Levi went to Bavel and commented that what they thought was the 11th of Tishrei was actually Yom Kipur as declared in Eretz Yisrael.
He refused to testify to that as he hadn't heard Beis Din sanctify Rosh Chodesh.
R. Yochanan announced that any place which is reached by the messengers of Nisan but not by those of Tishrei must also keep two days of Pesach; he rebuked his students who did otherwise.
Rava would observe two days of Yom Kipur in case Elul was a full month, and he was once found justified.
R. Nachman was once told at the end of Yom Kipur that the following day was being observed as Yom Kipur in Eretz Yisrael.
He blamed the informer for being the death of him.
CALCULATING NISAN AND PESACH
If the winter season extends to the 16th of Nisan, the year should be made into a leap year.
The reason is that the Pasuk instructs us to ensure that Nisan falls in spring.
R. Nachman instructed sailors to burn the chametz when the moon shines until sunrise.
Question: Surely that happens on the 15th of Nisan, and the chametz must be burned on the 14th?
Answer: On the open ocean, it is seen on the 14th.
MISHNAH: TRANSGRESSING SHABBOS FOR TESTIMONY
Witnesses of the new moon can transgress Shabbos to inform Beis Din only for the months of Nisan and Tishrei.
Messengers would be sent to Suria and the dates of the festivals were contingent on these.
When the Beis ha'Mikdash was standing, they could transgress Shabbos for all months, so that the Rosh Chodesh Korbanos could be correctly brought.
TRANSGRESSING SHABBOS FOR MESSAGE DELIVERY
Question: Would messengers be sent to Suria only for these two months?
In an earlier Mishnah, it stated that they are sent for six months!
Answer: They are sent for six months, but they may only transgress Shabbos for two.
This is because on these two months they may only leave after hearing Beis Din formally sanctify the month, which shortens the available journey time.
A Pasuk teaches us that the festivals must be declared on time, even at the cost of transgressing Shabbos (by witnesses).
One might think that we may also transgress Shabbos to ensure that they are observed on time (i.e. to send messengers).
But the Pasuk states that it is only for their declaration that Shabbos is transgressed.
Originally, witnesses would transgress Shabbos for all festivals; after the destruction of the Beis ha'Mikdash, R. Yochanan said that since there are no Korbanos, Shabbos may only be transgressed for Nisan and Tishrei.
MISHNAH: CLEAR MOONS
Whether or not the moon was clearly visible to all, the witnesses may transgress Shabbos.
(R. Yosi): If it was visible to all, they may not transgress Shabbos.
Forty pairs of witnesses once came but were detained by R. Akiva at Lod.
R. Gamliel warned that this would discourage them from coming in the future.
THE GATES OF UNDERSTANDING
A Pasuk is brought to show that the Mishnah is referring to the moon being clearly visible.
The end of the Pasuk speaks of seven times seven.
This refers to the 49 of 50 gates of understanding that were given to Moshe.
Koheles wanted that also, but was told that nobody could be like Moshe.
(Alternate opinion): There could only be no prophet like Moshe, but there could be a king like Moshe.
Koheles' request was to pass judgments without eyewitnesses and warnings, but was instructed otherwise.