The Cave and the Light: Plato Versus Aristotle, and the Struggle for the Soul of Western Civilization

by Arthur Herman

The first was the geometer Euclid. We know almost nothing about his life. He may have been from Athens; he may even have studied at Plato’s Academy. We do know he was teaching mathematics in Alexandria about the time Strato arrived and that Ptolemy I once asked him if there wasn’t an easier way to study geometry than by taking Euclid’s classes. Euclid is supposed to have answered, “There is no royal road to geometry.” Anyone who has read his or her way through the Elements knows what he meant. As a textbook, Euclid’s Elements is peerless in its clarity and majesty of progression from first principles, or axioms, to particular demonstrations, or proofs. They move from the definition of line and point on the first pages to Pythagoras’s famous theorem and beyond. For more than two thousand years, they have been the model of a priori reasoning at its most dignified. The subtext of the Elements is also the principle Euclid may have absorbed from his teachers at the Academy: that mathematics and geometry are reason’s direct insight into the mind of the “supreme geometer,” God Himself. Indeed, Euclid’s principal works could almost have served as textbooks for Plato’s Rulers in the Republic. They were the Elements for arithmetic and geometry; the Conic Sections for the study of proportion and harmony; and his Phaenomena as a guide to astronomy, since it focuses on the theory of uniformly rotating spheres.13