mathematics: Western Developments from the Twelfth to Eighteenth Centuries

Western Developments from the Twelfth to Eighteenth Centuries

Word of the Chinese and Middle Eastern works began to reach the West in the 12th and 13th cent. One of the first important European mathematicians was Leonardo da Pisa (Leonardo Fibonacci), who wrote on arithmetic and algebra (Liber abaci, 1202) and on geometry (Practica geometriae, 1220). With the Renaissance came a great revival of interest in learning, and the invention of printing made many of the earlier books widely available. By the end of the 16th cent. advances had been made in algebra by Niccolò Tartaglia and Girolamo Cardano, in trigonometry by François Viète, and in such areas of applied mathematics as mapmaking by Mercator and others.

The 17th cent., however, saw the greatest revolution in mathematics, as the scientific revolution spread to all fields. Decimal fractions were invented by Simon Stevin and logarithms by John Napier and Henry Briggs; the beginnings of projective geometry were made by Gérard Desargues and Blaise Pascal; number theory was greatly extended by Pierre de Fermat; and the theory of probability was founded by Pascal, Fermat, and others. In the application of mathematics to mechanics and astronomy, Galileo and Johannes Kepler made fundamental contributions.

The greatest mathematical advances of the 17th cent., however, were the invention of analytic geometry by René Descartes and that of the calculus by Isaac Newton and, independently, by G. W. Leibniz. Descartes's invention (anticipated by Fermat, whose work was not published until later) made possible the expression of geometric problems in algebraic form and vice versa. It was indispensable in creating the calculus, which built upon and superseded earlier special methods for finding areas, volumes, and tangents to curves, developed by F. B. Cavalieri, Fermat, and others. The calculus is probably the greatest tool ever invented for the mathematical formulation and solution of physical problems.

The history of mathematics in the 18th cent. is dominated by the development of the methods of the calculus and their application to such problems, both terrestrial and celestial, with leading roles being played by the Bernoulli family (especially Jakob, Johann, and Daniel), Leonhard Euler, Guillaume de L'Hôpital, and J. L. Lagrange. Important advances in geometry began toward the end of the century with the work of Gaspard Monge in descriptive geometry and in differential geometry and continued through his influence on others, e.g., his pupil J. V. Poncelet, who founded projective geometry (1822).

Sections in this article:

Introduction
In the Twentieth Century
In the Nineteenth Century
Western Developments from the Twelfth to Eighteenth Centuries
Chinese and Middle Eastern Advances
Greek Contributions
Development of Mathematics
Applied Mathematics
Geometry
Analysis
Algebra
Foundations
Bibliography

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