Bunch-Hellemans, Non-Euclidean geometry

Even in Hellenistic times, Euclid’s Fifth Postulate was viewed suspiciously. Euclid himself arranged his Elements so that the Fifth Postulate was not used in the first 25 propositions, although the 16th assumes something equivalent to it. The suspect postulate was much more complex and less self-intuitive than the other postulates, which are in the nature of, “through two distinct points one and only one straight line can be drawn.” The Fifth Postulate states, “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.” One objection to the Fifth Postulate was that the meeting point could be as far as one liked from the original line; in essence, the point could be infinitely far away. By the time of Proclus, 750 years after Euclid, geometers were bent on getting rid of this objectionable postulate.

There were essentially two strategies used: replacing the Fifth Postulate with a different equivalent postulate or establishing it as a mere theorem, a result proved from the other postulates. For the first strategy, the most common version of the postulate used is known today as Playfair’s Postulate after the 18th-century version of British mathematician John Playfair. It states, “Through a given point, not on a given line, only one parallel can be drawn to the given line.” Although Playfair’s Elements of Geometry popularized this postulate, Proclus had also used it as an alternative to the Fifth Postulate in the fifth century. Playfair’s postulate is a little easier to understand than Euclid’s Fifth Postulate, but because the two postulates are equivalent, it is just as suspect. Two postulates are equivalent when each one implies the other.

The second strategy also involves equivalent postulates. For each time that someone tried to prove the Fifth Postulate as a theorem, it was found that he had to invoke a result that was equivalent to the Fifth Postulate. These equivalent postulates include “a line that intersects one of two parallel lines intersects the other”; “the sum of the angles of a triangle is 180°”; “for any triangle there exists a similar triangle not congruent to the first triangle”; and “there exists a circle passing through any three noncollinear points.” Invoking any of these in a proof of the Fifth Postulate amounts to circular reasoning since each is equivalent to the Fifth Postulate.

In the 18th century, Girolamo Saccheri suggested a third strategy, indirect proof of the Fifth Postulate by contradiction. He assumed that one of the equivalents to the Fifth Postulate is not true. There are two ways in which this equivalent postulate could be not true, so Saccheri needed to deal with two assumptions. Saccheri developed a number of geometric theorems from each of these assumptions combined with Euclid’s other postulates, looking for a contradiction between two of these new theorems.

Finding such a contradiction from one assumption would mean that either the Fifth Postulate or the other assumption had to be true. Saccheri convinced himself that he had found such contradictions for both assumptions, leaving the truth of the Fifth Postulate the only remaining possibility. His report of his efforts was called Euclides ab omni aevo vindicatus (sometimes translated into English as Euclid Freed of Every Fleck). When he was 15, Karl Friedrich Gauss began to work on the problem of the Fifth Postulate. Gauss recapitulated the approaches of others with the same results, except that he was better than they had been at seeing that his attempted proofs did not work. After 25 years of working on the problem, he evidently reached the conclusion that the Fifth Postulate is independent of the others. This means that a contradiction to the Fifth Postulate can be used to develop a consistent geometry. Gauss proceeded to do this to his own satisfaction, but did not publish his work, although he told a few friends about his conclusion.

Shortly after (during the late 1820s), two other gifted mathematicians also reached the conclusion that the Fifth Postulate is independent of the others. Both Nikolai Ivanovich Lobachevski and Janos Bolyai independently discovered and published their non-Euclidean geometries. In all three versions (including that of Gauss), the mathematicians assumed that more than one line passing through the same point could be parallel to another line.

In 1854 Bernhard Riemann suggested that there were several non-Euclidean geometries. For example, one could avoid the contradictions of Saccheri’s first assumption by changing Euclid’s first and second postulates along with the fifth. The result appeared to be consistent –– that is, no contradictory theorems can arise.

In short order, Eugenio Beltrami was able to prove that the original non-Euclidean geometry of Lobachevski and Bolyai is consistent if Euclidean geometry is itself consistent. Felix Klein soon showed that two different versions of Riemann’s geometry were as consistent as Euclid’s. The method employed in each case was to find a model of the non-Euclidean geometry that was within Euclid’s geometry. The easiest such model to understand takes the surface of a sphere as the plane, which is undefined in the modern treatment of Euclidean geometry. Then if great circles are taken to be lines, they obey such postulates as “two distinct points determine at least one line” (replaces Euclid’s First Postulate); “a straight line is finite in length” (replaces Euclid’s Second Postulate), and “no two lines are parallel” (replaces Euclid’s Fifth Postulate). The new postulates not only describe Riemann’s form of non-Euclidean geometry, they also accurately describe geometry on a sphere (with the understanding that a line is really a great circle); but geometry on a sphere can also be described by Euclid’s original postulates when great circles are understood as circles, not lines.

Despite the mathematical validity of non-Euclidean geometries, they were not deemed of much practical value until 1915, when Albert Einstein and Hermann Minkowski showed that gravity could be explained by treating space as a four-dimensional Riemann-type geometry. In other words, space itself is non- Euclidean, despite our impression of it. This is because we view it on a small scale. On a small scale, Earth looks flat. On a larger scale, with a different definition of line, it is non-Euclidean.

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