Greek mathematics refers to mathematics texts written during and ideas stemming from the Archaic through the Hellenistic periods, extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by Greek culture and the Greek language. The word "mathematics" itself derives from the grc||máthēma , meaning "subject of instruction". The study of mathematics for its own sake and the use of generalized mathematical theories and proofs is an important difference between Greek mathematics and those of preceding civilizations.

Origins of Greek mathematics

The origin of Greek mathematics is not well documented. The earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilizations, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, they left behind no mathematical documents. Though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Unlike the flourishing of Greek literature in the span of 800 to 600 BC, very little is known about Greek mathematics in this early period—nearly all of which was passed down through later authors, beginning in the mid-4th century BC.Boyer & Merzbach (1991) pp. 43–61

Archaic and Classical periods

Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus (ca. 624–548 BC). Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, 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. Thales was the first person to receive credit for specific mathematical discoveries. Much of the knowledge he obtained was gained during his travels to Babylon, which is why he was regarded as "a pupil of the Egyptians and the Chaldeans." Thales' Theorem may have been learned during his time in Babylon and it is because of this Theory that Thales was called the first true mathematician by Proclus. 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. 580–500 BC). Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar,Heath (2003) pp. 36–111 but settled in Croton, Magna Graecia. While more information is available about Pythagoras, his actual achievements are still shrouded in mystery due to the loss of original works from that period. Pythagoras started the Pythagorean order with the motto of the school being "All is number" and a five-pointed star being the symbol of the school.. 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 came many achievements. It has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclid's ''Elements''. Distinguishing the work of Thales and Pythagoras from that of later and earlier mathematicians is difficult since none of their original works survive, except for possibly the surviving "Thales-fragments", which are of disputed reliability. However, many historians, such as Hans-Joachim Waschkies and Carl Boyer, have argued that much of the mathematical knowledge ascribed to Thales was 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. The reason 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. Some of these later works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments. 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. He is also credited by tradition with having made the first proof of two geometric theorems—the "Theorem of Thales" and the "Intercept theorem" described above. 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. 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, whereas others claim it was a proof for the theorem that he discovered. 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. They are credited with numerous mathematical advances, such as the discovery of irrational numbers. 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.

Hellenistic and Roman periods

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. 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. Greek mathematics and astronomy reached an advanced level during the Hellenistic and early Roman period, represented by scholars such as Euclid, Archimedes, Apollonius, Hipparchus, and Ptolemy. There is also evidence of combining mathematical knowledge with high levels of technical expertise, such as found in the construction of simple analogue computers like the Antikythera mechanism.The major difference between Greek and the other mathematics was the Greeks idea of proofs and being able to prove the math while also applying it.Sialaros, Michalis (2018). ''Revolutions and Continuity in Greek Mathematics''. DE GRUYTER. ISBN 978-3-11-056365-8. The Greeks where among the first to come up with the idea of infinity, specifically Zeno of Elea, whom explains his Achilles and Tortoise Paradox which deals with infinity. Near 150 AD Ptolemy wrote the almagest and in this important astronomical manual The Sector theorem a powerful mathematical tool, suited to determine arcs of a great circle on the surface of a sphere, was used 17 times and was a very important result of Greek spherical trigonometry. Also around the year 200 AD another important Mathematician during this time period was Diophantus, and his work in ''Arithmetica'' was a one of the first works on what is known as pre - modern algebra. 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). Most of the mathematical texts written in Greek have been found in Greece, Egypt, Asia Minor, Mesopotamia, and Sicily.

Achievements

Greek mathematics constitutes an important period in the history of mathematics: fundamental in respect of geometry and for the idea of formal proof. Greek mathematics also contributed importantly to ideas on number theory, mathematical analysis, combinatorics, applied mathematics, and, at times, approaches close to integral calculus. Eudoxus of Cnidus developed a theory of proportion that bears resemblance to the modern theory of real numbers using the Dedekind cut, developed by Richard Dedekind, who acknowledged Eudoxus as inspiration. Euclid, fl. 300 BC, collected many previous results and theorems in the ''Elements'', a canon of geometry and elementary number theory for many centuries. Archimedes was able to use infinitesimals in a way that anticipated modern ideas of the integral calculus. Using a technique dependent on a form of proof by contradiction he could reach answers to problems with an arbitrary degree of accuracy, while specifying the limits within which the answers lay. This technique is known as the method of exhaustion, and he employed in several of his works, such as to approximate the value of π (''Measurement of the Circle''). In ''The Quadrature of the Parabola'', Archimedes proved that the area enclosed by a parabola and a straight line is times the area of a triangle with equal base and height. He expressed the solution to the problem as an infinite geometric series, whose sum was . In ''The Sand Reckoner'', Archimedes set out to name the number of grains of sand that the universe could contain. 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. The most characteristic product of Greek mathematics may be the theory of conic sections, which was largely developed in the Hellenistic period, primarily by Apollonius. The methods used made no explicit use of algebra, nor trigonometry, the latter appearing around the time of Hipparchus.

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. The two major sources are * Byzantine codices, written some 500 to 1500 years after their originals, and * Syriac or Arabic translations of Greek works and Latin translations of the Arabic versions. Nevertheless, despite the lack of original manuscripts, the dates of Greek mathematics are more certain than the dates of surviving Babylonian or Egyptian sources because a large number of overlapping chronologies exist. Even so, many dates are uncertain; but the doubt is a matter of decades rather than centuries. Reviel Netz has counted 144 ancient exact scientific authors, of these only 29 are extant in greek language: Aristarchus, Autolycus, Philo of Byzantium, Biton, Apollonius, Archimedes, Euclid, Theodosius, Hypsicles, Athenaeus, Geminus, Hero, Apollodorus, Theon of Smyrna, Cleomedes, Nicomachus, Ptolemy, Gaudentius, Anatolius, Aristides Quintilian, Porphyry, Diophantus, Alypius, Damianus, Pappus, Serenus, Theon of Alexandria, Anthemius, Eutocius. Some works are extant only in Arabic translations:Toomer, G.J. Lost greek mathematical works in arabic translation. The Mathematical Intelligencer 6, 32–38 (1984). https://doi.org/10.1007/BF03024153 *Apollonius, ''Conics'' books V to VII *Apollonius, ''De Rationis Sectione'' *Archimedes, ''Book of Lemmas'' *Archimedes, ''Construction of the Regular Heptagon'' *Diocles, ''On Burning Mirrors'' *Diophantus, ''Arithmetica'' books IV to VII *Euclid, ''On Divisions of Figures'' *Euclid, ''On Weights'' *Hero, ''Catoptrica'' *Hero, ''Mechanica'' *Menelaus, ''Sphaerica'' *Pappus, ''Commentary on Euclid's Elements book X'' *Ptolemy, ''Optics'' *Ptolemy, ''Planisphaerium''

** See also **

* Greek numerals
* Chronology of ancient Greek mathematicians
* History of mathematics
* Timeline of ancient Greek mathematicians

Notes

References

* * * * * * * * * *

External links

Vatican Exhibit

Famous Greek Mathematicians

{{DEFAULTSORT:Greek Mathematics

Origins of Greek mathematics

The origin of Greek mathematics is not well documented. The earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilizations, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, they left behind no mathematical documents. Though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Unlike the flourishing of Greek literature in the span of 800 to 600 BC, very little is known about Greek mathematics in this early period—nearly all of which was passed down through later authors, beginning in the mid-4th century BC.Boyer & Merzbach (1991) pp. 43–61

Archaic and Classical periods

Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus (ca. 624–548 BC). Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, 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. Thales was the first person to receive credit for specific mathematical discoveries. Much of the knowledge he obtained was gained during his travels to Babylon, which is why he was regarded as "a pupil of the Egyptians and the Chaldeans." Thales' Theorem may have been learned during his time in Babylon and it is because of this Theory that Thales was called the first true mathematician by Proclus. 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. 580–500 BC). Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar,Heath (2003) pp. 36–111 but settled in Croton, Magna Graecia. While more information is available about Pythagoras, his actual achievements are still shrouded in mystery due to the loss of original works from that period. Pythagoras started the Pythagorean order with the motto of the school being "All is number" and a five-pointed star being the symbol of the school.. 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 came many achievements. It has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclid's ''Elements''. Distinguishing the work of Thales and Pythagoras from that of later and earlier mathematicians is difficult since none of their original works survive, except for possibly the surviving "Thales-fragments", which are of disputed reliability. However, many historians, such as Hans-Joachim Waschkies and Carl Boyer, have argued that much of the mathematical knowledge ascribed to Thales was 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. The reason 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. Some of these later works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments. 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. He is also credited by tradition with having made the first proof of two geometric theorems—the "Theorem of Thales" and the "Intercept theorem" described above. 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. 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, whereas others claim it was a proof for the theorem that he discovered. 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. They are credited with numerous mathematical advances, such as the discovery of irrational numbers. 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.

Hellenistic and Roman periods

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. 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. Greek mathematics and astronomy reached an advanced level during the Hellenistic and early Roman period, represented by scholars such as Euclid, Archimedes, Apollonius, Hipparchus, and Ptolemy. There is also evidence of combining mathematical knowledge with high levels of technical expertise, such as found in the construction of simple analogue computers like the Antikythera mechanism.The major difference between Greek and the other mathematics was the Greeks idea of proofs and being able to prove the math while also applying it.Sialaros, Michalis (2018). ''Revolutions and Continuity in Greek Mathematics''. DE GRUYTER. ISBN 978-3-11-056365-8. The Greeks where among the first to come up with the idea of infinity, specifically Zeno of Elea, whom explains his Achilles and Tortoise Paradox which deals with infinity. Near 150 AD Ptolemy wrote the almagest and in this important astronomical manual The Sector theorem a powerful mathematical tool, suited to determine arcs of a great circle on the surface of a sphere, was used 17 times and was a very important result of Greek spherical trigonometry. Also around the year 200 AD another important Mathematician during this time period was Diophantus, and his work in ''Arithmetica'' was a one of the first works on what is known as pre - modern algebra. 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). Most of the mathematical texts written in Greek have been found in Greece, Egypt, Asia Minor, Mesopotamia, and Sicily.

Achievements

Greek mathematics constitutes an important period in the history of mathematics: fundamental in respect of geometry and for the idea of formal proof. Greek mathematics also contributed importantly to ideas on number theory, mathematical analysis, combinatorics, applied mathematics, and, at times, approaches close to integral calculus. Eudoxus of Cnidus developed a theory of proportion that bears resemblance to the modern theory of real numbers using the Dedekind cut, developed by Richard Dedekind, who acknowledged Eudoxus as inspiration. Euclid, fl. 300 BC, collected many previous results and theorems in the ''Elements'', a canon of geometry and elementary number theory for many centuries. Archimedes was able to use infinitesimals in a way that anticipated modern ideas of the integral calculus. Using a technique dependent on a form of proof by contradiction he could reach answers to problems with an arbitrary degree of accuracy, while specifying the limits within which the answers lay. This technique is known as the method of exhaustion, and he employed in several of his works, such as to approximate the value of π (''Measurement of the Circle''). In ''The Quadrature of the Parabola'', Archimedes proved that the area enclosed by a parabola and a straight line is times the area of a triangle with equal base and height. He expressed the solution to the problem as an infinite geometric series, whose sum was . In ''The Sand Reckoner'', Archimedes set out to name the number of grains of sand that the universe could contain. 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. The most characteristic product of Greek mathematics may be the theory of conic sections, which was largely developed in the Hellenistic period, primarily by Apollonius. The methods used made no explicit use of algebra, nor trigonometry, the latter appearing around the time of Hipparchus.

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. The two major sources are * Byzantine codices, written some 500 to 1500 years after their originals, and * Syriac or Arabic translations of Greek works and Latin translations of the Arabic versions. Nevertheless, despite the lack of original manuscripts, the dates of Greek mathematics are more certain than the dates of surviving Babylonian or Egyptian sources because a large number of overlapping chronologies exist. Even so, many dates are uncertain; but the doubt is a matter of decades rather than centuries. Reviel Netz has counted 144 ancient exact scientific authors, of these only 29 are extant in greek language: Aristarchus, Autolycus, Philo of Byzantium, Biton, Apollonius, Archimedes, Euclid, Theodosius, Hypsicles, Athenaeus, Geminus, Hero, Apollodorus, Theon of Smyrna, Cleomedes, Nicomachus, Ptolemy, Gaudentius, Anatolius, Aristides Quintilian, Porphyry, Diophantus, Alypius, Damianus, Pappus, Serenus, Theon of Alexandria, Anthemius, Eutocius. Some works are extant only in Arabic translations:Toomer, G.J. Lost greek mathematical works in arabic translation. The Mathematical Intelligencer 6, 32–38 (1984). https://doi.org/10.1007/BF03024153 *Apollonius, ''Conics'' books V to VII *Apollonius, ''De Rationis Sectione'' *Archimedes, ''Book of Lemmas'' *Archimedes, ''Construction of the Regular Heptagon'' *Diocles, ''On Burning Mirrors'' *Diophantus, ''Arithmetica'' books IV to VII *Euclid, ''On Divisions of Figures'' *Euclid, ''On Weights'' *Hero, ''Catoptrica'' *Hero, ''Mechanica'' *Menelaus, ''Sphaerica'' *Pappus, ''Commentary on Euclid's Elements book X'' *Ptolemy, ''Optics'' *Ptolemy, ''Planisphaerium''

Notes

References

* * * * * * * * * *

External links

Vatican Exhibit

Famous Greek Mathematicians

{{DEFAULTSORT:Greek Mathematics