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Maybe it is not surprising if one thinks that parabolas, hyperbolas, circles, and ellipses are relatives because they all have kind of the same form of equations, i.e.,

$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, $$

where different values of $A, B, C, D, E,$ and $F$, as well as their relations, give different kinds of curves.

But how did people figure out they were all conic sections (how could one see a cone from that second-order equation)?

Or was it the case that someone were playing with cones and somehow wondered how a cone could be cut in different ways and figured out that the edges from those cuts were all related through the above equation?

Intelligenti pauca
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Petch Puttichai
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  • There are some answers already given on this site. Intersection of equation of plane and cone in 3D is a second degree equation. – jonsno Apr 07 '17 at 05:18
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    @samjoe: That's the modern answer; but for the history, I thought this stuff was known long before we used algebra like that to do geometry. –  Apr 07 '17 at 05:19
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    Conic sections were studied by the Greeks, long before coordinate geometry was introduced. – user49640 Apr 07 '17 at 05:20
  • Oh I thought he was talking about relation of conic with the equation. – jonsno Apr 07 '17 at 05:23

2 Answers2

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It's the other way around: in ancient Greece parabolas, hyperbolas and ellipses were defined as sections of a cone. From that definition, one can easily derive the analogous of a modern cartesian equation: as far as I remember, that was done for the first time by Apollonius of Perga in 3rd Century b.C. See here for a derivation in the case of a parabola and here for the same thing in the original by Apollonius (translation by T.L. Heath).

But of course in ancient times those equations were not regarded as fundamental, nor of course people realized that any quadratic equation would lead to a conic section: cartesian coordinates would appear many centuries later. On the other hand, often non-orthogonal coordinates were used. Moreover a cone, for Apollonius, was not necessarily a right cone: it was defined as a surface formed by the lines joining the points of a circle with a point outside the plane of the circle.

Intelligenti pauca
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I think they knew it long before analytic geometry.

You can prove those things using elementary geometry. The proves are elegant, but they rarely appear in modern books; however, you can still find them in some comprehensive textbooks or online. For example, http://www.cut-the-knot.org/proofs/conics.shtml

But I'm not aware of the exact proves or reasoning that ancient Greeks used.

Ken Draco
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