COUNTING THE DAYS OF THE YEAR
It is possible to observe natural phenomena which repeat each year, whether it be the position of a sun or a star or even the day on which the ice breaks free in a frozen river.
If the observed event occurs with sufficient regularity then it is logical to create a time relative to that regularity and to count the intervening number of days - ultimately called a year.
Having done so, it is easy to suggest that the year could be divided into segments. Two equal segments might be obtained by suggesting that the middle of summer is exactly one-half of the days of the year from the middle of the winter - or vice versa. A further extension might divide those in half to determine the arrival of Spring, and the arrival of the first day of Fall.
Today, based on measurements of the sun, we call the days which divide these parts of the year the Solstices (the longest and shortest days of the year) and Equinoxes (the two days where the sun is directly above the equator causing the length of the day to be equal to the length of the night). The Summer and Winter Solstices occur on June 21 and December 22, respectively, while the Spring (Vernal) and Fall (Autumnal) Equinoxes occur on March 21 and September 23, respectively.
Thus, a determination of the Summer Solstice by observing when the sun makes its highest track through the sky would then allow an extrapolation to the date of the Winter Solstice by merely adding 1/2 of the days of the year.
While the regularity of the event is also subject to the method of observation that is used, the repeated conduct of the activity over many years could allow mathematical averaging of any deviation.
Of course, as we know today, the time for the earth to complete its orbit around the sun is not 365 days, but is approximately 365 and 1/4 days. Hence, early calculations were in error until we learned to make a one day correction every Leap Year. (We now also make a smaller correction every 400 years since it is not off by exactly 1/4 day.)
Further, the length of the first half of the year is not the same as the length of the second half of the year due to the fact that the path of the earth around the sun is an ellipse rather than a circle. December 22 to June 21 is 181 days while June 21 to December 22 is 184 days. (Even more precisely, the solstices do not occur at the same hour of the day, making additional small differences.) See Links To Other Sites for more detailed technical information.
Thus, it is possible to understand why early observers in different cultures, using differing events to monitor, and different observation techniques, may have come up with differing solutions for the exact date of the winter solstice - and the resultant selection of the date for their own winter holiday celebration.