In the 11th century a new phase of mathematics began with the translations from Arabic. Scholars throughout Europe went to Toledo, Córdoba, and elsewhere in Spain to translate into Latin the accumulated learning of the Muslims. Along with philosophy, astronomy, astrology, and medicine, important mathematical achievements of the Greek, Indian, and Islamic civilizations became available in the West. Particularly important were Euclid’s Elements, the works of Archimedes, and al-Khwārizmī’s treatises on arithmetic and algebra. Western texts called algorismus (a Latin form of the name al-Khwārizmī) introduced the Hindu-Arabic numerals and applied them in calculations. Thus, modern numerals first came into use in universities and then became common among merchants and other laymen. It should be noted that, up to the 15th century, calculations were often performed with board and counters. Reckoning with Hindu-Arabic numerals was used by merchants at least from the time of Leonardo of Pisa (beginning of the 13th century), first in Italy and then in the trading cities of southern Germany and France, where maestri d’abbaco or Rechenmeister taught commercial arithmetic in the various vernaculars. Some schools were private, while others were run by the community.

The universities

Mathematics was studied from a theoretical standpoint in the universities. The Universities of Paris and Oxford, which were founded relatively early (c. 1200), were centres for mathematics and philosophy. Of particular importance in these universities were the Arabic-based versions of Euclid, of which there were at least four by the 12th century. Of the numerous redactions and compendia which were made, that of Johannes Campanus (c. 1250; first printed in 1482) was easily the most popular, serving as a textbook for many generations. Such redactions of the Elements were made to help students not only to understand Euclid’s textbook but also to handle other, particularly philosophical, questions suggested by passages in Aristotle. The ratio theory of the Elements provided a means of expressing the various relations of the quantities associated with moving bodies, relations that now would be expressed by formulas. Also in Euclid were to be found methods of analyzing infinity and continuity (paradoxically, because Euclid always avoided infinity).

Studies of such questions led not only to new results but also to a new approach to what is now called physics. Thomas Bradwardine, who was active in Merton College, Oxford, in the first half of the 14th century, was one of the first medieval scholars to ask whether the continuum can be divided infinitely or whether there are smallest parts (indivisibles). Among other topics, he compared different geometric shapes in terms of the multitude of points that were assumed to compose them, and from such an approach paradoxes were generated that were not to be solved for centuries. Another fertile question stemming from Euclid concerned the angle between a circle and a line tangent to it (called the horn angle): if this angle is not zero, a contradiction quickly ensues, but, if it is zero, then, by definition, there can be no angle. For the relation of force, resistance, and the speed of the body moved by this force, Bradwardine suggested an exponential law. Nicholas Oresme (died 1382) extended Bradwardine’s ideas to fractional exponents.

Another question having to do with the quantification of qualities, the so-called latitude of forms, began to be discussed at about this time in Paris and in Merton College. Various Aristotelian qualities (e.g., heat, density, and velocity) were assigned an intensity and extension, which were sometimes represented by the height and bases (respectively) of a geometric figure. The area of the figure was then considered to represent the quantity of the quality. In the important case in which the quality is the motion of a body, the intensity its speed, and the extension its time, the area of the figure was taken to represent the distance covered by the body. Uniformly accelerated motion starting at zero velocity gives rise to a triangular figure (see the figure). It was proved by the Merton school that the quantity of motion in such a case is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion; in modern formulation, s = 1/2at² (Merton rule). Discussions like this certainly influenced Galileo indirectly and may have influenced the founding of coordinate geometry in the 17th century. Another important development in the scholastic “calculations” was the summation of infinite series.

Basing his work on translated Greek sources, about 1464 the German mathematician and astronomer Regiomontanus wrote the first book (printed in 1533) in the West on plane and spherical trigonometry independent of astronomy. He also published tables of sines and tangents that were in constant use for more than two centuries.

The Renaissance

Italian artists and merchants influenced the mathematics of the late Middle Ages and the Renaissance in several ways. In the 15th century a group of Tuscan artists, including Filippo Brunelleschi, Leon Battista Alberti, and Leonardo da Vinci, incorporated linear perspective into their practice and teaching, about a century before the subject was formally treated by mathematicians. Italian maestri d’abbaco tried, albeit unsuccessfully, to solve nontrivial cubic equations. In fact, the first general solution was found by Scipione del Ferro at the beginning of the 16th century and rediscovered by Niccolò Tartaglia several years later. The solution was published by Gerolamo Cardano in his Ars magna (Ars Magna or the Rules of Algebra) in 1545, together with Lodovico Ferrari’s solution of the quartic equation.

By 1380 an algebraic symbolism had been developed in Italy in which letters were used for the unknown, for its square, and for constants. The symbols used today for the unknown (for example, x), the square root sign, and the signs + and − came into general use in southern Germany beginning about 1450. They were used by Regiomontanus and by Fridericus Gerhart and received an impetus about 1486 at the University of Leipzig from Johann Widman. The idea of distinguishing between known and unknown quantities in algebra was first consistently applied by François Viète, with vowels for unknown and consonants for known quantities. Viète found some relations between the coefficients of an equation and its roots. This was suggestive of the idea, explicitly stated by Albert Girard in 1629 and proved by Carl Friedrich Gauss in 1799, that an equation of degree n has n roots. Complex numbers, which are implicit in such ideas, were gradually accepted about the time of Rafael Bombelli (died 1572), who used them in connection with the cubic.

Apollonius’s Conics and the investigations of areas (quadratures) and of volumes (cubatures) by Archimedes formed part of the humanistic learning of the 16th century. These studies strongly influenced the later developments of analytic geometry, the infinitesimal calculus, and the theory of functions, subjects that were developed in the 17th century.