Sod Ha'ibur
— Origin of the Hebrew Calendar
— Conflict between Rabban Gamliel and Rabbi Yehoshua

תלמוד בבלי מסכת ראש השנה דף כד עמוד ב  משנה.  שבאו שנים, ואמרו: ראינוהו שחרית במזרח וערבית במערב. אמר רבי יוחנן בן נורי: עדי שקר הם. כשבאו ליבנה קיבלן רבן גמליאל. Two [witnesses] came to the court and said that they had seen [the old moon] in the morning in the east and [the new moon] in the west in the evening. R' Yochanan ben Nuri says they are false witnesses. They [the witnesses] went to Yavneh and Rabban Gamliel accepted then [as valid witnesses].

And everyone get's into a huff because it is common knowledge that you can not see the old moon in the morning and the new moon that evening. It does not make sense the Rabban Gamliel who is expert on all things of Kiddush HaChodesh should fail here. He doesn't! When R' Yochanan said that the witnesses were lying that was true. Rabban Gamliel knew they were lying as well. In the gemora Rosh Hashanah 20a it was noted:

הוו יודעין שכל ימיו של רבי יוחנן היה מלמדנו: מאיימין על העדים על החדש שלא נראה בזמנו לקדשו, אף על פי שלא ראוהו - יאמרו ראינו! Know that  R' Yochanan always taught we intimidate witnesses over a moon that was not seen in its time to sanctify it. Even though they didn't see it they should say we saw it!

If the start of Rosh HaShanah will be maneuvered for the crops or for the burial of the dead would it not be maneuvered so that the Jews in Bavel will be spared from a two day fast?

תלמוד ירושלמי מסכת ראש השנה פרק א דף נז טור ב /ה"ד חשין לצומא רבה תרין יומין אמר לון רב חסדא למה אתם מכניסין עצמכם למספק הזה חזקה שאין ב"ד מתעצלין בו.  In [Bavel] they are troubled by [the date] of the great fast [of Yom Kippur] of two days. Rav Chisda said to them why do you enter into this great doubt? The Beit Din [Calendar Court in Jerusalem] will fulfill their responsibilities [and declare Elul to have 29 days].

So what was the problem?

 תלמוד בבלי מסכת ראש השנה דף כ עמוד ב 

האי - מיחזי כשקרא. 

The problem is the appearance of falsehood. The Calculated Hebrew Calendar was not created by Hillel in 358 CE. It was created by the Rabbis many centuries before but was to be used in secret. 

On the morning of the 30th day people who enjoyed going to a large free feast would come and say they saw the new moon; even though they didn't. This time however the old moon was very visible in the morning sky of the 29th of Elul and these witnesses saw it and mentioned it. Likely R' Yehoshua saw it or knew of others who had. This is what I referred to as a three day discrepancy. 

From my chart of 2000 Years of Data we can see that it happened at 120 CE, a date consistent with Rabban Gamliel's retirement.

When Rabban Gamliel explained:

תלמוד בבלי מסכת ראש השנה דף כה עמוד א  כך מקובלני מבית אבי אבא: אין חדושה של לבנה פחותה מעשרים ותשעה יום ומחצה ושני שלישי שעה ושבעים ושלשה חלקים.  I have received a tradition from the house of my father’s father: The rebirth of the moon is not less than 29 and a half days and 793 parts.

What Rabban Gamliel was really saying that I have been doing like those before me and using a calculation based on an average lunar cycle of 29 and a half days and 793 parts to choose my witnesses.