# Forget trigonometry, 'cos Babylonians did it better 3,700 years ago – by counting in base 60!

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 …

#### Re: So much for digital

@Joe Harrison

You are using ternary (or trinary) digits. Confusingly abbreviated as "tits."

#### Re: So much for digital

"why has the modern world moved so far towards pure binary (and powers of 2 in specific contexts)?"

Because it is easier and more efficient to build computers that use binary.

#### Re: So much for digital

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)

#### Re: So much for digital

While living in SE Asia, we came across the kati (= 4/3 lbs) and the tahil (one sixteenth of a kati). Beer was sold in bottles containing 4/3 pints.

#### Re: So much for digital

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.

"why has the modern world moved so far towards pure binary (and powers of 2 in specific contexts)?"
Because of binary logic:

True/False

0/1

Yes/No...

Simples :-)

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.

Digital is just very fast analogue with switching thresholds

"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.

Ayup, that must be why Fuzzy Logic never went anywhere.

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 ?

#### Strangely enough ...

... everybody's ::koff::favorite::koff:: wiki has an arguably more readable image.

https://en.wikipedia.org/wiki/File:Plimpton_322.jpg

I'm pretty sure the scribble at the top right corner says "Three shirts, no starch, Wednesday late".

#### Re: Strangely enough ...

FWIW Wikipedia isn't public domain - we can't just lift their stuff.

#### Re: Strangely enough ...

"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.

#### Re: Strangely enough ...

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.

#### Re: Strangely enough ...copyright complications

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.)

#### Re: Strangely enough ...copyright complications

"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.

#### "depending on Babylonian copyright laws"

What Hammurabi stele says about it? Also, beware punishment for copyright infringement could be quite harsh, compared to our times...

"I'm pretty sure the scribble at the top right corner says "Three shirts, no starch, Wednesday late"."
There ya go! And I always thought it said: "Not tonight; it's my pyramid!"

This post has been deleted by its author

#### Three-valued logic

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.

Three-valued logic

#### Re: Three-valued logic

Yes, three valued logic for binary computers has already been described in high-impact journals!

#### Re: Three-valued logic

Three-valued (and even higher-valued) logics are nice but the thing is, you need to know what your logic operations are going to express, and there are quite a few of them (16 for binary, 19683 for ternary)

As for formulas, undecidablility arrives fast, too fast.

#### Re: Three-valued logic

The USSR did quite a lot of work on 'trinary' computing, back in the 1950s:

https://dev.to/buntine/the-balanced-ternary-machines-of-soviet-russia

#### We use base 10 for a reason

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...

#### Re: We use base 10 for a reason

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 ...

#### Re: We use base 10 for a reason

> There were 12 shillings (and 240 pennies in the pound)

Errm, we real oldies remember 12 pennies in a shilling and 20 shillings to the pound. Moreover, you could subdivide a pound into 960 farthings.

#### Re: We use base 10 for a reason

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!

#### Re: We use base 10 for a reason

"Moreover, you could subdivide a pound into 960 farthings."
And they were still legal tender... Save up enough and you could buy a penny ha'penny's-worth of sweets on the way home with your dad's Sunday bottle of cider from the offie.

#### Re: We use base 10 for a reason

"Errm, we real oldies remember 12 pennies in a shilling and 20 shillings to the pound. Moreover, you could subdivide a pound into 960 farthings."

And isn't it funny that everyone wrote out half-crowns as 2s 6d (2 and 6) rather than as, well, half a crown.

#### Re: We use base 10 for a reason

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.

#### Re: We use base 10 for a reason

>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.

#### Interesting factoid not in the article

Is that the person who originally found/stole this artifact is the guy on whom Indiana Jones was loosely based.

Edgar J Banks. Quite an interesting chap.

#### Re: Interesting factoid not in the article

Indiana Jones was based on the "archeologist" Langdon Warner (1881-1955) who stole manuscripts and Buddha's from the Dunhuang caves in China for the Harvard University Fogg Art Museum in the early 20th C.

#### Special cases

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.

#### Re: Special cases

"Pythagoras noted a number of special triangles, such as 3:4:5, that have integer ratios in base 10."
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.

#### Re: Special cases

The point is, the special cases were well known in the ancient world, pre-Greek. So this tablet simply confirms that and provides more evidence for how far back it was known.

It is not 'better' than sin/cos/tan, it's simply a collection of useful special cases.

#### Re: that have integer ratios in base 10

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

#### Re: that have integer ratios in base 10

Yes, sorry, I realised that after I posted. I was focussing on the bigger issue...

#### "It is not 'better' than sin/cos/tan, it's simply a collection"

You missed the whole point of the article, about a trigonometry not based on angles and sin/cos/etc functions....

#### Re: Special cases

"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.

#### Re: Special cases

> ... such as 3:4:5, that have integer ratios in base 10

It's nothing to do with the base (either in your post or the main article).

They're integer ratios in every base. (Excepting the deranged case of non-integer bases, but you don't use those, I hope.)

#### Re: Special cases

Well said, Wiggers.

The original academic article makes clear that the tablet was effectively equivalent to the ready reckoners and log tables familiar to the oldies among readers of El Reg.

#### Re: "It is not 'better' than sin/cos/tan, it's simply a collection"

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...

#### Re: Special cases

Home building and the 3:4:5

Three cases of beer for each 5 builders.

Works on a Friday.

#### Re: Special cases

One of my first programs in Fortran was to create a table of logarithms in base π as it made geometric problems a snap. This was before the first calculators came out. Naturally my junior high school maths teacher loved it.

#### Re: Special cases

Yes, but humanity "forgot" about it when the Great Library of Nabu was destroyed in the massive eruption of the great volcano, Sohcahtoa

#### Lissajous

sin and cos -

http://mathworld.wolfram.com/LissajousCurve.html

Paris is welcome to come over and investigate osculation by fiddling with my oscilloscope

#### Re: Special cases

Naturally my junior high school maths teacher loved it.

First day of trigonometry, my maths teacher wrote on the board...

"Some Of His Children Are Having Trouble Over Algebra"

sin = O/H, cos = A/H, tan = O/A

#### Re: Special cases

""Some Of His Children Are Having Trouble Over Algebra""
Better:

Smiles Of Happiness

Come After Having

Tankards Of Ale!!!

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