Counting from the new moon, and where Passover lands
Whether the month opens at the conjunction or at the first visible crescent moves every appointed day by a day or two. Here is what the arithmetic says over four thousand years.
The rule sounds unambiguous: the month begins at the new moon, and month I is the one holding the first full moon after the spring equinox. Pesach is the 14th. Compute the new moons, count fourteen days, done.
It is not quite done, because “new moon” means two different things, and the difference is systematic.
Two new moons
The conjunction is the instant the moon passes the sun in longitude. It is a precise astronomical event, computable to within seconds for recent centuries and to within a minute or so for antiquity. At that moment the moon is invisible: it is between us and the sun, its lit face turned away.
The first visible crescent is the first evening the new moon can actually be seen — a thread of light low in the west just after sunset. That is one to two days after conjunction, sometimes more. It depends on the elongation between sun and moon, on how far the moon is above the horizon when the sky darkens, on latitude, season and air. Below roughly seven degrees of elongation no crescent can be seen at all under any conditions.
A calendar that opens the month at conjunction and one that opens it at first sighting run one to two days apart, all year, every year.
What the arithmetic says
The calendar here uses the conjunction. Running it across all 4,001 years from 1001 BCE to 3000 CE, and asking on which day of month I the first post-equinox full moon lands:
| Full moon falls on | Years | Share |
|---|---|---|
| I 14 | 21 | 0.5% |
| I 15 | 1,538 | 38% |
| I 16 | 1,860 | 46% |
| I 17 | 582 | 15% |
Nearly half the time it falls on the 16th, and almost never on the 14th. That is a systematic offset, not noise, and it has a plain cause: the count starts at an event nobody can observe, roughly a day and a half before the moon first appears.
Open the month at first sighting instead and everything shifts one to two days later, which pulls that full moon back onto the 14th or 15th — where Pesach is.
Which is not the same as saying which is right
The arithmetic does not settle the question. It sharpens it.
If you hold that the appointed day should coincide with the full moon, the distribution above is an argument for the crescent: a rule that puts the full moon on the 14th in one year out of two hundred is a rule doing something other than what it appears to be doing.
If you hold that the month begins when the moon renews — an astronomical fact rather than an observation dependent on weather, eyesight and a clear western horizon — then conjunction is the more defensible anchor, and the full moon falling on the 16th is simply where it falls.
Both are coherent. What is not coherent is choosing one and quoting dates to the minute without saying which.
The borderline years
There is a second, smaller uncertainty. Month I is fixed by the first full moon after the March equinox. In 64 of the 4,001 years, that full moon follows the equinox by less than half a day. In 2019 the gap was three hours and forty-two minutes; in 1962, five hours and twenty-four minutes. In the tightest case in the whole span, fifteen minutes.
In those years the answer depends on whether “after the equinox” is measured between instants, between civil days, or between sunset-to-sunset days — and the three readings do not agree. Getting it wrong moves the entire year one lunation, about twenty-nine and a half days. The calendar flags those years rather than answering quietly.
What the calendar does about it
It states its choices instead of hiding them. Months open at the conjunction. “After the equinox” is measured instant to instant. Sunset is computed for Jerusalem, and every day runs from sunset to sunset. Times are Universal Time with ΔT subtracted, which matters enormously for ancient dates — ΔT is about four and three-quarter hours in 500 BCE.
Change any one of those and the dates change. Which is the honest position for a calculation of this kind: the astronomy is settled to within a minute, and the reckoning built on top of it is a set of decisions that a reader is entitled to see.