Babylonian trigonometry "simpler, more accurate... has clear advantages over our own."

[eShirl]

http://popular-archaeology.com/issue/june-2013/article/a-remarkable-ancient-babylonian-tablet-and-why-it-matters

UNIVERSITY OF NEW SOUTH WALES—UNSW Sydney scientists have discovered the purpose of a famous 3700-year old Babylonian clay tablet, revealing it is the world's oldest and most accurate trigonometric table, possibly used by ancient mathematical scribes to calculate how to construct palaces and temples and build canals.

"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 3000 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," he says.

photo of Plimpton 322

[eppur_se_muova]

1. "only completely accurate trigonometric table" -- an exceptional claim, requiring exceptional proof.

None is provided.

Trigonometry is not a science of approximations, but of identities. There can't BE a "more accurate" version.

[Wounded Bear]

2. Maybe they figured out how to make pi3...

[eShirl]

4. I defer to those who know trig

[Jim__]

5. Did you watch the video?

They give their rationale for the claim toward the end of this short video:

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[Binkie The Clown]

6. It's the kind of sci-babble that happens when they try to explain something to the uneducated

Why do reporters have to sensationalize and exaggerate everything they write for the general public? They could have simply said that the Babylonians took a slightly different approach and arrived at a pretty good approximation for their time. We also use ratios when we teach school children that pi is approximately equal to 22/7. The ratio 355/113 is even better. But neither of them is "more accurate" than the actual irrational (ir-rational) number that is pi.

That they has such an accurate table is newsworthy. The rest is just silliness, (like "maybe space aliens taught it to them&quot .

[hunter]

8. It wasn't an approximation, the tables ARE exact.

Last edited Sat Aug 26, 2017, 08:11 PM - Edit history (1)

There are simply no entries for angles that can't be described exactly with known fractions.

To solve problems with angles that are not exact fractions, the user of the table might make a "good enough" interpolation between the known fractions by adding a familiar (to them) sexagesimal fraction, 1/60, 2/60, 3/60.... etc.

The Babylonians handled pi in a similar fashion, using the fraction 25/8 knowing it was a little short but good enough for most work.

For more critical work 25/8 + 1/60 was a pretty good approximation.

25/8 + 1/60 = 3.141666......

[muriel_volestrangler]

9. I want to know what they mean by 'simplified' values

This is by the paper's authors:

This precise arithmetic of the Babylonians also influenced their geometry, which they preferred to be exact. They were able to generate a wide variety of right-angled triangles within exact ratios b/l and d/l, where b, l and d are the short side, long side and diagonal of a rectangle.
...
Fundamentally a trigonometric table must describe three ratios of a right triangle. So we throw away sin and cos and instead start with the ratios b/l and d/l. The ratio which replaces tan would then be b/d or d/b, but neither can be expressed exactly in sexagesimal.

Instead, information about this ratio is split into three columns of exact numbers. A squared index and simplified values of b and d to help the scribe make their own approximation to b/d or d/b.

https://theconversation.com/written-in-stone-the-worlds-first-trigonometry-revealed-in-an-ancient-babylonian-tablet-81472

"The first five rows of Plimpton 322, with reconstructed columns and numbers written in decimal."

Now, of course, tan is not "b/d or d/b", since tan does not involve a diagonal. tan is b/l, or l/b (depending on whether you're looking at angles less than or greater than 45 degrees). So these guys are pretty awful at expressing whatever idea they think they have.

But what does "simplified values of b and d" mean? And how were the reconstruction columns and numbers actually written - as the precise fractions - eg for the 5th row (other value of the triplet not shown is 72), so b/l=65/72, d/l=97/72, (d/l) squared=9409/5184? If so, then I suggest a table with its 4th row as 12709/13500, 18541/13500 (or perhaps 1 and 5041/13500), and 343768681/182250000 is not that practical.

[Jim__]

11. They do clarify that in the paper.

I apologize for the formatting, but I cannot get subscripting to work - most of these n's should be subscripts. The actual text is contained in section: 2. Plimpton 322 of the paper. From the paper:

We are not the first to propose that the missing fragment of P322 contained the ratios ?n=bn/ln and ?n=dn/ln. But we do make the novel proposition that the extant contents of P322 should be viewed as a description of the third ratio ?n/?n or ?n/?n required for a trigonometric table. Due to the OB preference for precision, this ratio has been replaced by three columns so that it can be presented without approximation. The squared value in column I? is an index, and columns II' and III? contain bn and dn, which are the simplified values ?n and ?n with regular common factor 1/ln removed, to help the scribe approximate bn/dn=?n/?n or dn/bn=?n/?n.

[hunter]

12. I think it made more sense to someone having an intimate familiarity with sexagesimal fractions

Converting the table to a decimal format doesn't make things clearer, starting with the first number .99166666 which is 119/120, but probably would have been thought of as the familiar fraction fifty-nine-and-a-half, much as we'd say 59-and-a-half seconds when talking about time. Users of that math would have an intuitive feeling for this fraction.

I tried briefly to use the table in a practical manner, but I'm lazy and not going to learn to sight read (without translation to decimal) sexagesimal numbers, fractions, and all the math shortcuts and abbreviated representations the Babylonians surely used.

I remember as a kid, crazy about computers, how learning to "think in" and do math in binary, octal, or hexadecimal numbers was not a trivial exercise.

[FiveGoodMen]

7. I would also like to know what they meant by that

[DetlefK]

3. Sounds to me like they used the next-more-complicated version of fractional numbers.

"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. It is a fascinating mathematical work that demonstrates undoubted genius.

In a fractional number, you give the ratio a:b

I guess, the babylonian trigonometry was based on a system of ratios a:b:c

And, just to be clear on this, our trigonometry is also based on ratios.
Sine and Cosine are both defined as ratios.
The equation sin² + cos² = 1 is exactly the same as the equation a²+b²=c²

[Jim__]

10. The full article in Historia Mathematica is available

Here - I’m not sure whether or not it will remain available.

[NNadir]

13. The clear advantage of the trigonometry I use over that used by the Babylonians, is that I can...

...calculate any value I need for any engineering or scientific task in nanoseconds on a computer.

The statement that the Babylonians had "simpler, more accurate" trigonometry is just silly.