Euclid:
The Measure of the World

By Matt McKeown

If we speak of standing on the shoulders of giants, few are more gigantic than Euclid.

As a person, Euclid is one of the more obscure figures on our author bank: hardly any contemporary references to his life survive, even in writers like Plutarch or Diogenes Laërtius who chronicled many ancient men’s lives. He is known to have been born sometime in the late fourth century BC—exactly what year, or where, is not clear—and to have worked at the famous Library of Alexandria for some decades. Writing over six hundred years after his death, the Neoplatonist philosopher Proclus related that Euclid was a Platonist who lived under the reign of Ptolemy I of Egypt, the immediate successor of Alexander the Great and founder of the Ptolemaic Egyptian realm and dynasty. Archimedes and Apollonius of Perga are both thought to have learned from Euclid’s pupils, probably after his death, which occurred around 270 BC; and that is about as much as we know. (Some scholars have gone so far as to doubt Euclid’s historical existence, but this hypothesis is not widely accepted.)

But unlike his person, what we do know a good deal about is Euclid’s work. He wrote extensively on mathematics, the sciences, and logic, and several of his works survive, complete or in fragments. His influence on ancient astronomy was significant; besides penning a book of his own on the subject (Phænomena), he also proved in his Optics that, for any two objects of unequal size, there is some vantage point from which they appear to be equal, a proof with important implications for the measurement of the stars and planets, and which was taken up by Claudius Ptolemy in his authoritative work on the subject. Euclid also wrote a book on mirrors, a (lost) introduction to fallacies, and a work on conic sections that may survive as the first half of Apollonius’ treatise on the subject.

But what Euclid is best remembered for is his work on geometry. The Elements, his guide to geometry, remained the standard textbook for two thousand years, and was displaced in the twentieth century only by the radical reimagining of the nature of space that Einstein‘s theory of general relativity required. The text also delves into number theory, with particular attention to prime, perfect, and irrational numbers.

There is no shortcut to geometry.

Euclid

His quote (reproduced here) that there is no shortcut—literally, no “royal road”—to geometry is reportedly what he told Ptolemy I, when the latter asked whether there were any simpler way to learn the discipline. Euclid’s sarcasm is understandable: the opening of the Elements is about as simple and intuitive as a work could possibly be, demanding of its student such gargantuan efforts of understanding as “Things which are equal to the same thing are equal to each other” and “A circle can be drawn with any center and any radius.” From principles as simple as these, he derives his entire system of geometry, using nothing but logical, step-by-step proofs. (It is hard not to see the influence of Plato here, if we compare the methodical chapters of the Elements to the dialogue between Socrates and the slave boy in the Meno.) Nor, in terms of technical equipment, is Euclidean geometry demanding; nothing more than a flat surface, a stylus of some kind, a compass, and a straightedge allow the most amateur student to reproduce every one of Euclid’s geometric proofs.

One genuine weakness, of a kind, is present in the Elements. Euclid’s fifth postulate, also called the parallel postulate, states that if line a crosses lines b and c, and the interior angles at the junctions a-b and a-c add up to less than 180 degrees, then lines b and c will (if produced far enough) eventually cross each other on the same side of a as those interior angles. What’s strange about this postulate is that it has turned out to be wrong, or rather, incomplete; many scholars attempted to prove it over the centuries, only for their proofs to be discovered—sometimes after being accepted for hundreds of years—to be flawed, most often by being unintentionally circular arguments. It was finally determined in the nineteenth century that coherent geometries could be constructed which contradicted the parallel postulate. The reason for this is that the parallel postulate only describes a specific kind of space, namely, a “flat” space; geometry on a curved surface is radically different, and based on Einstein’s theory of gravitation, space itself is, for lack of a better way of putting it, curved—distorted by the pull of any object with mass, and the more distorted the greater the mass. Euclidean geometry was thus shown to be rather like Newton’s laws, inaccurate at the cosmic level but perfectly good for most practical purposes. As the mathematician and historian W. W. Rouse Ball dryly put it, “The fact that for two thousand years the Elements was the usual textbook on the subject raises a strong presumption that it is not unsuitable for that purpose.”

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Every week, we publish a profile of one of the figures from the CLT author bank. For an introduction to classic authors, see our guest post from Keith Nix, founder of the Veritas School in Richmond, VA.

If you enjoyed this essay, take a look at some of our other posts here at the Journal, like this profile of Louisa May Alcott or this analysis of the twin ideas of good and evil. And be sure to check out our Journey Through the Author Bank seminar series, featuring guest lectures from academics across the country.

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