A full moon computed from the year number

Easter is defined astronomically: the first Sunday after the first full moon on or after the spring equinox. It is not computed astronomically, and never has been. The Moon that fixes it is a table, a nineteen-year cycle of lunar dates chosen because nineteen tropical years are almost exactly 235 lunar months, and the equinox that fixes it is a date, 21 March, whatever the Sun is doing. What the churches calculate every year is not the sky but the output of a small, completely specified piece of arithmetic that takes the year number as its only input.

That arithmetic is worth looking at as an object in its own right. It is one of the oldest algorithms in continuous use; it was the reason European astronomy was funded for a thousand years; and it behaves in ways its designers intended and in ways they did not.

Ten lines with no Moon in them #

The Gregorian computus, in the form a modern programmer would write it, takes a year $Y$ and produces a date. The golden number $G = (Y \bmod 19) + 1$ says where the year falls in the nineteen-year lunar cycle. The epact — the age of the tabulated Moon on 1 January — follows from it by adding eleven days a year, the amount by which twelve lunar months fall short of a solar year:

$$E = (11G + 20 + Z - X) \bmod 30.$$

The paschal full moon is then $44 - E$ days after the end of February, pushed into the next lunation if that falls before 21 March; and Easter is the Sunday after it, found from the year number by the same arithmetic that finds the day of the week of any date.

Two quantities in the formula are not in the Julian original, and they carry all the astronomy the reform put back. $X$ is the solar equation, $\lfloor 3C/4\rfloor - 12$ with $C$ the century: it counts the leap days the Gregorian calendar has dropped, three in every four hundred years, each of which moves every date a day earlier with respect to the Moon. $Z$ is the lunar equation, $\lfloor(8C+5)/25\rfloor - 5$: it adds a day eight times in every 2,500 years, because nineteen Julian years are about an hour and a half longer than 235 real lunar months, and a table that assumes they are equal drifts late by a day every three centuries. Both are integer functions of the century and nothing else. The computus changes only at century years, and a table of Easters printed in 1900 served unaltered until 2099.

Then there are two exceptions. An epact of 24 is replaced by 25, and an epact of 25 is replaced by 26 when the golden number exceeds 11. They look like patches and they are: the thirty possible epacts would give thirty possible full-moon dates, from 21 March to 19 April, and the reformers wanted the latest paschal full moon to be 18 April, as it had been in the Julian computus.

An equinox that is a date #

The other astronomical input is replaced even more bluntly. The spring equinox is the instant the Sun crosses the celestial equator northward, and its date moves: the Gregorian leap-year cycle carries it back and forth across 19, 20 and 21 March, drifting earlier by about three-quarters of a day through each century and snapping back at the century years that are not leap years. In the twenty-first century it falls on 20 March in most years and on 19 March in some.

The computus ignores all of that. Its equinox is 21 March by definition, the date the Council of Nicaea is supposed to have observed it in 325, and the reform of 1582 was designed precisely to put the real equinox back near that date by removing the ten days the Julian calendar had accumulated. The equinox in the computus is therefore a date the calendar is built to keep near the real equinox, not a measurement of it; and the sky’s equinox itself is a moving target, since the equinoxes precess round the ecliptic once in 26,000 years and the tropical year the calendar tracks is defined by that moving point rather than by the stars.

The date the table chooses more often #

The exceptions leave a mark in the output that has nothing to do with the sky.

Over the full cycle every tabulated full-moon date from 21 March to 16 April occurs almost equally often, about one year in thirty: the nineteen-year cycle steps the epact through its thirty values, and the century corrections, drifting slowly, spread the steps evenly over millions of years. The thirtieth value is not allowed. Whenever the arithmetic would put the full moon on 19 April, the first exception moves it to the 18th, which therefore receives its own share and most of its neighbour’s. The second exception, which moves some years’ moons from the 18th to the 17th, exists so that no nineteen-year cycle contains the same lunar date twice, and it takes back part of the excess.

Adding a Sunday converts the full-moon distribution into the Easter distribution of the first figure. Each full-moon date sends its Easter to one of the seven following days, one-seventh of the time each, so the flat middle of the full-moon distribution becomes the flat middle of the Easter one, and each end becomes a ramp: 22 March requires the earliest possible full moon, on 21 March, to fall on a Saturday, and happens in less than half a per cent of years. The excess at 18 April becomes an excess on the seven dates after it, and most visibly on 19 April, which is also the latest date the plateau reaches: 19 April is the commonest date of Easter because of a rule written to keep the full moon off it.

The whole pattern repeats exactly after 5,700,000 years, the least common multiple of the cycles buried in the arithmetic: nineteen years for the golden number, four hundred for the leap-year rule, and the 2,500-year pattern of the lunar equation combined with the thirty values of the epact. No calendar will be used for that long. The number is a property of the algorithm, not of the Moon, and it is the length of the machine’s own period.

A table that keeps pace with the Moon #

The purpose of the two equations was to make a table that would not drift, and the test is to compare the table’s full moons with real ones.

The comparison uses the mean Moon — full moons spaced exactly one mean lunation apart, 29.530588861 days, from a known epoch — which leaves out the real Moon’s wobble of about fourteen hours either way but not its long-term rate. Against it, the Julian table’s full moons drift steadily later: they fell about a day early in the fourth century, when the table was fixed, and have fallen later by a day every 321 years since, so that they are now about four and a half days late and will be thirteen days late by the year 5000. That drift, plus the thirteen-day offset between the Julian and Gregorian calendars, is why the Eastern churches’ Easter now falls weeks later in some years than the astronomical definition would give.

The Gregorian table does what it was designed to do. Averaged over each century its full moons sit on the mean Moon to within about half a day, with a saw-tooth of a day’s amplitude: the lunar drift accumulates continuously, and the lunar equation corrects it a whole day at a time, eight times in 2,500 years. That rate, one day in 312 years, was fixed in 1582 from the best lunar tables available, and against the modern mean lunation it is within a few per cent of right; the error it leaves would amount to a day only after several tens of thousands of years, by which time the lengthening of the day by tidal friction will have made the civil calendar itself unrecognisable.

Two centuries of corrections, and a net that goes both ways #

The solar and lunar equations push the epact in opposite directions, and the net movement is small and irregular.

Each dropped leap day makes the calendar a day earlier relative to the Moon, so the Moon’s tabulated age on 1 January falls by one; each lunar correction makes the tabulated Moon a day later, so its age rises by one. Over three thousand years the solar equation removes about twenty-three days from the epact and the lunar equation restores ten, a net of thirteen, and in any single century the net is minus one, zero or plus one. It is the calendar’s leap-year rule — the fraction chosen to approximate the tropical year — and the Metonic cycle’s small error, each corrected separately and combined by subtraction.

The design has a pedigree worth stating. The epact system was proposed by Aloysius Lilius, a physician from Calabria, and worked out and defended in detail by Christopher Clavius, the Jesuit mathematician who spent the rest of his life answering its critics. What they built was not a better table but a table with an update rule: instead of fixing the lunar cycle once and letting it drift, they parameterised it by the century and specified exactly when and how it would be corrected. The computus is a small program with two slowly varying constants, and the reform was the recognition that a table meant to last needs its own maintenance written into it.

Two Easters, four possible gaps #

The Eastern churches kept the Julian computus and the Julian calendar, and the difference between the two Easters is a direct readout of everything above.

The two dates can differ only by whole weeks, since both are Sundays, and only four gaps occur. When the Julian full moon — four and a half days later than the Gregorian one, and computed in a calendar thirteen days behind — falls in the same week, the two Easters coincide; when it falls past the next Sunday, the Eastern Easter is a week later. When the Gregorian full moon falls between 21 March and 3 April, the Julian computus sees a date before its own equinox — which is 21 March in the Julian calendar, 3 April in the Gregorian — and waits for the next full moon, a lunation later, and the gap is four or five weeks. The Eastern Easter is never the earlier of the two, because both of its differences — the calendar and the lunar drift — point the same way.

The gaps will change. The Julian calendar gains a day on the Gregorian every century year that is not a Gregorian leap year, and the Julian table’s moons drift later by a day in three centuries; both push the Eastern date later, and in tens of thousands of years, if nothing is changed, Eastern Easter will fall in summer.

A sky the computus never consults #

In 1997 a consultation of churches meeting in Aleppo proposed that both traditions compute Easter astronomically — the real equinox and the real full moon, reckoned for the meridian of Jerusalem — so that the two Easters would coincide and both would be tied to the sky. The proposal has not been adopted. Had it been, the two computuses would each disagree with it in some years, and 2019 was one: the astronomical full moon fell in the early hours of 21 March, a few hours after the equinox of 20 March, so the astronomical rule gives Easter on 24 March. The Gregorian table placed its full moon on 20 March — before its fixed equinox of 21 March — and waited a lunation, to 18 April, giving Easter on 21 April, four weeks later.

Such years are unavoidable, and not because the table is poor. The real full moon wanders by up to fourteen hours from the mean one, the Moon’s phases being a geometry of three bodies rather than a clock; the real equinox moves across 19, 20 and 21 March with the leap-year cycle; and a full moon within a day of the equinox is decided by hours. A table that uses a fixed equinox and a mean Moon must disagree with the sky whenever the sky is close to its boundary, and every few decades it is.

The algorithm that paid for astronomy #

For most of the first millennium the computus was the only application of mathematics that every institution in Latin Europe needed, and the people who could do it were the people who knew any astronomy at all. The Easter tables of Dionysius Exiguus in 525 are the reason years are counted from the birth of Christ: he numbered his table’s years that way instead of by an emperor’s reign. Bede’s treatise on the reckoning of time, in 725, taught the computus and in passing gave the first clear account in Europe of how the tides follow the Moon, because a monk calculating lunar dates had noticed that the sea kept the same schedule.

The reform itself needed measurements of the equinox good enough to show how far the Julian calendar had drifted, and the Church paid for them in its own buildings. Meridian lines were laid into the floors of cathedrals — the longest, in San Petronio in Bologna, in 1655 — with a small hole high in the vault letting a spot of sunlight cross the line at local noon, so that the Sun’s height, and with it the date of the equinox and the length of the year, could be measured to seconds of time. Those instruments, built to check a calendar, measured the varying distance of the Sun precisely enough to test Kepler’s orbit against the circle-and-equant alternatives. The computus’s demand for a correct equinox made cathedrals into some of the best solar observatories of the seventeenth century.

The arithmetic was put into closed form by Carl Friedrich Gauss in 1800, as a set of formulas that had to be patched in 1816 because they failed for some years in the far future; the version drawn here, a compact restatement of Clavius’s rules, is the one Donald Knuth set as an exercise in the first volume of The Art of Computer Programming, where it stands as an example of an algorithm older than the word — a procedure, specified exactly, run by hand every year for fifteen centuries, whose correctness mattered to millions of people who never saw it. The same kind of near-coincidence that makes nineteen years a lunar cycle also makes eighteen years and eleven days an eclipse cycle, and the medieval computists used both.

What the figures leave out #

The comparison with the sky is with the mean Moon, not the real one, and it uses civil dates with no time zone; the real Moon would scatter each century’s offset by most of a day around the mean line drawn. The distributions are computed for the Gregorian rules as they stand, run forward for millions of years as though the calendar would be kept that long, which it will not. The Eastern computus is taken as the Julian one converted to the Gregorian calendar, which is what most of the Orthodox churches use; a few use a revised calendar for fixed feasts while keeping the Julian Easter. And the figures describe what the computus produces, not the historical arguments about it, which filled much of the first millennium and turned in part on precisely the questions the arithmetic answers: which cycle, which equinox, and which moon.

Still open: whether the table will ever be retired #

The Gregorian computus is good enough that its errors are smaller than the Moon’s own irregularities, and its disagreements with the sky are a structural consequence of using a fixed equinox and a mean Moon rather than a failure of accuracy. What it cannot do is agree with the Julian computus, and the only fix proposed — computing from the sky — replaces an arithmetic that anyone can do with an ephemeris only astronomers can provide, and a definition that has never changed with one that depends on where on the Earth the Moon is watched. Whether the churches will ever share a date is not an astronomical question. The astronomy is settled: the arithmetic is ten lines, it repeats every 5.7 million years, and on 19 April it produces an Easter that the Moon has never asked for.