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A reexamination of a Babylonian tablet has found what may be the first appearance of trigonometry:
Consisting of four columns and 15 rows of numbers inscribed in cuneiform, the famous P322 tablet was discovered in the early 1900s in what is now southern Iraq by archaeologist, antiquities dealer, and diplomat Edgar Banks, the inspiration for the fictional character Indiana Jones.
Now stored at Columbia University, the tablet first garnered attention in the 1940s, when historians recognized that its cuneiform inscriptions contain a series of numbers echoing the Pythagorean theorem, which explains the relationship of the lengths of the sides of a right triangle. (The theorem: The square of the hypotenuse equals the sum of the square of the other two sides.) But why ancient scribes generated and sorted these numbers in the first place has been debated for decades.
Mathematician Daniel Mansfield of the University of New South Wales (UNSW) in Sydney was developing a course for high school math teachers in Australia when he came across an image of P322. Intrigued, he teamed up with UNSW mathematician Norman Wildberger to study it. "It took me 2 years of looking at this [tablet] and saying 'I'm sure it's trig, I'm sure it's trig, but how?'" Mansfield says. The familiar sines, cosines, and angles used by Greek astronomers and modern-day high schoolers were completely missing. Instead, each entry includes information on two sides of a right triangle: the ratio of the short side to the long side and the ratio of the short side to the diagonal, or hypotenuse.
Mansfield realized that the information he needed was in missing pieces of P322 that had been reconstructed by other researchers. "Those two ratios from the reconstruction really made P322 into a clean and easy-to-use trigonometric table," he says. He and Wildberger concluded that the Babylonians expressed trigonometry in terms of exact ratios of the lengths of the sides of right triangles [open, DOI: 10.1016/j.hm.2017.08.001] [DX], rather than by angles, using their base 60 form of mathematics, they report today in Historia Mathematica. "This is a whole different way of looking at trigonometry," Mansfield says. "We prefer sines and cosines ... but we have to really get outside our own culture to see from their perspective to be able to understand it."
Also at the University of New South Wales.
(Score: 0) by Anonymous Coward on Saturday August 26 2017, @10:24AM (3 children)
Tablet? Does it have Android 8.0 Oreo?
(Score: 0) by Anonymous Coward on Saturday August 26 2017, @10:55AM (1 child)
Look carefully. It has rounded corners, that means it's an *Apple* product.
Anyway...
"the ratio of the short side to the long side" -> min(tan,cot), or just tan if angle < 45deg.
"the ratio of the short side to the diagonal, or hypotenuse" -> min(sin,cos), or just sin if angle < 45deg.
So it's a table that lists pairs (sin, tan) for angles < 45 degrees (though it doesn't give angles in the table).
If it's really Babylonian trig, it's amazing enough without bullshit "different way of thinking" claims...
(Score: 2, Interesting) by Anonymous Coward on Saturday August 26 2017, @05:22PM
But that's the big thing, and it is a different way of thinking.
It's well-known that you only need to calculate one trigonometric function the hard way (approximating an infinite series, for example), and that only out to π/4; you can write more-or-less tidy closed-form solutions for the rest of that function, and any others you might need, in terms of that data.
Lengthy example, for those unfamiliar with the idea:
Apparently, the Babylonians did the same thing in reducing it to one two-column table; sure, they used ratios instead of decimals, and their columns were sin and tan, not θ and sin(θ), but it's the same amount of information, and I get why it looks the same.
What's not the same isn't the ratios themselves, or the fact that only two of them show up (I'm sure they knew equivalents to all six trigonometric functions -- a triangle has three sides, so there are exactly six ratios to be formed), but the fact that rather than expressing the ratios as functions of angle -- sin(θ), cos(θ), tan(θ) with θ in radians -- they seem to have dealt only with ratios -- x, cos(x), tan(x), where x is what we would call sin(θ) and θ is implicit. for most applications of trig (e.g. surveying), that should work just as well (I do think it would cause problems in calculus), but it really is a very different way of thinking about it.
(Score: 0) by Anonymous Coward on Saturday August 26 2017, @11:13AM
Anonymous Nimrod!
(Score: 0) by Anonymous Coward on Saturday August 26 2017, @12:53PM (1 child)
Doesn't no one know where this tablet came from, so the dating is based on the style of cuneiform alone... meaning it could be easily a fake?
(Score: 0) by Anonymous Coward on Sunday August 27 2017, @05:51AM
(Score: 1, Interesting) by Anonymous Coward on Saturday August 26 2017, @04:56PM
It's easy to get all excited and wonder if these ancients had trig, other fancy math, even mathematicians. But my guess is that a thoughtful builder (perhaps a privileged temple builder) worked out how to keep track of angles for roof pitch, stairs and the like. To save re-deriving it every time there was a need, a scribe was engaged to write it down for reference. Possibly similar to the rafter tables that are stamped on framing squares, a very handy reference once you understand it--but "greek" if you don't have the instructions: https://inspectapedia.com/roof/Framing_Square_Table_Use.php [inspectapedia.com]
Don't overthink these things... I seem to remember that many of these clay tablets are simple accounting, who bought and how much they paid, really mundane stuff, like an old check book register.
(Score: 2) by hendrikboom on Saturday August 26 2017, @05:07PM (3 children)
It's at most pretrigonometry. Just an ordered table of Pythoagorean triples. What distinguishes this from true trig is the ability to add angles to each other.
(Score: 2) by jimtheowl on Sunday August 27 2017, @05:06AM (2 children)
Babylonians knowledge of trigonometry is long established. Their sexagesimal numeric system is believed to be the reason that we still use 360 degrees to divide a circle.
(Score: 0) by Anonymous Coward on Sunday August 27 2017, @01:27PM (1 child)
I thought we used 360 degrees because that is roughly the same as the days in a year. The stars move about one degree per day.
(Score: 2) by PartTimeZombie on Sunday August 27 2017, @10:58PM
The only reason I use 360 degrees is because Mr. Newell told me I had to in 6th form maths.