410 



SCIENCE. 



[N. S. Vol. XIX. No. 480. 



which he shows scant warrant, expresses 

 himself after the following naive fashion: 



A clear understanding of the essential limita- 

 tions of systematic physics is important to the 

 engineer; it is I think equally important to the 

 biologist, and it is of vital importance to the 

 physicist, for, in the case of the physicist, to 

 raise the question as to limitations is to raise the 

 question as to whether his science does after all 

 deal with realities, and the conclusion which must 

 force itself on his mind is, I think, that his sci- 

 ence, the systematic part of it, comes very near 

 indeed to being a science of unrealities. 



Of course, we deeply sympathize with 

 this seemingly sad perception, with its 

 accompanying 'simple weeps,' 'trailing 

 weeps ' and ' steady weeps, ' but are tempted 

 to prescribe a tonic or bracer in the form 

 of a correspondence course in non-Euclid- 

 ean geometry. 



At least in part, space is a creation of 

 the human mind entering as a subjective 

 contribution into every physical experi- 

 ment. Experience is, at least in part, 

 created by the subject said to receive it, 

 but really in part making it. 



In rigorously founding a science, the 

 ideal is to create a sj^stem of assumptions 

 containing an exact and complete descrip- 

 tion of the relations between the element- 

 ary concepts of this science, its statements 

 following from these assumptions by pure 

 deductive logic. 



18. GEOMETRY NOT EXPERIMENTAL. 



Now, geometry, though a natural sci- 

 ence, is not an experimental science. If 

 it ever had an inductive stage, the experi- 

 ments and inductions must have been made 

 by our pre-human ancestors. 



Says one of the two greatest living 

 mathematicians, Poincare, reviewing the 

 work of the other, Hilbert 's transcendently 

 beautiful ' Grundlagen der Geometric ' : 



What are the fundamental principles of geom- 

 etry? What is its origin; its nature; its scope? 

 These are questions which have at all times en- 



gaged the attention of mathematicians and think- 

 ers, but which took on an entirely new aspect, 

 thanks to the ideas of Lobachevski and of Bolyai. 



For a long time we attempted to demonstrate 

 the proposition known as the postulate of Euclid; 

 we constantly failed; we know now the reason for 

 these failures. 



Lobachevski succeeded in building a logical edi- 

 fice as coherent as the geometry of Euclid, but in 

 which the famous postulate is assumed false, and 

 in which the sum of the angles of a triangle is 

 always less than two right angles. Riemann de- 

 vised another logical system, equally free from 

 contradiction, in which this sum is on the other 

 hand always greater than two right angles. These 

 two geometries, that of Lobachevski and that of 

 Riemann, are what are called the non-Euclidean 

 geometries. The postulate of Euclid then can not 

 be demonstrated ; and this impossibility is as abso- 

 lutely certain as any mathematical truth whatso- 

 ever. * * * 



The first thing to do was to enumerate all the 

 axioms of geometry. This was not so easy as one 

 might suppose; there are the axioms which one 

 sees and those which one does not see, which are 

 introduced unconsciously and without being no- 

 ticed. 



Euclid himself, whom we suppose an impeccable 

 logician, frequently applies axioms which he does 

 not expressly state. 



Is the list of Professor Hilbert final? We may 

 take it to be so, for it seems to have been drawn 

 up with care. 



But just here this gives us a startling 

 incident : the two greatest living mathe- 

 maticians both in error. In my own class 

 a young man under twenty, E. L. Moore, 

 proved that of Hilbert 's ' betweenness ' as- 

 sumptions, axioms of order, one of the five 

 is redundant, and by a proof so simple and 

 elegant as to be astonishing. Hilbert has 

 since acknowledged this redundancy. 



The same review touches another funda- 

 mental point as follows : 



Hubert's Fourth Book treats of the measure- 

 ment of plane surfaces. If this measurement can 

 be easily established without the aid of the prin- 

 ciple of Archimedes, it is because two equivalent 

 polygons can either be decomposed into triangles 

 in such a way that the component triangles of the 

 one and those of the other are equal each to each 

 (so that, in other words, one polygon can be con- 



