The problem of predicting the moment when, after conjunction, the new crescent will become visible is both astronomical and physical. Although realized already by Ptolemy, the actual solution never did go much beyond the well-established Babylonian rule of thumb that the moon cannot be seen earlier than one day after conjunction. This paper first discusses the importance of the sighting to the peoples of Islam and then mentions the criteria which control the phenomenon. This is followed by an outline of the theoretical solution given by the early Arab astronomers. It then proceeds to give a more accurate treatment, according to modern methods, which leads to rules by which the appearance and disappearance of the crescent can be predicted to within five minutes of time. The second part of the paper presents translated extracts on the subject from the oldest sources, using modern astronomical nomenclature. These are taken from the Hindu compendia Surya Siddhanta and Pancha Siddhantika, and from al-Battani's Handbook of Astronomy.