History of Mathematics Project | Babylonian Inner Diagonal Tablet

History of
Mathematics

Project

around 1900–1600 BCE

Babylonian Inner Diagonal Tablet

The Pythagorean theorem in three dimensions

This Babylonian cuneiform tablet fragment, dated to around 1900–1600 BCE, consists of the upper-left quarter of a full tablet. The presence of a detailed summary of its contents on the surviving portion indicates it originally contained 16 different geometric exercises concerning circles, diagonals, chords and a city gate.

One of the most interesting of the tablet's surviving problems concerns finding the "inner diagonal" drawn from the inner-left bottom corner to the outer-right top corner of a cuboid. This involves finding a solution in integers to the three-dimensional generalization of the Pythagorean theorem more than a millennium before Pythagoras lived.

Artifact dimensions

11.5 cm × 6.4 cm × 2.2 cm

Original artifact location

Uruk (historical name), Al-Muthannia, Iraq (current name)

Current artifact location

Oslo, Norway

Catalog number

MS 3049

Timeline

Interactive Content

Computational Explanation

The text of section 5 (§ 5) in Babylonian tablet MS 3049 is relatively well preserved, with only the last portions of the first three or four lines missing:

Unfortunately, the missing portion corresponds to the crucial beginning of the solution procedure, which is doubly unfortunate, both because there is no known parallel text in the Old Babylonian mathematical corpus and because the terminology used is partly new and hard to understand (Friberg 2007, p. 301).

Throughout the text, Babylonian numerals consisting of base-60 numbers written in cuneiform script are used. The following table illustrates such cuneiform representations for the whole numbers from 1 to 40:

In[]:=

TextGridFlatten/@TransposePartitionTablen,

[◼]	AncientNumberRepresentation

[n,"Babylonian"],{n,40},10,DividersAll

Out[]=

1	11	21	31
2	12	22	32
3	13	23	33
4	14	24	34
5	15	25	35
6	16	26	36
7	17	27	37
8	18	28	38
9	19	29	39
10	20	30	40

The surviving cuneiform text has the following transliteration and translation (Friberg 2007, p. 302):

In lines 4–5, the gate is stated to have height 26 40, width 8 53 20 and thickness 8 53 20, and line 13 gives the computed "inner cross-over" length as 28 53 20. As it turns out, the dimensions have not been arbitrarily chosen, but rather selected as integer multiples of the regular (meaning having a terminating expansion) sexagesimal fraction 1/27:

In[]:=

RealDigits

,60[[1]]

Out[]=

{2,13,20}

… with the multiples being 3, 4, 12 and 13 (Friberg 2007, p. 303):

The set of integers (3, 4, 12, 13) corresponds to a Pythagorean quadruple, which is a set of positive integers

(a,b,c,d)

such that



. Plugging in the values above shows them to indeed satisfy this relation:

In[]:=

{3^2+4^2+12^2,13^2}

Out[]=

{169,169}

The complete list of Pythagorean quadruples having no duplicate values and with largest member less than 20 is given by:

In[]:=

{a,b,c,d}/.{ToRules[Reduce[a^2+b^2+c^2d^2&&a<b<c<d<20,{a,b,c,d},PositiveIntegers]]}

Out[]=

{{1,4,8,9},{1,6,18,19},{2,3,6,7},{2,5,14,15},{2,6,9,11},{2,8,16,18},{2,10,11,15},{3,4,12,13},{4,6,12,14},{6,10,15,19},{8,9,12,17}}

The quadruple (3, 4, 12, 13) is therefore the unique such Pythagorean quadruple having thickness 3.

Summarizing the dimensions given on the tablet:

In[]:=

dimensions=Thread[{"thickness","width","height","inner cross-over"}->{3,4,12,13}/27]

Out[]=

thickness

,width

,height

,inner cross-over



Out[]=

dimension	value	base-60 representation
thickness	1 9	6 40
width	4 27	8 53 20
height	4 9	26 40
inner cross-over	13 27	28 53 20

The most amazing part of the tablet concerns the computation of the "inner cross-over" based on the other dimensions. In particular, in lines 6–11, the squares of these three cuboid dimensions are computed, and in line 12, the sum of their three squares is given in the form 13 54 34 14 26 40:

Out[]=

dimension	value	base-60 representation	square value	base-60 representation
thickness	1 9	6 40	1 81	44 26 40
width	4 27	8 53 20	16 729	1 19 0 44 26 40
height	4 9	26 40	16 81	11 51 6 40
inner cross-over	13 27	28 53 20	169 729	13 54 34 4 26 40

To see how the Babylonians arrived at these values, consider a cuboid with thickness t, width w, height h, inner cross-over d and lower-face diagonal

Out[]=

The connection between thickness, width, height and inner cross-over is given by two applications of the Babylonian diagonal rule in two dimensions (i.e. the Pythagorean theorem):



Adding the two equations to each other gives:



Subtracting

from both sides then gives:



… which is precisely the Pythagorean theorem (or equivalently, Euclidean distance formula) in three dimensions.

Note that finding d requires determining the square root of the six-digit sexagesimal number 13 54 34 4 26 40, a feat that is by no means easy:

In[]:=

RealDigits[Sqrt[FromDigits[{{13,54,34,4,26,40},0},60]],60][[1]]

Out[]=

{28,53,20}

So not only did Old Babylonian mathematicians know about the Pythagorean theorem in three dimensions, but they also knew how to construct exact integer solutions to it, both especially amazing considering that this knowledge and proficiency with calculation existed no less than 3,600–3,900 years ago!

Other Resources

Wolfram Demonstration
Babylonian Numerals

Wolfram Demonstration
Pythagorean Theorem

Additional Reading

Friberg, J. "MS 3049, § 5. The Inner Diagonal of a Rectangular Gate in a Wall" §11.1 d in A Remarkable Collection of Babylonian Mathematical Texts: Manuscripts in the Schøyen Collection: Cuneiform Texts I. New York: Springer, pp. 301–304 and 495, 2007.