Five million seven hundred thousand

The week closes, and it did not have to

Four hundred Gregorian years contain 97 leap days, so 400 × 365 + 97 = 146097 days. That number is divisible by seven, and nothing in the leap rule was aiming at it: Lilius and Clavius were fitting the year to the sun, not to the week. It is the reason every date pattern in the calendar — the 13th, your birthday, the ISO week numbers — repeats on a 400-year loop rather than a 2800-year one, and, at the bottom of this page, the reason Easter's cycle is five point seven million years instead of thirty-nine point nine.

Nobody arranged this either. There are 4800 thirteenths in a cycle and 4800 is not a multiple of seven, so the counts cannot come out level; the only question was which weekday would collect the extra three. It is Friday, by one, and it has been Friday since 1582 and will be until the calendar is replaced.

The moon takes longer

The lunar half of the calendar is a table, not an observation. Nineteen years is 235 lunations to within a couple of hours — the Metonic cycle — so a 19-year table of moon ages, the epact, would run forever if nothing drifted. Two things drift, and the reform wrote both down as rates:

The solar equation. Three times in four centuries the Gregorian rule throws away a leap day the Julian rule would have kept. A day removed from the year makes the moon one day older against the calendar, so the epact moves by one, three times per 400 years. The lunar equation. Nineteen years is not exactly 235 lunations, it is about an hour and a half longer, which is a day every 310 years; the reform rounded that to eight days in 2500 years, in the other direction.

From there the number falls out with no astronomy in it at all. Ten thousand years is the first span in which both rates are whole numbers, and the epact lives on a ring of thirty:

300000 × 19 = 5 700 000

All thirty-five of them

Easter can fall on 35 dates, 22 March to 25 April, and the cycle above is long enough to be the only honest denominator for how often each one comes up. The page runs every year of it in your browser and counts. It is 5.7 million iterations of a fourteen-line function; the time it took is printed with the result, because a claim about a whole cycle is worth more when you can watch it be made.

The shape is not a bell and not a slope. It is the moon's 19-year table printed through a week: each of the nineteen golden numbers hands Easter a paschal full moon, and each moon can be followed by a Sunday one to seven days later, so the interior dates collect from several golden numbers at once and the two ends collect from almost none. March 22 needs the moon on the 21st and a Saturday, which is why it is the rarest date on the wall — eight times in a thousand years, 1761, 1818, and then not again until 2285.

Is it really the smallest?

A period that works is easy; a period that is least is the claim worth checking, and it is cheap here. 5,700,000 factors as 2⁵ · 3 · 5⁵ · 19, and any proper divisor of it divides one of its four maximal divisors, so four counter-examples settle it. The page looks for them, and prints the first year where each shorter span fails.

A year, in full

Everything above, for one year at a time: the golden number that picks the row of the table, the moon that row gives you, the Sunday after it, and — for comparison — what the sky and the eastern churches say about the same spring.

The moon it uses is not the moon

The ecclesiastical moon is an arithmetic fiction and was always meant to be one: a table anyone can compute, in any century, without an observatory. The question is how good the fiction is. Two ways to ask.

Here the page computes the real March equinox and the real full moons from Meeus' series, reads both on the meridian the churches proposed at Aleppo in 1997 — Jerusalem — and works out the Easter an astronomer would have declared. The two rules disagree far more often than the calendar's reputation suggests, and when they disagree they usually disagree by a whole month.

The year it is aiming at is not the year you were taught

The Gregorian mean year is 146097 / 400 = 365.2425 days, and the textbook line is that this is 27 seconds too long, so the calendar slips a day every 3200 years. That is the arithmetic against the mean tropical year, which averages the four seasons. But the reform was fixing one thing only — the March equinox, so that Easter would not walk into summer — and the interval between two March equinoxes is longer than the mean tropical year, because the earth's orbit is not a circle and the equinox creeps around it. Measured against what it was aiming at, the calendar is more than twice as good.

Neither number is a promise. The tropical year is itself shrinking, by about half a second per century, so any statement of the form "one day in N years" is a snapshot of a moving target — and the whole question is due to be overtaken long before then by the slowing of the earth's rotation, which lengthens the day itself.

The other Easter

The eastern churches never took the reform, and their computus still runs on the Julian calendar with the uncorrected 19-year table — so the two Easters drift apart, meet occasionally, and will eventually stop meeting altogether. They fell together in 2025, on 20 April, which happened to be the seventeen-hundredth anniversary of the council at Nicaea that set the rule both of them are still following.

What this page is prepared to be wrong about

Every line below is computed here, now, and compared with a number that came from somewhere else: an almanac, a reform document, a printed Easter table, an eclipse somebody timed. If one of them goes red, the page is wrong and says so.