*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 termsâ€”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 assumptions except this one, in place of which he substituted a contradictory statement of his own making; he hazarded the novel assertion that: Through a given point there *can be 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 conclusion 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

was substituted for it and a new geometry developed without encountering 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 connection with any other, can not be proved at all. Moreover, this achievement, 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 definition of what parallels are, namely, if both 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