“Machines, too simple to get out of order, are they not more trustworthy than the average human brain crowded nowadays with the perplexities of modern civilization?” –Rikitaro Fujisawa in the Appendix (1912) to The Development of Mathematics in China and Japan
My global survey of the history of mechanical calculating devices is well underway. I began my research with the history of Chinese mathematics, and three weeks later, I have just completed my research into the computational devices of East and South Asia.
Reading about the history of math in China alone took a week and a half, despite being a subject I have researched, for Chinese mathematics is both well-documented and long-established. Even the more specific field of calculating devices has a long history; Li Shu-t’ien (1958) claims, “The genesis of physical computing devices in ancient China dates back to about 1100 BC in the early [Z]hou Dynasty, insofar as authentic classics and dynastical books of history have recorded.” Certainly, numbers are first seen scratched into bones from c. 1600 BCE (Katz, 2004), presumably used for divination purposes, and for the following sixteen centuries mathematical knowledge grew to an impressive extent, as evidenced in a summarizing work, Zhou bi suan jing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven), from the Han Dynasty (206 BCE-220 CE) (Boyer, 1968/1991; Katz, 2004; Li and Du, 1987). In order to facilitate performing math, the very ancient Chinese, like most civilizations, used finger-reckoning (Needham, 1959) and knots (Deng, 2010/2011; Li & Du, 1987; Kim, 1973; Needham, 1959), but the Chinese mathematicians are best remembered for their peculiar “counting rods”. By the Warring States period (403-221 BCE) of the Zhou Dynasty (Cheng, 1925; Li & Du, 1987; Needham, 1959), “counting rods” made of bamboo or ivory were used in columns and rows on “counting boards” to do calculations (Katz, 2004; Li, 1958; Li & Du, 1987; Mikami, 1913/1974; Needham, 1959). This method spread to Korea, where it remained in popular use until the relatively recent introduction of Western mathematics and methods (Kim, 1973).
In China and Japan, the method of counting sticks was supplanted by the abacus, which seems to have exploded into popularity in 15th-century China (Cheng,1925; Li & Du, 1987; Needham, 1959) and 16th– or 17th-century Japan (Kojima, 1954; Mikami, 1913/1974; Smith & Mikami, 1914). Evidence suggests, however, that the abacus or a precursor thereof had existed quietly for centuries (Li, 1958; Mikami, 1913/1974; Needham, 1959). The origins of the abacus are not precisely known: Some scholars believe that the abacus originated elsewhere and was only altered by the Chinese (Deng, 2010/2011; Needham, 1959); some think it developed independently, from or with the counting rods (Li & Du, 1987).
Despite geographic proximity and the exchange of astronomical understanding between China and India, human computers from the latter seem not to have utilized counting rods or abaci. Rather, Indian mathematicians* used dustboards—wooden tables covered with dust or sand in which written calculations were performed with a stylus (Datta, 1928; Levey & Petruck, 1965; Li & Du, 1987). Indian numerals, the predecessors of our own, were some of the few ancient number systems fit to be worked with directly (Pullan, 1969). Indian dustboard calculations were later adapted by Arabic mathematicians to fit pen and paper (Joseph, 2000).
The Malayo-Polynesian world volunteered a mechanical calculating device to my study as well. Since World War II, the use of the “Sungka Board” in mathematics has largely disappeared, now relegated to a popular board game and sometimes divination. The Sungka board resembles a mancala board, where pits (ignoring the ones on the end from mancala boards) represent place values; pits in the same column are of the same place value, but place value increases from right to left, so numbers will be read left to right as they are in the West. Calculations are performed by moving cowries, seeds, or pebbles between these pits (Manansala, 1995).
Having completed my research into East and South Asia, I will next move on to the mechanical calculating devices of the Middle East. This means Mesopotamian and Islamic mathematics, which are highly significant in the history of mathematics since the geographic centrality of the Middle East made it the perfect place for the exchange of ideas as well as products. Following the Middle East, I will examine briefly African math, especially Egyptian, before moving on to the broad topic of European mathematics.
* In the term “Indian mathematics” or “Indian mathematicians”, historians in fact mean to refer to the mathematics or mathematicians of all South Asia: India, Nepal, Bangladesh, Pakistan, and Sri Lanka, mainly (Plofker, 2007).
Boyer, C. B. (1991). China and India. In U. C. Merzbach (Ed.), A history of mathematics (2nd ed., pp. 195-224). New York, NY: John Wiley & Sons. (Original work published 1968)
Datta, B. (1928). The science of calculation by the board. The American Mathematical Monthly, 35(10), 520-529.
Deng, Y. (2011). Ancient Chinese inventions. (Wang P., Trans.). Cambridge, UK: Cambridge University Press. (Original work published in 2010).
Joseph, G. G. (2000). The crest of the peacock: Non-European roots of mathematics. Princeton, NJ: Princeton University Press.
Katz, V. J. (2004). A history of mathematics: Brief version. Boston, MA: Pearson Education.
Kim, Y.-W. (1973). Introduction to Korean mathematical history. Korea Journal, 13(7), 16-23; (8), 26-32; (9), 35-39.
Kojima, T. (1954). The Japanese abacus: Its use and theory. Rutland, VT: Charles E. Tuttle.
Levey, M., & Petruck, M. (1965). Introduction. In K. ibn Labban, Principles of Hindu reckoning (M. Levey & M. Petruck, Trans., pp. 3-41) [Introduction]. Madison, WI: University of Wisconsin Press.
Li, S.-T. (1959). Origin and development of the Chinese abacus. Journal of the Association for Computing Machinery, 6, 102-110.
Li, Y. & Du, S. (1987). Chinese mathematics: A concise history. (J. N. Crossley & A. W.-C. Lun, Trans.). Oxford, UK: Clarendon Press.
Manansala, P. (1995). Sungka mathematics of the Philippines. Indian Journal of History of Science, 30(1), 13-29.
Mikami, Y. (1974). The development of mathematics in China and Japan (2nd ed.). New York, NY: Chelsea Publishing Company. (Original work published in 1913)
Needham, J. (1959). Science and civilization in China (Vol. 3). Cambridge, UK: Cambridge University Press.
Plofker, K. (2007). Mathematics in India. In V. Katz (Ed.), The mathematics of Egypt, Mesopotamia, China, India, and Islam: A sourcebook (pp. 385-514). Princeton, NJ: Princeton University Press.
Pullan, J. M. (1969). The history of the abacus. New York, NY: Frederick A. Praeger.
Smith, D. E. & Mikami, Y. (1914). A history of Japanese mathematics. Chicago, IL: Open Court Publishing.
Edited 7/22/2011: Added information about Sungka mathematics and citations.