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I'm curious about the historical development of the idea that parabolas redirect incoming light to the focus. I'm curious how this was discovered, and whether it was discovered mathematically or experimentally. While it is fairly easy to show that a parabola has this property, going the other way (i.e., figuring out that such a shape would be a parabola) seems more difficult. I was able to derive a parabolic equation (i.e., a quadratic) from the meaning of "reflecting all incoming rays towards a focus", but it wound up utilizing some differential equations that I'm sure were not available when this property of parabolas was first noticed in antiquity.

Anyway, I'm curious about the history of this idea, and how it was discovered/established prior to calculus.

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Diocles was the first prove this in "On burning mirrors". There are no surviving records but fragments were found in Eutocius's notes on Archimedes "Spheres and Cylinders". Conic sections were well-studied in antiquity including parabolas but records were mostly kept by Arabic mathematicians. Ibn Sahl has a surviving proof which likely follows Diocles' methods.

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