On a 3700-year-old clay tablet that has been hidden in plain sight in an Istanbul museum for over a century, a UNSW mathematician has unearthed the beginnings of applied geometry.
The tablet, known as Si.427, was unearthed in what is now central Iraq in the late 1800s, but its importance was unknown until today, according to the UNSW scientist's investigation.
Most intriguing of all, Si.427 is likely to be the oldest known example of applied geometry – and a gripping human storey of land surveying is revealed in the study published today in Foundations of Science.
"Si.427 dates from the Old Babylonian (OB) period — 1900 to 1600 BCE," says UNSW Science's School of Mathematics and Statistics principal researcher Dr Daniel Mansfield.
"It's the only known cadastral document from the OB period, which is a surveyor's plan for defining land borders. In this case, it provides legal and geometric information on a field that has been divided after some of it has been sold."
Because the surveyor employs what are now known as "Pythagorean triples" to produce exact right angles, this is a noteworthy artefact.
"The tablet's discovery and study have significant consequences for the history of mathematics," says Dr Mansfield. "This is nearly a thousand years before Pythagoras was born, for example."
Hot on the heels of another world-first find
Dr Mansfield hypothesised in 2017 that Plimpton 322, a remarkable object from the same period, was a unique type of trigonometric table.
"It is widely assumed that the ancient Greeks invented trigonometry - the area of mathematics concerned with the study of triangles – in the second century BCE while examining the night sky," says Dr Mansfield.
"However, the Babylonians devised their own 'proto-trigonometry' to handle problems involving the measurement of the ground rather than the sky."
The tablet discovered today is assumed to have existed before Plimpton 322; in fact, surveying issues are thought to have influenced Plimpton 322.
"There's a whole zoo of different-shaped right triangles. However, Babylonian surveyors can only employ a restricted number of them. Plimpton 322 is a methodical investigation of this zoo in order to identify valuable shapes "Dr Mansfield agrees.
Tablet purpose revealed: surveying land
In 2017, the researchers theorised on the Plimpton 322, hypothesising that it was most likely employed for practical purposes such as building palaces and temples, canal construction, or surveying fields.
"With this new tablet, we can finally see why they were interested in geometry in the first place: to set down accurate land boundaries," says Dr Mansfield.
"This comes from a time when the property was beginning to become private, and people began to conceive of land in terms of land and your land,' with the goal of establishing an appropriate boundary in order to have pleasant neighbourly relationships. And this is exactly what this tablet says right away. It's a field that's being divided, and new limits are being established."
There are also hints regarding the stories behind these boundaries found on other tablets from the historical period.
"Another tablet mentions a conflict between Sin-bel-apply – a significant character referenced on numerous tablets, including Si.427 – and a wealthy female landowner," adds Dr Mansfield.
"The quarrel is over a grove of valuable date palms that runs between their two homes. To settle the dispute, the local administrator agrees to send out a surveyor. It's easy to see how precision was crucial in resolving conflicts between such powerful people."
According to Dr. Mansfield, the manner these boundaries are drawn demonstrates true geometric understanding.
Nobody anticipated the Babylonians to use Pythagorean triples in this fashion, according to Dr. Mansfield. "It's more akin to pure mathematics, and it's based on real-world problems."
It's easier said than done to create right angles.
Making a rectangle with sides 3 and 4 and a diagonal of 5 is a simple approach to construct an accurate right angle. The 3-4-5 "Pythagorean triple" is formed by these unusual numbers, and a rectangle with these measures has mathematically perfect right angles. This was crucial to ancient surveyors, and it is still in use today.
"The ancient surveyors who created Si.427 did something even better: they constructed correct right angles using a variety of Pythagorean triples, both as rectangles and right triangles," Dr. Mansfield explains.
In the base 60 Babylonian number system, however, working with prime numbers larger than 5 is challenging.
"This poses a very specific challenge — because of their unique base 60 number system, only a few Pythagorean forms may be used," Dr Mansfield explains.
"It appears that the author of Plimpton 322 looked through all of the Pythagorean shapes to discover the ones that were beneficial.
"The term 'proto-trigonometry' refers to a thorough and numerical grasp of the practical usage of rectangles, but it is not the same as modern trigonometry, which uses sin, cos, and tan."
Hunting down Si.427
Dr Mansfield first learnt about Si.427 from excavation records — the tablet was discovered during the Sippar expedition in 1894, in what is now Iraq's Baghdad province.
"It was extremely difficult to track down the tablet from these records and locate it physically – the report stated that the tablet was taken to the Imperial Museum of Constantinople, which no longer exists.
"Using that piece of information, I set out on a journey to find it, speaking with a variety of officials at Turkish government agencies and institutions until a photo of Si.427 arrived in my inbox in mid-2018.
"It was then that I discovered it was on display at the museum. Even after finding the piece, it took months to completely comprehend its significance, so it's quite wonderful to be able to finally share that narrative."
Dr Mansfield's next goal is to discover what other uses the Babylonians had for their proto-trigonometry.
Dr Mansfield has still to solve one mystery: on the back of the tablet, at the bottom, the sexagesimal number '25:29' is written in large print - think of it as 25 minutes and 29 seconds.
"I have no idea what these figures imply – it's a complete mystery to me. I'd love to talk to any historians or mathematicians who might have an idea what these statistics are attempting to tell us! ".
