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See Exodus 12: Passover is supposed to be celebrated on the 14th day of the month, counting from the new moon. Since they went by when the lunar crescent could be actually seen in the sky, it would probably be the 15th or 16th day of the lunar month by astronomical reckoning. That’s also why Easter is always on the Sunday following a full moon.

I have read Exodus 12 to refresh my memory and I found nothing associating the date of Passover being calculated by a crescent moon. There was the word month used but unless I read over it never did find the word lunar or moon.

Now when exactly did Easter become part of Christ? I mean this as a literal Biblical instruction. Christ only observed the Passover and even Paul instructs us to continue to observe Passover in ICorinthians 5:7 Purge out therefore the old leaven, that ye may be a new lump, as ye are unleavened. For even Christ our passover is sacrificed for us:

Therefore let us keep the feast, not with old leaven, neither with the leaven of malice and wickedness; but with the unleavened bread of sincerity and truth.

In Exodus they don't have to say that the month begins with the new moon because everyone knew it--most ancient societies had lunar calendars (inserting an extra month every second or third year to keep the year in line with the solar year so that seasonal feasts would fall at the right time of year). The Muslims still have a lunar calendar, but Muhammed forbade inserting the extra months, so their months come about 11 days earlier every year compared to ours.

See Isaiah 1.14--"Your new moons and your appointed feasts my soul hateth..."--the lunar calendar was taken for granted.