Hinton - The Fourth Dimension.pdf

56
THE FOURTH DIMENSION
although that is not the form in which they stated their
results.
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, or something that must
be assumed. The words of his ninth postulate are these:
“If 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 this 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 possible 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.
C
D
A B
Fig. 31.
Take a line AB and a point C.
We say and see and know that
through C can only be drawn one
line parallel to AB.
But Bolyai said: “I will draw two.” Let CD be parallel