Reflections on the principle of continuity on the basis of Ibn al-Haytham's commentary on Proposition I.7 of Euclid's Elements

Abstract

After his refutation of the doubts concerning Proposition I.7 (in the Book of solving doubts), Ibn al-Haytham mentions three possible ways in which circles may intersect, submitting them to the following “intuitive” argument: one part of one of the two circles is situated inside of the other circle, and its other part is situated outside of it. One is therefore tempted to believe that the commentator accepts the principle of continuity in the case of circles, since his argument has the following meaning: if a circle is divisible into two parts (or, again, passes through two points), one of which (or one of the two points) is situated inside the other circle, and the other outside of it, then the two circles cut one another. The author of this article proposes to establish the limits of this belief, on the basis of the following reflections: 1). It will be noted first of all that what could be called the ‘principle of the intersection of circles’ does not constitute ipso facto a principle in the mind of Ibn al-Haytham: no allusion is made to it in the commentary on Proposition I.1, among others. 2) It will be established later on that if one accepts (according to the explanation of Ibn al-Haytham in his Commentary on the premisses) that a line is the result of the movement of a point, the principle of continuity should be considered by him as something which is obvious by itself, without being stated. This conclusion will be based on an analysis of the notion of continuity in its classical meaning, and on Ibn al-Haytham’s commentary on Proposition X.1. 3) On the other hand, we should note the presence of a ‘sketch’ of topological language, which Ibn al-Haytham develops for the notion of a circle (particularly in the Commentary): one could say in this context that his reflection constitutes an important, if not principal, stage in the process which was to lead to the explicit formulation of the principle of continuity.