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There is evidence that Gauss
had anticipated some of J. Bolyai's discoveries, in fact, that Gauss had been working on
non-Euclidean geometry since the age of 15, i.e., since 1792 (see Bonola, 1955, Chapter
3). In 1817, Gauss wrote to W. Olbers: "I am becoming more and more convinced that
the necessity of our [Euclidean] geometry cannot be proved, at least not by human reason
or for human reason. Perhaps in another life we will be able to obtain insight into the
nature of space, which is now inattainable." In 1824, Gauss answered F. A. Taurinus,
who had attempted to investigate the theory of parallels:
In regard to your attempt, I have nothing (or not much) to say except that it is
incomplete. It is true that your demonstration of the proof that the sum of the three
angles of a plane triangle cannot be greater than 180° is somewhat lacking in geometrical
rigor. But this in itself can easily be remedied, and there is no doubt that the
impossibility can be proved most rigorously. But the situation is quite different in the
second part, that the sum of the angles cannot be less than 180°; this is the critical
point, the reef on which all the wrecks occur. I imagine that this problem has not engaged
you very long. I have pondered it for over thirty years, and I do not believe that anyone
can have given more thought to this second part than I, though I have never published
anything on it.
The assumption that the sum of the three angles is less than 180° leads to a
curious geometry, quite different from ours [the Euclidean], but thoroughly consistent,
which I have developed to my entire satisfaction, so that I can solve every problem in it
with the exception of the determination of a constant, which cannot be designated a
priori. The greater one takes this constant, the nearer one comes to Euclidean geometry,
and when it is chosen infinitely large the two coincide. The theorems of this geometry
appear to be paradoxical and, to the uninitiated, absurd; but calm, steady reflection
reveals that they contain nothing at all impossible.
For example, the three angles of a triangle become as small as one wishes, if only
the sides are taken large enough; yet the area of the triangle can never exceed a definite
limit, regardless of how great the sides are taken, nor indeed can it never reach it.
All my efforts to discover a contradiction, an inconsistency, in this non-Euclidean
geometry have been without success, and the one thing in it which is opposed to our
conceptions is that, if it were true, there must exist in space a linear magnitude,
determined for itself (but unknown to us). But it seems to me that we know, despite the
say-nothing word-wisdom of the metaphysicians, too little, or too nearly nothing at all,
about the true nature of space, to consider as absolutely impossible that which appears to
us unnatural. If this non-Euclidean geometry were true, and it were possible to compare
that constant with such magnitudes as we encounter in our measurements on the earth and in
the heavens, it could then be determined a posteriori. Consequently, in jest I have
sometimes expressed the wish that the Euclidean geometry were not true, since then we
would have a priori an absolute standard of measure.
I do not fear that any man who has shown that he possesses a thoughtful
mathematical mind will misunderstand what has been said above, but in any case consider it
a private communication of which no public use or use leading in any way to publicity is
to be made. Perhaps I shall myself, if I have at some future time more leisure than in my
present circumstances, make public my investigations.
It is amazing that, despite his great reputation, Gauss was actually afraid to make
public his discoveries in non-Euclidean geometry. He wrote to F. W. Bessel in 1829 that he
feared "the howl from the Boeotians" if he were to publish his revolutionary
discoveries. He told H. C. Schumacher that he had "a great antipathy against being
drawn into any sort of polemic."
The "metaphysicians" referred to by Gauss in his letter to Taurinus were
followers of Immanuel Kant, the supreme European philosopher in the late eighteenth
century and much of the nineteenth century. Gauss' discovery of non-Euclidean geometry
refuted Kant's position that Euclidean space is inherent in the structure of our mind. In
his Critique of Pure Reason (1781) Kant declared that "the concept of [Euclidean]
space is by no means of empirical origin, but is an inevitable necessity of thought."
Another reason that Gauss withheld his discoveries was that he was a perfectionist;
one who published only completed works of art. His devotion to perfected work was
expressed by the motto on his seal, pauca sedmatura ("few but ripe"). There is a
story that the distinguished mathematician K. G. J. Jacobi often came to Gauss to relate
new discoveries, only to have Gauss pull out some papers from his desk drawer that
contained the very same discoveries. Perhaps it is because Gauss was so preoccupied with
original work in many branches of mathematics, as well as in astronomy, geodesy, and
physics (he contented an improved telegraph with W. Weber), that he did not have the
opportunity to put his results on non-Euclidean geometry into polished form. The few
results he wrote down were found among his private papers after his death.
Gauss has been called "the prince of mathematicians" because of the range
and depth of his work. (See the biographies by Bell, 1934; Dunnington, 1955; and Hall,
1970.)
Another actor in this historical drama came along to steal the limelight from both
J. Bolyai and Gauss: the Russian mathematician Nikolai Ivanovich Lobachevsky (1792-1856).
He was the first to actually publish an account of non-Euclidean geometry, in 1829.
Lobachevsky initially called his geometry "imaginary," then later
"pangeometry." His work attracted little attention on the continent when it
appeared because it was written in Russian. The reviewer at the St. Petersburg Academy rejected it, and a Russian literary journal
attacked Lobachevsky for "the insolence and shamelessness of false new
inventions" (Boeotians howling, as Gauss predicted). Nevertheless, Lobachevsky
courageously continued to publish further articles in Russian and then a treatise in 1840
in German, which he sent to Gauss. In an 1846 letter to Schumacher, Gauss reiterated his
own priority in developing non-Euclidean geometry but conceded that "Lobachevsky
carried out the task in a masterly fashion and in a truly geometric spirit." At
Gauss' recommendation, Lobachevsky was elected to the Gottingen Scientific Society. (Why
didn't Gauss recommend Janos Bolyai?)
Lobachevsky openly challenged the Kantian doctrine of space as a subjective
intuition. In 1835 he wrote: "The fruitlessness of the attempts made since Euclid's time . . . aroused in me the suspicion that the truth .
. . was not contained in the data themselves; that to establish it the aid of experiment
would be needed, for example, of astronomical observations, as in the case of other laws
of nature." (Gauss privately agreed with this view, having written to Olbers that
"we must not put geometry on a par with arithmetic that exists purely a priori but
rather with mechanics." The great French mathematicians J. L. Lagrange (1736 -1813)
and J.B.Fourier (1768 -1830) tried to derive the parallel postulate from the law of the
lever in statics.)
Lobachevsky has been called "the great emancipator" by Eric Temple Bell;
his name, said Bell, should be as familiar to every schoolboy as
that of Michelangelo or Napoleon. Unfortunately, Lobachevsky was not so appreciated in his
lifetime; in fact, in 1846 he was fired from the University of Kazan, despite 20 years of outstanding service as a teacher and
administrator. He had to dictate his last book in the year before his death, for by then
he was blind.
It is amazing how similar are the approaches of J. Bolyai and Lobachevsky and how
different they are from earlier work. Both developed the subject much further than Gauss.
Both attacked plane geometry via the "horosphere" in three-space (it is the
limit of an expanding sphere when its radius tends to infinity). Both showed that geometry
on a horosphere, where "lines" are interpreted as "horocycles" (limits
of circles), is Euclidean. Both showed that Euclidean spherical trigonometry is valid in
neutral geometry and both constructed a mapping from the sphere to the non-Euclidean plane
to derive the formulas of non-Euclidean trigonometry (including the formula Taurinus
discovered — see Theorem 10.4, Chapter 10, for a simpler derivation using a plane
model). Both had a constant in their formulas that they could not explain; the later work
of Riemann showed it to be the curvature of the non-Euclidean plane.
SUBSEQUENT
DEVELOPMENTS
It was not until
after Gauss' death in 1855, when his correspondence was published, that the mathematical
world began to take non-Euclidean ideas seriously. (Yet, as late as 1888 Lewis Carroll was
poking fun at non-Euclidean geometry.) Some of the best mathematicians (Beltrami, Klein,
Poincare, and Riemann) took up the subject, extending it, clarifying it, and applying it
to other branches of mathematics, notably complex function theory. In 1868 the Italian
mathematician Beltrami settled once and for all the question of a proof for the parallel
postulate. He proved that no proof was possible. He did this by exhibiting a Euclidean
model of non-Euclidean geometry. (We will discuss his model in the next chapter.)
Bernhard Riemann, who was a student of Gauss, had the most profound insight into
the geometry, not just the logic. In 1854, he built upon Gauss' discovery of the intrinsic
geometry on a surface in Euclidean three-space. Riemann invented the concept of an
abstract geometrical surface that need not be embeddable in Euclidean three-space yet on
which the "lines" can be interpreted as geodesies and the intrinsic curvature of
the surface can be precisely defined. Elliptic (and, of course, spherical) geometry
"exist" on such surfaces that have constant positive curvature, while the
hyperbolic geometry of Bolyai and Lobachevsky "exists" on such a surface of
constant negative curvature. That is the view of geometers today about the
"reality" of those non-Euclidean planes. We will describe Gauss and Riemann's
idea only in Appendix A, since it is too advanced for the level of this text. A further
generalization of that idea provided the geometry for Einstein's general theory of
relativity.
Interestingly, a direct relationship between the special theory of relativity and
hyperbolic geometry was discovered by the physicist Arnold Sommerfeld in 1909 and
elucidated by the geometer Vladimir Varicak in 1912. A model of hyperbolic plane geometry
is a sphere of imaginary radius with antipodal points identified in the three-dimensional
space-time of special relativity, vindicating Lambert's idea (see Rosenfeld, 1988, pp. 230
and 270; or Yaglom, 1979, p. 222 ff.). Moreover, Taurinus' technique of substituting
effort go from spherical trigonometry to hyperbolic trigonometry received a structural
explanation in 1926-1927 when Elie Cartan developed his theory of Riemannian symmetric
spaces: The Euclidean sphere of curvature 1/r2 is "dual" to the hyperbolic plane
of curvature — 1/r2 (see Helgason, 1962, p. 206).
HYPERBOLIC
GEOMETRY
Let us return to
our elementary investigation of the particular non-Euclidean geometry discovered by Gauss,
J. Bolyai, and Lobachevsky, nowadays called hyperbolic geometry (see Appendix A for a
discussion of elliptic geometry and other geometries discovered by Riemann).
Hyperbolic geometry is, by definition, the
geometry you get by assuming all the axioms for neutral geometry and replacing Hilbert's
parallel postulate by its negation, which we shall call the "hyperbolic axiom."
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