224 Prof. Stevelly on the Doctrine of Parallel Lines. 



Proposition 28 of Euclid's First Book 

 Follows at once, as in Euclid, from Prop. 27. 



Prop, 29 of Euclid's First Book. 



Any line (ccy) which falls upon two parallel lines {wx and yz) 

 makes (1) the alternate angles equal (as wxy=zyx)', (2) any 

 external angle equal to the internal, remote on the same side ; 

 and (3) the two internal, on the same side, together equal to two 

 right angles. (1.) From x 



let fall xz perpendicular to 

 yz ; and from y, yw perpen- 

 dicular to wx. These lines 

 are also respectively equal 

 to each other, and perpen- 

 dicular, the first to wXy the 

 latter to yz (Cor. I. and III. Lemma II.). Hence in the two 

 triangles, three sides of one equal respectively three sides of the 

 other. Hence (8th of Euclid's first book) the angle wxy=zyx. 

 The alternate angles are equal. From which the other two parts 

 follow obviously as in Euclid. 



The other propositions follow as in Euclid, without any aid 

 from the 12th axiom, which then becomes an easy deduction 

 from the above, thus (as in the notes to Elrington's Euclid). 



Euclid's 12ih Axiom. 



If angles xwA -H wAz are together less than two right angles, 

 the lines shall, if produced, at length meet on that side. 



Draw k^ so as to make xwA'\-wA^-=- two right angles, kz^ 

 is parallel to wx (as proved above). Let fall AB perpendicular 

 to WXy and take Al = 12 = 23 = &c. Let fall 1«, 2a', 3«", &c. 

 perpendiculars on A-?', and draw 16, 2c, &c. perpendicular to 

 these. 



It then is obvious from Cor. III. Lemma II., and from 26th 

 of first book of Euclid, that 2a' is twice as long as 1 a, 3a" three 

 times as long, and so on. Hence as we assume equal distances 

 along kzy the perpendiculars must at length become longer than 

 BA, and therefore these points and also the line kz must at 

 length pass to the other side of wx. Q.E.D. 



