Indiana Banks and the Tablet of Trig

Edgar James Banks (1866 –1945) was an American diplomat, antiquarian, archeologist, author, and novelist.  He was a professor of Oriental languages and archeology at the University of Toledo, climbed Mt. Ararat in search of Noah’s Ark, started two movie companies, and was a consultant on Cecil B. DeMille’s Bible epics.  He is often mentioned as the real-life inspiration for Indiana Jones, and indeed he bore a strong resemblance to actor Harrison Ford.

In 1898, Banks was the American consul in Baghdad, near the ancient sites of Babylon and Nineveh.  When the locals discovered that Banks was willing to pay for clay tablets with cuneiform inscriptions, they brought him hundreds of them, dug up from nearby mounds called “tells.”  The Ottoman government did not attempt to regulate trade in minor antiquities, and Banks purchased more cuneiform inscriptions from a dealer in Istanbul. He re-sold these artifacts in small batches to museums, libraries, universities, and private collectors across America.

One of Banks’ private-collector clients was the well-heeled New York publisher George Arthur Plimpton (1855 – 1936), scion of Massachusetts iron magnates and the grandfather of noted writer George Plimpton. Just before his death, Plimpton donated his extensive collection, which included medieval and renaissance manuscripts as well as near-eastern antiquities, to Columbia University.  One of the clay tablets Plimpton purchased from Banks, reportedly for $10.00, is the artifact known as P322 (denoting that it is the 322nd item in the Plimpton catalog).

P322 dates to approximately 1,800 BC.  It has four columns and 15 rows of numbers inscribed in cuneiform.  Some of the numbers are positive integers that fit the rule:  a squared + b squared = c squared , as, for example, do the numbers 3, 4, 5 or 6, 8, 10, or 5, 12, 13. Mathematicians call such patterns Pythagorean triples, because they echo the Pythagorean theorem, which is that “the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse.”

But if P322 shows knowledge of the Pythagorean theorem, and is concerned with the discipline of trigonometry, there would seem to be a problem of anachronism: Pythagoras was born in 570 BC, and the Greek astronomer Hipparchus, long regarded as the father of trigonometry, was born in 190 BC.  Both lived well over a thousand years after P322 was produced. 

Mathematician Daniel Mansfield of the University of New South Wales in Sydney, Australia, was developing a course for high school math teachers when he came across a picture of P322.  He and a colleague, Norman Wildberger, decided to study it.  Because of the Pythagorean triples, they had a hunch that the tablet was related to trigonometry, but the familiar sines and cosines of spherical trigonometry, used by Greek astronomers and modern high school students, were nowhere to be found. Instead, each entry includes data on two sides of a right triangle: the ratio of the short side to the long side and the ratio of the short side to the diagonal or hypotenuse.

The table contains exact values of the sides for a range of right triangles. The top row of the table yields relatively equal ratios that specify nearly equilateral triangles. Moving down the table, the ratios decrease and so do the triangles’ inclination, specifying narrower triangles.

Mansfield and Wildberger conclude that the Babylonians expressed trigonometry in terms of exact ratios of the lengths of the sides of right triangles, rather than by the angles used by spherical trigonometry.

Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles. It is a fascinating mathematical work that demonstrates undoubted genius.  The tablet not only contains the world’s oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.

The greater accuracy is a function of the fact that Babylonian mathematics used a base 60, or sexagesimal system, rather than the base 10, or decimal system we use. Because 60 is far easier to divide by three, there are fewer fractions, or non-integers, and the calculations are easier and more accurate.  Says Mansfield:

This is a whole different way of looking at trigonometry.  We prefer sines and cosines . . . but we have to really get outside our own culture to see from their perspective to be able to understand it.  Plimpton 322 was a powerful tool that could have been used for surveying fields or making architectural calculations to build palaces, temples or step pyramids.

P322 not only contains the earliest evidence of trigonometry, notes Mathieu Ossendrijver, a historian of ancient science at Humboldt University in Berlin, it represents a more exact form of the discipline than the estimated numerical values sines and cosines provide.

This story attracted my attention because it is yet more evidence against the Darwinian narrative of human history, which holds that we evolved from apes and have been steadily gaining in technology and science ever since.  The reality is very different.  The evidence often points to initial brilliance followed by decline, the pattern that creationists would expect.  In this instance, extremely ancient people pioneered a form of trigonometry that is more exact than the form used by later Mesopotamian nations and by the Greeks, Romans, and later Western Civilization. 

There are many examples.  Students of ancient Greek learn that biblical Greek is not advanced over the classical Greek of 400 years earlier.  To the contrary, Koine Greek has lost shades of meaning that were capable of being communicated in classical Greek.  And it may come as a shock to us English-speakers that Koine Greek is capable of conveying meaning that cannot be conveyed as economically, if at all, in English.  (We English-speakers do not even have a second-person plural pronoun different from our second-person singular; southerners have coined “y’all” to circumvent this omission in our language.)

Another example is ancient Egyptian civilization, which was as brilliant as it would ever be right at the beginning.  John Anthony West writes:

Every aspect of Egyptian knowledge seems to have been complete at the very beginning. The sciences, artistic and architectural techniques and the hieroglyphic system show virtually no signs of a period of “development”; indeed, many of the achievements of the earliest dynasties were never surpassed, or even equaled, later on. This astonishing fact is readily admitted by orthodox Egyptologists, but the magnitude of the mystery it poses is skillfully understated, while its many implications go unmentioned.

Despite this pattern repeating itself over and over, many academics still manifest unbelief when confronted with evidence of ancient brilliance.  The Germans, who are good at coining one-word descriptions of unflattering human traits (e.g., schadenfreude), have a word for this prejudice against the past: “urdummheit” meaning, roughly, primeval stupidity.  Urdummheit is baked into the Darwinian weltanschauung.  

Hence, whenever anyone discovers something like P322, expect urdummheit to rear its ugly head and cast aspersions on that discovery. Right on cue, we have Jöran Friberg, a retired Swedish academic, thundering that the Babylonians “knew NOTHING about ratios of sides!” What are you going to believe, urdummheit or your lying eyes?  And historian of mathematics Christine Proust, of the French National Center for Scientific Research, calls Mansfield and Wildberger’s findings “a very seductive idea,” but argues that no known Babylonian texts suggest that the tablet was used to solve or understand right triangles.  That’s like saying we found swords buried in a Babylonian tell, but we have no documentary evidence that the Babylonians ever used swords as weapons.  Urdummheit.

Here is a four-minute discussion of P322:

And here is a 20-minute video in which Mansfield and Wildberger discuss their findings.