EINSTEIN'S NEW THEORY — INFELD 193 



ever entered physics. It is impossible to grasp the importance of 

 Einstein's achievement without being aware of this point. TVe know 

 tJie properties of a Euclidean space from our high-scliool days: 

 through a point outside a line, we can draw one, and only one, line 

 parallel to the given one. But since the nineteenth century we know 

 that Euclidean geometry is only one of the many possible geometries. 

 The simplest case of a non-Euclidean geometry would be, for example, 

 the one experienced by two-dimensional creatures living on the surface 

 of a sphere. They would find that a journey "straight ahead" (that 

 is, along a great circle for you) leads them to their point of departure; 

 that the ratio of the circumference of a circle to its diameter is smaller 

 than TT. 



The background of our physical events is a four-dimensional world. 

 There is nothing mysterious about it. Every event, like the death 

 of Julius Caesar, is characterized by the "place" and time in which 

 it took place. The "place" of an event is characterized by three num- 

 bers; hence, together with time, we have four. The totality of all 

 possible events forms one four-dimensional world. All this has been 

 known and successfully applied since 1908 when the great mathe- 

 matician PL Minkowski gave the beautiful four-dimensional mathe- 

 matical form to Einstein's Special Kelativity Theory. 



Yet General Relativity Theory goes one important step farther. 

 We ask: Is our four-dimensional world fiat, like the plane in two 

 dimensions ? Or is it curved, like a curved surface of two dimensions ? 

 The difficulty with these questions is that, whereas we can easily 

 visualize a two-dimensional flat or curved space, it is difficult to do 

 so if the space is four-dimensional. But where our intuition stops, 

 mathematics does not. Even before Einstein's time, the mathematics 

 describing many-dimensional curved space was known, though it 

 developed fully only under the impetus of relativity. The develop- 

 ment of this branch of mathematics is connected with the names of 

 Gauss, Lobachevski, Bolyai, Riemann, Ricci, Levi-Civita, and others. 

 Let us say here only that a four-dimensional space is characterized 

 by 10 functions; that, once we know these functions, M-e know the 

 geometry of such a space; we know whether such a space is curved 

 and how its geometry changes from point to point. 



In my room I can characterize the position of the end of my pencil 

 by quoting its distances from the ceiling and two perpendicular walls. 

 Or, generally, the position of a point is designated by three numbei-s 

 in a given coordinate system. In a town, the names of streets and 

 house numbers form two coordinates denoting with sufficient accuracy 

 the positions of its inhabitants on a piece of a surface (at least when 

 they stay at home). Similarly, in our four-dimensional world of 

 events, we must have a coordinate system so as to name the four 



