A year that is not a whole number of days

A calendar has one job, and it is arithmetical rather than astronomical: to keep a whole number of days in step with a quantity that is not a whole number of days. The tropical year — the interval between successive March equinoxes, which is what the seasons run on — is 365.24219 days. The fractional part is the entire problem, and every calendar ever kept is a proposal about how to approximate it with a rule a person can apply.

Which raises a question with a clean answer. Given that the target is 0.24219, what are the best fractions? And is the one in use one of them?

The fraction, and what a good one means #

Approximating a real number by a fraction has a complete theory, and it is one of the few places where “best” is not a matter of taste.

The continued-fraction expansion of $0.2421897$ begins $[0; 4, 7, 1, 3, 24, \ldots]$, and truncating it at each step gives the convergents:

$$\frac14,\quad \frac{7}{29},\quad \frac{8}{33},\quad \frac{31}{128},\quad \frac{845}{3489}.$$

The theorem that makes these interesting is that each convergent is a best approximation in the strong sense: no fraction with a denominator less than or equal to its own is closer to the target. That is not an asymptotic statement or a generic one — it is a guarantee about every competing fraction.

Each convergent is a recognisable calendar.

• $\tfrac14$ is Julian: one leap year in four. Adopted 45 BC.
• $\tfrac{7}{29}$ is nobody’s, and it errs the other way.
• $\tfrac{8}{33}$ is the Persian calendar, in use since 1079 and still the civil calendar of Iran and Afghanistan.
• $\tfrac{31}{128}$ is a rule proposed in the nineteenth century and never adopted anywhere: drop one Julian leap day every 128 years. Its error is one day in 455,000 years.

That last one is not a separate invention. The Julian error is exactly one day per 128 years, and $\tfrac14 - \tfrac1{128} = \tfrac{31}{128}$: the 128-year rule is the Julian rule with its own drift subtracted, which is why it is so much better and why it is so tempting.

Why 97/400 anyway #

The Gregorian rule is: every fourth year is a leap year, except century years, except that century years divisible by 400 are leap years after all. That is 97 leap days in 400 years.

It is not a convergent and it was not chosen to be. It was chosen because a clerk in 1582 could apply it from the digits of the year, with no table and no memory of where in a cycle the current year falls. “Divisible by 4” is a glance at the last two digits; “divisible by 100” and “by 400” are glances at the trailing zeros. The Persian rule requires knowing the year’s position in a 33-year cycle, which requires either a table or a reference epoch, and in an institution running on parish registers that is a different order of difficulty.

The commission also had a second constraint that the arithmetic alone does not show: the reform had to restore the equinox to 21 March, which is where it had been at the Council of Nicaea in AD 325 rather than where it had been at the calendar’s adoption. Ten days were dropped in October 1582 for that reason, and the choice of which historical date to restore is not a question the continued fraction has an opinion about.

The other calendar problem, and it is worse #

Everything above concerns the year. A lunisolar calendar has to match two incommensurable periods at once — the synodic month of 29.530589 days and the tropical year — and there is no exact solution to that either.

The nearest miss is the Metonic cycle: 235 synodic months is 6,939.688 days and 19 tropical years is 6,939.602. The two agree to two hours in nineteen years, which is one of the sharper coincidences in the solar system, and it is the basis of the Hebrew calendar, of the computation of Easter, and of the Greek and Babylonian calendars before them — every one of them a scheme for inserting a thirteenth month in seven years out of nineteen.

Two hours in nineteen years is one day in about 219 years, so the Metonic cycle drifts — and the ecclesiastical rules that compute Easter from it inherit the drift, which is why the ecclesiastical full moon is now up to two days away from the astronomical one.

The calendar that gave up on the year #

There is one widely used calendar that declines the problem altogether, and it is worth looking at because it shows what the problem costs.

The Islamic calendar counts twelve lunar months and stops. Twelve synodic months is 354.367 days, so its year is about 10.88 days short of the tropical year and its months move steadily backwards through the seasons, completing a circuit in roughly 33 years. Ramadan falls in every season in the course of a lifetime.

That is not a failure of the calendar; it is a decision about which period is authoritative. A calendar that tracks the Moon exactly is one whose months begin at the correct phase, which is what a lunar calendar is for, and the price is that it says nothing about when to plant. A calendar that tracks the Sun exactly — the Gregorian, the Persian, the Julian — has months that are pure bookkeeping and have not corresponded to the Moon for two thousand years. Every calendar is a choice of which of two incommensurable periods to keep, and the lunisolar ones are the attempt to keep both.

The arithmetic bears the decision out. Matching the year alone needs a fraction near 0.2422 and there are excellent ones. Matching year and month together needs a pair of integers $(m, y)$ with $m \times 29.530589 \approx y \times 365.2422$, and the best small solution is Meton’s 235/19 — with a residual of two hours, which is a hundred times worse in relative terms than 8/33 is for the year alone.

What was actually measured #

Two quantities underlie everything above, and both are measurements with histories.

The tropical year. Hipparchus obtained 365.24667 days around 130 BC, by comparing his own equinox timings with those recorded 150 years earlier — a difference method, and the error in it is one part in 4,000 across a baseline he did not control. The Gregorian commission used 365.2425, from the Alfonsine tables. The modern value at J2000 is 365.242190, and it is not a constant: it shortens by about half a second per century, so the “year” the Gregorian rule was fitted to is not quite the one it is now approximating. The shortening is a consequence of the same lunisolar torque that turns the pole, acting on the rate at which the equinox slips.

Worse, there is not one tropical year. The interval from March equinox to March equinox differs from the December-solstice interval by about eight minutes, because the Earth’s orbital speed varies round its eccentric orbit while the equinoxes precess. The 365.24219 figure is an average over the four cardinal points, and a calendar tied to the March equinox — which the Gregorian rule effectively is — is approximating 365.242374 instead. The target the rule is aimed at is itself uncertain at the level of the rule’s own error. The eight minutes between one cardinal interval and another are the same non-uniformity that makes the Sun a bad clock within a single year, read over a year rather than over a day.

The day. The mean solar day is not constant either. Tidal friction lengthens it by about 1.8 milliseconds per century, which is the same braking torque that is pushing the Moon away and which will eventually make a day five hours long into a day fifty hours long, and irregular exchanges of angular momentum with the atmosphere and core move it by a millisecond either way on decade timescales.

What a rule is worth, in one number #

It is worth putting the whole comparison on one line, because the numbers are more surprising than the argument.

The Revised Julian rule, adopted by several Orthodox churches in 1923, is the interesting entry: it keeps century years leap only when the year divided by 900 leaves 200 or 600, which is nearly as digit-friendly as the Gregorian rule and ten times more accurate. It agrees with the Gregorian calendar until 2800, at which point the two diverge — and there is a date, four centuries off, when a difference that has been purely notional becomes a difference of one day.

Below the 128-year rule the exercise stops meaning anything, because the tropical year’s own drift of half a second per century is 0.0000058 days per year — larger than the 31/128 rule’s error. A calendar accurate to one day in 455,000 years is approximating a target that will have moved by then, which is the point at which the arithmetic hands the problem back to the astronomy.

The reform that took three hundred and forty years #

A calendar is not adopted by being correct, and the Gregorian reform’s history is the clearest demonstration of it in the record.

The bull was issued in 1582 and took effect immediately in the Catholic states: Spain, Portugal, the Italian states and Poland dropped ten days that October. The Protestant and Orthodox states did not, and the reason was not arithmetical — it was that the reform came from a papal decree, and accepting it was a political act.

Britain and its colonies changed in 1752, by which time the discrepancy had grown to eleven days. Sweden attempted a gradual transition, skipping leap days one at a time over forty years, abandoned it partway through after a war interrupted the schedule, and had to insert a 30 February in 1712 to get back onto the Julian calendar before switching properly in 1753. Russia changed in 1918, by which time the gap was thirteen days — which is why the October Revolution is commemorated in November.

The practical consequence for anybody reading historical records is a persistent trap. A date in a document between 1582 and 1918 is ambiguous unless the calendar is specified, and correspondence between countries on different calendars routinely carried both. Newton was born on 25 December 1642 in England and on 4 January 1643 in most of Europe, and both are correct.

There is a second ambiguity in the same period that has nothing to do with the reform. England began its legal year on 25 March rather than 1 January until 1752, so a date in January, February or most of March carried a year number one lower than the modern convention. That is why historical dates from that period are often written with a double year.

A calendar’s job is agreement, and the interval during which the world did not agree lasted longer than the interval since it did, which is a reasonable measure of how much a calendar is a social object rather than an astronomical one.

Where the model stops #

A leap-year rule is an approximation to an average, and it says nothing about any individual year. The equinox in the Gregorian calendar wanders over about 53 hours — from late on 19 March to early on 21 March — because the rule corrects in whole days and the error accumulates and is discharged in lumps. A calendar in perfect long-term agreement still has an equinox that moves around, and the two are different properties.

Nor does any of the above touch the daily problem: a calendar fixes which day it is and a clock fixes the time within it, and the two are joined by the mean solar day, which is a fiction. The Sun does not cross the meridian at the same interval on successive days. There is a smaller consequence that still causes trouble. Astronomers avoid the whole problem by counting days rather than dating them: the Julian day number is a continuous count from a fixed epoch in 4713 BC, with no months, no years and no reform in it, and every observation in every catalogue is timestamped that way. A difference between two observations is then a subtraction rather than a calendrical computation, which is the only reliable way to do it across a period in which the calendar changed and the countries disagreed about when.

Both halves of the arithmetic are worth reading over shorter spans, because a calendar rule is judged by the error it accumulates within a civilisation’s memory rather than over ten thousand years.

The one cycle with nothing behind it #

Every period in this essay corresponds to something in the sky. There is one unit of calendrical time that does not, and its persistence is worth a section because it is the exception that shows what the others are.

The week has no astronomical basis whatever. It does not divide the year, it does not divide the month, and it corresponds to no motion of anything. Seven days is about a quarter of a synodic month, which is the usual explanation offered, and the correspondence is poor: a quarter of 29.53 days is 7.38, so four weeks fall a day and a half short of a lunation and the two drift apart within a couple of months.

What the week does have is a continuous count. The cycle of days has run without interruption for at least two thousand years and probably longer — through the Gregorian reform, which removed ten calendar days and did not disturb the sequence of weekdays at all. The Thursday after 4 October 1582 was 15 October, and it was a Friday.

That is a deliberate choice and an informative one. The reform’s designers could have adjusted the week to keep some correspondence with the date; they did not, because the week’s whole function is that it is uninterrupted. A cycle used for scheduling obligations is useful in proportion to its never having a gap.

The consequence is that the day of the week is the one calendrical quantity that can be computed backwards indefinitely without knowing anything about which calendar was in force. Everything else — the date, the month, the year — requires knowing which rules a place was using, and the weekday does not.

A unit with no natural period behind it turns out to be the most stable one in the system, precisely because there is nothing for it to drift against.

One more consequence follows from the week having no astronomical anchor, and it is the reason the twentieth century’s several attempts at calendar reform all failed. Every proposal that made the year tidy — thirteen months of twenty-eight days, or twelve months of thirty with five days left over — had to insert days belonging to no week in order to close the year. That is arithmetically trivial and was, every time, the objection that sank the proposal: a scheme that is better in every respect except that it interrupts a cycle with no astronomical content is a scheme nobody adopts.

And the lunisolar problem over a shorter span, since it is the one whose residual is large enough to matter within a single lifetime.

Where this ladder goes next #

This rung is about matching a count to a period. The rungs above it ask what happens when the period being matched is not stable, and what is done then.

The nearest is the leap second, and the argument now under way about abolishing it. Since 1972, UTC has been kept within 0.9 s of the Earth’s rotation by inserting seconds on an announced schedule; the schedule is irregular because the Earth’s rotation is irregular, and irregularity is expensive for computer systems in a way that a calendar’s is not. The decision taken in 2022 was to stop inserting them by 2035 and let the difference accumulate — which is exactly the Julian calendar’s strategy, and it will need exactly the Julian calendar’s eventual reform.

Beyond it lies the question of what a “day” and a “year” even are for a body that is not the Earth, where the tidally locked and the resonantly locked cases make the ordinary definitions come apart, and where a calendar would have to choose which of several incompatible periods to count.