their sum is a square”), but their solutions lack general ap-
plication or formal proof.
Mathematics came to hold tremendous symbolic impor-
tance in Greek culture. Pythagoras saw in numbers a model for
the underlying harmony of the universe, and the mysticism of
his worldview infl uenced many later philosophers, including
Plato. Th e famous story that the entrance to Plato’s Academy
bore the inscription “Let no one who is ignorant of geometry
enter” may be of doubtful authenticity; nevertheless, it refl ects
well the Platonic attitude toward the centrality of mathematics
in the education of the ideal citizen or philosopher.
ROME
BY JUSTIN CORFIELD
Th e system of numbers and counting used in Rome and the
Roman Empire was developed from Etruscan numerals.
Th ese numerals, in turn, were adapted from the Greek At-
tic numerals. Although there are only a few archaeological
fi nds that have Etruscan numerals, there are enough to show
that the letter I represented 1, an inverted V represented 5,
and X represented 10. Th e fi rst two of these fi gures can be
seen on surviving Etruscan six-sided dice, on which the op-
posite sides add up to seven, as in the dice of today. Several
Roman dice also have survived, again with the opposite sides
adding up to seven. Six small ivory dice were found at Pom-
peii, all the numbers denoted with small dots; the Romans
oft en played with two dice as opposed to the Greek system, in
which three dice were used.
It has been suggested that the original use of numbers
possibly came from notches on a tally stick, with the Romans
using a V for 5. Apart from the ordinary people who needed
to count out money and commodities as well as tell time
and work out the calendar, the Romans also had tax collec-
tors and mathematicians who needed to use larger numbers.
Th e Romans used the letters I for 1, V for 5, X for 10, L for
50, C for 100, D for 500, and M for 1,000; they also indicated
large numbers by placing a bar over a numeral to indicate a
numeral in the thousands. Above one million there was no
standard format, with fi ve million denoted either by placing a
double bar over a letter V or by underlining, as in V.
Th ere is confl icting evidence regarding the use of some
Roman numerals, especially subtractive ones, whereby IV
represents one less than 5 (that is, 4). Older Roman material
tends to have IIII for 4, making mathematics easier. However,
gradually the subtractive notion tended to be used more and
more, eventually becoming the accepted form despite the fact
that IV was also similar to initials representing the god Jupi-
ter (the name in Latin being IVPITER).
It was also from Roman numerals that the number 666
became known as the devil’s number. In Roman numerals
666 was represented by the letters DCLXVI, using every basic
numeral except the letter M. Th ose familiar with this phe-
nomenon must have seen a bad omen when the Great Fire of
London broke out in 1666 (MDCLXVI).
Interestingly, the Romans did not have a concept of a
zero. Indeed the fi rst Roman mathematician who is known
to have used the concept was Dionysius Exiguus in 525 c.e.,
though it seems probable that the concept of zero was in use
before then. Later the letter N for nullus (nothing) was used
to denote zero.
While many Greek mathematicians are household
names, there are no details about Roman mathematicians,
and it seems likely that the Romans had little interest in pure
mathematics. Indeed, it was said to have been a Roman soldier
who killed the Greek mathematician Archimedes of Syracuse
in 212 b.c.e., and most mathematicians in the Roman Em-
pire, such as Diophantus of Alexandria, were actually Greek.
However, the Romans must certainly have had people who
studied applied mathematics to work out architectural plans,
load bearings, astronomy, and the supervision of the raising
of revenue. Th e poor notation of the Roman numbers did,
however, handicap any major advances in mathematics. For
counting, the Romans used small bronze counters on occa-
sions, these having on them pictures of hands with a particu-
lar number of fi ngers raised. Cicero refers to these counters,
which eff ectively were early “counting boards.” Gradually the
Romans started using the abacus for more complicated sums,
the abacus also oft en being called a “counting board.” Many
mathematical instruments have been found in the excava-
tions at Pompeii.
Surviving surveying manuals show the combination
of arithmetic, geometry, and optics for the Romans’ work.
Weights and measures followed what now appear to have
been arbitrary calculations but must have had some mean-
ing at the time. For example, the pes, a Roman foot (in mea-
surement), was just over 11.5 inches, making their pace 4 feet,
10.25 inches. With a thousand of these feet making a Roman
mile, the mile was 1,618 yards. Roman numerals have been
used many times since the ancient world; they are regularly
found in Christian ecclesiastical documents and in European
accounts and are still used in calendars, clocks, and watches.THE AMERICAS
BY PENELOPE OJEDA DE HUALA
Very little information exists regarding the function of num-
bers and counting in the ancient Americas. We know that the
inhabitants of the Americas probably used the lunar and solar
cycles to track the change of seasons, as these changes were
vital to their survival. Archaeological remains show evidence
of early engineering, which required forms of mathematics.
However, by the time of European contact, numbering sys-
tems existed throughout the Americas.
Th e Archaic Period (7000–1800 b.c.e.) in North America
saw a near extinction of big-game animals, forcing native
groups to rely on local resources to supplement their diets.
While full-scale agriculture would not be introduced until
about 800 c.e., around 3000 b.c.e. some forms of early cul-
tivation occurred throughout North America. Agriculturalnumbers and counting: The Americas 805