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Note that the succession of year types also follows a pattern in all cases. If the year type for a given occurrence of the pattern is n, then the year type for the following occurrence four years later is given by the expression (n 2) with the proviso that numbers less than one have seven added to them. Thus, occurrence 1827/1877 has year Type 2, so the following occurrence 1831/1881 has type (2 2) = 0, which is less than one, so seven is added, and 0 + 7 = 7. |
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This rule works because the four-year period between leap years contains a total of 1,461 days (365 + 365 + 365 + 366). This period of 1,461 days is hereafter referred to as a quad (since its period is four years). The number 1,461 divided by seven leaves a remainder of five. Hence, if a given year starts (January 1) on a given DOW, the year four years later will start on the DOW five days later in the cycle of seven days per week. Within the weekly cycle of seven days, moving forward five days is the same as moving backward two days. The rule can be written just as well as (n + 5), where numbers greater than seven have seven subtracted from them. |
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Up to now, I have described what the calendar does, but I have not attempted to determine why it does it. The following discussion shows that the cycles depend upon the number of days between dates, not upon calendar years, and also shows that there may be a large number of cycles operating at any given time. |
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The first thing you must realize is that the calendar year is not the critical factor in determining the period of the cycles. Clearly, if the DOW of 1 January 1827 matches the DOW of 1 January 1877, then the DOW of the previous day must also match for both years, so the DOW of 31 December 1826 must also match the DOW of 31 December 1876. Once an agreement is established between two date sequences, some event must occur to interrupt the match of dates and DOWs in order for the matching to fail. The only condition that interrupts the DOW sequence is the intercalation of a day, and in the Gregorian calendar, this only occurs between 28 February and 1 March, according to the leap year rule. Thus, the boundary that matters is not the boundary between calendar years, but the boundary between computational years at 28 February and 1 March in leap years, since that is the only time that the correspondence between dates and DOWs can be changed. |
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At this point, I will use the Julian Day number to simplify matters. The JD#, as it is called, is a sequential count of days that have elapsed since 1 January 4713 B.C. Using this number allows us to easily count the number of days that have elapsed between two dates. Calculating the JD#s for the two dates of 1 January 1827 and 1 January 1877, I find that they are separated by 18,263 days. As long as two dates are separated by this number of days, under certain conditions, their DOWs will match. Table 9.4 shows how this works for the dates around 1 January 1827 and 1877. |
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Checking the DOWs for all the dates shown and the intervening dates, you can verify that the DOWs match for all dates separated by 18,263 days and do not match for dates separated by 18,262 days. |
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This discovery reveals several things. First, the match does not just cover the years 1827 and 1877. These are the complete years that match, but the matching between |
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