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Looking for clarification on how ancient timekeepers set the day of Passover. I understand that it would begin on the 14th of first month of new year. I’m not clear if the new moon that starts the month had to occur after the equinox. Wikipedia says:

Technically, its New Years Day is the day after the New Moon closest to (within fifteen days before or after) the Vernal Equinox (Spring Equinox, when the day and the night is of equal length, set at March 21 in the Gregorian Calendar). It begins the first month, named Nisanu/Nisan/Abib. The month of Nisan is important since it begins the Spring Feasts of Israel.

What was the official rule in the days of Sanhedrin?

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Following is based on a summary of what I recall from Ramba"m hilchot Birkat Hachodesh. I will, iy"h, add more specific info, later.

Ramba"m mentions that the primary concern is to have Pesach (15 Nissan) occur in the spring (i.e., after the equinox) since the Torah refers to Pesach as Chodesh Ha'aviv. The concern is not regarding the 1st day of Nissan, so, in most cases, that date occurred before the equinox. Sanhedrin would declare a leap year if the 15th of Nissan occurred before the equinox, Since this adds an extra month, by doing this, Rosh Hodesh Nissan now most likely occurs after the equinox, as well.

Sanhedrin could also declare a leap year for other reasons such as the barley has not yet ripened. Barley was needed for the omer offering on the 16th of Nissan. If heavy rains knocked out bridges so that it became too difficult for many people to make the pilgrimage to Jerusalem, Sanhedrin may declare that year a leap year, too even if without it, 15 Nissan would be after the equinox, anyway.

Currently, we are on a fixed calendar. In most non-leap years, Rosh Hodesh Nissan occurs before the equinox. When Pesach is at its earliest (I think the earliest Gregorian date I have seen is March 26) it is barely after the equinox. Of course, if it is not, that year, on the fixed calendar becomes a leap year.

If you look at the Judaic fixed calendar historically, you will notice a "seasonal shift" in that the 1st day of Pesach is slowly getting later on the Gregorian calendar. (There's a separate M.Y. question discussing this, if you're interested.) Thus, currently year 8 in the 19 year Judaic calendar cycle has the latest Gregorian date for Pesach. This year is a leap year, which, currently, could be eliminated as without it, Pesach would still be after the equinox. However, when the fixed calendar was created millennia ago, that year needed to be a leap year. We haven't changed the fixed calendar system at all. However, assuming that we kept it this way eternally, we would need to adjust things as at some point Pesach would occur in the fall season.

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In the Babylonian calendar, which Jews used in the late pre-exilic, exilic, and early post-exilic periods, the month of Nisanu always began after the equinox by our definition, though not always by the Babylonians' definition.

In the Herodian period, according to Josephus, it is the 14th of Nisan that falls on or after the equinox:

In the month of Xanthicus, which with us is called Nisan and begins the year, on the fourteenth day by lunar reckoning, the sun being then in Aries, our lawgiver, seeing that in this month we were delivered from bondage to the Egyptians, ordained that we should year by year offer the same sacrifice which...we offered then on departure from Egypt. Antiquities 3.10.5

The Tosefta gives the rule that the 16th of Nisan was to fall on or after the equinox:

The year is not to be intercalated unless the spring equinox is still distant the greater part of a month. How much is the greater part of a month? Sixteen days. R. Jehuda says: two-thirds of a month, twenty days. Tosefta Sanhedrin 2.7

The fixed Rabbinic calendar has already been covered by DanF in his answer.

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