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Did they calculate a numerical value for the "extreme and mean ratio" or did they just have ways to construct it geometrically? If so, what value did they use and how did they calculate it?

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    $\begingroup$ They could relate it to $sqrt{5}$. Indeed, Euclid shows in general how to solve a quadratic equation. But these were a "lengths of line segments", not "numbers". $\endgroup$ Jul 11, 2023 at 1:12
  • $\begingroup$ @GeraldEdgar, thank you. Do you know of a resource which indicates this? I know Hippasus is credited with the discovery of irrational numbers and Theodorus of Cyrene proved that $ \sqrt{5} $ was irrational, but I've never found a reference to a numerical value for the golden ratio. $\endgroup$ Jul 11, 2023 at 4:51
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    $\begingroup$ You could be more specific what you mean by "numerical". You probably don't expect them to use a decimal expansion? As far as I remember, Greeks knew how to derive continuous fraction expansions for numbers (or ratios) like these, which is a valid representation of a number. $\endgroup$ Jul 11, 2023 at 19:45
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    $\begingroup$ If you go to google books or jstor.org and search for golden ratio, a ton of resources will pop up. Some of them must have the answer to your question. $\endgroup$
    – DJohnson
    Jul 12, 2023 at 7:03
  • $\begingroup$ @MichałMiśkiewicz, yes I mean a fraction $\endgroup$ Jul 12, 2023 at 21:01

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First of all, Greeks were not fascinated with Golden ratio as we are. Modern golden-ratio hype started about from the time of Leonardo da Vinci. Second, Greek mathematicians were not very interested in numeric values in general. They believed that calculations were a low job of merchants. They were more interested in geometric construction. They knew how to construct "extreme and mean ratio" with a ruler and a compass, and that was enough for them. One exception was Archimedes. He did not consider himself above dirty calculations. For example, he estimated $\pi$ to be between $3 \frac{10}{71}$ and $3\frac{1}{7}$. In the process he solved quadratic equations and calculated square roots. Of course he could use the same method to calculate $\phi$. If he did not do that, then it's not because it was too hard, but because it was too easy: everyone could do that.

Now, this is not true for Greek astronomers. They had to do calculations. In particular, they needed trigonometric tables. How did they calculate trigonometric tables? They knew chords of simple school angels like $60^\circ$, $45^\circ$. They also knew that chord $36^\circ$ is the "extreme and mean ratio" (note, that they used chord instead of sin: chord $x = 2\sin \frac{x}{2}$). From these simple angles they calculated chords of other angles using angle addition and subtraction theorems.

So, to answer you question you need to look at Ptolemy's trigonometric table. And lo and behold you will find that according to Ptolemy chord $36^\circ= 37;04;55$ in sexagesimal. And this is absolutely correct: $\frac{\sqrt 5 - 1}{2} = 0; 37; 04; 55; 20...$. Ptolemy's precision was 2.6e-6. So the answer is yes, they did calculate at least three sexagesimal digits of the golden ratio.

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