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CYCLUS DECEMNOVENNALIS DIONYSII
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last modified 2003-08-25 copyright (C) Michael Deckers 2003
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NINETEEN YEAR CYCLE OF DIONYSIUS
The following translation is as literal as I could do it in order to
reflect the style and diction of the original. Subsequent comments
contain a translation of the calendrical content into modern
algebraic notation.
The Latin text has been transcribed and edited by Rodolphe Audette
http://hermes.ulaval.ca/~sitrau/calgreg/denys.html
I have profited from the learned comments and helpful suggestions
by Joe Kress, A R Tom Peters, and Robert H van Gent.
Any typos in the Latin text and errors in the translation
and the comments are mine.
Incipit cyclus decemnovennalis, quem Græci Enneacaidecaeterida vocant,
constitutus a sanctis Patribus, in quo quartas decimas paschales omni tempore
sine ulla reperies falsitate; tantum memineris annis singulis, qui cyclus lunæ
et qui decemnovennalis existat. In præsenti namque tertia indictio est,
consulatu Probi junioris, tertius decimus circulus decemnovennalis, decimus
lunaris est.
The nineteen year cycle begins, which the Greek call Enneakaidekaeterida
(nineteen yearly), established by the holy [Church] Fathers, in which
you shall find fourteen paschal[ moon]s each time without error;
you shall just bear in mind, in each of the years, which cycle of the moon
and which nineteen year [cycle] prevails. In the present [year], in the
consulship of Probus Junior, it is the thirteenth of the nineteen
year cycle, and the tenth lunar one.
---------- ------ ------- ---- ------- ------------- ------------ ----------
ANNI quæ epac- con- quotus quæ sit dies Domi- quota sit
DIOCLE sint tæ, cur- sit luna XIIII nicæ luna
TIANI indic- id est ren- lunæ paschalis festivitatis ipsius
tiones adjec- tes circu- diei
tiones dies lus dominici
lunæ
---------- ------ ------- ---- ------- ------------- ------------ ----------
epacts, which
what ie, which date of is the
are incre- con is the day 14 day age of
YEARS the ments cur cycle of the of the the moon
OF indic of the rent of the paschal Sunday of this
DIOCLETIAN tions moon days moon moon festival Sunday
---------- ------ ------- ---- ------- ------------- ------------ ----------
CCXXVIIII vi nulla i xvii non.Apr. vii id.Apr. xvi
229 0513 Apr 05 Apr 07
CCXXX vii xi ii xviii viii k.Apr. iii k.Apr. xviiii
230 0514 Mar 25 Mar 30
CCXXXI viii xxii iii xviiii id. Apr. xiii k.Maii xx
231 0515 Apr 13 Apr 19
CCXXXII viiii iii v i iiii non.Apr. iii non.Apr. xv
232 0516 Apr 02 Apr 03
CCXXXIII x xiiii vi ii xi k.Apr. vii k.Apr. xviii
233 0517 Mar 22 Mar 26
CCXXXIIII xi xxv vii iii iiii id.Apr. xvii k.Maii xviiii
234 0518 Apr 10 Apr 15
CCXXXV xii vi i iiii iii k.Apr. ii k.Apr. xv
235 0519 Mar 30 Mar 31 *
CCXXXVI xiii xvii iii v xiiii k.Maii xiii k.Maii xv ogd.
236 0520 * Apr 18 Apr 19 *
CCXXXVII xiiii xxviii iiii vi vii id.Apr. iii id.Apr. xviii
237 0521 Apr 07 Apr 11
CCXXXVIII xv viiii v vii vi k.Apr. iii non.Apr. xxi
238 0522 Mar 27 Apr 03 *
CCXXXVIIII i xx vi viii xvii k.Maii xvi k.Maii xv
239 0523 Apr 15 Apr 16 *
CCXL ii i i viiii ii non.Apr. vii id.Apr. xvii
240 0524 Apr 04 Apr 07
CCXLI iii xii ii x viiii k.Apr. iii k.Apr. xx
241 0525 Mar 24 Mar 30
CCXLII iiii xxiii iii xi ii id.Apr. xiii k.Maii xxi
242 0526 Apr 12 Apr 19 *
CCXLIII v iiii iiii xii k.Apr. ii non.Apr. xvii
243 0527 Apr 01 Apr 04
CCXLIIII vi xv vi xiii xii k.Apr. vii k.Apr. xviiii
244 0528 * Mar 21 Mar 26
CCXLV vii xxvi vii xiiii v id.Apr. xvii k.Maii xx
245 0529 Apr 09 Apr 15
CCXLVI viii vii i xv iiii k.Apr. ii k.Apr. xvi
246 0530 Mar 29 Mar 31
CCXLVII viiii xviii ii xvi xv k.Maii xii k.Maii xvii hend.
247 0531 Apr 17 Apr 20
---------- ------ ------- ---- ------- ------------- ------------ ----------
ANNI quæ epac- con- quotus quæ sit dies Domi- quota sit
DOMINI sint tæ, cur- sit luna XIIII nicæ luna
NOSTRI indic- id est ren- lunæ paschalis festivitatis ipsius
JESU tiones adjec- tes circu- diei
CHRISTI tiones dies lus dominici
lunæ
---------- ------ ------- ---- ------- ------------- ------------ ----------
epacts, which
YEARS what ie, which date of is the
OF OUR are incre- con is the day 14 day age of
LORD the ments cur cycle of the of the the moon
JESUS indic of the rent of the paschal Sunday of this
CHRIST tions moon days moon moon festival Sunday
---------- ------ ------- ---- ------- ------------- ------------ ----------
B DXXXII x nulla iiii xvii non.Apr. iii id.Apr. xx
0532 Apr 05 Apr 11
DXXXIII xi xi v xviii viii k.Apr. vi k.Apr. xvi
0533 Mar 25 Mar 27
DXXXIIII xii xxii vi xviiii id.Apr. xvi k.Maii xvii
0534 Apr 13 Apr 16
DXXXV xiii iii vii i iiii non.Apr. vi id.Apr. xx
0535 Apr 02 Apr 08
B DXXXVI xiiii xiiii ii ii xi k.Apr. x k.Apr. xv
0536 Mar 22 Mar 23 *
DXXXVII xv xxv iii iii iiii id.Apr. ii id.Apr. xvi
0537 Apr 10 Apr 12
DXXXVIII i vi iiii iiii iii k.Apr. ii non.Apr. xviiii
0538 Mar 30 Apr 04
DXXXVIIII ii xvii v v xiiii k.Maii viii k.Maii xx ogd.
0539 * Apr 18 Apr 24
B DXL iii xxviii vii vi vii id.Apr. vi id.Apr. xv
0540 Apr 07 Apr 08 *
DXLI iiii viiii i vii vi k.Apr. ii k.Apr. xviii
0541 Mar 27 Mar 31
DXLII v xx ii viii xvii k.Maii xii k.Maii xviiii
0542 Apr 15 Apr 20
DXLIII vi i iii viiii ii non.Apr. non.Apr. xv
0543 Apr 04 Apr 05 *
B DXLIIII vii xii v x viiii k.Apr. vi k.Apr. xvii
0544 Mar 24 Mar 27
DXLV viii xxiii vi xi ii id.Apr. xvi k.Maii xviii
0545 Apr 12 Apr 16
DXLVI viiii iiii vii xii k.Apr. vi id.Apr. xxi
0546 Apr 01 Apr 08 *
DXLVII x xv i xiii xii k.Apr. viiii k.Apr. xvii
0547 * Mar 21 Mar 24
B DXLVIII xi xxvi iii xiiii v id.Apr. ii id.Apr. xvii
0548 Apr 09 Apr 12
DXLVIIII xii vii iiii xv iiii k.Apr. ii non.Apr. xx
0549 Mar 29 Apr 04
DL xiii xviii v xvi xv k.Maii viii k.Maii xxi hend.
0550 Apr 17 Apr 24 *
---------- ------ ------- ---- ------- ------------- ------------ ----------
DLI xiiii nulla vi xvii non.Apr. v id.Apr. xviii
0551 Apr 05 Apr 09
B DLII xv xi i xviii viii k.Apr. ii k.Apr. xx
0552 Mar 25 Mar 31
DLIII i xxii ii xviiii id.Apr. xii k.Maii xxi
0553 Apr 13 Apr 20 *
DLIIII ii iii iii i iiii non.Apr. non.Apr. xvii
0554 Apr 02 Apr 05
DLV iii xiiii iiii ii xi k.Apr. v k.Apr. xx
0555 Mar 22 Mar 28
B DLVI iiii xxv vi iii iiii id.Apr. xvi k.Maii xx
0556 Apr 10 Apr 16
DLVII v vi vii iiii iii k.Apr. k.Apr. xvi
0557 Mar 30 Apr 01
DLVIII vi xvii i v xiiii k.Maii xi k.Maii xvii ogd.
0558 * Apr 18 Apr 21
DLVIIII vii xxviii ii vi vii id.Apr. id.Apr. xx
0559 Apr 07 Apr 13
B DLX viii viiii iiii vii vi k.Apr. v k.Apr. xv
0560 Mar 27 Mar 28 *
DLXI viiii xx v viii xvii k.Maii xv k.Maii xvi
0561 Apr 15 Apr 17
DLXII x i vi viiii ii non.Apr. v id.Apr. xviiii
0562 Apr 04 Apr 09
DLXIII xi xii vii x viiii k.Apr. viii k.Apr. xv
0563 Mar 24 Mar 25 *
B DLXIIII xii xxiii ii xi ii id.Apr. id.Apr. xv
0564 Apr 12 Apr 13 *
DLXV xiii iiii iii xii k.Apr. non.Apr. xviii
0565 Apr 01 Apr 05
DLXVI xiiii xv iiii xiii xii k.Apr. v k.Apr. xxi
0566 * Mar 21 Mar 28 *
DLXVII xv xxvi v xiiii v id.Apr. iiii id.Apr. xv
0567 Apr 09 Apr 10 *
B DLXVIII i vii vii xv iiii k.Apr. k.Apr. xii
0568 Mar 29 Apr 01
DLXVIIII ii xviii i xvi xv k.Maii xi k.Maii xviii hend.
0569 Apr 17 Apr 21
---------- ----- ------ ---- ------- ------------ ------------ ----------
DLXX iii nulla ii xvii non.Apr. viii id.Apr. xv
0570 Apr 05 Apr 06 *
DLXXI iiii xi iii xviii viii k.Apr. iiii k.Apr. xviii
0571 Mar 25 Mar 29
B DLXXII v xxii v xviiii id.Apr. xv k.Maii xviii
0572 Apr 13 Apr 17
DLXXIII vi iii vi i iiii non.Apr. v id.Apr. xxi
0573 Apr 02 Apr 09 *
DLXXIIII vii xiiii vii ii xi k.Apr. viii k.Apr. xvii
0574 Mar 22 Mar 25
DLXXV viii xxv i iii iiii id.Apr. xviii k.Maii xviii
0575 Apr 10 Apr 14
B DLXXVI viiii vi iii iiii iii k.Apr. non.Apr. xx
0576 Mar 30 Apr 05
DLXXVII x xvii iiii v xiiii k.Maii vii k.Maii xxi ogd.
0577 * Apr 18 * Apr 25 *
DLXXVIII xi xxviii v vi vii id.Apr. iiii id.Apr xvii
0578 Apr 07 Apr 10
DLXXVIIII xii viiii vi vii vi k.Apr. iiii non.Apr.xx
0579 Mar 27 Apr 02
B DLXXX xiii xx i viii xvii k.Maii xi k.Maii xx
0580 Apr 15 Apr 21
DLXXXI xiiii i ii viiii ii non.Apr. viii id.Apr. xvi
0581 Apr 04 Apr 06
DLXXXII xv xii iii x viiii k.Apr. iiii k.Apr. xviiii
0582 Mar 24 Mar 29
DLXXXIII i xxiii iiii xi ii id.Apr. xiiii k.Maii xx
0583 Apr 12 Apr 18
B DLXXXIIII ii iiii vi xii k.Apr. iiii non.Apr.xv
0584 Apr 01 Apr 02 *
DLXXXV iii xv vii xiii xii k.Apr. viii k.Apr. xviii
0585 * Mar 21 Mar 25
DLXXXVI iiii xxvi i xiiii v id.Apr. xviii k.Maii xviiii
0586 Apr 09 Apr 14
DLXXXVII v vii ii xv iiii k.Apr. iii k.Apr. xv
0587 Mar 29 Mar 30 *
B DLXXXVIII vi xviii iiii xvi xv k.Maii xiiii k.Maii xv hend.
0588 Apr 17 Apr 18 *
---------- ----- ------ ---- ------- ------------ ------------ ----------
DLXXXVIIII vii nulla v xvii non.Apr. iiii id.Apr. xviiii
0589 Apr 05 Apr 10
DXC viii xi vi xviii viii k.Apr. vii k.Apr. xv
0590 Mar 25 Mar 26 *
DXCI viiii xxii vii xviiii id.Apr. xvii k.Maii xvi
0591 Apr 13 Apr 15
B DXCII x iii ii i iiii non.Apr. viii id.Apr. xviii
0592 Apr 02 Apr 06
DXCIII xi xiiii iii ii xi k.Apr. iiii k.Apr. xxi
0593 Mar 22 Mar 29 *
DXCIIII xii xxv iiii iii iiii id.Apr. iii id.Apr. xv
0594 Apr 10 Apr 11 *
DXCV xiii vi v iiii iii k.Apr. iii non.Apr. xviii
0595 Mar 30 Apr 03
B DXCVI xiiii xvii vii v xiiii k.Maii x k.Maii xviii ogd.
0596 * Apr 18 Apr 22
DXCVII xv xxviii i vi vii id.Apr. xviii k.Maii xxi
0597 Apr 07 Apr 14 *
DXCVIII i viiii ii vii vi k.Apr. iii k.Apr. xvii
0598 Mar 27 Mar 30
DXCVIIII ii xx iii viii xvii k.Maii xiii k.Maii xviii
0599 Apr 15 Apr 19
B DC iii i v viiii ii non.Apr. iiii id.Apr. xx
0600 Apr 04 Apr 10
DCI iiii xii vi x viiii k.Apr. vii k.Apr. xvi
0601 Mar 24 Mar 26
DCII v xxiii vii xi ii id.Apr. xvii k.Maii xvii
0602 Apr 12 Apr 15
DCIII vi iiii i xii k.Apr. vii id.Apr. xx
0603 Apr 01 Apr 07
B DCIIII vii xv iii xiii xii k.Apr. xi k.Apr. xv
0604 * Mar 21 * Mar 22 *
DCV viii xxvi iiii xiiii v id.Apr. iii id.Apr. xvi
0605 Apr 09 Apr 11
DCVI viiii vii v xv iiii k.Apr. iii non.Apr. xviiii
0606 Mar 29 Apr 03
DCVII x xviii vi xvi xv k.Maii viiii k.Maii xx hend.
0607 Apr 17 Apr 23
---------- ----- ------ ---- ------- ------------ ------------ ----------
B DCVIII xi nulla i xvii non.Apr. vii id.Apr. xvi
0608 Apr 05 Apr 07
DCVIIII xii xi ii xviii viii k.Apr. iii k.Apr. xviiii
0609 Mar 25 Mar 30
DCX xiii xxii iii xviiii id.Apr. xiii k.Maii xx
0610 Apr 13 Apr 19
DCXI xiiii iii iiii i iiii non.Apr. ii non.Apr. xvi
0611 Apr 02 Apr 04
B DCXII xv xiiii vi ii xi k.Apr. vii k.Apr. xviii
0612 Mar 22 Mar 26
DCXIII i xxv vii iii iiii id.Apr. xvii k.Maii xviiii
0613 Apr 10 Apr 15
DCXIIII ii vi i iiii iii k.Apr. ii k.Apr. xv
0614 Mar 30 Mar 31 *
DCXV iii xvii ii v xiiii k.Maii xii k.Maii xvi ogd.
0615 * Apr 18 Apr 20
B DCXVI iiii xxviii iiii vi vii id.Apr. iii id.Apr. xviii
0616 Apr 07 Apr 11
DCXVII v viiii v vii vi k.Apr. iii non.Apr. xxi
0617 Mar 27 Apr 03 *
DCXVIII vi xx vi viii xvii k.Maii xvi k.Maii xv
0618 Apr 15 Apr 16 *
DCXVIIII vii i vii viiii ii non.Apr. vi id.Apr. xviii
0619 Apr 04 Apr 08
B DCXX viii xii ii x viiii k.Apr. iii k.Apr. xx
0620 Mar 24 Mar 30
DCXXI viiii xxiii iii xi ii id.Apr. xiii k.Maii xxi
0621 Apr 12 Apr 19 *
DCXXII x iiii iiii xii k.Apr. ii non.Apr. xvii
0622 Apr 01 Apr 04
DCXXIII xi xv v xiii xii k.Apr. vi k.Apr. xx
0623 * Mar 21 Mar 27
B DCXXIIII xii xxvi vii xiiii v id.Apr. xvii k.Maii xx
0624 Apr 09 Apr 15
DCXXV xiii vii i xv iiii k.Apr. ii k.Apr. xvi
0625 Mar 29 Mar 31
DCXXVI xiiii xviii ii xvi xv k.Maii xii k.Maii xvii hend.
0626 Apr 17 Apr 20
---------- ------ ------- ---- ------- ------------- ------------ ----------
The leftmost column in this table gives a year number pertaining
to the feast of Easter described in each line, and to the indiction:
column 1 =(in the second part of the table:)
the year number Y of the incarnation
=(in the first part of the table:)
the Diocletian year number D = Y - 284
Column 1 has the prefix "B" (for bissextum) if Y mod 4 = D mod 4 = 0.
The other columns are all related to Y, as follows:
column 2 = 1 + (2 + Y)mod 15 (see [Argumentum 2])
column 3 = ((Y mod 19)·11) mod 30 (see [Argumentum 3])
column 4 = 1 + (3 + floor( Y·5/4 ))mod 7 (see [Argumentum 4])
column 5 = 1 + (Y - 3)mod 19 (see [Argumentum 6])
column 6 = March 21 + (15 - 11·( Y mod 19))mod 30 d
(see [Argumentum 14])
Column 8 has the postfix "ogd." (for ogdoadas) if Y mod 19 = 8 - 1
and the postfix "hend." (for hendeka) if Y mod 19 = 11 + 8 - 1.
We have indicated extreme values in columns 6, 7, 8 with asterisks.
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ARGUMENTA PASCHALIA
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Incipiunt argumenta de titulis paschalibus Ægyptiorum investigata solertia ut
præsentes indicent.
This begins the argumenta on the determination of Easter by the Egyptians,
carefully investigated as shown in the following.
Argumentum primum. De annis Christi.
Si nosse vis quotus sit annus ab incarnatione Domini nostri Jesu Christi,
computa quindecies XXXIV, fiunt DX; iis semper adde XII regulares, fiunt DXXII;
adde etiam indictionem anni cujus volueris, ut puta, tertiam, consulatu Probi
junioris, fiunt simul anni DXXV. Isti sunt anni ab incarnatione Domini.
First Argumentum. On the years of Christ.
If you want to find out which year it is since the incarnation of our
Lord Jesus Christ, compute fifteen times 34, yielding 510; to these
always add the correction 12, yielding 522; also add the indiction of
the year you want, say, in the consulship of Probus Junior, the third,
yielding 525 years altogether. These are the years since the
incarnation of the Lord.
That the year numbers Y employed here agree with the usual Julian year
numbers J is shown by the formulae for the indiction in [Argumentum 2]
(using Y mod 15), for the epacts in [Argumenta 3 and 11] (using Y mod
19), and for the day of the week in [Argumentum 4] (using Y mod 28).
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 epactæ, 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 septimanæ 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. Duæ sunt
epactæ solis, id est concurrentes septimanæ 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].
Argumentum VI. De cyclo lunari.
Si vis scire quotus cyclus lunæ 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 lunæ 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 tertiæ 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. -
Si vero mense Aprili Pascha celebramus, computa menses a Septembri usque ad
Martium, fiunt VII. His semper adjice II, fiunt IX. Adde epactas lunæ anni cujus
volueris, ut puta, indictionis IV, XXIII, qui fiunt XXXII, et diem mensis quo
Pascha celebramus, id est Aprilis XIX, qui simul fiunt LI; deduc XXX, remanent
XXI. Luna XXI est in die resurrectionis Domini.
In the month of April. -
If however we celebrate Easter in the month of April, compute the months from
September to March, yielding 7. To this always add 2, yielding 9. Add the
lunar epacts of the year you want, say 23 for indiction 4, yielding 32, and
the day of the month in which we celebrate Easter, that is April 19, which
together yield 51; deduct 30, 21 are left over. The age of the moon is 21 on
the day of the resurrection of the Lord.
This amounts to the same formula as above for the remaining year numbers:
age of the moon on( Julian date(Y, April, D) )
= ( age of the moon on( Julian date(Y, March, 22) ) + 9 + D )mod 30
Thus, the age of the moon is supposed to increase by 1 for each day
throughout the 35 day interval from March 21 until April 25 in which
these formulae are applicable; this agrees with columns 6 and 8 in the
table above. The year number for the example could be 0526.
Si requiras a Septembri usque ad Decembrem, tres semper in his IV
mensibus regulares adjicias: in bissexto autem solummodo anno duos regulares
suprascriptis mensibus adnumerabis, et pro XXXI die, XXXII annis singulis
Decembri mense assumes in fine.
If you need it from September to December, you should always add the
correction three in these 4 months: only in a leap year you also
shall add the correction two for these months described above, and
finally in non-leap years, for day 31 in the month of December you
should assume 32.
This is probably meant as a recipe similar to the two above for the
age of the moon on( Julian date(Y, January, D) )
= ( (Y mod 19)·11 + 4 + 3 + (1 or 2) + D )mod 30
where the "4" acts as the number of months from September to December,
"3" is the the correction in every year, and the "(1 or 2)" comes either
from the correction 2 for leap years, or, for non-leap years, it is an
interpretation of the effect of assuming 32 days in December.
The interpretation above is consistent with [Argumentum 11] since:
Julian date( Y, January, 00 )
~= Julian date( Y, March, 22 ) - 3 synodic month + (7.6 or 8.6) d
(with 8.6 instead of 7.6 for leap year numbers Y).
Argumentum X. De die septimanæ sanctæ feria paschali.
Si vis cognoscere quotus dies septimanæ est, sume dies a Januario usque ad
mensem quem volueris, ut puta, ad XXX diem mensis Martii, fiunt LXXXIX. His
adjicies semper unum, fiunt XC; et semper adde epactas solis, id est
concurrentes septimanæ dies cujus volueris anni, ut puta II, indictionis III,
fiunt simul XCII. Hos partire per VII, remanet una: ipsa est dominica paschalis
festi. Sic quamlibet diem a calendis Januarii usque ad XXXI diem mensis
Decembris, quota feria fuerit, invenies computando, ut regularem unum et
concurrentes, quæ a Januario mense semper incipiunt, pariter assumas.
Argumentum 10. On the day of the holy week of the feast of Easter.
If you want to learn which day of the week it is, add the days since
January until the month you want, say until March 30, there are 89. To this
always add one, yielding 90; and always add the solar epacts, that is, the
concurrents of the seven day week for the year you want, say
2 for the indiction 3, yielding 92 altogether. Divide those by 7, one is
left over: this is the Sunday of the feast of Easter.
In this way, if you venture to compute which day of the week it is for any
day from the first of January until the 31st of the month of December, you
should equally assume the correction one and the concurrents which always
begin in the month of January.
The example date could be Julian date( 0525, March, 30 ), as can be
seen from the table above. The example shows that the
number of days from January to( Julian date( Y, January, 01 ) + D d)
is meant to be D + 1 rather than D ("Roman inclusive counting").
With the solar epacts of [Argumentum 4], the formula given amounts to
day of the week number( Julian date( Y, January, 01) + D d )
= ( D + 1 + 1 + 4 + Y + floor(Y / 4) )mod 7
= ( D + (Y - 1) + floor(Y / 4) )mod 7
which agrees with the correct formula of [Argumentum 12] only
if Y is not divisible by 4, and otherwise is one day ahead.
Argumentum XI. De luna citimi paschalis.
Si vis scire quota luna sit in XI calendas Aprilis, sume annos
incarnationis Domini nostri Jesu Christi, ut puta, DCLXXV. Hos partire per
XIX, remanent X; et multiplica decem per
XI, fiunt CX. Partire tricesima, remanent XX: vicesima luna est in XI calendas
Aprilis. Si autem VII, septima; si asse, prima.
Argumentum 11. On the moon closest to Easter.
If you want to know which moon it is on March 22, add the years since
the incarnation of our Lord Jesus Christ, say 675. Divide those by
19, 10 are left over; and multiply ten by
11, yielding 110. Divide by 30, 20 are left over: it is the twentieth day
of the moon on March 22. And if 7 [is left over], then the seventh, if one,
the first.
Only with the suggested correction (and allowing for remainders of zero)
this yields
age of the moon in year( Y )on March 22 = ((Y mod 19)·11) mod 30
which are the epacts of [Argumentum 3]. Thus, Dionysius Exiguus uses
Julian date( Y, March, 22 )
as "sedes epactorum" (seating of the epacts).
Besides these so-called Dionysian epacts, several other epacts
have been used in computs for the same or a different Easter date,
such as the Alexandrian epacts (8 + (Y mod 19)·11) mod 30,
which, according to [Argumentum 13], would give the nominal age of
the moon on the day preceding January 01.
Argumentum XII.
Si vis nosse diem calendarum Januarii, per singulos annos,
quota sit feria, sume annos incarnationis Domini nostri Jesu Christi, ut puta,
annos DCLXXV. Deduc assem, remanent DCLXXIV. Hos per quartam partem partiris, et
quartam partem, quam partitus es, adjicies super DCLXXIV, fiunt simul DCCCXLII.
Hos partiris per VII, remanent II. Secunda est dies calendarum Januarii. Si V,
quinta feria; si asse, dominica; si nihil, sabbatum.
Argumentum 12.
If you want to find out which day of the week it is on the first day of
January, for non-leap years, then add the years since the incarnation of our
Lord Jesus Christ, say 675 years. Subtract one, 674 are left over. Divide
those into the fourth part, and add the fourth part obtained by the division
to 674, yielding 842 altogether. Divide those by 7, 2 are left over. It is
Monday on the first of January. If 5 [are left over] then [it is] Thursday,
if one, then Sunday; if nothing, Saturday.
This amounts to
day of the week number of year( Y )on January 01
= ( (Y - 1) + floor( (Y - 1)/4 ) )mod 7
with 0 for Saturday, which is in fact the number for
day of the week(Julian date( Y, January, 01)).
This is true for leap year numbers Y as well.
Argumentum XIII. De luna calendarum Januarii.
Si vis scire quota luna sit calendis Januarii, scito quotus lunaris cyclus sit,
verbi gratia cyclus XV. Tene tibi unum, id est ipsas calendas Januarii, et duces
quinquies quinquies decies: faciunt LXXV; quos adjicies super unum, et fiunt
LXXVI. Item duces sexies decies quinquies, faciunt XC; quos adjicies super
LXXVI, et sic summa numerorum CLXVI; in quibus partiris tricesima, remanent XVI.
Sexta decima luna est calendis Januarii, et puncti XVI. Isto modo per XIX cyclos
lunares computabis semper, et calendis Januarii, quota sit luna, absque errore
reperies.
Argumentum 13. On the age of the moon on the first of January.
If you want to know which moon it is on January 01, knowing which lunar cycle
it is, for instance cycle 15. Retain one, which is for the same January 01,
and take five fifteen times: yielding 75; to which you always add one, thus
yielding 76. Now take six fifteen times, making 90, which you add to 76, thus
the sum of the numbers is 166; divide these into the thirtieth [part], 16 are
left over. It is the sixteenth moon on January 01, and 16 puncti. In this way
you can always compute for the 19 cycles of the moon, and you will obtain
without error the age of the moon on January 01.
For year numbers Y, and with the lunar cycle L = 1 + (-3 + Y)mod 19
from [Argumentum 6], this computation yields
age of the moon on( Julian date( Y, January, 01 ) )
= ( L·5 + 1 + L·6 )mod 30
= (12 + ((-3 + Y)mod 19)·11 )mod 30
=(for Y mod 19 >= 3:) (9 + (Y mod 19)·11)mod 30
up to to the puncti (to be discussed below).
Unless Y is divisible by 4, this agrees with the formula suggested
at the end of [Argumentum 9].
Dum autem veneris ad XVII cycli lunaris, et duxeris quinquies decies
septies, super calendas Januarii, qui faciunt LXXXV, si partiris sexagesima, et
adjicies ipsum assem, fiunt LXXXVI. Deinde ducis sexies decies septies, fiunt
CII. Eos adjicies super LXXXVI, et fiunt CLXXXVIII.
Adiicies unum, fiunt CLXXXVIIII.
Partire ibi tricesima, remanent IX. Nona luna est calendis Januarii, et puncti
XXVI. Sic et in XVIII et XIX cyclo facies. A primo vero cyclo lunari, usque in
sextum decimum, non partiris sexagesimam, ne in errorem incidas.
As soon as you shall come to lunar cycle 17, then take five times seventeen,
after January 01, which makes 85, if you divide into the sixtieth [part], and
add the resulting one to it, this yields 86. Meanwhile take six times
seventeen, yielding 102. Those add to 86, and it yields 188.
Add one, yielding 189.
Divide this by thirty, 9 are left over. It is the ninth moon on January 01,
and 26 puncti. In this way you also compute in cycles 18 and 19. From the
first lunar cycle until the sixteenth you do not divide by 60 so as not to
make an error.
For the Julian year number Y, this computation is said to apply if
L = 1 + (-3 + Y)mod 19 is 17, 18, or 19, that is, if L = Y mod 19 + 17.
With the addition of 1 as amended above in brackets, it yields
age of the moon on( Julian date( Y, January, 01 ) )
= ( L·5 + floor(L/12) + L·6 + 1 )mod 30
=(for Y mod 19 < 3:) (9 + (Y mod 19)·11)mod 30
resulting in the same formula as above for the remaining year numbers Y.
Apparently, a separate formula is given for 17 <= L <= 19 because of
the term floor(L/12). Of course, floor(L/17) would have worked
for all L; this would have required the remainder modulo 85
instead of modulo 60. With the 19 year cycle (as in [Argumentum 5]),
a single (and simpler) formula would do.
The separate multiplication by 5 in both computations above is very
likely due to a formula of the type
fractional age of the moon on( Julian date( Y, January, 01 ) )
= ( A + (Y - B)·30·235/19 )mod 30
= ( A + ((Y - B) mod 19)·(5·(1 + 1/95) + 6) ) mod 30
derived directly from the Metonic value for the synodic month. For
integral B and suitable A it gives values for the age of the moon that
are integral multiples of 1/95. The number 1/95 is close to a 1/96
= (1 punctus)/(1 d) (see [Argumentum 16]) which would explain the
appearance of puncti in the age of the moon.
Unfortunately, the text is not explicit about the computation of the
puncti, and the two examples leave many possibilities open, such as:
age of the moon on( Julian date( Y, January, 01 ) )in days and puncti
= ( 11 + 42/96 + ((Y - 3) mod 19)·(5·(1 + 1/96) + 6) ) mod 30
or = ( 37/96 + ((Y - 2) mod 19)·(5·(1 + 1/96) + 6) ) mod 30
or = ( 19 + 32/96 + ((Y - 1) mod 19)·(5·(1 + 1/96) + 6) ) mod 30
And if we assume that the second example is meant to yield an age
of the moon of 8 (rather than 9) plus 16 puncti, then we could have
age of the moon on( Julian date( Y, January, 01 ) )in days and puncti
= ( 8 + 27/96 + (Y mod 19)·(5·(1 + 1/96) + 6) ) mod 30
In all these formulae, the age of the moon increases by 11 + 5/96 per
year except for the "saltus" of 11 + 6/96 once every 19 years.
Thus, this Argumentum incompletely describes a kind of Alexandrian
epacts that apparently already had been described more fully elsewhere;
I do not know such a source, however.
Argumentum XIV. Quota feria luna XIV incidat cycli decemnovennalis anno primo.
Incipit calculatio quomodo reperiri possit quota feria singularis anni decima
quarta luna paschalis, id est primi circuli decemnovennalis.
Argumentum 14. On which day of the week the fourteenth moon falls in the
first year of the nineteen year cycle.
The calculation begins whereby one can find out on which day of the week
the fourteenth paschal moon falls in a single year, this one being for the
first circle of nineteen.
Anno primo, quia non habet epactas lunares, pro eo quod cum noni decimi
inferioris anni XVIII, et suis XI epactis, addito etiam ab Ægyptiis die una,
fiunt XXX, id est luna mensis unius integra, et nihil remanet de epactis, et
quod in Aprili mense incidit eo anno luna paschalis XIV, tene regulares in eo
semper XXXV, subtrahe XXX, id est ipsa luna integra, et remanent V. Quinto die a
calendis, hoc est nonis Aprilis, occurrit luna paschalis XIV. Tene suprascriptos
V, adde et concurrentes ejusdem anni IV, fiunt IX. Adde et regulares in eodem
semper mense Aprili VII, fiunt XVI. Hos partire per VII, id est bis septeni XIV,
remanent II. Secunda feria occurrit luna paschalis XIV, et dominicus festi
paschalis dies luna XX.
In the first year, which does not have lunar epacts, because to those 18 from
the previous nineteenth year, and its 11 epacts, one day is added by the
Egyptians, yielding 30, that is one full lunar month, so that nothing remains
from the epacts, and so that in this year the 14th paschal moon falls in the
month of April, in this year always take the correction 35, subtract 30, that
is this full month, and 5 remains. The 14th paschal moon occurs five days
from the Kalends, which is April 05. Take the 5 from above, and add the
concurrents 4 of this year, yielding 9. And always add to this the
correction 7 in the month of April, yielding 16. Divide those by 7, that
is, two times seven is 14, 2 are left over. On Monday occurs the 14th
paschal moon, and the Sunday of the Easter holiday on the day of the
20th moon.
For Julian year number Y, the rule first given is meant to be
day of the 14th paschal moon in year( Y )
= April 35 - ( 16 + (-16 + (Y mod 19)·11 )mod 30 ) d
for the special case Y mod 19 = 0. For all integral Y, this is equal to
March 21 + ( 15 - (Y mod 19)·11 )mod 30 d
which is in fact the first date >= Julian date( Y, March, 21 ) whose age
of the moon is 14 according to [Argumentum 11], and using an increase in
the age of the moon of 1 mod 30 per day. With the numbering of the days of
the week as in [Argumentum 12], the second rule is:
( 35 - 16 - (-16 + (Y mod 19)·11 )mod 30
+ day of the week( Julian date( Y, March, 24 ) + 7)mod 7
= ( day of the week( Julian date( Y, March, 24 ) - 3
+ ( 15 - (Y mod 19)·11 )mod 30 )mod 7
= day of the week( day of the 14th paschal moon in year( Y ) )
The addition of 7 in this rule is of course unnecessary; it just
ensures that X mod 7 is never evaluated with X < 7.
The first sentence describes the "saltus lunæ" when the epacts increase
by 12 rather than 11 from year (Y - 1) to year Y with Y mod 19 = 0.
Anno secundo.
Item præfati circuli annus secundus, a quo sumunt exordium epactæ XI. Incidit in
eo anno luna paschalis XIV mense Martii. Tene XXXVI regulares in eo semper,
subtrahe semper epactas XI, remanent XXV. Vicesimo quinto die a calendis Martii,
quod est VIII calendas Aprilis, occurrit luna paschalis XIV. Tene suprascriptos
XXV, adde concurrentes ejusdem anni V, fiunt XXX. Adde semper in fine hujus
mensis regulares IV, hos partire per VII, id est septies quaterni XXVIII,
remanent VI. Sexta feria occurrit luna XIV paschalis, et dominicus festi
paschalis dies luna XVI.
In the second year.
Now to the second year of the above mentioned circle, for which the epacts
add up to 11 to begin with. In this year, the 14th paschal moon occurs in
the month of March. In this [month], always take the correction 36, always
subtract the epacts 11, 25 are left over. The 14th paschal moon occurs
twenty five days from the beginning of March, that is, on March 25. Take
the 25 from above, add the concurrents 5 for this year, yielding 30. Finally
always add the correction 4 for this month, divide those by 7, that is
four times seven or 28, 6 remain. The 14th paschal moon occurs on Friday,
and the Sunday of the feast of Easter is the day of the 16th moon.
For Julian year number Y, the rule given first is
day of the 14th paschal moon in year( Y )
= March 36 - ( (Y mod 19)·11 )mod 30 d
and if the 14th mooon is in March, then this is again equal to
March 21 + ( 15 - (Y mod 19)·11 )mod 30 d
because ( (Y mod 19)·11 )mod 30 <= 15 in this case (see [Argumentum 7]).
The second rule also amounts to the same as above.
Anno tertio. -
Item mense Aprili sæpe dicti circuli primi anno tertio. Tene semper in eo mense
imprimis regulares XXXV. Subtrahe epactas ejusdem anni XXII, remanent XIII.
Tertio decimo die mensis, id est idibus Aprilis, occurrit luna paschalis XIV.
Tene hos XIII, adde concurrentes VI, fiunt XIX. Adde in Aprili semper inferius
regulares VII, fiunt XXVI. Hos partire per VII ter septeni, XXI, remanent
quinque. Quinta feria erit decima quarta luna paschalis, et dominicus dies
paschalis festi luna XVII.
In the third year.
In the third year of said first cycle, [the 14th moon occurs] also always in
the month of April. For this month, always take first the correction 35.
Subtract the epacts 22 of this year, 13 are left over. The 14th paschal moon
occurs on the thirteenth day of the month, that is on April 13.
Take those 13, add the concurrents 6, yielding 19. Then always add in April
the correction 7, yielding 26. Divide those by 7, three times 7 [are] 21,
five are left over. On Thursday was the fourteenth paschal moon, and the
Sunday of the feast of Easter on the 17th moon.
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 præterea 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 æquinoctii et solstitii.
Qua die natus est Dominus Jesus Christus secundum carnem ex Maria Virgine in
Bethlehem, in qua incipit crescere dies. Æquinoctium primum est in VIII calendas
Aprilis, in qua æquatur dies cum nocte. Eodem die Gabriel nuntiat sanctæ Mariæ,
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.
Æquinoctium 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
anni 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 years
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 præteriit. Sed ab hoc dicitur
bissextus, quod in unumquemque mensem punctus unus accrescit. Punctus vero unus
quarta pars horæ est. IV vero puncti unam horam faciunt; XII vero puncti III
horas explicant. Ergo in VI annis ternæ horæ, quæ 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,
unus 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
horæ III DC. Iterum facis: decies sexageni DC, et bis sexageni CXX. Fiunt ergo
in sexagenis diebus horæ 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 horæ IIII CCCLXXX, et alias tantas in nocte, fiunt simul
dierum et noctium totius anni VIII DCCLX horæ. 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. Istæ tres horæ
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
one 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.
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