Mathematics

From the decline of the Roman Empire (including Greece) Euclid was quiescent mathematically, though the Carolingian kingdom inspired some work, at least in education. The revival dates from around the late twelfth century, when universities also began to be formed. The major source for mathematics was Latin translations of Greek and Arabic writings (and re-editions of Roman writers, especially Boethius). In addition, the Italian Leonardo Fibonacci (c. 1170–c. 1240) produced a lengthy Liber Abbaci in 1202 that reported in Latin many parts of Arabic arithmetic and algebra (including the Indian numerals); his book was influential, though perhaps less than is commonly thought. The Italian peninsula was then the most powerful region of Europe, and much commercial and "research" mathematics was produced there; the German states and the British Isles also came to boast some eminent figures. In addition, a somewhat distinct Hebrew tradition arose—for example, in probability theory.

A competition developed between two different methods of reckoning. The tradition was to represent numbers by placing pebbles (in Latin, calculi) in determined positions on a flat surface (in Latin, abacus, with one b), and to add and subtract by moving the pebbles according to given rules. However, with the new numerals came a rival procedure of calculating on paper, which gradually supervened; for, as well as also allowing multiplication and division, the practitioner could show and check his working, an important facility unavailable to movers of pebbles.

Mathematics rapidly profited from the invention of printing in the late fifteenth century; not only were there printed Euclids, but also many reckoning books. Trigonometry became a major branch in the fifteenth and sixteenth centuries, not only for astronomy but also, as European imperialism developed, for cartography, and the needs of navigation and astronomy made the spherical branch more significant than the planar. Geometry was applied also to art, with careful studies of perspective; Piero della Francesca (c. 1420–1492) and Albrecht Dürer (1471–1528) were known not only as great artists but also as significant mathematicians.

Numerical calculation benefited greatly from the development of logarithms in the early seventeenth century by John Napier (1550–1617) and others, for then multiplication and division could be reduced to addition and subtraction. Logarithms superseded a clumsier method called "prosthaphairesis" that used certain trigonometrical formulas.

In algebra the use of special symbols gradually increased, until in his Géométrie (1637), René Descartes (1596–1650) introduced (more or less) the notations that we still use, and also analytic geometry. His compatriot Pierre de Fermat (1601–1665) also worked in these areas and contributed some theorems and conjectures to number theory. In addition, he corresponded with Blaise Pascal (1623–1662) on games of chance, thereby promoting parts of probability theory.

In mechanics a notable school at Merton College, Oxford, had formed in the twelfth century to study various kinds of terrestrial and celestial motion. The main event in celestial mechanics was Nicolaus Copernicus's (1473–1543) De revolutionibus (1453; On the revolutions), where rest was transferred from the Earth to the sun (though otherwise the dynamics of circular and epicyclical motions was not greatly altered). In the early seventeenth century the next stages lay especially with Johannes Kepler's (1571–1630) abandonment of circular orbits for the planets and Galileo Galilei's (1564–1642) analysis of (locally) horizontal and vertical motions of bodies.