IS EUCLID'S GEOMETRY MERELY A THEORY? 559 



the larger the triangle, the less the sum of its internal angles. Logically 

 pursued, the largest triangle possible would have all its sides parallel 

 and all its angles zero. 



This last statement has touched the verge of infinity and that is no 

 doubt treacherous territory. Coming back to our real universe — How 

 can we prove that Euclid is right about it? The real universe is large. 

 If, like the adversary in the Book of Job, we could go to and fro in the 

 great world and walk up and down in it, then we might decide the 

 controversy. But the limit of man's present astronomical measure- 

 ments is only about 30 light-years — 176 millions of millions of miles. 

 Within this compass he has observed no drift or change in the direction 

 of rays of light. If in our real space parallels are not exactly and 

 everywhere equidistant, Euclid's geometry is incorrect. The slightest 

 deviation in parallels would give the victory to Lobachevski or else to 

 the third competitor, Eiemann. The three justly claim equal con- 



FiG. 3. 



sideration in the light of present knowledge. Could such drift, if real, 

 escape human observation ? Yes ; first, because our instnmients are not 

 absolutely accurate ; and secondly, because eyesight is no infallible test. 

 If the human eye could survey a suflBciently tremendous expanse, 

 then parallels running through it might present the appearance of the 

 hyperbolic curves limiting the black and white areas in Fig. 4. These 

 curves may represent, and in certain respects they do simulate, the 

 parallels of Lobachevski. They are, to be sure, not parallels, for par- 

 allels are by definition straight; however, by placing the eye an inch 

 and a half above the center of this figure, these lines can be made to 

 look straight — a fact that confirms the statement that eyesight is not 

 an infallible test of straightness. 



