MathsInfo

MathematicsInfo

III

MEDIEVAL AND RENAISSANCE MATHEMATICS

Following the time of Ptolemy, a tradition of study of the mathematical masterpieces of the preceding centuries was established in various centers of Greek learning. The preservation of such works as have survived to modern times began with this tradition. It was continued in the Islamic world, where original developments based on these masterpieces first appeared.

Islamic and Indian Mathematics

After a century of expansion in which the religion of Islam spread from its beginnings in the Arabian Peninsula to dominate an area extending from Spain to the borders of China, Muslims began to acquire the results of the “foreign sciences.” At centers such as the House of Wisdom in Baghdād, supported by the ruling caliphs and wealthy individuals, translators produced Arabic versions of Greek and Indian mathematical works.

By the year 900 ad the acquisition was complete, and Muslim scholars began to build on what they had acquired. Thus mathematicians extended the Hindu decimal positional system of arithmetic from whole numbers to include decimal fractions, and the 12th-century Persian mathematician Omar Khayyam generalized Hindu methods for extracting square and cube roots to include fourth, fifth, and higher roots. In algebra, al-Karaji completed the algebra of polynomials of Muhammad ibn Mūsā al-Khwārizmī. Al-Karaji included polynomials with an infinite number of terms. (Al-Khwārizmī's name, incidentally, is the source of the word algorithm, and the title of one of his books is the source of the word algebra.) Geometers such as Ibrahim ibn Sinan continued Archimedes' investigations of areas and volumes, and Kamal al-Din and others applied the theory of conic sections to solve optical problems. Using the Hindu sine function and Menelaus's theorem, mathematicians from Habas al-Hasib to Nasir ad-Din at-Tusi created the mathematical disciplines of plane and spherical trigonometry. These did not become mathematical disciplines in the West, however, until the publication of De Triangulis Omnimodibus by the German astronomer Regiomontanus.

Finally, a number of Muslim mathematicians made important discoveries in the theory of numbers, while others explained a variety of numerical methods for solving equations. The Latin West acquired much of this learning during the 12th century, the great century of translation. Together with translations of the Greek classics, these Muslim works were responsible for the growth of mathematics in the West during the late Middle Ages. Italian mathematicians such as Leonardo Fibonacci and Luca Pacioli, one of the many 15th-century writers on algebra and arithmetic for merchants, depended heavily on Arabic sources for their knowledge.

Western Renaissance Mathematics

Although the late medieval period saw some fruitful mathematical considerations of problems of infinity by writers such as Nicole Oresme, it was not until the early 16th century that a truly important mathematical discovery was made in the West. The discovery, an algebraic formula for the solution of both the cubic and quartic equations, was published in 1545 by the Italian mathematician Gerolamo Cardano in his Ars Magna. The discovery drew the attention of mathematicians to complex numbers and stimulated a search for solutions to equations of degree higher than 4. It was this search, in turn, that led to the first work on group theory at the end of the 18th century, and to the theory of equations developed by the French mathematician Évariste Galois in the early 19th century.

The 16th century also saw the beginnings of modern algebraic symbolism, as well as the remarkable work on the solution of equations by the French mathematician François Viète. His writings influenced many mathematicians of the following century, including Pierre de Fermat in France and Isaac Newton in England.