Science

Backgrounder: Plimpton 322 and trigonometry

Plimpton 322. Image: UNSW/Andrew Kelly
Friday, 25 August, 2017
Deborah Smith

What is trigonometry?

Trigonometry comes from the Greek terms trigonon, for triangle, and metron, for measure.

Trigonometry is the systematic study of triangles. In general, a trigonometric table allows you to determine the shape of a right-angle triangle based on any ratio of two sides.

Measurement of angles and circular functions were first introduced by Greek astronomers about 300 BC and have been the foundation of modern trigonometry ever since.

The Greek astronomer Hipparchus who lived about 120 years BC has long been regarded as the father of trigonometry, with his “table of chords” considered the oldest trigonometric table.

Chords are the base of triangles which have one point at the centre of a circle and the other two points on the circle.

Today, high school students work out the angles of triangles and lengths of sides using trigonometric functions such as sine, cosine and tangent, which are ratios of sides.

A modern trigonometric table contains a list of right-angle triangles with diagonal 1 and approximate side lengths sin θ and cos θ, along with the ratio tan θ  = sin θ/cos θ. It is indexed by the angle θ.

Modern trigonometry relies solely on approximation.

Plimpton 322

The tablet Plimpton 322 is named after its owner George Arthur Plimpton, and measures 12.7 cm by 8.8 cm. He bought it from Edgar Banks in about 1922.

Based on a comparison of writing styles with other Babylonian tablets, Plimpton 322 has been dated to between 1822 to 1762 BC, which is around the time of the Babylonian King Hammurabi.

Vertical column lines are also drawn on the back of the tablet, which is otherwise empty.

In 1945, the Austrian mathematician Otto Neugebauer and his associate Abraham Sachs were the first to note that Plimpton 322 has 15 pairs of numbers forming parts of Pythagorean triples.

A Pythagorean triple consists of three whole numbers a, b and c such that a2 + b2 = c2. The integers 3, 4 and 5 are a well-known example of a Pythagorean triple, but the values on Plimpton 322 are often considerably larger with, for example, the first row referencing the triple 119, 120 and 169.

Plimpton 322 – what do the numbers represent?

Column 1

Column 2

Column 3

Column 4

Column 5

Column 6

β = b/l

δ = d/l

δ2

base b

diagonal d

Row number

0.99166666

1.40833333

1.98340277

119

169

1

0.97424768

1.39612268

1.94915855

3367

4825

2

0.95854166

1.38520833

1.91880212

4601

6649

3

0.94140740

1.37340740

1.88624790

12709

18541

4

0.90277777

1.34722222

1.81500771

65

97

5

0.88611111

1.33611111

1.78519290

319

481

6

0.84851851

1.31148148

1.71998367

2291

3541

7

0.83229166

1.30104166

1.69270941

799

1249

8

0.80166666

1.28166666

1.64266944

481

769

9

0.76558641

1.25941358

1.58612256

4961

8161

10

0.75

1.25

1:5625

45

75

11

 

Plimpton 322 has four columns and 15 rows of numbers written on it in the cuneiform script of the time using a base 60, or sexagesimal, system.

The left-hand edge of the tablet is broken and the UNSW researchers build on previous research to present new mathematical evidence that there were originally 6 columns and that the tablet was meant to be completed with 38 rows.

The table above lists the numbers in the first 11 rows of the completed version of Plimpton 322, written in our decimal number system, approximated to eight decimal points. (For more rows see the published paper).

UNSW Sydney mathematicians have shown that the completed tablet contains the three ratios of sides of a right-angle triangle – but written exactly.

A known ratio of sides of a right-angle triangle could be compared to these lists to work out the unknown ratios.

For a right-angle triangle with base b, long side l, and diagonal d.

Column 1 contains the exact ratio b/l.

Column 2 contains the exact ratio d/l.

A modern trigonometric table would also contain the ratio b/d or d/b. But this ratio could not be written exactly by the Babylonians in base 60.

So, to maintain the exact nature of the table, they split this ratio into three parts.

Column 3 contains the exact ratio (d/l)2 and was used as an index into the table when b/d or d/b was already known.

Columns 4 and 5 contain the numbers b and d with common factors removed and was used so the scribe could construct their own approximation to b/d or d/b.

Column 6 is just the row number

An example of how P322 could have been used

Suppose that a ramp leading to the top of a ziggurat, or stepped pyramid, is 56 cubits long, and the vertical height of the ziggurat is 45 cubits.

What is the distance x from the outside base of the ramp to the point directly below the top?

Solution using Plimpton 322

For this right-angle triangle, l = 45 and d = 56.

This means δ = d/l = 56/45 = about 1.2444.

On the tablet, the closest value of δ in column 2 is found in row 11, where δ = 1.25.

From columns 4 and 5 in row 11 you can work out that the corresponding ratio for this triangle is b/d = 45/75 = 3/5 = 0.6.

Thus x approximately equals 56 x 0.6 = 33.6

Using a calculator today, we can determine that x approximately equals the square root of 56 2

452 = 33.3317.

The UNSW mathematicians show the Babylonian approach, which avoids calculating square roots, was more accurate than a trigonometric sine table approach devised by the Indian mathematician Madhava more than 3000 years later.

Why is P322 the most accurate trigonometric table and why is Babylonian mathematics relevant today?

Babylonians counted in base 60, with digits 1 to 59, rather than the base ten we use, with digits 1 to 9, as well as place holder of 0.

Unlike us, they also only used exact numbers, and avoided irrational numbers that cannot be expressed as a ratio of two integers.

Mathematics in Babylonian days was sophisticated, but very practical. Scribes were required to master a number of procedures and tables, and they were able to make very large calculations.

For example, they had exact tables for multiplication, reciprocals, squares and square roots.

P322 is a trigonometric table that is the world’s only completely accurate one, because it only includes Pythagorean triangles with side lengths that are whole numbers. This ensures that the ratios of sides and squared ratios can be represented in the table without approximation.

We still use base 60 for measuring time – there are sixty seconds in a minute and sixty minutes in an hour.

The advantage of this is that 60 has three prime factors (2, 3, and 5) so an hour can be divided exactly into many factors: 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes and 1 minute.

In base ten there are fewer exact factors: 5, 2 and 1.

Computers work in base 2, which has only one exact factor: 1

The price for the simplicity of working in base 2 is precision. A computer can’t even divide three properly and must rely on an approximation to do calculations.

Working in base 60 could have great advantages for modern computing, because a lot of computing energy is spent making calculations using inexact numbers, and errors are also introduced when approximations are made.

“If computers could be programmed to work in base 60 it might be possible to increase accuracy and decrease cost,” says Dr Mansfield. “This would be particularly beneficial for work requiring high accuracy, such as surveying, or scientific calculations, or where power is an issue, such as in space probes or where speed is an issue, such as in computer graphics.”

It may also be possible to teach students to think like Babylonians, and this approach to geometrical measurements could be easier than the way trigonometry is taught now using irrational numbers and transcendental functions such as sine and cosine.