438 PROFESSOR KELLAND ON THE LIMITS OF OUR KNOWLEDGE 



Prop. VII. If a straight line falling upon two other st7'aigJit lines makes the alternate 

 angles equal to one another, these two straight lines are parallel (Euc. I. 27.) 

 And from any given j)oint in one of them there can he drawn one and only one 

 straight line to the other, mhieh shall make the alternate angles equal. 



Let EF make with AB and CD the alternate angles AEF, EFD, equal to one 

 another. 



Let G be any given point in AB : make FH = EG ; the alternate angles which 

 the straight line GH makes Avith AB and CD, are equal. 



Bisect EF in K, join GK, KH : the triangles GEK, ^ 



KFH, are equal in every respect (Euc. I. 4) : therefore 

 ..^GKE = HKF; to each of these add EKH, then the , 



F L H ^ 



angles GKE and EKH together are equal to HKF and 



EKH ; but HKF and EKH are equal to two riglit angles ; therefore GKE and 

 EKH are equal to two right angles, and GKH is a straight line. But the angles 

 EGK, KHP, are equal, and they are alternate angles; therefore, through the 

 given point G, a straight line GH has been drawn to CD, making the alternate 

 angles equal to one another. 



Also, from the same point G, no other straight line can be drawn to CD which 

 shall make the alternate angles equal. 



For, if possible, let GL be a straight line which makes the alternate angles 

 BGL, GLC, equal to one another. Then, because BGH = GHC, by subtraction 

 LGH is equal to the difference between GLC and GHC ; that is, one of the interior 

 opposite angles of a triangle is equal to the difference between the exterior angle 

 and the other interior opposite angle, which is impossible (Ax. Cor. 1.) 



Prop. VIII. If from a given point without a given straight line, straight lines he 

 drawn to the line, the angles which they make with it hecome less and less as the 

 distance from the pei'pendicular increases ; and a. straight line may he drawn 

 from the given point to the given straight line, so as to make with it an angle 

 less than any assignable angle, hy proceeding far enough. 



Let A be the given point ; BC the given straight line ; AB perpendicular to 

 BC ; AD any other line drawn from 

 A to BC. Make DE= AD ; join AE, 



and makeEF = AE, and so on. The 



angles ADB, AEB, AFB, continually 



diminish (Euc. 1. 16). And by proceeding far enough a line may be drawn making 



with the given linean angle less than any assignable angle. 



