The two primary examples that have plagued all complex cultures—the fractional number of days and lunations in the solar year—arise from the same source: nature’s stubborn refusal to work by simple numerical relations in the very domain where such regularity would be most useful to us. Nature, apparently, can make a gorgeous hexagon, but she cannot (or did not deign to) make a year with a nice even number of days or lunations.
What a bummer. Both our practical requirements (to know the seasons for hunting or agriculture, and the tides for fishing or navigation, not to mention that great bugaboo of Christian history, the calculation of Easter), and our intrinsic mental need to seek numerical regularity as one way of ordering a confusing world, drive us to keep track of the three great natural cycles—the days of the earth’s rotation, the lunations of the moon’s revolution, and the years of the earth’s revolution. (Our other major cycles, from weeks to millennia, do not map astronomical events, and arise for more complex and contingent reasons of human history.)
If any of these three natural cycles worked as an even multiple of any other, we could have such a nice, easy, and recurrent calendar—making life ever so much more convenient. Nature, however, gives us nothing but fractionality to innumerable and nonending decimal places—and so it goes. We may best gauge how this inconvenient construction of reality has affected human history by tracing how Western society has treated the two great calendrical complexities imposed by nature’s noncoincident cycles.
The Days of the Solar Year
365 days, 5 hours, 48 minutes, and 45.96768 … seconds! What hath God wrought? The Egyptians found out, so did the Chinese, and so did the Mayans—all independently, and all to their dismay. 365.25—exactly 365 and an extra quarter day precisely—would have been bad enough. We would still face the inconvenience of a leap year every four years, with all the attendant lore—including a variable February that gobbles up a full two-thirds of the six-line ditty that once taught every schoolchild the lengths of the months:
Thirty days hath September
April, June and November.
All the rest have thirty-one,
Except for February alone
Which hath but twenty-eight, in fine,
Till leap year grants it twenty-nine.
The Last Judgement (1808), William Blake. (illustration credit 3.2)
The power of such doggerel can be daunting. To this day, I cannot separate the 30s from the 31s without intoning the first two lines in their entirety.
More rational solutions can easily be devised to regularize the intermediary units that many calendars utilize and that we call months, even though they run out-of-whack with true lunations (for good reasons discussed in the next section). Several societies independently hit upon the idea of dividing 360 days into equal divisions (18 “months” of 20 days each for the vigesimal Mayans; 12 newly named months of 30 days each for the revolutionary French in their wipe-the-slate-clean-and-start-again calendar of 1792)—and then proclaiming five special days to round out the year (viewed as especially unlucky by the Mayans, but as a grand excuse for a long party by the French). Fair enough, but you still have to deal with that pesky extra quarter day each year. So the French added an extra special day, for six in toto, every four years.
The riddles of leap year can provoke endless complexity and wondrously trivial discussion. Just consider all the birthday lore, and the tales of great characters both actual and fictional. Take the case of the ever-youthful composer Rossini, who recently celebrated his forty-eighth birthday on February 29, 1992, just after the earth completed its two hundredth circuit around the sun since his birth in 1792. (Yes, his forty-eighth, not his fiftieth birthday; hang on a bit, for this vexatious little item requires the next level of calendrical complexity, discussed in the next section, for a resolution.)
And consider the poor pirate ’prentice Frederick, indentured to the notorious Pirates of Penzance until his twenty-first birthday. Gilbert and Sullivan’s comic opera of the same name begins with celebrations for Frederick’s forthcoming release. But the poor lad was born on February 29, so he is only “five and a little bit over.” The opera bears the subtitle “The Slave of Duty”—so you can figure out that Frederick agrees to stay until the contractually appointed time. He then importunes his fiancée Mabel: “In 1940 I of age shall be; I’ll then return and claim you, I declare it.” Mabel replies, “It seems so long,” but then promises to wait. Poor Mabel. The situation is bad enough already, but Gilbert—as we shall see—made the same mistake as folks who thought that Rossini had celebrated his fiftieth birthday in 1992. Mabel must really wait until 1944, when Frederick will be a spry eighty-eight, not 1940, at her beau’s distinctly more youthful eighty-four chronological years.
The first modern reform of the Western calendar, introduced by Julius Caesar himself in 45 B.C., didn’t recognize the additional irregularity of 365-and-a-teeny-little-bit-less-than-a-quarter-of-a-day (365.242199 … to be precise) and used exactly 365-and-a-quarter instead. Can we possibly need to worry about such a minor rounding-off that overestimates the true solar year by a mere eleven minutes and change? Thus, the Julian calendar operated in a maximally simple manner (given the undeniable reality of that fractional day after the full 365). That is, the Julian calendar makes one correction, and one correction only—and this correction follows an invariable rule. On every fourth or “leap” year, the calendar adds an extra day to make a year of 366 days. Since we cannot abide fractional days in a rational calendar, an endlessly repeating sequence of 365, 365, 365, and 366 will serve as a good whole-day version of a solar year that actually runs for 365 and a quarter days.
Except for the inconvenient additional complexity that the solar year doesn’t quite reach 365 and a quarter days. The year falls short of this fractional regularity by those pesky eleven minutes and change. The minor overestimate of the Julian calendar will not matter much at first, but those eleven extra minutes do begin to add up after a while, and Caesar did live a rather long time ago. Eventually, the calendar will start to accumulate noticeable extra days (seven every thousand years, in fact), and the process must continue indefinitely, forcing the Julian calendar more and more out of whack with the true solar year. That is, if we want the vernal equinox to fall on about the same day, March 21 or so, every year (an enormous convenience for all manner of people, from priests to farmers, and a pressing necessity, as we shall see, for the crucial determination of Easter), then the Julian calendar gets progressively worse as the centuries roll. The vernal equinox (and any other fixed date) begins to creep farther and farther up the calendar. And this blot on Caesar’s reputation, rather than Brutus’s wound, may turn out to be the most unkindest cut of all.
Pope Gregory XIII therefore made a kind and rational cut instead. By the sixteenth century, this inexorable overestimate, ticking along at eleven minutes and fourteen seconds per year, had accumulated ten extra days. This sloppiness had begun to generate some serious consequences, particularly for priests and astronomers charged with the solemn and sacred duty of determining the date for Easter. So Gregory followed a strategy favored from time immemorial—he convened a committee and appointed a very smart chairman, the eminent Jesuit mathematician Christopher Clavius. This committee, beginning its work in 1578, came up with one of those lovely, practical solutions that has absolutely no mandate in elegant or highfalutin theory but possesses the cardinal virtue of working pretty damned well. Pope Gregory proclaimed the new rules in a papal bull issued on February 24, 1582. We call his correction the Gregorian reform, and the improved calendar—the one we still use today—the Gregorian calendar.
Clavius’s committee faced two separate problems and solved them in different ways. First of all, the old Julian calendar was now running ten days ahead and had to be brought back into alignment with the solar year (so that equinoxes and solstices would fall at their traditional times—and stay put). This problem could only be solved by old-fashioned damage control—of a fairly radical sor
t, but what else could they do? Clavius recommended that ten days be dropped into oblivion by official proclamation, and Pope Gregory did so—just like that, and by fiat! In 1582, October 5 through 14 simply disappeared and never occurred at all! The date following October 4 became October 15, and the calendar came back into sync.
This solution strikes many people as bizarre, if not monstrous—an affront both to nature and to human dignity. How can any arbitrary earthly power make days disappear at a whim? Now I don’t deny that Gregory’s solution imposed some problems (salaries, bank interest—if such a concept existed—ages, birthdays, and so on), though probably nothing on the scale of the forthcoming debacle (which we will try to prevent at great expense) when computers, on January 1, 2000, read their two-digit 00 year code as 1900 rather than 2000, and promptly go berserk with confusion. (I am, at least, looking forward to a hefty check for interest on an account that my bank has just read as on deposit for a hundred years.)
Los (1794), William Blake, from The Book of Urizen. (illustration credit 3.3)
In fact, Gregory’s solution of dropping days was not monstrous in the slightest but eminently wise and practical. The day records a true astronomical cycle, but the date that we affix to each day is only a human convention. October 5–14 were always part of an invented human system, not a natural reality. If we need to excise these dates in order to bring our artificial system into conformity with a natural cycle of equinoxes and solstices, then we may do so at will, and without guilt.
Second, Clavius and company had to devise a new calendrical rule that would avoid the creeping inaccuracy of the Julian system. They accomplished this goal by devising a year of 365.2422 days, much closer to astronomical reality than the calculationally simpler Julian solution of 365.25. To institute this new year, they made a second-order correction to the old rule of leap years—thus setting a more complex rule that we still use today. The Julian calendar had included too many leap years, so Clavius devised a neat little way to drop an occasional leap year in a regular manner that would give the entire system an appearance of wisdom and principle (thus hiding the purely practical problem that only required a workable and arbitrary solution). Clavius suggested that we drop the leap year at century boundaries, every hundred years.
But, as I argued at the outset of this section, natural cycles impose a numerical muddle—the very opposite of the adamantine order that Galileo or Jeans or Joyce wished to attribute to the cosmos. Simple rules rarely work, and the decision to drop leap years at century boundaries required yet another correction—third-order this time, with the Julian leap year as a first-order correction for the fractionality of days, the century drop as a second-order correction for the Julian overestimate, and this final rule as a third-order correction to the century drop.
Clavius recognized that if the Julian solution added too much, the century-dropping correction took away too much—requiring that something be put back every once in a while. Clavius therefore suggested that the leap year be restored every fourth century. He then expressed this procedure as a rule: Remove leap years at century boundaries, but put them back at century boundaries divisible by 400. (As I said, this may sound like a principled decision, but really represents no more than a codified rule of thumb.)
This third-order correction isn’t perfect either, but it does bring the Gregorian calendar—that is, our calendar—into pretty fair sync with the solar year. In fact, the Gregorian year now departs from the solar year by only 25.96 seconds—accurate enough to require a correction of one day only once every 2,800 years or so. Finally, the discrepancy has become small enough not to matter in any practical way. (Or will these become famous last words as our technological society becomes ever more needful of precision?)
In summary, the Gregorian reform of 1582 revised the Julian calendar by dropping those “extra” ten days, and then promulgating a new rule of leap years to prevent any substantial future inaccuracy: Proclaim a leap year every four years, except for three out of four century boundaries; institute this rule by retaining the leap year at century boundaries divisible by 400. This Gregorian rule has an interesting consequence for the forthcoming millennial year 2000. What a special time, and what a privilege for all of us! Not only do we get to witness a millennial transition, but we also get to live in that rare year that comes only once in four hundred—a century boundary with a February 29. Yes, 2000 will be a leap year—and our lives will include the special bonus of an extra day that comes only once every four hundred years. Use it in good health!
As a final footnote to the subject of Gregorian corrections, this century rule explains the paradox of Rossini being only forty-eight years old after two hundred chronological years (1800 and 1900 were not leap years, so he didn’t have a birthday), and poor Mabel’s additional four years of waiting for Frederick to come of age (1900 was not a leap year).
So much for astronomy, but we also have to deal with the foibles of human history and human xenophobia. The truly improved Gregorian calendar was quickly accepted throughout the Roman Catholic world. But in England, the whole brouhaha sounded like a Popish plot, and the Brits would be damned if they would go along. Thus, England kept the Julian calendar until 1752, when they finally succumbed to reason and practicality—by which time yet another “extra” day had accumulated in the Julian reckoning, so Parliament had to drop eleven days (September 3–13, 1752) in order to institute the belated Gregorian reform.
When you know this history, some puzzling little footnotes in our common chronology gain an easy explanation (trivial in one sense, to be sure, but ever so frustratingly annoying if you don’t know the reason). George Washington’s birthday, for example, is sometimes given, particularly by contemporary sources in colonial America, as February 11, 1731—rather than the February 22, 1732, that we used to celebrate on time, before all our public holidays moved to convenient Mondays and we decided to split the difference between Lincoln and Washington with a common Presidents’ Day. As an English colony, America still used the Julian calendar at Washington’s birth. The eleven days had not yet been dropped (so Gregorian February 22 still counted as Julian February 11 in the British world). Moreover, the Julian year began in March (at least in England), so Washington was born a year early as well.
Similarly, many people used to puzzle every year at the Soviet celebration of the “October Revolution” in November. (Remember all those tanks parading through Red Square past the Politburo on the balcony?) Russia did not adopt the Gregorian calendar until 1918, when the secularists ousted the orthodoxy. So the Julian October revolution had actually occurred in Gregorian November—those extra days again! Finally, since the enemy within is always more dangerous than the enemy without, the Eastern Orthodox church has still not accepted the Gregorian calendar—a Romish plot, no doubt. The Julian calendar still lives, but “there is a tide in the affairs of men …”
The Inconvenient Noncoincidence of the Lunar and Solar Year
The day and the solar year fail to come into sync only by that tiny little bit less than a quarter of a day—but look at all the trouble so caused! When we turn to the moon, the situation deteriorates and, in fact, could hardly be worse.
The moon takes 29 and a half days to circle the earth (29.53059 days, to be more precise)—giving the natural month a horrendous fractionality when counted in days or factored into years. No regular “year” of lunations can therefore come even close to the solar year—the nearest approximation being twelve lunations of 354 total days (354.36706, to be more precise again), falling short of the solar year by almost eleven days.
This discrepancy might not matter if complex societies did not need to reconcile the lunar and solar years. But unfortunately though inevitably, they do—for the two reasons that have circulated throughout this text. First of all, practicality demands (for solar and lunar cycles are both so eminently, and differently, useful); second, reason also delights. (We are, for better or for worse, conscious creatures who wonder about our surroun
dings; we can scarcely observe the moon in its cycling phases and not ponder their regularity and their correlation with the other great cycles of days and years. We really have no choice.)
Many major societies in human history, notably imperial China, Judaism, and Islam, use a predominantly lunar calendar but also must establish reconciliation with the solar year. How can this be done? First of all, lunar months can’t have fractional days, so you solve the problem of 29 and a half days per month by granting some of the twelve months 29 days (called “defective” or “hollow”) and giving others (called “full”) 30 days—all to make a full lunar year of 354 days. But what can be done about the eleven-day shortfall?
All societies with lunar calendars have struggled with this question, and all have discovered some variation of the so-called Metonic Cycle—another of those rough-and-ready rules of thumb disguised to look more like a principled law than a practical solution. When operating on such a coarse scale of 29- or 30-day lunar months, the simplest correction for the eleven-day shortfall in a year of lunations just adds an extra lunation—a “leap month,” if you will—to make an occasional long year of thirteen lunar months or 384 days, whenever the accumulating shortfalls become troubling.
The Metonic Cycle, named for the fifth-century B.C. Athenian astronomer Meton (but discovered earlier and independently in China and then again by Babylon, and thence into the Jewish calendar), recognizes the shortest sequence of years that can bring the solar and lunar calendars into nearly perfect alignment. The Metonic Cycle runs for nineteen years and requires the insertion of a leap month in any seven of those nineteen. (Actually, Meton’s original version still included a discrepancy of five days after nineteen years, but this minor problem could easily be corrected by a variety of ad hoc solutions, including the addition of an extra day to five of those leap months.) The Metonic Cycle may sound rough and arbitrary; but it works, and nothing simpler can be devised. Therefore, nearly all lunar calendars follow this system of inserting leap months into seven of every nineteen years. The modern Jewish calendar, for example, intercalates a thirteenth month of 30 days in the third, sixth, eight, eleventh, fourteenth, seventeenth, and nineteenth years of a Metonic Cycle.
Questioning the Millennium Page 10