Parallel postulate

Summary: The parallel postulate led to thousands of years of investigation and debate.

One of humanity’s greatest intellectual achievements occurred in approximately 300 b.c.e. when the axiomatic method was born. The classic text Elements, written by the great Greek geometer Euclid of Alexandria, is a work that shaped the nature of mathematics and stands to this day as an example of the beauty and elegance of reasoning and proof.

Euclid was among the first people to understand that abstract mathematics is based on reasoning, from assumptions to general conclusions. From a very modest set of assumptions—his five postulates (called “axioms”)—Euclid set out to argue the truth of a large number of propositions (called “theorems”) in geometry.

The first four of Euclid’s postulates appear reasonable enough: (1) any two points determine a unique line; (2) any line segment can be extended to an infinite line; (3) given any center and radius, a circle can be constructed; and (4) all right angles are congruent. But the fifth postulate stands out for its comparative complexity:

If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

This fifth postulate has come to be known as the “parallel postulate,” in part for its very content, but also for the key role it plays in proving certain propositions about parallel lines.

From an historical perspective, Euclid himself seemed a bit uncomfortable with his fifth postulate. This discomfort is evidenced by the order of his work in Book I of Elements, where, on his way to eventually proving 48 propositions, he waited until proposition 29 to use the parallel postulate. The first 28 results rely only on the first four postulates and theorems that can be proven using those assumptions.

Attempts to Prove the Parallel Postulate as a Theorem

As subsequent mathematicians studied the Elements, most were troubled in some way by the parallel postulate. Because of its complexity, as well as its “if-then” format, it struck most mathematicians that Euclid’s fifth postulate really ought to be a theorem. In other words, the parallel postulate ought to be a consequence of the first four postulates, and this fact ought to be provable, using only those four postulates and any theorems that could be derived from them.

Thus, many mathematicians set out to prove the parallel postulate as a theorem. It is one of the great tales of the history of mathematics that every single mathematician who attempted to prove the parallel postulate failed. Early on, many of these esteemed intellects made a common error that the rules of logic forbid—they assumed precisely what they were attempting to prove. Clearly, if the goal is to prove a statement S, one should never be allowed to simply assume that S is true. While certainly no mathematician was so dull as to say, “To prove the parallel postulate, I will assume the parallel postulate,” many people did make the mistake of making the assumption that certain “obvious” statements were true. For example, they may have assumed statements such as the following:

• Parallel lines are everywhere equidistant.
• The sum of the measures of the interior angles of a triangle is 180 degrees.
• If a line intersects one of two parallel lines, then it must also intersect the other.
• There exists a rectangle (a quadrilateral having four right angles).

Remarkably, each of the above statements (along with many others) is equivalent to the parallel postulate. Said differently, if one of the above statements is called P and the statement of the parallel postulate is called S, then it turns out that P is true if and only if S is true—the truth of one implies the truth of the other, and vice versa.

Hence, when a mathematician said, “Using the fact that any triangle’s angle sum is 180 degrees,” and then went on to “prove” the parallel postulate, this argument was like saying “the parallel postulate is true because the parallel postulate is true.” These errors came to be well understood by the end of the eighteenth century, perhaps most prominently in G. S. Klugel’s 1763 doctoral dissertation in which he debunked 43 flawed “proofs” of the parallel postulate.

Girolamo Saccheri’s Developments

Of course, even though nobody had found a valid proof of the parallel postulate did not mean that one could not be found, and many continued the search. Around the turn of the eighteenth century, a Jesuit priest named Girolamo Saccheri (1677–1733) made a lasting contribution to the study of the parallel postulate in particular, and to the history of mathematics in general. Saccheri considered the unthinkable, as part of his effort to prove the parallel postulate through a contradiction argument: what if the parallel postulate is false?

It was well understood by Saccheri’s time that an equivalent statement of the parallel postulate was Playfair’s Postulate, which states that

For any line l and any point P not on l, there exists a unique line through P parallel to l.

A contradiction argument works by assuming that the statement one wants to prove true is actually false and showing that some contradiction follows. Thus, it is natural to consider Playfair’s Postulate and suppose that there is not be a unique line through P parallel to l. That is, one would assume that either there is not any line through P parallel to l, or there is more than one line through P parallel to l. Saccheri considered a similar scenario where he had transformed the problem about parallels to an equivalent one about quadrilaterals (now called “Saccheri quadrilaterals”) in which the quadrilateral has two congruent sides perpendicular to the base. Fundamentally, Saccheri was trying to prove that a rectangle existed by showing that the summit angles of his quadrilateral were also right angles. After proving that the summit angles were congruent, he realized that there were three possibilities: the summit angles were each right angles, each was less than a right angle, or each was more than a right angle.

While Saccheri was able to rule out the possibility that the summit angles were obtuse by assuming that they were obtuse and finding a contradiction, when he assumed that the summit angles were acute, he could not find a contradiction. From this assumption, he went on to prove many strange and unusual theorems. Unknowingly, Saccheri had discovered a whole new geometry, one that another mathematician named Janos Bolyai would call “a strange, new universe” in his own investigations. What both of these mathematicians, along with others such as Carl Gauss, started to realize is that there actually exists a geometry in which there is more than one line through a point P not on line l such that each is parallel to l. This realization stands as one of the greatest accidental discoveries in the history of the human intellect: Saccheri did not find what he set out to prove, but instead developed a collection of ideas that would radically change mathematics.

Bibliography

Dunham, Douglas. “A Tale Both Shocking and Hyperbolic.” Math Horizons 10 (April 2003).

Greenberg M. Euclidean and Non-Euclidean Geometries: Development and History. New York: W. H. Freeman and Co., 2007.

Socrates Bardi, Jason. The Fifth Postulate: How Unraveling A Two Thousand Year Old Mystery Unraveled the Universe. Hoboken, NJ: Wiley, 2008.