“Accurate reckoning. The entrance into the knowledge of all existing things and all obscure secrets.” –Introduction to the Rhind Mathematical Papyrus (as quoted by Victor J. Katz in A History of Mathematics: Brief Version)
In my last blog post, I discussed the mechanical calculating devices I had at that time identified within East and South Asia. The literature on China, Japan, and Korea proved to be relatively forth-coming, nearly verbose, on the subject of how calculations were performed by ancient and medieval mathematicians. India, however, was not so straightforward. After several days of “digging”, I discovered only the dustboard—while reading a chapter on Arabic mathematics as a break from that which was frustrating me. (See my previous blog post for more information on the above information, as well as for references.)
I had hoped, after completing the first blog post, that India was merely an anomaly, that somehow its unique climate prevented preservation of calculating devices or documents thereon or that its unique numeral system (pretty much perfect for carrying out written operations) rendered in this one part of the world calculating devices less necessary. Looking forward to Mesopotamia and Arabia, so significant in the history of science as centers of global exchange, I hoped, even expected to some extent, to return to East-Asian levels of information.
I settled for “better than India”.
For the most part, it seems, Mesopotamian (traditionally called Babylonian) mathematicians performed arithmetic operations mentally or with the help of tables (Carrucio, 1964; Katz, 2004). Whether the abacus ever existed alongside these tables is not definitively known. As prolific historian of mathematics David Eugene Smith said in Volume 2 of his History of Mathematics, “We have as yet no direct evidence of a Babylonian abacus. The probabilities are, however, that the Babylonians, like their neighbors, made use of it.” Like Smith, most scholars seem to make this assumption (Carruccio, 1964; Ifrah, 2000, 2001; Lilley, 2002; O’Conner & Robertson, 2000; Ziavras, 2002); in fact, Georges Ifrah (2000, 2001) suggests a reconstruction of an abacus from Sumer.
A few centuries later, Islamic expansionism enabled access to various cultural traditions, facilitating a new intellectual interest in the region (Berggren, 2007; Boyer, 1991). During this period, India’s dustboard method was adapted to Arabic uses (Joseph, 2000), and Hindu numerals were introduced (Katz, 2004; Kunitzsch, 2003; Levey & Petruck, 1965; Turner, 1997). Arabic mathematician al-Uqlidisi (c. 952) eventually translated dustboard techniques to pen-and-paper in order to enable true scholars and the upper-class to distance themselves from “street astrologers and other ‘good-for-nothings’ [who used a dustboard] to earn a living!” This transition to pen-and-paper calculations became part of European cultural traditions several centuries later as a result (Joseph, 2000).
Returning to the “ancients”, Ancient Egypt was among the “neighbors” mentioned by Smith in the above quote. Again, there is limited physical evidence for a calculating device (Sugden, 1950), but in this case this deficiency is remedied by references in literature to Egyptians manipulating pebbles like counters on an abacus. Herodotus (c. 425), the Greek historian, mentioned in passing the Egyptian tradition of calculating with pebbles (Needham, 1959; “Counters”, 1950). Furthermore, on the back of a papyrus now housed in the British Museum, one finds ten columns of ten dots where a line separates the top five rows from the bottom five. This unique diagram, of which there is only one other example among surviving Egyptian papers, would enable someone to add or subtract and could even act as a multiplication table (Pullan, 1969; Sugden, 1981). “Since no more than two examples of the Egyptian dot diagram are known it cannot be suggested that there was more than occasional use of such a device; but their appearance fits in well with the belief that Egyptian calculators of the time used a combination of mental arithmetic and practical methods” (Pullan, 1969).
After Egypt, I moved on to the rest of Africa, followed by North and South America (before European influence). The information I collected on these parts of the world was on generic methods, by which I mean methods that can be seen in the histories of nearly every culture. Such general topics I intend to discuss in a separate post.
Finally, I looked at European devices. Unsurprisingly, there is a great deal recorded on this subject and thus a great deal to be said, so that subject as well may receive its own post.
Berggren, J. L. (2007). Mathematics in medieval Islam. In V. J. Katz (Ed.), The mathematics of Egypt, Mesopotamia, China, India, and Islam: A sourcebook (pp. 513-675). Princeton, NJ: Princeton University Press.
Boyer, C. B. (1991). The Arabic hegemony. In U. C. Merzbach (Ed.), A history of mathematics (2nd ed., pp 225-245). New York, NY: John Wiley & Sons. (Original work published in 1968)
Carruccio, E. (1964). Pre-Hellenic mathematics. In I. Quigly (Trans.), Mathematics and logic in history and in contemporary thought (pp. 13-19). Chicago, IL: Aldine.
Counters; Computing if you can count to five. (1950, November). The Mathematics Teacher, 43, 368-370.
Ifrah, G. (2000). The universal history of numbers: From prehistory to the invention of the computer (D. Bellos, E. F. Harding, S. Wood, & I. Monk, Trans.). New York, NY: John Wiley & Sons.
Ifrah, G. (2001). The universal history of computing: From the abacus to the quantum computer (E. F. Harding, Trans.) New York, NY: John Wiley & Sons.
Joseph, G. G. (2000). The crest of the peacock: Non-European roots of mathematics. Princeton, NJ: Princeton University Press. (Original work published in 1991)
Katz, V. J. (2004). A history of mathematics: Brief version. Boston, MA: Pearson Education.
Kunitzsch, P. (2003). The transmission of Hindu-Arabic numerals reconsidered. In J. P. Hogendijk & A. I. Sabra (Eds.), The enterprise of science in Islam: New perspectives (pp. 3-21). Cambridge, MA: The MIT Press
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.
Lilley, P. (2002). Timelines of technology. In Hacked, attacked and abused: Digital crime exposed (pp. 2-11). London, UK: Kogan Page.
Needham, J. (1959). Mathematics. In Science and civilization in China (Vol. 2, pp. 1-168). London, UK: Cambridge University Press.
O’Conner, J. J. & Robertson, E. F. (2000, December). An overview of Babylonian mathematics. In MacTutor. Retrieved July 21, 2011, from School of Mathematics and Statistics, University of St Andrews website: http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_mathematics.html
Pullan, J. M. (1969). The history of the abacus. New York, NY: Frederick A. Praeger.
Sugden, K. F. (1981, Fall). A history of the abacus. Accounting Historians Journal, 8(2), 1-22.
Smith, D. E. (1958). History of mathematics. New York, NY: Dover Publications.
Turner, H. R. (1997). Mathematics: Native tongue of science. In Science in medieval Islam. University of Texas Press: Austin, TX.
Ziavras, S. G. (2002). History of computation. Retrieved July 21, 2011, from New Jersey Institute of Technology website: http://web.njit.edu/~ziavras/Ziavras-history.pdf