The use of Astronomical Tables by Albertus Magnus

Biblia vulgata : “A sexta autem hora tenebrae factae sunt super universam terram usque ad horam nonam” (Secundum Matthaeum 27: 45); “Et facta hora sexta, tenebrae factae sunt per totam terram usque in horam nonam” (Secundum Marcum 15: 33). (All Biblical references in this paper are to the Latin Vulgate Bible).

“Erat autem fere hora sexta, et tenebrae factae sunt in universam terram usque in horam nonam. Et obscurantus est sol …” (Secundum Lucam 23: 44–45). Two other accounts describe the Moon as turned “into blood” at the time of an eclipse of the Sun (in a quotation from the prophet Joel, ascribed to Peter in the Acts of the Apostles 2:20, “Sol convertetur in tenebras, Et luna in sanguinem …” and in the apocryphal Report of the Crucifixion by Pilate).

Drs Colin Humphreys and Graeme Waddington concluded that the crucifixion must have taken place on 3 April, a.d. 33 ( The [Toronto] globe and mail , 19 April 1984, 18A).

These would be interesting, for example, as indicators of the historical periods in which the questions were posed.

This would include tables such as those for the ‘Eighth Sphere’.

Pseudo-Dionysius the Areopagite was a Greek writer purporting to be the disciple of Paul mentioned in the Acts of Apostles (17:34). The real identity of the author is still not established; his works seem, however, to date from the sixth and not the first century a.d.

“Quid dicis de defectu facto in cruce salvatoris?”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 508, line 59. Unless otherwise noted, all references to the writing of Albertus Magnus will stem from his Opera omnia , ed. by Institutum Magni Alberti Coloniense (Munster in Westfalen, 1951–), identified throughout as: (ed. Colon).

All of Albertus's numerical data have been recorded and tabulated in Appendix 1.

The only Scriptural evidence the twentieth-century Oxford scientists used to fix the date of the crucifixion was the fact that it occurred during the reign of Pontius Pilate as procurator of Judea. His reign is known to have been between a.d. 26 and 36.

The twentieth-century Oxford scientists also accepted as fact the exegetical information that the crucifixion occurred on a Friday.

Albertus seems to have quoted from a “Kalendarium” as if it were his most direct source for information which he felt was ultimately derived from the exegesis of Biblical texts, In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 34–37 and 510, lines 14–20. Cf. also Albertus's commentary on Matthew, Super Matthaeum 26:2 in Opera omnia , ed. by Borgnet A. (Paris, 1890–99), xxi, 148b–149a; any reference to Albertus's works as edited by Borgnet are identified as: (ed. Borgnet). The Biblical texts which served as his sources are in Matthew (Secundum Matthaeum 21:12 and 26:2), John (Secundum Ioannem 12:1) and Exodus (12:3).

“Secundum Kalendarium autem vigesima quinta die Martii passus fuit dominus in meridie”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon .), xxxvii/ii, 509, lines 26–28. “et concordat Kalendario, in quo scribitur, quod octavo kalendas Aprilis passus est dominus, qui est dies vicesimus quintus Martii”, ibid. , lines 33–36. It is somewhat curious that Albertus, a Dominican friar, would have had to consult a calendar to find the traditional date of the crucifixion, but perhaps he was simply helping his readers to follow his logic.

Cf. “VIII Kal. Annuntiatio virginis Mariae. Dominus crucifixus est.”, Pseudo-Bede, Ephemeris, Kalendarium m. Martii in Patrologia Latina, xc, 763–64.

Roger of Hereford, for example, who flourished around 1180, had united his interests in revising both the calendar and the Toledan Tables, with the not-too-surprising result that both often circulated together. Examples of this are to be found in Oxford, Bodleian Library, MS Digby 40 (computus and calendar), London, British Library, MS Arundel 377 (fragments of his Tables and a calendar), and Cambridge, University Library, MS Kk 1.1 (computus and calendar). Other versions of the Toledan Tables were also accompanied by calendars, such as a complete set of mean motion tables for Julian years set up for the meridian of Paris, epoch March, a.d. 1180 with a prefatory “Kalendarium” in Oxford, Bodleian Library, MS Savile 20. Details of such tables are covered by Mercier R. , “Astronomical tables in the twelfth century”, in Adelard of Bath , ed. by Burnett C. (London, 1986), 87–118.

The numbers Albertus mentioned later in his Dionysian exegesis are, however, yet to be found in such a work. Cf. Ferdinand Piper , Karls des Grossen Kalendarium und Ostertafel (Berlin, 1858; reprinted Walluf bei Wiesbaden, 1974), 98.

Albertus neither cited his Biblical sources nor drew any explicit conclusion. The information must be gathered from numerous scattered lines: re. a.d. 34, cf. In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 8–16; re Friday , cf. ibid. , 510, lines 14–20; re March, cf. ibid., 509, lines 8–16; re 25th at noon, cf. ibid. , 509, lines 26–28.

The evidence indicates that he did not enter the date as recorded. There were, however, contemporary eclipse tables in which it was the practice to enter the cardinal instead of the ordinal numbers, i.e. the date, or the current year, month and day, rather than a quantity of elapsed time. Cf. Neugebauer O. , The astronomical tables of Al-Khwarizmi (Copenhagen, 1962), 208, f. 142r of Oxford, Corpus Christi College, MS 283. A twelfth-century copy of al-Khwarizmi's astronomical tables contains just such an instruction.

“annus secundum astronomos incipit a Martio”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 10–11. Albertus's specification, “secundum astronomos”, reflects his awareness that the year of astronomical tables differed from the civil year which might have begun at Christmas or Easter, or at any of several other dates, depending on the time and place in question. He does not seem to have been aware, however, that the start of the astronomical year was not universally the same. Cf. the calendar in Oxford, Bodleian Library, MS Sa vile 23 (dated mid-thirteenth century), f. 1r at the entry for 1 January, “Anni ab incarnatione domini secundum usum astronomorum hie mutantur”, and f. 2r at the entry for 25 March, “Anni ab incarnatione domini secundum vulgarem usum hic mutantur et incipit annus emergens. In principio Januar annus usualis”. The evidence from astronomical tables themselves indicates variety as well. While the standard radix for the Toledan Tables was 28/2, (29/2) or 1/3, among its variations can be found 1/1 for Hereford and Marseille (Mercier, correspondence).

“dies secundum astronomos incipiat in meridie”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 21–22. Here, as above, Albertus's specification, “secundum astronomos”, reflects his awareness that the day of astronomical tables differed from the civil day which usually began at midnight. Again, apparently unknown to Albertus, there are some exceptions to the generalization he made about the astronomers' day, most notably in some Toledan Table variants, such as Hereford and Marseille where the epoch is midnight (Mercier, correspondence).

Several are extant: Paris, Bibliothèque Nationale, MS Lat. 14704 and Oxford, Bodleian Library, MS Bodley 613. Cf. Toomer G. J. , “A survey of the Toledan Tables”, Osiris , xv (1968), 5–174, Appendix C, #24 and #77. Whenever the present study demands citations from the Toledan Tables, if possible, Toomer's “Survey” has been used.

“Et quia dominus noster Iesus Christus in Martio noscitur esse passus trigesimo quarto aetatis suae anno et annus secundum astronomos incipit a Martio, in quo ipse de anno trigesimo quarto transiverat tres menses, dimisi tres menses et a Martio primi anni nativitatis suae computando accepi triginta tres annos perfectos, quia hoc est idem, ac si acciperentur a die nativitatis triginta tres [anni] et tres menses”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 8–16.

For all its triviality the calculation offers internal evidence for the year of Albertus's composition of the commentary on the Dionysian letters, an important historical detail which the work might not otherwise have revealed. Cf. Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, p. xxv, line 8.

“Ad hoc autem verificandum in revolutionibus annorum secundum doctrinam Ptolemaei reduxi stellas quantum ad modum coniunctionis et praeventionis luminarium abiectis omnibus circulis integris, qui fuerunt in mille ducentis sedecim annis”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 4–8.

Neugebauer O. , A history of ancient mathematical astronomy (Berlin, 1975), ii, 614–26.

Here Ptolemy is quoting from Aristarchus. Albertus's own source was probably not the Almagest directly, for the doctrine is mentioned in only this passage of the Almagest and here only in passing. What source he did use is not yet ascertained.

Cf. Lane Reginald Poole , Medieval reckonings of time (London, 1918).

If he were to have pursued this line further, Albertus could have learned just which year(s) of the Metonic cycle a.d. 34 and a.d. 1250 were. The Golden Number or year of the Metonic cycle is obtained by dividing 19 into the Julian or civil year. The Number, in this case 8, is the remainder of the quotient plus one. Thus Albertus's conclusion, that the two years share the same Golden Number, would have been confirmed as correct. For some recent reflections on the medieval calendar, cf. Gregorian Reform of the calendar: Proceedings of the Vatican conference to commemorate its 400th anniversary, 1582–1982 , ed. by Coyne G. V. Hoskin M. A. Pedersen O. (Vatican City, 1983).

The Latin Church calendar had incorporated the Dionysian Easter-Day cycle of 532 years which governed the return of the date of Easter. A much shorter 28-year cycle (532/19 = 28) was sufficient to establish weekday repetition in the same calendar. Reference to this later cycle was made by means of a chart of the Dominical Letter or the day of the week on which the first day of the year falls, accompanying a calendar. It is not likely that Albertus would have made such an error of day, using the chart. He did, however, come into error somehow on the days of the week in March, a.d. 34, and his maintaining the Easter Tables' identity of the two years, a.d. 34 and a.d. 1250, might be a possible explanation for his confusion. The Dominical Letter for a.d. 34 was C, i.e. the first day of the year was a Friday; for a.d. 1250, it was B, i.e. the first day of the year was a Saturday. This yields the dates of Easter Sunday in each year as 28 March and 27 March respectively.

Albertus was definitely not aware of the conflict between his conclusions from exegesis and his information from the civil calendar, the very conflict which has made it difficult for centuries to determine the calendric year of Jesus's death.

“Inveni autem in Tabulis Praeventionis Solis et Lunae tempus oppositionis lunae aequatum et diversum; nullem mensem, viginti tres dies, quindecim horas, quinquaginta quinque minuta horae et quattuordecim secunda”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 17–21.

Cf. Nicholas of Lynn, The ‘Kalendarium’ , ed. by Eisner S. (Athens, Georgia, 1980), 78–79, for an example of a true conjunction table, and 25–26, for the explanation of its use.

Hypothetically he could have referred simply to a calendar for a.d. 1250 because, as mentioned above (ref. 27), those years shared the same Golden Number. He could have read the Golden Number into the new and full moon table for the year a.d. 1250, at the month of March, and derived the day and time of either a conjunction or opposition. Such calendric tables often started from a noon epoch. It would have been equally possible for Albertus to have used a calendar for a.d. 1230, perhaps the Calendarium of Robert Grosseteste accompanying a revision of his Computus. The dating of the recensions is still under discussion. See J. McEvoy for a date c. 1225–30 of Recension I, “The chronology of Robert Grosseteste's writings on nature and natural philosophy”, Speculum , lviii (1983), 618–22, and a subsequent discussion by Southern R. W. dating the same recension (now labelled III) c. 1215–20, Robert Grosseteste: The growth of an English mind in medieval Europe (Oxford, 1986), 102, 120, 127–31, 139.

It is, however, safe to say that the table the Cologne editors of Albertus allude to could not, as will be seen, have been his source: “Cf. Tabula praeventionis sive oppositionis solis et lunae per medium cursum ad annos Christi ad medium diem Tolosae (Paris. Bibl. nat. lat. 16658 f. 80r–81v) (Poulle)”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, note to lines 17–18; cf. also ed. cit. (Introduction, p. xxx).

This is the suggestion of the editors of Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii. Cf. the preceding reference.

“sol habet circulum excentricum, in quo habet duos motus, scilicet aequalem, quem astronomi vocant medium, et motum augis”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 508, lines 20–22.

“Habet [sol] etiam circulum signorum, sub quo movetur, ad quem cum appropinquet elevando et deprimendo plus et minus, necesse est, quod diversimode moveatur, quod vocatur motus eius diversus”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 508, lines 27–30.

To accept this interpretation does require understanding by “tempus aequalis” that Albertus meant the time of the lunar “motus aequalis”. (“Tempus aequalis” subsumes the value of the mean time to which corrections are made in order to obtain the true time.) As for Albertus's term “diversus”, it appears to have its source in the standard title of a correction table, see an example in the Toledan Tables for obtaining the true hourly velocity of the Moon, “Tabula equationis diversi motus lunae in una hora” (Toomer, “Survey” (ref. 20), 85, Table 57) and another in the Toulouse Tables, “Tabula diversi motus solis et lunae in una hora” (Paris, Bibliothèque Nationale, MS Lat. 16658, f. 82r).

Inspection of the pertinent astronomical tables to which the editors refer, Paris, Bibliothèque Nationale, MS Lat. 16658, found on f. 70r and f. 81r, do not yield the same values as cited here. The manuscript's value for mean opposition time is 23^d1^h36^m41^s. R. Mercier has analysed the manuscript to find that the values it gives are wrong, i.e. inconsistent with the longitude values provided. In other words, this manuscript contains errors in its mean motion tables' values throughout. Neither the manuscript's nor Mercier's value agrees with Albertus's own; I cite here the internally consistent value provided by Dr Mercier.

Goldstine Herman H. , New and full moons 1001 b.c. to a.d. 1651 (American Philosophical Society Memoirs, xciv; Philadelphia, 1973), 87.

Cf. Paris, Bibliothèque Nationale, MS Lat. 16658, f. 79r: “Radix: Menses 11; dies 23 [recte: 28]; horem 1” which means the 23rd [recte: 28th] day of February, the 11th month. Cf. also ibid. , f. 78r and f. 81f.

Cf. Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 507, lines 9–13, and Toomer , “Survey” (ref. 20), 134–5, Table 83; In VII epistulam Dionysii (loc. cit., lines 15–20) and Toomer , ibid. , 42–43; In VII epistulam Dionysii (loc. cit., lines 16–20) and Toomer , ibid. , 40, Table 23.

“Illo autem tempore verus motus solis sine motu octavi fuit nihil in signis, octo in gradibus, tria in minutis et viginti sex in secundis”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 37–39.

“motus autem lunae in eodem tempore novem horis ante meridiem vicesimae quintae diei fuit sex signa, octo gradus, tria minuta et viginti sex secunda”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 39–42. Albertus goes on to note the names of the zodiacal signs to which the numbers refer: “ergo sol in nono gradu arietis fuit et luna in nono gradu librae; ergo diametraliter opponebantur” (lines 42–44).

Mercier, correspondence. Cf. Paris, Bibliothèque Nationale, MS Lat. 16658, f. 80a and f. 81a.

“motus autem octavi illo tempore habuit octogradus et quattuordecim minuta et unum secundum, et secundum doctrinam Thebit fuit ex arcubus, qui diminuuntur in meridie, hoc est caput arietis ultra aequinoctialem in meridie fuit secundum quantitatem praedicti spatii”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 47–52.

During Albertus's lifetime the location of the star which served to demarcate this conventional point was nearer σ Piscium.

Although Albertus did not seem to realize it, all he really needed for his purposes was the time of conjunction or opposition, and that is independent of the choice of longitude-measuring convention.

Mercier, correspondence.

The oldest reference to Thabit as the author, appears to be in the work of Abraham ibn Ezra, c. 1150. Mercier also held it to be Thabit's creation until 1979 (correspondence). Cf. Mercier R. , “Studies in the medieval conception of precession”, Archives internationales d'histoire des sciences , xxvi, nos 99 (December 1976), 197–220, and 100 (June 1977), 33–71, esp. no. 99, 209–20. The refutation of Thabit's authorship is explained briefly in Mercier, “Astronomical tables” (ref. 14), 106.

Cf. Albertus Magnus , De proprietatibus elementis , I, tr. 2, c. 3 (ed. Colon.), v/ii, 65, line 67–66, line 9; De quatuor coaequaevis, tr. 3, q. 12, art. 2 (ed. Borgnet), xxxiv, 426b; Super IV Sententiarum, dist. 44, A, art. 3 (ed. Borgnet), xxx, 549a.

Cf. Weisheipl James A. , “The life and works of St. Albert the Great”, in Albertus Magnus and the sciences , ed. by Weisheipl J. A. (Toronto, 1980), 13–51, p. 22.

He wrote that the tables were compiled according to a “second zodiacal motion of points on the Sphere of the Fixed Stars”. The motion is briefly described as that of the first points of the zodiacal sign, Aries and of the zodiacal sign, Libra moving backward and forward on a circle 8° in diameter.

“Secundum Thebith autem est secundum motum capitis Arietis et Librae super circulum, cujus diameter est octo gradum accedendo, et recedendo, secundum quod etiam compositae sunt tabulae accessus et recessus octavae sphaerae”, Albertus Magnus , De quatuor coaequaevis , tr. 3, q. 12, art. 2 (ed. Borgnet), xxxiv, 426b.

For an edition of this work, cf. The astronomical works of Thabit B. Qurra , ed. and trans. by Carmody F. J. (Berkeley, 1960), 84–113.

The table is found in Paris, Bibliothèque Nationale, MS Lat. 16658, ff. 82v–83v.

Mercier, correspondence.

“Et ideo diminuendus praedictus motus fuit a motibus stellarum; qui si diminuantur a motu solis superius habito, remanebit motus solis undecim signa et viginti novem [triginta, var.] gradus, novendecim minuta et viginti quinque secunda. Motus autem lunae, si ab eo removeatur praedictus motus, erit quinque signa, viginti novem [xxxix, var.] gradus, novemdecim minuta et viginti quinque secunda”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 53–60.

“ergo secundum hoc sol fuit in trigesime gradu piscium et luna in trigesimo virginis”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 60–61.

“et tunc luna non cecidit ab angulo plenitudinis nisi motu novem hoirarum, qui est motus quattuor graduum, quinquaginta sex minutorum et viginti octo secundorum”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 28–31.

Toomer , “Survey” (ref. 20), 48, Table 29.

“motus enim eius diurnus est tredecim gradus et decem minuta et triginta quinque secunda”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 507, lines 33–34.

“Coniunctio autem in eodem mense praecesserat solis et lunae, quae quidem fuit secundum diligentissimam aequationem undecimo die martii; habuit enim tempus nullum mensem, decem dies, quindecim horas, triginta sex minuta et viginta tria secunda”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, lines 63–67.

This conjunction was in fact the occasion of a total solar eclipse visible in the South Pacific (Mercier, correspondence).

“Luminaria autem ambo simul sine ablatione motus octavi habuerunt undecim signa, viginti duos gradus, quadraginta sex minuta, quinquaginta unum secunda; ergo simul steterunt in vigesimo tertio gradu piscium”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 509, line 67–510, line 1.

Paris, Bibliothèque Nationale, MS Lat. 16658 reads 11^s20°34′36′. The values cited here come from Mercier (correspondence).

“Ablato autem octavo fuit motus luminarium undecim signa, quattuordecim gradus, triginta duo minuta et quinquaginta secunda, et secundum hoc in quinto decimo gradu piscium simul steterunt”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 510, lines 1–5. In this case the subtraction is correctly done!.

“Cum autem tertius decimus dies terminetur in meridie quartae decimae diei, patet, quod aetas lunae fuit a conjunctione ad tempus, in quo dominus pependit in cruce, quindecim dies fere, et ad noctem captivitatis quattuordecim”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 510, lines 8–12.

The editors of In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 510, lines 14–15 annotate this reference as follows: “Glossa ord. super Matth. 21,12 et Ioh. 12,2–Exod. 12,3”.

Albertus Magnus In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 510, lines 15–20.

The twentieth-century Oxford scholars, on the other hand, found ambiguity in the Gospel records as to whether the crucifixion actually occurred on the 15th day of Nisan or on the day before, the 14th day of Nisan, the day of Passover on which the sacrificial lambs were slaughtered. To assume that Jesus's death occurred on the 14th of Nisan would be consistent with some exegetical material Albertus did not consider; Jesus is frequently referred to as the sacrificial lamb.

“Secundum aequationem autem diligentem latitudo lunae in illo tempore fuit sex signa, quattuordecim gradus, viginti quinque minuta et quattuor secunda, quod dicitur in Astronomia argumentum meridionale minus”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 511, lines 8–12.

Toomer , “Survey” (ref. 20), 78–79, Tables 52–55.

Cf. the Canons of a perpetual Almanac translated from Arabic into Latin in 1307, particularly “de latitudine planetarum”, for a slightly different, but analogous way of designating the quarters of the lunar latitudinal path, J. Ma Millás-Vallicrosa, Estudios sobre Azarquiel (Madrid-Granada, 1943–50), 400.

This value is in disagreement with the value he would have derived from adding a value for the Moon's mean nodal position to his value of the mean Moon at true opposition, i.e. 6^s1°40′44′.

“In hoc autem argumento luminaria non sunt infra gradus eclipsis, nisi latitudo habeat sex signa et minus duodecim gradibus”, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 511, lines 12–14. Poulle has emended the text to read “duodecim”. The manuscripts offer “XIII” or “XIV”. Cf. ref. 78 below for the reason why the manuscript value of “XIII” ought to be retained.

Ptolemy , Almagest VI, 5.

Cf. Pedersen O. , A survey of the Almagest (Odense, 1974), 228–31.

For the formula that represents the nodal extreme distance of the Moon, w = (pm + 0;33,20) × 11;30 + ps, where pm and ps are the relative parallaxes of the Moon and Sun, 0;33,20 the sum of the luminaries' greatest radii, and 11;30 derived from the planar configuration of the Moon's inclination. In the case of a latitude of 32° (Jerusalem's), w = (0;33 + 0;33,20) × 11;30 + 1;6,20 = 13;20,20. The eclipse limit would then be 180° plus less than 13° (or 14°) as Albertus's text reads. Cf. Poulle's emendation, Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 511, line 14. For more on the calculation of eclipse limits, cf. Neugebauer, History (ref. 24), i, 125–9.

For the kind of table, cf. Toomer , “Survey” (ref. 20), 86, Table 58, esp. “Tabula eclipsis solis … portio latitudinis meridiane”, 2nd and 3rd columns.

“Quod si daretur, quod tunc fuisset infra gradus eclipsis, cum argumentum sit meridionale, tenebrae eclipsis non fuissent directe nisi in meridie, et ad septentrionalem nullae fuissent in quarto climate et quinto et in tertio et fortisan parvae et paucae in secundo et primo; est autem Ierusalem in quarto climate; ergo nullae tenebrae ibi apparuissent”, Albertus Magnus In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 511, lines 17–23.

Neugebauer , History (ref. 24), i, 132–3.

Albertus described the Sun's visible shadows at those latitudes as “parvae et paucae”, In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 511, lines 21–22.

Albertus Magnus , In VII epistulam Dionysii (ed. Colon.), xxxvii/ii, 510, lines 31–64.