These are the same rules as for the first year.
Ita singulis annis a primo usque ad nonagesimum quintum annum calculabis. Si quando mense Martio XIV luna paschalis incurrit, XXXVI regulares imprimis teneas, ex quibus epactas cujus volueris anni deducas, et concurrentes adjicias, et in fine: semper IV regulares augmentes. Aprili vero mense semper XXXV in capite tene, ex quibus, ut supradictas epactas, et adjectos ejusdem anni concurrentibus suis regulares in fine VII augmenta. Facilius namque et brevius omnia argumenta paschalia calculabis. Hoc tamen praeterea lectori sit cognitum, quoties in utrosque menses suprascriptos in prima regula contigerit, ut deductas epactas, amplius a XXX remaneant, dimitte XXX. Quod si unus aut duo, vel amplius superfuerint, tot dies ipsius mensis a calendis Januarii [ ? Aprilis ] sit luna paschalis XIV. Quando autem (post) deductas epactas infra XXX [? XXI] ut puta XX, seu amplius minusve remanserit, quod semel in XIX annis accidere manifestum est, XXX die Aprilis erit luna paschalis XIV.
In this way you calculate for each year from the first to the nineteenth year. When the 14th moon occurs in the month of March, then you first take the correction 36, from which you deduct the epacts of the year you want, and add the concurrents, and finally you always add the correction 4. But in April you keep 35 in mind, from which [you take] the epacts mentioned above, and finally add the correction 7 increased by the concurrents of the same year. Thus you will calculate all the argumenta for Easter more easily and faster. Above all, let the reader know that, whenever it happens that more than 30 are left over when the epacts are deducted for any of the months described above to which the first rule applies, then dismiss 30. When one or two or more are left over, then so many days from January 01 [ ? April 01 ] is the 14th paschal moon. And if less than 30 [ ? 21 ] should be left over (after) the epacts have been deducted, say 20, or more or less, which is bound to happen once in 19 years then the 14th paschal moon will be on the 30th day of April.
The first part repeats the rules which have already been applied in the preceding paragraphs to the cases where Y mod 19 is <= 2.
The last portion of the text contains several errors and seems to deal with the case when the "first" formula March 36 - ( (Y mod 19)*11 )mod 30 d is applied when the 14th paschal moon is in April, in which case it yields a date after March 31 or (a wrong one) before March 21. Thus, if 36 - ( (Y mod 19)*11 )mod 30 is > 30 then it is also > 31 and the 14th paschal moon is on April 00 + (5 - ( (Y mod 19)*11 )mod 30) d rather than on April 00 + (6 - ( (Y mod 19)*11 )mod 30) d as asserted in the text. And if 36 - ( (Y mod 19)*11 )mod 30 is < 21 then the 14th paschal moon is again in April. Note that the example where it is supposed that 36 - ( (Y mod 19)*11 )mod 30 equals 20 cannot occur (because ( (Y mod 19)*11 )mod 30 is never 16 for integral Y). The largest values < 21 that can occur are 19 (for Y mod 19 = 7) and 18 (for Y mod 19 = 18). And, of course, a 14th paschal moon never is on April 30.
Argumentum XV. De die aequinoctii et solstitii.
Qua die natus est Dominus Jesus Christus secundum carnem ex Maria Virgine in Bethlehem, in qua incipit crescere dies. Aequinoctium primum est in VIII calendas Aprilis, in qua aequatur dies cum nocte. Eodem die Gabriel nuntiat sanctae Mariae, dicens: Spiritus sanctus superveniet in te, et virtus altissimi obumbrabit te. Propterea quod ex te nascetur, vocabitur Filius Dei. In qua etiam passus est Christus secundum carnem. Solstitium secundum est VIII calendas Julii, quando etiam natus est sanctus Joannes Baptista ex quo incipit decrescere dies. Aequinoctium secundum est VIII calendas Octobris, in qua die conceptus est Joannes Baptista. Et hinc jam minor efficitur dies nocte, usque ad natalem Domini Salvatoris. Ex VIII calendas Aprilis et in VIII calendas Januarii, dies numerantur CCLXXI. Unde secundum numerum dierum conceptus est Christus Dominus noster in die dominica VIII calendas Aprilis, et natus est in III feria XIII calendas Januarii Christus Dominus noster. In die qua passus est, fiunt anni CXXXIII [ ? XXXIII ] et menses III, qui sunt dies XII CCCCXIIII. Unde secundum numerum dierum ejus stat cum III feria natum, et passum VI feria: natum VIII calendas Januarii, passum VIII calendas Aprilis. Ex quo baptizatus est Jesus Christus Dominus noster, fiunt anni II, et dies numerantur XC, qui fiunt DCCCXX, cum bissextis diebus suis, ac sic baptizatur VIII idus Januarii die, V feria, et passus est, ut superius dixi, VIII calendas Aprilis, VI feria. Cum bissextis diebus suis fiunt simul dies XII CCCCXV, et (ab) VIII idus Januarii in VIII calendas Aprilis dies XC.
Argumentum 15. On the day of the equinox and the solstice.
The day on which the Lord Jesus Christ was born into flesh from the Virgin Mary in Bethlehem is the one on which the day begins to increase. The first equinox is on March 25, when day is equal with night. On this very day Gabriel annunciates to Holy Mary, saying: The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee. Therefore also that which shall be born of thee shall be called the Son of God. [Luke 1.35, courtesy King James] Also on this day Christ has suffered in the flesh. The second solstice is on June 24, from which the day starts to decrease, and also when Saint John the Baptist was born. The second equinox is on September 24, on which day John the Baptist was conceived. And right from then on until the birth of the Lord and Saviour, the day becomes shorter than the night. From March 25 and until December 25, the days number 271. And that number of days after our Lord Christ was conceived on Sunday March 25, our Lord Christ was born on Tuesday December 20. On the day on which he has suffered death, 133 [? 33] years and 3 months have elapsed, which are 12 [thousand] 414 days. And that number of days after his birth took place on a Tuesday, he suffered death on a Friday: he was born on December 25 and suffered death on March 25. From when our Lord Jesus Christ was baptized, there were 2 years and the days numbered 90, yielding 820, with its leap days, and so he was baptized on the day January 06, a Thursday, and suffered death, as I said above, on March 25, a Friday. With its leap days this yields 12 [thousand] 415 days altogheter, and 90 days (from) January 06 to March 25.
This Argumentum does not concern the determination of Easter but certain ecclesiastical dates connected with the life of Jesus. Moreover, the numbers and dates in the text of this Argumentum are not consistent with the rest of the liber. They are even inconsistent among themselves, and there is no obvious reading that would make them consistent. In fact, the inconsistencies are so easy to spot that we may assume that the author of this Argumentum was not even concerned with chronological correctness nor consistency with the preceding Argumenta. The rest of this comment indicates some of the inconsistencies.
The date of birth of Jesus is given as December 25 several times, and once as December 20; there is also a reference to January 06 which is another popular date for nativity. The number of 271 days from conception to birth, as given in the text, would fit one of these, using "Roman inclusive counting": December 20 - preceding March 25 = 270 d = 38*7 d + 4 d but it does not fit December 25. On the other hand, if conception is on March 25 and on a Sunday, and birth is on a Tuesday, then birth has to be on December 25.
The next time interval mentioned is given as 12 414 d and as 12 415 d. One has 12 414 d = 1773*7 d + 3 d = 34*365.25 d - 4.5 d = Julian date( Y + 34, March, 21) - Julian date( Y, March, 25) or = Julian date( Y + 34, March, 20) - Julian date( Y, March, 25) depending on whether floor( Y/2 ) is even or odd. This could be a miswritten value (some 4 d too small) for the time interval from conception to death, but it certainly is not any integral number of years plus 3 months as pretended. Assuming a different scribal error, it could also be meant as the time interval from birth to death: 11 414 d = 1630*7 d + 4 d = 31*365.25 d + 91.25 d = Julian date( Y + 32, March, 25) - Julian date( Y, December, 25) or = Julian date( Y + 32, March, 25) - Julian date( Y, December, 24) depending on whether Y is divisible by 4 or not. This can be said to be 32 years (but not 33 years) plus 3 months. If 11 413 d were actually meant ("Roman inclusive counting"), this would even be compatible with the days of the week Friday and Tuesday for death and birth (the other reading would not).
However, these days of the week are inconsistent with the numbering of years since the incarnation: the year numbers closest to 0 yielding a Sunday for March 25 are Julian date( -0003, March, 25 ) and Julian date( 0003, March, 25 ) as can be seen easily from the table above and also from [Argumentum 4]. (We use the astronomical numbering of years .., -0001, 0000, +0001,.. for which the formula of [Argumentum 4] is always valid).
The next time interval mentioned is 820 d = 117*7 d + 1 d = 2*365.25 d + 89.5 d = Julian date( Y + 3, March, 25) - Julian date( Y, December, 25) or = Julian date( Y + 3, March, 25) - Julian date( Y, December, 26) depending on whether or not Y is divisible by 4. While this could be considered as 2 years and 90 days "with its leap days", it is not consistent with the date January 06 for the baptism. 810 days would be consistent with that date but not with the day of the week Thursday for the baptism.
The last time interval mentioned is Julian date( Y, March, 25) - Julian date( Y, January, 06) = 79 d or = 78 d = 11*7 d + 1 d depending on whether Y is divisible by 4 or not; only in the latter case can the two dates be a Friday and a Thursday. This is incorrectly given as 90 d = 12*7 d + 6 d, which happens to be Julian date( Y, March, 25) - Julian date( Y - 1, December, 25) unless Y is divisible by 4.
Using the Easter dates of the table above for year numbers around 0562 it is also easy to see that March 25 never was a Good Friday in the years with numbers around 0030; Julian date( 0034, March, 26 ) is the closest.
Argumentum XVI. De ratione bissexti.
Bissextum non ob illum diem fieri, ut quidam putant, quo Josua oravit solem stare, credendum est: quia dies ille et fuit, et praeteriit. Sed ab hoc dicitur bissextus, quod in unumquemque mensem punctus unus accrescit. Punctus vero unus quarta pars horae est. IV vero puncti unam horam faciunt; XII vero puncti III horas explicant. Ergo in VI annis ternae horae, quae sunt XII, diem faciunt I, qui addatur Februario, cum VI calendas Martii habuerit, ut in crastino sic habeat. Verbi causa, si hodie VI calendas Martii additur ille dies in IV anno expleto; nihilominus et crastino VI calendas Martii habeatur. Et ideo bissextus dicitur, quia bis VI calendas Martii habet Februarius.
Argumentum 16. On the rationale of the leap day.
One must not believe what some people maintain, that the leap day has arisen from that day on which Joshua commanded the sun to stand still: that day has been and is long gone. But it is called leap day because it gains one punctus in each month. The punctus is indeed the fourth part of an hour. And 4 puncti make one hour; and 12 puncti explain 3 hours. Hence in 4 years three hours each, which are 12, making 1 day which is added to February, so that when it is February 24, it is the same the next day. For instance, if today is February 24 and that day is added if 4 years are complete; then it will nevertheless be February 24 tomorrow. And it is called bisextile because February has two times the 6th of the calends of March.
In this "explanation", a leap day accumulates from 1/48 d per month. Because 1 d is taken to be 12 h, the 1/48 d per month is taken to be 1 punctus = 1/4 h = 1/96 d = 1/48*12 h per month.
Sex diebus fecit Deus mundum, septimo requievit. Ut ergo plenius intelligatur, computa quot horas habeat unus dies [ ? annus ], et divides illas in VII partes, et quantus remanet, exinde sit bissextus. Primum computa dies CCC, quomodo horas habent, decies tricenteni sunt tria millia. Iterum facis: bis tricenteni, sexcenteni: fiunt in tricentis diebus horae III DC. Iterum facis: decies sexageni DC, et bis sexageni CXX. Fiunt ergo in sexagenis diebus horae DCXX [DCCXX]. Iterum facis: decies quini L, et bis quini X. Ecce habes in quinque diebus horas LX. Fiunt simul integro anno in diebus CCCLXV horae IIII CCCLXXX, et alias tantas in nocte, fiunt simul dierum et noctium totius anni VIII DCCLX horae. Divide in illas VII partes. Primum facis: septies milleni VII, remanent I DCCLX. Item facis: septies ducenti, fiunt I CCCC, remanent CCCLX. Item facis: septies quinquageni, fiunt CCCL, remanent X. Item facis: septies as VII, remanent III. Istae tres horae faciunt in IV annis diem.
In six days God created the world, on the seventh he rested. So that this can be more fully understood, compute the number of hours one day [? year] has, and divide those into 7 parts, and the leap day shall come from what is left over. First compute how many hours 300 days have, ten times three hundred are three thousand. Then do: two times three hundred, six hundred: yielding 3600 hours in three hundred days. Then do: ten times six [is] 60, and two times sixty [is] 120. Thus, this yields 620 [720] hours in sixty days. Then do: ten five times [is] 50, and two times five [is] 10. Thus you have 60 hours in five days. Together, a whole year in 365 days yields 4 [thousand] 380 hours, and as many also in the night, yielding with day and night together 8760 hours. Divide those into 7 parts. First do: seven times thousand [is] 7[000], 1 [thousand] 760 are left over. Then do: seven times two hundred yield 1400, 360 are left over. Then do: seven times fifty yield 350, 10 are left over. Then do: seven times one [is] 7, 3 are left over. These three hours make a day in 4 years.
Here, a leap day accumulates from 1/4 d per year. And 1/4 d per year is "explained" with numerology: 1/4 d is taken to be 3 h (assuming that 1 d is 12 h) and explained as (365 d) mod (7 h) = (8760 h) mod (7 h) = 3 h which is correct only if we assume that 1 d is 24 h.
This text was translated by Michael Deckers, 2006, who kindly placed it in the public domain. This file and all material on this page is in the public domain - copy freely.