Babylonian Trig

Evelyn Lamb in Scientific American:

Don’t Fall for Babylonian Trigonometry Hype

Separating fact from speculation in math history

You may have seen headlines about an ancient Mesopotamian tablet. “Mathematical secrets of ancient tablet unlocked after nearly a century of study,” said the Guardian. “This mysterious ancient tablet could teach us a thing or two about math,” said Popular Science, adding, “Some researchers say the Babylonians invented trigonometry—and did it better.” National Geographic was a bit more circumspect: “A new study claims the tablet could be one of the oldest contributions to the study of trigonometry, but some remain skeptical.” Daniel Mansfield and Norman Wildberger certainly did a good job selling their new paper in the generally more staid journal Historia Mathematica. I’d like to help separate fact from speculation and outright nonsense when it comes to this new paper.

What is Plimpton 322?

Plimpton 322, the tablet in question, is certainly an alluring artifact. It’s a broken piece of clay roughly the size of a postcard. It was filled with four columns of cuneiform numbers around 1800 BCE, probably in the ancient city of Larsa (now in Iraq) and was removed in the 1920s. George Plimpton bought it in 1922 and bequeathed it to Columbia University, which has owned it since 1936. Since then, many scholars have studied Plimpton 322, so any picture you might have of Mansfield and Wildberger on their hands and knees in a hot, dusty archaeological site, or even rummaging through musty, neglected archives and unearthing this treasure is inaccurate. We’ve known about the artifact and what was on it for decades. The researchers claim to have a new interpretation of how the artifact was used, but I am skeptical.

Scholars have known since the 1940s that Plimpton 322 contains numbers involved in Pythagorean triples, that is, integer solutions to the equation a2+b2=c2. For example, 3-4-5 is a Pythagorean triple because 32+42=9+16=25=52. August 15 of this year was celebrated by some as “Pythagorean Triple Day” because 8-15-17 is another, slightly sexier, such triple.

The far right column consists of the numbers 1 through 15, so it’s just an enumeration. The two middle columns of Plimpton 322 contain one side and the hypotenuse of a Pythagorean triangle, or a and c in the equation a2+b2=c2. (Note that a and b are interchangeable.) But these are a little brawnier than the Pythagorean triples you learn in school. The first entries are 119 and 169, corresponding to the Pythagorean triple 1192+1202=1692. The far left column is a ratio of squares of the sides of the triangles. Exactly which sides depends slightly on what is contained in the missing shard from the left side of the artifact, but it doesn’t make a huge difference. It’s either the square of the hypotenuse divided by the square of the remaining leg or the square of one leg divided by the square of the other leg. In modern mathematical jargon, these are squares of either the tangent or the secant of an angle in the triangle.

We can interpret one of the columns as containing trigonometric functions, so in some sense, it is a trig table. But despite what the headlines would have you believe, people have known that for decades. The mystery is what purpose the tablet served in its time. Why was it created? Why were those particular triangles included in the table? How were the columns computed? In a 1980 paper titled “Sherlock Holmes in Babylon,” R. Creighton Buck implied that through mathematics and cunning observation, one could sleuth out the meaning of the tablet and offered an explanation he thought fit the data. But Eleanor Robson, in “Neither Sherlock Holmes nor Babylon,” writes, “Ancient mathematical texts and artefacts, if we are to understand them fully, must be viewed in the light of their mathematico-historical context, and not treated as artificial, self-contained creations in the style of detective stories.” It’s arrogant and will probably lead to incorrect conclusions to look at ancient artifacts primarily through the lens of our modern understanding of mathematics.

What did it do?

There are a few theories about how Plimpton 322 was created and used by the person or people who made it. Mansfield and Wildberger are not the first to believe it’s some sort of trig table. On the other hand, some believe it links the Pythagorean theorem (known by these ancient Mesopotamians and many other civilizations long before Pythagoras) with the method of completing the square to solve a quadratic equation, a common problem in mathematical texts from that time and place. Some believe the triples were generated using different numbers not included in the table in a “number theoretic” way. Some believe the numbers came from so-called reciprocal pairs that were used for multiplication. Some think the tablet was a pedagogical tool, perhaps a source of exercises for students. Some believe it was used in something more like original mathematical research. Academic but readable information about these interpretations can be found in articles by Buck in 1980, Robson in 2001 and 2002, and John P. Britton, Christine Proust, and Steve Shnider from 2011.

If it is a trigonometry table, is it better than modern trigonometry tables?

Mansfield and Wildberger’s contribution to scholarship on Plimpton 322 seems to be speculation that the artifact could be used to do trigonometry in a more exact way than we do now. In a publicity video by UNSW that must have accompanied the press releases sent to many math and science journalists (but not to me—what gives, UNSW?), Mansfield makes the claims that this table is “superior in some ways to modern trigonometry” and the “only completely accurate trigonometry table.”

It’s hard to know where to start with this part of their claims. For one, the tablet contains some well-known errors, so claims that it is the most accurate or exact trig table ever are just not true. But even a corrected version of Plimpton 322 would not be a revolutionary replacement for modern trig tables….

Is base 60 better than base 10?

Perhaps the utility of different types of trig tables is a matter of opinion, but the UNSW video also has some outright falsehoods about accuracy in base 60 versus the base 10 system we now use. Around the 1:10 mark, Mansfield says, “We count in base 10, which only has two exact fractions: 1/2, which is 0.5, and 1/5.” My first objection is that any fraction is exact. The number 1/3 is precisely 1/3. Mansfield makes it clear that what he means by 1/3 not being an exact fraction is that it has an infinite (0.333…) rather than a terminating decimal. But what about 1/4? That’s 0.25, which terminates, and yet Mansfield doesn’t consider it an exact fraction. And what about 1/10 or 2/5? Those can be written 0.1 and 0.4, which seem pretty exact.

Indefensibly, when he lauds the many “exact fractions” available in base 60, he doesn’t apply the same standards. In base 60, 1/8 would be written 7/60+30/3600 which is the same idea as writing 0.25, or 2/10+5/100, for 1/4 in base 10. Why is 1/8 exact in base 60 but 1/4 not exact in base 10? It’s hard to believe this is an honest mistake coming from a mathematician and instead makes me even more suspicious that his work is motivated by an agenda.

Plimpton 322 is a remarkable artifact, and we have much to learn from it. When I taught math history, I loved opening the semester by having my students read a few papers about it to show how much scholarship has gone into understanding such a small document and how accomplished scholars can disagree about what it means. It demonstrates differences in the way different cultures have done mathematics and outstanding computational facility. It has raised questions about how ancient Mesopotamians approached calculation and geometry. But using it to sell a questionable pet theory won’t get us any closer to the answers.

Different Sources

One of the maddening things about the history recorded in the Hebrew Bible is that it is not corroborated by many other primary sources. There is no Egyptian evidence, for instance, that the Hebrews were ever enslaved in Egypt – and no archaeological evidence that 100,000 of them were wandering around the Sinai Desert for forty years. Some scholars even dispute the existence of King David. But as we move forward in time some events are attested in other sources. Two of my favorites are below – and they provide very interesting examples of differing perspectives on the same event. The first takes place in 701 BC, when Sennacherib, king of Assyria, invaded the kingdom of Judah and besieged Jerusalem (his predecessor Shalmaneser had already defeated and deported the northern kingdom of Israel in 722 BC).

2 Kings 19: King Sennacherib sent messengers to Hezekiah with this word: “Say to Hezekiah king of Judah: Do not let the god you depend on deceive you when he says, ‘Jerusalem will not be handed over to the king of Assyria.’ Surely you have heard what the kings of Assyria have done to all the countries, destroying them completely. And will you be delivered? Did the gods of the nations that were destroyed by my forefathers deliver them?… Hezekiah received the letter from the messengers and read it. Then he went up to the temple of the LORD and spread it out before the LORD. And Hezekiah prayed to the LORD: “O LORD, God of Israel, enthroned between the cherubim, you alone are God over all the kingdoms of the earth. You have made heaven and earth. Give ear, O LORD, and hear; open your eyes, O LORD, and see; listen to the words Sennacherib has sent to insult the living God… That night the angel of the LORD went out and put to death a hundred and eighty-five thousand men in the Assyrian camp. When the people got up the next morning – there were all the dead bodies! So Sennacherib king of Assyria broke camp and withdrew. He returned to Nineveh and stayed there.

“185,000 men” is  probably an exaggeration, but we can be pretty certain that this actually occurred, for one of the documents dug up at the Assyrian capital of Nineveh in the nineteenth century was Sennacherib’s Annals, recorded in Akkadian cuneiform on three hexagonal clay prisms. The relevant bits:

As to Hezekiah, the Jew, he did not submit to my yoke, I laid siege to his strong cities, walled forts, and countless small villages, and conquered them by means of well-stamped earth-ramps and battering-rams brought near the walls with an attack by foot soldiers, using mines, breeches as well as trenches. I drove out 200,150 people, young and old, male and female, horses, mules, donkeys, camels, big and small cattle beyond counting, and considered them slaves. Himself I made a prisoner in Jerusalem, his royal residence, like a bird in a cage. I surrounded him with earthwork in order to molest those who were at his city’s gate. Thus I reduced his country, but I still increased the tribute and the presents to me as overlord which I imposed upon him beyond the former tribute, to be delivered annually.

“Himself I made a prisoner in Jerusalem, [and] surrounded him with earthwork.” And…? The reader is expecting a third clause, something about taking Jerusalem, putting the inhabitants to the sword, and razing the Temple. But no such thing is mentioned! Cleary he didn’t manage to take Jerusalem for some reason – and a plague breaking out among his men is as likely a reason as any. (I doubt that it had much of a long-term effect on his state, though, and as he brags he did quite a bit of damage to Judah otherwise.)

It was the Babylonians, of course, who succeeded where the Assyrians failed, who took Jerusalem in 587 and then deported its inhabitants to Babylon as slaves. When Cyrus the Great conquered Babylon in 539, he famously allowed the Jews to return home. Here is what the book of Ezra has to say about it, the first of our second pair of documents:

In the first year of Cyrus king of Persia, in order to fulfill the word of the LORD spoken by Jeremiah, the LORD moved the heart of Cyrus king of Persia to make a proclamation throughout his realm and to put it in writing: “This is what Cyrus king of Persia says: ‘The LORD, the God of heaven, has given me all the kingdoms of the earth and he has appointed me to build a temple for him at Jerusalem in Judah. Anyone of his people among you – may his God be with him, and let him go up to Jerusalem in Judah and build the temple of the LORD, the God of Israel, the God who is in Jerusalem. And the people of any place where survivors may now be living are to provide them with silver and gold, with goods and livestock, and with freewill offerings for the temple of God in Jerusalem.’” Then the family heads of Judah and Benjamin, and the priests and Levites – everyone whose heart God had moved – prepared to go up and build the house of the LORD in Jerusalem. 6 All their neighbors assisted them with articles of silver and gold, with goods and livestock, and with valuable gifts, in addition to all the freewill offerings.

If you think that it’s odd that Cyrus should be so inspired by Yahweh, the God of the Jews… you’re probably right. From the Kurash Prism, a more direct source for the king’s motivations:

I am Kurash [“Cyrus”], King of the World, Great King, Legitimate King, King of Babilani, King of Kiengir and Akkade, King of the four rims of the earth… whose rule Bel and Nebo love, whom they want as king to please their hearts. When I entered Babilani as a friend and when I established the seat of the government in the palace of the ruler under jubilation and rejoicing, Marduk, the great lord, induced the magnanimous inhabitants of Babilani to love me, and I was daily endeavoring to worship him…. As to the region from as far as Assura and Susa, Akkade, Eshnunna, the towns Zamban, Me-turnu, Der as well as the region of the Gutians, I returned to these sacred cities on the other side of the Tigris the sanctuaries of which have been ruins for a long time, the images which used to live therein and established for them permanent sanctuaries. I also gathered all their former inhabitants and returned them to their habitations. Furthermore, I resettled upon the command of Marduk, the great lord, all the gods of Kiengir and Akkade whom Nabonidus had brought into Babilani to the anger of the lord of the gods, unharmed, in their former temples, the places which make them happy.

No mention of the Jews or their God at all! Instead, Cyrus is working to please Marduk, the god of the Babylon he has just conquered – and he ordered everyone to return to their former habitations, not just Jews.

Methinks that the author of Ezra has succumbed to the very human desire to believe that It’s All About Me.

(Both of these pairs [1, 2] may be found in the Internet Ancient History Sourcebook.)

Code of Hammurabi

It should really be the Code of Marduk, of course. Just look at the bas-relief sculpture at the top of the stele:


American Historical Association.

According to people more informed than I am, King Hammurabi (ruled in Babylon, in Mesopotamia, in the early 18th century BC) is the one standing in the posture of respect to the god, who is sitting, holding a symbol of authority, and telling the king which the laws he wants instituted. The classic comparison here is Moses atop Mount Sinai getting instruction from Yahweh; in both cases we see how religion helps to justify the system. These laws aren’t arbitrary! Some human, as important as he was, didn’t just make them up on the toilet one morning. They come from a god, so you’d better obey them.

The two notable features of the Code of Hammurabi are retributive justice (“an eye for an eye”), and differentiated punishment, based on one’s social status. Generally, if you harm a social equal, you suffer the same harm back, but if you harm someone beneath you in the social hierarchy (that is, if you’re an aristocrat and you hurt a commoner, or if you’re a commoner and you hurt a slave), you can always pay a fine and get out of being hurt yourself.* Nowadays we’re appalled by this, of course: as the bumper sticker says, “an eye for an eye leaves everyone blind!” And really, if someone harmed me, I wouldn’t be satisfied by getting to inflict the same harm on him – I’d much rather have some cold hard cash for compensation instead. But I have a theory about the lex talionis. That is, nowadays murder, assault, battery, grievous bodily harm, etc. are all crimes – crimes against the state, whose subjects have been harmed and whose peace has been disturbed. Thus does the state reserve for itself the right to punish such actions. This wasn’t always the case – in many other times and places hurting someone was something between you and him – or more accurately, between your people and his people. The state was much more self-interested and self-preservative. Harming someone else was like putting up a fence three feet beyond your actual property line and trying to claim a bit of your neighbor’s yard. This is not something he can call the police about, and it’s not something that anyone will even enforce save for him complaining about it. Violent revenge, in other times and places, was legitimate in a way that it is not in the present-day United States. What retributive justice did, therefore, is to impose a ceiling on the amount of revenge you could take. It’s an eye for an eye – not two eyes, seven teeth, and an ear. It is natural to escalate, to inflict far more suffering than you have suffered, but even ancient states had an interest in stopping such cycles of violence. Thus the proportional (and limited) violence allowed.

* As far as the formal laws were concerned, of course. We have no idea whether these laws were actually enforced, or merely rhetorical. (Nevertheless, laws remain a good historical source for a given society, because no one passes a law against something that isn’t happening.)

Some of my favorite laws from the Code of Hammurabi:

2. If any one bring an accusation against a man, and the accused go to the river and leap into the river, if he sink in the river his accuser shall take possession of his house. But if the river prove that the accused is not guilty, and he escape unhurt, then he who had brought the accusation shall be put to death, while he who leaped into the river shall take possession of the house that had belonged to his accuser.

This is a wonderful description of trial by ordeal. The river knows! But don’t make your accusation lightly (the stakes are a little higher here than losing a civil suit and having to pay your opponent’s court costs, as is the case in Canada.)

I hope that enterprising Babylonians taught themselves how to swim.

53. If any one be too lazy to keep his dam in proper condition, and does not so keep it; if then the dam break and all the fields be flooded, then shall he in whose dam the break occurred be sold for money, and the money shall replace the corn which he has caused to be ruined.

Agriculture was dependent on irrigation – but everyone had to pull together to make sure that it worked.

57. If a shepherd, without the permission of the owner of the field, and without the knowledge of the owner of the sheep, lets the sheep into a field to graze, then the owner of the field shall harvest his crop, and the shepherd, who had pastured his flock there without permission of the owner of the field, shall pay to the owner twenty gur of corn for every ten gan.

Oh, the farmer and the shepherd should be friends! It’s interesting how you either raised crops or you raised animals – rarely did people do both. It’s also interesting how raising crops seems to be the more important activity here – in contrast to ancient Hebrew society, which seemed to favor pastoralism (viz. the gifts of Cain and Abel in Genesis).

104. If a merchant give an agent corn, wool, oil, or any other goods to transport, the agent shall give a receipt for the amount, and compensate the merchant therefor. Then he shall obtain a receipt from the merchant for the money that he gives the merchant.

Here we see evidence of long-distance trade carried out by merchants and their employees – and the perennial temptation to cheat.

108. If a tavern-keeper (feminine) does not accept corn according to gross weight in payment of drink, but takes money, and the price of the drink is less than that of the corn, she shall be convicted and thrown into the water.

110. If a “sister of a god” open a tavern, or enter a tavern to drink, then shall this woman be burned to death.

Then as now bars were disreputable places. They watered down the liquor! They were so bad otherwise that the Babylonian equivalent of nuns were forbidden to enter them. I wonder if they weren’t associated with prostitution, with the feminine tavern-keeper playing the role of the madam.

142. If a woman quarrel with her husband, and say: “You are not congenial to me,” the reasons for her prejudice must be presented. If she is guiltless, and there is no fault on her part, but he leaves and neglects her, then no guilt attaches to this woman, she shall take her dowry and go back to her father’s house.

It’s definitely a man’s world in ancient Babylon, but I like how women have some rights. Here, she can actually initiate divorce, and as long as she is “guiltless,” she can leave.

215. If a physician make a large incision with an operating knife and cure it, or if he open a tumor (over the eye) with an operating knife, and saves the eye, he shall receive ten shekels in money.

218. If a physician make a large incision with the operating knife, and kill him, or open a tumor with the operating knife, and cut out the eye, his hands shall be cut off.

So Babylon had professional physicians. I wonder if the ten shekels was a floor or a ceiling – that is, was it an especially generous reward for competence, or was it a maximum, to prevent the greedy physician from charging even more? Note that you were punished exceedingly if you failed. That’s just the way it is in the Code of Hammurabi!

Astronomical Geometry

From the National Post:

Clay tablets reveal Babylonians invented astronomical geometry 1,400 years before Europeans

The medieval mathematicians of Oxford, toiling in torchlight in a land ravaged by plague, managed to invent a simple form of calculus that could be used to track the motion of heavenly bodies. But now a scholar studying ancient clay tablets suggests that the Babylonians got there first, and by at least 1,400 years.

The astronomers of Babylonia, scratching tiny marks in soft clay, used surprisingly sophisticated geometry to calculate the orbit of what they called the White Star — the planet Jupiter.

These tablets are quite incomprehensible to the untrained eye. Thousands of clay tablets — many unearthed in the 19th century by adventurers hoping to build museum collections in Europe, the United States and elsewhere — remain undeciphered.

But they are fertile ground for Mathieu Ossendrijver of Humboldt University in Berlin, whose remarkable findings were published Thursday in the journal Science.Ossendrijver is an astrophysicist who became an expert in the history of ancient science.

For a number of years he has puzzled over four particular Babylonian tablets housed in the British Museum in London.

“I couldn’t understand what they were about. I couldn’t understand anything about them, neither did anyone else. I could only see that they dealt with geometrical stuff,” he said this week in a phone interview from Germany.

Then one day in late 2014, a retired archaeologist gave him some black-and-white photographs of tablets stored at the museum. Ossendrijver took notice of one of them, just two inches across and two inches high. This rounded object, which he scrutinized in person in September 2015, proved to be a kind of Rosetta Stone.

Officially named BH40054 by the museum, and dubbed Text A by Ossendrijver, the little tablet had markings that served as a kind of abbreviation of a longer calculation that looked familiar to him. By comparing Text A to the four previously mysterious tablets, he was able to decode what was going on: This was all about Jupiter. The five tablets computed the predictable motion of Jupiter relative to the other planets and the distant stars.

“This tablet contains numbers and computations, additions, divisions, multiplications. It doesn’t actually mention Jupiter. It’s a highly abbreviated version of a more complete computation that I already knew from five, six, seven other tablets,” he said.

Most strikingly, the methodology for those computations used techniques that resembled the astronomical geometry developed in the 14th century at Oxford. The tablets have been authoritatively dated to a period from 350 B.C. to 50 B.C.

The people of Mesopotamia — what is now Iraq — developed mathematics about 5,000 years ago. Among them were the Babylonians who wrote in cunieform script and, over time, adopted a sexagesimal (base 60) numbering system. Early mathematics was essentially a form of counting, and the things being counted were mostly sheep and the like.

Mathematics progressed, as did the sharing of knowledge in the wake of Alexander the Great’s conquering journeys across Asia. The ancient Greek astronomer Aristarchus of Samos argued for a heliocentric universe — one in which the Earth orbited the sun, contrary to what seems to be the case when one looks at the sky. That view was shared by another astronomer, possibly Greek as well, who lived in Mesopotamia on the Tigris River and was known as Seleucus of Seleucia.

But Ossendrijver said nothing in the newly decoded computations suggests that the ancient scientist or scientists who etched the tablets understood that heliocentric model. The calculations merely describe Jupiter’s motion over time as it appears to speed up and slow down in its journey across the night sky. Those calculations are done in a surprisingly abstract way — the same way the Oxford mathematicians would do them a millennium and a half later.

“It’s geometry, which is itself old, but it’s applied in a completely new way, not to fields, or something that lives in real space, but to something that exists in completely abstract space,” Ossendrijver said. “Anybody who studies physics would be reminded of integral calculus.”

Which was invented in Europe in 1350, according to historians.

“In Babylonia, between 350 and 50 B.C., scholars, or maybe one very clever guy, came up with the idea of drawing graphs of the velocity of a planet against time, and computing the area of this graph — of doing a kind of computation that seems to be thoroughly modern, that is not found until 1350,” he said.

Alexander Jones, a professor at New York University’s Institute for the Study of the Ancient World, praised Ossendrijver’s research, which he said shows the “revolutionary brilliance of the unknown Mesopotamian scholars who constructed Babylonian mathematical astronomy during the second half of the first millennium BC.”


An article from the BBC, courtesy my friend Alex Lesk Blomerus:


Post-traumatic stress ‘evident in 1300BC’

By James Gallagher

Evidence of post-traumatic stress disorder can be traced back to 1300BC – much earlier than previously thought – say researchers.

The team at Anglia Ruskin University analysed translations from ancient Iraq or Mesopotamia.

Accounts of soldiers being visited by “ghosts they faced in battle” fitted with a modern diagnosis of PTSD.

The condition was likely to be as old as human civilisation, the researchers concluded.

Prof Jamie Hacker Hughes, a former consultant clinical psychologist for the Ministry of Defence, said the first description of PTSD was often accredited to the Greek historian Herodotus.

Referring to the warrior Epizelus during the battle of Marathon in 490BC he wrote: “He suddenly lost sight of both eyes, though nothing had touched him.”

But Prof Hughes’ report – titled Nothing New Under the Sun – argues there are references in the Assyrian Dynasty in Mesopotamia between 1300BC and 609BC.

In that era men spent a year being toughened up by building roads, bridges and other projects, before spending a year at war and then returning to their families for a year before starting the cycle again.

Prof Hughes told the BBC News website: “The sorts of symptoms after battle were very clearly what we would call now post-traumatic stress symptoms.

“They described hearing and seeing ghosts talking to them, who would be the ghosts of people they’d killed in battle – and that’s exactly the experience of modern-day soldiers who’ve been involved in close hand-to-hand combat.”

A diagnosis and understanding of post-traumatic stress disorder emerged after the Vietnam War. It was dismissed as shell shock in World War One.

Prof Hughes said: “As long as there has been civilisation and as long as there has been warfare, there has been post-traumatic symptoms. It’s not a 21st Century thing.”


It would have been nice if they had named and actually quoted the sources pointing to this. Note that the headline claims PTSD was “evident in 1300 BC”, and the article says it was during the Assyrian empire, “between 1300 and 609 BC.” My hunch is that the sources are probably from the later end of that span, perhaps contained in some of the 20,000 cuneiform tablets excavated at Nineveh and now in the British Museum.

(Groundbreaking work on the topic of historical PTSD was done by Jonathan Shay in his 1994 book, Achilles in Vietnam, which examined veterans of combat in Vietnam in light of the characters in Homer’s Iliad.)

Speaking of cuneiform, I recommend Irving Finkel’s The Ark Before Noah: Decoding the Story of the Flood, which my wife got me for Christmas.