Mathematical secrets of ancient tablet unlocked after nearly a century of study

[brooklynite]

Source: The Guardian

At least 1,000 years before the Greek mathematician Pythagoras looked at a right angled triangle and worked out that the square of the longest side is always equal to the sum of the squares of the other two, an unknown Babylonian genius took a clay tablet and a reed pen and marked out not just the same theorem, but a series of trigonometry tables which scientists claim are more accurate than any available today.

The 3,700-year-old broken clay tablet survives in the collections of Columbia University, and scientists now believe they have cracked its secrets.

The team from the University of New South Wales in Sydney believe that the four columns and 15 rows of cuneiform – wedge shaped indentations made in the wet clay – represent the world’s oldest and most accurate working trigonometric table, a working tool which could have been used in surveying, and in calculating how to construct temples, palaces and pyramids.

...snip...

“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 3,000 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.”

Read more: https://www.theguardian.com/science/2017/aug/24/mathematical-secrets-of-ancient-tablet-unlocked-after-nearly-a-century-of-study

[Achilleaze]

1. republicans claim Babylonians used proveable math

and besides were a bunch of pointy headed libs with beards who ate olives and humus and flatbread, and who wore sandals like Jesus Christ.

Consequently, repubes prefer to keep on using the same kind of sucky 2+2 = 0 Alternative Mathematics that they used in colluding with the evil empire of russia to overthrow American democracy and install a god-forsaken Five Time Republican Draft Dodger with a reputation for colossal a-holery.

[iluvtennis]

12. LOL

[Plucketeer]

2. We MUST find the other tablets!

The ones where such wise folks as these would tell us how to solve the problems of abject apathy and determined ignorance!

[struggle4progress]

3. "... other experts on .. Plimpton 322 .. say the new work is speculative at best ..."

This ancient Babylonian tablet may contain the first evidence of trigonometry
By Ron Cowen
Aug. 24, 2017 , 2:00 PM
http://www.sciencemag.org/news/2017/08/ancient-babylonian-tablet-may-contain-first-evidence-trigonometry

[GeoWilliam750]

8. I love science and Science. Always something fascinating to learn.

[dalton99a]

32. Mansfield & Wildberger agree - from their paper:

"There is no known historical evidence that confirms how P322(CR) was actually used. This is a question that must be answered by archeology, or by further discoveries on existing tablets."

[eppur_se_muova]

4. "more accurate than any available today" ?? Um, yeah, right.

Bad science journo ! Bad !

[MountainFool]

6. confusing, but not wrong ....

The embedded video explains that a side-effect of their using base-60 was fewer problems with inexact fractions. So their trig table has more "exactly right" answers than a table using base-10 digits.

And while the tech is available today, I can't say anyone has used it ... yet

Check out the embedded video, starting at about the 1 minute mark:

[rickford66]

10. Kinda late to change, except for maybe a standalone application.

[aggiesal]

27. This can be ...

programmed into computers & calculators.
You'll never know what algorithm is used,
yet you'll get more accurate values.

[TreasonousBastard]

14. He's right, sorta, but we've been using base 10 for a loooong time and are used to it...

We've also known that the Babylonians used base 60, and could have used it ourselves for years, but for some strange reason we don't.

We do use octal, though.

[WinkyDink]

15. 360 degrees; 60 minutes; 60 seconds; months of 12, days of 30, hours of 24. The key number was SIX,

as well as sixty.

[LudwigPastorius]

34. Babylonian Schoolhouse Rock!

[Sgent]

19. Most high level trig

is done in units of Pi (radians) anyway -- not base 10 or base 60. We have known about base number systems for a LONG time, commonly used are 2, E, PI, 8, 10, 16, 32, 64 -- all of those are each responsible for billions if not 10's of trillions of activity. I don't know what this adds to our knowledge.

[DallasNE]

20. That Would Have Been One Awkward Keyboard

And, boy, the multiplication tables would have sucked.

[WinkyDink]

13. And you would know differently, how, exactly?

[caraher]

21. Trig tables are basically obsolete

Any trig function can be calculated to essentially arbitrary precision by a computer. Base 10 vs. Base 60 is not a limitation.

This is something one would know by having an elementary understanding of modern computational techniques

[hunter]

33. Simple microprocessors don't have floating point math coprocessors.

On these simple processors floating point math is a large subroutine and it takes a long time.

Integer math is much faster. Early video games used integer math exclusively, even going so far as to use "good enough" approximations, for example the fast inverse square root algorithm.

It wouldn't surprise me if the Babylonians used some very sophisticated shortcuts in their mathematical calculations that are long forgotten. shortcuts that might not ever be rediscovered in a world where dirt cheap microprocessors can do floating point math in hardware.

Even so, exact trigonometry using integer math and tables might still be very useful in some applications.

[Dr. Strange]

23. Yeah, "more accurate".

Here's another article to peruse.

Something to note:

The Babylonian approach is also much simpler because it only uses exact ratios. There are no irrational numbers and no angles, and this means that there is also no sin, cos or tan or approximation.

So the familiar 45-45-90 triangle? Doesn't exist in this trigonometry. So great, it's more accurate! But it doesn't allow for much of what our geometry does. Draw a circle and a triangle so they cross each other. Do they intersect? In this Babylonian geometry, probably not. Not unless the coordinates of the point of intersection just happen to be rational numbers.

Screw you, ancient mathematicians! You shan't take my irrational numbers from me!

[eppur_se_muova]

25. Making quantities easier to calculate, not necessarily more accurate.

i.e. the usual tradeoff.

[hunter]

41. I'm guessing the isosceles right triangle was a special case, with known useful approximations.

Pi was likewise a special case. At the time this tablet was made Babylonians used 25/8 for pi, knowing very well it was an approximation but good enough for most work.

This same 25/8 approximation appears frequently in integer math computer graphics. It reduces much of the math to simple binary shift instructions. I've seen it used in traditional English measure machine work too. It's only half a percent short. Machinists who used this fraction probably got an intuitive feeling for making up the difference by the 32nd or 64th of an inch, just as a Babylonian might have had a feeling for the base 60 fraction.

[Progressive dog]

43. You are obviously correct

The Pythagorean theorem can be used to obtain any accuracy desired.

[keithbvadu2]

5. Fake math! - So sad!

(sarcasm thingie goes here)

[dembotoz]

7. Wonder if any of this was taught as NEW math

[AngryAmish]

9. As long as they only have one of the Millennium items...

One must play the Egyptian God Cards with care.

[Kingofalldems]

11. Aliens.

[progressoid]

36. +1

[kimbutgar]

16. Isn't Babylon what is modern day IranIraq in a the Middle East ?

Most people there were brown mmmm. I thought the master intelligent race was white! What gives?


Snark!

[yardwork]

45. Iraq. Iran was Persia. Two very different cultures.

[Jim__]

17. As an aside: One of these researchers is N J Wildberger - he has a History Of Math course online.

It’s a pretty good course - a brief description:

Starting with the ancient Greeks, we discuss Arab, Chinese and Hindu developments, polynomial equations and algebra, analytic and projective geometry, calculus and infinite series, Stevin's decimal system, number theory, mechanics and curves, complex numbers and algebra, differential geometry, topology, the origins of group theory, hyperbolic geometry and more. Meant for a broad audience, not necessarily mathematics majors. good course

The course is here

[thbobby]

39. Thanks Jim!

I love mathematics. Appreciate your link to this course. Fascinating to me how intelligent ancient people were. Truly humbles me (a good thing!). Love the early Babylonian approach to science. To think we cannot learn from them seems to me to be arrogant and ignorant.

[tblue37]

18. K&R & thanks. nt

[left-of-center2012]

22. "The 3,700-year-old broken clay tablet ..."

So, that's 1,683 B.C. or B.C.E.?

[Orrex]

24. It's impolite to ask a tablet's age

[Pacifist Patriot]

30. Coffee meet keyboard.

[struggle4progress]

26. Plimpton 322 poses several challenges

It is a short table consisting of several columns of numbers. Unfortunately, the first column has been damaged, which has produced controversy about how to reconstruct it. The second and third columns are:

(119,169)
(3367,4825)
(4601,6649)
(12709,18541)
(65,97)
(319,481)
(2291,3541)
(481,769)
(4961,8161)
(45,75)
(1679,2929)
(161,289)
(1771,3229)
(56,106)

It was recognized quite long ago that these are always the leg and hypotenuse of a right triangle:

(119,120,169)
(3367,3456,4825)
(4601,4800,6649)
(12709,13500,18541)
(65,72,97)
(319,360,481)
(2291,2700,3541)
(481,600,769)
(4961,6480,8161)
(45,60,75)
(1679,2400,2929)
(161,240,289)
(1771,2700,3229)
(56,90,106)

This leads to another question: namely, how did the Babylonians find these right triangles? Here, we may have less difficulty, since we do have information about what could be done in classical antiquity, and some of those methods are easy enough that they might have been known earlier

A curious feature of the tablet is that (in all but one case) the difference between the third and second column is twice a square:

169-119=50=2x5x5
4825-3367=1458=2x27x27
6649-4601=2048=2x32x32
18541-12709=5832=2x54x54
97-65=32=2x4x4
481-319=162=2x9x9
3541-2291=1250=2x25x25
769-481=288=2x12x12
8161-4961=3200=2x40x40
75-45=30=2x15
2929-1679=1250=2x25x25
289-161=128=2x8x8
3229-1771=1458=2x27x27
106-56=50=2x5x5

In other words, all but one triple is an instance of the identity u^2 + (4ua^2 + 4 a^4) = (u + 2a^2)^2: here (except for the one exceptional case) u is the second column entry and u + 2a^2 is the third column entry

It is easy to see (by drawing pebble pictures) that (4ua^2 + 4 a^4) = (2a^2)^2 x (u + a^2) -- and it is also easy to see (again by drawing pebble pictures) that this will be a square if (u + a^2) is a square

Setting aside for a moment the exceptional row (45,75), let us ask for each of 119, 3367, 4601, 12709, 65, 319, 2291, 481, 4961, 1679, 161, 1771, 56: what is the smallest square a^2 that can be added to each to make the result a square?

The answers are:

119 + 5^2 = 12^2
3367+27^2 = 64^2
4601+32^2=75^2
12709+54^2=125^2
65+4^2=9^2
319+9^2=20^2
2291+25^2=54^2
481+12^2=25^2
4961+40^2=81^2
1679+25^2=48^2
161+8^2=15^2
1771+27^2=50^2
56+5^2=9^2

And in fact -- still with the exception of the one row (45,75) -- the entries in the third column of the tablet are exactly what we might have expected, namely u + 2a^2:

119+2x25=169
3367+2x729=4825
4601+2x1024=6649
12709+2*2916=18541
65+2x16=97
319+2x81=481
2291+2x625=3541
481+2x144=769
4961+2x1600=8161
1679+2x625=2929
161+2x64=289
1771+2x729=3229
56+2x25=106

The presence of several instances of the same a^2 here suggests that the actual problem solved in finding these triples might not have been "given u find a^2 so that u + a^2 is a square" but rather "given a^2 find u so that u + a^2 is a square." There is a simple way to solve this problem that was known in antiquity: it consists of starting with a square array of pebbles (n x n) and extending it to a larger square by adding a "gnomen" of n + n + 1 pebbles: to illustrate, if we start with a 4 x 4 array

....
....
....
....

then adding 4 + 4 more pebbles gives

.....
.....
.....
.....
....

so adding one more pebble gives a 5 x 5 array

.....
.....
.....
.....
.....

This shows that 4^2 + 9 = 5^2 because after adding 4 + 4 + 1 pebbles we have a 5x5 array.

This method can easily be extended by the obvious iteration: if we start with a n x n square and add n + n + 1 pebbles, then add n + n + 3 pebbles, then add n + n + 5 pebbles, and so on until we tire, we always get a sequence of squares. So if we starts with the nth square (n x n) and add the consecutive odd numbers n + n + 1, n + n + 3, n + n + 5, ... , n + n + (2k - 1) we will finally obtain another square, (n + 2k - 1) x (n + 2k - 1). To use this method effectively, we must know how to sum

{n + n + 1}+{n + n + 3}+{n + n + 5}+ ... +{n + n + (2k - 1)}

but again, a solution was known in antiquity: simply average the first and last term and multiply by the number of terms, so the sum of

{n + n + 1}+{n + n + 3}+{n + n + 5}+ ... +{n + n + (2k - 1)}

is simply (n + n + k) x k. To illustrate again, suppose we want to find numbers that added to 5x5 will give a perfect square; such numbers are

11 x 1 = 11
12 x 2 = 24
13 x 3 = 39
14 x 4 = 56
~snip~
17 x 7 = 119
~snip~

and we quickly discover the 56 and 119 appearing in the tablet. As another illustration, suppose we want to find numbers that added to 25x25 will give a perfect square; such numbers are

51x1=51
52x2=104
53x3=159
~snip~
73x23=1679
~snip~
79x29=2291
~snip~

and so we could find the 1679 and 2291 appearing in the tablet

This leaves the exceptional row (45,75) corresponding to the Pythagorean triple (45,60,75) -- but it is just the multiple (by 15) of the simplest possible triple (3,4,5) which was well-known in the ancient world

[muriel_volestrangler]

28. Which raises the question of why did they multiply the 3,4,5 triplet by 15?

If they'd left it as (3,5) it would be 5-3=2=2x1x1 - twice a square. Or, if they didn't like using one as a square, multiply it by 4, to use (12,20) with 20-12=8=2x2x2 etc. It's not as if the sizes of other numbers used seem to be in any particular range. They could have used (48,80) to have a pair in almost the same range if that mattered for some reason.

[struggle4progress]

29. On inspection, the wild fluctuations (of the sizes in the second and third columns)

seems to be explained by the fact that the numbers in the first column of the tablet appear to be the ratios

(hypotenuse)^2 divided by (leg)^2

but with the rows sorted so that these ratios are in order

That, of course, does not answer your question about the multiplication by 15: it has however been suggested that the tablet represents exercises for a student, which would remove that mystery

[mahatmakanejeeves]

35. It's homework, then, and waaaaaaaaaaaaaaaaay overdue.

"The saber-tooth tiger ate my homework."

[muriel_volestrangler]

37. I found one of the earlier papers investigating this

It suggested that perhaps they didn't think of it as a '3,4,5' triangle, but a '45,60,75' one, and so they left that as the ratio they were used to.

They didn't speculate why they might 'commonly' know it with those numbers, but I will: with a number system based on 60, they might have thought it 'neat' that a 45,60,75 triangle was right-angled - "take your rule divided into 60, mark off one side with it, mark off another with 15 divisions less, then another with 15 divisions more, and you've got yourself a right angle!"

[struggle4progress]

40. I suppose that's possible but it's very easy to see, from pebble pictures, that

if (a,b,c) is a Pythagorean triple then so is (n*a,n*b,n*c)

For example, if (3,4,5) is Pythagorean so is (45,60,75)

The "proof" is very simple. Start with (say) a 3 x 3 square of red pebbles and a 4 x 4 square of blue pebbles. The red and blue pebbles can be rearranged as a 5 x 5 square. Now replace every red pebble with a 15 x 15 square of orange pebbles and every blue pebble with a 15 x 15 square of green pebbles. The 3 x 3 square becomes a 45 x 45 square of orange pebbles; and the 4 x 4 square similarly becomes a 60 x 60 square of green pebbles pebbles. Any process of rearranging the red and blue pebbles into a 5 x 5 square implies a process of rearranging 15 x 15 orange squares and 15 x 15 green squares into a 75 x 75 square

The converse -- "if (n*a,n*b,n*c) is a Pythagorean triple then so is (a,b,c)" -- is likewise obvious from pebble pictures

So it seems unreasonable to imagine that competent Babylonian mathematicians were unaware of the relation between (3,4,5) and (45,60,75)

Moreover this relationship is relevant to the first column of Plimpton 322; since here it does not actually matter which of the two common reconstructions of the first column we use, I will illustrate with the simple assumption that the first column gives S^2/L^2 where S and L are the short and long legs of the triple: using more modern concepts, this is the square of the tangent of the smaller acute angle in the triangle. One might want a table of such ratios for astronomical purposes (say); and a natural way to construct such a table would be to compute the ratio for various known right triangles, the easiest being (3,4,5)

To compute 3^2/4^2 in Babylonian sexagesimal notation, one should like the denominator a power of 60: this is accomplished by first multiplying by 15^2/15^2 to obtain (3x15)^2/(4x15)^2 = (45)^2/(60)^2; then dividing 45^2 = 2025 by 60 to obtain a quotient and remainder 2025 = 33 x 60 + 45; and finally noting (33 x 60 + 45)/60^2 = 33/60 + 45/60^2 which in the Babylonian notation is 33 45 -- exactly as reported by the tablet

Similar methods will work whenever L divides a power of 60: that is, whenever L is a product of 2s, 3s, and 5s -- which is the case for EVERY triple in the tablet. For example, the triple (119, 120, 169) could lead to the calculation

119^2/120^2 = (119^2*15)/(120^2*15) = 212415/60^3 = (59 x 60^2 + 15)/60^3 = 59/60 + 15/60^3 which corresponds to the Babylonian notation 59 0 15 reported

Sometimes rather tedious computation with large numbers is required, but the ability involved is mechanical. A more interesting question might be how the Babylonians actually found triples having L a product of 2s, 3s, and 5s: I am not enough of a number theoretician to be sure, but I suspect a hard theory might lie here so that we should think the Babylonians found such triples by trial and error

[Pacifist Patriot]

31. This is where I love being hopelessly liberal arts and mathematically challenged.

I'm grooving on the historical coolness of this and couldn't care less about the math being debated above. We're deciphering something so freaking old!

[defacto7]

38. This makes me wonder about human intellect

and its potential throuout history. Have we had this level of potential for analytical thought from the beginning of civilization or were these just anomalies appearing now and then? If the potential was there, what inhibitor put the breaks on analytical thinking? Was it sociopolitical, natural cataclysmic, or maybe fluctuations in our brains evolution and devolution? Lots of questions.

[hunter]

42. That's why anti-intellectual traditions scare me.

I've never had any success arguing with Christian Creationists, the sorts of people who think the universe is a few thousand years old and all the dinosaurs drowned because they didn't listen to god and refused passage on Noah's ark.

The only success I've had is keeping Creationists out of public school classrooms. I'll sadly confess it's sometimes been by raw intimidation, which probably left them feeling persecuted and perversely pleasured. These sorts of Christians love to feel persecuted.

We had a rock guy in town who claimed to be a geologist. He had a lovely collection of rocks and an entertaining way of presenting them to kids, which made him a darling guest of teachers and administrators who were scientifically illiterate. I was on his ass for about two years before he gave up on public schools.

He's still making the rounds of private "Christian" schools, but I can't really do anything about that.

[Dr. Strange]

44. Don't Fall for Babylonian Trigonometry Hype

Good response from Evelyn Lamb here: https://blogs.scientificamerican.com/roots-of-unity/dont-fall-for-babylonian-trigonometry-hype/