Daily Archives: August 8, 2021

How ancient Babylonian land surveyors developed a unique form of trigonometry — 1,000 years before the Greeks

This stone tablet records the restoration of certain lands by the Babylonian king Nabu-apla-iddina to a priest. Babylonian, circa 870 BCE. From Sippar (Tell Abu Habbah)

Daniel Mansfield, UNSWOur modern understanding of trigonometry harks back to ancient Greek astronomers studying the movement of celestial bodies through the night sky.

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.

Read more:
Written in stone: the world’s first trigonometry revealed in an ancient Babylonian tablet


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.

Si.427 shows a surveyor’s plan of a field.
Author provided

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.

Surveyor with modern tools.
In modern times, surveyors measure land with specialised GPS tools.
Chris Arnison

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.

Plimpton 322 in the Rare Book and Manuscript Library at Columbia University in New York.
UNSW/Andrew Kelly

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.

In this short video I summarise my findings, explaining how the ancient clay tablet Si.427 is the oldest known and most complete example of applied geometry.

Read more:
The weird world of one-sided objects

Cc bcThe Conversation

Daniel Mansfield, Senior lecturer, UNSW

This article is republished from The Conversation under a Creative Commons license. Read the original article.

Jawaharlal Nehru’s vision for a just and equitable post-colonial world, with India leading the way

Wikimedia Commons, CC BY

Ian Hall, Griffith UniversityThis piece is part of a new series in collaboration with the ABC’s Saturday Extra program. Each week, the show will have a “who am I” quiz for listeners about influential figures who helped shape the 20th century, and we will publish profiles for each one. You can read the other pieces in the series here.

Jawaharlal Nehru was not just the architect of modern India and the country’s first prime minister. He also played a central role in the discrediting of European imperialism and gave a voice to people across Asia and Africa struggling for self-determination and racial equality.

An unlikely revolutionary, Nehru was born in 1889 into wealth and privilege. His father was a Kashmiri, a high caste Brahmin and a successful barrister, able to fund the best education for the young Jawaharlal the British system could offer.

After attending Harrow School and Cambridge University, Nehru, too, became a lawyer and could easily have settled into a comfortable life.

Instead, Nehru was swept by the enigmatic Mahatma Gandhi into the campaign against British rule in India. For the next 25 years, he dressed in homespun cotton, endured long terms in prison and campaigned relentlessly for the cause.

Jawaharlal Nehru (left) sharing a joke with Mahatma Gandhi during a meeting of the All India Congress in 1946.
Wikimedia Commons

Successes and failures

Once the British were overthrown and he rose to power, Nehru quickly set about ensuring his vast, impoverished and hugely diverse country was governed by democratically elected leaders and the rule of law.

In parallel, he tried to make India economically self-reliant, so that it could no longer be exploited or manipulated by foreign powers.

Perhaps inevitably, given the scale of the challenges involved, the results of these efforts were mixed.

Nehru’s hopes for a peaceful transition from British rule were dashed by the horrific violence that accompanied partition — the division of the British colony into the separate states of India and Pakistan. Hundreds of thousands died in clashes between Hindus and Muslims, and millions more were displaced and traumatised.

Jawaharlal Nehru signing the first Indian constitution in 1950.
Wikimedia Commons

Nehru did succeed in the herculean effort of transforming India into a constitutional democracy, but his ambitious plans to modernise the economy proved harder to realise.

To be sure, India avoided mass famines like those that ravaged Bengal in mid-1940s and China during the so-called “Great Leap Forward” in the late 1950s and early 1960s. But most Indians did not see major improvements in their standard of living.

A vision for a post-colonial world

Where Nehru really shone was on the world stage. Urbane, well-read, charismatic and eloquent, he was convinced India had a special role to play in international politics, despite its poverty and relative weakness.

And to ensure that happened, Nehru served as his own foreign minister and ambassador-at-large.

Initially, Nehru’s principal concern was the struggle against European imperialism, especially in Asia. Britain, France and the Netherlands all reasserted control over their colonial possessions in the region after the second world war. In response, Nehru and Gandhi rallied anti-colonial leaders, holding the Asian Relations Conference in New Delhi in 1947 to chart the way forward for the continent.

Read more:
Gandhi is still relevant – and can inspire a new form of politics today

In Nehru’s view, Asia’s newly liberated or soon-to-be liberated states should show the world a different way to conduct international relations.

They need not be suspicious of each other’s intentions, nor greedy for each other’s territory, he argued. And they should not waste their scarce resources on building armies or atom bombs. Committed to social and economic development and to treating others with mutual respect, they could — and should — create a more just and peaceful world.

Nehru was highly adept in using new platforms like the United Nations and the global media to promote this vision. He delivered passionate speeches and charmed foreign journalists in long interviews.

He campaigned against nuclear weapons, calling in 1954 for the superpowers to halt their tests of increasingly destructive bombs. This paved the way for a partial ban on testing in 1963.

He called for an end to racial discrimination, most notably in South Africa. He also commissioned India’s diplomats to offer their services to mediate in a series of disputes, including the Korean war and France’s disastrous attempt to cling to its colonial possessions in Indochina.

The birth of non-alignment

Throughout, Nehru made the case for what became known as “non-alignment” — perhaps his greatest contribution to the 20th century world.

India and other post-colonial states, he argued, had no good reason to take sides in the Cold War and plenty of reasons to maintain cordial relations with both the US and Soviet Union.

Jawaharlal Nehru (right) with Egyptian President Gamal Abdel Nasser and Yugoslavian President Josip Broz Tito at the Conference of Non Aligned Nations in Belgrade in 1961.
Wikimedia Commons

Allying with one or the other was too costly and compromising. It brought obligations to build armies and fight distant wars. And it meant renouncing the ability to criticise your ally when it did things with which you disagreed.

Non-alignment annoyed Cold Warriors in both Moscow and Washington. But it proved popular elsewhere, especially among newly independent states.

It helped inspire a series of major meetings intended to promote African and Asian cooperation in the shadow of US-Soviet competition, including the Bandung Conference in 1955, as well as the creation of the Non-Aligned Movement in the 1970s.

Today, 120 states belong to the movement, though India’s interest in the bloc has waned as it has grown stronger and wealthier.

Read more:
The ‘Bandung Divide’: Australia’s lost opportunity in Asia?

Nehru’s greatest failure

Nehru helped delegitimise imperialism and usher in a new world no longer dominated by the Europe powers. He laid out principles that he hoped would encourage mutual respect in international relations – principles eagerly embraced, if not always followed, by other post-colonial leaders.

It is ironic, then, that arguably Nehru’s greatest failure – the one that irreparably tarnished his leadership and broke his health – concerned foreign policy.

Convinced China would abide by the “Five Principles of Peaceful Coexistence” agreement it struck with India in the mid-1950s, Nehru failed to anticipate a military conflict over the long-disputed frontier. When Mao Zedong ordered a surprise attack in 1962, India’s forces were humiliated. And to Nehru’s dismay, neither the UN, nor the superpowers, intervened.

To critics, the Sino-Indian war exposed Nehru’s naivety and the limits of non-alignment. It compelled India to retrench and rearm, and laid bare his dream of an Asia free from “power politics”.

Indian soldiers surrender to Chinese forces during Sino-Indian War in 1962.
Wikimedia Commons

India has travelled far since Nehru’s time and left much of his legacy by the wayside. It now possesses the world’s second largest military – after China – and a nuclear arsenal. It has forged a strong security partnership with the United States.

But New Delhi still remains wary of alliances or anything that might compromise independence of voice or action. And it is as convinced today as when Nehru was in power that India is destined to play a special role in the world.The Conversation

Ian Hall, Deputy Director (Research), Griffith Asia Institute, Griffith University

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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