June 33, 1913] 



SCIENCE 



891 



seen the length of a ray contains exactly as 

 many miles as there are integers or terms 

 in (8). The question, then, is: has the 

 totality of these terms a definite assign- 

 able number? The answer is, yes. To 

 show it, look sharply at the following fact, 

 a bit difficult to see only because it is so 

 obvious, being writ, so to speak, on the 

 very surface of the eye. I wish, in a word, 

 to make clear what is meant by the cardi- 

 nal number of any given class of things. 

 The fingers of my right hand constitute a 

 class of objects ; the fingers of my left hand, 

 another class. We can set up a one-to-one 

 correspondence between the classes, pair- 

 ing the objects in the one with those in the 

 other. Any two classes admitting of such 

 a correlation are said to be equivalent. 

 Now given any class K, there is another 

 class C composed of all the classes each of 

 which is equivalent to K. C is called the 

 cardinal number of K, and the name of C, 

 if it have received a name, tells how many 

 objects are in E. Thus, if K is the class of 

 the fingers of my right hand, the word five 

 is the name of the class of classes each 

 equivalent to K. Now to the application. 

 The terms of (8) constitute a class JS^ (of 

 terms). Has it a definite number? Yes. 

 What is it? It is the class of all classes 

 each equivalent to E. Has this number- 

 class received a name of its own ? Yes, and 

 it has, like many other numbers, received 

 a symbol, namely, ^o, read Aleph null. It 

 is, then, this cardinal number Aleph, not 

 familiar, indeed, but perfectly definite as 

 denoting a definite class, it is this that tells 

 us how many terms are in (8) and there- 

 with tells us the length of the rays of our 

 sheaf. Herewith the concept that was 

 forming is now completely formed: space 

 is a sphere centered at P. 



But is the sphere, as Pascal asserts, an 

 infinite sphere ? We may easily see that it 

 is. Again consider the sequence (8) and 



with it the similar sequence (<S"), 



(S) 1, 2, 3, 4, 5, 6, 7, ... , 

 (S') 2, 4, 6, 8,10,12,14, .... 



Observe that all the terms in {8') are in 

 {8) and that (8) contains terms that are 

 not in {8'). {8') is, then, a proper part 

 of (8). Next observe that we can pair 

 each term in (8) with the term below it in 

 {8'). That is to say: the whole, (8), is 

 equivalent to one of its parts, {8'). A 

 class that thus has a part to which it is 

 equivalent is said to be infinite, and the 

 number of things in such a class is called 

 an infinite number. Aleph is, then, an 

 infinite number, and so we see that the 

 rays of our sheaf, the radii of our sphere, 

 are infinite in length: space is an infinite 

 sphere entered at P. 



Finally, what of the phrases, center 

 everywhere, surface nowhere? Can we 

 give them a meaning consistent with com- 

 mon usage and common sense? We can, 

 as follows. Let be any chosen point 

 somewhere in your neighborhood. By say- 

 ing that the center P is everjrwhere we 

 mean that P may be taken to be any point 

 within a sphere centered at and having 

 a fijiite radius, a radius, that is, whose 

 length in miles is expressed by any integer 

 in {8). And by saying that the surface 

 of our infinite sphere is nowhere we mean 

 that no point of the surface can be reached 

 by traveling out from P any finite number, 

 however large, of miles, by traveling, that 

 is, a number of miles expressed by any 

 number, however large, in {8). 



Here we have touched our primary goal : 

 we have demonstrated that men and wo- 

 men whose education, in respect of space, 

 has been of normal type, believe pro- 

 foundly, albeit unawares, that the space of 

 our universe is an infinite sphere of which 

 the center is everywhere and the surface 

 nowhere. Such is the beautiful conception, 



