“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 15^{th}-century China (Cheng,1925; Li & Du, 1987; Needham, 1959) and 16^{th}– or 17^{th}-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).

References

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* (2^{nd} 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.

Wow, who knew there was so much material in the way of mechanical calculating devices around the world to research. The Chinese counting rods and numerical oracle bones are unsurprising, but I find it interesting that some areas basically drew in the dirt with a stick. Are you going to analyze the development of calculating devices from ancient times to the advent of modern electrical devices, or just conduct a survey of the ancient and early modern world? Either way, it’s surprisingly interesting stuff.

To answer your question, Graham, my project this summer is a simple survey. The more research I do, the more I am convinced that attempting to analyze the development of calculating devices would be far too complicated and enormous a project to be tackled in one summer, and in any case, a lack of evidence at present means that any analysis I might undertake would be largely speculation. Mostly, the extent of communication between coexisting cultures prior to (this is my rough estimate) the sixteenth century is poorly understood, so it cannot be factored appropriately into hypotheses.

Greetings, Ms. Wood. I hope the research is going well! I have no doubt that the combination of mathematics and history is a perfect niche for you. I have a couple questions:

The abacus seems to be the iconic ancient calculating device. How the generic abacus work, or is there such a thing as a generic abacus?

My second question is a bit more specific:

I took a class on the History of South Asia and one of the themes was the amalgamation of cultural, political, and economic entities. Did such a thing happen in South Asia (Indo-Arabic) and if so, when did this interaction occur to change the history of calculating devices?

Thanks, and I hope the research continues to go well!

I apologize, Michael, for my tardiness in responding to your comment, for its length now that it’s here, and for the fact that I’m going to postpone answering your first question in depth. My brief answer is that no, there is not a “generic abacus”. In fact, there are three categories of abaci, within each of which there are several types with individual histories and characteristics. I made a few attempts at being more specific and addressing the question of an abacus’ operation, but apparently it is impossible for me to be clear, thorough,

andbrief; I kept writing small essays. That, plus the fact that a few people have expressed interest to me verbally, I think I will explain how to use an abacus in a later blog post.As for “the amalgamation of cultural, political, and economic entities” and its effect on the history of mechanical calculating devices—for the sake of honesty, I feel compelled to start with a disclaimer: My research this summer has not focused on the social/cultural environments of mechanical calculating devices, but instead on their existence and operation. That said, from all the reading I have done and the fact that the effects of social/cultural environments on mathematics is what I hope to specialize in, I can give a rough idea. Basically, the year 622 marks the beginning of the Islamic era, which brought about huge changes in global economic structure, cultural interactions, balance of political power, etc. For mathematics, the Islamic/Arabic empire centralized intellectual industry and influence, and it connected distant parts of the world. The Arabic world is especially well known in mathematics for conveying the numerals used in India to the Western world, and for translating the Hindu dustboard methods of manipulating those numerals into algorithms for pen and paper. This enabled the (gradual) transition from abacus calculation to algorismic calculation favored now by most of the world. (And I do mean

gradual: the full adoption of algorisms over abaci took almost a thousand years.)I hope that answers your question or even addresses it in any way. And thanks for asking! Feeling perpetually pressed for time, I haven’t really had a chance to read about or contemplate the cultural environments of the devices I am reading and writing about, and this response now has enabled me to take a lovely, enjoyable stroll into that part of the history of mathematics.