Any straight line
through point P is also by definition a great circle - and all great
circles on a sphere intersect one another. Thus on the surface of
a sphere there are no straight lines through a point that do not
meet the original line. Where on the plane there was one line that
never met, now we have a geometry in which there are none.
How do we know that there may not be other options?
For two thousand years mathematicians sought to prove that the
parallel postulate must be true, in the sense that there could never
be more than one straight line though a point that did not meet
a given line. The idea that this might not be true struck terror
into their Euclidean hearts offending rational sensibilities and
evoking a sense of moral outrage.
In order to prove the parallel postulate mathematicians resorted
primarily to the method of reductio ad absurdum in which one begins
by assuming the inverse of what ones hope to prove and then showing
how this leads to a contradiction. That is, they assumed the parallel
postulate was false and tried to show how that led inevitably to
What they discovered was a host of absurdities – but, infuriatingly,
no outright contradictions. Girolamo Saccheri, an eighteenth century
Jesuit priest who devoted his life to the problem of parallels,
went to his Maker a failure in his own eyes, unable to demonstrate,
after Sisyphean effort, a single contradiction.
Finally, in the nineteenth century the effort to prove the parallel
postulate exhausted itself, as mathematicians accepted the mounting
evidence for the existence of a geometry based upon its absence.
“I have created a new and different world,” Janos Bolyai
wrote to his father in 1823. In Russia, Nickolai Lobachevsky came
to a similar insight. An alternative to Euclid, however disturbing,
was logically undeniable. To put it into Playfair’s terms,
mathematicians were compelled to acknowledge that there exists a
space in which given a line and an external point P, there are many
lines that go through P yet do not meet the original line.