A very famous [theorem of Poncelet](https://en.wikipedia.org/wiki/Poncelet%27s_closure_theorem) states that for an elliptic billiard all $n$-periodic trajectories are tangent to some ellipse. As far as I know, Poncelet proved this theorem while sitting in Russian jail so he didn't write it down. Could anybody give me a reference on a first book or article where the proof was actually given? 


I want to make a correct reference in the article I am writing. Moreover, I am also thinking about mentioning a greek mathematician Proclus who lived in the 5th century BC and proved that once a line in an ellipse is tangent to some other smaller ellipse, its reflection is tangent to the smaller ellipse too. 



Reference question: Poncelet theorem
Poncelet published his theorem ("Poncelet's porism) in 1822, after he returned to France following his captivity as war prisoner in Russia:


J.V. Poncelet, Traité des propriétés projectives des figures (Paris, 1822).


The book has been scanned and can be read [here](https://docnum.unistra.fr/digital/collection/coll7/id/71696) or on [archive.org](https://archive.org/details/traitdesproprit00goog/page/n7/mode/2up).


