238 CONTEMPORARY SCIENCE 



airplane. This means that whereas with reference to 

 axes fixed to the earth the path of the drop is vertical; 

 with reference to other axes, the path is not. Or, stating 

 the conclusion in general language, changing the axes of 

 reference (or effecting a mathematical transformation) 

 in general changes the shape of any line. If one imagines 

 the line forming a part of the space, it is evident that if 

 the space is deformed by compression or expansion the 

 shape of the line is changed, and if sufficient care is take^n 

 it is clearly possible, by deforming the space, to make the 

 line take any shape desired, or better stated, any shape 

 ^pecified by the previous change of axes. It is thus pos- 

 sible to picture a mathematical transformation as a defor- 

 mation of space. Thus I can draw a line on a sheet of 

 paper or of rubber and by bending and stretching the 

 sheet, I can make the line assume a great variety of 

 shapes; each of these new shapes is a picture of a suit- 

 able transformation. 



Now, consider world-lines in our four dimensional 

 space. The complete record of all our knowledge is a 

 series of sequences of intersections of such lines. By 

 analogy I can draw in ordinary space a great number of 

 intersecting lines on a sheet of rubber; I can then bend 

 and deform the sheet to please myself ; by so doing I do 

 not introduce any new intersections nor do I alter in the 

 least the sequence of intersections. So in the space of our 

 world-lines, the space may be deformed in any imaginable 

 manner without introducing any new intersections or 

 changing the sequence of the existing intersections. It is 

 this sequence which gives us the mathematical expression 

 of our so-called experimental laws ; a deformation of our 

 space is equivalent mathematically to a transformation 

 of axes, consequently we see why it is that the form of 

 our laws must be the same when referred to any and all 

 sets of axes, that is, must remain unaltered by any mathe- 

 matical transformation. 



Now, at last we come to gravitation. We can not imag- 



