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.
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.
In this series, we look at under-acknowledged women through the ages.
“It belongs in a museum.” With these words, Indiana Jones, the world’s best-known fictional archaeologist, articulated an association between archaeologists, antiquities, and museums that has a very long history. Indeed, even Jones himself would likely marvel at the historic setting of the world’s first “museum,” and the remarkable woman who is believed to have been its curator, the Mesopotamian princess, Ennigaldi-Nanna.
Ennigaldi-Nanna was the priestess of the moon deity Sin, and the daughter of the Neo-Babylonian king, Nabonidus. In the ancient Mesopotamian city of Ur, around 530BCE, a small collection of antiquities was gathered, with Ennigaldi-Nanna working to arrange and label the varied artefacts.
This collection was considered by the British archaeologist, Sir Charles Leonard Woolley, to be the earliest known example of a “museum”.
In 1925, Woolley and his team were excavating at Ur (now in the Dhi Qar governate of southern Iraq). They discovered a curious collection of artefacts among the ruins of a Babylonian palace. Especially unusual was that while the items were from different geographical areas and historical settings, they were neatly assembled together.
The items ranged in dates from around 2100 BCE to 600 BCE. They included part of a statue of the famous early king, Shulgi of Ur, who ruled around 2058 BCE, a ceremonial mace-head made of stone, and some texts. The statue, Woolley observed, had been carefully restored to preserve the writing.
There was also a Kassite boundary stele (called a “kudurru”), a written document used to mark boundaries and make proclamations. The stele was dated to around 1400 BCE, and contained, Woolley noted, a “terrific curse” on anyone who removed or destroyed the record it contained.
Many items were accompanied by labels giving details about the artefacts. These were written in three languages, including Sumerian. The labels have been described in modern scholarship as early examples of the “metadata” that is so critical to the preservation of antiquities and the historical record.
The museum, over 2,500 years old, was centred on cultural heritage, and it is thought to have perhaps had an educational purpose. Along with her other roles, Ennigaldi-Nanna is believed to have run a scribal school for elite women.
When considering the discovery, Woolley noted that the discovery of a museum associated with the priestess was not unexpected, given the close connection between religious specialists and education. He also commented on the “antiquarian piety” of the time of the museum’s construction — an interest in history was a common feature among monarchs from the Neo-Babylonian period.
A family fascination with history
Indeed, Ennigaldi-Nanna’s appreciation for the past seems to have been a family trait. Her father Nabonidus had a fascination with history which led him to conduct excavations and discover lost texts. Many of the items in the collection were discovered by him, with Nabonidus sometimes described in the modern day as the world’s first archaeologist.
Nabonidus was the last king of the Neo-Babylonian Empire, and a religious reformer. His eldest son, Belshazzar, ruled as his regent for many years, but is perhaps best known for his appearance in the biblical Book of Daniel. In a famous scene, the unfortunate regent sees the end of the Neo-Babylonian kingdom coming when it is foretold through the writing of a disembodied hand on a wall.
King Nabonidus’ interest in history didn’t end with archaeology. He also worked to revive ancient cultic traditions relating to the moon deity, Sin (Sumerian Nanna). His daughter Ennigaldi was an important part of these efforts, indeed, her name is an ancient Sumerian one, meaning “the priestess, the desire of the Moon god.”
The appointment of Ennigaldi as high priestess in Ur reinvigorated a historical trend made famous by Sargon of Akkad, who installed his daughter, the poetess Enheduanna, in the role over 1000 years earlier.
By the time of Ennigaldi-Nanna’s appointment, the religious role she would inhabit had long been unoccupied, and the rituals associated with the post had been forgotten. Nabonidus, however, describes finding an ancient stela belonging to Nebuchadnezzar I, and using it to guide his actions.
The historic aspects of the appointment of Ennigaldi-Nanna were further emphasised by Nabonidus when noting his research into the requirements of her role. The king describes consulting the writings of a previous priestess, a sister of the ruler Rim-Sin named En-ane-du.
Rim-Sin reigned over 1200 years before Nabonidus came to power. While some scholars doubt Nabonidus’ discovery of the stela of Nebuchadnezzar I, his recovery of the writings of the priestess, En-ane-du, has greater acceptance.
Little known today
Ennigaldi is largely unknown in the modern day. An exception to her modern anonymity may be found in the luxury fashion line, Ennigaldi, which creates pieces inspired by ancient Babylonian architecture.
While relatively little is known of the life of Ennigaldi, there are other well-known women in her family tree. Ennigaldi’s grandmother, Adad-guppi, was also a powerful priestess involved in the political world of her son, Nabonidus. Adad-guppi is best known in the present day from her “autobiography,” a cuneiform account of her life, written in the first person. Adad-guppi’s autobiography records the blessings she received from the moon deity such as living to the age of 104 with a sound mind and body.
The city of Ur and its museum were abandoned around 500 BCE, due to deteriorating environmental conditions. These included a severe drought, along with changing river and silt patterns. The prevalence of drought has also been cited as a likely cause of the falls of many earlier kingdoms from the Bronze Age.
In this series, we look at under-acknowledged women through the ages.
The world’s first known author is widely considered to be Enheduanna, a woman who lived in the 23rd century BCE in ancient Mesopotamia (approximately 2285 – 2250 BCE). Enheduanna is a remarkable figure: an ancient “triple threat”, she was a princess and a priestess as well as a writer and poet.
The third millennium BCE was a time of upheaval in Mesopotamia. The conquest of Sargon the Great saw the development of the world’s first great empire. The city of Akkad become one of the largest in the world, and northern and southern Mesopotamia were united for the first time in history.
In this extraordinary historical setting, we find the fascinating character of Enheduanna, Sargon’s daughter. She worked as the high priestess of the moon deity Nanna-Suen at his temple in Ur (in modern-day Southern Iraq). The celestial nature of her occupation is reflected in her name, meaning “Ornament of Heaven”.
Enheduanna composed several works of literature, including two hymns to the Mesopotamian love goddess Inanna (Semitic Ishtar). She wrote the myth of Inanna and Ebih, and a collection of 42 temple hymns. Scribal traditions in the ancient world are often considered an area of male authority, but Enheduanna’s works form an important part of Mesopotamia’s rich literary history.
Enheduanna’s status as a named poet is significant given the anonymity surrounding works of even earlier authors. Yet she is almost entirely unknown in the modern day, and her achievements have been largely overlooked (a notable exception is the work of Jungian analyst Betty De Shong Meador).
Her written works are deeply personal in subject, containing numerous biographical features.
Enheduanna’s cycle of temple hymns concludes with an assertion of the work’s originality and its authorship:
The compiler of the tablets was En-hedu-ana. My king, something has been created that no one has created before.
While clearly asserting ownership over the creative property of her work, Enheduanna also comments on the difficulties of the creative process — apparently, writer’s block was a problem even in ancient Mesopotamia.
Long hours labouring by night
In her hymns, Enheduanna comments on the challenge of encapsulating divine wonders through the written word. She describes spending long hours labouring over her compositions by night, for them then to be performed in the day. The fruits of her work are dedicated to the goddess of love.
Enheduanna’s poetry has a reflective quality that emphasises the superlative qualities of its divine muse, while also highlighting the artistic skill required for written compositions.
Enheduanna’s works were written in cuneiform, an ancient form of writing using clay tablets but have only survived in the form of much later copies from around 1800 BCE, from the Old Babylonian period and later. The lack of earlier sources has raised doubts for some over Enheduanna’s identification as the author of myths and hymns and her status as a religious official of high rank. However, the historical record clearly identifies Enheduanna as the composer of ancient literary works, and this is undoubtedly an important aspect of the traditions surrounding her.
Aside from poetry, other sources for Enheduanna’s life have been discovered by archaeologists. These include cylinder seals belonging to her servants, and an alabaster relief inscribed with her dedication. The Disk of Enheduanna was discovered by British archaeologist Sir Charles Leonard Woolley and his team of excavators in 1927.
The Disk was discarded and apparently defaced in antiquity, but the pieces were recovered through excavations and the scene featuring the writer successfully restored. The scene depicts the priestess at work: along with three male attendants, she observes a libation offering being poured from a jug.
Enheduanna is situated in the centre of the image, with her gaze focused on the religious offering, and her hand raised in a gesture of piety. The image on the Disk emphasises the religious and social status of the priestess, who is wearing a cap and flounced garment.
Art imitates life
Enheduanna’s poetry contains what are thought to be autobiographical elements, such as descriptions of her struggle against a usurper, Lugalanne. In her composition The Exaltation of Inanna, Enheduanna describes Lugalanne’s attempts to force her from her role at the temple.
Enheduanna’s pleas to the moon god were apparently met with silence. She then turned to Inanna, who is praised for restoring her to office.
The challenge to Enheduanna’s authority, and her praise of her divine helper, are echoed in her other work, such as in the myth known as Inanna and Ebih.
In this narrative, the goddess Inanna comes into conflict with a haughty mountain, Ebih. The mountain offends the deity by standing tall and refusing to bow low to her. Inanna seeks help from her father, the deity Anu. He (understandably) advises her against going to war with the fearsome mountain range.
Inanna, in typically bold form, ignores this instruction and annihilates the mountain, before praising the god Enlil for his assistance. The myth contains intriguing parallels with the conflict described in Enheduanna’s poetry.
In the figure of Enheduanna, we see a powerful figure of great creativity, whose passionate praise of the goddess of love continues to echo through time, 4000 years after first being carved into a clay tablet.
On April 10 2003, the first looters broke into the National Museum of Iraq. Staff had vacated two days earlier, ahead of the advance of US forces on Baghdad. The museum was effectively ransacked for the next 36 hours until employees returned.
While the staff – showing enormous bravery and foresight – had removed and safely stored 8,366 artefacts before the looting, some 15,000 objects were taken during that 36 hours. While 7,000 items have been recovered, more than 8,000 remain unaccounted for, including artefacts thousands of years old from some of the earliest sites in the Middle East.
The looting is regarded as one of the worst acts of cultural vandalism in modern times, but much more of Iraq’s rich cultural history has been destroyed, damaged or stolen in the years since. Indeed the illegal trade in looted antiquities is growing.
One of the museum objects that remains lost is a black stone weight shaped like a duck made around 2070 BC and excavated from the ancient city of Ur. Another is a fluted gold and lapis bowl from a royal cemetery in the same city.
The museum’s collection of cylinder seals (used to print images, usually into clay) was hit especially hard as they were easy to conceal and transport and had a ready market overseas. Of the 5144 taken, just over half have been returned. The museum reopened in 2014, somewhat a shadow of its former self.
Some high value items looted from the museum were so recognisable that they could not possibly appear on the open market, suggesting they were taken with buyers already lined up. In contrast to this was the opportunistic looting undertaken by locals: in some galleries copies were stolen but genuine pieces ignored.
Global outrage at the looting did lead to immediate action. One of the most successful programs was an amnesty granted by authorities that saw almost 2,000 items returned by January 2004, and a further thousand items seized by Iraqi and US investigators.
Initial returns were largely local. One early success was the famous Lady of Warka, dated to around 3100 BC; she was recovered by investigators at a nearby farm following a tip off.
Others have come home following international investigations (a large number of objects seem to have travelled through London and New York in the aftermath), such as a statue of Assyrian king Argon II seized in New York in 2008 and returned to the museum in 2015.
While destruction and looting of cultural heritage has been a by-product of war for thousands of years, the scale of the looting of the Iraq Museum was staggering. Particularly frustrating were the neglected warnings that such an incident could happen, and the immediate response from the Bush administration that “stuff happens”.
The museum looting should have been a clarion call for the need for better protection of antiquities in conflict zones, both from combatants and local populations. Sadly, this has not been the case. There has been subsequent destruction of archaeological sites and museums in Syria and Libya, ISIS selling antiquities to finance weapons, and increases in thefts from both private and public collections and from archaeological sites.
Part of the problem with halting the illegal global trade of stolen antiquities is the scale of the market. In late 2017, an investigation by the Wall Street Journal presented the sobering assessment that over 100,000 antiquities are offered for sale online daily, of which up to 80% are likely to be faked or looted.
In order to halt looting, it is essential that private collectors and institutions only purchase antiquities with a legal provenance to dry up the demand.
Ironically, centuries after many of the remains of these ancient cultural entities were looted by European colonial forces in order to fill grand national museums, we are seeing a 21st century version of cultural colonialism. Private collectors are enabling an entire economy of illegal activities.
The loss of these sites and artefacts is disastrous for humanity. The Baghdad looting has shown that in times of conflict, not even a museum can necessarily provide a sanctuary, without meaningful policies of protection. Sadly, it appears we have not learnt the lessons of April 2003.
It is a little-known piece of history that Saddam Hussein was a great fan of ancient Mesopotamian literature. His enthusiasm for epics written in cuneiform – the world’s oldest known form of writing – can be seen in his own efforts at writing political romance novels and poetry. Hussein’s first novel, Zabibah and the King, blended the Epic of Gilgamesh with the 1001 Nights, and was adapted into a television series and a musical.
Indeed, the Iraqi dictator was said to be so immersed in his novel-writing that he left much of the military strategising to his sons leading up to the 2003 war. He continued writing in prison, using a card table as a writing desk. This example from the modern genre of “dictator literature” provides an unusual insight into the diverse reception of cuneiform literature in the modern day.
The decipherment of cuneiform in the late 18th century, a tale of academic virtuosity and daring, revealed a “forgotten age” and challenged the traditional, biblical view of history. One scholar was even put on trial for heresy for the wonders he uncovered in the translated script.
For over 3,000 years, cuneiform was the primary language of communication throughout the Ancient Near East (roughly corresponding to the Middle East today) and into parts of the Mediterranean. The dominance of the cuneiform writing style in antiquity has led scholars to refer to it as “the script of the first half of the known history of the world”. Yet it disappeared from use and understanding by 400 CE, and the processes and causes of the script’s vanishing act remain somewhat enigmatic.
Cuneiform is composed of wedge-shaped characters and was written on clay tablets (often likened to marks made by a chicken scratching in the mud). Unlike other ancient writing media, such as the papyri or leather scrolls used in Ancient Greece and Rome, cuneiform tablets survive in great abundance. Hundreds of thousands of tablets have been recovered from ruined Mesopotamian cities.
The discoveries yielded from the recovery of cuneiform writing continue to unfold in unexpected and exciting ways. In August this year, mathematicians at an Australian university made international headlines with their discovery involving a 3,700-year-old clay tablet containing a trigonometric table. The researchers said the cuneiform table reveals a sophisticated understanding of trigonometry — in some ways more advanced than in modern-day mathematics!
Lost in translation
It is difficult to overstate the influence of cuneiform literature in the ancient world. Many languages throughout a vast geographical span over thousands of years were written in cuneiform, including Sumerian, Hittite, Hurrian and Akkadian. Among these, Akkadian (an early cognate of Hebrew and Arabic) became the lingua franca of the Near East, including Egypt, during the Late Bronze Age.
Cuneiform was used to preserve the official royal correspondences between leaders of empires, but also simple transactions and record-keeping that were part of daily life. Over time, the skill of writing moved outside the main institutions of cities, such as temples and scribal schools, into the hands of citizens, as well as into private homes.
Despite its dominance in antiquity, the use of cuneiform ceased entirely at some point between the first and third centuries CE. The great empires of the Ancient Near East experienced a long decline over many centuries, which ultimately resulted in the loss of Egyptian hieroglyphs and cuneiform as written languages.
Cuneiform’s sphere of influence shrank after the sixth century BCE, before vanishing entirely. The disappearance of cuneiform accompanied, and likely facilitated, the loss of Mesopotamian cultural traditions from the ancient and modern worlds.
There are several schools of thought surrounding the disappearance of cuneiform, including competition with alphabetic languages (where letters correspond to sounds) such as Aramaic and Greek, and the decline of writing traditions. However, the process of the transition from cuneiform to alphabet is yet to be clearly understood.
Deciphering the code
The resurrection of cuneiform writing systems was described by legendary Sumerologist Samuel Noah Kramer as an “eloquent and magnificent achievement of 19th century scholarship and humanism”.
In the 15th century, cuneiform inscriptions were observed in Persepolis (in modern-day Iran). The script’s patterned dashes were not immediately recognised as writing. The name “cuneiform” (a Latin-based word meaning “wedge-shaped”) was given to the undeciphered writings by Oxford professor Thomas Hyde in 1700.
Hyde viewed the cuneiform markings as decorative rather than conveying language — a widely held view in academic circles of the 18th century. Despite some efforts to popularise the name “arrow writing”, “cuneiform” gained general acceptance. Yet cuneiform remained cryptic, and its ancient masterpieces buried and inscrutable.
The modern-day decipherment of cuneiform owes a great debt to the rulers of the Persian Achaemenid dynasty, who reigned in what is modern-day Iran in the first millennium BCE. These rulers made cuneiform inscriptions recording their achievements.
The most important of these inscriptions for the decipherment of cuneiform was the Behistun inscription, which recorded the same message in three languages: Persian, Elamite and Akkadian. This trilingual inscription was carved into the face of a cliff in Behistun in what is now western Iran.
Detailing the successes of King Darius I of Persia, the Behistun inscription was inscribed on rock some 100 metres off the ground around 520 BCE. In 1835, Henry Creswicke Rawlinson was training troops of the Shah of Iran when he encountered the inscription. In order to reach the writings and transcribe them, Rawlinson needed to dangle from the cliffs, or to stand on the very top rung of a long ladder. From these precarious positions, he copied as much of the inscription as possible.
A “Kurdish boy”, whose name seems to be lost to history, assisted the daring endeavour. The boy was said to have used pegs dug into the rock wall as anchors to swing across the cliffs and reach the most inaccessible parts of the writing. Returning home, Rawlinson began working to unlock the secret of the lost script, perhaps with his pet lion cub by his side.
Of the three languages, the Old Persian was the first to be decoded by Rawlinson. Scholars working on deciphering the script gained a sense of the chronological placement of the inscription and recognised some repeated signs, thereby gleaning something of the content and structure of the writings.
The presence of king lists in the Behistun inscription, which could be compared with lists in Herodotus’ Histories, provided a point of reference for deciphering the signs. Other Greek historians, and the Bible, were also consulted in the process. Through the contributions of a number of scholars in the first half of the 19th century, cuneiform slowly began to reveal its secrets.
The significance of the Behistun inscription in the translation of cuneiform is often likened to the importance of the Rosetta Stone for deciphering Egyptian hieroglyphs. In recent years, the inscription has been the focus of restorative efforts, after sustaining various types of damage — notably when Allied troops used the inscription for target practice during World War II. It is now a UNESCO World Heritage site.
As the deciphering went on, divisions developed in the academic community over whether efforts to unravel cuneiform had proven successful. Part of the controversy stemmed from the extreme intricacy of the writing system. Cuneiform languages are made up of a collection of signs, and the meaning of these signs shows a great deal of variety.
In the Akkadian language, for example, a cuneiform sign may have a phonetic value — but not always the same phonetic value — or it may be a logogram, symbolising a word (such as “temple”), or a determinative sign, such as for a place or an occupation. This gives the translation of cuneiform a puzzle-like quality. The translator must select the value of the sign that appears best suited to the context.
Some scholars probably had sensible reasons for questioning the deciphering of cuneiform. Others held the inaccurate view that ancient Assyrians would have lacked the capacity to comprehend such a difficult writing system. To resolve the controversy, the British scientist W.H. Fox Talbot suggested a kind of cuneiform competition.
The British Royal Asiatic Society held the contest in 1857. Four scholars – Fox Talbot, Rawlinson and a Dr Hincks and a Dr Oppert – made unique translations of a single, previously unseen, cuneiform inscription. Each scholar then sent their translation in strict confidence to the society for comparison. After opening the sealed letters and examining the four translations, the society decided that the similarities between them were sufficiently compelling to declare cuneiform deciphered.
The rediscovery of cuneiform literature was not without further controversy. Fierce debates were conducted in eloquent handwritten letters over who had contributed to the discovery and decipherment of texts, and who deserved credit for the achievement.
As well as this, the content of the literature caused friction in the academic communities of the 19th century. Prior to the rediscovery of cuneiform, the most prominent source for the Ancient Near East was the Hebrew Bible. The ability of cuneiform literature to provide a new perspective on the rich history of Egypt and Mesopotamia was embraced by many, but viewed with suspicion by others. For some, the translation of the long-forgotten writings raised the possibility of conflict between cuneiform sources and biblical literature.
Perhaps one of the most overt examples of these tensions in scholarly circles can be seen in the career of Nathaniel Schmidt from Colgate University. Schmidt was tried for heresy in 1895, due to the view that many of his translations of cuneiform appeared contrary to biblical traditions. He was dismissed from his position at Colgate in 1896. Following his dismissal, the eminent scholar was recruited by Cornell University (his controversial departure from Cornell made his appointment something of a “bargain”), where he taught Hebrew, Arabic, Aramaic, Coptic, Syriac and many other ancient languages.
From cuneiform to the stars
The recovery of cuneiform has provided access to an embarrassment of textual riches, including hundreds of thousands of legal and economic records, magico-medical texts, omens and prophecies, wisdom literature and lullabies.
Cuneiform has also aided scientific mysteries. Babylonian records of a solar eclipse, written in cuneiform, have helped astronomers figure out how much Earth’s rotation has slowed.
The decipherment of the cuneiform script has reopened a timeless dialogue beyond ancient and modern civilisations, providing continued opportunities to better understand the world around us, and beyond.
Note: This essay contains details from the article “Comparative Translations”, in the Journal of the Royal Asiatic Society 18, 1861. My grateful thanks to the Royal Asiatic Society for generously allowing access to their collection.
The ancient Babylonians – who lived from about 4,000BCE in what is now Iraq – had a long forgotten understanding of right-angled triangles that was much simpler and more accurate than the conventional trigonometry we are taught in schools.
Our new research, published in Historia Mathematica, shows that the Babylonians were able to construct a trigonometric table using only the exact ratios of sides of a right-angled triangle. This is a completely different form of trigonometry that does not need the familiar modern concept of angles.
At school we are told that the shape of a right-angled triangle depends upon the other two angles. The angle is related to the circumference of a circle, which is divided into 360 parts or degrees. This angle is then used to describe the ratios of the sides of the right-angled triangle through sin, cos and tan.
But circles and right-angled triangles are very different, and the price of having simple values for the angle is borne by the ratios, which are very complicated and must be approximated.
This approach can be traced back to the Greek astronomer and mathematician Hipparchus of Nicaea (who died after 127 BCE). He is said to be the father of trigonometry because he used his table of chords to calculate orbits of the Moon and Sun.
But our new research shows this was not the first, or only, or best approach to trigonometry.
The Babylonians discovered their own unique form of trigonometry during the Old Babylonian period (1900-1600BCE), more than 1,500 years earlier than the Greek form.
Remarkably, their trigonometry contains none of the hallmarks of our modern trigonometry – it does not use angles and it does not use approximation.
The Babylonians had a completely different conceptualisation of a right triangle. They saw it as half of a rectangle, and due to their sophisticated sexagesimal (base 60) number system they were able to construct a wide variety of right triangles using only exact ratios.
The sexagesimal system is better suited for exact calculation. For example, if you divide one hour by three then you get exactly 20 minutes. But if you divide one dollar by three then you get 33 cents, with 1 cent left over. The fundamental difference is the convention to treat hours and dollars in different number systems: time is sexagesimal and dollars are decimal.
The Babylonians knew that their sexagesmial number system allowed for many more exact divisions. For a more sophisticated example, 1 hour divided by 48 is 1 minute and 15 seconds.
This precise arithmetic of the Babylonians also influenced their geometry, which they preferred to be exact. They were able to generate a wide variety of right-angled triangles within exact ratios b/l and d/l, where b, l and d are the short side, long side and diagonal of a rectangle.
The ratio b/l was particularly important to the ancient Babylonians and Egyptians because they used this ratio to measure steepness.
The Plimpton 322 tablet
We now know that the Babylonians studied trigonometry because we have a fragment of a one of their trigonometric tables.
Plimpton 322 is a broken clay tablet from the ancient city of Larsa, which was located near Tell as-Senkereh in modern day Iraq. The tablet was written between 1822-1762BCE.
In the 1920s the archaeologist, academic and adventurer Edgar J Banks sold the tablet to the American publisher and philanthropist George Arthur Plimpton.
Plimpton bequeathed his entire collection of mathematical artefacts to Columbia University in 1936, and it resides there today in the Rare Book and Manuscript Library. It’s available online through the Cuneiform Digital Library Initiative.
In 1945 the tablet was revealed to contain a highly sophisticated sequence of integer numbers that satisfy the Pythagorean equation a2+b2=c2, known as Pythagorean triples.
This is the fundamental relationship of the three sides of a right-angled triangle, and this discovery proved that the Babylonians knew this relationship more than 1,000 years before the Greek mathematician Pythagoras was born.
Plimpton 322 has ruled space on the reverse which indicates that additional rows were intended. In 1964, the Yale based science historian Derek J de Solla Price discovered the pattern behind the complex sequence of Pythagorean triples and we now know that it was originally intended to contain 38 rows in total.
The tablet also has missing columns, and in 1981 the Swedish mathematics historian Jöran Friberg conjectured that the missing columns should be the ratios b/l and d/l. But the tablet’s purpose remained elusive.
The surviving fragment of Plimpton 322 starts with the Pythagorean triple 119, 120, 169. The next triple is 3367, 3456, 4825. This makes sense when you realise that the first triple is almost a square (which is an extreme kind of rectangle), and the next is slightly flatter. In fact the right-angled triangles are slowly but steadily getting flatter throughout the entire sequence.
So the trigonometric nature of this table is suggested by the information on the surviving fragment alone, but it is even more apparent from the reconstructed tablet.
This argument must be made carefully because modern notions such as angle were not present at the time Plimpton 322 was written. How then might it be a trigonometric table?
Fundamentally a trigonometric table must describe three ratios of a right triangle. So we throw away sin and cos and instead start with the ratios b/l and d/l. The ratio which replaces tan would then be b/d or d/b, but neither can be expressed exactly in sexagesimal.
Instead, information about this ratio is split into three columns of exact numbers. A squared index and simplified values of b and d to help the scribe make their own approximation to b/d or d/b.
The most remarkable aspect of Babylonian trigonometry is its precision. Babylonian trigonometry is exact, whereas we are accustomed to approximate trigonometry.
The Babylonian approach is also much simpler because it only uses exact ratios. There are no irrational numbers and no angles, and this means that there is also no sin, cos or tan or approximation.
It is difficult to say why this approach to trigonometry has not survived. Perhaps it went out of fashion because the Greek approach using angles is more suitable for astronomical calculations. Perhaps this understanding was lost in 1762BCE when Larsa was captured by Hammurabi of Babylon. Without evidence, we can only speculate.
We are only beginning to understand this ancient civilisation, which is likely to hold many more secrets waiting to be discovered.