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Ancient Greek astronomer Aristarchos who lived around 300 BC found a method to compute the Eart-Sun distance by applying geometry.

In this video about the cosmic distance ladder, at 26:30, Terence Tao explains that Aristarchos was basically limited by nonexisting clock technology. So he was not able to accurately measure the time difference between the time of (1/2 (new moon - fullmoon)) and time at which sun-moon distance was exactly at 90 degrees to the intersection of earthorbit - moonorbit. See screenshots.

Aristarchos 1

Aristachos 2

Why was his method never refined in more modern times? Someone could have independently discovered the computation by a similar way of reasoning, but with better clocks.

Or did some European Astronomer else do this after mechanical clocks were invented, and it was just not accurate enough so they went for the transit-of-venus-method?

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The video is very long and you might want to skip to 6:30 where the information presention starts. –  Kent Byerley Feb 21 '12 at 21:09
you can see from the screenshots that the information is presented at 26:30 –  knb Feb 21 '12 at 21:18

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