Now, 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 assumptions, 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 coffin, 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 statement 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, Poincaré, about eighteen centuries 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 Académie 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 argument. 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.
- Poincaré, "Science and Hypothesis," transl. by G. B. Halsted, p. 30; where a clear account of these geometries will be found.