TRANSLATOR S INTRODUCTION. vii 



prove the inverse, that parallels cut by a trans 

 versal make equal alternate angles, he brings 

 in the unwieldy assumption thus translated by 

 Williamson (Oxford, 1781) : 



&quot;11. And if a straight line meeting two 

 vStraight lines make those angles which are in 

 ward 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.&quot; 



As Staeckel says, &quot;it requires a certain 

 courage to declare such a requirement, along 

 side the other exceedingly simple assumptions 

 and postulates.&quot; But was courage likely to 

 fail the man who, asked by King Ptolemy if 

 there were no shorter road in things geometric 

 than through his Elements? answered, &quot;To 

 geometry there is no special way for kings!&quot; 



In the brilliant new light given by Bolyai 

 and Lobachevski we now see that Euclid un 

 derstood the crucial character of the question 

 of parallels. 



There are now for us no better proofs of the 

 depth and systematic coherence of Euclid s 

 masterpiece than the very things which, their 

 cause unappreciated, seemed the most notice 

 able blots on his work. 



