Astronomy & Celestial Mechanics
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Calendars — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · sky
  2. 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.

    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.

    part 2 · sky

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