Pseudo-Euclid on the Position of the Image in Reflection: Interpretations by an Anonymous Commentator, by Pena, and by Kepler

Abstract

In theorem 16 of the pseudo-Euclidean treatise Catoptrica the position of the image of a point reflected in a plane mirror is found as the intersection of the ray between eye and mirror and the perpendicular from the object to the plane of the mirror.¹ In Fig. 1, which is taken from pseudo-Euclid, with Roman letters replacing Greek, A is the object, B the eye, D the point of reflection, and E the image. E is found as the intersection of BD and the perpendicular AG. In theorems 17 and 18 the same is demonstrated for convex and concave spherical mirrors, the perpendicular being replaced by the line joining the object and the centre of the sphere. In the three theorems Pseudo-Euclid gives the standard ancient and medieval procedure to find the position of the image in catoptrics: Ptolemy,² Alhazen,³ Grosseteste⁴ and Witelo⁵ all give this procedure. It is also to be found in Risner’s own work on optics⁶ as well as in his editions of Alhazen and Witelo. Decades after the publication of Kepler’s Paralipomena in Vitellionem(1604) the principle was discussed by Tacquet and Barrow.⁷ It even survived into the eighteenth century.⁸ In many of these authors the principle is discussed together with a similar principle for refraction, but only reflection will be considered here.