xii TRANSLATOR S INTRODUCTION. 



tl^is plain, straight forward, simple theorem: 

 \ho&amp;gt;e straights which arc produced indefin 

 itely from less than two right angles meet.&quot; 

 [This is the form which occurs in the Greek 



of Ku. i. : ( ).] 



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 a&amp;gt; flawless in the chapter 

 which Stolx devotes to it in 1885 as when 

 through Newton it first gave us our pre&amp;gt;mt 

 continuous number-system; 



But what fortune had this genius iu the light 

 with its self -chosen simple theorem? Was it 

 found to he deducible from all the definitions, 

 and tin- nine &quot;Common Notions,&quot; and the five 

 otlu-r Postulates of the immortal Elements? 

 Not so. But meantime Euclid went ahead 

 without it through twenty-eight propositions, 

 more than half his first hook. But at last 

 came the practical pinch, then as now the tri 

 angle .- angle-sum. 



He gets it by his twenty-ninth theorem: 44 A 

 Straight falling upon two parallel straights 

 make&amp;gt; the alternate angle- e&amp;lt;jiial.&quot; 



But for the proof of thi- he needs that re 

 calcitrant proposition which has how long 

 been keeping him awake nights and waking 



