Parallel postulate

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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 that:

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 the angles sum to less than 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 cannot hold 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 absolute geometry (or, in some places, neutral geometry).

1 Converse of Euclid's parallel postulate
2 Logically equivalent properties
3 History
4 References

[edit] Converse of Euclid's parallel postulate

If the sum of the two interior angles equals 180°, the lines are parallel and will never intersect.

Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry. The Elements contains the proof of an equivalent statement: Any two angles of a triangle are together less than two right angles. The proof depends on an earlier proposition: In a triangle ABC, the exterior angle at C is greater than either of the interior angles A or B. This in turn depends on Euclid's unstated assumption that two straight lines meet in only one point, a statement not true of elliptic geometry.

[edit] 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. There are, in addition, properties that are equivalent to the conjunction of Euclid's parallel postulate and its converse, and thus can be used to distinguish Euclidean geometry from both elliptic geometry and hyperbolic geometry simultaneously. 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.

It is possible that Euclid chose not to use Playfair's axiom because it doesn't say how to construct the unique parallel line. With Euclid's original axiom, the construction of the parallel line is given as a proposition. The ancient Greeks declared objects "not to exist" if a construction cannot be found for them^[1]

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:

The sum of the angles in every triangle is 180°.
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 of 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.
Given two parallel lines, any line that intersects one of them also intersects the 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. [1]

However, the alternatives which employ the word "parallel" cease appearing so simple when one is obliged to explain which of the three common definitions of "parallel" is meant - constant separation, never meeting or same angles where crossed by a third line - since the equivalence of these three is itself one of the unconsciously obvious assumptions equivalent to Euclid's fifth postulate.

[edit] History

For two thousand years, the parallel postulate was suspected by some mathematicians to be a theorem which could be proven 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 that 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 proven by Euclid in Book I of the Elements (Proposition 17).

Archimedes, in his treatise On Parallel Lines, defined parallel lines as those equidistant to each other everywhere. From this, the parallel postulate can be "proven" if you are willing to accept that a "line" equidistant to a straight line everywhere is in fact a straight line.

Omar Khayyám (1050-1123) recognized that three possibilities arose from omitting Euclid's Fifth; if two perpendiculars to one line cross another line, judicious choice of the last can make the internal angles where it meets the two perpendiculars equal (it is then parallel to the first line). If those equal internal angles are right angles, we get Euclid's Fifth; otherwise, they must be either acute or obtuse. He persuaded himself that the acute and obtuse cases lead to contradiction, but had made a tacit assumption equivalent to the fifth to get there. Girolamo Saccheri (1667-1733) pursued the same line of reasoning more thoroughly, correctly obtaining absurdity from the obtuse case (proceeding, like Euclid, from the implicit assumption that lines can be extended indefinitely and have infinite length), but failed to debunk the acute case (but managed to wrongly persuade himself that he had).

Where Saccheri and Khayyám had attempted to prove Euclid's fifth by disproving the only possible alternatives, the nineteenth century finally saw mathematicians exploring those alternatives and discovering the logically consistent geometries which result. In 1829, Nikolai Ivanovich Lobachevski published an account of acute geometry in an obscure Russian journal (later re-published in 1840 in German). In 1831, János Bolyai included, in a book by his father, an appendix describing acute geometry, which, doubtlessly, he had developed independently of Lobachevski. It is probable that Carl Friedrich Gauss had actually studied the problem before that, but if so, he didn't publish any of his results. The resulting geometries were later developed by Lobachevsky, Riemann and Poincaré into hyperbolic geometry (the acute case) and spherical geometry (the obtuse case).

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.