Babylonian mathematics
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Background
‘In any event, it is important to remember that, however faithfully I have attempted to render the original sources into English, what you will be reading
are my translations. and thus interpretations which are open to doubt and challenge. They are not the primary sources themselves, but an early twenty-first
century representation of them which is far removed from the originals. You wlll
be reading alphabetic texts from a printed book in a familiar language, perhaps
seated at a desk in a library or an office, a physical experience far removed from
squatting on the ground in bright sunlight to pore over a clay tablet held in
your hand. It will never be possible to fully comprehend this mathematics as it
was mean to be read, for we cannot entirely escape our own twenty-first century
lives. . . But even if the enterprise is ultimately doomed to failure, that does not
mean it is not rewarding and satisfying to try.’ (Robson (2008), p. 68)
Eleanor Robson’s remarks can stand at the beginning both of this lecture and
of the course as a warning about what the limitations of the history of mathematics, as well as a specific comment on the ‘Babylonian problem’. And it’s a
good exercise to think about what she says, and then to reflect about what you
think of ‘the history of mathematics’. What can it tell us?
Until about a hundred years ago, the ‘history of mathematics’ was thought to
begin with the classical Greeks, i.e. about 500 B.C.E. However, archaeological
discoveries (unearthing texts) and work on translating and interpreting them
have shown us that there was a great deal of mathematics done much earlier
— in Egypt (the Nile valley) and Iraq (the valleys of the rivers Tigris and
Euphrates). In both cases the ‘start date’ for the development of a number
system and an accompanying arithmetic are around 3000 B.C.E. — that is, 2500
years before the earliest Greek mathematics. While you could do well to research
the Egyptians, this course will study ancient Iraqi (also called Mesopotamian,
or Babylonian) mathematics, mainly because:
(i) There is much more material, so more to study;
(ii) The mathematicians of Iraq reached a higher level of ‘difficulty’ in their
methods, and in the problems they could solve;
(iii) The system of notation for writing numbers was strikingly sophisticated
(see later);
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(iv) Accordingly, more historians have written about Iraq than abut Egypt, and
in a more interesting way.
For an idea of where the ‘history’ took place, and so where the sources come
from, see the map in Fig. 1.
However, if you want to study Babylonian mathematics you face serious difficulties. There are thousands of ‘mathematical documents’ which come from
ancient Iraq, and there’s a great deal of evidence about the people who wrote
them (again archaeological). But the documents are in a language, and a script,
which are long dead and hard to read. Understanding has come slowly, but the
main ideas for how we read these texts were in place by the 1930s; and while
there have been important improvements in how we read, our ideas on the
number system (for example) have not changed. Today quite a large number
of scholars are now at work on understanding what they say, but they don’t
always agree. You can find pictures of the original documents at the end, to
which I’ll refer.
Such a ‘document’ is a tablet of baked clay, square or rectangular, with characters on it; these were ‘written’ on it, or pressed with a stylus and the baking
preserved the writing. A typical tablet will contain words and number-signs.
The one given in Figs. 2 and 3 is a good example; it calculates the area of a
square. Its side is much too small to be a field, and — this is typical — it looks
as if it’s set as a school exercise, in multiplying and in converting a complicated
system of units.
My plan in this handout is to begin with the mathematics and how we understand it — what was done — and then to go on to why it was done. This isn’t
the usual order, but it seems to make sense here.
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The sexagesimal number system
The Internet will tell you, without much trouble, that ‘the Babylonians’ wrote
their numbers using base 60, whereas we use base 10. This system — which
arrived in its final form around 2000 B.C.E., after the said Babylonians had been
doing mathematics for over 1000 years — was an impressive one. There were
signs for ‘one’ and for ‘ten’, which combined to form the signs for the numbers
1 through 59. (See fig. 4.) To write larger numbers you used a ‘place value’
system as shown in fig. 5. The two signs at the top left of fig. 2 therefore both
mean ‘1 45’. The fact that this is a place value system means that they are to
be interpreted as ‘1 × 60 + 45 = 105’. Scholars, in transcribing documents, will
write them as ‘1, 45’ with the comma to help you read them. And here are some
sums for you to check, to see that you can handle this way of writing numbers:
14 × 14 = 3, 16
1, 40 × 1, 40 = 2, 46, 40
(note that 1,40 means ‘100’ in decimal).
Among the simplest tablets are those which tell you the area of a square; if the
side is 20, the area is 6,40 (OK?). If the side is 16,40, the area is 4,37,46,40.
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However — and this is important — the symbol for ‘1’ could mean not only 1, or
60, or 602 = 3600 (depending on its place); but it could also be a fraction, 60−1
or 60−2 ,. . . This is what in computer science is called ‘floating’ point’ arithmetic
— very sophisticated, and easy to get wrong, one would think. The best example
is the very common symbol ‘30’, which may mean 30, but may equally well mean
30
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1
60 = 2 . Similarly, ‘20’ can mean 3 and so on.
Why was this not completely confusing? We have to understand the place of
mathematics in society, which was basically practical. If your question was
how many bricks you needed to build a wall, in the first place you would know
whether the answer was more likely to be 60 or 3600. You would use the
sexagesimal numbers to arrive at an answer like ‘1 45’; and you’d use — perhaps
— other systems (see section 4) to interpret what your answer meant.
I suggest that you now consider, as an exercise the following examples from
the table of reciprocals; lines 5, 6 and 7 read:
[Of 60] the 6th part is 10
the 7,12th part is 8,20
the 7,30th part is 8
See if you can (i) read the numbers in the picture and (ii) follow the calculations.
The above brief sketch will explain basically how the sexagesimal system worked.
It was understood quite early by researchers, and is at the centre of the classic
work [Neugebauer and Sachs 1945]. These researchers were mathematicians,
and they liked the sexagesimal system because it was mathematically advanced
and simple. Accordingly, as later workers have stressed, they tended to leave
out work which didn’t ‘fit in’ — of which more later.
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Examples.
Here are two examples of ‘problems’ from Babylonian tablets, which I’ve chosen
because they are just problems in sexagesimal ‘algebra’. These were important
in the main period which gets studied, the so-called ‘Old Babylonian’, say 20001600 BCE.
1. Tablet BM 34568, problem 9.
Length and breadth added is 14 and 48 is the area. The sizes are not known. 14
times 14 is 3,16. 48 times 4 is 3,12. You subtract 3,12 from 3,16 and it leaves 4.
What times what should I take in order to get 4? 2 times 2 is 4. You subtract
2 from 14 and it leaves 12. 12 times 30 is 6. 6 is the width. To 2 you shall add
6, it is 8. 8 is the length.
A modern ‘translation’ into algebra would set length =l, width =w; so
l + w = 14, lw = 48
From this, using a standard method, you’d get (in our version a quadratic
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equation w2 − 14w + 48 = 0) — and so
w=
�
�
1�
(l + w) − (l + w)2 − 4lw
2
This is why you square 14, and subtract four times the area from the square.
It’s the right answer, but there’s no explanation.
Exercise. Can you think of how you might explain the method?
A few notes on my text (for those who care). This is copied from Robson
(2008) p. 279, which is interesting. She gives a ‘classical’ translation (Neugebauer) and a ‘modern’ one (Høyrup) to compare how ways of translating have
changed and why; I’ve chosen Neugebauer’s one, which is easier to understand.
But I have written ‘30’ instead of ‘0; 30’, which both translations do, and which
I think is quite wrong as there’s no ‘0’ on the tablet.
2. Tablet AO6484.
A number and its reciprocal add to 2; 03. Multiply by 0;30 so that 1;01,30. Multiply 1;01, 30 by 1;01,30 so that 1;03,02,15. Diminish by 1: remains 0;03,02,15.
What times what should I take in order to get 0;03,02,15? 0;13,30 times 0;13,30
is 0;03,02,15. Increase 0;13,30 by 1;1,30. 1;15 the reciprocal. (Source is Robson
in Katz (), p. 175.)
Here the text has even more problems. As you can see, the editor has had to
add 0’s in several places to make sense of it, as well as semi-colons — none of
which were in the original. This — if you think about it — is a problem with
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— has a
quadratic equations. The problem which is being set — x + x1 = 2 20
nice answer, but the same is not true if you read ‘2 03’ as ‘63’. The original
students (and teachers) must have had to read in an intelligent way.
That said, the ‘algebra’ underlying the solution is again simple. From x + x1 =
2; 03 we deduce x2 − 2; 03x + 1 = 0, so
(x − 1; 01, 30)2 = 1 : 01, 302 − 1 = 0; 03, 02, 15
And you can check that the solution (x = 54 , x−1 = 45 ) is right. (Exercise. Try
to solve the equation using modern methods.)
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Scribes and Bureaucracy.
‘You wrote a tablet, but you cannot grasp its meaning. You wrote a letter, but
that is the limit for you! Go to divide a plot, and you are not able to divide
the plot; go to apportion a field, and you cannot even hold the tape and rod
properly; the field pegs you are unable to place; you cannot figure out its shape,
so that when wronged men have a quarrel you are not able to bring peace but
you allow brother to attack brother. Among the scribes you (alone) are unfit for
the clay. What are you fit for? Can anybody tell us?’ (‘The Dialogue between
Girini-isag and Enki-manshum, in Robson 2008, p. 349)
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The mathematics of the ‘Old Babylonian’ period described above is so striking,
that one might think that it was done for its own sake, and that Babylonians
thought it was a good thing in itself to solve quadratic equations. Far from it!
They were used, like other mathematical methods, by the so-called ‘scribes’, an
essential part of the Babylonian state system from a quite early date. In our
terms, scribes were the state bureaucracy; as the above quote shows, they made
tax calculations, wrote letters, surveyed land, settled lawsuits and much else —
including religious ceremonial. The mathematics which they used was not pure,
indeed it was not very like the above examples; it was more like the text of the
tablet shown in Figs. 2 and 3. This is explained by Robson in ((2008), p.15) —
it’s a practical procedure for measuring a square area, and uses different systems
of numeration. The top left hand corner contains the sexagesimal number ‘1,45’
(in decimal 105) written twice. The tablet reads:
0;00 01 45
0;00 01 45
A square is 13 cubit, 12 finger on each side.
What is its area?
Its area
is 9 grains and a 5th of a grain.
The first thing to notice is that there are three units of measurement here. Two
(cubits and fingers) refer to length, the third (grains) refers to area. Part of
your training as a scribe was to learn how these related — and the tables were
complicated, and not sexagesimal. To be precise: 12 cubits = 1 rod; 30 fingers
= 1 cubit. And
60 shekels = 1 square rod; 180 grains = 1 shekel.
At this point you may be starting to glaze over, and I don’t blame you. A Babylonian scribe didn’t have that option. He’d translate the units into sexagesimal
3600
by writing a finger as 12×30
, that is 10 3600ths of a rod. (Or some equivalent.)
So a cubit, in these fractional units, is 300; and a third of a cubit is 100 or 1,40.
While half a finger is half ten, i.e. 5. The length of the side of the square is
1,45. (Robson has inserted all those 0’s to emphasize that you’re dealing with
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units of 60×60
) This, finally, explains the number ‘1 45’.
We now have to find the square area — easy, it’s 1, 45 × 1, 45 = 3, 03, 45 square
rods. That is, with several 0’s and the ‘sexagesimal point’ or semicolon inserted
in the right place.
I’ll spare you the remaining details of how the result comes out as 9 15 grains —
it’s in Robson if you want to look. But I hope I’ve made the main point, that
ancient Babylonian scribes were required to do formidably complex calculations,
involving juggling different systems of measurement units, as a day-to-day exercise. (By the way, a rod is about 6 m., so a cubit is 0.5 m and a finger, about
1.6 cm.)
What was the point of all this learning? To put it simply: to exploit, as efficiently as possibly, the mass of labouring peasants for the benefit of the rich,
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the élite rulers of the temple. Høyrup (1991, p. 11-13) describes the ‘formation
of the state’ as resulting — already about 3000 BCE — in city-states ‘dominated by a number of large temples, which can only have been built because
of the existence (and availability to the theocratic rulers) of a large agricultural surplus’; where ‘to keep track of tribute and other deliveries and of the
products of public agriculture and herding, and also in order to calculate the
rations of officials, workmen and domestic animals, techniques for accounting
and computation were developed’. To be crude, and brief: taxation plus redistribution and rationing plus a caste system leads you to mathematics and a
caste of mathematicians, the ‘scribes’, who are the people who can do it — and
all this happened before there was a properly formed written language!
I’ve appended Høyrup’s paper, on the early developments which is useful if too
long, to the reading matter on KEATS. But more entertaining is the now classic
text of Nissen, Damerow and Englund, Archaic Bookkeeping (see bibliography),
which concentrates on the role of scribes in extracting taxes, particularly in the
early period. As all sources agree, the peak of the process was reached under
the dreadful ‘Third Dynasty of Ur’ (2100-2000 BCE, there are disputes about
the exact dates), where ‘royal estates, governmental trade and governmental
workshops and even textile factories worked by slaves were all-important. The
precise booking of rations, work-days, and of flight, illness and death within
the work-force allotted to each overseer also reveals an extremely harsh regime.’
(Høyrup p. 26.) (And, as under the Stalinist plans of the 1930s, peasants
were required to produce a certain amount and penalized for failing.) The
classic text describing this system is Englund’s ‘Hard Work: Where will it Get
You?’, which I’ve also posted on KEATS; it has some good illustrations, and
a detailed description of how the system worked. I also recommend Englund
(2009) or Lafont (2013) for a description of how early mathematics worked in
enumerating and describing slaves.
The Third Dynasty collapsed — perhaps it tried too hard. And it seems that
the use of mathematics after that, in the ‘Old Babylonian’ period, from which
most of the school texts come, was not so brutal. Still, taxes had to be gathered
and rations assigned, and mathematicians, or scribes, were trained to do the
work. There were periods when the subject seems to have been neglected, but
some version was still in place nearly two thousand years later. After which,
the cuneiform script in which it was written disappeared, we don’t know why.
However, by that time, as we shall see, many of the mathematical methods had
become part of ‘culture’.
Question. What aspects of our society’s use of mathematics, if any, are similar
too the uses to be found in ancient Babylon — and how?
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Bibliography
Englund, R., (1991) ‘Hard Work: Where will it Get You?’ , posted on KEATS
—————- (2009),‘The Smell of the Cage’, online at http://cdli.ucla.edu/pubs/
cdlj/2009/cdlj2009004.html
Høyrup, J. (1991) ‘Mathematics and early state formation’, posted on KEATS
Katz, V, (2007), The Mathematics of Egypt, Mesopotamia, China, India, and
Islam: A Sourcebook, Princeton, NJ: Princeton University Press.
Lafont, B., (2013), ‘State employment of women during the Ur III period’ Second
and third REFEMA workshops, Tokyo (June 2013) and Carqueiranne (September 2013); online at http://refema.hypotheses.org/976
Neugebauer, O. and Sachs, A. (1945) Mathematical Cuneiform Texts, New
Haven CN: American Oriental Society.
Nissen H., Damerow, P. and Englund, R., (1993), Archaic Bookkeeping: Early
Writing and Techniques of Economic Administration in the Ancient Near East
Chicago IL: University of Chicago Press.
Robson, E. (2008), Mathematics in Ancient Iraq, Princeton NJ: Princeton University Press.
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