Hipparchus

Hipparchus of Nicaea was a Greek astronomer and mathematician. He is considered the founder of trigonometry but is most famous for his incidental discovery of precession of the equinoxes. Hipparchus was born in Nicaea and died on the island of Rhodes, Greece, he is known to have been a working astronomer at least from 162 to 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity, he was the first whose accurate models for the motion of the Sun and Moon survive. For this he made use of the observations and the mathematical techniques accumulated over centuries by the Babylonians and by Meton of Athens, Aristyllus, Aristarchus of Samos and Eratosthenes, among others, he developed trigonometry and constructed trigonometric tables, he solved several problems of spherical trigonometry. With his solar and lunar theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipses.

His other reputed achievements include the discovery and measurement of Earth's precession, the compilation of the first comprehensive star catalog of the western world, the invention of the astrolabe of the armillary sphere, which he used during the creation of much of the star catalogue. There is a strong tradition that Hipparchus was born in Nicaea, in the ancient district of Bithynia, in what today is the country Turkey; the exact dates of his life are not known, but Ptolemy attributes astronomical observations to him in the period from 147–127 BC, some of these are stated as made in Rhodes. His birth date was calculated by Delambre based on clues in his work. Hipparchus must have lived some time after 127 BC because he analyzed and published his observations from that year. Hipparchus obtained information from Alexandria as well as Babylon, but it is not known when or if he visited these places, he is believed to have died on the island of Rhodes, where he seems to have spent most of his life.

It is not known what Hipparchus's economic means were nor how he supported his scientific activities. His appearance is unknown: there are no contemporary portraits. In the 2nd and 3rd centuries coins were made in his honour in Bithynia that bear his name and show him with a globe. Little of Hipparchus's direct work survives into modern times. Although he wrote at least fourteen books, only his commentary on the popular astronomical poem by Aratus was preserved by copyists. Most of what is known about Hipparchus comes from Strabo's Geography and Pliny's Natural History in the 1st century. Hipparchus was amongst the first to calculate a heliocentric system, but he abandoned his work because the calculations showed the orbits were not circular as believed to be mandatory by the science of the time. Although a contemporary of Hipparchus', Seleucus of Seleucia, remained a proponent of the heliocentric model, Hipparchus' rejection of heliocentrism, supported by ideas from Aristotle, remained dominant for nearly 2000 years until Copernican heliocentrism turned the tide of the debate.

Hipparchus's only preserved work is Τῶν Ἀράτου καὶ Εὐδόξου φαινομένων ἐξήγησις. This is a critical commentary in the form of two books on a popular poem by Aratus based on the work by Eudoxus. Hipparchus made a list of his major works, which mentioned about fourteen books, but, only known from references by authors, his famous star catalog was incorporated into the one by Ptolemy, may be perfectly reconstructed by subtraction of two and two thirds degrees from the longitudes of Ptolemy's stars. The first trigonometric table was compiled by Hipparchus, now known as "the father of trigonometry". Hipparchus was in the international news in 2005, when it was again proposed that the data on the celestial globe of Hipparchus or in his star catalog may have been preserved in the only surviving large ancient celestial globe which depicts the constellations with moderate accuracy, the globe carried by the Farnese Atlas. There are a variety of mis-steps in the more ambitious 2005 paper, thus no specialists in the area accept its publicized speculation.

Lucio Russo has said that Plutarch, in his work On the Face in the Moon, was reporting some physical theories that we consider to be Newtonian and that these may have come from Hipparchus. According to one book review, both of these claims have been rejected by other scholars. A line in Plutarch's Table Talk states that Hipparchus counted 103049 compound propositions that can be formed from ten simple propositions. 103049 is the tenth Schröder–Hipparchus number, which counts the number of ways of adding one or more pairs of parentheses around consecutive subsequences of two or more items in any sequence of ten symbols. This has led to speculation that Hipparchus knew about enumerative combinatorics, a field of mathematics that developed independently in modern mathematics. Earlier Greek astronomers and mathematicians were influenced by Babylonian astronomy to some extent, for instance the period relations of the Metonic cycle and Saros cycle may have come from Babylonian sources (see "Babylonian astron

Hypatia

Hypatia was a Hellenistic Neoplatonist philosopher and mathematician, who lived in Alexandria, Egypt part of the Eastern Roman Empire. She was a prominent thinker of the Neoplatonic school in Alexandria where she taught philosophy and astronomy, she is the first female mathematician. Hypatia was renowned in her own lifetime as a wise counselor, she is known to have written a commentary on Diophantus's thirteen-volume Arithmetica, which may survive in part, having been interpolated into Diophantus's original text, another commentary on Apollonius of Perga's treatise on conic sections, which has not survived. Many modern scholars believe that Hypatia may have edited the surviving text of Ptolemy's Almagest, based on the title of her father Theon's commentary on Book III of the Almagest. Hypatia is known to have constructed astrolabes and hydrometers, but did not invent either of these, which were both in use long before she was born. Although she herself was a pagan, she was tolerant towards Christians and taught many Christian students, including Synesius, the future bishop of Ptolemais.

Ancient sources record that Hypatia was beloved by pagans and Christians alike and that she established great influence with the political elite in Alexandria. Towards the end of her life, Hypatia advised Orestes, the Roman prefect of Alexandria, in the midst of a political feud with Cyril, the bishop of Alexandria. Rumors spread accusing her of preventing Orestes from reconciling with Cyril and, in March 415 AD, she was murdered by a mob of Christians led by a lector named Peter. Hypatia's murder shocked the empire and transformed her into a "martyr for philosophy", leading future Neoplatonists such as Damascius to become fervent in their opposition to Christianity. During the Middle Ages, Hypatia was co-opted as a symbol of Christian virtue and scholars believe she was part of the basis for the legend of Saint Catherine of Alexandria. During the Age of Enlightenment, she became a symbol of opposition to Catholicism. In the nineteenth century, European literature Charles Kingsley's 1853 novel Hypatia, romanticized her as "the last of the Hellenes".

In the twentieth century, Hypatia became seen as an icon for women's rights and a precursor to the feminist movement. Since the late twentieth century, some portrayals have associated Hypatia's death with the destruction of the Library of Alexandria, despite the historical fact that the library no longer existed during Hypatia's lifetime. Hypatia was the daughter of the mathematician Theon of Alexandria. According to classical historian Edward J. Watts, Theon was the head of a school called the "Mouseion", named in emulation of the Hellenistic Mouseion, whose membership had ceased in the 260s AD. Theon's school was exclusive prestigious, doctrinally conservative. Theon rejected the teachings of Iamblichus and may have taken pride in teaching a pure, Plotinian Neoplatonism. Although he was seen as a great mathematician at the time, Theon's mathematical work has been deemed by modern standards as "minor", "trivial", "completely unoriginal", his primary achievement was the production of a new edition of Euclid's Elements, in which he corrected scribal errors, made over the course of nearly 700 years of copying.

Theon's edition of Euclid's Elements became the most widely-used edition of the textbook for centuries and totally supplanted all other editions. Nothing is known about Hypatia's mother, never mentioned in any of the extant sources. Theon dedicates his commentary on Book IV of Ptolemy's Almagest to an individual named Epiphanius, addressing him as "my dear son", indicating that he may have been Hypatia's brother, but the Greek word Theon uses does not always mean "son" in the biological sense and was used to signal strong feelings of paternal connection. Hypatia's exact year of birth is still under debate, with suggested dates ranging from 350 to 370 AD. Many scholars have followed Richard Hoche in inferring that Hypatia was born around 370. According to a description of Hypatia from the lost work Life of Isidore by the Neoplatonist historian Damascius, preserved in the entry for her in the Suda, a tenth-century Byzantine encyclopedia, Hypatia flourished during the reign of Arcadius. Hoche reasoned that Damascius's description of her physical beauty would imply that she was at most 30 at that time, the year 370 was 30 years prior to the midpoint of Arcadius's reign.

In contrast, theories that she was born as early as 350 are based on the wording of the chronicler John Malalas, who calls her old at the time of her death in 415. Robert Penella argues that both theories are weakly based, that her birth date should be left unspecified. Hypatia was a Neoplatonist, like her father, she rejected the teachings of Iamblichus and instead embraced the original Neoplatonism formulated by Plotinus; the Alexandrian school was renowned at the time for its philosophy and Alexandria was regarded as second only to Athens as the philosophical capital of the Greco-Roman world. Hypatia taught students from all over the Mediterranean. According to Damascius, she lectured on the writings of Aristotle, he states that she walked through Alexandria in a tribon, a kind of cloak associated with philosophers, giving impromptu public lectures. According to Watts, two main varieties of Neoplatonism were taught in Alexandria during the late fourth century; the first was the overtly pagan religious Neoplatonism taught at the Serapeum, influenced by the teachings of Iamblichus.

The second variety was the more moderate and less polemical variety c

Democritus

Democritus was an Ancient Greek pre-Socratic philosopher remembered today for his formulation of an atomic theory of the universe. Democritus was born in Abdera, around 460 BC, although there are disagreements about the exact year, his exact contributions are difficult to disentangle from those of his mentor Leucippus, as they are mentioned together in texts. Their speculation on atoms, taken from Leucippus, bears a passing and partial resemblance to the 19th-century understanding of atomic structure that has led some to regard Democritus as more of a scientist than other Greek philosophers. Ignored in ancient Athens, Democritus is said to have been disliked so much by Plato that the latter wished all of his books burned, he was well known to his fellow northern-born philosopher Aristotle. Many consider Democritus to be the "father of modern science". None of his writings have survived. Democritus was said to be born in the city of Abdera in Thrace, an Ionian colony of Teos, although some called him a Milesian.

He was born in the 80th Olympiad according to Apollodorus of Athens, although Thrasyllus placed his birth in 470 BC, the date is more likely. John Burnet has argued that the date of 460 is "too early" since, according to Diogenes Laërtius ix.41, Democritus said that he was a "young man" during Anaxagoras's old age. It was said that Democritus's father was from a noble family and so wealthy that he received Xerxes on his march through Abdera. Democritus spent the inheritance which his father left him on travels into distant countries, to satisfy his thirst for knowledge, he traveled to Asia, was said to have reached India and Ethiopia. It is known that he wrote on Meroe, he himself declared that among his contemporaries none had made greater journeys, seen more countries, met more scholars than himself. He mentions the Egyptian mathematicians, whose knowledge he praises. Theophrastus, spoke of him as a man who had seen many countries. During his travels, according to Diogenes Laërtius, he became acquainted with the Chaldean magi.

"Ostanes", one of the magi accompanying Xerxes, was said to have taught him. After returning to his native land he occupied himself with natural philosophy, he traveled throughout Greece to acquire a better knowledge of its cultures. He mentions many Greek philosophers in his writings, his wealth enabled him to purchase their writings. Leucippus, the founder of atomism, was the greatest influence upon him, he praises Anaxagoras. Diogenes Laertius says, he may have been acquainted with Socrates, but Plato does not mention him and Democritus himself is quoted as saying, "I came to Athens and no one knew me." Aristotle placed him among the pre-Socratic natural philosophers. The many anecdotes about Democritus in Diogenes Laërtius, attest to his disinterest and simplicity, show that he lived for his studies. One story has him deliberately blinding himself, he was cheerful, was always ready to see the comical side of life, which writers took to mean that he always laughed at the foolishness of people.

He was esteemed by his fellow citizens, because as Diogenes Laërtius says, "he had foretold them some things which events proved to be true," which may refer to his knowledge of natural phenomena. According to Diodorus Siculus, Democritus died at the age of 90, which would put his death around 370 BC, but other writers have him living to 104, or 109. Popularly known as the Laughing Philosopher, the terms Abderitan laughter, which means scoffing, incessant laughter, Abderite, which means a scoffer, are derived from Democritus. To his fellow citizens he was known as "The Mocker". Most sources say that Democritus followed in the tradition of Leucippus and that they carried on the scientific rationalist philosophy associated with Miletus. Both were materialist, believing everything to be the result of natural laws. Unlike Aristotle or Plato, the atomists attempted to explain the world without reasoning as to purpose, prime mover, or final cause. For the atomists questions of physics should be answered with a mechanistic explanation, while their opponents search for explanations which, in addition to the material and mechanistic included the formal and teleological.

Greek historians consider Democritus to have established aesthetics as a subject of investigation and study, as he wrote theoretically on poetry and fine art long before authors such as Aristotle. Thrasyllus identified six works in the philosopher's oeuvre which had belonged to aesthetics as a discipline, but only fragments of the relevant works are extant; the theory of Democritus held that everything is composed of "atoms", which are physically, but not geometrically, indivisible. Of the mass of atoms, Democritus said, "The more any indivisible exceeds, the heavier it is". However, his exact position o

Diophantus

Diophantus of Alexandria was an Alexandrian Hellenistic mathematician, the author of a series of books called Arithmetica, many of which are now lost. Sometimes called "the father of algebra", his texts deal with solving algebraic equations. While reading Claude Gaspard Bachet de Méziriac's edition of Diophantus' Arithmetica, Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, noted in the margin without elaboration that he had found "a marvelous proof of this proposition," now referred to as Fermat's Last Theorem; this led to tremendous advances in number theory, the study of Diophantine equations and of Diophantine approximations remain important areas of mathematical research. Diophantus coined the term παρισότης to refer to an approximate equality; this term was rendered as adaequalitas in Latin, became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician.

In modern use, Diophantine equations are algebraic equations with integer coefficients, for which integer solutions are sought. Little is known about the life of Diophantus, he lived in Alexandria, during the Roman era from between AD 200 and 214 to 284 or 298. Diophantus has variously been described by historians as either Greek, non-Greek, Hellenized Egyptian, Hellenized Babylonian, Jewish, or Chaldean. Much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the problems states:'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old:'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'This puzzle implies that Diophantus' age x can be expressed as x = x/6 + x/12 + x/7 + 5 + x/2 + 4which gives x a value of 84 years.

However, the accuracy of the information cannot be independently confirmed. In popular culture, this puzzle was the Puzzle No.142 in Professor Layton and Pandora's Box as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first. Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics, it is a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arabic books discovered in 1968 are by Diophantus; some Diophantine problems from Arithmetica have been found in Arabic sources. It should be mentioned here. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus. “Our author not the slightest trace of a general, comprehensive method is discernible. For this reason it is difficult for the modern scholar to solve the 101st problem after having studied 100 of Diophantos’s solutions”.

Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the so-called Dark Ages, since the study of ancient Greek, literacy in general, had declined. The portion of the Greek Arithmetica that survived, was, like all ancient Greek texts transmitted to the early modern world, copied by, thus known to, medieval Byzantine scholars. Scholia on Diophantus by the Byzantine Greek scholar John Chortasmenos are preserved together with a comprehensive commentary written by the earlier Greek scholar Maximos Planudes, who produced an edition of Diophantus within the library of the Chora Monastery in Byzantine Constantinople. In addition, some portion of the Arithmetica survived in the Arab tradition. In 1463 German mathematician Regiomontanus wrote: “No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the flower of the whole of arithmetic lies hidden....”Arithmetica was first translated from Greek into Latin by Bombelli in 1570, but the translation was never published.

However, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander; the best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition, available. Pierre de Fermat owned a copy, studied it, made notes in the margins; the 1621 edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy: “If an integer n is greater than 2 an + bn = cn has no solutions in non-zero integers a, b, c. I have a marvelous proof of this proposition which this margin is too narrow to contain.”Fermat's proof was never found, the problem of finding a proof for the theorem went unsolved for centuries. A proof was found in 1994 by Andrew Wiles after working on it for seven years, it is believed that Fermat did not have the proof he claimed to

Hippasus

Hippasus of Metapontum, was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers; the discovery of irrational numbers is said to have been shocking to the Pythagoreans, Hippasus is supposed to have drowned at sea as a punishment from the gods for divulging this. However, the few ancient sources which describe this story either do not mention Hippasus by name or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere; the discovery of irrationality is not ascribed to Hippasus by any ancient writer. Some modern scholars though have suggested that he discovered the irrationality of √2, believed to have been discovered around the time that he lived. Little is known about the life of Hippasus, he may have lived in the late 5th century BC, about a century after the time of Pythagoras. Metapontum in Italy is referred to as his birthplace, although according to Iamblichus some claim Metapontum to be his birthplace, while others the nearby city of Croton.

Hippasus is recorded under the city of Sybaris in Iamblichus list of each city's Pythagoreans. He states that Hippasus was the founder of a sect of the Pythagoreans called the Mathematici in opposition to the Acusmatici. Iamblichus says about the death of Hippasus It is related to Hippasus that he was a Pythagorean, that, owing to his being the first to publish and describe the sphere from the twelve pentagons, he perished at sea for his impiety, but he received credit for the discovery, though it all belonged to HIM. According to Iamblichus in The life of Pythagoras, by Thomas Taylor There were two forms of philosophy, for the two genera of those that pursued it: the Acusmatici and the Mathematici; the latter are acknowledged to be Pythagoreans by the rest but the Mathematici do not admit that the Acusmatici derived their instructions from Pythagoras but from Hippasus. The philosophy of the Acusmatici consisted in auditions unaccompanied with demonstrations and a reasoning process. Memory was the most valued faculty.

All these auditions were of three kinds. Aristotle speaks of Hippasus as holding the element of fire to be the cause of all things. Diogenes Laërtius tells us that Hippasus believed that "there is a definite time which the changes in the universe take to complete, that the universe is limited and in motion." According to one statement, Hippasus left no writings, according to another he was the author of the Mystic Discourse, written to bring Pythagoras into disrepute. A scholium on Plato's Phaedo notes him as an early experimenter in music theory, claiming that he made use of bronze disks to discover the fundamental musical ratios, 4:3, 3:2, 2:1. Hippasus is sometimes credited with the discovery of the existence of irrational numbers, following which he was drowned at sea. Pythagoreans preached that all numbers could be expressed as the ratio of integers, the discovery of irrational numbers is said to have shocked them. However, the evidence linking the discovery to Hippasus is confused. Pappus says that the knowledge of irrational numbers originated in the Pythagorean school, that the member who first divulged the secret perished by drowning.

Iamblichus gives a series of inconsistent reports. In one story he explains how a Pythagorean was expelled for divulging the nature of the irrational. In another account he tells how it was Hippasus who drowned at sea for betraying the construction of the dodecahedron and taking credit for this construction himself. Iamblichus states that the drowning at sea was a punishment from the gods for impious behaviour; these stories are taken together to ascribe the discovery of irrationals to Hippasus, but whether he did or not is uncertain. In principle, the stories can be combined, since it is possible to discover irrational numbers when constructing dodecahedrons. Irrationality, by infinite reciprocal subtraction, can be seen in the Golden ratio of the regular pentagon; some modern scholars prefer to credit Hippasus with the discovery of the irrationality of √2. Plato in his Theaetetus, describes how Theodorus of Cyrene proved the irrationality of √3, √5, etc. up to √17, which implies that an earlier mathematician had proved the irrationality of √2.

Aristotle referred to the method for a proof of the irrationality of √2, a full proof along these same lines is set out in the proposition interpolated at the end of Euclid's Book X, which suggests that the proof was ancient. The method is a proof by contradiction, or reductio ad absurdum, which shows that, if the diagonal of a square is assumed to be commensurable with the side the same number must be both odd and e

Apollonius of Perga

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, he brought them to the state they were in just before the invention of analytic geometry, his definitions of the terms ellipse and hyperbola are the ones in use today. Apollonius worked including astronomy. Most of the work has not survived except in fragmentary references in other authors, his hypothesis of eccentric orbits to explain the aberrant motion of the planets believed until the Middle Ages, was superseded during the Renaissance. 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, states: “Apollonius, the geometrician... came from Perga in Pamphylia in the times of Ptolemy Euergetes, so records Herakleios the biographer of Archimedes....” Perga at the time was a Hellenized city of Pamphylia in Anatolia.

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, 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; the figure Specific birth and death years stated by the various 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. During the last half of the 3rd century BC, Perga changed hands a number of times, being alternatively under the Seleucids and under the Kingdom of Pergamon to the north, ruled by the Attalid dynasty. Someone designated "of Perga" might well be expected to have worked there.

To the contrary, if Apollonius was identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived and wrote in Alexandria. A letter by the Greek mathematician and astronomer Hypsicles was part of the supplement taken from Euclid's Book XIV, part of the thirteen books of Euclid's Elements. "Basilides of Tyre, O Protarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of the bond between them due to their common interest in mathematics. And on one occasion, when looking into the tract written by Apollonius about the comparison of the dodecahedron and icosahedron inscribed in one and the same sphere, to say, on the question what ratio they bear to one another, they came to the conclusion that Apollonius' treatment of it in this book was not correct, but I myself afterwards came across another book published by Apollonius, containing a demonstration of the matter in question, I was attracted by his investigation of the problem.

Now the book published by Apollonius is accessible to all. "For my part, I determined to dedicate to you what I deem to be necessary by way of commentary because you will be able, by reason of your proficiency in all mathematics and in geometry, to pass an expert judgment upon what I am about to write, because, on account of your intimacy with my father and your friendly feeling towards myself, you will lend a kindly ear to my disquisition. But it is time to have done with the preamble and to begin my treatise itself." Apollonius lived toward the end of a historical period now termed the Hellenistic Period, characterized by the superposition of Hellenic culture over extensive non-Hellenic regions to various depths, radical in some places, hardly at all in others. The change was initiated by Philip II of Macedon and his son, Alexander the Great, subjecting all of Greece is a series of stunning victories, went on to conquer the Persian Empire, which ruled territories from Egypt to Pakistan. Philip was assassinated in 336 BC.

Alexander went on to fulfill his plan by conquering the vast Persian empire. The material is located in the surviving false “Prefaces” of the books of his Conics; these are letters delivered to influential 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 requested by a house guest at Alexandria, the geometer, otherwise unknown to history. 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”, he intended releasing each one as it was completed. Hearing of this plan from Apollonius himself on a subsequent visit of the latter to Pergamon, Eudemus had insisted Apollonius send him each book before release; the circumstances imply that at this stage Apollonius was a young geometer seeking the company and advice of established professionals. Pappus states. Euclid was long gone; this stay had been the final stage of Apollonius’ education.

Eudemus was a senior figure in his earlier education at Pergamon.

Isidore of Miletus

Isidore of Miletus was one of the two main Byzantine Greek architects that Emperor Justinian I commissioned to design the cathedral Hagia Sophia in Constantinople from 532 to 537. The creation of an important compilation of Archimedes' works has been attributed to him; the spurious Book XV from Euclid's Elements has been attributed to Isidore of Miletus. Isidore of Miletus was a renowned mathematician before Emperor Justinian I hired him. Isidorus taught stereometry and physics at the universities, first of Alexandria of Constantinople, wrote a commentary on an older treatise on vaulting. Eutocius together with Isidore studied Archimedes work. Isidore is renowned for producing the first comprehensive compilation of Archimedes' work, the Archimedes palimpsest survived to the present. Emperor Justinian I appointed his architects to rebuild the Hagia Sophia following his victory over protesters within the capital city of his Roman Empire, Constantinople; the first basilica was completed in 360 and remodelled from 404 to 415, but had been damaged in 532 in the course of the Nika Riot, “The temple of Sophia, the baths of Zeuxippus, the imperial courtyard from the Propylaia all the way to the so-called House of Ares were burned up and destroyed, as were both of the great porticoes that lead to the forum, named after Constantine, houses of prosperous people, a great deal of other properties.”The warring factions of Byzantine society, the Blues and the Greens, opposed each other in the chariot races at the Hippodrome and resorted to violence.

During the Nika Riot, more than thirty thousand people died. Emperor Justinian I ensured that his new structure would not be burned down, like its predecessors, by commissioning architects that would build the church out of stone, rather than wood, “He compacted it of baked brick and mortar, in many places bound it together with iron, but made no use of wood, so that the church should no longer prove combustible.”Isidore of Miletus and Anthemius of Tralles planned on a main hall of the Hagia Sophia that measured 70 by 75 metres, making it the largest church in Constantinople, but the original dome was nearly 6 metres lower than it was constructed, “Justinian suppressed these riots and took the opportunity of marking his victory by erecting in 532-7 the new Hagia Sophia, one of the largest, most lavish, most expensive buildings of all time.”Although Isidore of Miletus and Anthemius of Tralles were not formally educated in architecture, they were scientists that could organize the logistics of drawing thousands of labourers and unprecedented loads of rare raw materials from around the Roman Empire to create the Hagia Sophia for Emperor Justinian I.

The finished product was built in admirable form for the Roman Emperor, “All of these elements marvellously fitted together in mid-air, suspended from one another and reposing only on the parts adjacent to them, produce a unified and most remarkable harmony in the work, yet do not allow the spectators to rest their gaze upon any one of them for a length of time.”The Hagia Sophia architects innovatively combined the longitudinal structure of a Roman basilica and the central plan of a drum-supported dome, in order to withstand the high magnitude earthquakes of the Marmara Region, “However, in May 558, little more than 20 years after the Church’s dedication, following the earthquakes of August 553 and December 557, parts of the central dome and its supporting structure system collapsed.” The Hagia Sophia was cracked by earthquakes and was repaired. Isidore of Miletus’ nephew, Isidore the Younger, introduced the new dome design that can be viewed in the Hagia Sophia in present-day Istanbul, Turkey.

After a great earthquake in 989 ruined the dome of Hagia Sophia, the Byzantine officials summoned Trdat the Architect to Byzantium to organize repairs. The restored dome was completed by 994. Cakmak, AS. "The Structural Configuration of the First Dome of Justinian's Hagia Sophia: An Investigation Based on Structural and Literary Analysis". Soil Dynamics and Earthquake Engineering. 29. Krautheimer, Richard. Early Christian and Byzantine Architecture. Baltimore: Penguin Books. ISBN 978-0-300-05294-7. Mango, Cyril A.. The Art of the Byzantine Empire, 312-1453: Sources and Documents. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0-8020-6627-5. Maranci, Christina. "The Architect Trdat: Building Practices and Cross-Cultural Exchange in Byzantium and Armenia". The Journal of the Society of Architectural Historians. 62: 294–305. Doi:10.2307/3592516. Prokopios. Anthony Kaldellis, ed; the Secret History: With Related Texts. Indianapolis: Hackett Publishing. ISBN 978-1-60384-180-1. Watkin, David. A History of Western Architecture.

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