Bardi, Jason Socrates. The Fifth Postulate: How Unraveling a Two-Thousand-Year-Old Mystery Unraveled the Universe. Hoboken, N.J.: John Wiley Press, 2009. Pp. xi + 253. Hardcover, $27.95.

The fifth postulate was stated by Euclid: two lines that are not parallel will cross if they are in the same plane.  A proof was elusive, despite much work, until the early nineteenth century.  The resolution created non-Euclidean geometry, which allowed mathematicians “to pursue a logical mathematical system that was divorced from reality.”  Ironically, it turns out that non-Euclidean geometry does have relevance to the physical world. The approach is at the heart of the experimental confirmation of the General Theory of Relativity, done by measuring the effect of the sun’s gravitational field on the direction of light rays.  The story centers around Carl Friedrich Gauss, certainly the most distinguished mathematician in the first half of the nineteenth century, who actually published hardly any of his own non-Euclidean work but gave credibility by noting the importance of the work done by others. The two mathematicians who did publish were Janos Bolyai, an army officer in Hungary, and Nikolai Lobachevsky, a professor in a remote Russian university.  None of the three principals ever learned before publication of what the others were doing and, in fact, never met. The publications were, in both cases, almost totally ignored during the authors’ lifetimes, which especially disheartened Bolyai.  Gauss was working on many other problems, including wireless transmission, and may have been reluctant to enter into the inevitable controversy about a radical departure from traditional geometry—to “stir the nests of wasps.” Lobachevsky became Rector of his distant university and seems never to have entered the mainstream of European mathematics.  From their experiences the peril of creating an entirely new approach to a well-studied subject comes out clearly.  Gauss points out that he could never have understood Lobachevsky unless he had already thought long and hard about the problem, and Gauss’s colleagues were simply unprepared for the new paradigm. The narrative gives insight into how research was done in a time without a close scientific network, nor external pressures to publish and seek grants.  The history of attempts to prove the fifth postulate includes many, perhaps most, of the major figures in mathematics in the last two thousand years, and Bardi includes stories about many of these people.  The exposition is clear and accessible, even suspenseful.  A drawing or two would have helped me, as well as slightly longer explanations of the mathematics, but such might have been intimidating to the general reader.  All in all, the book is a good read, about an unfamiliar story, one with some worthwhile lessons.