348 11. POST-EUCLIDEAN GEOMETRY
three-dimensional Euclidean space that was a perfect model for the Lobachevskii-
Bolyai plane, in the sense that its geodesies corresponded to straight lines and
lengths and angles were measured in the ordinary way, remained open until Hubert,
in an article "Uber Flachen von konstanter Gaufischer Kriimmung" ("On surfaces
of constant Gaussian curvature"), published in the Transactions of the American
Mathematical Society in 1901, showed that no such surface exists.
5.6. Foundations of geometry. The problem of the parallel postulate was only
one feature of a general effort on the part of mathematicians to improve on the
rigor of their predecessors. This problem was particularly acute in the calculus, but
the parts of calculus that raised the most doubts were those that were geometric
in nature. Euclid, it began to be realized, had taken for granted not only the
infinitude of the plane, but also its continuity, and had not specified in many cases
what ordering of points was needed on the line for a particular theorem to be true.
If one attempts to prove these theorems without drawing any figures, it becomes
obvious what is being assumed. It seemed obvious, for example, that a line joining
a point inside a circle to a point outside the circle must intersect the circle in
a point, but that fact could not be deduced from Euclid's axioms. A complete
reworking of Euclid was the result, expounded in detail in Hubert's Grundlagen
der Geometrie (Foundations of Geometry), published in 1903. This book went
through many editions and has been translated into English (Bernays, 1971). In
Hubert's exposition the axioms of geometry are divided into axioms of incidence,
order, congruence, parallelism, and continuity, and examples are given to show what
cannot be proved when some of the axioms are omitted.
One thing is clear: No new comprehensive geometries are to be expected by pur-
suing the axiomatic approach of Hubert. In a way, the geometry of Lobachevskii and
Bolyai was a throwback even in its own time. The development of projective and
differential geometry would have provided—indeed, did provide—non-Euclidean
geometry by a natural expansion of the study of surfaces. It was Riemann, not
Lobachevskii and Bolyai, who showed the future of geometry. The real "action" in
geometry since the early nineteenth century has been in differential and projective
geometry. That is not to say that no new theorems can be produced in Euclidean
geometry, only that their scope is very limited. There are certainly many such
theorems. Coolidge, who undertook the herculean task of writing his History of
Geometric Methods in 1940, stated in his preface that the subject was too vast to
be covered in a single treatise and that "the only way to make any progress is by a
rigorous system of exclusion." In his third chapter, on "later elementary geometry,"
he wrote that "the temptation to run away from the difficulty by not considering
elementary geometry after the Greek period at all is almost irresistible." But to
attempt to build an entire theory as Apollonius did, on the synthetic methods and
limited techniques in the Euclidean tool kit, would be futile. Even Lobachevskii and
Bolyai at least used analytic geometry and trigonometry to produce their results.
Modern geometries are much more algebraic, as we shall see in Chapter 12.
6. Questions and problems
11.1. The figure used by Zenodorus at the main step in his proof of the isoperi-
metric inequality had been used earlier by Euclid to show that the apparent size of
objects is not inversely proportional to their distance. Prove this result by referring
