Those Clever Babylonians and Astronomical Geometry

Hola darlings!

Saw this on my round of online newspaper reading, article at The Washington Post:

Clay Tablets Reveal Babylonians Discovered Astronomical Geometry 1,400 Years Before Europeans

By Joel Achenbach
January 28, 2016

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.

Text A. (Trustees of the British Museum/Mathieu Ossendrijver)

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.

Could this be an ancient Egyptian protractor?

This is absolutely fascinating.  Who is right, the mathematician, or the archaeologist?  Notice also the implication in the last sentence of the article: this is the only one found and therefore it cannot be a measuring instrument...

From New Scientist
Egyptian tomb mystery may be world's first protractor
16:17 29 July 2011 by Jo Marchant

The bizarre object to the right was found in the tomb of an ancient Egyptian architect. For over 100 years, it has languished while archaeologists debated its function.

Now, a physicist has thrown her hat into the ring, arguing that it is the world's first known protractor. The intriguing suggestion – which has drawn scepticism from archaeologists – is based on the numbers encoded within the carvings on its surface.

The architect Kha helped to build pharaohs' tombs during the 18th dynasty, around 1400 BC. His own tomb was discovered intact in 1906 by archaeologist Ernesto Schiaparelli in Deir-al-Medina, near the Valley of the Kings. Among Kha's belongings were measuring instruments including cubit rods, a levelling device that resembles a modern set square, and what appeared to be an oddly shaped empty wooden case with a hinged lid.

Schiaparelli thought this last object had held another levelling instrument. The museum in Turin, Italy, where the items are now exhibited identifies it as the case of a balancing scale.

But Amelia Sparavigna, a physicist at Turin Polytechnic, suggests that it was a different architectural tool – a protractor. The key, she says, lies in the numbers encoded in the object's ornate decoration, which resembles a compass rose with 16 evenly spaced petals surrounded by a circular zigzag with 36 corners.

Sparavigna says that if the straight bar part of the object were laid on a slope, a plumb line would revealed its inclination on the circular dial (as illustrated in this graphic).

Significant numbers

The fraction of one-sixteenth features in a calculus system the Egyptians used, says Sparavigna, and they also identified 36 star groups called the decans, which later formed the basis of a star clock. She suggests the object was "a protractor instrument with two scales, one based on Egyptian fractions, the other based on decans".

But Kate Spence, an archaeologist at the University of Cambridge who specialises in ancient Egyptian architecture, is not convinced and maintains the object is simply a decorative case. She says that unlike those on known measuring instruments, the markings in question are not particularly accurate: "When the Egyptians want to be precise, they are." She says the Egyptians tended to define angles by measuring the two sides of a rectangle, and that no similar instrument is known.

Ancient Egyptian Mathematicians Flexed Their Muscles with Puzzles

From The New York Times
Math Puzzles’ Oldest Ancestors Took Form on Egyptian Papyrus
By PAM BELLUCK
Published: December 6, 2010

“As I was going to St. Ives I met a man with seven wives. ...”

You may know this singsong quiz,

But what you might not know is this:

That it began with ancient Egypt’s

Early math-filled manuscripts. 

The Rhind Papyrus, more than 3,600 years old

 It’s true. That very British-sounding St. Ives conundrum (the one where the seven wives each have seven sacks containing seven cats who each have seven kits, and you have to figure out how many are going to St. Ives) has a decidedly archaic antecedent.

An Egyptian document more than 3,600 years old, the Rhind Mathematical Papyrus, contains a puzzle of sevens that bears an uncanny likeness to the St. Ives riddle. It has mice and barley, not wives and sacks, but the gist is similar. Seven houses have seven cats that each eat seven mice that each eat seven grains of barley. Each barley grain would have produced seven hekat of grain. (A hekat was a unit of volume, roughly 1.3 gallons.)

The goal: to determine how many things are described. The answer: 19,607.

The Rhind papyrus, which dates to 1650 B.C., is one of several precocious papyri and other artifacts displaying Egyptian mathematical ingenuity. There is the Moscow Mathematical Papyrus (held at the Pushkin State Museum of Fine Arts in Moscow), the Egyptian Mathematical Leather Roll (which along with the Rhind papyrus is housed at the British Museum) and the Akhmim Wooden Tablets (at the Museum of Egyptian Antiquities in Cairo).

They include methods of measuring a ship’s mast and rudder, calculating the volume of cylinders and truncated pyramids, dividing grain quantities into fractions and verifying how much bread to exchange for beer. They even compute a circle’s area using an early approximation of pi. (They use 256/81, about 3.16, instead of pi’s value of 3.14159....)

It all goes to show that making puzzles is “the most ancient of all instincts,” said Marcel Danesi, a puzzle expert and anthropology professor at the University of Toronto, who calls documents like the Rhind papyrus “the first puzzle books in history.”

Dr. Danesi says people of all eras and cultures gravitate toward puzzles because puzzles have solutions.

“Other philosophical puzzles of life do not,” he continued. “When you do get it you go, ‘Aha, there it is, damn it,’ and it gives you some relief.”

But the Egyptian puzzles were not just recreational diversions seeking the comforting illusion of competence. They were serious about their mission. In the Rhind papyrus, its scribe, known as Ahmes, introduces the roughly 85 problems by saying that he is presenting the “correct method of reckoning, for grasping the meaning of things and knowing everything that is, obscurities and all secrets.”

And the documents were practical guides to navigating a maturing civilization and an expanding economy.

“Egypt was going from a centralized, structured world to partially being decentralized,” said Milo Gardner, an amateur decoder of Egyptian mathematical texts who has written extensively about them. “They had an economic system that was run by absentee landowners and paid people in units of grain, and in order to make it fair had to have exact weights and measures. They were trying to figure out a way to evenly divide the hekat so they could use it as a unit of currency.”

So the Akhmim tablets, nearly 4,000 years old, contain lists of servants’ names, along with a series of computations concerning how a hekat of grain can be divided by 3, 7, 10, 11 and 13.

The Egyptian Mathematical Leather Roll, also from about 1650 B.C., is generally considered a kind of practice test for students to learn how to convert fractions into sums of other fractions.

The Rhind papyrus contains geometry problems that compute the slopes of pyramids and the volume of various-shaped granaries. And the Moscow papyrus, from about 1850 B.C., has about 25 problems, including ways to measure ships’ parts and find the surface area of a hemisphere and the area of triangles. Especially interesting are problems that calculate how efficient a laborer was by how many logs he carried or how many sandals he could make and decorate. Or the problems that involve a pefsu, a unit measuring the strength or weakness of beer or bread based on how much grain is used to make it.

One problem calculates whether it’s right to exchange 100 loaves of 20-pefsu bread for 10 jugs of 4-pefsu malt-date beer. After a series of steps, the papyrus proclaims, according to one translation: “Behold! The beer quantity is found to be correct.”

The problems in these ancient texts are not difficult by modern mathematical standards. The challenge for scholars has come in deciphering what the problems are saying and checking their accuracy. Some of the numerical equivalents are written in a symbolic system called the Eye of Horus, based on a drawing representing the eye of the sky god Horus, depicted as a falcon. Sections of the falcon’s eye are used to represent fractions: one-half, one-quarter and so on, up to one sixty-fourth.

Scholars have found a few errors in the problems, and Ahmes even wrote an incorrect number in his St. Ives problem. But over all, the equations are considered remarkably accurate.

“The practical answers are solved,” Mr. Gardner said. “What is unsolved about them is the actual thinking in the scribe’s head. We don’t know exactly how he thought of it.”

Pythagoras' Math More Than 1,000 Years Before He Was Born

From Art Daily:

Saturday November 13, 2010
Ancient Tablets Reveal Mathematical Achievements of Ancient Babylonian Culture

Tablet Plimpton 322, a table of Pythagorean triangles
a(2) + b(2) = c(2) 1,000 years before the Theorem existed

NEW YORK, NY.- An illuminating exhibition of thirteen ancient Babylonian tablets, along with supplemental documentary material, opens at New York University’s Institute for the Study of the Ancient World (ISAW) on November 12, 2010. Before Pythagoras: The Culture of Old Babylonian Mathematics reveals the highly sophisticated mathematical practice and education that flourished in Babylonia—present-day Iraq—more than 1,000 years before the time of the Greek sages Thales and Pythagoras, with whom mathematics is traditionally said to have begun.

The tablets in the exhibition, at once beautiful and enlightening, date from the Old Babylonian Period (ca. 1900–1700 BCE). They have been assembled from three important collections: the Columbia Rare Book and Manuscript Library, Columbia University; the University of Pennsylvania Museum of Archaeology and Anthropology; and the Yale Babylonian Collection, Yale University.

Before Pythagoras has been curated by Alexander Jones, ISAW Professor of the History of the Exact Sciences in Antiquity, and ISAW visiting scholar Christine Proust, historian of mathematics and ancient sciences at the Institut Méditerranéen de Recherches Avancées, in Marseille. The exhibition remains on view at ISAW through December 17, 2010.

Jennifer Chi, ISAW director for exhibitions and public programs, states, “It has long been widely recognized that many of the critical achievements of Western Civilization, including writing and the code of law that is the basis for our present-day legal system, developed in ancient Mesopotamia. However, the stunningly advanced state of mathematics in this region has largely been known only to scholars. By demonstrating the richness and sophistication of ancient Mesopotamian mathematics, Before Pythagoras adds an important dimension to the public knowledge of the history of historic cultures and attainments of present-day Iraq.”

Babylonian mathematics is known to the modern world through the work of scribes, primarily young men who, coming from wealthy families in which literacy and professional expertise were handed down through generations, were formally trained in reading and writing. Destined to work in such fields as accounting, building-project planning, and other professions in which mathematics is essential, the scribes learned and practiced mathematics by copying symbols and solving problems—some practical, others theoretical—such as those seen in the tablets in the exhibition.

Alexander Jones notes, “The evidence we have for Old Babylonian mathematics is amazing not only in its abundance, but also in its range, from basic arithmetic to really challenging problems and investigations. And since the documents are the actual manuscripts of the scribes, not copies selected and edited by later generations, we feel as if we were looking over their shoulders as they work; we can even see them getting confused and making mistakes. Recent research has made this human dimension very vivid, using archeological evidence to re-imagine the schools and the process of teaching and learning. Moreover, the contents of the tablets are still recognizable, as they continue to be taught in contemporary mathematics.”

The tablets in Before Pythagoras, inscribed in cuneiform script, cover the full spectrum of mathematical activity, from arithmetical tables copied by scribes-in-training to sophisticated work on topics that today would be classified as number theory and algebra. In so doing, they illuminate three major themes: arithmetic exploiting a notation of numbers based entirely on two basic symbols; the scribal schools of Nippur, which was the most prestigious center of scribal education; and advanced mathematical training.

Many of the problems solved by scribes at the advanced level of training were in fact much more difficult than any they would have to deal with in their careers, and their solutions depended on principles that, before the rediscovery of the Babylonian tablets, were believed to have been discovered by the Greeks of the sixth century BCE and later. One of the tablets, for example, is an extremely unusual diagram showing a square with its two diagonals and three numbers that demonstrate that what we call the Pythagorean Theorem existed 1,000 years before Pythagoras lived. The content of other tablets ranges from mathematical tables for training, to practical calculations for professionals, to abstract algebraic problems.

The meaning of these and other tablets from the Old Babylonian Period were first elucidated by mathematician and historian of science Otto Neugebauer (1899–1990), who spent some two decades, beginning in the 1920s, transcribing and interpreting tablets that had come to light in ancient Mesopotamia since the nineteenth century. It is his pioneering research, as well as the work of his associates, rivals, and successors, that revealed to modern scholars the period’s rich culture of mathematics. Before Pythagoras includes a selection of manuscripts and correspondence, on loan from the Institute for Advanced Study (Princeton, New Jersey), that offers a glimpse of Neugebauer’s methods and his central role in this “heroic age” of scientific discovery.

In order to enable visitors to appreciate the cuneiform tablets more fully, ISAW has developed an extended exhibition pamphlet that will guide viewers in reading cuneiform numbers. The exhibition features a content-rich website.