The Return of Mathematics to Europe

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The Return of Mathematics to Europe


Early medieval mathematics was based on only a few classical texts, since the majority of ancient mathematical knowledge had been lost after the fall of Rome. Slowly, over many centuries, these texts were reintroduced to Europe through contact with Arab mathematicians, who had preserved and extended classical learning. Social and economic changes in Europe created a demand for a newly sophisticated mathematical learning.


Mathematics flourished in the Greek world from 600 b.c. to a.d. 300 in what has been called the Golden Age of Mathematics. The rise of the Roman Empire saw mathematical philosophy take a back seat to practical methods, since the Romans, in general, preferred language studies to abstract mathematics, but Greek learning was still preserved and studied.

With the fall of Rome and the collapse of the empire in the fourth century, however, many ancient mathematical works were lost or destroyed. Europe endured a period of anarchy and political fragmentation. Trade became localized, and towns and cities shrank in importance and size. These economic and political changes all reduced the role of mathematics in society.

What little knowledge from the ancient world remained was preserved in the Byzantine Empire (roughly modern-day Turkey), or in monasteries scattered across Europe where they were stored and copied over the centuries. While the Byzantine Empire used little of this knowledge themselves, they shared it with neighboring Arab lands; in this way, much of Greek learning was translated into Arabic.

In order to preserve these ancient texts over many centuries as parchment aged and crumbled, frequent copies had to be made. In Europe monks tended to concentrate their efforts on theological and philosophical texts, not mathematical or scientific ones; as a result, many great works literally crumbled into dust. Scholars came to rely on a small selection of Latin texts, many compiled into large encyclopedias in the fifth and sixth centuries by scholars such as Boethius (480-524). These large collections simplified complicated concepts for the European audience, which often meant deleting mathematical figures and calculations. In addition, medieval writers frequently did not use numbers, so even books on technical subjects such as glassmaking and jewelry tended to contain nonspecific quantities, such as "a medium-sized piece" or "a bit more."

This kind of simplification was necessary because education in the Middle Ages contained almost no higher mathematics. Although arithmetic was taught as part of the seven liberal arts—the quadrivium (arithmetic, music, geometry, and astronomy) and the trivium, (grammar, rhetoric, and dialectic)—it was merely the theory of numbers, not the calculation of problems we associate with the subject today. Medieval mathematicians focused on basic properties, such as odd and even numbers, ratios, proportions, and the harmony of numbers. Addition, subtraction, multiplication, and division were separate subjects, collectively called computus, which were rarely taught.

Furthermore, the chaotic political and economic situation in Europe, combined with the preference for religious and philosophical texts, severely limited the scope and application of mathematical ideas in medieval society. Mathematical computation was restricted to that needed for the small-scale trade of the era. Over time, however, the economic and political stability of Europe began to improve, and mathematics slowly revived to meet the needs of the changing society.


One of the major factors in medieval life was the church. The Christian tradition was ambivalent toward numbers. Parts of the Bible seemed to support mathematics, such as the use of apocalyptic numbers in Daniel and Revelation. However, there were also some sections that appeared hostile. An abbot in 1130 stopped his men counting their provisions to see if they would survive an impending crisis, as this would suggest they did not trust God to see them safely through. He referred them to the biblical story of King David, who was punished for counting the people in his kingdom.

However, two religious trends helped spur the return of mathematics: the revival of numeric apocalyptic prophecy, and the need to calculate the correct date for Easter. Calendar reform became a major religious issue in the fifth century, and many popes recruited the best mathematical minds from across Europe to ensure that Christ's resurrection would be celebrated on the correct date.

The reign of Charlemagne (742-814) produced a short-lived revival of numerical knowledge, as well as increased trade and localized political stability. Irish monks, whose libraries of ancient texts were well-preserved, brought their knowledge to Charlemagne's court, and scholars from across Europe flocked there to study mathematics. However, Viking invasions ended the calm, and few texts survived to influence later scholars. One important lasting innovation did come from the so-called Carolingian Renaissance: the foundation of cathedral and monastery schools across Europe, some of which evolved into universities.

Through contact with the Arab world, Europeans rediscovered the mathematical heritage of the ancients, and added a few discoveries of their own. However, it was an uneven and frustratingly slow process. Gerbert of Aurillac (946-1003) attempted to introduce a number of Eastern innovations into European mathematics in the tenth century, but with limited success. Gerbert studied in southern Spain, then in Islamic lands. He learned Arabic, and was particularly impressed with the abacus and the Hindu-Arabic numeral system, the basis of the numerals we use today. Gerbert brought these innovations back to Italy.

By contrast, the European abacus (or counting board) that was in use at the time had severe limitations, and was much slower than the string-bead abacus that can still be seen in use in the Middle East today. By its very nature the abacus discouraged mathematical writing, as computations were confined to a particular time and space, and the transitory steps of calculation were not recorded, only the result. Very little information on the medieval abacus remains today, and only a few examples survive.

Hindu-Arabic numerals did not prove popular, despite later attempts by individuals such as such as Adelard of Bath (1090-1150) and Leonardo of Pisa, also known as Fibonacci, (1175-1250) to popularize them. The Hindu-Arabic system included the number zero, which caused some philosophical problems for many Europeans. It appeared to represent a mystical quantity, entirely abstract and somewhat frightening. Some condemned the zero as heretical. Europeans preferred to use Roman numerals until the sixteenth century, although strange blends of both systems did occur. The maker of one calendar in 1430 wrote the length of a year as "ccc and sixty days and 5 and sex odde howres." Even more confusing was an inscription of IVOII to represent the 1502. Europeans were also slow to recognize the usefulness of mathematical symbols, like +, -, and =.

In the twelfth century the Crusades accelerated exposure to the Arab world. Many important trade and intellectual contacts were made, and more lost Greek mathematical texts were found and translated into Latin. These works, however, often suffered from multiple translations from Greek to Arabic to Latin, and many errors had crept in. It was also a large task, and it took a handful of dedicated translators until the middle of the fifteenth century to complete it. Copies were few, and extremely expensive, as they had to be painstakingly handwritten. Only with the late-fifteenth century printing revolution was the continued survival of the ancient mathematical texts ensured, with cheap, plentiful copies.

Individuals attempted and failed to introduce mathematical innovations because mathematics had little relevance to European life. However, as the political, economic, and social structures developed a new need for mathematics, mathematical learning began to emerge. The rise of commerce fueled a need for more numerate clerks and scribes in the business sector. There was a corresponding rise in the importance of trading centers, and strategic urban towns began to grow rapidly. Towns became the focal points of the training and employment of the newly numerate. The relative peace and stability of Europe of the twelfth century also led to rise in political administration, and a new interest in counting everything from money to soldiers. Prosperous times helped fund new buildings, and architecture absorbed the ancient Greek mathematical models, giving rise to the splendor of medieval cathedrals.

Mathematics began to invade all areas of medieval life. Clocks began to divide the day into regular intervals. Alchemists used numbers for the supposed mystical properties they contained. By the end of the thirteenth century mathematics was reentering medieval life apace. New translations recovered much that had been lost a thousand years before, and new ideas were discovered in Arabic and Hindu scholarship. The abacus helped revive the art of calculation, and the teaching of mathematics was being demanded by many sectors of medieval society.

However, the growth of the economy and of urban centers was dramatically interrupted in the fourteenth century, with wars and the Black Death killing as much as one third of the European population, and stunting the intellectual revival. Yet the impulses that had begun the revival of mathematics still remained, and with a new period of relative peace and stability from the mid-fifteenth century mathematics once again flourished. Mathematics spread into more fields, such as the application of geometry in painting to produce perspective. The development of printing ensured that ancient learning could no longer be lost through lack of copies. The mathematicians of the Renaissance would later help forge the way for the eventual marriage of mathematics and science; one of the fundamental characteristics of the modern age.


Further Reading

Burton, David M. The History of Mathematics: An Introduction. Allyn and Bacon, Inc., Newton, Massachusetts, 1985.

Cooke, Roger. The History of Mathematics: A Brief Course. John Wiley and Sons, Inc, New York, 1997.

Crosby, Alfred W. The Measure of Reality, Quantification and Western Society, 1250-1600. Cambridge University Press, Cambridge, 1997.

Murray, Alexander. Reason and Society in the Middle Ages. Clarendon Press, Oxford, 1978.

Wagner, David L. (ed.). The Seven Liberal Arts in the Middle Ages. Indiana University Press, Bloomington, 1983.

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The Return of Mathematics to Europe

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The Return of Mathematics to Europe