History of Mathematics Project | Babylonian Geometrical Problem Tablet

History of
Mathematics

Project

around 1900–1600 BCE

Babylonian Geometrical Problem Tablet

Clay tablet with geometrical problems accompanied by diagrams

Tablet BM 15285, held by the British Museum, has been instrumental in the understanding of Old Babylonian line geometry. It consists of 40 problems with diagrams but no solutions. A third of the tablet is missing.

This tablet, dated to around 1900–1600 BCE, contains problems involving figures drawn inside squares. The problems are ordered by increasing complexity. The first 12 problems concern squares, circles and right triangles; many of the remainder consider more complicated figures.

Artifact dimensions

26.2 cm × 30.8 cm

Original artifact location

Babylonia (historical name), Iraq (current name)

Current artifact location

British Museum, London

Catalog number

BM 15285

Timeline

Interactive Content

Computational Explanation

Introduction and Background

The problems in tablet BM 15285 are believed to be a list of exercises for children. Its goal was probably to train students in the art of computing areas. The configuration used for all the problems on the tablet consisted of a unit square divided into a

4×4

grid into which geometric shapes were successively added or removed:

Out[]=

A number of basic geometric shapes were ubiquitous in Mesopotamian visual culture and probably played the role of ancient emoji, symbolizing, for example, the lyre, bow, ox's brow, barleycorn and crescent moon. One such shape is the so-called apsamikku (see Robson 2008), which has the following appearance:

Out[]=

While it is visually similar to the curve known as the astroid, it was defined instead as the interior region of four identical circles placed together symmetrically so as to be mutually tangent:

Out[]=

Since the apsamikku is highly reminiscent of the shape of bovids' noses on Babylonian lyres, some Assyrian experts refer to it as a cow's nose:

Embedding a cow's nose inside a unit square in such a way that the lengths of the external arcs are

, the diagonal, running corner-to-corner, has length

, and the short transversal, running across the narrowest part of the figure, has length

Out[]=

The exact value for its area computed using the Wolfram Language is:

In[]:=

AreaRegionDifferenceRectangle[{1/4,1/4},{3/4,3/4}],RegionUnionTableDiskpt,

,{pt,{{0,1/2},{1/2,1},{1,1/2},{1/2,0}}}

Out[]=

4-π

The Babylonians estimated the area of a cow's nose to be 4/9 of the arc squared, i.e.



. Using that fact gives an estimate for π:

In[]:=

Solve

4-pi



,pi

Out[]=

pi



3.11111...

Another favorite geometric figure was the crescent moon, which is simply a half-disk. While the method for computing the area of a cow's nose is not known, the Babylonians certainly noticed that a unit square is equal to the sum of two cow's noses plus four crescent moons (two disks of diameter

Out[]=



Taking three tangent crescent moons and subtracting the middle one gives another Babylonian favorite, a bow figure:

Out[]=

⟶

A fascinating result that was part of Babylonian mathematical folklore is the following case of the lunes of Hippocrates, named after Hippocrates of Chios who lived about 450 BCE. Considering the right triangle depicted below, the area of the lune created by subtracting a disk of diameter

to a crescent moon of diameter

is equal to the area of the right triangle:

Out[]=

In[]:=

Area[Triangle[{{0,1/2},{1/4,3/4},{1/4,1/4}}]]

Out[]=

In[]:=

AreaRegionDifferenceDisk14,

,

,Disk0,

,



Out[]=

The area of this lune, termed here a squarable lune, is therefore equivalent to a

square:

Out[]=

In[]:=

Area[Rectangle[{0,0},{1/4,1/4}]]

Out[]=

By applying this mathematical result to different geometrical configurations, the Babylonians were able to compute exact solutions to some complex geometrical figures, such as problem #32 in tablet BM 15285.

Out[]=



A rearrangement of the pair of white lunes gives a pair of squarable lunes in gray. These can therefore be replaced by right triangles or

squares without increasing or decreasing the total area in gray:

Out[]=



Notice that the latter rearrangement is composed of

squares, of which there are 14 in total. A unit square divided in a

4×4

grid has 16 squares, so the total area is



. Therefore, the geometrical region in problem #32 is squarable and its area is equal to a unit square divided in an

8×8

grid with its diagonal removed, i.e. 7/8:

Out[]=



There are no solutions in tablet BM 15285, but if there were, then the solution to #32 would be:

In[]:=

babylonian[x_]:=Modulerat=x/.π



,RowExpand[x],If[Element[x,Rationals],"","⟶"],

[◼]	AncientNumberRepresentation

[rat,"Babylonian"],Spacer[2]

In[]:=

babylonian[7/8]

Out[]=



(In the above, the Babylonian cuneiform representation uses the inferred Babylonian rational approximation for π, as well as a similarly inferred value for the irrational number

Babylonian Homework

After having reviewed some of the figures appearing on this tablet, imagine that you were sent back in time and are now a student in an Old Babylonian school whose teacher has assigned you the collection of fun geometrical puzzles comprising tablet BM 15285 as homework. It is your task to recreate the figures and compute the relevant areas. While this assignment can of course be done by hand, the following sections illustrate the solution to each problem using a modern secret weapon, the Wolfram Language, which allows each problem to be visualized and solved in just a few lines of simple code.

To simplify the task of the student in navigating the subtleties, ambiguities, and archaic language in a literal transliteration of the original cuneiform, problem statements below are based on Robson (2007) but rewritten using simplified and disambiguated wording.

Problem 1

[We have a unit square. We extend a border (1/4) on each side and then we draw a second square. What is its area?]

Graphics

styles

,Rectangle[],Rectangle[{1/4,1/4},{3/4,3/4}]

Out[]=

In[]:=

Area[Rectangle[{1/4,1/4},{3/4,3/4}]]//babylonian

Out[]=



Problem 2

We have a unit square. We extended a border (1/4) on each side and then we draw a second square. Inside the square I draw a circle. What is its area?

Graphics

,Rectangle[],Rectangle[{1/4,1/4},{3/4,3/4}],Disk@@Insphere[Rectangle[{1/4,1/4},{3/4,3/4}]]

Out[]=

In[]:=

Area[Disk@@Insphere[Rectangle[{1/4,1/4},{3/4,3/4}]]]//babylonian

Out[]=

⟶

Problem 3

We have a unit square. We extend a border (1/4) on each side and we draw a circle. What is its area?

Graphics

,Rectangle[],Disk[{1/2,1/2},1/4]

Out[]=

In[]:=

Area[Disk[{1/2,1/2},1/4]]//babylonian

Out[]=

⟶

Problem 4

We have a unit square. Inside it we draw a square and a circle. The circle that we draw touches the square. What is its area?

The difference between #2 and #4 is that in the former exercise a circle is inscribed in a square, while in the latter exercise a square is circumscribed around a circle!

In[]:=

Area[Disk[{1/2,1/2},1/4]]//babylonian

Out[]=

⟶

Problem 5

[We have a unit square.] <We draw> a second [square. Inside the second square we draw] 4 wedges and 1 circle. What are their areas?

Identical to #2 and #4.

In[]:=

Area[Polygon[{{1/4,1/4},{1/4,3/4},{3/4,3/4},{3/4,1/4}}]]//babylonian

Out[]=



In[]:=

Area[Disk[{1/2,1/2},1/4]]//babylonian

Out[]=

⟶

Problem 6

[We have a unit square. <We draw> a second square. Inside the second square] we draw [4 squares and 1 circle.] What are their areas?

Graphics

,Rectangle[],Table[Rectangle[{i,j},{i+1/4,j+1/4}],{i,1/4,1/2,1/4},{j,1/4,1/2,1/4}],Disk[{1/2,1/2},1/4]

Out[]=

In[]:=

Area[Rectangle[{1/4,1/4},{1/2,1/2}]]//babylonian

Out[]=



In[]:=

Area[Disk[{1/2,1/2},1/4]]//babylonian

Out[]=

⟶

Problem 7 (and 8)

We have a unit square. Inside it we draw a second square. The square that we draw touches the outer square. What is its area?

Graphics

,Rectangle[],Parallelogram[{0,1/2},{{1/2,1/2},{1/2,-1/2}}]

Out[]=

In[]:=

Area[Parallelogram[{0,1/2},{{1/2,1/2},{1/2,-1/2}}]]//babylonian

Out[]=



Problem #8 is identical to problem #7.

Problem 9

We have a unit square. Inside it we draw a square. The square that we draw touches the <first> square. Inside the second square we draw a third square. <The square> that we draw touches the <second> square. What is its area?

Graphics

,Rectangle[],Parallelogram[{0,1/2},{{1/2,1/2},{1/2,-1/2}}],Rectangle[{1/4,1/4},{3/4,3/4}]

Out[]=

In[]:=

Area[Rectangle[{1/4,1/4},{3/4,3/4}]]//babylonian

Out[]=



As noted by Joran Friberg (2007), the situation in problems #7–8, #10–11 and #18 resembles the famous geometric passage in Plato's Meno, where Socrates tries to get a slave boy to figure out on his own how a square can be constructed that is twice as large as a given square.

Problem 10 (and 11)

We have a unit square. Inside it we draw 8 wedges. What are their [areas]?

Graphics

,Table[Rectangle[{i,j},{i+1/2,j+1/2}],{i,0,1/2,1/2},{j,0,1/2,1/2}],Parallelogram[{0,1/2},{{1/2,1/2},{1/2,-1/2}}]

Out[]=

In[]:=

Area[Triangle[{{0,1/2},{1/2,1/2},{1/2,1}}]]//babylonian

Out[]=



Problem #11 is identical to problem #10.

Problem 12

We have a unit square. Inside it [we draw] 16 wedges. What are their areas?

Out[]=

In[]:=

Area[Triangle[{{0,0},{0,1/2},{1/4,1/4}}]]//babylonian

Out[]=



Problem 13

We have a unit square. Inside it we draw 4 ox's brows and 2 wedges. What are their areas?

Out[]=

Area for an ox's brow:

In[]:=

Area[Triangle[{{1/4,1/2},{1/2,1/2},{1/2,1}}]]//babylonian

Out[]=



Area of its complement:

In[]:=

4Area[Polygon[{{0,1/2},{1/4,1/2},{1/2,1},{0,1}}]]//babylonian

Out[]=



Problem 14

We have a unit square. We extend half on each side and we draw a square-side. Inside the second square-side we draw a third square-side. What is its area?

In[]:=

Graphics

,Rectangle[],Rectangle[{1/8,1/8},{7/8,7/8}],Parallelogram[{1/8,1/2},{{3/8,3/8},{3/8,-3/8}}]

Out[]=

In[]:=

Area[Parallelogram[{1/8,1/2},{{3/8,3/8},{3/8,-3/8}}]]//babylonian

Out[]=



Problems 15–16 (missing)

Diagrams and text missing.

Problem 17

We have a unit square. <Inside it> we draw 12 wedges and 4 squares. What are their areas?

Graphics

,Rectangle[],Parallelogram[{0,1/2},{{1/2,1/2},{1/2,-1/2}}],Rectangle[{1/4,1/4},{3/4,3/4}],Line[{{0,0},{1/4,1/4}}],Line[{{0,1},{1/4,3/4}}],Line[{{1,1},{3/4,3/4}}],Line[{{1,0},{3/4,1/4}}],Table[Rectangle[{i,j},{i+1/4,j+1/4}],{i,1/4,1/2,1/4},{j,1/4,1/2,1/4}]

Out[]=

In[]:=

Area[Triangle[{{0,0},{0,1/2},{1/4,1/4}}]]//babylonian

Out[]=



In[]:=

Area[Rectangle[{1/4,1/4},{1/2,1/2}]]//babylonian

Out[]=



Problem 18

We have a unit square. Inside it we draw 4 wedges. What are their areas?

In[]:=

Graphics

,Rectangle[],Triangle[{{0,0},{0,1},{1/2,1/2}}],Triangle[{{1,0},{1,1},{1/2,1/2}}]

Out[]=

In[]:=

1/4//babylonian

Out[]=



Problems 19–22 (missing)

Diagrams and text missing.

Problem 23

We have a unit square. Inside it <we draw> 4 squares, 4 rectangles and 4 wedges. What are their areas?

In[]:=

Graphics

,Rectangle[],Rectangle[{1/4,1/4},{3/4,3/4}],Table[Rectangle[{i,j},{i+1/4,j+1/4}],{i,0,3/4,3/4},{j,0,3/4,3/4}],Triangle[{{1/4,1/4},{1/4,3/4},{1/2,1/2}}],Triangle[{{3/4,1/4},{3/4,3/4},{1/2,1/2}}]

Out[]=

In[]:=

Area[Rectangle[{1/4,0},{3/4,1/4}]]//babylonian

Out[]=



In[]:=

Area[Triangle[{{1/4,1/4},{1/2,1/2},{3/4,1/4}}]]//babylonian

Out[]=



Problem 24

We have a unit square. Inside it we draw 16 squares. What are their areas?

Graphics

,Table[Rectangle[{i,j},{i+1/4,j+1/4}],{i,0,3/4,1/4},{j,0,3/4,1/4}]

Out[]=

In[]:=

1/(44)//babylonian

Out[]=



Problem 25

The... is 1 cable... the width.

In[]:=

Graphics

,Rectangle[],Disk[{3/8,1/2},1/4,{Pi/2,3Pi/2}],Triangle[{{3/8,1/4},{3/8,3/4},{7/8,1/2}}]

Out[]=

In[]:=

Area[RegionUnion[Disk[{3/8,1/2},1/4,{Pi/2,3Pi/2}],Triangle[{{3/8,1/4},{3/8,3/4},{7/8,1/2}}]]]//babylonian

Out[]=

⟶

Problems 26–27 (missing)

Diagrams and text missing.

Problem 28

We have a unit square. We extend a border on each side and we draw a square-side. Inside the square that [we drew is] 1 cow's nose. What is [its area]?

In[]:=

Graphics

,Rectangle[],Rectangle[{1/4,1/4},{3/4,3/4}],DiskSegment[{1/4,1/4},1/4,{0,Pi/2}],DiskSegment[{3/4,1/4},1/4,{Pi/2,Pi}],DiskSegment[{3/4,3/4},1/4,{Pi,3Pi/2}],DiskSegment[{1/4,3/4},1/4,{3Pi/2,2Pi}]

Out[]=

In[]:=

Area[RegionDifference[Rectangle[{1/4,1/4},{3/4,3/4}],RegionUnion[Disk[{1/4,1/4},1/4,{0,Pi/2}],Disk[{3/4,1/4},1/4,{Pi/2,Pi}],Disk[{3/4,3/4},1/4,{Pi,3Pi/2}],Disk[{1/4,3/4},1/4,{3Pi/2,2Pi}]]]]//babylonian

Out[]=

⟶

Problem 29

Text missing.

In[]:=

Graphics

,Rectangle[],Rectangle[{1/4,1/4},{3/4,3/4}],Table[Rectangle[{i,j},{i+1/4,j+1/4}],{i,0,3/4,3/4},{j,0,3/4,3/4}],DiskSegment[{1/4,1/4},1/4,{0,Pi/2}],DiskSegment[{3/4,1/4},1/4,{Pi/2,Pi}],DiskSegment[{3/4,3/4},1/4,{Pi,3Pi/2}],DiskSegment[{1/4,3/4},1/4,{3Pi/2,2Pi}]

Out[]=

Assume the missing problem asks for the darker gray area with pieces summing up to a whole disk of radius

and a

square. Hence the total area is:

In[]:=

^2+

//babylonian

Out[]=

⟶

Out[]=

Problem 30

We have a unit square. We extend a border on each side and we draw the shape of a lyre. What is its area?

In[]:=

Graphics

,Rectangle[],Rectangle[{1/4,0},{0,1}],Rectangle[{1,0},{3/4,1}],DiskSegment{1/2,-1/4},12

,{45Degree,135Degree}

Out[]=

In[]:=

2Area[Rectangle[{1/4,0},{0,1}]]+AreaDiskSegment{1/2,-1/4},12

,{Pi/4,3Pi/4}//babylonian

Out[]=