46. The ogdoas and the hendecas
THE RECKONING OF TIME
hold true because of the intervention of the Paschal Moon, whose ¢nal days not infrequently fall on the kalends of May and can also last until the second or third day after the kalends, but which is nonetheless the Moon of April and not of May, as it ought ever to be said to be.
46. THE OGDOAS AND THE HENDECAS
There is another division of the aforementioned [19-year] cycle by which the years are separated into the ogdoas and the hendecas. This happens in the third and the sixteenth year [of the lunar cycle], because /423/ in the eighth and eleventh year [of the 19-year cycle],13 the Paschal Moon reaches the furthest limits of its nativity,14 and in an anomalous manner a single common year precedes the embolismic year in both cases. Alternately, [this came to pass] because the ancients saw that eight solar years equals the number of days in the same number of lunar years.15 Later on, more learned doctors proved that this is impossible unless eleven years were added. The entire cycle of the Moon is distributed through the ogdoas and the hendecas, and demonstrates both observations. For at one time the Greeks, whom the ancient Romans are also said to have imitated, when they had computed eight common lunar years of equal length, intercalated three embolismic months. For should you wish to calculate 8 times 11ˆ~˘, you will make 90 days, that is, three months. In fact, there are now people who think that 8 solar years equal 8 lunar years, and similarly that 11 solar years make up the same number of lunar years, with an equal number of days.16
At this point it should be stated, ¢rst of all, without any contradiction, that if 8 years of both planets correspond, then this must always be the case, /424/ and it is impossible for 11 years [of both planets to correspond]; but if 11 years of both planets begin and end together, they will always do so, nor will there be any time when the correspondence of 8 years can be restored.17 In order to hit upon this number, it should be observed that 8 solar years, excluding the leap-year days, make 2,920
13 On the di¡erence between the lunar cycle and the 19-year cycle, see below, ch. 56.
14 I.e. the new Moon of Easter occurs at its latest possible calendar date.
15 On the ancient eight-year cycle or octae« teris, see Introduction, section 3.
16 ''The entire cycle of the Moon . . . equal number of days.'' Cf. Macrobius, Saturnalia 1.13.8^10 (62.23^63.9)
17 I.e. one cannot have both an 8-year and an 11-year luni-solar cycle.
TRANSLATION
days, for 8 times 365 equals 2,920. But note how many days that number of lunar years has ^ 8 times 354 is 2,832 ^ and then add the 90 days of the three embolismic months, and these make 2,922. But the two days lacking in the ogdoas of the Sun can expect to be supplied. Let us look at the hendecas of both planets, and see if perhaps there the Sun can summon the help of the leap-year day, for both periods of time [ogdoas and hendecas] must be bound by the same rule [concerning leap years].
Eleven times 365 makes 4,015. Likewise 11 times 354 makes 3,894. Add 120 days for the embolismic months, and it makes 4,014. Take away the saltus, and there remains 4,013. See, then, whether here the hendecas can make use of the assistance /425/ of the leap-year day, so that by adding two or three days, it will su⁄ce to compensate the lunar hendecas.
Certainly it does not need the help of the quarter-day in order to equal the lunar hendecas, for it already exceeds it by two days; rather, it replaces what is missing in the ogdoas by these excess days. So it is plain that in harmonizing their periods, there is nothing which prejudices the leapyear days, for however many occur in the 19-year cycle, they bene¢t both planets equally, as we have said.18
Now to gut the bowels of this question! In the ¢rst year of the Paschal cycle the epact is zero, because the Moon ¢nishes its course on the 11th kalends of April [22 March]. Again in the ninth year of the cycle, that is, after 2,922 days, on the 9th kalends of April [24 March] the Moon is 30 days old, which the sequence of the epacts (which are then 28) plainly shows. What good was it /426/ to add two days of the Sun's course ^ that is, the 10th and 9th kalends of April [25 and 24 March] ^ in order to complete the lunar ogdoas, if two leap-year days could have completed them? Therefore the two days by which the ogdoas of the Sun lags behind the ogdoas of the Moon are not supplied by the intercalation of the leap-year days, but by the two days in which the Sun's hendecas exceeds the Moon's. Nor does the saltus lunae harm it, which indeed some place in the ogdoas,19 but which we have stated should be placed in the hendecas. But wherever you position it, the question we posed is solved at the same point. The 19-year cycle has 228 solar months and 235 lunar months, making 6,935 days, excluding leap-year days: 19 times 12 is 228, plus 7 is 235; again 19 times 365 is 6,935, and 19 times
354 is 6,726, plus 210 days for the 7 embolismic months, making 6,936.
18 See above, ch. 41.
19 I.e. Victorius.