A study of Computus, the reckoning of the date of Easter. Part of my Final Year Project as a Mathematics & Computer Science student. I gave this presentation on Thursday, the 4th of April, 2019.

1. A mathematical
study of Computus
By Taidgh Murray, BSc

2. What is Computus?
• Historical calculation of the Date of Easter
• Sunday after full moon, on/after March Solstice
• Mathematical approximation, dependent on Lunar Cycle
• Topic of ecclesiastical controversy
• Acknowledgements – Immo Warntjes & Jacopo Bisagni
• Slides on slideshare (slideshare.com/TaidghMurray)
• But first…

3. • Gregorian Calendar – Improvement of Julian Calendar (46 BCE)
• Lunisolar Calendar
• Pope Gregory XIII, 1583
• E.g 1700, 1800, 1900 are not leap years
• But 2000 is
Our Modern Calendar

4. 19 Year Metonic Cycle (Lunar vs Solar)
• Lunar (Synodic) Month ~= 29.5 days
• 235 Synodic Months are 6939 days, 14.5 hours
• Solar Year ~= 365.25 days
• 19 Solar years are 6939 days, 16.5 hours
• Epact (age of moon on 1st Jan) will be periodic
every 19 years

5. CF Gauss’ Easter Algorithm
• Carl Fredrich Gauss (1777 – 1855)
• Developed a general algorithm in 1800, essentially
the one in use today
• Julian or Gregorian Easter date

6. Dionysius Exiguus’ Easter Table
• 6th Century Monk
• Standardised the reckoning of the date of Easter
• Along with the AD dating System
• Generated an ‘Easter Table’ to achieve this

7. Dionysius Exiguus’ Easter Table
22-Mar 24-Mar
AD Indiction Epact Concurrent Day Lunar CycleEaster Full Moon Lunar 14 Pasc MoonPasc Moon Easter Month Lun Age
559 7 28 2 Mon 6 April 7 2 Mon 13 April 20
…
572 5 22 5 Thur 19 April 13 4 Wed 17 April 18
…
582 15 12 3 Tues 10 March 24 3 Tues 29 March 19
…
597 15 28 1 Sun 6 April 7 1 Sun 14 April 21
…

8. Dionysius Exiguus’ Easter Table
• y is year
• Indiction
• 15-year Roman fiscal cycle for taxation purposes
• Epact
• Age of the Moon on 22nd of March (Equinox)
• 30 values
• Concurrent
• Weekday on 24th of March
AD Indiction Epact Concurrent Conc. Day Lun. Cycle Easter Full Moon Lunar 14 Pasc MoonPasc Moon Easter Month Lun Age
559 7 28 2 Mon 6 April 7 2 Mon 13 April 20

9. Dionysius Exiguus’ Easter Table
• Lunar Cycle (19 year)
• Lunar 14
• Date on which the Full Moon (Lunar 14 value) occurs
• If epact<5 or >15, full moon will fall in April
• Easter Date
• Between 22nd of March and 25th of April
• Online Calculator (Click Me)
AD Indiction Epact Concurrent Conc. Day Lun. Cycle Easter Full Moon Lunar 14 Pasc MoonPasc Moon Easter Month Lun Age
559 7 28 2 Mon 6 April 7 2 Mon 13 April 20

10. Latercus Easter Table
• Used by British & Irish churches in 4/5th century
• Abandoned in 716 AD
• Believed to be lost
• Found in a manuscript in Padua in the 1980s
• Dáibhí Ó Cróinín & Daniel McCarthy

13. Latercus Easter Table
• Conglomerate March and April for Easter Calculation
• Do the same for March and February for Initium
• Generate Index, Feria, and Epact
• These values cannot be inferred from year alone
• Calculate March 1st Feria
• Determine date and Lune of Easter
• Work backwards to find Initium date and Lune

14. *Click here*

15. Thank you!

16. • Suplimentary slides from here on, designed to give a bit more
background for anyone curious

17. Our Modern Calculator
• Julian Calendar - 46 BCE
• Drift of
• ‘Roman Republic Calendar’
• 12 months with intercalary month of 23 days
• ‘Legendary 10 Month Calendar’ – Earliest point
• 304 days, 61 unorganised ‘Winter’ days

19. CF Gauss’ Easter Algorithm
• Lunar Cycle (a)
• Leap Year (b)
• Epact Feria (c)
• Century (k)
• ‘Metonic Correction’ (p)
• Originally, corrected to
• Correcting Gregorian leap year rule (q)

20. CF Gauss’ Easter Algorithm
• Lunar 14 distance (M)
• Julian: Cycle year 1, Lunar 14 was always on 36th of March
(5th of April), so M is 15
• Gregorian value changes due to Leap Year rule
• Feria distance (N)
• Julian: distance was always 6
• Corrections need to be made for Gregorian calendar

21. CF Gauss’ Easter Algorithm
• Distance from 21st March and Lunar 14 (d)
• 19 has little to do with 19 year Lunar Cycle
• Corrects 11 day difference between calendars
• 19 mod 30 is the same as taking 11 away
• Distance from Lunar 14 to following Sunday (e)
• Similar idea for the coefficients outlined previously
• Lunar 14 Feria changes by either 1 or 2 each year
• 6 mod 7 is the same as taking 1 away

22. CF Gauss’ Easter Algorithm
• Assumes March & April are one month during
calculations
• Epact is 1st of January
• If d = 29, e = 6, April 26th becomes April 19th
• 26th is regarded as being too late for Easter by church
• If d = 28, e = 6, a > 10 April 25th becomes April
19th
• Let’s look at an example

23. Latercus Easter Table
• Cycle index
• Year
• 1 Jan. – Feria on January 1st
• Epact – Age of moon on Jan 1st / March 1st
• Easter date
• Lune – Age of moon on Easter
• Initium – Beginning of Lent
• Lune – Age of moon on Initium