Mathematical mystery of ancient Babylonian clay tablet solved

August 24, 2017, University of New South Wales
The 3,700-year-old Babylonian tablet Plimpton 322 at the Rare Book and Manuscript Library at Columbia University in New York. Credit: UNSW/Andrew Kelly

UNSW Sydney scientists have discovered the purpose of a famous 3700-year old Babylonian clay tablet, revealing it is the world's oldest and most accurate trigonometric table, possibly used by ancient mathematical scribes to calculate how to construct palaces and temples and build canals.

The new research shows the Babylonians beat the Greeks to the invention of trigonometry - the study of triangles - by more than 1000 years, and reveals an ancient mathematical sophistication that had been hidden until now.

Known as Plimpton 322, the small tablet was discovered in the early 1900s in what is now southern Iraq by archaeologist, academic, diplomat and antiquities dealer Edgar Banks, the person on whom the fictional character Indiana Jones was based.

It 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.

"Plimpton 322 has puzzled mathematicians for more than 70 years, since it was realised it contains a special pattern of numbers called Pythagorean triples," says Dr Daniel Mansfield of the School of Mathematics and Statistics in the UNSW Faculty of Science.

"The huge mystery, until now, was its purpose - why the ancient scribes carried out the complex task of generating and sorting the numbers on the tablet.

"Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles. It is a fascinating mathematical work that demonstrates undoubted genius.

"The tablet not only contains the world's oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.

"This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3000 years, but it has possible practical applications in surveying, computer graphics and education.

"This is a rare example of the ancient world teaching us something new," he says.

The new study by Dr Mansfield and UNSW Associate Professor Norman Wildberger is published in Historia Mathematica, the official journal of the International Commission on the History of Mathematics.

A trigonometric table allows you to use one known ratio of the sides of a right-angle triangle to determine the other two unknown ratios.

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

"Plimpton 322 predates Hipparchus by more than 1000 years," says Dr Wildberger. "It opens up new possibilities not just for modern mathematics research, but also for mathematics education. With Plimpton 322 we see a simpler, more accurate trigonometry that has clear advantages over our own."

"A treasure-trove of Babylonian tablets exists, but only a fraction of them have been studied yet. The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us."

Dr Mansfield read about Plimpton 322 by chance when preparing material for first year mathematics students at UNSW. He and Dr Wildberger decided to study Babylonian mathematics and examine the different historical interpretations of the tablet's meaning after realizing that it had parallels with the rational trigonometry of Dr Wildberger's book Divine Proportions: Rational Trigonometry to Universal Geometry.

The 15 rows on the tablet describe a sequence of 15 right-angle triangles, which are steadily decreasing in inclination.

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.

They also demonstrate how the ancient scribes, who used a base 60 numerical arithmetic similar to our time clock, rather than the base 10 number system we use, could have generated the numbers on the tablet using their mathematical techniques.

The UNSW Science mathematicians also provide evidence that discounts the widely-accepted view that the tablet was simply a teacher's aid for checking students' solutions of quadratic problems.

"Plimpton 322 was a powerful tool that could have been used for surveying fields or making architectural calculations to build palaces, temples or step pyramids," says Dr Mansfield.

The , which is thought to have come from the ancient Sumerian city of Larsa, has been dated to between 1822 and 1762 BC. It is now in the Rare Book and Manuscript Library at Columbia University in New York.

A Pythagorean triple consists of three, positive 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.

The name is derived from Pythagoras' theorem of right-angle triangles which states that the square of the hypotenuse (the diagonal side opposite the right angle) is the sum of the squares of the other two sides.

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Caliban
4.6 / 5 (10) Aug 24, 2017
The new research shows the Babylonians beat the Greeks to the invention of trigonometry - the study of triangles - by more than 1000 years, and reveals an ancient mathematical sophistication that had been hidden until now.


So this system was already in situ 3700 years ago.

One has to wonder just how anciently it was initially derived, given that it is fully developed at this late date, and this is known only because of the fluke preservation of a cuneiform tablet that enabled the discovery to be made at all.

Prior to this, it was accepted that Joe Pythagoras was responsible, and at a much more recent date, as stated in the article.

Given that Babylon was pretty much a direct -and, some would say, degenerate- descendant of ancient Sumer, it's probably a pretty safe bet that this knowlegde is considerably older, and perhaps way older, than Babylon.

Mighty difficult to confirm that theory, of course.
Shootist
1 / 5 (1) Aug 24, 2017
one wonders how it got lost.
Dr_McFoo
2.3 / 5 (3) Aug 24, 2017
It's been taught for a long time that Pythagoras theorem was already know in the middle East at the time of the Greeks. Not sure why they would highlight that in this article. It would be more interesting if they compared to ancient Chinese and Indian mathematics.
marko
2 / 5 (4) Aug 24, 2017
If you want to see more and read more about these discoveries, then go to Professor Norman Wildbergers youtube channel.

The Babylonian mathematics series starts here:

Old Babylonian Mathematics and Plimpton 322: A new perspective

https://www.youtu..._BAGa2Yk
SlartiBartfast
2.3 / 5 (3) Aug 24, 2017
Wildberger is a known crank, which isn't to say that everything he does is necessarily wrong, but he does have some really strange ideas that normal mathematicians pretty well laugh at -- mainly to do with his apparent distaste for infinite sets. Here's a (good) blog post about him:

http://goodmath.s...of-math/

And a somewhat wandering reddit thread that still has some really good observations thrown in:

https://www.reddi...ms_with/

--

I've even tried to deal with the guy in comments on his YouTube videos, and he's incredibly dishonest. My advice is to just smile and nod, and let the rest of the math world move on without him.

Mastoras
1 / 5 (2) Aug 25, 2017
Plimpton 322 is Babylonian exact sexagesimal trigonometry

Daniel F. MANSFIELD, N. J. WILDBERGER

2017

http://www.scienc...17300691

https://doi.org/1...7.08.001
jibbles
3.8 / 5 (4) Aug 25, 2017
total hyperbole regarding its modern mathematical significance. who is this jackass?
marko
1 / 5 (4) Aug 25, 2017
Modern mathematics hey ...

Definitions of Limit, Set and Real Number please ....
Turgent
1 / 5 (2) Aug 25, 2017
By what path did they come to sexagesimal? Counting sheep. Seems it would be much more difficult to work with than Roman Numerals. Wonder if something backed them into the choice of sexagesimal? Do the primes fall out in some kind of pattern?
SlartiBartfast
3.7 / 5 (3) Aug 25, 2017
Modern mathematics hey ...

Definitions of Limit, Set and Real Number please ....


There are multiple kinds of limits depending on the context. Here's the def for a sequence of numbers: We say the limit of a sequence (a_n) is L if for any ε > 0, there exists N (possibly depending on ε) such that if n > N, then |a_n - L| < ε.

The real numbers are the unique (up to isomorphism) complete, ordered field. They can be constructed a number of ways, the most common being Cauchy completion of the rational numbers or via Dedekind cuts.

As far as sets go, that would be a bit much for the 400 or so characters I have left. If only there were some sort of world-wide network with the sum of human knowledge at your fingertips...but alas...

And something tells me that none of this is going to make you happy, but it is what it is.
TheGhostofOtto1923
4.4 / 5 (7) Aug 25, 2017
Discovering such revelations in what little is left of the ancient past makes one wonder what was lost when the church burned the library of Alexandria, and the 20,000 Mayan books leaving only 3.
SwamiOnTheMountain
3.3 / 5 (3) Aug 25, 2017
That's maths for you. No matter how original you think your discovery is...
somebody else probably discovered it LONG beforehand. =)
Whydening Gyre
5 / 5 (3) Aug 26, 2017
one wonders how it got lost.

I believe the Sumerians called these tablets socks. One day they put this one in a dryer, and....
Da Schneib
2.3 / 5 (3) Aug 26, 2017
Here's me just wondering how people who don't "believe in" calculus and infinities think Sir Edmund Halley predicted the return of Halley's Comet correctly using Isaac Einstein's calculus equations that use infinities.

A couple hundred years ago.

I mean, just sayin'. How dumb you gotta be?
Whydening Gyre
5 / 5 (3) Aug 26, 2017
Here's me just wondering how people who don't "believe in" calculus and infinities think Sir Edmund Halley predicted the return of Halley's Comet correctly using Isaac Einstein's calculus equations that use infinities.

A couple hundred years ago.

I mean, just sayin'. How dumb you gotta be?

Don't you mean "Newton's" calculus...?
Da Schneib
5 / 5 (3) Aug 26, 2017
Going back to the original article, it's clear that a number system based on 2s and 3s instead of 2s and 5s like ours works better for trig. You get exact results, instead of nonterminating decimals. This is obvious to anyone who works much with trig. So are the reasons why.
Da Schneib
5 / 5 (2) Aug 26, 2017
LOL, yes, Whyde, I am still drinking my first cup of coffee. Above all one must retain the ability to laugh at one's own mistakes.
Whydening Gyre
5 / 5 (1) Aug 26, 2017
Going back to the original article, it's clear that a number system based on 2s and 3s instead of 2s and 5s like ours works better for trig. You get exact results, instead of nonterminating decimals. This is obvious to anyone who works much with trig. So are the reasons why.

the real "why" appears to be - why did a more organic (and simple) system disappear...
marko
1 / 5 (2) Aug 27, 2017

There is a video for this too.

https://www.youtu...Ayn-oK4M

Modern mathematics hey ...

Definitions of Limit, Set and Real Number please ....


There are multiple kinds of limits depending on the context. Here's the def for a sequence of numbers: We say the limit of a sequence (a_n) is L if for any ε > 0, there exists N (possibly depending on ε) such that if n > N, then |a_n - L| < ε.

The real numbers are the unique (up to isomorphism) complete, ordered field. They can be constructed a number of ways, the most common being Cauchy completion of the rational numbers or via Dedekind cuts.

As far as sets go, that would be a bit much for the 400 or so characters I have left. If only there were some sort of world-wide network with the sum of human knowledge at your fingertips...but alas...

And something tells me that none of this is going to make you happy, but it is what it is.

Da Schneib
not rated yet Aug 27, 2017
the real "why" appears to be - why did a more organic (and simple) system disappear...
Because the Romans used a system based on counting their fingers. That's why there are 5s in it. When Arabic numbering became more prevalent, its use was biased in favor of these 5s, rather than the more, as you say, "organic" systems that had preceded the Roman system.
KBK
1 / 5 (1) Aug 27, 2017
It was possibly a dedicated math for the creation or measurement of physical structure. Which would be the nature of most critical Royal measurements.

It may have been considered to be an architectural math system, and possibly only to be used by the kings, or some such connection.

Being that is is perfect for further subdivision, after it's initial usage. A secret 'higher' math, shown only to initiates. Only for the king and temple crew. Accuracy and secrets help keep them in charge.

Possible, as IIRC, there is only this one tablet found. One example means...No possibility to disprove or prove my hypothesis.
Whydening Gyre
not rated yet Aug 28, 2017
It was possibly a dedicated math for the creation or measurement of physical structure. Which would be the nature of most critical Royal measurements.
It may have been considered to be an architectural math system, and possibly only to be used by the kings, or some such connection.
...

The most critical math system to a ruling authority was for use with taxation. Architecture or geometry came 2nd. Those were just for showing off how good he was at the first...

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