xii TRANSLATOR'S INTRODUCTION. 



this plain, straightforward, simple theorem: 

 ^rhose straights which are produced indefin- 



itely from less than two right angles meet. 

 [This is the form which occurs in the Greek 

 of Eu.1.29.] 



Let us not underestimate the subtle power 

 of that old Greek mind. We can produce no 

 Venus of Milo. Euclid's own treatment of 

 proportion is found as flawless in the chapter 

 which StoU devotes to it in 1885 as when 

 through Newton it first gave us our present 

 continuous number-system. 



But what fortune had this genius in the fight 

 with its self-chosen simple theorem? Was it 

 found to be deducible from all the definitions, 

 and the nine "Common Notions/' and the five 

 other Postulates of the immortal Elements? 

 Not so. But meantime Euclid went ahead 

 without it through twenty-eight propositions, 

 more than half his first book. But at last 

 came the practical pinch, then as now the tri- 

 angle's angle-sum. 



He gets it by his twenty-ninth theorem: "A 

 straight falling upon two parallel straights 

 makes the alternate angles equal." 



But for the proof of this he needs that re- 

 calcitrant proposition which has how long 

 been keeping him awake nights and waking 



