6 4 o POPULAR SCIENCE MONTHLY. 



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 demon- 

 strated, 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 geom- 

 etry 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 thatsiichlichen 

 Richtigkeit des Satzes fiir unsern Raum gezweifelt habe, so wenig, wie an der 

 der Dreidimensionalitat des Raumes, obwohl er audi hier das logisch Hypoth- 

 etische 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 Wil- 

 liamson (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 Mathe- 

 matiques,' 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. 



