1.
Euclid of Megara
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Euclid of Megara was a Greek Socratic philosopher who founded the Megarian school of philosophy. He was a pupil of Socrates in the late 5th century BCE and he held the supreme good to be one, eternal and unchangeable, and denied the existence of anything contrary to the good. Editors and translators in the Middle Ages often confused him with Euclid of Alexandria when discussing the latters Elements, Euclid was born in Megara, but in Athens he became a follower of Socrates. He is represented in the preface of Platos Theaetetus as being responsible for writing down the conversation between Socrates and the young Theaetetus many years earlier, Socrates is also supposed to have reproved Euclid for his fondness for eristic disputes. He was present at Socrates death, after which Euclid returned to Megara, in Megara, Euclid founded a school of philosophy which became known as the Megarian school, and which flourished for about a century. Euclids pupils were said to have been Ichthyas, the leader of the Megarian school, Eubulides of Miletus, Clinomachus. Thrasymachus was a teacher of Stilpo, who was the teacher of Zeno of Citium, Euclid himself wrote six dialogues — the Lamprias, the Aeschines, the Phoenix, the Crito, the Alcibiades, and the Amatory dialogue — but none survive. The main extant source on his views is the summary by Diogenes Laërtius. Euclids philosophy was a synthesis of Eleatic and Socratic ideas, Socrates claimed that the greatest knowledge was understanding the good. The Eleatics claimed the greatest knowledge is the one universal Being of the world, mixing these two ideas, Euclid claimed that good is the knowledge of this being. Therefore, this good is the thing that exists and has many names but is really just one thing. He identified the Eleatic idea of The One with the Socratic Form of the Good and this was the true essence of being, and was eternal and unchangeable. The idea of a universal good also allowed Euclid to dismiss all that is not good because he claimed that good covered all things on Earth with its many names. Euclid adopted the Socratic idea that knowledge is virtue and that the way to understand the never-changing world is through the study of philosophy. Euclid taught that virtues themselves, however, were simply the knowledge of the one good, or Being. As he said, The Good is One, but we can call it by names, sometimes as wisdom, sometimes as God, sometimes as Reason, and he declared. Euclid was also interested in concepts and dilemmas of logic, Euclid and his Megarian followers used dialogue and the eristic method to defend their ideas. The eristic method allowed them to prove their ideas by disproving those of the one they were arguing with and he also rejected argument from analogy
2.
Alexandria
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Alexandria is the second largest city and a major economic centre in Egypt, extending about 32 km along the coast of the Mediterranean Sea in the north central part of the country. Its low elevation on the Nile delta makes it vulnerable to rising sea levels. Alexandria is Egypts largest seaport, serving approximately 80% of Egypts imports and exports and it is an important industrial center because of its natural gas and oil pipelines from Suez. Alexandria is also an important tourist destination, Alexandria was founded around a small Ancient Egyptian town c.331 BC by Alexander the Great. Alexandria was the second most powerful city of the ancient world after Rome, Alexandria is believed to have been founded by Alexander the Great in April 331 BC as Ἀλεξάνδρεια. Alexanders chief architect for the project was Dinocrates, Alexandria was intended to supersede Naucratis as a Hellenistic center in Egypt, and to be the link between Greece and the rich Nile valley. The city and its museum attracted many of the greatest scholars, including Greeks, Jews, the city was later plundered and lost its significance. Just east of Alexandria, there was in ancient times marshland, as early as the 7th century BC, there existed important port cities of Canopus and Heracleion. The latter was rediscovered under water. An Egyptian city, Rhakotis, already existed on the shore also and it continued to exist as the Egyptian quarter of the city. A few months after the foundation, Alexander left Egypt and never returned to his city, after Alexanders departure, his viceroy, Cleomenes, continued the expansion. Although Cleomenes was mainly in charge of overseeing Alexandrias continuous development, the Heptastadion, inheriting the trade of ruined Tyre and becoming the center of the new commerce between Europe and the Arabian and Indian East, the city grew in less than a generation to be larger than Carthage. In a century, Alexandria had become the largest city in the world and and it became Egypts main Greek city, with Greek people from diverse backgrounds. Alexandria was not only a center of Hellenism, but was home to the largest urban Jewish community in the world. The Septuagint, a Greek version of the Tanakh, was produced there, in AD115, large parts of Alexandria were destroyed during the Kitos War, which gave Hadrian and his architect, Decriannus, an opportunity to rebuild it. On 21 July 365, Alexandria was devastated by a tsunami, the Islamic prophet, Muhammads first interaction with the people of Egypt occurred in 628, during the Expedition of Zaid ibn Haritha. He sent Hatib bin Abi Baltaeh with a letter to the king of Egypt and Alexandria called Muqawqis In the letter Muhammad said, I invite you to accept Islam, Allah the sublime, shall reward you doubly. But if you refuse to do so, you bear the burden of the transgression of all the Copts
3.
Hellenistic Egypt
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The Ptolemaic Kingdom was a Hellenistic kingdom based in Egypt. Alexandria became the city and a major center of Greek culture. To gain recognition by the native Egyptian populace, they named themselves the successors to the Pharaohs, the later Ptolemies took on Egyptian traditions by marrying their siblings, had themselves portrayed on public monuments in Egyptian style and dress, and participated in Egyptian religious life. The Ptolemies had to fight native rebellions and were involved in foreign and civil wars led to the decline of the kingdom. Hellenistic culture continued to thrive in Egypt throughout the Roman and Byzantine periods until the Muslim conquest. The era of Ptolemaic reign in Egypt is one of the most well documented periods of the Hellenistic Era. In 332 BC, Alexander the Great, King of Macedon invaded the Achaemenid satrapy of Egypt and he visited Memphis, and traveled to the oracle of Amun at the Oasis of Siwa. The oracle declared him to be the son of Amun, the wealth of Egypt could now be harnessed for Alexanders conquest of the rest of the Persian Empire. Early in 331 BC he was ready to depart, and led his forces away to Phoenicia and he left Cleomenes as the ruling nomarch to control Egypt in his absence. Following Alexanders death in Babylon in 323 BC, a crisis erupted among his generals. Perdiccas appointed Ptolemy, one of Alexanders closest companions, to be satrap of Egypt, Ptolemy ruled Egypt from 323 BC, nominally in the name of the joint kings Philip III and Alexander IV. However, as Alexander the Greats empire disintegrated, Ptolemy soon established himself as ruler in his own right, Ptolemy successfully defended Egypt against an invasion by Perdiccas in 321 BC, and consolidated his position in Egypt and the surrounding areas during the Wars of the Diadochi. In 305 BC, Ptolemy took the title of King, as Ptolemy I Soter, he founded the Ptolemaic dynasty that was to rule Egypt for nearly 300 years. All the male rulers of the dynasty took the name Ptolemy, while princesses and queens preferred the names Cleopatra, Arsinoe and Berenice. Because the Ptolemaic kings adopted the Egyptian custom of marrying their sisters, many of the kings ruled jointly with their spouses and this custom made Ptolemaic politics confusingly incestuous, and the later Ptolemies were increasingly feeble. The only Ptolemaic Queens to officially rule on their own were Berenice III, Cleopatra V did co-rule, but it was with another female, Berenice IV. Cleopatra VII officially co-ruled with Ptolemy XIII Theos Philopator, Ptolemy XIV, and Ptolemy XV, upper Egypt, farthest from the centre of government, was less immediately affected, even though Ptolemy I established the Greek colony of Ptolemais Hermiou to be its capital. But within a century Greek influence had spread through the country, nevertheless, the Greeks always remained a privileged minority in Ptolemaic Egypt
4.
Euclidean geometry
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Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
5.
Euclid's Elements
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Euclids Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
6.
Euclidean algorithm
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It is named after the ancient Greek mathematician Euclid, who first described it in Euclids Elements. It is an example of an algorithm, a procedure for performing a calculation according to well-defined rules. It can be used to reduce fractions to their simplest form, the Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example,21 is the GCD of 252 and 105, since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the two numbers. By reversing the steps, the GCD can be expressed as a sum of the two numbers each multiplied by a positive or negative integer, e. g.21 =5 ×105 + ×252. The fact that the GCD can always be expressed in this way is known as Bézouts identity, the version of the Euclidean algorithm described above can take many subtraction steps to find the GCD when one of the given numbers is much bigger than the other. A more efficient version of the algorithm shortcuts these steps, instead replacing the larger of the two numbers by its remainder when divided by the smaller of the two. With this improvement, the algorithm never requires more steps than five times the number of digits of the smaller integer and this was proven by Gabriel Lamé in 1844, and marks the beginning of computational complexity theory. Additional methods for improving the algorithms efficiency were developed in the 20th century, the Euclidean algorithm has many theoretical and practical applications. It is used for reducing fractions to their simplest form and for performing division in modular arithmetic, finally, it can be used as a basic tool for proving theorems in number theory such as Lagranges four-square theorem and the uniqueness of prime factorizations. This led to abstract algebraic notions such as Euclidean domains. The Euclidean algorithm calculates the greatest common divisor of two numbers a and b. The greatest common divisor g is the largest natural number that divides both a and b without leaving a remainder, synonyms for the GCD include the greatest common factor, the highest common factor, the highest common divisor, and the greatest common measure. The greatest common divisor is often written as gcd or, more simply, as, although the notation is also used for other mathematical concepts. If gcd =1, then a and b are said to be coprime and this property does not imply that a or b are themselves prime numbers. For example, neither 6 nor 35 is a prime number, nevertheless,6 and 35 are coprime. No natural number other than 1 divides both 6 and 35, since they have no prime factors in common
7.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
8.
Bramante
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Donato Bramante, born as Donato di Pascuccio dAntonio and also known as Bramante Lazzari, was an Italian architect. He introduced Renaissance architecture to Milan and the High Renaissance style to Rome and his Tempietto marked the beginning of the High Renaissance in Rome when Pope Julius II appointed him to build a sanctuary over the spot where Peter was allegedly crucified. Bramante was born under the name Donato dAugnolo, Donato di Pascuccio dAntonio, here, in 1467, Luciano Laurana was adding to the Palazzo Ducale an arcaded courtyard and other Renaissance features to Federico da Montefeltros ducal palace. Around 1474, Bramante moved to Milan, a city with a deep Gothic architectural tradition, and built several churches in the new Antique style. The Duke, Ludovico Sforza, made him virtually his court architect, beginning in 1476, space was limited, and Bramante made a theatrical apse in bas-relief, combining the painterly arts of perspective with Roman details. There is a sacristy, surmounted by a dome. In Milan, Bramante also built the tribune of Santa Maria delle Grazie, other works include the Cloisters of SantAmbrogio, Milan. However, in 1499, with his Sforza patron driven from Milan by an invading French army, Bramante made his way to Rome, in Rome, he was soon recognized by Cardinal Della Rovere, shortly to become Pope Julius II. Despite its small scale, the construction has all the proportions and symmetry of Classical structures, surrounded by slender Doric columns. According to an engraving by Sebastiano Serlio, Bramante planned to set it within a colonnaded courtyard. In November 1503, Julius engaged Bramante for the construction of the grandest European architectural commission of the 16th century, the cornerstone of the first of the great piers of the crossing was laid with ceremony on 17 April 1506. Very few drawings by Bramante survive, though some by his assistants do, Bramantes plan envisaged four great chapels filling the corner spaces between the equal transepts, each one capped with a smaller dome surrounding the great dome over the crossing. So Bramantes original plan was much more Romano-Byzantine in its forms than the basilica that was actually built. Bramante also worked on other commissions. Among his earliest works in Rome, before the Basilicas construction was under way, is the cloister of Santa Maria della Pace near Piazza Navona, Santa Maria presso San Satiro, Milan, ca. Palazzo Caprini, Rome, started around 1510 Leon Battista Alberti Giorgio Vasari Davies, Paul, Bramante, Donato, The Dictionary of Art, Vol. IV, New York, Grove, pp. 642–653, ISBN9781884446009. Bramante, Encyclopædia Britannica, 9th ed. Vol. IV, New York, Charles Scribners Sons,1878, Bramante, Encyclopædia Britannica, 11th ed. Vol. IV, Cambridge, Cambridge University Press,1911, p.418. Donato Bramante Source Information, Pictures & Documentaries about Donato
9.
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
10.
School of Athens
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The School of Athens is one of the most famous frescoes by the Italian Renaissance artist Raphael. It was painted between 1509 and 1511 as a part of Raphaels commission to decorate the rooms now known as the Stanze di Raffaello, the picture has long been seen as Raphaels masterpiece and the perfect embodiment of the classical spirit of the Renaissance. The School of Athens is one of a group of four main frescoes on the walls of the Stanza that depict distinct branches of knowledge, accordingly, the figures on the walls below exemplify Philosophy, Poetry, Theology, and Law. The traditional title is not Raphaels, indeed, Plato and Aristotle appear to be the central figures in the scene. However, all the philosophers depicted sought knowledge of first causes, many lived before Plato and Aristotle, and hardly a third were Athenians. The architecture contains Roman elements, but the general semi-circular setting having Plato, compounding the problem, Raphael had to invent a system of iconography to allude to various figures for whom there were no traditional visual types. For example, while the Socrates figure is immediately recognizable from Classical busts, aside from the identities of the figures depicted, many aspects of the fresco have been variously interpreted, but few such interpretations are unanimously accepted among scholars. The popular idea that the gestures of Plato and Aristotle are kinds of pointing is very likely. Aristotle, with his four-elements theory, held that all change on Earth was owing to motions of the heavens, in the painting Aristotle carries his Ethics, which he denied could be reduced to a mathematical science. Finally, according to Vasari, the scene includes Raphael himself, however, as Heinrich Wölfflin observed, it is quite wrong to attempt interpretations of the School of Athens as an esoteric treatise. The all-important thing was the motive which expressed a physical or spiritual state. An interpretation of the fresco relating to hidden symmetries of the figures, the identities of some of the philosophers in the picture, such as Plato or Aristotle, are certain. Beyond that, identifications of Raphaels figures have always been hypothetical, to complicate matters, beginning from Vasaris efforts, some have received multiple identifications, not only as ancients but also as figures contemporary with Raphael. Vasari mentions portraits of the young Federico II Gonzaga, Duke of Mantua, leaning over Bramante with his hands raised near the bottom right and he was writing over 40 years after the painting, and never knew Raphael, but no doubt reflects what was believed in his time. Many other popular identifications of portraits are very dubious, luitpold Dussler counts among those who can be identified with some certainty, Plato, Aristotle, Socrates, Pythagoras, Euclid, Ptolemy, Zoroaster, Raphael, Sodoma and Diogenes. Other identifications he holds to be more or less speculative, both figures hold modern, bound copies of their books in their left hands, while gesturing with their right. Plato holds Timaeus, Aristotle his Nicomachean Ethics, Plato is depicted as old, grey, wise-looking, and bare-foot. By contrast Aristotle, slightly ahead of him, is in manhood, handsome, well-shod and dressed with gold
11.
Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
12.
Euclides of Megara
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Euclid of Megara was a Greek Socratic philosopher who founded the Megarian school of philosophy. He was a pupil of Socrates in the late 5th century BCE and he held the supreme good to be one, eternal and unchangeable, and denied the existence of anything contrary to the good. Editors and translators in the Middle Ages often confused him with Euclid of Alexandria when discussing the latters Elements, Euclid was born in Megara, but in Athens he became a follower of Socrates. He is represented in the preface of Platos Theaetetus as being responsible for writing down the conversation between Socrates and the young Theaetetus many years earlier, Socrates is also supposed to have reproved Euclid for his fondness for eristic disputes. He was present at Socrates death, after which Euclid returned to Megara, in Megara, Euclid founded a school of philosophy which became known as the Megarian school, and which flourished for about a century. Euclids pupils were said to have been Ichthyas, the leader of the Megarian school, Eubulides of Miletus, Clinomachus. Thrasymachus was a teacher of Stilpo, who was the teacher of Zeno of Citium, Euclid himself wrote six dialogues — the Lamprias, the Aeschines, the Phoenix, the Crito, the Alcibiades, and the Amatory dialogue — but none survive. The main extant source on his views is the summary by Diogenes Laërtius. Euclids philosophy was a synthesis of Eleatic and Socratic ideas, Socrates claimed that the greatest knowledge was understanding the good. The Eleatics claimed the greatest knowledge is the one universal Being of the world, mixing these two ideas, Euclid claimed that good is the knowledge of this being. Therefore, this good is the thing that exists and has many names but is really just one thing. He identified the Eleatic idea of The One with the Socratic Form of the Good and this was the true essence of being, and was eternal and unchangeable. The idea of a universal good also allowed Euclid to dismiss all that is not good because he claimed that good covered all things on Earth with its many names. Euclid adopted the Socratic idea that knowledge is virtue and that the way to understand the never-changing world is through the study of philosophy. Euclid taught that virtues themselves, however, were simply the knowledge of the one good, or Being. As he said, The Good is One, but we can call it by names, sometimes as wisdom, sometimes as God, sometimes as Reason, and he declared. Euclid was also interested in concepts and dilemmas of logic, Euclid and his Megarian followers used dialogue and the eristic method to defend their ideas. The eristic method allowed them to prove their ideas by disproving those of the one they were arguing with and he also rejected argument from analogy
13.
Greek mathematics
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Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
14.
Ptolemy I
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Ptolemy I Soter I, also known as Ptolemy Lagides, was a Macedonian Greek general under Alexander the Great, one of the three Diadochi who succeeded to his empire. Ptolemy became ruler of Egypt and founded a dynasty which ruled it for the three centuries, turning Egypt into a Hellenistic kingdom and Alexandria into a center of Greek culture. He assimilated some aspects of Egyptian culture, however, assuming the title pharaoh in 305/4 BC. The use of the title of pharaoh was often situational, pharaoh was used for an Egyptian audience, like all Macedonian nobles, Ptolemy I Soter claimed descent from Heracles, the mythical founder of the Argead dynasty that ruled Macedon. Ptolemy was one of Alexanders most trusted generals, and was among the seven somatophylakes attached to his person and he was a few years older than Alexander and had been his intimate friend since childhood. He was succeeded by his son Ptolemy II Philadelphus, Ptolemy served with Alexander from his first campaigns, and played a principal part in the later campaigns in Afghanistan and India. Ptolemy had his first independent command during the campaign against the rebel Bessus whom Ptolemy captured and handed over to Alexander for execution. During Alexanders campaign in the Indian subcontinent Ptolemy was in command of the guard at the siege of Aornos. When Alexander died in 323 BC, Ptolemy is said to have instigated the resettlement of the made at Babylon. Ptolemy quickly moved, without authorization, to subjugate Cyrenaica, by custom, kings in Macedonia asserted their right to the throne by burying their predecessor. Ptolemy then openly joined the coalition against Perdiccas, Perdiccas appears to have suspected Ptolemy of aiming for the throne himself, and may have decided that Ptolemy was his most dangerous rival. Ptolemy executed Cleomenes for spying on behalf of Perdiccas — this removed the check on his authority. In 321 BC, Perdiccas attempted to invade Egypt only to fall at the hands of his own men, Ptolemys decision to defend the Nile against Perdiccass attempt to force it ended in fiasco for Perdiccas, with the loss of 2000 men. This failure was a blow to Perdiccas reputation, and he was murdered in his tent by two of his subordinates. Ptolemy immediately crossed the Nile, to provide supplies to what had the day before been an enemy army, Ptolemy was offered the regency in place of Perdiccas, but he declined. Ptolemy was consistent in his policy of securing a power base and his first occupation of Syria was in 318, and he established at the same time a protectorate over the petty kings of Cyprus. When Antigonus One-Eye, master of Asia in 315, showed dangerous ambitions, Ptolemy joined the coalition against him, in Cyprus, he fought the partisans of Antigonus, and re-conquered the island. A revolt in Cyrene was crushed the same year, in 312, Ptolemy and Seleucus, the fugitive satrap of Babylonia, both invaded Syria, and defeated Demetrius Poliorcetes, the son of Antigonus, in the Battle of Gaza
15.
History of mathematics
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Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322, the Rhind Mathematical Papyrus, All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Greek mathematics greatly refined the methods and expanded the subject matter of mathematics, Chinese mathematics made early contributions, including a place value system. Islamic mathematics, in turn, developed and expanded the known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, from ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, the origins of mathematical thought lie in the concepts of number, magnitude, and form. Modern studies of cognition have shown that these concepts are not unique to humans. Such concepts would have part of everyday life in hunter-gatherer societies. The idea of the number concept evolving gradually over time is supported by the existence of languages which preserve the distinction between one, two, and many, but not of numbers larger than two. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. The Ishango bone, found near the headwaters of the Nile river, may be more than 20,000 years old, common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10, predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian, Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. The majority of Babylonian mathematical work comes from two widely separated periods, The first few hundred years of the second millennium BC, and it is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics, in contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and they developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period
16.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
17.
Perspective (visual)
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Perspective in the graphic arts is an approximate representation, on a flat surface, of an image as it is seen by the eye. If viewed from the spot as the windowpane was painted. Each painted object in the scene is thus a flat, scaled down version of the object on the side of the window. All perspective drawings assume the viewer is a distance away from the drawing. Objects are scaled relative to that viewer, an object is often not scaled evenly, a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening, Perspective drawings have a horizon line, which is often implied. This line, directly opposite the viewers eye, represents objects infinitely far away and they have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous to the Earths horizon, any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a vanishing point, usually directly opposite the viewers eye. All lines parallel with the line of sight recede to the horizon towards this vanishing point. This is the standard receding railroad tracks phenomenon, a two-point drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of lines that are at an angle relative to the plane of the drawing. Perspectives consisting of parallel lines are observed most often when drawing architecture. In contrast, natural scenes often do not have any sets of parallel lines, the only method to indicate the relative position of elements in the composition was by overlapping, of which much use is made in works like the Parthenon Marbles. Chinese artists made use of perspective from the first or second century until the 18th century. It is not certain how they came to use the technique, some authorities suggest that the Chinese acquired the technique from India, oblique projection is also seen in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga. This was detailed within Aristotles Poetics as skenographia, using flat panels on a stage to give the illusion of depth, the philosophers Anaxagoras and Democritus worked out geometric theories of perspective for use with skenographia. Alcibiades had paintings in his house designed using skenographia, so this art was not confined merely to the stage, Euclids Optics introduced a mathematical theory of perspective, but there is some debate over the extent to which Euclids perspective coincides with the modern mathematical definition
18.
Conic section
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In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them
19.
Spherical geometry
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Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry that is not Euclidean, two practical applications of the principles of spherical geometry are navigation and astronomy. In plane geometry, the concepts are points and lines. On a sphere, points are defined in the usual sense, the equivalents of lines are not defined in the usual sense of straight line in Euclidean geometry, but in the sense of the shortest paths between points, which are called geodesics. On a sphere, the geodesics are the circles, other geometric concepts are defined as in plane geometry. Spherical geometry is not elliptic geometry, but is rather a subset of elliptic geometry, for example, it shares with that geometry the property that a line has no parallels through a given point. An important geometry related to that of the sphere is that of the projective plane. Locally, the plane has all the properties of spherical geometry. In particular, it is non-orientable, or one-sided, Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. Higher-dimensional spherical geometries exist, see elliptic geometry, the earliest mathematical work of antiquity to come down to our time is On the rotating sphere by Autolycus of Pitane, who lived at the end of the fourth century BC. The book of unknown arcs of a written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the law of sines. The book On Triangles by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe, however, Gerolamo Cardano noted a century later that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah. L. Euler, De curva rectificabili in superficie sphaerica, Novi Commentarii academiae scientiarum Petropolitanae 15,1771, pp. 195–216, Opera Omnia, Series 1, Volume 28, pp. 142–160. L. Euler, De mensura angulorum solidorum, Acta academiae scientarum imperialis Petropolitinae 2,1781, p. 31–54, Opera Omnia, Series 1, vol. L. Euler, Problematis cuiusdam Pappi Alexandrini constructio, Acta academiae scientarum imperialis Petropolitinae 4,1783, p. 91–96, Opera Omnia, Series 1, vol. L. Euler, Geometrica et sphaerica quaedam, Mémoires de lAcademie des Sciences de Saint-Petersbourg 5,1815, p. 96–114, Opera Omnia, Series 1, vol. L. Euler, Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta academiae scientarum imperialis Petropolitinae 3,1782, p. 72–86, Opera Omnia, Series 1, vol
20.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
21.
Pappus of Alexandria
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Pappus of Alexandria was one of the last great Alexandrian mathematicians of Antiquity, known for his Synagoge or Collection, and for Pappuss hexagon theorem in projective geometry. Nothing is known of his life, other than, that he had a son named Hermodorus, Collection, his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a range of topics, including geometry, recreational mathematics, doubling the cube, polygons. Pappus flourished in the 4th century AD, in a period of general stagnation in mathematical studies, he stands out as a remarkable exception. In this respect the fate of Pappus strikingly resembles that of Diophantus, in his surviving writings, Pappus gives no indication of the date of the authors whose works he makes use of, or of the time at which he himself wrote. If no other information were available, all that could be known would be that he was later than Ptolemy, whom he quotes, and earlier than Proclus. The Suda states that Pappus was of the age as Theon of Alexandria. A different date is given by a note to a late 10th-century manuscript, which states, next to an entry on Emperor Diocletian. This works out as October 18,320 AD, and so Pappus must have flourished c.320 AD. The great work of Pappus, in eight books and titled Synagoge or Collection, has not survived in complete form, the first book is lost, and the rest have suffered considerably. The Suda enumerates other works of Pappus, Χωρογραφία οἰκουμενική, commentary on the 4 books of Ptolemys Almagest, Ποταμοὺς τοὺς ἐν Λιβύῃ, Pappus himself mentions another commentary of his own on the Ἀνάλημμα of Diodorus of Alexandria. Pappus also wrote commentaries on Euclids Elements, and on Ptolemys Ἁρμονικά and these discoveries form, in fact, a text upon which Pappus enlarges discursively. Heath considered the systematic introductions to the books as valuable, for they set forth clearly an outline of the contents. From these introductions one can judge of the style of Pappuss writing, heath also found his characteristic exactness made his Collection a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us. The portions of Collection which has survived can be summarized as follows and we can only conjecture that the lost Book I, like Book II, was concerned with arithmetic, Book III being clearly introduced as beginning a new subject. The whole of Book II discusses a method of multiplication from a book by Apollonius of Perga. The final propositions deal with multiplying together the values of Greek letters in two lines of poetry, producing two very large numbers approximately equal to 2*1054 and 2*1038. Book III contains geometrical problems, plane and solid, on the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure
22.
Tyre, Lebanon
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Tyre, sometimes romanized as Sour, is a city in the South Governorate of Lebanon. There were approximately 117,000 inhabitants in 2003, however, the government of Lebanon has released only rough estimates of population numbers since 1932, so an accurate statistical accounting is not possible. Tyre juts out from the coast of the Mediterranean and is located about 80 km south of Beirut, the name of the city means rock after the rocky formation on which the town was originally built. The adjective for Tyre is Tyrian, and the inhabitants are Tyrians, Tyre is an ancient Phoenician city and the legendary birthplace of Europa and Dido. Today it is the fourth largest city in Lebanon and houses one of the major ports. The city has a number of ancient sites, including its Roman Hippodrome which was added to UNESCOs list of World Heritage Sites in 1979. Tyre originally consisted of two urban centres, Tyre itself, which was on an island just off shore. Alexander the Great connected the island to the mainland by constructing a causeway during his siege of the city, the original island city had two harbours, one on the south side and the other on the north side of the island. The harbour on the side has silted over, but the harbour on the north side is still in use. Tyre was founded around 2750 BC according to Herodotus and was built as a walled city upon the mainland. Phoenicians from Tyre settled in houses around Memphis, south of the temple of Hephaestus in a called the Tyrian Camp. Tyres name appears on monuments as early as 1300 BC, philo of Byblos quotes the antiquarian authority Sanchuniathon as stating that it was first occupied by Hypsuranius. Sanchuniathons work is said to be dedicated to Abibalus king of Berytus—possibly the Abibaal who was king of Tyre, there are ten Amarna letters dated 1350 BC from the mayor, Abimilku, written to Akenaten. The subject is often water, wood and the Habiru overtaking the countryside of the mainland, the commerce of the ancient world was gathered into the warehouses of Tyre. The city of Tyre was particularly known for the production of a rare and extraordinarily expensive sort of dye, produced from the murex shellfish. The colour was, in ancient cultures, reserved for the use of royalty or at least the nobility, Tyre was often attacked by Egypt, besieged by Shalmaneser V, who was assisted by the Phoenicians of the mainland, for five years. From 586 until 573 BC, the city was besieged by Nebuchadnezzar II until it agreed to pay a tribute. The Achaemenid Empire conquered the city in 539 BC and kept it under its rule until Alexander the Great laid siege to the city, in 315 BC, Alexanders former general Antigonus began his own siege of Tyre, taking the city a year later
23.
Library of Alexandria
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The Royal Library of Alexandria or Ancient Library of Alexandria in Alexandria, Egypt, was one of the largest and most significant libraries of the ancient world. It was dedicated to the Muses, the nine goddesses of the arts, the library was part of a larger research institution called the Musaeum of Alexandria, where many of the most famous thinkers of the ancient world studied. The library was created by Ptolemy I Soter, who was a Macedonian general, most of the books were kept as papyrus scrolls. It is unknown precisely how many such scrolls were housed at any given time, arguably, this library is most famous for having been burned down resulting in the loss of many scrolls and books, its destruction has become a symbol for the loss of cultural knowledge. Sources differ on who was responsible for its destruction and when it occurred, the library may in truth have suffered several fires over many years. Possible occasions for the partial or complete destruction of the Library of Alexandria include a set by the army of Julius Caesar in 48 BC. After the main library was destroyed, scholars used a library in a temple known as the Serapeum. The library may have finally been destroyed during the Muslim conquest of Egypt in AD642, the library itself is known to have had an acquisitions department and a cataloguing department. A hall contained shelves for the collections of papyrus scrolls known as bibliothekai, according to popular description, an inscription above the shelves read, The place of the cure of the soul. The library was but one part of the Musaeum of Alexandria, in addition to the library, the Musaeum included rooms for the study of astronomy, anatomy, and even a zoo containing exotic animals. The classical thinkers who studied, wrote, and experimented at the Musaeum include the names of mathematics, astronomy, physics, geometry, engineering, geography, physiology. These included notable thinkers such as Euclid, Archimedes, Eratosthenes, Herophilus, Erasistratus, Hipparchus, Aedesia, Pappus, Theon, Hypatia and it is not possible to determine the collections size in any era with any certainty. A single piece of writing might occupy several scrolls, and this division into self-contained books was an aspect of editorial work. King Ptolemy II Philadelphus is said to have set 500,000 scrolls as an objective for the library. The librarys index, Callimachus Pinakes, was lost with the rest of the library and this library, with the largest holdings of the age, acquired its collection by laborious copying of originals. Galen spoke of how all ships visiting the city were obliged to surrender their books for immediate copying, the owners received a copy while the pharaohs kept the originals in the library within their museum. As a research institution, the library filled its stacks with new works in mathematics, astronomy, physics, natural sciences and its empirical standards applied in one of the first and certainly strongest homes for serious textual criticism. As the same text often existed in different versions, comparative textual criticism was crucial for ensuring their veracity
24.
Alexander the Great
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Alexander III of Macedon, commonly known as Alexander the Great, was a king of the Ancient Greek kingdom of Macedon and a member of the Argead dynasty. He was born in Pella in 356 BC and succeeded his father Philip II to the throne at the age of twenty and he was undefeated in battle and is widely considered one of historys most successful military commanders. During his youth, Alexander was tutored by Aristotle until the age of 16, after Philips assassination in 336 BC, he succeeded his father to the throne and inherited a strong kingdom and an experienced army. Alexander was awarded the generalship of Greece and used this authority to launch his fathers Panhellenic project to lead the Greeks in the conquest of Persia, in 334 BC, he invaded the Achaemenid Empire and began a series of campaigns that lasted ten years. Following the conquest of Anatolia, Alexander broke the power of Persia in a series of battles, most notably the battles of Issus. He subsequently overthrew Persian King Darius III and conquered the Achaemenid Empire in its entirety, at that point, his empire stretched from the Adriatic Sea to the Indus River. He sought to reach the ends of the world and the Great Outer Sea and invaded India in 326 BC and he eventually turned back at the demand of his homesick troops. Alexander died in Babylon in 323 BC, the city that he planned to establish as his capital, without executing a series of planned campaigns that would have begun with an invasion of Arabia. In the years following his death, a series of civil wars tore his empire apart, resulting in the establishment of several states ruled by the Diadochi, Alexanders surviving generals, Alexanders legacy includes the cultural diffusion which his conquests engendered, such as Greco-Buddhism. He founded some twenty cities that bore his name, most notably Alexandria in Egypt, Alexander became legendary as a classical hero in the mold of Achilles, and he features prominently in the history and mythic traditions of both Greek and non-Greek cultures. He became the measure against which military leaders compared themselves, and he is often ranked among the most influential people in human history. He was the son of the king of Macedon, Philip II, and his wife, Olympias. Although Philip had seven or eight wives, Olympias was his wife for some time. Several legends surround Alexanders birth and childhood, sometime after the wedding, Philip is said to have seen himself, in a dream, securing his wifes womb with a seal engraved with a lions image. Plutarch offered a variety of interpretations of dreams, that Olympias was pregnant before her marriage, indicated by the sealing of her womb. On the day Alexander was born, Philip was preparing a siege on the city of Potidea on the peninsula of Chalcidice. That same day, Philip received news that his general Parmenion had defeated the combined Illyrian and Paeonian armies, and it was also said that on this day, the Temple of Artemis in Ephesus, one of the Seven Wonders of the World, burnt down. This led Hegesias of Magnesia to say that it had burnt down because Artemis was away, such legends may have emerged when Alexander was king, and possibly at his own instigation, to show that he was superhuman and destined for greatness from conception
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Plato
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Plato was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world. He is widely considered the most pivotal figure in the development of philosophy, unlike nearly all of his philosophical contemporaries, Platos entire work is believed to have survived intact for over 2,400 years. Along with his teacher, Socrates, and his most famous student, Aristotle, Plato laid the foundations of Western philosophy. Alfred North Whitehead once noted, the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato. In addition to being a figure for Western science, philosophy. Friedrich Nietzsche, amongst other scholars, called Christianity, Platonism for the people, Plato was the innovator of the written dialogue and dialectic forms in philosophy, which originate with him. He was not the first thinker or writer to whom the word “philosopher” should be applied, few other authors in the history of Western philosophy approximate him in depth and range, perhaps only Aristotle, Aquinas and Kant would be generally agreed to be of the same rank. Due to a lack of surviving accounts, little is known about Platos early life, the philosopher came from one of the wealthiest and most politically active families in Athens. Ancient sources describe him as a bright though modest boy who excelled in his studies, the exact time and place of Platos birth are unknown, but it is certain that he belonged to an aristocratic and influential family. Based on ancient sources, most modern scholars believe that he was born in Athens or Aegina between 429 and 423 BCE. According to a tradition, reported by Diogenes Laertius, Ariston traced his descent from the king of Athens, Codrus. Platos mother was Perictione, whose family boasted of a relationship with the famous Athenian lawmaker, besides Plato himself, Ariston and Perictione had three other children, these were two sons, Adeimantus and Glaucon, and a daughter Potone, the mother of Speusippus. The brothers Adeimantus and Glaucon are mentioned in the Republic as sons of Ariston, and presumably brothers of Plato, but in a scenario in the Memorabilia, Xenophon confused the issue by presenting a Glaucon much younger than Plato. Then, at twenty-eight, Hermodorus says, went to Euclides in Megara, as Debra Nails argues, The text itself gives no reason to infer that Plato left immediately for Megara and implies the very opposite. Thus, Nails dates Platos birth to 424/423, another legend related that, when Plato was an infant, bees settled on his lips while he was sleeping, an augury of the sweetness of style in which he would discourse about philosophy. Ariston appears to have died in Platos childhood, although the dating of his death is difficult. Perictione then married Pyrilampes, her mothers brother, who had served many times as an ambassador to the Persian court and was a friend of Pericles, Pyrilampes had a son from a previous marriage, Demus, who was famous for his beauty. Perictione gave birth to Pyrilampes second son, Antiphon, the half-brother of Plato and these and other references suggest a considerable amount of family pride and enable us to reconstruct Platos family tree
26.
Ptolemy I Soter
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Ptolemy I Soter I, also known as Ptolemy Lagides, was a Macedonian Greek general under Alexander the Great, one of the three Diadochi who succeeded to his empire. Ptolemy became ruler of Egypt and founded a dynasty which ruled it for the three centuries, turning Egypt into a Hellenistic kingdom and Alexandria into a center of Greek culture. He assimilated some aspects of Egyptian culture, however, assuming the title pharaoh in 305/4 BC. The use of the title of pharaoh was often situational, pharaoh was used for an Egyptian audience, like all Macedonian nobles, Ptolemy I Soter claimed descent from Heracles, the mythical founder of the Argead dynasty that ruled Macedon. Ptolemy was one of Alexanders most trusted generals, and was among the seven somatophylakes attached to his person and he was a few years older than Alexander and had been his intimate friend since childhood. He was succeeded by his son Ptolemy II Philadelphus, Ptolemy served with Alexander from his first campaigns, and played a principal part in the later campaigns in Afghanistan and India. Ptolemy had his first independent command during the campaign against the rebel Bessus whom Ptolemy captured and handed over to Alexander for execution. During Alexanders campaign in the Indian subcontinent Ptolemy was in command of the guard at the siege of Aornos. When Alexander died in 323 BC, Ptolemy is said to have instigated the resettlement of the made at Babylon. Ptolemy quickly moved, without authorization, to subjugate Cyrenaica, by custom, kings in Macedonia asserted their right to the throne by burying their predecessor. Ptolemy then openly joined the coalition against Perdiccas, Perdiccas appears to have suspected Ptolemy of aiming for the throne himself, and may have decided that Ptolemy was his most dangerous rival. Ptolemy executed Cleomenes for spying on behalf of Perdiccas — this removed the check on his authority. In 321 BC, Perdiccas attempted to invade Egypt only to fall at the hands of his own men, Ptolemys decision to defend the Nile against Perdiccass attempt to force it ended in fiasco for Perdiccas, with the loss of 2000 men. This failure was a blow to Perdiccas reputation, and he was murdered in his tent by two of his subordinates. Ptolemy immediately crossed the Nile, to provide supplies to what had the day before been an enemy army, Ptolemy was offered the regency in place of Perdiccas, but he declined. Ptolemy was consistent in his policy of securing a power base and his first occupation of Syria was in 318, and he established at the same time a protectorate over the petty kings of Cyprus. When Antigonus One-Eye, master of Asia in 315, showed dangerous ambitions, Ptolemy joined the coalition against him, in Cyprus, he fought the partisans of Antigonus, and re-conquered the island. A revolt in Cyrene was crushed the same year, in 312, Ptolemy and Seleucus, the fugitive satrap of Babylonia, both invaded Syria, and defeated Demetrius Poliorcetes, the son of Antigonus, in the Battle of Gaza
27.
Apollonius of Perga
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Apollonius of Perga was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic and his definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. Apollonius worked on other topics, including astronomy. Most of the work has not survived except in references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, for such an important contributor to the field of mathematics, scant biographical information remains. The 6th century Palestinian commentator, Eutocius of Ascalon, on Apollonius’ major work, Conics, states, “Apollonius, the geometrician. Came from Perga in Pamphylia in the times of Ptolemy Euergetes, the ruins of the city yet stand. It was a center of Hellenistic culture, Euergetes, “benefactor, ” identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession. Presumably, his “times” are his regnum, 246-222/221 BC, times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father. The identity of Herakleios is uncertain, the approximate times of Apollonius are thus certain, but no exact dates can be given. The figure Specific birth and death years stated by the scholars are only speculative. Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt, never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. Someone designated “of Perga” might well be expected to have lived and worked there, to the contrary, if Apollonius was later identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived, studied and wrote in Alexandria, philip was assassinated in 336 BC. Alexander went on to fulfill his plan by conquering the vast Iranian empire, the material is located in the surviving false “Prefaces” of the books of his Conics. These are letters delivered to friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that Conics was initially requested by a house guest at Alexandria, Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation” and he intended to verify and emend the books, releasing each one as it was completed
28.
Nicolas Bourbaki
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With the goal of grounding all of mathematics on set theory, the group strove for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, in 1934, young French mathematicians from various French universities felt the need to form a group to jointly produce textbooks that they could all use for teaching. André Weil organized the first meeting on 10 December 1934 in the basement of a Parisian grill room, Bourbakis main work is the Elements of Mathematics series. This series aims to be a completely self-contained treatment of the areas of modern mathematics. Assuming no special knowledge of mathematics, it takes up mathematics from the beginning, proceeds axiomatically. The volume on spectral theory from 1967 was for almost four decades the last new book to be added to the series, after that several new chapters to existing books as well as revised editions of existing chapters appeared until the publication of chapters 8-9 of Commutative Algebra in 1983. Then a long break in publishing activity occurred, leading many to suspect the end of the publishing project, however, chapter 10 of Commutative Algebra appeared in 1998, and after another long break a completely re-written and expanded chapter 8 of Algèbre was published in 2012. More importantly, the first four chapters of a new book on algebraic topology were published in 2016. Besides the Éléments de mathématique series, lectures from the Séminaire Bourbaki also have been published in monograph form since 1948. Notations introduced by Bourbaki include the symbol ∅ for the empty set and a dangerous bend symbol ☡, and the terms injective, surjective, and bijective. The emphasis on rigour may be seen as a reaction to the work of Henri Poincaré, the impact of Bourbakis work initially was great on many active research mathematicians world-wide. For example, Our time is witnessing the creation of a monumental work and it provoked some hostility, too, mostly on the side of classical analysts, they approved of rigour but not of high abstraction. This led to a gulf with the way theoretical physics was practiced, Bourbakis direct influence has decreased over time. This is partly because certain concepts which are now important, such as the machinery of category theory, are not covered in the treatise. It also mattered that, while especially algebraic structures can be defined in Bourbakis terms. On the other hand, the approach and rigour advocated by Bourbaki have permeated the current mathematical practices to such extent that the task undertaken was completed and this is particularly true for the less applied parts of mathematics. The Bourbaki seminar series founded in post-WWII Paris continues, it has going on since 1948. It is an important source of survey articles, with sketches of proofs, the topics range through all branches of mathematics, including sometimes theoretical physics
29.
Oxyrhynchus
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Oxyrhynchus is a city in Middle Egypt, located about 160 km south-southwest of Cairo, in the governorate of Al Minya. It is also a site, considered one of the most important ever discovered. Among the texts discovered at Oxyrhynchus are plays of Menander, fragments from the Gospel of Thomas, Oxyrhynchus lies west of the main course of the Nile, on the Bahr Yussef, a branch of the Nile that terminates in Lake Moeris and the Fayum oasis. It was the capital of the 19th Upper Egyptian Nome, after the conquest of Egypt by Alexander the Great in 332 BC, the city was reestablished as a Greek town, called Oxyrrhynkhoupolis. In Hellenistic times, Oxyrhynchus was a regional capital, the third-largest city in Egypt. After Egypt was Christianized, it became famous for its churches and monasteries. Oxyrhynchus remained a prominent, though declining, town in the Roman. After the Arab invasion of Egypt around 641, the system on which the town depended fell into disrepair. Today the town of El Bahnasa occupies part of the ancient site, for more than 1,000 years, the inhabitants of Oxyrhynchus dumped garbage at a series of sites out in the desert sands beyond the town limits. When the canals dried up, the water fell and never rose again. The area west of the Nile has virtually no rain, so the garbage dumps of Oxyrhynchus were gradually covered with sand and were forgotten for another 1,000 years, private citizens added their own piles of unwanted papyri. Because papyrus was expensive, papyri were often reused, a document might have farm accounts on one side, the Oxyrhynchus Papyri, therefore, contained a complete record of the life of the town, and of the civilizations and empires of which the town was a part. It is also likely that there were military buildings, such as barracks, since the city supported a military garrison on several occasions during the Roman, during the Greek and Roman periods, Oxyrhynchus had temples to Serapis, Zeus-Amun, Hera-Isis, Atargatis-Bethnnis and Osiris. There were also Greek temples to Demeter, Dionysus, Hermes, in the Christian era, Oxyrhynchus was the seat of a bishopric, and the modern town still has several ancient Coptic Christian churches. When Flinders Petrie visited Oxyrhynchus in 1922, he remains of the colonnades. Now only part of a single column remains, everything else has been scavenged for building material for modern housing, in 1882, Egypt, while still nominally part of the Ottoman Empire, came under effective British rule, and British archaeologists began the systematic exploration of the country. My first impressions on examining the site were not very favourable, the rubbish mounds were nothing but rubbish mounds. However, they soon realized what they had found
30.
Papyrus Oxyrhynchus 29
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Papyrus Oxyrhynchus 29 is a fragment of the second book of the Elements of Euclid in Greek. It was discovered by Grenfell and Hunt in 1897 in Oxyrhynchus, the fragment was originally dated to the end of the third century or the beginning of the fourth century, although more recent scholarship suggests a date of 75–125 CE. It is housed in the library of the University of Pennsylvania, the text was published by Grenfell and Hunt in 1898. The manuscript was written on papyrus in sloping irregular uncial letters, with no iota adscript, the fragment measures 85 by 152 mm. The fragment provides a statement of the 5th proposition of Book 2 of the Elements, together with a diagram. No part of the proof is provided, Oxyrhynchus Papyri Papyrus Oxyrhynchus 28 Papyrus Oxyrhynchus 30 This article incorporates text from a publication now in the public domain, B. P. Grenfell, A. S. Hunt
31.
Mathematical proof
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In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, in principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies, Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture, Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical practice, quasi-empiricism in mathematics. The philosophy of mathematics is concerned with the role of language and logic in proofs, the word proof comes from the Latin probare meaning to test. Related modern words are the English probe, probation, and probability, the Spanish probar, Italian provare, the early use of probity was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, the development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales proved some theorems in geometry, eudoxus and Theaetetus formulated theorems but did not prove them. Aristotle said definitions should describe the concept being defined in terms of other concepts already known and his book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Further advances took place in medieval Islamic mathematics, while earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, division and he used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures, there is no longer an assumption that axioms are true in any sense, this allows for parallel mathematical theories built on alternate sets of axioms
32.
Perfect numbers
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In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. Equivalently, a number is a number that is half the sum of all of its positive divisors i. e. σ1 = 2n. This definition is ancient, appearing as early as Euclids Elements where it is called τέλειος ἀριθμός. Euclid also proved a formation rule whereby q /2 is a perfect number whenever q is a prime of the form 2 p −1 for prime p —what is now called a Mersenne prime. Much later, Euler proved that all even numbers are of this form. This is known as the Euclid–Euler theorem and it is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first perfect number is 6 and its proper divisors are 1,2, and 3, and 1 +2 +3 =6. Equivalently, the number 6 is equal to half the sum of all its positive divisors, the next perfect number is 28 =1 +2 +4 +7 +14. This is followed by the perfect numbers 496 and 8128, in about 300 BC Euclid showed that if 2p−1 is prime then 2p−1 is perfect. The first four numbers were the only ones known to early Greek mathematics. Philo of Alexandria in his first-century book On the creation mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by Origen, and by Didymus the Blind, st Augustine defines perfect numbers in City of God in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number. The Egyptian mathematician Ismail ibn Fallūs mentioned the next three numbers and listed a few more which are now known to be incorrect. Euclid proved that 2p−1 is a perfect number whenever 2p −1 is prime. Prime numbers of the form 2p −1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, for 2p −1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2p −1 with a prime p are prime, in fact, Mersenne primes are very rare—of the 9,592 prime numbers p less than 100,000, 2p −1 is prime for only 28 of them. Nicomachus conjectured that every number is of the form 2p−1 where 2p −1 is prime. Ibn al-Haytham circa 1000 AD conjectured that every perfect number is of that form
33.
Mersenne primes
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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing
34.
Fundamental theorem of arithmetic
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For example,1200 =24 ×31 ×52 =3 ×2 ×2 ×2 ×2 ×5 ×5 =5 ×2 ×3 ×2 ×5 ×2 ×2 = etc. The requirement that the factors be prime is necessary, factorizations containing composite numbers may not be unique. This theorem is one of the reasons why 1 is not considered a prime number, if 1 were prime. Book VII, propositions 30,31 and 32, and Book IX, proposition 14 of Euclids Elements are essentially the statement, proposition 30 is referred to as Euclids lemma. And it is the key in the proof of the theorem of arithmetic. Proposition 31 is proved directly by infinite descent, proposition 32 is derived from proposition 31, and prove that the decomposition is possible. Book IX, proposition 14 is derived from Book VII, proposition 30, indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. Article 16 of Gauss Disquisitiones Arithmeticae is a modern statement. < pk are primes and the αi are positive integers and this representation is commonly extended to all positive integers, including one, by the convention that the empty product is equal to 1. This representation is called the representation of n, or the standard form of n. For example 999 = 33×37,1000 = 23×53,1001 = 7×11×13 Note that factors p0 =1 may be inserted without changing the value of n, allowing negative exponents provides a canonical form for positive rational numbers. However, as Integer factorization of large integers is much harder than computing their product, gcd or lcm, these formulas have, in practice, many arithmetical functions are defined using the canonical representation. In particular, the values of additive and multiplicative functions are determined by their values on the powers of prime numbers, the proof uses Euclids lemma, if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b. We need to show that every integer greater than 1 is either prime or a product of primes, for the base case, note that 2 is prime. By induction, assume true for all numbers between 1 and n, if n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = ab and 1 < a ≤ b < n, by the induction hypothesis, a = p1p2. pj and b = q1q2. qk are products of primes. But then n = ab = p1p2. pjq1q2. qk is a product of primes, assume that s >1 is the product of prime numbers in two different ways, s = p 1 p 2 ⋯ p m = q 1 q 2 ⋯ q n. We must show m = n and that the qj are a rearrangement of the pi, by Euclids lemma, p1 must divide one of the qj, relabeling the qj if necessary, say that p1 divides q1
35.
Integer factorization
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In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these integers are further restricted to numbers, the process is called prime factorization. When the numbers are large, no efficient, non-quantum integer factorization algorithm is known. However, it has not been proven that no efficient algorithm exists, the presumed difficulty of this problem is at the heart of widely used algorithms in cryptography such as RSA. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, not all numbers of a given length are equally hard to factor. The hardest instances of these problems are semiprimes, the product of two prime numbers, many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, the RSA problem. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure, by the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. If the integer is then it can be recognized as such in polynomial time. If composite however, the theorem gives no insight into how to obtain the factors, given a general algorithm for integer factorization, any integer can be factored down to its constituent prime factors simply by repeated application of this algorithm. The situation is complicated with special-purpose factorization algorithms, whose benefits may not be realized as well or even at all with the factors produced during decomposition. For example, if N =10 × p × q where p < q are very large primes, trial division will quickly produce the factors 2 and 5 but will take p divisions to find the next factor. Among the b-bit numbers, the most difficult to factor in practice using existing algorithms are those that are products of two primes of similar size, for this reason, these are the integers used in cryptographic applications. The largest such semiprime yet factored was RSA-768, a 768-bit number with 232 decimal digits and this factorization was a collaboration of several research institutions, spanning two years and taking the equivalent of almost 2000 years of computing on a single-core 2.2 GHz AMD Opteron. Like all recent factorization records, this factorization was completed with an optimized implementation of the general number field sieve run on hundreds of machines. No algorithm has been published that can factor all integers in polynomial time, neither the existence nor non-existence of such algorithms has been proved, but it is generally suspected that they do not exist and hence that the problem is not in class P. The problem is clearly in class NP but has not been proved to be in, or not in and it is generally suspected not to be in NP-complete. There are published algorithms that are faster than O for all positive ε, i. e. sub-exponential, the best published asymptotic running time is for the general number field sieve algorithm, which, for a b-bit number n, is, O. For current computers, GNFS is the best published algorithm for large n, for a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time
36.
Greatest common divisor
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In mathematics, the greatest common divisor of two or more integers, when at least one of them is not zero, is the largest positive integer that is a divisor of both numbers. For example, the GCD of 8 and 12 is 4, the greatest common divisor is also known as the greatest common factor, highest common factor, greatest common measure, or highest common divisor. This notion can be extended to polynomials and other commutative rings, in this article we will denote the greatest common divisor of two integers a and b as gcd. What is the greatest common divisor of 54 and 24, the number 54 can be expressed as a product of two integers in several different ways,54 ×1 =27 ×2 =18 ×3 =9 ×6. Thus the divisors of 54 are,1,2,3,6,9,18,27,54, similarly, the divisors of 24 are,1,2,3,4,6,8,12,24. The numbers that these two share in common are the common divisors of 54 and 24,1,2,3,6. The greatest of these is 6 and that is, the greatest common divisor of 54 and 24. The greatest common divisor is useful for reducing fractions to be in lowest terms, for example, gcd =14, therefore,4256 =3 ⋅144 ⋅14 =34. Two numbers are called relatively prime, or coprime, if their greatest common divisor equals 1, for example,9 and 28 are relatively prime. For example, a 24-by-60 rectangular area can be divided into a grid of, 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, therefore,12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can be divided into a grid of 12-by-12 squares, in practice, this method is only feasible for small numbers, computing prime factorizations in general takes far too long. Here is another example, illustrated by a Venn diagram. Suppose it is desired to find the greatest common divisor of 48 and 180, first, find the prime factorizations of the two numbers,48 =2 ×2 ×2 ×2 ×3,180 =2 ×2 ×3 ×3 ×5. What they share in common is two 2s and a 3, Least common multiple =2 ×2 × ×3 ×5 =720 Greatest common divisor =2 ×2 ×3 =12. To compute gcd, divide 48 by 18 to get a quotient of 2, then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, note that we ignored the quotient in each step except to notice when the remainder reached 0, signalling that we had arrived at the answer. Formally the algorithm can be described as, gcd = a gcd = gcd, in this sense the GCD problem is analogous to e. g. the integer factorization problem, which has no known polynomial-time algorithm, but is not known to be NP-complete. Shallcross et al. showed that a problem is NC-equivalent to the problem of integer linear programming with two variables, if either problem is in NC or is P-complete, the other is as well
37.
Non-Euclidean geometry
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, when the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are many lines through A not intersecting ℓ, while in elliptic geometry. In elliptic geometry the lines curve toward each other and intersect, the debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclids work Elements was written. In the Elements, Euclid began with a number of assumptions. Other mathematicians have devised simpler forms of this property, regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclids other postulates,1. To draw a line from any point to any point. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī and these theorems along with their alternative postulates, such as Playfairs axiom, played an important role in the later development of non-Euclidean geometry. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Another example is al-Tusis son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusis later thoughts and he essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements. His work was published in Rome in 1594 and was studied by European geometers and he finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, in 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral and he quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle