In 2017 the teacher Daniel mansfield The University of New South Wales (UNSW, Australia) discovered that a Babylonian rectangular tablet called Plimpton 322, about 3,800 years old, was the oldest and most accurate trigonometric table in the world.
Now, Mansfield has published in the journal Foundations of Science a new study with more details on that piece of clay and on another round one – a hand tablet to carry comfortably in the hand – that had been forgotten for more than a century in the past. Istanbul Archaeological Museum (Turkey).
The tablet, labeled Si.427 and dated to Ancient Babylon between 1900 and 1600 BC, is the only known example of a cadastral document from that time, a blueprint used by surveyors to define land boundaries.
It is labeled as Yes. 427 and was collected in 1894 by the French archaeologist Jean-Vincent Scheil in Sippar, a city of the Lower Mesopotamia located in what is now the province of Baghdad, in Iraq. According to Mansfield, it is the oldest known example of applied geometry.
Si.427 dates from the Ancient Babylonian period (between the 1900 and 1600 BC) ”, Affirms this mathematician, and as confirmed to SINC,“ we know it because the tablet refers to a field owned by Sîn-bêl-app, a landowner who lived in Sippar at that time, and also the style of the language confirms it ”.
“It is the only known example of a cadastral document from that period –he emphasizes–, a plan used by surveyors to define the boundaries of the lands. In this case, it informs us of the legal and geometric details of some lands (swamps or wetlands) divided after the sale of a part ”.
Pythagorean triples before Pythagoras
But if there is something that stands out in this ancient clay, it is the use of what today is known as Pythagorean triples to make precise right angles. Mansfield points out that this discovery has important implications for the history of mathematics, since this geometric figure appears a thousand years before Pythagoras was born.
This discovery has important implications for the history of mathematics, since Pythagorean triples appear a thousand years before Pythagoras was born.
Daniel Mansfield (UNSW, Australia)
“A Pythagorean triple is a right triangle (or rectangle) of very simple measures that satisfies the Pythagorean theorem,” he explains. Most rectangles and right triangles have awkward lengths like 1.4142135623730951 …, but these triples have very simple measurements like 3, 4, and 5. This makes them an easy way to build perpendicular lines. It was used in ancient India as early as 800 BC, but we now know that it was also used in Babylonian topography in 1900 BC. C., about a thousand years before ”.
In the case of the round tablet, written in cuneiform script, the sold field is drawn with superimposed horizontal and vertical grid lines, which allowed the surveyor to make a precise subdivision. Specifically, the tablet contains the triples 5-12-13 (twice) and 8-15-17, which were used as a theoretical basis for the perpendicular lines of the grid.
Other tablets of the time also mention disputes of Sîn-bêl-apli over issues related to their lands, as in one where a lawsuit with another wealthy landowner is mentioned over the valuable date palms that grew between their two properties. A local administrator agreed to send a surveyor to resolve the conflict, another example of how important it was even then to establish cadastral boundaries.
“We knew that the Babylonians knew the right triangles and Pythagorean triples or ‘triples’, but we did not know why and why,” says the researcher, “but now the new Si.427 tablet shows us that they used these shapes to accurately measure the terrain, and this helps us understand other tablets from the same period, such as the more famous Plimpton 322 ”. In 2017 it was already proposed that it had some practical purpose to build palaces, temples, canals … or in the topography of fields.
Mansfield considers that the way the boundaries are established between the lands reveals a true geometric understanding: “No one expected the Babylonians to use Pythagorean triples in this way. It is something more like pure math, inspired by the practical problems of the time ”.
It is generally accepted that trigonometry (the branch of mathematics that deals with the study of triangles) was developed by the ancient Greeks when studying the night sky in the 2nd century BC, but studies like this show that the Babylonians developed their own ‘proto-trigonometry’ alternative to solve, in this case, problems related to the measurement of the ground, not the sky.
Creating right angles: easier said than done
An easy way to create an exact right angle is to make a rectangle with sides 3 and 4, and the diagonal 5. These special numbers form the Pythagorean triple 3-4-5, and a rectangle with these measurements has mathematically perfect right angles. This was important to surveyors in ancient times and is still used today.
“The ancient surveyors who recorded Si.427 did something even better: they used a variety of different Pythagorean triples, both in the form of rectangles and right triangles, to build precise right angles ”, says the Australian mathematician.
However, it is difficult to work with prime numbers greater than 5 in the Babylonian number system which was base 60. “This poses a very particular problem: its unique base 60 number system means that only some Pythagorean forms can be used,” he explains. Mansfield, “and it seems that the author of Plimpton 322 went through all those forms to find those useful.”
The mystery of 25:29
There is only one mystery that the author has not yet managed to unravel: on the back of the tablet, at the bottom, the sexagesimal number 25:29 appears in large print, and the mathematician thinks it is 25 minutes and 29 seconds.
“But I can’t understand what they mean, it’s quite an enigma,” he admits. I’m looking forward to discussing any clues with historians or mathematicians who might have a hunch about what these numbers are trying to tell us. “