🏛️ Euclid

Euclid of Alexandria was active around 300 BC, which places him at the intellectual center of the ancient world: the great library city of Alexandria in Egypt, founded by Alexander the Great in 331 BC and rapidly becoming the most important center of scholarship in the Mediterranean. We know almost nothing about his life — where he was born, who his parents were, whether he was Greek by birth or education — and yet his influence on the history of mathematics is so enormous that he has been called the Father of Geometry, a title that underestimates him only because it focuses on a single branch of knowledge when his true achievement was the invention of a method.

That method is the axiomatic proof. Euclid's Elements — a 13-book treatise on geometry, number theory, and proportion — does not begin with facts. It begins with definitions, postulates, and common notions: the basic terms and assumptions from which all subsequent results are derived by pure logical reasoning. The five postulates of Euclidean geometry are among the most famous sentences in the history of thought. Four of them are simple and intuitive: a straight line can be drawn between any two points; any straight line can be extended indefinitely; a circle can be drawn with any center and radius; all right angles are equal. The fifth — the parallel postulate — is different: through a point not on a given line, exactly one line can be drawn parallel to the given line. This fifth postulate is longer, less self-evident, and for two thousand years mathematicians attempted to prove it from the other four, until the nineteenth century demonstrated that they could not: by denying it, they could construct consistent geometries different from Euclid's, opening the door to non-Euclidean geometry, Riemannian manifolds, and ultimately Einstein's general relativity.

The Elements covers an astonishing range of mathematics. Books I through VI deal with plane geometry — triangles, rectangles, circles, proportion, and similarity. Books VII through IX cover number theory: the definition of prime numbers, the proof that there are infinitely many primes, and the algorithm for finding the greatest common divisor of two numbers — the Euclidean algorithm, still taught in computer science courses today as one of the oldest algorithms in existence. Book X is devoted entirely to irrational numbers, building on the Pythagorean discovery that √2 cannot be expressed as a fraction. Books XI through XIII cover solid geometry, culminating in the proof that there are exactly five regular (Platonic) solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

The proof of the infinitude of primes — found in Book IX, Proposition 20 — is a masterpiece of brevity and elegance that has been reproduced in mathematics textbooks for twenty-three centuries and will presumably be reproduced for as long as mathematics is taught. Assume there are finitely many primes. Multiply them all together and add one. The result is either prime (contradicting our assumption) or divisible by a prime not in our list (also contradicting our assumption). Therefore there are infinitely many primes. The argument fits in a paragraph and is absolutely airtight. It is the purest possible demonstration of what mathematical proof can do: establish something with absolute certainty through reasoning alone.

King Ptolemy I of Egypt, according to a story told by Proclus, once asked Euclid whether there was a shorter road to geometry than through the Elements. Euclid replied: "There is no royal road to geometry." Whether or not the exchange actually happened, it captures something true about his vision of mathematics: that there is no shortcut, no privileged access, no authority that substitutes for proof. Anyone who follows the chain of reasoning can verify every step. The king gets no special treatment. This democratization of mathematical truth — the idea that a proof is correct regardless of who proves it or who accepts it — is one of the most radical intellectual contributions of the ancient world.

Euclid's Elements was translated into Arabic in the eighth century AD, when Islamic scholars preserved and extended Greek learning during the European Dark Ages. It was translated from Arabic into Latin in the twelfth century, giving medieval European scholars their first access to its contents. It was one of the first books printed in Europe after Gutenberg — the 1482 Venice edition of the Elements is among the most beautiful scientific books ever printed. By the nineteenth century it had gone through more than a thousand editions. Abraham Lincoln reportedly taught himself Euclidean geometry from a copy of the Elements to improve his powers of reasoning. As late as 1900, Euclid was the standard geometry textbook in British schools. No other scientific text has had a longer run in active use.