The Fifth Postulate

Euclid is known as The Father of Geometry. His definitive book on the subject written in 300 BCE described and proved most of the facts that students learn today in school. However, his monumental work of 13 books called The Elements achieved something far more than a list of geometric facts. It described the nature of mathematics in a way that wouldn’t be fully understood until the 20th century. Euclid began by introducing fundamental geometric ideas in Book One as definitions and postulates. With such a program, all geometric facts are the result of these postulates. He stated 5 postulates that act like a constitution for laws of geometry. All but the last are straightforward and simple. However, the fifth postulate appears as though it should be a consequence of the others and hence redundant to be stated as a fundamental postulate. Therefore many mathematicians since the time of Euclid attempted to determine this redundancy without success. Yet the success of their failures would reveal a whole new geometry and description of space and time.

Euclid’s fifth postulate: 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.

The following is excerpted, with permission of the publisher, from The Fifth Postulate

By Jason Socrates Bardi

In the second decade of the nineteenth century, Gauss began to record and discuss his sense of the futility of continuing to try to prove the fifth postulate. Taking him at face value is difficult because he never drew his ideas together into a single cogent argument. All we have are scattered notes, inferences, and occasional remarks to friends in letters. In 1813, he wrote in his notes, "We are now no further than Euclid was. This is the partie honteuse [shameful part] of mathematics, which sooner or later must get a very different form."

A few years later, in 1816, Gauss showed that he had even darker thoughts about geometry. He wrote to his student Christian Gerling that he thought there was nothing absurd in denying the fifth postulate. A year later, he wrote to Olbers, "I am becoming more and more convinced that the necessity of our geometry cannot be proved." In the same letter he said, "Perhaps only in another life will we attain another insight into the nature of space, which is unattainable to us now."

Still, with the exception of these and a few other letters, Gauss declined to publish anything at all on the matter—for reasons that historians are at a loss to explain. He was satisfied to leave it be, avoid controversy, and go on with other work.

Perhaps Gauss was so prolific in so many other areas of mathematics that he simply may not have had time to finish his work in geometry. Other universities were still trying to lure him away, even after he had settled at Göttingen. Or it may be that he was not satisfied enough with his material to let it out into the world. Plus he now had the added responsibility of raising a family and attending to the administrative duties of being a professor. Like one of his idols, Isaac Newton, Gauss was not very fond of the administrative day-to-day.

Gauss idolized Newton because Newton, like Gauss, was an agile genius whose mathematical mind was unrivaled in his day. And like Newton, Gauss was a great believer in perfection in his work. He sought it—needed it—in what he wrote. He idolized Newton for his celebrated works, which were products of years of hard work and effort.

Unlike Newton, whose genius was not truly celebrated until he was middle-aged, Gauss was barely into his twenties when his scientific celebrity rippled throughout Europe. This may have had a profound effect on him by setting the bar even higher. He was driven by a complete love of perfection—perhaps even hampered by his need to attain it.

Possibly there was another reason for Gauss’s reluctance to publish anything on non-Euclidean geometry. He may have been reluctant to go against two thousand years of established wisdom—indeed, against reality itself. This was a reality to which, by all indications, Gauss himself subscribed. He saw mathematics as a basic method for understanding nature. Non-Euclidean geometry flew in the face of that, and it flew in the face of the most influential German philosopher of Gauss’s day—Immanuel Kant.

Kant’s latest work, the Critique of Practical Reason, was still just a few years off the presses when Gauss started college. It followed his monumental Critique of Pure Reason, which had a profound effect in intellectual circles throughout Europe, especially in Germany. Göttingen was no exception. There, Kant was king. He asserted in his Critique of Pure Reason that the universe is Euclidean. Gauss read the Critique five times because he was interested in how Kant conceptualized space. Kant said that Euclid was right. Who was Gauss to say otherwise?

He may even have been afraid of the uncertainty the controversy would stir. Certainly some of the commentators who tried to make sense of this after he died put it down to cowardice on his part. Why would he discover something as novel as non-Euclidean geometry and not tell anyone anything about it? Gauss himself said that he feared the "clamor of the Boetians," a reference to the ancient people of Boetia, who are said to have been dull-witted and boorish.

Maybe this was not so much fear as it was the desire to continue his work in peace. This clamor, had it erupted, would surely have driven Gauss to distraction. He was a great mathematician—and a historic figure—but in the end he was a simple man given to the simple taste of hard work. He applied his vast theoretical mathematical skills to some of the greatest problems of his day, and he was happy to do his work in peace. Little did he know that thousands of miles to the east, a talented mathematician at a remote Russian university was about to disturb the peaceful Euclidean universe.

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