Lobachevsky & the mysterious case of Postulate 5

By Bilsana Bibic

Seniors are just beginning Lobachevsky this week. We are moving away from classical Euclidean geometry towards the unknown regions of the thrilling but extremely challenging modern geometry – the land where the parallel lines meet.

Lobachevsky will establish a system of geometry whose foundation avoids Postulate 5 of Euclid’s Elements in order to define parallel lines. The challenge of “proving” the truth of Postulate 5 (since it does not seem as intuitive as the other postulates) was around for centuries. Euclid does not see the necessity of proving it directly but bases a lot of his proofs on this controversial postulate. For example, the idea that the sum of all angles in a right triangle are less that two right angles is based on this Postulate. But is Postulate 5 really necessary for the rest of Euclid? If this Postulate is not truly a Postulate, wouldn’t we still be able to use the other propositions without relying on it?

After a failed attempt to prove the truth of this Postulate (this attempt took him his whole life) a Hungarian mathematician F. Bolyai writes to his son, J. Bolyai, who is trying to prove it and says that he should “leave the science of parallels alone” because he has “accomplished monstrous, enormous labors” and has “turned back when [he] saw that no man can reach the bottom of the night.” F. Bolyai claims that he “turned back unconsoled, pitying [himself] and all mankind” when he failed in his attempt to prove Postulate 5 and parallelism as its direct consequence.

Postulate 5, as you can see, is a big deal.

With this background in mind, Lobachevsky proposes a whole new system of geometry based on the idea of avoiding the Postulate 5, as its intuitiveness is controversial. My senior Math class has just began sailing in these turbulent waters.