It is now understood that the parallel postulate is independent of the other axioms of Euclidean geometry. This was first shown during the first half of the 19th century, and involved Lobachevsky, Bolyai, and Gauss. The following is our own informal proof.

• Let us use an Aristotelian style version of the parallel postulate: Through each point not on a line, there passes exactly one line which does not intersect the original line. (Such a new line is said to be parallel to the original line.)
• Most readers of Euclid, probably Euclid himself, could not help but think of the points and lines referred to in the axiomatization, as strictly referring no `real' points and lines in the space we see around us; this intuitive space is even referred to as Euclidean space. But the `points' and 'lines' could refer to other objects, like chairs and tables, as long as the chairs and tables satisfy the axiomatic statements made about them. Here is a successful example:
  • Consider a perfect spherical surface.
  • Each imaginary line through the center, produces two `antipodal' dots on the surface of the sphere. Now we let the word `point' refer to such antipodal pairs of dots on the sphere.
  • Each imaginary planar surface through the center of the sphere, produces a so-called great circle when it cuts through the sphere. Now we let the word `line' refer to such great circles on the sphere.
  • The word `circle' refers to an antipodal pair of circles of equal size on the surface of the sphere.
  • These alternate so-called points, lines, and circles, satisfy a rich theory which obeys all Euclidean axioms and more, but not the parallel postulate. The parallel postulate fails because all lines (that is, great circles on the sphere) intersect. So there are no `parallel' lines at all.
  If the parallel postulate were rigorously provable from the other Euclidean axioms, then the model above would have to satisfy the parallel postulate. Since the model above does not satisfy the parallel postulate, the parallel postulate can not be derivable from the other Euclidean axioms.

By modern standards, Euclid's axiomatization is not complete. Its proofs contain hidden assumptions.

In 1898, David Hilbert's book Grundlagen der Geometrie, Foundations of Geometry, appeared. It set new standards of rigor for the axiomatic method. It is the primary example for the twentieth century. Axiomatic approaches are introduced for old as well as for new areas of mathematics, for algebra, for analysis, and for all of twentieth century mathematics as a whole.