The Beauty of the Gregorian Calendar Heiner Lichtenberg and Peter H. Richter November 1998 Introduction The Gregorian calendar was developed in the later part of the 16th century, mainly by Aloysius Lilius and Christophorus Clavius [2]. It was named after Pope Gregory XIII who decreed its implementation in 1582 [3]. By that time the Julian calendar had run out of step with the astronomical data in two ways. In its solar part, it had accumulated an error of ten days; the true average vernal equinox fell on March 11 rather than March 21 as the calendar assumed. This was corrected by omitting the ten calendar days October 5 through October 14, 1582. In its lunar part, the Julian calendar was wrong by three days; the true average age of the moon (the number of days elapsed since the last new moon) was three days larger than its calendar prediction. This was corrected by a sudden increase of the epact (see below) by three. After that one-time correction, a new algorithm was put in effect which is about twenty times more accurate in its solar part, and four times more accurate in its lunar part. We are using the Gregorian calendar to this very day [15]. However, it does not seem to be widely known that the moon plays a role in our calendar. Most people believe the Gregorian calendar to be a pure solar calendar, as opposed to the pure lunar calendar of the Arab world. But, in fact, Lilius and Clavius were careful to preserve a tradition that goes back at least to Babylonean times, not unlike Rabbi Hillel II. who, in the middle of the fourth century, reformed the Jewish calendar so as to pay respect to astronomical knowledge on the one hand, and to the dignity of 1 sun and moon as the two major celestial bodies on the other. In the Jewish calendar, the lunar part is more conspicuous than in the Christian, because it determines the average length of months and the beginning of a new year. In the Christian calendar, Julian or Gregorian, the moon has been associated with the date of Easter, i. e., with the holiest day of the Christian religion (related to the Jewish calendar through the role of Pessah in the evangelical record). For Lilius and Clavius this was a matter of such obvious importance that their main intellectual effort concerned the question of how, in the light of improved astronomical data, the synodic lunar period might be reconciled with the length of a tropical year – the fundamental problem of any lunisolar calendar. The problem, of course, is the incommensurability of the average peri- ods of sun and moon with the length of a day and with each other: year, month, and day are time units with irrational ratios. Even worse: in the long run, their ratios are not even constant. Calendars are algorithms which try to overcome this incommensurability in terms of more or less satisfac- tory rational approximations. The following is an attempt to recall the basic principles underlying the Gregorian solution of this task. Using the fact that continued fractions provide optimal rational approximations to given irra- tional numbers, we assess the relative accuracy of various possible calendars. Astronomical accuracy was an important aspect of the Gregorian reform, but not the only one. It will be argued that beauty and wisdom are contained in two principles which have not received much attention in the unending dis- cussions of possible new calendars. One is the principle of secularity which decrees Julian calendar rules for all years that are no secular years (divisible by 100); corrections may only be applied at turns of centuries. This guaran- tees a minimum of changes with respect to well established traditions. The other principle is the openness for adjustments as need arises. Contrary to a general misconception, the Gregorian system is not fixed once and for all, bound to eventually run out of phase with the astronomical data. Lilius and Clavius were conscious of the fact that the knowledge of their time might be limited. They designed their calendar as perpetual. The history of the adoption of the Gregorian calendar has been a com- plicated one and will not be discussed here [15]. To this very day, it has not been accepted by the Orthodox Churches. In an attempt to unify Eastern and Western calendars, the World Council of Churches has recently been discussing a compromise in which the lunar part of the Gregorian calendar would effectively be sacrificed; the date of Easter would be determined by the lunar ephemerides for the location of Jerusalem [16]. We feel that before any such decision is made, a fair evaluation of the merits of the Gregorian system ought to be undertaken. A cultural asset of its caliber should not 2 lightly be disposed of. Basic principles of calendar design Any lunisolar calendar has to be based on three incommensurate astronom- ical periods, - the mean solar day dsol, i. e., the average time between two successive lower transits of the sun across a meridian; - the synodic month msyn, i. e., the average time between two successive new moons; - the tropical year atrop, i. e., the average time between two successive vernal equinoxes. Taking the day as a convenient unit of time, there are only two ratios that matter, m a M := syn = 29.530 589 ... and Y := trop = 365.242 19 ... (1) dsol dsol When the year is measured in numbers of months, the relevant ratio is Y N := = 12.368 266 ... (2) M These numbers are best fits to long term astronomical observations, but they are not constant over long times. Their secular variations are of the order of 1 second per century, corresponding to drifts in the last digit, mostly due to tidal friction between earth and moon. (The short time variations of M and Y are much larger: perturbations by other planets affect the length of the year in the range of minutes, while months vary on the time scale of hours due to the complexity of the lunar orbit.) All cyclic lunisolar calendar algorithms make use of rational approxima- tions to these numbers. As it is well known that optimal rational approxi- mations to real numbers X are obtained from their continued fraction rep- resentation [5] 1 X = x + =: [x , x , x ,... ] , (3) 0 1 0 1 2 x1 + x2 + ··· 3 we give here the corresponding expressions for Y and N, as they will be needed later on: Y = [y , y , y ,... ] = [365, 4, 7, 1, 3,... ] , 0 1 2 (4) N = [n0, n1, n2,... ] = [12, 2, 1, 2, 1, 1, 17,... ] . Let us recall a few elementary facts on rational approximations by continued fractions. Truncating the representation (3) of X at the kth level, one obtains a rational number pk Xk = [x0, x1, x2, . , xk] =: , (5) qk with integers pk and qk which are easily generated by the recursion pk = pk−1xk + pk−2 , qk = qk−1xk + qk−2 , (6) starting with (p−1, q−1) = (1, 0) and (p0, q0) = (x0, 1). The number X has the exact representation p + p R X = k k−1 k , (7) qk + qk−1Rk where the remainder Rk is defined as 1/Rk = [xk+1, xk+2,... ] and obeys the recurrence relation 1 Rk = . (8) xk+1 + Rk+1 As a consequence, it may be concluded (Liouville 1851) that the distance |X − Xk| decreases as the square of the denominators qk, 1 |X − Xk| ≤ 2 , (9) xk+1qk and that no rational number p/q with q ≤ qk comes closer to X than does Xk. (In contrast, the approximation of X by decimal numbers improves only with the first power of the denominators 10k.) Note that the kth approximation is particularly good when xk+1 is a large number. Let us analyze the numbers Y and N from this point of view, and consider their best rational approximations at increasing levels of precision. We start with the tropical year Y and its first continued fraction approximations Yk = dk/ak. They are listed in the following Table: 4 The last column is 10 times the difference Y − Yk; it tells us how many days in 10 000 years a calendar runs ahead (+) or lags behind (−) the true 4 4 level k yk dk ak (Y − Yk) · 10 0 365 365 1 +2 421.9 1 4 1 461 4 −78.1 2 7 10 592 29 +8.1 3 1 12 053 33 −2.3 4 3 46 751 128 +0.0 th tropical year if it distributes dk days over ak calendar years. The 0 approx- imation reflects the solar calendar of ancient Egypt; it runs too fast by 2 422 days in 10 000 years, or by one year in 1 461 (the Sothis period, see [14]). The 1st approximation gives the Julian calendar with its well known leap year rule to let every fourth year have 366 days; this calendar stays behind the true sun by 78 days in 10 000 years. The 4th approximation suggests an excellent fit to the natural length of the year by distributing 46 751 days over 128 years: just omit every 32nd leap year (only 31 leaps in 128 years). However, Lilius and Clavius, the fathers of the Gregorian calendar, had good reasons not to choose this particular improvement of the Julian calendar, as will be explained below. The number N = [n0, n1, n2,... ] of synodic months per year is approxi- mated by Nk = µk/αk according to the following table: 4 level k nk µk αk (Nk − N) · 10 0 12 12 1 −3 682.66 1 2 25 2 +1 317.34 2 1 37 3 −349.33 3 2 99 8 +67.34 4 1 136 11 −46.30 5 1 235 19 +1.55 6 17 4 131 334 −0.03 4 The number (Nk − N) · 10 tells us how many months a lunar calen- dar which distributes µk months over αk years, runs ahead (+) or lags behind (−) the true moon’s motion in 10 000 years.