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It is one thing to assert axioms such as 'parallel lines never meet', an assertion for which there was no obvious falsification for over 2,000 years, as did Euclid.
That's not even an axiom, it is a definition. There are no parallel lines on a sphere. The axiom of parallels is something entirely else:

  • There is at most one line that can be drawn parallel to another given one through an external point. (Playfair's axiom)
  • The sum of the angles in every triangle is 180° (triangle postulate).
  • There exists a triangle whose angles add up to 180°.
  • The sum of the angles is the same for every triangle.
  • There exists a pair of similar, but not congruent, triangles.
  • Every triangle can be circumscribed.
  • If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.
  • There exists a quadrilateral in which all angles are right angles.
  • There exists a pair of straight lines that are at constant distance from each other.
  • Two lines that are parallel to the same line are also parallel to each other.
  • In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem).
  • There is no upper limit to the area of a triangle. (Wallis axiom)
  • The summit angles of the Saccheri quadrilateral are 90°.
  • If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. (Proclus' axiom)


A society committed to the notion that government is always bad will have bad government. And it doesn't have to be that way. — Paul Krugman
by Migeru (migeru at eurotrib dot com) on Wed Jan 15th, 2014 at 11:05:46 AM EST
[ Parent ]
Perhaps falsification was not the proper choice of words.

"It is not necessary to have hope in order to persevere."
by ARGeezer (ARGeezer a in a circle eurotrib daught com) on Thu Jan 16th, 2014 at 12:56:57 AM EST
[ Parent ]

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