The fourth dimension

2 56 THE FOURTH DIMENSION

7. although that is not the form in which they stated their

(- / results.

f The way in which they were led to these results was the

following. Euclid had stated the existence of parallel lines

as a postulate—putting frankly this unproved proposition

—that one line and only one parallel to a given straight

line can be drawn, as a demand, as something that must

be assumed. The words of his ninth postulate are these:

“Tf a straight line meeting two other straight lines

makes the interior angles on the same side of it equal

to two right angles, the two straight lines will never meet.”

The mathematicians of later ages did not like this bald assumption, and not being able to prove the proposition they called it an axiom—the eleventh axiom.

Many attempts were made to prove the axiom; no one doubted of its truth, but no means could be found to

~~ demonstrate it. At last an Italian, Sacchieri, unable to find a proof, said: ‘‘Let us suppose it not true.” He deduced the results of there being possibly two parallels to one given line through a given point, but feeling the waters too deep for the human reason, he devoted the latter half of his book to disproving what he had assumed in the first part.

Then Bolyai and Lobatchewsky with firm step entered on the forbidden path. There can be no greater evidence of the indomitable nature of the human spirit, or of its manifest destiny to conquer all those limitations which bind it down within the sphere of sense than this grand assertion of Bolyai and Lobatchewsky.

é Take a line aB and a pointc. We

say and see and know that through c 3 can only be drawn one line parallel Fig. 31. to AB. But Bolyai said: “1 will draw two.” Let cp be parallel