The epistemological and ontological background of the debate on superposition between Jacques Peletier du Mans (1517-1582) and Christophorus Clavius (1538-1612)

Angela Axworthy

Max Planck Institute for the History of Science Berlin

Abstract: 
The general aim of my paper is to present the conditions and limits of the admission of motion in geometry in the commentaries on Euclid’s Elements of Jacques Peletier du Mans and Christophorus Clavius by considering the debate which arose between them regarding the validity of superposition as a method to demonstrate the congruence of figures. The starting point of this debate was Peletier’s rejection of superposition in his commentary of Prop. I.4 of the Elements in 1557. In this context, Peletier rejected superposition as too mechanical and as improper to convey the dignity of geometry. Clavius criticized Peletier’s position on this issue in the 1589 edition of his own commentary on the Elements, stating that Euclid would not have understood superposition as an actual and concrete translation of a figure onto another, contrary to how Peletier would have interpreted it, but rather as a purely intellectual, and therefore non-mechanical, mode of comparison of figures. Starting from this debate, my aim will be to examine its epistemological and ontological background in order, first of all, to establish more precisely how Peletier and Clavius understood superposition in the context of the Elements (notably in which sense it may or may not be called mechanical) and to determine, secondly, how their opinions on this issue were related to their conceptions regarding the introduction of motion in geometry. In this framework, I will not only look at the way Peletier and Clavius conceived the motion of figures supposedly entailed by Euclid’s use of superposition, but also see how they considered the cinematic definitions of the circle and of the “solids of revolution” (the sphere, the cone and the cylinder, such as defined in Book XI of the Elements), as well as the notion of flux or flow of points, lines and surfaces, which was commonly brought forth by ancient geometers to teach the modes of “generation” of lines and of the various plane and solid figures. By analysing the conceptions of Peletier and Clavius on these various kinds of geometrical motions (which differ from each other as regards their ontological implications and epistemological function), I will not only set forth the conditions for Peletier’s refusal and Clavius’ admission of superposition, but also give a general outline of their attitudes towards the use of motion in the definition and study of geometrical figures.