Why Not Relate the Conic Sections to the Cone?

THIS article is addressed to those who believe that analytic geometry still has an important place in a mathematics se quence and that the study of the conic sections is an important part of analytic geometry. If this premise be accepted, then the various conies should be defined, their equations derived, and their properties investigated. It seems unfortunate to this writer that most, if not all, modern text books give three different definitions for the ellipse, parabola, and hyperbola, whereas a single definition would be his torically, aesthetically, and pedagogically sounder. The definitions usually given for the ellipse and hyperbola are really prop erties of those curves?properties impor tant both mathematically and physically, but still only properties. In the usual present-day treatment of conies, eccentricity of an ellipse or a hyperbola is discussed after the equations are derived. The term "eccentricity" may not even be mentioned in connection with the parabola. Whether the authors of cur rent textbooks intend it or not, this treat ment of eccentricity appears to the student as a sort of afterthought, one of no great importance. It is here submitted that eccentricity is a basic parameter of a conic section. There seems to be no good reason for taking what could be regarded as enor mous liberties with the original notion of the conic sections. Is it argued that a uni fied treatment is too difficult for the stu dents concerned? This writer holds that it is not. The algebra involved in deriving the equations of the ellipse and hyperbola, starting with a focus, a directrix, and an eccentricity, is (if anything) easier than that involved in deriving them from the other definitions. This writer believes, moreover, that starting with the cone and using some elementary solid geometry to demonstrate the physical significance of the focus, directrix, and eccentricity adds greatly to the elegance of the treatment, at little or no extra cost in time and effort either for the instructor or for the student.