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posted by Fnord666 on Saturday August 26 2017, @09:54AM   Printer-friendly
from the overrated-greeks dept.

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.


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  • (Score: 2, Interesting) by Anonymous Coward on Saturday August 26 2017, @05:22PM

    by Anonymous Coward on Saturday August 26 2017, @05:22PM (#559499)

    (though it doesn't give angles in the table)

    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:

    So if you start with a table of sines from 0 to π/4, you can compute cos from 0 to π/4 as sqrt(1-sin2); you can use complementary-angle relations to cover π/4 to π/2 as sin(θ)=cos(π/2-θ), cos(θ)=sin(π/2-θ), and so on. Once you've got the whole 0 to 2π (or ±π, if you prefer) range sorted, you can get tan as sin/cos, sec as 1/cos, csc as 1/sin, and cot as cos/sin.

    So a table like this is all we really need, and all we'd carry around if each table was a stone/ceramic tablet:
    angle | sine
      0.1 | 0.0998
      0.2 | 0.1987
      0.3 | 0.2955
      0.4 | 0.3894
      0.5 | 0.4794
      0.6 | 0.5646
      0.7 | 0.6442

    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.

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