SPACE AND GEOMETRY 39 



have come to us only like a fossil of ancient 

 thought, or like some brilliant butterfly im- 

 paled in a museum case. 



If we now claim a better understanding of 

 geometry than was possessed by the Greeks, it 

 is not through any lack of admiration of their 

 stupendous achievement; nor is it that we be- 

 lieve that we are better thinkers, or that our 

 ideas have yet reached anything like comple- 

 tion. I suspect that it is only the very young 

 vine that says, "When I grow big I shall reach 

 the sun." The older thinks only, "I climb a 

 little higher day by day." 



The geometry of Euclid is an excellent exam- 

 ple of a structure based upon several clear-cut 

 and distinct logical levels. In his "common no- 

 tions," such as that the whole is greater than 

 any one of its parts, Euclid presents a number 

 of general rules of thought (or rules of lan- 

 guage) which are useful in many other fields 

 beside the limited field of geometry. In his pos- 

 tulates he states the particular rules of his 

 geometry; and finally in his definitions he dis- 

 plays the subject matter, — points, lines, tri- 

 angles, circles and the like. Nowadays it is the 

 fashion to leave these elements undefined, but 

 that is a matter of taste. No word can be com- 



