Trigonometry from The 18th Century BC!

Plimpton 322, a mathematical table that is thought to have been made about 1800 BC.

Back in 1922, G.A. Plimpton bought the tablet shown above from an archaeologist named Edgar Banks, and it has become known as “Plimpton 322.” According to an analysis of the writing, it is of Babylonian origin and probably dates back to the 18th-century BC. It has been known for a while that Plimpton 322 is a mathematical table that contains ratios related to triangles. However, there were aspects of the table that didn’t make sense, at least until recently. According to a study published in Historia Mathematica, it is actually the world’s oldest trigonometry table!

For those of you who didn’t take (or don’t remember) trignonometry, it is a branch of mathematics that deals with triangles. I was first introduced to it in high school, as part of my “college preparatory” mathematics education. One thing that initially struck me about this branch of mathematics was the fact that there were times you had to use a lookup table (or a calculator) in order to get the results you needed. I had never before done math like that. Sure, calculators made some math faster and certainly cut down on errors. However, for some trigonometry problems, you simply couldn’t get the answer without looking up numbers in a table or using a calculator.

Once I studied chemistry and physics at university, trigonometry became a pretty constant companion. In physics, you use it to analyze vectors, which are one of the most fundamental aspects of that scientific discipline. In chemistry, you use it to study molecular structure. Over time, I got really adept at using my calculator to solve trigonometry-related problems. Interestingly enough, however, this tablet represents a completely different means by which you can do trigonometry.

In the trigonometry we use today, angles are important. However, in this Babylonian trigonometry, angles aren’t used. Instead, only the length of a triangle’s sides are used. Indeed, in figuring out what this tablet means, the authors actually state:

First we abandon the notion of angle, and instead describe a right triangle in terms of the short side, long side and diagonal of a rectangle.

I am sure it is easy for mathematicians to do such a thing, but it is hard for me to wrap my head around the idea that you can do trigonometry without angles. However, they show that when you look at the tablet that way, it is clearly a lookup table that gives you everything you need to make specific trigonometry calculations. Why would the Babylonians be using such a table? The authors refuse to speculate, but most likely, the Babylonians were using trigonometry to survey fields and build structures like temples and palaces.

There are two other really interesting things about this table. First, it predates what historians thought was the earliest form of trigonometry, which was developed by the Greek mathematician Hipparchus in the second century BC. That means the Babylonians, not the Greeks, can be credited as the first to investigate this mathematical discipline.

Second, this roughly 3,700-year-old tablet represents, in some ways, an improvement on the trigonometry that we do. That’s because our trigonometry, which is based on lookup tables and calculators, is really an approximation. For example, if the answer is one-third, a lookup table or a calculator will usually have something like 0.3333. While that’s an excellent approximation of one-third, it is not exact. Only the fraction 1/3 is an exact representation of one-third. Well, it turns out that this table has only exact representations. While there is no practical difference between a really good approximation and an exact value, from a mathematical point of view, an exact value is more elegant. So if nothing else, this table is more mathematically elegant than our modern tables and our modern calculators.

If you’ve ever had the misguided notion that ancient people were “ignorant savages,” finds like this one should make you change your mind!

12 Comments

  1. John says:

    Do you think architecture was the only application for ancient trig?

    1. Jay Wile says:

      Well, it could have been used in surveying. It’s very useful, so there probably are applications I am not thinking of.

      1. John D. says:

        I’m just wondering what drives an ancient race to develop the math. I suppose if you add up architecture, surveying, and astronomy you get there soon enough.

        1. Jay Wile says:

          I don’t know much about the Babylonians, but the standard Greek thought (based on Plato and his followers) was that the realm of numbers and logic was “perfect,” while the material world was “corrupt.” As a result, the Greeks investigated mathematics to learn more about this “perfect realm.” I am not sure if a similar view was held by Babylonians or not, but it’s at least possible that they were simply investigating the properties of triangles and didn’t have anything practical in mind.

  2. Alaska Nivanuatu says:

    And if the Babylonians were using trigonomentry 37 hundrend years ago, who knows how many other ancient peoples used it as well! The Israelites could have used it while rebuilding Jerusalem during the return to Zion!!!

    1. Alaska Nivanuatu says:

      Actually, if this tablet is dated from the 18th B.C., it’s very possible that the Israelites might have also had knowledge of Trig well before the exile to Babylon. It would seem almost certain, however, that the Israelites could have brought this knowledge back to Israel from Babylon after the exile, since they had lived there for 70 years, and since certain Israelite men had been trained in Babylonian language and literature in order to serve King Nebuchadnezzer (Daniel 1:3-5)

  3. Bill Murphy says:

    Bet the Babylonians didn’t know the meaning of 42. 🙂

    1. Jay Wile says:

      Hehe. Perhaps they understood the question. After all, you can’t understand both the question AND the answer…

  4. John D. says:

    That’s pretty interesting. It brings to mind a conversation I had with someone a while back about whether he liked the ideas being put forward about granular space. I remember him saying something to the effect of “Yes, space is granular – it’s probably a crystal lattice. Plato was almost right”

  5. martin says:

    Good article, and no mathematician! Learning is fun, about a place many of us want to know more of. Abraham, I was taught, was from Ur of Chaldea. They were famous for astronomy and math. It’s still out but many believe Abraham introduced Middle East math to Egypt. If so, then when they began to build mathematically and astrologically correct edifices shows when he was there. Peace!

    1. Alaska Nivanuatu says:

      That’s very interesting! Do you have some resources/links you could post?