340 11. POST-EUCLIDEAN GEOMETRYFIGURE 11. (a) Lines through A that intersect BE and those that
share a common perpendicular with BE are separated by a line
(AL) that is asymptotic to BE. (b) The angle defect of AAB'C
is more than twice the defect of AABC.
two lines cannot enclose an area, so that Saccheri can hardly be faulted for dealing
with only one Euclidean postulate at a time. Since the parallel postulate implies
that the summit and base of a Saccheri quadrilateral must meet on both sides of the
quadrilateral under the hypothesis of the obtuse angle, even a severe critic should
be inclined to give Saccheri a passing grade when he rejects this hypothesis.
Having disposed of the hypothesis of the obtuse angle, Saccheri then joined
battle (his phrase) with the hypothesis of the acute angle. Here again, he proved
some basic facts about what we now call hyperbolic geometry. Given any quadri-
lateral having right angles at the base and acute angles at the summit, it follows
from continuity considerations that the length of a perpendicular dropped from
the summit to the base must reach a minimum at some point, and at that point
it must also be perpendicular to the summit. Saccheri analyzed this situation in
detail, describing in the process a great deal of what must occur in what is now
called hyperbolic geometry. In terms of Fig. 11(a),^20 he considered all the lines
like AF through the point A such that angle BAF is acute. He wished to show
that they all intersected the line BE.
Saccheri proved that there must be at least one angle 0n for which the line AL
making that angle neither intersects BE nor has a common perpendicular with it.
This line, as Saccheri showed in Proposition 23, must approach BE asymptotically
as we would say. At that point he made the small slip that had been warned
against even in ancient times, assuming that "approaching" implies "meeting." His
intuition for hyperbolic geometry was very good, as he imagined a line perpendicular
to BE moving away from AB and the lines from A perpendicular to it rotating
any length. It is not necessary to require that the extension never overlap the portion already
present.
(^20) Since the flat page is not measurably non-Euclidean, and wouldn't be even if spread out to
cover the entire solar system, the kinds of lines that occur in hyperbolic geometry cannot be
drawn accurately on paper. Our convention is the usual one: When asymptotic properties are
not involved, draw the lines straight. When asymptotic properties need to be shown, draw them
as hyperbolas. Actually, if the radius of curvature of the plane were comparable to the width of
the page, two lines with a common perpendicular would diverge from each other like the graphs
of cosh £ and — coshi, very rapidly indeed.