When calculating moon phases and prayer times found acceptance among the scholars?

After posting the question https://islam.meta.stackexchange.com/questions/1767/are-questions-on-the-historical-inauguration-of-some-utilities-or-methods-on on meta I finally decided to ask ask it here: I'm pretty sure that calendars or methods to calculate the moon phases have been developed in the golden ages of islamic caliphates, as there have been lots of scientists such al-Biruni (even Imam Malik has been said to have written a book about that) and many others. But what I'd like to know is: When and where scholars or clerics introduced or accepted them as a utility for declaring or calculating important dates like 1st day of Ramadan? Any details on this historical inauguration would be fine! I think of information like: how was the general acceptance at the time, who was the cleric, who was the person or scientists whom's caclulation methods or calendar was accepted ...