Ladies and Gentlemen:

Today I am going to discuss the problem of curved space and its relation to the phenomena of gravitation. I have no doubt that any one of you can easily imagine a curved line or a curved surface, but at the mention of a curved, three-dimensional space your faces grow longer and you are inclined to think that it is something very unusual and almost supernatural. What is the reason for this common ‘horror’ for a curved space, and is this notion really more difficult than the notion of a curved surface? Many of you, if you will think a little about it, will probably say that you find it difficult to imagine a curved space because you cannot look on it ‘from outside’ as you look on a curved surface of a globe, or, to take another example, on the rather peculiarly curved surface of a saddle. However, those who say this convict themselves of not knowing the strict mathematical meaning of curvature, which is in fact rather different from the common use of the word. We mathematicians call a surface curved if the properties of geometrical figures drawn on it are different from those on a plane, and we measure the curvature by the deviation from the classical rules of Euclid. If you draw a triangle on a flat piece of paper the sum of its angles, as you know from elementary geometry, is equal to two right angles.