Rosh HaShanah 25a:
רבי חייא חזייא לסיהרא דהוה קאי בצפרא דעשרים ותשעה, שקל קלא פתק ביה, אמר: לאורתא בעינן לקדושי בך, ואת קיימת הכא? זיל איכסי!
Professor Isaiah M. Gafni of the Hebrew University of Jerusalem in his book Land, Center and Diaspora refers to "a particular sage scolding the (outgoing) moon (of the previous month) for having the temerity to continue its appearance an extra day, contrary to rabbinic calculations. ...the moon is forced to acquiesce to the authority of the rabbinic court, and forthwith disappears."
Professor Gafni has painted R. Hiya as a buffoon chasing the moon from the sky with a clod of dirt.
According to Rashi the events of R. Hiyah took place at the time of Rosh HaShanah.
לאורתא בעינן לקדושיך - צריכין אנו לעשות הלילה יום טוב של ראש השנה.
The implication of the R' Hiyah story is that the Beit Din is establishing the New Month on some basis other than witnesses, i.e. a calculation. Rashi disputes this idea and is instead saying that this is a case where the Beit Din is maneuvering the start of Rosh HaShanah to prevent Yom Kippur from occurring on the day before Shabbat. According to Rashi The Beit Din did not want Tishrei to begin on Wednesday because that would cause Yom Kippur to be on Friday, i.e. vegetable and burial issues. In this scenario the 29th day of Elul is Monday and the Beit Din was scheduled to sanctify the month that night, but the old moon was visible. Santifying the next night after making Elul full would mean that Yom Kippur would be on Friday a circumstance which the Rabbis sought to avoid.
All this makes sense until one checks the lunar facts. We need a comparable date when the conjunction is midpoint between approximately 5:30 AM for the sighting of the "old moon" and 6:30 PM one day later for the sighting of the new moon. Such a moon is only 18 and a half hours old and only marginally visible which makes this scenario already suspect because the underlying rule of the lunar calendar system is that one must always seen the new moon if 30 days have passed since seeing the previous one. We can use the actual lunar conjunction of September 10, 1999, at 0:02 a.m to further illustrate these facts. At sunset September 10, the moon will be at at 0.72% visibility. According to the van Gent chart it would be visible with a small telescope under perfect conditions. The old moon at moonrise September 9 4:35 AM would be at .8% visibility and soon be completely lost in the 5:19 sunrise. None of this is likely to been seen by anyone, neither the old moon or the new moon. In September, in Jerusalem, it will be impossible to see the old moon in the morning and the new moon at night but also nearly impossible to see the old moon in the morning and the new moon the night following.
From the Yerushalmi this issue becomes clearer:
רבי חייה רבה הילך לאורו של ישן ארבעת מיל רבי אבון משדי עלוי צררין ואמר לה לא תבהית בני מריך ברמשא אנן בעיין תיתחמי מיכא ואת מיתחמי מיכא מיד איתבלע.
R' Hiyah the Great walked by the light of the old moon 4 miles. Rabbi Avon threw stones at (the old moon). And said to it: “Do not upset the children of you Master (i.e., the Jews), tonight we have to see you from this side (i.e., the new moon), and you are seen from here (the old moon is still visible)”. Immediately it disappeared.
This moon is clearly the moon I discussed in Molad Zaqen. It is the moon of a three day discrepancy. R' Hiyya lived between 180 CE and 230 CE. He was born in Bavel where it was thought he came under the influence of Shmuel but moved to Palestine. According to the chart there is a possible date for him to experience a three day descrepancy, The story becomes more than a story as we know it happened on the morning of Monday, September 18, 198 CE.
The Calendrical Court in 198 CE was using the Hebrew Calculated Calendar.