# A new angle on ancient trigonometry

This article first appeared in Cosmos Weekly on 6 August 2021. For more stories like this, subscribe to Cosmos Weekly.

For some people, the word “trigonometry” conjures up images of right-angled triangles, or maybe even our old friends sine, cosine and tangent. And that may mean tears of blood, as “trigonometry” is a trigger for many from their school days.

But without “trig”, architects would botch your new extension, GPS wouldn’t exist – and I would hate to see what a CT scan would do to you if we didn’t understand trigonometric functions as we shot X-rays at your flesh.

You may recall an old Greek chap, Pythagoras, and some of the many discoveries attributed to his name. Most famously, perhaps, is Pythagoras’ theorem, which gives us this Pythagorean equation: a2 + b2 = c2 . This tells us that if we have any right-angled triangle, the sum of the squares of the perpendicular sides is equal to the length of the hypotenuse squared.

Any three positive integers that satisfy the Pythagorean equation are known as “Pythagorean triples”. For example, (3,4,5) is a Pythagorean triple because 32 + 42 = 52. A less obvious Pythagorean triple is (140, 171, 221), and we know that, as often happens in maths, there is an infinite number of these things.

However, over a thousand years before Pythagoras even set foot on Earth, the Babylonians were not only aware of Pythagorean triples, they were using them in sophisticated ways.

Enter Plimpton 322, a Babylonian clay tablet that was found in the ancient Sumerian city of Larsa (in modern-day Iraq) and has been dated back to between 1822 and 1762 BCE. This partially crumbled mathematical document, approximately 3,700 years old, is simply a table of numbers (written in cuneiform script) comprising four columns and 15 rows. The wondrous thing about this tablet is that not only does each row contain two of the three numbers in a series of Pythagorean triples, but these numbers are also written in sexagesimal (base 60) notation, which provides mathematical advantages for writing and calculating fractions.

We express numbers in a decimal system, or base 10, in which we use the 10 unique digits, or symbols, 0, 1, 2, …, 9. The sexagesimal number system used by the Babylonians has unique symbols (or unique combinations of symbols) to represent the numbers 1 to 59.

The reason that 60 is an advantageous base has to do with the number of common factors it has. The number 60 is what we would refer to as a “superior highly composite number”, which you know must be good due to the double superlative. It can be divided exactly by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60, whereas 10 can only be divided exactly by 1, 2, 5 and 10. And this becomes important when we’re looking at the information contained on the tablet.

To simplify the discussion, let’s use a right-angled triangle with shorter perpendicular side s, longer perpendicular side l, and diagonal d, such that s2 + l2 = d2 .

Columns two and three of Plimpton 322 simply contain values for s and d respectively for the series of Pythagorean triples. Column four is just a list of the numbers 1 to 15, so we can remember which row we’re up to. But column one represents the ratio d2 / l2, and since we’re given the value of d in column three, we can calculate l, and voila … a complete Pythagorean triple (s,l,d) is revealed!

That’s how Plimpton 322 works. However, there has been some debate as to the purpose of this tablet. Many historians believed that it was simply a teaching aid. But at the root of this debate is the idea that there is a precise (and computationally simpler) description of trigonometry that predates the ancient Greeks, to whom the discovery of trigonometry is usually attributed. Right-angled triangles on the Plimpton 322 tablet: the world’s first trigonometric table. Credit: UNSW/Andrew Kelly.

In 2017, two researchers from the University of New South Wales, Dr Daniel Mansfield and Professor Norman Wildberger, proclaimed that the numerical complexity of Plimpton 322 makes it historically and mathematically significant because it is both the first trigonometric table and the only trigonometric table that is precise.

However, some historians were unconvinced that this tablet was anything other than a simple scribal school text. That scepticism set Mansfield on a mission. “What were the Mesopotamians doing?” he asked.

The one really obvious answer he could come up with was: surveying.

Like a modern-day Indiana Jones, he went searching in museums, private collections and libraries around the world for similar types of mathematical documents from the same period that also contained Pythagorean triples. As it happens, the missing piece of the puzzle was hiding in plain sight, on display at the Istanbul Archaeological Museum.

Si. 427, a lenticular tablet from the Old Babylonian period, contained no less than three Pythagorean triples – (5, 12, 13) twice, and (8, 15, 17) – and was used by surveyors to apportion private land in an exceptionally accurate way, by constructing perpendicular field boundaries using those Pythagorean triples.

Previously, surveying documents had simply been agricultural estimates. But people started becoming concerned about exact boundaries when they started owning land privately.

This discovery allowed Mansfield to arrive at a new hypothesis as to the true purpose of Plimpton 322: a theoretical investigation to find rectangles with regular sides, possibly inspired by the need for such rectangles in surveying. “Regular” here means that the length of the side is a factor of 60.

“To use Pythagorean triples for surveying, they need to know what rectangles are usable,” explains Mansfield.

So, a simple Pythagorean triple such as (5, 12, 13) is not particularly useful because 13 is not a factor of 60 and, hence, it does not appear on Plimpton 322. “They’ve gone through all the Pythagorean triples they could possibly think of and then looked at them to say which sides are regular,” says Mansfield.

This revelation about Plimpton 322 supports both the idea that it may have simply been a theoretical investigation to find rectangles with regular sides, and that this theoretical work was possibly motivated by the practical need of such rectangles in surveying.

Pure maths and applied maths coming together, like the true friends they should be.

This article first appeared in Cosmos Weekly on 6 August 2021. To see more in-depth stories like this, subscribe today and get access to our weekly e-publication, plus access to all back issues of Cosmos Weekly. 