## Cuneiform tablet is oldest (& newest) trigonometric table

Mathematicians at the University of New South Wales (UNSW) in Sydney, Australia, in think they’ve cracked a cuneiform code that has given rise to vigorous debate in the mathematical community for almost a century. The bone, or in this case clay tablet, of contention is known as Plimpton 322 and has been in the Columbia University collection since it was bequeathed to them in 1936 by the wealthy publisher and avid collector of historical written materials George Arthur Plimpton. (His grandson was George Plimpton, author, literary critic and one of the shrinks Matt Damon chases away in *Good Will Hunting*.) George Arthur left the tablet and the rest of his exceptional collection of rare books and manuscripts to the Butler Library; they are now in the university’s Rare Book and Manuscript Library.

The tablet is five inches wide, 3.5 inches high and .8 inch thick and features a table of cuneiform numbers four columns across and 15 rows long. The outer left edge is broken, possibly in the modern era because there are remains of glue indicating a repair attempt with the now-missing piece. This break took some of the first column figures with it, forcing mathematicians to have to extrapolate what the complete numbers were based on the extant ones. Experts think the complete table was six columns wide and 38 rows long. All of the numbers are in Babylonian sexagesimal (base 60) notation.

Its origins are obscure because the tablet was one of hundreds acquired by roving antiquarian/adventurer/novelist/diplomat Edgar James Banks starting in the late 19th century. He sold it to Plimpton in around 1922, reportedly for $10. Neither of them realized the significance of the cuneiform text. Based on the writing style and formatting, researchers believe the tablet was created in the Sumerian city of Larsa in what is today southern Iraq between 1822 and 1762 B.C., right around the time of Hammurabi (c. 1810-1750 B.C.), 6th king of the First Babylonian Dynasty and promulgator of the law code that bears his name. Several other tablets with inscriptions of Babylonian mathematics came from Larsa.

Mathematicians have been studying Plimpton 322 since it was first published at the end of World War II and there have been rollicking debates on the purpose of the table, whether it’s a complex accounting tool, a mathematical table (if so what kind), a teacher’s edition list of answers for math students or something else entirely. Austrian mathematician and historian of science Otto Neugebauer recognized that the numbers on the table are Pythagorean triples, two different integers that squared and added together equal the square of the third integer, but to what end did ancient scholars take the immense trouble to compile the lists?

The UNSW team posits that it is indeed a trigonometric table, but one that takes a previously unknown approach to calculation.

“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,” said Dr Daniel Mansfield of the School of Mathematics and Statistics in the UNSW Faculty of Science.

“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.

“This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3000 years, but it has possible practical applications in surveying, computer graphics and education. This is a rare example of the ancient world teaching us something new.”

Until now, the Greek mathematician Hipparchus has been granted the title the Father of Trigonometry because he designed a table in a circle that was long credited with being the oldest known trigonometric table. If the scribe who wrote Plimpton 322 had signed his work, Hipparchus would have had to take that crown off his own head and put it on his Sumerian predecessor’s.

“Plimpton 322 predates Hipparchus by more than 1000 years,” says Dr Wildberger.

“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.” […]

“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,” added Dr Mansfield.

Their research has been published in the journal *Historia Mathematica* and can be read in its entirety here (download it now before it ends up behind a paywall).

This may be the nerdiest web of nerdery I’ve ever woven (which is saying something), but when I first saw this story I couldn’t help but imagine the following dialogue.

**Plimpton 322**: I’m an idiot because I can’t make a lamp?

**Diogenes**: No, you’re a genius because you can’t make a lamp.

**Plimpton 322**: What do you know about trigonometry?

**Diogenes**: I could care less about trigonometry.

**Plimpton 322**: Well did you know without trigonometry there’d be no ziggurats?

**Diogenes**: Without lamps, there’d be no honest men.

I’ll see myself out.

B.E.S.T. = Baby. Exact. Sex. Trig.

And they say maths is boring.

I see a plot line in there somewhere for the Big Bang Theory show.

ki 4

(“Row 4”)__1.53.10.29.32.52.16 __3.31.49__ 5.09.01—————

diagonal d = 1+9*60+5*60^2 = 18541

base b = 49+31*60+3*60^2=12709

—————

Indeed, (12709^2+13500^2)^.5=18541

However, 13500^2 = 1.8225e+08

Here, “1.53.10.29.32.52.16” should be related to 13500, which implies that you puzzle around with the 1 and multiples of 60 … AND -of course- there are the “ratios” that are key here.

We ancient cattle farmers and astronomers tended to have similar problems – May they be in the Outback of Australia or Southern Iraq.

I’m rather sceptical of the claim “it is also the only completely accurate trigonometric table”. In one sense it must be true: if you use ratios of integers rather than decimal fractions then you have perfect accuracy. On the other hand it’s an absurd claim because if you have to interpolate in the tables to get the solution to a particular problem then your perfect accuracy has just vanished.

“True but bogus” might be a fair description.

———————–

babylon2dez:=proc(bab)

begin

dez:=0;

for i from 1 to nops(bab) do

dez:=dez+bab[i]*60^(i-1)

end_for;

return(dez)

end_proc:

———————–

babylon2dez([1,09,5]) = 18541

babylon2dez([49,31,3]) = 12709

babylon2dez([16,52,32,29,10,53,1]) = 88004782336

13500^2 = 182250000

———————

b=12709, d=18541, l=13500

———————

The “Ratios”:

b/l = 56.29.04 = 56/60+29/60^2+4/60^3

d/l = 1.22.24.16 = 1+22/60^1+24/60^2+16/60^3

1/(56/60+29/60^2+4/60^3)*12709 = 13500

1/(1+22/60^1+24/60^2+16/60^3)*18541 = 13500

Those “Ratios”, when you look at their first 500 digits (no idea if there are 500, I did not count here), do look rather

strange– at least to a cattle herder like me, they do:———-

0.94140740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740740741

… and …

1.3734074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074074

…whereas, when divided, the “bogus” number becomes …

0.68545385901515560110026427916509357639825252143897308667277924599536163097999029178577207270373766247775200906099994606547651151502076479154306671700555525591931395286122647106412814842780864031066285529367348039480071193571004800172590475163151933552667062186505582223181058195350844075292594789925031012351005878863060244862736637721805727846394477104794779138126314654010031821368858206137748772989590636966722399007604767811876382072164392427592902216708915376732646567067579957931071678981716197

I’m simply glad to see Austin Powers has a new gig.