Australasian Science Magazine

Daniel Mansfield holds the Plimpton 322 tablet at the Rare Book and Manuscript Library at Columbia University in New York. Credit: Andrew Kelly/UNSW

By Daniel Mansfield

By decoding an ancient stone tablet, researchers have realised that the Babylonians employed a form of trigonometry that is very different to our own.

Plimpton 322 is one of the most famous tablets from the Old Babylonian period (19th–16th century BCE). It was obtained, perhaps illegally, in the southern Iraq desert where it found its way into the hands of Edgar Banks (the inspiration for the fictional character Indiana Jones).

In about 1922 Banks sold the tablet to the famous publisher and antiquities collector George Arthur Plimpton. Shortly before his death, Plimpton bequeathed his entire collection to the University of Columbia, where the artefact known as “Plimpton 322” remains today.

The tablet itself contains a fragment of a larger table: four columns with headings, and 15 rows of numbers remain. Neither Banks nor Plimpton knew any more than that. In 1945 the surviving column headings were translated, and the numbers were revealed to describe the sides of right triangle of steadily decreasing slope, along with a squared ratio of sides.

This presented quite a conundrum: angles and trigonometry as we know it would not be invented for another 1500 years, and yet this appears to be some kind of trigonometric table. The answer, which was recently published in Historia Mathematica, is that the Babylonians had a form of trigonometry that was so very different to our own that it has taken us more than 70 years just to recognise it.

We inherited our concepts regarding right triangles from ancient Greece. There is a hypotenuse, an angle θ, and the opposite and adjacent sides relative to that angle. We prefer the trigonometric ratios sin(θ), cos(θ) and tan(θ), which are almost always approximated. Table 1 is an example of a trigonometric table from modern times.

The Babylonians were much more ancient, but also very familiar with the geometry of right triangles. They knew about Pythagoras’ theorem, similar triangles, ratios and area. Furthermore, they worked in a sexagesimal (base-60) number system and were able to perform sophisticated numerical operations, often exactly.

The Babylonians described a right triangle as half of a rectangle with only a “short side”, “long side” and “diagonal”. They also used ratios, but not the ones described above. Their preferred ratio appears to be the “short side”/“long side”, which they called the fruit of the triangle. Interestingly, the same concept is also found in ancient Egyptian mathematics.

Let’s take a closer look at the tablet itself. The column headings are a good place to start because they should tell us something about the function of the tablet. The headings can be translated as “The square of the diagonal, subtract 1 to obtain the square of the width”, “The ib-si of the short side”, “The ib-si of the diagonal”, and “row number”. The word ib-si is difficult to interpret; it usually means the result of some operation. The final heading simply reads “Its name”.

Now let us examine the first row of numbers written in the sexagesimal system of the Babylonians:

1.59.00.15     1.59     2.49     row 1

Using our more familiar decimal system, these numbers are:

28,561/14,400     119     169     row 1

According to the column heading, 28,561/14,400 is a square number: (169/120)². But this is no ordinary square number: if we subtract one then we should see another square number and, as promised:

28,561/14,400 – 1 = 14,161/14,400 = (119/120)²

The heading tells us that we should interpret (169/120)² as the square of the diagonal and (119/120)² as the square of the short side. So, we are looking at a right triangle with short side 119/120, diagonal 169/120 and long side 1. The next two numbers appear to be the numerator of the short side and diagonal, which is consistent with the column headings. The final column literally says that this is “row 1”. Note that the short side and long side are almost equal in length; indeed tan^-1(119/120) is approximately 44.76, so this triangle is almost half a square.

The next row of Plimpton 322 contains (in decimal):

(4825/3456)²     3367     4825     row 2

Performing the same analysis as before, we can interpret from this row a description of a right triangle with short side 3367/3456, long side 1 and diagonal 4825/3456. Because tan^-1(3367/3456) is approximately 44.25, this right triangle is slightly flatter than the one described in row 1.

The Babylonians would not have thought about these triangles in terms of their angles, but it is instructive for us to notice that the 15 right triangles listed on Plimpton 322 are decreasing from almost 45° to almost 28° at roughly 1° decrements.

The purpose of a trigonometric table is to describe the relationship between the sides of a right triangle. A modern trigonometric table achieves this using the concept of angle.

But what the Babylonians have showed us through this tablet is really quite remarkable: the concept of angle is not a necessary part of this framework. Another way to construct a trigonometric table is to use a squared ratio of sides as an index. The advantage is that it is computationally simple to use the squared ratio rather than take a square root.

This is quite significant if you are living a computationally restricted life in ancient times. Given any two sides of a right triangle, the square of the unknown side can be computed by Pythagoras’ theorem. Once you know the square of all three sides, you can then compute the ratio of the squares by division. Consider the calculation of the ratio (diagonal/long side)2 in the following three scenarios.

Case 1. You have a right triangle with long side 4 and diagonal 5. What is the ratio (diagonal/long side)²?

5²/4² = 1.5625

Case 2. You have a right triangle with short side 3 and diagonal 5. What is the ratio (diagonal/long side)²?

5²/(5²–3²) = 1.5625

Case 3. You have a right triangle with short side 3 and long side 4. What is the ratio (diagonal/long side)²?

(3² + 4²)/(4²) = 1.5625

If the purpose of the first column is to serve as an index, what is the purpose of the next two columns? This is the “data” of the table.

For example, suppose you are a Babylonian surveyor and your task is to measure a field in the shape of a right triangle (we will work in decimal for convenience). You measure the long side of the field to be 121, and the diagonal to be 170. What is the length of the short side?

First you compute the index 170²/121² ≈ 1.97, and look up this value in column 1 of Table 2.

The number 1.9834028 found in row 1 is the best match for this triangle, and using the ib-si of the width and the ib-si of the diagonal you can approximate the length of the short side as:

119/169 × 170 ≈ 119.7

In modern times we would use Pythagoras’ theorem to find the square of the short side 1702 – 1212 = 14,259, and then use some numerical tool to more accurately approximate the length of the short side as √14,259 ≈ 119.4.

But the modern method is more computationally intensive and produces a level of accuracy that is beyond the needs of ancient surveyors.

This new interpretation makes Plimpton 322 the world’s oldest trigonometric table, predating Hipparchus’ table of chords by more than 1500 years. Furthermore, each of the numbers in Plimpton 322 is exact – there is no approximation in this table whatsoever. This makes it also the world’s only completely accurate trigonometric table.

But how does it stack up compared with more modern tables? Before answering this question, we must address the problem that we only see a fragment of the full table.

In 1964 de Solla Price was able to reconstruct the missing rows of Plimpton 322 and showed that the complete table would contain 38 rows describing triangles ranging from almost 45° all the way down to 0°. This reconstruction adds powerfully to the trigonometric interpretation. Not only does the reconstructed table cover the entire range of right triangles, but such a table would be superior in power to Madhava’s table of sines 3000 years later.

There is a saying that mathematics is a universal language. There may well be some truth in the idea, but if it is a language then it is spoken by people, and so mathematics cannot be entirely isolated from people and their history. Trigonometry based on angles is part of the mathematical culture we inherited from the ancient Greeks, and today we consider this as the only form of trigonometry. The Babylonians however, were more ancient and they thought very differently to the Greeks.

In Plimpton 322 we see a different form of exact trigonometry that was well ahead of its time. This shows us more than just a nifty new way of studying right triangles. It highlights our cultural assumptions about trigonometry that were thought to be universal.