“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.

References

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* (2^{nd} 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

Hello, Deborah!

After reading your posts, I have two separate questions for you.

First, you have mentioned several times that the Hindu numerals, later adopted by the Arabs, have a distinctive advantage in terms of mental calculation over the numerals used by other cultures of the time. Being rather ignorant of the history of mathematics, I was wondering whether a civilization’s ease of mental calculation was in any way related to the it’s overall understanding of the theories of mathematics? It would seem that a superior numeral system would result in a more accurate understanding of mathematics and thus an increased proficiency in the field. Or, on the other hand, does having a more visual representation increase the likelihood that a mathematician would catch something somebody using only his mind would miss?

Second, what relationship have you observed between a civilization’s overall mathematical ability and the sources written about its devices? Intuitively, I would think that a civilization that had a more stellar mathematical history would have more sources written about it.

Thanks!

Ashley

Hey, Ashley!

I apologize for not finding your comment until just now. I’m so absent-minded; without getting notifications about comments on my posts, I typically forget to check. It’s so bad of me. Anyway, I hope my response adequately answers your question. Forgive me if it’s a bit vague because I have not had a chance to recollect my sources.

When reading about “ease of mental calculation”, I always at least get the impression from the author, if I’m not told explicitly, that the advantage of Hindu numerals in this area is that they are abstract symbols that individually summarize a lot of information that concrete representations must explicate. For a simple example, if I want to add 397 and 483 in my head, I need to remember what symbol or number goes where and what that means for the sum. If I want to add 397 and 483 in my head using a mental abacus, then, just to set the problem up, instead of (4)(8)(3), I have to remember (no upper beads, four lower beads)(one upper bead, three lower beads)(no upper bead, three lower bead) and that I will be adding 397. Then, when I make the first addition in the units place, I have to think (no upper beads, four lower beads)(one upper bead, three lower beads)(no upper bead,

~~three lower bead+seven beads~~ten beads)=(no upper beads, four lower beads)(one upper bead,~~three+one~~four lower beads)(no upper beads, no lower beads). Basically, the advantage of Hindu numerals in mental calculation is remembering only “4” instead of “(no upper beads, four lower beads)”–there is less translation from idea to object and back–but there is no proof to argue one way or the other on whether this relates to overall understanding of mathematical theory. Of course, one is probably better off doing the calculation physically on a tangible abacus than with Hindu numerals in one’s head, for the increased likelihood of catching mistakes.From my research into the history of math, efficiency of numeral system does not actually seem to correlate with mathematical accomplishments. Indian mathematics, for example, despite the advantage of easier mental calculation, did not achieve staggeringly more than other civilizations. On the other hand, Japanese mathematicians using a

sorobanmay have developed calculus in the 18th century.Regarding the relationship between a civilization’s mathematical progress and sources about its devices, there definitely is a connection. Civilizations famous for making amazing discoveries are more frequently written on, so by virtue of how much information I have access to, I am more likely to learn about their calculating devices. However, greater contributing factors include whether the civilization had/has the time/interest to write about their own mathematical tradition(s) and how close their geographic/cultural descendants feel to them. For example, I can find a TON on European devices because (1) it’s a civilization that hasn’t really died out, so Europeans had and continue to have plenty of time to write about themselves, (2) nationalism in Europe creates an interest in writing about themselves and their predecessors, and (3) sources on European devices tend to be in European languages, thus making them more accessible to me. I find much literature on Chinese devices because (1) again, the civilization continues, (2) China is also motivated by a nationalistic desire to remind Europe they exist and did/do impressive work. Unfortunately, most literature on Chinese math is in Chinese; only the translated works or the few pieces done by European scientists on Chinese scientists are available to me. As for ancient civilizations that do not continue to the present, I’ll contrast ancient Egyptian and Babylonian mathematics: both did impressive work, but I find the Babylonians a bit more inspiring; however, knowledge about Babylonian calculating devices can be summarized as “They probably had an abacus”, whereas knowledge about Egyptian computation could be written as a paragraph. This disparity arises (in my opinion) because of Egypt’s later connection to Greece and therefore the West as a whole, while Babylon was considered Eastern and therefore not worth much attention in Europe.

Geez. This is really long, and it’s mostly speculation. Sorry! I hope I answered your questions in there somewhere…

Thanks for asking!

Deborah