Nowadays, we take the 5th Postulate of Euclid for granted. It is briefly mentioned in most middle-school geometry textbooks, and off they steam to compass-and-ruler constructions. However, it took humanity nearly 2,000 years to realize that the 5th is indeed a postulate and cannot be derived from the rest of the axioms. For comparison, Fermat's Great Theorem needed a bit less than 360 years to get a proof (1637 - 1995). The Poincare conjecture stood open for less than a century (1904 - 2002). Among the people who have invested some considerable and unsuccessful effort in an attempt to understand the situation with the fifth postulate, we see many of the best minds humanity has produced, from Euclid and Eratosthenes to Euler, Lagrange, and Gauss. The latter got quite close to the answer, but did not publish his findings. And for a good reason! The first person who published a paper that laid foundation for non-Euclidean (hyperbolic) geometry, Nikolai Lobachevsky of Russia, was declared insane. The first person to solve the problem and the second to publish, Janos Bolyai of Hungary, ended up living in constant depression. It took his father Farkas Bolyai, himself a renowned mathematician, 24 years to understand his son's work. The rest of the scientific community (Gauss excluded) gave him only posthumous recognition.
In this talk, we shall see what makes the fifth postulate so tricky. We will begin with deriving the theorems about the sums of angles of a triangle in the Euclidean and elliptic planes from the axioms and proceed to Boy's surface and geometry of the Universe at large. This is an expository talk that mixes together mathematics, history of science, cosmology, and philosophy. As such, it will be accessible to undergraduate and graduate students, and working mathematicians.