Advances in the Study of Curves and Surfaces
Overview
Eighteenth-century mathematicians enjoyed a vastly expanded set of techniques that could be applied to the study of curves and surfaces and a vastly expanded set of reasons to study them. Problems of projectile and planetary motion required a renewed understanding of the conic sections. Problems from engineering and the need for accurate maps of Earth's curved surface drew special attention to the general problems ofrepresenting curves and surfaces by equations. The researches of Jacob Hermann, Leonhard Euler, Gaspard Monge, and others would lead to the new disciplines of descriptive and differential geometry.
Background
The ancient Greeks had a good understanding of those curves generated by the conic sections—the hyperbola, parabola, ellipse, and circle—but from a geometrical perspective only. With the invention of analytic geometry by the French mathematician and philosopher René Descartes (1596-1650) and French mathematician Pierre de Fermat (1601-1665), such curves were all understood to be described by algebraic equations. The study of curves and curved surfaces received new impetus in the eighteenth century from both science and technology. Given the laws of motion and gravitation of Sir Isaac Newton (1642-1727), it was now possible to calculate the trajectory of projectiles and planets with accuracy.
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