In our times we have accurate calenders and we can easily say on which day a new month starts and when we celebrate the holidays. But in the times of the temple, people didn’t know when the new month starts. People who saw the the moon renewed, came to the temple to testify in front of a court, and only then a new month could be announced.
The declaration of a new month ראש חודש Rosh Chodesh had major significance because according to this announcement, it was determined when the holidays of the month were to be celebrated and people could gather together.
In the Mishna and in the Gemara we read stories of how the judges used to inquire the witnesses regarding the location of the appearing new moon and its accurate shape.
Here is a part of this description:
Rabban Gamliel had shapes of the moon on a board on a wall in his attic,
that with them he used to show to the simpleton saying: Did you see this ?
or did you see that ?
(Please note that those moon shapes apply only to the north part of the globe, in the south part of the globe the shapes work opposite, meaning: left curve of the moon is the beginning of the new month, and right curve of the moon is the end of the month.)
source: Talmud > Mishna > Mo’ed order > tractet Rosh Hashana, chapter b, paragraph 8
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