Conic Section
The three conic sections are the parabola, hyperbola, and ellipse (the circle is considered a special case of an ellipse). They first arose in the fourth century b.c. in the work of the Greek mathematician Menaechmus who was trying to solve the problem of doubling the cube. Menaechmus constructed the conic sections through the intersection of a plane with three different types of a right circular cone.
Euclid (c.300 b.c.) is known to have written a four-volume work, now lost, on conic sections. The conics had been known for almost a century when Archimedes (287 b.c.-212 b.c.) computed the area of an ellipse, the area of sectors of a parabola, and the volumes of segments of the solids of revolution of conic sections using a method similar to that of integral calculus. The third great mathematician of the golden age of Greek mathematics, Apollonius of Perga (262 b.c.-190 b.c.), wrote an eight-volume collection on the conic sections called the Conics, the first seven of which have survived. These books consolidated most of the previous work on the subject and added new results about the theory and properties of these curves. Apollonius was the first to describe how all three conics could be obtained from a single double-napped cone by varying the inclination of the cutting plane. He also introduced the names ellipse, parabola, and hyperbola in the Conics.
In a.d. 320 Pappus of Alexandria produced an eight-volume work called the Mathematical Collection. In Book VII he showed that the ratio of the distance of any point on a conic from a fixed point (the focus) and a fixed line (the directrix) is a constant (the eccentricity). The conic is an ellipse, a parabola, or a hyperbola if the eccentricity is, respectively, less than 1, equal to 1, or greater than 1.
The reprinted editions of Apollonius' Conics in the 16th century contributed to the rebirth of interest in geometry. During this time the conic sections began to be defined as a locus of points in the plane rather than as sections of a plane intersecting a cone. For example, in 1579 Guidobaldo del Monte defined the ellipse as the locus of points such that the sum of the distances from the two foci is a constant. The development of analytic geometry furthered the study of the conics from an analytic rather than geometric point of view. Both Issac Newton and Leonard Euler extended Descartes' ideas to the study and classification of the conic sections. In 1656, the English mathematician John Wallis gave the first analysis of the conic sections as curves of the second degree.
The conic sections have numerous applications. Apollonius developed the hemicyclium, a type of sundial, by using a surface based on a conic section upon which the hour lines were inscribed. During the 10th century mathematicians in India investigated the optical properties of mirrors made from conic sections. If a source of radiation or sound is placed at the focus of a concave reflector with elliptical sections, then the radiation will converge onto the other focus. Likewise, radiation or sound emitted from a source at the focus of a reflector with a parabolic section is reflected in a parallel beam. This concept is employed today in car headlights, searchlights, and electric heaters. Omar Khayyam (1048-1131) found a geometric method to solve cubic equations by using conic sections. It was not until 1545 that Girolamo Cardano published an algebraic formula for solving the general cubic equation. Finally, the first of Johann Kepler's three laws of planetary motion, developed empirically in 1609 from years of astronomical observations, stated that the path of each planet is an ellipse with the sun at one focus. Almost 80 years later, Newton proved in his Principia that the ellipse is the path of any heavenly body moving around another in a closed orbit in accordance with the inverse square law of gravitation.
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