J W Downs - Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas

In conics I can floor peculiarities parabolous. Gilbert and Sullivan, Pirates of Penzance

Since the time of Alexander the Great, there has been an interest in cones and conic sections. Until the time of Apollonius of Perga (the Great Geometer) in the third century B.C., it was thought that ellipses were taken from acute cones, parabolas from right angle cones, and hyperbolas from obtuse cones, as shown in .

Apollonius demonstrated that all conic curves could be obtained by sectioning any cone, acute or obtuse, simply by changing the angle at which the plane intersects the cone. Although he wrote eight books (chapters) on conics and named ellipses, parabolas, and hyperbolas, he and the others of his time missed the most important attributes of conic sections, and no thought was given to any practical use of this unique family of curves.

shows a pair of cones joined at the apex. Lines AB, AC, AC , and AD represent planes pivoting at point A and cutting one or both cones. As the plane cuts through the cones, it will encounter ellipses within the sector where it passes completely through the cone ( AB ). When the plane is parallel to one side of the cone ( AC or AC ), the plane will no longer pass completely through the cone but can be extended forever. At this point the figure is a parabola. Beyond this angle the plane will never leave the cone ( AD ), and it will intersect the other cone, generating the two halves of a hyperbola.

Anthemius of Thrales, the architect of the Church of the Holy Faith in Istanbul in the sixth century, wrote a book called Concerning Remarkable Mechanical Devices, in which he discussed burning mirrors and the reflective properties of solid conic curves.

Archimedes probably experimented with parabolic reflectors, although there is no reference to this in his surviving works. The story that he devised burning mirrors to set the Roman ships on fire can only be legend unless the captains of the Roman ships were uncommonly accommodating. The focal length of the reflector (or lens) cannot be more than approximately twice its diameter to gather enough solar energy to start a fire. The Roman ships would have to have been docked within a few feet of the mirrors and carefully positioned with relation to the sun for burning mirrors to work.

As the science of astronomy developed more accurate ways of measuring planetary motion, it became evident that the planets were not moving in circular orbits around the sun. Kepler discovered that planetary motion was elliptical, with the sun at one focus.

In the seventeenth century, Ren Descartes devised a means of plotting conic (and other) curves on graph paper. This did wonderful things for mathematics, but it removed conic curves from the realm of pure geometry and placed them in the realm of mathematics. At present, nearly everyone thinks of these curves in terms of columns of numbers and dots on graph paper. We are sometimes amazed to find that they all occur in nature. The planets move freely in elliptical orbits without once thinking that they must move over so many x units and down so many y units. They simply move freely in balance with the forces influencing their motion. The same applies to the hyperbola described by a high-speed comet making a single pass by the sun or the parabola described by a nearsighted monkey that misses the branch for which it was jumping. The natural occurrences of conic curves are numerous. Lamps cast hyperbolic patterns against living room walls, and a soccer ball casts an elliptical shadow as it is thrown in the early morning sunlight. The frequency at which these curves occur naturally and the diverse way of generating them may substantiate the belief of the ancients that conic curves were of divine origin.