Page:Popular Science Monthly Volume 67.djvu/646

This most celebrated, most notorious of all postulates, Euclid's parallel-postulate, is not used for his first 28 propositions. When at length used, it is seen to be the inverse of a proposition already demonstrated, the seventeenth, as Proklos remarked, therefrom, according to Lambert, arguing its demonstrability. Moreover, its one and only use is in proving the inverse of another proposition already demonstrated, the twenty-seventh. No one had a doubt of the necessary external reality and exact applicability of the postulate. The Euclidean geometry was supposed to be the only possible form of space-science; that is, the space analyzed in Euclid's axioms and postulates was supposed to be the only non-contradictory sort of space. Even Gauss never doubted the actual reality of the parallel-postulate for our space, the space of our external world, according to Dr. Max Simon, who says in his 'Euclid,' 1901, p. 36:

Nur darf man nicht glauben, dass Gauss je an der thatsächlichen Richtigkeit des Satzes für unsern Raum gezweifelt habe, so wenig, wie an der der Dreidimensionalität des Raumes, obwohl er audi hier das logisch Hypothetische erkannte.

But could not this postulate be deduced from the other assumptions and the 28 propositions already proved by Euclid without it? Euclid had among these very propositions demonstrated things more axiomatic by far. His twentieth, 'Any two sides of a triangle are together greater than the third side,' the Sophists said, even donkeys knew. Yet, after he has finished his demonstration, that straight lines making with a transversal equal alternate angles are parallel, in order to prove the inverse, that parallels cut by a transversal make equal alternate angles, he brings in the unwieldy assumption thus translated by Williamson (Oxford, 1781):

11. And if a straight line meeting two straight lines makes those angles which are inward and upon the same side of it less than two right angles, the two straight lines being produced indefinitely will meet each other on the side where the angles are less than two right angles.

This ponderous assertion is neither so axiomatic nor so simple as the theorem it is used to prove. As Staeckel says:

It requires a certain courage to declare such a requirement, alongside the other exceedingly simple axioms and postulates.

Says Baden Powell in his 'History of Natural Philosophy,' p. 34: The primary defect in the theory of parallel lines still remains.

This supposed defect an ever renewing stream of mathematicians tried in vain to remedy. Some of these merely exhibit their profound ignorance, like Ferdinand Hoefer, who in his 'Histoire des Mathematiques,' Paris, 1874, p. 176, says:

Certain defects with which Euclid is reproached may be explained by simple transpositions. Such is the case of the Postulatum V.