Mathematician Finds Pythagorean Triples On Ancient Babylonian Tablet
A University of New South Wales, Sydney mathematician has discovered the oldest example of applied geometry ever recorded, the university’s newsroom reports . The tablet illustrates the use of Pythagorean triples in dividing land, 1,100 years before the geometric principle was recorded by the Greek mathematician Pythagoras.
On a 3,700-year-old Babylonian clay tablet map recovered in Iraq more than 100 years ago, Dr. Daniel Mansfield has identified an advanced form of mathematics that was used to divide a large plot of land into geometrically precise sections. Amazingly, this applied geometry relied on a mathematical principle that wouldn’t be officially discovered until another 11 centuries had passed.
Dr. Mansfield, holding the Babylonian clay tablet with evidence of Pythagorean Triples usage. ( UNSW Sydney )
Pythagorean Triples, As It Turns Out, Are A Babylonian Idea!
The Babylonian clay tablet in question, which is known as Si. 427 , has been kept on display at a museum in Istanbul since it was first discovered in the 19th century. It has been linked to the time of the First or Old Babylonian Empire , which first established and then ruled the ancient state of Babylonia in Mesopotamia (modern-day Iraq and Syria) from the 19th century BC through the 16th century BC.
Inscriptions on the reverse side revealed that the tablet had been used as a kind of map, specifically of plots of land owned by various individuals. But the true story of its creation had remained hidden, until Daniel Mansfield came along and looked at it with a fresh perspective.
The front (obverse side) of the Si.427 Babylonian clay tablet. Inscriptions on the reverse side revealed that the tablet had been used as a kind of map, specifically of plots of land owned by various individuals and this helped Dr Daniel Mansfield to figure out that Pythagorean triples were actually invented long before Pythagoras got credit for them. Photograph by and courtesy of the İstanbul Arkeoloji Müzeleri. ( Foundations of Science )
As Dr. Mansfield explains in an article published in the latest edition of Foundations of Science , the Babylonian mapmaker/land surveyor was making use of an important mathematical principle, known as the Pythagorean triple, to perform his work correctly.
A Pythagorean triple consists of three whole numbers, the first two of which squared will equal the third squared (i.e., 3² + 4² = 5², or 9 + 16 = 25). This principle can be used to create precise right-angled triangles, with the vertical and horizontal lines equaling the first two numbers and the diagonal line equaling the third. This is why the concept behind Pythagorean triples (the famed Pythagorean Theorem) is classified as a principle of trigonometry (the geometry of triangles).
So, if perpendicular lines that meet to form a right angle are drawn to lengths of three and four units, the diagonal line that connects their ends to form a triangle will be exactly five units long, every time. It is important to note that Pythagorean triples can be used to make exact rectangles as well as exact triangles, since a rectangle can be divided into two right-angled triangles placed one on top of the other.
This is not just an abstract mathematical formula. It offers a way to create exact shapes from perfectly perpendicular lines in real-world situations.
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Among its many practical uses, Pythagorean triples can be used to divide a section of land into smaller triangular or rectangular sections of equal sizes and shapes. That’s what the person who created the Si.427 clay tablet was trying to do. And with the assistance of an advanced mathematical principle, he was able to do an outstanding job.
This mathematical concept was allegedly first discovered by Pythagoras, the legendary Greek philosopher , mathematician, and mystic who lived in the sixth century BC. As Dr. Mansfield’s research makes clear, what Pythagoras actually did was rediscover a principle that the Babylonians already understood and used 1,100 years before he was born.
An ancient cadastral map from 1780 of Laghestel di Piné, Trentino-Alto Adige Region in Italy (From Catasto Teresiano, Archive of "Libro fondiario e Catasto", Trento). Based on Dr. Mansfield’s research he was able to prove that Babylonian cadastral maps relied on Pythagorean triples! ( Mapping Synusiae book chapter)
Problem Solving Through Trigonometry in Ancient Babylonian
The S.427 clay tablet land map was not created for simple record keeping purposes. “Si.427 is about a piece of land that’s being sold,” Dr. Mansfield explained. “It’s the only known example of a cadastral document from the OB [Old Babylonian] period, which is a plan used by surveyors to 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.”
Details about the purpose of the tablet are revealed in ancient cuneiform script written on the tablet’s reverse side. This was decoded long ago, and it turned out to be a description of the agricultural land being divided and some of its features.
Many documents from the Old Babylonian era have been recovered and decoded, and in several mention is made of an important landowner named Sin-bel-apli, who apparently owned at least some of the property depicted on the face of Si.427.
A Babylonian clay tablet map of the world, 700-500 BC. (Gary Todd / CC0)
“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 said. “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,” he continued. “Much like we would today, you’ve got private individuals trying to figure out where their land boundaries are, and the surveyor comes out, but instead of using a piece of GPS equipment, they use Pythagorean triples.”
Was Si.427 created to help solve a disagreement over who owned a grove of date palms? Or was this document used to facilitate another land transaction involving the powerful and influential Sin-bel-apli?
This question can’t be answered definitively. But what can be said with certainty is that the ancient Babylonians used mathematics to develop rational solutions to thorny problems. While people could lie, they must have reasoned, numbers never would.
“Nobody expected that the Babylonians were using Pythagorean triples in this way,” Dr. Mansfield marveled. “It is more akin to pure mathematics, inspired by the practical problems of the time.”
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The Plimpton 322 Babylonian clay tablet first alerted Dr. Mansfield to the fact that Mesopotamians were interested in trigonometry and that they understood Pythagorean triples. Photograph courtesy of the Rare Books and Manuscripts Library, Columbia University. (Andrew Kelly / Foundations of Science )
Decoding the Mathematical Genius of Babylonian Builders
Daniel Mansfield first discovered the ancient Babylonians’ interest in trigonometry through his study of another clay tablet that was created during the time of the Old Babylonian Empire.
This tablet, known as Plimpton 322 , featured several columns of numbers that revealed the Babylonian’s interest in trigonometry, and their awareness of the Pythagorean triple. Since artifact collector George Plimpton had donated this tablet for study in the 1930s, however, academics had struggled to explain its true meaning and purpose.
This changed in 2017, when Dr. Mansfield and his UNSW Sydney colleague Norman Wildberger published an article in the journal Historia Mathematica explaining that what the tablet really meant and how it had been used. Plimpton 322’s intricately prepared trigonometric table was designed not as a teaching device, but for use in real-world projects that required precise calculations and measurements.
Based on Dr. Mansfield’s research he was able to show that problems like the one above was being solved in Mesopotamian times. Problem: Suppose that a ramp leading to the top of a ziggurat wall is 56 cubits long, and the vertical height of the ziggurat is 45 cubits. What is the distance x from the outside base of the ramp to the point directly below the top? ( Historia Mathematica journal)
“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,” Dr. Mansfield told an interviewer from the Guardian at that time.
Excavations in the area where the Babylonian Empire once reigned have uncovered many building and infrastructure projects that demonstrate Babylon’s impressive achievements in architecture and engineering. This includes projects initiated by the rulers of the Old Babylonian Empire and by their successors (the Empire as a whole endured for more than 1,200 years). Their most notable accomplishment was the creation of the Hanging Gardens of Babylon , one of the seven wonders of the ancient world that archaeologists have been trying to locate for centuries.
In Dr. Mansfield’s new article in Foundations of Science , he explains how his discoveries with respect to Plimpton 322 steered him in the right direction while examining Si. 427. Once the first had been properly interpreted, the truth about the second became crystal clear.
“Once you understand what Pythagorean triples are, your society has reached a particular level of mathematical sophistication,” Dr. Mansfield said, expressing admiration for what the ancient Babylonians were able to accomplish.
They not only understood this important principle 1,100 years before Pythagoras supposedly “invented” it, but they knew how to apply it in the real world. In the complex society the ancient Babylonian empire builders were aspiring to create rational approaches to problem-solving would have been essential to their success.
Top image: Babylonian clay tablet shows Pythagorean Triples were used 3,700 years ago. Source: University of New South Wales
By Nathan Falde