Dionysian Paschal Table

Generate a 95-year Dionysian table and extend it by another 95 Julian years.

Extension rules

General rules
1A common source year requires no correction Copy the Luna XIV date, the Luna XIV weekday, and the Easter Sunday date from the source year to the target year unchanged.
2A bissextile source year requires correction Use the bissextile rules.
Bissextile rules
1Luna XIV date Copy the Luna XIV date from the source year to the target year unchanged.
2Luna XIV weekday Set the Luna XIV weekday of the target year to the day immediately preceding the source year’s Luna XIV weekday (Monday becomes Sunday).
3Easter Sunday
3a If the source Luna XIV falls on Monday through Saturday, set the Easter Sunday date of the target year one calendar day later than the Easter Sunday date of the source year.
3b If the source Luna XIV falls on Sunday, set the Easter Sunday date of the target year six calendar days earlier than the Easter Sunday date of the source year.

Starting year

Choose a Julian year and generate a 95-year source table and a 95-year extension table.

Extension exercise

Use the extension rules above to fill in the blanks and complete the extension table.

Source table Extension table
Source year Year type Luna XIV date Luna XIV weekday Easter Sunday Target year Luna XIV date Luna XIV weekday Easter Sunday Result and
rule feedback
About this exercise

All dates are Julian; Gregorian dates are not used

The source table and answer key follow Dionysius Exiguus’s Alexandrian 19-year lunar cycle together with the Julian 28-year solar cycle. For AD 532–626, the generated source table reproduces the Luna XIV and Easter Sunday dates in Dionysius’s historical 95-year table; outside that range, the Dionysian 95-year table is projected backward or forward using the same Alexandrian 19-year lunar cycle and Julian weekday reckoning.

Dionysius and the extension rules

In his prefatory letter to Petronius, Dionysius writes that his 95-year table does not repeat unchanged after 95 years in every respect. The epact, the lunar cycle, and the Luna XIV date retain their regular recurrence; the concurrent (the annual number used to calculate weekdays), the Easter Sunday date, and the lunar age on Easter Sunday do not share the same regularity.

When writing about the weekday cycle, he gives the annual rule for advancing the concurrent:

“In quo per annos singulos unum numerare curabis; in eo tantummodo anno, in quo bissextus fuerit, duos adicies.” (Dionysius Exiguus, prefatory letter to Petronius, in Bruno Krusch’s edition (1938), p. 64).

“In [the weekday cycle] you will take care to add one for each successive year; only in the year in which there is a leap day will you add two.”

The exercise applies this annual rule over 95 years. If the source year is common, Luna XIV returns on the same weekday. If the source year is bissextile, Luna XIV falls one weekday earlier, because that interval contains one fewer leap day. For a bissextile source year, the 95-year interval between the source and target Luna XIV dates contains 23 leap days, compared with 24 for a common source year. For a common source year, the accumulated weekday advance is 95 + 24 = 119 = 17 × 7, so Luna XIV returns on the same weekday 95 years later. For a bissextile source year, it is 95 + 23 = 118 = 17 × 7 − 1, so Luna XIV falls one weekday earlier.

Extending the table

Dionysius identifies the extra weekday advance in bissextile years as the reason why his 95-year table does not repeat unchanged in every respect. He then expressly directs the reader to return to his table’s first 19-year cycle (AD 532–550) after completing its last (AD 608–626):

“Post expletionem XCV annorum … non ad quintum cyclum sancti Cyrilli … sed ad nostrum primum vigilanter excurrat; et ordine, quo diximus, per eos, qui firmum cursum retinent, eorum progressum, qui videntur titubare, sustentet.” (Dionysius Exiguus, prefatory letter to Petronius, in Bruno Krusch’s edition (1938), p. 64).

“After the completion of ninety-five years … let him carefully turn, not to Saint Cyril’s fifth cycle … but to our own first cycle; and, in the order we have described, let him use those which maintain a firm course to sustain the progress of those which appear to falter.”

The exercise follows this procedure by extending each source row by 95 years. Since 95 years contain five complete 19-year lunar cycles, the Luna XIV date is retained. Dionysius’s annual rule for advancing the concurrent, applied over 95 years, determines whether the Luna XIV weekday remains unchanged or moves one day earlier. The extension rules give an explicit arithmetic form to Dionysius’s instruction to use those elements of his table which maintain a firm course after 95 years to compute the elements which do not. In this exercise, the recurring Luna XIV dates in Dionysius’s table provide the basis for computing Luna XIV dates, Luna XIV weekdays, and Easter Sunday dates for both common and bissextile source years.

Voigt credit

The exercise is inspired by Ulrich Voigt’s analysis of the Dionysian Watum and its 247–95–95–95 interval sequence. Voigt writes: “After a 95-year period, the Easter date changes when and only when the source year is bissextile: by −6 days when Luna XIV in that source year falls on a Sunday, and by +1 day otherwise” (Ulrich Voigt, Zyklen und Perioden (2013), p. 82; translated from German).