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I know that Greenwich mean time was established in 1675, as an aid to navigation. Portuguese and Spanish navigators in the 15th century also appeared to be aware of the usefulness of timekeeping for longitudinal navigation, though they lacked sufficiently accurate clocks to implement the technique.

My question is this: when was the fundamental principle discovered? When did travelers first notice that, if you go to the west, the Sun rises and sets later?

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Well, the potential for using time to measure longitude was certainly understood by Hipparchus in the second century BC.

He proposed that longitudes of distant places could be calculated by measuring the local solar time of lunar eclipses, which are visible over half the Earth's surface. However, the available means of timekeeping weren't sufficiently accurate for this to be useful.

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Essentially when they understood that the Earth is round, and the Sun rotates about it with a period one day. Ancient Greeks credited this discovery to Pythagoras. Modern scientists consider Pythagoras a somewhat legendary figure, so the name of the first person who said this is unknown.

Anyway this was a common knowledge in the (educated) Hellenistic circles long before Hipparchus. The first attempt to measure the size of the Earth is associated with Eratosthenes (3d century b.c.), and at this time it was certainly clear (to educated people) that the local time depends on the longitude.

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    Are you sure the Sun rotates about the Earth & it's not due to the Earth spinning with a period of one day? – Fred Mar 22 '17 at 8:04
  • I think you mean Eratosthenes instead of Erathosphenes? – bof Mar 22 '17 at 10:44
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    @Fred: For me, this is a choice of coordinate system. The ancient preferred the coordinate system where Earth does not move. – Alex Mar 22 '17 at 12:50
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When the eight remaining crewmembers of Magellan's circumnavigation of the earth reached the Cape Verde Islands in July of 1522, they were puzzled by their apparent loss of one day with respect to the actual date on the islands.

https://www.staff.science.uu.nl/~gent0113/idl/idl_discovery.htm

So, the knowledge of change in sunrise time with longitude would appear to be after this date.

We can of course ignore Phileas Fogg's similar confusion in the late 1800s as Verne's dramatic license...

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    Knowing time of day changes doesn't necessarily lead to understanding that a loop changes a whole day. and 1522 is after the 15th century. – user22111 Mar 21 '17 at 18:28
  • I agree with @notstoreboughtdirt -- how would they know they were a day behind what they thought it should be? – Jeff Mar 21 '17 at 18:40
  • They had kept a careful log, and had counted the seven days of each week to properly observe Sunday religious activities. When they reached shore, the current date in their log differed from the shore date, and their respective day of the week did not agree... – DJohnM Mar 21 '17 at 18:49
  • @DJohnM: While that makes sense, I wonder how consistent "shore dates" were around the world or how the shore date got be reckoned as a day different from from I guess Portugal's. – Jeff Mar 21 '17 at 18:52
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    Sailors did not anticipate the whole earth effect, but that is poor evidence that nobody knew of the effect on smaller scales. @Jeff Those are good but separate questions. The sailors and the islands were right to have dates off by one, the solution was the international dateline which hadn't been invented yet. – user22111 Mar 21 '17 at 18:58
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Its likely pretty much everyone who understood that the Earth was a sphere also understood this. All you'd need to see it is a sphere of your own to act as a model.

Eratosthenes in the third century BC not only understood this, but used it to calculate the circumference of the earth.

Basicly, he knew what day the sun was directly overhead at noon where he lived (Alexadria). So he had someone in another city a known distance away calculate the angle of the sun at noon on that same day, did a bit of trigonometry, and came up with an actual number (that wasn't off by a whole lot) for the circumference of the Earth.

enter image description here

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    Eratosthene use an angle distance along a meridian. The question was about time difference in the orthogonal East-West direction. So, even if he understood it, he did not use this property to calculate the circumference of the earth. – Frédéric Grosshans Mar 22 '17 at 17:58

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