It is not clear from the text, however, when the years of the incarnation are supposed to begin. The formulae imply that Y and J agree on January 01 ([Argumentum 12]), on the leap day ([Argumentum 8]), and around Easter; and [Argumentum 2] suggests that Y and J agree until September 01. The year 0001 of the era of Diocletianus, which Dionysius wants to replace with era domini, is usually taken to start at the sunset epoch Julian date( 0284, August, 28.75 ).
Some of the assertions on the birthday of Jesus in [Argumentum 15] below would render "anni ab incarnatione Domini" a misnomer.
Argumentum II. De indictione.
Si vis scire quota est indictio, ut puta, consulatu Probi junioris, sume annos ab incarnatione Domini nostri Jesu Christi DXXV. His semper adjice III, fiunt DXXVIII. Hos partire per XV, remanent III. Tertia est indictio. Si vero nihil remanserit, decima quinta indictio est.
Argumentum 2.
On the indiction. If you want to know which indiction it is, say in the consulship of Probus Junior, then add the years since the incarnation of our Lord Jesus Christ, 525. To this always add 3, yielding 528. Divide these by 15, 3 are left over. It is the third indiction. But if nothing would be left over then it is the fifteenth indiction.
For year number Y this gives indiction( Y ) = 1 + (2 + Y)mod 15 as is confirmed by the second column in the table above. This is in fact the cycle of indiction number for Julian year Y from January 01 until that number changes later in the year (on September 01 or some time later).
Argumentum III. De epactis.
Si vis cognoscere quot sint epactae, id est adjectiones lunares, sume annos ab incarnatione Domini nostri Jesu Christi, quot fuerint DXXV. Hos partire per XIX, remanent XII. Per XI multiplica, fiunt CXXXII. Hos item partire per XXX, remanent XII. Duodecim sunt adjectiones lunares.
Argumentum 3. On the epacts.
If you want to learn the number of epacts, that is, of the lunar increments, then add the years since the incarnation of our Lord Jesus Christ, of which 525 have passed. Divide those by 19, 12 are left over. Multiply by 11, yielding 132. And then divide those by 30, 12 are left over. Twelve is the lunar increment.
For year number Y this is meant to describe the formula epacts( Y ) = ((Y mod 19)*11) mod 30 (where both modulo operations can yield zero) as is confirmed by the third column in the table above. Since Y is integral, this is the same as epacts( Y ) = floor( Y*(235/19)*30 ) mod 30 which shows that the formula uses the Metonic value of (calendar year)/(synodic month) ~= 235/19 ~= (365.25 d)/(29.530 85 d). This estimate of the synodic month exceeds modern estimates by only 1 d in about 300 years.
The formula for the epacts does in fact extend the epacts given for the Diocletian year numbers D = Y - 284 in the table above. Note that ( Y - D ) mod 19 = 18, which makes the formula for Y somewhat simpler to express verbally than that for D (because no "regulares" are needed): epacts for Diocletian year number( D ) = ((D - 1)mod 19)*11) mod 30.
The formula for the epacts remains the same if the year number Y is replaced by the year number S = Y + 38 since the Spanish era (this count may have been known to Dionysius).
Argumentum IV. De concurrentibus.
Si vis scire adjectiones solis, id est concurrentes septimanae dies, sume annos ab incarnatione Domini quot fuerint, ut puta DXXV; per indictionem tertiam et annorum qui fuerint quartam partem semper adjice, id est, nunc CXXXI, qui simul fiunt DCLVI. His adde IV, fiunt DCLX. Hos partire per VII, remanent II. Duae sunt epactae solis, id est concurrentes septimanae dies, per suprascriptam indictionem, consulatu Probi junioris.
Argumentum 4. On the concurrents.
If you want to know the solar increments, that is the concurrent days of the week, add the years since the incarnation of the Lord that have passed, say 525; for the third indiction and the years that have passed until then always add the fourth part, which is now 131, these yield 656 altogether. To these add 4, yielding 660. Divide those by 7, 2 are left over. Two are the epacts of the sun, that is, the concurrent days of the week, for the indiction described above, in the consulship of Probus Junior.
For year number Y, this is intended to give concurrentes( Y ) = 1 + (3 + Y + floor(Y / 4)) mod 7 as is confirmed by column four of the table above.
With the numbering of [Argumentum 12] for the days of the week (but with 7 instead of 0 for Saturday), this amounts to concurrentes( Y ) = day of the week(Julian date( Y, March, 24 )) which agrees with the concurrents for year number Y as defined by Bede about 200 years later.
Argumentum V. De cyclo decemnovennali.
Si vis scire quotus sit annus circuli X et IX annorum, sume annos Domini, ut puta, DXXV, et unum semper adjice, fiunt DXXVI. Hos partire per X et IX, remanent XIII. Tertius decimus est annus cycli decemnovennalis. Quod si nihil remanserit, IX decima est.
Argumentum 5. On the cycle of nineteen years.
If you want to know which year it is in the circle of 10 plus 9 years, add the years of the Lord, say 525, and always add one, yielding 526. Divide those by 10 plus 9, 13 are left over. The year is the thirteenth in the nineteen year cycle. If nothing would be left over, it is the 9teenth.
Thus, for year number Y, cycle of nineteen years( Y ) = 1 + ( Y mod 19 ), which is also known as the Numerus Aureus of the year. It is used only in [Argumentum 14].