Makch 11, 1904.] 



SCIENCE. 



405 



inches from the line are farther apart than 

 their points one inch from the line. 



In the Euclidean geometry every three 

 points are either on a straight line or a 

 circle. In this non-Euclidean geometry 

 there are triplets of paints which are nei- 

 ther costraight nor concyclic. Thus three 

 points each one inch above a straight line are 

 neither on a straight line nor on a circle. 



7. MISTAKE OF THE INEXPEET. 



These seeming paradoxes could be multi- 

 plied indefinitely, and they form striking 

 examples of this new geometry. They 

 seem so bizarre, that the first impression 

 produced on the inexpert is that the tradi- 

 tional geometry could easily be proved, as 

 against this rival, by careful experiments. 

 Into this error have fallen Professors An- 

 drew W. Phillips and Irving Fisher, of 

 Yale University. In their 'Elements of 

 Geometry,' 1898, page 23, they say: "Lo- 

 bachevski proved that we can never get rid 

 of the parallel axiom without assuming the 

 space in which we live to be very different 

 from what we know it to be through ex- 

 perience. Lobachevski tried to imagine a 

 different sort of universe in which the 

 parallel axiom would not be true. This 

 imaginary kind of space is called non- 

 Eiiclidean space, whereas the space in 

 which we really live is called Euclidean, 

 because Euclid (about 300 B. C.) first 

 wrote a systematic geometry of our space." 



Now, strangely enough, no one, not even 

 the Tale professors, can ever prove this 

 naive assertion. If any one of the possible 

 geometries of uniform space could ever be 

 proved to be the system actual in our ex- 

 ternal physical world, it certainly could 

 not be Euclid's. 



Experience can never give, for instance, 

 such absolutely exact metric results as pre- 

 cisely, perfectly two right angles for the 

 angle sum of a triangle. As Dr. E. W. 

 Hobson says : " It is a very significant fact 



that the operation of counting, in connec- 

 tion with which numbers, integral and 

 fractional, have their origin, is the one and 

 only absolutely exact operation of a mathe- 

 matical character which we are able to 

 undertake upon the objects which we per- 

 ceive. On the other hand, all operations 

 of the nature of measurement which we can 

 perform in connection with the objects of 

 perception contain an essential element of 

 inexactness. The theory of exact measure- 

 ment in the domain~of the ideal objects of 

 abstract geometry is not immediately de- 

 rivable from intuition." 



8. THE ARTIFICIALLY CREATED COMPONENT 

 IN SCIENCE. 



In connecting a geometry with experi- 

 ence there is involved a process which we 

 find in the theoretical handling of any em- 

 pirical data, and which, therefore, should 

 be familiarly intelligible to any scientist. 



The results of any observations are al- 

 ways valid only within definite limits of 

 exactitude and under particular conditions. 

 When we set up the axioms, we put in place 

 of these results statements of absolute pre- 

 cision and generality. In this idealization 

 of the empirical data our addition is at first 

 only restricted in its arbitrariness in so 

 much as it must seem to approximate, must 

 apparently fit, the supposed facts of ex- 

 perience, and, on the other hand, must in- 

 troduce no logical contradiction. Thus our 

 actual space to-day may very well be the 

 space of Lobachevski or Bolyai. 



If anything could be proved or disproved 

 about the nature of space or geometry by 

 experiments, by laboratory methods, then 

 our space could be proved to be the space 

 of Bolyai by inexact measurements, the 

 only kind which will ever be at our dis- 

 posal. In this way it might be known to 

 be Mom-Euclidean. It never can be known 

 to be Euclidean. 



