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As I was being observed today, an administrator asked me for a practical application of parabolas. I responded by talking about objects in free-fall. Afterwards as I was re-thinking this conversation it occurred to me that an object in orbit is also in a sort of free-fall, but its path is not parabolic but elliptical.

And a question began to form.

We can only turn to these contexts to illustrate conic sections because we live in the modern world. And math books often illustrate conic sections with technological examples like satellite dishes.

But during the very long interval between the first ancient Greek discussions of conic sections and the work of Kepler and Newton, what did everyone think that conic sections were good for? Were they interested in them purely as ideas, without thought of utility? That would be oddly beautiful and somewhat surprising.

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    $\begingroup$ I would suggest looking into the works of Apollonius of Perga and Archimedes. One example apparently from him is the idea of the parabolic mirror focusing light. Needless to say, it's unlikely the Greeks had some applications of them in our modern sense. $\endgroup$
    – kcrisman
    Nov 22 '19 at 2:16
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    $\begingroup$ On a separate note, you may wish to check out the math and science history Stackexchange site instead for this, or (alternately) if you have a more directly teaching-related question you may want to clarify the educational direction your question is going to take (as opposed to just for your own edification). $\endgroup$
    – kcrisman
    Nov 22 '19 at 2:17
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From Wikipedia on Menaechmus, ~350 BC:

Menaechmus is remembered by mathematicians for his discovery of the conic sections and his solution to the problem of doubling the cube. Menaechmus likely discovered the conic sections, that is, the ellipse, the parabola, and the hyperbola, as a by-product of his search for the solution to the Delian problem.

The Delian problem is the problem of constructing the side length of a cube which has double the volume of a given cube. If the given cube is a unit cube, the problem is to construct $2^\frac{1}{3}$.

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  • $\begingroup$ I didn't know this, and I'm interest to learn it. But I was asking about practical uses rather than purely academic or intellectual interest. This Delian problem seems to me a case of the latter. $\endgroup$
    – Chaim
    Nov 22 '19 at 19:16
  • $\begingroup$ @Chaim: You've asked an interesting question, and I look forward to other answers more relevant to your interests. $\endgroup$
    – Joseph O'Rourke
    Nov 22 '19 at 20:55

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