Predicting the first visibility of the young lunar crescent from a given location is an astronomical problem which has challenged astronomers
and mathematicians for more than four millennia. Already in the second half of the first millennium BCE Babylonian priest-astronomers developed
sophisticated numerical algorithms for predicting the motion of the moon and the times of its synodic phases and the first visibility of the
lunar crescent above the western horizon just after sunset. More accurate methods were devised in later ages by Hindu, Islamic and Jewish scholars
and the refinement of these methods has continued up until our own time.

In the past the dates of first lunar visibility were crucial parameters in regulating the religious and administrative calendars of many
cultures. In the present time several communities still depend on lunar calendars but these are computed according to well-defined astronomical
and mathematical rules and no longer depend on an actual first sighting of the lunar crescent. Examples of such calendars are the Chinese calendar,
the Jewish calendar, the Christian Easter reckoning and some variants of the Islamic calendar.

For many Muslims, however, the first sighting of the lunar crescent at the begin of each month is still a matter of importance, debate and
dispute. Especially around the begin and the end of Ramaḍān (the Islamic month of fasting and abstinence) and the begin of Dhu
ʾl-Ḥijja (the Islamic month of pilgrimage), the first sighting of the lunar crescent is eagerly awaited by Muslims from the four
corners of the world.

Please note that
planetarium programs such as Google Sky,
Stellarium or SkyViewCafé cannot be used to determine the first visibility of the lunar crescent.
These programs can be useful for computing where the lunar crescent is but
they cannot predict whether it will be visible or not.