David Pierce // Matematik (Mathematics) // M.S.G.S.Ü.

my other courses at the Nesin Matematik Köyü

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: x2 = 4py for the parabola, or x2/a2 ± y2/b2 = 1 for the ellipse or hyperbola. In justification of the name “conic sections,” a book may 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:

  1. “Conic sections as such (first draft)”
  2. “Conic sections as such (second draft)”
  3. “Parabolas”

Son değişiklik: Tuesday, 03 April 2012, 12:56:18 EEST