The History of Mathematics: A Brief Course

Chapter 11. Post-Euclidean Geometry

A certain dullness came over Greek geometry from the beginning of the second

century BCE. The preceding century had seen the beginning of Roman expansion,

whose early stage took the life of the aged Archimedes. Julius Caesar (100-44

BCE), who did more than anyone before him to turn Rome from a republic into

an empire, took an army to Egypt to fight his rival Pompey and incidentally help

Cleopatra, the last of the heirs of Ptolemy Soter, defeat her brother in a civil war.

In pursuing his aim he sent fire ships into the harbor of Alexandria to set it ablaze.

Although he himself naturally says nothing about any destruction of the city, later

writers, such as Plutarch in his Life of Caesar and Gellius in his Attic Nights,

say that the fire damaged the Library. Gellius claims that 700,000 books were

destroyed. After Caesar's heir Octavian defeated Mark Antony and Cleopatra in

31 BCE, Egypt became a province of the Roman Empire. Whether because of this

disruption or from limitations inherent in the Pythagorean philosophy, the level

of brilliant achievements of Euclid, Archimedes, and Apollonius was not sustained.

Nevertheless, geometry did not die out entirely, and some of the later commentators

are well worth reading. Very little new geometry was written in Greek after the sixth

century, however. From the ninth century to the fifteenth the Euclidean tradition

in geometry was pursued by Muslim mathematicians. Since these mathematicians

were also interested in the philosophy of Aristotle, in their work mathematics once

again began to be mixed with philosophy, as it was in the time just before Euclid.

When the Roman Empire was vigorous, all upper-class Romans understood

Greek, and many seemed to prefer it to Latin. The Emperor Marcus Aurelius,

for example, who ruled from 161 to 180, wrote his meditations in Greek. Af-

ter the Emperor Diocletian (284-305) split the empire into eastern and western

halves to make it governable and the eastern Emperor Constantine (307-337), who

proclaimed Christianity the official religion of the Empire, moved his capital to

Constantinople, knowledge of Greek began to decline in the western part of the

empire. Many books were translated into Latin, or replacements for them were

written in Latin.

The repeated ravaging of Italy by invaders from the north caused an irreversible

decline in scholarship there. In the east, which fared somewhat better, scholarship

continued for another thousand years, until the Turkish conquest of Constantinople

in 1453. The eastern Emperor Justinian (525-565) managed to reassert his rule over

part of Italy, but this project proved too expensive to sustain, and Italy was soon

once again beyond the control of the Emperor. For several centuries before the reign

of Justinian an entirely new civilization based on Christianity had been replacing

the ancient Greco-Roman world, symbolically marked by the Justinian's closing of

the pagan Academy at Athens in 529.

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The History of Mathematics: A Brief Course, Second Edition

by Roger Cooke