The roots of modern maths - from the Middle Ages to the Enlightenment - A Mathematical Mystery Tour

UNESCO Courier, Nov, 1989 by Catherine Goldstein, Jeremy Gray

THE mathematics teaching offered by the European universities around the time when they were founded in the late twelfth and early thirteenth centuries, was largely inherited from the traditions of classical Antiquity. At Paris, Oxford and Bologna, the curriculum covered the quadrivium of arithmetic, geometry, music and astronomy. However as the study of divinity, leading to the elucidation of the basic Christian texts, was considered to be of more importance, the arts faculties, which ran mathematics courses, enjoyed much less prestige than the faculties of law, medicine and above all theology.

The content of maths courses was rudimentary, consisting of the elementary principles of arithmetic and the first books of Euclid's treatise on geometry, the Elements. Principles of motion were also studied in the works of Aristotle, and at Oxford and Paris, mathematical methods were beginning to be applied to the study of natural phenomena. The fourteenth-century French bishop and economist Nicole Oresme, for example, attempted to make a graphical representation of variations in speed in a given period of time in certain types of movement.

Measuring the land and charting the heavens

In the late Middle Ages, mathematics was being used by two distinct groups of people, in some cases graduates of the same universities: on the one hand there were tradesmen such as surveyors, engineers and accountants, and on the other there were physicians and astrologers, who had a higher social status.

Economic change in the fourteenth century, the development of the towns and of trade, favoured the rise of mathematicians in society. The increasing complexity of trade called for specialized calculating and accounting skills. Mathematicians gave consultations in "abacus shops", where they drafted contracts and calculations in layman's terms, and resolved banking problems concerning interest rates, the exchange, circulation and value of currency, and the distribution of profits.

In order to make a decent living, these tradesmen had to prove their worth against their competitors and convince their employers or clients that their skills and know-how were useful. Their mastery of the algebraic techniques borrowed from the Arab world was one reason for their professional success.

At first, algebra was not formulated as it is today, by using symbols to express relationships between numbers. it served instead to classify the possible relationships between unknown values and to describe general methods of determining them. These methods were usually set out in treatises as practical examples. The adoption of a system of decimal notation in Europe was both part of and a great incentive to the advancement of mathematics. Texts packed with approximations of roots and complex fractions proliferated, their publication facilitated by the advent of printing.

Thus the growth of mathematics as a skilled profession, whose prestige was linked to that of its chief clients, the merchants, was a corollary of changing economic and social priorities.

New perspectives

Not only the financial world was in a state of upheaval. A number of practical inventions, such as the compass and binoculars, appeared in the West towards the end of the Middle Ages, and descriptions of measuring instruments and how they worked were sometimes published with texts on algebra. The voyages of discovery, in fact all commercial sea traffic, popularized the use of the astrolabe and the quadrant. Part practical, part theoretical treatises on perspective were written by artists, cartographers and architects, associated with what is now known as geometrical optics. There was no hard and fast division between mathematical theorists and practitioners.

The late fifteenth century saw the rise of the Humanist movement, which concentrated on the study and dissemination of the texts of classical Antiquity. At first the movement did not really favour mathematical methods. If Humanist scholars in the late fifteenth century showed any interest in arithmetic, it was because of their study of Plato or of Pythagoras, but on the whole they despised arithmetical calculation.

Intellectual life, of which the Reformation was one of the most spectacular manifestations, was more concerned with the study of texts than contact with craftsmen. Humanist scholars produced a new batch of translations of key works of Antiquity, from Arabic, then directly from Greek. In going back to the Greek geometers, they were responsible for a revival of interest in mathematics.

The mathematical heritage of Antiquity flourished because it did not fall on barren soil. European rulers increasingly sought the services of scientists, such as the astronomers Tycho Brahe and Johannes Kepler who were attached to the court of Rudolf II in Prague, not only to cast their horoscopes but to construct fortifications or resolve ballistic problems as interest grew in the application of mathematical skills to warfare. Mathematics began to find a place in the largely Humanist education of the aristocracy, and some military engineers were even ennobled as a reward for their services. The growing interest in mathematics in the sixteenth century is typified in the works of such men as john Dee, the English alchemist and astrologer who wrote the preface to the first English translation of Euclid; Gerolamo Cardano, the Italian physician and astrologer whose book on the "Great Art" is one of the cornerstones of the history of algebra; or Francois Viete, the French jurist who specialized in cryptography and introduced the first systematic algebraic notation.