# Babylonian tablet 3,700 years old, first trigonometry

August 25, 2017 6:59 PM Subscribe

Babylonian tablet predates the Pythagorean Theorum by a millennium The Plimpton 322 tablet, offers Pythagorean Geometry, and a base sixty trigonometry used for measurement and planning, more concise than what we have been using in modern time. The Guardian covers it nicely. The Atlantic has a different take on it, not so fond of the Astrology they used the Astronomy for, however their Astronomical predictions grew more accurate.

They are the inventors of their own base 60 trigonometry used in building the Hanging Gardens of Babylon, long called one of The Seven Wonders of the World. Babylonians tracked five of the planets and knew of the periodic nature of eclipses. Here is yet another take on it from Smithsonian Magazine.

They are the inventors of their own base 60 trigonometry used in building the Hanging Gardens of Babylon, long called one of The Seven Wonders of the World. Babylonians tracked five of the planets and knew of the periodic nature of eclipses. Here is yet another take on it from Smithsonian Magazine.

It's 322, I think they got the number wrong. Here is the Wikipedia article. The work they're referring to in this article is likely ref 7 on the Wikipedia page for 322.

For what it's worth there is a prior interpretation of this tablet not as trig but instead as a set of exercises for school for solving quadratic equations; see the links to Robson in the Wikipedia page for more.

posted by nat at 7:38 PM on August 25, 2017

For what it's worth there is a prior interpretation of this tablet not as trig but instead as a set of exercises for school for solving quadratic equations; see the links to Robson in the Wikipedia page for more.

posted by nat at 7:38 PM on August 25, 2017

I think there are two different discoveries described in the linked articles—I was reading the

But the OP articles dated 2016, the

posted by XMLicious at 7:40 PM on August 25, 2017

*Guardian*article about the Plimpton 322 tablet last night which was just published in the last few days and discusses a "3,700-year-old broken clay tablet", which would place its origin in the early 2nd millennium BC.But the OP articles dated 2016, the

*Atlantic*and*Smithsonian Magazine*ones, refer to "a cuneiform tablet dating to between 350 and 50 B.C.", so late 1st millennium, easily 1500 years afterwards.posted by XMLicious at 7:40 PM on August 25, 2017

phys.org :P

also btw...

John Baez: how to multiply using trig - "Back in the 1500's, people on long sea journeys navigated using the stars. They needed big tables of trig functions to do this! These tables were made by astronomers. Those folks did thousands of calculations. Often they needed to multiply large numbers! That was tiring... but around 1580, they figured out a clever way to approximately multiply large numbers using tables of trig functions."

posted by kliuless at 7:56 PM on August 25, 2017 [6 favorites]

also btw...

John Baez: how to multiply using trig - "Back in the 1500's, people on long sea journeys navigated using the stars. They needed big tables of trig functions to do this! These tables were made by astronomers. Those folks did thousands of calculations. Often they needed to multiply large numbers! That was tiring... but around 1580, they figured out a clever way to approximately multiply large numbers using tables of trig functions."

posted by kliuless at 7:56 PM on August 25, 2017 [6 favorites]

Sorry for bad tablet naming, I am not so good with making posts. Forgive plz.

posted by Oyéah at 8:10 PM on August 25, 2017 [2 favorites]

posted by Oyéah at 8:10 PM on August 25, 2017 [2 favorites]

I was hoping to track down details of the more strident claims mentioned in the

posted by XMLicious at 8:12 PM on August 25, 2017 [2 favorites]

*Guardian*article—*...*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.

*—but I'm noticing that some of the phys.org comments on kliuless's link assert that one of the authors of the paper is a crank.*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.

posted by XMLicious at 8:12 PM on August 25, 2017 [2 favorites]

I have discovered a truly marvelous proof for this theorem, which this cuneiform tablet is drying too quickly to contain

posted by Itaxpica at 8:15 PM on August 25, 2017 [27 favorites]

posted by Itaxpica at 8:15 PM on August 25, 2017 [27 favorites]

oh and babylonian math/astronomy previously :P

posted by kliuless at 8:19 PM on August 25, 2017 [3 favorites]

posted by kliuless at 8:19 PM on August 25, 2017 [3 favorites]

I love this notion that Babylonian trig was superior (for architectural purposes) because it used base 60 and a geometry using ratios, instead of base 10 with angles, in complete opposition to "modern" sensibilities.

posted by polymodus at 8:58 PM on August 25, 2017 [2 favorites]

posted by polymodus at 8:58 PM on August 25, 2017 [2 favorites]

Well, if they'd come up with some way to express quantities like √2 with some other mathematical structure, because they'd never heard of irrational numbers or the concept of zero or anything like that, it would be pretty significant. But you'd expect to find a discovery of such import in more Google results, rather than just tucked behind a paywall in a historical journal and glossed over in mainstream press articles.

posted by XMLicious at 9:19 PM on August 25, 2017

posted by XMLicious at 9:19 PM on August 25, 2017

That Atlantic article bugs the hell out of me. It calls attention to the critique of Whiggish history, and then insists that because the Babylonians didn't study the night sky for the right reasons, they didn't really study the night sky. They only

The Babylonians, like the Greeks and the Egyptians, made observations about the world around them and interpreted their findings according to prevailing ideas about the universe. If those prevailing ideas should change, you can't look back and say "see, modern Science is objective and value-free, unlike what those idiots were doing." Modern, Western science is as susceptible to influence from social and political forces as anything else, and I really wish people would stop pretending otherwise.

posted by shapes that haunt the dusk at 9:34 PM on August 25, 2017 [35 favorites]

*resembled*scientists. I absolutely hate it when people say "yeah, they discovered stuff, but only by accident while they were looking for other stuff." Yeah, doesn't sound at all like science to me.The Babylonians, like the Greeks and the Egyptians, made observations about the world around them and interpreted their findings according to prevailing ideas about the universe. If those prevailing ideas should change, you can't look back and say "see, modern Science is objective and value-free, unlike what those idiots were doing." Modern, Western science is as susceptible to influence from social and political forces as anything else, and I really wish people would stop pretending otherwise.

posted by shapes that haunt the dusk at 9:34 PM on August 25, 2017 [35 favorites]

*"But you'd expect to find a discovery of such import in more Google results, rather than just tucked behind a paywall in a historical journal and glossed over in mainstream press articles."*

The recent paper on Plimpton 322 is actual open access. Here's the link for those who want to read it.

posted by Proofs and Refutations at 10:04 PM on August 25, 2017 [5 favorites]

Ah, I guess blocking browser cookies tripped me up. Thank you very much, Proofs and Refutations!

posted by XMLicious at 10:09 PM on August 25, 2017

posted by XMLicious at 10:09 PM on August 25, 2017

It seems strange to call it a "trigonometric table" when there aren't angles. The relationship between sides and angles in a triangle is kind of the whole point of trigonometry. If you only work with ratios of sides, I don't think it's fair to call it trigonometry, and it definitely isn't fair to compare it to trigonometric tables that involve angles. The transcendental relationship between angles and ratios of sides is what makes trigonometry messy (in the sense of requiring approximations for so many practical calculations). So crowing about exact results when you aren't dealing with angles seems a bit silly.

I don't really doubt that Babylonians might have used this tablet in the way the authors describe. I just think that the problems they suggest the Babylonians solved with this kind of tablet aren't actually trigonometry problems. You can use trigonometry to solve them (as the authors show in section 6.2), but who would do that? All the problems come down to using the Pythagorean theorem and extracting a square root. As far as I know (which is admittedly not very far), cultures that worked out trig already knew about the Pythagorean theorem.

posted by samw at 12:53 AM on August 26, 2017

I don't really doubt that Babylonians might have used this tablet in the way the authors describe. I just think that the problems they suggest the Babylonians solved with this kind of tablet aren't actually trigonometry problems. You can use trigonometry to solve them (as the authors show in section 6.2), but who would do that? All the problems come down to using the Pythagorean theorem and extracting a square root. As far as I know (which is admittedly not very far), cultures that worked out trig already knew about the Pythagorean theorem.

posted by samw at 12:53 AM on August 26, 2017

Sanw: I make bevel gears for a living... a radical shift from previous jobs in the IT world... one of the things required is to calculate the roll ratio... how much to rotate the cradle (containing a pair of reciprocating cutting tools which generate the involute profile) vs the roll of the part being cut.

All the the formula in the book reference the sine function.... but just for fun I wanted to check the ratio... and my $2 calculator doesn't have trig... so I figured out how to do it without using the sine function... strictly by the ratio of the sides of triangles.... which gives the required value, without having to determine the angle at all...

Is it still trig?

Also, sin(x)= x-x^3/3!+x^5/5!-x^7/7!... is actually not too hard to work out with an 8 digit calculator and a sheet of note paper.

PS: Avoiding Tau or Pi and the irrationality of angles in radians may be why they avoided direct measurement of angles.

posted by MikeWarot at 2:55 AM on August 26, 2017 [3 favorites]

All the the formula in the book reference the sine function.... but just for fun I wanted to check the ratio... and my $2 calculator doesn't have trig... so I figured out how to do it without using the sine function... strictly by the ratio of the sides of triangles.... which gives the required value, without having to determine the angle at all...

Is it still trig?

Also, sin(x)= x-x^3/3!+x^5/5!-x^7/7!... is actually not too hard to work out with an 8 digit calculator and a sheet of note paper.

PS: Avoiding Tau or Pi and the irrationality of angles in radians may be why they avoided direct measurement of angles.

posted by MikeWarot at 2:55 AM on August 26, 2017 [3 favorites]

*All the problems come down to using the Pythagorean theorem and extracting a square root.*

I mean, part of what is notable about this is that it predates the Pythagorean theorem by what, 1200 years?

posted by shapes that haunt the dusk at 3:12 AM on August 26, 2017 [4 favorites]

That

The author cautions that sifting through the past "for morsels that can be called precursors of science makes for bad history," but in many ways the Ancient Greeks have been the biggest beneficiaries of such an approach. These were people, after all, who dogmatically believed in humors and the four elements -- they weren't practicing modern era, peer reviewed, empirical science. Yet, the Western canon of history, and particularly the history of science, emphasizes their works, no matter how tangential or partial. Part of it, of course, is a colonial era conceptoin of history which formulated a modern view of "Europe" as a thing apart from the "East" and firmly rooted in the supposed rational thinking of the Ancient Greek philosophers (making them both distinct from their contemporaries and antecedents of the Enlightenment thinkers).

Another large part, however, comes down to the fluke of history wherein Alexander ended up Hellenizing large parts of Eurasia, which led to Greek writings being preserved and replicated for centuries both by Romans and then, notably, Arabs. So there seems to be a bias in the author in trying to dismiss and ignore the implications of the tablet, which suggests much more opportunities for discovery in the field of ancient Mesopotamian mathematics and astronomy the author seems singularly uninterested in engaging with. The disengagement is particularly apparent when contrasting the *Atlantic* piece with one of the academics quoted in the Guardian article, who notes that:

posted by Panjandrum at 7:10 AM on August 26, 2017 [10 favorites]

*Atlantic*article seems both unnecessarily defensive about the Greeks and overly pedantic about the definition of "scientist." The invocation, and subsequent disparagement, of Whig History in order to defend the Greeks as the true birth of science in the Western canon against the possibility of earlier predecessors seems not just incongruous, but fairly Whiggish in itself. There's also the fact that the Ancient Greeks were not exempt from reading religous omens in the sky; they did, as the author notes, assemble the foundation for modern day astrology (borrowing heavily from Babylonian work).The author cautions that sifting through the past "for morsels that can be called precursors of science makes for bad history," but in many ways the Ancient Greeks have been the biggest beneficiaries of such an approach. These were people, after all, who dogmatically believed in humors and the four elements -- they weren't practicing modern era, peer reviewed, empirical science. Yet, the Western canon of history, and particularly the history of science, emphasizes their works, no matter how tangential or partial. Part of it, of course, is a colonial era conceptoin of history which formulated a modern view of "Europe" as a thing apart from the "East" and firmly rooted in the supposed rational thinking of the Ancient Greek philosophers (making them both distinct from their contemporaries and antecedents of the Enlightenment thinkers).

Another large part, however, comes down to the fluke of history wherein Alexander ended up Hellenizing large parts of Eurasia, which led to Greek writings being preserved and replicated for centuries both by Romans and then, notably, Arabs. So there seems to be a bias in the author in trying to dismiss and ignore the implications of the tablet, which suggests much more opportunities for discovery in the field of ancient Mesopotamian mathematics and astronomy the author seems singularly uninterested in engaging with. The disengagement is particularly apparent when contrasting the *Atlantic* piece with one of the academics quoted in the Guardian article, who notes that:

A treasure trove of Babylonian tablets exists, but only a fraction of them have been studied yet. The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us.I have my own biases, of course. I've studied more than my fair share of Mesoamerican history, and the recent eclipse in the US led me to brush up on works like the *Dresden* and *Borgia* codices, works with day counts in association with eclipse imagery which provide a mathematical basis for predicting eclipses on 148 and 177 day cycles. These cycles are factors of the 18-yearish saros astronomical cycle. It would be easy to dismiss such works, with their attendant imagery of Sun and Venus deities, as "unscientific," and I would not disagree; the pre-modern era was profoundly unscientific. To disparage one poorly documented and poorly understood astronomical tradition as a bunch of superstitious garbage in while touting the much better documented Greeks, with their interesting, if often incorrect or marginally correct conclusions, as paragons of scientific inquiry is to ignore the realities of historical records in favor of well... a Whig explanation of history.

posted by Panjandrum at 7:10 AM on August 26, 2017 [10 favorites]

*All the the formula in the book reference the sine function.... but just for fun I wanted to check the ratio... and my $2 calculator doesn't have trig... so I figured out how to do it without using the sine function... strictly by the ratio of the sides of triangles.... which gives the required value, without having to determine the angle at all... Is it still trig?*

Got a link to an explanation of the formula? If you didn't have to use an angle, then usually it isn't what I'd think of an trig, or there's a hidden approximation somewhere that sweeps the trig under the rug.

*Also, sin(x)= x-x^3/3!+x^5/5!-x^7/7!... is actually not too hard to work out with an 8 digit calculator and a sheet of note paper.*

This actually demonstrates my point. If you're calculating the sine of an angle, your calculation involves two approximations: one for the angle (either you know the angle you want geometrically, as an eighth of a full turn, say, in which case you have to approximate x, or you have an exact value for x in mind, in which case you can only approximately draw the angle!) and one when you decide when to stop adding up the series.

*PS: Avoiding Tau or Pi and the irrationality of angles in radians may be why they avoided direct measurement of angles.*

It seems a little implausible, without some solid evidence, that the Babylonians were aware of the irrationality of pi.

This discussion reminded me of a thread from long ago were some guy came up with a replacement for trigonometry that avoided transcendental trig functions at the expense of using angle and length measurements that weren't proper measures. I tracked it down and it's the same Norman Wildberger guy! That explains a lot. The way the Babylonians seem to have worked with triangles is pretty close to his own ideas, so he ascribes to them many of this own arguments and intents. That's my take on it, anyway.

He seems like a crank, to be honest. For example, see this (which was linked in the comments on the phys.org page kliuless posted). I have no problem with strict constructivists as long as they're honest about their true objections to standard foundations. But this Wildberger guy seems to always try to hide the real issues behind a bunch of linguistic slight of hand and excessive incredulity.

posted by samw at 8:36 AM on August 26, 2017

They used zero as a place holder in larger numbers, not theoretically. They were practical users of their calculations. Practical for their time.

posted by Oyéah at 8:40 AM on August 26, 2017

posted by Oyéah at 8:40 AM on August 26, 2017

Well northern Europeans did not discover this, or use this. 5300 years later at the time of Shakespeare, who was certainly able in the humanities, northern Europeans were still throwing their crap out of windows, and down onto the street. Not so with the Babylonians, and their sumptuous hanging gardens. It is nice to find that humans have a richer, longer, deeper intellectual footprint on this world. I suspect it will go back farther and farther if we can bring ourselves to interpret, or if we are able to interpret their work. Every week that goes by the human history on our world, the history of the physical forms of our species grows to be much longer than we initially reckoned. This is by hundreds of thousands of years, if not eventually millions.

I found this interesting because of math I studied in junior high. We worked with base twelve, and the teacher showed us how if we had only an extra digit on both hands, math would have been much easier with so many divisible numbers. Ultimately a computer program will read these tablets like we read the papers. Our trove of understanding will astonish us then.

posted by Oyéah at 8:48 AM on August 26, 2017 [1 favorite]

I found this interesting because of math I studied in junior high. We worked with base twelve, and the teacher showed us how if we had only an extra digit on both hands, math would have been much easier with so many divisible numbers. Ultimately a computer program will read these tablets like we read the papers. Our trove of understanding will astonish us then.

posted by Oyéah at 8:48 AM on August 26, 2017 [1 favorite]

Mod note: Corrected the number 332 to 322 in the post; carry on.

posted by LobsterMitten (staff) at 9:06 AM on August 26, 2017 [1 favorite]

posted by LobsterMitten (staff) at 9:06 AM on August 26, 2017 [1 favorite]

It took a while to understand what they meant by "exact". They are talking about how the ratios terminate in a base 60 representation, but are repeating in base 10. The additional divisor of 3 certainly makes it easier to find ratios that terminate, but I don't see why it would be impossible to make a similar table with terminating values in decimal.

I don't see any regularity to the table. They have come up with some useful values, but it's not clear what you are supposed to do if you want to work with a triangle between two rows. Angular trig has the benefit that you can work with arbitrary triangles, though it may require a bit of work.

posted by Horselover Fat at 10:30 AM on August 26, 2017

I don't see any regularity to the table. They have come up with some useful values, but it's not clear what you are supposed to do if you want to work with a triangle between two rows. Angular trig has the benefit that you can work with arbitrary triangles, though it may require a bit of work.

posted by Horselover Fat at 10:30 AM on August 26, 2017

*The author cautions that sifting through the past "for morsels that can be called precursors of science makes for bad history," but in many ways the Ancient Greeks have been the biggest beneficiaries of such an approach.*

Yes yes yes, flagged as fantastic. It drives me up the wall when people act like we were all mindless savages until the noble Greeks came along (and of course, it's very interesting to see the contexts where Greece is considered a part of the West, and the contexts where it's considered bizarre and foreign -- ancient philosophy = western;

old woman roasting sheep bones to read the cracks for omens = foreign).

I took an entire course on Greek and Egyptian science and magic, and one of the first things we talked about is how you simply cannot nearly separate those categories. That was the especially frustrating thing about the Atlantic article, because he sort of offhandedly mentioned that the Greeks practiced divination, almost as if it was due to the negative influence of the Babylonians. When the Babylonians practice divination, it's the opposite of science, but when the Greeks do it, it's just a quirk for these otherwise-wise masters.

(Also that class was

*amazing*. If I remember right, the theory behind Greek divination was not so much that the stars exerted influence over the world, but that they reflected a great chain of events that included events on Earth. Patterns in the movements of the stars would correspond with patterns here on Earth, just on a different scale. It makes sense in its context, even if it doesn't seem like science now. The accuracy of your measurements would therefore mean greater accuracy in divination.

I mean, then you have a whole bunch of other practices, like standing on a tripod and casting stones, but there are similar principles at work. It's a fascinating belief system, and it's especially interesting to see how far it's legacy traveled -- "Greek medicine" in India, the influence of Neoplatonism on Christian science and medicine, etc.)

posted by shapes that haunt the dusk at 12:39 PM on August 26, 2017 [7 favorites]

Tables of Pythagorean triplets are called "roofing tables" in the construction industry (in the US, and probably other Anglophone countries) because you need them to figure out the measurements for rafter cuts. I'm sitting about four feet from a framing square that has a roofing table engraved on the side of it. They're also necessary for other things like measuring and cutting the components that make up staircases.

Mathematics is very unforgiving in the building trades. We live an a universe with three spatial dimensions. The laws geometry and trigonometry are laws about how things fit together in our universe. If you're working in construction and you make a math error, the pieces won't fit. There's no way around this.

Ancient Mesopotamians didn't make buildings with pitched roofs and rafter systems. But they did build buildings with sloped sides and stairs. As a former construction worker, I'm

For people in ancient Mesopotamia to have figured out that you could use the same mathematical principles in astronomy rather than, say, laying out staircases, is non-obvious and pretty impressive.

posted by nangar at 1:37 PM on August 26, 2017 [9 favorites]

Mathematics is very unforgiving in the building trades. We live an a universe with three spatial dimensions. The laws geometry and trigonometry are laws about how things fit together in our universe. If you're working in construction and you make a math error, the pieces won't fit. There's no way around this.

Ancient Mesopotamians didn't make buildings with pitched roofs and rafter systems. But they did build buildings with sloped sides and stairs. As a former construction worker, I'm

*intimately*familiar with things they would have and almost certainly did use tables like this for.For people in ancient Mesopotamia to have figured out that you could use the same mathematical principles in astronomy rather than, say, laying out staircases, is non-obvious and pretty impressive.

posted by nangar at 1:37 PM on August 26, 2017 [9 favorites]

I found this interesting because of math I studied in junior high. We worked with base twelve, and the teacher showed us how if we had only an extra digit on both hands, math would have been much easier with so many divisible numbers.

According to this, maybe even without the extra digits:

On the subject of a measurement system based on twelve, Georges Ifrah seems more persuasive. He argues that the twelve-part numbering system, like the related sixty-part numbering system, goes back to a long-established and widely used finger-counting method. As he puts it, the “duodecimal finger-counting method used in India, Indochina, Pakistan, Afghanistan, Iran, Turkey, Iraq, Syria and Egypt” involves counting to twelve. One does this by using the thumb of the right hand, beginning with the outermost of the three bones at the tip of the little finger. The sexagesimal finger-counting method, still used in most of the same countries, is complementary to the duodecimal one. It uses the left hand to indicate twelve, twenty-four, thirty-six, forty-eight, and sixty by closing down each of the five fingers, starting with the little finger and finishing with the thumb. Understandably, Ifrah feels that the duodecimal finger-counting method may have been a factor in leading the ancient Egyptians to divide day and night each into twelve unequal or temporal hours. It may also have “led the Sumerians, and the Assyrians and Babylonians after them, to divide the cycle of day and night into twelve equal parts (calledposted by XMLicious at 2:00 PM on August 26, 2017 [2 favorites]danna, each equivalent to two of four hours), to adopt for the ecliptic and the circle a division into twelveberu(30° each), and to give the number 12, as well as its divisors and multiples, a preponderant place in their various measurements.”

I'm coming across as really negative. The paper is an interesting look into how the Babylonians likely solved these sorts of right triangle problems. It's too bad the authors over-sell it so much.

posted by samw at 3:04 PM on August 26, 2017

posted by samw at 3:04 PM on August 26, 2017

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