The King of Infinite Space by David Berlinski

Author:David Berlinski [Berlinski, David]
Language: eng
Format: epub, mobi
ISBN: 9780465033461
Publisher: Basic Books

THE PROPOSITIONS THAT Euclid demonstrated in the Elements ascend by number, and the numbers are a reasonable guide to their difficulty. Euclid’s twenty-seventh proposition retains something of the obvious. It encourages the student (or the reader) to a concurrent grunt of affirmation. The theorem is dramatic nevertheless in its reach and power. It draws a connection between a pair of equal angles on the one hand and a pair of parallel lines on the other.

A look is enough to gauge the character of an angle, but no look, however lingering, does much to determine the character of parallel lines. Straight lines are parallel if they never meet. Within the Euclidean plane, never goes on and on. How is the geometer to establish that lines that never met will never meet? Once they have passed the last point of public inspection, lines that seem parallel might willfully undertake an unexpected decision to draw close after all.

But equal angles are equal locally, visible in the here and now. By checking the angles made by certain straight lines, the geometer may determine their parallel character once and for all. There is no need to track them to the back of beyond.

This theorem is interesting without in any way being extraordinary. What is extraordinary is what is so often hidden in the Elements: the rich ensemble of instruments that Euclid has employed to serve his ends. The proof of proposition twenty-seven is entirely a matter of the scant few lines needed to move with logical assurance from Euclid’s premises to his conclusions. But like an army, every one of Euclid’s theorems carries a long logistical tail: its apparatus of propositions, axioms, definitions, common notions, and rules of inference. And its illustrations, those diagrams that provide an intuition that is “rapid, unconscious, and not entirely certain.”

No part of this immense logistical tail by itself compels belief or elicits assent. It was Euclid’s genius to grasp the whole and to trust in the reader to follow what he had grasped.

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