344 11. POST-EUCLIDEAN GEOMETRY
Lobachevskii's geometry. Lobachevskii connected the parts of a hyperbolic triangle
through his formula for the angle of parallelism, which is the angle è 0 referred to
above, as a function of the length AB. He gave this formula as
where á denotes the length AB and F(a) the angle 6Q. Here e could be any
positive constant, since the radius of curvature of the hyperbolic plane could not
be determined. However, Lobachevskii found it convenient to take this constant to
be e = 2.71828 — In effect, he took the radius of curvature of the plane as the
unit of length. Lobachevskii gave the Pythagorean theorem, for example, as
Of the two nearly simultaneous creators of hyperbolic geometry and trigonom-
etry, Lobachevskii was the first to publish, unfortunately in a journal of limited
circulation. He was a professor at the provincial University of Kazan' in Russia and
published his work in 1826 in the proceedings of the Kazan' Physico-Mathematical
Society. He reiterated this idea over the next ten years or so, developing its impli-
cations. Like Gauss, he drew the conclusion that only observation could determine
if actual space was Euclidean or not. As luck would have it, the astronomers were
just beginning to attempt measurements on the interstellar scale. In particular, by
measuring the angles formed by the lines of sight from the Earth to a given fixed
star at intervals of six months, one could get the base angles of a gigantic triangle
and thereby (since the angle sum could not be larger than two right angles, as every-
one agreed) place an upper bound on the size of the parallax of the star (the angle
subtended by the Earth's orbit from that star). Many encyclopedias claim that
the first measurement of stellar parallax was carried out in Konigsberg by Bessel
in 1838, and that he determined the parallax of 61 Cygni to be 0.3 seconds. Rus-
sian historians credit another Friedrich Wilhelm, namely Friedrich Wilhelm Struve
(1793-1864), who emigrated to Russia and is known there as Vasilii Yakovlevich
Struve. He founded the Pulkovo Observatory in 1839. Struve determined the par-
allax of the star Vega in 1837. Attempts to determine stellar parallax must have
been made earlier, since Lobachevskii cited such measurements in an 1829 work
and claimed that the measured parallax was less than 0.000372", which is much
smaller than any observational error.^22 As he said (see his collected works, Vol. 1,
p. 207, quoted by S.N. Kiro, 1967, Vol. 2, p. 159):
At the very least, astronomical observations prove that all the lines
amenable to our measurements, even the distances between ce-
lestial bodies, are so small in comparison with the length taken
as a unit in our theory that the equations of (Euclidean) plane
trigonometry, which have been used up to now must be true with-
out any sensible error.
The vast distances between stars make terrestrial units of length inadequate. The light-year
(about 9.5 · 10^12 km) is the most familiar unit now used, particularly good, since it tells us "what
time it was" when the star emitted the light we are now seeing. Stellar parallax provides another
unit, the parsec, which is the distance at which the radius of the Earth's orbit subtends an angle
of 1". A parsec is about 3.258 light-years.
sin F(a) sin F(b) = sin F(c).
