554 THE POPULAR SCIENCE MONTHLY 



IS EUCLID'S GEOMETEY MEKELY A THEORY? 



By EDWARD MOFFAT WETER, Ph.D. 



PROFESSOR OF PHILOSOPHY, WASHINGTON AND JEFFERSON COLLEGE 



AS there are creeds in religion, so there are creeds in geometry. 

 Most of tis 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 battle- 

 ments 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 as- 

 sumed 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 per- 

 ceives 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. Eor by that time the master would 

 know that geometers no longer blindly accept his teachings : that, more- 

 over, our real space holds mysteries of which he never dreamed. When 

 finally he should discover that experts have arisen who would under- 

 take 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 Eiemann — no mere 

 card-castles, but valid in their application to every known space-con- 

 dition of the universe. Like the rest of us, Euclid would ask himself : 

 In which of these varieties of space does our actual universe belong? 



