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Did James Cook find out how long it took Venus to go around the sun?

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No, that was known before. However, in 1766 the The Royal Society of London hired James Cook to measure a certain Venus transit before the Sun. The purpose was to determine the distance from Earth to Venus by the parallax method, and that required measurements from various places on Earth, therefore James Cook was required to measure from a foreign and distant place from London.

The measurements failed because of the Black drop effect, and it failed everywhere. The atmosphere of Venus makes all start and end measurements of Venus transits impossible.

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