March 11, 1904.] 



SCIENCE. 



407 



criticism in Science, of McCosh's disciple 

 Ormond, 'we may well doubt whether a 

 thinker, standing with one foot firmly 

 planted on the Rock of Ages and the other 

 pointing heavenward, has struck the atti- 

 tude most conducive to progress.' 



Kant, supposing that we knew Euclid's 

 geometry and Aristotle's logic to be true 

 absolutely and necessarily, accounted for 

 the paradox by teaching that this seem- 

 ingly universal synthetic knowledge was in 

 reality particular, being pai't of the appa- 

 ratus of the human mind itself. 



But now the very foundations are cut 

 away from under the Kantian system of 

 philosophy by this new geometry which is 

 in simple and perfect harmony with ex- 

 perience, with experiment, with the prop- 

 erties of the solid bodies and the motions 

 about us. Thus this new geometry has 

 given explanation of what in the old geom- 

 etry was accepted without explanation. 



12. WHAT GEOMETRY IS. 



At last we really know what geometry is. 

 Geometry is the science created to give 

 understanding and mastery of the external 

 relations of things; to make easy the ex- 

 planation and description of such relations, 

 and the transmission of this mastery. 

 Geometry is the most perfect of the sci- 

 ences. It precedes experiment and is safe 

 above all experimentation. 



The pure idea of a plane is something we 

 have made, and by aid of which we see 

 surfaces as perfectly plane, over-riding 

 imperfections and variations, which them- 

 selves we can see only by help of our self- 

 created precedent idea. Just so the 

 straight line is wholly a creation of our 

 own. 



13. ARE THERE ANY LINES? 



I was once consulted by an eminent 

 theologian and a powerful chemist as to 

 whether there are really any such things 

 as lines. I drew a chalk-mark on the 



blackboard, and used the boundary idea. 

 Along the sides of the chalk-mark is there 

 a common boundary where the white ends 

 and the black begins, neither white nor 

 black, but common to both? 



Said the theologian, yes. Said the chem- 

 ist, no. 



Though lines are my trade, I sympa- 

 thized with the chemist. 



There is nothing there until I create a 

 line and then see it there, if I may say I 

 see what is an invisible creation of my 

 mind. 



Geometry is in structure a system of 

 theorems deduced in pure logical way 

 from certain unprovable assumptions pre- 

 created by auto-active animal and human 

 minds. 



14. THE REQUIREMENT OP RIGOR IN 

 REASONING. 



Some unscientific minds have a personal 

 antipathy to 'a perfect logical system,'" 

 'deduced logically from simple funda- 

 mental truths.' But as Hilbert says; 

 "The requirement of rigor, which has be- 

 come proverbial in mathematics, corre- 

 sponds to a universal philosophic necessity 

 of our understanding; and, on the other 

 hand, only by satisfying this requirement 

 do the thought content and the suggestive- 

 ness of the problem attain their full effect. 

 Besides, it is an error to believe that rigor 

 in the proof is the enemy of simplicity. 

 On the contrary, we find it confirmed by 

 numerous examples that the rigorous 

 method is at the same time the simpler 

 and the more easily comprehended. The 

 very effort for rigor forces us to find out 

 simpler methods of proof. 



"Let us look at the principles of an- 

 alysis and geometry. The most suggestive 

 and notable achievements of the last cen- 

 tury in this field are, as it seems to me, 

 the arithmetical formulation of the concept 



