Argument

In this paper I offer some reflections on the thirty-second proposition of Book I of Euclid’s Elements, the assertion that the three interior angles of a triangle are equal to two right angles, reflections relating to the character of the theorem and the reasoning involved in it, and especially on its historical background. I reject a common view according to which there was at some time a petitio principii in the theory of parallels and argue that certain kinds of skepticism about Proclus’ historical reports are excessive. The evidence we have makes it reasonable to suppose that the so-called common notions were made explicit in the earlier fourth century BCE and the postulates, including the parallel postulate, somewhat later than that. On the other hand, it seems clear that some proof of I,32 was available by the mid-fifth century. I attempt to describe what such a proof would have been like and reflect on its significance for our understanding of early Greek geometry.