**Greek mathematics** refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly attested from the late 7th century BC to the 6th century AD, around the shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire region, from Anatolia to Italy and North Africa, but were united by Greek culture and the Greek language.^{ [1] } The development of mathematics as a theoretical discipline and the use of proofs is an important difference between Greek mathematics and those of preceding civilizations.^{ [2] }^{ [3] }

Greek *mathēmatikē* ("mathematics") derives from the Ancient Greek : μάθημα , romanized: *máthēma*, Attic Greek: [má.tʰɛː.ma] Koine Greek: [ˈma.θi.ma] , from the verb *manthanein*, "to learn". Strictly speaking, a *máthēma* could be any branch of learning, or anything learnt; however, since antiquity certain *mathēmata* (mainly arithmetic, geometry, astronomy, and harmonics) were granted special status.^{ [4] }^{ [5] }

The origins of Greek mathematics are not well documented.^{ [6] }^{ [7] } The earliest advanced civilizations in Greece and 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.^{ [8] }^{ [9] }^{ [6] } Unlike the flourishing of Greek literature in the span of 800 to 600 BC, not much is known about Greek mathematics in this early period—nearly all of the information was passed down through later authors, beginning in the mid-4th century BC.^{ [10] }^{ [11] }

Greek mathematics allegedly began with Thales of Miletus (c. 624–548 BC). Very little is known about his life, although it is generally agreed that he was one of the Seven Wise Men of Greece. According to Proclus, he traveled to Babylon from where he learned mathematics and other subjects, coming up with the proof of what is now called Thales' Theorem.^{ [12] }^{ [13] }

An equally enigmatic figure is Pythagoras of Samos (c. 580–500 BC), who supposedly visited Egypt and Babylon,^{ [11] }^{ [14] } and ultimately settled in Croton, Magna Graecia, where he started a kind of brotherhood. Pythagoreans supposedly believed that "all is number" and were keen in looking for mathematical relations between numbers and things.^{ [15] } Pythagoras himself was given credit for many later discoveries, including the construction of the five regular solids. However, Aristotle refused to attribute anything specifically to Pythagoras and only discussed the work of the Pythagoreans as a group.^{ [16] }^{ [17] }

Almost half of the material in Euclid's * Elements * is customarily attributed to the Pythagoreans, including the discovery of irrationals, attributed to Hippasus (c. 530–450 BC) and Theodorus (fl. 450 BC).^{ [18] } The greatest mathematician associated with the group, however, may have been Archytas (c. 435-360 BC), who solved the problem of doubling the cube, identified the harmonic mean, and possibly contributed to optics and mechanics.^{ [18] }^{ [19] } Other mathematicians active in this period, not fully affiliated with any school, include Hippocrates of Chios (c. 470–410 BC), Theaetetus (c. 417–369 BC), and Eudoxus (c. 408–355 BC).

Greek mathematics also drew the attention of philosophers during the Classical period. Plato (c. 428–348 BC), the founder of the Platonic Academy, mentions mathematics in several of his dialogues.^{ [20] } While not considered a mathematician, Plato seems to have been influenced by Pythagorean ideas about number and believed that the elements of matter could be broken down into geometric solids.^{ [21] } He also believed that geometrical proportions bound the cosmos together rather than physical or mechanical forces.^{ [22] } Aristotle (c. 384–322 BC), the founder of the Peripatetic school, often used mathematics to illustrate many of his theories, as when he used geometry in his theory of the rainbow and the theory of proportions in his analysis of motion.^{ [22] } Much of the knowledge about ancient Greek mathematics in this period is thanks to records referenced by Aristotle in his own works.^{ [11] }^{ [23] }

The Hellenistic era began in the late 4th century BC, following 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 regions. Greek became the * lingua franca * of scholarship throughout the Hellenistic world, and the mathematics of the Classical period merged with Egyptian and Babylonian mathematics to give rise to Hellenistic mathematics.^{ [25] }^{ [26] }

Greek mathematics and astronomy reached its acme during the Hellenistic and early Roman periods, and much of the work represented by authors such as Euclid (fl. 300 BC), Archimedes (c. 287–212 BC), Apollonius (c. 240–190 BC), Hipparchus (c. 190–120 BC), and Ptolemy (c. 100–170 AD) was of a very advanced level and rarely mastered outside a small circle.^{ [27] } There is also evidence of combining mathematical knowledge with technical or practical applications, as found for instance in the work of Menelaus of Alexandria (c. 70–130 AD), who wrote a work dealing with the geometry of the sphere and its application to astronomical measurements and calculations (*Spherica*).^{ [28] } Similar examples of applied mathematics include the construction of analogue computers like the Antikythera mechanism,^{ [29] }^{ [30] } the accurate measurement of the circumference of the Earth by Eratosthenes (276–194 BC), and the mechanical works of Hero (c. 10–70 AD).^{ [31] }

Several centers of learning appeared during the Hellenistic period, of which the most important one was the Mouseion in Alexandria, Egypt, which attracted scholars from across the Hellenistic world (mostly Greek, but also Egyptian, Jewish, Persian, among others).^{ [32] }^{ [33] } Although few in number, Hellenistic mathematicians actively communicated with each other; publication consisted of passing and copying someone's work among colleagues.^{ [34] }

Later mathematicians in the Roman era include Diophantus (c. 214–298 AD), who wrote on polygonal numbers and a work in pre-modern algebra (* Arithmetica *),^{ [35] }^{ [36] } Pappus of Alexandria (c. 290–350 AD), who compiled many important results in the *Collection*,^{ [37] } Theon of Alexandria (c. 335–405 AD) and his daughter Hypatia (c. 370–415 AD), who edited Ptolemy's * Almagest * and other works,^{ [38] }^{ [39] } and Eutocius of Ascalon (c. 480–540 AD), who wrote commentaries on treatises by Archimedes and Apollonius.^{ [40] } Although none of these mathematicians, save perhaps Diophantus, had notable original works, they are distinguished for their commentaries and expositions. These commentaries have preserved valuable extracts from works which have perished, or historical allusions which, in the absence of original documents, are precious because of their rarity.^{ [41] }^{ [42] }

Most of the mathematical texts written in Greek survived through the copying of manuscripts over the centuries, though some fragments dating from antiquity have been found above all in Egypt, and to a much lesser extent in Greece, Asia Minor, Mesopotamia, and Sicily.^{ [27] }

Greek mathematics constitutes an important period in the history of mathematics: fundamental in respect of geometry and for the idea of formal proof.^{ [43] } Greek mathematicians also contributed to number theory, mathematical astronomy, combinatorics, mathematical physics, and, at times, approached ideas close to the integral calculus.^{ [44] }^{ [45] }

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.^{ [46] }^{ [47] }^{ [48] }^{ [49] }

Euclid, who presumably wrote on optics, astronomy, and harmonics, collected many previous mathematical results and theorems in the * Elements *, a canon of geometry and elementary number theory for many centuries.^{ [50] }^{ [51] }^{ [52] }

Archimedes made use of a technique dependent on a form of proof by contradiction to reach answers to problems with an arbitrary degree of accuracy, while specifying the limits within which the answers lay. Known as the method of exhaustion, Archimedes employed it in several of his works, including an approximation to π (* Measurement of the Circle *),^{ [53] } and a proof 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 (* Quadrature of the Parabola *).^{ [54] } Archimedes also showed that the number of grains of sand filling the universe was not uncountable, devising his own counting scheme based on the myriad, which denoted 10,000 (* The Sand-Reckoner *).^{ [55] }

The most characteristic product of Greek mathematics may be the theory of conic sections, which was largely developed in the Hellenistic period, starting with the work of Menaechmus and perfected primarily under Apollonius in his work * Conics *.^{ [56] }^{ [57] }^{ [58] } The methods employed in these works made no explicit use of algebra, nor trigonometry, the latter appearing around the time of Hipparchus.^{ [59] }^{ [60] }

Ancient Greek mathematics was not limited to theoretical works but was also used in other activities, such as business transactions and in land mensuration, as evidenced by extant texts where computational procedures and practical considerations took more of a central role.^{ [9] }^{ [61] }

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.^{ [62] } 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.

Netz has counted 144 ancient authors in the mathematical or exact sciences, from whom only 29 works are extant in Greek: 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, and Eutocius.^{ [63] }

The following works are extant only in Arabic translations:^{ [64] }^{ [65] }

- 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*(extant in Latin from an Arabic translation of the Greek) - Ptolemy,
*Planisphaerium*

- Al-Mansur – 2nd Abbasid caliph (r. 754–775)
- Chronology of ancient Greek mathematicians
- Greek numerals – System of writing numbers
- History of geometry – Historical development of geometry
- History of mathematics
- Timeline of ancient Greek mathematicians – Timeline and summary of ancient Greek mathematicians and their discoveries

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**Archimedes of Syracuse** was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.

**Euclid** was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the *Elements* treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics.

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).

The **history of mathematics** deals with the origin of discoveries in mathematics and 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, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.

**Eudoxus of Cnidus** was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original 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.

**Conon of Samos** was a Greek astronomer and mathematician. He is primarily remembered for naming the constellation Coma Berenices.

**Archytas** was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist affiliated with the Pythagorean school and famous for being the reputed founder of mathematical mechanics and a friend of Plato.

**Euclid's Elements** is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates, propositions, and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines.

**Sir Thomas Little Heath** 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.

**Apollonius of Perga** was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. Gottfried Wilhelm Leibniz stated “He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.”

**Pappus of Alexandria** was one of the last great Greek mathematicians of antiquity; he is known for his *Synagoge* (Συναγωγή) or *Collection*, and for Pappus's hexagon theorem in projective geometry. Nothing is known of his life, other than what can be found in his own writings: that he had a son named Hermodorus, and was a teacher in Alexandria.

The timeline below shows the date of publication of possible major scientific breakthroughs, theories and discoveries, along with the discoverer. This article discounts mere speculation as discovery, although imperfect reasoned arguments, arguments based on elegance/simplicity, and numerically/experimentally verified conjectures qualify. The timeline begins at the Bronze Age, as it is difficult to give even estimates for the timing of events prior to this, such as of the discovery of counting, natural numbers and arithmetic.

**Science in classical antiquity** encompasses inquiries into the workings of the world or universe aimed at both practical goals as well as more abstract investigations belonging to natural philosophy. Classical antiquity is traditionally defined as the period between 8th century BC and the 6th century AD, and the ideas regarding nature that were theorized during this period were not limited to science but included myths as well as religion. Those who are now considered as the first scientists may have thought of themselves as natural philosophers, as practitioners of a skilled profession, or as followers of a religious tradition. Some of the more widely known figures active in this period include Hippocrates, Aristotle, Euclid, Archimedes, Hipparchus, Galen, and Ptolemy. Their contributions and commentaries spread throughout the Eastern, Islamic, and Latin worlds and contributed to the birth of modern science. Their works covered many different categories including mathematics, cosmology, medicine, and physics.

**Ancient Greek astronomy** is the astronomy written in the Greek language during classical antiquity. Greek astronomy is understood to include the Ancient Greek, Hellenistic, Greco-Roman, and Late Antiquity eras. It is not limited geographically to Greece or to ethnic Greeks, as the Greek language had become the language of scholarship throughout the Hellenistic world following the conquests of Alexander. This phase of Greek astronomy is also known as **Hellenistic astronomy**, while the pre-Hellenistic phase is known as **Classical Greek astronomy**. During the Hellenistic and Roman periods, much of the Greek and non-Greek astronomers working in the Greek tradition studied at the Museum and the Library of Alexandria in Ptolemaic Egypt.

Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.

This is a **timeline of mathematicians in ancient Greece**.

A timeline of **algebra** and **geometry**

* The Ancient Tradition of Geometric Problems* is a book on ancient Greek mathematics, focusing on three problems now known to be impossible if one uses only the straightedge and compass constructions favored by the Greek mathematicians: squaring the circle, doubling the cube, and trisecting the angle. It was written by Wilbur Knorr (1945–1997), a historian of mathematics, and published in 1986 by Birkhäuser. Dover Publications reprinted it in 1993.

* A History of Greek Mathematics* is a book by English historian of mathematics Thomas Heath about history of Greek mathematics. It was published in Oxford in 1921, in two volumes titled Volume I, From Thales to Euclid and Volume II, From Aristarchus to Diophantus. It got positive reviews and is still used today. Ten years later, in 1931, Heath published

- Boyer, Carl B. (1985),
*A History of Mathematics*, Princeton University Press, ISBN 978-0-691-02391-5 - Boyer, Carl B.; Merzbach, Uta C. (2011),
*A History of Mathematics*(3rd ed.), John Wiley & Sons, Inc., ISBN 978-0-471-54397-8 - Jean Christianidis, ed. (2004),
*Classics in the History of Greek Mathematics*, Kluwer Academic Publishers, ISBN 978-1-4020-0081-2 - Cooke, Roger (1997),
*The History of Mathematics: A Brief Course*, Wiley-Interscience, ISBN 978-0-471-18082-1 - Derbyshire, John (2006),
*Unknown Quantity: A Real And Imaginary History of Algebra*, Joseph Henry Press, ISBN 978-0-309-09657-7 - Stillwell, John (2004),
*Mathematics and its History*(2nd ed.), Springer Science + Business Media Inc., ISBN 978-0-387-95336-6 - Burton, David M. (1997),
*The History of Mathematics: An Introduction*(3rd ed.), The McGraw-Hill Companies, Inc., ISBN 978-0-07-009465-9 - Heath, Thomas Little (1981) [First published 1921],
*A History of Greek Mathematics*, Dover publications, ISBN 978-0-486-24073-2 - Heath, Thomas Little (2003) [First published 1931],
*A Manual of Greek Mathematics*, Dover publications, ISBN 978-0-486-43231-1 - Szabo, Arpad (1978) [First published 1978],
*The Beginnings of Greek Mathematics*, Reidel & Akademiai Kiado, ISBN 978-963-05-1416-3

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