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.