442 



PROFESSOR KELLAND ON THE LIMITS OF OUR KNOWLEDGE 



be drawn at right angles to CD, and produced to meet AB in G, then LH=KF. 

 Now (Prop. X.) GH:::^EF, therefore LG^KE. Also, since E, F, and H are right 

 angles, LGE is less than a right angle, and therefore LGR greater than a right 

 angle. Cut off the right angle LGI ; then the triangles LGI, KEP have two 

 angles of the one equal to two angles of the other, but the side LG adjacent to 

 equal angles in the one greater than KE in the other ; therefore LI is greater than 

 KP (Prop.^V.), but LR^Ll .-. LR is much greater than KP, and RS:^PQ. 



Prop. XIV. If through the middle point of the common perpendicular to two parallel 

 straight lines, a straight line he drawn perpendicular to it, this line will bisect 

 at right angles all lines drawn so as to maize equal angles towards the same side 

 with the given parallels. 



Let EF be the common perpendicular to the two parallel straight lines AB, 

 CD. Through X, the point of bisection of EF, let XY 

 be drawn perpendicular to EF, meeting GH in Y. XY 

 bisects GH perpendicularly. For the triangle EXY= 



FXY ; therefore EY=FY, -^XEY=XFY and -:^EYX= ^ 



FYX : hence ^YEG= YFH, and triangle YEG= YFH ; therefore GY = HY, that 

 is, GH is bisected by XY. 



It is also bisected perpendicularly; for ^EYX=FYX and ^EYG=FYH; 

 therefore -=:::XYG=XYH, and each of them is a right angle. 



Cor. 1. All straight lines drawn so as to make equal angles respectively with 

 each of two parallel straight lines towards the same side are themselves parallel. 



Cor. 2. And the straight line which bisects them all is parallel to the given 

 parallels. 



These propositions appear to draw the connection between parallelism and 



rectilinearity very close. Props. X., XL, and XII. tally very well with the (po- 

 pular) view of parallels as circles ofvery large and equal radius ; for instance, the 

 equality of the angles at which any of the lines parallel to that which is perpen- 



