# Babylonian Trigonometry again? **URL:** \ **Category:** Factual Questions\ **Created:** [August 25, 2017, 4:16pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663 "2017-08-25T16:16:29Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [August 25, 2017, 4:16pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/1 "2017-08-25T16:16:29Z")
An article in the Guardian, [Mathematical secrets of ancient tablet unlocked after nearly a century of study](https://www.theguardian.com/science/2017/aug/24/mathematical-secrets-of-ancient-tablet-unlocked-after-nearly-a-century-of-study), makes claims about a newly deciphered Babylonian tablet. > [@](#): > > Daniel Mansfield, of the university’s school of mathematics and statistics, described the tablet which may unlock some of their methods as “a fascinating mathematical work that demonstrates undoubted genius” – with potential modern application because the base 60 used in calculations by the Babylonians permitted many more accurate fractions than the contemporary base 10… > > “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.” I understand that different techniques can make calculations easier. I’m more dubious at the claim that base 60 permits more accurate fractions than base 10. The original paper in full can be found [here](http://www.sciencedirect.com/science/article/pii/S0315086017300691). My glance at the original shows they praise the Babylonians for using exact ratios for calculation, but that method also had problems dealing with irrationals. Could one of mathperts explain the significance of approaching trig this way? ---
**Author:** ![GreenWyvern](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/greenwyvern/32/2834_2.png) [@GreenWyvern](https://boards.straightdope.com/u/GreenWyvern)\ **Post date:** [August 25, 2017, 4:25pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/2 "2017-08-25T16:25:44Z")
Another article: > **[Were the Babylonians better mathematicians than us?](https://www.abc.net.au/news/2017-08-25/babylonian-tablet-unlocks-simpler-trigonometry-mathematics/8841368)** > > A small clay tablet that's 3,700 years old may be about to make studying trigonometry much easier, thanks to the work of an Australian mathematician. And a 2 minute video by Dr. Mansfield: [![](https://img.youtube.com/vi/i9-ZPGp1AJE/maxresdefault.jpg "Ancient Babylonian tablet - world's first trig table") ](https://www.youtube.com/watch?v=i9-ZPGp1AJE) **Conclusion** (from the paper linked above) > [@](#): > > Artifacts from ancient civilizations offer us a glimpse at their complex social and scientific achievements. Sophisticated artifacts offer us a larger window, and P322 is undeniably complex. Through P322 we see a forgotten ancient approach to trigonometry that is based on exact sexagesimal ratios. > > The approximate nature of modern trigonometry is so culturally enshrined that we give it no second thought. It comes as a complete surprise, then, to find that the OB culture developed a trigonometry that actually only contains exact information, and so any imprecision is a consequence of using the table rather than inherent in the table itself. > > … > > 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. > > This new interpretation of P322 significantly elevates the status of Babylonian mathematics, and the vast number of untranslated tablets are likely to contain many more surprises waiting to be found. 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. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [August 25, 2017, 4:41pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/3 "2017-08-25T16:41:26Z")
Seems hard to say anything without a transcript of the tablet in question. The first linked article claims it is a table of Pythagorean triples. The significance seems to be that it is the oldest such tablet known; I do not see that recording the numbers in base 60 is in itself significant. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [August 25, 2017, 5:11pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/4 "2017-08-25T17:11:25Z")
The big advantage of base 10 over base 60 is that base-10 numbers can be written using a set of ten simple symbols, distinct from alphabetic letters. To come up with 60 such distinct symbols would probably be unwieldy; instead [the Babylonians built up 59 numeral symbols from just two symbols](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Babylonian_numerals.svg/450px-Babylonian_numerals.svg.png): one for 1 and one for 10. (Eventually they had a zero symbol also.) Another area where base-60 would seem very inconvenient is multiplication. We memorize the 10x10 multiplication table and go from there. A 60x60 table would be hard to memorize and would require a huge amount of clay bricks to carry around! _ **However the Babylonians did have a very elegant method of multiplication.** _ As shown [at Wikipedia](https://en.wikipedia.org/wiki/Babylonian_mathematics#Arithmetic), as early as 2000 BC Babylonians apparently multiplied numerals using two table lookups, two subtractions and one addition. In my own experiments I have found that programming their method allowed _multiplication significantly faster than the multiply instruction on many processors_, including some of the late 1980’s (_), and that today, 4000 years later, very few programmers are familiar with the technique. (_ - Most processors have fast multiply hardware by now.) ---
**Author:** ![GreenWyvern](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/greenwyvern/32/2834_2.png) [@GreenWyvern](https://boards.straightdope.com/u/GreenWyvern)\ **Post date:** [August 25, 2017, 6:00pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/5 "2017-08-25T18:00:33Z")
> [@DPRK](#): > > Seems hard to say anything without a transcript of the tablet in question. The first linked article claims it is a table of Pythagorean triples. > > The significance seems to be that it is the oldest such tablet known; I do not see that recording the numbers in base 60 is in itself significant. You can always download and read the research [paper](http://www.sciencedirect.com/science/article/pii/S0315086017300691) if you want to do more than speculate. It’s open access. It’s a long and complex paper that’s not quick or easy to digest, or reach an opinion on. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 25, 2017, 6:02pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/6 "2017-08-25T18:02:17Z")
Utter nonsense. It’s possible to produce a perfectly precise trig table in any base, provided that you limit yourself to certain particular angles. And if you don’t limit yourself to those angles, then it’s not possible in any base. ---
**Author:** ![GreenWyvern](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/greenwyvern/32/2834_2.png) [@GreenWyvern](https://boards.straightdope.com/u/GreenWyvern)\ **Post date:** [August 25, 2017, 6:10pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/7 "2017-08-25T18:10:38Z")
> [@Chronos](#): > > Utter nonsense. It’s possible to produce a perfectly precise trig table in any base, provided that you limit yourself to certain particular angles. And if you don’t limit yourself to those angles, then it’s not possible in any base. It’s better to base our opinions on what the research paper actually says, rather than on superficial and distorted news reports. See [The Science News Cycle](http://phdcomics.com/comics/archive.php?comicid=1174) from PHD Comics ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 25, 2017, 6:17pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/8 "2017-08-25T18:17:26Z")
Like, the part of the paper where it says that this is “the only trigonometric table that is precise”? Or where it says that “Irrational numbers… are not actually necessary for trigonometry”? ---
**Author:** ![Larry\_Borgia](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/larry_borgia/32/156_2.png) [@Larry\_Borgia](https://boards.straightdope.com/u/Larry_Borgia)\ **Post date:** [August 25, 2017, 6:22pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/9 "2017-08-25T18:22:27Z")
> [@GreenWyvern](#): > > It’s better to base our opinions on what the research paper actually says, rather than on superficial and distorted news reports. > > See [The Science News Cycle](http://phdcomics.com/comics/archive.php?comicid=1174) from PHD Comics How about basing our opinions on well established mathematical truths, like\*\* Chronos\*\* did? He’s not saying anything that’s remotely in doubt. You might as well claim that the square root of 2 is a rational number. ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [August 25, 2017, 6:35pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/10 "2017-08-25T18:35:30Z")
So I did read the paper and I am an honest to goodness real mathematician and everything. The table contains a list of pythagorean triples (a, b, c) where a, b, c are integers, a \< b \< c, and a^2 + b^2 = c^2. (I’m not in the mood for sup tags) The interesting part is that b is always a number that can be written 2^x 3^y 5^z. This means that 1/b has a finite representation in base 60. This also means that the tangent (a/b) and the secant (c/b) have finite representations in base 60. That’s the whole story. It is pretty impressive that ancient people were able to generate these, but the authors of the paper are seriously blowing the importance of this out of proportion. And, as **Chronos** alluded to earlier, it really is the authors of the paper doing this puffery and not just the journalists sensationalizing it. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [August 25, 2017, 6:50pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/11 "2017-08-25T18:50:27Z")
Wait a minute here … “… with potential modern application because the base 60 used in calculations by the Babylonians permitted many more accurate fractions than the contemporary base 10 …” I grabbed my book of trig tables published in the 1950’s and the angle measures are all in base-60 … minutes- and seconds-of-arc … were we really that backwards in the 20th Century? … ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [August 25, 2017, 7:14pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/12 "2017-08-25T19:14:38Z")
I suppose it IS interesting that 1000 years before Hipparchus, there existed a table of pythagorean triples such that useful trigonometric ratios could be precisely calculated. Presumably useful in building things utilizing right triangles if you weren’t all focused on a particular **angle**. Clearly, using base-60 helps in accomplishing such a table, since 60 has so many more integer factors than 10 does. Of course, as long as this particular cuneiform tablet’s information isn’t ever connected to anything else we know about the Babylonians (for example: do their structures show that they used these triples in building things?), then it’s just a quaintly curious bit of calculation by someone with way too much time on their hands. 😃 ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [August 25, 2017, 7:34pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/13 "2017-08-25T19:34:59Z")
One more thing. The bit about the ‘only’ exact trig table is straight from the authors and pure bullshit. When I taught calculus, I would tell the students early in the semester that I expected them to know have at least some trigonometry immediately recallable to their brain. This included some a small number of values for trig functions for common angles. Then I would put this trig table on the board as an easy way to remember those values. ```auto t sin t cos t 0 sqrt(0/4) sqrt(4/4) pi/6 sqrt(1/4) sqrt(3/4) pi/4 sqrt(2/4) sqrt(2/4) pi/3 sqrt(3/4) sqrt(1/4) pi/2 sqrt(4/4) sqrt(0/4) ``` That is, without a doubt, an exact trig table and it’s been around a long time. I learned it in high school and I’m sure it wasn’t new then. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [August 25, 2017, 9:18pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/14 "2017-08-25T21:18:12Z")
Base 60 gives access to more exact fractions, since 60 is evenly divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. And I would say that all the angles we care about most fit in a 60 degree circle (hell, they fit in a 12 degree circle). You know, degrees of 30, 45, 60, 90: angles used in the simplest right triangles. So, using integers, it makes sense that they may have been able to calculate more triples exactly. It would just basically be trying them out. It would be inherent to measure things in groups of 60, due to their system, just like we measure things in groups of 10. Maybe this was the most exact table **up until that point in history**. That would make sense. And, yes, in a way, they are doing some very basic trigonometry without the use of irrational numbers. The first might be a huge deal, but the second not so much. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [August 25, 2017, 9:35pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/15 "2017-08-25T21:35:10Z")
> [@Lance\_Turbo](#): > > So I did read the paper and I am an honest to goodness real mathematician and everything. > > The table contains a list of pythagorean triples (a, b, c) where a, b, c are integers, a \< b \< c, and a^2 + b^2 = c^2. (I’m not in the mood for sup tags) > > The interesting part is that b is always a number that can be written 2^x 3^y 5^z. This means that 1/b has a finite representation in base 60. This also means that the tangent (a/b) and the secant (c/b) have finite representations in base 60. > > That’s the whole story. It is pretty impressive that ancient people were able to generate these, but the authors of the paper are seriously blowing the importance of this out of proportion. > > And, as **Chronos** alluded to earlier, it really is the authors of the paper doing this puffery and not just the journalists sensationalizing it. Thank you! ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [August 25, 2017, 9:49pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/16 "2017-08-25T21:49:49Z")
Speaking of going back to the original, I am still curious about the tablet itself. Various descriptions and transcriptions of it present a table with four columns: a ratio, two integers, and finally a row number. (There are also a few errors in the table!!) But, looking at color and black-and-white photographs, I see what looks like five columns, and the table is headed by a couple of rows of text at the top. Can anyone describe/transcribe/translate the additional bits? ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [August 25, 2017, 11:38pm UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/17 "2017-08-25T23:38:56Z")
One more thing. At least one of the paper’s authors, N. J. Wildberger, is a real math professor. However, rational trigonometry is a fringe idea that is a bit of a hobby horse for him. Twelve years ago he wrote [Divine Proportions: Rational Trigonometry to Universal Geometry](http://www.wildegg.com/divineproportions-rationaltrig.html). I didn’t find anything about the other author, but I also didn’t look very hard. ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [August 26, 2017, 12:22am UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/18 "2017-08-26T00:22:10Z")
> [@Lance\_Turbo](#): > > One more thing. The bit about the ‘only’ exact trig table is straight from the authors and pure bullshit. > > When I taught calculus, I would tell the students early in the semester that I expected them to know have at least some trigonometry immediately recallable to their brain. This included some a small number of values for trig functions for common angles. Then I would put this trig table on the board as an easy way to remember those values. > > ```auto > > t sin t cos t > 0 sqrt(0/4) sqrt(4/4) > pi/6 sqrt(1/4) sqrt(3/4) > pi/4 sqrt(2/4) sqrt(2/4) > pi/3 sqrt(3/4) sqrt(1/4) > pi/2 sqrt(4/4) sqrt(0/4) > > ``` > > That is, without a doubt, an exact trig table and it’s been around a long time. I learned it in high school and I’m sure it wasn’t new then. It’s not “exact” because, while sqrt 2 is an exact number, it’s not a number you can use to create an “exact” answer with (since it’s irrational). I’m presuming they mean that the table is “exact” in the sense that all the values in the table are rational. ---
**Author:** ![Mangosteen](https://avatars.discourse-cdn.com/v4/letter/m/6a8cbe/32.png) [@Mangosteen](https://boards.straightdope.com/u/Mangosteen)\ **Post date:** [August 26, 2017, 12:58am UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/19 "2017-08-26T00:58:15Z")
Didn’t the Babylonians use 360 as a base not 60? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 26, 2017, 1:01am UTC](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663/20 "2017-08-26T01:01:29Z")
But if the sines and cosines are rational, then the angles won’t be, in any system of measuring angles you care to name. The whole point of trigonometry is that you have both the ratios and the angles. [Next page](https://boards.straightdope.com/t/babylonian-trigonometry-again/794663.md?page=2)