Copyright
The Babylonians knew a thing or two about copyright as well if the tablet is still protected after over 3000 years.
Those of you who can remember trigonometry can feel free to forget it, because ancient Babylonian mathematicians had a better way of doing it – using base 60! That's the conclusion of a new paper, Plimpton 322 is Babylonian exact sexagesimal trigonometry, in the new issue of the journal Historia Mathematica. The “Plimpton 322 …
While I did upvote....
technically, I don't think it's the tablet that is protected by copyright, it's the images of the tablet that are copyrighted.
If you had access to the physical tablet, you could take your own photos, and you would own the copyright in those photos (or release them public domain if you so choose).
"If you had access to the physical tablet, you could take your own photos, and you would own the copyright in those photos (or release them public domain if you so choose)."
There are images already in the public domain, so not much point. I suspect HkraM was only trying to get a laugh...
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Symon: "Yep. http://cyberlaw.stanford.edu/blog/2007/03/fairy-use-tale"
Wow, thank you. That is absolutely brilliant. And quite possibly the best example of passive-aggressive trolling I have ever seen.
@Antron Argaiv
With puns like that, don't be surprised if somebody hits you with a COSH. Then you'll end up in a hospital COT.
BTW, what's a nortna? The second part of your name is obvious, but not the first (not to me, anyway, I expect about 200 replies pointing out what I've missed).
"I'll watch the video again, and this time, I'll try to ignore the SECs and pay more ATAN-tion to the skin tones..."
There's a direct connection to ATAN insomuch every programmer unfortunate enough to have dabbled in geometry-handling maths knows it's an unreliable bitch - you really should use ATAN2 instead...
I don't know... Something to do with Napoleon possibly?
Whilst I work (measure and cut) in mm, I estimate in feet and inches. However I may be missing a trick because of the way that 12 can be easily divided by 3 and by 4 (and obviously 2 and 6). For centuries, carpenters have been able to make beautiful pieces without a unit of measurement by means of dividers - it is only important that they can express a length as a rational of another.
"Thought that was why there was 60 seconds / minutes in a hour."
I think that's also derived from the Babylonians as do the divisions of a circle. But, of course, it was they who had the wit to use a number base that was convenient for integer division rather than an inconvenient one based simply on counting their fingers.
> Thought that was why there was 60 seconds / minutes in a hour. Because it was easy to work out divide by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30
And by 37⅙, if you're not too bothered about integer results (or your time source shows serious short-term stability issues).
Correct. Numbers like 12 and 60 were used by street market traders due to their high level of divisibility.
This was especially important when buying selling items that could not be easily divided such as fruit/vegetables which would spoil once cut. A cart of 60 apples could be split into 10 different uniform units.
Obvious. Using base 10 evolved because we have 10 digits (most of us). Using base two reflects the decline in education standards, which means that many people count hands rather than fingers.
Of course, binary is also handy for smart-arse techies because we can count to 1023 on our fingers.
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Interesting. With that method, using one hand to count units (up to 12), and the other hand to count the groups of 12... it's possible to count up to 60 using your hands.
Hadn't realised that before, but it makes sense as to a practical way to count up to 60 in early historical times.
"why has the modern world moved so far towards pure binary (and powers of 2 in specific contexts)?"
Imperial measurement made considerable use of binary. Weights from pounds down to drachms were binary as were volumes from gallons down to gills. In general they seem to have been based on measures which were a convenient size for some purpose with a strong inclination to subdivide on a binary basis. It's a natural thing to do. If you have a standard of weight, for instance, you can weigh out that amount of sand, flour or whatever on scales and then, using the same scales, divide that into two equal portions and subdivide further.
The problem arises when two different scales of measurement overlap and we end up with a stone of 14 pounds. Other stones were available - I've seen reference to a stone of 15lbs in the C18th - but I suppose a atone of 16lbs would have required too much adjustment to reconcile with the larger scales in use for other purposes.
The commercial world, which calls itself the real world, runs on binary coded decimal
S'truth. Today on java.lang.BigDecimal until JSR 354 comes out. (actually you need a BigRational too)
See also: Why not use Double or Float to represent currency?
For some applications, that's not entirely true anymore. Multi-Level Cells in FLASH are not 1 or 0, there's several inbetween.
In asynchronous (i.e. clock-less) electronics there's been some different approaches too: 1, 0, or not sure yet...
Binary logic dominates in CPUs because with volts/no volts representing 1 and 0, there's practically no calibration to incorporate into a manufacturing process. With multi-level logic, e.g. 1, 2/3, 1/3, 0 (2 bits) suddenly it all becomes a lot harder to build.
However, in communications multi-level representation of data is pretty common. QAM (quadrature amplitude modulation, see Wikipedia) involves multiple signal amplitudes, a far remove from simple binary on/off keying. In communications such schemes are used to get more data through a given signal bandwidth. It's the equivalent of storing more than 1 bit in a single memory cell (which is what MLC Flash is doing).
Of course, the reason such signalling isn't used on, say, the memory bus between RAM and CPU is because it takes a lot of electronics to generate and receive such a signal; not good for speed / power. RAM buses these days are complicated enough, what with their propagation de-skew delay lines, more than 1 bit on the PCB trace at a time, etc. But complex modulations are used on links like Thunderbolt, USB3, Ethernet. There's way more than one bit on the wire at any single point in time.
Really these days most high speed buses inside and outside of computers are RF data links, not just a single voltage level on a PCB trace.
"For some applications, that's not entirely true anymore. Multi-Level Cells in FLASH are not 1 or 0, there's several inbetween..."You are of course correct; I was being a bit glib. In truth we have found quite a few uses for multivalent logics in recent time so we are moving away from binary logic rather than towards it. William of Ockham (ca. 1287–1347) did a bit of work on trivalent logic without finding a use for it. Probably too busy counting angels on pinheads and neglecting to write down the results.
To get the Slurp handle ... Microsoft, in their infinite wisdom, have extended boolean to a tri-state, which has 5 states (Yeahhh, don't get me started), of which two, exactly true and false, are supported ... the other 3 ain't ... just useless ... seriously, no joke, here's the doc:
https://msdn.microsoft.com/en-us/library/microsoft.office.core.msotristate.aspx
When you do Office automation, Word accepts standard $true & $false and casts those to msotristate:msotrue/false, respectively, where as other Office applications, I am looking at YOU PowerPoint don't ... behavior changes depending on which language (VBA, VBS, PowerShell) you use... what a pile, hey ?
"Wikipedia isn't public domain - we can't just lift their stuff."The photograph of the Plimpton tablet isn't "their stuff"; it's in the public domain. Wikipedia lifting stuff from the public domain doesn't mean it's copyright Wikimedia Foundation no matter what Jimmy Wale might believe.
The photograph of the Plimpton tablet isn't "their stuff"; it's in the public domain.
Image copyright is horribly complicated. The maker of the image owns copyright in the image. If someone is permitted to take a photo of the image then they have copyright of the new image, but so (usually) does the original photographer!
Artworks are even more fun. If I own a picture by a modern artist, and take a photo of it, unless the artist has assigned all rights to me as part of the sale, I can't publish that photo (on the web or anywhere) without the permission of the artist (or other copyright holder).
So, does this tablet count as an original artwork? Although after 3700 years I suspect the carver's copyright has expired (depending on Babylonian copyright laws - are they in the Berne convention? After all, Disney characters are copyright for all time.)
"Image copyright is horribly complicated. The maker of the image owns copyright in the image. If someone is permitted to take a photo of the image then they have copyright of the new image, but so (usually) does the original photographer!"Ain't that the truth? I'm a member of the Australian Copyright Council and beneficiary of royalties therefrom.
The photograph in question is old enough the photographer is more than likely dead. Nobody knows who the photographer was, or if they do they've stayed schtum for a very long time.
FWIW Wikipedia isn't public domain - we can't just lift their stuff.
True, but most of it you can lift, subject to their various licences, most of which permit use with attribution. Images all state what the copyright position is. Hell, there are people selling Print-on-demand books on Amazon that are just a printout of Wikipedia articles on a topic.
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Meant to add this to my remark above. The Wiki-bloody-pedia's quite good on this. My last university class was taught by a bloke who did his PhD on 3-value logics.
Yes, three valued logic for binary computers has already been described in high-impact journals!
We can count off on 8 fingers and two thumbs (alright we can go to 20 in warmer climates if we can use our toes). Some people in the world still count in 60s using the same 8 fingers and two thumbs. If you are predominantly right handed, use your right thumb to count to 3 with the top, middle and lower phalanx of your right hand little finger, then three more with the ring finger joints, then the middle finger, then the index finger to give 12. Extend the little finger of your left hand to count off the first 12, the repeat for another 12 with the right hand and extend the ring finger for 24, then count another 12, and use the left hand middle finger for 36, then the left index finger for 48, and finally the left thumb for 60.
It may be one reason why old farts like me were taught the duodecimal system. We bought things in dozens and paid for them in shillings and pence - Also ten is only divisible by the integers 1,2, and 5; twelve is divisible by 1,2,3,4, and 6; and sixty is divisible by 1,2,3,4,5,6,10,12,15, 20, and 30 - So very "handy" when selling items or dividing them up between people. There were 12 shillings (and 240 pennies in the pound), so we could divide a pound by 16, 24, 30, 40, and 60 as well.
"Uphill both ways, in the snow, barefoot" might allow a higher count...
When my daughter was learning to count (age 4ish), I taught her to count to 15 on four fingers. She added the thumb, and then the other hand, on her own. In highschool, she "invented" three extra digits on each extremity, for full 32-bit compatibility ... with her right eye as a carry-bit.
Teach your kids alternates to decimal numbers early and often ...
I did O-levels (pre GCSE!) in the good old pounds, shillings and pence days. I had to learn how to add, subtract, multiply and divide sums in pounds, shillings and pence. However, the only sensible way to do compound interest calculations was to convert to pounds and decimals of a pound, do the calculation, then convert back! However, there were a lot of useful hacks for less demanding arithmetic, some of which I can still remember - the dozen rule, for example (things were often sold in multiples of 12 in those days, so knowing that the price of 12 items in shillings = price of one item in pennies was very useful), the score rule (same in shillings and pounds), and several others I've now forgotten!
>I did O-levels (pre GCSE!) in the good old pounds, shillings and pence days. I had to learn how to add, subtract, multiply and divide sums in pounds, shillings and pence
I was a kid in the 1970s and just started school the year before decimalisation. I remember our 'times tables' books went up to 12, but we only ever were taught multiplication up to 10 times.
I guess learning to multiply by twelve was, all of a sudden, no longer a required skill, bit they hadn't updated the text books yet.
BTW: being a child of the decimal era, I always just thought that pre-decimal coinage was just another example of the previous generation making things unnecessarily complicated. It wasn't til much later in adult life that it suddenly occurred to me one day that 12 is divisible by 2,3,4 and 6, whilst 10 is only divisible by 2 and 5 —and I realised there was a method to the old folks' madness, after all.
And isn't it funny that everyone wrote out half-crowns as 2s 6d (2 and 6) rather than as, well, half a crown.
Is it? Did they?
I used to write 2/6, just as I'd have written 4/9 for four shillings and ninepence ... it's quicker than "half a crown" and more consistent ... and "1/2 crown" might have been confused with one and tuppence.
I never wrote "one florin" when I meant 2/-, either.
Pythagoras noted a number of special triangles, such as 3:4:5, that have integer ratios in base 10.
https://en.wikipedia.org/wiki/Pythagorean_triple
The Babylonians had similar special cases that can be written in base 60 as integers as shown on the tablet. Very similar techniques.
The breakthrough with sin/cos/tan is that any triangle can be described and calculated because they are continuous transcendental functions.
From the original article, which many of course didn't read...
"P322 is historically and mathematically significant because it is both the first trigonometric table and also the only trigonometric table that is precise. Irrational numbers and their approximations are seen as essential to classical metrical geometry, but here we have shown they are not actually necessary for trigonometry. If the dice of history had fallen a different way, and the deep mathematical understanding of the scribe who created P322 not been lost, then very possibly ratio-based trigonometry would have developed alongside our angle-based approach."
"The discovery of trigonometry is attributed to the ancient Greeks, but this needs to be reconsidered in light of the much earlier, computationally simpler and more precise Babylonian style of exact sexagesimal trigonometry. In addition to being historically significant, P322 also brings the founding assumptions of our own mathematical culture into perspective. Perhaps this different and simpler way of thinking has the potential to unlock improvements in science, engineering, and mathematics education today."
"The novel approach to trigonometry and geometrical problems encapsulated by P322 resonates with modern investigations centered around rational trigonometry both in the Euclidean and non-Euclidean settings"
What is important is not just the tables in the tablet itself - it's the way the table may have been computed, and the underlying assumptions.
It's also interesting the article refers to some Knuth's article - it looks he was also interested in Babylonian algorithms...
"The Egyptians beat Pythagoras to it; they used 3:4:5 for land surveying. Heck, I used it a lot when building my home 14 years ago."
When we moved into our home some years ago after my parents had dies I wondered what became of the 3:4:5 wooden triangle my dad made to set out the walls when he built the house. A year or so ago I found it propped up against a boundary wall when I was cutting back a holly. The joints attaching the hypotenuse had rotted but I still have the right angle.
If it has an integer ratio - then it is irrelevant what base the number is in e.g. 3:4:5 is same whether it is decimal or hexadecimal and if binary is still an integer just has more digits 11:100:101, all a different base number does is change which numbers are easier to divide by
Off-topic with respect to this article:
sin and cos -
http://mathworld.wolfram.com/LissajousCurve.html
Paris is welcome to come over and investigate osculation by fiddling with my oscilloscope
Because Jobs was a descendant of Gilgamesh (remember when his father came from...) - which was in fact a quite nasty king oppressing his people - and also was looking for "ethernal life".
The sexy, round corner design comes from Shamhat - which after all tamed the wild Enkidu and lead him to his death...
Scribe Ziqquratberg made people record the story on its Faceclay application...
Well, Gilgamesh isn't Babylonian and he's been dead over 70 years.
Have a beer.
"If Gilgamesh existed, he probably was a king who reigned sometime between 2800 and 2500 BC. The Sumerian King List claims that Gilgamesh ruled the city of Uruk for 126 years."
He probably did exist, but unlikely to have been as described in the Epic Sumerian/Akkadian poem developed over 100s of years (In that sense a bit like King Arthur, though evidence for him is more tenuous and most of what you read/see is stuff made up 100s of years after original Welsh legends.)
"And not actually a king either, assuming he existed."From the OED:
" In OE. the title appears first as the name of the chiefs of the various Anglian and Saxon ‘kins’, tribes, or clans, who invaded Britain, and of the petty states founded by them, as well as of the native British chiefs or princes with whom they fought, and of the Danish chiefs who at a later time invaded and occupied parts of the country. Among the Angles and Saxons the kingship was not strictly hereditary, according to later notions; but the cyning was chosen or accepted in each case from a recognized kingly or royal cynn or family (usually tracing its genealogy up to Woden). With the gradual ascendancy and conquests of Wessex in the 9th and 10th c., the king of the West Saxons became the king of the Angelcynn, Angelþéode, or English (Angligenarum, gentis Angligenæ, Anglorum), and the tribal kings came to an end. But there still remained a King of Scotland, and several petty kings in Ireland. In European and other more or less civilized countries, king came to be the title of the ruler of an independent organized state called a kingdom; but in mediæval times, as subsequently in the German Empire, some kings were really or nominally subordinate to the Emperor (as ostensibly representing the Roman Cæsar or Imperator), and a King was held to rank below an Emperor. In reference to ancient times the name is applied, like L. rex, Gr. βασιλεύς, Heb. melek, to the more or less despotic rulers not only of great dominions like Assyria, Persia, Egypt, but of petty states or towns such as Jericho, Ai, Mycenæ, Ithaca, Syracuse, and Rome. It is still applied to the native rulers of petty African states, towns, or tribes, Polynesian islands, and the like. "
"From the OED:" etc
Yup, but AFAIK the historical thinking is that the title doesn't apply as no such ruler is known but there may have been a military commander of that name. Or maybe there wasn't. All the references are considerably later, and have a strong whiff of myth about them. The only battle attributed to him in these sources which can be matched in earlier sources is Mons Badonis and that earlier source, Gildas, doesn't attach any participant's names at all. In fact, although he says it was a siege it doesn't even say who besieged whom.
"All the references are considerably later, and have a strong whiff of myth about them."More accurately, there are few references because not much writing has survived from the period. The heroic poem Y Gododdin is dated to anywhere between the 7th and 11th Century and that mentions Arthur only in passing. None of the sources are at all reliable of course. I sometimes wonder what future historians will make of today when they pore over fragments of The Grauniad, and The Daily Fail.
Babylonia culture directly descended from Sumerian/Akkadian one - one of the sources of the epic is from Babylonia.
Babylonians would rule upon Uruk in certain periods. So it's easy to make some confusion, and some old text may refer to Gilgamesh as Babylonian.
People have come up with all kinds of clever ways of counting in various different bases using their fingers and/or other body parts, but there's no evidence that any base system was developed because of the ability to do so. Indeed, the fact that it's possible to count in so many different bases rather suggests no such preference even makes much sense. The best explanation for why different bases have been preferred at different times is exactly the same as why different languages, alphabets, and so on have also been used at different times - coincidence and habit. A language or numerical system or whatever evolves naturally, and then people keep using it because it's what they're used to, until it evolves into something else or gets pushed out by a new system for a variety of different reasons.
As for the article itself, the tablet is interesting but the comparison to trigonometry makes no sense at all. The whole point of trigonometric functions is that they are the ratios of the sides of triangles (OK, it gets a bit more complicated, but that's how they were first developed). In fact, the Babylonian method is clearly more primitive since they only address special cases, much like the Egyptians and others who also knew a bit about triangles long before Pythagoras*. The big deal with trigonometry is that you're no longer stuck with a few special cases and laborious tables, but can instead use general functions to handle any case you like.
* Pythagoras himself quite possibly knowing nothing about triangles at all, with no evidence linking him with mathematics at all until over five centuries after he died.
Everyone acts like this was brand new 3700 year ago - but stuff like this doesn't appear overnight like some mushroom. The chances are that it was well establish much earlier and we simply haven't found the evidence yet. Look at Göbekli Tepe ... 12,000 years ago - you think that was built by people counting on their fingers?
This is a misconception. Based on long timescales it was "over night", but a LOT of the math and theory for GR and QM were already there. Einstine was around for the finalling of the theory, the observation of evidence, but his work was built on the backs of giants.
But those are two examples of very fine detail observations and mechanics. Building and construction, astronomy and planetary path prediction, medicine and health are all things every day people can make observations with. Math including. Which means it can get independent discovery, or just plain "common knowledge" use (such as Aboriginals in Australia not knowing what bacteria is, but knowing not to kill and eat meat near the camp for risk of disease).
Not talking about symbols, names.
In base ten, in English, nine plus one gives us not "One-zero", but "ten". Double that and we don't get "Two-zero", we get "twenty".
There are 21 individual names for numbers between 0 and 20. There are another 12 names if you want to count up beyond 1 000 000 000.
Now, how many names do you need for the sexadecimal digits that make life so much "easier".
AFAIK, number names are still subject to debate. That's because you need to find a phonetic representation of the number besides its symbolic representation (i.e. "two" <-> "2"). In some instance numbers were also used to refer to gods...
Also AFAIK, they had no distinct names for numbers 1-60, but like those counting in base 10, had names for numbers up to 20, and then names for tens and so on. Remember the mathematics and positional system was base 60, the number symbols were not.
...it's about circles.
The hypotenuse is the radius of a unit circle, and the sin and cosine are the height and width of that line (the vertical and horizontal components of that radial line). As the tip of the radius follows the circumference of the circle, those horizontal and vertical components generate the sine and cosine waves at 90 degree phase to each other. Hence trigonometry is really about how circular motion relates to oscillating motion (think pistons and crankshafts); the stuff about triangles is all fine, but misses the bigger picture.
This is also why the Babylonian table of ratios in the tablet are not going to replace Greek trigonometry any time soon, contrary to the assertion of the historian in the video - ratios of triangle sides miss out the actual point of what trigonometry is about.
If trigonometry really were about circles, then why the "tri"?
I will agree with you, though, that there's a surprisingly strong connection between circles and triangles. Thus if you were to plot a graph of the Pythagoream Theorem (x^2 + y^2 = z^2), you get a perfect circle of radius z.
PS. Interestingly, if you generalize the equation to x^n + y^n = z^n and plot their graphs for greater values of n, you find the graph morphs from a circle to a square as n approaches infinity.
sin, cos, tan and the reciprocals and derivative are used heavly in design and describing AC circuits which includes audio and radio circuits and basically any circuit which has a signal. After all sin describes a pure AC signal or audio wave hence sine wave.
To be honest I found the maths far to difficult just have a look at wikipedia https://en.wikipedia.org/wiki/Trigonometric_functions and scroll down
I found using matrices to solve simultaneous equations with more than two unknowns to be much easier to get my head round not that I can remember how now.
For example GPS uses trigonometric functions for the radio waves aspect, trigonometry to calculate your position using simultaneous equations to solve the 4 unknowns of your position in x, y, z and time.
The Coward is right - and the diagrams show it.
The authors seem a little confused...
"it does not use angles and it does not use approximation": " A squared index and simplified values of b and d to help the scribe make their own approximation to b/d or d/b" - so did they approximate or not?
As has been pointed out, the examples are just special cases of right angle triangle ratios, only relevant when processing those triangles, or the "half a rectangle", whereas the sine/cosine/tan ratio mechanism is not restricted to right-angled triangles, just to angles. Even better if you further generalize to the circle view and bring in radians...
Then they say "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."
Well a 30 degree angle, which has a lovely sine value of 0.5, would have an inconvenient "ratio" expression that is irrational in any base. So much for exact calculation.
The only reason those examples are exact and don't involve irrationals is because they cannot handle the cases where irrationals are needed and so restrict themselves to a few special cases.
There's a reason why we don't do things their way, and haven't for a long time. And it isn't because the ancients had a deeper understanding of trigonometry than we do... two thousand years ago the Greeks knew you can't square the circle.
Yay, and another one for the brief appearance of Cuisenaire rods in the video
The ancient Babylonians used this trigonometry for their advanced building projects. In the Biblical timeline, 3700 years ago is around the youth of Isaac, when Abraham was over 100 years old. So the Babylonians certainly had knowledge of this when Abraham was a young man, which was at the time when the Babylonians were building the Tower of Babel, according to the traditional Biblical timeline in the classical book "Seder Olam." How else could such a difficult engineering project have been undertaken, without such an advanced trigonometric knowledge of how to design it?
Also, base-60 trigonometry would be a natural system for accurate star-charting in ancient astronomy and astrology (60 minutes per hour in a twelve hour day, during which the earth makes a full 360 degree axial rotation). The Babylonians were advanced in this knowledge at that time, as we know that Abraham was an expert astrologer. (See Rashi on Genesis 13:5,)
Genesis 13:5
And Lot also, who went with Abram, had flocks, and herds, and tents.?
Rashi interpretation,
What brought this about? The fact that he traveled with Avram.
I was curious so thought I would look it all up however it makes no sense to me in your context, I would like to ask why it's took 3700 years for this to now be known, surely records must have been kept by the scholars over the years.
Talking about copyright and mathematical tables, set my mind off on a train of thought.
What if the publisher of a set of mathematical tables incorporated deliberate errors to allow easy spotting of plagiarism? Cartographers are said to have done similar things with maps ..... And then what if one day, the designer of a bridge stumbled upon one of these deliberate errors whilst doing an innocent engineeering calculation, and ended up underspecifying something and then as a consequence, one day, the bridge collapses and a train falls into a river?
What if they checked their calculations using a different book of tables, which happened to be a pirate copy of the original and therefore included the same mistake?
It's probably a good job there were no modern-style lawyers in those days. (Another thought: Perhaps it's precisely because we have so many labour-saving devices at our disposal today, it is entirely possible for some people literally to have nothing better to do with their time .....)
"What if the publisher of a set of mathematical tables incorporated deliberate errors to allow easy spotting of plagiarism?"The preface to my Chambers' mathematical tables states there are known errors that would trap any plagiarism. The errors are presumably in the last decimal place and consequently unlikely to prove problematic.
The book's in storage but would be forty years old at least.
A simple way to identify tables that are printed as graphics is to sprinkle single yellow pixel codes over the image area. This was invented by Xerox for the early colour printers. The government insisted that the serial number of each machine be printed on the copies to make it possible to track money counterfeiters.
Single yellow pixels are almost impossible to see visually. Privacy violation started a long time ago.
The deliberate errors are fairly easy to spot - a simple difference between the last two figures of two values will show an out of trend value quite quickly - something any anally retentive 12 year old would know half way through the first lesson they finished 30 minutes before the rest.
"Cartographers are said to have done similar things with maps"
Cartographers have definitely done similar things with maps - I used a non-existent barn in Bedfordshire (OS landranger sheet 165) as a tie breaker question in a club mini-rally (What's special about the barn at grid reference XXXYYY?), and also been caught out instructing my driver to turn left at the barn while rallying somewhere in Scotland (that was in the late 1990s; the driver besmirched my map-reading skills, but much to my delight there was an item in the national press a few days after the rally confirming legal action between two road atlas manufacturers on similar grounds, vindicating my original instruction; IIRC, it was the AA who'd been copying OS maps, including the errors).
The details of the tablet - and its mathematical significance - were published by Otto Neugebauer in 1945, Neugebauer being a professor of astronomy, a mathematician AND sufficiently well educated in classics as to translate it directly.
And republished https://arxiv.org/pdf/1004.0025.pdf
Tsk Tsk Reg, almost as bad as Pythagoras himself.
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