Constructive and Non-constructive Methods in Mathematics

Abstract

In the previous chapter we started by tacitly assuming that any given pair of line segments could necessarily be measured exactly by some (suitably small) common measure. But, when this assumption was applied to the pair of segments AB, AC in Figure 6, we derived a contradiction. We were thus forced by the principles of logic to admit that our tacit assumption was untenable: there clearly exist some pairs of segments (such as AB, AC in Figure 6) which have no common measure at all.