What are straight lines? On the plane, the straight lines are the locus of the point where the direction of motion does not change.

On the sphere, we can regard any given circle as the circle of latitude (the equator is a special circle of latitude). The point along the circle of latitude movement, is the east-west direction of movement, that is, the movement does not change direction. So circles on the sphere are straight lines . Great circles are straight lines, and small are straight lines.

So circles are all straight lines on the sphere, so,Through a given point, only one line can be drawn parallel to a given line.

On the sphere, what is the tangent of the circle of latitude? The tangent of the circle of latitude is the circle of latitude itself. So the slope of circle of latitude equals zero. On the plane, the slope of the straight line parallel to the X axis of the coordinate is equal to zero.

  • $\begingroup$ As has been told to you in this question, only the great circles are straight on a sphere. The blue curves on this picture are not straight. Only the red ones (and one of the blue ones, the equator) are. $\endgroup$ – Arthur Nov 17 '17 at 15:07
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    $\begingroup$ If you allow "small circles" to be "straight lines" on a sphere there are circles which do not meet and it is quite easy to construct transversals which don't cross them at right-angles and which contradict your purported parallel property. $\endgroup$ – Mark Bennet Nov 17 '17 at 15:08
  • $\begingroup$ "If a straightline intersects two straightlines forming two interior angles on the same side that sum to less than two right angles, then the two straightlines, Close to each other on that side on which the angles sum to less than two right angles." This is not a sentence. Then the two straightlines do what? $\endgroup$ – fleablood Nov 17 '17 at 15:26
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    $\begingroup$ "the straight lines is circles on a sphere, the great circles and small circles are all straight lines." If I defined straight lines as being Panda Bears then I could prove that straight lines eat bamboo. Would that be useful to you? $\endgroup$ – fleablood Nov 17 '17 at 15:27
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    $\begingroup$ If lesser circles are defined to be be straight lines, there will be no difference between spherical and plane geometries definitions of lines and circles. Every condition of a small circle on a sphere applies to a small circle in a plane. The parralel postulate does not apply to circles. $\endgroup$ – fleablood Nov 17 '17 at 16:01

Let alone that small circles are not straight lines on the sphere, and put the OP's statement the following way:

Consider two great circles $a$ and $b$ that are intersected by a third one, $c$. Let the "interior angles" be denoted by $M$ and $N$ and let $M+N<2R$. $a$ and $b$ will meet on the side at which the sum of the angles is less the $2R$ ($Q$). So the Euclidean parallel postulate is met on the sphere. As shown below.

enter image description here

If one looks at the figure the one will discover that the straight lines meet at $L$ as well. That is, on the side at which the sum of the external angles $M'+N'$ is greater than $2R$.

The essence of the parallel postulate is that the straights at stake meet on that side at which the sum of the angles... and not on the the side at which...

If we consider the opposite points to be the same, as it is done in spherical geometry then the expression "on that side" will loose its meaning because the straight lines meet on booth sides of $c$ in the same point that can be called $QL$.

So, the Euclidean postulate of parallels is not met on the surface of the sphere.

  • $\begingroup$ “If we consider the opposite points to be the same, as it is done in spherical geometry…” – you mean elliptic geometry, right? $\endgroup$ – MvG Nov 20 '17 at 16:47
  • $\begingroup$ @MvG: right. Right. 15 chars... $\endgroup$ – zoli Nov 20 '17 at 19:51
  • $\begingroup$ @zoli Thank you for your detailed answer. I've revised my question. I hope you'll continue to comment. $\endgroup$ – enbin zheng Nov 21 '17 at 13:01
  • $\begingroup$ @zoli You only consider the big circle, but the small circle is straight line, so the parallel exists, on the sphere. $\endgroup$ – enbin zheng Nov 21 '17 at 14:05
  • $\begingroup$ The small circles are not straight lines: two points do not determine an exact small circle. (They determine one great circle!) $\endgroup$ – zoli Nov 21 '17 at 14:51

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