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The Parallel Postulate, Euclidean Geometry, and Non-Euclidean Geometry
Absolute Geometry
Euclid stated five postulates in The Elements. Postulates 1 – 4 are fairly straightforward and caused little
controversy. All of the theorems whose proofs are based only on these four postulates make up what is
known as Absolute Geometry.
Postulate 1. A straight line can be drawn from any point to any point.
Postulate 2. It is possible to extend a finite straight line indefinitely.
Postulate 3. A circle can be drawn with any point as center and any distance as radius.
Postulate 4. All right angles are equal.
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Euclidean Geometry
The fifth postulate is called the Parallel Postulate. It is not nearly as straightforward as the first four, and
Euclid and many other mathematicians have tried to prove it as a theorem using the other postulates. But
nobody could ever do it. It took about 2000 years, but it was finally shown to be impossible.
Postulate 5, the Parallel Postulate
If a straight line meets two straight lines, so as to make the two interior angles on the same side
of it taken together less than two right angles, these straight lines, being continuously produced,
shall at length meet on the side on which are the angles that are less than two right angles.
Alternative, but equivalent, version of the Parallel Postulate
Given a line l and a point P not on l, there is only one line m containing P such that l || m.
This alternative version is the most commonly used version, but there are several others. It is interesting
that one of the other equivalent versions is the statement that “the sum of the angles in a triangle is 180°”.
We prove this as a theorem, and the alternative version of the Parallel Postulate will be a very important
piece of the proof. They are equivalent because if you started with the “the sum of angles in triangle is
180°” you could prove the parallel postulate. All theorems whose proofs rely on the Parallel Postulate,
plus the Absolute Geometry theorems, are what is known as Euclidean Geometry or Flat Geometry.
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Non-Euclidean Geometry
One way of attempting to prove the Parallel Postulate is the method of indirect proof, by assuming that
the conclusion of the postulate is false and coming up with a contradiction. This results in two new
Parallel Postulates which end up not only failing to produce a contradiction, but producing two new
systems of Non-Euclidean Geometry.
1.
Elliptical Parallel Postulate
Given a line l and a point P not on l, there is no line m containing P such that l || m.
This produces Spherical Geometry, or the more general Elliptical Geometry, where “the sum of the of
the angles in a triangle is greater than 180°”.
2.
Hyperbolic Parallel Postulate
Given a line l and a point P not on l, there is at least two lines m containing P such that l || m.
This produces Hyperbolic Geometry where “the sum of the angles in a triangle is less than 180°”.