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Mathematics in Civilization

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Guide to Discrete Mathematics

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This chapter considers the contributions of early civilizations to the computing field, including the achievements of the Babylonians, Egyptians, Greeks and Romans, and the Islamic world. The Babylonian civilization flourished in Mesopotamia (in modern Iraq) from about 2000 B.C. until about 500 B.C., and they made important contributions to mathematics. The Egyptian Civilization developed along the Nile from about 4000 B.C., and their knowledge of mathematics allowed them to construct the pyramids at Giza. The Greeks made major contributions to Western civilization including mathematics, logic and philosophy. The Golden Age of Islamic civilization was from 750 A.D. to 1250 A.D., and enlightened caliphs recognized the value of knowledge, and sponsored scholars to come to Baghdad to gather and translate the existing world knowledge into Arabic.

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  1. 1.

    Of course, it is essential that the population of the world moves towards more sustainable development to ensure the long-term survival of the planet for future generations. This involves finding technological and other solutions to reduce greenhouse gas emissions as well as moving to a carbon neutral way of life. The solution to the environmental issues will be a major challenge for the twenty-first century.

  2. 2.

    Tutankhamun was a minor Egyptian pharaoh who reigned after the controversial rule of Akhenaten. Tutankhamun’s tomb was discovered by Howard Carter in the Valley of the Kings, and the tomb was intact. The quality of the workmanship of the artefacts found in the tomb is extraordinary, and a visit to the Egyptian museum in Cairo is memorable.

  3. 3.

    The origin of the word ‘democracy’ is from demos (\(\updelta \upeta \upmu {\text{o}}\varsigma\)) meaning people and kratos (\({\upkappa \uprho \upalpha \uptau }{\text{o}}\varsigma\)) meaning rule. That is, it means rule by the people and it was introduced into Athens following the reforms introduced by Cleisthenes. He divided the Athenian city-state into 30 areas, where 20 of these areas were inland or along the coast and ten were in Attica itself. Fishermen lived mainly in the ten coastal areas; farmers in the ten inland areas; and various tradesmen in Attica. Cleisthenes introduced ten new clans where the members of each clan came from one coastal area, one inland area on one area in Attica. He then introduced a Boule (or assembly) which consisted of 500 members (50 from each clan). Each clan ruled for 1/10 th of the year.

  4. 4.

    The Athenian democracy involved the full participations of the citizens (i.e. the male adult members of the city-state who were not slaves), whereas in representative democracy the citizens elect representatives to rule and represent their interests. The Athenian democracy was chaotic and could be easily influenced by individuals who were skilled in rhetoric. There were teachers (known as the Sophists) who taught wealthy citizens rhetoric in return for a fee. The origin of the word ‘sophist’ is the Greek word \(\upsigma{\text{o}}\upvarphi {\text{o}}\varsigma \) meaning wisdom, and one of the most well known of the sophists was Protagoras. The problems with the Athenian democracy led philosophers such as Plato to consider alternate solutions such as rule by philosopher kings. This totalitarian utopian state is described in Plato’s Republic.

  5. 5.

    The Elgin marbles are named after Lord Elgin who was the British ambassador to Greece (which was then part of the Ottoman Empire), and he removed them (at his own expense) over several years from the Parthenon in Athens to London during the first decade of the nineteenth century. The marbles show the Pan-Athenaic festival that was held in Athens in honour of the goddess Athena after whom Athens is named.

  6. 6.

    The origin of the word Hellenistic is from Hellene (‘\({\text{E}}{\uplambda \uplambda \upeta \upnu }\)) meaning Greek.

  7. 7.

    The hanging gardens of Babylon were one of the seven wonders of the ancient world.

  8. 8.

    Henry Rawlinson played an important role in the decipherment of cuneiforms, and especially for making a copy of the large Behistun inscription (the equivalent of the Rosetta stone for Assyriologists) that recorded the victory of the Persian king Darius I over those who rebelled against him in three languages. Edward Hicks and others played a key role in deciphering the inscription.

  9. 9.

    A positional numbering system is a number system where each position is related to the next by a constant multiplier. The decimal system is an example: e.g. 546 = 5* 102 + 4* 101 + 6.

  10. 10.

    The decorations of the tombs in the Valley of the Kings record the life of the pharaoh including his exploits and successes in battle.

  11. 11.

    The cartouche surrounded a group of hieroglyphic symbols enclosed by an oval shape. Champollion’s insight was that the group of hieroglyphic symbols represented the name of the Ptolemaic pharaoh “Ptolemy”.

  12. 12.

    The Rhind papyrus is sometimes referred to as the Ahmes papyrus in honour of the scribe who wrote it in 1832 B.C.

  13. 13.

    The length of a side of the bottom base of the pyramid is b1, and the length of a side of the top base is b2..

  14. 14.

    Plato’s Republic describes his utopian state, and seems to be based on the austere Spartan model.

  15. 15.

    The Pythagoreans took a vow of silence with respect to the discovery of incommensurable numbers. However, one member of the society is said to have shared the secret result with others outside the sect, and an apocryphal account is that he was thrown into a lake for his betrayal and drowned. The Pythagoreans obviously took Mathematics seriously back then.

  16. 16.

    The library in Alexandria is predated by the Royal library of Ashurbanipal which was established in Nineveh (the capital of the Assyrian Empire) in the seventh century B.C. The latter contained over 30,000 cuneiform tablets including the famous “Epic of Gilgamesh”, and the laws in Hammurabi’s code.

  17. 17.

    The town of Aswan is famous today for the Aswan high dam, which was built in the 1960s. There was an older Aswan dam built by the British in the late nineteenth century. The new dam led to a rise in the water level of Lake Nasser and flooding of archaeological sites along the Nile. Several archaeological sites such as Abu Simbel and the temple of Philae were relocated to higher ground.

  18. 18.

    Syracuse is located on the island of Sicily in Southern Italy.

  19. 19.

    The origin of the word “odometer” is from the Greek words ‘\({\text{o}}\updelta {\text{o}}\upzeta \)’ (meaning journey) and \({\upmu \upvarepsilon \uptau \uprho }{\text{o}}{\upnu }\) meaning (measure).

  20. 20.

    The figures given here are for the distance of one Roman mile. This is given by \(\uppi {4}*{4}00 = {12}.{56}*{4}00 = {5}0{24}\) (which is less than 5280 feet for a standard mile in the Imperial system).

  21. 21.

    Socrates was a moral philosopher who deeply influenced Plato. His method of enquiry into philosophical problems and ethics was by questioning. Socrates himself maintained that he knew nothing (Socratic ignorance). However, from his questioning it became apparent that those who thought they were clever were not really that clever after all. His approach obviously would not have made him very popular with the citizens of Athens. Chaerephon (a friend of Socrates) had consulted the oracle at Delphi to find out who the wisest of all men was, and he was informed that there was no one wiser than Socrates. Socrates was sentenced to death for allegedly corrupting the youth of Athens, and the sentence was carried out by Socrates being forced to take hemlock (a type of poison). The juice of the hemlock plant was prepared for Socrates to drink.

  22. 22.

    Chrysippus was the head of the Stoics in the third century B.C.

  23. 23.

    Aquinas’ (or St. Thomas’) most famous work is Summa Theologica.

  24. 24.

    The Aeneid by Virgil suggests that the Romans were descended from survivors of the Trojan war, and that Aeneas brought surviving Trojans to Rome after the fall of Troy.

  25. 25.

    Carthage was located in Tunisia, and the wars between Rome and Carthage are known as the Punic wars. Hannibal was one of the great Carthaginian military commanders, and during the second Punic war, he brought his army to Spain, marched through Spain and crossed the Pyrenees. He then marched along southern France and crossed the Alps into Northern Italy. His army also consisted of war elephants. Rome finally defeated Carthage and destroyed the city.

  26. 26.

    The Celtic period commenced around 1000 B.C. in Hallstatt (near Salzburg in Austria). The Celts were skilled in working with Iron and Bronze, and they gradually expanded into Europe. They eventually reached Britain and Ireland around 600 B.C. The early Celtic period was known as the “Hallstaat period” and the later Celtic period is known as “La Téne”. The later La Téne period is characterised by the quality of ornamentation produced. The Celtic museum in Hallein in Austria provides valuable information and artefacts on the Celtic period. The Celtic language would have similarities to the Irish language. However, the Celts did not employ writing, and the Ogham writing used in Ireland was developed in the early Christian period.

  27. 27.

    The origin of the word ‘Calculus’ is from Latin and means a small stone or pebble used for counting.

  28. 28.

    Modular arithmetic is discussed in chapter seven.

  29. 29.

    Augustus was the first Roman emperor, and his reign ushered in a period of peace and stability following the bitter civil wars. He was the adopted son of Julius Caesar and was called Octavian before he became emperor. The earlier civil wars were between Caesar and Pompey, and following Caesar’s assassination civil war broke out between Mark Anthony and Octavian. Octavian defeated Anthony and Cleopatra at the battle of Actium, and became the first Roman emperor, Augustus.

  30. 30.

    The origin of the word ‘Moor’ is from the Greek work \(\upmu \upupsilon {\text{o}}\uprho {\text{o}}\upzeta \) meaning very dark. It referred to the fact that many of the original Moors who came to Spain were from Egypt, Tunisia and other parts of North Africa.

  31. 31.

    The Moorish influence includes the construction of various castles (alcazar), fortresses (alcalzaba) and mosques. One of the most striking Islamic sites in Spain is the palace of Alhambra in Granada, and it represents the zenith of Islamic art.

  32. 32.

    The Catholic Monarchs refer to Ferdinand of Aragon and Isabella of Castille who married in 1469. They captured Granada (the last remaining part of Spain controlled by the Moors) in 1492.


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O’Regan, G. (2021). Mathematics in Civilization. In: Guide to Discrete Mathematics. Texts in Computer Science. Springer, Cham.

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