556 THE POPULAR SCIENCE MONTHLY 



Let no one suppose, however, that the least dispute has ever arisen 

 as to what parallel lines are. Euclid defined them as: Straight lines 

 which are in the same plane, and which, being produced ever so far 

 both ways, do not meet. All geometers accept this definition, just as at 

 whist every one agrees on the meaning of certain terms — calls a spade 

 a spade, and so forth. To the brilliant young Lobachevski no good 

 reason presented itself why through a given point there should be only 

 one such parallel to a given straight line. He accepted all Euclid's as- 

 sumptions except this one, in place of which he substituted a contra- 

 dictory statement of his own making; he hazarded the novel assertion 

 that: Through a given point there can he two parallels to the same 

 straight line. On this foundation he erected a new geometry, building 

 proposition upon proposition until he had reared an edifice as coherent 

 and in every respect as perfect as the geometry of Euclid. What con- 

 clusion may we draw? This: had Euclid's postulate been eternally 

 true, then to deny it while holding to his other axioms would have led 

 Lobachevski into endless inconsistencies. But the fact that its contrary 



FiO. 1. 



was substituted for it and a new geometry developed without encounter- 

 ing any logical obstacle shows that the postulate rests on nothing more 

 fundamental than itself; shows that it swings, so to speak, in mid-air, 

 unaffected by Euclid's other assertions. No statement can be proved 

 by itself alone ; consequently, this statement, having no logical connec- 

 tion with any other, can not be proved at all. Moreover, this achieve- 

 ment, broadly comprehended, set the entire Euclidean system aswing 

 without support; its supposed connection with the solid earth was a 

 fact only of the imagination. 



I promised, in the next place, to show that Euclid's postulate lacks 

 self-evidence. In Fig. 1 there is a point P, lying without a straight 

 line CD. Another straight line AB passes through this point, and we 

 shall imagine both AB and CD to be produced ever so far both ways. 



Now AB will be parallel to CD, if they conform to Euclid's defi- 

 nition of what parallels are, namely, if l)oth lines are straight, and in 

 the same plane, and being produced indefinitely, do not meet. In that 

 position the lines would be parallel, but let us start from the position 



