Easter by hand: the Julian column

Easter by hand: the Julian column

A follow-up to the Gregorian table (easter-moons.md, on the earlier note). The Orthodox churches reckon Easter on the Julian calendar, with the older Paschal table.

Method

Meeus's Julian algorithm: a = Y mod 19, d = (19a + 15) mod 30. The table's Paschal full moon is Julian March 21 + d. In 1900–2099 the Julian calendar runs 13 days behind, so in Gregorian dates that's April 3 + d, measured from an equinox of April 3. Easter is the next Sunday after it.

Sky: USNO full moons (UTC). All twenty Orthodox dates below come out of the hand computation and match published lists.

Table, 2019–2038

| year | a | d | Julian PFM (Greg.) | sky moon, same lunation (UTC) | Julian − sky | Orthodox | Western | |---|---|---|---|---|---|---|---| | 2019 | 5 | 20 | Apr 23 | Apr 19 11:12 | +4 | Apr 28 | Apr 21 | | 2020 | 6 | 9 | Apr 12 | Apr 8 02:35 | +4 | Apr 19 | Apr 12 | | 2021 | 7 | 28 | May 1 | Apr 27 03:31 | +4 | May 2 | Apr 4 | | 2022 | 8 | 17 | Apr 20 | Apr 16 18:55 | +4 | Apr 24 | Apr 17 | | 2023 | 9 | 6 | Apr 9 | Apr 6 04:34 | +3 | Apr 16 | Apr 9 | | 2024 | 10 | 25 | Apr 28 | Apr 23 23:49 | +5 | May 5 | Mar 31 | | 2025 | 11 | 14 | Apr 17 | Apr 13 00:22 | +4 | Apr 20 | Apr 20 | | 2026 | 12 | 3 | Apr 6 | Apr 2 02:12 | +4 | Apr 12 | Apr 5 | | 2027 | 13 | 22 | Apr 25 | Apr 20 22:27 | +5 | May 2 | Mar 28 | | 2028 | 14 | 11 | Apr 14 | Apr 9 10:26 | +5 | Apr 16 | Apr 16 | | 2029 | 15 | 0 | Apr 3 | Mar 30 02:26 | +4 | Apr 8 | Apr 1 | | 2030 | 16 | 19 | Apr 22 | Apr 18 03:20 | +4 | Apr 28 | Apr 21 | | 2031 | 17 | 8 | Apr 11 | Apr 7 17:21 | +4 | Apr 13 | Apr 13 | | 2032 | 18 | 27 | Apr 30 | Apr 25 15:09 | +5 | May 2 | Mar 28 | | 2033 | 0 | 15 | Apr 18 | Apr 14 19:17 | +4 | Apr 24 | Apr 17 | | 2034 | 1 | 4 | Apr 7 | Apr 3 19:19 | +4 | Apr 9 | Apr 9 | | 2035 | 2 | 23 | Apr 26 | Apr 22 13:20 | +4 | Apr 29 | Mar 25 | | 2036 | 3 | 12 | Apr 15 | Apr 10 20:22 | +5 | Apr 20 | Apr 13 | | 2037 | 4 | 1 | Apr 4 | Mar 31 09:53 | +4 | Apr 5 | Apr 5 | | 2038 | 5 | 20 | Apr 23 | Apr 19 10:36 | +4 | Apr 25 | Apr 25 |

What it shows

• The Julian table's moon is 3 to 5 days late every year: +4 in fourteen, +5 in five, +3 in one. Two of the +5s (2024, 2027) are late-evening UTC moons and would be +4 a few time zones east.
• Its equinox, April 3, is 14 days late against the sky's March 20.
• Orthodox Easter matches Western in 6 of the 20 years. It is a week later in 9, because of the late moon. It is four or five weeks later in 5 (2021, 2024, 2027, 2032, 2035), because the late equinox skips a lunation.
• The two errors partly cancel. A lunation is skipped when its moon falls before April 3, and a moon that runs 4 days late clears that line more often. With the real moon and the Julian equinox, 2026 (sky moon Apr 2), 2029 (Mar 30) and 2037 (Mar 31) would also skip a month. That makes 8 jumps in 20 years, where the table gives 5. In effect the boundary is about 14 − 4 = 10 days late, which is why the jump happens for d of 21 or more and not over a wider band.
• In 2037 the late moon is the only reason the two churches agree. The Julian moon lands on Saturday April 4, Easter is April 5 both ways, and a moon on time would have sent the Orthodox date into May.
• This mirrors 2019 and 2038 in the Gregorian table. There, two one-day errors pointed the same way and cost a month. Here, two large errors point opposite ways and save one, three times in twenty years.

Drift (my arithmetic, unsourced)

Nineteen Julian years are 6939.75 days, and 235 mean lunations are 6939.688, so the table's moon falls a day later about every 308 years. The Julian year is 0.0078 days too long, so the equinox falls a day later about every 128 years. The equinox outruns the moon, so the boundary's net error grows by a day about every 220 years. Each extra day adds roughly one year in thirty to the month-late Easters.