*THE POPULAR SCIENCE MONTHLY*

IS EUCLID'S GEOMETRY MERELY A THEORY? |

By EDWARD MOFFAT WEYER, Ph.D.

PROFESSOR OF PHILOSOPHY, WASHINGTON AND JEFFERSON COLLEGE

AS there are creeds in religion, so there are creeds in geometry. Most of us pin our faith upon Euclid, whose masterpiece of reasoning is still, after twenty-two centuries, the wonder of the world. The system that Euclid founded stands, as it were, four-square and solid; it meets every need in the only kind of space that we practically know. Over against this edifice, the modern geometries of hyperspace have been reared, from foundations which we Euclideans regard as fantastic. They are intangible structures, like the towers and battlements of a region of dreams.

The present writer holds no brief in favor of a fourth dimension of space. Hypothetical realms, wherein the dimensions of space are assumed to be greater in number than three, yield strange geometries, which are only card-castles, products of a sort of intellectual play, in the construction of which the laws of logic supply the rules of the game. The character of each system is determined by whatsoever assumptions its builder lays down at the start. The illustrious Euclid himself, whom none would rank as visionary, would probably set no great store by these hypergeometries. If he were to return to earth to-day, his interest in them would be that of a retired chess-champion who perceives that his old style of play has given rise to new varieties of the game. Nothing from out this fairyland of thinking could endanger Euclid's prestige; he might contemplate retirement on a professor's old-age pension.

Nevertheless, as soon as Euclid had viewed modern geometry throughout its entire range, the mere suggestion of a pension would in all likelihood ruffle his spirit. For by that time the master would know that geometers no longer blindly accept his teachings; that, moreover, our real space holds mysteries of which he never dreamed. When finally he should discover that experts have arisen who would undertake to instruct him at his own game, he would investigate the massive non-Euclidean systems—the Lobachevski-Bolyai or pseudospherical geometry, and another, the spherical, invented by Riemann—no mere card-castles, but valid in their application to every known space-condition of the universe. Like the rest of us, Euclid would ask himself: In which of these varieties of space does our actual universe belong?