The Nineteenth Century. In the history of knowl-
edge, even more complex than the era of Enlighten-
ment is an era following it. It is, schematically, the
half-century centered in the year 1800, that is the
half-century 1776-1825, which has been called the Age
of Eclosion by Bochner. During this era there was a
great and sudden outburst of knowledge in all academic
disciplines; and in historical studies of various kinds,
there emerged a “critical” approach to the evaluation
of source material, which is sometimes called “higher
criticism.” In mathematics there was no such sudden
increase of activity, but an analogue to the “higher
criticism” did begin to manifest itself. A critical ap-
proach to so-called foundations of mathematics began
to spread, and even novel patterns of insight were
emerging. Thus, C. F. Gauss in his Disquisitiones arith-
meticae, which appeared in 1801, explicitly formulated
the statement that any integer can be represented as
a product of prime numbers, and uniquely so. Implic-
itly the statement is already contained, between the
lines, in Book 7 of Euclid's Elements. But explicitly
the statement is a kind of “existence” and “uniqueness”
theorem, for which even number theorists like Fermat,
Wallis, Euler, and Lagrange were “not ready” yet.
Afterwards, in the course of the nineteenth century,
mathematics filled all those “gaps” in Greek mathe-
matics, which the Greeks themselves could not or
would not close. Thus, the nineteenth century finally
elucidated the role of Euclid's axiom on parallels, and
the role of axioms in general, as well as in geometry;
and it finally constructed a “Euclidean” space as a
background space for (Greek) mathematical figures and
astronomical orbits. In a peculiar logical sense, Greek
mathematics was “completed” only around 1900, but
by the developments which brought this about, it was
also rendered “antiquarian” by them. On the other
hand, “humanistic” Greek works like those of Homer
and Aeschylus, Herodotus and Thucydides, and Plato
and Aristotle were not “completed” in a similar sense,
but they have, on the other hand, not been rendered
“antiquarian,” as shelves full of reissues of these works
in paperback can attest.
This completes our preliminary observations, and we
now turn to two topics from antiquity for a somewhat
less summary analysis.
