The theory of parallel lines in the age of Leibniz

pp. 7-19

in: Vincenzo Risi, Vincenzo De Risi, Leibniz on the parallel postulate and the foundations of geometry, Basel, Birkhäuser, 2016

Abstract

It may be useful to give a picture of the discussions on the Parallel Postulate in the age of Leibniz, as well as a list of his mathematical sources on the topic. The definition of parallel lines, in Euclid's wording, is that of 'straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction". The Parallel Postulate appears as Postulate Five in the First Book of the Elements, and states that "if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side on which are the angles less than the two right angles". In Early Modern editions of Euclid, however, the Postulate was normally arranged among the axioms.