An ancient Babylonian tablet may contain the first evidence of trigonometry

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".

Thanks for that, it was a good quick introduction to what's going on on the tablet.

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.

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.

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.

... Except we use it all the time.

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.

> 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.

As well as:

https://news.ycombinator.com/item?id=15098189

https://news.ycombinator.com/item?id=15097301

https://news.ycombinator.com/item?id=15094190