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.
If you, like me, did not grow up using trig tables, they are fantastic tools when you don’t have a computer that does calculations with 10 digits of accuracy in a split second. A trig table would include columns with the sine, cosine, tangent, and possibly other trigonometric functions of angles. Someone or a group of someones would do these painstaking computations by hand, and then you could just look up the value when, say, cos(24°) came up in a computation. Today, computers generally use formulas for trig functions rather than calling up a list of all the values, and humans don’t need to know many values at all. These formulas are based on calculus and can be as precise as necessary. Need the correct answer to 50 digits? Your computer can do it, probably pretty quickly.
If you memorized “soh cah toa” or a mnemonic about “some old hippie,” you might remember that the basic trig functions are ratios of side lengths of triangles. The sine of an angle is the opposite side divided by the hypotenuse, the cosine is the adjacent divided by the hypotenuse, and the tangent is the opposite divided by the adjacent. The values of trig functions of most angles are not rational numbers. They can’t be written as the ratio of two whole numbers, so the entries you’ll find in trig tables are truncated after some number of decimal points. Mansfield and Wildberger seem to have homed in on the observation that when the side lengths of a right triangle are all integers, these ratios are all rational. Plimpton 322 is an “exact” trigonometric table because it only has trig functions based on triangles that have integer side lengths. (And in fact, the creator of the table set it up so the denominators of all the fractions are easy to represent in base 60.)
Modern trig tables are based on angles that increase at a steady rate. They might give the sines of 1°, 2°, 3°, and so on, or 0.1°, 0.2°, 0.3°, and so on, or even finer gradations of angles. Because like other ancient Mesopotamians, the people who produced Plimpton 322 thought of triangles in terms of side lengths rather than angles, the angles do not change steadily. That’s the difference between this seen as a trig table and modern trig tables. Neither way is inherently superior. If we wanted to make modern trig tables with angles that had only rational trig functions, we could, but it wouldn’t make computations dramatically more accurate. Either way, we could get the accuracy we needed for any particular application.
A little digging shows that Wildberger has a pet idea called “rational trigonometry.” He seems to be somewhat skeptical of things involving infinity, including irrational numbers, which have infinite, nonrepeating decimal representations. From a cursory reading of a chapter he’s written on rational trigonometry, I don’t see anything blatantly wrong with the theory, but it seems like a solution to a problem that doesn’t exist. The fact that most angles have irrational sines, cosines, and tangents doesn’t bother the vast majority of mathematicians, physicists, engineers, and others who use trig. It’s hard not to see their work on Plimpton 322 as motivated by a desire to legitimize an approach that has almost no traction in the mathematical community.
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.