IS EUCLID'S GEOMETRY MERELY A THEORY? 555 



Xow, Euclid's geometry is, of course, something more than a game. 

 Its rules — the twelve axioms and five postulates — taken as a group, 

 Euclid might well be proud of. Most people believe that the whole 

 body of his proof rests upon them as upon an eternally established 

 foundation. Eternity is too long to contemplate, but we are certain 

 that to the present time, in no instance, have these seventeen assump- 

 tions, when correctly used, ever led to a detectable error. His reasoning 

 is always consistent in itself, always in perfect accord with the known 

 laws of mechanics. One does not feel that the system is a Mahomet's 

 coflBn, hovering unsupported in mid-air; it seems to rest on the solid 

 earth, and very firmly. Where, then, is any weakness in the foundation 

 that he laid ? 



Possibly, in his sleep through the centuries, Euclid has turned over 

 once or twice at doubts, first raised by Ptolemy in the second century 

 A.D., who never became quite convinced that a certain momentous state- 

 ment was perfectly self-evident, a statement which Euclid used without 

 proving. Apparently it was an afterthought with Euclid in the first 

 place, for not until he had reasoned himself well into the heart of his 

 subject did the need for it, or for something like it, become imperative. 

 Then he asserted quite dogmatically that: Through the same point 

 there can not be two parallels to the same straight line. Ptolemy, 

 hoping to strengthen Euclid's foundation, tried to prove this parallel 

 postulate, but concerning the outcome, Poincare, about eighteen cen- 

 turies later, has recently said : " What vast effort has been wasted in 

 this chimeric hope is truly unimaginable. Finally, in the first quarter 

 of the nineteenth century, and almost at the same time, a Hungarian 

 and a Russian, Bolyai and Lobachevski, established irrefutably that this 

 demonstration is impossible: they have almost rid us of geometries 

 * sans postulatum ' ; since then the Academic des Sciences receives only 

 about one or two new demonstrations a year." ^ The parallel postulate, 

 then, is a weak spot in the Euclidean system. The demonstration that 

 beyond all doubt no proof of its correctness can be devised was an 

 epoch-making discovery. Bolyai's share in this event took concrete 

 form in a brief appendix to a work by his father, published in 1831. 

 Halsted characterizes this document as " the most extraordinary two 

 dozen pages in the history of human thought." 



Our chief concern for the next few moments will be to comprehend 

 why the truth of the Euclidean postulate can not be established by argu- 

 ment. After that, I shall try to show why it is not self-evident. These 

 steps taken, I believe that the reader will agree with me that, since it 

 can not be proved, one may freely choose some other postulate in its 

 stead, and thus develop a different geometry every whit as trustworthy. 



' Poincar^, ' ' Science and Hypothesis, ' ' transl. by G. B. Halsted, p. 30 ; 

 where a clear account of these geometries will be foiind. 



