Babylonians used trigonometry

[shunyadragon]

From: An ancient clay tablet shows that Babylonian scholars might have invented trigonometry

An ancient clay tablet shows that Babylonian scholars might have invented trigonometry

This form of trigonometry is very different from what is currently used

"A new interpretation into the nature of an ancient clay tablet known as Plimpton 322 claims that ancient Babylonians might have developed an advanced form of trigonometry — long before Greek mathematicians are commonly believed to have invented the concept.

That’s the theory put forward by two mathematicians from the University of New South Wales, Daniel F. Mansfield and Norman Wildberger, who published their study in the latest issue of Historia Mathematica. They claim that the tablet demonstrates a sophisticated understanding of mathematics, and that modern assumptions of the field should be reexamined in light of the interpretation.

The tablet in question is approximately five inches wide by three inch tall, and dates back to somewhere between 1822 and 1762 BCE. It was discovered by an American archeologist and diplomat named Edgar Banks in Larsa (what is now in southern Iraq) in the early 1920s. Banks sold the tablet to New York publisher George Arthur Plimpton, who later bequeathed it and his collection to Columbia University."

Actually, the use of trigonometry by the Babylonians has been known since the at least the 1980's, but this new complete translation describes how Babylonians constructed their trigonometry based on based 60 math.

I believe engineering for building in ancient civilizations as the 'mother of invention' for math like trigonometry.

 

[Polymath257]

From: An ancient clay tablet shows that Babylonian scholars might have invented trigonometry

An ancient clay tablet shows that Babylonian scholars might have invented trigonometry

This form of trigonometry is very different from what is currently used

"A new interpretation into the nature of an ancient clay tablet known as Plimpton 322 claims that ancient Babylonians might have developed an advanced form of trigonometry — long before Greek mathematicians are commonly believed to have invented the concept.

That’s the theory put forward by two mathematicians from the University of New South Wales, Daniel F. Mansfield and Norman Wildberger, who published their study in the latest issue of Historia Mathematica. They claim that the tablet demonstrates a sophisticated understanding of mathematics, and that modern assumptions of the field should be reexamined in light of the interpretation.

The tablet in question is approximately five inches wide by three inch tall, and dates back to somewhere between 1822 and 1762 BCE. It was discovered by an American archeologist and diplomat named Edgar Banks in Larsa (what is now in southern Iraq) in the early 1920s. Banks sold the tablet to New York publisher George Arthur Plimpton, who later bequeathed it and his collection to Columbia University."

Actually, the use of trigonometry by the Babylonians has been known since the at least the 1980's, but this new complete translation describes how Babylonians constructed their trigonometry based on based 60 math.

I believe engineering for building in ancient civilizations as the 'mother of invention' for math like trigonometry.

There are several issues with this interpretation.

The most basic issue is that Plimpton 322 consists of columns of numbers. The question is where those numbers came from and what the authors were thinking. One difficulty is that the headers for the columns are not complete or damaged.

It is clear that this tablet forms a precursor to trigonometry, but I am not at all clear that it deserves the name just yet. Here is why:

First, trigonometry is based on finding the ratios of the sides of right triangles given an angle in the triangle. it has long been known that if x and y are the two shorter sides of a right triangle, and z is the long side, then the Pythagorean relationship x^2 +y^2 =z^2 holds. The Plimpton 322 tablet seems to show ways of finding x,y, and z for 'base 60' triangles. But there does NOT appear to be a strong connection between the values found and the actual angles of the resulting triangle. Instead, some very crafty numerical tricks are used involving base 60 reciprocals to find triples of x,y, and z that correspond to *some* triangle.

Part of the trick is to find reciprocal pairs that are both 'nice' base 60. And analogy for base 10 would be, say, 8 and .125. Both of these are terminating decimals. Compare them to 3 and .33333.... where the second is NOT a terminating decimal. The main value of working base 60 is that there are more such 'nice' pairs.

From such pairs, the tablet seems to show a method for constructing the x, y, and z of a right triangle. All of this is purely numerical with very little obvious geometry involved, let alone geometry related to trigonometry, which would involve the angles of the resulting triangle.

In any case, this seems to be a new translation of the tablet, which is a good thing. And while the tablet is very interesting in how it does things, it isn't clear what it does should be called trigonometry.

 

[sun rise]

Controversy aside, this is a very neat breakthrough.

 

[Ingledsva]

From: An ancient clay tablet shows that Babylonian scholars might have invented trigonometry

An ancient clay tablet shows that Babylonian scholars might have invented trigonometry

This form of trigonometry is very different from what is currently used

"A new interpretation into the nature of an ancient clay tablet known as Plimpton 322 claims that ancient Babylonians might have developed an advanced form of trigonometry — long before Greek mathematicians are commonly believed to have invented the concept.

That’s the theory put forward by two mathematicians from the University of New South Wales, Daniel F. Mansfield and Norman Wildberger, who published their study in the latest issue of Historia Mathematica. They claim that the tablet demonstrates a sophisticated understanding of mathematics, and that modern assumptions of the field should be reexamined in light of the interpretation.

The tablet in question is approximately five inches wide by three inch tall, and dates back to somewhere between 1822 and 1762 BCE. It was discovered by an American archeologist and diplomat named Edgar Banks in Larsa (what is now in southern Iraq) in the early 1920s. Banks sold the tablet to New York publisher George Arthur Plimpton, who later bequeathed it and his collection to Columbia University."

Actually, the use of trigonometry by the Babylonians has been known since the at least the 1980's, but this new complete translation describes how Babylonians constructed their trigonometry based on based 60 math.

I believe engineering for building in ancient civilizations as the 'mother of invention' for math like trigonometry.

Yep, I read another article, and watched a video on this today. Very interesting.

I love how they keep pushing things back farther and farther.

*

 

[Polymath257]

Yep, I read another article, and watched a video on this today. Very interesting.

I love how they keep pushing things back farther and farther.

*

The problem is that this really isn't new. I've been including a discussion of Plimpton 322 in my history of math course for ages.

It does show that the Babylonians of this time period had some algebraic and numerical sophistication that surprises many.

But the history of mathematics is very uneven: some concepts that we find very basic didn't appear until quite late (the concept of a number line, for example), while others that we consider to be quite sophisticated (like proving certain ratios to be irrational) appear quite early.

 

[Revoltingest]

Base 60?
I'd never have learned to count.
(I don't have enuf fingers & toes.)

 

[Polymath257]

Base 60?
I'd never have learned to count.
(I don't have enuf fingers & toes.)

Actually, you use it all the time. Second, minutes, hours. So, literally, whenever you use time, you are using base 60. And the reason we use that system for time is ultimately because of the Babylonian base 60 system.

In fact, the Babylonians used a combination system, counting ones and tens up to 60 and then flipping. Just like our seconds, minutes, and hours.

 

[Revoltingest]

Actually, you use it all the time. Second, minutes, hours. So, literally, whenever you use time, you are using base 60. And the reason we use that system for time is ultimately because of the Babylonian base 60 system.

In fact, the Babylonians used a combination system, counting ones and tens up to 60 and then flipping. Just like our seconds, minutes, and hours.

But 60 seconds is still counted in base 10.
In base 60, what would the additional 50 numbers be?
One
Two
Three
Four
Five
Six
Seven
Eight
Nine
Bleen
Floop
Biffel
Twerdlit
Poly
????

 

[Brickjectivity]

The explanation I read said it looked like a table of numbers which let you take the ratio of two sides of one right-triangle to find the ratio between two other sides. Is this what is meant by 'Reciprocals' ? The sine is the ratio of the rising side to the hypotenuse, and the cosine is the ratio of the running side to the hypotenuse. What is in the table?

 

[Polymath257]

But 60 seconds is still counted in base 10.
In base 60, what would the additional 50 numbers be?
One
Two
Three
Four
Five
Six
Seven
Eight
Nine
Bleen
Floop
Biffel
Twerdlit
Poly
????

But that's what I am saying about the mixed system. The Babylonians would have written 53, say, as 5 symbols for ten and 3 symbols for 1.

 

[Polymath257]

The explanation I read said it looked like a table of numbers which let you take the ratio of two sides of one right-triangle to find the ratio between two other sides. Is this what is meant by 'Reciprocals' ? The sine is the ratio of the rising side to the hypotenuse, and the cosine is the ratio of the running side to the hypotenuse. What is in the table?

I'll run through an example.

Take two 'nice' reciprocals, say 4 and .25. Alternately add and subtract them to get 4.25 and 3.75. Now divide both by 2 to get 2.125 and 1.875. Now, both can be multiplied by 8 to give whole numbers: 2.125*8=17 and 1.875*8=15.

These are two sides of a right triangle. The third side is given by y^2=17^2 -15^2 =289-225=64, so y=8. This is also the number that needs to be multiplied before.

So we get a right triangle with sides x=15, y=8, and z=17.

One of the ratios on the tablet was the 2.125. This does correspond to the cosecant of the angle, but the tablet doesn't measure the angle: only the three sides.

And it is the correspondence between the angle and the ratios that makes up trigonometry.

 

[Revoltingest]

But that's what I am saying about the mixed system. The Babylonians would have written 53, say, as 5 symbols for ten and 3 symbols for 1.

I could live with that.

 

[Polymath257]

A bit more about Plimpton 322:

Plimpton 322 - Wikipedia

/E: This goes into how to write numbers in Babylonian notation and some about Plimpton 322.

Plimpton 322

 

[Polymath257]

From: An ancient clay tablet shows that Babylonian scholars might have invented trigonometry

I watched the video on this link and found it to be rather unsatisfying. The speaker makes several claims that are simply false.

For example, he claims that base 10 only has two 'nice' pairs: 2<->.5 and 5<->.2.

There are a couple of problems with this claim. The first problem is that the Babylonians would have considered this to be a *single* pair 5<->2 with the 'decimal place' changed.

The second is that it ignores such pairs as 4<->25 and 8<->125, etc.

Now, it is true that base 60 allows for more such pairs. Instead of multiplying to be a power of 10, the pair has to multiply to be a power of 60.

So, for example, 2<->30, 3<->20, 4<->15, 5<->12, 6<->10 would all be *single* base 60 pairs.

But we can also get 8<->450, 36<->100, etc.

Next, the speaker claims this is the only example of a perfectly accurate trigonometry table. This is, again, just wrong. And it is wrong for two reasons: one is that every trig student memorizes a perfectly accurate table for the standard angles. And the second is that *any* table of Pythagorean triples would produce a table of exact values for the trig ratios.

Finally, the speaker makes a claim that this is new. Like I said before, I have been teaching about this tablet and its connection to Pythagorean triples for many years now. It is hardly news that this tablet is a table of such triples and their ratios.

Smells like someone out for getting their name in the popular news rather than actual scholarly news.

 

[Ingledsva]

But 60 seconds is still counted in base 10.
In base 60, what would the additional 50 numbers be?
One
Two
Three
Four
Five
Six
Seven
Eight
Nine
Bleen
Floop
Biffel
Twerdlit
Poly
????

Here is some info.

 

[Paraselsus]

Here is my theory: The Plimpton 322 cuneiform tablet had an astronomical use. The Plimpton 322 cuneiform tablet inscription appears to have been written in the heavens by drawing connecting lines between circumpolar stars where it could be seen and memorized. Plimpton 322 follows the astronomical text format of documents used by the temple and palace administrators of ancient Larsa.

Archeoastronomy Animations: Plimpton 322

 

[shunyadragon]

Here is my theory: The Plimpton 322 cuneiform tablet had an astronomical use. The Plimpton 322 cuneiform tablet inscription appears to have been written in the heavens by drawing connecting lines between circumpolar stars where it could be seen and memorized. Plimpton 322 follows the astronomical text format of documents used by the temple and palace administrators of ancient Larsa.

Archeoastronomy Animations: Plimpton 322

OK