A SUMERIAN NUMERICAL PALIMPSEST

1

A SUMERIAN NUMERICAL PALIMPSEST D.A.R. DeSegnac A draft for an essay “Every sentence I utter must be understood not as an affirmation, but as a question.” Niels Henrik David Bohr

INTRODUCTION There is no doubt that the Sumerians of the second half of the 4th millennium BC were exceptionally numerate. Their ability to mostly mentally compute in the multiples of 60 with units of a rather large variety of systems of measures (Nissen et al. Archaic Bookkeeping, The Chicago U Press, 1993,) provided the foremost intellectual basis that, in its absence, it would be impossible to comprehend the course of the rapid evolution and expansion of their civilization. The content and an evolved manner of exposition of the archaic numerical texts such as in the Figs. 1, from the last few centuries of the 4th millennium BC and onward, attest to this by the scores.

Sexa-denary numerical system

Implied land measures system

Fig. 1. CDLI photograph and line art W(arka) 19408,76+ ca. 3350-3200 BC (Photo: http://cdli.ucla.edu/cdlisearch/search/ipadweb/ showcase; line art: CDLI P 003118)

Fig. 2. Author’s drawing of the W 19408,76 based on Damerow/Englund, ABKK pg. 55/58 (From. D.A.R. DeSegnac “A Mesopotamian surveyor’s puzzle” A draft for an essay, academia.edu)

The early archaic tablets like the W19408,76 show that the high level of numeracy could have existed for some time before the second half of the 4th millennium, that is, “… counting in multiples of sixty, must have been in existence before the invention of writing.” M. A. Powell Jr., Thesis 1971, UMI 1984 A linear presentation of the Sumerian counting in multiples of 60 is given in Fig. 3.

Fig. 3. A presentation of the archaic numerals based 60 The structure of the Sumerian numeration system is succinctly given by Powell (ibid pg. 81,) (Fig. 4.) “The central axis of Sumerian numeration is the number sixty, which represents the culmination of one sequence and the beginning of another.”

2

Fig. 4. Archaic Sumerian counting sequences based on the multiples of 60 All three sequences in Fig. 4, are constructed by the ×6, ×10, ×1 rule, and all end in multiples of 60(ḡeš). The first sequence begins with the count of 1(diš) that after ×10 became 10(u) which by ×6 became 60(ḡeš.) The second sequence begins with 1(ḡeš) and ends up in 60(ḡeš) Sumerian šar2, or “everything”. The third is a replica of the second, beginning by 1(šar2 = 60 ḡeš) and ending by 60(šar2) Sumerian šar2gal, or “big everything.” The ×6, ×10, ×1 factors effecting each sequence are, essentially, the spine of the archaic 60 based system, along which specific numerical condensation occurs in each sequence (Fig. 4.) About the ×10 and ×6 factors Powell writes (ibid, pg. 81): “The role of ten is obvious, but the role of six is not to be discerned in the number words themselves. It only appears to be a factor when the system is formally analyzed. Thus, šar, which begins a new order of numeration … is theoretically composed of sixty times ten times six, but six is not deciding factor which determines the new series of numerals. The essential factor in the creation of a new order of numerals is the numeral sixty itself. In this process six plays only an incidental role. …” In the Fig. 3 & 4, I included the factor ×1 as the beginning factor of a new sequence, thus ×6 generates the end of a sequence. At this stage of discussion it is not possible to offer a sufficient reason for why the ×6, ×10, ×1 factors may need to be considered one of the fundamental concepts of the 60 based counting. I hope to offer a simple and yet a plausible reason in the second part of the article. NUMERALS OBSCURED The initial sequence (Figs. 4 & 5,) ending in 60(ḡeš) deserves some additional attention because all other sequences are structuraly only its replicas in multiples of 60. Importantly, it assumes individual counts from 1(diš) through 10(u) and then the multiples of 10 up to 60(ḡeš).

Fig. 5. The primary archaic 60 based numeration sequence

3 “Of the numerals … only ‘one’ through ‘ten’ … and the numerals for ‘twenty,’ ‘thirty,’ ‘forty,’ ‘fifty,’ and ‘sixty’ are attested in the lexical lists … .” (Powell, ibid pg.47)

Fig. 6. The numerals of the primary base 60 sequence While it may appear that the reading of the numerals in the Fig. 6, is a straightforward affair, that is one numeral, one name, one meaning, that actually is valid only for 9 numerals in the sequence, namely: 1(diš) through 5(ia), 10(u), 8(ussu), 20(niš) and 60(ḡeš). The other numeral names are amalgams of the other numerals. How these numeral names should be read is given by Powell (ibid pg. 82) “It is upon the lexical texts … that we must depend for any certain knowledge of the form of Sumerian numerals. The system of notation employed to write the numerals can be very deceiving. Who would have suspected, without linguistic evidence preserved in lexica, that stood for ‘five + one,’ or threes,’

‘two twenties,’ and

for ‘ five + two,’ or

for ‘five + four,’

for ‘ten

for ‘two twenties ten.”

Even when the numerals in the above quote were shown in the much earlier, archaic notation, centuries older than the cuneiform, the discrepancy remains: “…Who would have suspected, without linguistic evidence preserved in lexica, that stood for ‘five + one,’ or for ‘five + two,’ or for ‘five + four,’ for ‘ten threes,’

‘two twenties,’ and

for ‘two twenties ten.”

However, the reading of the given numerals reveals their peculiar structure that may be much older not only than the archaic notations but even older than the 60 based numeration itself. In other words, it seems that another manner of counting, dissimilar in genesis and form, preceded the counting in multiples of 60, and could have been the foundation on which already quite numerate Sumerians built a more effective system of reckoning. Of the difficulties with Sumerian numeration Powell writes (ibid, pg. 82): “ …The basic obstacle in studying the origin of Sumerian numeration is that the system in its fundamental form, i.e., counting in multiples of sixty, must have been in existence before the invention of writing. Since we are entirely dependent upon written documentation for the study of Sumerian numeration, we may never expect to obtain evidence contemporary with the genesis of that numeration. …” (Powell, ibid, pg. 82)” COUNTING BY HANDS Powell suggested the use of a hand for counting (Powell, ibid, pg. 37): “… the numerals below ‘ten’ are based - at least in part – upon the fingers of the hand.” And, indeed, when the Sumerian names of numerals from 1(diš) to 10(u) are displayed along depiction of two hands, one cannot overlook the fact that the readings of the numerals 6(aš) to 10(u), save the 8(ussu), confirms counting base 5 (Fig. 7.)

4

Fig. 7. A pair of hands – the first natural counting device Counting by two hands may not be exhausted at 10(u) (Fig. 8.) For example, the manner of doubling the 20 to obtain 40, and then using 40 with added 10 to obtain 50 suggests the base 20 counting but save the three numerals nothing else supports such a thought. And there is 10 3’s (ušu ) that, with all the other numerals, due to pragmatism of Sumerians, suggests an eclectic count forming manners. Thus, it is hard to assume that the names of the numerals from 10 through 50 in Fig. 8, did not originated before the adoption of 60 based counting - their form does not fit the concept of the 60 based naming of numerals.

Fig. 8. Sumerian numerals 20 through 60 vs. archaic notation But counting in multiples of 10 ends abruptly with uniquely named 60(ḡeš): “The etymology of the word for ‘sixty’ is not known …” (Powell, ibid, pg. 53.) Following the naming convention of the numerals larger than 10(u) we may expect, for example, 3 20’s, or 2 30’s, but instead we got ×6 10 (Fig. 5,) a forceful bundling of the 6 of 10’s into the single value that ends the primary 60 based counting sequence and provides the first 60(ḡeš), the basis of the counting in multiples of sixty.

Although being “built” by it, 60(ḡeš) is a drastic, someone may even say an “ideological,” break with the old manner of counting by hands and fingers (Fig. 8.) In what may seem to appear without a cause, the 60(ḡeš) determines nothing else and nothing less than exactly 60 še, that is, 60 barleycorns. In that sense all of the numerals encountered earlier became measures in še, and that is what the Sumerian counting entails: counting in barleycorns, thus še diš, še u, and še ḡeš.

5 This new thoughts introduce an ironclad system of the še ḡeš ← še u ← še diš sequences based on the ×6, ×10, ×1 rule (Fig. 4,) effectively obscuring, like on a palimpsest, the old mélange of the numerals’ naming and counting by hand(s). CONCLUSION This draft is the first part of the originally conceived essay “Sumerian Counting Devices.” From the entire discussion of this, the first part, it is discernible that the digits and hand(s) may, or rather, could have been for some time used as a natural device for counting of the inhabitants of the alluvial Mesopotamia, probably, very long before the the 4th millennium BC, and could have shaped the ways of naming and constructing their numerals. The names of the numerals from 1(diš) through 50(nimmu) were used to generate the primary 60 based counting sequence and that nomenclature was retained in the new, 60 based, counting mode. The crown of the new counting manner was 60(ḡeš). The structure of the primary counting sequence (Fig. 5,) that is, ×6, ×10, ×1 shows that the counts from 1(diš) through 10(u) were condensed first into 10(u) and then the six of these “condensates” bundled into 60(ḡeš), thus the sixty ← ten ← one (Figs. 7&8.) Sumerians were not mathematicians but practical people and they could not abstract that kind of counting system without employing a physical model which, of its structure, is closely reflected in the še sixty ← še ten ← še one / ×6, ×10, ×1 mode of counting. With some hindsight I can say that such a model must have been ubiquitous thus always ready for inspection, of exceptional significance to Sumerians, and, in one or another way, used for some time if only as an approximate grain measure, and, above all, provide for easy visual verification of concrete counting in parts and multiples of še ḡeš, in other words, in barleycorns. That conjures two questions: Why ḡeš for 60? and: Why še, that is barleycorns? For both must have been inseparable to be the integral parts of the Sumerian counting base 60. Any reasonable answer to either question may satisfy both, but also Powell’s: “Why did Sumerians count in the multiples of sixty? This is a question for which there exists no answer at the present time.” (Powell, ibid pg. 82.)

LITERATURE 1. 2. 3. 4. 5.

M. A. Powell Jr., Thesis 1971, UMI 1984 Nissen et al. “Archaic Bookkeeping” The Chicago U Press, 1993 Jøran Friberg, “A Remarkable Collection if Babylonian Mathematical Texts” Springer 2007 Denise Schmandt-Basserat “Before Writing” University of Texas Press, 1982 Karl Menninger “Number World and Number Symbols” A Cultural History of Numbers, Translation: Paul Boneer, The MIT Press 1977 6. G. Ifrah, “From One to Zero – A Universal History of Numbers” Transl. L. Bair Penguin Books 1987