The second postulate states that it's possible to extend this line segment as far as you wish. This would mean that the length of the resulting line segment can be as large as you want. Again, the ruler makes this obviously true on a sheet of paper, but in spherical geometry, it's obviously false. The maximum length of a spherical line segment is the circumference of the sphere, so it's impossible to extend a line segment beyond this length. So, what Euclid was trying to establish with this postulate was that the geometry he was working with was unbounded.

 

However, this does raise some concerns for the Poincare Disk Model, because the Poincare Disk Model is very clearly bounded to the unit circle. What makes the Disk Model slip through this postulate is that while the disk is finite, it's the projection of something infinite. So in the Poincare distance metric, the boundary of the unit circle is infinitely far away. Extending hyperbolic lines segments into hyperbolic lines also reveals some neat geometric properties. Specifically the hyperbolic line passing through two points is actually the arc of a Euclidean circle, which is the circle passing through the two points that intersects the unit circle at right angles. This is demonstrated in the next interactive: