Because of the lack of original sources there exist different conjectures on how Theodorus might have proved geometrically the irrationality of √ 3 (and some other roots), as well as on how Archimedes could have found his approximating values for √ 3. We claim that their results can be obtained similarly to those of √ 2. Thus we start with case √ 2, where, unlike to other interpretations, we consider two cases of approximations. Moreover, we use one figure to study both the irrationality and the approximation problems. In our argumentation, only concepts and methods from ancient Greek geometry are applied, namely antanairesis, propositions II.9 and II. 10 of the Elements, as well as some basic geometrical facts.
This geometrical approach helps the teacher to form a descriptive idea on the abstract concept of irrational numbers, and even on Cantor’s axiom.