ARITHMOS, YBC 7289, ORESME and Laplace

Yes, the title is opaque, so let me explain. ARITHMOS is the acronym for a seminar entitled Arithmos: Readings In The History of Mathematics from Original Sources. This reading group was originally organized by Rob Bradley (Adelphi University) and Ed Sandifer (now Professor Emeritus at Southern Connecticut State University) and first met in 2001. The meetings, three or four per year, are usually held at SCSU in Danbury, CT, on a Saturday afternoon and Sunday morning. You can consult the web site
to see what topics have been discussed in the past.

We read aloud an English translation of a previously selected mathematical text, usually with some people following along in the original language, other translations, and other editions. We move around the table, each person reading a paragraph or so. Often the reader is interrupted with questions and comments and there is more discussion after they finish. This may seem ponderous and tedious, but it is not, for the discussion is always enlightening. We are fortified with tea and chocolate, and this approach works well for us. Almost always at the end, everyone feels that they have gained a rich appreciation of the text.

As we approached our fortieth meeting, we discussed the possibility of doing something different. After considerable discussion we decided to visit the Beineke Library at Yale to look at some cyphering books and other treasures in their collection, but the timing did not work out (so now it is on our agenda for a future summer meeting). We also wanted to visit the Yale Babylonian Collection and one of our members, Ross Gingrich (Southern Connecticut State University in New Haven), arranged for our visit. Associate Curator Dr. Ulla Kasten gave a lovely general description of the largest collection of tablets in the Americas. She showed us a tablet with an envelope, an empty bulla, and several cylinder seals. These were very costly for they are made of stone (not clay) and were passed from generation to generation.

Dr. Kasten discussed some of the objects in the cases around the small classroom in the library where we met. There was the The Yale Gilgamesh Tablet (Gilg. Y vi 262), one of the earliest works of literature, an epic on creation of the world (NBC 11595). Sadly, it was damaged by an excavator's pickaxe. We also saw the first work whose author is identified by name (YBC 4671), which dates from 2260 BC. The author was the poet Enheduanna, a woman.

We were amazed by the small writing on the tablets. We learned that the clay tablets shrink by about 5% when dried and another 3% when fired, but even so, the writing was tiny. It took effort to make out the numerals on the tablets. We were allowed to take photographs for our personal use but it was difficult to get a good raking shot to make the writing legible.

Dr. Kasten let us play with clay and stylus to make our own sample tablets and practice the difference between regular vertical wedges and the Winkelhaken, the symbol whose value is 10. We all had trouble making a Winkelhaken.

For this meeting we were honored to be joined by Duncan Melville of St. Lawrence University who led the discussion of the 25 mathematical tablets that we saw. As a graduate student at Yale he learned a great deal about mathematics in Mesopotamia and continues to do research in the area. In fact, he has a very interesting web page on Mesopotamian Mathematics.

We were surprised that we were permitted to handle the tablets without wearing gloves. For me, it was most exciting to hold YBC 7289. Because of its size it is called a "hand tablet" and it fit very nicely in my hand. This is the famous tablet that contains the number 1,24,51,10, which is a sexagesimal approximation for the square root of 2. What I did not know before is that this number also appears on an Old Babylonian coefficient list. This means that the schoolboy who created the tablet most likely looked up the value, rather than computing it himself.

We saw multiplication tables, reciprocal tables, geometric algebra tables and a tablet dealing with the area of a circle. Most interesting was one dealing with the survey of a field (YBC 3900, 2045 BCE). To find the area the field was broken up into rectangles, trapezoids and right triangles, and the areas of these regions is then added. This is the same technique that George Washington used in 1750 as a professional surveyor.

Towards the end of the meeting a historical and linguistic overview was provided by Elizabeth (Lee) Payne, the Conservator of the Babylonian Collection.

The next day we read portions from Eleanor Robson's treatise on Mesopotamian Mathematics in Victor Katz's The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook that has the texts of the tablets we saw. What makes this seminar work so well is that people have read the material in advance and so know what their questions are. The discussion was spirited, enlightening, and fun.

The only other reading group of this nature that we are aware of is the ORESME (Ohio River Early Sources in Mathematical Exposition) group which has been meeting since 1998 in Southern Ohio and Northern Kentucky. The organizers are Daniel J. Curtin (Northern Kentucky University) and Daniel E. Otero (Xavier University). It is amusing that both groups first made up their acronyms and then found words to fit them. Reports on earlier meetings can be found on the ORESME web site:

Their most recent meeting was held at Xavier University in Cincinnati, Ohio, on September 21-22, 2012. The topic for discussion was Pierre Simon Laplace's Théorie Analytique des Probabilités which was published just two hundred years ago. We discussed Part 2, Chapter 4, section 18, which contains a proof of the Central Limit Theorem. This proved to be a very tough read but fortunately Richard J. Pulskamp of Xavier University has just published a discussion of the work, "The Legacy of the Théorie Analytique des Probabilités." He also prepared the translation that we used. Although difficult, this was a rewarding text to read for we could follow Laplace working through ever more general cases.

I have attended the ORESME seminar several times and regularly attend ARITHMOS. The ORESME group meets for lunch at a Mexican restaurant on Saturday and then reads, discusses, and argues about the mathematics from 2:00 to 6:00, at which time we go to dinner. We reconvene on Sunday at 9:00 and work till about 1:00 and then go to lunch. This provides time for lots of discussion of the topic of the meeting, chat about many other topics in the history of mathematics, and social interaction. ORESME meets on Friday evening for dinner and then works from 8:00 to 10:00 PM and then again Saturday morning from 10:00 to noon.

At the most recent meetings there were eight historians in attendance. This is a large enough group to deal with all of the mathematical and historical questions that arise, but small enough that there can be close interaction. We find this to be an ideal number.

I would encourage you to try to form groups like these whenever you can gather a cadre of individuals that live within, say, an hours drive. After the word got around about how interesting these sessions were, some people were willing to drive three or four hours and stay overnight just to participate. Like them, you will find it to be a rewarding experience.

A few references:

Dave Richeson explains how to typeset Babylonian numerals in TeX:

Pulskamp, Richard, "The Legacy of the Théorie Analytique des Probabilités,"Boletim ISBrA (The official bulletin of the Brazilian section of the International Society for Baysian Analysis), vol. 5, no. 1, July 2012, pp. 9-19.

Stephens, Ferris J., "A surveyor's map of a field," Journal of Cuneiform Studies, vol. 7, no. 1 (1953), pp. 1-4.

The Cuneiform Digital Library Initiative has images of many tablets:

For an on-line version of this note, with many links, see

V. Frederick Rickey