Author: Maria Popova / Source: Brain Pickings
“Euclid alone has looked on Beauty bare,” Edna St. Vincent Millay wrote in her lovely ode to how the father of geometry transformed the way we see and comprehend the world. But although the ancient Alexandrian mathematician provided humanity’s only framework for understanding space for centuries to come, shaping both science and art, his beautiful system was wormed by one ineluctable flaw: Euclid’s famous fifth postulate, known as the parallel postulate — which states that through any one point not belonging to a particular line, only one other line can be drawn that would be parallel to the first, and the two lines, however infinitely they may be extended into space, will remain parallel forever — is not a logical consequence of his other axioms.
This troubled Euclid. He spent the remainder of his life trying to prove the fifth postulate mathematically, and failing. Generations of mathematicians did the same for the next two thousand years. It even stumped Gauss, considered by many the greatest mathematician of all time. It took a Hungarian teenager to solve the ancient quandary.
In 1820, more than two millennia after Euclid’s death, the seventeen-year-old János Bolyai (December 15, 1802–January 27, 1860) told his father — the mathematician Wolfgang Bolyai, who had introduced his son to the enchantment of mathematics four years earlier — about his obsession with the parallel postulate.
In the exchange that followed, recounted in George E. Martin’s classic 1975 primer The Foundations of Geometry and the Non-Euclidean Plane (public library), Bolyai Senior responded with the opposite of encouragement, writing to his son:
Don’t waste an hour on that problem. Instead of reward, it will poison your whole life. The world’s greatest geometers have pondered the problem for hundreds of years and not proved the parallel postulate without a new axiom. I believe that I myself have investigated all the possible ideas.
But the young man persisted. On November 3, 1823, the twenty-one-year-old mathematical maverick wrote to his father while serving as an artillery officer in the Hungarian army:
I have resolved to publish a work on the theory of parallels as soon as I have arranged the material and my circumstances allow it. I have almost been overwhelmed by them, and it would be the cause of constant regret if they were lost. When you see them, my dear father, you too will understand. At present I can say nothing except this: I have created a new universe from nothing. All that I have sent to you till now is but a house of cards in comparison with a tower. I’m fully persuaded that…
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