Greek mathematics
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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. The word "mathematics" itself derives from the ancient Greek μαθημα (mathema), meaning "subject of instruction".[1]
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[edit] 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. 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). 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]
[edit] Classical Period
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 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 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. 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. Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, 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.
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. 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 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. 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. 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. [2][4]
[edit] 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. 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.
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.[7]
Most of the mathematical texts written in Greek have been found in Greece, Egypt, Asia Minor, Mesopotamia, and Sicily.
Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. 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. This technique is known as the method of exhaustion, and he employed it to approximate the value of π (Pi). 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. He expressed the solution to the problem as an infinite geometric series, whose sum was 4/3. In The Sand Reckoner, Archimedes set out to calculate 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.
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.
[edit] Achievements
Greek mathematics constitutes a major period in the history of mathematics, fundamental in respect of geometry and the idea of formal proof. Greek mathematics also contributed importantly to ideas on number theory, mathematical analysis, applied mathematics, and, at times, approached close to integral calculus.
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.
The most characteristic product of Greek mathematics may be the theory of conic sections, largely developed in the Hellenistic period. The methods used made no explicit use of algebra, nor trigonometry.
[edit] 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. 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. For instance, the oldest surviving Latin version of Euclid's Elements is a 12th century translation from Arabic.
[edit] See also
[edit] Footnotes
- ^ Heath. A Manual of Greek Mathematics, 5.
- ^ a b c Boyer & Merzbach (1991) pp. 43-61
- ^ Hans-Joachim Waschkies, "Introduction" to "Part 1: The Beginning of Greek Mathematics" in Classics in the History of Greek Mathematics, pp. 5-11
- ^ a b Heath (2003) pp. 36-111
- ^ Such as Hans-Joachim Waschkies and Carl Boyer
- ^ Hans-Joachim Waschkies, "Introduction" to "Part 1: The Beginning of Greek Mathematics" in Classics in the History of Greek Mathematics, pp. 11-12
- ^ George G. Joseph (2000). The Crest of the Peacock, p. 7-8. Princeton University Press. ISBN 0691006598.
[edit] References
- Boyer, Carl B. (1985), A History of Mathematics, Princeton University Press, ISBN 0691023913
- Boyer, Carl B. & Uta C. Merzbach (1991), A History of Mathematics (Second Edition ed.), John Wiley & Sons, Inc., ISBN 0471543977
- Jean Christianidis, ed. (2004), Classics in the History of Greek Mathematics, Kluwer Academic Publishers, ISBN 1-4020-0081-2
- Cooke, Roger (1997), The History of Mathematics: A Brief Course, Wiley-Interscience, ISBN 0471180823
- Derbyshire, John (2006), Unknown Quantity: A Real And Imaginary History of Algebra, Joseph Henry Press, ISBN 030909657X
- Stillwell, John (2004), Mathematics and its History (Second Edition ed.), Springer Science + Business Media Inc., ISBN 0387953361
- Burton, David M. (1997), The History of Mathematics: An Introduction (Third Edition ed.), The McGraw-Hill Companies, Inc., ISBN 0070094659
- Heath, Thomas Little (1981), A History of Greek Mathematics, Dover publications, ISBN 0486240738, ISBN 0486240746 (first published 1921).
- Heath, Thomas Little (2003), A Manual of Greek Mathematics, Dover publications, ISBN 048643231-9 (first published 1931).
- Szabo, Arpad The Beginnings of Greek Mathematics (Tr Ungar) Reidel & Akademiai Kiado, Budapest 1978 ISBN 963 05 1416 8
[edit] External links
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