Overview: Mathematics 700-1449

During the classical period of mathematics in the ancient world, many fundamental branches of mathematics originated and were developed to a remarkable degree. However, due to various catastrophes, much of this ancient mathematical learning was lost, particularly in medieval Europe. The period between 700 and 1449 was a time of recovery for European mathematics, with many ancient texts restored, copied, and translated. However, it was also a period of innovation in many other parts of the world, and the mathematical knowledge of many cultures were exchanged. By chance, as well as design, European mathematics was to become the greatest benefactor of this process of recovery and melding, and social and cultural forces would give European mathematics the impetus to rapidly develop in the following centuries.

Mathematics has a long history, dating back at least as far as the Mesopotamian culture (in modern-day Iraq) over 4,000 years ago. Soon after, many other ancient cultures also began to invent addition, multiplication, and develop their mathematics. The Babylonians created a number system, sexagesimaal (base-60), developed a practical geometry, elementary algebra-like calculations, and fractions. The Egyptians used a decimal system (base-10), and developed a practical geometry that enabled them to construct and determine the areas of many simple figures, most famously demonstrated by the construction of the pyramids. Greek mathematics went beyond the practical elements of other ancient traditions and looked at abstract, philosophical, and mystical aspects of numbers. Greek scholars developed mathematics to levels unsurpassed for many centuries. The influence of Greek ideas was so dominant that many of the mistakes in their writings went unquestioned by later thinkers. The Roman Empire was primarily concerned with practical forms of mathematics, but preserved Greek writings. However, with the collapse of the Roman Empire in the fourth century much of the ancient mathematical knowledge of the Greeks was destroyed, lost, or scattered.

In other parts of the world mathematics was still actively studied. Chinese mathematics began to advance greatly from the seventh century on. Cubic and quadratic equations, as well as astronomical and applied mathematics, were significantly developed. In the thirteenth century, a number of mathematicians wrote important summaries of Chinese mathematics. Ch'in Chiu-Shao (1202?-1261?) wrote on the solution of numerical equations. Yang Hui (c. 1299) recorded work on arithmetical progressions, proportions, simultaneous linear equations, quadratic equations, and others areas. Chu Shih-Chieh (fl. 1280-1303) summarized many earlier discoveries in his writings. The Chinese did not study in isolation, absorbing many Hindu ideas and influencing Japanese mathematics. However, after the thirteenth century a number of political and social factors discouraged further mathematical development in this region.

On the Indian subcontinent there was also a long tradition of mathematical study. Mathematics was mainly studied as an aid to astronomy, and there were also mystical and poetical elements. There is evidence of a strong Greek influence on early Hindu mathematics, but the stress was on arithmetic, rather than geometry. Hindu innovation and development appears to have peaked before 700, but there were a number of important books and individuals in the following centuries. The writings of mathematicians such as Mahavira (c. 850) and Sridhara (c. 850) display the wide range and complexity of Hindu mathematics, including fractions, finding squares and cubes, and the calculation of areas. In the twelfth century Atscharja Bhaskara's (1114-1185) work was only a little more advanced than earlier texts, and represents the last great summary of Hindu mathematics before the Muslim conquest of India. While Hindu mathematics waned, the influence of their developments spread far and wide. Most famously, the numeric system we use today is derived from Indian mathematics, as is the concept of the zero.

In the years between 700 and 1449, the most dynamic and flourishing mathematical tradition was that of the Arab world. The expansion of the Islamic faith, by conquest and conversion, went as far west as Spain, and as far east as India. Arab rulers encouraged specialists of all types, and their courts became centers of learning and research in medicine, astronomy, mathematics, and other disciplines. Arab scholars were ideally placed at the crossroads of many other mathematical cultures, and they absorbed ideas from the Babylonians, the Greeks, the Hindus, and others. Muhammad ibn-Musa al-Khwarizmi (780?-850?) helped popularize Hindu mathematical concepts such as the zero, fractions, and the Hindu numeral system. His writing was so influential that when Hindu numerals were introduced to Europe, by way of translations of his work, they were called Arabic numbers. Al-Khwarizmi also gave us the word algebra, and his early work in this field was carried on by many later court mathematicians, such as al-Karkhi (c. 1020). The influence of Greek mathematics led to developments in the fields of geometry, astronomy, and trigonometry by scholars such as Abu al-Wafa (940-998).

The city of Baghdad was the earliest center of Arab mathematical study, but with the expansion of the Muslim world other important centers sprang up in Egypt, Morocco, and Spain. In Egypt, important work on the volumes of paraboloids was done. Moroccan mathematicians excelled in the fields of conics and astronomy. But by far the biggest Arab center of learning outside Baghdad was in Cordova, Spain. Schools and libraries were founded there in the tenth century, and Spain became the main point of contact for Arab and European scholars. However, by the end of the fifteenth century the Spanish territories had been lost, and political and social concerns in the Arab world led to a decline in mathematical research.

In Europe little mathematical knowledge remained after the fall of Rome. Over the centuries more texts were lost, to disasters and the decay of time. European mathematicians came to rely on a small collection of Latin translations. Translators, such as Boethius (480-524), simplified the ancient Greek texts, leaving out difficult material, which often included explanatory figures, numbers, and calculations, making the texts hard to follow. In the Middles Ages mathematical studies included arithmetic, music, geometry, and astronomy, but were not popular subjects. However, as the economic and political stability of Europe improved, European society developed a new need for mathematics. The lost ancient mathematical texts were reintroduced by contact with the Arab world. However, many of these recovered Greek writings suffered from the multiple translation process they had undergone over the centuries, from Greek, to Arabic, and finally to Latin. Yet slowly mathematical knowledge caught up with that of the ancient Greeks.

New innovations were also introduced from the East, though not without resistance. Gerbert of Aurillac (946?-1003) attempted to introduce the abacus and the Hindu-Arabic numeral system. Later attempts to popularize the Hindu-Arabic numeral system by Adelard of Bath (1090-1150) and Leonardo of Pisa (Fibonacci) (1175-1250) also met with resistance, but eventually they replaced Roman numerals due to the ease and quickness of calculation they afforded.

New fields of study were created in Europe, such as the study of observed motion (kinematics). A number of fourteenth-century scholars used mathematical ideas to help describe physical actions. Thomas Bradwardine (1295-1349) wrote on accelerated motion, instantaneous velocity, and force, and his work was expanded upon by Nicole of Oresme (1323-1382), and others. From this simple beginning, the eventual integration of physics and mathematics would become a cornerstone of modern scientific knowledge.

European trade began to grow rapidly late in the Middle Ages, producing a need for more numerate clerks and administrators. New innovations, such as the plus (+) and minus (-) signs were introduced to make bookkeeping easier. Mathematics was also pursued for pleasure. Recreational mathematics provided intellectual entertainment, and helped popularize the study of numbers. Even chess was considered a mathematical pursuit, and one worthy of the noblest gentleman.

The years immediately after 1450 saw Europeans rapidly develop many areas of learning, including mathematics. The Renaissance, as this period has come to be known, is considered by many the beginning of the modern period of world history, with advances in science and technology that enabled European domination of much of the globe. However, without the recovery of ancient knowledge that occurred in the years between 700 and 1449, the spectacular developments of the Renaissance could not have occurred. Through contact with other cultures, and especially the absorption of Arab ideas and innovations, European learning in fields such as mathematics was able to go beyond the work of ancient scholars. New fields of study unknown to the Greeks were opened, leading to such developments as the calculus of Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716), which would revolutionize both mathematics and science.

DAVID TULLOCH