798 numbers and counting: Africa good example is the cubit, a unit of length from a person’s elbow and the tip of the middle finger Sometimes, these units varied depending on what was being measured The common cubit was used for everyday purposes, but the “royal cubit” was used for public construction projects For counting purposes, the fingers 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 The ancients 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 The development of systems of numbers and counting was bound up with the development of writing For this reason, 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 The development of a sense of quantification was basic to the construction of number and counting systems in Africa The concept of numbers and counting represented a certain apprehension of reality and became the foundations of mathematics, 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 different ways of thinking about mathematics and numerals The sense of quantification, however, had universal application to reality These diverse groups had numeral systems that ranged from the simple to the complex The Yoruba of West Africa, for instance, adopted an intricate system woven around addition, subtraction, and multiplication They formulated different terms for numbers from to 10 as well as for 20, 30, 200, and 400; the rest were multiples or compounds Thus 11, 12, 13, and 14 were reckoned as 10 plus or plus up to 14, while 15 to 20 were reckoned as 20 less to 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 civilizations, 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 fingers and toes As in other ancient cultures, standards of length, numbers, and measures existed with reference to parts of the human body To several groups, fingers became an important instrument of calculation Ancient Africans developed the quinary and denary scale of numeration, or counting by the fingers of one or both hands Toes and fingers and their multiples were used at different stages of enumeration Part of the culture of counting that ancient Africans transmitted to their offspring was the capacity to handle the various kinds of currency in use Similarly, 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 astronomy became a significant tool in this regard Conversely, numbers and counting also became major tools for understanding nature Thus, the celestial order helped in the creation of an earthly order This philosophy led to a number of complex number symbolisms The 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 daylight than the winter In various parts of Africa wood, bone, and stone were adopted as instruments of counting The bone was a counting tool for simple arithmetical procedures There 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 fibula with 29 notches, dating to around 35,000 b.c.e and found between South Africa and Swaziland, is the oldest-known mathematical 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 This small animal bone, which has been dated to around 20,000 b.c.e., is inscribed with markings thought to represent numbers It is assumed 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 often as accounting 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 This 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 became