Question

The negation of which property leads to a logically consistent geometry called spherical geometry?

Answer 1

Euclid's fifth postulate states, rather wordily, that:

if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

If that sounds like a mouthful to you, you're not alone. Geometers throughout history found that postulate incredibly awkwardly-worded compared with his other four, and many in the 19th century rejected it outright and created a number of interesting new geometries from its ashes.

Euclid's fifth, put another way, states that two lines that aren't parallel will eventually meet, which consequently implies that two parallel lines will never meet. Without intending it, this property defines the space of Euclid's geometry to be an infinite flat plane.

If we take that parallel postulate and throw it out, then we've defined a spherical space for our geometry. Now, it doesn't matter where we draw our lines; all of them will meet at some point. If you need any convincing of this, take a look at the attached image. The longitude lines seem parallel at first, but they all eventually meet at the north and south poles.