66 THE ANATOMY OF SCIENCE . 



ture and the top represents a period of one 

 year. 



A complete picture of the motions of bodies 

 in a space of three dimensions would require a 

 four-dimensional construction, which we might 

 represent by its projection upon a three-dimen- 

 sional space, as we have represented our three- 

 dimensional model by its projection upon a 

 plane, but such a construction is difficult to 

 make and also to visualize. 



Now under what circumstances are we to 

 think of a set of figures such as I have shown 

 as constituting a geometry? I have puzzled 

 over this perplexing question, and it seems to 

 me that the answer is simply the one suggested 

 in the last chapter. A geometry is more than a 

 set of figures; it involves also the operations 

 which permit us to compare one figure with 

 another and thus establish rules of measure- 

 ment. If we can think of any operations and 

 transformations to which we can subject the 

 figures that are before us, analogous to the 

 transformations of ordinary geometry, and if 

 these operations give interesting and useful re- 

 sults, then and only then may we be said to 

 have a geometry. 



I am going to apply this criterion to our pic- 



