The paternity of mathematics is traditionally attributed to the Greeks, and in particular to Euclid, Thales and Pythagoras. However, the theorem that bears the name of the latter was already known and used by the Mesopotamians in order to draw up maps and organize the land.
According to Daniel Mansfield, the Si.427 tablet testifies to the use of Pythagorean triples by the Mesopotamians, in order to establish precise plans.
UPDATE. This article, originally published on August 23, 2021 on the Sciences et Avenir website, was updated September 7 with comments from math historian Christine Proust.
If Pythagoras, a famous Greek mathematician of the 6th century BC, was the first to demonstrate "his" theorem, the Babylonian civilization, established in Mesopotamia between the 2nd millennium BC- C. and the beginning of our era, was already using it. And we now know one of its uses.
The Babylonians were fine mathematicians
Mathematics occupied a prominent place in Mesopotamia, where the practice of calculation emerged from the 4th millennium BC. From the end of the 3rd millennium, the teaching of mathematics was based on a sexagesimal numeration, that is to say a numeration with a base of 60, from which we take our measurement of time (60 minutes make up an hour, 60 seconds make up a minute). The writing of numbers is based on two signs - the nails (for the sixties and the units) and the chevrons (for the tens) - and a position system - that is to say that according to its position, a nail can denote 60, or 1. For example, the product 8 x 8 is noted with a nail on the left to count the sixties, four on the right to count the units.
The calculation tablets we have today are often exercise tablets. If these are not rare objects - around 2000 so-called "school" tablets have been exhumed - some tell us better than others about the level of knowledge of the Mesopotamians. In particular, Plimpton 322, named after the collector who acquired it in 1922, became the subject of much study and discussion after a list of Pythagorean triples was spotted in 1945, i.e. a list of sets of three numbers satisfying the famous equality a² + b² =c². In an article published in Images of Mathematics , Christine Proust, historian of mathematics, explains that the tablet presents a double-entry table in which are inscribed sexagesimal numbers, and that above one of the columns, one can read:"The square of the diagonal, from which 1 is subtracted, and from which the width is derived" . By taking a rectangle of length a, equal to 1, this sentence translates:c² - 1² =b², i.e. c² =b² + 1², and therefore… a² + b² =c²!
The many studies of Plimpton 322 have not revealed its nature, which is still the subject of discussion. In fact, knowledge about the use of Pythagorean triples by Mesopotamians is quite limited, although hypotheses have been proposed. But the recent discovery of a new tablet finally unveils one of the uses of Pythagorean triples.
Si.427 is proof of a practical use of mathematics
In an article published in Foundations of Science , Daniel Mansfield, Australian mathematician and researcher at the University of New South Wales (UNSW), reveals what he presents as the oldest example of applied mathematics. It is a tablet dated 3700 years, which corresponds to the Paleo-Babylonian period (late 19th - late 16th century BC). Discovered in 1894, in Iraq, then described by the archaeologists in charge of the excavations, the tablet, named Si.427, then disappeared from radar after the fall of the Ottoman Empire. Daniel Mansfield went looking for it, and was not disappointed when the object was found in Istanbul, on the shelves of one of the city's archeology museums. The mathematician would then have been confronted with the first cadastral type document - aimed at delimiting landed properties - from the paleo-Babylonian period.
The tablet represents a field, subdivided into several geometric shapes (rectangles, trapezoids, right-angled triangles) in order to facilitate drawing. Two Pythagorean triples are inscribed:(5, 12, 13) and (8, 15, 17). This is a partition of land following the sale of part of the field, and therefore, for Daniel Mansfield, evidence of the use of Pythagorean triples in response to empirical issues. This coincides with the beginnings of the privatization of land, and overlaps with the information contained in other tablets, which reveal the underside of the establishment of borders by recounting the disputes which gave rise to the establishment of cadastres. "It is easy to understand that the precision of the measurements was essential in order to put an end to disputes between high-ranking individuals" explains the researcher in a press release, before confiding that "no one expected to learn that the Babylonians used Pythagorean triples in this way" .
In reality, for Grégory Chambon, historian of mathematics, it does not seem so surprising that Mesopotamian geometry could have been used for this purpose: "It has long been known that Mesopotamian geometry has used to establish cadastres” he confides to Sciences et Avenir. Even before the establishment of the Paleo-Babylonian Empire, under the Akkadian Empire (24th - 22nd century BC), then under the Dynasty of Ur (22nd - 21st century BC) , surveyors used theoretical principles of geometry to establish plans of properties. If no document of a cadastral nature from the Babylonian period had been unearthed until now, this is explained by their dispersion for the historian:"At the time of the empire of Akkad , the state was very centralized, very bureaucratic, so a lot of the established documents fell under the public administration, while in the Paleo-Babylonian era there is a social, ethnic and cultural change with the arrival of Bedouins who create kingdoms (Babylon, Mari, Larsa) in which there were many more private records" .
"'Applied mathematics', particularly the application of geometry to surveying, is about as old as mathematics itself, and arguably dates back to the end of the 4th millennium BC" confirms Christine Proust, contacted by Sciences et Avenir . The historian also specifies that “this tablet was known and accessible, but it did not attract the attention of historians”.
The majority of known ancient administrative documents having to do with accounting or public inquiries, Assyriologists had already brought to light the importance of numbers in the administration of power and economic life. Within the community of specialists in Mesopotamian mathematics, a consensus has gradually been established around the economic, political and social function of mathematics, and the fact that it is inseparable from its intellectual nature.
"Mansfield did not discover cadastres, nor the existence of the Si 427 cadastre, nor applied math in Mesopotamia, nor the use of the Pythagorean property in mathematical texts cuneiform. But he found Pythagorean triples in a cadastre. It's not nothing, even if it's not as surprising as he claims summarizes Christine Proust. To better visualize the constitution of the Si.427 tablet:https://news.unsw.edu.au/en/australian-mathematician-reveals-oldest-applied-geometry