Mapping Time

reviewed by John P. Pratt

© 1999 American Mathematical Monthly, 107 (Jan. 2000), 92-99.
reprinted with permission. All rights reserved.
Index, Home

1. Predictability
2. Accuracy
3. Simplicity
4. Day Count
5. Nested Patterns
6. Enoch Calendar
At the beginning of the year 2000, it is appropriate to consider the roots of our Gregorian calendar, and of calendars in general. Mapping Time: The Calendar and Its History by E. G. Richards may be the best overall general review of the history and astronomical basis of the principal calendars that have been used throughout history all around the world. The book strikes an excellent balance between covering both the subject's breadth, which spans mathematics, astronomy, history, and anthropology, and its depth. Richards covers several subjects beyond what might be expected, including a history of arithmetic, algorithms for calendar conversion, and the details of how to calculate the date of Easter. The work is a valuable contribution because it summarizes the results of extensive research, done credibly by one trained in critical thinking (Richards has a biophysics background). The only disappointing feature is the lack of footnotes (no doubt the publisher's requirement for a lay audience), but the extensive annotated bibliography, arranged by subject, helps fill that void.

The Calendrical Problem. There is one major problem at the heart of all calendars that track astronomical cycles: namely, how best to approximate irrational cycle lengths by an integral number of days. One aspect of a good solution is long-term accuracy: a good calendar should not drift away from the astronomical phenomenon it is attempting to follow. The "best" solution to the calendar problem, however, should balance mathematical accuracy with a simple and aesthetically pleasing pattern which virtually everyone can remember.

In this review, I propose five principles on which an excellent calendar should be based and I rate calendars by awarding one "star" for each priciple followed. I also discuss how our current Gregorian calendar could be improved from a 3 1/2-star rating to become a full 5 stars.

A 5-Star Calendar Rating System

After studying the many calendars used throughout history, and the many proposals to reform existing calendars, one can distill out a few desirable features for an ideal calendar. The following five fundamental calendrical criteria have been employed in some of the most successful calendars. Each of the five principles corresponds to one star in a 5-star rating system.

1. Predictability. One of the main functions of a calendar is to enable the scheduling of future events. Indeed, one definition of the word "calendar" today is a verb meaning "to schedule". If future dates depend on observations yet to be made, then one cannot plan with any precision. As Richards points out, some calendars currently in use in the world, such as one version of the Islamic calendar, still depend on observations. This means that tenants may not know exactly when the rent is due, and that travelers may not know when to buy airline tickets for a future vacation. Such dependencies can lead to legal disputes.

To merit the first star, a calendar must be entirely predictable, reproducible solely by mathematical calculations or tables without any reference to actual observations of the sky. Of course, astronomers should have first made their most accurate, precise observations of the cycles that the calendar is designed to track. The whole idea of a calendar, however, is to replace observations, preferably with simple patterns. If new observations prove that the current calendar lacks sufficient accuracy, then it can be reformed. A calendar that depends on continual observations is really being continually reformed. Because such a calendar would not be predictable, it would not be awarded this first star.

2. Long-term Accuracy. Another important feature of a calendar is that it not slowly drift away from whatever celestial body it is tracking. Ensuring long-term accuracy is a principal reason that some calendars require continual observations.

Calendar reform usually is a mechanism to improve long-term accuracy. The "Old Style" Julian calendar, which had a leap year every four years in order to make years average 365.25 days, was replaced with our modern Gregorian calendar when it was discovered that the year is really only about 365.2422 days long. By skipping leap years in three out of four century years, the Gregorian calendar averages 365.2425 days per year, a much better approximation. The century years 1700, 1800, and 1900 were not leap years, but 2000 is. The awarding of the second star is a judgment call, but, for example, the Julian calendar would not merit this star for long-term accuracy, whereas the Gregorian calendar would.

Just what constitutes an acceptable average value? Richards has done his homework in finding equations, which he supplies in an appendix, for how the average length of the year varies with time. Unfortunately, those equations are not useful for designing calendars because they give the length of the year in terms of fixed units of time, based on an atomic clock. While that is a convenient unit for astronomers, calendars need to be based on the length of the mean solar day, which is itself slowly lengthening. If the day is lengthening, then any fixed unit of time will appear to be decreasing when measured in days.

The rotating Earth makes an extremely accurate clock: the mean solar day increases in length by only about 1.5 milliseconds per century. Nevertheless, that seemingly minuscule effect sometimes needs to be taken into account in calendrical design. For example, in the case of the lunar month, which is the average length of the cycle of the moon's phases, it makes the difference in whether the length of the lunar month is increasing or decreasing.

The moon causes tides on the Earth, but the reaction of those tides on the moon accelerates it into a higher energy orbit, causing it to recede slowly from the Earth, which in turn increases the orbital period. The equation for this increasing period in terms of fixed-length days using the International System of Units (SI), based on an atomic clock, is:

29.5305888531 + 2.1621 × 10-7T - 3.64 × 10-10T2,

where T is measured in Julian centuries of 36525 days after A.D. 2000 [1]. The plus sign on the second term indicates that the moon's orbital period in absolute days is increasing. However, the Earth's rotation rate is slowing down, mostly because of frictional heat loss from the sloshing tides, so the ratio of a mean solar day dM to the ephemeris day, dE, based on the atomic second, is:

dM/dE = 1 + 1.8 × 10-8 (T +1)

which means that the day is lengthening by about 1.5 milliseconds per century [2]. Combining these two equations implies that the length of the lunar month in mean solar days is

29.53058883 - 3.2 × 10-7 T.

Thus, the length of the month in mean solar days is actually decreasing because the length of the day is increasing faster than the length of the month. Another way to look at it is that the lengthening of the moon's orbital period partially compensates for the increasing length of the day, so that a calendar based on the lunar month is intrinsically extremely stable [3].

Although the variation in the length of the lunar month is small enough that it can be ignored for most calendrical considerations; it is important to select a good average value over historical times. For recorded history over the last 5,000 years, a good value is 29.530596, which is surprisingly close to the traditional Hebrew value of 29.530594. Thus, the Hebrew lunar cycle definitely gets a star for no long-term drift. It has never been clear just where the extremely accurate Hebrew value came from.

On the other hand, in the case of the year, variations in the Earth's rotational rate and orbit combine additively. The Earth encounters particles in its orbit which, along with other effects, remove energy and cause the orbit to decrease very slowly in size. Kepler's equations dictate that the Earth will therefore speed up slightly, so the length of the year is slightly decreasing in days of absolute length. The length in SI days is [4]

365.2421896698 - 6.15359 × 10-6T - 7.29 × 10-10T2 + 2.64 × 10-10T3.

But the slow increase in the length of the day is of about the same magnitude as the first-order correction and adds to this effect, so the mean length of the year in mean solar days is about

365.242183 - 1.3 × 10-5T.

Thus, a calendar based on the moon's phases, such as the Hebrew calendar, is intrinsically more stable than any solar calendar.

The result of these calculations is that even though most books on calendars focus on the modern year length of 365.2422 days, a better value for a calendar designed to be used throughout recorded history would be 365.2425 days (the Gregorian value).

3. Simplicity. Equally important as long-term accuracy, and much more important than short-term accuracy, is the need for a calendar to have a simple repeating pattern. An extremely accurate calendar might be so complicated that the layman would not be able to understand it. On the other hand, if one clings exclusively to a very simple pattern, which might be only a crude approximation, then the calendar will probably suffer long-term drift. This is a classic case where art and science meet; the best calendars strike a balance between simplicity and accuracy. Some examples are in order.

The Julian calendar, which has a leap year every four years, certainly gets a star for a simple pattern. Note, however, that when the Julian calendar began in 45 B.C., "every fourth year" was misunderstood. At that time, it was common to count inclusively; for example, the Bible states that Christ was resurrected on the third day after Friday, meaning Sunday. Similarly, "every fourth year" of the Julian calendar was interpreted to mean what we would call "every third year". This error went undetected for 36 years, during which 12 rather than 9 leap days were inserted. Augustus corrected the mistake by omitting leap days from 8 B.C. to A.D. 4.

The requirement for simplicity is less important when correcting for long-term drift. The Gregorian calendar skips the leap day in three out of every four centuries, but this is not grounds for denying it a star. Since most people are totally unaware of this minor correction until a new century approaches, it does not detract from the overall simplicity of the calendar.

An example of good long-term accuracy with a poor pattern is the following possible correction to the Julian calendar. Inserting one extra year after every eighth leap year creates blocks of 33 years which contain 8 leap days, leading to an average year length of (33 × 365 +8)/33 = 365.2424, which rivals the Gregorian approximation for accuracy. But the price of this increased accuracy is the loss of the simple rule of dividing a year by four to know if it is a leap year. Whether a given calendar earns a star for simplicity is a judgment call, but the Gregorian calendar wins this star and the 33-year calendar does not.

A calendar that tracks more than one celestial cycle should be rated on each cycle separately. The Hebrew calendar tracks both the sun and moon, reckoning months as beginning with the new moon, and years as beginning in the fall season of the sun's annual cycle. It employs months of either 29 or 30 days to get an average lunar month of 29.53 days, and years with either 12 or 13 months to get an average solar year of 365.24 days. Such a calendar is called a lunisolar calendar because it attempts to integrate both the lunar and solar cycles. As for the lunar part, it gets a star for accuracy because its traditional value of 29.530594 days can hardly be improved upon at all, being accurate to within one sixth of a day during all of recorded history. It gets no star, however, for a simple pattern, because it has no pattern at all. Every year has to be calculated using a complicated set of rules. Refreshingly, Richards actually understands the mathematical basis of those rules, which he includes in full. On the other hand, the solar part of the Hebrew calendar gets the star for a fairly simple pattern of intercalating years (called the 19-year Metonic cycle), but it suffers from a long-term drift similar to that of the Julian calendar, so it fails to get the long-term accuracy star for the solar cycle.

4. Alignment with an uninterrupted day count. One feature of the most advanced calendars is an uninterrupted day count that is aligned with other cycles. Simply having such a fixed day count earns half a star, and aligning the count with other cycles, such as the lunar month and the year, earns the other half star.

A day count provides a double check on any given date and on long-term calculations, especially for calendars which are not perfectly predictable. An example of an uninterrupted day count is the 7-day week; it was not broken even during the transition from the Julian to the Gregorian calendar. If one includes the day of the week with the date, then that redundant information serves as a double check to verify accuracy.

This extra information is especially useful for dates during the transition period from the Julian to the Gregorian calendar, which in some Eastern Orthodox countries lasted until the 1940s. For example, George Washington's birthday used to be celebrated as a national holiday in the United States on February 22, but was that his birthday (in 1732) in the "Old Style" Julian calendar or in the "New Style" Gregorian calendar, which was not adopted in the British colonies until 1752? Knowing that Washington was born on Friday, February 22, 1732 provides enough information to determine that the date is on the Gregorian calendar. Before 1752 the new year began on March 25 for the British. That is, the day before March 25, 1732 was March 24, 1731. Because much of the rest of Europe used the Gregorian calendar, dates between January 1 and March 24 were usually printed with both years, so Washington's birth date would have been recorded as Friday, February 11, 1731/32.

Washington's birthday was the 11th on the Julian calendar and the 22nd on the Gregorian because the Julian calendar had suffered a long-term drift of eleven days since the Council of Nicaea in A.D. 325 which prescribed the date of Easter. The drift of the celebration of Easter into summer is what prompted the Gregorian reform, and this explains why the reform was done by the Pope rather than by the government. Richards devotes the entire fourth section of his book to the calculation of the date of Easter.

An example of the second use of the fixed day count is Sir Isaac Newton observation that the 7-day week provides a tool to propose an exact date of the crucifixion of Jesus Christ [5]. The Judean observational lunisolar calendar at that time had an uncertainty of about one or two days because the new month was determined by actually observing the thin new crescent moon. All four gospels record that the death of Jesus occurred on a Friday, which was called "the preparation" [6] (for the Saturday sabbath). Because the 7-day week has been religiously kept as an uninterrupted cycle since several centuries B.C., one can calculate exactly which years are candidates for years in which the given Judean date of the Crucifixion (14 Nisan) could have fallen on a Friday. Newton narrowed the field to A.D. 33 or 34, preferring 34; most modern scholars narrow it to A.D. 30 or 33. The evidence now favors A.D. 33 [7], but the reason this date can be determined at all is the continuous day count of the week.

The best calendars are cyclically aligned with the day count. The major complaint against the Gregorian calendar is that it is not aligned with the week. Many businesses would find it advantageous to have every quarter begin on the same day of the week, and to have the same number of weeks in each quarter. Because it met this criterion, the "World Calendar" proposed earlier this century was hailed even by modern calendar experts, such as Anthony Aveni, as the "technically flawless" calendar [8]. Every quarter would begin on a Sunday and have exactly 13 weeks. This magic was attained, however, at the cost of adding one or two days each year that were not reckoned as days of the week, to bring the total from 364 days (52 weeks) up to 365 in a regular year or 366 in a leap year. Far from being a brilliant innovation, these extra days destroy most of the mathematical usefulness of a continuous 7-day cycle.

My calendrical rating system awards the Gregorian calendar the first three stars plus half a star for including the week as an uninterrupted day count, but not the half star for being aligned with the week. The World Calendar does not have an uninterrupted day count, but does have quarters aligned with its pseudo-week, so it also receives 3 1/2 stars. As the two calendars receive the same score, there is not sufficient reason to adopt the World Calendar over the Gregorian.

The Hebrew calendar gets a full star on this point because it is fully integrated with the 7-day weekday count. Every year may begin on only four days of the week, and every day of each month during the 7-month festival season can only occur on only one of four days of the week. The leap days and leap months were purposely designed to be inserted in the less important part of the year.

The most advanced calendars have multiple fixed day counts. I award no extra rating stars for this feature, but it is instructive to look at an example to appreciate the idea. The Mesoamerican calendar was used in one form or another by Mayans, Aztecs, and most Native American tribes, especially in Central America (Mesoamerica). The two principal day counts in the Mesoamerican system are a 13-day cycle of days numbered from 1 to 13 and 20-day cycle of days represented by figures or glyphs. For example, one of the glyphs was a jaguar, and the next figure in the series was an eagle. The numbers 13 and 20 have no common factors, and it is usually best if the lengths of different day counts are relatively prime. Each cycle progresses independently, so that the day 1 Jaguar is followed by 2 Eagle, and 13 Jaguar is followed by 1 Eagle. The interaction of the numbers and names can be thought of as a 13-toothed gear meshing with a 20-toothed gear. In our calendar, the analogous effect is that Monday the 1st day of the month is followed by Tuesday the 2nd. In the Mesoamerican calendar, the two cycles realign after 13 × 20 = 260 days, the "Sacred Round" that forms the heart of the calendar.

The advantage of having two or more day counts is that their lengths can be judiciously chosen to interlock in clever ways. In the Mesoamerican system, a convenient unit of time was forty days, because two of the 20-day cycles contained one more day than 3 of the 13-day cycles. On a day such as 4 Jaguar, one would know that 40 days later the day would be 5 Jaguar. That is, the day glyph would be the same after two 20-day cycles, while the day number would increase by one. This 40-day unit of length was so useful that it was given a special name, the "foot", presumably because it could be used to step off time, much like pacing off a distance.

The Mesoamericans used these interlocking cycles to great advantage, perhaps the most clever use being to name a year for the first day. Their year had 365 days, which is 1 more than a multiple of 13. If a given year began on a day with the day number 1 of the 13-day count, then the next year would begin on day number 2. Thus, the years would automatically be counted and bundled into groups of 13. Because the 365 days in a year and the 260 days in a Sacred Round have a common factor of 5, the year begins with the same day number and day glyph after 52 (that is, 260/5) years, four of the 13-year bundles. This is the origin of the well known 52-year cycle of the Aztec calendar.

5. Nested Patterns. Perhaps the most advanced feature of certain calendars is the use of nested cycles, "wheels within wheels,". The best method is to use the very same pattern to reckon increasingly larger cycles of time, which usually are not linked to any physical phenomena. According to Anthony Aveni, the many ancient nations that used such cycles were not necessarily primitive for having conceived of time cyclically, rather than linearly as we do [9]. While he argues that such nested cycles are equally as good as our systems, my rating system asserts that nested cycles are superior and deserve a fifth star.

The most common use of nested cycles is to count years in the same way as days. For example, the Hebrews count days by sevens and also count years by sevens. Moreover, there is one especially holy year which was to be reckoned, according to the law of Moses, in exactly the same manner as one especially holy day: the jubilee year was the fiftieth year and the feast of Pentecost was the fiftieth day.

I award half a star if the calendar has any nested cycles at all, which usually means any grouping of years, such as the weeks and jubilees of the Hebrew calendar; and the other half star if the larger pattern is essentially the same as the smaller. Again the Hebrew calendar would probably get a full star simply because weeks of years is the same pattern as weeks of days. It would be better, however, if the calendar had Big Years averaging about 365 years each.

The 5-star Enoch Calendar. Let us see what would have to be done to the Gregorian calendar to achieve a five-star rating. The following discussion is based on the calendar in the Book of Enoch, which is old enough to have been accepted as extremely ancient at the time of Christ [10]. The calendar was based on the week as a fixed 7-day count, and it was designed to align with the week. It divided the year into four seasons of 91 days, each composed of three 30-day months followed by an equinox or solstice day, which could be considered the 31st day of the last month of a quarter. Each season had 13 weeks, presumably beginning on a Sunday and ended on a Saturday. Thus a regular year had 364 days, with every year beginning on Sunday.

The Enoch calendar has been criticized as hopelessly primitive because, with only 364 days, it would get out of sync with the seasons so quickly: in only 25 years the seasons would arrive an entire month late. Such a gross discrepancy, however, merely indicates that the method for intercalation has been omitted. Richards notes the clever quarters of the basic Enoch calendar, along with a proposed modification by Searle to intercalate entire weeks to exactly equal the accuracy of the Gregorian calendar, boosting the calendar to a 4-star rating [11]. It turns out, however, that there is a better way to intercalate entire weeks so that the calendar will merit a 5-star rating.

First, count years by weeks of sevens to begin the nesting pattern of counting years in the same way as days. Thus, years could be thought of as named "Sunday" through "Saturday," in repeating groups of seven. Each year except Saturday years would have 364 days. Most Saturday years would contain an entire extra week, for a total of 371 days, so that a week of years would usually average 365 days in length, since 365 = 6 × 364 + 371)/7. That extra week could be placed at a convenient vacation time, not counted in the quarter of most businesses, just as many businesses currently shut down during the week of Christmas. This is an example of the short-term accuracy of a calendar being subordinate to a simple pattern aligned with a fixed day count.

But what about long-term accuracy? Using the intercalary unit of an entire week, in order to synchronize with the short-term pattern, it is easy to correct for long-term accuracy. Each fourth Saturday year would contain two extra weeks, for a total of 378 days, so that the average length of the year in a 28-year period would be 365.25 days. Such long years might seem unusual at first, but they pose far fewer problems than we have with our current calendar. The main reason to reckon years at all is to track the seasons, especially for agriculture, and for such purposes a precision of two weeks rarely poses a problem. In order to keep aligned with the moon, the Hebrew calendar inserts an entire extra month about every third year, with a maximum year length of 385 days, and that calendar has been used for agriculture for thousands of years.

The years in this Enoch calendar could be bundled in the same way as days, into groups of 364. Each Great Year would be divided into 12 Great Months of 30 years each, with 4 special years for Great Seasonal markers. The simple pattern of 28 years aligns with the Great Year since 364 is evenly divisible by 28.

The final long-term correction is that in every set of five Great Years, two of the extra weeks ending the 28-year cycle would be skipped, one in the third and another in the fifth Great Year. In those two years there would be only one extra week of years rather than two. That correction results in an average year length of 365.2423 days, which better approximates the current year length of 365.2422 days than does the Gregorian year of 365.2425 days.

Conclusion. Under my rating system, most of the calendars discussed in Richards's book receive about a 3-star rating. I propose this system as a way to measure and to appreciate different features of the wide variety of calendars that have been devised over the millennia, and that are so well summarized in Mapping Time.


  1. L. E. Doggett, Calendars, Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac, 1992. Unfortunately, Richards has a typographical error in this equation.
  2. C. W. Allen, Astrophysical Quantities, University of London, 1976, p. 18.
  3. The effect is noted by Nachum Dershowitz and Edward M. Reingold, Calendrical Calculations, University of Cambridge Press, 1997, p. 152.
  4. From Doggett, op. cit.
  5. J. P. Pratt, Newton's Date for the Crucifixion, Quar. Jour. Roy. Astr. Soc. 32 (1991) 301-304.
  6. Mat. 27:62, Mark 15:42, Luke 23:54, John 19:42. Some scholars try to force an interpretation that "preparation" means only the day 14 Nisan, which was the preparation day for the Passover (John 19:14) and could fall on other days of the week, but the usage at the time was that "preparation" meant what we now call "Friday."
  7. Harold Hoehner, Chronological Aspects of the Life of Christ, Zondervan, Grand Rapids, 1977, p. 111.
  8. Anthony Aveni, Empires of Time, Basic Books, New York, 1989, p. 162.
  9. Aveni, Anthony, op. cit., pp. 330-333.
  10. It was so quoted in Jude 14 as having actually been written by Enoch. It is also called 1 Enoch and is found in the collection of apocryphal writings called the Pseudepigrapha.
  11. Mapping Time, pp. 116, 119.