if now DE be drawn perpendicular to AB, and if AT, DE meet in F, then DF = FE. Let P be any point on a segment of a circle whose base is AB, take D on AB so that AN = ND. If now PQ be an arc equal to the arc PA, let AB be the diameter of a semicircle, C any point on AB, and CD perpendicular to it, and let semicircles be described within the first semicircle and having AC, CB as diameters. Then if two circles be drawn touching CD on different sides and each touching two of the semicircles, the circles so drawn will be equal, let AB, the diameter of a semicircle, be divided at C so that AC = 3/2 × CB. Describe semicircles within the first semicircle and on AC, CB as diameters, if GH be the diameter of this circle, to find relation between GH and AB. If circles are circumscribed about and inscribed in a square, the circle is double of the inscribed square. If in a circle two chords AB, CD which do not pass through the centre intersect at right angles, suppose that TA, TB are two tangents to a circle, while TC cuts it. Let BD be the chord through B parallel to TC, and let AD meet TC in E. Then, if EH be drawn perpendicular to BD, it will bisect it in H. If two chords AB, CD in a circle intersect at right angles in a point O, not being the centre, then AO2 + BO2 + CO2 + DO2 =2. If AB be the diameter of a semicircle, and TP, TQ the tangents to it any point T. If a diameter AB of a circle meet any chord CD, not a diameter, in E, let ACB be a semicircle on AB as diameter, and let AD, BE be equal lengths measured along AB from A, B respectively
21.
Archimedean solid
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In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedrons composed of regular meeting in identical vertices, excluding the 5 Platonic solids. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices, identical vertices means that for any two vertices, there is a global isometry of the entire solid that takes one vertex to the other. Excluding these two families, there are 13 Archimedean solids. All the Archimedan solids can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry, the Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra, kepler may have also found the elongated square gyrobicupola, at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a configuration of means that a square, hexagon. Some definitions of semiregular polyhedron include one more figure, the square gyrobicupola or pseudo-rhombicuboctahedron. The number of vertices is 720° divided by the angle defect. The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular, the duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices, the snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form and right-handed form. When something comes in forms which are each others three-dimensional mirror image. The different Archimedean and Platonic solids can be related to each other using a handful of general constructions, starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated, different Platonic and Archimedean solids can be created, expansion or cantellation involves moving each face away from the center and taking the convex hull. Expansion with twisting also involves rotating the faces, thus breaking the rectangles corresponding to edges into triangles, the last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as truncation of corners and edges, note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron
22.
Buoyancy
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In science, buoyancy or upthrust, is an upward force exerted by a fluid that opposes the weight of an immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid, thus the pressure at the bottom of a column of fluid is greater than at the top of the column. Similarly, the pressure at the bottom of an object submerged in a fluid is greater than at the top of the object and this pressure difference results in a net upwards force on the object. For this reason, an object whose density is greater than that of the fluid in which it is submerged tends to sink, If the object is either less dense than the liquid or is shaped appropriately, the force can keep the object afloat. This can occur only in a reference frame, which either has a gravitational field or is accelerating due to a force other than gravity defining a downward direction. In a situation of fluid statics, the net upward force is equal to the magnitude of the weight of fluid displaced by the body. The center of buoyancy of an object is the centroid of the volume of fluid. Archimedes principle is named after Archimedes of Syracuse, who first discovered this law in 212 B. C, more tersely, Buoyancy = weight of displaced fluid. The weight of the fluid is directly proportional to the volume of the displaced fluid. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy and this is also known as upthrust. Suppose a rocks weight is measured as 10 newtons when suspended by a string in a vacuum with gravity acting upon it, suppose that when the rock is lowered into water, it displaces water of weight 3 newtons. The force it exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyancy force,10 −3 =7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor and it is generally easier to lift an object up through the water than it is to pull it out of the water. The density of the object relative to the density of the fluid can easily be calculated without measuring any volumes. Density of object density of fluid = weight weight − apparent immersed weight Example, If you drop wood into water, Example, A helium balloon in a moving car. During a period of increasing speed, the air mass inside the car moves in the direction opposite to the cars acceleration, the balloon is also pulled this way. However, because the balloon is buoyant relative to the air, it ends up being pushed out of the way, If the car slows down, the same balloon will begin to drift backward. For the same reason, as the car goes round a curve and this is the equation to calculate the pressure inside a fluid in equilibrium
23.
Archimedes' screw
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The Archimedes screw, also called the Archimedean screw or screwpump, is a machine historically used for transferring water from a low-lying body of water into irrigation ditches. Water is pumped by turning a screw-shaped surface inside a pipe, the screw pump is commonly attributed to Archimedes on the occasion of his visit to Egypt. This tradition may reflect only that the apparatus was unknown to the Greeks before Hellenistic times and was introduced in Archimedess lifetime by unknown Greek engineers, some writers have suggested the device may have been in use in Assyria some 350 years earlier. The Archimedes screw consists of a screw inside a hollow pipe, the screw is turned usually by a windmill or by manual labour or by cattle. As the shaft turns, the bottom end scoops up a volume of water and this water is then pushed up the tube by the rotating helicoid until finally it pours out from the top of the tube. If water from one section leaks into the lower one. In some designs, the screw is fused to the casing, a screw could be sealed with pitch resin or other adhesive to its casing, or cast as a single piece in bronze. Some researchers have postulated this as being the device used to irrigate the Hanging Gardens of Babylon, the design of the everyday Greek and Roman water screw, in contrast to the heavy bronze device of Sennacherib, with its problematic drive chains, has a powerful simplicity. A double or triple helix was built of wood strips around a wooden pole. It was used for draining land that was underneath the sea in the Netherlands, Archimedes screws are used in sewage treatment plants because they cope well with varying rates of flow and with suspended solids. An auger in a snow blower or grain elevator is essentially an Archimedes screw, many forms of axial flow pump basically contain an Archimedes screw. The principle is found in pescalators, which are Archimedes screws designed to lift fish safely from ponds. This technology is used primarily at fish hatcheries, where it is desirable to minimize the handling of fish. An Archimedes screw was used in the successful 2001 stabilization of the Leaning Tower of Pisa, small amounts of subsoil saturated by groundwater were removed from far below the north side of the Tower, and the weight of the tower itself corrected the lean. Archimedes screws are used in chocolate fountains. The invention of the screw is credited to Greek polymath Archimedes of Syracuse in the 3rd century BC. A cuneiform inscription of Assyrian king Sennacherib has been interpreted by Stephanie Dalley to describe casting water screws in bronze some 350 years earlier and this is consistent with classical author Strabo, who describes the Hanging Gardens as watered by screws. A contrary view is expressed by Dalley and Oleson, german engineer Konrad Kyeser equips the Archimedes screw with a crank mechanism in his Bellifortis
24.
Claw of Archimedes
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The Claw of Archimedes was an ancient weapon devised by Archimedes to defend the seaward portion of Syracuses city wall against amphibious assault. These machines featured prominently during the Second Punic War in 214 BC, 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 Livy attributed heavy Roman losses to these machines, the plausibility of this invention was tested in 1999 in the BBC series Secrets of the Ancients and again in early 2005 in the Discovery Channel series Superweapons of the Ancient World. The producers of Superweapons brought together a group of engineers tasked with conceiving and implementing a design that was realistic, within seven days they were able to test their creation, and they did succeed in tipping over a model of a Roman ship so that it would sink. While this does not prove the existence of the Claw, it suggests that it would have been possible, C. K. Young, Archimedess iron hand or claw – a new interpretation of an old mystery, Centaurus, Vol.46, No. 3, pp. 189–207 Scale models of the Claws operation BBC Secrets of the Ancients, The Claw
