# Conic sections *à la* Apollonius of Perga

This course takes inspiration from the editor of *Matematik
Dünyası,*
who wrote on page
1 of the fall 2004
issue,

Bana ne biliyorsun diye sorsalar, neyi bilmediğimi biliyorum, derim diye düşünüyordum, en azından matematikte (If they ask me what I know, I thought I would say, “I know what I don't know, at least in mathematics”)…

Textbooks today often define the conic sections as plane loci of points
whose distances from certain points (*foci*)
or a line (*directrix*) have certain relations. Then equations for such loci are immediately derived:
*x ^{2} = 4py* for the parabola, or

*x*for the ellipse or hyperbola. In justification of the name “conic sections,” a book may

^{2}/a^{2}± y^{2}/b^{2}= 1*state*that these curves can also be obtained by slicing cones with a planes; but a modern book rarely

*proves*this. So a lot of students may end up

*thinking*that they know why the conic sections are so called, although they really

*don't*know the connection between the conic sections and the equations that are supposed to define them.

The connection *can* be seen in some books today. An example is
Hilbert and Cohn-Vossen's *Geometry and the Imagination* (first
German publication, 1932), whose first chapter does show how the conic
sections can indeed by obtained from cones. Or see *Matematik
Dünyası* 2005-III, pp. 33–37. But in these places, the
cones considered are all *right* (*dik*).

For Apollonius of Perga (active around 200 BCE), a cone is determined by
a circle and a point not in its plane. The straight lines through the
point and the circumference of the circle trace out the surface of a
cone. The circle is then the base of the cone, and the point is its
apex. The cone is *right* if the straight line through the apex
and the center of the base is perpendicular to the base. But Apollonius
works with arbitrary cones. The beginnings of what Apollonius does can
be seen in Matematik
Dünyası 2005-II, pp. 54–61. Apollonius goes on to
demonstrate some beautiful properties of conic sections that are little
known today. One purpose of this course is to demonstrate those
properties.

Another purpose is to raise the question: What has René
Descartes (1596–1650) done to mathematics? In his *Geometry* (original French publication, 1937) Descartes shows how to describe curves with equations. Then properties of curves can be proved by algebraic manipulation of equations. With experience, this manipulation becomes almost automatic, and one hardly has to think. This may be convenient, but is it always good? The arguments of Apollonius can be difficult, but they are visual: you can see what is happening. In the *Rules for the Direction of the Mind* (written in Latin, published posthumously) Descartes himself suggests that the Ancients did have algebraic methods like his own, though they kept them hidden. If so, did they have good reason?

In preparing for this course, I wrote some notes. None were ever finished; but here they are: