342 11. POST-EUCLIDEAN GEOMETRY
to become negative, which was surely impossible. Unfortunately, the possibility of
repeated doubling that he had in mind was just one of those small points mentioned
by Lambert that turn out to be equivalent to the parallel postulate. In fact, it is
rather easy to see that such is the case, since (Fig. 11(6)) the possibility of drawing
a line B'C through a point A' inside the angle CAB that intersects both AB and
AC is simply another way of saying that the lines AB and AC must both intersect
some line through A', that is, AC cannot be parallel to every line through A' that
intersects AB.
5.3. Gauss. The true situation in regard to the parallel postulate was beginning
to be understood by the end of the eighteenth century. Gauss, who read Lambert's
work on parallels (which had been published posthumously), began to explore this
subject as a teenager, although he kept his thoughts to himself except for letters to
colleagues and never published anything on the subject. His work in this area was
published in Vol. 8 of the later edition of his collected works. It is nicely summarized
by Klein (1926, pp. 58-59). In 1799 he wrote to Farkas Bolyai (1775-1856), his
classmate from Gottingen, that he could prove the parallel postulate provided that
triangles of arbitrarily large area were admitted. Such a confident statement can
only mean that he had developed the metric theory of hyperbolic geometry to a
considerable extent. Five years later he wrote again to explain the error in a proof
of the parallel postulate proposed by Bolyai. Gauss, like Lambert, realized that a
non-Euclidean space would have a natural unit of length, and mentioned this fact
in a letter of 1816 to his student Christian Ludwig Gerling (1788-1864), proposing
as unit the side of an equilateral triangle whose angles were 59° 59' 59.99999.. .".^21
To Gauss' surprise, in 1818 he received from Gerling a paper written by Ferdinand
Karl Schweikart (1780-1859), a lawyer then in Marburg, who had developed what
he called astral geometry. It was actually hyperbolic geometry, and Schweikart had
gone far into it, since he knew that there was an upper bound to the area of a triangle
in this geometry, that its metric properties depended on an undetermined constant
C (its radius of curvature), and that it contained a natural unit of length, which he
described picturesquely by saying that if that length were the radius of the earth,
then the line joining two stars would be tangent to the earth. Gauss wrote back
to correct some minor points of bad drafting on Schweikart's part (for example,
Schweikart neglected to say that the stars were assumed infinitely distant), but
generally praising the work. In fact, he communicated his formula for the limiting
area of a triangle:
7TC^2{\n(l + V2)Y'
By coincidence, Schweikart's nephew Franz Adolph Taurinus (1794-1874), also
a lawyer, who surely must have known of his uncle's work in non-Euclidean geom-
etry, sent Gauss his attempt at a proof of the parallel postulate in 1824. Gauss
explained the true situation to Taurinus under strict orders to keep the matter se-
cret. The following year, Taurinus published a treatise Geometric prima elementa
(First Elements of Geometry) in which he accepted the possibility of other ge-
ometries. Gauss wrote to the astronomer -mathematician Friedrich Wilhelm Bessel
(^21) In comparison with the radius of curvature of space, this would be an extremely small unit of
length; however, if space is curved negatively at all, its radius of curvature is so enormous that in
fact this unit might be very large.