Calendars, Calibration and Culture: Or, If You Want to Know the Time, Ask an HistorianAuthor(s): Robert PooleSource: Mathematics in School, Vol. 27, No. 5 (Nov., 1998), pp. 14-16Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215398 .

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This article, which can be read as a companion piece to Jack Oliver's article on page 2 in this issue, explains that while mathematics is essential for calculating time, the calendar is in the end an historical construct. It shows why there is no one 'true' way of measuring calendar time, and looks at the historical roots of the western calendar. The date of Easter and the vexed question of when the millennium begins provide examples. There is a

short guide to reliable literature on calendar calculations. The essential point is that the calendar is an historical, not a mathematical, entity.

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Or, if you want to know the time, ask an historian

by Robert Poole

"Time is always transient,

in a continual tlux, neither to be seen, nor Ielt,

nor reserved; but only

measured by an act of the mind."

This was the problem that exercised the scientific writer William Holder when he wrote his 'Discourse Concerning Time' in 1694. Holder was a Fellow of the Royal Society, the world's first national scientific academy, during the 'scientific revolution'. It was the age of measurement: if only accurate measures could be made of the physical world, it was be- lieved, it could thereby be understood, and the secrets of creation revealed. Holder shared the belief of his day that in contemplating, measuring, and understanding creation, we were also contemplating God. 'God made all things in num- ber, weight and measure,' proclaimed Holder. Of all forms of measurement, 'the measure of time' was the highest. But how to measure this ethereal thing called timea

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Ideally, Holder reasoned, we would measure the motion of the primum mobile-the 'prime mover', or outermost sphere of heaven, wherein dwelt God. 'But that being difficult to measure, we ... take our measures for common use from those heavenly bodies which carry light along with them'- the stars. 'The celestial motions are the original measures of time ... they have some stamp of authority from the almighty Lord of heaven and earth ... [But] we cannot properly call the celestial motions, standard measures, because we cannot make any such standing measures to be reserved, and kept for public use, and produced when we please.' In other words, you can't measure time itself, any more than you can bottle sunshine. You can make a yard, but you can't make a year. You just have to make do with tracking the movements of the solar system, which move conformably with the invis- ible flow of true time, like twigs floating on a river.

Whether or not we follow Holder's religious view of crea- tion, his point lies at the heart of the calendar problem. There is no absolute time, no celestial umpire to which all disputes about the true time can be referred. We pick our standard from a whole range of heavenly motions. The choice we make is in the end a cultural one; scientific measurement of the calendar, however refined, is always secondary to this cultural choice. Even in this age when a second is defined as 9,192,631,770 vibrations of a caesium-133 atom (I am not making this up), there are still leap seconds added every now and then in order to ensure that time still conforms to its ultimate standard, the slowing rotation of the earth.

There is no absolute, scientifically correct calendar, although talk of measuring it leads us to assume that there is. Before you can measure the calendar, you have to agree what exactly has to be measured, and this is where the real argu- ments begin. As one Tudor writer put it, 'the world was merrier before measuring was used'.

Measuring the Incommensurable Jack Oliver's article sets out the complications of the world's main calendars, with their different choice of lunar and solar variables and their different religious and cultural rules about crucial markers such as Easter and Passover. I won't

Mathematics in School, November 1998

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repeat these here. The essential point is that none of the standards of calendar time we adopt is clear and unambigu- ous. The year can be either solar (measured against the sun) or tropical (measured against the stellar background-a day longer because of the extra revolution the earth makes on its axis when circling the sun). The month is lunar, but in practice most cultures follow solar months, rounded to the nearest day or two. The week is a regular seven days, but only because it is entirely a human creation; there are no weeks in the heavens. Aliens trying to work out where the week comes from might reasonably conclude we had seven fingers.

The day seems simpler, but as the sundial users of the past knew, clock time and solar time are two different things. The earth appears to move faster or slower against the stellar background according to whether it is moving along the flatter section or the bend of its elliptical orbit. Because of this, solar time can be anything up to sixteen minutes differ- ent from clock time, depending on the season. Our day is merely an average day, accurate only four times a year. The mechanical hour, in any case, was only invented with the mechanical clock. In the Middle Ages, the time between sunrise and sunset was always twelve hours long whatever the season. The hours of the day were indeed longer in the summer. Even when mechanical hours did become standard, local time was used in England until Greenwich time was made the law of the land in 1880. While travel was slow this was not a problem, but latterly stagecoaches travelling from London to Bristol carried clocks set to lose ten minutes as they followed the sun west, and then swapped them for clocks set to gain ten minutes as they returned east.

We use the universe as a clock, but the universe doesn't run like clockwork. None of the measures of time we use divides neatly into one another: a lunar month is about

291/2 days, a year is about 3651/4 days, and so on. This awkwardness of creation flies in the face of time-honoured beliefs about the perfection of the heavens. Medieval and Renaissance thinkers modelled the cosmos on the belief that the movements of the heavens moved in regular ways: circles for Ptolemy and Copernicus, ellipses for Kepler, whose uni- verse of platonic harmonies was perhaps mathematics' great- est heroic failure. Alas, the laws of motion might be simple, but their results are as complex as the ripples on the ocean. The solar system wobbles around like an ancient orrery, with rusty bearings and bent pivots.

It is ironic that it is only man-made measures of time that divide neatly, like physical measures, whereas those taken from the heavens are a muddle: seven days in a week, twelve months in a year, but 291/2 days in a lunar month, 3651/4 days in a year, and so on. To measure time meaningfully, we have to adopt our own, cultural standards of measurement. This is particularly obvious when we are dealing with calendars, where a great variety of epochs and events, both historical and astronomical, is used in the world's many calendars.

Calendrical Calculations

Calendars provide numerous examples of the way that time measurement is culturally, rather than scientifically, based: the Egyptian and solar calendars, based on worship of the sun; the Jewish and Islamic lunar calendars, based upon observation of the moon; and the Christian luni-solar calen- dar, based on the Roman calendar but crucially organized around the need to find an acceptable date for Easter, depending on both sun and moon. Easter provides a good illustration of this point.

The Gregorian calendar of 1582, which is now effectively the western calendar, was driven not by the desire to measure time more accurately (that had been possible for centuries) but by the theological importance of celebrating Easter at the correct time: the first Sunday on or after the first full moon after the vernal equinox. The aim was to re-assert the author-

Mathematics in School. November 1998

ity and accuracy of the Roman Catholic church in an age when both had been challenged by the Protestant reforma- tion. A corrected Easter had to rest within a corrected calen- dar. Astronomical measurement was a means to these ends, not an end in itself. As Kepler (a protestant) put it, 'Easter is a feast, not a planet. You do not measure it to hours, minutes and seconds.' There is no mathematically correct date for Easter, and if the Church's rules change (as they may), Easter will change.

Easter, in fact, is governed by such a complex set of religious rules (such as the correct longitude from which to define the new moon, and the need for Easter not to clash with the Passover) that it was 1876 before a mathematical formula capable of describing it was arrived at (based on a rule devised by the German mathematician Carl Friedrich Gauss in 1800). But this formula can only mimic Easter; it does not determine it, something which can only be done by reference to the cyclus, the complex set of tables and cycles set out by the papal astronomer Clavius at the time of the Gregorian reform of 1582. It is a little-known fact that British legislation contains no official date for Easter after 2199.

Recently, an entire book of Calendrical Calculations has been published, by two mathematicians from the department of computer science at the University of Illinois (Dershowitz and Reingold, 1997). The authors have devised algorithms to describe fourteen of the world's major calendars, including not only Julian, Gregorian, Hebrew and Islamic but also Chinese, Mayan, Coptic, Persian, old and new Hindu, and those quaint museum pieces the French Revolutionary and the modern ISO commercial calendars. I have no doubt that their mathematics are impeccable, and that their algorithms will be of enduring value. But we must remember that these formulae are not themselves the calendars; they only mimic them.

Especially when projected far into the future, mathemati- cal formulae for calendars are always liable to fall foul of religious and cultural rules with no rational mathematical basis. Thus, both Lotus 1-2-3 and Microsoft Excel regard 1900 as a leap year, and only the sheer fluke that the year 2000 happens to be the one century year in four which is a leap year prevents the millennium bug from being even worse than it is. The authors of Calendrical Calculations are able to point out numerous flaws in the calendar guides of other authors, and have themselves established a website to carry their own errata.

Astronomers are particularly ill-positioned to tell the rest of us what calendars to use. Astronomers began their day at midnight until 1925, to avoid a change in the middle of a night's observing. And for long-range purposes, astronomers continue to use the Julian calendar, with its regular but inaccurate sequence of leap years, rather than the more accurate but irregular Gregorian. My copy of the planetarium CD Red Shift, for example, is calibrated in Julian years.

Millennium Bugs

When does the millennium enda This favourite argument of the moment provides us with a final illustration of the cultural basis of time measurement. Jack Oliver provides us with the common, scientific view of the issue: there was no year zero, decades end in ten, so the third millennium does not begin until 1 January 2001. I cannot fault the logic, but calendars are produced by history, not by logic.

It is, of course, true that there was no year zero. But nor was there a year one, two, or three, or for that matter a year one hundred, two hundred or three hundred. In these early Christian centuries, years were reckoned according to the reigns of Roman emperors (as with Egyptian dynasties), or according to the fifteen-year cycle of the Roman indiction, or tax year (Roman taxes being one of the most durable and predictable events in the early Christian world). The system of dating by the 'Year of Our Lord', starting from the birth

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of Christ, was devised by a sixth-century monk named Dionysius. On the best evidence available, he decided that Christ had been born some 525 years before, and numbered the years retrospectively. The Church agreed, and we've been using it ever since.

Dionysius, inevitably, got it wrong. His historical evidence about the reigns of Roman emperors was incomplete, and he miscounted. Christian historians and astronomers in later centuries decided Christ had probably been born around 4 BC, a date which astronomical deductions from the Gospels tend to favour--but the Gospels in turn are historical docu- ments, full of patches, guesswork and inconsistencies. But even if we are being literal-minded about all this, the millen- nium will not mark 2000 years from the birth of Christ, for the year begins not with the birth of Christ (25 December) but with the circumcision (1 January). Perhaps what we should be celebrating is 2000 years since the conception (25 March), the official start of year for much of the middle ages

(and in England until 1752)a The case for a millennium on 1 January 2001 is ironically

based on the assumption that all these religious and historical imponderables can be neatly reduced to a single date for the start of the Christian era: the entirely hypothetical first of January, AD 1. The 'correct' millennium on 1 January 2001 turns out to be an historical rule ofthumb, not a mathematical absolute. The most obvious transition is the one everyone is anticipating: when the digital chronometers flip over, with magnificent serenity, from 1999 to 2000. The millennium is a human construction, and that is as good a millennium as any.

The date of the millennium is, of course, a question of little interest to Jews, who will be starting the year 5761 in the autumn of the year 2000, or Moslems, who will be starting the year 1421 in the spring. Perhaps it is not quite that important to Christians, either. If the best calculations are correct and Christ was born in 4 BC, we can relax already: the third millennium has already begun.

If there is a single conclusion, it is this: the calendar is an historical, not a mathematical, entity. If you want to know the time, ask an historian. 1-]

Reading

There is a plethora of books of variable quality on calendars, time and the millennium; encyclopaedias and computer- based material seem particularly prone to errors and wrong assumptions. I offer here just a few reliable starting points.

C. R. Cheney, A Handbook of Dates for Students of English History (Boydell & Brewer, for the Royal Historical Society, 1991) is the standard guide to the calendar in England. ISBN 0 901050 10 5.

The 'Datella' is an ingenious perpetual calendar which can show the day of the week for any date from AD 1 to 2499, Julian until 1800 and Gregorian from 1582. It can be obtained for a2 inc, post from Dr Marcus Merri- man, History Dept, Lancaster University, Lancaster LA1 4YG (m.mer- riman@lancaster.ac.uk).

N. M. Dershowitz and E. M. Reingold, Calendrical Calculations (Cambridge University Press, 1997), as discussed in the text, offers mathematical explanations with detailed algorithms for all the world's main calendars. ISBN 0 521 56474 3 (paperback). Their website is at: http:emr. cs. uiuc.edu/home/reingold/calendar-book/index.html

Robert Poole, Time's Alteration: Calendar Reform in Early Modern England (UCL Press/Taylor & Francis, 1998), offers accounts of the Gregorian calendar reform, the English calendar reform of 1752, and the famous 'Give us our eleven days' riots which are held to have accompanied it. ISBN 1 85728 622 7.

G. J. Whitrow, Time in History (Oxford, 1988) is a readable and accurate survey of the whole field, with mathematical appendices. ISBN 0 19 285211 (paperback).

Author Dr Robert Poole, History Dept, University College of St Martin, Lancaster LA1 3JD. e-mail: r.poole@ucsm.ac.uk

A documentary on the calendar, provisionally entitled: Apocalypse Whena - the Mathematics of the Millennium will be transmitted in the Channel 4 Equinox series this coming Spring.

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