Calendar drift

Why do human calendars need leap days and extra months at all?

A calendar has one job that matters: to say the same thing about the sky every year. Plant when it says to plant; hold the festival when it says to hold it. If the sky has moved on, the calendar has failed, whatever it prints.

And here is the difficulty, which no amount of administrative competence can dissolve. The day is one rotation of the Earth. The year is one orbit around the Sun. Nothing arranged for these to fit inside each other neatly, and they do not. So what happens to a calendar built on the assumption that they do?

Start with the measurement. The interval that governs the seasons — the tropical year, from one March equinox to the next — is about 365.2422 days. Not 365. The remainder is roughly 0.2422 of a day, near enough a quarter.

Build a 365-day calendar and follow it. After one year the calendar is about a quarter-day ahead of the Sun; after four, a full day. That is small, and here is the trap: it is small enough that no single generation notices. After a century the calendar has slipped roughly twenty-four days — and midsummer has moved into what the calendar calls a different part of the year. Drift is not an error you catch, it is an error you inherit.

So insert a day every four years. That is the Julian reform, taken up under Caesar in 46 BC, and it treats the year as 365.25 days long. Does it fix the problem? Almost. Compare: 365.25 against 365.2422 is an overshoot of about 0.0078 of a day — roughly eleven minutes a year.

Eleven minutes sounds like nothing, and that is exactly why the Julian calendar survived so long. But the arithmetic is patient. Eleven minutes a year is about three days in four centuries. The Council of Nicaea in AD 325 pegged the calculation of Easter to the equinox falling on 21 March; by 1582, twelve and a half centuries later, the accumulated overshoot had reached very nearly ten days, and the real equinox was arriving on the eleventh. Easter, an astronomical rule administered by a drifting calendar, was slipping toward summer.

Gregory's reform did two separate things, and they are worth distinguishing. It deleted ten days once — 4 October 1582 was followed by the 15th — to put the equinox back where Nicaea had left it. And it changed the rule going forward, so the deletion would not have to be repeated: century years are leap years only when divisible by 400. So 1600 and 2000 were leap years; 1700, 1800 and 1900 were not. That drops three leap days every four centuries, giving an average year of 365.2425 days — an error of about 0.0003 of a day, which takes something on the order of three thousand years to accumulate into a single day.

Now, one more question that changes the shape of everything. Why does anyone use a lunar calendar, which drifts against the seasons far worse?

Because the Moon offers something the Sun does not: a visible, countable marker. Anyone can see a new crescent; nobody can see an equinox without instruments and records. A month is the honest unit for a society without astronomers. But twelve lunar months — the synodic month is about 29.53 days — come to roughly 354 days, some eleven days short of the solar year. A purely lunar calendar therefore walks backwards through the seasons, completing a circuit in about 33 years, which is why Ramadan falls in every season across a lifetime.

Anyone who wants both the Moon's months and the Sun's seasons faces a harder problem than mere drift: two incommensurable cycles to satisfy at once. The lunisolar answer is to add a whole extra month in some years, as the Hebrew and traditional Chinese calendars do. The arithmetic that makes it tractable is the Metonic cycle — 19 solar years come to almost exactly 235 lunar months, a coincidence accurate to within a couple of hours — so seven extra months distributed across nineteen years keeps both masters roughly satisfied.

The natural misreading is that leap days are a patch for sloppy measurement — that better astronomy would have removed the need. It would not. The tropical year is not a whole number of days and never will be, so any calendar of whole days must drift; leap rules do not eliminate the drift, they only cancel it in periodic lumps. The Gregorian calendar still drifts, just slowly enough that nobody alive need care.

Two honest complications. First, the tropical year is not a fixed constant to eight decimal places: it varies slightly depending on which equinox or solstice you measure from, and it changes gradually over millennia, so quoted values differ a little between sources. Second, the length of the day itself is not perfectly constant — tidal friction slowly lengthens it — which is a separate problem handled by leap seconds rather than leap days, and one that has no bearing on the seasonal drift discussed here.

A final refinement: the ten days removed in 1582 were a one-off correction of accumulated debt, not part of the ongoing rule. Adoption was staggered across Europe — Britain waited until 1752, by which time eleven days had to be dropped.

• How Easter's date is actually computed, and why it still uses an idealised Moon rather than the real one.
• What happens after the Gregorian rule's error accumulates, and whether anyone has proposed a successor.

Every term the collection defines is gathered in the glossary.