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Euclid's fifth postulate states, rather wordily, that:
<em>if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, </em><span><em>the two straight lines, if produced indefinitely, meet on that side </em><em>on</em><em> which are the angles less than the two right angles.
</em></span>
<em />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 <em>two parallel lines will never meet</em>. Without intending it, this property defines the space of Euclid's geometry to be <em>an infinite flat plane</em>.
If we take that parallel postulate and throw it out<em>, </em>then we've defined a <em>spherical space</em> for our geometry. Now, it doesn't matter where we draw our lines; <em>all of them will meet at some point</em>. If you need any convincing of this, take a look at the attached image. The longitude lines <em>seem </em>parallel at first, but they all eventually meet at the north and south poles.
<em>if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, </em><span><em>the two straight lines, if produced indefinitely, meet on that side </em><em>on</em><em> which are the angles less than the two right angles.
</em></span>
<em />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 <em>two parallel lines will never meet</em>. Without intending it, this property defines the space of Euclid's geometry to be <em>an infinite flat plane</em>.
If we take that parallel postulate and throw it out<em>, </em>then we've defined a <em>spherical space</em> for our geometry. Now, it doesn't matter where we draw our lines; <em>all of them will meet at some point</em>. If you need any convincing of this, take a look at the attached image. The longitude lines <em>seem </em>parallel at first, but they all eventually meet at the north and south poles.