886 



SCIENCE 



[N. S. Vol. XXXVII. No. 968 



those days, though I was not aware of it 

 nor became aware of it till after many- 

 years, that there were then coming into 

 mathematics, just entering the fringe, so to 

 speak, or the vestibule of the science, certain 

 striking ideas which, as I venture to hope 

 we may see, were destined, if not indeed to 

 enable us to answer the questions with cer- 

 tainty, at all events to clarify them, to en- 

 rich their meaning and to make it possible 

 to discuss them profitably. It has not been 

 my fortune to meet many persons who had 

 seriously propounded the questions to them- 

 selves or who seemed to be immediately in- 

 terested in them when propounded by 

 others — not many, even among astron- 

 omers, whose minds, it may be assumed, 

 are specially " accustomed to contempla- 

 tion of the vast." And so I have been 

 forced to the somewhat embarrassing con- 

 clusion that my own long interest in the 

 questions has been due to the fact of my 

 being of a specially practical turn of mind. 

 Quite seriously I venture to say that we are 

 here engaged in a practical enterprise. 

 For even if the questions were in the na- 

 ture of the case unanswerable, which we do 

 not admit, who does not know how great 

 the boons that have come to men through 

 pursuit of the unattainable? And who 

 does not know that, as Mr. Chesterton has 

 said, if you wish really to know a man, the 

 most practical question to ask is, not about 

 his occupation or his club membership or 

 his party or church affiliations, but what 

 are his views of the all-embracing world? 

 What does he think of the universe? Do 

 but fancy for a moment that in somewise 

 men should come to know the exact shape 

 or figure and especially the exact size or 

 dimensions of the all-immersing space of 

 our universe. It requires but little imagi- 

 nation, not much reflection, no extensive 

 knowledge of cosmogonic history and spec- 



ulation, no very profound insight into the 

 ways of truth to men ; it needs, I say, but 

 little philosophic sense to see that such 

 knowledge would in a thousand ways, di- 

 rect and indirect, react powerfully upon 

 our whole intelligence, upon all our atti- 

 tudes, sentiments and views, transforming 

 our theology, our ethics, our art, our relig- 

 ion, our philosophy, our literature, our sci- 

 ence, and therewith affect profoundly the 

 whole sense and manner, the tone, color and 

 meaning, of all our institutions and the 

 affairs of daily life. Nothing is quite so 

 practical, in the sense of being effectual and 

 influential, as the views men hold, con- 

 sciously or unconsciously, regarding the 

 great locus of their lives and their cosmic 

 home. 



In order to discuss the questions before 

 us intelligibly and profitably it is not neces- 

 sary by way of clearing the ground to enter 

 far into metaphysical speculation or into 

 psychological analysis with a view to ascer- 

 taining what it is that we mean or ought to 

 mean by space. We are not obliged to dis- 

 pute, much less decide, whether space is 

 subjective or objective or both or indeed 

 something that, as Plato in the "Timeeus" 

 acutely contends, is neither the one nor the 

 other. We may or may not agree with the 

 contention of Kant that space is, not an ob- 

 ject, but the form, of outer sense; we may 

 or may not agree with the radically differ- 

 ent contention of Poincare that (geometric 

 as distinguished from sensible) space is 

 nothing but what is known in mathematics 

 as a group, of which the concept "is im- 

 posed on us, not as form of our sense, but 

 as form of our understanding." It is, I 

 say, not necessary for us, in the interest of 

 soundness and intelligibility, to try to com- 

 pose such differences or to attempt a settle- 

 ment of these profound and important 

 questions. As to the distinction between 



