History of Mathematics Project | Babylonian Metric Algebra Problems Tablet

History of
Mathematics

Project

around 1900–1700 BCE

Babylonian Metric Algebra Problems Tablet

Metric algebra problem texts

Each line of the MS 5112 tablet denotes a different mathematical problem. The problems include two ways to extend a square with a solution to complete the square (line 1) and a rectangle's area that is equal to its length plus the front solved with metric algebra (line 8).

Babylonian Metric Algebra Problems Tablet

While the tablet originally contained 23 metric algebra problem texts, only 13 full problems and 3 partly preserved problems remain intact. However, even in fragmentary form, the tablet attests to the proficiency of the ancient Babylonians at solving simple systems of linear and quadratic equations.

Artifact dimensions

8.9 cm × 9.8 cm × 2.7 cm

Original artifact location

Babylonia (historical name), Iraq (current name)

Current artifact location

Oslo, Norway

Catalog number

MS 5112

Timeline

Interactive Content

Computational Explanation

As illustrated by Friberg 2007, while the tablet originally contained 23 metric algebra problem texts, only 13 full problems and 3 partly preserved problems remain intact:

Friberg (2007) gives a detailed analysis of these problems as well as many schematic and illustrative figures. The discussion of the first two problems included here closely follows that discussion.

Problem 1

Problem 1 on MS 5112 consists of a single exercise spread over four lines of text:

The best available interpretation of this problem's text states:

"The area (of a square) and two sides of the square are added. The result is 2 (times 60). What are the area and the sides?"

Restating using modern notation and letting a be the side length of the square:

+2a260

… where the total value has been multiplied by 60 to allow integer solutions, a step that is possible since the base-60 numeral system of the Babylonians did not fix place values, meaning the interpretation of written numbers is defined only up to powers of 60.

Solving the algebraic equation:

In[]:=

soln=Solve[

+2a260,a,PositiveReals]//Flatten

Out[]=

{a10}

… so the square has side length 10 and area:

In[]:=

FullSimplify[a^2/.soln]

Out[]=

100

However, a Babylonian mathematician would almost certainly interpret the question literally as referring to a square field and would approach the solution in geometric terms instead of formulating and solving it algebraically. Therefore, when the text says that the area of a square and two sides of the square together are equal to 2 (00) (meaning

260120

), what is meant is that a square field is extended by a unit in two directions (Friberg 2007, p. 310):

Out[]=

The Babylonians would therefore set up the problem as:

+2peAe1A2(00)

… (where 2 00 expresses

260+00120

in sexagesimal) and would solve it as:

(p+e)

A+

2(00)+1201p+e

201

11p11-e11-110.

Problem 2

The second problem is posed as follows:

Rephrasing in modern language states the problem as:

"The sum of the areas of two squares is 21 40. The side of the first square is 30. What is the side of the second square?"

… and the solution procedure as:

"Remove the square of the side from the sum of the squares of the two sides. The square side of the remainder is the second side."

The sexagesimal number 21 40 corresponds to:

In[]:=

FromDigits[{21,40},60]

Out[]=

1300

… as a decimal number, so the problem asks to find the value of b, solving the equation:

1300

… giving solution:

In[]:=

Reduce[

1300,b,PositiveIntegers]

Out[]=

b20

Other Resources

Wolfram Demonstration
Babylonian Numerals

Additional Reading

Friberg, J. "MS 5112. A Text with Equations for Squares and Rectangles." § 11.2 in A Remarkable Collection of Babylonian Mathematical Texts: Manuscripts in the Schøyen Collection: Cuneiform Texts I. New York: Springer, pp. 308–341, 2007.

Babylonian Metric Algebra Problems Tablet

Timeline

Interactive Content

Computational Explanation

Problem 1

Problem 2

Other Resources

Wolfram Demonstration

Additional Reading

Additional Links