Mathematics

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Mathematics from 1600 to 1900

The scientific revolution of the 17th century spurred advances in mathematics as well. The founders of modern science—Nicolaus Copernicus, Johannes Kepler, Galileo, and Isaac Newton—studied the natural world as mathematicians, and they looked for its mathematical laws. Over time mathematics grew more and more abstract as mathematicians sought to establish the foundations of their fields in logic.

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17th-Century Mathematics

The 17th century saw the greatest advances in mathematics since the time of the ancient Greeks. The major invention of the century was calculus. Although two great thinkers—Sir Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany—have received credit for its invention, they built on the work of others. As Newton noted, “If I have seen further, it is by standing on the shoulders of giants.” Major advances also were made in numerical calculation and geometry.

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Calculation

The 17th century opened with the discovery of logarithms by Scottish mathematician John Napier and Swiss mathematician Justus Byrgius. Logarithms enabled mathematicians to extract the roots of numbers and simplified many calculations by basing them on addition and subtraction rather than on multiplication and division. Napier, who was interested in simplification, studied the systems of the Indian and Islamic worlds and spent years producing the tables of logarithms that he published in 1614. Kepler’s enthusiasm for the tables ensured their rapid spread.

So useful did logarithms prove that, two centuries after their discovery, French astronomer Pierre Simon Laplace said Napier had doubled the lifetime of astronomers by halving their labors. The need for logarithm tables disappeared with the widespread use of electronic calculators and computers in the second half of the 20th century.

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Analytic Geometry

The most important development in geometry during the 17th century was the discovery of analytic geometry by René Descartes and Pierre de Fermat, working independently in France. Analytic geometry makes it possible to study geometric figures using algebraic equations.

Descartes wished to overcome the limitations of Euclidean geometry, and he did so by applying algebra to geometry. In his publication Discours de la méthode (1637; Discourse on Method), Descartes showed how to use developments in algebra since the Renaissance to investigate the geometry of curves. Descartes maintained that an acceptable curve is one that can be expressed by a unique algebraic equation in x and y. His introduction of x and y coordinates was a major step. This made possible the classification of equations by the shape of the curves they made when graphed, and it opened the study of curves. As a consequence, many new curves important for science, including the cycloid and catenary, were introduced in the 17th and 18th centuries.

Descartes’s discoveries in geometry led to a reversal of the historical roles of geometry and algebra. French mathematician Joseph Louis Lagrange observed in the 18th century, “As long as algebra and geometry proceeded along separate paths their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection.”

Descartes’s book, together with short treatises published along with it, provided the impetus and basis for Newton’s mathematical work later in the century. Fermat, however, regarded his own work on what became known as analytic geometry as a reformulation of Apollonius’s treatise on conic sections. That treatise had provided the basic work on the geometry of curves from ancient times until Descartes.

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Projective Geometry

Other developments came in the field of projective geometry—that is, the properties of geometric figures projected onto another plane. French engineer Gérard Desargues published his discovery of projective geometry in 1639. The work was much appreciated by Descartes and Blaise Pascal, another French philosopher and mathematician of the 17th century. But eccentric terminology and the excitement of Descartes’s earlier publication on analytic geometry delayed the development of the ideas in this work until the early 19th century.

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