This calendar is the Athenian calendar, updated to be a calendar for the modern world. It follows the spirit of the ideal calendar, being lunisolar and based on the visibility of the crescent. But it compromises by using a calculation and by basing it on visibility not at Athens itself, but with adjustment so that the visibility somewhere in the populated world is the criterion.
In Athens they weren't very concerned with calendar accuracy, and setting the calendar was a political act. So it is unlikely that any date in this calendar corresponds to a historical date in Athens.
Basic principles of the calculation:
Any international calendar faces the problem of the date line. We use an international date line which corresponds to the current international Gregorian date line, in the Pacific Ocean, since this offers a geographically convenient dividing line.
The day and month begins based on local sunset on the same international day as it begins in Athens. Thus the day can always be said to begin at sunset on a given Gregorian date. For example, Hekatombaion of the 4th year of the 694th olympiad begins at sunset on July 2, 2000.
Rather than attempt to estimate the visibility of the new moon at Athens, we adopt an approximation useful through most of the world. This approximation uses aproach used for the lunar crescent visibility prediction algorithm developed at the South African Astronomical Observatory, http://www.saao.ac.za/sky/vishome.html which is based on the altitude of the moon distance from the sun along the horizon at the sunset at a location on the earth. Their method estimates when the sighting is impossible, and when it is improbable.
The SAAO criterion for visibility is that sighting the crescent will be "IMPROBABLE" below an ALTITUDE of about 8 degrees if the moon is directly above the sun at sunset. For the international calendar, we use a more relaxed criterion and we use a location 1/3 of the world to the west of Athens for the calculation (that is, a time about 8 hours after sunset in Athens, at a location in the center of North America). If the moon is more than 10 degrees from the sun at that time, it should be visible somewhere in the Americas, at least, if not for more of the world. If the separation is less than 10 degrees, it may be visible in parts of the west, but with difficulty.
The intent of using this criterion is to avoid having the crescent moon noticably visible at sunset on the last day of the month.
Sometimes the crescent will be visible in Athens on that date, sometimes not.
For people concerned with local visibility, the time of the astronomical new moon is given in the calendar, and for people concerned with visibility in Athens, the "Noumenia kata Selene", new moon according to the moon, are given as well. If the moon is older than 15 hours at a location, it may be visible, though this is unlikely. So if you wish to maintain a strict visibility criterion for your location, you may want to calculate visibility yourself when the moon is more than 24+15 hours old at sunset in your location at the start of that month. See http://www.moonsighting.com/ for more information.
The inputs for the calculation are:
The data for the printed calendar, including additional data is:
Solstices and equinoxes are calculated from algorithms in Astronomical Algorithms by Jean Meeus. Astronomical cross quarters are 1/2 way between solstices and eqinoxes. New, full, and quarters also calculated from this source.
The program Xephem, by Elwood Downey, was used to produce tables of: Julian date, local date, sun azimuth, moon azimuth and altitude, moon-sun angular distance for local sunset at Athens and at latitude 37:58/longitude 96:44 (120 degrees west of Athens).
Events which take place at the same instant throughout the world, e.g. a solstice, are placed in the calendar on the day it takes place on in Athens.