An ancient Babylonian tablet may contain the first evidence of trigonometry(sciencemag.org)
sciencemag.org
An ancient Babylonian tablet may contain the first evidence of trigonometry
http://www.sciencemag.org/news/2017/08/ancient-babylonian-tablet-may-contain-first-evidence-trigonometry
26 comments
I made a little activity from the demonstration given in the paper,
https://teacher.desmos.com/activitybuilder/custom/59a05b5f50...
I agree that the actual paper is way better than most press I've read about it. (others have linked to it, but http://www.sciencedirect.com/science/article/pii/S0315086017... )
I found the surprise appearance by Knuth super interesting!
https://teacher.desmos.com/activitybuilder/custom/59a05b5f50...
I agree that the actual paper is way better than most press I've read about it. (others have linked to it, but http://www.sciencedirect.com/science/article/pii/S0315086017... )
I found the surprise appearance by Knuth super interesting!
Knuth is worth quoting in full, so here is the very first paragraph of Ancient Babylonian Algorithms (1972) by Donald E. Knuth:
One of the ways to help make computer science respectable is to show that it is deeply rooted in history, not just a short-lived phenomenon. Therefore it is natural to turn to the earliest surviving documents which deal with computation, and to study how people approached the subject nearly 4000 years ago. Archeological expeditions in the Middle East have unearthed a large number of clay tablets which contain mathematical calculations, and we shall see that these tablets give many interesting clues about the life of early "computer scientists".
One of the ways to help make computer science respectable is to show that it is deeply rooted in history, not just a short-lived phenomenon. Therefore it is natural to turn to the earliest surviving documents which deal with computation, and to study how people approached the subject nearly 4000 years ago. Archeological expeditions in the Middle East have unearthed a large number of clay tablets which contain mathematical calculations, and we shall see that these tablets give many interesting clues about the life of early "computer scientists".
Thanks for that, it was a good quick introduction to what's going on on the tablet.
> And science historian Jöran Friberg, retired from the Chalmers University of Technology in Sweden, blasts the idea. The Babylonians “knew NOTHING about ratios of sides!” he wrote in an email to Science.
“When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.” — Arthur C. Clarke
“When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.” — Arthur C. Clarke
Please be aware that I haven't read the paper (http://ac.els-cdn.com/S0315086017300691/1-s2.0-S031508601730...), so some of the things I say below may be wrong.
According to http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.htm..., the first column of the tablet is the squared ratio of two sides. It gives a full transcription of the base-60 fractions. This squared ratio, or ratio of squares, is monotonic in both the ratio of the sides and the angle over the relevant interval. It may be more convenient to calculate with than the raw ratio because it is rational when the sides in question are integers (or even rational), as they are in this table.
From my point of view, the interesting thing about this is that the table is sorted by this first column, which means it's sorted in angle order. This means that if you have a measured angle and you want to do some calculations about the relevant triangle, you can reasonably easily look up the nearest two rows in this table and go from there.
It's true that you have to convert your angle measurement into this weird form of the ratio of the squares of two sides, essentially cos²θ (or sin²θ if you're thinking about the complementary angle). A thing I haven't seen mentioned in the other comments is that the paper's second author, Wildberger, has spent the last 12 years trying to convince modern-day people that just such a ratio of squares would be a good way to measure angles (he calls it "spread") because, among other things, it allows your trigonometry theorems to apply to any field of characteristic >2, rather than just to ℝ. He's published a book, founded a book publishing company to publish it, and uploaded a long series of foundations-of-mathematics lectures to YouTube on the subject, all based on a finitistic formulation of mathematics which has been out of fashion for a century, largely thanks to Hilbert's passionate advocacy of not abandoning Cantor's paradise.
So of course Wildberger would be tremendously excited to discover evidence that suggests that the Babylonians were doing trigonometric calculations 3800 years ago using a system very similar to the one he is advocating today.
And, in a sense, Friberg would be precisely correct — the tablet contains no ratios of side lengths, only ratios of "quadrances", in Wildberger's terminology.
According to http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.htm..., the first column of the tablet is the squared ratio of two sides. It gives a full transcription of the base-60 fractions. This squared ratio, or ratio of squares, is monotonic in both the ratio of the sides and the angle over the relevant interval. It may be more convenient to calculate with than the raw ratio because it is rational when the sides in question are integers (or even rational), as they are in this table.
From my point of view, the interesting thing about this is that the table is sorted by this first column, which means it's sorted in angle order. This means that if you have a measured angle and you want to do some calculations about the relevant triangle, you can reasonably easily look up the nearest two rows in this table and go from there.
It's true that you have to convert your angle measurement into this weird form of the ratio of the squares of two sides, essentially cos²θ (or sin²θ if you're thinking about the complementary angle). A thing I haven't seen mentioned in the other comments is that the paper's second author, Wildberger, has spent the last 12 years trying to convince modern-day people that just such a ratio of squares would be a good way to measure angles (he calls it "spread") because, among other things, it allows your trigonometry theorems to apply to any field of characteristic >2, rather than just to ℝ. He's published a book, founded a book publishing company to publish it, and uploaded a long series of foundations-of-mathematics lectures to YouTube on the subject, all based on a finitistic formulation of mathematics which has been out of fashion for a century, largely thanks to Hilbert's passionate advocacy of not abandoning Cantor's paradise.
So of course Wildberger would be tremendously excited to discover evidence that suggests that the Babylonians were doing trigonometric calculations 3800 years ago using a system very similar to the one he is advocating today.
And, in a sense, Friberg would be precisely correct — the tablet contains no ratios of side lengths, only ratios of "quadrances", in Wildberger's terminology.
> Friberg would be precisely correct — the tablet contains no ratios of side lengths, only ratios of "quadrances"
Haha. Someone should email him to say so. :-P
Haha. Someone should email him to say so. :-P
To be fair another law is in play
> Sci-fi authors often get it super wrong. Look back at how many sci-fi works correctly predicted the ubiquitousness of cell phones and Internet-capable phones.
> Sci-fi authors often get it super wrong. Look back at how many sci-fi works correctly predicted the ubiquitousness of cell phones and Internet-capable phones.
Submitted that link earlier, probably the most detailed analysis I have seen (with explanation of Babylonian floating point!)
http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.htm...
http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.htm...
What I find more interesting is how form follows function in this numbering system. It is the clay "writing" system of pressing a kind of wedge into the clay in a layered fashion that gave form to the numbering system as a matter of practicality from the medium. The clay dictated how to "write" the numbers, it wasn't simply a matter of a writing system that was simply recorded in the medium of clay. It is precisely why that "writing" system did not survive the transition tho papyrus or parchment. Or, in other words, it was deemed to have advantageous characteristics that were then adapted to through the numbering system.
For what it’s worth, cuneiform writing lasted about as long or longer than any other writing system in history, though the Chinese writing system is pretty close by now.
I think the main reason it went out of style is that the languages written in cuneiform were replaced by others which already had their own alphabets. I’m not remotely an expert on this though.
I think the main reason it went out of style is that the languages written in cuneiform were replaced by others which already had their own alphabets. I’m not remotely an expert on this though.
The Sumerians knew about logarithms: https://blog.kareldonk.com/planet-x-nibiru-an-analysis-of-ak...
And vedi trigonometry? http://veda.wikidot.com/tip:ganita
One of the publicity videos for this talks about the Babylonian base-60 system allowing them to write down fractions more accurately than we can. This is bunk. You can't write down one-third exactly as a base 10 decimal, but you can work with it exactly by writing it down as 1/3. It's more laborious, and probably the Babylonians found base-60 very useful, but given we have obscenely powerful computers it's probably useless to us as a practical matter.
Notice that nothing like our modern notation for fractions existed at the time (the first such notation is from India in ~500 CE), and calculations throughout the ancient world were done using mental arithmetic, finger counting, or physical manipulation of tokens either in piles or on some kind of counting board, rather than using symbolic manipulation with columns of written digits the way we learn in schools today.
In particular, long division is a real pain compared to multiplication, so being able to use a reciprocal table to turn division problems into multiplication problems would have been a big help.
* * *
Here’s my Reddit comment showing a bit about how sexagesimal can be nice for doing exact computations with numbers that can be exactly reciprocated:
As a simple example, 0.4 (base 10) is a much more precise way of writing 2/5 than ~0.31 (base 8).
In base sixty you get many more divisors which result in terminating positional fraction expansions: 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, etc. (any number which can be written as powers of 2, 3, or 5)
Whereas in base ten, you only have 2, 4, 5, 8, 10, 16, 20, 25, etc.
If you want to fill out a table of all the possible divisors between 1.0 and 2.0 which are “decimally smooth” and use up to 4 decimal digits (10000 possibilities) you get:
1.0, 1.024, 1.25, 1.28, 1.5625, 1.6, 1.6384, 2.0
If you fill out a table of all the divisors between 1:00 and 2:00 which are “sexagesimally smooth” and use up to 2 sexagesimal digits (3600 possibilities) you get:
1:00, 1:00:45, 1:04, 1:04:48, 1:06:40, 1:08:16, 1:12, 1:12:54, 1:15, 1:16:48, 1:20, 1:21, 1:23:20, 1:25:20, 1:26:24, 1:30, 1:33:45, 1:36, 1:37:12, 1:40, 1:41:15, 1:42:24, 1:44:10, 1:46:40, 1:48, 1:49:21, 1:55:12, 2:00 (I might have missed a couple here.)
As you can see this is a much richer set of divisors, more usefully spaced throughout the interval.
In particular, long division is a real pain compared to multiplication, so being able to use a reciprocal table to turn division problems into multiplication problems would have been a big help.
* * *
Here’s my Reddit comment showing a bit about how sexagesimal can be nice for doing exact computations with numbers that can be exactly reciprocated:
As a simple example, 0.4 (base 10) is a much more precise way of writing 2/5 than ~0.31 (base 8).
In base sixty you get many more divisors which result in terminating positional fraction expansions: 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, etc. (any number which can be written as powers of 2, 3, or 5)
Whereas in base ten, you only have 2, 4, 5, 8, 10, 16, 20, 25, etc.
If you want to fill out a table of all the possible divisors between 1.0 and 2.0 which are “decimally smooth” and use up to 4 decimal digits (10000 possibilities) you get:
1.0, 1.024, 1.25, 1.28, 1.5625, 1.6, 1.6384, 2.0
If you fill out a table of all the divisors between 1:00 and 2:00 which are “sexagesimally smooth” and use up to 2 sexagesimal digits (3600 possibilities) you get:
1:00, 1:00:45, 1:04, 1:04:48, 1:06:40, 1:08:16, 1:12, 1:12:54, 1:15, 1:16:48, 1:20, 1:21, 1:23:20, 1:25:20, 1:26:24, 1:30, 1:33:45, 1:36, 1:37:12, 1:40, 1:41:15, 1:42:24, 1:44:10, 1:46:40, 1:48, 1:49:21, 1:55:12, 2:00 (I might have missed a couple here.)
As you can see this is a much richer set of divisors, more usefully spaced throughout the interval.
thanks for posting this. what does "decimally smooth" mean in this context?
I just mean a number such that both itself and its reciprocal is a terminating decimal (rather than a repeating decimal like the reciprocal of 3; 1/3 = 1.33333...). That is, some rational number made up of 2 to some power times 5 to some power (either or both of the exponents could be negative).
“Decimally regular” might be a more standard term.
“Decimally regular” might be a more standard term.
> but given we have obscenely powerful computers it's probably useless to us as a practical matter.
Babylonian arithmetic methods have been used to give us our obscenely powerful computers.
Babylonian arithmetic methods have been used to give us our obscenely powerful computers.
Wildberger should be pleased with himself.
The original sin.
Why do people believe this sort of stuff still?
We know humans pattern match. It gives us gods, rain dances, lucky charms.
But we are in the 21st century and this isn't Reddit.
It aint going to be trig.
We know humans pattern match. It gives us gods, rain dances, lucky charms.
But we are in the 21st century and this isn't Reddit.
It aint going to be trig.
I thought the Arabs invented al-gebra...
Also see the authors’ (ongoing) series of Youtube video lectures about their work and its context: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWU...
Here’s what page 2 of the paper says about the paper’s contents:
> We present an improved approach to the generation and reconstruction of the table which concurs with Britton, Proust and Shnider (2011) on the likely missing columns. [...] We show that in principle the information on P322 is sufficient to perform the same function as a modern trigonometric table using only OB techniques, and we apply it to contemporary OB questions regarding the measurements of a rectangle.
> We then exhibit the impressive mathematical power of P322 by showing that it holds its own as a computational device even against Madhava’s sine table from 3000 years later. This is a strong argument that the essential purpose of P322 was indeed trigonometric: suggesting that an OB scribe unwittingly created an effective trigonometric table 3000 years ahead of its time is an untenable position.
> [...] Further research is required to investigate the historical and mathematical possibilities we are suggesting. On the historical side the question arises of how the Babylonians might have used such a table, and we do not attempt to answer this question here. On the mathematical side it is becoming increasingly clear that the OB tradition of step-by-step procedures based on their concrete and powerful arithmetical system is much richer than we formerly imagined. Perhaps the understanding of this ancient culture can help inspire new directions in modern mathematics and education.