Van Aubel's theorem and the theory of equipollences

Van Aubel's theorem states that if squares are constructed on the sides of a quadrilateral then the line segments joining the centres of the squares on opposite sides are equal and perpendicular. The main goal of this paper is to provide a translation of (Van Aubel, H, [‘Note concernant les centres de carrés construits sur les ĉotés d’un polygon quelconque’, Nouvelle Correspondance Mathématique, 4 (1878), 40–44].) paper in which this and other results are proven. The paper is interesting in its use of the method of equipollences developed by Giusto Bellavitis in 1835. From a modern perspective, the method combines vectors and complex numbers. Bellavitis knew about complex numbers but developed his system to avoid them as a given and instead provide a method that was fundamentally geometrical. What it means to be geometrical is certainly an educational issue arising naturally from reading Van Aubel's original paper. In that way, it can be very useful to teachers interested in using original sources in mathematics classrooms.