Tag Archives: ancient
How ancient Babylonian land surveyors developed a unique form of trigonometry — 1,000 years before the Greeks
But in 2017, I showed the ancient Babylonians likely developed their own kind of “proto-trigonometry” more than 1,000 years before the Greeks. So why were the Babylonians interested in right-angled triangles? What did they use them for?
I have spent the past few years trying to find out. My research, published today in Foundations of Science, shows the answer was hiding in plain sight.
Many thousands of clay tablets have been retrieved from the lost cities of ancient Babylon, in present-day Iraq. These documents were preserved beneath the desert through millennia. Once uncovered they found their way into museums, libraries and private collections.
One example is the approximately 3,700-year-old cadastral survey Si.427, which depicts a surveyor’s plan of a field. It was excavated by Father Jean-Vincent Scheil during an 1894 French archaeological expedition at Sippar, southwest of Baghdad. But its significance was not understood at the time.
It turns out that Si.427 — which has been in Turkey’s İstanbul Arkeoloji Müzeleri (Istanbul Archaeological Museums) for several decades and is currently on display — is in fact one of the oldest examples of applied geometry from the ancient world. Let’s look at what makes it so special.
A brief history of Babylonian surveying
The ancient Babylonians valued land, much as we do today. Early on, large swathes of agricultural land were owned by institutions such as temples or palaces.
Professional surveyors would measure these fields to estimate the size of the harvest. But they did not establish field boundaries. It seems those powerful institutions did not need a surveyor, or anyone else, to tell them what they owned.
The nature of land ownership changed during the Old Babylonian period, between 1900 and 1600 BCE. Rather than large institutional fields, smaller fields could now be owned by regular people.
This change had an impact on the way land was measured. Unlike institutions, private landowners needed surveyors to establish boundaries and resolve disputes.
The need for accurate surveying is apparent from an Old Babylonian poem about quarrelling students learning to become surveyors. The older student admonishes the younger student, saying:
Go to divide a plot, and you are not able to divide the plot; go to apportion a field, and you cannot even hold the tape and rod properly. The field pegs you are unable to place; you cannot figure out its shape, so that when wronged men have a quarrel you are not able to bring peace, but you allow brother to attack brother. Among the scribes, you (alone) are unfit for the clay.
This poem mentions the tape and rod, which are references to the standard Babylonian surveying tools: the measuring rope and unit rod. These were revered symbols of fairness and justice in ancient Babylon and were often seen in the hands of goddesses and kings.
Babylonian surveyors would use these tools to divide land into manageable shapes: rectangles, right-angled triangles and right trapezoids.
Earlier on, before surveyors needed to establish boundaries, they would simply make agricultural estimates. So 90° angles back then were good approximations, but they were never quite right.
Right angles done right
The Old Babylonian cadastral survey Si.427 shows the boundaries of a small parcel of land purchased from an individual known as Sîn-bêl-apli.
There are some marshy regions which must have been important since they are measured very carefully. Sounds like a normal day at work for a Babylonian surveyor, right? But there is something very distinct about Si.427.
In earlier surveys, the 90° angles are just approximations, but in Si.427 the corners are exactly 90°. How could someone with just a measuring rope and unit rod make such accurate right angles? Well, by making a Pythagorean triple.
A Pythagorean triple is a special kind of right-angled triangle (or rectangle) with simple measurements that satisfy Pythagoras’s theorem. They are easy to consturct and have theoretically perfect right angles.
Pythagorean triples were used in ancient India to make rectangular fire altars, potentially as far back as 800 BCE. Through Si.427, we now know ancient Babylonians used them to make accurate land measurements as far back as 1900 BCE.
Si.427 contains not one, but three Pythagorean triples.
Crib notes for surveyors
Si.427 has also helped us understand other tablets from the Old Babylonian era.
Not all Pythagorean triples were useful to Babylonian surveyors. What makes a Pythagorean triple useful are its sides. Specifically, the sides have to be “regular”, which means they can be scaled up or down to any length. Regular numbers have no prime factors apart from 2, 3 and 5.
Plimpton 322 is another ancient Babylonian tablet, with a list of Pythagorean triples that look similar to a modern trigonometric table. Modern trigonometric tables list the ratios of sides (sin, cos and tan anyone?).
But instead of these ratios, Plimpton 322 tells us which sides of a Pythagorean triple are regular and therefore useful in surveying. It is easy to imagine it was made by a pure mathematician who wanted to know why some Pythagorean triples were usable while others were not.
Alternatively, Plimpton 322 could have been made to solve some specific practical problem. While we will never know the author’s true intentions, it is probably somewhere between these two possibilities. What we do know is the Babylonians developed their own unique understanding of Pythagorean triples.
This “proto-trigonometry” is equivalent to the trigonometry developed by ancient Greek astronomers. Yet it is different because it was developed in response to the problems faced by Babylonian surveyors looking not at the night sky — but at the land.
The weird world of one-sided objects
The discovery of the lost city of ‘the Dazzling Aten’ will offer vital clues about domestic and urban life in Ancient Egypt
Built by Amenhotep III and then used by his grandson Tutankhamen, the ruins of the city were an accidental discovery. In September last year, Hawass and his team were searching for a mortuary temple of Tutankhamen.
Instead, hidden under the sands for almost three and a half millennia, they found the Dazzling Aten, believed to be the largest city discovered in Egypt and, importantly, dated to the height of Egyptian civilisation. So far, Hawass’ excavations have unearthed rooms filled with tools and objects of daily life such as pottery and jewellery, a large bakery, kitchens and a cemetery.
The city also includes workshops and industrial, administrative and residential areas, as well as, to date, three palaces.
Ancient Egypt has been called the “civilisation without cities”. What we know about it comes mostly from tombs and temples, whilst other great civilisations of the Bronze Age, such as Mesopotamia, are famous for their great cities.
The Dazzling Aten is extraordinary not only for its size and level of prosperity but also its excellent state of preservation, leading many to call it the “Pompeii of Ancient Egypt”.
The rule of Amenhotep III was one of the wealthiest periods in Egyptian history. This city will be of immeasurable importance to the scholarship of archaeologists and Egyptologists, who for centuries have struggled with understanding the specifics of urban, domestic life in the Pharaonic period.
Foundations of urban life
I teach a university subject on the foundations of urban life, and it always comes as a surprise to my students how little we know about urbanism in ancient Egypt.
The first great cities, and with them the first great civilisations, emerged along the fertile valleys of great rivers in Mesopotamia (modern day Iraq), the Indus Valley (modern day India and Pakistan) and China at the beginning of the Bronze Age, at least 5,000 years ago.
Just like cities today, they provided public infrastructure and roads, and often access to sanitation, education, health care and welfare. Their residents specialised in particular professions, paid taxes and had to obey laws.
But the Nile did not support the urban lifestyle in the same way as the rivers of other great civilisations. It had a reliable flood pattern and thus the second longest river in the world could be easily tamed, allowing for simple methods of irrigation that did not require complex engineering and large groups of workers to maintain. This meant the population didn’t necessarily need to cluster in organised cities.
Excavations of Early Dynastic (c. 3150-2680 BCE) Egyptian cities such as Nagada and Hierakonpolis have provided us with a plethora of information regarding urban life in the early Bronze Age . But they are separated from the Dazzling Aten by some 1,600 years — as long as separates us from the Huns of Attila attacking ancient Rome.
One city closer in age to the Dazzling Aten we do know a little more about is the short-lived capital of Amenhotep’s III son, Akhenaten, known as the “Horizon of the Aten”, or Tell el-Amarna. Amarna was functional for only 14 years (1346-1332 BCE) before being abandoned forever. It was first described by a travelling Jesuit monk in 1714 and has been excavated on and off for the last 100 years.
Very few other Egyptian cities from the Early Dynastic Period (3150 BCE) to the Hellenistic period (following Alexander the Great’s conquest of Egypt in 332 BCE), have been excavated. This means that domestic urban life and urban planning have long been contentious research areas in the study of Pharaonic Egypt.
The scientific community is impatiently waiting for more information to draw comparisons between Akhenaten’s city and the newly discovered capital founded by his father.
The magnificent pharaoh
Amenhotep III, also known as Amenhotep the Magnificent, ruled between 1386 and 1349 BCE and was one of the most prosperous rulers in the Egyptian history.
During his reign as the ninth pharaoh of the 18th Dynasty, Egypt achieved the height of its international power, climbing to an unprecedented level of economic prosperity and artistic splendour. His vision of greatness was immortalised in his great capital, which is believed to have been later used by at least Tutankhamen and Ay.
In 2008, for the first time in history, the majority of world’s inhabitants lived in the cities. Yet, with globalisation, the differences between the “liveability” of modern cities are striking.
As a society we need to understand where cities come from, how have they formed and how they shaped the development of past urban communities to learn lessons for the future. We look forward to research and findings being published from the ancient city of Amenhotep III to enlighten us about the daily lives of ancient Egyptians at their height.
The idea of “news” is a pretty new thing. So is the concept of “fake news”, as in false or misleading information presented as news. Accordingly, we don’t expect to understand the term outside of our own epoch.
Most people identify “fake news” with Donald Trump, as he used the term widely to challenge mass media coverage of his 2016 presidential campaign. Trump ran as much against the “fake news” of the New York Times and CNN as against Hillary Clinton and the Democrats.
For sure, it’s a long way from Trump to Thucydides, the famous Athenian historian and general. There was no “news” in the ancient world, unless we consider the scuttlebutt in the agora (city square) as a kind of Athens Times or some such.
And poor Thucydides would probably cringe at being compared to Trump. Yet there seems to be a meaningful analogy between Trump and fake news, and Thucydides and myth. More on that in a moment.
Mistrust and misinformation
By news, we mean something like truth, facts about the world. In that sense, fake news is an oxymoron. News can be false, of course. But we’d like to believe that untrue in this case really means a mistake, a gaffe that in some sense is always correctable. News agencies can and do retract stories and reporters file corrections.
News suggests the default is truth or a commitment to truth. If they are true to their profession, journalists demonstrate a higher commitment or calling, to get stories right, or at least not to fake it. Intentional falsification results in professional suicide.
Which brings us back to Trump and Thucydides. Trump’s brilliance, if we can call it that, was his grasp of a certain presentiment in the American electorate that proved strong enough to catapult him to victory in 2016.
People’s mistrust in institutions seems to be at an all-time high. They feel they are being gaslighted, that there exists a cabal of smug elites who hold them in contempt. As Trump would have it, that cabal includes a press corps, threatened by new media, that has sold out and joined with the deep state and the Democratic Party.
Trump realised he could not become president by preaching to Republicans only, to those who never or almost never voted Democratic. He needed those whose distrust of institutions was compounded by a sense of betrayal.
The point of all of this is the importance of truth. Real fake news (as opposed to the claim that all news is fake) is about serving up falsehood as truth. No news or fake news in a democracy can be extremely pernicious, as representative government relies on information.
In the US today, a fundamentally ill-informed public produces inferior laws and weak administration. Over time it may well bring about the ultimate disintegration of the democratic regime altogether.
So, too, went the argument in ancient Athens 26 centuries ago.
There was no Trump or (fake) news. But there was Thucydides (and Plato) and a democracy that needlessly destroyed itself. By engaging in the disastrous Peloponnesian War, the Athenians forfeited their empire, upended their democracy and lost their freedom.
Thucydides and Plato lived through the crisis of Athenian democracy and, not unlike Trump, informed posterity that the fate of their beloved Athens resulted from the systematic misinformation and mis-education of the citizens.
The wrong myths
Demagogues easily manipulated the Athenian demos (common people), precisely because they had mistaken the fake for the real, because they had been systematically mis-educated. Of course, neither blamed the press or journalists. They blamed the poets.
Athenians read, or had read to them, Homer and the stories of epic heroes and war trophies and great victories on the battlefield. Thucydides and Plato decried Homer as the fake news of the ancient world. These heroes were the wrong kind and the myths containing their stories had to go.
Plato seemed desperate to displace Homer. His teacher Socrates was offered as an antidote to the sullen, self-centred, violent heroes of Homer’s Iliad and Odyssey. Socrates was a new hero for a new time, a hero of logos (reason) for a new era where the reed would be mightier than the sword.
So too with Thucydides. Throughout his history of war and plague, he demonstrated with scientific observation the futility of appealing to gods and myths. What good did sacrifices to the gods do the Athenians? How did faith in a higher justice serve the Melians or the people of Mytilene?
Homeric fake news doomed the citizenry of Athens to war and decline. Salvation depended on the people dis-enthralling themselves. Survival entailed embracing the logos and adopting a science of society.
The Athenians instead exiled Thucydides and offered Socrates a hemlock milkshake. Trump got off lightly, being merely impeached twice.
This story is based on the author’s public lecture, “Fake news in ancient times: Thucydides, Plato and the expense of truth”, University of Canterbury, February 25.