Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate since it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry. It states:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. A geometry where the parallel postulate is violated is known as a non-Euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the first four postulates) is known as affine geometry.

Logically equivalent properties

Several properties of Euclidean geometry are logically equivalent to Euclid's parallel postulate, meaning that they can be proven in a system where the parallel postulate is true, and that if they are assumed as axioms, then the parallel postulate can be proven. One of the most important of these properties, and the one that is most often assumed today as an axiom, is Playfair's axiom, named after the Scottish mathematician John Playfair. It states:

Exactly one line can be drawn through any point not on a given line parallel to the given line.

Some of the other statements that are equivalent to the parallel postulate appear at first to be unrelated to parallelism. Some even seem so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates. Here are some of these results:

1. The sum of the angles in a triangle is 180°.
2. There exists a triangle whose angles add up to 180°.
3. The sum of the angles is the same for every triangle.
4. There exists a pair of similar, but not congruent, triangles.
5. Every triangle can be circumscribed.
6. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.
7. There exists a quadrilateral of which all angles are right angles.
8. There exists a pair of straight lines that are at constant distance from each other.
9. Two lines that are parallel to the same line are also parallel to each other.
10. Given two parallel lines, any line that intersects one of them also intersects the other.
11. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem).

History

For two thousand years the parallel postulate was suspected by some mathematicians to be a theorem which could be proved using Euclid's first four postulates. A great many attempts were made to provide such a proof, constituting one of the largest collections of writings on any single topic in mathematics.

The main reason such a proof was so highly sought after was that while Euclid's other postulates appeared self-evident and intuitively obvious, the fifth postulate essentially described the intersection of lines at potentially infinite distances, a concept that could hardly be called self-evident. In addition, the converse of the fifth postulate is a theorem that was proved by Euclid in Book I of the Elements (Proposition 17).

Janos Bolyai (and probably Carl Friedrich Gauss before him) realized that the negation of the fifth postulate leads to logically consistent geometries which were later developed by Lobachevsky, Riemann and Poincaré into hyperbolic geometry and spherical geometry.

The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868. For more information, see the history of non-Euclidean geometry.da:Euklids aksiomer de:Parallelenaxiom nl:Parallellenpostulaat pl:Równoległość sv:Parallellaxiomet zh:平行公設