1
$\begingroup$

Recently I came across a variant of the Parallel Postulate known as Playfair's Axiom:

In Euclidean (planar) geometry there is at most one line that can be drawn parallel to another given one through an external point.

However, in the case of spherical geometry there exists no such line which can be drawn parallel.

I am having a doubt in this concept. Because if the surface is spherical in nature, then there always will be a point a diametrically opposite on the surface from where a parallel line can be drawn. So, why is it not taken into consideration or if it is then, am I missing an important concept?

$\endgroup$
0
$\begingroup$

That there is no such line in spherical geometry is not part of Playfair's axiom and, as you point out, is false. If you want to clearly differentiate between Euclidean and spherical geometry you have to reword the axiom, for instance,

For every line and point not on the line, there exists exactly one line passing though that point and parallel to the given line.

There is something else going on here: spherical geometry obeys Playfair's axiom, as you've written it. But it obeys the parallel postulate as well!

This raises a pressing question: which of Euclid's axioms is violated by spherical geometry? For instance proposition 27 is false in spherical geometry. It may not be obvious at a glance from the classical wording of the five postulates.

The key thing to understand is that Euclid's arguments, though tremendously influential, are not rigorous or complete by modern standards. If you follow Euclid's arguments very carefully you will find propositions where his arguments break down in spherical geometry, because of unproven "obvious" assertions that do not appear in his list of axioms. For instance proposition 16, a key step in Euclid's treatment of parallel lines, makes a claim about the relative magnitude of angles that is true in the plane but not on the sphere.

$\endgroup$
  • $\begingroup$ Thank you for the explanation sir but my doubt still persists. In this [link] (en.wikipedia.org/wiki/Parallel_postulate#History) and in one of the lecture series video (MIT6_042JF10_lec01_300k) of the MITOpencourseware by Prof Tom Leighton there is a description which says Given a line, L , and a point p (not on L) there is no line through p parallel to L. But I fail to understand this because there does exist a line parallel to L at a point diametrically opposite to the line L. $\endgroup$ – durwasa.chakraborty Jun 12 '15 at 8:05
  • $\begingroup$ @durwasa.chakraborty I don't understand what you mean. What does it mean to be diametrically opposite to a line L? If you take a point $p$ and draw a great circle through it, the point diametrically opposite to $p$ is guaranteed to lie on it. $\endgroup$ – user7530 Jun 12 '15 at 15:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.