NON-EUCLIDEAN GEOMETRY 157 



occupy us considerably in later chapters. Before leaving 

 the subject of gravitation I wish to say a little about 

 the meaning of space-curvature and non-Euclidean 

 geometry. 



Non-Euclidean Geometry. I have been encouraging you 

 to think of space-time as curved; but I have been careful 

 to speak of this as a picture, not as a hypothesis. It is 

 a graphical representation of the things we are talking 

 about which supplies us with insight and guidance. 

 What we glean from the picture can be expressed in a 

 more non-committal way by saying that space-time has 

 non-Euclidean geometry. The terms "curved space" 

 and "non-Euclidean space" are used practically synony- 

 mously; but they suggest rather different points of view. 

 When we were trying to conceive finite and unbounded 

 space (p. 81) the difficult step was the getting rid of 

 the inside and the outside of the hypersphere. There is 

 a similar step in the transition from curved space to 

 non-Euclidean space — the dropping of all relations to 

 an external (and imaginary) scaffolding and the holding 

 on to those relations which exist within the space itself. 

 If you ask what is the distance from Glasgow to New 

 York there are two possible replies. One man will tell 

 you the distance measured over the surface of the 

 ocean; another will recollect that there is a still shorter 

 distance by tunnel through the earth. The second man 

 makes use of a dimension which the first had put out 

 of mind. But if two men do not agree as to distances, 

 they will not agree as to geometry; for geometry treats 

 of the laws of distances. To forget or to be ignorant of 

 a dimension lands us into a different geometry. Dis- 

 tances for the second man obey a Euclidean geometry 

 of three dimensions; distances for the first man obey 



