

ON GEOMETRY 
621 



like a sphere. Try as we may, we shall be unable to build on a plane any 00ordinate system which will fit the last formula. Thus we arrive at an important conclusion; namely, that from measurements we have a structural hint as to the kind of world we are in. 



Section B. On the notion of the 'Internal Theory of Surfaces'.
Let us imagine some twodimensional beings confined to their surface and unable to have a look at that surface from our third dimension. For them our third dimension would be 'unthinkable', and therefore the surface of a sphere like our earth which is curved in the. third dimension would also be 'unthinkable' or 'beyond them'. Should they conduct some measurements in their 'world' and find that these measurements did not fit any of the first three formulae but only the fourth, they would have to reconstruct radically their 'world conception' and conclude that their world was a spherical surface. Our own situation does not differ radically from the situation of the inhabitants of this hypothetical twodimensional world,
If we find ourselves so restricted as not to know whether we are finally dealing with a flat or spherical surface, we can select a point 0 and with a definite radius R describe from this point a circle ABC. Then we can measure the circumference of this circle. Now we know from geometry that in the plane the circumference of the circle 



where R is the radius of the circle and ... If our surface is flat (ABCD), our measurement of L and R will satisfy the relation expressed in the formula. But if the surface is curved, our R = OA will be larger than R' =AO', and we shall find that our x is not 3.1415 . . . , but smaller. We see once more that the metrical proper 


ties of our world throw some light on its structural character.
We should notice also that the curvature of a two 



dimensional surface is in the third dimension and that it is the means of giving us data about the surface without our leaving the surface and going into a third dimension. It is easy to convince oneself about these facts by taking 12 wires or strings of equal length and constructing the figure shown in Fig. 8. If we build it on a flat surface the 12 equal wires will fit exactly. 




But if we try this experiment on a curved surface, for 
Fig. 8 



instance on a pillow, or saddleshaped surface, the last closing wire will not fit, and will be too short or too long depending on the kind of surfaces we have. 


