### Main Text

# Editor’s NoteCritical Apparatus1Editor’s NoteCritical Apparatus10. Capitulum decimum

- 2Primum plenilunium post equinoctium vernale, etiamsi in ipso die
- Critical Apparatus3equinoctii contigerit, est terminus designans pascha, quia proxima
- 4dominica post illud plenilunium celebratur dies pasche. Et in ipso die
- 5plenilunii, hoc est luna 14
^{a}, occidebatur in lege agnus paschalis. Et nos- 6dicimus lunam 14
^{am}proximam post vernale equinoctium terminum- Critical Apparatus7pasche, quia non illo die celebramus pascha, sed ille dies determinat
- Critical Apparatus8nobis, quod in proxima dominica sequente est pascha celebrandum. Et
- Critical Apparatus9ex iam dictis patet, quod luna debet dici 14
^{a}in ipso die plenilunii. Et- Critical Apparatus10inde patet, quod ipsa debet dici prima non in die coniunctionis eius
- 11cum sole, sed in die proximo sequente, quo possibile est, ut primo
- 12appareat accensa a sole, quia, si inciperet tempus primationis ab hora
- 13coniunctionis lune cum sole, necessario diceretur luna 15
^{a}in die- 14plenilunii.
- Critical Apparatus15Que cum ita sint, manifestum est, quod iam accidit nobis error in
- 16sumptione termini paschalis et etiam in sumptione diei pasche duabus
- 17de causis. Nos enim ponimus equinoctium vernale esse 12° Kalendas
- 18Aprilis et ibi ponimus infimum terminum pasche. Et manifestum est
- 19tam per instrumentum considerationis quam per tabulas astronomi-
- Critical Apparatus20cas, ibi non esse equinoctium in hoc tempore nostro, sed secundum
- Critical Apparatus21tabulas Toletanas fundatas supra quantitatem anni et motum octave
- Critical Apparatus22spere, quos posuit Thebit, equinoctium vernale in hoc nostro tempore
- Critical Apparatus23est pridie Idus Martii. Et constat, quod, si equinoctium fuit 12
^{o}- pg 16224Kalendas Aprilis in tempore priorum doctorum, qui primo tradide-
- 25runt doctrinam de inveniendo termino pasche, quod in hoc tempore
- 26nostro non est equinoctium eodem die, immo necesse est nunc
- Critical Apparatus27equinoctium precedere diem illum per rationem eandem, quam
- 28diximus in capitulo primo de antecessione solstitii hiemalis.
- 29Item, ut supra ostensum est, erramus in positione quantitatis
- Critical Apparatus30lunationis equalis. Ex quo errore, ut ostendimus, accidet per longitu-
- 31dinem temporis lunam dici primam, quando erit plena et quando erit
- 32in quavis distantia a sole. Et ita non dicetur 14
^{a}in die plenilunii. Et- Critical Apparatus33nunc in hoc nostro tempore est error iste manifestus, quia numquam
- Critical Apparatus34est nunc plenilunium, quando dicimus lunam 14
^{am}, sed quando ipsa- Critical Apparatus35est 13
^{a}vel 12^{a}. Et hoc manifestum est per eclipses lunares, que nunc- Critical Apparatus36semper accidunt antequam luna sit 14
^{a}. Propterea preter hoc, quod- 37erramus in ponendo equinoctium 12° Kalendas Aprilis, erramus etiam
- 38in sumptione termini paschalis. Et cum erramus in termino pasche,
- Critical Apparatus39necesse est nobis frequenter errare in sumptione ipsius diei pasche et
- 40ceterorum festorum mobilium.
- 41Modus autem verificandi hunc errorem est, ut verificetur anni
- 42quantitas et verificata ponatur in kalendario. Vel etiam absque verifica-
- 43tione quantitatis anni cognoscatur semper dies equinoctii vernalis per
- 44instrumentum considerationis vel per tabulas astronomicas verificatas.
- 45Et per id, quod supra docuimus de sumptione primationum secun-
- 46dum veritatem, cognoscatur dies plenilunii primi post illud equinoc-
- 47tium et dies ille ponatur terminus pasche. Verumtamen, quia sancta
- 48ecclesia nondum mutavit antiquam doctrinam de inveniendis festis
- 49mobilibus, illam doctrinam explanabimus.
- 50Dicimus ergo, quod equinoctium vernale est 12
^{o}Kalendas Aprilis.- Editor’s Note51Ibi enim secundum Rabanum fuit equinoctium in primo anno seculi.
- Critical Apparatus52Et credo, quod ibi fuerit equinoctium in tempore doctorum, qui istam
- 53doctrinam primo tradiderunt, ad quos nondum pervenit scientia vere
- 54quantitatis anni nec scientia antecessionis solstitii et equinoctii. Et
- 55ponimus plenilunium, cum luna est 14
^{a}. Et cum ibi ponamus locum- Critical Apparatus56equinoctii, ibidem ponendus est infimus terminus pasche. Quapropter
- pg 16457necesse est, ut primum pascha ponatur 11° Kalendas Aprilis, quia,
- 58quando terminus pasche accidit in die sabbati, erit dies pasche in
- 59crastino. Et primatio signata per 16 8
^{o}Idus Martii erit prima incensio- 60lune paschalis, quia luna ibidem existente prima erit eadem 14
^{a}12°- 61Kalendas Aprilis.
- 62Et quia 19 primationes consequenter scripte in kalendario ab 8°
- 63Idus Martii usque ad Nonas Aprilis omnes et sole habent in die 14
^{a}- 64lune plenilunium proximum post equinoctium vernale, primatio
- 65signata per 8 in Nonis Aprilis erit ultima incensio lune paschalis et 14°
- 66Kalendas Maii, in quo die luna huius primationis est semper 14
^{a}, erit- 67ultimus terminus pasche. Et cum iste terminus accidit die dominica,
- 68erit dominica sequens 7° Kalendas Maii dies pasche. Quapropter 7°
- 69Kalendas Maii est ultimum pascha.
- Critical Apparatus70Et quando accidit lunam esse primam super aliquam 17 primatio-
- Critical Apparatus71num, que sunt inter 8° Idus Martii et Nonas Aprilis, fiat similiter
- 72semper computatio a loco primationis, usque dum occurrat luna 14
^{a},- 73et ibi ponatur terminus pasche. Et proxima dominica sequente
- Critical Apparatus74celebretur pascha. Hanc itaque artem inveniendi terminum paschalem
- Critical Apparatus75retinent per hos versus:

- Editor’s NoteCritical Apparatus76 Post Martis Nonas, ubi sit nova luna, requiras,
- 77 que postquam fuerit bis septima, pascha patebit.

- Critical Apparatus78Habitis autem termino pasche et die pasche de facili habentur
- Critical Apparatus79termini et loca vera aliorum festorum mobilium, scilicet septuagesime,
- Critical Apparatus80quadragesime, rogationum et pentecostes, quia terminus septuagesime
- 81precedit terminum pasche 9 integris septimanis et similiter dominica
- 82septuagesime precedit diem pasche totidem septimanis et terminus et
- Critical Apparatus83dies quadragesime precedunt terminum et diem pasche 6 integris
- 84septimanis et terminus et dies pasche precedunt terminum et diem
- 85rogationum 5 septimanis integris et iterum terminus et dies pasche
- Critical Apparatus86precedunt terminum et diem pentecostes 7 integris septimanis.
- Critical Apparatus87Si autem primo vis invenire terminum septuagesime et per illum
- 88reliquos terminos sequentium festorum mobilium, considera, quota
- pg 16689est luna in die epiphanie, et computato numero etatis eius super diem
- 90epiphanie procedas continue computando usque ad 40 et diem, super
- Critical Apparatus91quem occurrit 40, ponas terminum septuagesime. Et sequenti
- Critical Apparatus92dominica celebrabitur septuagesima. Si autem fuerit annus bisextilis et
- Critical Apparatus93dies, super quem occurrit 40, fuerit dies sabbati, tunc non erit dies
- 94crastinus dominica septuagesime, sed erit terminus septuagesime et
- Critical Apparatus95secunda dominica a die, super quem occurrit 40, erit dominica septua-
- Critical Apparatus96gesime. Et istam artem retinent per hos versus:

- Editor’s Note97 A festo stelle numerando perfice lune
- 98 quadraginta dies et septuagesima fiet.
- Critical Apparatus99 Linque diem Domini primum retinendo secundum,
- Critical Apparatus100 si cadat in feriam septenam sitque bisextus.

- Critical Apparatus101Est autem et alius modus inveniendi festa mobilia, scilicet per
- 102claves terminorum. Et est clavis termini alicuius festi mobilis
- 103numerus quidam positus in loco determinato in kalendario. A quo
- Critical Apparatus104loco facta computatione per dies consequenter scriptos in kalendario
- 105usque ad finem illius numeri, ubi terminatur numerus ille, ibi ponitur
- 106terminus illius festi mobilis. Et quia omnes termini festorum
- 107mobilium periciunt periodum suam in 19 annis et redeunt ad loca
- 108prima, unicuique anno cycli decennovennalis appropriatus est
- Critical Apparatus109numerus unus, qui est clavis terminorum illius anni.
- Critical Apparatus110Locus autem clavium termini septuagesime est super primum G
- Critical Apparatus111Ianuarii, scilicet 7° Idus illius, et locus clavium termini quadragesime
- Critical Apparatus112est super ultimum G Ianuarii, scilicet 5° Kalendas Februarii, et locus
- Critical Apparatus113clavium termini pasche est super secundum G Martii, scilicet 5° Idus
- 114illius. Locus autem clavium terminorum rogationum est super tertium
- Critical Apparatus115G Aprilis, scilicet 17° Kalendas Maii, et locus clavium terminorum
- 116pentecostes est super ultimum G Aprilis, scilicet 3
^{o}Kalendas Maii.- 117Clavis autem terminorum in primo anno cycli decennovennalis est
- 11826, quia, si incipiamus computare secundum numerum naturalem a 5°
- 119Idus Martii per dies continue sequentes usque ad 26, ubi occurrit 26,
- pg 168120erit terminus pasche in primo anno cycli, et hoc est in Nonis Aprilis.
- Critical Apparatus121Ibi enim est eodem anno luna 14
^{a}. Et per consimilem computationem- Critical Apparatus122a 7° Idus Ianuarii factam invenitur eodem anno terminus septuagesime
- 123in Kalendis Februarii. Et penitus eodem modo inveniuntur eodem
- Critical Apparatus124anno terminus rogationum et terminus pentecostes facta computa-
- Critical Apparatus125tione a 17
^{o}Kalendas Maii et 3° Kalendas Maii usque ad 26.- Critical Apparatus126Ex clave primi anni cycli decennovennalis formatur clavis secundi
- Critical Apparatus127anni et ex clave secundi anni formatur clavis tertii anni, et ita deinceps
- Critical Apparatus128semper ex clave precedentis anni formatur clavis anni sequentis hoc
- Critical Apparatus129modo: Si clavis alicuius anni sit maior 21, subtrahatur ab ipso 11 et,
- Critical Apparatus130quod relinquitur, erit clavis sequentis anni. Si vero clavis alicuius anni
- Critical Apparatus131fuerit minor 21, addantur ei 19 et aggregatum erit clavis sequentis
- Critical Apparatus132anni. Ratio autem, quare debeant subtrahi 11 vel addi 19, est, quod,
- 133quotiens primatio paschalis sequentis anni precedit primationem
- 134paschalem proximo antecedentis anni, precedit eam 11 diebus, et
- 135quotiens primatio paschalis sequentis anni sequitur primationem
- Critical Apparatus136paschalem antecedentis anni, sequitur eam 19 diebus.
- 137Posset autem quivis numerus minor quam 26, dummodo esset
- Critical Apparatus138maior quam 15, poni pro clave primi anni cycli. Et similiter quivis
- Critical Apparatus139maior numerus quam 26 prepositis vel postpositis locis clavium
- 140secundum quantitatem differentie numeri positi pro clave ad 26.
- 141Si autem in anno bisextili ceciderit terminus septuagesime in die
- 142sabbati, non est celebranda septuagesima in crastino, sed in die nono a
- 143termino, quia a termino septuagesime sunt 9 septimane integre in
- 144kalendario usque ad terminum pasche. Et cum in anno bisextili fuerit
- 145terminus septuagesime die sabbati, erit terminus pasche die dominica
- 146et celebrabitur pascha 8° die post eius terminum. Unde cum dies
- 147bisextilis inter initium septuagesime et pascha computetur et a die
- Critical Apparatus148septuagesime usque ad diem pasche sint tantum 9 septimane, necesse
- 149est, ut septuagesima celebretur 9
^{o}die post terminum eius.- Critical Apparatus150Loca autem clavium retinent per hos versus:

- pg 170Editor’s Note151 In Iano prima supremaque, Marte secunda,
- 152 tertia G monstrat Aprilis et ultima claves.

- Critical Apparatus153Formatio vero clavium habetur per hos versus:

- Editor’s NoteCritical Apparatus154 Si clavis fuerit vicena minorve, sequenti
- Critical Apparatus155 huic pro clave monos addito bisque novem.
- 156 Undenas tollas, si sit vicena secunda
- 157 vel maior, numerus proxima clavis erit.

### Translation

# Chapter 10

The first full moon after the vernal equinox, even if it falls on the day of the equinox itself, is the boundary indicating the date of Easter, because the Easter day is celebrated on the nearest Sunday after this full moon. And on the day of the full moon itself, that is, the 14th day of the Moon, the paschal lamb was killed in accordance with the law. And we refer to the 14th day of the Moon that comes closest after the vernal equinox as the 'paschal boundary', because this is not the day on which we celebrate Easter, but instead it informs us that Easter must be celebrated on the nearest Sunday after it. And from what has been said already it is clear that the Moon must be called 'the 14th' when it is the day of the full moon. And from this it is plain that it must be called 'the first' not on the day of its conjunction with the Sun, but on the nearest day after it, on which it is possible for [the Moon] to become visible for the first time, having been kindled by the Sun. For if the time of the new moon were to start from the hour of the Moon's conjunction with the Sun, the Moon would necessarily have to be called 'the 15th' when it is the day of the full moon.

With this being so, it is obvious that we already commit an error in the way we locate the paschal boundary and also in the way we determine the date of Easter, which happens for two reasons. For we assume the vernal equinox to be on 21 March and put there the lowest paschal boundary. And it is obvious from observational instruments as much as from astronomical tables that the equinox is not found there at our present time, but according to the Toledan Tables, which are based on the length of the year and the motion of the eighth sphere assumed by Thābit, the pg 163vernal equinox in our present time falls on 14 March. And it is evident that, if the equinox was on 21 March at the time of the first teachers, who were the first to transmit the rule of finding the paschal boundary, the equinox is no longer on the same date in our present time. Much rather, it is necessary that the equinox now precedes this day for the reason we discussed in the first chapter with regard to the anticipation of the winter solstice.

Likewise, as shown above, we err in assuming the length of the mean lunation. We have shown that this error over long periods of time leads us to call the Moon 'the first' when it is already full or when it will be at any other distance from the Sun. And this way it is not called 'the 14th' on the day of the full moon. And this error is evident in our present time, for nowadays there is never a full moon when we call the Moon 'the 14th', but when [the calendrical Moon] is in its 12th or 13th day. And this is conspicuous from lunar eclipses, which nowadays always occur before the Moon is in its 14th day. For this reason, we err not just in placing the equinox on 21 March, but also in our calculation of the paschal boundary. And if we err about the paschal boundary, it is necessary for us to frequently err in calculating the day of Easter itself and of the other mobile feasts.

The way to correct this error is to determine the true length of the year and to adjust the calendar accordingly. Or even without determining the true length of the year, one can always find the day of the vernal equinox with the help of observational instruments or accurate astronomical tables. And one should use what we taught above concerning the accurate calculation of new moon dates to determine the day of the first full moon after this equinox and put this day down as the paschal boundary. Yet since Holy Church has not yet changed the ancient rule of finding the mobile feasts, we shall explain this rule.

We say, then, that the vernal equinox is on 21 March. For this is where the equinox was in the first year of the world according to Rabanus. And I believe that this was the date of the equinox at the time of the teachers who first transmitted this rule, whom knowledge of the true length of the year had not yet reached, nor [did they have] knowledge of the anticipation of the solstice and equinox. And we assume that there is a full moon whenever the Moon is in its 14th day. And when we put the equinox on the same date, we must place there the lowest boundary date for Easter. It pg 165is hence necessary for the first Easter to fall on 22 March, for when the paschal boundary falls on a Saturday, the day after will be the Easter day. And the new moon marked by the number 16 [next to] 8 March will be the first kindling of the paschal lunation, for if the Moon is in its first day on this date, it will be in its 14th day on 21 March.

And since the 19 new moons that are written in sequence in the calendar from 8 March to 5 April are the only ones where the corresponding 14th day is the full moon that comes next after the vernal equinox, the new moon that is indicated by 8 on 5 April will be the latest new moon of the paschal lunation and 18 April, on which day the Moon that belongs to this new moon is always in its 14th day, will be the latest paschal boundary. And whenever this boundary falls on a Sunday, the following Sunday, on 25 April, will be the Easter day. This is why 25 April is the latest Easter.

And whenever it happens that the Moon reaches the first day on any of the 17 new moons that are between 8 March and 5 April, there shall always be the same calculation from the date of the new moon until the 14th day of the Moon has been reached, which is where the paschal boundary must be placed. And Easter shall be celebrated on the Sunday that follows next. They memorize this method of finding the paschal boundary with the help of these verses:

- Search for where the Moon is new after 7 March,
- once [this Moon] is 14 [days old], Easter will be revealed.

Now, once one has found the paschal boundary and the date of Easter, it is easy to obtain the boundaries and true dates of the other mobile feast days, which are Septuagesima, Quadragesima, Rogation Sunday, and Pentecost. For the boundary of Septuagesima precedes the paschal boundary by nine whole weeks and, likewise, Septuagesima Sunday precedes the date of Easter by just as many weeks and the boundary and date of Quadragesima precede the boundary and date of Easter by six whole weeks and the boundary and date of Easter precede the boundary and date of Rogation Sunday by five whole weeks and, again, the boundary and date of Easter precede the boundary and date of Pentecost by seven whole weeks.

But if you wish first to find the boundary of Septuagesima Sunday and through it the remaining boundaries of the following mobile feasts, you must take into account the age of the Moon on the day of Epiphany, and pg 167once you have counted the number corresponding to its age on the day of Epiphany, you must proceed to count continuously up to 40, and the day on which the number 40 falls you shall make the boundary of Septuagesima. And on the following Sunday Septuagesima will be celebrated. In cases, however, where the year is bissextile and the day on which the number 40 falls is a Saturday, the following day will not be Septuagesima Sunday, but it will be the boundary of Septuagesima and the second Sunday from the day on which the number 40 falls will be Septuagesima Sunday. And they memorize this method with the help of these verses:

- Start counting lunar days from the feast of the star and complete
- 40 days and the result will be Septuagesima.
- Forsake the first Sunday and retain the second one
- if [the day in question] falls on Saturday and there is a bissextile day.

There is, however, another method of finding the mobile feasts, namely that of using boundary keys. And the boundary key of any mobile feast is a certain number placed on a fixed date in the calendar. And if one counts from this position the days written in sequence in the calendar up to the end of this number, the date on which this number finishes will be where the boundary of this mobile feast is placed. And since all boundaries of mobile feasts complete their period in 19 years and return to their original places, each year of the 19-year cycle has a number to itself that is the boundary key of this year.

The place for the boundary keys of Septuagesima is the first date in January with the letter G, which is 7 January, and the place for the boundary keys for Quadragesima is the last date in January with the letter G, which is 28 January, and the place for the boundary keys of Easter is the second date in March with the letter G, which is 11 March. The place for the boundary keys of Rogation Sunday is the third date in April with the letter G, which is 15 April, and the place for the boundary keys of Pentecost is the last date in April with the letter G, which is 29 April.

Now, in the first year of the 19-year cycle the boundary key is 26, for if we start counting according to natural numbers from 11 March up to 26 through the days that follow in continuous sequence, the date where 26 falls will be the boundary of Easter in the first year of the cycle, and that pg 169is on 5 April. For there the Moon is in its 14th day in the year in question. And by making the same kind of count from 7 January one will find in the same year the boundary of Septuagesima on 1 February. And in exactly the same manner one can find in the same year the boundary of Rogation Sunday and the boundary of Pentecost, if the count up to 26 is made from 15 April and from 29 April.

From the key of the first year of the 19-year cycle the key of the second year is formed and from the key of the second year the key of the third year is formed, and so on, such that the key of the following year is always formed from the key of the preceding year, in the following way: if the key of a given year is greater than 21, one must subtract 11 from it and what remains will be the key of the following year. If, on the other hand, the key of a given year is smaller than 21, one must add 19 to it, and the sum will be the key of the following year. The reason why one must subtract 11 or add 19 is that whenever the Easter new moon of the following year falls earlier than the Easter new moon of the year immediately before it, it precedes it by 11 days; and every time the Easter new moon of the following year falls later than the Easter moon of the preceding year, it follows 19 days after it.

One could use any number smaller than 26 as the key of the first year of the cycle, as long as it is greater than 15. And, similarly, [one could use] any number greater than 26, if the places for the keys are pre- or postponed according to the difference between 26 and the number that has been chosen as the key.

But if the boundary of Septuagesima falls on a Saturday in a bissextile year, Septuagesima must not be celebrated on the following day, but on the ninth day from the boundary, because there are nine whole weeks from the boundary of Septuagesima to the boundary of Easter. And in cases where the boundary of Septuagesima falls on a Saturday in a bissextile year, the boundary of Easter will be on a Sunday, and Easter will be celebrated on the eighth day after its boundary. Inasmuch as the bissextile day is counted between the beginning of Septuagesima and Easter and there are only nine weeks from the day of Septuagesima to the day of Easter, it is hence necessary for Septuagesima to be celebrated on the ninth day after its boundary.

The places for the keys are memorized with the help of these verses:

- pg 171 In January it is the first and the last [G], in March the second,
- in April the third and the final G that show the keys.

The method of forming the keys is had with the help of these verses:

- If the key is 20 or smaller than that
- you must add 19 to get the key that follows it.
- Take away 11 if it is 22
- or higher. The [resulting] number will be the next key.

### Notes

^{157}

1220 | 14 March, 04:06h |

1221 | 14 March, 10:01h |

1222 | 14 March, 15:56h |

1223 | 14 March, 21:51h |

1224 | 14 March, 03:47h |

1225 | 14 March, 09:42h |

1226 | 14 March, 15:37h |

1227 | 14 March, 21:33h |

1228 | 14 March, 03:28h |

1229 | 14 March, 09:24h |

1230 | 14 March, 15:19h. |

^{158}Taking the term 'full moon' (

*plenilunium*) literally, he interpreted this as the moment when Sun and Moon were in opposition to each other (ll. 9–14). In order for this moment to occur on the 14th day of the Moon, the lunation itself could not be reckoned from the day of mean conjunction, as the mean interval between conjunction and opposition exceeded 14d 18h and was hence greater than 14 full days. It was therefore apposite to choose the second option already proposed in chapter 4 (ll. 195–200), which was to begin the calendrical lunation on the day that followed the mean conjunction.

*computus*had not yet exceeded two days, the current predicament led to numerous instances where Easter Sunday was celebrated in the wrong week. According to criterion (ii) Easter Sunday always had to follow the 14th day of the Moon but was excluded from occurring in the final quarter of the lunation, i.e., any later than the 21st day. Yet this was exactly what happened whenever the conventional

*computus*showed Easter Sunday as falling on

*luna*20 or 21, as in such cases the 'true' age of the Moon had already reached

*luna*22 or higher. In ad 1220, for example, Easter Sunday was celebrated on 29 March, which the standard

*computus*designated as the 21st day of the lunation of April. A use of the conversion tables Grosseteste included in ch. 5 would have shown this date to be equivalent to the 23rd day of the Arabic month of Muḥarram, two days ahead of the ecclesiastical reckoning. Even if one chose to begin the lunation on the day after the mean conjunction, and hence on the second day of the Arabic month, the 'true' age of the Moon on 29 March still turned out to be

*luna*22. Now and again the two calendrical errors could even compound. In ad 1207, for instance, the 14th day of the Arabic month of Sha'bān fell on Thursday, 15 March and hence after the 'true' equinox indicated by the Toledan Tables. The astronomically valid date of Easter Sunday accordingly should have been 18 March, whereas the Church placed the paschal boundary (

*terminus paschalis*) a whole month later, on 15 April. This was a Sunday, necessitating a shift of the Easter date to 22 April (

*luna*21). On the Arabic lunar count, 15 April would have been the 16th day of Ramaḍān, revealing that Easter Sunday should have been celebrated at least a week, but in fact 35 days, earlier.

*Compotus*Grosseteste had expressed some weak support for a solution that suppressed the bissextile day once every century in line with al-Battānī's year-length of 365¼ – $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$100$}\right.$ days. Such a solution could look futile, however, in the light of the access-and-recess model that characterized the Toledan Tables and the treatise

*De motu octave spere*, which juxtaposed a fixed sidereal year with a continuously changing tropical one.

^{159}Although he provided no further detail on this point, one feasible option would have been to use a quadrant or the alidade (sighting tool) on the back of an astrolabe to observe the noon altitude of the Sun for a series of days close to the equinox. The correct time could then be inferred from the difference in degrees between the observed noon altitude and the local co-latitude, which corresponds to the noon altitude the Sun will theoretically reach at crossing the celestial equator. Successful measurements of the time of the equinox that worked along these very lines were carried out in Paris some decades after Grosseteste's

*Compotus*by Peter of Limoges and William of Saint-Cloud, the latter of whom used the result to critique the Toledan Tables in the introduction to his

*Almanach planetarum*of 1292.

^{160}

*Compotus*.

^{161}Three problems in particular would have compromised the practical feasibility of the plan he lightly sketched in ch, 10. The first concerns the suggested reliance on the 30-year cycle of the Arabs, which did not constitute a cycle relative to the Julian calendar and therefore lacked one of the key features that had made the use of the Alexandrian 19-year cycle attractive to the Church. The second concerns the complications wrought by an astronomical method of finding the vernal equinox, which would have forced Latin Christians to settle on a prime meridian relative to which the date of Easter was calculated. Even with such a meridian in place, calculations or observations carried out by different astronomers were liable to yield non-identical results, which could in turn cause discrepancies of a whole month in fixing the date of Easter. Thirdly, and perhaps most importantly, there was the change in boundary-dates that Grosseteste's approach necessarily entailed. The official

*computus*of the Church allowed Easter Sunday to fall on any of the 35 dates between 22 March and 25 April (ll. 62–76), whereas a calculation that used the astronomical equinox on 14 March would have had to shift this range to 15 March–18 April. Further shifts were bound to happen in the not-too-distant future, as Grosseteste's suggestions contained no measures to halt the continuing drift of the equinoxes and solstices, for instance by omitting the bissextile day in certain years. What made such changes to the accustomed boundaries unattractive were the inevitable knock-on effects they had on all the mobile feast days, which ended up losing their accustomed calendrical relation to the fixed feast days.

^{162}Other than Easter Sunday itself, computistical texts of the high Middle Ages tended to pay special attention to four mobile Sundays of the liturgical year, to wit:

Septuagesima Sunday | 18 January–22 February (nine weeks before Easter) |

Quadragesima Sunday | 8 February–14 March (six weeks before Easter) |

Rogation Sunday | 26 April–30 May (five weeks after Easter) |

Pentecost/Whitsunday | 10 May–13 June (seven weeks after Easter). |

*Compotus Petri*, whose author devoted a whole chapter to their rationale and liturgical significance.

^{163}In showing how to fix the dates of the five mobile Sundays, Peter made an effort to be as comprehensive as possible, by outlining four principal ways of obtaining the same results.

^{164}These are: (i) a conventional computation based on epacts and regulars or on Golden Numbers that uses a different range of

*termini*or boundary-dates for each mobile feast; (ii) a unified system of

*claves terminorum*or 'boundary-keys'; (iii) the so-called

*versus angelici*;

^{165}(iv) a bipartite Easter table. Grosseteste took a more selective approach, as he severely curtailed the discussion of (i), ignored (iii) altogether, and hived off (iv) to the following chapter. His treatment of the mobile feast days began by outlining the simplest available method, which was to use the Golden Number to identify the new moon of the paschal lunation, which could fall no earlier than 8 March and no later than 5 April (ll. 70–7). The corresponding mnemonic (

*Post Martis nonas … pascha patebit*) could have been drawn from the

*Compotus Petri*, as is the case with Grosseteste's subsequent reminder that the dates of the remaining four Sundays can be treated as a function of their interval from Easter Sunday (ll. 78–86).

^{166}

*terminus*for Septuagesima Sunday (17 January to 14 February). Both Peter and Grosseteste use the same two lines of verse (ll. 97–8

*A festo stelle*etc.) to highlight the principle according to which the interval between the two dates will be equivalent to the difference between the age of the Moon on Epiphany and 40 days.

^{167}The

*terminus*of Septuagesima precedes the

*terminus*of Easter by nine weeks or 63 days, which will ordinarily contain one 30-day lunation (ending in January or March) and one 29-day lunation (ending in February or April). Subtracting these two lunations from 63 days leaves 4 days (63 – 30 – 29 = 4) unaccounted for, which explains why the age of the Moon on the

*terminus*of Septuagesima is for most years defined as

*luna*10 rather than

*luna*14. Exceptions to this rule occur in Julian leap years as well as in years 8 and 19 of the 19-year cycle, which feature two consecutive lunar months of 30 days—the lunation ending in March followed by an embolismic month—whose combined length causes the

*terminus*of Septuagesima to end up on the 11th day of the Moon. These exceptions were carefully taken into account by Peter, who admonished his readers to count up to 41 rather than 40 days from the lunar age of Epiphany, not just in bissextile years, but also in years 8 and 19 of each lunar cycle.

^{168}Grosseteste, by contrast, merely reminds us that Septuagesima Sunday will have to be postponed by a week if it is a bissextile year and the 40th inclusive day from Epiphany comes to fall on a Saturday. This exception is the subject of two additional lines of mnemonic verse (ll. 99–100

*Linque diem*etc.) that already appear (in reverse order) in the

*Compotus ecclesiasticus*as well as the

*Massa compoti*.

^{169}What is completely missing from Grosseteste's account, however, is any hint that the same shift will become necessary in certain embolismic years, regardless of whether they are bissextile or not.

*terminus*of Septuagesima Sunday on the basis of the

*claves terminorum*, which carried the advantage of being applicable to more than one mobile feast (ll. 101–57). Unlike the epact-and-regular system, the

*claves*had been unknown to early medieval authorities on the computus such as Bede and Rabanus Maurus.

^{170}Their popularity only soared during the twelfth century, when textbook authors began to devote entire sections to explaining their use and underlying rationale.

^{171}Similar to the lunar epact, the

*clavis terminorum*was a function of the year in the 19-year cycle, as the value of the

*clavis*changed by –11 or +19 depending on whether the current lunar year was common or embolismic. Besides the annual value of the

*clavis*, computists had to memorize five reference dates or

*sedes clavium*, which in the standard system fell 10 days ahead of the earliest permissible

*terminus*of the corresponding feast day. Hence the

*sedes*for Septuagesima was 7 January, that for Quadragesima was 28 January, that for Easter was 11 March, that for Rogation Sunday was 15 April, and that for Pentecost fell on 29 April. From these dates, which all shared the same dominical letter (G), it followed automatically that the value of the

*clavis*in the first year of the 19-year cycle had to be 26. Counting forward the corresponding number of days in an inclusive manner landed the

*terminus*for Septuagesima Sunday on 2 February and that of Easter Sunday on 5 April, where they respectively belonged. The

*termini*for the other three mobile Sundays were calculated in the same way, such that the interval represented by the

*clavis*always remained inclusive, beginning on the date of the

*sedes clavium*and ending on the date of the corresponding

*terminus*.

*claves terminorum*is arbitrary in the sense that it can differ from 26 as long as the

*sedes*dates are shifted accordingly. Grosseteste is correct in remarking, however, that there is at least one constraint in that the

*clavis*of the first year has to be greater than 15 (ll. 137–40). No explanation is given in the text, but it is easy to recognize that 16 would be the starting-point of the sequence if all five

*sedes*were moved downwards by ten days, in which case the

*sedes clavium*for Easter would merge with the earliest possible

*terminus paschalis*. In this scenario, the

*clavis*for the 16th year of the 19-year cycle would drop to 1 instead of the conventional 11, indicating that the

*sedes*and

*terminus*fall on the same date.

^{172}

^{157}The times listed here are those predicted by the simulation of the Toledan Tables in Raymond Mercier's program

*Deviations*(available at http://www.raymondm.co.uk), with times rounded to the nearest minute.

^{158}Grosseteste's interpretation seems to agree with ch. 62 in Bede's

*De temporum ratione*, which defines Easter as the 'Sunday following the full moon which falls on or after the equinox' (trans. Wallis 1999: 148). Note, however, that Bede elsewhere insists that the full moon of Easter must rise

*after*the vernal equinox, while in ch. 17 he unambiguously states that the Moon is in opposition to the Sun on the 15th day. See Bede,

*De temporum ratione*, ch. 6, 17, 61, 62 (ed. Jones, CCSL 123B: 291, ll. 24–32; 338, ll. 21–6; 451, ll. 20–5; 453, ll. 27–8).

^{159}The gloss in

*Vu*, fol. 167r, interprets the instruments in question as

*armillae*.

*Lh3*, fol. 105v, inserts this gloss into the main text.

^{160}See the editions of the relevant passage in the

*Almanach planetarum*in Delorme 1936: 560–1, and Pedersen 2014: 9–10, and the discussions in Duhem 1916: 16–17; Millás Vallicrosa 1943–1950: 393–4; Harper 1966: 41–3; Lejbowicz 1997: 209–10; Nothaft 2017b: 219–20. An observational note in MS Vatican City, BAV, Reg. lat. 1191, fol. 139r, edited by Delorme (1936: 559) as stemming from William of Saint-Cloud, is in fact in the hand of Peter of Limoges. Another version of the same note, found in MS Vienna, ÖNB, 2311, fol. 96v, confirms that one of the observers was named

*Petrus*.

^{161}On the Gregorian reform, see Coyne, Hoskin, and Pedersen 1983; Steinmetz 2011. Its medieval prehistory is dealt with in more detail in Kaltenbrunner 1876; Nothaft 2018.

^{162}See on this point Nothaft 2018: 12, 206, 215, 223, 233, 256–7, 288, 294.

^{163}

*Compotus Petri*, ch. 68, fols. 43vb–44rb.

^{164}Ibid., ch. 69, fols. 44rb–46v. This chapter is entitled 'Quot modis pasca et alie quatuor sollempnitates moviles, sive per terminos, sive per claves terminorum, sive per versus angelicos, sive per tabulam inveniantur.'

^{165}Van Wijk 1936: 102–3; Jones 1943a; Ó Cróinín 1982; Wiesenbach 1986: 60; Borst 1998: 495–6; Roger of Hereford,

*Compotus*, ch. 19 (ed. Lohr, CCCM 272: 183–4).

^{166}

*Compotus Petri*, ch. 69, fol. 44vb.

^{167}See on this mnemonic Cordoliani 1960–1, pt. 1: 115; pt. 2: 182.

^{168}

*Compotus Petri*, ch. 69, fol. 45rb. See also Cunestabulus,

*Compotus*, ch. 24 (ed. Lohr, CCCM 272: 86, ll. 27–31; 88, ll. 99–101).

^{169}

*Compotus ecclesiasticus*, ch. 23 (ed. Moreton 2015: 60);

*Massa compoti*, ll. 148–9 (ed. van Wijk 1936: 56).

^{170}According to Borst (1998: 426), their earliest recorded appearance is in MS Paris, BnF, lat. 9432, fols. 3r, a manuscript from Amiens. The calendar found on fols. 3r–8v of this manuscript was written in the late ninth or early tenth century, but the references to the

*claves*on fol. 3r may have been added later. See Borst 2001: 138–9.

^{171}Honorius Augustodunensis,

*Imago mundi*, ch. 111 (ed. Flint 1982: 120); MS Vienna, ÖNB, 275, fol. 29r;

*Compotus Petri*, ch. 69, fols. 44vb–45ra; Roger of Hereford,

*Compotus*, ch. 2 (ed. Lohr, CCCM 272: 139–40);

*Compotus ecclesiasticus*, ch. 23 (ed. Moreton 2015: 58–61).

^{172}A sequence of

*claves*starting on 16 rather than 26 was known as the

*minores claves*. See Roger of Hereford,

*Compotus*, ch. 2 (ed. Lohr, CCCM 272: 140).

**10,1**Capitulum decimum] Capitulum (

*om.*

*Cu5*) decimum (Decimum capitulum

*Lh2*). De ostensione erroris nostri in sumptione terminorum et locorum festorum mobilium (et de modo sumendi terminos et loca festorum mobilium secundum doctrinam kalendarii nostri

*add.*

*Cu5*

*Os*

*Ou*)

*Cu5*

*Lh2*

*Os*

*Ou*,

*om.*

*Od9*

*Pb5*

*Tc*

**3**contigerit] contingeret

*Od9*

*Vu*

*Lh3*

*Ny*

*Od9*

*Ou*

*Pb5*

*Tc*

*Vu*

**7**quia non] non quia

*Cu5*

*Ou*

*praem.*

*Pb5*

*Vu*, in illa

*Tc*

**8**Et]

*om.*

*Cu5*

*Ny*

*Od9*

*Pb5*

**9**dici]

*om.*

*Cu5*

*Vu*

**10**inde] ex hoc

*Cp*

*Os*

*Ou*

**15**Que…sint] Quod cum ita sit

*Cu5*

*Tc*

**20**in]

*om.*

*Od9*

*Tc*

*Vu*

**21**supra] super

*Od9*

*Pb5*

*Tc*

*Vu*

**22**nostro tempore] tempore nostro

*Lh2*

*Lh3*

*Od9*

**23**fuit] fuerit

*Od9*

*Tc*

**10,27**rationem eandem] eandem rationem

*Lh2*

*Ou*

**30**lunationis equalis] equalis lunationis

*Cp*

*Os*

*Lh3*

*Ny*

*Od9*

*Tc*

*Vu*

**33**hoc]

*om.*

*Lh3*

*Ny*

*Od9*

*Ou*

*Pb5*

*Tc*

*Vu*

*om.*

*Lh3*

*Pb5*

**34**nunc]

*om.*

*Lh2*

*Lh3*

*Pb5*

**35**que] qui

*Cp*

*Os*, quia

*Lh3*

*Ny*

*Od9*

*Ou*

*Pb5*

*Tc*

*Vu*

**36**Propterea] Preterea

*Cu5*

*Lh2*

*Lh3*

*Tc*

*Vu*

**39**ipsius]

*om.*

*Ny*

*Od9*

*Pb5*

**10,51**Ibi…seculi] cf. Rabanus Maurus,

*De computo*23, 53 (ed. Stevens, CCCM 44, p. 227, ll. 14–19; pp. 263–4, ll. 15–17).

**52**fuerit] fuit

*Cp*

*Ny*

*Pb5*

*Tc*

**56**infimus terminus] terminus infimus

*Tc*

*Vu*

**10,70**aliquam] aliam

*Pb5*

*Tc*

*Vu*

**71**similiter] consimiliter

*Cp*

*Os*,

*om.*

*Ny*

**74**celebretur] celebrabitur

*Lh3*, celebratur

*Od9*

*Tc*

*Vu*

**75**retinent] compotiste

*add.*

*Lh2*

*Lh3*

**76**Martis Nonas] Nonas Martis (Martii

*Lh3*)

*Cu5*

*Lh2*

*Lh3*

*Pb5*

*Tc*

**76**Post…77 patebit] cf.

*Compotus Petri*, c. 69, fol. 44vb.

**78**Habitis]

*nov. cap. incip.*

*Od9*

*Ou*

*Tc*

**79**septuagesime] et

*add.*

*Cp*

*Os*

**80**quadragesime] et

*add.*

*Cp*

*Os*

**83**terminum…11,81 notulam] deest

*Ou*

**86**integris septimanis] septimanis integris

*Lh2*

*Pb5*

**87**primo vis] volueris primo

*Cp*

*Os*

**10,91**quem] quam

*Cp*

*Lh3*

*Os*

*Tc*

*Vu*

*Cp*

*Ny*, occurrerit

*Lh3*sequenti] sequente

*Cp*

*Od9*

**92**celebrabitur] celebratur

*Cp*

*Od9*

*Vu*

**93**quem] quam

*Lh2*

*Lh3*

*Ny*

*Tc*

*Ny*

*Tc*

**95**quem] quam

*Lh2*

*Lh3*

*Tc*

*Lh3*

*Ny*

**96**retinent] compotiste

*add.*

*Lh2*

*Lh3*

**97**A…100 bisextus] cf.

*Compotus ecclesiasticus*, c. 23 (ed. Moreton 2015: 60); cf.

*Massa compoti*, ll. 144–145, 148–149 (ed. van Wijk 1936: 55–6).

**99**retinendo] retineque

*Lh2*

*Tc*

*Vu*, capiendo

*Lh3*

*Ny*

*Od9*

*Pb5*

**100**cadat] cadit

*Cp*

*Cu5*

*Ny*

*Os*

*Pb5*

*Tc*

**101**et]

*om.*

*Cu5*

*Os*

*Vu*

**104**scriptos] scriptas

*Cp*

*Os*

**109**numerus unus] unus numerus

*Lh2*

*Os*

*Pb5*

*Tc*

*Vu*

*om.*

*Cp*

*Lh3*, unus numerus

*Lh2*

*Vu*

**110**termini] (et

*add.*

*Lh2*) terminorum

*Cp*

*Lh2*

**111**termini] terminorum

*Cp*

*Os*

*om.*

*Cp*

*Os*

**112**est]

*om.*

*Cu5*

*Lh2*

*Od9*

*Pb5*

*Vu*

**113**termini] terminorum

*Cp*

*Os*

*om.*

*Tc*

*Vu*

**115**scilicet] id est

*Cp*,

*om.*

*Od9*

*Tc*

**10,121**eodem] in

*praem.*

*Lh3*

*Od9*

*Tc*

*Vu*

**122**invenitur] invenietur

*Cp*

*Cu5*

*praem.*

*Ny*

*Od9*

**124**anno] modo

*Lh2*

*Lh3*

**125**Kalendas

^{1}] Kalendis

*Ny*

*Od9*

*Os*

*praem.*

*Lh3*

*Ny*

**126**clave] clavi

*Cp*

*Os*

**127**clave] clavi

*Cp*

*Os*

**128**clave] clavi

*Cp*

*Os*anni sequentis] sequentis (subsequentis

*Lh2*) anni

*Cp*

*Lh2*

*Os*

**129**sit] fuerit

*Cp*

*Os*subtrahatur] subtrahantur

*Os*

*Tc*

*Od9*

*Tc*

*Vu*, ea

*Lh2*

**130**clavis

^{2}]

*om.*

*Cp*

*Pb5*

**131**addantur] addatur

*Lh3*

*Ny*

*Od9*

*Pb5*

*Vu*

*om.*

*Cp*

*Tc*

**132**vel] et

*Cp*

*Os*

**136**antecedents anni] anni antecedentis

*Od9*

*Os*

**138**clave] clavi

*Cp*

*Od9*

*Os*

*Pb5*

*Tc*

**139**maior numerus] numerus maior

*Lh2*

*Lh3*

**148**sint] sunt

*Lh2*

*Os*

**150**retinent] compotiste

*add.*

*Lh2*

*Lh3*, retinentur

*Tc*

**151**In…152 claves] cf.

*Compotus ecclesiasticus*, c. 23 (ed. Moreton 2015: 59).

**10,153**vero]

*om.*

*Lh3*, autem

*Pb5*

*Tc*

**154**minorve] minorque

*Cp*

*Lh2*

*Od9*

**154**Si…157 erit] cf.

*Compotus ecclesiasticus*, c. 23 (ed. Moreton 2015: 59).

**155**monos] nonas

*Lh2*

*Od9*