What were the applications of ellipses before elliptical orbits were discovered?

I'm interested in the history of ellipses. When were they discovered, what uses (if any) did they have before the true shape of orbits were found (by Kepler I think)?

There are some interesting ways to construct an ellipse. The common way is to anchor a string at the two focii and trace a circle with a pencil, but there is another way called a trammel of Archimedes that looks like it might be useful for something.

Did the ancients ever construct ellipses or ellipsoids for anything useful? Did they want to but were just too difficult to build?

This question has been discussed on Math overflow:

https://mathoverflow.net/questions/116627/useless-math-that-became-useful/116653#116653

This is about conic sections in general. In my answer I gave them as example of "useless mathematics which became useful centuries later". Some tried to challenge this statement.

One of the conjectures mentioned that conic sections were discovered when the Greeks tried to develop a theory of sundials. (Normally you get a hyperbola rather than ellipse, an ellipse will arise only in polar regions on the days when Sun never sets). This conjecture was much discussed in the history literature but on my opinion this is an unlikely explanation how conic sections were discovered. Another conjecture says that Menachem used conic sections for doubling the cube. This conjecture is nore plausible. See the references in the link above.

EDIT. There is of course a well-known legend about Archimedes burning Roman ships with parabolic mirrors, and it is clear that this is only a legend: no serious scientist believes that this can be actually performed. (But you can see this in the movies).

• Hmm, yes I forgot about conic sections. But was the ellipse really a useful one back then? I think I heard parabolic dishes were invented a long time ago to focus sound, but can ellipses also focus sound or light like that? – DrZ214 Jan 29 '16 at 13:45
• Ellipses can make "echo rooms", where if you stand at one focus, you can hear someone whispering at the other focus. But I don't know if that is before elliptical orbits or not. – Gerald Edgar Jan 29 '16 at 17:26
• @Gerald Edgar: yes, of course. And there were indeed some echo rooms. But I have never read about ELLIPTICAL echo rooms before Kepler. – Alexandre Eremenko Jan 29 '16 at 19:58
• @AlexandreEremenko I guess I'm confused with ellipses and parabolas. If both can focus sound to a single point, then I suppose the ancients (and probably us) would only build whichever one was easier. – DrZ214 Jan 29 '16 at 23:30

I am not going to trace back to the antiquity the whole history of ellipses and the instruments that drew them; I trust that more knowledgeable users would comment on Meneachmus, Proclus and others. Instead, let me offer some clear purpose for which ellipses were (and still are) used, namely for depicting circles in perspective in paintings and drawings. One of the pioneers of this use was Albrecht Dürer (1471-1528; hence he comes before Kepler, who was born in 1571). He even found a method of constructing an ellipse as a section of a right circular cone (which however was based on a mistaken assumption). More information about this, along with pictures from Dürer's "Treatise on mensuration" can be found e.g. in the article by Frank J. Swetz and Victor J. Katz,"`Mathematical treasures. Albrecht Dürer's treatise on mensuration", at this link to the (free) online magazine "Convergence" published by MAA: