II 



Space and Geometry 



MANY ideas are associated in our minds 

 with the word 'space.' The man on the 

 street will say that space is what you put things 

 into or take things out of. He is making a sort 

 of abstraction or idealization of barrels, chests, 

 wardrobes and the like. It would be easy to say 

 that this is not the right concept of space, but 

 how absurd such a statement would be. By the 

 rules of language a word means what any large 

 body of people suppose it to mean, and when we 

 speak of space the idealized chests and ward- 

 robes are surely in the background of our 

 thought. But science demands some refinement 

 of common ideas, and often a more limited and 

 more technical use of language. Mathematics is 

 even more exacting, and when we analyze the 

 highly refined concept of space used by the 

 mathematicians we find it to be quite similar to 

 the concept of number. As number has its two 

 aspects, ordinal and cardinal, answering the 

 questions "In what order.?" and "How many.?" 

 so we have two ideas of space, one involving 



