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It is currently accepted that a year is equal to the time it takes for the earth to revolve around the sun. However around Roman times, Ptolemy's geocentric model was the widely accepted view of nature. Under the geocentric model the sun and all the stars moved around the earth once per day while the earth stayed fixed. But around the same time they used the Julian calendar which considered a year as 365.25 days long, a highly accurate number for the length of a year.

So my question is how exactly did they arrive at this figure? What exactly did a year mean to them if it wasn't the sun revolving around the earth? If the earth was allowed to rotate while the sun revolved around the earth, then I can see how they could calculate that. But it seems like the earth rotating was not an acceptable proposition at that time.

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First of all, whether the earth is rotating or everything else is rotating around it, is completely irrelevant for the question. This is just the matter of point of view.

As seen from the earth, the fixed stars rotate with period day+night. On this background, the Sun describes a circle slowly with period "one year". When one looks closely, one sees that there are TWO approximately equal (but not exactly equal) periods of the sun:

a) The time in which the Sun returns to the same position WITH RESPECT TO THE STARS. This is called "sideral year", and

b) The time in which the Sun travels from one spring equinox to another. This is called "tropical year".

The difference is roughly 50 angular seconds per year. The difference was discovered by Hypparchus in the 2-nd century BC.

For practical purposes (agriculture) tropical year is relevant (it is the period of change of seasons). To compute its length in days, one has to observe equinoxes (or solstices). The difference in days from say a spring equinox to the next one is the tropical year. Ancients could not observe and time them with high precision, but there is a way around this.

Suppose that you observe a solstice and another one, not the next one but $m$-th one (so that $m$ years passed from one to another). And you count days and you find that $n$ days passed from the first solstice to the other one. Then the length of the year will be approximately $n/m$ days. If $m$ is very large, you obtain a very good approximation. Larger the $m$, more precise the approximation is.

Systematic observations of sky were made by Babylonians since 8 century BC. At least this is the time from which continuous records survive. So at the time of Hipparchus the data for about 6 centuries were available. (In principle. It is disputed how much did he use or know Babylonian observations. He apparently had in his possessions the Greek records of solstices for a few centuries. Practically nothing survived of Hipparchus writings. We know about them from Ptolemy who wrote 3 centuries later). This is how the very good approximation of the length of the tropical year was found.

Then making a calendar is a technical (but non-trivial) problem. Egyptians had a calendar with exactly 365 days in the year. So the beginning of the year was floating among the seasons. A reform was made in the 1-st century AD (Julian calendar) which decreed the year of 365 1/4 days. Though a more precise value was already known. More than a millenium later the need for a better calendar was recognized, and eventually the modern calendar (called Gregorian) was introduced.

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  • $\begingroup$ This is a good answer. If I may sharpen one point: The Julian calendar is not a "reform" of the Egyptian calendar, but of the Roman republican calendar. The Greeks in the meanwhile had a luni-solar year based on the Metonic cycle, which keeps pace with the tropical year quite nicely. $\endgroup$ – fdb Jul 7 '15 at 20:12
  • $\begingroup$ @fdb: In the Roman empire, astronomy was mostly practiced in Alexandria (now in Egypt). So from the point of view of Roman citizens living in Alexandria (and these are most relevant for our discussion) the Julian reform affected traditional Egyptian calendar. $\endgroup$ – Alexandre Eremenko Oct 17 '17 at 3:42
  • $\begingroup$ In the Almagest Ptolemy uses the old Egyptian 365-day year (i.e., no leap years) and does not even mention the Julian calendar (which by then was three centuries old). In his book on weather prediction (Phaseis) he uses the “Alexandrian” calendar with 12 months of 30 days each, 5 epagomenai in a common year and 6 in a leap year. This can be described as modification of the old Egyptian calendar and coordination of the same with the Julian calendar. The Alexandrian calendar was later adopted also by the Egyptian (Coptic) church. $\endgroup$ – fdb Oct 17 '17 at 9:54
  • $\begingroup$ @fdb: even later astronomers preferred the 365 days Egyptian year because of its convenience for astronomy. $\endgroup$ – Alexandre Eremenko Oct 17 '17 at 13:25
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The ancients distinguished two separate motions of the sun. On the one hand, the sun and all the stars seem to rotate around the Earth from East to West once every day. This was explained by the daily rotation of the sphere of the fixed stars. At the same time, the sun, moon and planets move in the opposite direction, along the ecliptic, around the Earth, each at its own pace. In the case of the sun, it takes one year to return to the same position with regards to the ecliptic. By taking an average of the apparent time required for this rotation Hipparchus (2nd century BC) calculated the tropical year as 365.24667 days (though he expressed this figure not with decimal fraction, but with a sexagesimal fraction). This is more accurate than the figure later adopted by the Julian calendar.

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