Argumentum VI. De cyclo lunari.
Si vis scire quotus cyclus lunae est, qui decemnovennali circulo continetur, sume annos Domini, ut puta, DXXV, et subtrahe semper II, et remanent DXXIII. Hos partire per X et IX, remanent X. Decimus cyclus lunae est decemnovennalis circuli. Quoties autem nihil remanet, nonus decimus est.
Argumentum 6. On the lunar cycle.
If you want to know which cycle of the moon it is, that is contained in the nineteen year circle, add the years of the Lord, say 525, and always subtract 2, and 523 are left over. Divide those by 10 plus 9, 10 are left over. It is the tenth lunar cycle in the nineteen year circle. And whenever nothing is left over, it is the nineteenth.
Thus, for year number Y, lunar cycle( Y ) = 1 + (Y - 3)mod 19, which is also known as the Jewish lunar cycle number machzor. Besides (Y mod 19) as used in [Argumentum 3] and the "cycle of nineteen years" of [Argumentum 5], it is the third function essentially equivalent to (Y mod 19). It is used only in [Argumentum 13] to compute a kind of Alexandrian epacts.
Argumentum VII. De luna decima quarta in mense Martio.
Si vis nosse quibus annis decemnovennalis circuli Martio mense, XIV luna paschalis incurrat: anno II, V, VII, X, XIII, XVI, XVIII, hos suprascriptos VII annos in Martio mense reperies; residuos vero XII, secundum regulam subter annexam, Aprili mense indubitanter calculabis.
Argumentum 7. On the fourteenth moon in the month of March.
If you want to find out in which years of the nineteen year circle the 14th paschal moon occurs in the month of March: in the year 2, 5, 7, 10, 13, 16, 18, in these 7 years above you shall see it in the month of March; but in the remaining 12 you will calculate it without doubt in the month of April, according to the rule appended below.
These are in fact all the numbers Y of years in which the age of the moon computed with the rule in [Argumentum 9] is 14 on some day from March 21 to March 31 (with (Y + 9)mod 19 mod 8 mod 3 = 0). The referenced rule probably is the one in [Argumentum 9] for April.
Argumentum VIII. De bissexto.
Si vis scire quando bissextus dies sit, sume annos Domini, ut puta DXXV. Partire hos per IV. Si nihil remanserit, bissextus est. Si I aut II, vel III, remanent, bissextus non est.
Argumentum 8. On the leap day.
If you want to know when the leap day is, add the years of the Lord, say 525. Divide those by 4. If nothing should be left over, there is a leap day. If 1 or 2 or 3 are left over, there is no leap day.
This says that Y is the number of a leap year iff Y mod 4 = 0.
Ne tibi forsitan aliqua caligo erroris occurrat, per omnem computum per quem ducis, si nihil superfuerit, eumdem computum esse per quem ducis agnosce, ut puta, si per X et IX ducis, et nihil superfuerit, XIX esse; si per XV, quindecimum, et, si per VII, septimum.
So that any unclarity does not possibly lead you into error, for all divisions you do, if nothing is left over, you should consider this computation to yield that by which you divide, thus for instance, if you divide by 10 plus 9, and nothing would remain, you should consider it to be 19; if by 15, then fifteen, and if by 7, then seven.
This rule says that the remainder of( A )upon division by( B ) = 1 + (A - 1)mod B rather than just A mod B. This rule, however, is not always applied: (a) The remainder operations by 19 and by 30 in [Argumentum 3] and [Argumentum 11] must yield 0, so that, for Y mod 19 = 0, the epact is 0 (as asserted in [Argumentum 14] and the table above) and not 29; (b) in [Argumentum 12], the remainder upon division by 7 can be "nihil".
Argumentum IX. De luna paschali mense Martio.
Si vis cognoscere quota luna festi paschalis occurrat; si Martio mense Pascha celebratur, computa menses a Septembri usque ad Februarium, fiunt VI. His semper adjice regulares II, fiunt VIII; adde epactas, id est adjectiones lunares cujus volueris anni, ut puta, indictionis tertiae XII, fiunt XX; et diem mensis qua Pascha celebratur, id est Martii XXX, fiunt simul L. Deduc XXX, remanent XX; vicesima est in die resurrectionis Domini.
Argumentum 9. On the Easter moon in the month of March.
If you want to learn which moon it is on which the feast of Easter occurs; if Easter is celebrated in the month of March, compute the months from September to February, yielding 6. To this always add the correction 2, yielding 8; add the epacts, that is, the lunar increments of the year you want, say 12 for the third indiction, yielding 20; and the day of the month on which Easter is celebrated, that is March 30, yielding together 50. Deduct 30, 20 are left over; the twentieth [moon] is on the day of the resurrection of the Lord.
This amounts to age of the moon on( Julian date(Y, March, D) ) = ( (Y mod 19)*11 + 6 + 2 + D )mod 30 = ( age of the moon on(Julian date(Y, March, 22)) - 22 + D )mod 30 if Easter is Julian date(Y, March, D). But of course it works for any day number D between 22 and 31, and for all year numbers Y, not just those of [Argumentum 7]. The year number for the example could be 0525.
In this calculation and the following one for dates in April, Dionysius suggests that the epacts for year Y not only give the age of the moon at March 22, as stated in [Argumentum 11], but also at some day around September of year (Y - 1). Only late August and late September would work: Julian date(Y, March, 22) ~= 7 synodic month + Julian date( Y - 1, August, 27.29 or 28.29 ) ~= 6 synodic month + Julian date( Y - 1, September, 25.82 or 26.82 ) (where the second day numbers are to be taken iff Y is divisible by 4).
Mense Aprili. -