June 13, 1913] 



SCIENCE 



893 



to note and bear that in mind. Observe 

 that this sphere — this surface — is a kind of 

 room. It is a kind of space, region or 

 room where certain things, as points, circle 

 ares and countless other configurations can 

 be and move. These things, confined to 

 this surface, which is their world, their 

 universe of space, if you please, enjoy a 

 certain amount, an immense amount, of 

 freedom : the points of this world can move 

 in it hither, thither and yonder; they can 

 move very far, millions and millions of 

 miles, even in the same direction, if only 

 the sphere be taken large enough. I see 

 no reason why we should not, for the sake 

 of vividness, fancy that spherical world 

 inhabited by two-dimensional intelligences 

 conformed to their locus and home just as 

 we are conformed to our own space of 

 three dimensions. I see no reason why we 

 should not fancy those creatures, in the 

 course of their history, to have had their 

 own Democritus and Epicurus, to have had 

 their own Roman republic or empire and 

 in, it to have produced the brilliant an- 

 alogues of our own Vergil, Cicero and 

 Lucretius. Do but note attentively — for 

 this is the point — that their Lucretius 

 could have said about their space precisely 

 what our own said about ours. Their 

 Lucretius could have said to his fellow- 

 inhabitants of the sphere: "starting at any 

 point, go as far as ever you please in any 

 straight line" — such line would of course 

 (as we know) be a great circle of the 

 sphere — "and then hurl your javelin" — 

 the javelin would, as we know, be only an 

 arc of a great circle — ' ' either it will go on, 

 in which case, etc.; or it will not, etc."; 

 thus giving an argument identical with 

 that of our own Lucretius. But what 

 could it avail ? We know what would hap- 

 pen to the javelin when hurled as supposed 

 in the surface : it would go on for a while, 

 there being nothing to prevent it. But 



whether it went on or not, it could not be 

 logically inferred that the surface, the 

 space in question, is infinite, for we know 

 that the surface is finite, just so many, a 

 finite number of, square miles. The fal- 

 lacy, at length, is bare. It consists — the 

 fact has been recently often pointed out — 

 in the age-long failure to distinguish ade- 

 quately between unbegrenzt and unendlich 

 — between "boundless and infinite as ap- 

 plied to space. What our fancied Lucre- 

 tius proved is, if anything, that the sphere 

 is boundless, but not that it is infinite. 

 What our real Lucretius proved is, if any- 

 thing, that the space of our universe is 

 boundless, but not that it is infinite. That 

 a region or room may be boundless without 

 being infinite is clearly shown by the 

 sphere (surface). How evident, once it is 

 drawn, the distinction is. And yet it was 

 never drawn, in thinking about the dimen- 

 sions of space, until in 1854 it was drawn 

 by Riemann in his epoch-marking and 

 epoch-making Hahilitationschrift on the 

 foundations of geometry. 



What, then, is the fact? Is space finite, 

 as Riemann held it may be? Or is it in- 

 finite, as Lucretius and Pascal deliberately 

 asserted, and as the normally educated 

 mind, however unconsciously, yet firmly 

 believes ? No one knows. The question is 

 one of the few great outstanding scientific 

 questions that intelligent lajonen may, with 

 a little expert assistance, contrive to grasp. 

 Shall we ever fibid the answer? Time is 

 long, and neither science nor philosophy 

 feels constrained to haul down the flag and 

 confess an ignorabimus. Neither is it 

 necessary or wise for science and philos- 

 ophy to camp indefinitely before a problem 

 that they are evidently not yet equipped to 

 solve. They may proceed to related prob- 

 lems, always reserving the right to return 

 with better instruments and added light. 



In the present instance, let us suppose. 



