Astronomy & Celestial Mechanics
the sky, computed
The observed sky

A month that has to be tabulated

A lunisolar calendar reconciles two periods that share no common multiple, so every historical cycle is a convergent of one continued fraction. Meton's nineteen years is wrong by two hours, which is a day in two hundred and nineteen years — and that is why the moon that fixes Easter is a table rather than the sky.

Assumes Calendars, Phases and eclipses and Timescales.

18 min read 9 figures The sky as a clockSmall perturbations, large consequences

The rung below fitted days into a year and found that the problem is a rational approximation: 365.2422 is not a whole number, and the whole history of the solar calendar is a sequence of fractions getting closer to its fractional part.

A lunisolar calendar has to do that twice at once. It has to keep its months in step with the Moon, so that the first of a month is a new moon, and its years in step with the Sun, so that a named month falls in the same season. And the two requirements are incompatible, because a year is 12.368266 synodic months and no whole number of months makes a whole number of years.

The Metonic cycle slips a day in 219 years, and the next rule needs 334. Accumulated disagreement between a lunisolar rule and the sky, against elapsed time, for the continued-fraction convergents of the 12.368266 synodic months in a tropical year. Each line is one historical cycle: 2 years to 25 months, 3 years to 37 months, 8 years to 99 months, 11 years to 136 months, 19 years to 235 months, 334 years to 4131 months. Every line has slope exactly 1, because an error made once per cycle accumulates linearly, so the only thing that distinguishes the rules is where they start. The two that were actually used are the octaeteris — eight years, 1.59 days out per cycle, useless within a generation — and Meton's nineteen, which is out by 2.1 hours per cycle and therefore takes 219 years to slip a single day. The reason nineteen is so much better than eleven is a number in the continued fraction rather than anything about the Moon: the partial quotient that follows 235/19 is 17, and a convergent's error is bounded by one over the next quotient times the square of its denominator — so a large quotient there is exactly a good approximation here, and the next improvement costs 334 years of cycle for a rule nobody could keep. What follows from the 219 years is the whole character of a lunisolar calendar: it is a table rather than an observation. The ecclesiastical moon that fixes Easter is computed from a Metonic cycle, not looked at, and the Julian version of that computus — still used to date Easter in the Eastern churches — has slipped four to five days from the sky since it was fixed in the fourth century, exactly as this plot says it must. A calendar's job is agreement, and agreement and accuracy are different requirements that diverge at a rate the arithmetic predicts.
Fig. 1 The whole problem and every historical answer to it. Accumulated disagreement between a cycle and the sky, against elapsed time, for the continued-fraction convergents of 12.368266. Every line has slope exactly 1 — an error made once per cycle accumulates linearly — so the only thing separating the rules is where they start. Meton’s nineteen years to 235 months is out by two hours per cycle and takes 219 years to slip a whole day; the next improvement costs 334 years of cycle for a rule nobody could keep.

The two periods, and why they are what they are

The synodic month is 29.530589 days: the time from one new moon to the next, which is not the time the Moon takes to go round the Earth. The sidereal month is 27.32 days, and the difference is the same effect that makes a solar day longer than a sidereal one — the Sun has moved on, and the Moon has to catch up.

The tropical year is 365.2422 days: equinox to equinox, which is again not the orbital period, because the equinox itself moves.

Neither number is constant. Tidal friction is lengthening the month and the day; the year is shortening very slowly; and the changes are at the fifth decimal place over a millennium, which is large enough to matter for a rule quoted to four figures and small enough that no calendar has ever been designed around them.

Continued fractions, and why the historical cycles are exactly the convergents

The problem is to approximate 12.368266 by a fraction p/qp/q: qq years containing pp months. The tool for that is the continued fraction, and its convergents are best approximations in a strict sense — no fraction with a smaller denominator is closer.

Expanded, the ratio is

12+12+11+12+11+11+117+…12 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{17 + \dots}}}}}}

and the convergents are 12/1, 25/2, 37/3, 99/8, 136/11, 235/19, 4131/334.

Two of those have names. 99 months in 8 years is the octaeteris, used in archaic Greece; 235 months in 19 years is the Metonic cycle, attributed to Meton of Athens in 432 BC and independently in use in Babylon a century earlier.

They are not coincidences of history. They are what the arithmetic offers. Any society that keeps a lunisolar calendar for long enough discovers the octaeteris within a lifetime — the sort of near-commensurability that also makes eclipses repeat — the error is a day and a half per cycle, so it is unmistakable inside twenty years — and then discovers that nineteen years works spectacularly better.

The best fractions, and the one that was adopted instead. The error of a leap-year rule against the denominator it needs, on logarithmic axes. The marked points are the continued-fraction convergents of 0.2421897 — 1/4, 7/29, 8/33, 31/128 — and each is a best approximation in the strict sense: no fraction with a denominator that small is closer, which is checked here by enumerating every one of them rather than by appeal to the theorem. The Gregorian 97/400 is drawn apart from the staircase because it is not on it. It is beaten by 8/33, which is 1.32 times closer at 12.1 times the denominator, and which the Persian calendar had been using for five hundred years when the Gregorian reform was made. The reason for choosing it anyway is in the shape of the rule and not in this plot: 400 is four centuries, and "drop the leap day in three century years out of four" is a rule a clerk can apply from the digits of the year, while "eight leap years in thirty-three" requires knowing where in a 33-year cycle the year falls.
Fig. 2 The same machinery applied to the solar calendar alone, for comparison. The convergents of the year’s fractional part, each checked here to be a best approximation by enumerating every fraction with a smaller denominator, and the Gregorian 97/400 marked apart from the staircase because it is not one of them. A rule can be chosen for reasons other than accuracy, and this is what that looks like on a plot: beaten by a fraction with a quarter of the denominator, and adopted anyway because its arithmetic can be done from the digits of the year.

Why nineteen is so much better than eleven

The jump in quality between 136/11 and 235/19 is a factor of thirty, and the reason is not about the Moon at all.

A convergent’s error is bounded by

∣pq−x∣<1an+1q2,\left|\frac{p}{q} - x\right| < \frac{1}{a_{n+1}q^2},

where an+1a_{n+1} is the next partial quotient in the expansion. The quotient following 235/19 is 17 — much larger than the 1s and 2s around it — so the bound on 235/19’s error is seventeen times tighter than a generic convergent of that size would give.

A large partial quotient means the previous convergent was already very good, which is the same statement seen from the other side. It is why a Metonic cycle is out by two hours in nineteen years while its neighbour is out by a day and a half in eleven, and it is why the improvement after it is unusable: 4131 months in 334 years is accurate to a day in eleven thousand years, and requires a rule spanning three centuries.

Nineteen years is a human interval. Somebody can be told which year of the cycle it is and check it against a grandparent. Three hundred and thirty-four years is a document.

The Metonic cycle slips a day in 219 years, and the next rule needs 334. Accumulated disagreement between a lunisolar rule and the sky, against elapsed time, for the continued-fraction convergents of the 12.368266 synodic months in a tropical year. Each line is one historical cycle: 2 years to 25 months, 3 years to 37 months, 8 years to 99 months, 11 years to 136 months, 19 years to 235 months. Every line has slope exactly 1, because an error made once per cycle accumulates linearly, so the only thing that distinguishes the rules is where they start. The two that were actually used are the octaeteris — eight years, 1.59 days out per cycle, useless within a generation — and Meton's nineteen, which is out by 2.1 hours per cycle and therefore takes 219 years to slip a single day. The reason nineteen is so much better than eleven is a number in the continued fraction rather than anything about the Moon: the partial quotient that follows 235/19 is 17, and a convergent's error is bounded by one over the next quotient times the square of its denominator — so a large quotient there is exactly a good approximation here, and the next improvement costs 334 years of cycle for a rule nobody could keep. What follows from the 219 years is the whole character of a lunisolar calendar: it is a table rather than an observation. The ecclesiastical moon that fixes Easter is computed from a Metonic cycle, not looked at, and the Julian version of that computus — still used to date Easter in the Eastern churches — has slipped four to five days from the sky since it was fixed in the fourth century, exactly as this plot says it must. A calendar's job is agreement, and agreement and accuracy are different requirements that diverge at a rate the arithmetic predicts.
Fig. 3 The same lines over two millennia rather than twenty, with the 334-year rule left off as unkeepable — which is the view a society choosing a calendar actually has. On this scale the octaeteris has slipped a fortnight before anyone using it has died, 11 years to 136 months is visibly worse than 19 to 235 within a century, and the Metonic line is nearly flat. The ranking is unmistakable after a few generations of records, which is the answer to how three independent civilisations found the same fraction: it does not take a continued-fraction algorithm to find a best approximation, only a long enough series of observations and the patience to compare two rules against them.
Five rules, and how fast each one leaves the seasons behind. The accumulated difference between a calendar and the seasons, for five leap-year rules, over 4000 years. Each is a straight line whose slope is the rule's fraction minus the tropical year's 0.2421897, and nothing else about a calendar matters to this plot — not which months are long, not where the year starts, not what anything is called. The Julian quarter is out by 11.25 minutes a year, which is one day every 128 years and is why ten days had to be removed in 1582. The Gregorian rule takes 3223 years to lose a day and the Persian 4264, in the other direction — the older rule is the better one, at a denominator of 33 against 400. The 128-year rule is level on this scale: 454545 years to a day, which is longer than any calendar has been kept — and it is not a separate invention but the Julian rule with its own error subtracted, since 1/4 − 1/128 = 31/128 exactly and the Julian drift is exactly one day per 128 years. What the figure cannot show is that the tropical year is itself shortening, by about half a second a century, so the slopes drawn here are the ones at the present epoch and every line is very slightly curved.
Fig. 4 The same accumulation for the solar problem alone, over four thousand years. A rule that is out by a fixed amount per year drifts linearly, and the only reason the Julian calendar’s ten days of error took sixteen centuries to become intolerable is that eleven minutes a year is a small number. The lunisolar problem is this one twice over, with the added difficulty that the two errors are independent: a cycle can be right about the Moon and wrong about the Sun, and most of them are.

What a lunisolar calendar does with the cycle

Nineteen tropical years is 6939.60 days and 235 synodic months is 6939.69. So the rule is: nineteen years contain 235 months, which is twelve years of twelve months and seven years of thirteen.

Which seven is a matter of convention, and every lunisolar calendar has made a different choice. The Hebrew calendar intercalates in years 3, 6, 8, 11, 14, 17 and 19 of its cycle, by a fixed rule with no observation in it. The Chinese calendar intercalates the month in which no principal solar term falls, which is an observational rule with the same nineteen-year average. The Babylonian scheme was fixed by decree in the fourth century BC after several hundred years of intercalating by inspection.

The Metonic cycle slips a day in 219 years, and the next rule needs 334. Accumulated disagreement between a lunisolar rule and the sky, against elapsed time, for the continued-fraction convergents of the 12.368266 synodic months in a tropical year. Each line is one historical cycle: 2 years to 25 months, 3 years to 37 months, 8 years to 99 months, 11 years to 136 months, 19 years to 235 months. Every line has slope exactly 1, because an error made once per cycle accumulates linearly, so the only thing that distinguishes the rules is where they start. The two that were actually used are the octaeteris — eight years, 1.59 days out per cycle, useless within a generation — and Meton's nineteen, which is out by 2.1 hours per cycle and therefore takes 219 years to slip a single day. The reason nineteen is so much better than eleven is a number in the continued fraction rather than anything about the Moon: the partial quotient that follows 235/19 is 17, and a convergent's error is bounded by one over the next quotient times the square of its denominator — so a large quotient there is exactly a good approximation here, and the next improvement costs 334 years of cycle for a rule nobody could keep. What follows from the 219 years is the whole character of a lunisolar calendar: it is a table rather than an observation. The ecclesiastical moon that fixes Easter is computed from a Metonic cycle, not looked at, and the Julian version of that computus — still used to date Easter in the Eastern churches — has slipped four to five days from the sky since it was fixed in the fourth century, exactly as this plot says it must. A calendar's job is agreement, and agreement and accuracy are different requirements that diverge at a rate the arithmetic predicts.
Fig. 5 Five millennia, which is longer than any of these calendars has run and long enough to show what the choice of intercalation pattern does not affect. Every rule with the same fraction lies on the same line whatever its pattern: the Hebrew fixed rule and the Chinese observational one both average 235 months to 19 years, so they drift identically against the sky and differ only in which month of a given year is doubled. The pattern decides where a festival falls within its season; the fraction decides where the season itself has got to. Only the second is on this plot, and it is the only part of a lunisolar calendar that arithmetic can improve.

The moon that is not the Moon

Here is where the arithmetic turns into something stranger, and it is the point of this rung.

The date of Easter is defined as the first Sunday after the first full moon on or after the vernal equinox. All three of those terms were replaced by tables at the Council of Nicaea and afterwards, and the replacement is not an approximation to the sky — it is a substitution.

The equinox is fixed at 21 March by definition, regardless of where the actual equinox falls.

The full moon is the ecclesiastical full moon, the fourteenth day of a lunar month reckoned by a Metonic cycle through a quantity called the epact — the age of the moon on 1 January. It is computed from the year number by arithmetic, and it is never checked against the sky.

And the Sunday is the only part of the definition that is what it says.

The reason is not carelessness. A festival observed simultaneously across an empire cannot depend on whether a moon was visible from any particular place on any particular evening, and observation of a new crescent is notoriously dependent on weather, on horizon, and on how good the observer’s eyes are. A calendar’s job is agreement rather than accuracy, and once that is accepted the table is not a compromise but the correct design.

The mean moon and the true one

Every rule above uses a mean month, and the Moon does not keep one. A calendar has to decide whether to follow the mean or the actual sky, and the two traditions that made opposite choices are still in use.

The arithmetic tradition takes the mean. A month begins when the rule says it begins, computed from a cycle, and the date is known centuries in advance and is the same everywhere. Its failure mode is that the calculated new moon and the observed one can differ by up to about half a day, so the calendar’s first of the month occasionally falls on the wrong evening.

The astronomical tradition takes the true conjunction. The Chinese calendar defines a month to begin on the day containing the actual new moon, computed for a specific meridian, and its intercalary month is placed by a rule involving the Sun’s position rather than by a cycle. That removes the half-day error entirely and replaces it with two costs.

The first is that the calendar cannot be computed by a person with a table. It requires an ephemeris — a genuine calculation of two bodies’ positions, accurate to a fraction of a day, for every month. That is why the astronomical rule was historically a state function, maintained by an office, and why it changed whenever the official theory of the Moon’s motion was revised.

The second is that the answer depends on where the observer is. A conjunction happening at 23:50 local time in one place happens at 00:50 the next day a fifteen degrees to the east, so the month begins on different dates in different countries using nominally the same calendar. The Chinese rule fixes a meridian by convention, which resolves it by decree rather than by astronomy.

The choice is between a calendar that is wrong occasionally and everywhere alike, and one that is right always and differs by a day between cities, and each tradition has taken the trade it found more tolerable. The arithmetic tradition’s cost is an error; the astronomical tradition’s is a disagreement.

What the cycle is worth as an observation

It is worth asking how good the Metonic cycle is as a measurement, because it was one before it was a rule.

Nineteen years of observation, with the day of a new moon recorded to within a day, determines the ratio of the two periods to about one part in seven thousand. That is roughly what the discrepancy of two hours per cycle corresponds to, and it is the precision at which a Babylonian record-keeper working with a horizon, a tablet and no instrument at all was operating.

Running the cycle for several repetitions improves it. Four Metonic cycles is the Callippic cycle, 76 years, and its purpose was not a better calendar — it is the same fraction, 940/76=235/19940/76 = 235/19 — but a better number: Callippus dropped one day from the 27,759 that four Metonic cycles nominally contain, which is a correction to the assumed length of the year rather than to the ratio of the periods. Hipparchus later dropped one more day from four Callippic cycles, 304 years, and the length of the tropical year he inferred was wrong by about six minutes.

Six minutes in 365 days is one part in ninety thousand, from naked-eye observation, and it stood as the best available value for seventeen centuries.

The Metonic cycle slips a day in 219 years, and the next rule needs 334. Accumulated disagreement between a lunisolar rule and the sky, against elapsed time, for the continued-fraction convergents of the 12.368266 synodic months in a tropical year. Each line is one historical cycle: 2 years to 25 months, 3 years to 37 months, 8 years to 99 months, 11 years to 136 months, 19 years to 235 months, 334 years to 4131 months. Every line has slope exactly 1, because an error made once per cycle accumulates linearly, so the only thing that distinguishes the rules is where they start. The two that were actually used are the octaeteris — eight years, 1.59 days out per cycle, useless within a generation — and Meton's nineteen, which is out by 2.1 hours per cycle and therefore takes 219 years to slip a single day. The reason nineteen is so much better than eleven is a number in the continued fraction rather than anything about the Moon: the partial quotient that follows 235/19 is 17, and a convergent's error is bounded by one over the next quotient times the square of its denominator — so a large quotient there is exactly a good approximation here, and the next improvement costs 334 years of cycle for a rule nobody could keep. What follows from the 219 years is the whole character of a lunisolar calendar: it is a table rather than an observation. The ecclesiastical moon that fixes Easter is computed from a Metonic cycle, not looked at, and the Julian version of that computus — still used to date Easter in the Eastern churches — has slipped four to five days from the sky since it was fixed in the fourth century, exactly as this plot says it must. A calendar's job is agreement, and agreement and accuracy are different requirements that diverge at a rate the arithmetic predicts.
Fig. 6 The full extent of the arithmetic, over fifty thousand years, which is the scale on which the ranking of the convergents is complete and on which none of it means anything. At the far right the 334-year rule is still under a fortnight out and every shorter cycle has slipped by months — but the synodic month and the tropical year have both changed by more than the difference between the two best rules long before the plot ends. The figure is drawn to its arithmetic limit rather than its physical one deliberately: the point at which the lines separate cleanly is well past the point at which the quantity they approximate has stopped being the same number.

What the drift has actually done

The Metonic cycle is out by 0.0868 days per nineteen years, in the direction that makes the calculated moon late. That is a day in 219 years.

The Julian computus — the one still used by the Eastern churches to date Easter — was fixed in the fourth century and has never been corrected. Sixteen hundred years at a day per 219 gives about seven days of accumulated lunar error, of which the reformed Gregorian computus removed most for the Western churches by introducing two corrections: a solar equation that skips a leap day in three century years out of four, and a lunar equation that advances the moon by a day eight times in 2500 years. Between them they hold the ecclesiastical moon within a couple of days of the real one indefinitely, which is as close as an arithmetic rule can get without becoming an ephemeris.

The two churches’ Easters therefore differ by up to five weeks, and the difference is almost entirely a thirteen-day offset in the assumed equinox plus a few days of accumulated lunar drift. It is a measurement of the arithmetic in the first figure of this essay, carried out over sixteen centuries by institutions that were not trying to measure anything.

Five rules, and how fast each one leaves the seasons behind. The accumulated difference between a calendar and the seasons, for five leap-year rules, over 8000 years. Each is a straight line whose slope is the rule's fraction minus the tropical year's 0.2422000, and nothing else about a calendar matters to this plot — not which months are long, not where the year starts, not what anything is called. The Julian quarter is out by 11.23 minutes a year, which is one day every 128 years and is why ten days had to be removed in 1582. The Gregorian rule takes 3333 years to lose a day and the Persian 4459, in the other direction — the older rule is the better one, at a denominator of 33 against 400. The 128-year rule is level on this scale: 80000 years to a day, which is longer than any calendar has been kept — and it is not a separate invention but the Julian rule with its own error subtracted, since 1/4 − 1/128 = 31/128 exactly and the Julian drift is exactly one day per 128 years. What the figure cannot show is that the tropical year is itself shortening, by about half a second a century, so the slopes drawn here are the ones at the present epoch and every line is very slightly curved.
Fig. 7 The solar half of that divergence, over eight thousand years, judged against 365.2422 — the four-figure value every textbook quotes rather than the eight-figure one. The Julian line reaches ten days in sixteen centuries, which is the offset the Gregorian reform removed and the Eastern churches did not, and it keeps going: another eight days by the year 3000. The Gregorian line is nearly flat and is not quite flat, and its residual slope on this drawing is a real thing rather than a rounding artefact — 97/400 is not the best fraction available and the section above shows what it was chosen for instead.

The exceptions a rule accumulates

An arithmetic calendar is a rule, and a rule applied to a society acquires exceptions that have nothing to do with astronomy. The Hebrew calendar is the clearest case, and it is worth setting out because it shows what a lunisolar rule looks like once it has been used for long enough.

The base is the Metonic cycle with a fixed intercalation pattern, and a month length taken from a value for the synodic month accurate to a fraction of a second — a number transmitted from Babylonian astronomy and better than anything measured in Europe for another fifteen centuries. From that alone the calendar would be fully determined.

It is not, because four postponement rules sit on top. A new year may not begin on certain days of the week, because doing so would place a fast day adjacent to a sabbath in a way the law does not permit. A new year computed to fall after a particular hour of the day is pushed to the next. And two further rules exist solely to prevent the year lengths that the first two would otherwise produce, since a year is permitted only four possible lengths and the postponements would occasionally generate a fifth.

The last two are the interesting ones: they are corrections to corrections, introduced because a rule adopted for a religious reason had an arithmetic consequence nobody wanted. Their effect is that the calendar’s structure is periodic with a period of 689,472 years rather than nineteen, which is the smallest interval over which every combination of cycle position, weekday and postponement recurs.

None of that is astronomy. The underlying month is excellent and the year is inherited from the Metonic ratio, so the calendar drifts against the seasons at the rate the first figure gives — about a day in two centuries, which over the sixteen centuries since the rules were fixed has moved the spring festival measurably later. The astronomical error is a slow linear drift and the complexity is entirely social, which is a fair description of most calendars that have lasted.

There is a general point in the shape of it. A rule with a fixed period is easy to state and easy to check, and every constraint added on top of it multiplies the period of the whole system rather than adding to it. Four postponements with periods of seven days, of a fraction of a day, and of a year between them turn a nineteen-year cycle into one repeating on a scale nobody will ever observe — and the calendar remains perfectly usable, because what a user needs is next year’s dates rather than the length of the grand cycle.

That is worth holding against the astronomical tradition’s difficulty. An arithmetic calendar’s complexity is bounded and computable and its error is a slow drift; an observational one has no drift and no bound on the disagreement between two places using it. Neither has solved the problem the first figure states, which is that two periods with no common multiple cannot both be tracked exactly by any rule whatever.

What the arithmetic cannot fix

Three things.

The periods are not constant. The synodic month is lengthening by about 0.04 seconds per century, and the tropical year is shortening by about 0.5 seconds. Over the eleven thousand years that 4131/334 would nominally be good for, those drifts are larger than the cycle’s own error, so the better convergent is not actually better on its own timescale. There is a horizon past which arithmetic precision is pointless, and it is around two thousand years.

The best fractions, and the one that was adopted instead. The error of a leap-year rule against the denominator it needs, on logarithmic axes. The marked points are the continued-fraction convergents of 0.2421897 — 1/4, 7/29, 8/33, 31/128 — and each is a best approximation in the strict sense: no fraction with a denominator that small is closer, which is checked here by enumerating every one of them rather than by appeal to the theorem. The Gregorian 97/400 is drawn apart from the staircase because it is not on it. It is beaten by 8/33, which is 1.32 times closer at 12.1 times the denominator, and which the Persian calendar had been using for five hundred years when the Gregorian reform was made. The reason for choosing it anyway is in the shape of the rule and not in this plot: 400 is four centuries, and "drop the leap day in three century years out of four" is a rule a clerk can apply from the digits of the year, while "eight leap years in thirty-three" requires knowing where in a 33-year cycle the year falls.
Fig. 8 The convergent staircase drawn out to a denominator of eight hundred, so that the rules past the useful horizon are on the plot with the ones before it. Each step down is a genuine improvement in the arithmetic and, past 128 years or so, an improvement in nothing at all — the quantity being approximated has moved by more than the gain before a rule of that period could be checked once. The Gregorian point sits off the staircase, beaten by a fraction with a quarter of its denominator, and it is the only rule here that has ever been kept by anybody. Accuracy stops being the criterion some way to the left of where the arithmetic stops offering it.

The Moon does not keep a mean month. The interval between successive new moons varies by up to thirteen hours either side of the mean, because the Moon’s orbit is eccentric and the Sun perturbs it. A calendar built on the mean month will therefore predict a new moon that has already happened, or has not yet, by half a day fairly regularly — irrespective of how good the mean is. And visibility is not the same as conjunction. A calendar that begins its month at the first sighting of the crescent — which several still do — is tracking a quantity that depends on the observer’s latitude, the season, the Moon’s ecliptic latitude and the clarity of the air, and which can differ by a day between two cities on the same evening. That is not an error in the arithmetic. It is a different definition, and no amount of arithmetic reconciles two definitions.

One more range of denominators shows how far the continued fraction has to run before it improves on the cycles anybody uses.

The best fractions, and the one that was adopted instead. The error of a leap-year rule against the denominator it needs, on logarithmic axes. The marked points are the continued-fraction convergents of 0.2421897 — 1/4, 7/29, 8/33, 31/128 — and each is a best approximation in the strict sense: no fraction with a denominator that small is closer, which is checked here by enumerating every one of them rather than by appeal to the theorem. The Gregorian 97/400 is drawn apart from the staircase because it is not on it. It is beaten by 8/33, which is 1.32 times closer at 12.1 times the denominator, and which the Persian calendar had been using for five hundred years when the Gregorian reform was made. The reason for choosing it anyway is in the shape of the rule and not in this plot: 400 is four centuries, and "drop the leap day in three century years out of four" is a rule a clerk can apply from the digits of the year, while "eight leap years in thirty-three" requires knowing where in a 33-year cycle the year falls.
Fig. 9 The convergents with denominators up to two thousand. Better approximations exist and every one of them requires a cycle longer than any administration has ever operated — which is why the calendars in use are the short convergents plus a correction applied by hand.

The lunar month resists every rule anybody has proposed for it, and that is the reason a lunisolar calendar is tabulated rather than computed — the arithmetic that would generate it does not close.

Where this ladder goes next

This rung has taken the reconciliation of two incommensurable periods and shown that the historical answers are exactly the convergents of one continued fraction, with the quality of each set by a single integer further along the expansion.

The rung above is the machinery that turned those cycles into an algorithm: the computus proper, the epact, the golden number and the dominical letter, which together compute a date from a year with no astronomy in the arithmetic at all — and which are a worked example of what happens when a periodic natural phenomenon is replaced by a finite-state calculation.

Beside it lies the eclipse cycles, which are the same continued-fraction problem with three periods instead of two, and where the near-coincidence that matters is a triple one.

And below it, the habit: an approximation’s quality is set by the term after the one being used. The Metonic cycle is not good because nineteen is a special number; it is good because the number after it in the expansion is seventeen, and that fact is invisible in the cycle itself and visible immediately in the fraction it came from.

Where this goes next

Five rules, and how fast each one leaves the seasons behind. The accumulated difference between a calendar and the seasons, for five leap-year rules, over 4000 years. Each is a straight line whose slope is the rule's fraction minus the tropical year's 0.2421897, and nothing else about a calendar matters to this plot — not which months are long, not where the year starts, not what anything is called. The Julian quarter is out by 11.25 minutes a year, which is one day every 128 years and is why ten days had to be removed in 1582. The Gregorian rule takes 3223 years to lose a day and the Persian 4264, in the other direction — the older rule is the better one, at a denominator of 33 against 400. The 128-year rule is level on this scale: 454545 years to a day, which is longer than any calendar has been kept — and it is not a separate invention but the Julian rule with its own error subtracted, since 1/4 − 1/128 = 31/128 exactly and the Julian drift is exactly one day per 128 years. What the figure cannot show is that the tropical year is itself shortening, by about half a second a century, so the slopes drawn here are the ones at the present epoch and every line is very slightly curved. A year that is not a whole number of days The tropical year is 365.24219 days, so every calendar is a fraction chosen to approximate 0.24219. The continued fraction says which fractions are best, and the one in use is not among them. Phases are a viewing angle, not a shadow. A satellite at eight points of its orbit. Exactly half of it is lit at every one of them; what changes is how much of the lit half faces the centre. Nothing is in shadow except at an eclipse. Phases are not shadows, and eclipses are Half the Moon is lit at every instant of every month. The phase is which part of the lit half faces the Earth — and confusing that with a shadow is the commonest error in astronomy. Three periods that nearly share a multiple. How far the draconic and anomalistic months are from a whole number, after a whole number of synodic months, in hours. At 223 synodic months — 6585.321 days — both are within an hour of closing, which is what makes an eclipse repeat. The draconic residual is 0.87 hours and the anomalistic 5.19 hours. The eclipse that repeats a third of a world away Three lunar periods nearly share a multiple after 6,585 days. The word "nearly" is what makes eclipses predictable, and the leftover third of a day is what moves each repeat a third of the way round the Earth.

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ComputusContinued fractionConvergentEcclesiastical moonEpactIntercalationLunisolar calendarMetonic cycleOctaeterisPartial quotientSynodic monthTropical year