### Cuneiform
Cuneiform, originating around 3300 B.C., marked th...
In a sexagesimal system you use 60 as the base, unlike our modern b...
In other words, if you saw the two digit sexagesimal number 2,20 in...
Again remember that they inferred the exponent depending on the con...
In this context, a table of reciprocals is a lookup table that prov...
Here is an example of an actual Cuneiform tablet with a reciprocals...
This formula can be derived from the context of a quadratic equatio...
A cubit and a gar are units of measurement used in ancient Babyloni...
Linear interpolation is a method used to estimate values between tw...
The Plimpton tablet is a Babylonian clay tablet that dates back to ...
Ancient
Babylonian
Algorithms
Donald E. Knuth
Stanford University
The early origins of mathematics are discussed,
emphasizing those aspects which seem to be of greatest
interest from the standpoint of computer science. A
number of old Babylonian tablets, many of which have
never before been translated into English, are quoted.
Key Words and Phrases: history of computation,
Babylonian tablets, sexagesimal number system, sorting
CR Categories: 1.2
One of the ways to help make computer science re-
spectable is to show that it is deeply rooted in history,
not just a short-lived phenomenon. Therefore it is natu-
ral to turn to the earliest surviving documents which
deal with computation, and to study how people ap-
proached the subject nearly 4000 years ago. Archeo-
logical expeditions in the Middle East have unearthed a
large number of clay tablets which contain mathematical
calculations, and we shall see that these tablets give
many interesting clues about the life of early "computer
scientists."
Copyright @ 1972, Association for Computing Machinery, Inc.
General permission to republish, but not for profit, all or part
of this material is granted, provided that reference is made to this
publication, to its date of issue, and to the fact that reprinting
privileges were granted by permission of the Association for Com-
puting Machinery.
Author's address: Stanford University, Computer Science De-
partment, Stanford, CA 94305. The preparation of this paper was
supported in part by the National Science Foundation, under grant
GJ-992.
671
Introduction to Babylonian Mathematics
The tablets in question come from the general area of
Mesopotamia (present day Iraq), between the Tigris and
Euphrates rivers, centered more or less about the ancient
city of Babylon (near present-day Baghdad). They are
covered with cuneiform (i.e. "wedge-shaped") script, a
form of writing which goes back to about 3000 B.C. The
tablets of greatest mathematical interest were written
about the time of the Hammurabi dynasty, about 1800-
1600 B.c., and we shall be primarily concerned with
texts that date from this so-called Old-Babylonian pe-
riod.
It is well known that Babylonians worked in a
sexagesirnal,
i.e. radix 60, number system, and that our
present sexagesimal units of hours, minutes, and seconds
are vestiges of their system. But it is less widely known
that the Babylonians actually worked
withfloating-point
sexagesimal numbers, using a rather peculiar notation
that did not include any exponent part. Thus, the two-
digit number
2,20
stood for 2 × 60 + 20 = 140, and for 2 + 20/60 = 2~,
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and for 2/60 + 20/3600, and in general for 140 X 60 ",
where n is any integer.
At first sight this manner of representing numbers
may look very awkward, but in fact it has significant
advantages when multiplication and division are in-
volved. We use the same principle when we do calcula-
tions by slide rule, performing the multiplications and
divisions without regard to the decimal point location
and then supplying the appropriate power of 10 later.
A Babylonian mathematician computing with numbers
that were meaningful to him could easily keep the ap-
propriate power of 60 in mind, since it is not difficult to
estimate the range of a value within a factor of 60. A
few instances have been found where addition was per-
formed incorrectly because the radix points were im-
properly aligned [7, p. 28], but such examples are sur-
prisingly rare.
As an indication of the utility of this floating-point
notation, consider the following table of
reciprocals:
2 30 16 3,45 45 1,20
3 20 18 3,20 48 1,15
4 15 20 3 50 1,12
5 12 24 2,30 54 1,6,40
6 10 25 2,24 1 1
8 7,30 27 2,13,20 1,4 56,15
9 6,40 30 2 1,12 50
10 6 32 1,52,30 1,15 48
12 5 36 1,40 1,20 45
15 4 40 1,30 1,21 44,26,40
Dozens of tablets containing this information have been
found, some of which go back as far as the
"Ur
III
dynasty" of about 2250 B.c. There are also many mul-
tiplication tables which list the multiples of these num-
bers; for example, division by 81 = 1,21 is equivalent to
multiplying by 44,26,40, and tables of 44,26,40 × k for
1 < k < 20 and k = 30,40,50 were commonplace. Over
two hundred examples of multiplication tables have
been catalogued.
Babylonian "Programming"
The Babylonian mathematicians were not limited
simply to the processes of addition, subtraction, mul-
tiplication, and division; they were adept at solving
many types of algebraic equations. But they did not
have an algebraic notation that is quite as transparent as
ours; they represented each formula by a step-by-step
list of rules for its evaluation, i.e. by an algorithm for
computing that formula. In effect, they worked with a
"machine language" representation of formulas instead
of a symbolic language.
The flavor of Babylonian mathematics can best be
appreciated by studying several examples. The transla-
tions below attempt to render the words of the original
texts as faithfully as possible into good English, without
extensive editorial interpretation. Several remarks have
been added in parentheses, to explain some of the things
that were originally unstated on the tablets. All numbers
are presented Babylonian-style, i.e. without exponents,
so the reader is warned that he will have to supply an
appropriate scale factor in his head; thus, it is necessary
to remember that 1 might mean 60 and 15 might mean ¼.
The first example we shall discuss is excerpted from
an Old-Babylonian tablet which was originally about
5 X 8 × 1 inches in size. Half of it now appears in the
British Museum, about one-fourth appears in the
Staatliche Museen, Berlin, and the other fourth has ap-
parently been lost or destroyed over the years. The
original text appears in [3, pp. 193-199; 4, Tables 7, 8,
39, 40; and 8, pp. 11-21].
A (rectangular) cistern.
The height is 3,20, and a volume of 27,46,40 has been
excavated.
The length exceeds the width by 50. (The object is to find the
length and the width.)
You should take the reciprocal of the height, 3,20, obtaining 18.
Multiply this by the volume, 27,46,40, obtaining 8,20. (This
is the length times the width; the problem has been reduced
to finding x and y, given that x -- y = 50 and
xy
= 8,20.
A standard procedure for solving such equations, which
occurs repeatedly in Babylonian manuscripts, is now used.)
Take half of 50 and square it, obtaining 10, 25.
Add 8,20, and you get 8,30, 25. (Remember that the radix point
position always needs to be supplied. In this case, 50 stands
for 5/6 and 8,20 stands for 8], taking into account the
sizes of typical cisterns!)
The square root is 2,55.
Make two copies of this, adding (25) to the one and subtracting
from the other.
You find that 3,20 (namely 3-~) is the length and 2,30 (namely
2½) is the width.
This is the procedure.
The first step here is to divide 27,46,40 by 3,20; this is
reduced to multiplication by the reciprocal. The mul-
tiplication was done by referring to tables, probably by
manipulating stones or sand in some manner and then
writing down the answer. The square root was also
computed by referring to tables, since we know that
many tables of n vs. n ~ existed. Note that the rule for
computing the values of x and y such that x -- y = d
and
xy = p
was to form
sqrt((d/2) ~ + p) 4-
(d/2).
The calculations described in Babylonian tablets are
not merely the solutions to specific individual problems:
they actually are general procedures for solving a whole
class of problems. The numbers shown are merely in-
cluded as an aid to exposition, in order to clarify the
general method. This tact is clear because there are
numerous instances where a particular case of the gen-
eral method reduces to multiplying by 1 ; such a multi-
plication is explicitly carried out, in order to abide by
the general rules. Note also the stereotyped ending,
"This is the procedure," which is commonly found at
the end of each section on a tablet. Thus the Babylonian
procedures are genuine algorithms, and we can com-
mend the Babylonians for developing a nice way to ex-
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plain an algorithm by example as the algorithm itself
was being defined.
Here is another excerpt from the same tablet, this
time involving only a linear equation:
A cistern.
The length (in cubits) equals the height (in gars, where 1 gar =
12 cubits).
A certain volume of dirt has been excavated.
The cross-sectional area (in square cubits) plus this volume (in
cubic cubits) comes to 1,10 (namely 1-~).
The length is 30 (namely ½ cubit). What is the width?
You should multiply the length, 30, by 12, obtaining 6; this is the
height (in cubits instead of gars).
Add 1 to 6, and you get 7.
The reciprocal of 7 does not exist; what will give 1,10 when
multiplied by 7? 10 will.
(Hence 10, namely ~, is the cross-sectional area in square cubits.)
Take the reciprocal of 30, obtaining 2.
Multiply 10 by 2, obtaining the width, 20 (namely a x cubit).
This is the procedure.
Note the interesting way in which the Babylonians dis-
regarded units, blithely adding area to volume; similar
examples abound, showing that the
numerical
algebra
was of primary importance to them, not the physical or
geometrical significance of the problems. At the same
time they used conventional units of measure (cubits,
even "gars" and the understood relation between gars
and cubits), in order to set the scale factors for the
parameters. And they "applied" their results to practical
things like cisterns, perhaps because this would make
their work appear to be socially relevant.
In this problem it was necessary to divide by 7, but
the reciprocal of 7 didn't appear on the tables because
it has no finite reciprocal. (There is an infinite repeating
expansion 1/7 - 8,34,17,8,34,17,..., but we have no
evidence that the Babylonians knew this.) In such cases
where the reciprocal table was of no avail, the text
always says, in effect, "What shall I multiply by a in
order to obtain b?" and then the answer is given. This
wording indicates that a multiplication table is to
be used backwards; for example, the calculation of
11,40 - 35 = 20 [3, p. 329] could be read off from a
multiplication table. For more difficult divisions, e.g.
1,26,40 - 43,20 = 15 [3, p. 246; 5, p. 8], a slightly
different wording was used, indicating perhaps that a
special division procedure was employed in such cases.
At any rate we know that the Babylonians were able to
compute
7 + 2,6; 28,20-- 17; 10,12,45 + 40,51;
and so on. One Old-Babylonian table of reciprocals is
known that gives reciprocals of irregular numbers to
three sexagesimal places, but it is not extremely accurate
[3, p. 16].
Further Examples
We have noted that general algorithms were usually
given, accompanied by a sample calculation. In rare
instances such as the following text (again from the
British Museum), the style is somewhat different [5, p.
19]:
The sum of length, width, and diagonal is 1,1 and 7 is the area.
What are the corresponding length, width, and diagonal?
The quantities are unknown.
1,10 times 1,10is 1,21,40.
7 times 2 is 14.
Take 14 from 1,21,40 and 1,7,40 remains.
1,7,40 times 30 is 33,50.
By what should 1,10 be multiplied to obtain 33,50?
1,10 times 29 is 33,50.
29 is the diagonal.
The sum of length, width, and diagonal is 12 and 12 is the area.
What are the corresponding length, width, and diagonal?
The quantities are unknown.
12 times 12 is 2,24.
12 times 2 is 24.
Take 24 from 2,24 and 2 remains.
2 times 30 is 1.
By what should 12 by multiplied to obtain 17
12 times 5 is 1.
5 is the diagonal.
The sum of length, width, and diagonal is 1 and 5 is the area.
Multiply length, width, and diagonal times length, width, and
diagonal.
Multiply the area by 2.
Subtract the products and multiply what is left by one-half.
By what should the sum of length, width, and diagonal be
multiplied to obtain this product?
The diagonal is the factor.
This text comes from the considerably later "Seleucid"
period of Babylonian history (see below), which may
account for the difference in style. It treats a problem
based on the rather remarkable formula
d -- ½((1 q- w q- d) ~ - 2A)/(l q- w W d),
where
A = lw
is the area of a rectangle,
d -- x/(l 2 -b w 2) is the length of its diagonals.
(There is ample evidence from other texts that the Old-
Babylonian mathematicians knew the so-called Pythago-
rean theorem, over I000 years before the time of
Pythagoras.) The first two sections quoted above work
out the problem for the cases (1, w, d) = (20, 21, 29) and
(3, 4, 5) respectively, but without calculating l and w;
we know from other texts that the solution tox -b y = a,
x ~ q_ y2 = b was well known in ancient times. The de-
scription of the calculation in these two sections is un-
usually terse, not naming the quantities it is dealing
with. On the other hand, the third section gives the
same
procedure entirely
without
numbers. The reason for this
may be the fact that the stated parameters 1 and 5 can-
not possibly correspond to the length-F width-Fdiagonal
and the area, respectively, of any rectangle, no matter
what powers of 60 are attached! Viewed in this light,
teachers of computer science will recognize that the
above text reads very much like the solution to an ex-
amination in which an impossible problem has been
posed. (Note also that the second section follows the
673
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general procedure, as stated in the third section, very
faithfully when it comes to dividing 1 by 12, instead of
using the reciprocal of 12.)
Instances of algorithms without accompanying num-
bers are very rare; here is another one, this time an
Old-Babylonian text from the Louvre [4, p. 39; 8, p. 71]:
Length and width is to be equal to the area.
You should proceed as follows•
Make two copies of one parameter•
Subtract 1.
Form the reciprocal.
Multiply by the parameter you copied.
This gives the width,
In other words, if x + y =
xy,
it is possible to compute
y by the procedure y = (x - 1) -1 x. The fact that no
numbers are given made this passage particularly hard
to decipher, and it was not properly understood for
many years (see [9, pp. 73-74]); hence we can see the
advantages of numerical examples.
The above procedure reads surprisingly like a pro-
gram for a "stack machine" like the Burroughs B5500l
Note that both in this example and in the very first one
we discussed we are told to make two copies of some
number; this indicates that actual numerical calcula-
tions generally destroyed the operands in the process of
finding a result. Similarly we find in other texts the in-
struction to "Keep this number in your head" [6, pp.
50-51], a remarkable parallelism with today's notion
that a computer stores numbers in its "memory." In
another place we read, in essence, "Replace the sum of
length and width by 30 times itself" [3, p. 114], an
ancient version of the assignment statement "x :=
x/2".
Conditionals and Iterations
So far we have seen only "straight-line" calculations,
without any branching or decision-making involved. In
order to construct algorithms that are really nontrivial
from a computer scientist's point of view, we need to
have some operations that affect the flow of control.
But alas, there is very little evidence of this in the
Babylonian texts. The only thing resembling a condi-
tional branch is implicit in the operation of division,
where the calculation proceeds a little differently if the
reciprocal of the divisor does not appear in the table.
We don't find tests like "Go to step 4 if x < 0",
because the Babylonians didn't have negative numbers;
we don't even find conditional tests like "Go to step 5
if x = 0", because they didn't treat zero as a number
either! Instead of ha,~ing such tests, there would effec-
tively be separate algorithms for the different cases. (For
example, see [3, pp. 312-314] for a case in which one
algorithm is step-by-step the same as another, but sim-
plified since one of the parameters is zero.)
Nor are there many instances of iteration. The basic
operations underlying the multiplication of high-preci-
sion sexagesimal numbers obviously involve iteration,
and these operations were clearly understood by the
Babylonian mathematicians; but the rules were ap-
parently never written down. No examples showing in-
termediate steps in multiplication have been found.
The following interesting example dealing with com-
pound interest, taken from the Berlin Museum collec-
tion, is one of the few examples of a "DO I = 1 TO N" in
the Babylonian tablets that have been excavated so far
[3, pp. 353-365; 4, Tables 32, 56, 57; 5, p. 59; 8, pp.
118-120]:
I invested 1 maneh of silver, at a rate of 12 shekels per maneh (per
year, with interest apparently compounded every five years).
I received, as capital plus interest, 1 talent and 4 manehs.
(Here 1 maneh = 60 shekels, and 1 talent = 60 manehs.)
How many years did this take?
Let 1 be the initial capital.
Let 1 maneh earn 12 (shekels) interest in a 6 (= 360) day year.
And let 1,4 be the capital plus interest.
Compute 12, the interest, per 1 unit of initial capital, giving 12
as the interest rate.
Multiply 12 by 5 years, giving 1.
Thus in five years the interest will equal the initial capital.
Add 1, the five-year interest, to 1, the initial capital, obtaining 2.
Form the reciprocal of 2, obtaining 30.
Multiply 30 by 1,4, the sum of capital plus interest, obtaining 32.
Find the inverse of 2, obtaining 1. (The" inverse" here means the
logarithm to base 2; in other problems it stands for the value
of n such that a given valuef(n) appears in some table.)
Form the reciprocal of 2, obtaining 30.
Multiply 30 by 30 (the latter 30 apparently stands for 32 -- 2, for
otherwise the 32 would never be used and the rest of the
calculation would make no sense), obtaining 15 ( = total
interest without initial capital if the investment had been
cashed in five years earlier).
Add 1 to 15, obtaining 16.
Find the inverse of 16, obtaining 4.
Add the two inverses 4 and 1, obtaining 5.
Multiply 5 by 5 years, obtaining 25.
Add another 5 years, making 30.
Thus, after the 30th year the initial capital and its interest will
be 1,4.
... (Here about 4 lines of the text have broken off. Apparently
there is now a question of checking the previous answer.)
•..
giving 12 as the interest rate.
Multiply 12 by 5 years, giving 1.
Thus in five years the interest will equal the initial capital•
Add 1, the five-year interest, to 1, the initial capital, obtaining 2,
the capital and its interest after the fifth year.
Add 5 years to the 5 years, obtaining 10 years.
Double 2, the capital and its interest, obtaining 4, the capital
and its interest after the tenth year.
Add 5 years to the 10 years, obtaining 15 years.
Double 4, the capital and its interest, obtaining 8, the capital
and its interest after the fifteenth year.
Add 5 years to the 15 years, obtaining 20 years.
Double 8, obtaining 16, the capital and its interest after the
twentieth year.
Add 5 years to the 20 years, obtaining 25 years.
Double 16, the capital and its interest, obtaining 32, the capital
and its interest after the twenty-fifth year.
Add 5 years to the 25 years, obtaining 30 years.
Double 32, the capital and its interest, obtaining I, 4, the capital
and its interest after the thirtieth year.
This long-winded and rather clumsy procedure reads
almost like a macro expansion !
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A more sophisticated example involving compound
interest appears in another section of the Louvre tablet
quoted earlier. The same usurious rate of interest (20
percent per annum) occurs, but now compounded an-
nually:
One kur (of grain) has been invested; after how many years will
the interest be equal to the initial capital?
You should proceed as follows.
Compound the interest for four years.
The combined total (capital + interest) exceeds 2 kur.
What can the excess of this total over the capital plus interest
for three years be multiplied
by
in order to give the four-year
total minus 2?
2,33,20 (months).
From four years, subtract 2,33,20 (months), to obtain the desired
number of full years and days.
Translated into decimal notation, the problem is to de-
termine how long it would take to double an investment.
Since
1.728 = 1.23 < 2 < 1.24 = 2.0736,
the answer lies somewhere between three and four years.
The growth is linear in any one year, so the answer is
1.24 -- 2 33 20
1.24 _ 1.23 X 12 = 2 q- ~ q- 36---~
months less than four years. This is exactly what was
computed [5, p. 63].
Note that here we have a problem with a nontrivial
iteration, like a "WHILE" clause: The procedure is to
form powers of I q- r, where r is the interest rate, until
finding the first value of n such that (1 + r)" >_ 2; then
calculate
12((1 -F r)" -- 2)/((1 q- r)" -- (1 -Jr- r)"-1),
and the answer is that the original investment will
double in n years minus this many months.
This procedure suggests that the Babylonians were
familiar with the idea of linear interpolation. Therefore
the trigonometric tables in the famous "Plimpton tab-
let" [6, p. 38-41] were possibly used to obtain sines and
cosines in a similar way.
ample, a symbol for zero was now used within numbers,
instead of the blank space that formerly appeared. The
following excerpts from a text in the Louvre Museum [3,
pp. 96-103; 8, p. 76] indicate some of the other ad-
vances:
From 1 to
10, sum
the powers (literally the
"ladder")
of 2.
The last term you add is 8,32.
Subtract 1 from 8,32, obtaining 8,31.
Add 8,31 to 8,32, obtaining the answer 17,3.
The squares from 1 X 1 = 1 to 10 X 10 = 1,40; what is their
sum?
Multiply 1 by 20, namely by one-third, giving 20.
Multiply 10 by 40, namely by two thirds, giving 6,40.
6,40 plus 20 is 7.
Multiply 7 by 55 (which is the sum of 1 through 10), obtaining
6,25.
6,25 is the desired sum.
Here we have correct formulas for the sum of a geo-
metric series
~-~2 k = 2nq - (2"-- 1)
k=l
and for the sum of a quadratic series
kffil ~ n k.
These formulas have not been found in Old-Babylonian
texts.
Moreover, this same Seleucid tablet shows an in-
creased virtuosity in calculation; for example, the roots
to complicated equations like
xy=
1, xq-y= 2,0,0,33,20
(solution: x = 1,0,45 and y = 59,15,33,20) are com-
puted. Perhaps this problem was designed to demon-
strate the use of the new zero symbol.
An extremely impressive example of the Seleucid era
calculating ability appears in another Louvre Museum
tablet [3, pp. 14-22]. It is a 6-place table of reciprocals,
which begins thus:
By the power of Anu and Antum, whatever I have made with my
hands, let it remain intact.
The Seleueids
Old-Babylonian mathematics has several other in-
teresting aspects, but a more elaborate discussion is be-
yond the scope of this paper. Very few tablets have been
found that were written after 1,600 B.c., until approxi-
mately 300 B.c. when Mesopotamia became part of the
empire of Alexander the Great's successors, the "Seleu-
cids." A great number of tablets from the Seleucid era
have been found, mostly dealing with astronomy which
was highly developed. A very few pure mathematical
texts of this era have also been found; these tablets
indicate that the Old-Babylonian mathematical tradition
did not die out during the intervening centuries.
Indeed, some noticeable progress was made; for ex-
Reciprocal 1
Reciprocal 1,0,16,53,53,20
Reciprocal 1,0,40, 53,20
Reciprocal 1,0,45
and so on, ending with
2,57,8,49,12
Reciprocal 2,57,46,40
2,59,21,40,48,54
is 1
59,43,10,50,52,48
59,19,34,13,7,30
59,15,33,20
20,19,19,34,45,35,48,8,53,20
20,15
20,4,16,22,28,44,14,57,40,4,
56,17,46,40
Reciprocal 3 is 20
First part; results for 1 and 2, incomplete.
Table of Nidintum-Anu, son of Inakibit-Anu, son of Kuzu,
priests of Anu and Antum in Uruk. Author Inakibit-Anu.
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Apparently Inakibit-Anu (whom we shall call Inaki-
bit for short) was the author of this remarkable table;
and his son made a copy. Another tablet or tablets, now
lost, continued the table to numbers beginning with
3, 4, ....
There are exactly 231 sexagesimal numbers of six
digits (i.e. six sexagesimal places) or less which have a
finite reciprocal and which begin with 1 or 2. This table
contains every one of them, without exception. And 20
further entries, giving reciprocals of numbers that have
more than six digits, are also included. It is not clear
how these 20 extra numbers were selected. (See the Ap-
pendix to this paper for further discussion.)
How did Inakibit prepare this table? The simplest
procedure would be to start with the pair of numbers
(1, 1) and then to go repeatedly from (x, y) to (2x, 30y),
(3x, 20y), and (5x, 12y) until no more numbers x of six
or less digits are possible. (In fact this procedure can be
simplified further if we note that only those values x of
the form 2~3~'5 k need to be considered where either i _< 1
orj --- 0 or k = 0; other numbers are multiples of 60.)
There is some evidence that this is exactly what he did;
for example, several tables are known that start with
some pair of reciprocals and then repeatedly apply one
of these three operations [6, p. 13-16]. An even more
convincing argument for this hypothesis is that Inaki-
bit's values for 3 -22 and 3 -23 are both wrong; and most of
the errors in 3 -2a are accounted for if we assume that he
calculated 3 -~8 from the incorrect value of 3 -~2.
The complete table requires that 721 pairs (x, y)
must be generated, and of course it is very laborious to
work with such high-precision numbers. Moreover, even
after all these pairs (x, y) have been computed, the work
is far from done; it is still necessary to sort them into
order ! Inakibit's table is the earliest known example of a
large file that has been sorted; and this is one of the
reasons his work is so impressive, as anyone who has
tried to sort over 700 cards by hand will attest. To get
some idea of the immensity of this task, consider that it
takes many hours to sort 700 large numbers by hand
nowadays; imagine how difficult it must have been to do
this job in ancient times! Yet Inakibit must have done it,
since there is no simple way to generate such a table in
order. (As we might expect, he made a few mistakes;
there are three pairs of lines which should be inter-
changed to bring the table into perfect order.)
Thus, Inakibit seems to have the distinction of being
the first man in history to solve a computational prob-
lem that takes longer than one second of time on a
modern electronic computer !
Suggestions for Further Reading
If you have been captivated by Babylonian mathe-
matics, there are several good books on the subject
which give further interesting details. The short intro-
ductory text Episodes from the Early History of Mathe-
matics by A.A. Aaboe [1] can be recommended, as can
B.L. van der Waerden's well-known treatise Science
Awakening [9]. Much of the deciphering of Babylonian
mathematical texts was originally due to Otto Neuge-
bauer, who has written an authoritative popular ac-
count The Exact Sciences in Antiquity [7]; see especially
his fascinating discussion, pp. 59-63; 103-105, of the
problems that plague historical researchers in this field.
For more detailed study, it is fun to read the original
source material. Neugebauer published the texts of all
known mathematical tablets,, together with German
translations, in a comprehensive series of three volumes,
during the period 1935-1937 [3, 4, 5]. A French edition
of the texts [8] was published in 1938 by F. Thureau-
Dangin, an eminent Assyriologist. Then in 1945, Neuge-
bauer and A. Sachs published a supplementary volume
[6], which includes all mathematical tablets discovered
in the meantime (mostly in American museums). The
Neugebauer-Sachs volume is written in English, but un-
fortunately these tablets are not quite as interesting as
the ones in Neugebauer's original German series. A list
of developments since 1945 appears in [7, p. 49].
Most of the Babylonian mathematical tablets have
never been translated into English. The translations
above have been made by comparing the German of
[3, 4, 5] with the French [8]; but these two versions ac-
tually differ in many details, so the Akkadian and
Sumerian vocabularies published in [4, 8, 6] have been
consulted in an attempt to give an accurate rendition.
Since only a tiny fraction of the total number of clay
tablets has survived the centuries, it is obvious that we
cannot pretend to understand the full extent of Babylo-
nian mathematics. Neugebauer points out that the job
of discovering what they knew is something like trying
to reconstruct all of modern mathematics from a few
pages that have been randomly torn out of the books in
a modern library. We can only place "lower bounds" on
the scope of Babylonian achievements, and speculate
about what they did not know.
What about other ancient developments? The Egyp-
tians were not bad at mathematics, and archeologists
have dug up some old papyri that are almost as old as
the Babylonian tablets we have discussed. The Egyptian
method of multiplication, based essentially in the binary
number system (although their calculations were deci-
mal, using something like Roman numerals)~ is espe-
cially interesting; but in other respects, their use of
awkward "unit fractions" left them far behind the
Babylonians. Then came the Greeks, with an emphasis
on geometry but also on such things as Euclid's al-
gorithm; the latter is the oldest nontrivial algorithm
which still is important to computer programmers. (See
[7, 9] for the history of Egyptian mathematics, and [1, 7,
9] for Greek mathematics. A free translation of Euclid's
algorithm in his own words, together with his incom-
plete proof of its correctness, appears in [2, p. 294-296].)
And then there are the Indians, and the Chinese; it is
clear that much more can be told.
676
Communications July
1972
of Volume 15
the ACM Number 7
Acknowledgment.
I wish to thank Professor Abra-
ham Seidenberg for his courtesy in helping me obtain
copies of [3] and [8] when I needed them.
Appendix
The 20 additional entries included in Inakibit's table are some-
what mysterious. In 19 of the cases, the number has a reciprocal
with six digits or less; the exception is 3 z3 = 2,1,4,8,3,0, 7, whose
reciprocal has 17 sexagesimal digits.
Let's say that a sexagesimal number is a
Q-number
if it has
six or less digits, while its reciprocal is finite and has more than
six digits and begins with 1 or 2. There are 132 Q-numbers in
all, only 19 of which appear in Inakibit's table. Five of these are
217, 223, 311, 3
TM,
and 32z; they constitute all Q-numbers of the forms
2 ~, 3 ., or 5 ~, and it is likely that such numbers appeared in special
tables. However, the Q-number 611 is not included, so it is not
simply a matter of perfect powers being included. The three-
digit Q-numbers 2131° and 2239 are excluded, so it not a matter of
including the smallest cases. The Q-numbers which do appear,
besides the five listed above, are 3951, 3105 a, 31155; 213951, 2131'52,
213135 a (but not
2131554);
31851, 2339, 2731°, 212311, 2183
TM,
2203 ~, 29259,
2'2452. It is perhaps noteworthy that 31153 does not appear, but its
multiple 3u5 ~ does.
Since so many Q-numbers are missing, we may conclude that
Inakibit continued his table by giving the reciprocals of all six-
digit numbers up to 59,43,10, 50, 52,48, not taking advantage of
symmetry. Hence the complete table contained the reciprocals of
at least 721 six-digit numbers, and it probably filled three clay
tablets in all.
References
1°
Aaboe, Asger A.
Episodes from the Early History of Mathematics.
Random House, New York, 1964, 133 pp.
2.
Knuth, Donald E.
Seminumerical Algorithms.
Addison-Wesley,
Reading, Mass., 1971 (second printing), 624 pp.
3.
Neugebauer, O. Mathematische keilschrift-texte. In
Quellen und
Studien zur Geschichte der Mathematik, Astronomie, und
Physik,
Vol. A3, Pt. 1, 1935, 516 pp.
4.
Neugebauer, O. Mathematische keilschrift-texte. In
Quellen und
Studien zur Geschichte der Mathematik, Astronomie, und
Physik,
Vol. A3, Pt. 2, 1935, 64 pp. plus 69 reproductions of
tablets.
5.
Neugebauer, O. Mathematische keilschrift-texte. In
Quellen und
Studien zur Geschiehte der Mathematik, Astronomie, und
Physik,
Vol. A3, Pt. 3, 1937, 83 pp. plus 6 reproductions of
tablets.
6.
Neugebauer, O., and Sachs,
A. Mathematical Cuneiform Texts.
American Oriental Society, New Haven, Conn., 1945, 177 pp.
plus 49 reproductions of tablets.
7.
Neugebauer, O.
The Exact Sciences in Antiquity.
Brown U. Press,
Providence, R.I., 1957 (second ed.), 240 pp. plus 14
photographic plates.
8.
Thureau-Dangin, F.
Textes Math~matiques Babyloniens.
E.J.
Brill, Leiden, The Netherlands, 1938, 243 pp.
9.
van der Waerden, B.L.
Science Awakening.
Tr. by Arnold Dresden.
P. Noordhoff, Groningen, The Netherlands, 1954, 306 pp.
677
Communications
of
the ACM
July 1972
Volume 15
Number 7
This table is taken apparently verbatim from reference [7, Neugebauer]. It appears on page 32 of the Dover edition I have in my library. This was published in 1969 and the front matter states is " ... an abridged and slightly corrected republication of the second edition, published in 1957 by Brown University Press."
In other words, if you saw the two digit sexagesimal number 2,20 in a tablet, it would be representing:
$$ 140 \times 60^n$$
Where you’d have to infer $n$ based on the context.
So 2,20 could have meant any of the following (or more):
$$2 \times 60^1 + 20 \times 60^0 = 140$$
$$2 \times 60^{0} + 20 \times 60^{-1} = 2 + 20 \times \frac{1}{60} = 2 + \frac{1}{3}$$
$$2 \times 60^{-1} + 20 \times 60^{-2} = \frac{2}{60} + \frac{20}{3600} = \frac{7}{180} $$
Here is an example of an actual Cuneiform tablet with a reciprocals table
Again remember that they inferred the exponent depending on the context of the problem. Therefore 1 can mean $1\times 60^{1}$ therefore meaning 60, and 15 can mean $15\times 60^{-1}= \frac{15}{60}$ therefore meaning $\frac{1}{4}$
A cubit and a gar are units of measurement used in ancient Babylonian mathematics:
1. **Cubit**: A cubit is primarily a unit of length. In Babylonian measurements, a cubit was approximately equivalent to the length of a forearm, roughly 18 to 20 inches (45.7 to 50.8 cm). However, the exact length varied depending on the region and period.
2. **Gar**: A gar is a unit of length that equates to 12 cubits, so it's similar to our modern concept of a "dozen." This means that one gar is about 216 to 240 inches (548.6 to 609.6 cm) based on the 18 to 20 inch approximation for a cubit.
### Cuneiform
Cuneiform, originating around 3300 B.C., marked the start of writing in early Mesopotamian cities. Initially, proto-cuneiform pictograms were drawn on clay tablets. Over time, these pictograms evolved into "wedge-shaped" cuneiform signs impressed on the surface. Cuneiform extended from Syria's Mediterranean coast to western Iran, adapting to at least fifteen languages. The last cuneiform text dates to A.D. 75, but the script lingered for another two centuries.
*Cuneiform tablet*
Linear interpolation is a method used to estimate values between two known points on a line. Given two known data points, (x1, y1) and (x2, y2), linear interpolation provides an easy way to estimate the value of y for any value of x between x1 and x2.
The formula for linear interpolation is:
\[ y = y1 + \frac{(x - x1)(y2 - y1)}{x2 - x1} \]
In simple terms, linear interpolation assumes that the change between two points is linear and uses this assumption to predict values within the range of the known points.
This formula can be derived from the context of a quadratic equation. Given the problem's constraints:
1. \(x - y = d\)
2. \(xy = p\)
From the first equation (1), we can express \(y\) in terms of \(x\) and \(d\):
\[ y = x - d \]
Substitute this expression for \(y\) into the second equation (2) to get:
\[ x(x - d) = p \]
Expanding and rearranging gives:
\[ x^2 - dx - p = 0 \]
Plugging these values into the quadratic formula, we get:
\[ x = \frac{d \pm \sqrt{d^2 + 4p}}{2} \]
We can then get:
\[ x = \frac{d \pm 2\sqrt{\left(\frac{d}{2}\right)^2 + p}}{2} \]
Finally:
\[ x = \frac{d}{2} \pm \sqrt{\left(\frac{d}{2}\right)^2 + p} \]
In this context, a table of reciprocals is a lookup table that provides the multiplicative inverse (or reciprocal) of a number in the sexagesimal system. In mathematics, the reciprocal of a number $x$ is $\frac{1}{x}$. The product of a given number and its reciprocal will always give the value 1.
For example, in our familiar base-10 system:
- The reciprocal of 2 is 0.5 (or 1/2).
- The reciprocal of 3 is 0.333... (or 1/3).
In the Babylonian sexagesimal system, this concept is the same, but the numbers are represented in base-60.
To illustrate:
- For the number 2, the reciprocal is 30 (in base-60, which in this case represents $30 \times 60^{-1}$)
$$2 \times \frac{30}{60} = 1 $$
Such tables were incredibly valuable because they saved the mathematician from having to compute these reciprocals each time they were needed, which was frequently, especially in division operations. By having a table of reciprocals, the Babylonians could turn a division problem into a multiplication one, which was often easier to handle.
In a sexagesimal system you use 60 as the base, unlike our modern base-10 (decimal) system. Here are some quick notes:
1. **Base-60:** 60's choice comes from its excellent divisibility properties. 60 can be evenly divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, making it versatile for fractions.
2. **Place Value:** As in our decimal system, position matters. However, values increase in powers of 60. For instance, in our decimal system, the number $210$ means $2\times10^2 + 1\times10^1 + 0\times10^0$. In sexagesimal, $21$ would mean $2\times60^1 + 1\times60^0 = 120 + 1 = 121$ in decimal.
3. **Representation:** The Babylonians used two cuneiform symbols:
- A wedge for ones
- A corner wedge for tens
The Plimpton tablet is a Babylonian clay tablet that dates back to around 1800 BC. It contains a table of Pythagorean triples, which are sets of three positive integers a, b, and c such that \( a^2 + b^2 = c^2 \).
One could argue that the concepts present in the Plimpton tablet are early precursors to trigonometry. The Pythagorean triples can be seen as primitive trigonometric values since they describe the relationships between the sides of right triangles. If one were to normalize the hypotenuse (the diagonal) to have a length of 1 unit, the other two sides could arguably be seen as providing cosine and sine values, respectively. They could then use linear interpolation to estimate cosine and sine values for other angles.
*Plimpton 322 Tablet*