Three thousand years of sexagesimal numbers in Mesopotamian mathematical texts
 39 Downloads
Abstract
The Mesopotamian system of sexagesimal counting numbers was based on the progressive series of units 1, 10, 1·60, 10·60, …. It may have been in use already before the invention of writing, with the mentioned units represented by various kinds of small clay tokens. After the invention of protocuneiform writing, c. 3300 BC, it continued to be used, with the successive units of the system represented by distinctive impressed cup and diskshaped number signs. Other kinds of “metrological” number systems in the protocuneiform script, with similar number signs, were used to denote area numbers, capacity numbers, etc. In a handful of known mathematical cuneiform texts from the latter half of the third millennium BC, the ancient systems of sexagesimal counting numbers and area numbers were still in use, alongside new kinds of systems of capacity numbers and weight numbers. Large area numbers, capacity numbers, and weight numbers were counted sexagesimally, while each metrological number system had its own kind of fractional units. In the system of counting numbers itself, fractions could be expressed as sixtieths, sixtieths of sixtieths, and so on, but also in terms of small units borrowed from the system of weight numbers. Among them were the “basic fractions” which we would understand as 1/3, 1/2, and 2/3. In a very early series of metromathematical division exercises and an equally early metromathematical table of squares (Early Dynastic III, c. 2400 BC), “quasiintegers” of the form “integer plus basic fraction” play a prominent role. Quasiintegers play an essential role also in a recently found atypical cuneiform table of reciprocals. The invention of sexagesimal numbers in placevalue notation, in the NeoSumerian period c. 2000 BC, was based on a series of innovations. The range of the system of sexagesimal counting numbers was extended indefinitely both upward and downward, and the use of quasiintegers was abolished. Sexagesimal placevalue numbers were used for all kinds of calculations in Old Babylonian mathematical cuneiform texts, c. 1700 BC, while traditional metrological numbers were retained in both questions and answers of the exercises. Examples of impressive computations of reciprocals of manyplace regular sexagesimal placevalue numbers, with no practical applications whatsoever, are known from the Old Babylonian period. In the Late Babylonian period (the latter half of the first millennium BC), such computations were still popular, performed by the same persons who constructed the manyplace sexagesimal tables that make up the corpus of Late Babylonian mathematical astronomy.
1 Preliterate number tokens possibly representing sexagesimal numbers
^{1}It is now well known that people in Mesopotamia and neighboring regions were using small clay figurines, socalled tokens, for as much as 5 millennia before the invention of writing (around 3300 BC), almost certainly for some kind of communication and archiving.^{2} Also known is that, a relatively short time before the invention of writing, groups of such tokens started to be enclosed in hollow clay balls, known as bullae, sometimes with indications on the outside about the contents. Such bullae have been found not only in the ancient Mesopotamian city Uruk, but also much further east, in the ancient city Susa in what is now Iran.
The inscription on the outside of Sb 1927 appears to be a description of the tokens inside the bulla, with the three round holes on the outside resulting from one of the smaller cones being pushed into the clay with the point first. Interestingly, the inscription on the outside seems to have been imprinted from right to left, just as the protoElamite script was inscribed from right to left.

1(10 · 60) 3(60) 3(10), in other words 13 · 60 + 30.

13 1/2 · 60 = (1 – 1/10) · 15(60).
2 Other tentatively identified preliterate systems of number notations
In examples like the one above, rods, lenses, cones, and a punched cone which have been found together in bullae can be compared to the units 1, 10, 60, and 10 · 60 of the welldocumented protocuneiform/protoElamite system S of sexagesimal counting numbers. Note the alternation of factors 10 and 6 in the factor diagram for system S.

3(100) 2(10) 4 = 27 · 12 = (1 – 1/10) · 6(60).

2d of barley for (1 – 1/10) month of work, which is the same as (1 + 1/9) · 2d of barley for 1 month’s work.
3 Sexagesimal or bisexagesimal counting numbers and whole or fractional capacity units in a protocuneiform school text
MSVO 4, 66 
MSVO 4, 66, shown above, is a wellknown protocuneiform “breadandbeer text” from the socalled Jemdet Nasr period, about 3000 BC. It is remarkable because it makes use of several different protocuneiform systems of number notations. In addition, it contains quite sophisticated calculations.^{6}
The lack of names and titles of persons in charge of the account in the empty box 1:iv on the reverse of MSVO 4, 66 indicates that this is a school text rather than a normal administrative protocuneiform text. It is easy to see that the text lists expenses of barley for large and small rations of bread and beer, but the presence of both unusually large numbers and of rations of many different sizes confirms the expression that this is a rather advanced “metromathematical” exercise aiming to train students both in proper attention to metrological questions and in performing complicated mathematical operations.

1(60) · M = 12 c = 2d, 1(2 · 60) · m2 = 12 c = 2d, 1(2 · 60) · m3 = 8 c = 1d 2c,

2(2 · 60) 1(60) · m4 = 5(60) · m4 = 15 c = 2d 3c, 5(2 · 60) · m5 = 1(2 · 60) · M = 4d,

5(10 · 2 · 60) · m6 (not explicitly indicated) = 5 · 40 c = 1C 3d 2c.
In the three text boxes 2:i to 2:iii on the obverse of MSVO 4, 66, jars of beer of three different (presumably known but not explicitly mentioned) strengths are counted in terms of sexagesimal counting numbers, 2(60), 3(60), and 5(60).
1(60) + 1(2 · 60) + 1(2 · 60) + 2(2 · 60) 1(60) + 5(2 · 60) + 5(10 · 2 · 60) = 1(60 · 2 · 60).  (rev. 1:ia1:iia) 
2(60) + 3(60) + 5(60) = 1(10 · 60).  (rev. 1:iiia) 

2C = 6(60)d = 360 c = 1800M, 1c 4M = 9M, so that 1C 2D 9d 4c 1M = (1 – 1/200) · 2C.
This can hardly be by accident, but why the grand total is of this kind is hard to explain.
4 Counting with loan and interest in a protocuneiform text

1c 2M m2 = 7M m2 = 15 m2, of which 1/10 is 15 dxs = (10 + 4 + 1) dxs = m2 m5 dxs.

(1 + 1/10) · 1c 2M m2 = (1 + 1/10) · 30 · m2.
In text box iii, the expense for five months is computed (somewhat incorrectly) as 12c 1M m2 (instead of 12c 1M m4).
Note the use of protocuneiform number signs for time measures, here 1 and 5 months. Note also that no cuneiform sign for “1/10” is used here, only the sign _{_} ^{_} for “interest.”
5 The historical development of the system of sexagesimal numbers
Above have been mentioned several known protocuneiform and/or protoElamite number systems, the systems S and C of sexagesimal counting numbers and capacity numbers, the system B of bisexagesimal counting numbers, and the system D of decimal counting numbers (protoElamite only). In addition, there was a protocuneiform system E of (probably) weight numbers, a system T of time numbers, and a system A of protocuneiform area numbers, the latter intimately related to length numbers (sexagesimal multiples of a certain basic length unit).^{8}
The decimal system D disappeared when the shortlived protoElamite script ceased to exist. As a matter of fact, in Sumerian and Akkadian texts from the Early Dynastic III period half a millennium after the time of the last protocuneiform texts, only systems S and A remained of the many early systems of numbers. The system A of area numbers was still in use in the Old Babylonian period in the first half of the second millennium BC, while sexagesimal numbers continued to be used until the end of the cuneiform script, at least in mathematical (and astronomical) cuneiform texts.
They were probably related to notations for fractions in an Old Sumerian and Old Akkadian system of weight numbers.
The fourth factor diagram above, finally, refers to sexagesimal numbers in floating placevalue notation, which were in use, almost exclusively, in NeoSumerian and Babylonian mathematical texts from about 2000 BC and onward (and Late Babylonian astronomical texts from the second half of the first millennium BC):
6 An Old Sumerian (Early Dynastic III) division exercise, around 2600 BC

The barley in a full granary of (apparently) known capacity has to be divided into individual rations of 7 sìla each. (The sìla was a capacity unit equal to around 1 L.) Find the number of rations.

1 gur.mah = 8 barig, 1 barig = 6 bán, 1 bán = 10 sìla.
1 bán  =  1  ration  plus  3 sìla  
1 barig  =  6 · 1 bán =  8  rations  plus  4 sìla 
1 gur.mah  =  8 · 1 barig =  1(60) 8  rations  plus  4 sìla 
10 gur.mah  =  10 · 1 gur.mah =  11(60) 25  rations  plus  5 sìla 
1(60) gur.mah  =  6 · 10 gur.mah =  1(60 · 60) 8(60) 34  rations  plus  2 sìla 
10(60) gur.mah  =  10 · 1(60) gur.mah =  11(60 · 60) 25(60) 42  rations  plus  6 sìla 
40(60) gur.mah  =  4 · 10(60) gur.mah =  45(60 · 60) 42(60) 51  rations  plus  3 sìla 
This presumed algorithmic computation is modeled after the explicitly performed metromathematical algorithmic division of 1 ríba (= 10,000) níg.sagshu by 24 níg.sagshu in the roughly contemporary text TM.75.G.2346 from the site Ebla in presentday Syria.^{9}
7 The second oldest known metromathematical theme text, c. 2300 BC

1 ma.na = 1(60) gín.

The selling price for beads is 1 ^{2}/_{3} 5 (gín) times the buying price.

Beads were sold for the price 1 (ma.na) – ^{1}/_{2} gín. For how much were they bought?

Answer: 1/2 ma.na 4 gín.

1 ^{2}/_{3} 5 gín · n = 1 ma.na – ^{1}/_{2} gín, n = ?

1 ^{2}/_{3} 5 gín · n* = 1 ^{2}/_{3} · 4 gín + 5 · 1/60 · 4 gín = 6 ^{2}/_{3} gín + ^{1}/_{3} gín = 7 gín.

1 ma.na – ^{1}/_{2} gín = 59 ^{1}/_{2} gín = 8 ^{1}/_{2} · 7 gín.

n = n* · 8 1/2 = 8 1/2 · 4 gín = 34 gín = 1/2 ma.na 4 gin.
An interesting, hidden meaning of this exercise is that it really, like the exercise discussed above in Sect. 6, may have been concerned with the problem of division by the sexagesimally nonregular number 7. The concept of sexagesimally regular numbers is well known and easy to understand in the case of sexagesimal numbers in placevalue notation. Namely, a sexagesimal number n in placevalue notation is said to be regular if there exists another number n´ of the same kind such that n · n´ = some power of 60. (Otherwise it is nonregular.) Then n´ can be referred to as rec. n (the reciprocal of n).

1 ^{2}/_{3} · 36 = 36 + 24 = 60.

7 · 8 ^{1}/_{2} = 56 + 3 ^{1}/_{2} = 59 ^{1}/_{2} = 1(60) – ^{1}/_{2}.

3 ^{1}/_{2} · 17 = 1(60) – ^{1}/_{2}, 1 ^{2}/_{3} 5 gín · 34 = 1(60) – ^{1}/_{2}, and so on

1 ^{2}/_{3} 5 gín · ^{1}/_{2} ma.na 4 gín = 1 ma.na – ^{1}/_{2} gín.

Potash(?) can be bought at a market rate of 1 ^{2}/_{3} sìla of potash for 1 sìla of barley.

The amount of potash bought was 2 barig. How much was paid for the potash?

1 barig = 6 bán, 1 bán = 6 sìla, 1 sìla = 60 gín.

1 ^{2}/_{3} · p = 2 barig, p = ?

5 p = 6 barig,

p = 1/5 of 6 barig = 1 1/5 barig = 1 barig 1 1/5 bán = 1 barig 1 bán 1 1/5 sìla = 1 barig 1 bán 1 sìla 12 gín.

p = 1/5 of 2 barig = 2 bán + 1/5 of 2 bán = 2 bán 2 sìla + 1/5 of 2 sìla = 2 bán 2 sìla ^{1}/_{3} (sìla) 4 gín of barley.
This is the amount of barley recorded at the end of the first column on the obverse of the text.
Exercise # 5

Someone borrowed or invested ^{2}/_{3} ma.na of silver. The silver was returned at a rate of 1 ^{2}/_{3} gín of silver for each original gín. How much silver was returned?

1 ^{2}/_{3} · 2/3 ma.na = 1 ^{2}/_{3} · 40 gín = 40 gín + 26 ^{2}/_{3} gín = 1 ma.na 6 ^{2}/_{3} gín.
For a modern reader, it is surprising to see how complicated such seemingly simple division and multiplication exercises could be at a time when it was no trivial matter to divide by quasiintegers, and when it was important to take account of the units and conversion factors in the systems of measures in terms of which the metromathematical problem was stated.
8 An Early Dynastic cuneiform text with several tables of areas of squares

100 sar = 1 iku, 6 iku = 1 èshe, and 3 èshe = 1 bùr.

1 shár.kid = 60 · 60 · 60 · 3 · 6 · 100 sar = c. 14,000 square kilometers.
1 nindan  square (is)  1 sar (of)  area 
10  square  1(iku)  area 
1(gésh)  square  2(bùr)  area 
10(gésh)  square  3(shár) 20(bùr)  area 
1(shár)  square  2(shár).gal  area 
10(shár)  square  3(shár).kid 20(shár).gal  area 
sq. 10(gésh) =  10 · 10 · 2(bùr) =  10 · 20(bùr) =  3(shár) 20(bùr) 
sq. 1(shár) =  6 · 6 · 3(shár) 20(bùr) =  6 · 20(shár) =  2(shár).gal 
sq. 10(shár) =  10 · 10 · 2(shár).gal =  10 · 20(shár).gal =  3(shár).kid 20(shár).gal. 

Subtable B is a table of squares of multiples of 1 nikkas (= 1/4 nindan).

Subtable C is a table of squares of multiples of 1 kùsh.numun (= 1/6 nindan).

Subtable D is a table of squares of multiples of 1 gish.bad = (1/12 nindan).

Subtable E is a table of squares of multiples of 1 shu.bad = (1/24 nindan).
In the partial transliteration below is shown, as an example, 2 of the 10 columns on the clay tablet. (There are in all 7 columns on the obverse and 3 on the reverse.)

B: sq. 1 nikkas = sq. (1/4 nindan) = 1/4 of 1/4 sar = 1/4 of 15 gín = 3 ^{2}/_{3} gín 5 (gín.bi)

C: sq. kùsh.numun = sq. (1/6 nindan) = 1/6 of 1/6 sar = 1/6 of 10 gín = 1 ^{2}/_{3} gín

D: sq. 1 gish.bad = sq. (1/12 nindan) = 1/12 of 1/12 sar = 1/12 of 5 gín = ^{1}/_{3} gín 5 gín.bi

E: sq. 1 shu.bad = sq. (1/24 nindan) = 1/24 of 1/24 sar = 1/24 of 2 ^{1}/_{2} gín = 6 gín.bi 15 gín.ba.gín.

1 gín = 1/60 of 1 sar, 1 gín.bi = 1/60 of 1 gín, and 1 gín.ba.gín = 1/60 of 1 gín.bi.
Evidently, what we see here is an attempt to extend the system A of area numbers downward sexagesimally, so that it comes to include also a limited number of sexagesimal fractions!
9 An old Akkadian geometric division exercise, c. 2250 BC

s = 1 nindan 5 kùsh 2 shu.dù.a 3 shu.si ^{1}/_{3} shu.si.

1(60) 7 ^{1}/_{2} nindan · s = 1(60) 40 sar (sq. nindan).

1 nindan = 12 kùsh (cubits), 1 cubit = 3 shu.dù.a, 1 shu.dù.a = 10 shu.si (fingers).

1(60) 7 ^{1}/_{2} = 27 · 2 ^{1}/_{2}, where 27 and 2 ^{1}/_{2} are sexagesimally regular integers or quasiintegers.
1(60) 40 sq. nindan  =  2 ^{1}/_{2} nindan ·  40 nindan 
=  7 ^{1}/_{2} nindan ·  13 nindan 4 cubits  
=  22 ^{1}/_{2} nindan ·  4 nindan 5 ^{1}/_{3} cubits  
=  1(60) 7 ^{1}/_{2} nindan ·  1 nindan 5 ^{2}/_{3} cubits 3 ^{1}/_{3} fingers 
Interestingly, it can be said that the problem in this exercise had a simple solution precisely because the given number 1(60) 7^{1}/_{2}was sexagesimally regular, and because so were the conversion factors 12, 3, and 10 in the Old Akkadian system of length measures.
10 A table of reciprocals without sexagesimal placevalue numbers
SM 2685 (shown below in conform transliteration) is a clay tablet from the Sulaymaniyah Museum in the Kurdistan region of Northern Iraq.^{14} The clay tablets in the Sulaymaniyah Museum were acquired in the antiquities market and are therefore unprovenanced, but in most cases they are probably from Old Babylonian Larsa. The writing on SM 2685 itself is such that the text can be either from the NeoSumerian Ur III period (c. 2150–2000 BC), or somewhat younger, from the Early Old Babylonian period. However, the content of the text has no known Old Babylonian parallels. Indeed, it is likely that the clay tablet is a copy of a much older text, certainly from before the invention around 2000 BC of sexagesimal numbers in placevalue notation.
The ubiquitous presence in the text of the phrase igi n gál.bi clearly shows that SM 2685 is some kind of table of reciprocals. The term igi n, or igi n gál, has the meaning “the reciprocal of n” or “the nth part.” It is otherwise known, for instance, from the early curious table of areas of squares A 681 (Adab, Early Dynastic III), where igi 4 appears twice,^{15} and from an equally early and even more curious lexical text for weight measures with multiples and fractions of the ma.na (also ED III), which mentions igi 3 gál, igi 4 gál, and igi 6 gál.^{16} The Sumerian term igi n gál, literally meaning “it has n eyes,” may have been a surviving reminiscence of the protocuneiform number signs m2, m3, …, m6, for the fractions 1/2M, 1/3M, …. 1/6M (in system C(pc), see Sect. 3).

igi 13 ^{1}/_{3} gál.bi 4 ^{1}/_{2} which can be understood as meaning rec. 13 ^{1}/_{3} = 4 ^{1}/_{2}.

13 ^{1}/_{3} · 4 ^{1}/_{2} = 13 · 4 + ^{1}/_{3} · 4 + 13 · ^{1}/_{2} + ^{1}/_{3} · ^{1}/_{2} = 52 + 1 ^{1}/_{3} + 6 ^{1}/_{2} + 1/6 = 59 ^{1}/_{2}^{1}/_{3}^{1}/_{6} = 1(60).

igi 13 ^{1}/_{2} gál.bi 4 ^{1}/_{3} 6 ^{2}/_{3} which can be understood as meaning rec. igi 13 ^{1}/_{2} = 4 ^{1}/_{3} 6 ^{2}/_{3} gín.
Moreover, even if it can be shown, through a series of laborious multiplications, that the product of the pair of numbers in each line of the table is equal to 1(60) (which is the case), then it remains to be shown how each such pair of numbers had been found. It can hardly have been done through trial and error, testing all kinds of weird quasiintegers to see whether there existed a corresponding more or less complicated reciprocal number.
Instead, it is likely that the pairs of numbers in the table of reciprocals SM 2685 were constructed in a systematic series of quite simple computations, in the following way:
All these 25 pairs seem to have been present in the table of reciprocals SM 2685 (before it was damaged so that some of the initial pairs were lost). On the other hand, these 25 pairs are only about one half of all the pairs in the table (before it was damaged).
The remaining pairs in the table are such that n is a quasiinteger, but not an integer. Now, it is quite obvious how the pairs were constructed when the fraction in the quasiinteger for n is one of the basic fractions ^{2}/_{3}, ^{1}/_{2}, or ^{1}/_{3}, namely by the kind of reciprocal compensation mentioned above. Namely, if n, rec. n is one of the pairs in the brief table above such that n · rec. n = 1(60), and if n is multiplied by ^{2}/_{3}, ^{1}/_{2}, or ^{1}/_{3}, while rec. n is multiplied by 1 ^{1}/_{2}, 2, or 3, respectively, then the new pair is still such that the product of the two numbers is 1(60).
Altogether, 10 + 7 + 2 = 19 new pairs can be constructed in this way, if duplicates are omitted. Of these 19 new pairs, 17 are present in the table on SM 2685. Only 2 are missing, for one reason or another.
53 ^{2}/_{3} · 1 7 ^{1}/_{2} gín =  (8 · 6 ^{2}/_{3}) · (7 ^{1}/_{2} · 9 gín) =  (8 · rec. 9) · (rec. 8 · 9 gín) =  1(60) 
56 15 gín · 1 4 gín =  (15 · 3 ^{2}/_{3} 5 gín) · (4 · 16 gín) =  (15 · rec. 16) · (rec. 15 · 16 gín) =  1(60) 
57 ^{1}/_{2} 6 gín · 1 2 ^{1}/_{2} gín =  (24 · 2 ^{1}/_{3} 4 gín) · (2 ^{1}/_{2} · 25 gín) =  (24 · rec. 25} · (rec. 24 · 25 gín) =  1(60) 
Here (8, 9), (15, 16), and (24, 25) are three examples of what may be called sexagesimally regular twins of the type (n, n – 1), where both n and n – 1 are sexagesimally regular. Other examples are (2, 3), (3, 4), (4, 5), (5, 6), (9, 10), but using them in a similar way would not produce new pairs of reciprocal numbers in SM 2685.

1(60) 21, ^{2}/_{3} [x x x].
This is a twice extended quasiinteger!
Also these two exceptional lines were probably added to the table for pedagogical reasons.
1(60).da ^{2}/_{3}.bi 40.àm  “of 1(60), its ^{2}/_{3} is 40” 
shu.ri.a.bi 30.àm  “its half is 30” 
It is probably significant that in SM 2574 the first sign in line 1 is an oversize vertical wedge, a clear indication that it stands for “1(60),” and not for “1.” Therefore, the mentioned first lines of the table can be interpreted as saying that 2/3 of 1(60) is 40, one half of 1(60) is 30. This is potentially important, in view of the observation above that n · rec. n = 1(60) in each line of the atypical table of reciprocals SM 2865.
11 A New Explanation of the Set of Attested Head Numbers in Old Babylonian Multiplication Tables

50, 48, 45, 44 26 40, 40, 36, 30, 25, 24, 22 30, 20, 18, 16 40, 16, 15, 12 30, 12, 10, 9, 8 20, 8, 7 30, 7 12, 7, 6 40, 6, 5, 4 30, 4, 3 45, 3 20, 3, 2 30, 2 24, 2 15, 2, 1 40, 1 30, 1 20, 1 15, 1 12.
Several attempts have been made to explain this list of head numbers, all with very limited success.^{20} A basic assumption has usually been that this list of head numbers in some way was derived from the list of reciprocals in the Old Babylonian/NeoSumerian standard table of reciprocals. However, this assumption explains readily only the italicized head numbers in the list above. Now, with the discovery of the atypical table of reciprocals SM 2685, a simple and much more satisfactory explanation of the attested set of head numbers is available. Indeed, the mentioned list of head numbers can be explained as being derived from the list of reciprocal numbers in an atypical table of reciprocals very much like, but not identical with, SM 2685!^{21}
Interestingly, the need of a combined multiplication table derived from the list of reciprocal numbers in a table of reciprocals like SM 2685 may have been felt for the first time by the one who constructed the impressive Early Dynastic III combined table of areas of squares CUNES 5008001 (see Sec. 8 above). Take, for instance, subtable C in CUNES 5008001, which was constructed, essentially, by first computing 1/6 · 1/6 · 1 sq. nindan = 1 ^{2}/_{3} gín and then multiplying this (area) number by the square numbers 4, 9, 16, etc. In the same way, subtable B was constructed by first computing 1/4 · 1/4 · 1 sq. nindan = 3 ^{2}/_{3} gín 5 (gín.bi) and then multiplying by 4, 9, 16, etc.
It is not inconceivable that the earliest precursor of the atypical table of reciprocals SM 2685 was not just a hypothetical atypical NeoSumerian table of reciprocals, but even a table of reciprocals from the Early Dynastic III period, contemporary with CUNES 5008001. For that matter, it is not inconceivable even that the earliest precursor of the NeoSumerian/Old Babylonian combined multiplication tables, preceded by a standard table of reciprocals, may have been an Early Dynastic III combined multiplication table for quasiintegers, preceded by an atypical table of reciprocals like SM 2685!
12 Counting with sexagesimal placevalue numbers in old Babylonian mathematical exercises
It is now well known that the invention of placevalue notation for sexagesimal numbers appears to have been NeoSumerian. On the other hand, it is not known at all if also socalled metrological tables were a NeoSumerian invention, or if such tables were introduced a hundred or a few hundred years later, in the Old Babylonian period. A metrological table is a table for conversion into sexagesimal placevalue numbers (multiples of some suitable “basic unit”) of a systematically arranged growing list of capacity numbers, weight numbers, area numbers, or length numbers.^{22} In the elementary first part of the education of students in Old Babylonian scribe schools, the learning of such metrological tables, alongside tables of reciprocals and multiplication tables, played a fundamental role.^{23}

The market rate rose, and I bought 30 gur of barley, the market rate fell, and I bought 30 gur of barley.

I added my market rates, it was 9.

I added the silver for market rates, it was 1 ma.na 7 ^{1}/_{2} gín.

What were my market rates?

Note that the silver was 1 07 30. Compute the reciprocal of 1 07 30, then 53 20 comes up.

Multiply 53 20 that came up with 9 for the market rates, then 8 comes up.

Multiply by 2 30 for your market rates, then 20 comes up.

……

1 gur = 5 barig = 5 · 6 bán = 5 · 6 · 10 sìla = 5(60) sìla, so that

30 gur = 30 · 5(60) sìla = 2 30 · 1(60) sìla = ”2 30.”
Then, the computation starts with these converted data. Unfortunately, the last part of the text is lost, but it is clear that the result of the computation of the two market rates must have been “5” and “4” in floating sexagesimal numbers. In the final answer, this result would have been converted back into the numerical measures 5(60) sìla = 1 gur, and 4(60) sìla = 4 barig of barley, respectively, in exchange for 1 gín of silver.
There are other types of known Old Babylonian mathematical texts containing no metrological data at all, but instead concerned exclusively with totally abstract sexagesimal placevalue numbers. Typical examples are interesting computations of square roots of given “manyplace” regular sexagesimal numbers as the products of the square roots of factors of the given numbers, or computations of reciprocals of given manyplace regular sexagesimal numbers as the product of the reciprocals of factors of the given numbers. Both kinds of computations may be called “factorization algorithms.”
Particularly interesting are algorithms for the construction of new pairs of mutually reciprocal regular sexagesimal placevalue numbers with departure from already known such pairs. The method used in such algorithms is a further development of the method used for the construction of the table of reciprocals SM 2865 in Sec. 10 above, a “reciprocal compensation algorithm.”
One such algorithm is used in the text CBS 1215. See the transliteration below.^{25} The algorithm takes it departure from the given pair of reciprocals (2 05, 28 48), where 2 05 (= 125) = 5 · 5 · 5 and 28 48 = 12 · 12 · 12. In the first text box of CBS 1215, in column i, the mentioned factorization method is used in order to show that rec. 2 05 = 28 48, and that, conversely, rec. 28 48 = 2 05. In the next text box, the pair of reciprocals is (4 10, 14 24), obtained by use of a “doubling and halving algorithm,” a kind of reciprocal compensation. And so on, until in the last text box, in column vi, the final pair of reciprocals is (10 06 48 53 20, 5 55 57 25 18 45).
13 Floating sexagesimal placevalue numbers in late Babylonian mathematical texts
A number of Late Babylonian texts from the second half of first millennium BC reintroduced and developed further the mentioned Old Babylonian algorithmic methods for the computation of reciprocals of given regular sexagesimal placevalue numbers or for the systematic construction of enormously comprehensive tables of reciprocals of manyplace regular sexagesimal numbers.
Another interesting example is the extended table of reciprocals AO 6456^{27} (see the photographs below), explicitly dated to the Seleucid (Hellenistic) period, c. 200 BC. It can be shown that the table of reciprocals was constructed by use of reciprocal compensation, just like the NeoSumerian atypical table of reciprocals SM 2685 in Sec. 10 above, although with sexagesimal placevalue numbers.

rec. 1 29 40 50 24 27 = 40 08 32 44 57 28 29 55 20 09 52 35 33 20.
This reciprocal pair, where n is a 6place and rec. n a 14place regular sexagesimal number, is inscribed in the third row from the end of the second column on the obverse of the clay tablet. It can be shown that 1 29 40 50 24 27 is the 19th power of 3.

rec. 1 29 12 19 26 34 23 19 49 38 08 36 52 20 44 26 40 = 40 21 22 41 (00) 09.

AO 6456 © Musée du Louvre, dist. RMNGP/Raphaël Chipault
Note that the construction of the extended table of reciprocals AO 6456 was a really awesome achievement. It began with the laborious computation, by use of the doublingandhalving algorithms described above, of some 300 pairs (n, rec. n) of regular “manyplace” sexagesimal numbers in floating placevalue notation, probably recorded on quite a few small clay tablets. The numbers n constructed in this way then had to be interpreted as sexagesimal numbers between 1 and 1(60), and ordered by size, another extremely laborious operation. Finally, the corresponding pairs (n, rec. n) had to be copied onto (probably) two large clay tablets, of which AO 6456 was the first. According to a badly understood endnote on AO 6456, the table of reciprocals goes from “1” to “2”. Indeed, the table of reciprocals begins with n = 1, n = 1 00 16 53 53 20 and ends with n = 2 57 46 40, n = 3.
It is no wonder that Late Babylonian priests/mathematicians who were able to construct such magnificent manyplace sexagesimal tables of reciprocals in some cases were identical with the persons who constructed the various kinds of even more aweinspiring manyplace sexagesimal tables belonging to the genre of Late Babylonian mathematical astronomy.^{28}
14 Conclusion: The invention of sexagesimal placevalue numbers
Interestingly, this factor diagram shows that two of the key features of the fullblown Sumerian system of sexagesimal counting numbers (see the factor diagrams in Sec. 5 above) were present from the beginning, namely the repeating alternation between the factors 10 and 6, and the fact that the sign for 60 was just a bigger version of the sign for 1.
The form of the system of sexagesimal counting numbers stayed essentially the same until the end of the third millennium BC. However, it is worth noticing that the system apparently did not involve any specific signs for fractions. Take, for instance, a look at the text HS 815 in Sec. 9 above, which shows that although multiples of the length measure nindan were counted sexagesimally, the fractions of the nindan were expressed in terms of special notations for small length measures. Similarly, multiples of the Sumerian capacity gur were counted sexagesimally, but the fractions of the gur were expressed in terms of notations for small capacity measures. The situation was the same in the case of the Sumerian weight unit ma.na and (essentially) in the case of the Sumerian area unit bùr.
The Sumerian cuneiform signs for the weight numbers ^{1}/_{3}, ^{1}/_{2}, and ^{2}/_{3} ma.na were borrowed into all other Sumerian/Old Babylonian systems of number notations as cuneiform signs for the “basic fractions” ^{1}/_{3}, ^{1}/_{2}, ^{2}/_{3}. An early example is the Early Dynastic III table of areas of squares A 681, where ^{1}/_{3} sar, ^{1}/_{2} sar, and ^{2}/_{3} sar are notations for fractions of the area unit sar.^{30} In the Early Dynastic III metromathematical theme text CUNES 5218035 (see Sec. 7 above), the mentioned basic fractions are used to denote fractions of both the ma.na and the gín.
In Old Babylonian metrological table texts, as well as in questions and answers in Old Babylonian mathematical exercises, the gín and the barleycorn (she), both borrowed from system M(S), were used as signs for the fractions 1/60 and 1/180 of 1/60, respectively. This situation could have been the end for any smooth development of sexagesimal fractions.
Luckily, there was another way open. This was demonstrated in Sec. 8 above, in the case of the large Early dynastic table of areas of squares CUNES 5008001, in which appeared two kinds of fractions of the area unit sar. On the one hand, there were the basic fractions ^{1}/_{3}, ^{1}/_{2}, ^{2}/_{3}, and on the other hand the fractions gín = 1/60 and gín.bi = 1/60 gín (and even, in subtable E, gín.ba.gín = 1/60 gín.bi)). More precisely, there appeared in this table of areas of squares quasiintegral multiples of the area units sar, gín, and gín.bi. (The notion of quasiintegers was introduced in Sec. 10 above, in connection with an attempted explanation of the atypical table of reciprocals SM 2685.)

The accidental circumstance that the cuneiform sign for “1(60)” could not easily be distinguished from the similar sign for “1.”

The observation that if also the special cuneiform sign for “10 · 60” was replaced by the cuneiform sign for “10,” then the repeating alternation between the factors 10 and 6 in the factor diagram for the first few units of the system of sexagesimal counting numbers could be continued forever.

The abandonment of the use of quasiintegers, in favor of integral multiples of the gín, gín.bi, and gín.ba.gín.

The observation that the repeating alternation between the factors 10 and 6 in the factor diagram for sexagesimal counting numbers could be imitated in a similar factor diagram for sexagesimal fractions.

The invention of metrological tables, which securely linked the various traditional systems of measure units to abstract counting numbers in the form of sexagesimal “floating” placevalue numbers.
Even if the inventors of sexagesimal placevalue numbers did not argue directly in terms of “factor diagrams,” they must reasonably have had in mind something similar.
Footnotes
 1.
An updated version of Friberg (2014).
 2.
SchmandtBesserat (1992).
 3.
See Friberg (2007), pp. 380–384.
 4.
For the concept of “almost round numbers”, see Friberg (1997/98).
 5.
See Friberg (2007) A Remarkable Collection, p. 382.
 6.
 7.
 8.
See Nissen et al. (1993), Fig. 50.
 9.
See Friberg (2007) A Remarkable Collection, App. 6.6.
 10.
W 19408, 76. See Nissen et al. (1993), Fig. 50.
 11.
Published in Bartash (2011).
 12.
See Friberg (2007) A Remarkable Collection, App. 7.
 13.
Westenholz (1975).
 14.
Recently discovered by my coworker professor Farouk AlRawi and published in Friberg and AlRawi (2016), Ch. 13.
 15.
See Friberg (2007) A Remarkable Collection, Apps. A.1.3 and A 7.2.
 16.
CUNES 4712176. See Friberg (2007) op. cit, App. A.7.4.
 17.
Op. cit., p. 427.
 18.
Compare with the equation 1 ^{2}/_{3} 5 gín · ^{1}/_{2} ma.na 4 gín = 1 ma.na – ^{1}/_{2} gín in CUNES 5218035, Sect. 7 above, where 1 ^{2}/_{3} 5 gín was explained as ^{1}/_{2} · ^{1}/_{2} · 7 and ^{1}/_{2} ma.na 4 gín as 2 · 2 · 8 ^{1}/_{2} gín.
 19.
This clay tablet was just like SM 2685 found by F. N. AlRawi in the Sulaymaniyah Museum.
 20.
See, for instance, Friberg (2007) A Remarkable Collection, Sec. 2.6 f.
 21.
See Friberg and AlRawi (2016), Sec. 13.8.
 22.
See, for instance, Friberg (2007) A Remarkable Collection, Ch. 3 and App. 5.
 23.
See, for instance, Proust (2007), Chs. 5 and 8.
 24.
 25.
Borrowed from Friberg (2007) A Remarkable Collection, App. 3.
 26.
Friberg and AlRawi (2016), Sec. 2.1.7.
 27.
See Friberg and AlRawi (2016), Sec. 1.5.
 28.
 29.
See, in particular, the Old Akkadian lexical text CUNES 4712176, a decreasing list of weight measures, in Friberg (2007) A Remarkable Collection, App. A.7.4.
 30.
Friberg (2007) A Remarkable Collection, App. A.1.3.
Notes
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
References
 Bartash, V. 2011. Miscellaneous Early Dynastic and Sargonic Texts in the Cornell University Collections. Google Scholar
 Englund, R.K. 1996. ProtoCuneiform Texts from Diverse Collections.Google Scholar
 Friberg, J. 1997/98. Round and almost round numbers in protoliterate metromathematical field texts. Archiv für Orientforschung 44/45: 1–58.Google Scholar
 Friberg, J. 1999. Counting and accounting in the protoliterate Middle East. Journal of Cuneiform Studies 51: 107–137.CrossRefGoogle Scholar
 Friberg, J. 2007. A Remarkable Collection of Babylonian Mathematical Texts.Google Scholar
 Friberg, J., and A. George. 2010. Six more mathematical cuneiform texts in the Schøyen Collection. In: Papyri Graecae Schøyen, eds. Minutoli, D., and R. Pintaudi, (PSchøyen II), 155.Google Scholar
 Friberg, J. 2014. Tretusen år med sexagesimala tal. Aigis Suppl. III(FS Taisbak): 1–23.Google Scholar
 Friberg, J., and F.N. AlRawi. 2016. New Mathematical Cuneiform Texts.Google Scholar
 Monaco, W.F. 2012. Loan and interest in the archaic texts. Zeitschrift für Assyriologie 102: 165–178.Google Scholar
 Neugebauer, O., and A. Sachs. 1945. Mathematical Cuneiform Texts.Google Scholar
 Neugebauer, O. 1955. Astronomical Cuneiform Texts, I–III. Google Scholar
 Nissen, H.J., P. Damerow, and R.K. Englund. 1993. Archaic Bookkeeping.Google Scholar
 Ossendrijver, M. 2012. Babylonian Mathematical Astronomy: Procedure Texts. Google Scholar
 Ossendrijver, M. 2019. Scholarly Mathematics in the Rēš Temple. In: Scholars and Scholarship in Late Babylonian Uruk. Why the Sciences of the Ancient World Matter eds. Proust, C., and J. Steele 2:187–217.Google Scholar
 Proust, C. 2007. Tablettes mathématiques de Nippur.Google Scholar
 SchmandtBesserat, D. 1992. Before Writing, vol. 1: From Counting to Cuneiform. Google Scholar
 Westenholz, A. 1975. Early Cuneiform Texts in Jena. Google Scholar
Copyright information
OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.