# A closer look at Gregory's calendar

Jan 1, 2003To the editor:

A new millennium draws near, and the navigator who peeks behind this momentous calendar event for a closer look at the Gregorian calendar will find a rich treasure of historical interest, some useful tips, and a fascinating way of telling the day of the week of any date, past or present, all without even looking at a calendar.But first a bit of the basics.

From antiquity man has been aware of the never-ceasing cycles in the universe around us. With the advent of agriculture, and thus government, religion, and business, fixing, scheduling, and predicting dates of interest became important to civilization. The calendar is the device for rationalizing these rhythms of nature: day, month, and year. Of course, there is no absolute way of making a non-varying calendar, since the cycles do not neatly interlace. A solar day, the natural calendar unit, is the daily rotation of the earth around its axis relative to the sun. There are 365.242190 solar days in one orbit of the earth around the sun (the tropical year), which leaves a devilishly inconvenient (and approximate) five hours, 48 minutes, 45 seconds more than a whole number of days.

The moon is even more fickle. Complex interactions of gravitational influences cause the moon's orbit to vary in so-called nodal cycles of about 19 years, during which the mean lunar orbit is 29.5306 days. (Actually, it is somewhat more of a problem, since the Earth's spin and orbit, as well as the moon's orbit, vary over time.)The history of calendars is the history of mankind's attempts to force these three square pegs into the round holes of a fixed, neatly repetitive calendar. The first calendars were moon-based. At least as long ago as 30,000 years, pre-Neolithic men were notching what is almost certainly a moon-phase diagram into animal bones, perhaps to let them know when nights would be lit by a bright moon. When agriculture arose, the year became an important cycle.

In Egypt, calendars were in existence 3,500 years ago to predict the annual flooding of the Nile, a critical event signaling the planting season. The early Egyptians also sought to coordinate both the lunar and solar cycles, as has been done in the Arabian and Jewish calendars, which are still used today. In all cases, however, the lack of whole numbers of days in the various orbital cycles meant that the calendars could not be permanent. A calendar year of 365 days would outpace the actual sun. Months, too, moved out of synch with the actual moon. In time the discrepancies in the calendar cumulated until the seasons shifted from their accustomed places in the calendar year, and great complications arose in trying to figure out a system of months.

Both the Babylonians and the Egyptians knew that a 365-day calendar year was in error. Their calendars inserted (intercalated) additional days from time to time to correct the calendar drift. The Roman calendar in use long before Julius Caesar had dropped the leap-year intercalation. In 47 BC, Julius Caesar, accepting the suggestion of the Greek astronomer Sosigenes, re-introduced the leap-year day every four years. (Alas, nothing was done to straighten out the months. Indeed, months were added, named after various notables, like Julius Caesar, July, and Caesar Augustus, August. This left us with the anomaly of having October, November, and December, whose names derive from the Latin for eight, nine, and 10, as the 10th, 11th, and 12th months.)Of course, the leap-year fix only reduces the error, it doesn't eliminate it. An extra day added every four years makes the average year equal exactly 365.25 days. In fact, this is an overcorrection, making the calendar year slightly longer than a tropical year.

By 1582, the error had again cumulated until the calendar was 11 days behind the earth's actual position in its orbit. The calendar thus indicated a vernal equinox well in advance of its original March 21 datecreating problems in determining the date for Easter, which was set as the first Sunday following the first full moon following the vernal equinox. This left the Catholic Church, which adopted the Julian calendar in 365 AD, with a need to reform the calendar so that Christendom's most solemn observance could be more precisely determined.This Pope Gregory XIII did in 1582, by skipping three leap years every 400 years. He decreed that centurial years (i.e., those ending in 00, like 1700, 1800, etc.) are not non-leap years unless exactly divisible by 400. This created a 400-year cycle of three centuries each with 76 common years of 365 days and 24 leap years of 366 days; plus one century with 75 common years and 25 leap years. This is a cycle of 146,097 days.

The Gregorian scheme has some interesting features. First, a leap year in a centurial year is rare. The upcoming year 2000 is a centurial leap year, a once-in-four-hundred-year event. Further, while the Gregorian calendar tracks the earth's motion more closely than did the Julian calendar, it, too, is not exact. It now is faster than the actual tropical year by about 29 seconds. Fortunately, it will take another 2,600 years or so for this error to amount to a full day, so a further correction will not be required for a long time.

A serendipitous feature of the Gregorian calendar is that its 400-year cycle contains an exact number of weeks20,871, to be precise. Thus, each cycle will start on the same day of the week. New Year's Day 2000 will fall on a Saturday, and so will each succeeding centurial leap year, of which the next is 2400. Less helpful is the fact that, in between cycles, each New Year's Day falls on a series of weekdays that repeat every 28 years. This is why you need to change your calendar each year, if you want a calendar that indicates days of the week. This inconvenience results from the fact that a 365-day year is 52 weeks and one day. New Year's Day in the next year will start one weekday later. That is, if a common year begins on a Sunday, the next year begins on a Monday. It is different for a leap year, due to its extra day. If a leap year begins on a Sunday, the next year begins on a Tuesday. That is, the leap year causes the next year to skip (get it, leap) over one weekday.

The week is another man-made cycle, wholly unrelated to any natural celestial event. Combined with the now inconsistent variation in the numbers of days in the months, finding the day of the week for any given date appears to be a formidable undertaking. Actually, it is not. There are simple but little-known algorithms for finding the day of the week for any Gregorian date. One form that I have developed, which is harder to describe than to use, can be worked in your head with nothing more than simple arithmetic. The only difficulty is the need to memorize a one-digit key number for each of the 12 months of the year. Once that is done, you can dispense with calendars and give your shipmates the impression you have total recall.

The principal is elementary: since weeks are recurring seven-day cycles, all you need to know is how many days have elapsed between a reference date (whose day of the week is known) and the date in question, divide by seven, and look only for the remainder. The number of whole weeks doesn't matter. For example, New Year's Day 1995 was a Sunday. Nine days later (i.e., one week and two days), must fall on a Tuesday, and so must any date that is a whole number of weeks and two days later. Actually, you do not need to count the number of days between the reference date and the date in question. You simply break up the intervening period into segments (years, months, and days) and keep track of the partial weeks. When partial weeks in these segments add to more than seven days, you can cast out more 7s until left with only a remainder of 0 through 7.

The algorithm works like this:

1. Take the last two digits of the year. Divide this by 4, and ignore any remainder. Add the second number to the first. Example: May 15, 1935. The last two digits are 35. One-fourth of 35, ignoring the remainder, is 8. The sum of these (35 + 8) is 43.

2. Add the day of the month to the number just calculated. In the example, this is 43 + 15 = 58.

3. Take the key number for the month in question from the table below and add it to the number calculated in Step 2. For May (month 5) the key number is 1, so 58 + 1 = 59. Month 1 2 3 4 5 6 7 8 9 10 11 12 Key 0 3 3 6 1 4 6 2 5 0 3 5

4. Divide by 7 and retain only the remainder. That is, cast out 7s. Thus, 59 divided by 7 gives 8, remainder 3, of which only the 3 is relevant. This number indicates the weekday. Monday is 1, Tuesday is 2,Sunday is 7 (or 0, both work). So, May 15, 1935, fell on a Wednesday.

5. For dates before February 29 of a leap year, a correction is needed. Subtract 1 from the remainder obtained in step 4.

6. For Gregorian dates other than in the 1900s, a century correction is needed. For dates in the 1800s add a correction of 2; for the 1700s, add 4; for the 1600s and 2000s, add 6. Indeed, the correction is itself a repeating cycle of 6, 4, 2, 0. Example: July 4, 1776. Steps 1-4 give a weekday number of 0, to which we add 4. Thus, the first Independence Day was a Thursday.

See if you can deduce that December 7, 1941, was a Sunday.

Why the algorithm works is worth knowing. Step 1 adds up the extra days in the years from the beginning of the century. (Remember? There is one extra day for each common year and yet one more for each leap year.) Step 3 keeps track of the extra days that cumulate during the year. The key-numbers represent the leftover days up to the beginning of the month in question. The leap year correction in Step 5 takes account of the fact that the calendar adds the extra day at the end of February, but Step 1 effectively adds it at January 1. Step 6 merely adjusts the different starting weekday for each of the four centurial years in the 400-year cycle.

Memorizing the monthly keys for Step 3 is no more difficult than remembering a long-distance telephone number. In fact, all you need to keep in mind is 0336-1462-5035, which is the string of the key numbers. To tie these to the month, just relate them to the month number (e.g., August, which is the 8th month, is the 8th key in the table: 2).

Another tip for using the algorithm is that casting out sevens can be done at any step. This can reduce every carry forward to a one-digit number. Applying this to the example given above, the 43 in step 1 is carried forward as merely 1. In Step 2, 15 becomes a 1, which then makes a carry forward of 2, and so on.

Since most quests for a day of the week involve the current year, another convenient thing to do is to reduce step 1 to its one-digit number and just keep it in mind during the year. For example, 1999 produces a carry forward of 4 in Step 1. Remembering this for 1999 lets you avoid the arithmetic in Step 1 for all of 1999.

The Gregorian calendar is not necessarily the best calendar. A number of other systems have been suggested from time to time. It would be convenient to have a calendar that more conveniently meshed the days of the week, moon phase, and months, but all modern calendar reforms designed to improve the Gregorian calendar have failed. When you realize it was adopted even before Galileo demonstrated that Earth orbits the sun, not vice-versa, we owe a grudging admiration to this durable calendar. It still forms the time line by which we list the positions of celestial landmarks in the nautical almanacs.

Armed with the knowledge presented here, you should be able to master the nuances of this relic of the Renaissance.