346 11. POST-EUCLIDEAN GEOMETRY
pseudonym Lobachevskii, since "it is hardly likely that two or even three people
knowing nothing of one another would produce almost the same result by different
routes."
5.5. The reception of non-Euclidean geometry. Some time was required for
the new world revealed by Lobachevskii and Bolyai to attract the interest of the
mathematical community. Because it seemed possible—even easy—to prove that
parallel lines exist, or equivalently, that the sum of the angles of a triangle could not
be more than two right angles, one can easily understand why a sense of symmetry
would lead to a certain stubbornness in attempts to refute the opposite hypothesis
as well. Although Gauss had shown the way to a more general understanding with
the concept of curvature of a surface, which could be either negative or positive, in
the 1825 paper on differential geometry (published in 1827, to be discussed in detail
in Sect. 3 of Chapter 12), it took Riemann's inaugural lecture in 1854 (published
in 1867, also discussed in detail in Sect. 3 of Chapter 12), which made the crucial
distinction between the unbounded and the infinite, to give the proper perspective.
After that, acceptance of non-Euclidean geometry was quite rapid. In 1868, the year
after the publication of Riemann's lecture, Eugenio Beltrami (1835-1900) realized
that Lobachevskii's theorems provide a model of the Lobachevskii-Bolyai plane in
a Euclidean disk. This model is described by Gray (1989, p. 112), as follows.
Imagine a directed line perpendicular to the Lobachevskii-Bolyai plane in
Lobachevskii-Bolyai three-dimensional space. The entire set of directed lines that
are parallel (asymptotic) to this line on the same side of the plane generates a
unique horosphere tangent to the plane at its point of intersection with the line.
Some of the lines parallel to the given perpendicular in the given direction intersect
the original plane, and others do not. Those that do intersect it pass through the
portion of the horosphere denoted Ù in Fig. 12. Shortest paths on the horosphere
are obtained as its intersections with planes passing through the point at infinity
that serves as its "center." These paths are called horocycles. But there is only one
horocycle through a given point in Ù that does not intersect a given horocycle, so
that the geometry of Ù is Euclidean. As a result, we have a faithful mapping of the
Lobachevskii-Bolyai plane onto the interior of a disk Ù in a Euclidean plane, under
which lines in the plane correspond to chords on the disk. This model provides
an excellent picture of points at infinity: they correspond to the boundary of the
disk Ù. Lines in the plane are parallel if and only if the chords corresponding to
them have a common endpoint. Lines that have a common perpendicular in the
Lobachevskii-Bolyai plane correspond to chords whose extensions meet outside the
circle. It is somewhat complicated to compute the length of a line segment in the
Lobachevskii-Bolyai plane from the length of its corresponding chordal segment in
Ù or vice versa, and the angle between two intersecting chords is not simply related
to the angle between the lines they correspond to.^23 Nevertheless these computa-
tions can be carried out from the trigonometric rules given by Lobachevskii. The
result is a perfect model of the Lobachevskii-Bolyai plane within the Euclidean
plane, obtained by formally reinterpreting the words line, plane, and angle. If
(^23) It can be shown that perpendicular lines correspond to chords having the property that the
extension of each passes through the point of intersection of the tangents at the endpoints of
the other. But it is far from obvious that this property is symmetric in the two chords, as
perpendicularity is for lines.