June 13, 1913] 



SCIENCE 



889 



feel that what is new is, not an added vast- 

 ness, but a discovery of a vastness that al- 

 ways was and always will be. Let us trust 

 this feeling and, regarding space as con- 

 stant from everlasting to everlasting, let us 

 take the questions in their natural intent 

 and form: what are the dimensions and 

 what is the figure of our universe of space? 

 If you propound these questions to a 

 normal student of natural science, say to a 

 normal astronomer, his response will be — 

 what? If you appear to him to be quite 

 sincere and if, besides, he be in an amiable 

 mood, his response will, not improbably, be 

 a significant shrug of the shoulders, de- 

 signed to intimate that his time is too 

 precious to be squandered in considering 

 questions that, if not meaningless, are at 

 all events unanswerable. I maintain, on 

 the contrary, that this same student of nat- 

 ural science and, indeed, all other normally 

 educated men and women, have, as a part 

 of their intellectual stock in trade, per- 

 fectly definite answers to both of the ques- 

 tions. I do not mean that they are aware 

 of possessing such wealth nor shall I under- 

 take to say in advance whether their an- 

 swers be correct. What I am asserting and 

 what, with your assistance, I shall endeavor 

 to demonstrate, is that perfectly precise, 

 very intelligent and perfectly intelligible 

 answers to both of the questions are log- 

 ically involved in what every normally edu- 

 cated mind regards as the securest of its in- 

 tellectual possessions. In order to show 

 that such answers are to be found embedded 

 in the content of the normally educated 

 mind and in order to lay them bare, it will 

 be necessary to have recourse to the process 

 of explication. Explication, however, is 

 nothing strange to an academic audience. 

 It is true, indeed, that we no longer derive 

 the verb, to educate, from educere, but it is 

 yet a fact, as every one knows, that a large 

 part of education is eduction — the leading 



forth into light what is hidden in the fa- 

 miliar content of our minds. 



What are those answers? I shall present 

 them in the familiar and brilliant words of 

 one who in the span of a short life achieved 

 a seven-fold immortality: immortality as a 

 physicist, as a philosopher, as a mathema- 

 tician, as a theologian, as a writer of prose, 

 as an inventor and as a fanatic. From 

 this brief but "immortal" characteriza- 

 tion I have no doubt that you detect the au- 

 thor at once and at once recall the words : 

 Space is an infinite sphere whose center is 

 everywhere and whose surface is nowhere. 



You will observe that, without change of 

 meaning, I have substituted "space" for 

 "universe" and "surface" for "circum- 

 ference." This brilliant mot of Blaise 

 Pascal, as every one knows, has long been 

 valued throughout the world as a splendid 

 literary gem. I am not aware that it has 

 been at any time regarded seriously as a 

 scientific thesis. It may, however, be so 

 regarded. I propose to show, with your 

 cooperation, that this exquisite saying of 

 Pascal expresses with mathematical pre- 

 cision the firm, albeit unconscious, con- 

 viction of the normally educated mind re- 

 specting the size and the shape of the space 

 of our universe. Be good enough to note 

 carefully at the outset the cardinal phrases : 

 infinite sphere, center everywhere, surface 

 nowhere. 



If you are told that there is an object 

 completely enclosed and that the object is 

 equally distant from all parts of the en- 

 closing boundary or wall, you instantly and 

 rightly think of a sphere having that object 

 as center. Let me ask you to think of some 

 point, any convenient point, P, together 

 with all the straight lines or rays — caUed a 

 sheaf of lines or rays — that, beginning at 

 P, run out from it as far as ever the nature 

 of space allows. We ask : do all the rays of 

 the sheaf run out equally far? It seema 



