Greek mathematics refers to mathematics texts and advances written in Greek, developed 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 culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word "mathematics" itself derives from the Ancient Greek: μάθημα, romanized: máthēma Attic Greek: [má.tʰɛː.ma] Koine Greek: [ˈma.θi.ma], meaning "subject of instruction". The study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations.
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. 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.
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. The two earliest mathematical theorems, Thales' theorem and Intercept theorem are attributed to Thales. The former, 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, 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 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.
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 a rather advanced stage during the Hellenistic and Roman period, represented by scholars such as Hipparchus, Apollonius and Ptolemy, to the point of constructing simple analogue computers such as the Antikythera mechanism.
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.
Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. Using a technique dependent on a form of proof by 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 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.
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.
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
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.
The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system. The exact origin of the abacus is still unknown. Today, abacuses are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of wood, stone, or metal.
Abacuses come in different designs. Some designs, like the bead frame consisting of beads divided into tens, are used mainly to teach arithmetic, although they remain popular in the post-Soviet states as a tool. Other designs, such as the Japanese soroban, have been used for practical calculations even involving several digits. For any particular abacus design, there are usually numerous different methods to perform a certain type of calculation, which may include basic operations like addition and multiplication, or even more complex ones, such as calculating square roots. Some of these methods may work with non-natural numbers (numbers such as 1.5 and 3⁄4).
Although today many use calculators and computers instead of abacuses to calculate, abacuses still remain in common use in some countries. Merchants, traders and clerks in some parts of Eastern Europe, Russia, China and Africa use abacuses, and they are still used to teach arithmetic to children. Some people who are unable to use a calculator because of visual impairment may use an abacus.Attic numerals
The Attic numerals are a symbolic number notation used by the ancient Greeks. They were also known as Herodianic numerals because they were first described in a 2nd-century manuscript by Herodian; or as acrophonic numerals (from acrophony) because the basic symbols derive from the first letters of the (ancient) Greek words that the symbols represented.
The Attic numerals were a decimal (base 10) system, like the older Egyptian and the later Etruscan, Roman, and Hindu-Arabic systems. Namely, the number to be represented was broken down into simple multiples (1 to 9) of powers of ten — units, tens, hundred, thousands, etc.. Then these parts were written down in sequence, in order of decreasing value. As in the basic Roman system, each part was written down using a combination of two symbols, representing one and five times that power of ten.
Attic numerals were adopted possibly starting in the 7th century BCE, and were eventually replaced by the classic Greek numerals around the 3rd century BCE. They are believed to have served as model for the Etruscan number system, although the two were nearly contemporary and the symbols are not obviously related.Euclid's orchard
In mathematics, informally speaking, Euclid's orchard is an array of one-dimensional "trees" of unit height planted at the lattice points in one quadrant of a square lattice. More formally, Euclid's orchard is the set of line segments from (i, j, 0) to (i, j, 1), where i and j are positive integers.
The trees visible from the origin are those at lattice points (m, n, 0), where m and n are coprime, i.e., where the fraction m/ is in reduced form. The name Euclid's orchard is derived from the Euclidean algorithm.
If the orchard is projected relative to the origin onto the plane x + y = 1 (or, equivalently, drawn in perspective from a viewpoint at the origin) the tops of the trees form a graph of Thomae's function. The point (m, n, 1) projects to
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean
a number represented as a discrete r-dimensional regular geometric pattern of r-dimensional balls such as a polygonal number (for r = 2) or a polyhedral number (for r = 3).
a member of the subset of the sets above containing only triangular numbers, pyramidal numbers, and their analogs in other dimensions.Greek numerals
Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those in which Roman numerals are still used elsewhere in the West. For ordinary cardinal numbers, however, Greece uses Arabic numerals.Hippasus
Hippasus of Metapontum (; Greek: Ἵππασος ὁ Μεταποντῖνος, Híppasos; c. 530 - c. 450 BC), 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, and Hippasus is supposed to have drowned at sea, apparently 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 (e.g. Pappus) or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer. Some modern scholars though have suggested that he discovered the irrationality of √2, which is believed to have been discovered around the time that he lived.History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, together with Ancient Egypt and Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy and to formulate calendars and record time.
The most ancient mathematical texts available are from Mesopotamia and Egypt - Plimpton 322 (Babylonian c. 1900 BC), the Rhind Mathematical Papyrus (Egyptian c. 2000–1800 BC) and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples and so, by inference, the Pythagorean theorem, seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a "demonstrative discipline" begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction". Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics. Although they made virtually no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya numerals.
Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century. At the end of the 19th century the International Congress of Mathematicians was founded and continues to spearhead advances in the field.Jan Hogendijk
Jan Pieter Hogendijk (born 21 July 1955) is a Dutch mathematician and historian of science. Since 2005 he is professor of history of mathematics at the University of Utrecht.
Hogendijk became a member of the Royal Netherlands Academy of Arts and Sciences in 2010.Hogendijk has contributed to the study of Greek mathematics and mathematics in medieval Islam; he provides a list of Sources on his website (below).
In 2012, he was awarded the inaugural Otto Neugebauer Prize for History of Mathematics, by the European Mathematical Society, "for having illuminated how Greek mathematics was absorbed in the medieval Arabic world, how mathematics developed in medieval Islam, and how it was eventually transmitted to Europe."A bibliography of Hogendijk's publications is included in his website.Lemniscate
In algebraic geometry, a lemniscate is any of several figure-eight or ∞-shaped curves. The word comes from the Latin "lēmniscātus" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons", or which alternatively may refer to the wool from which the ribbons were made.Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli, and the lemniscate of Gerono. The study of lemniscates (and in particular the hippopede) dates to ancient Greek mathematics, but the term "lemniscate" for curves of this type comes from the work of Jacob Bernoulli in the late 17th century.List of mathematics history topics
This is a list of mathematics history topics, by Wikipedia page. See also list of mathematicians, timeline of mathematics, history of mathematics, list of publications in mathematics.
Archimedes' use of infinitesimals
Arithmetization of analysis
Edinburgh Mathematical Society
Fermat's last theorem
Thomas Little Heath
History of topos theory
Italian school of algebraic geometry
Kraków School of Mathematics
Law of Continuity
Lwów School of Mathematics
Seven bridges of Königsberg
Unifying theories in mathematics
Warsaw School of MathematicsMathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity, structure, space, and change. It has no generally accepted definition.Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.Mathematics in medieval Islam
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as the full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra (named for The Compendious Book on Calculation by Completion and Balancing by scholar Al-Khwarizmi), and advances in geometry and trigonometry.Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries.Salamis Tablet
The Salamis Tablet is a marble counting board (an early counting device) dating from around 300 B.C. that was discovered on the island of Salamis in 1846. A precursor to the abacus, it is thought that it represents an ancient Greek means of performing mathematical calculations common in the ancient world. Pebbles (Latin: calculi) were placed at various locations and could be moved as calculations were performed. The marble tablet itself has dimensions of approximately 150 × 75 × 4.5 cm.Tetractys
The tetractys (Greek: τετρακτύς), or tetrad, or the tetractys of the decad is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. As a mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was also associated with planetary motions and music.Theon of Smyrna
Theon of Smyrna (Greek: Θέων ὁ Σμυρναῖος Theon ho Smyrnaios, gen. Θέωνος Theonos; fl. 100 CE) was a Greek philosopher and mathematician, whose works were strongly influenced by the Pythagorean school of thought. His surviving On Mathematics Useful for the Understanding of Plato is an introductory survey of Greek mathematics.Thomas Heath
Thomas Heath may refer to:
Thomas Little Heath (1861–1940), British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer
Thomas Heath (cricketer) (1806–1872), cricketer
Thomas Kurton Heath (1853–1938), vaudeville actor
Tommy Heath (born 1947), musician
Tommy Heath (baseball) (1913–1967), American catcher, scout and baseball manager
Thomas Heth (fl. 1583), also Heath, English mathematicianThomas Little Heath
Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath translated works of Euclid of Alexandria, Apollonius of Perga, Aristarchus of Samos, and Archimedes of Syracuse into English.Timeline of ancient Greek mathematicians
This is a timeline of ancient Greek mathematicians (see also Chronology of ancient Greek mathematicians).Wilbur Knorr
Wilbur Richard Knorr (August 29, 1945 – March 18, 1997) was an American historian of mathematics and a professor in the departments of philosophy and classics at Stanford University. He has been called "one of the most profound and certainly the most provocative historian of Greek mathematics" of the 20th century.
Ancient Greek mathematics