numbers up to powers of 10^80 , but such numbers were limited
to ritual use. Th e Chinese common or standard numeral sys-
tem and the computational rod system were adopted and read-
ily used in Japan, Korea, and Annam. Chinese mathematical
techniques were also adopted by these cultures, as were the
Chinese classical manuals describing applications of mathe-
matics. Th e most famous and infl uential of these manuals was
the Jiu zhang suan shu (Nine Chapters on the Mathematical
Art), written in about 100 b.c.e.; this was a bureaucratic hand-
book of the mathematics needed to run an empire.
Th e earliest epigraphic evidence of a counting system
in India can be traced back to the Maurya Empire (321–185
b.c.e.) and to the activities of its greatest ruler, Asoka (r. ca.
286–231 b.c.e.), who constructed cast-iron pillars listing the
precepts of his rule. On these structures is found evidence
of a decimal system of numeration, the Kharosthi numer-
als. Th ese numerals are basically tally symbols but conceived
around the use of base 10. Th e tally symbols eventually
evolved into a cursive form that could be written down eas-
ily. Like the rod numerals of China, this system also left an
empty space to indicate the lack of an enumerator for a power
of 10 (a “zero”). Numerals were written horizontally from left
to right with the digit representing the highest power of 10
occupying the position farthest to the left. Th is later Brahmi
form spread throughout the Hindu sphere of infl uence.
Indian scholars also devised a collection of computational
algorithms to accompany these numerals. Th ese algorithms
could be undertaken using the writing materials at hand and
did not require an auxiliary computing device. Early Hindu
traders and missionaries, brought this knowledge and these
techniques to their clients and converts throughout Asia, spe-
cifi cally into the Malay Archipelago, including the present-
day Philippines, Sri Lanka, the Srivijaya maritime empire of
present-day Sumatra, the Khmer kingdom of Angkor (Cam-
bodia), the Champa kingdom (central Vietnam), Majapahit
(Java), and among the Mon people (Th ailand). Th is decimal
system of enumeration and its variants became very popular
along the trade routes of the East Indies, where it was also
adopted and perpetuated by traveling Arab merchants who
would eventually introduce it into Europe.
Before the Common Era an extensive trade system was
established among the peoples of the Pacifi c region both be-
tween themselves and with foreigners. It is most likely that
these contacts promoted various systems of counting and
enumeration. Language similarities within the traditional
counting systems of the Chamorro people of the Mariana
Islands, the Ilocano and Kapampangans of the Philippines,
the Napu and Batak of Indonesia, and the inhabitants of Fiji
and Samoa all support this conclusion. For example, all their
ancient, traditional languages share the same word for the
number 5, lima, a proto-Malay word for hand, referring to
the origin of fi nger counting. Similarly, Tongans and Tahi-
tians employed a decimal system of counting, one that could
accommodate enumeration into the millions. Th e Hawaiians,
descendants of these people, also employed a base 10 count-
ing system, one in which groupings of four—4, 40, 400, and
so on—bore special signifi cance in counting specifi c items
such as taro and coconuts.
Th is early prevalence of decimal notation, in many cases,
depended upon exposure to foreign, external infl uences im-
parted by traders and missionaries. Th us, coastal peoples
and inhabitants of administrative urban centers may have
used decimal counting, but their isolated hinterland peers
employed other systems. For most of these remote peoples
there is little historical evidence in existence as to their an-
cient counting practices, but anthropological research among
traditional peoples of the Pacifi c region, especially those of
Papua New Guinea, supplies insights into their ancestors’
mathematical systems. Findings indicate that probably these
isolated peoples had limited number vocabularies and em-
ployed a variety of counting bases, among which the base 5
was popular in emulating the number of digits on the human
hand. Many of their number terms would have been more
qualitative than quantitative, comparable to words such as
“couple,” “herd,” and “fl ock” that not only designate a gen-
eral number but also convey further information about the
objects under consideration. Th ese people also might have
employed several counting systems, that is, using diff erent
number words for counting special items, such as yams, pigs,
and canoes, and for doing time reckoning. Some of these so-
cieties were known to have used four or fi ve such diff erent
systems for their counting purposes.EUROPE
BY AMY HACKNEY BLACKWELL
Ancient Europeans did not develop mathematics and count-
ing to nearly the degree that the peoples of the Mediterranean
region did. Most ancient European peoples outside Greece
and the Roman Empire did not write at all, so almost no writ-
ten records remain that could show the level of mathematical
knowledge in Europe. People in Mesopotamia and the Medi-
terranean used mathematics to trade and build objects, but
the Celts, Germans, and other European peoples did not leave
traces of numerical systems for accounts, records, or architec-
tural calculations. One of the earliest uses of numbers was to
keep track of days and seasons. Bones dating from as early as
11,000 b.c.e. contain regular scratches that may have corre-
sponded to the phases of the moon. People early gained a sense
of the numbers that were important for keeping track of time,
such as 28 to 30 days per month and 12 months per year.
Prehistoric European peoples were good enough at astro-
nomical calculations to build stone structures precisely engi-
neered to catch the sun at a particular time on a certain day.
For example, the tomb at Newgrange in Ireland, built around
3500 b.c.e., is designed so that the rising sun on the winter
solstice (approximately December 21) falls on a design deep
within a grave chamber. Stonehenge and other giant stone
sites also are placed in such a way as to catch the sunlight on
particular days of the year.802 numbers and counting: Europe