FARA No.50 REVISITED

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FARA No. 50 - REVISITED M ENTAL COMPUTING OF FARA No.50 – AN EASY WAY

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

Fig. 1. A grain goddess gifting a barley spike Cylinder seal, Akkadian, ca. 2350-2150 BCE Drawing c S. Beaulieu, after Boehmer 1965: Plate XLVI, #538

INTRODUCTION Computing Fara No.50 surely appears intimidating due to the very large numerical values involved. But that impression may disappear once we try to think harder of some possible mental reckoning that the Fara scribes may have developed or inherited from generations of Sumerian reckonners before them. The manner of mental computing that I am following here is, indeed, pleasantly straightforward thus so simple and quick that it is hard to think that it may not have been known to the astute scribes of the Fara No.50 tablet times. That splendid, so accurate and precise account, they endeavored to impress into a humble chunk of wet clay only to be discovered millennia later. The initial values of a gur-mah = 480 sila and of “granary” = 2400 gur-mah, strongly indicate that the task was considered as a routine mental reckoning with no need for any tracking of the intermediate results, if even that, given the reckoning proficiency of the scribes to whom the task was intended in the first place. Notorious division of ĝeš(60) by imin(7) is at the heart of the task making it, along with the very large numerical values involved (1152000 sila,) appear frightening indeed. But, in fact, the Fara No.50 task used the common grain measures and most common numerical values that mental processing of their multiples and fractions would not be a reckoning hindrance for even an apprentice, much less for a seasoned journeyman scribe. COMPUTING OF FARA No.50 – AN EASY WAY When in the sexagesimal reckoning 7 is a divisor, the mental tasks prove to be difficult. But, as in so many different cases of archaic computing, a slight adjustment of the initial numerical values, followed at some point by appropriate corrections, could make a seemingly difficult mental task turn into a breeze. My articles: M SVO 1, 002 A POSSIBLE CASE FOR AN ARCHAIC M ESOPOTAM IAN M ENTAL ARITHM ETIC (academia.edu) and ARCHAIC M ESOPOTAM IAN COMPUTING M ANNERS -TABLET AO 6060, Ur III, ca. 2100 2000 BC, A CASE FOR M ENTAL ARITHM ETIC (academia.edu) may further illustrate the mental arithmetic ways of archaic computing over a long time span.

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Relevant data about Fara No.50 tablet could be obtained from the article published in Historia Mathematica 9 (1982) 19-36, Investigation of an early Sumerian Division Problem, c. 2500 B,C, by Jens Høyrup, which information I am using here. The result of the computing task states: “The contents of one storehouse of grain are distributed to number of men, each man receiving a ration of 7 sila. Consequently, 45,42, 51 rations (i.e., 164571 rations) are distributed, and a remainder of 3 sila is left over. “

Instead of using the sexagesimal value place notation it would be far more instructive to display the number of 7sila rations in the numerals and similar fashion as depicted on the tablet, thus:

Fig. 1. Fara No.50 tablet numerals and a scale of the Fara numerical values (Author’s sketches after picture and data in the J. Høyrup’s article)

Because the result of 164571 of 7 sila allotments with the 3 sila remnant is already given in the right column on the tablet, the challenge is to find a possible way of computing by which that result of great accuracy and precision was accomplished. The starting point is 1(diš) granary that, in Fara times, contained

4(geš-u) (2400) gur-mah of

8(geš) (480) sila (~0.8 liters) of barley.

To tackle the task, as a first step, we need (I think) to find the quantity of 7sila allotment (7si-al) in either the granary of 2400 gur-mah or only in 1 gur-mah. Neither of these initially given values, the 4(geš-u) (2400), nor the (600) is enlarged by

8(geš) (480) is divisible by

3(u), or the

8(geš) (480) is enlarged by

Here I will opt for enlarging a gur-mah by

1(u) (10) to obtain

7(diš). However, if

1(geš-u)

1(u) (10), then it can be done. 8(geš)1(u) (490) sila.

Now, a division of 8(geš)1(u) (490) by 7(diš) will yield directly 1(geš)1(u) (70) 7si-al. When that result multiplies 4(geš-u) (2400) we will obtain the 7si-al‘s in the whole granary. That can be done mentally by splitting 70 into 60 and 10 and then do computing knowing well that

1(geš)(60) ×

1(geš-u) (600) makes

1(geš)(60) 7si-al/gur-mah ×

1(šar-u), thus:

4(geš-u) (2400) gur-mah =

4(šar-u) 7si-al

3 All of these operations, enlarging gur-mah, dividing it by 7, and multiplying the granary content by 70 could be done in a snap because of a scribe’s experience in working such common values innumerable times in many different contexts. To this result we need add the really easy mental task because in the

1(u) (10) 7si-al/gur-mah × the gur-mah’s of the granary – a 1(u) (10) ×

3(geš-u) (1800) there are

5(šar) and

in the 1(u) (10) × 1(geš-u) (600) there is 1(šar)4(geš-u) (6000). When both amounts (24000) of 7si-al are added to the previous one, we obtain an easy to remember value of:

4(šar-u)6(šar)4(geš-u) (168000) 7si-al The whole mental computing process would not take no more than a couple of minutes, if that, using (maybe) the fingers of one hand to keep track of the steps and remembering what was computed. Or a few pokes in wet clay of a spare tablet may have been all what was necessary to save the result for later. A focused Sumerian reckonner must have been quite familiar with these basic operations involving only standard grain measures and the multiples thereof.

CORRECTING FOR ENLARGED gur-mah It is now necessary to correct the obtained result for the

1(u) (10) sila we added initially to the

8(geš) (480) sila of each gur-mah, with which to facilitate quick division by again, requires multiplication of of sila and gur-mah, therefore:

1(u) (10) sila ×

1(u) (10) ×

7(diš). That,

4(geš-u) (2400) to obtain 24000 but now in terms

4(geš-u) (2400) gur-mah =

6(šar)4(geš-u) (24000) sila

This 24000 sila we need to convert into 7si-al dividing it by 7(diš). Mentally splitting 24000 into 21000, 2800, and 140, each divisible by 7, we can obtain 3420. Dividing the remaining 60 sila/7 we get 8 7si-al + 4 sila remainder. Together, we need to subtract 3428 from 168000 thus:

4(šar-u)6(šar)4(geš-u) (168000) 7si-al 5(geš-u)7(geš)8(diš) (3428) 7si-al There is not enough in the 2400 si-al at the end of the 168000 si-al, from which to subtract 3428. Thus 1 šar of 3600 si-al will be borrowed from the 168000, to with 2400 make 6000. There remains

(162000) 7si-al 4(šar-u)

5(šar)

4 Subtracting 3400 of 3428, from 6000, leaves 2600. Subtracting 28 from that makes 2572 si-al. Adding that to the 162000 si-al makes 164572 of 7si-al:

(164572) 7si-al 4(šar-u)

5(šar) 4(geš-u) 2(geš) 5(u) 2(diš)

It should be emphasized again that the kind of the operations above must have been experienced mentally by the Fara scribes many times in their schooling as well as in the long practicing of their trade. To apply such experience to this subtraction task most probably would not be considered even mildly difficult. We are not trained to work the numbers mentally and for sure not to use the archaic Fara numerals. The closest we come to the kind of doing sexagesimal reckoning is when computing angles and hours in their peculiar parts. It is sure that we may be intimidated by the reckoning of the Fara No.50 and similar tasks, but the experienced scribes of yore will perform such tasks in strides. Finally we need to take care of the 4 sila that remains after dividing 60 si-al by 7. That we could accomplish by borrowing 1 si-al from the 164572 si-al and using it as 7 sila from which to subtract 3 sila thus 3 sila is left over. 1(diš) si-al -

4(diš) sila =

7(diš) sila -

4(diš) sila =

3(diš) sila

The end result is, of course:

(164571) 7si-al 4(šar-u)

5(šar)

4(geš-u) 2(geš) 5(u) 1(diš)

3(diš) sila remains CONCLUSION This is my second attempt to discern if mental reckoning may have been used to compute the result displayed on the so elegant Fara No.50 tablet. I think that there should be no doubt that a mental computing process could have been used here because of the demonstrated simplicity by which the accurate result could be attained. To appreciate sexagesimal mental arithmetic one ought to practice it for a little while in order to recognize the benefits of computing manners by which the scribes of the Mesopotamian yore could have been able to, with a little mental effort, accomplish tasks as one of the Fara No.50. It would, probably, take far more time to create so elegant a tablet than to compute the result via mental reckoning as described in this article.