History of Mathematics Project | Susa Mathematical Tablets

History of
Mathematics

Project

around 1900–1600 BCE

Susa Mathematical Tablets

Babylonian approximation to π based on a regular hexagon

In 1933, a total of 26 mathematical tablets were excavated some two hundred miles from Babylon in the ancient Elamite city of Susa. The tablets are believed to date from between 1900 and 1600 BCE and contain a number of interesting geometric figures and mathematical constants arising from geometric computations. Partial translations of the tablets were published in 1950, but the first detailed analysis of three particularly interesting tablets did not appear until 1961.

The tablets referred to as TMS 2 and TMS 3 are of particular interest. TMS 2 is relatively straightforward, containing a diagram of a regular hexagon and markings that give an approximation to the area of an equilateral triangle. TMS 3, on the other hand, contains a tabular list of geometric constants corresponding to many shapes, including polygons but also featuring many more complicated geometric shapes, including the semicircle (termed a crescent) and lens-shaped figures known to the Babylonians as the ox-eye and grain figure (or grain-field, or barleycorn). The tabulation includes 68 constants spread over the 71 lines. On line 30, the perimeter-to-circumcircle circumference ratio for a regular hexagon is given as 24/25, which implies perhaps the world's oldest recorded approximation to π with value 3 + 1/8 = 3.125.

Original artifact location

Susa (historical name), Shush, Iran (current name)

Current artifact location

The Louvre, Paris

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Computational Explanation

The Susa tablets contain a fascinating array of mathematical constants corresponding to a variety of geometric shapes studied by the Babylonians. In fact, the first line of of tablet TMS 3 claims that the tablet contains "coefficients for absolutely everything." Examples from two of these tablets are discussed in the following.

TMS 2: Equilateral Triangle

Tablet TMS 2, despite missing a wedge-shaped portion of its left side, contains a clearly drawn diagram of a regular hexagon divided into six equal equilateral triangles:

The cuneiform symbols carved into the tablet near the left and bottom edges of one of the triangles denote the number 30 in the Babylonian numeration system:

In[]:=

[◼]	AncientNumberRepresentation

[30,"Babylonian"]

Out[]=

… while the more complicated-looking cuneiform digits inside the triangle correspond to 6 33 45:

In[]:=

[◼]	AncientNumberRepresentation

[{6,33,45},"Babylonian"]

Out[]=





Since Babylonian numerals do have an equivalent of a decimal point, they are ambiguous up to powers of 60. The digits 6 33 45 could therefore stand for any of:

In[]:=

Table[

-n

(63600+3360+45),{n,0,2}]

Out[]=

23625,

1575

105



Based on the location of these numbers inside the triangle, a natural assumption is that they correspond to the area of the triangle. This can be checked by noting that area of an equilateral triangle with side length a is given by (as can be derived using the Pythagorean theorem):

A

Plugging in

a30

then gives:

A

225

Since

does not have a terminating representation in base 60 (or base 10, for that matter), the Babylonians used rational approximations. To see how such an approximation can be constructed, consider its so-called continued fraction:

In[]:=

ContinuedFraction[

,10]

Out[]=

{1,1,2,1,2,1,2,1,2,1}

Taking only the first five terns, the continued fraction representation [1; 1, 2, 1, 2] represents the number:

In[]:=

Out[]=

… and is an approximation to the original number

. Such approximations can be computed by taking an increasing number of terms in the continued fraction, giving a sequence of numbers termed convergents:

Such numbers can be computed more directly as:

In[]:=

Convergents[

,5]

Out[]=

1,2,



In this case, these convergents have numerical values:

In[]:=

N[%]

Out[]=

{1.,2.,1.66667,1.75,1.72727}

… which can be seen to approach ever closer to

In[]:=

]

Out[]=

1.73205

The Babylonians used the approximation

≈7/41.75

corresponding to the fourth convergent, which has the convenient property of having a finite base-60 expansion. Returning now to the expression for the area of the triangle and replacing

with its approximation then gives:

A225

≈225



1575

Examining the base-60 representation of the number:

In[]:=

RealDigits

1575

,60

Out[]=

{{6,33,45},2}

… confirms that value in the center of the triangle gives the area of the triangle expressed using a rational approximation to

TMS 3 Lines 7–9: Semicircle

Tablet TMS 3 begins with the proclamation that it contains "coefficients for absolutely everything." One of its first groups of coefficients considers a semicircle (which the tablet terms a crescent):

In[]:=

With[{r=1},Graphics[Circle[{0,0},r,{0,π}]]]

Out[]=

The arc length a of a semicircle is given by half the circumference length of a full circle:

In[]:=

ArcLength[Circle[{0,0},r,{0,π}]]

Out[]=

πr

… so in terms of the circle radius r and diameter d:

aπr

πd

Solving for d and r in terms of a then gives:

d

r

Similarly, the area A of the half-disk bounded by the semicircle on top and the x axis on the bottom is given by:

In[]:=

Area[Disk[{0,0},r,{0,π}]]

Out[]=

… so:

A



Turning now to lines 7–9 of the tablet:

… they can be seen to give the constants

in:

Aad

da

ra

… by reading off the values 1/4, 2/π and 1/π from the equations above and making the approximation

π≈3

In[]:=

ColumnRealDigits[#,60][[1]]&/@

/.π3

Out[]=

{15}

{40}

{20}

TMS 3 Lines 26–28: Regular Polygons

Lines 26–28 of TMS 3 give the areas of unit regular polygons (called n-fronts by the Babylonians) for

n5

, 6, 7:

In modern notation, the formula for the area of a regular n-gon with side length a is given by:



ncot

… where cot(x) denotes the cotangent function

cot(x)1/tan(x)cos(x)/sin(x)

. Taking

a1

, corresponding to polygons of unit edge length, and plugging in n from 5 to 7 gives the following values:

In[]:=

Columnexact=Tablen,

nCot

,

nCot

//N,{n,5,7}

Out[]=

5,

,1.72048

6,

,2.59808

7,

Cot

,3.63391

In lines 26–28 of the tablet, the sexagesimal values 1 40, 2 7 30 and 3 41 are written:

In[]:=

Column[babylonian={#,FromDigits[{#,1},60],FromDigits[{#,1},60]//N}&/@{{1,40},{2,37,30},{3,41}}]

Out[]=

{1,40},

,1.66667

{2,37,30},

,2.625

{3,41},

221

,3.68333

Comparing these the exact expressions and the rational approximations obtained by the Babylonians show them to be in relatively close agreement:

Out[]//TraditionalForm=

n	exact	numeric	Babylonian sexigesimal	Babylonian exact	Babylonian numeric
5	5 4 1+ 2 5	1.72048	{1,40}	5 3	1.66667
6	3 3 2	2.59808	{2,37,30}	21 8	2.625
7	7 4 cot π 7	3.63391	{3,41}	221 60	3.68333

The differences arise from the fact that the Babylonians made rational approximations for numbers such as π and

that are not expressible in closed form using base 60. For example, the unit regular hexagon can be made of six equilateral unit triangles, each of height

/2, giving an exact area of:

In[]:=

6

×1×

Out[]=



Using the already-obtained approximation

≈7/4

then results in:

≈



In sexagesimal, this corresponds to the digits:

In[]:=

RealDigits

,60[[1]]

Out[]=

{2,37,30}

… precisely matching the numbers written on line 27.

TMS 3 Lines 16–18: Grain Figure

Consider a unit circle with an inscribed square. The square partitions the interior of the circle into four congruent regions on the exterior of the square. Position the figure so that an edge of the square lies along the y axis and is centered on the x axis, and consider the region to the left of the y axis:

Withsquare=RegularPolygon

,0,1,4,Graphics[{{LightYellow,DiskSegment[square[[1]],1,{3π/4,5π/4}]},Circumsphere[square],FaceForm[],EdgeForm[Black],square},AxesTrue]

Out[]=

The figure formed by combining the highlighted circular segment with its reflection across the y axis was known as a grain figure (or grain-field, or barleycorn) by the Babylonians:

Out[]=

Consider now the grain figure made from arcs with unit lengths. In base 60, "unit" can be taken to be 60 since both 1 and 60 have the same representation in cuneiform. (In general, numbers that differ by a power of 60 cannot be distinguished when expressed in cuneiform.) The circumference of a full circle corresponding to

n4

arcs is then:

C60n460240.

The square on which the arcs are erected then has circumradius:

R

2π



120

Using the approximation

π≈3

then gives:

R

120

≈

120

40.

From the Pythagorean theorem, the side length of the square is then given by:

a



Using the previously found approximation to R and taking

≈17/12

, which can be obtained from the example using continued fraction convergents as previously discussed:

In[]:=

Convergents[

,5]

Out[]=

1,



… then gives:

a

R≈

40

170

Similarly, the inradius of the square can be found using the Pythagorean theorem as:

rR-

R1-

… which, after plugging in the previous approximations, results in:

rR1-

≈401-



The height of the grain figure is therefore given by:

h2r≈

… the width by:

w2(R-r)≈240-



170

… the area enclosed by the circle by:

disk

π

≈3

4800

… and the area of the square (calculated from the approximate circumradius R as opposed to using the side length a) is:

square

4

2

≈2

3200.

The area of four half-grain figures is therefore given by the difference

disk

square

Out[]=



… so multiplying by 2/41/2 gives the area of a single grain figure:

grainfigure



(

disk

square

)≈

(4800-3200)800

This provides all the information needed to identify the constants

in:

grainfigure



grainfigure



grainfigure



… as given by lines 16–18 of the tablet:

… as:

In[]:=

babylonian={#[[1]],FromDigits[#,60],FromDigits[#,60]//N}&/@{{{13,20},2},{{56,40},1},{{23,20},1}};

The exact values can be found by plugging in the unapproximated formulas previously noted:

In[]:=

exact=

(π

-2

),2(R-r),2r/.rR1-

/.R

120

//Simplify

Out[]=



7200(-2+π)

120

120(-2+

)



Summarizing;

Out[]//TraditionalForm=

parameter	exact	numeric	Babylonian sexigesimal	Babylonian exact	Babylonian numeric
grain figure area	7200(π-2) 2 π	832.806	{13,20}	800	800.
grain figure width	120 2 π	54.019	{56,40}	170 3	56.6667
grain figure height	- 120 2 -2 π	22.3754	{23,20}	70 3	23.3333

TMS 3 Lines 19–21: Ox-Eye

Now consider a unit circle with an inscribed equilateral triangle. The triangle partitions the interior of the circle into three congruent regions on the exterior of the triangle. Position the figure so that an edge of the triangle lies along the y axis and is centered on the x axis, and consider the region to the left of the y axis:

In[]:=

Withtriangle=RegularPolygon

,0,{1,0},3,Graphics[{{LightYellow,DiskSegment[triangle[[1]],1,{2π/3,4π/3}]},Circumsphere[triangle],FaceForm[],EdgeForm[Black],triangle},AxesTrue]

Out[]=

The figure formed by combining the highlighted circular segment with its reflection across the y axis was known as an ox-eye by the Babylonians:

Out[]=

Consider the ox-eye, made as before from circular arcs of unit length. The circumference of a full circle corresponding to

n3

arcs is then:

C60n360180.

The equilateral triangle on which the arcs are erected then has circumradius:

R

2π



180

2π



≈

30,

and inradius:

r

≈15

The Pythagorean theorem then gives the half-edge length of the triangle as:

v

≈

15

≈

157



105

The height of the ox-eye is given by

h2v30

, or approximately by:

h2v≈

105

Similarly, its width is:

w2(R-r)R≈30

The area enclosed by the circle is:

disk

π

≈3

2700

… and that of the triangle is:

triangle



(2v)(r+R)

w(r+R)≈

105

(15+30)

4725

The area of three half-ox-eyes is therefore the difference

disk

triangle

Out[]=



… so multiplying by 2/3 gives:

ox-eye





disk

triangle

≈

2700-

4725



2025

This provides all the information needed to identify the constants

in:

ox-eye



ox-eye



ox-eye



… as given by lines 19–21 of the tablet (where the value 53 20 is a typographical error for 52 30 in BR 20):

… as:

In[]:=

babylonian={#[[1]],FromDigits[#,60],FromDigits[#,60]//N}&/@{{{16,52,30},2},{{52,30},1},{{30},1}};

The exact values can be found by plugging in the unapproximated formulas previously noted:

exact=

π

(r+R),2

,R/.r

/.R

//Simplify

Out[]=

-

1350(3

-4π)



Summarizing;

Out[]//TraditionalForm=

parameter	exact	numeric	Babylonian sexigesimal	Babylonian exact	Babylonian numeric
ox-eye area	- 13503 3 -4π 2 π	1008.12	{16,52,30}	2025 2	1012.5
ox-eye height	90 3 π	49.6196	{52,30}	105 2	52.5
ox-eye width	90 π	28.6479	{30}	30	30.

TMS 3 Line 30: π from the Regular Hexagon

Line 30 of the table:

… contains the cuneiform representation of the digits 57 36:

In[]:=

[◼]	AncientNumberRepresentation

[#,"Babylonian"]&/@{57,36}

Out[]=





Interpretation of the additional cuneiform writing indicates that this line gives the ratio of the perimeter of a regular hexagon:

In[]:=

HexagonPerimeter=Perimeter[RegularPolygon[R,6]]

Out[]=

… to the circumference of its circumcircle:

In[]:=

CircumcirclePerimeter=Perimeter[Disk[{0,0},R]]

Out[]=

2πR

The exact value is:

In[]:=

HexagonPerimeter

CircumcirclePerimeter

Out[]=

This value is close to 1:

In[]:=

N[%]

Out[]=

0.95493

In fact, as previously seen, Babylonian computations commonly make the approximation

π≈3

, a practice that would reduce this ratio to precisely 1. In contrast, the approximation to this ratio given in line 30 is 57 36, which corresponds to:

In[]:=

FromDigits[{{57,36},0},60]

Out[]=

Therefore, using this value implies a value for π given by:

Solve

HexagonPerimeter

CircumcirclePerimeter

/.Pipi

,pi

Out[]=

pi



… or

π3+1/83.125

, a value much closer to the true value

π≈3.1416

than given by taking

π≈3

Other Resources

History of Mathematics
Babylonian Geometrical Problem Tablet

Wolfram Demonstration
Babylonian Numerals

Additional Reading

Beckmann, P. A History of Pi. New York: Barnes & Noble, pp. 21–22, 1993.
Bruins, E. M. "Quelques textes mathématiques de la Mission de Suse." Proc. Roy. Dutch Acad. Sci., Vol. 53, pp. 1025–1033, 1950.
Bruins E. M. and Rutten, M. Mémoires de la Mission Archéologique en Iran, Tome XXXIV: Textes mathématiques de Suse. Paris: Librairie Orientaliste Paul Geuthner, pp. 26 and 33 and Pls. IV and 4, 1961.
Friberg, J. A Remarkable Collection of Babylonian Mathematical Texts: Manuscripts in the Schøyen Collection: Cuneiform Texts I. New York: Springer, pp. 217–218, 2007.
Friberg, J. and Al-Rawi, F. N. H. New Mathematical Cuneiform Texts. Springer, pp. 256–257, 325, and 396, 2016.
Harper, P. O.; Aruz, J. and Tallon, F. (Eds.). The Royal City of Susa: Ancient Near Eastern Treasures in the Louvre. New York: The Metropolitan Museum of Art, pp. 276–278, 1992.
Nemet-Nejat, K. R. Cuneiform Mathematical Texts as a Reflection of Everyday Life in Mesopotamia. New Haven, CT: American Oriental Society, p. 274, 1993.
Neugebauer, O. The Exact Sciences in Antiquity, 2nd ed. New York: Dover, pp. 46–48, 1969.
Park, J. "Cultural and Mathematical Meanings of Regular Octagons in Mesopotamia: Examining Islamic Art Designs." Journal of History Culture and Art Research, Vol. 7, No. 1, pp. 301–318, 2018.
Robson, E. Mesopotamian Mathematics, 2100-1600 BC: Technical Constants in Bureaucracy and Education. Oxford, England: Clarendon Press, pp. 19–21, 48–50 and 199–201, 1999.
Sarton, G. Ancient Science Through the Golden Age of Greece. New York: Dover, p. 74, 1993.

Additional Links

Cuneiform Digital Library Initiative: P254835