46 THE ANATOMY OF SCIENCE 



we set up a body of rules according to which 

 we agree to be governed as long as we are play- 

 ing a particular game of geometry. Thus we 

 make rules for what we call a transformation 

 whereby a figure is reproduced in (rather than 

 moved to) another part of our diagram, and we 

 ordinarily make the rules such that the figure 

 thus produced has the same intrinsic metrical 

 properties as its original. Thus the area, the 

 length of a certain side, the angle between two 

 sides, we shall agree to call the same in the new 

 figure as in the old. It will suffice to consider 

 two kinds of transformation, one of which we 

 may call parallel shift, and the other, rotation. 



The first of these is so defined that every line 

 produced by the parallel shift is parallel to its 

 original. This is illustrated in Figure 2, where 

 ABDC is the original figure, and A'B'D'C is 

 its reproduction. Such a transformation may be 

 made in a single step, as shown in the figure, 

 or in a succession or "group" of steps, each of 

 which is itself a parallel shift. 



Since in any parallel shift a line goes over 

 into another line of equal length, it is possible 

 not onl}^ to compare the length of any part of 

 the line AB with the length of any other part 

 of it, but also with the length of any part of 



