A Bottomless Night and a New World

Euclid’s Elements is undoubtedly the most famous single mathematical work, renowned since the 3rd century BC as a model of logical reasoning. Much of this fame comes from its five simple postulates, from which all subsequent theorems are derived. (Modern mathematicians tend to use the term axiom for fundamental premises like this, though the Elements needs a few more subtle assumptions to be fully rigorous). Euclid’s postulates are:

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 the segment as radius and one endpoint as center.
4. All right angles are congruent.
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.

The first four postulates are elegant and obvious; the fifth, or parallel postulate, is much less so. This bothered many admirers of the Elements, such as the Greek philosopher Proclus (412-485). He was typical in thinking it should be “struck out of the postulates altogether” and proved as a theorem from the other four. Such a proof, however, failed to appear. Most of the numerous attempts only produced replacements for the parallel postulate. (A simple, useful example is that the angles of any triangle sum to 180 degrees.)

Much more creative was the failed proof of Giovanni Saccheri (1667-1733). It began by showing that only three possibilities could be consistent with the first four postulates: the “hypothesis of the right angle,” which is equivalent to the parallel postulate; the “hypothesis of the obtuse angle,” in which the angles of a triangle always sum to more than 180 degrees; and the “hypothesis of the acute angle,” in which the angles of a triangle always sum to less than 180 degrees. A contradiction in both of the last two cases would prove the parallel postulate, and Saccheri accurately demonstrated that the case of the obtuse angle was impossible. But with the acute angle, he went wrong. Ultimately unable to escape his intuitions of how lines and points “should” behave, he derived a result “repugnant to the nature of the straight line.” Though he thought it incompatible with the first four postulates, the fifth postulate is actually necessary to produce a true contradiction. 

Saccheri published all this in his treatise Euclid Freed of Every Flaw, dying shortly thereafter. He is now considered a major forerunner of non-Euclidean geometry, but the mathematicians who truly created the field knew nothing of him. Nor did they know much of each other; this was one of those times in the history of mathematics when ideas were simply ready to be discovered. All this makes it difficult to assign credit. The three who generally get it, however, are Gauss, Bolyai, and Lobachevsky.   

By the 18th century, some had begun to doubt the provability of the parallel postulate, but Carl Friedrick Gauss (1777-1855) was the first to question whether it described the real world at all. He speculated the universe could follow a geometry which appeared Euclidean without actually being such, just as the surface of the earth appears to be a plane without actually being one. Unwilling, however, to start a controversy, Gauss kept all this unpublished. Euclidean geometry had a strong reputation; the philosopher Immanuel Kant (1724-1804), for instance, had invoked it as absolute truth. While such caution may have been suitable to someone of Gauss’s fame, it would prove a mistake in 1832, when Farkas Bolyai (1775-1856), an old university friend who had decades ago tried unsuccessfully to prove the parallel postulate, sent him the research of his son János.¹

János Bolyai (1802-1860), a young Hungarian artillery officer, had begun investigating the parallel postulate despite his father’s dire warning: “Do not try the parallels…I have measured that bottomless night, and all the light and joy of my life went out there.” He persisted nonetheless, ending up with something very different—absolute geometry, which followed Euclid’s first four postulates while making no assumptions about the veracity of the fifth. Bolyai used standard geometrical reasoning similar to Saccheri’s, but, like Gauss, clearly stated that standard Euclidean geometry was a particular instance of the “case of the acute angle.” And he was open-minded to weirdness. The “singular” results he derived, such as non-equidistant parallel lines, did not drive him to the conclusion that his new geometry was invalid. 

Gauss was amazed to learn that someone else had come up with ideas so like his own. But while he privately praised Bolyai as a “genius of the first rank,” he did nothing in the way of publicity. János Bolyai would only live to see his work published in an obscure textbook of his father’s. Towards the end of his life, like Galois, he became very bitter and resentful at the lack of recognition. 

Neither Gauss nor the Bolyais were then aware of Nikolai Ivanovich Lobachevsky (1792-1856), a Russian isolated both geographically and linguistically. Back in 1826, he had published a paper on pangeometry. This pangeometry is the same as Bolyai’s absolute geometry, but Lobachevsky took a somewhat different approach, defining lines in terms of circles and focusing much more on the values of angles. Gauss would come across a later publication on the topic in the 1840s; as with Bolyai, he was impressed yet did little to publicize it. Lobachevsky, too, would die unaware of his work’s significance.

In 1868, Eugenio Beltrami (1835-1900) found a concrete application of absolute geometry which proved its consistency. He demonstrated that absolute geometry describes lines and points on a pseudosphere (pictured). This space has negative curvature; setting the curvature to zero produces a cylinder, which, like a plane, satisfies all five of Euclid’s postulates. Only a few years later, Bernhard Riemann (1826-1866) realized that removing the second postulate makes the case of the obtuse angle consistent (though not without weirdnesses of its own—parallel lines, for instance, do not exist). Now called elliptic geometry, it models surfaces with a positive curvature, such as spheres. The quest for a proof of the parallel postulate was finally at an end. Its most lasting legacy is the realization that, even if no connection with the real world is immediately obvious, interesting sets of consistent axioms are very much worth investigating.

1. Farkas and János Bolyai are also known by the Germanized names Wolfgang and Johann. 

Sources:

https://www.wikipedia.org/

Ewald, William. From Kant to Hilbert: readings in the foundations of mathematics. New York: Oxford University Press Inc., 1996.

Hall, Tord. Carl Friedrich Gauss: A Biography. Translated by Albert Froderberg. Cambridge: M.I.T. Press, 1970.

Kline, Morris. Mathematics in Western Culture. London: George Allen and Unwin Ltd, 1953.

Smith, David Eugene. A source book in Mathematics, by David Eugene Smith. New York: McGraw-Hill Book C°, 1929. 

Stewart, Ian. Significant Figures. New York: Hatchette Book Group, 2017.

Image Credit:

https://sunfaceman.medium.com/math-and-magic-euclid-defines-space-ea987f61709c

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https://www.wikiwand.com/en/Constructions_in_hyperbolic_geometry

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