Euclidean Geometry, the Parallel Postulate, and the Work of Lambert and Legendre

Abstract

This chapter switches to the history of Euclidean geometry, and especially the issue of the parallel postulate. This 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 that side on which are the angles less than two right angles. Alone among the postulates of Euclid’s Elements it lacks intuitive credibility, and many attempts were made to deduce it from the other postulates. We look at the ideas of Gerolamo Saccheri, who showed in 1733 that a geometry based on all of Euclid’s postulates except the parallel postulate must be one of at most three kinds, which are distinguished by the angle sums of triangles. Either every triangle has an angle sum greater that π, or equal to π, or less than π. He then showed that the first case cannot occur, but his attempts to show that the third case also leads to a contradiction, which would have left Euclidean geometry as the only possibility, failed. Saccheri was followed by Johann Heinrich Lambert, who noted some more unusual features of a geometry based on the third of Saccheri’s hypotheses. Then came Adrien-Marie Legendre, who also made several unsuccessful attempts to refute the third hypothesis, one of which is considered in detail. Extract: Lambert on the consequences of a non-Euclidean parallel postulate.