Detest it as lewd intercourse, it can deprive you of all your leisure, your health, your rest, and the whole happiness of your life.
Do not try the parallels in that way: I know that way all along. I have measured that bottomless night, and all the light and all the joy of my life went out there.
-Letters from Farkas Bolyai to his son, János, attempting to dissuade him from his investigations of the Parallel Postulate.
In our previous post, cataloging various notions of mathematical independence, we introduced the idea of logical independence. Briefly, given a consistent set of axioms, , a sentence is independent from if it can be neither proven nor disproven from the sentences in . Today, we discuss one of the most prominent and interesting instances of logical independence: Euclid’s Parallel Postulate.
Among the most famous sets of axioms (top 5, certainly) are Euclid’s postulates, five statements underpinning (together with 23 definitions and five other statements putting forth the properties of equality) the mathematical system of Euclidean geometry set forth in the Elements and still taught in high school classrooms to this day. (We should note here that, from a modern viewpoint, Euclid’s proofs do not always strictly conform to the standards of mathematical rigor, and some of his results rely on methods or assumptions not justified by his five postulates. This has been fixed, for example by Hilbert, who gave a different set of axioms for Euclidean geometry in 1899. Now that we have noted this, we will proceed to forget it for the remainder of the post.)
Euclid’s first four postulates are quite elegant in their simplicity and self-evidence. Reformulated in modern language, they are roughly as follows:
- Given any two points, there is a unique line segment connecting them.
- Given any line segment, there is a unique line (unbounded in both directions) containing it.
- Given any point and any radius , there is a unique circle of radius centered at .
- All right angles are congruent.
The fifth postulate, however, which is known as the Parallel Postulate, is, quite unsatisfyingly, markedly more complicated and less self-evident:
- If two lines 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 must intersect one another on that side.
A picture might help illustrate this postulate:
This postulate doesn’t seem to be explicitly about parallel lines, so the reader may be wondering why it is often called the Parallel Postulate. The reason becomes evident, though, when considering the following statement and learning that, in the context of the other four postulates, it is in fact equivalent to the Parallel Postulate:
- Given any line and any point not on , there is exactly one line through that is parallel to .
(Recall that two lines are parallel if they do not intersect.) This reformulation of the Parallel Postulate is often named Playfair’s Axiom, after the 18th-century Scottish mathematician John Playfair, though it was stated already by Proclus in the 5th century.
The Parallel Postulate was considered undesirably unwieldy and less satisfactory than the other four postulates, even by Euclid himself, who made a point of proving the first 28 results of the Elements without recourse to the Parallel Postulate. The general opinion among mathematicians for the next two millennia was that the Parallel Postulate should not be an axiom but rather a theorem; it should be possible to prove it using just the other four postulates.
Many attempts were made to prove the Parallel Postulate, and many claimed success at this task. Errors were then inevitably discovered by later mathematicians, many of whom subsequently put forth false proofs of their own. The aforementioned Proclus, for example, after pointing out flaws in a purported proof of Ptolemy, gives his own proof, which suffers from two instructive flaws. The first is relatively minor: Proclus assumes a consequence of Archimedes’ Axiom, which essentially states that, given any two line segments, there is a natural number such that times the length of the shorter line segment will exceed the length of the longer. (We encountered Archimedes’ Axiom in a previous post, about infinitesimals, which the reader is invited to revisit.) Archimedes’ Axiom seems like an entirely reasonable axiom to assume, but it notably does not follow from Euclid’s postulates.
Proclus’ more serious error, though, is that he makes the assumption that any two parallel lines have a constant distance between them. But this does not follow from the first four postulates. In fact, the statement, “The set of points equidistant from a straight line on one side of it form a straight line,” known as Clavius’ Axiom, is, in the presence of Archimedes’ Axiom and the first four postulates, equivalent to the Parallel Postulate. Proclus’ proof is therefore just a sophisticated instance of begging the question.
In the course of the coming centuries’ attempts to prove the Parallel Postulate, a number of other axioms were unearthed that are, at least in the presence of Archimedes’ Axiom and the first four postulates, equivalent to the Parallel Postulate. In addition to Playfair’s Axiom and Clavius’ Axiom, these include the following:
- (Clairaut) Rectangles exist. (A rectangle, of course, being a quadrilateral with four right angles.)
- (Legendre) Given an angle and a point in the interior of the angle, there is a line through that meets both sides of the angle.
- (Wallis) Given any triangle, there are similar triangles of arbitrarily large size.
- (Farkas Bolyai) Given any three points, not all lying on the same line, there is a circle passing through all three points.
A key line of investigation into the Parallel Postulate was carried out, probably independently, by Omar Khayyam, an 11th-century Persian mathematician, astronomer, and poet, and by Giovanni Gerolamo Saccheri, an 18th-century Italian Jesuit priest and mathematician. For concreteness, let us consider Saccheri’s account, which has the wonderful title, “Euclid Freed from Every Flaw.”
Saccheri and Khayyam were, similarly to their predecessors, attempting to prove the Parallel Postulate. Their method of proof was contradiction: assume that the Parallel Postulate is false and derive a false statement from it. To do this, they considered figures that came to be known as Khayyam-Saccheri quadrilaterals.
To form a Khayyam-Saccheri quadrilateral, take a line segment (say, BC). Take two line segments of equal length and form perpendiculars, in the same direction, at B and C (forming, say, AB and DC). Now connect the ends of those two line segments with a line segment (AD) to form a quadrilateral. A picture is given below.
By construction, the angles at B and C are right angles, but the angles at A and D are unclear. Saccheri proves that these two angles are equal. He also proves that, if these angles are obtuse, then they are obtuse for every such quadrilateral; if they are right, then they are right for every such quadrilateral; and, if they are acute, then they are acute for every such quadrilateral. This then naturally divides geometries into three categories: those satisfying the Obtuse Hypothesis, those satisfying the Right Hypothesis, and those satisfying the Acute Hypothesis. (These types of geometries subsequently became known as semielliptic, semieuclidean, and semihyperbolic, respectively.)
At this point, Saccheri attempts to prove that the Obtuse Hypothesis and the Acute Hypothesis both lead to contradiction. (Note that a geometry satisfying all five of Euclid’s postulates must satisfy the Right Hypothesis. The converse is not true, so even a successful refutation of the Obtuse and Acute Hypotheses would not be enough to establish the Parallel Postulate.) Saccheri is able to prove (in the presence of Archimedes’ Axiom) that the Obtuse Hypothesis leads to the conclusion that straight lines are finite, thus contradicting the second postulate. He is unable to obtain a logical contradiction from the Acute Hypothesis, though. Instead, he derives a number of counter-intuitive statements from it and then concludes that the Acute Hypothesis must be false because it is “repugnant to the nature of a straight line.”
The next big steps towards the establishment of the independence of the Parallel Postulate were made by Nikolai Lobachevsky and János Bolyai (who fortunately did not heed his father’s letters quoted at the top of this post), 19th-century mathematicians from Russia and Hungary, respectively. (Similar work was probably done by Gauss, as well, though it was never published.) Their work entailed a crucial shift in perspective – rather than attempt to prove the Parallel Postulate from the others, the mathematicians seriously considered the possibility that it is not provable and thought of non-Euclidean geometries (i.e., those failing to satisfy the Parallel Postulate) as legitimate objects of mathematical study in their own right. In particular, they were interested in hyperbolic geometry, in which the Parallel Postulate is replaced by the assertion that, given any line and any point not on the line, there are at least two distinct lines passing through and parallel to . (Not surprisingly, considering the nomenclature, hyperbolic geometries are semihyperbolic, i.e., they satisfy the Acute Hypothesis.) This viewpoint was vindicated when, in 1868, Eugenio Beltrami produced a model of hyperbolic geometry. This shows that, as long as Euclidean geometry is consistent, then the Parallel Postulate is independent of the other four postulates: all five postulates are true, for example, in the Cartesian plane, while the first four are true and the Parallel Postulate is false in any model of hyperbolic geometry.
A number of other models for hyperbolic geometry are now known. In our next post, we will look at a particularly elegant one: the Poincaré disk model.
Cover Image: Michael Tompsett, Parallel Lines
For more information on this and many other geometric topics, I highly recommend Robin Hartshorne’s excellent book, Geometry: Euclid and Beyond.