The study of mathematics in medieval Latin Europe,¹ after the eleventh century, was stimulated by Latin translations from the Greek and, especially, from the Arabic. They were prepared in those places where the contact between the Christian and the Islamic civilizations was the most intimate, notably in Sicily, Southern Italy, and Spain. Some of the most prolific translators were Adelard of Bath, Robert of Chester, and Gerhard of Cremona in the twelfth century and Johannes Campanus in the thirteenth. In this way Latin Europe became acquainted with the geometry of Euclid (c. 300 b.c.), originally composed in Greek (and previously...

We have seen that mathematical studies in late medieval Latin Europe were stimulated by Latin translations from the Arabic. An important source of information was the treatise on equations written by Mohammed Al-Khwārizmī (c. a.d. 825). The Latin translation of its Arabic title,Liber algebrae et almucabala, gave the namealgebrato the theory of equations until the nineteenth century; since that time the term has been used in a much wider sense.

Al-Khwārizmī’s book dealt with linear and quadratic equations only. Sixteenth-century Italian mathematicians added the numerical solution of cubic and biquadratic equations. Gradually the study of equations came...

Medieval geometry was mostly taken from Euclid, whose work was partially known through the widely circulating works of Boethius (sixth century) and which became available through translations by Gerhard of Cremona, Johannes Campanus, and others in the twelfth and thirteenth centuries. The first printed Euclid, in the Latin version of Campanus, was that of Erhard Ratdolt in Venice (1482), a beautiful piece of work with many figures, and from that time on full or partial editions, in the original Greek and in translations, appeared in several countries. During the sixteenth century published editions of the works of Archimedes, Apollonius, and...

During the Middle Ages mathematical meditation on the infinitely great and infinitesimally small usually took the form of speculation on ideas of Aristotle and Plato concerning the relation of point to line, the nature of the incommensurable, the paradoxes of Zeno, the existence of the indivisible, the potentially and the actually infinite. Occasionally an infinite series, or a simple “integration,” appears. A good account of these speculations can be found in C. B. Boyer,The history of the calculus and its conceptual development(Dover, New York, 1959), chap. 3, which includes also direct quotations. On the study of Archimedes in...

The final discovery of the calculus required the assimilation of the geometric methods of Cavalieri and Barrow with the analytic methods of Descartes, Fermat, and Wallis; it also required the understanding of the relation between the search for tangent constructions and quadratures. This fundamental step was taken by Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716). Newton’s “golden period” of discovery fell between 1664 and 1668 (see Selection III.8); Leibniz also had such a “golden period,” from 1672 to 1676, when he resided in Paris, met with the twenty-one-years older Huygens, and twice made trips to London to...