Easter Algorithms from the
Six Canons of Christopher Clavius, etc.

General Notes

This page is still under development, but the main substance dates from Nov/Dec 2008.

Source Material

The Bull

The original authorisation for the current dates of Easter Sunday is, as for the current Leap Year rules, Pope Gregory XIII's Papal Bull Inter Gravissimas of 1582. For more links about the Bull, including text and translation on the Web, see my Leap Years and Date Links pages.

The Six Canons

Although the Bull determines Leap Years completely, it does not contain the Easter Rules, merely referring to an "explanation". The Six Canons, published in 1582, contain some details and enable the dates of Gregorian Easter to be determined up to AD 4999. They lack an explanation of the Lunar Correction, which is needed to obtain perpetual dates.

Some Web pages by Rodolphe Audette, Les textes fondateurs du calendrier grégorien (+) ff., now hosted by Henk Reints, provide more legible Latin, with a parallel French translation, for at least the most important parts. Follow the links to the Six Canons and see Table des matières de l'Explicatio de Clavius, extraite du tome 5 de ses Opera Mathematica (+). One can get the French material in something resembling English by using Google Translate.

The Explicatio

The full explanation, published about twenty years later, is in the lengthy Romani calendarij à Gregorio XIII. P. M. restituti explicatio (1603) by Christopher Clavius, which is included with the Bull and the Canons in the fifth volume of Opera Mathematica (imaged) (1612).

European Cultural Heritage Online (ECHO) has a much better copy of the Romani Calendarii a Gregorio XIII. P. M. Restituti Explicatio (URL from Al Petrofsky) (+). Compared to Notre Dame's copy, this version is better focused, in higher resolution, and in colour, and it also has the benefit of being the original 1603 typesetting, rather than the 1612 reprint in the Opera Mathematica. The page numbering and chapter numbering are different.

Text Here

Text on this page should express the essence of the necessary Latin, referring back to the corresponding parts of the original material, and from those the algorithms are derived.

Text Source View

Text with this background colour will show enough, when toggled by click, to locate the appropriate part of the original Latin (and so the French). Try here. The Latin. Paragraph boundaries are indicated thus : ¶.

The Methods

This page derives JavaScript algorithms for the Date of Gregorian Easter Sunday, and for the (Julian) Date of Julian Easter Sunday, in "Literal" and "Optimised" versions.

Literal Methods

Easter is determined by using representations of Clavius's Tables from the Six Canons. The code reads copies of the .innerHTML strings of the relevant parts of the displayed Tables - which is equivalent to reading the Tables from the screen - and follows Clavius's instructions closely.

Those representations are calculated, for convenience; but their validity is based not on the calculation but on the visible agreement between the displayed representation and Clavius's Tables. They have been re-shaped or extended to facilitate their calculation, display, and/or usage, so that their string form can easily be read by code.

In Canons 2 & 4, (Epacts & Sunday Letters), Tables need one entry per centade. Here they are used directly for AD 0 to 9999. For Canon 2, the corresponding part of the arithmetic method for the Epact (which was used to generate the Table) is used after 9999. For Canon 4, the year is used modulo 10000, since it is clear enough from the Bull and the lengths of common and bissextile years that the week pattern repeats every 400 years. ( 400×365 + 100 - 4 + 1 = 146097 = 7×20871 )

Material with this background is used by the Literal Gregorian method.

Optimised Methods

These use arithmetic, yet are traceable to Clavius.

Material with this background is used by the Optimised Gregorian method.

Julian Methods

Material with this background is used only for Julian Easter.

Abbreviations

• BCP - The Book of Common Prayer of the Church of England, as decreed by the British Calendar Act of 1751, Section 9, Tables and Rules for the Feasts and Fasts through the whole Year
• GN - Golden Number
• PFM - Paschal Full Moon
• SC - Solar Cycle (1 to 28)
• SL - Sunday Letter (A to g)
• SN - Sunday Number (0 to 6)

Code Notes

I now write and develop code mainly with Firefox 3, and do long tests with Opera.

Suffix 1, 2, ... is applied to global identifiers for "Literal" methods, and ..., 9 to those for "Optimised" methods.

Code which reads Tables may assume a four-character year format.

JavaScript : X|0 is equivalent (for X<2³¹) to Math.floor(X), so X/Y|0 is integer division; X\Y in some languages. And X%Y is X mod Y, remainder of division

Material in ? is computed by code on this page.

Utility Code

These routines are specific to this page; for others, see JavaScript Include Files.

The Essence of Christopher Clavius's Text

The Six Canons of 1582, with the Bull, provided all that was needed to give the Date of Easter Sunday up to AD 4999. Extension requires only the knowledge that the Lunar Correction repeats every 2500 years; all else can be deduced from Bull and Canons.

The Compendium COMPENDIVM NOVÆ RATIONIS

... This repeated comparison applies every 300 years, as said before, but not every 2400 years, because it is not exactly 300 years before there is a difference of one day, but twelve years and six months more, so that after 2400 years would be too much by a third of a day, therefore the application of the comparison should be carried forward to the next century year, ... ... Ea autem multiplex est ; una quæ 300. annis debitur ut diximus : Altera quæ bis millenis & quadringenis : quoniam non exacte 300. illis annis unus debetur dies. 12. enim anni & sex menses desiderantur, qui in 2400. annis tertiam fere diei partem conficiunt, & in sequentum centesimum æquationem transferunt ; ...

Canons of the Perpetual Gregorian Calendar CANONES IN CALENDARIVM GREGORIANVM PERPETVVM

It seems that the Canons were released in 1582.

Tables in Canons 2 & 4 here are computed to cover AD 0 to 9999. The extension for Canon 2 to cover AD 0 to 9999 requires justification of function Greg_CYPHER (which is also required for Table II of the Calendar Act Annexe). It is evidently to be found in liber novæ rationis restituendi calendarii Romani. The extension for Canon 4 back to AD 0 is obvious.

Canon 1
The Nineteen-Year Cycle of the Golden Number DE CYCLO DECENNOVENNALI AVREI NVMERI

The whole of this Canon is for both Julian and Gregorian.

The Golden Number is cyclic, 1 to 19, year 1577 has Golden Number = 1. ¶ Cyclus decennovennalis aurei numeri ...
... qui dicitur aureus, est 1. ...
... anno 1596 aureus numerus sit rursus 1 et anno 1597 sit 2, etc.

From that, one can get GN = 1 + ( (Y-1577) mod 19 ) As 1577 mod 19 = 0   GN = 1 + Y mod 19  

New Moon dates repeat, approximately, every 19 years. Continet autem hic cyclus aurei numeri annos 19 quia post 19 annos solares elapsos revertuntur novilunia ad eosdem dies mensium, licet non omnino præcise, sed aliqua diei particula citius, ut a computistis et in libro novæ rationis restituendi calendarii Romani ostenditur.

The Golden Number changes at the Dec/Jan change; 1..19, repeated. Then some history. Quilibet vero annus aurei numeri ...
... idemque magis ac magis in dies futurus sit inutilis, ...

Acknowledges the loss of 10 days in October 1582 and the new Leap Year Rules. ... tum propter decem dies ex mense Octobri anni 1582 auferendos, tum etiam propter tres bissextos omittendos quibusque quadringentis annis,

Would need 30 calendars. nisi in 30 ordines redigatur, hoc est, nisi 30 calendaria construantur, ut ex illis seligatur semper illud, quod certo cuidam tempori congruit: quæ res quantas perturbationes, quantosque sumptus personis præsertim ecclesiasticis esset allatura, nemo non videt.

Sumulationtion of 30 calendars selected by epoch. 30 Epacts. Epact depends on Golden Number. Hoc incommodum ut vitetur, substitutus est in locum aurei numeri in calendario, cyclus epactarum constans ex 30 numeris epactalibus; ...
... locumque amplius non habeat ad novilunia festaque mobilia invenienda. ¶

The following table is to find Golden Number for any year from 1582. ¶ Igitur ut aureus numerus quolibet anno proposito inveniatur, ... duratque in perpetuum. ¶

 ----------------------------------------------------------------------
| vj vij viij ix x xj xij xiij xiv xv xvj xvij xviij xix j ij iij iv v |
 ----------------------------------------------------------------------

Short explanation of tedious use of that table for Golden Number. ¶ Ex ea enim aureus numerus cuiuslibet ...
... propositus cadit dabit aureum numerum quæsitum. ¶

Previous is tedious far from 1582, so another Table for any year. ¶ Sed quoniam valde laboriosum ... 1582 invenietur, hac arte. ¶

Examples. Reason for adding 1. Notes that AD 1 has Golden Number=2. Construction of Table. ¶ Exemplis res fiet illustrior. Sit inveniendus aureus numerus anni 700. ... numero anni 1595. ¶ Additur autem semper ... numerus 3, etc. ¶ Compositio quoque huius tabulæ ... et præterea anno 1000 respondet aureus numerus 12, etc. ¶

Canon 2
On Epacts and New Moons DE EPACTIS, ET NOVILVNIIS

Table of the Equation of the Perpetual Cycle of Epacts Tabula æquationis cycli epactarum perpetui

Let us illustrate this by examples. ¶ Exemplis planum id faciemus. Anno 1582 ...
... Atque hoc modo epactam cuiuslibet anni invenies in perpetuum. ¶

On constructing short Tables. ¶ Ex his facile quivis tabellam componere poterit ... et ab epacta XIII illius anni sic stabit. ¶

Table for 2200 up to 2300 Tabella epactarum respondentium aureis numeris ab anno 2200 inclusive usque ad annum 2300 exclusive.

On constructing short Tables. Use of perpetual table instead. ¶ Sed eædem hæ epactæ facilius ... restituti reperies. ¶

Get the Sunday Letter from the Solar Cycle for Julian in Canon 4.

Canon 4
The Sunday Letter DE LITERA DOMINICALI

The days of each year, omitting February 29th, are given the letters Abcdefg cyclically. Thus January 1st and December 31st are both always lettered A. I use 0123456 as corresponding Sunday Numbers.

The missing 10 days and the missing leap years now break the old system. ¶ Quoniam tum propter decem dies ablatos ex mense Octobri anni 1582 ...
... usque ad annum 1700 exclusive. ¶

Table of Sunday Letters from the Ides of October 1582, year of the reform, (after the suppression of 10 days), to 1700 exclusive. Not used here Tabella literarum dominicalium ab Idibus Octobris anni correctionis 1582 (detractis prius X diebus) usque ad annum 1700 exclusive.

To use that table : the Sunday Letters sequence as before. Late 1582 has letter c, 1583 has b, 1584 has Ag, up to 1700 (not leap). ¶ Usus huius tabellæ hic est. Anno correctionis 1582 post Idus Octobris (detractis prius X diebus) tribuatur litera c primæ cellulæ; ...
... ad festum S. Matthiæ, inferior autem b, in reliqua parte anni. ¶

To help, the following table of years like 1582. ¶ Verum ut in annis, qui parum ab anno 1700 distant, ...
... anno 1700 ad quem usus tabellæ literarum dominicalium non pervenit. ¶

Years in which the Sunday Letter table starts. Not used here. 1582 1610 1638 1666 1694 Anni a quibus tabella literarum dominicalium incipit.

Use of that Table. ¶ Itaque si annus cuius litera dominicalis quæritur, ...
... quæ tertia est post bissextum et dominicalis eo anno. ¶

After 1699, as follows. ¶ Finito autem anno 1699 ...
... hoc modo. ¶

For 1700 onwards, seek in the larger Table whichever of I II III is against the highest number less than or equal to the year. Seek that in the Table of letters, count forwards from that from the number just found to the actual year, circularly, take its (lower) letter. Inventurus literam dominicalem cuiuslibet anni, ...
... quemadmodum in aliis annis bissextilibus. ¶

Example. ¶ Exemplum. Anno 1710 ...
... cum in tabula æquationis non contineatur. ¶

Not needed? ¶ Hic autem utendum erit quoque artificio supra descripto, ...
... in cellulam duarum literarum b, A, ut prius. ¶

Not needed. 3500 3528 3556 3584 3612 3640 3668 3696 -

Not needed? ¶ Immo eadem hæc tabella cuilibet ... 3500 proxime insequitur. ¶

Not needed? ¶ Facillima porro est constructio tabulæ æquationis. ... numeri antiqui I, II, III ordine repetuntur. ¶

Not needed? ¶ Ex his non difficile erit cuilibet ex nostra tabella perpetua ...
... Sequenti deinde anno 1801 litera dominicalis sit d, etc. ¶

Table of Sunday Letters from 1800 to 1900 exclusive. Not needed Tabella literarum dominicalium ab anno 1800 usque ad annum 1900 exclusive.

To find Julian Sunday Letter of any year ¶ Expedite quoque eamdem literam dominicalem cuiusque anni ... sit bissextilis et tunc idem numerus antiquus repetatur, ita se habet. ¶

Sunday Number and Letter Arithmetically

It is clear that the Sunday Letter is [gfedcbA] and cyclically decrements at each New Year and each Leap Day. From that, the arithmetical method for the Sunday Letter applying to Eastertide is deducible for either Calendar.

Test Sunday Letters in 3rd, 12th/13th. and 22nd centuries; all letter columns should agree :
 

Canon 5
The Indiction DE INDICTIONE

Irrelevant for Easter; and rather like the Golden Number.

Canon 6
The Moveable Feasts DE FESTIS MOBILIBVS

Nicæa : Easter is the Sunday next following the 14th day of the lunar month, which is the month having its 14th day on or after March 21st. ¶ Quoniam ex decreto sacri concilii Nicæni Pascha, ex quo reliqua festa mobilia pendent, celebrari debet die dominico qui proxime succedit XIV lunæ primi mensis (is vero apud Hebræos vocatur primus mensis cuius XIV luna vel cadit in diem verni æquinoctii, quod die XXI mensis Martii contingit, vel propius ipsum sequitur) efficitur ...

Which is like saying that Easter is the Sunday next following the Full Moon which is or after March 21st (and that the Full Moon is on the 14th day of the lunar month).

Get the Epact from Canon 2, find it in the CALENDARIVM (Printed pp.36-47; pp.38,39 shown below) (+) between March 8th and April 5th inclusive, add 14 days to reach PFM), go to the Sunday next following for Easter Sunday. ... ut si epacta cuiusvis anni inveniatur ex canone 2 ... sit dies Paschæ. ¶

Example. 1583, April 10th; 1585, April 21st; 1592, March 29th. ¶ Exemplum. Anno 1583 iam emendato epacta est VII et litera dominicalis b. ...
... celebrabitur eo anno Pascha die 29 Martii. ¶

After having found Easter, one can easily find the other moveable feasts. ... paragraph of no present interest. ¶ Invento autem die Paschæ, facile alia festa mobilia invenientur. ...
... investigare paulo infra. ¶

We have also constructed the two following Paschal tables, one old, one new, to help find moveable feasts. ¶ Cæterum ut facilius omnia festa mobilia inveniantur, compositæ sunt duæ sequentes tabulæ paschales, una antiqua et nova altera. ...

Whether one uses the old or the new Easter table, all moveable feasts must be found, in Leap Years, using the second Sunday Letter ... Little interest ¶ Sed sive antiqua, sive nova tabula ...
... quare nihil additur, etc. ¶

Little interest ¶ Rursus anno 3784 bissextili epacta erit ...
... numerentur usque ad Septuagesimam. ¶

Little interest ¶ Adventus Domini celebratur semper ...
... quia ibi est litera g in calendario, etc. ¶

Little interest ¶ Numerus quoque dominicarum inter ...
... inter Pentecostem et Adventum Domini. ¶

From the above one sees how to construct the two general Easter tables. ¶ Ex his omnibus facile intelligi potest, qua ratione utraque tabula paschalis composita sit. ¶

They are preceded by a table for several years. ¶ Utrique tabulæ paschali generali ...
... pro quibuscumque annis. ¶

In the reformed old table, Golden Numbers are added to allow finding Easter etc. from the Council of Nicæ to 1582. Find the Golden Number, move down to the Sunday Letter to find the dates of feasts. ¶ In priori porro tabula paschali antiqua ...
... et sic de cæteris. ¶

Audette has the following Tables in his Canon 6 page; Clavius separates them.

Following :
Associated Tables TABELLA TEMPORARIA FESTOrum Mobilium
TABVLA PASCHALIS Antiqua Reformata
TABVLA PASCHALIS NOVA REFORMATA

Dates for 1582 to 1631, p.32. Not needed here. Not in Audette. TABELLA TEMPORARIA FESTOrum Mobilium

Other Parts

Testing These Computer Algorithms

Tests have covered years 0 to 5700000+ Gregorian.

The following form has several parts :-
• Greg_Test1 Gregorian, literal, following Clavius
• Greg_Test9 Gregorian, arithmetical, to be traceable to Clavius
• Juln_Test1 Julian, literal
• Juln_Test9 Julian, arithmetical
Results are compared with known-good values :

Code for four tests :-

Long tests are best done in a browser which does not give "busy" warnings; I often use Opera for this.

Notes

• CAPUT XXIIII must be a refutation of the expanded calendar of Francis Vieta.

Pending ?

• Needed : Location of Lunar Correction in Explicatio
• Improvement of "Optimised" ?
• Julian Easter using Epact
• Easter Repeat Interval
• Animated literal calculation ??
• ?