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different ways, depending upon whether leap years start with the first, second, third, or fourth year within the 19-year cycle: |
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Leap Year
Starts in Year | Leap Years Are | | 1, 5, 9, 13, 17 | | 2, 6, 10, 14, 18 | | 3, 7, 11, 15, 19 | | 4, 8, 12, 16 |
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In three cases there are five leap years, and in one case there are only four leap years. Thus, on average, in every four cycles, five days are added three times, and four days are added once. The total number of days in these lunations is thus (3(6,935 + 5) + (6,935 + 4))/4 = 6,939.75 days, which is exactly the number of days in 19 Julian years calculated above. |
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This scheme, therefore, allows us to create a lunar cycle table in which the phase of the moon on any given day in a given year of the cycle will be the same in the equivalent year of the next cycle. In short, a table showing the phases of the moon for 19 Julian years can be used for any year by simply determining the number of the year in the 19-year cycle. This number is the golden number, or numerus aureus. |
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The carefully constructed relationship between lunations and solar year, however, is based upon oversimplified and incorrect values for both the length of the tropical year and the synodic lunar period. The true synodic lunar period is 29.530588 days, which in 235 lunations is 6,939.68818 days, not the assumed 6,939.75 days. The error is 0.06182 days, which amounts to one day in about 307 years. After this period, new moons occur one day earlier than indicated by the golden number. |
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As part of the Gregorian reform, then, the golden number was dropped as a useful tool for calculating Easter date. In its place, the epact was adopted. The term epact came to be used to signify the age of the moon at the first of the year. The idea of epact is quite simple. If a new moon falls on 1 January of one year, then it will be 11 days old on 1 January of the following year, since the difference between the solar year and lunar year is (365 354 =) 11 days. The distortions of leap years and the inaccuracy of the lunar cycle, however, greatly complicate this situation. |
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Lilius applied his considerable talents to the required calculations and discovered that the omissions of leap year in century years had the effect of decreasing the value of epact by unity, whereas the errors in the lunar cycle had the effect of increasing the value of epact by unity, with the result that the two effects tended to cancel each other out. |
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After deliberation, it was decided that corrections for errors in the lunar cycle should be made at the end of 300 years. It was assumed that the necessary correction was one day in 312.5 years, which is eight days in 2,500 years, and a schedule of |
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