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Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly The 6th century BC started the first day of 600 BC and ended the last day of 501 BC. The 5th century is the period from 401 to 500 in accordance with the Julian calendar in Anno Domini / Common Era. The word "mathematics" itself derives from the ancient Greek μαθημα (mathema), meaning "subject of instruction". [1]

Contents

Origins of Greek Mathematics

Greek mathematics has origins that are presumed to go back to the early Thalassic Age, but are not easily documented. It is generally believed that Greek traders, scholars, and businessmen brought back to Greece the mathematics of the Babylonians and Egyptians. Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital Ancient Egypt was an Ancient Civilization in eastern North Africa, concentrated along the lower reaches of the Nile River in what is now Between 800 BC and 600 BC Greek mathematics generally lagged behind Greek literature, and there is very little known about Greek mathematics from this period—nearly all of which was passed down through later authors, beginning in the mid-4th century BC. [2]

The use of generalized mathematical theories and proofs is usually regarded as the key difference between Greek mathematics and what came before. However, Babylonian mathematics was not entirely without generalizations and formalized mathematical knowledge; historian Jens Høyrup and other have pointed to the example of "Problem 20" in the Babylonian cuneiform text BM 85 194 as evidence of important precursors to Greek proofs, e. g. , of the Pythagorean theorem. The problem explains a method for calculating the length of a chord of a circle, given the circumference and length of the "arrow" of the chord (the perpendicular distance from the chord's mid-point to the circle); this problem seems to rely implicitly on a premise equivalent to the Pythagorean theory (although in a different form, since the concept of an angle was absent from pre-Greek mathematics). A chord of a Curve is a geometric Line segment whose endpoints both lie on the curve In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Høyrup has suggested that pre-Greek cultures failed to develop proofs in part because the mathematical knowledge was passed down in schools for training scribes, in which the student's ultimate goal was to become an administrator capable of solving complex numerical problems; there was no need, in that context, to state the general premises used for solving the problems. [3]

Classical Period

Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus (ca. Thales of Miletus According to Bertrand Russell, "Philosophy begins with Thales Miletus (mī lē' təs ( Ancient Greek: Μίλητος literally Transliterated Milētos, Latin Miletus) was an Ancient 624 - 548 BC). Little is known about the life and work of Thales, so little indeed that his day of birth and death are estimated from the eclipse of 585 BCE, which probably occurred while he was in his prime. Despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The Seven Sages (of Greece or Seven Wise Men (Greek οἱ ἑπτά σοφοί hoi hepta sophoi c The Theorem of Thales, which states that an angle inscribed in a semicircle is a right angle, may have been learned by Thales while in Babylon but tradition attributes to Thales a demonstration of the theorem. 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. Although it is not known whether or not Thales was the one who introduced into mathematics the logical structure that is so ubiquitous today, it is known that within two hundred years of Thales the Greeks had introduced logical structure and the idea of proof into mathematics.

Another important figure in the development of Greek mathematics is Pythagoras of Samos (ca. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. 580 - 500 BC). Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar,[2][4] but settled in Croton, Magna Graecia. Croton may also refer to a plant genus See Croton (genus. Or to the NY village Croton-on-Hudson. 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. Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced And since in antiquity it was customary to give all credit to the master, Pythagoras himself was given credit for the discoveries made by his order. 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 moral basis for the conduct of life. Indeed, the words "philosophy" (love of wisdom) and "mathematics" (that which is learned) are said to have been coined by Pythagoras. From this love of knowledge can many achievements. It has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclid's Elements. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek

Distinguishing the work of Thales and Pythagoras from that of later and earlier mathematicians is difficult since none of their original works survives, except for possibly the surviving "Thales-fragments", which are of disputed reliability. However, many[5] historians have argued that much of the mathematical knowledge ascribed to Thales was in fact developed later, particularly the aspects that rely on the concept of angles, while the use of general statements may have appeared earlier, such as those found on Greek legal texts inscribed on slabs. [6] The reason that it is not clear exactly what either Thales or Pythagoras actually did is that almost no contemporary documentation has survived. The only evidence comes from traditions recorded in works such as Proclus’ commentary on Euclid written centuries later. Proclus Lycaeus ( February 8, c 411 &ndash April 17, 485) called "The Successor" or "Diadochos" ( Greek Próklos Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Some of these later works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced

Thales is supposed to have used geometry to solve problems such as calculating the height of pyramids based on the length of shadows, and the distance of ships from the shore. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position He is also credited by tradition with having made the first proof of a geometric theorem. He is said to have demonstrated that an angle inscribed in a semi-circle is a right angle, which is known as the Theorem of Thales. Pythagoras is widely credited with recognizing the mathematical basis of musical harmony, and according to Proclus' commentary on Euclid he discovered the theory of proportionals and constructed regular solids. In Western music, harmony is the use of different pitches simultaneously and chords actual or implied in Music. In Geometry, a Platonic solid is a convex Regular polyhedron. Some modern historians have questioned whether he really constructed all five regular solids, suggesting instead that it is more reasonable to assume that he constructed just three of them. Some ancient sources attribute the discovery of the Pythagorean theorem to Pythagoras, where as others claim it was a proof for the theorem that he discovered. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Modern historians believe that the principle itself was known to the Babylonians and likely imported from them. The Pythagoreans regarded numerology and geometry as fundamental to understanding the nature of the universe and therefore central to their philosophical and religious ideas. Numerology is any of many Systems Traditions or Beliefs in a mystical or Esoteric relationship between Numbers and physical They are credited with numerous mathematical advances, such as the discovery of irrational numbers. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction Historians credit them with a major role in the development of Greek mathematics (particularly number theory and geometry) into a coherent logical system based on clear definitions and proven theorems that was considered to be a subject worthy of study in its own right, without regard to the practical applications that had been the primary concern of the Egyptians and Babylonians. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes [2][4]

Hellenistic

The Hellenistic period began in the 4th century BC with Alexander the Great's conquest of the eastern Mediterranean, Egypt, Mesopotamia, the Iranian plateau, Central Asia, and parts of India, leading to the spread of the Greek language and culture across these areas. This article focuses on the historical aspects of the Hellenistic age for the cultural aspects see Hellenistic civilisation. The 4th century BC started the first day of 400 BC and ended the last day of 301 BC. Alexander the Great ( or, Mégas Aléxandros; July 20 356 BC June 10 or June 11 323 BC also known as Alexander III of Macedon (el Ἀλέξανδρος Γ' Ancient Egypt was an Ancient Civilization in eastern North Africa, concentrated along the lower reaches of the Nile River in what is now Mesopotamia (from the Greek meaning "land between the rivers" is an area geographically located between the Tigris and Euphrates rivers largely corresponding The Iranian Plateau, also known as the Persian plateau is a Geological formation in Southwest Asia, Southern Central Asia is a region of Asia from the Caspian Sea in the west to central China in the east and from southern Russia in the north to northern Pakistan in the south India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country Greek became the language of scholarship throughout the Hellenistic world, and Greek mathematics merged with Egyptian and Babylonian mathematics to give rise to a Hellenistic mathematics. Egyptian mathematics refers to the style and methods of Mathematics performed in Ancient Egypt. Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (ancient Iraq) from the days of the early Sumerians to the fall of

The most important centre of learning during this period was Alexandria in Egypt, which attracted scholars from across the Hellenistic world, mostly Greek and Egyptian, but also Jewish, Persian, Phoenician and even Indian scholars. Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια Ancient Egypt was an Ancient Civilization in eastern North Africa, concentrated along the lower reaches of the Nile River in what is now This article is about the contemporary North African ethnic group PLEASE TAKE NOTE************ layout and formatting it should ensure no clashes with the top of the infobox Phoenicia ( Phoenician: Phoenician nunsvg|12px|נ]]Phoenician nun This article is about the history of South Asia prior to the Partition of British India in 1947 [7]

Most of the mathematical texts written in Greek have been found in Greece, Egypt, Asia Minor, Mesopotamia, and Sicily. This article is about the country of Egypt For a topic outline on this subject see List of basic Egypt topics. Anatolia (Anadolu Ανατολία Anatolía) or Asia minor, comprising most of modern Turkey, is the geographic region bounded by the Black Mesopotamia (from the Greek meaning "land between the rivers" is an area geographically located between the Tigris and Euphrates rivers largely corresponding Sicily ( Italian and Sicilian: Sicilia) is an autonomous region of Italy.

The Antikythera mechanism, an ancient mechanical calculator.
The Antikythera mechanism, an ancient mechanical calculator. The Antikythera mechanism (ˌæntɪkɪˈθɪərə an-ti-ki- theer -uh is an ancient mechanical Calculator (also described as the first known " mechanical

Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space By assuming a proposition to be true and showing that this would lead to a contradiction, he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield This technique is known as the method of exhaustion, and he employed it to approximate the value of π (Pi). The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 times the area of a triangle with equal base and height. The Quadrature of the Parabola is a treatise on Geometry, written by Archimedes in the 3rd century B In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line He expressed the solution to the problem as an infinite geometric series, whose sum was 4/3. In Mathematics, a geometric series is a series with a constant ratio between successive terms. In The Sand Reckoner, Archimedes set out to calculate the number of grains of sand that the universe could contain. The Sand Reckoner ( Greek: Ψαμμίτης Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number In doing so, he challenged the notion that the number of grains of sand was too large to be counted, devising his own counting scheme based on the myriad, which denoted 10,000. Myriad is a classical Greek name for the Number 104 = 10000. In modern English the word refers to an unspecified large quantity

Greek mathematics and astronomy reached a rather advanced stage during Hellenism, with scholars such as Hipparchus, Posidonius and Ptolemy, capable of the construction of simple analogue computers such as the Antikythera mechanism. This article focuses on the historical aspects of the Hellenistic age for the cultural aspects see Hellenistic civilisation. Hipparchus ( Greek; ca 190 BC &ndash ca 120 BC was a Greek Astronomer, Geographer, and Mathematician of the Hellenistic Posidonius ( Greek: Ποσειδώνιος / Poseidonios "of Apameia " (ὁ Απαμεύς or "of Rhodes " (ὁ Ρόδιος (ca Claudius Ptolemaeus ( Greek: Klaúdios Ptolemaîos; after 83 &ndash ca The Antikythera mechanism (ˌæntɪkɪˈθɪərə an-ti-ki- theer -uh is an ancient mechanical Calculator (also described as the first known " mechanical

Achievements

Greek mathematics constitutes a major period in the history of mathematics, fundamental in respect of geometry and the idea of formal proof. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position see also Mathematical proof, Proof theory, and Axiomatic system. Greek mathematics also contributed importantly to ideas on number theory, mathematical analysis, applied mathematics, and, at times, approached close to integral calculus. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Analysis has its beginnings in the rigorous formulation of Calculus. Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

Well-known figures in Greek mathematics include Pythagoras, a shadowy figure from the isle of Samos associated partly with number mysticism and numerology, but more commonly with his theorem, and Euclid, who is known for his Elements, a canon of geometry for many centuries. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. Samos (Σάμος is a Greek island in the North Aegean Sea, south of Chios, north of Patmos and the Dodecanese, and off Numerology is any of many Systems Traditions or Beliefs in a mystical or Esoteric relationship between Numbers and physical In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek

The most characteristic product of Greek mathematics may be the theory of conic sections, largely developed in the Hellenistic period. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface The methods used made no explicit use of algebra, nor trigonometry. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

Transmission and the manuscript tradition

Although the earliest Greek language texts on mathematics that have been found were written after the Hellenistic period, many of these are considered to be copies of works written during and before the Hellenistic period. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Nevertheless, the dates of Greek mathematics are more certain than the dates of earlier mathematical writing, since a large number of chronologies exist that, overlapping, record events year by year up to the present day. Even so, many dates are uncertain; but the doubt is a matter of decades rather than centuries.

During the Middle Ages, Europe derived much of its knowledge of Greek mathematics via Islamic mathematics. The texts of Greek mathematics were for the most part preserved and transmitted in the Muslim world. The term Muslim world (or Islamic world) has several meanings For instance, the oldest surviving Latin version of Euclid's Elements is a 12th century translation from Arabic. Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek

See also

Footnotes

  1. ^ Heath. The area of study known as the history of mathematics is primarily an investigation into the origin of new discoveries in Mathematics and to a lesser extent an investigation A Manual of Greek Mathematics, 5.  
  2. ^ a b c Boyer & Merzbach (1991) pp. 43-61
  3. ^ Hans-Joachim Waschkies, "Introduction" to "Part 1: The Beginning of Greek Mathematics" in Classics in the History of Greek Mathematics, pp. 5-11
  4. ^ a b Heath (2003) pp. 36-111
  5. ^ Such as Hans-Joachim Waschkies and Carl Boyer
  6. ^ Hans-Joachim Waschkies, "Introduction" to "Part 1: The Beginning of Greek Mathematics" in Classics in the History of Greek Mathematics, pp. 11-12
  7. ^ George G. Joseph (2000). The Crest of the Peacock, p. 7-8. Princeton University Press. The Princeton University Press is an independent publisher with close connections to Princeton University. ISBN 0691006598.

References

External links

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