The Easter table of Dionysius Exiguus begins with the 19-year cycle from 532 CE.:
(Cycle from the Incarnation of our Lord organised according to Abbot Dionysius)
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.
Firstly, there is the nineteen-year cycle, which the Greeks call Enneakaidekaeterida, which was established by the holy [church] fathers and according to which the fourteenth days of the moon prove to be faultless at all times; one need only remember which lunar cycle and which nineteen-year cycle exists in each year. In the present year, namely the third of the indiction and year of the consulship of Probus junior, it is the tenth moon in the thirteenth nineteen-year cycle.
[...]
ANNI DO- QUAE EPA- CON- QUO- QUOTA SIT DIES DO- LUNA
MINI SINT CTAE CUR- TUS LUNA XIIII MINICAE IPSIUS
NOSTRI INDI- I.E. REN- SIT- PASCHALIS FESTIVI- DIEI
IESU CTIO- ADIE- TES LUNAE TATIS DOMI-
CHRISTI NES CTIO- DIES CIR- NICI
NES CU-
LUNA- LUS
RES
B DXXXII X nulla IIII XVII NON.APR. III ID.Apr. XX
DXXXIII XI XI V XVIII VIII KAL.APR. VI KAL.APR. XVI
DXXXIIII XII XXII VI XVIIII ID.Apr. XVI KAL.MAI. XVII
DXXXV XIII III VII I IIII NON.APR. VI ID.Apr. XX
B DXXXVI XIIII XIIII II II XI KAL.APR. X KAL.APR. XV
DXXXVII XV XXV III III IIII ID.Apr. II ID.Apr. XVI
DXXXVIII I VI IIII IIII III KAL.APR. II NON.APR. XVIIII
DXXXVIIII II XVII V V XIIII KAL.MAI. VIII KAL.MAI. XX ogd.
B DXL III XXVIII VII VI VII ID.Apr. VI ID.Apr. XV
DXLI IIII VIIII I VII VI KAL.APR. II KAL.APR. XVIII
DXLII V XX II VIII XVII KAL.MAI. XII KAL.MAI. XVIIII
DXLIII VI I III VIIII II NON.APR. NON.APR. XV
B DXLIIII VII XII V X VIIII KAL.APR. VI KAL.APR. XVII
DXLV VIII XXIII VI XI II ID.Apr. XVI KAL.MAI. XVIII
DXLVI VIIII IIII VII XII KAL.APR. VI ID.Apr. XXI
DXLVII X XV I XIII XII KAL.APR. VIIII KAL.APR. XVII
B DXLVIII XI XXVI III XIIII V ID.Apr. II ID.Apr. XVII
DXLVIIII XII VII IIII XV IIII KAL.APR. II NON.APR. XX
DL XIII XVIII V XVI XV KAL.MAI. VIII KAL.MAI. XXI hend.
[DE2]
For the following calculation rules is defined: function mod1(a, b) { m = a mod b; return m ? m : b; }
Heading: Cycli incipiunt anno DXXXII = The cycles begin in the year 532
Column 1
(without column caption): Display whether the year is a leap year
(B,
according to the leap day bissextus,
the doubled day VI
Kal. Mar.) or not ( ).
Calculation: sj = ( j mod
4 ) ? 0 : 1
Column
2: Anni Domini
Nostri Iesu Christi = The years of our Lord Jesus Christ
The year (hereinafter: y ) is assumed to be given.
Column
3: Quae sint
Indictiones = What are the indictions
Calculation:
ind = mod1( j+3, 15 )
[DE3, Argument 2]
Column 4:
Epactae
i.e. adiectiones lunares =
epacts, ie increments of the moon.
Calculation: epa = ((
j mod 19 ) * 11 ) mod 30
[DE3, Argument 3]
Column
5: Concurrentes
dies = concurrent days
Calculation: con = mod1( Floor( j *
5 / 4 ) + 4, 7 )
[DE3, Argument 4]
Column
6: Quotus sit
lunae circulus = which is the circle of the moon
Calculation:
luc = mod1( j - 2, 19 )
[DE3, Argument 6]
Intermediate step
a: The age of the moon on 22 March(XI Kal. Apr.)
Calculation:
luXIKA = mod1(( j mod 19 ) * 11, 30 )
[DE3, Argument 11]
Column
7: Quota sit
luna XIIII paschalis = date of day 14 of the paschal moon
Calculation: lu14 = luXIKA <= 15 ? 36 – luXIKA : 66 –
luXIKA
lu14 assumes the values 21 to 50,
21 to 31 is the date in March,
otherwise (lu14 – 31) is the date in April.
Intermediate step
b: Weekday of the Easter moon
Calculation: wt14 = mod1( 60 +
lu14 + con, 7 )
wt14 assumes the values 1 to 7, 1 => Sun, 2 =>
Mon, ..., 7 => Sat
[DE3, Argument 10]
Intermediate step
c: Weekday of the Easter moon, alternative Calculation
Calculation:
if ( lu14 <= 31 ) { t = 36 – epa; if ( t > 30 ) t = t –
30; t += con + 4 }
else {
t = 35 – epa; if ( t > 30 ) t = t – 30; t += con + 7
}
wt14a = mod1( t, 7 )
[DE3, Argument 14]
Column
8: Dies
dominicae festivitatis = Easter Sunday
Calculation:
os = lu14 + 8 – wt14
os assumes the values 22 to 57,
22 to 31 is the date in March, otherwise (os – 31) is the date in April.
Column
9: Luna ipsius
diei dominici = Moon age of this Easter Sunday
Calculation: if ( os <= 31 ) { lu = 8 + epa + os } else { lu =
9 + epa + os – 31 }
luos = mod1( lu, 30
)
[DE3, Argument 9]
Column 10 (without column caption): indicates when the 8th year of the cycle ("ogd." = Ogdoad) and when the 11th year of thhe remaining years of the cycle ("hend." = Hendekad) is reached.
I ignore the displays in the columns without column captions (column 1 and column 10) as insignificant in the following programme examples.
(Here Microsoft Excel 2007, German licence)
Note: The alternative calculation of the weekday of the Easter moon is omitted here, because it is too much space-consuming.
|
Field |
Contents |
Meaning |
|
A1 |
'ADNIC |
Year number |
|
B1 |
'Indic |
Indiction |
|
C1 |
'IS |
Intermediate step |
|
D1 |
'Epact |
Epacts |
|
E1 |
'Concurr |
Concurrent days |
|
F1 |
'Lu Ci |
Lunar circle |
|
G1 |
'Lu XIKA |
Moon age on 22 March |
|
H1 |
'LuXIV |
Luna14 |
|
I1 |
'WtLuXIV |
Weekday of Luna14 |
|
J1 |
'Do Pasc |
Easter Sunday |
|
K1 |
'Lu Do Pasc |
Moon age ES 1st step |
|
L1 |
'Lu Do Pasc |
Moon age of Easter Sunday |
|
A2 |
532 |
532 |
|
B2 |
=WENN(REST(A2+3;15);REST(A2+3;15);15) |
|
|
C2 |
=REST(A2;19)*11 |
|
|
D2 |
=REST(C2;30) |
|
|
E2 |
=WENN(REST(GANZZAHL(A2*5/4+4);7);REST(GANZZAHL(A2*5/4+4);7);7) |
|
|
F2 |
=WENN(REST(A2+17;19);REST(A2+17;19);19) |
|
|
G2 |
=WENN(REST(C2;30);REST(C2;30);30) |
|
|
H2 |
=WENN(G2<=15;36-G2;66-G2) |
|
|
I2 |
WENN(REST(60+H2+E2;7); REST(60+H2+E2;7); 7)/P> |
|
|
J2 |
=H2+8-I2 |
|
|
K2 |
=WENN(J2<=31;8+D2+J2;9+D2+J2-31) |
|
|
L2 |
=WENN(REST(K2;30); REST(K2;30);30) |
|
|
A3 |
=A2+1 |
|
|
B3:L3 |
Copy from B2:L2 |
|
|
A4:L96 |
Copy from A3:L3 |
|
function DaynumberToDayAndMonth( daynumber ) {
// assert( daynumber > 0 && daynumber < 62 );
if ( daynumber <= 31 ) {
this.dd = daynumber;
this.mm = 3;
} else {
this.dd = daynumber - 31;
this.mm = 4;
}
return this;
}
function mod1( a, b ) {
var m = a % b ;
return m ? m : b ;
}
function ExiguusEasterTableLine( annus ) {
// assert( annus > 0 && annus <= 4999 );
var date;
this.an = annus;
this.ind = mod1( annus + 3, 15 );
this.epa = (( annus % 19 ) * 11 ) % 30;
this.con = mod1( floor( annus * 5 / 4 + 4), 7 );
this.luc = mod1( annus + 17, 19 );
this.luXIKA = mod1(( annus % 19 ) * 11, 30 );
this.lu14 = this.luXIKA <= 15 ? 36 – this.luXIKA : 66 – this.luXIKA;
this.wt14 = mod1( 60 + lu14 + con, 7 );
if ( lu14 <= 31 ) {
t = 36 -– epa;
if ( t > 30 ) t = t -– 30;
t += con + 4
} else {
t = 35 -– epa;
if ( t > 30 ) t = t –- 30;
t += con + 7
}
this.wt14a = mod1( t, 7 );
this.os = lu14 + 8 – this.wt14;
var lu = os <= 31 ? 8 + epa + os : 9 + epa + os – 31;
this.luos = mod1( lu, 30 );
return this;
}
function ExiguusEasterTable( annus, times ) {
// assert( annus > 0 && annus <= 4996 );
// assert( times >= 4 && times <= 532 );
// assert( annus + times <= 5000 );
for ( let j = annus; j < annus + times; j++ ) {
var line = ExiguusEasterTableLine( j );
generateExiguusEasterTableOutput( line );
}
}
If you click on the button below, a new calculation of the data appears for comparison with
the first 19-year section of the Easter table of Dionysius Exiguus shown above,
generated according to the algorithm just developed with the JavaScript functions shown,
whereby in this newly calculated table the first column with the indication of the leap year
and the last column with the indication of the Ogdoad and Hendekad are omitted as
insignificant for the calculation.
An Easter calculator with this algorithm, where you can choose the start year,
the number of years and the output format, can be found at
The Easter table of Dionysius Exiguus
as a table calculator.
DE2 = Dionysius Exiguus (525): Ciclus ab incarnatione domini secundum dionysium abbatem ordinatus; <https://web.archive.org/web/20220928223629/http://www.nabkal.de/osterstreit/anhang/dionysius_2.html> see also DELP, DEOE
DE3 = Dionysius Exiguus (525): Argumenta Paschalia Aegyptiorum; <https://web.archive.org/web/20221014111543/http://www.nabkal.de/osterstreit/anhang/dionysius_3.html> see also DELP, DEOE
DELP = Dionysius Exiguus (2003): Liber de Paschate; <http://henk-reints.nl/cal/audette/denys.html>
DEOE = Dionysius Exiguus (2003): On Easter, or, the Paschal Cycle; <https://www.tertullian.org/fathers/dionysius_exiguus_easter_01.htm>
The author is a mathematician and worked as a software developer.
Karl-Heinz Lewin, Haar: karl-heinz.lewin@t-online.de
Copyright © Karl-Heinz Lewin, 2024