A very famous theorem of Poncelet 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.