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Figure 41

Figure 42

Figure 43

If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it, would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years. So I remained quite stupefied. So far as my own work is concerned, of which up till now I have put little on paper, my intention was not to let it be published during my lifetime.

Let me parenthetically note that apparently Gauss feared that the radically new geometry would be regarded by the Kantian philosophers, to whom he referred as “the Boetians” (synonymous with “stupid” for the ancient Greeks), as philosophical heresy. Gauss then continued:

Figure 44

On the other hand it was my idea to write down all this later so that at least it should not perish with me. It is therefore a pleasant surprise with me that I am spared this trouble, and I am very glad that it is the son of my old friend, who takes the precedence of me in such a remarkable manner.

While Farkas was quite pleased with Gauss’s praise, which he took to be “very fine,” János was absolutely devastated. For almost a decade he refused to believe that Gauss’s claim to priority was not false, and his relationship with his father (whom he suspected of prematurely communicating the results to Gauss) was seriously strained. When he finally realized that Gauss had actually started working on the problem as early as 1799, János became deeply embittered, and his subsequent mathematical work (he left some twenty thousand pages of manuscript when he died) became rather lackluster by comparison.

There is very little doubt, however, that Gauss had indeed given considerable thought to non-Euclidean geometry. In a diary entry from September 1799 he wrote: “In principiis geometriae egregios progressus fecimus” (“About the principles of geometry we obtained wonderful achievements”). Then, in 1813, he noted: “In the theory of parallel lines 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 a letter written on April 28, 1817, he stated: “I am coming more and more to the conviction that the necessity of our [Euclidean] geometry cannot be proved.” Finally, and contrary to Kant’s views, Gauss concluded that Euclidean geometry could not be viewed as a universal truth, and that rather “one would have to rank [Euclidean] geometry not with arithmetic, which stands a priori, but approximately with mechanics.” Similar conclusions were reached independently by Ferdinand Schweikart (1780–1859), a professor of jurisprudence, and the latter informed Gauss of his work sometime in 1818 or 1819. Since neither Gauss nor Schweikart actually published his results, however, the priority of first publication is traditionally credited to Lobachevsky and Bolyai, even though the two can hardly be regarded as the sole “creators” of non-Euclidean geometry.

Hyperbolic geometry broke on the world of mathematics like a thunderbolt, dealing a tremendous blow to the perception of Euclidean geometry as the only, infallible description of space. Prior to the Gauss-Lobachevsky-Bolyai work, Euclidean geometry was, in effect, the natural world. The fact that one could select a different set of axioms and construct a different type of geometry raised for the first time the suspicion that mathematics is, after all, a human invention, rather than a discovery of truths that exist independently of the human mind. At the same time, the collapse of the immediate connection between Euclidean geometry and true physical space exposed what appeared to be fatal deficiencies in the idea of mathematics as the language of the universe.

Euclidean geometry’s privileged status went from bad to worse