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Euclid’s Elements is the most influential textbook in the history of western civilization, a model of reasoning not only in mathematics but in philosophy, theology, and politics. But Euclid’s geometry rests on postulates, one of which didn’t seem self-evident from the start. Mathematicians kept trying to prove that postulate, especially, as we’ll see, Lagrange. But meanwhile, the unchallenged authority of the Euclidean ideal was used by people like Newton, Voltaire, Euler, and Lagrange to support the Enlightenment world-view. In the nineteenth century, though, non-Euclidean geometries challenged the authority of mathematics, undermined fundamental philosophical and cultural ideas, and had a hand in the birth of modernism. Challenges came not only from scientists like Gauss, Lobachevsky, Helmholtz, and Einstein, but also from artists and philosophers. This story illustrates both how culture helps shape mathematics and how mathematics has shaped the modern world.