Easter, the moving target

The oldest running machine in the calendar. ← Calendar Contraptions

No other date is computed the way Easter is. It is the first Sunday after the first full moon of spring — except that neither the moon nor the spring is the real one. Both are schematic stand-ins, kept by arithmetic that has been patched, argued over and occasionally got wrong for seventeen centuries. Western and Eastern churches run different versions of the machine, so most years have two Easters. This page runs both, shows the gears, and explains where the machine bends.

The season around it

Everything else movable hangs off Easter at fixed distances — Lent ends, the great week arrives and Pentecost follows, all derived from the one number above.

Watch the machine run

This is the algorithm as it was submitted to the journal Nature in 1876 by an anonymous "New York correspondent" — the form reprinted by Butcher, Downing, Spencer Jones and Meeus, and the one this page runs. Ten divisions, no tables. The numbers in the right column are the picked year's actual values.

Thirty-five possible dates

The rule's fine print pins the paschal moon between 21 March and 18 April, so Easter can only land on the 35 days from 22 March to 25 April. Counted over the thousand years from 1583 to 2582, they are anything but evenly spread:

Two Easters, one rule

The Eastern church runs the same idea on the Julian calendar, whose equinox and moon have drifted the other way. Pick a range and watch the two machines agree and disagree:

Easter on a date

Which years put Easter on a day you care about? Pick one of the 35 possible dates — the finder counts every hit since the Gregorian machine started and lists the next ones.

The rule nobody wrote down at Nicaea

The story usually begins "the Council of Nicaea decreed in 325…" — but Nicaea decreed no such algorithm. Its surviving letter only says the feuding churches should celebrate "at the same time with the Romans and yourselves [the church of Alexandria] and all those who have observed Easter from the beginning." The how was left to Alexandria, whose 19-year tables became the standard over the fourth century. The fight that Nicaea was settling had been running for two centuries already: the Christians of Asia (Quartodecimans, from the Latin for "fourteenth") kept Easter on the 14th of Nisan, whenever the week put it, in step with Passover — Polycarp defended the custom to Rome's face, Polycrates defended it in writing a generation later, and both times the churches agreed to disagree. Two more schisms (Blastus in Rome, Polycrates in the East) are counted from the same quarrel.

The modern phrasing hides a trap. The rule is the first Sunday after the first ecclesiastical full moon on or after 21 March — and shortening that to "after 21 March" is the classic computus bug: a full moon landing exactly on 21 March must count, and the shortened phrase skips it. Britain's Calendar Act of 1750 words it beautifully: Easter-day is "always the first Sunday after the Full Moon, which happens upon, or next after the Twenty-first Day of March. And if the Full Moon happens upon a Sunday, Easter-day is the Sunday after." Parliament needed the rule worded that carefully for an unglamorous reason: Britain adopted the Gregorian Easter in 1752 while making sure — in writing — that it owed nothing to papal authority.

How tight is the rule? The paschal lunar month must begin between 8 March and 5 April, so its fourteenth day (the full moon) falls between 21 March and 18 April, and the Sunday after lands between 22 March and 25 April. Those bounds have held since 1583. Before the machine, there was only argument — Bede records that Queen Eanflæd of Northumbria once fasted on her Palm Sunday while her husband, King Oswiu, feasted on his Easter Sunday, because their two table books disagreed.

The moon that isn't there

The whole machine rests on one approximation: 19 years ≈ 235 moons (the Metonic cycle, named for an Athenian astronomer of 432 BC). Nature gets close: 19 average Gregorian years are 6,939.6075 days, 235 true lunations about 6,939.6884 — the moon outpaces the bookkeeping by roughly two hours each cycle. The church keeps a schematic moon instead: a year of twelve moons is 354 days, eleven short of the civil year, so the moon's age on any fixed date gains 11 days a year — a bookkeeping called the epact, corrected when the arithmetic demands. One correction is structural: 19 advances of 11 days add up to 209 ≡ 29 (mod 30), one short of a full turn — the saltus lunae, the "leap of the moon", which drops a day from the lunar month each cycle.

The Julian church still runs that uncorrected 19-year wheel, and the drift is real: the schematic moon falls behind the true one by a day every three centuries or so. By the 1500s it was four days out — the same drift that had moved the Julian equinox, and the same complaint that drove the Gregorian reform of piece 3. The Gregorian fix was typically baroque: the epact loses a day in three of every four century years (the "solar equation", mirroring the dropped leap days), gains one back eight times per 2,500 years (the "lunar equation", first applied in 1800, next in 2100 — and in 1800 and 2100 the two corrections cancel exactly). There is even a day with two labels, "xxv/25", and a rule for which label wins. The upshot: the Gregorian epact table is only valid for a century or three at a stretch — the current one covers the twentieth to twenty-second centuries, and 2100 will quietly reshuffle it.

Gauss shipped a bug

In 1800 Carl Friedrich Gauss, then 23, published a closed-form Easter algorithm — the first that needs no tables. It was wrong. The term for the century correction, p = floor(k/3), is incorrect; Gauss fixed it in 1816 and thanked his student Peter Paul Tittel for pointing it out. He simplified a condition in 1807, and in 1811 limited his formula to the 18th and 19th centuries, adding two manual patches: when the arithmetic says 26 April, celebrate 19 April instead, and in one rare configuration, 25 April becomes 18 April. The anonymous 1876 Nature algorithm above folds all of that into ten divisions, with the m row carrying the patch: when m = 1, the machine wanted a Sunday that history has forbidden.

Those forbidden Sundays are not mathematics — they are diplomacy. The Gregorian reform promised, for the sake of the Julian church, to keep Easter within the limits it already had, so the machine is clamped: 26 April can never be Easter, and 19 April — one week before it — comes up more often than its neighbours because it absorbs every clamped year. In 1981 and 1992 the raw machine said 26 April; the patch pulled both Easters back to 19 April. Three times since 1583 — 1954, 2049 and 2106 — the clamp has landed Easter exactly on the schematic full moon itself, a Sunday one week before the honest reading. The algorithm disagrees with the rule that defines it, on purpose, and nobody minds.

One Easter or two

The Eastern church never took the Gregorian patch set. Its paschal moon rides the uncorrected Julian wheel, and the Julian calendar's own drift adds 13 civil days (1900–2099) to every date. So the two Easters differ — but unevenly. Between 1900 and 2099 the two machines give the same Sunday 57 times, the East one week later 91 times, and four or five weeks later 52 times — and the Eastern Easter is never the earlier one (not once since 1583). The five-week gaps cluster on golden numbers 3, 8, 11, 14 and 19 — the years where the two schematic moons disagree most. The gap widens again in 2100, when the Julian calendar takes a leap day the Gregorian skips, and the East falls another day further behind.

The comparison table above counts it live; the short answer for the near future is: 2027 five weeks apart (28 March against 2 May), then together in 2028, 2031 and 2034. One curiosity: the Finnish Orthodox church is the one Eastern Orthodox church that computes Easter on the Gregorian machine — for the practical reason that Finland's public life runs on one calendar, and one Easter.

When the sky disagrees

Every so often the machine gives an answer the actual sky would not. The mismatches even have a name — Easter "paradoxies" — and a cataloguer: Ludwig Lange, who in 1928 sorted them and noted the last had been 1685 and 1924 and that "the next is expected in 1943". In those years the true first-full-moon-of-spring and the schematic one disagree by enough to move Easter by a month.

Twice, churches have nearly replaced the schematic moon with the real one. A 1923 Pan-Orthodox congress in Constantinople proposed an astronomical Easter on the new (Revised Julian) calendar — on paper it would have fallen a month before the Gregorian Easter in 1924, 1943 and 1962, and a week after it in 1927, 1954 and 1967; no church ever used it. In 1997 the World Council of Churches met in Aleppo and proposed the same idea again: first Sunday after the first true full moon after the true equinox, calculated for the meridian of Jerusalem. That version would have matched the Gregorian Easter every year from 2000 to 2025 — except 2019, when the real moon would have moved Easter a month earlier. The plan was to start in 2001. It never started; the schematic moon keeps the job.

The protestant detour

When the protestant states of the Holy Roman Empire and Denmark finally adopted the Gregorian calendar's days in 1700, they balked at the pope's moon. Their "improved calendar" (Verbesserte Kalender) took the Gregorian leap rules but computed Easter astronomically, from the lunar phases of Kepler's Rudolphine Tables — built on Tycho Brahe's observations from the island observatory of Uraniborg, in Uraniborg time. It even had a diplomatic clause: postpone a week if the date would collide with the Jewish Nisan 15 (which moved 1700, 1778 and 1798). The astronomical Easter disagreed with the Gregorian one all the same — a week early in 1724 and 1744 in Germany; Sweden, which joined in 1740, was a week early in 1744 and a week late in 1805, 1811 and 1818; Finland (part of Sweden until 1809) ran a week late in 1825, 1829 and 1845. The inconvenience outlasted the theology: Denmark and the Empire surrendered to the Gregorian Easter in 1776, Sweden in 1844 — Finland held the Kepler moon until 1869.

The Easter that almost got fixed

The other standing proposal is to stop moving entirely. Britain's Parliament passed the Easter Act in 1928: Easter would be the Sunday after the second Saturday in April — always between 9 and 15 April, Lent trimmed or stretched to fit. The Act has never been brought into force; it still sits on the statute book, waiting for the churches' consent that was made a condition. A fixed Sunday in April was also the West's first preference at the 1997 Aleppo table — given up, the WCC's record notes, to accommodate the Orthodox, who would rather share one moving Easter than accept two calendars. Further attempts followed in 2008–2009, and in 2016 the Archbishop of Canterbury hoped aloud for a fixed date within a decade. Statements around Nicaea's 1,700th anniversary in 2025 revived the goal — with, as ever, no algorithm attached. For now the moving target keeps moving.

The full cycle

Because the machine mixes a 19-year moon with a 28-year week cycle, the Julian Easter repeats exactly every 532 years — the paschal cycle tabulated by Victorius of Aquitaine in 457 and still the drumbeat of the Eastern church. The Gregorian machine's full period is absurd by comparison: the corrections repeat after 100 centuries, the epact mapping needs 3,000 of them, and the whole pattern of Easters comes round once every 5,700,000 years — 70,499,183 schematic lunations, 2,081,882,250 days, numbers first worked out by Magnus Georg Paucker in 1837. The mean month that machine counts, 29.53058690 days, is off by less than a second from the real moon's average — and it is still not the real moon, day by day, which is the whole story of this page.

What this page doesn't do

It computes the rules as the churches actually keep them, not the sky: the schematic moons here are the ecclesiastical ones, and the essays above list the years where a real-moon Easter would differ. The Western machine starts in 1583, when the Gregorian calendar did. The Eastern date is computed on the Julian calendar and shown in both forms — the church date (Julian) and the civil date (Gregorian), mapped through the real Julian–Gregorian gap of the year, so the mapping stays exact as the gap grows. If the churches ever agree on a fixed or astronomical Easter, this machine becomes a historical document — the finder above will still tell you when Easter last behaved.