A new analysis of a set of ancient clay tablets has revealed that ancient astronomers of Babylonia used advanced geometrical methods to calculate the position of Jupiter – a conceptual leap that was previously thought to have occurred in 14 ^{th} century Europe.

In a report published today in the journal Science, it was revealed that the ancient tablets containing the secret knowledge of the Babylonians, which date from 350 BC to 50 BC, had laid unnoticed in the cuneiform collection at London’s British Museum for decades. It was not until Astroarchaeologist Mathieu Ossendrijver of Humboldt University in Berlin conducted a reanalysis of the tablets from photographs, that the significance of the text was realized.

## Babylonian Astronomy

The history of astronomy in Babylonia (present-day Iraq) originated with the Sumerians who recorded their observations as early as 3500–3200 BC. Astronomical phenomena were important to the Sumerians, who equated planets with gods that held an important role in their mythology and religion. Jupiter, for example, was associated with their main god, Marduk, patron deity of the city of Babylon.

Sumerian astronomy had an important influence on the astronomy of the Babylonians, who produced their first star catalogues by around 1200 BC.

By the 8 ^{th} century BC, Babylonian astronomers had developed a new empirical approach to the prediction of planetary movements, an approach that was later adopted and further developed by the ancient Greeks.

*Astronomical phenomena were important to the Babylonians. Kudurru (stele) of King Melishipak I (1186–1172 BC): the king presents his daughter to the goddess Nannaya. The crescent moon represents the god Sin, the sun the Shamash and the star the goddess Ishtar. *

## Astronomical Calculations in Babylonia

Studies of their textual records found on clay tablets in cuneiform text, suggested that the Babylonian astronomers were using purely arithmetic methods to make their calculations and predictions. However, one of the tablets in the newly analyzed collection, made reference to a trapezoid shape while discussing Jupiter.

Ossendrijver discovered that the trapezoid drawing was being used to predict Jupiter’s place in the zodiac. The computations covered a period of 60 days, beginning on a day when Jupiter first appeared in the night sky just before dawn.

“By calculating the area inside the trapezoid, Babylonian astronomers could find where the planet would be in the sky – exploiting the same link between velocity and displacement taught in introductory calculus classes,” reports New Scientist . This constitutes the only known geometrical method used in Babylonian astronomy, a method thought to have been invented only in 14 ^{th}-century Oxford, Cambridge.

Historian Alexander Jones of New York University told ScienceMag that compared with the complex geometry embraced by the ancient Greeks a few centuries later, the Babylonian inscriptions reflect “a more abstract and profound conception of a geometrical object in which one dimension represents time. Such concepts have not been found earlier than in 14th century European texts on moving bodies. Their presence … testifies to the revolutionary brilliance of the unknown Mesopotamian scholars who constructed Babylonian mathematical astronomy.”

*The Babylonians used geometric methods to predict Jupiter’s place in the zodiac *

Back in 1974, historian A. Aaboe, said in his article ‘ Scientific Astronomy in Antiquity’ that Babylonian astronomy was:

"the first and highly successful attempt at giving a refined mathematical description of astronomical phenomena" and that "all subsequent varieties of scientific astronomy, in the Hellenistic world, in India, in Islam, and in the West—if not indeed all subsequent endeavour in the exact sciences—depend upon Babylonian astronomy in decisive and fundamental ways."

The latest study certainly supports Aaboe’s conclusion over three decades ago, revealing that Babylonian astronomy has made an extremely important contribution to the history of science.

*Featured image: The Clay Tablet that reveals the Babylonians were using calculus to track the path of Jupiter. Credit: Trustees of the British Museum/Mathieu Ossendrijver. *

## Babylonians tracked Jupiter with sophisticated geometrical math

Used geometry that hints at calculus 1,500 years before Europeans.

Trustees of the British Museum/Mathieu Ossendrijver

Even when a culture leaves behind extensive written records, it can be hard to understand their knowledge of technology and the natural world. Written records are often partial, and writers may have been unaware of some technology or simply considered it unremarkable. That’s why the ancient world can still offer up surprises like the Antikythera Mechanism, an ancient mechanical computer that highlighted the Greeks’ knowledge of math, astronomy, and the mechanical tech needed to tie them together.

It took several years after the discovery for the true nature of the Antikythera Mechanism to be understood. And now something similar has happened for the Babylonians. Clay tablets, sitting in the British Museum for decades, show that this culture was able to use sophisticated geometry to track the orbit of Jupiter, relying on methods that in some ways pre-figure the development of calculus centuries later.

We already knew that the Babylonians tracked the orbits of a variety of bodies. There are roughly 450 written tablets that describe the methods and calculations that we’re aware of, and they date from 400 to 50 BCE. Most of the ones that describe how to calculate orbital motion, in the words of Humboldt University’s Mathieu Ossendrijver, “can be represented as flow charts.” Depending on the situation, they describe a series of additions, subtractions, and multiplications that could tell you where a given body would be.

(Complicating matters, Babylonian astronomy worked in base-60, which leads to a very foreign-looking notation.)

The Babylonians did have a grasp of geometric concepts—Ossendrijver calls them “very common in the Babylonian mathematical corpus”—but none of them appeared in their known astronomical calculations.

In the British Museum, however, he located a tablet that hadn’t been formally described, and it contained parts of the procedure for tracking Jupiter. Combined with other tablets, it starts with Jupiter’s first morning rising, tracks it through its apparent retrograde motion, and finishes with its last visible setting at dusk. Again, it’s procedural. Different sections are used to predict the planet’s appearance at different segments of its orbit.

The trapezoid used to calculate the first 120 days of Jupiter’s orbit. The red line divides the first shape into two equal areas. Picture by John Timmer

Ossendrijver took the procedure for calculating the first 120 days and showed that calculating its daily displacement over time produces a trapezoid. In this case, the shape was largely a rectangle but with its top side angled downward over time in two distinct segments. A series of other tablets treated the calculations explicitly as producing a trapezoid.

Things get interesting in the next procedure, which is used to calculate when Jupiter reaches the midpoint in the first half of this stage of its motion. This procedure involved taking the left half of the trapezoid and dividing it into two pieces of equal area. The location of the dividing line (labelled vc above) then produces the answer. As Ossendrijver describes it, “They computed the time when Jupiter covers half this distance by partitioning the trapezoid into two smaller ones of ideally equal area.”

Figuring this out obviously required some sophisticated geometry. European scholars wouldn’t develop similar methods until the 14th century, when they became used at Oxford. The Greeks did use geometry for some astronomical work, but this involved calculations of actual space. The Babylonians here are working in an abstracted time-velocity space.

It’s also striking that this general approach is similar to some aspects of calculus. There, the area under a curve is calculated by mathematically creating an infinite number of small geometric figures and summing their areas. There’s no indication that the Babylonians were anywhere close to taking this intellectual leap given that they only divided this shape up a few times. But it does show that they recognised the value of the general approach.

## Clay Tablet Reveals Ancient Babylonians Used Calculus to Track Jupiter 1,500 Years before Europeans - History

Stone tablets (pictured) used by Babylonians reveal how they tracked the movements of Jupiter

Ancient Babylonians used geometry to track planets in the solar system more than 1,400 years before Europeans were believed to have devised the mathematical technique.

The findings come from analysis of four stone tablets, which reveal how the ancient civilisation tracked the movements of Jupiter.

Scientists had believed the mathematical method had been developed by scholars in Europe in the 14th century, but the tablets date to between 350 to 50 BC.

This could mean Babylonians were using geometrical methods as much as 1,700 years earlier.

Professor Mathieu Ossendrijver, a researcher in the history of ancient science at the Humboldt-Universität zu Berlin who deciphered the markings, said the tablets have redefined the history books.

Translation of the cuneiform tablets revealed they described a mathematical technique to calculate the positions of large bodies in space and time.

The markings depict measurements made to track Jupiter as it moved along the horizon, calculating the planet’s position at 60 days and then at 120 days.

From this, it appears the Babylonians used geometrical calculations based on a trapezoid – a four-sided shape in which the top and bottom lines are parallel – working out the area of the shape, along with its ‘long’ and ‘short’ sides.

The ancient astronomers also computed the time when Jupiter covered half of the 60-day distance by partitioning the trapezoid into two smaller shapes of equal area.

What makes the findings surprising is that it had previously been thought that Babylonian astronomers only used arithmetical concepts, and not geometry.

While ancient Greeks used geometrical figures to describe configurations in physical space, these Babylonian tablets use geometry in an abstract sense to define time and velocity.

The markings depict measurements made to track Jupiter as it moved along the horizon, calculating the planet’s position at 60 days and then at 120 days. From this, it appears the Babylonians used geometrical calculations based on a trapezoid (right) working out the area of the shape

The ancient astronomers also computed the time when Jupiter (pictured) covered half of the 60-day distance by partitioning the trapezoid into two smaller shapes of equal area. What makes the findings surprising is that it had previously been thought that Babylonian astronomers only used arithmetical concepts, and not geometry

The discovery of these calculations had historically been credited to scholars in Europe during the 14th century, at Merton College, Oxford as well as in Paris.

Analysis of four stone tablets has revealed that Babylonian astronomers used geometry to track the movement of Jupiter.

Translation of the cuneiform tablets revealed they described a mathematical technique to calculate the positions of large bodies in space and time.

The markings depict measurements made to track Jupiter as it moved along the horizon, calculating the planet’s position at 60 days and then at 120 days.

From this, it appears the Babylonians used geometrical calculations based on a trapezoid – a four-sided shape in which the top and bottom lines are parallel – working out the area of the shape, along with its ‘long’ and ‘short’ sides.

When the measurements were made, Jupiter would have first been seen rising at dawn and would have been visible for a number of days.

The astronomers would have then captured the ‘daily displacement’ of the gas giant as it moved across the sky, measured as degrees per day.

Scientists had believed that the mathematical method had been developed by scholars in Europe in the 14th century, but the tablets date to between 350 to 50BC, which could mean Babylonians were using geometrical methods as much as 1,700 years earlier.

Professor Ossendrijver explained: ‘The Babylonian trapezoid procedures can be viewed as a concrete example of the same computation.

‘They also show that Babylonian astronomers did, at least occasionally, use geometrical methods for computing planetary positions.’

The translations were prompted by a newly discovered tablet, approximately 4cm across, which shed new light on how the Babylonian astronomers used the trapezoid method.

When the measurements were made, Jupiter would have first been seen rising at dawn and would have been visible for a number of days.

The astronomers would have then captured the ‘daily displacement’ of the gas giant as it moved across the sky, measured as degrees per day.

Professor Ossendrijver wrote: ‘The idea of computing a body’s displacement as an area in time-velocity space is usually traced back to 14th-century Europe.

‘I show that in four ancient Babylonian cuneiform tablets, Jupiter’s displacement along the ecliptic is computed as the area of a trapezoidal figure obtained by drawing its daily displacement against time.

He added: ‘The tablets date from 350 to 50 BC.

‘The trapezoid procedures offer the first evidence for the use of geometrical methods in Babylonian mathematical astronomy, which was thus far viewed as operating exclusively with arithmetical concepts.’

Speaking to MailOnline, Professor Ossendrijve said: ‘The importance of the findings is that these tablets prove that the Babylonians were using geometry in a way that anticipates how we describe and compute motion nowadays.

‘That is, the construct figures that are actually graphs of the velocity (of a planet) against time, so time on one axis, velocity on the other. This is not found anywhere else in antiquity.

hen the measurements were made, Jupiter would have first been seen rising at dawn and would have been visible for a number of days. The astronomers would have then captured the ‘daily displacement’ of the gas giant as it moved across the sky, measured as degrees per day (stone pictured)

‘Any book on the history of math and physics will tell you this type of graph (velocity against time) was invented around 1350, in the middle ages. We now know the Babylonians already invented it.’

A spokesperson for the British Museum, which supplied the tablets, told MailOnline: ‘We always make our objects freely available to scholars and researchers and are truly delighted when one of these valiant workers makes an important breakthrough.

‘It enlarges our appreciation of what the ancient Babylonian astronomers were capable of all those centuries ago.’

The findings are published today in the journal Science.

The cuneiform stone tablets left behind by the Babylonians, like the ones translated in this study, provide a wealth of information on ancient Mesopotamian civilisation.

Last year an exhibition of ancient

clay tablets discovered in modern-day Iraq shed light on the daily life of Jews exiled to Babylon 2,500 years ago.

The cuneiform stone tablets left behind by the Babylonians, like the ones translated in this study, provide a wealth of information on ancient Mesopotamian civilisation. This stock image of cuneiform inscriptions from Babylonia are written in the Sumerian language. They date from the 15th century

More than 100 cuneiform tablets,

each no bigger than an adult’s palm, detailed transactions

and contracts between Judeans driven from, or convinced to move

from, Jerusalem by King Nebuchadnezzar around 600 BC.

Archaeologists got their first chance to see the tablets – acquired by a wealthy London-based Israeli collector – barely two years ago, and said they were ‘blown away’.

Nebuchadnezzar, a powerful ruler famed for the Hanging

Gardens of Babylon, came to Jerusalem several times as he sought

to spread the reach of his kingdom.

Each time he came – and one visit coincided with the

destruction of Jerusalem’s first temple in 586 BC – he either

forced or encouraged the exile of thousands of Judeans.

One exile in 587 BC saw around 1,500 people make the

perilous journey via modern-day Lebanon and Syria to the fertile

crescent of southern Iraq, where the Judeans traded, ran

businesses and helped the administration of the kingdom.

BABYLON AND ANCIENT MESOPOTAMIA

Mesopotamia is an ancient Greek term meaning ‘the land between rivers.’

The region was the name for the Tigris–Euphrates river system, corresponding to modern-day Iraq, northeastern Syria and southeastern Turkey and smaller parts of southwestern Iran and Kuwait.

Mesopotamia has been called the ‘cradle of civilisation’ because agriculture and domestication developed there earlier than anywhere else, around 8,000 years ago.

Bronze Age Mesopotamia included Sumer and the Akkadian, Babylonian, and Assyrian empires. By 3,000 BC, the Mesopotamians had already invented the wheel, developed writing, and created the world’s first cities.

Babylon was a significant city in the region, which was established in 2,300BC.

It is thought to have been the largest city on the world during its history, and spawned an advanced civilisation, known for its astronomical observations.

The city of Babylon was famed for its ‘hanging gardens’ but some have suggested they were not in Babylon at all but were in a Assyrian city called Ninevah, which was known as New Babylon.

Gate to Babylon: Babylon was a significant city in Mesopotamia, established in 2,300BC

The hanging gardens of Babylon were believed to have been built in the city because of its name. However, the gardens may have been found in the Assyrian capital of Nineveh, which was known as New Babylon

## Babylonians developed trigonometry 'superior' to modern day version 3,700 years ago

The Ancient Babylonians knew about a form of trigonometry more advanced than the modern-day version – about 1,000 years before its supposed invention by the Ancient Greeks, academics in Australia say.

The astonishing claim is based on a 3,700-year-old clay tablet inscribed with a table of numbers.

Known as Plimpton 322, it is already known to contain evidence that the Babylonians knew Pythagoras’ famous equation for right-angled triangles, long before the Greek philosopher gave his name to it.

### Recommended

And researchers at the University of New South Wales (UNSW) have claimed it also shows the Babylonians developed a highly sophisticated form of trigonometry – the system of maths used to describe angles that has tortured generations of school pupils with sine, cosine and tangent.

The city of Babylon in Mesopotamia, an early cradle of human civilisation in what is now Iraq, was famed for its Hanging Gardens, said to be one of the Seven Wonders of the ancient world.

And mathematician Dr Daniel Mansfield suggested its people developed trigonometry to help their architects design the city’s major buildings.

“Our research shows it’s a trigonometric table so unfamiliar and advanced that in some respects it’s superior to modern trigonometry,” he said.

“We’ve discovered these lines represent the ratios for a series of right-angled triangles ranging from almost a square to almost a flat line.

“This makes Plimpton 322 a powerful tool that could have been used for surveying fields or architectural calculations to build palaces, temples or step pyramids.”

Dr Mansfield explained that the Babylonians’ system of counting enabled them to perform complicated calculations more easily that mathematicians today.

“The Babylonians unique approach to arithmetic and geometry means this is not only the world’s oldest trigonometric table, it’s also the only completely accurate trigonometric table on record,” he said.

“Why? It all comes down to fractions. We count in base 10 which only has two exact fractions, one half, which is 0.5, and one fifth, which is 0.2.

“That’s problematic if you want to divide. For example, one dollar divided by three is 33 cents with one cent left over.

“The Babylonians counted in base 60, the same system we use for telling time. This has many more exact fractions.

“It doesn’t sound like much, but this allowed them to do a lot more exact division. One hour divided by three is 20 minutes – exactly.

“By using this system, the Babylonians were able to make calculations that completely avoided any inexact numbers, thereby avoiding any errors associated with multiplying those numbers.”

And the Babylonian system might actually have lessons for science today, he claimed.

“With this greater accuracy we think this system has enormous potential for application in surveying, computers and education,” Dr Mansfield said.

“It’s rare that the ancient world teaches us something new. After 3,000 years, Babylonian mathematics might just be coming back into fashion.”

Plimpton 322 was discovered in southern Iraq by the early 1900s by archaeologist, diplomat and antique dealer Edgar Banks, who was the inspiration for the character of Indiana Jones.

The tablet has numbers written in cuneiform script in four columns and 15 rows.

There were suggestions in the 1980s that the numbers showed knowledge of trigonometry, but this had been dismissed more recently.

But Dr Mansfield said their research revealed it was a “novel kind of trigonometry” that was based on ratios, rather than angles and circles.

“It is a fascinating mathematical work that demonstrates undoubted genius,” he said.

One problem with Plimpton 322 is the left-hand edge is broken.

The UNSW researchers presented mathematical evidence that it originally had six columns, rather than four, and 38 rows, not 15.

They believe ancient scribes could have generated numbers using the tablet, which they suggest was a teacher’s aid to checking students’ quadratic equations.

Hipparchus, a Greek astronomer who lived in about 120 BC, is traditionally regarded as the founder of trigonometry.

But Professor Norman Wildberger, who worked with Dr Mansfield, said: “Plimpton 322 predates Hipparchus by more than 1,000 years.

“It opens up new possibilities not just for modern mathematics research, but also for mathematics education. With Plimpton 322 we see a simpler, more accurate trigonometry that has clear advantages over our own.

“A treasure-trove of Babylonian tablets exists, but only a fraction of them have been studied yet. The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us.”

A paper about the research was published in *Historia Mathematica*, the official journal of the International Commission on the History of Mathematics.

## RELATED ARTICLES

The ancient astronomers also computed the time when Jupiter covered half of the 60-day distance by partitioning the trapezoid into two smaller shapes of equal area.

What makes the findings surprising is that it had previously been thought that Babylonian astronomers only used arithmetical concepts, and not geometry.

While ancient Greeks used geometrical figures to describe configurations in physical space, these Babylonian tablets use geometry in an abstract sense to define time and velocity.

The markings depict measurements made to track Jupiter as it moved along the horizon, calculating the planet's position at 60 days and then at 120 days. From this, it appears the Babylonians used geometrical calculations based on a trapezoid (right) working out the area of the shape

The ancient astronomers also computed the time when Jupiter (pictured) covered half of the 60-day distance by partitioning the trapezoid into two smaller shapes of equal area. What makes the findings surprising is that it had previously been thought that Babylonian astronomers only used arithmetical concepts, and not geometry

The discovery of these calculations had historically been credited to scholars in Europe during the 14th century, at Merton College, Oxford as well as in Paris.

### GEOMETRY AND ASTONOMY

Analysis of four stone tablets has revealed that Babylonian astronomers used geometry to track the movement of Jupiter.

Translation of the cuneiform tablets revealed they described a mathematical technique to calculate the positions of large bodies in space and time.

The markings depict measurements made to track Jupiter as it moved along the horizon, calculating the planet's position at 60 days and then at 120 days.

From this, it appears the Babylonians used geometrical calculations based on a trapezoid - a four-sided shape in which the top and bottom lines are parallel – working out the area of the shape, along with its 'long' and 'short' sides.

When the measurements were made, Jupiter would have first been seen rising at dawn and would have been visible for a number of days.

The astronomers would have then captured the 'daily displacement' of the gas giant as it moved across the sky, measured as degrees per day.

Scientists had believed that the mathematical method had been developed by scholars in Europe in the 14th century, but the tablets date to between 350 to 50BC, which could mean Babylonians were using geometrical methods as much as 1,700 years earlier.

Professor Ossendrijver explained: 'The Babylonian trapezoid procedures can be viewed as a concrete example of the same computation.

'They also show that Babylonian astronomers did, at least occasionally, use geometrical methods for computing planetary positions.'

The translations were prompted by a newly discovered tablet, approximately 4cm across, which shed new light on how the Babylonian astronomers used the trapezoid method.

The astronomers would have then captured the 'daily displacement' of the gas giant as it moved across the sky, measured as degrees per day.

Professor Ossendrijver wrote: 'The idea of computing a body's displacement as an area in time-velocity space is usually traced back to 14th-century Europe.

'I show that in four ancient Babylonian cuneiform tablets, Jupiter's displacement along the ecliptic is computed as the area of a trapezoidal figure obtained by drawing its daily displacement against time.

He added: 'The tablets date from 350 to 50 BC.

'The trapezoid procedures offer the first evidence for the use of geometrical methods in Babylonian mathematical astronomy, which was thus far viewed as operating exclusively with arithmetical concepts.'

Speaking to MailOnline, Professor Ossendrijve said: 'The importance of the findings is that these tablets prove that the Babylonians were using geometry in a way that anticipates how we describe and compute motion nowadays.

'That is, the construct figures that are actually graphs of the velocity (of a planet) against time, so time on one axis, velocity on the other. This is not found anywhere else in antiquity.

When the measurements were made, Jupiter would have first been seen rising at dawn and would have been visible for a number of days. The astronomers would have then captured the 'daily displacement' of the gas giant as it moved across the sky, measured as degrees per day (stone pictured)

'Any book on the history of math and physics will tell you this type of graph (velocity against time) was invented around 1350, in the middle ages. We now know the Babylonians already invented it.'

A spokesperson for the British Museum, which supplied the tablets, told MailOnline: 'We always make our objects freely available to scholars and researchers and are truly delighted when one of these valiant workers makes an important breakthrough.

'It enlarges our appreciation of what the ancient Babylonian astronomers were capable of all those centuries ago.'

The findings are published today in the journal Science.

The cuneiform stone tablets left behind by the Babylonians, like the ones translated in this study, provide a wealth of information on ancient Mesopotamian civilisation.

Last year an exhibition of ancient clay tablets discovered in modern-day Iraq shed light on the daily life of Jews exiled to Babylon 2,500 years ago.

The cuneiform stone tablets left behind by the Babylonians, like the ones translated in this study, provide a wealth of information on ancient Mesopotamian civilisation. This stock image of cuneiform inscriptions from Babylonia are written in the Sumerian language. They date from the 15th century

More than 100 cuneiform tablets, each no bigger than an adult's palm, detailed transactions and contracts between Judeans driven from, or convinced to move from, Jerusalem by King Nebuchadnezzar around 600 BC.

Archaeologists got their first chance to see the tablets - acquired by a wealthy London-based Israeli collector - barely two years ago, and said they were 'blown away'.

Nebuchadnezzar, a powerful ruler famed for the Hanging Gardens of Babylon, came to Jerusalem several times as he sought to spread the reach of his kingdom.

Each time he came - and one visit coincided with the destruction of Jerusalem's first temple in 586 BC - he either forced or encouraged the exile of thousands of Judeans.

One exile in 587 BC saw around 1,500 people make the perilous journey via modern-day Lebanon and Syria to the fertile crescent of southern Iraq, where the Judeans traded, ran businesses and helped the administration of the kingdom.

### BABYLON AND ANCIENT MESOPOTAMIA

Mesopotamia is an ancient Greek term meaning 'the land between rivers.'

The region was the name for the Tigris–Euphrates river system, corresponding to modern-day Iraq, northeastern Syria and southeastern Turkey and smaller parts of southwestern Iran and Kuwait.

Mesopotamia has been called the 'cradle of civilisation' because agriculture and domestication developed there earlier than anywhere else, around 8,000 years ago.

Bronze Age Mesopotamia included Sumer and the Akkadian, Babylonian, and Assyrian empires. By 3,000 BC, the Mesopotamians had already invented the wheel, developed writing, and created the world's first cities.

Babylon was a significant city in the region, which was established in 2,300BC.

It is thought to have been the largest city on the world during its history, and spawned an advanced civilisation, known for its astronomical observations.

The city of Babylon was famed for its 'hanging gardens' but some have suggested they were not in Babylon at all but were in a Assyrian city called Ninevah, which was known as New Babylon.

Gate to Babylon: Babylon was a significant city in Mesopotamia, established in 2,300BC

The hanging gardens of Babylon were believed to have been built in the city because of its name. However, the gardens may have been found in the Assyrian capital of Nineveh, which was known as New Babylon

## Ancient Babylonian Astronomy Text Changes History

This Babylonian tablet, written in cuneiform script, contains geometric calculations used to track the motions of Jupiter.**A newly-translated tablet reveals that ancient Babylonian astronomers were using methods far more advanced than imagined for that era.**

The ancient Babylonians were calculating planetary displacement arcs over 1,000 years before the method’s ‘invention’.

Analysis of the tablets reveals that the Babylonian star gazers were able to calculate the position of Jupiter using geometric techniques previously believed to have been first used some 1,400 years later in 14th century Europe.

The tablets, housed at the British Museum, are believed to have been unearthed from an archaeological dig in Mesopotamia, the present day Iraq, sometime in the 1800s.

Gizmodo reports: It’s a well-known fact that the Babylonians were skilled mathematical astronomers, who preserved their knowledge on hundreds of clay tablets. But when astroarchaeologist Matthieu Ossendrijver of Humboldt University in Berlin translated an unstudied text on Jupiter, he discovered something astonishing. To track the gas giant’s path across the sky, the Babylonians used a geometric technique—the so-called trapezoid procedure—that’s a cornerstone of modern calculus. Until now, this method was believed to have been developed in medieval Europe, some 1,400 years later.

“This shows just how highly developed this ancient culture was,” Ossendrijver, whose discovery appears in today’s Science, told Gizmodo. “I don’t think anybody expected something like this would be discovered in a Babylonian text.”

The text belongs to a collection of thousands of clay tablets, inscribed with cuneiform and excavated in Iraq during the 19th century. By translating and studying them over the past century, archeologists have learned a great deal about Babylonians, including their advanced system of astronomy, which grew out of the development of the zodiac around 400 BCE.

Also priests, Babylonian astronomers believed that all Earthly happenings—the weather, the price of grain, the level of the rivers—were connected to the motion of the planets and stars. And of all the forces influencing our world from above, none were as important as Marduk, the patron deity of Babylon. He was associated with Jupiter.

As Ossendrijver explains in his paper, approximately 340 known Babylonian astronomy tablets are filled with data on planetary and lunar positions, arranged in rows and columns like a spreadsheet. Another 110 are procedural, with instructions describing the arithmetical operations (addition, subtraction, and multiplication) used to compute the positions of celestial objects.

But one collection—a set of four tablets on the position of Jupiter—appears to preserve portions of a procedure for calculating the area under a curve. These texts are fragmentary, and for decades their astronomical significance went unnoted. In 2014, Ossendrijver discovered their instruction book: a tablet, he said, that “just fell through the cracks,” and has been collecting dust in the British Museum since 1881.

One of the fragmentary Babylonian texts (left) showing a portion of a calculaton for determining Jupiter’s displacement across the ecliptic plane as the area under a time-velocity curve (right). Via Mathieu Ossendrijver

The now-decoded “text A” describes a procedure for calculating Jupiter’s displacement across the ecliptic plane, the path that the Sun appears to trace through the stars, over the course of a year. According to the text, the Babylonians did so by tracking Jupiter’s speed as a function of time and determining the area under a time-velocity curve.

Until now, the earliest origin of this concept dated to mid 14th-century Europe. “In 1350, mathematicians understood that if you compute the area under this curve, you get the distance travelled,” Ossendrijver said. “That’s quite an abstract insight about connection between time and motion. What is shown by [these texts] is that this insight came about in Babylonia.”

In Ossendrijver’s view, it’s unlikely that this method survived the vast gulf of time between the disappearance of Babylonian culture and its emergence in medieval Europe. “I think it’s more likely they [Europeans] developed it independently,” he said, noting that the trapezoid procedure doesn’t appear to have been popular among Babylonian astronomers, and that much of their knowledge was lost when the culture died out around 100 A.D.

“Who knows what else is hidden in the thousands of tablets lying in in museums around the world?” Ossendrijver continued. “This is part of the history of science, and I hope it raises awareness of the value of protecting that heritage.”

## <em>Science</em>: Ancient Babylonians Used Advanced Geometry to Track Jupiter

Analysis of ancient Babylonian tablets reveals that the tablets' makers used geometry to calculate the position of Jupiter — using a technique that was previously believed to have been developed at least 1400 years later in 14 th century Europe. The findings are published in the 29 January issue of *Science*.

Babylon was an ancient and powerful epicenter in the Middle East. It was rich in many ways, including in scholars and particularly astronomers and mathematicians. Historians rely mostly on clay tablets that record the scholarly work of this era. While several hundred fragmented tablets exist, the analysis of just five of them reveals advanced geometry techniques used to calculate the position of Jupiter through time and space.

"All five tablets discussed in *Science* contain instructions on how to compute the celestial position of Jupiter for a specific period of 60 days after Jupiter becomes visible in the night sky," said Mathieu Ossendrijver of the Humboldt University of Berlin. "On four of them, these instructions mention a geometrical figure, a trapezoid. As it turns out, this trapezoid depicts how Jupiter's velocity changes with time over the 60 days."

Before these results were published, researchers knew that four of the tablets referred to a trapezoid shape, but the context for mentioning the shape was unclear. The deciphering of the fifth tablet was key to understanding the references to the trapezoid in the other tablets, revealing just how advanced ancient Babylonian astronomers were.

The texts contain geometrical calculations based on a trapezoid's area, and its long and short sides. The ancient astronomers also computed the time when Jupiter covers half of this 60-day distance by partitioning the trapezoid into two smaller ones of equal area.

The tablets were most likely written in Babylon between 350 and 50 BCE, making them the earliest known examples of using geometry to calculate positions in time and space. "Ancient Greek astronomers used a lot of geometrical techniques, but the geometrical figures that they use are always situated in a real space, with either two- or three-spatial dimensions," Ossendrijver explains. "The Babylonian geometrical methods discussed here involve figures that are defined in a more abstract mathematical space obtained by drawing velocity against time, almost in a modern fashion."

Since the ancient Greeks were using geometry but not to calculate time and velocity, researchers believed that Europeans in the 1300s were the first to develop such a combination of calculations. However, these tablets redefine the history books, revealing that European scholars in Oxford and Paris in the 14th century were, in fact, centuries behind their ancient Babylonian counterparts.

## RIELPOLITIK

Source – news.nationalpost.com, By Joel Achenbach

– **The astronomers, scratching tiny marks in soft clay, used surprisingly sophisticated geometry to calculate the orbit of the ‘White Star’ — the planet Jupiter:**

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.”

## Babylonians Tracked Jupiter with Fancy Math, Tablet Reveals

BERLIN — For a text that may rewrite the history of mathematics, it looks rather sloppy.

The brown clay tablet, which could fit in the palm of your hand, is scrawled with hasty, highly abbreviated cuneiform characters. And, according to science historian Mathieu Ossendrijver, it proves that the ancient Babylonians used a complex geometrical model that looks like a rudimentary form of integral calculus to calculate the path of Jupiter. Scientists previously thought this mathematical technique was invented in medieval Europe.

"It sounds minute for a layperson, but this geometry is of a very special kind that is not found anywhere else, for instance, in ancient Greek astronomy," Ossendrijver said. "It is an application in astronomy that was totally new. Thus far everybody thought Babylonian scholars only computed with numbers." [The 7 Most Mysterious Archaeological Finds on Earth]

**A sophisticated invention**

The tablet has long been in the collection at the British Museum in London, and it was likely created in Babylon (located in modern-day Iraq) between 350 and 50 B.C. Ossendrijver recently deciphered the text, and he described his discovery in an article that's featured on the cover of the journal Science this week.

From his office at Humboldt University here in Berlin, which is decorated with posters of both the Ishtar Gate and the Antikythera mechanism (thought to be the world's oldest known computer), he explained that the tablet plots the apparent decreasing velocity of Jupiter from the planet's first appearance along the horizon, to 60 days later, and then 120 days later. If drawn on a graph, this relationship is represented in the shape of two conjoined trapezoids. The area of each trapezoid describes Jupiter's total displacement (measured in degrees) along the ecliptic, or the path of the sun.

"It's not an actual trapezoid that describes the shape of a field, or some configuration of the planets in space," Ossendrijver told Live Science. "It's a configuration in a mathematical space. It's a highly abstract application."

Ancient Greek mathematicians and astronomers were using geometry around the same time, but only to make calculations involving real, 3D space, such as using circles torepresent the orbits of planets around Earth. Students of math might take it for granted today, but the abstract use of geometry was, until now, unheard of at the time.

"Anyone who has studied physics or a little bit of math is familiar with making graphs — plotting one quantity against time — but actually this had to be invented once," Ossendrijver said.

Current textbooks on the history of math say this invention took place around A.D. 1350. In the mid-14th century, mathematicians at Merton College in England who were referred to as the "Oxford Calculators," and another scholar collaborating with them in Paris, were interested in understanding the velocity and displacement of an object over time. They came up with the Merton mean speed theorem, which holds that the distance a uniformly accelerating body travels in a given interval of time is the same distance it would travel if it were moving at a constant velocity (with that constant velocity being the average of the accelerating body's initial and final velocity). [Images: The World's 11 Most Beautiful Calculations]

But the mean speed theorem now seems to be a reinvention of a lost model about 1,400 years earlier, it seems the Babylonians had their own technique to make calculations based on this principle.

"When I looked at the text, I was immediately convinced," said Jens Høyrup, an expert in Babylonian mathematics at Roskilde University in Denmark, who was not involved in the new study. "There are words that indisputably point to geometric understanding — not a geometric model of how the planets move, but a geometric technique to make some arithmetic calculations."

**Missing piece**

To build its collection in the 19th century, the British Museum gathered crates of clay tablets by methods that would not be considered scientifically sound today namely, buying artifacts that had been dug up around Babylon and Uruk without any archaeological context. Since the 1880s, scholars have been making sense of the astronomical concepts described on many of the tablets. [See Images of Ancient Babylonian Cuneiform Texts]

Four of these astronomical tablets had stumped historians, because they included computations mentioning trapezoids, even though the Babylonians had never been known to use geometry in their astronomical calculations.

The newly deciphered tablet was essentially the missing piece in the puzzle. Last year, a colleague handed Ossendrijver a stack of photographs, including an image of a tablet he had never seen before in the British Museum. In September, Ossendrijver went to London to hold it in his hand and read it himself, confirming what he already suspected the calculations describe.

"Actually, this particular tablet has ugly handwriting," Ossendrijver said. "It's slanted. It's like cursive if it were written very rapidly. It's very abbreviated. He left out everything that is not absolutely necessary to follow the computation."

Jupiter is not even mentioned in the newly deciphered tablet, but the computations it describes were already partly known from the other trapezoid tablets that do mention Jupiter, Ossendrijver said.

**Scholar-priests reading the skies**

The period between 400 and 200 B.C. might be considered the last period of innovations in Babylonian science, Ossendrijver said, and this is the time the zodiac and the horoscope were invented.

In general, the question of what the ancient Babylonians did with their astronomy is not fully answered, Ossendrijversaid. But they believed everything that happened on Earth was connected to whatever was happening in the sky.

"It was thought that if you are able to predict the motion of Jupiter, you would also be able to predict the price of grain, the weather, the level of the river Euphrates," Ossendrijver said. Part of his work also involves trying to understand the social context of Babylonian astronomers, and learning more about the kind of families and elite clans they belonged to. Certainly, the job description for an astronomer was much different 2,000 years ago.

Before the rise of personal horoscopes, astrology was a state affair. A court astrologer would have been called upon to interpret omens and predict plagues or other events, which could have real-world consequences. For instance, Høyrup said, if the astrologer was certain the king was going to die, the Babylonians could install a proxy king for six months, kill him at the end of his service, and let the original king return to the throne.

"The purpose of all this refined astronomy is astrology," Høyrup said. "They never speak about themselves in a way that suggests that they were pure astronomers or mathematicians their profession was to be scholar-priests."

Copyright 2016 LiveScience, a Purch company. All rights reserved. This material may not be published, broadcast, rewritten or redistributed.