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This paper emphasizes the great interest of the later Middle Ages in terms of burning mirrors, especially parabolic mirrors, which were considered as particularly efficient; the underlying theory was that of conic sections. This interest will be documented in the Appendix in the form of a short presentation of 29 treatises. The starting point is Paris BnF ms. lat. 9335, a collection of translations from the Arabic, copied in the early 13th century, which came into the possession of the Venetian physician and naturalist John Fontana (ca. 1395–1455). The optical part (Diokles, Apollonius, Alhazen) has been explained by him in a marginal note and a series of short definitions. These observations support the proposal by Horst Kranz, to change the date of an anonymous treatise on the construction of parabolic mirrors (Speculi almukefi compositio), previously dated by Marshall Clagett to the mid 14th century. We consider it now as a work of the famous Venetian naturalist John Fontana (ca 1430–1435).