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> And science historian Jöran Friberg, retired from the Chalmers University of Technology in Sweden, blasts the idea. The Babylonians “knew NOTHING about ratios

by rosser 9y ago

> And science historian Jöran Friberg, retired from the Chalmers University of Technology in Sweden, blasts the idea. The Babylonians “knew NOTHING about ratios of sides!” he wrote in an email to Science. “When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.” — Arthur C. Clarke

kragen 9y ago
Please be aware that I haven't read the paper (http://ac.els-cdn.com/S0315086017300691/1-s2.0-S0315086017300691-main.pdf?_tid=efc2933e-89db-11e7-92c1-00000aacb360&acdnat=1503696517_42abce1ada90093650a4c0374a39a2f7 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.html 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.
- jacobolus 9y ago
  > Friberg would be precisely correct — the tablet contains no ratios of side lengths, only ratios of "quadrances" Haha. Someone should email him to say so. :-P
27182818284 9y ago
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.

by rosser 9y ago

kragen 9y ago

Please be aware that I haven't read the paper (http://ac.els-cdn.com/S0315086017300691/1-s2.0-S0315086017300691-main.pdf?_tid=efc2933e-89db-11e7-92c1-00000aacb360&acdnat=1503696517_42abce1ada90093650a4c0374a39a2f7 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.html 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.

jacobolus 9y ago
> Friberg would be precisely correct — the tablet contains no ratios of side lengths, only ratios of "quadrances" Haha. Someone should email him to say so. :-P

27182818284 9y ago

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.