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Can You Really Derive Conic Formulae from a Cone? - Where Can You Go From Here?

Gary S. Stoudt

There are many wonderful and useful properties of conic sections; you are no doubt familiar with parabolic mirrors, elliptical "whispering rooms," and the like. Who first discovered these properties? Did Archimedes really use properties of conic sections to start enemy ships on fire? There is much more to investigate, but we must end our story here for now. You could dive right in and begin to investigate the wonders of Apollonius' Conics for yourself in [1] and [2]. Be aware that Apollonius is rough going, so you may want to start with a book on the history of mathematics, of which Katz [11] is the best. For a readable work on Archimedes' use of conic sections, check out [13]. For a look at conic sections from the point of view of geometric transformations, try [3]. Conic sections played an important role in Newton's theory of planetary motion in his Philosophia Naturalis Principia Mathematica [12].  Of course Newton's Principia is also very tough reading, so you might want to check out Dana Densmore's book [6], which stays true to the style of Newton's work, and contains plenty of material on conic sections; or Chandrasekhar's book [4], which uses modern mathematics to investigate the Principia. If all of this sounds intimidating, crack open a calculus book. They usually have a chapter on conic sections, where many of their useful properties are derived. A good calculus book with a historical spirit is Hahn's text [7].

In any case, there is much more to learn about conic sections, and why not learn it from the masters themselves? Good luck!

Gary S. Stoudt, "Can You Really Derive Conic Formulae from a Cone? - Where Can You Go From Here?," Loci (August 2010)