Posted on 08/04/2021 8:55:53 AM PDT by Red Badger
Si.427 is a hand tablet from 1900-1600 BC, created by an Old Babylonian surveyor. It’s made out of clay and the surveyor wrote on it with a stylus. Credit: Must credit UNSW Sydney
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A UNSW mathematician has revealed the origins of applied geometry on a 3700-year-old clay tablet that has been hiding in plain sight in a museum in Istanbul for over a century.
The tablet – known as Si.427 – was discovered in the late 19th century in what is now central Iraq, but its significance was unknown until the UNSW scientist’s detective work was revealed today.
Most excitingly, Si.427 is thought to be the oldest known example of applied geometry – and in the study released today (August 4, 2021) in Foundations of Science, the research also reveals a compelling human story of land surveying.
“Si.427 dates from the Old Babylonian (OB) period – 1900 to 1600 BCE,” says lead researcher Dr. Daniel Mansfield from UNSW Science’s School of Mathematics and Statistics.
“It’s the only known example of a cadastral document from the OB period, which is a plan used by surveyors define land boundaries. In this case, it tells us legal and geometric details about a field that’s split after some of it was sold off.”
This is a significant object because the surveyor uses what are now known as “Pythagorean triples” to make accurate right angles.
“The discovery and analysis of the tablet have important implications for the history of mathematics,” Dr. Mansfield says. “For instance, this is over a thousand years before Pythagoras was born.”
Hot on the heels of another world-first find In 2017, Dr. Mansfield conjectured that another fascinating artifact from the same period, known as Plimpton 322, was a unique kind of trigonometric table.
“It is generally accepted that trigonometry – the branch of maths that is concerned with the study of triangles – was developed by the ancient Greeks studying the night sky in the second century BCE,” says Dr. Mansfield.
“But the Babylonians developed their own alternative ‘proto-trigonometry’ to solve problems related to measuring the ground, not the sky.”
The tablet revealed today is thought to have existed even before Plimpton 322 – in fact, surveying problems likely inspired Plimpton 322.
“There is a whole zoo of right triangles with different shapes. But only a very small handful can be used by Babylonian surveyors. Plimpton 322 is a systematic study of this zoo to discover the useful shapes,” says Dr. Mansfield.
Tablet purpose revealed: surveying land Back in 2017, the team speculated about the purpose of the Plimpton 322, hypothesizing that it was likely to have had some practical purpose, possibly used to construct palaces and temples, build canals or survey fields.
“With this new tablet, we can actually see for the first time why they were interested in geometry: to lay down precise land boundaries,” Dr. Mansfield says.
“This is from a period where land is starting to become private – people started thinking about land in terms of ‘my land and your land’, wanting to establish a proper boundary to have positive neighborly relationships. And this is what this tablet immediately says. It’s a field being split, and new boundaries are made.”
There are even clues hidden on other tablets from that time period about the stories behind these boundaries.
“Another tablet refers to a dispute between Sin-bel-apli – a prominent individual mentioned on many tablets including Si.427 – and a wealthy female landowner,” Dr. Mansfield says.
“The dispute is over valuable date palms on the border between their two properties. The local administrator agrees to send out a surveyor to resolve the dispute. It is easy to see how accuracy was important in resolving disputes between such powerful individuals.”
Dr. Mansfield says the way these boundaries are made reveals real geometric understanding.
“Nobody expected that the Babylonians were using Pythagorean triples in this way,” Dr. Mansfield says. “It is more akin to pure mathematics, inspired by the practical problems of the time.”
Creating right angles – easier said than done One simple way to make an accurate right angle is to make a rectangle with sides 3 and 4, and diagonal 5. These special numbers form the 3-4-5 “Pythagorean triple” and a rectangle with these measurements has mathematically perfect right angles. This is important to ancient surveyors and still used today.
“The ancient surveyors who made Si.427 did something even better: they used a variety of different Pythagorean triples, both as rectangles and right triangles, to construct accurate right angles,” Dr. Mansfield says.
However, it is difficult to work with prime numbers bigger than 5 in the base 60 Babylonian number system.
“This raises a very particular issue – their unique base 60 number system means that only some Pythagorean shapes can be used,” Dr. Mansfield says.
“It seems that the author of Plimpton 322 went through all these Pythagorean shapes to find these useful ones.
“This deep and highly numerical understanding of the practical use of rectangles earns the name ‘proto-trigonometry’ but it is completely different to our modern trigonometry involving sin, cos, and tan.”
Hunting down Si.427 Dr. Mansfield first learned about Si.427 when reading about it in excavation records – the tablet was dug up during the Sippar expedition of 1894, in what’s the Baghdad province in Iraq today.
“It was a real challenge to trace the tablet from these records and physically find it – the report said that the tablet had gone to the Imperial Museum of Constantinople, a place that obviously doesn’t exist anymore.
“Using that piece of information, I went on a quest to track it down, speaking to many people at Turkish government ministries and museums, until one day in mid 2018 a photo of Si.427 finally landed in my inbox.
“That’s when I learned that it was actually on display at the museum. Even after locating the object it still took months to fully understand just how significant it is, and so it’s really satisfying to finally be able to share that story.”
Next, Dr. Mansfield hopes to find what other applications the Babylonians had for their proto-trigonometry.
There’s just one mystery left that Dr. Mansfield hasn’t unlocked: on the back of the tablet, at the very bottom, it lists the sexagesimal number ‘25:29’ in big font – think of it as 25 minutes and 29 seconds.
“I can’t figure out what these numbers mean – it’s an absolute enigma. I’m keen to discuss any leads with historians or mathematicians who might have a hunch as to what these numbers trying to tell us!”
Reference: “Plimpton 322: A Study of Rectangles” by Daniel F. Mansfield, 3 August 2021, Foundations of Science. DOI: 10.1007/s10699-021-09806-0
It turns out that I still have my CRC Handbook of Logarithmic and Trigonometric Tables! Yay, me!
G
Crib notes for Jerry’s move smuggled into the bedroom by a Babylonian George Costanza . . . at bottom is the swirl at the end.
I STILL HAVE MINE, TOO!......................... somewhere...................
Crib notes for Jerry’s move smuggled into the bedroom by a Babylonian George Costanza . . . at bottom is the swirl at the end.
I use the triplets all the time. I built my wife an 8x10 raised flower bed, I used 3, 4, 5 feet to make it square.
I’m still working on my 5,000 year project. I need an optics designer, a business manager, marketing and several other key positions filled.
Anyone who wants to be a part of the ultimate conservative humanitarian project FReep mail me.
You’ll need to sign an NDA (pretty mild, 0.01% of the project’s expected duration) for further details.
I’m working on the patent application and the attorney says I need to expand its scope, I need a bit of help on some of the finer details and the further enabling technology.
Oh, good Lord! I still have my precious Post Versalog slide rule two! Oh, just look at that beauty!
Freepmail me if you’re serious
60’s pretty handy, imagine dividing the spoils of war, the inheritance in a large family, or the grain on a farm evenly by 60, 30, 20, 15, 12, 10, 6, 5, 4, 3, 2 recipients.
Dead simple in base 60.
We have no trouble thinking of a quarter hour as 15 minutes, although we don’t actually think of it, we just know it.
You have FReepmail.
Dead simple in base 60.
We have no trouble thinking of a quarter hour as 15 minutes, although we don’t actually think of it, we just know it.
—
Ah, true! Don’t know why I didn’t think of 60 minutes in an hour, which we are use to. Although come to think of it, this has the implication that their base 60 system and our 60 minutes in an hour might have a common ancestor.
Wonder how that number 60 was arrived at. Interesting to think about. We could have, seemingly, used 10 “minutes” in an “hour” and 10 “hours” in a day, but somehow we have 60 minutes in an hour and 24 hours in a day. Gets kind of mysterious when you stop to think about it.
I still have mine on a bookshelf right behind me. My old slide rule is buried in a drawer somewhere.
Looks like a peanut butter cookie.
their unique base 60 number system means that only some Pythagorean shapes can be used,”
=== = = = = = = = = = = = = = = = =
This has been longly fascinating to me. Base 60 is so central to so much today. Pretty soon it will be called white supremacist, or racist.
60 minutes (both in time and in angles)
60 seconds (see above)
360 days in a year (close enough, and it is possible that the years got longer, IMO)
24 hours goes into 60 exactly 2.5 times (under the old math from my youth)
360 degrees in a circle
These people were not primitive and barbaric as we were frequently told. In fact, mankind has been going downhill from the Creation described in the Book of Genesis.
Good day!
All our modern clocks use Base 60....................
It is angle related. A circle is 360 degrees but you can think about it in minutes in seconds as well. So, if due north is 00:00 then due south is 30:30. Northeast is 07:30.
I do not know when it became time related.
Its why we have 60 minutes in an hour—and not 100.
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