Parallel Postulate | Research & Encyclopedia Articles

Parallel Postulate

The parallel postulate was Euclid's famous, and sometimes infamous, fifth postulate. Euclid's five postulates, or axioms, as they are sometimes called, appear in his masterpiece the Elements, one of the most influential mathematics books ever written. Euclid's program in theElements was to build a deductive geometry starting with the minimum number of postulates necessary. Since postulates are essentially assumptions stated without proof and since a deductive system is primarily about proving theorems, mathematicians from Euclid (c. 300 BC) to the present have desired to assume as little as necessary and prove as much as possible. Euclid thought that five, but no more than five, postulates were necessary to build up his deductive geometry. The first four of Euclid's postulates caused no controversy at all. They were four simple assumptions that nobody could doubt; and those are the properties, simplicity and certainty, for which one looks in a postulate or axiom. The first four of Euclid's five postulates are stated here for reference:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having that segment as its radius and one endpoint of the segment as its center.

4. All right angles are congruent.

These four postulates have very simple statements and are about as "self-evident" as statements get. Unfortunately, the same cannot be said of the fifth postulate:

5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Note that in this form, postulate 5 is neither simple nor obvious. This fact was to account for much consternation, as well as much good mathematics, during the next two millennia. The fifth postulate looks so out of place among the other four, that generations of mathematicians were troubled by its inclusion as a postulate in theElements. Several attempts were made to come up with a simpler, but equivalent, form of the postulate. The earliest on record was a version by the Greek philosopher and mathematician Proclus (410-485) which said, "If a line intersects one of two parallel lines, it must intersect the other line also." This is certainly a simpler statement than Euclid's and probably more "obvious," but, as we shall see, the use of words such as "obvious" and "self-evident" can sometimes depend upon subtleties of which we may be unaware. Another very simple statement that is equivalent to Euclid's fifth postulate is this, called the Equidistance Postulate: "Parallel lines are everywhere equidistant." Perhaps the most famous statement that is equivalent to the parallel postulate is due to John Playfair (1748-1819) and is called Playfair's Axiom: "Given a line and a point not on that line, there is exactly one line which passes through the given point and is parallel to the given line." This is perhaps a more complicated statement than the equidistant postulate, but, historically, Playfair's Axiom is the one most often used in discussions of the parallel postulate. Therefore, we will stay with that tradition in the remainder of this article.

Even with the simpler versions of Euclid's fifth postulate in place, many mathematicians remained uneasy about its presence with the other four postulates. A suspicion arose that the parallel postulate was not independent of the other four as it should be if it is truly needed as a postulate. If the parallel postulate were not independent of the other four, then it should be possible to prove it as a theorem by using some combination of the first four postulates. This became the program of a number of mathematicians of the seventeenth and eighteenth centuries, most notably the Italian mathematician Geralamo Saccheri (1667-1733). Saccheri, with great gusto, set out to "vindicate Euclid of every blemish" by showing that the parallel postulate could be proved using just the first four of Euclid's postulates. This would show that the fifth "postulate" was really a theorem and did not need to be assumed. Alas, Saccheri's "proof" at the end of a very long treatise was found to be fallacious. However, the enterprise was not a total waste of Saccheri's effort, for in the first part of his book, which was without error, he developed a geometry completely independent of the parallel postulate. This made Saccheri, unwittingly, the inventor of what is now called absolute geometry and has been a very fruitful area of mathematical research.

Following Saccheri's "accidental" development of absolute geometry, a host of eighteenth and nineteenth century mathematicians turned their attention to the question of what would happen if the parallel postulate were replaced by an axiom that was contradictory to it. This could take one of two directions. The first, proposed by the Russian mathematician Nikolai Lobachevsky (1792-1856), replaces the parallel postulate with this: "Given a line and a point not on that line, there are at least two lines which pass through the given point and are parallel to the given line." At first this seems counter-intuitive because we are used to thinking in Euclidean terms, i.e. the geometry of a "flat" plane. Of course, this is precisely the geometry which results from assuming the parallel postulate; but with the parallel postulate replaced by Lobachevsy's, a different geometry arises, which, nevertheless, is just as internally consistent as Euclidean geometry. The great French mathematician Henri Poncaré (1854-1912) produced a model of a geometry which satisfied Lobachevsky's axiom. In this model, unfamiliar things happen. For example, the sum of the measures of the angles of a triangle is always less than 180 degrees, whereas in Euclidean geometry, this sum is always equal to 180 degrees. Nevertheless, Poincaré's model showed that Lobachevskian geometry, also called hyperbolic geometry, is just as consistent as Euclidean geometry. The second direction one may take in denying the parallel postulate is to replace it with this axiom: "Given a line and a point not on that line, there are no lines which pass through the given point and are parallel to the given line." This was the direction taken by the great German mathematician Bernhard Riemann (1826-1866). Note that Riemann's replacement of the parallel postulate is equivalent to saying that no parallel lines exist in this geometry. At first this may seem absurd, but consider the surface of a sphere in which "lines" are defined to be the great circles, e.g., lines of longitude on the surface of the earth. Then all lines intersect at the north and south poles, meaning that none are parallel. In this "Riemannian" or "spherical" geometry, the sum of the angles of any triangle is always greater than 180 degrees and some triangles have two right angles. Yet this geometry is also just as consistent as Euclidean geometry. In fact, Riemannian geometry is the geometry of Albert Einstein's Relativity Theory, which changed the way physicists view the universe. Lobachevskian geometry and Riemannian geometry are examples of non-Euclidean geometries. Their developments are among the great results of nineteenth century mathematics and both are the result of denying the "obvious" parallel postulate of Euclid.