good example is the cubit, a unit of length from a person’s el-
bow and the tip of the middle fi nger. Sometimes, these units
varied depending on what was being measured. Th e common
cubit was used for everyday purposes, but the “royal cubit”
was used for public construction projects. For counting pur-
poses, the fi ngers and toes proved useful, giving rise to systems
of counting that were based on the number 20, as in ancient
Mesoamerica. Still today, many people in the British Isles use
“stone” as a measure of weight.
Counting was used for other purposes as well. Th e an-
cients devised systems of counting to create calendars, which
in turn required mathematical calculations of the movements
of heavenly bodies. As trade and commerce began to emerge,
systems of counting were needed to keep track of accounts.
Also required were primitive monetary systems, which began
as objects such as clay tokens used to keep track of quantities
and transactions and later evolved into true money.
Th e development of systems of numbers and counting
was bound up with the development of writing. For this rea-
son, the ancient Europeans fell far behind other regions of the
world; lacking systems of writing until much later, they had
no means of recording measurements. Some historians believe
that ancient scripts found in eastern Europe may have included
numbers, but so far they have been unable to decipher these
scripts, so historians remain uncertain. In contrast, the ancient
Asians and Mesopotamians had systems of writing that record
mathematical measurements, and historians have been able to
reconstruct these systems with some completeness.
AFRICA
BY OLUTAYO CHARLES ADESINA
Th e development of a sense of quantifi cation was basic to the
construction of number and counting systems in Africa. Th e
concept of numbers and counting represented a certain ap-
prehension of reality and became the foundations of math-
ematics, which in turn provided the key to existence. Despite
this awareness, however, ancient Africans never formulated
uniform numerals and systems of counting. Neither did they
devise a set nomenclature for native methods of counting.
Various cultures and ethnic groupings ultimately worked out
diff erent ways of thinking about mathematics and numerals.
Th e sense of quantifi cation, however, had universal applica-
tion to reality.
Th ese diverse groups had numeral systems that ranged
from the simple to the complex. Th e Yoruba of West Africa,
for instance, adopted an intricate system woven around ad-
dition, subtraction, and multiplication. Th ey formulated dif-
ferent terms for numbers from 1 to 10 as well as for 20, 30,
200, and 400; the rest were multiples or compounds. Th us 11,
12, 13, and 14 were reckoned as 10 plus 1 or plus 2 up to 14,
while 15 to 20 were reckoned as 20 less 1 to 5. But as there
were such groups as the Yoruba, the Galla, the Danakil, and
the Shiko whose numerical scale extended to 1,000, there also
existed those such as the San (Bushmen) of South Africa, who
possessed numerals not greater than 10. Among such groups,
and even among those with more highly developed civiliza-
tions, higher numbers were represented by the use of words
equivalent to “much” and “many.”
Evidence suggests that base 10 and base 20 counting
systems were popular among Africans, probably originating
from the number of human fi ngers and toes. As in other an-
cient cultures, standards of length, numbers, and measures
existed with reference to parts of the human body. To several
groups, fi ngers became an important instrument of calcu-
lation. Ancient Africans developed the quinary and denary
scale of numeration, or counting by the fi ngers of one or both
hands. Toes and fi ngers and their multiples were used at dif-
ferent stages of enumeration. Part of the culture of counting
that ancient Africans transmitted to their off spring was the
capacity to handle the various kinds of currency in use. Simi-
larly, games and puzzles formed part of the prehistoric system
of numeration.
With the adoption of a numeral system, ancient Africans
were able to develop a concept of mathematics. Ancient as-
tronomy became a signifi cant tool in this regard. Conversely,
numbers and counting also became major tools for under-
standing nature. Th us, the celestial order helped in the cre-
ation of an earthly order. Th is philosophy led to a number
of complex number symbolisms. Th e seconds, minutes, days,
and months derived from the celestial arrangement became
an integral part of a numeral system. Early humans divided
the day into temporal hours with length conditioned by the
time of the year, the summer having a longer period of day-
light than the winter.
In various parts of Africa wood, bone, and stone were
adopted as instruments of counting. Th e bone was a count-
ing tool for simple arithmetical procedures. Th ere existed
tally sticks, bones with orderly notches that represented the
number of days in a moon cycle or days spent by a group in
one geographic location. A piece of baboon fi bula with 29
notches, dating to around 35,000 b.c.e. and found between
South Africa and Swaziland, is the oldest-known mathemati-
cal artifact. Several groups in Africa adopted such bones as
calendar sticks. Other examples include the Ishango bone
found on the border of Zaire and Uganda. Th is small animal
bone, which has been dated to around 20,000 b.c.e., is in-
scribed with markings thought to represent numbers. It is as-
sumed to have been used as a counting tool for mathematical
purposes. Commodities also were used in this regard. Such
products as salt and slaves were used more oft en as account-
ing units or standards against which goods were valued. In
the western Sudan cowries (shells) adopted as currency were
strung together and used in counting.
Among ancient Africans, mathematics was concerned
with numbers and their operations, with which calculations
could be achieved. Th is belief translated into the “multitude”
and the “magnitude.” It encapsulated arithmetic, philosophy,
geometry, and stereometry (the measurement of volume).
Rather than being an abstract concept, mathematics became798 numbers and counting: Africa