44 THE ANATOMY OF SCIENCE 



Lobachevski and Bolyai, has as yet found ap- 

 plication only in minor ways, but its simplicity 

 and elegance make us almost certain that it 

 also will eventually be of great utility. 



These curved geometries are already familiar 

 to many of you, and I do not propose to discuss 

 them further, but rather to call your attention 

 to two geometries which have come into very 

 recent notice, of which one bids fair to rival 

 in importance Euclidean geometry itself. The 

 parallel postulate of Euclid has been the storm 

 center of geometry for centuries, but little 

 attention has been paid to his postulate regard- 

 ing circles. It is, however, by retaining the 

 parallel postulate and abandoning the circular 

 postulate that the two new geometries of which 

 I am about to speak are obtained.^ 



First, however, let me show how in one respect 

 modern geometrical thought has run ahead of 

 Euclid's. The chief problem of geometry is to 

 compare the metrical properties of one figure 

 with those of another. Language has its words 



3 The most important of these two geometries (the 

 geometry of asymptotic rotation) was clearly in the mind 

 of Minkowski, as shown by some of the figures in his Baum 

 und Zeit. The details of this geometry and of the other one 

 (the geometry of shear rotation) were worked out in some 

 detail by Professor E. B. Wilson and myself {Proceedings 

 American Academy of Arts and Sciences, 1912). 



