Dissolution of Ancient Civilization. By “ancient
civilization” we mean, as usual, the large conglom-
eration of component civilizations of so-called antiq-
uity that severally came into being in and around the
Mediterranean Littoral. In the last centuries B.C. and
first centuries A.D., the conglomeration merged into one
compound civilization, creating “one civilized world,”
the self-styled oikouméne. The passing-away of this
civilization constitutes one of history's greatest prob-
lems. The central part of this problem is the seemingly
“formal” question of determining when ancient civili-
zation terminated, if indeed it did “terminate”; or,
rather, since the termination was probably a process
of some duration, when the process of termination
began and when it ended.
It is widely accepted that the process ended towards
the close of the fifth century A.D., as symbolized by
the fact that in 476 A.D. “the last claimant to the
Roman throne in the West was deposed” (Kagan, p.
viii). An almost lone dissenter among historians, but
a leading one, is Henri Pirenne, who maintains that
ancient civilization was brought to an end—and then
rather violently so—only in the seventh century A.D.,
by the widespread militancy of Islam. Ancient civili-
zation was then not actually “killed,” but rather pushed
away from its Mediterranean habitat into more north-
ern parts of Europe, where it became isolated and
immobilized for centuries (Havighurst, 1958).
Another kind of dissent, a qualified one, is caused
inevitably by the fact that there was an Eastern Roman
Empire which lasted very long, until 1453 A.D. Most
Byzantinists of today find that this empire was very
viable in 476 A.D. and beyond, until the onset of the
Crusades at any rate. They do not deny that there was
a ruin of the Western Empire in the fifth century, but
most of them do not allow that this ruin came about,
to a meaningful degree, by decay from within. They
argue that the causes for decay from within would have
been present also in the Eastern half of the Empire,
and should have had the same destructive effect at the
time (Jones, Ch. 24; J. B. Bury, quoted in Kagan, pp.
7-10), so that, in their view, only military and other
external causes remain. A very outspoken champion
of this view is Lynn White, Jr., and he is severely
critical of the details, attitudes, and emphases in
Edward Gibbon's key work, Decline and Fall of the
Roman Empire (White, pp. 291-311).
With regard to the beginning of the process of the
decline of ancient civilization widely spaced dates have
been proposed, explicitly or by implication. F. W.
Walbank finds “... the germs of the illness of antiquity
already present in the Athens of the fifth century B.C.”
(Kagan, p. viii). This is a very early date indeed. Less
extreme is the finding of M. I. Rostovtzeff that “decline
began as early as the second century B.C.” (Kagan, p.
2). Finally “according to Gibbon, the Roman Empire
reached its zenith in the age of the Antonines [second
century A.D.] after which the decline set in” (White,
p. 25). Thus, by implication, in the view of Gibbon
the decline set in around 200 A.D., after the era of the
“Five Good Emperors” (90-180 A.D.).
We proceed now to review the cogency of the above
dates in the light of the history of mathematics.
The history of mathematics fully corroborates the
familiar textbook assertion that around 500 A.D. a
large-scale decline occurred. Mathematics as an intel-
lectual activity—as an academic discipline, so to
speak—was suddenly lost from sight, as if swallowed
up by a wave of a flood, or buried by a sandstorm.
In the Latinized West there had been a mathematics
bearing the telltale mark of a Greek heritage even
when dealing with non-Greek or extra-Greek topics
of the mathematical corpus. It was this kind of mathe-
matics that suddenly disappeared. Also in the West,
this kind of mathematics came fully to surface only
after seven centuries or so, in the famous Liber abaci
of Leonardo of Pisa (Fibonacci) around 1200 A.D. This
is not to say that in the intervening centuries mathe-
matics was unknown in the West. A mathematics of
a kind was of course included in school curricula. There
were some translations from the Arabic, especially
during the so-called twelfth-century Renaissance.
There was also a certain pursuit of “utilitarian” math-
ematics, even during the so-called Carolingian Renais-
sance (Smith, I, 175-220). But, before the work of
Fibonacci, this pursuit did not evince a quest for the
kind of originality, if only on a modest scale, with
which mathematics had been imbued since the sixth
century B.C., when the Greeks had begun to weave
mathematics into the texture of their rationality.
This decay of Greek mathematics did not spare the
Eastern Roman empire, which, by an official reckoning,
lasted from 529 to 1453, from Justinian to the fall of
Constantinople. As far as mathematics is concerned this
Empire might have never been. There is an encyclo-
pedic treatise which lists the extant Byzantine writings,
including works on mathematics and cognate subjects
(Krumbacher, pp. 620-26). The mathematical works
of the collection bespeak sterility and stagnation, and
only in the art of warfare were there some elements
of originality (Taton, p. 446).
Finally, we note that, in the judgment of a leading
student of chronological innovations (Ginzel, III,
178ff.), at the beginning of the sixth century A.D. the
Roman Canonist Dionysius Exiguus founded our
present-day system of designating years by A.D. and
B.C. This event, even if conceived very modestly by
its author, was, from our retrospect, mathematically
tinged, and it took place in Rome, after its “official”
fall, even if not long after it. Thus, this may be viewed
as a mathematical corroboration of the fact—which
was stressed, from very different approaches, by Henri
Pirenne, Lynn White, and probably others—that the
political fall of Rome in 476 A.D. was not, instantly,
also a social and intellectual disintegration.
This achievement of the Roman Canonist, modest
as it may appear to be when viewed in isolation, cannot
be overestimated as a determinant of history. The
Greeks never quite succeeded in introducing a com-
parable dating of years. Mathematically this would
have amounted to introducing a coordinate system on
the time axis, and the Greeks never achieved this.
Furthermore, it is remarkable that the “Christian” era
of Dionysius not only became a “common” era but
has turned out to be the most durable era ever. The
French Revolution of 1793, the Russian Revolution of
1917, the Italian Revolution of 1922, and the German
Revolution of 1933, each attempted to introduce a new
era beginning with itself, but none really succeeded
or even made the attempt in earnest. Our “common”
era, however “Christian” by origin, has become a
standard institution that cannot be tampered with.
We now turn to the question of mathematical evi-
dence for the beginning of the decline of ancient civi-
lization. Firstly, mathematics clearly concurs with the
assertion of Rostovtzeff that a general decay began in
the second century B.C. Secondly, mathematics can
offer no tangible corroboration of the fact, known from
general history, that life in the Greco-Roman world
was much bleaker in the third century A.D. than in
the preceding one. Thirdly, and finally, mathematics
can even corroborate the thesis of Walbank that germs
of some of the illnesses of antiquity can already be
found in the Athens of the fifth century B.C.; namely,
by a tour-de-force, we may elicit from the nature of
Greek mathematics some peculiar comment of the
thesis, which can be interpreted as a corroboration of
it, in a sense.
Greek mathematics built on a considerable body of
mathematics that had preceded it, but it was never-
theless a singular achievement of ancient civilization
as a whole, and a hallmark of its Hellenic aspect in
particular. Naively or boldly, the Greeks made a fresh
start. They were inspired to begin from a new begin-
ning and they succeeded. They erected an edifice of
mathematics that was a veritable “system” in our
present-day sense of the word. The bricks and stones
in the edifice may have been Egyptian, Babylonian,
or other, but the structure was Greek. This Greek
mathematics attained its intellectual acme in the
achievements of Archimedes. Isaac Newton even com-
posed his Principia in an Archimedean mise-en-scène,
but he acknowledged no indebtedness to an Egyptian
calculus of fractions, or even a Babylonian calculus of
quadratic equations, even if he knew anything about
them at all.
From what is known, this Greek mathematics
showed the first signs of being itself around 600 B.C.
It then grew and kept unfolding for about four centur-
ies, that is, till about 200 B.C., and the last of these
four centuries, that is the era from 300 B.C. to 200 B.C.,
was a culminating one. In fact, around 300 B.C. Euclid
composed his Elements, Archimedes flourished around
250 B.C., and around 200 B.C., Apollonius produced his
monumental Conics.
But after that, in the second century B.C., unexpect-
edly and inexplicably, as if on a signal, the development
of this mathematics came almost to a halt. After 200
B.C. it began to level off, to loose its impetus, and then
to falter, bringing to the fore only such works as those
of a Nicomedes, Dioclos, and Hypsicles. The phenom-
enon was no passing setback but a permanent recession,
the beginning of a decline. It was, for mathematics,
the beginning of the end in the conception of Ros-
tovtzeff, even if the true end itself, that is the final
extinction of Greek mathematics in its own phase,
came only considerably later, around 500 A.D., that is,
around the time of the fall of Rome.
It is noteworthy though, that the second century B.C.,
as if to almost show that it was not entirely down and
out, produced the astronomer Hipparchus, famed dis-
coverer of the precession of the equinoxes; and that
the second century A.D., as if to lay claim to being
indeed a “good” century, brought forth his great “suc-
cessor” Claudius Ptolemy, author of the majestic
Almagest, and of a Geography. It must be stated how-
ever, emphatically, that in “basic” mathematics
Ptolemy was in no wise farther along than Archimedes,
even if the Almagest, as an astronomical text, was a
live text still for Copernicus in the first half of the
sixteenth century and began to become antiquarian
only half a century later in consequence of the mathe-
matically articulated innovations of Kepler.
