It is one of those odd kinds of years that got me wondering wondering about the new moon. The molad [conjunction of moon and sun] was yesterday. Today is when a court could have seen it. But tomorrow is when everyone is doing rosh hashanah.
To know the exact time of the molad is not hard. Hipparchus brings it in his book that he got it from the ancient Babylonians. But that is not the same thing as getting the lunar and solar time to correspond at least to some degree.
The first time the calendar that is used was recorded was by Meton in Athens. That was considerably before the first mention of a Hebrew Calendar which was by Saadia Gaon. The Gemara never mentions any calendar at all. and it just says the sages knew when the molad was קביעא דירחא. But that has nothing to do with the calendar.
Nor is there any mention of Hillel II setting up any kind of calendar. And during the time of the geonim we have letters bringing dates that do not correspond to the calendar.
What clearly happened was after the troubles that occurred in Babylon when the yeshivas were closed that there was no central authority and people needed to know when to celebrate the festivals. The calendar that was already widely used to made a reasonable correspondence between the solar and lunar years was Meton's calendar.
So the best idea is to depend on Tosphot in Sanhedrin [10b] that says that the day of the Molad is Rosh Hodesh
Saadia Gaon says the calendar is from the "shamua" but writing in Arabic uses the word that he uses elsewhere to refer to the general body of tradition. Not Halacha to Moses from Sinai. Later people misunderstood Rav Saadia and thought he meant Halacha to Moshe from Sinai. But that not at all what Saadai wrote.
