Hellenistic Greece

In the context of ancient Greek art and culture, Hellenistic Greece corresponds to the period between the death of Alexander the Great in 323 BC and the annexation of the classical Greek Achaean League heartlands by the Roman Republic. This culminated at the Battle of Corinth in 146 BC, a crushing Roman victory in the Peloponnese that led to the destruction of Corinth and ushered in the period of Roman Greece. Hellenistic Greece's definitive end was with the Battle of Actium in 31 BC, when the future emperor Augustus defeated Greek Ptolemaic queen Cleopatra VII and Mark Antony, the next year taking over Alexandria, the last great center of Hellenistic Greece; the Hellenistic period began with the wars of the Diadochi, armed contests among the former generals of Alexander the Great to carve up his empire in Europe and North Africa. The wars lasted until 275 BC, witnessing the fall of both the Argead and Antipatrid dynasties of Macedonia in favor of the Antigonid dynasty; the era was marked by successive wars between the Kingdom of Macedonia and its allies against the Aetolian League, Achaean League, the city-state of Sparta.

During the reign of Philip V of Macedon, the Macedonians not only lost the Cretan War to an alliance led by Rhodes, but their erstwhile alliance with Hannibal of Carthage entangled them in the First and Second Macedonian War with ancient Rome. The perceived weakness of Macedonia in the aftermath of these conflicts encouraged Antiochus III the Great of the Seleucid Empire to invade mainland Greece, yet his defeat by the Romans at Thermopylae in 191 BC and Magnesia in 190 BC secured Rome's position as the leading military power in the region. Within two decades after conquering Macedonia in 168 BC and Epirus in 167 BC, the Romans would control the whole of Greece. During the Hellenistic period the importance of Greece proper within the Greek-speaking world declined sharply; the great centers of Hellenistic culture were Alexandria and Antioch, capitals of Ptolemaic Egypt and Seleucid Syria respectively. Cities such as Pergamon, Ephesus and Seleucia were important, increasing urbanization of the Eastern Mediterranean was characteristic of the time.

The quests of Alexander had a number of consequences for the Greek city-states. It widened the horizons of the Greeks, making the endless conflicts between the cities which had marked the 5th and 4th centuries BC seem petty and unimportant, it led to a steady emigration of the young and ambitious, to the new Greek empires in the east. Many Greeks migrated to Alexandria and the many other new Hellenistic cities founded in Alexander's wake, as far away as what are now Afghanistan and Pakistan, where the Greco-Bactrian Kingdom and the Indo-Greek Kingdom survived until the end of the 1st century BC; the defeat of the Greek cities by Philip and Alexander taught the Greeks that their city-states could never again be powers in their own right, that the hegemony of Macedon and its successor states could not be challenged unless the city states united, or at least federated. The Greeks valued their local independence too much to consider actual unification, but they made several attempts to form federations through which they could hope to reassert their independence.

Following Alexander's death a struggle for power broke out among his generals, which resulted in the break-up of his empire and the establishment of a number of new kingdoms. Macedon fell to Cassander, son of Alexander's leading general Antipater, who after several years of warfare made himself master of most of the rest of Greece, he founded a new Macedonian capital at Thessaloniki and was a constructive ruler. Cassander's power was challenged by Antigonus, ruler of Anatolia, who promised the Greek cities that he would restore their freedom if they supported him; this led to successful revolts against Cassander's local rulers. In 307 BC, Antigonus's son Demetrius captured Athens and restored its democratic system, suppressed by Alexander, but in 301 BC a coalition of Cassander and the other Hellenistic kings defeated Antigonus at the Battle of Ipsus, ending his challenge. After Cassander's death in 298 BC, Demetrius seized the Macedonian throne and gained control of most of Greece, he was defeated by a second coalition of Greek rulers in 285 BC, mastery of Greece passed to the king Lysimachus of Thrace.

Lysimachus was in turn defeated and killed in 280 BC. The Macedonian throne passed to Demetrius's son Antigonus II, who defeated an invasion of the Greek lands by the Gauls, who at this time were living in the Balkans; the battle against the Gauls united the Antigonids of Macedon and the Seleucids of Antioch, an alliance, directed against the wealthiest Hellenistic power, the Ptolemies of Egypt. Antigonus II ruled until his death in 239 BC, his family retained the Macedonian throne until it was abolished by the Romans in 146 BC, their control over the Greek city states was intermittent, since other rulers the Ptolemies, subsidised anti-Macedonian parties in Greece to undermine the Antigonids' power. Antigonus placed a garrison at Corinth, the strategic centre of Greece, but Athens, Rhodes and other Greek states retained substantial independence, formed the Aetolian League as a means of defending it. Sparta remained independent, but refused to join any league. In 267 BC, Ptolemy II persuaded the Greek cities to revolt against Antigonus, in what became the Chremonidian War, after the Athenian leader Chremonides.

The cities were defeated and Athens lost her independence and her democratic institutions. The Aetolian League was restricted to the Peloponnese, but on being allowed to gain control of Thebes in 245 BC became a

Eudoxus of Cnidus

Eudoxus of Cnidus was an ancient Greek astronomer, mathematician and student of Archytas and Plato. All of his works are lost, though some fragments are preserved in Hipparchus' commentary on Aratus's poem on astronomy. Sphaerics by Theodosius of Bithynia may be based on a work by Eudoxus, his name Eudoxus means "honored" or "of good repute". It is analogous to the Latin name Benedictus. Eudoxus's father Aeschines of Cnidus loved to watch stars at night. Eudoxus first travelled to Tarentum to study from whom he learned mathematics. While in Italy, Eudoxus visited Sicily. Around 387 BC, at the age of 23, he traveled with the physician Theomedon—who some believed was his lover—to Athens to study with the followers of Socrates, he attended lectures of Plato and other philosophers for several months, but due to a disagreement they had a falling-out. Eudoxus could only afford an apartment at the Piraeus. To attend Plato's lectures, he walked the 7 miles in each direction each day. Due to his poverty, his friends raised funds sufficient to send him to Heliopolis, Egypt, to pursue his study of astronomy and mathematics.

He lived there for 16 months. From Egypt, he traveled north to Cyzicus, located on the south shore of the Sea of Marmara, the Propontis, he traveled south to the court of Mausolus. During his travels he gathered many students of his own. Around 368 BC, Eudoxus returned to Athens with his students. According to some sources, around 367 he assumed headship of the Academy during Plato's period in Syracuse, taught Aristotle, he returned to his native Cnidus, where he served in the city assembly. While in Cnidus, he built an observatory and continued writing and lecturing on theology and meteorology, he had one son and three daughters, Actis and Delphis. In mathematical astronomy, his fame is due to the introduction of the astronomical globe, his early contributions to understanding the movement of the planets, his work on proportions shows insight into real numbers. When it was revived by Tartaglia and others in the 16th century, it became the basis for quantitative work in science for a century, until it was replaced by Richard Dedekind.

Craters on Mars and the Moon are named in his honor. An algebraic curve is named after him. Eudoxus is considered by some to be the greatest of classical Greek mathematicians, in all antiquity second only to Archimedes, he rigorously developed Antiphon's method of exhaustion, a precursor to the integral calculus, used in a masterly way by Archimedes in the following century. In applying the method, Eudoxus proved such mathematical statements as: areas of circles are to one another as the squares of their radii, volumes of spheres are to one another as the cubes of their radii, the volume of a pyramid is one-third the volume of a prism with the same base and altitude, the volume of a cone is one-third that of the corresponding cylinder. Eudoxus introduced the idea of non-quantified mathematical magnitude to describe and work with continuous geometrical entities such as lines, angles and volumes, thereby avoiding the use of irrational numbers. In doing so, he reversed a Pythagorean emphasis on number and arithmetic, focusing instead on geometrical concepts as the basis of rigorous mathematics.

Some Pythagoreans, such as Eudoxus' teacher Archytas, had believed that only arithmetic could provide a basis for proofs. Induced by the need to understand and operate with incommensurable quantities, Eudoxus established what may have been the first deductive organization of mathematics on the basis of explicit axioms; the change in focus by Eudoxus stimulated a divide in mathematics. In combination with a Greek intellectual attitude unconcerned with practical problems, there followed a significant retreat from the development of techniques in arithmetic and algebra; the Pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with the sides of the square. This discovery had heralded the existence of incommensurable quantities beyond the integers and rational fractions, but at the same time it threw into question the idea of measurement and calculations in geometry as a whole. For example, Euclid provides an elaborate proof of the Pythagorean theorem, by using addition of areas and only much a simpler proof from similar triangles, which relies on ratios of line segments.

Ancient Greek mathematicians calculated not with quantities and equations as we do today, but instead they used proportionalities to express the relationship between quantities. Thus the ratio of two similar quantities was not just a numerical value. Eudoxus was able to restore confidence in the use of proportionalities by providing an astounding definition for the meaning of the equality between two ratios; this definition of proportion forms the subject of Euclid's Book V. In Definition 5 of Euclid's Book V we read: Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever be taken of the first and third, any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike

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

Curve

In mathematics, a curve is speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line. Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition. A curve is a topological space, locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is its ending point—that is, a path from any of its points to the same point. Related meanings include the graph of a function and a two-dimensional graph. Interest in curves began; this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times.

Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach. The term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are today called lines from curved lines. For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length", while a straight line is defined as "a line that lies evenly with the points on itself". Euclid's idea of a line is clarified by the statement "The extremities of a line are points,". Commentators further classified lines according to various schemes. For example: Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard compass and straightedge construction; these curves include: The conic sections studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles and used as a method to double the cube.

The conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle and square the circle; the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century; this enabled a curve to be described using an equation rather than an elaborate geometrical construction. This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between curves that can be defined using algebraic equations, algebraic curves, those that cannot, transcendental curves. Curves had been described as "geometrical" or "mechanical" according to how they were, or could be, generated. Conic sections were applied in astronomy by Kepler. Newton worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways.

The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became accessible by means of differential calculus. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into'ovals'; the statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions. Since the nineteenth century there has not been a separate theory of curves, but rather the appearance of curves as the one-dimensional aspect of projective geometry, differential geometry; the era of the space-filling curves provoked the modern definitions of curve. In general, a curve is defined through a continuous function γ: I → X from an interval I of the real numbers into a topological space X. Depending on the context, it is either γ or its image γ, called a curve. In general topology, when non-differentiable functions are considered, it is the map γ, called a curve, because its image may look differently from what is called a curve.

For example, the image of the Peano curve fills the square. On the other hand, when one considers curves defined by a differentiable function, this is the image of the function, called a curve; the curve is said to be simple, or a Jordan arc, if γ is injective, i.e. if for all x, y in I, we have γ = γ

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

Eratosthenes

Eratosthenes of Cyrene was a Greek mathematician, poet and music theorist. He was a man of becoming the chief librarian at the Library of Alexandria, he invented the discipline of geography, including the terminology used today. He is best known for being the first person to calculate the circumference of the Earth, which he did by comparing angles of the mid-day Sun at two places a known North-South distance apart, his calculation was remarkably accurate. He was the first to calculate the tilt of the Earth's axis, again with remarkable accuracy. Additionally, he may have calculated the distance from the Earth to the Sun and invented the leap day, he created the first map of the world, incorporating parallels and meridians based on the available geographic knowledge of his era. Eratosthenes was the founder of scientific chronology. Eratosthenes dated The Sack of Troy to 1183 BC. In number theory, he introduced the sieve of Eratosthenes, an efficient method of identifying prime numbers, he was a figure of influence in many fields.

According to an entry in the Suda, his critics scorned him, calling him Beta because he always came in second in all his endeavors. Nonetheless, his devotees nicknamed him Pentathlos after the Olympians who were well rounded competitors, for he had proven himself to be knowledgeable in every area of learning. Eratosthenes yearned to understand the complexities of the entire world; the son of Aglaos, Eratosthenes was born in 276 BC in Cyrene. Now part of modern-day Libya, Cyrene had been founded by Greeks centuries earlier and became the capital of Pentapolis, a country of five cities: Cyrene, Berenice and Apollonia. Alexander the Great conquered Cyrene in 332 BC, following his death in 323 BC, its rule was given to one of his generals, Ptolemy I Soter, the founder of the Ptolemaic Kingdom. Under Ptolemaic rule the economy prospered, based on the export of horses and silphium, a plant used for rich seasoning and medicine. Cyrene became a place of cultivation. Like any young Greek, Eratosthenes would have studied in the local gymnasium, where he would have learned physical skills and social discourse as well as reading, arithmetic and music.

Eratosthenes went to Athens to further his studies. There he was taught Stoicism by its founder, Zeno of Citium, in philosophical lectures on living a virtuous life, he studied under Aristo of Chios, who led a more cynical school of philosophy. He studied under the head of the Platonic Academy, Arcesilaus of Pitane, his interest in Plato led him to write his first work at a scholarly level, inquiring into the mathematical foundation of Plato's philosophies. Eratosthenes investigated the art of poetry under Callimachus, he was a imaginative poet. He wrote poems: one in hexameters called Hermes, illustrating the god's life history, he wrote Chronographies, a text that scientifically depicted dates of importance, beginning with the Trojan War. This work was esteemed for its accuracy. George Syncellus was able to preserve from Chronographies a list of 38 kings of the Egyptian Thebes. Eratosthenes wrote Olympic Victors, a chronology of the winners of the Olympic Games, it is not known when he wrote his works.

These works and his great poetic abilities led the pharaoh Ptolemy III Euergetes to seek to place him as a librarian at the Library of Alexandria in the year 245 BC. Eratosthenes thirty years old, accepted Ptolemy's invitation and traveled to Alexandria, where he lived for the rest of his life. Within about five years he became Chief Librarian, a position that the poet Apollonius Rhodius had held; as head of the library Eratosthenes tutored the children of Ptolemy, including Ptolemy IV Philopator who became the fourth Ptolemaic pharaoh. He expanded the library's holdings: in Alexandria all books had to be surrendered for duplication, it was said that these were copied so that it was impossible to tell if the library had returned the original or the copy. He sought to maintain the reputation of the Library of Alexandria against competition from the Library of Pergamum. Eratosthenes created a whole section devoted to the examination of Homer, acquired original works of great tragic dramas of Aeschylus and Euripides.

Eratosthenes made several important contributions to mathematics and science, was a friend of Archimedes. Around 255 BC, he invented the armillary sphere. In On the Circular Motions of the Celestial Bodies, Cleomedes credited him with having calculated the Earth's circumference around 240 BC, using knowledge of the angle of elevation of the Sun at noon on the summer solstice in Alexandria and on Elephantine Island near Syene. Eratosthenes believed there was good and bad in every nation and criticized Aristotle for arguing that humanity was divided into Greeks and barbarians, that the Greeks should keep themselves racially pure; as he aged he contracted ophthalmia, becoming blind around 195 BC. Losing the ability to read and to observe nature plagued and depressed him, leading him to voluntarily starve himself to death, he died in 194 BC at 82 in Alexandria. Eratosthenes calculated the Earth's circumference without leaving Alexandria, he knew that at local noon on the summer solstice in Syene (modern Asw

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