Greek mathematics

Last updated
An illustration of Euclid's proof of the Pythagorean theorem. Pythagoras Euclid.svg
An illustration of Euclid's proof of the Pythagorean theorem.

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 Ancient Greek : μάθημα , romanized: máthēma Attic Greek:  [má.tʰɛː.ma] Koine Greek:  [ˈma.θ] , meaning "subject of instruction". [1] 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. [2]


Origins of Greek mathematics

The origin of Greek mathematics is not well documented. [3] 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. [3] Between 800 BC and 600 BC, Greek mathematics generally lagged behind Greek literature,[ clarification needed ] and very little is known about Greek mathematics from this period—nearly all of which was passed down through later authors, beginning in the mid-4th century BC. [4]

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. The two earliest mathematical theorems, Thales' theorem and the 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.

Statue of Euclid in the Oxford University Museum of Natural History EuclidStatueOxford.jpg
Statue of Euclid in the Oxford University Museum of Natural History

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, [4] [5] 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[ by whom? ] to have been coined by Pythagoras. From this love of knowledge came many achievements. It has been customarily said[ by whom? ] 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. [6] 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. [4] [5]

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 Roman period, represented by scholars such as Hipparchus, Apollonius and Ptolemy, who were able to construct 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). [7]

Most of the mathematical texts written in Greek have been found in Greece, Egypt, Asia Minor, Mesopotamia, and Sicily.

The Antikythera mechanism, an ancient mechanical calculator. NAMA Machine d'Anticythere 1.jpg
The Antikythera mechanism, an ancient mechanical calculator.

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 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 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 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, applied mathematics, and, at times, approached close to integral calculus.

Euclid, fl. 300 BC, collected the mathematical knowledge of his age in the Elements , a canon of geometry and elementary number theory for many centuries.

The most characteristic product of Greek mathematics may be the theory of conic sections, which was largely developed in the Hellenistic period. The methods used made no explicit use of algebra, nor trigonometry.

Eudoxus of Cnidus developed a theory of real numbers strikingly similar to the modern theory of the Dedekind cut, developed by Richard Dedekind, who acknowledged Eudoxus as inspiration. [8]

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. [9] 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.

See also


  1. Heath (1931). "A Manual of Greek Mathematics". Nature. 128 (3235): 5. Bibcode:1931Natur.128..739T. doi:10.1038/128739a0.
  2. Boyer, C.B. (1991), A History of Mathematics (2nd ed.), New York: Wiley, ISBN   0-471-09763-2. p. 48
  3. 1 2 Hodgkin, Luke (2005). "Greeks and origins". A History of Mathematics: From Mesopotamia to Modernity . Oxford University Press. ISBN   978-0-19-852937-8.
  4. 1 2 3 Boyer & Merzbach (1991) pp. 43–61
  5. 1 2 Heath (2003) pp. 36–111
  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. ISBN   0-691-00659-8.
  8. J J O'Connor and E F Robertson (April 1999). "Eudoxus of Cnidus". The MacTutor History of Mathematics archive. University of St. Andrews. Retrieved 18 April 2011.
  9. J J O'Connor and E F Robertson (October 1999). "How do we know about Greek mathematics?". The MacTutor History of Mathematics archive. University of St. Andrews. Retrieved 18 April 2011.

Related Research Articles

Euclid Greek mathematician, inventor of axiomatic geometry

Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I. His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour.

History of geometry Historical development of geometry

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).

History of mathematics Aspect of history

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 patterns in nature, the field of astronomy and to record time/formulate calendars.

Mathematics Field of study

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

Number theory Branch of mathematics

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers or defined as generalizations of the integers.

Pythagoras 6th century BC Ionian Greek philosopher and mystic

Pythagoras of Samos was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a gem-engraver on the island of Samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated for complete vegetarianism.

Thales of Miletus Ancient Greek philosopher and mathematician

Thales of Miletus was a Greek mathematician, astronomer and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him as the first philosopher in the Greek tradition, and he is otherwise historically recognized as the first individual in Western civilization known to have entertained and engaged in scientific philosophy.

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

Euclids <i>Elements</i> Mathematical treatise by Euclid

The 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. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century.

Thaless theorem Angle formed by a point on a circle and the 2 ends of a diameter is a right angle

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, who is said to have offered an ox, probably to the god Apollo, as a sacrifice of thanksgiving for the discovery, but it is sometimes attributed to Pythagoras.

The timeline below shows the date of publication of possible major scientific breakthroughs, theories and discoveries, along with the discoverer. For the purposes of this article, we do not regard mere speculation as discovery, although imperfect reasoned arguments, arguments based on elegance/simplicity, and numerically/experimentally verified conjectures qualify. We begin our timeline at the Bronze Age, as it is difficult to estimate the timeline before this point, such as of the discovery of counting, natural numbers and arithmetic.

Nicomachus of Gerasa was an important ancient mathematician best known for his works Introduction to Arithmetic and Manual of Harmonics in Greek. He was born in Gerasa, in the Roman province of Syria. He was a Neopythagorean, who wrote about the mystical properties of numbers.


Hippasus of Metapontum 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 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.

The history of science in early cultures refers to the study of protoscience in ancient history, prior to the development of science in the Middle Ages. In prehistoric times, advice and knowledge was passed from generation to generation in an oral tradition. The development of writing enabled knowledge to be stored and communicated across generations with much greater fidelity. Combined with the development of agriculture, which allowed for a surplus of food, it became possible for early civilizations to develop and spend more of their time devoted to tasks other than survival, such as the search for knowledge for knowledge's sake.

Ancient Greek astronomy

Greek astronomy is astronomy written in the Greek language in 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 Musaeum and the Library of Alexandria in Ptolemaic Egypt.

History of trigonometry

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler (1748).

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.

Geometry Branch of mathematics

Geometry is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.

Pythagorean theorem Equation relating the side lengths of a right triangle

In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":