logo To Foot
© J R Stockton, ≥ 2013-01-15

Clavius's Operum Mathematicorum
Tomus Quintus
.

No-Frame * Framed Index * Frame This
Links within this site :-

Incomplete. Unchecked. Continuation unlikely.

2010-10-02
I see that Google Translate now does Latin.

Notes

The original authority for the date 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 it, including text and translation on the Web, see my Leap Years and Date Links pages.

The Bull determines Leap Years completely. But it does not contain the full Easter Rules, citing other documents. It seems that the other documents were not properly published at the time.

The Text below is from the fifth volume of Christopher Clavius's Opera Mathematica (1612) - Images of Pages. It includes the Papal Bull, the Six Canons of 1582, and Romani calendarij à Gregorio XIII. P. M. restituti explicatio (1603).

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 1611/12 reprint in the Opera Mathematica. The page numbering and chapter numbering are different.

I found a Contents List, by Rodolphe Audette - "Table des matières de l'Explicatio de Clavius, extraite du tome 5 de ses Opera Mathematica" (+). The task appears to be about 30 times greater than it at first seemed, so I do not intend to continue with it.

Indeed, he has done much of the work that I had hoped to do, though into French. He also provides more legible Latin. One can see the French material, using Google Translate, in approximate English.

Henk Reints has the Compendium in Dutch, which Google can translate.

My Easter Algorithms from the Six Canons of Christopher Clavius, etc. includes quasi-translations of parts of the Canons.

Notation

Font face and size are very loosely approximated. Positive page numbers generally appear in the original; I've extended the sequence below 0 with roman numerals.

Page Numbering is as at Notre Dame.

These boxes vaguely indicate the appearance of the corresponding page.
These boxes may refer to the content of the corresponding page.

The Full Text, with Translation, Discontinued

Indexes to the 1603 and 1612 printings are at The Essence of Christopher Clavius's Text.


Fifth Volume - Endleaf Front - Page 1

Blank

Fifth Volume - Volume Title Page and verso - Page 1


CHRISTOPHORI
CLAVII BAMBER-
GENSIS E SOCIE-
T A T E   I E S V

OPERVM MATHEMA-
TICORVM
T O M V S   Q V I N T V S
Continens

ROMANI CALENDARII à Gregorio XIII. P. M. RESTITVTI Explicationem S. D. N. Clementis VIII. P. M. iussu editam.

NOVI CALENDARII ROMANI APOLOGIAM aduersus Michaelem Mæstlinum Gæppingensem in Tubingensi Academia Mathematicum duobus libris explicatam.

APPENDICEM AD NOVI CALENDARII ROMANI Apologiam, in qua Iosephus Scaliger, Georgius Germanus, & Franciscus Vieta, qui Calendariu, aliter instaurandum esse contenderunt, seorsim singuli confutantur. Accessit refutatio Cyclometriæ eiusdem Scaligeri.

**LOGO**
M O G V N T I Æ,
Sumptibus ANTONII Hierat, excludebat
IOANNES VOLMARI.
Cum Gratia & Priuilegio Sacræ Cæsar. Maiest.

Anno M. DC. XII.

Volume Title Page and verso - Page 2

[ BOOK-PLATE ]

Volume introduction - Page 3

REVERENDISSIMO AC
ILLVSTRISSIMO PRINCIPI
AC DOMINO, DOMINO IOANNI

[CONTINUED]

Volume introduction - Page 4

[CONCLUDED}

Roman Calendar of Gregory XIII - Title page - Page i


R O M A N I
CALENDARII
A GREGORIO XIII.
P. M. RESTITVTI
EXPLICATIO

S.D.N.CLEMENTIS VIII.
P. M. IVSSV EDITA.

A U C T O R E

CHRISTOPHORO CLAVIO
B A M B E R G E N S I   S O C I E -
T A T I S   I E S V

SANCTIS

Introduction - Page ii


SANCTISSIMO
PATRI AC DOMI-
NO NOSTRO
CLEMENTI VIII.

CHRISTOPHORVS CLAVIVS
E SOCIETATE IESV F.


Q
VANTI sit semper, P. B. ponderis ac momenti res habita, vt Christi liberatoris nostri reuiuiscentis memoria certis, ac statis anni temporibus recoleretur, vel ex eo facile intelligere licet, quod complures Romani Pontificies super ea re multa eaque grauissima, decreta extare voluere; quinetiam sæpius seueriffima anathematis denunciatone cauerunt, ne quis ad sanctissimum

[CONTINUED]

sanctissi-

Introduction - Page iii


Introduction - Page iv


Introduction - Page v


C L E M E N S
PAPA VIII.

AD PERPETVAM REI
Memoriam


Q
UAECUNQUE à Romanis Pontificibus Prædecessoribus nostris ad communem totius Catholica Ecclesiæ vtilitatem diuturno studio, & longa deliberatione,

[CONTINUED]

Introduction - Page vi

... Duodecimo.

M. Vestrius Barbianus.  

Readings - Page vii

L E C T O R I   S.

Readings - Page viii

[ CONCLUDED ]

Roman Calendar of Gregory XIII - Preamble - Page 1

1
P R O ΠM I V M

[ See - READ FIRST - Proœmium at ULaval. It omits the italic sidebar, which I find illegible. ]

Preamble - Page 2

[ CONCLUDED ]

Compendium - Page 3

3
C O M P E N D I V M
N O V Æ   R A T I O N I S

Restituendi Calendarium à Gregorio  XIII.
Pontifice Maximo


AD PRINCIPES CHRISTIANOS, ET CELE-
berrimas quasq. Academias missum anno Domini  1577.

P E R I T I S   M A T H E M A T I C I S.

... ... ...

Compendium - Page 4

[ TEXT CONCLUDED; LOGO ]

Compendium - Page 5

[ JAN FEB MAR ]

Compendium - Page 6

[ APR MAY JUN ]

Compendium - Page 7

[ JUL AUG SEP ]

Compendium - Page 8

[ OCT NOV DEC ]

Compendium - Page 9

Clavius's original Latin Rodolphe Audette's French Google Translate of Henk Reints' Dutch My interpretation of those
... 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 ; ... ... Cette équation est multiple; on doit d'abord l'appliquer tous les trois cents ans, comme nous l'avons dit; ensuite tous les deux mille quatre cents ans, parce que ce n'est pas en 300 ans exactement que se produit un jour d'écart, mais en 12 ans et six mois de plus, lesquels produisent en 2400 ans un tiers de jour d'écart environ et font reporter l'application de l'équation à l'année séculaire suivante; ... ... This comparison is multiple, we must apply it every 300 years, as I said, but not every two thousand four hundred years, because it is not exactly 300 years before the day is beaten, but twelve years and six months, in 2400 year third day would bring too much, so that the application of the comparison should be carried forward to the next century year, ... ... 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, ...
[ EPACT TABLE ; TEXT ]

Compendium - Page 10

[ TEXT & TWO TABLES ]

Compendium - Page 11

[ TEXT; FOUR EPACT TABLES; TEXT ]

Compendium - Page 12

[ TEXT & TWO TABLES ]

Roman Calendar of Gregory XIII - Calendar - Page 13

   13   


C A L E N D A R I V M
G R E G O R I A N V M
P E R P E T V V M.

Orbi Christiano vniuerso à GREGORIO XIII. P. M. pro-
positum. Anno M. D. LXXXII.


G R E G O R I U S   E P I S C O P U S
S E R V U S   S E R V O R U M   D E I
AD PERPETVAM REI MEMORIAM.

I
nter grauissimas Pastoralis officij nostri curas, ea postrema non est, vt quæ à sacro Tridentino Concilio Sedi Apostolica reseruata sunt, illa ad finum optatum, Deo adiutore perducantur. ... ...
A
mong the heaviest cares of our Pastoral office, not the least is, that which reserved by the holy Apostolic sitting of the Council of Trent, we wish to finish, with God's continuing help. ... ...

The Bull, in Latin, French, and English, may be found at
bluewaterarts
so I only show the beginning and the end here.

A better translation is at
Inter Gravissimas (Papal Bull) (1582) by Gregory XIII, translated by Wikisource

14

... ... ...

15

... ...

  Datum Tusculi Anno Incarnationis Dominicæ M. D. LXXXI. Sexto Calend. Martij, Pontificatus nostri Anno Decimo.

Cæ. Glorierius,  
A. de Alexijs.  

  Given at Tusculum, in the year of the Incarnation of the Lord 1581, the Sixth of the Kalends of March, in the tenth year of our Pontificate.

Cæ. Glorierius,  
A. de Alexijs.  

The year number above is Old Style, in which the year number changes at March 24/25; the date is Julian A.D. 1582-02-24.

The following paragraph seems not to be part of the Bull itself.

  Anno à Natuitate Domini nostri Iesu Christi Millesimo Quingentesimo Octuagesimo secundo, Indictione decima, Die vero Iouis prima mensis Martij, Pontificatus vero sanctissimi in Christo Patris, & D. N. Gregorij diuina prouidentia Papæ XIII. anno eius decimo: Retroscriptæ literæ Apostolicæ publicatæ & affixæ fuerunt in Valuis Principis Apostolorum de Vrbe & Cancellariæ Apostolicæ ac in acie Campi Floræ, vt moris est, per me Scipionem de Octauianis Apostolicim Curs.

Franciscus Baron Magister Cursorum.  

16

16 C A L E N D A R I V M   


C A N O N E S
I N   C A L E N D A R I V M
G R E G O R I A N V M
P E R P E T V V M

Canon   1.
D E   C Y C L O   D E C E N N O V E N A L I
A U R E I   N U M E R I

C
yclvs Decennouenalis Aurei numeri est resolutio numeri 19.annorum ab 1. vsque ad 19. qua reuolutione peracta, iterum ad vnitatem reditur. Verbi gratia, Anno 1577. Numerus cycli decennouennalis, qui dicitur Aureus, est 1. Anno sequenti 1578. est 2. & ita deinceps in sequentibus annis, vno semper amplius, vsque ad 19. qui Aureus numerus cadet in annum 1595. Post quem iterum ad unitatem redeundum est, ita vt anno 1596. Aureus numerus sit rursus 1. & anno 1597 sit 2. &c. Continet autem hic cyclus Aurei numeri annos 19. quia post 19. annos solares elapsos reuertuntur Nouilunia ad eosdem dies mensium, licet non omnino præcise, sed aliqua dici particula ci?ius, vt à Computistis, & in libro nouæ rationis restituendi Calendarij Romani ostenditur. Quilibet vero annus Aurei numeri terminatur in sine mensis Decembris, & in principio Ianuarij sequentis anni initium sumit alius annus Aurei numeri : quemadmodu etiam sit in annis Domini, qui semper terminantur in mense Decembri, & initium sumunt à Ianuario. Vt anno Dni 1582. annus cycli decennouennalis, qui dicitur Aureus numerus, est 6. terminaturq; simul cum ipso anno Domini 1582, in mense Decembri. In mense autem Ianuario initium sumit alius annus Domini, nempe 1583. & in eode mense Ianuario assumitur etiam alius annus Aurei numeri, nimirum 7, Et ita deinceps agendum erit in aliis annis, quousque perficiatur numerus 19. à quo iterum redeundum est ad vnitatem : atque hoc modo in perpetuum. Hoc cyclo decennouennali Aurei numeri per dies Calendarij distributo Ecclesia Romana ad hanc usque diem usa est tum ad coniunctiones Solis ac Lunæ inquiriendas, tum vero maxime ad inuiendum diem festum Paschae, & ad indaganda alia festa mobilia : propterea quod veteres putabant, Nouilunia, transacta spatio 19. annorum solarium, ad eundem prorsus diem, eandemque horam redire; quod verum non est, cum Nouilunia paulo citius, quam spatium 19. annorum Solarium compleatur, ad eandem sedem redeant, vt dictum est. Hinc factum est, ut Novilunia hoc tempore plus quam quatuor dies distent ab Aureo numero in veteri Calendario Romano & secundum illius normam Pascha sæpenumero post xxj Linam, contra Maiorum instituta, celebretur: adco vt cyclus hic Aurei numeri inutilis omnino iam sit inuentus ad Nouilunia, festaque mobilia indicanda, idemque magis ac magis in dies futuris sit inutilis tum propter decem dies ex mense Octibri anni 1582. auferendos, tum etiam propter tres Bissextos omittendos quibusque quadrigentis annis; nisi in 30. ordines redigatur, hoc est, nisi 30. Calendaria construantur, vt ex illis seligitatur sem per illud, quod certo cuidam tempori congruit: quæres, quantas perturbationes, quantosque sumptus, personis præsertim Ecclesiasticus, esset allatura, nemo non videt. Hoc incommodum vt vitetur, substitutus est in locum Aurei numeri in Calendario Cyclus Epactarum constans ex 30. numeris Epactalibus: qui quidem nihil aliud est quam cyclus decennouennalis Aurei numeri æquatus, ita vt sit instar Aurei numeri in 30. Calendaria, de quibis dictum est, distributi, vt in libro nouæ rationis restituendi Calendarij Romani declaratur. Aureo vero numero vtemur in posterum, non quidem ad Nouilunia, & festa mobilia inquirenda, vt ad hanc vsque diem factum est ab Ecclesia, sed solum ad inuestigandum Epactum cuiuslibet snni, ex qua & Nouilunia, & festa mobilia deinde reperiantur, vt in sequenti Canone docebimus: ita vt etiam numc necessarium omnini sit Aureum numerum quouis anno indagare, licet is de Calendario sit submotus, locumque amplius non habet ad Nouilunia, festaque mobilia inuenienda.

 IGITUR vt Aureus numerus quolibet anno proposito inueniatur, composita est sequens tabella Aureorum numerorum, cuius usus incipiy ab anno correctionalis 1582. inclusiue, duratque in perpetuum. Ex ea enim Aureus numerus cuiuslibet anni postannum 1582. reperietur hoc modo.

The Nineteen-year cycle of the Golden Numbers is ...

[CONTINUES]


Tabella cycli Aurei numeri initium sumens ab anno correctionis 1582.
Aur. num. vj vij viij ix x xj xij xiij xiiij xv xvj xvij xviij xix j ij iij iiij v
Small table of the cycle of Golden Numbers initium sumens from the year of correction 1582.
Gol. num. 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5

17

   G R E G O R I A N V M   17

Anno 1582 tribuatur primus numerus tabellae, quæ est vj. secundus autem, qui est vij. sequenti anno 1583, & ita deinceps in infinitum, donec ad annum, cuius Aureum numerum quæris, perueniatur, redeundo ad principum tabellæ,quotiescunque eam percurreris. Nam cellula, in quam annus propositius cadit, dabit Aureum numerum quæsitum.

  SED quoniam valde laboriosum est, ac molestum, tot annos in dicta tabella enumerare, eamque toties repetere, donec ad annum, cuius Aureus numerus quæritur, peruienatur, præsertim vero, si annus propositus procul ab anno 1582, absit, construximus hanc aliam dequentum tabulam, ex qua fino magno labore Aureus numerus cuiuscunque anni tam ante, quam post annum 1582. inuientur hac arte.

  Quæratur annus propositus in tabula sub annis Domini, qui si descriptus in ea fuerit, Aureus numerus ad dexteram ipsius collocatus, addita prius vnitate, vt in vertice tabulæ præcipitur, erit is, qui quæritur. Si vero annus propositus in tabuls non continentur, accipiatur annus in tabula contentus proxime minor, vna cum Aureo numero respondente: Deinde sumantur in eadem tabula anni, qui supersunt, vna cum aureo numero respondente, qui priori Aureo numero inuento addatur, reciianturque à composito numero 19. se reiici possunt. Et tandem vnitas adiiciatur. Componetur enim hac ratione Aureus numerus proposito anni. Quod si neque anni, qui supersuerunt, in tabula reperiantur, accipiendus erit rursum annus proxime minor, vna cum eius Aureo numero, qui priori Aureo numero inuento adiiciendus est, & à composito numero reiicienda 19. si reiici possunt: Idemque faciendum erit cum reliquis annus, qui supersunt, donec omnes in tabula inueneris; Et tandem vltimo Aureo numero ex Aureis in tabula repertis confecto ( reiectis prius 19. si reiici possunt, vt dictum est ) addenda vnitas. Conficietur enim hoc modo Aureus numerus anno propositi. Quod si post additionem vnitas numerus compositus fuertt 19. its vy, detractis 19. nihil remaneat, erit Aureus numerus 19.

  Exemplis res fiet illustrior. Sit inueniendus Aureus numerus anni 700. Quoniam hic annus in tabula reperitur, eique respondet Aureus numerus 16. si huic Aureo numero adiiciatur 1. erit anno 700. Aureus numeris 17. Rursus inueniendus proponatur Aureus numerus Anno 1583. Quoniam hic annus in tabula non existit, sumendus est annus 1000. in tabula proxime minor, eiusque Aureus numerus 12. Deinde accipiendi in tabula anni residui 583, qui quoniam in ea non continentur, capiendus iterum est annus 500. in tabula pro-

The year 1582 ...



Tabula ad Aureum numerum cuiuslibet anni inueniendum.
Anni
Domini
Aureus
Num
Adde
1
Anni
Domini
Aureus
Num
Adde
1
Anni
Domini
Aureus
Num
Adde
1
Anni
Domini
Aureus
Num
Adde
1
1
2
3
4
1
2
3
4
300
400
500
600
15
1
6
11
50000
60000
70000
80000
11
17
4
10
7000000
8000000
9000000
10000000
1
12
4
15
5
6
7
8
5
6
7
8
700
800
900
1000
16
2
7
12
90000
100000
200000
300000
16
3
6
9
20000000
30000000
40000000
50000000
11
7
3
18
9
10
20
30
9
10
1
11
2000
3000
4000
5000
5
17
10
3
400000
500000
600000
700000
12
15
18
2
60000000
70000000
80000000
90000000
14
10
6
2
40
50
60
70
2
12
3
13
6000
7000
8000
9000
15
8
1
13
800000
900000
1000000
2000000
5
8
11
3
100000000
200000000
300000000
400000000
17
15
13
11
80
90
100
200
4
14
5
10
10000
20000
30000
40000
6
12
18
5
3000000
4000000
5000000
6000000
14
6
17
9
500000000
600000000
700000000
800000000
9
7
5
3


xime minor, eiusque Aureus numerus 6. quo ad priorem Aureum numerum 12. inuentum adiecto, conficietur numerus 18. ... ...

[CONTINUES]

[CONTINUES]

18

Words for GN:=(Y+1) mod 19; if GN=0 then GN:=19;

[ CONCLUDED; Canon 2 / DE EPACTIS, ET NOVILVNIIS. ]

19

...

20

...

21

...

22

[ Canon 3 / DE CYCLO SOLARI, SIVE LITERARVM ]

23

...

24

...

25

[ Canon 4 / DE LITERA DOMINICALI. ]

26

...

27

   G R E G O R I A N V M   27

Cyclus Solaris, seu literarum Dominicalium antiquus 28. annorum perpetuus

[TABLE]

[TABLE]

  Inuenturus ergo literam Dominicalem quocunque anno dato; Vide in tabula æquationis, qui numerus ...

  Exemplum. ...

A.quæ ambæ Dominicales erunt eo anno cum bissextilis sit. Sed prior ratio expeditior est, cum Solari cyclo non egeat.

To be discovered therefore the Sunday Letter for a given year; look in the table of equations, which number ...

28

28 C A L E N D A R I V M   
C A N O N   V.

D E   I N D I C T I O N E

I
NDICTIO est ...

[CONTINUES]

The Indiction is ...

[CONTINUES]

29

[ CONCLUDED; Canon VI. / DE FESTIS MOBILIBVS. ]

...

And a great deal more ... a small but important part of which is in Easter Algorithms from the Six Canons of Christopher Clavius, etc.


Home Page
Mail: no HTML
© Dr J R Stockton, near London, UK.
All Rights Reserved.
These pages are tested mainly with Firefox and W3's Tidy.
This site, http://www.merlyn.demon.co.uk/, is maintained by me.
Head.