Changing Contexts and Criteria for the Justification of Computistical Knowledge and Practice

An insightful current discussion of computus and its literature is in the introduction to Faith Wallis , transl., Bede: The reckoning of time (Translated Texts for Historians, 29; Liverpool, 1999), pp. xv–lxxi.

Daniel McCarthy , “Origins of the Latercus Paschal Cycle in the insular Celtic churches”, Cambrian medieval Celtic studies , xxviii (1994), 25–49, pp. 27–30. An electronic version is available at http://www.cs.tcd.ie/Dan.McCarthy, pp. 4–5.

Concilia Galliae, A. 314 — A. 506 , ed. by de Clercq C. (CCSL cxlviii; Turnholt, 1963), 5.

Eusebius , De vita Constantini , 3.18–19, ed. by Migne J.-P. (Patrologiæ Græecæ, xx; Paris, 1857), cols 1073–8.

Cummian , De controversia Paschali , 107–10, ed. by Walsh Cróinín Ó , Cummian's letter De controversia paschali … together with a related Irish computistical tract, De ratione conputandi (Studies and Texts, lxxxvi; Toronto, 1988), 72–75; Bede , Ecclesiastical history of the English people , 3.25, transl. by Leo Sherley-Price (Harmondsworth, 1990), 189.

Cummian , De controversia Paschali , 277–83, ed. by Walsh Cróinín Ó , 92–95. In 631 the Roman Easter fell on 24 March; the Irish Easter fell four weeks later on 21 April. McCarthy , “Origins of the Latercus Paschal Cycle” (ref. 2), 8.

Van Dam Raymond , Leadership and community in late antique Gaul (Berkeley, 1985); Peter Brown , The cult of the saints: Its rise and function in Latin Christianity (Chicago, 1981).

Lilybaeum Paschasinus , Epistola III ad Leonem papam , ed. by Migne J.-P. (Patrologiæ Latinæ, liv; Paris, 1846), col. 609. Cassiodorus Senator reports another baptismal pool at Marcellianum in southern Italy that filled miraculously at Eastertime; Cassiodorus , Variarum , lib. 8, epist. 33 (King Athalaric to Severus, a.d. 527), transl. by Barnish S. J. B. (Liverpool, 1992), 110. As there is no report of the filling of this pool being used to test the proper date of Easter, it will not be considered further.

Gregory Tours , Glory of the martyrs , 23–25, transl. by Van Dam R. (Liverpool, 1988), 42–44.

Gregory Tours , The history of the Franks , V, 17, transl. by Lewis Thorpe (Harmondsworth, 1974), 274. Jones C. W. , “The Victorian and Dionysiac Paschal tables in the West”, Speculum , ix (1934), 408–21, p. 412, n. 3.

Gregory Tours , The history of the Franks (ref. 10), X, 23, transl. by Thorpe , 581–2.

Gregory Tours , Glory of the martyrs (ref. 9), 24, transl. by Van Dam , 43, 44. We might note that since Easter is a movable feast, varying by over a lunar month, the flow of the springs cannot be a natural, seasonally recurring phenomenon.

Bede , Ecclesiastical history (ref. 5), 2.2, transl. by Sherley-Price , 105.

Cummian , De controversia Paschali (ref. 5), 285–8, ed. by Walsh Cróinín Ó , 94–95.

Sancti Columbani opera , ed. & transl. by Walker C. S. M. (Dublin, 1957), 18–19.

Cummian , De controversia Paschali (ref. 5), 270–7, ed. by Walsh Cróinín Ó , 92–93.

Bede , Ecclesiastical history (ref. 5), 3.25, 5.19, transl. by Sherley-Price , 191–2, 306–7.

Bede , De temporum ratione , 30.71–76, 38.40–42, ed. by Jones C. W. , Bedae Opera de temporibus (hereafter cited as Jones , BOT; Cambridge, Mass., 1943), 237, 252; De temporibus , 10.10–18, ed. by Jones , BOT , 299; Ceolfrith , “Letter to Nechtan” (a.d. 710), in Bede , Ecclesiastical history (ref. 5), 5.21, transl. by Sherley-Price , 314. Ceolfrith's letter bears signs of Bede's influence, and may have been drafted by him.

Victorius Aquitaine , “Prologus Victorii Aquitaini ad Hilarum Archidiaconum”, 3, in Krusch Bruno (ed.), Studien zur christlich-mittelalterlichen Chronologie (Abhandlungen der Preußischen Akademie der Wissenschaften , Phil.-hist. Kl., 1937). Victorius does not describe any observations that would support his judgement of the truth of the nineteen-year cycle.

“Carefully correct the psalms, notation, chant, computus, grammar, and the Catholic books in every monastery and diocese, because often some desire to pray to God properly, but they pray badly because of uncorrected books.” Admonitio Generalis , cap. 72; Monumenta Germaniae Historica, Leges, Capitularia, I (Hanover, 1883), 60.

Alcuini Epistolae 126, 145 (Monumenta Germaniae historica, epistolarum IV; Berlin, 1895), 187, 234.

Epistolae variorum , 42 (Monumenta Germaniae historica, epistolarum IV; Berlin, 1895), 565–7.

The nature and feasibility of such an observation remains uncertain; compare Kenneth Harrison , “Easter cycles and the equinox in the British Isles”, Anglo Saxon England , vii (1978), 1–8, pp. 5–7; Wesley Stevens , Bede's scientific achievement (Jarrow Lectures, 1985; n.p., n.d.), 16, nn. 48–49; and Stephen McCluskey , Astronomies and cultures in early medieval Europe (Cambridge, 1998), 95–96. The word horologium, itself, is ambiguous, meaning a sundial, a table of the length of shadows at various times of day and seasons of the year, or a diagram depicting the daily movements of the sun at various seasons. Consideration of any one of these horologia would illustrate Bede's point, but only a sundial would provide an empirical test. In the tenth century Ælfric used the term dægmæl; sundial (literally day measure or day mark) when he rendered Bede's ambiguous horologica consideratione into Old English. Aelfric , De temporihus anni , 6.5, ed. by Heinrich Henel (Early English Text Society, O.S. ccxiii; London, 1942), 46–47.

Bede , De temporum ratione , 30.71–76; 38.41–48, ed. by Jones , BOT , 237; De temporibus , 10: 12–18, ed. by Jones , BOT , 299. The position of sunrise at the equinoxes changes by about 10 arc minutes each year; the change of over a full solar diameter after four years could easily be observed.

Rabanus Maurus , De computo , 53.6–22, 42–45, ed. by Wesley Stevens (Corpus Christianorum Continuatio Medievalis, xliv; Turnholt, 1989), 263–5; cf. Bede , De temporum ratione , 30.48–58, 71–76, ed. by Jones , BOT , 236–7.

Rabanus Maurus , De computo (ref. 25), 54.6–13, ed. by Stevens , 266–7; cf. Bede , De temporibus , 10.4–18, ed. by Jones , BOT , 298–9. Rabanus was no stranger to observation; see his apparent observation of the planets in De computo , 48.14–19, ed. by Stevens , 259. He just didn't use observations to demonstrate or evaluate computistical concepts.

Helperic , Liber de computo , 31, ed. by Migne J.-R (Patrologiæ Latinæ, cxxxvii; Paris, 1879), cols. 40–43.

On annalistic accounts see Schove D. J. , Chronology of eclipses and Comets: AD 1–1000 (Woodbridge, Suffolk, 1984).

Einhardi Annales , A. 807, Monumenta Germaniae historica, Scriptorum , i (1826, reprint New York, 1963), 194.

3. Id. Febr. fuit eclypsis solis media die, stante utroque sidere in vicesima quinta parte Aquarii. Einhardi Annales (ref. 29), A. 807, 194.

Modern values of solar longitude taken from Goldstine Herman H. , New and full moons: 1001 B.C. to A.D. 1651 (Memoirs of the American Philosophical Society , xciv; Philadelphia, 1973).

John McCulloh , “Martyrologium Excarpsatum: A new text from the early Middle Ages”, in King Margot H. Stevens Wesley M. (eds), Saints scholars and heroes: Studies in medieval culture in honour of Charles W. Jones (Collegeville, Minn., 1979), ii, 197–237; Arno Borst , Die karolingische Kalenderreform (Monumenta Germaniae Historica Schriften, xlvi; Hanover, 1998), 321.

Einhardi Annales (ref. 29), A. 810, 198. Given the contemporary concern with the dissemination of reliable computistical texts, we might ask whether this sudden cluster of eclipse observations could have been used to test the accuracy of computistical techniques. The principal parameter readily available for this use would be the age of the moon on the date of the eclipses.

An eclipse was visible from Aachen on 18 March at the second hour of the night, the time that was recorded in the annals for the eclipse of 5 March.

Bede , De temporum ratione , 27. 17–19, ed. by Jones , BOT , 230; Rabanus Maurus , De computo , 46. 4–6, ed. by Stevens (ref. 25), 257.

The eclipse of June 810 (greatest magnitude 27+) was visible only in the far South Pacific; the eclipse of July 810 (greatest magnitude 20+) was visible only in the North Polar regions. Circumstances computed using SkyMap Pro, version 8.0.

Pliny , Historia naturalis , 2. 10. 56, transl. by Rackham H. (Cambridge, Mass., 1949); Bede , De temporum ratione , 27. 17–18, ed. by Jones , BOT , 230; Rabanus Maurus , De computo , 46. 4–5, ed. by Stevens , 257.

Dungali Scotti Epistolae , 1 (Monumenta Germaniae Historica, Epistolarum IV; Berlin, 1895), 570–8; Eastwood Stansfield Bruce , “The astronomy of Macrobius in Carolingian Europe: Dungal's letter of 811 to Charles the Great”, Early medieval Europe , iii (1994), 117–34.

Dungali Scotti Epistolae (ref. 38), 576–7.

Eastwood , “The astronomy of Macrobius” (ref. 38), 125.

Alcuini Epistolae , 170 (ref. 21), 278–80.

Calculations of the appearance of the moon between sunset and moonset at Aachen using SkyMap Pro, version 8. Since this discussion related the observation to the date of Easter, Charlemagne was probably following computistical practice by reckoning the day from sunset. See Bede , De temporum ratione , 5. 114–16, ed. by Jones , BOT , 190: “populus israhel … festa tamen omnia sua, sicut et nos hodie facimus, vespere incipiens, vespere consummarit dicente legislatore: A vespera usque in vesperam celebratis sabbata vestra” (Lev. 23, 32); Rabanus Maurus , De computo (ref. 25), 22. 25–28, ed. by Stevens , 226. On reckoning from dawn see Bede , De temporum ratione , 5. 90–96, ed. by Jones , BOT , 189; Rabanus Maurus , De computo , 19, 10–11, ed. by Stevens , 222.

Alcuini Epistolae , 170 (ref. 21), 278–9; cf. Pliny , Historia naturalis , 2. 7. 48.

On the nature of the medieval geometrical and computistical traditions, and their convergence in the Carolingian court see McCluskey , Astronomies and cultures (ref. 23), pp. x–xii, 132–40.

Alcuini Epistolae , 170 (ref. 21), 280–1.

Alcuini Epistolae , 170 (ref. 21), 280.

Oxford, Bodleian Library, MS Auct. F.1.9. f. 86^v.

Ibid. , ff. 91^r, 95^v–96^r.

In the context of Aristotelian science one could ask whether the rival computistical traditions relied on universally valid demonstrations and whether the computists' understanding of the structure of the cosmos provides truly causal explanations. This point was suggested by the presentations of Gerhardt Endreß , “The language of demonstration”, and Jim Bennett , “Virtue, error and expertise: The attributes of instruments and the morality of operators”, both forthcoming in a special edition of Early science and medicine devoted to the Frankfurt Symposium.

Compotus est scientia numerationis et divisionis temporum. Numerantur enim tempora et signantur et dividuntur per signationes et differentias quas dant eis motus celestium corporum et differentias quas dant eis cultus religionum, et in hiis duobus modus numeracionis et divisionis completa est scientia que Compotus nominatur. Robert Grosseteste , Compotus … factus ad correctionum communis kalendarii nostri , ed. by Steele R. ( Opera Hactenus inedita Rogeri Baconi , Fasc. 6; Oxford, 1926), 213, 11. 6–11. I have followed the manuscript reading religionum against Steele's emendation, regionum. Sciencia de tempore est sciencia distinccionis et numeracionis temporum, exteriorum corporum motibus et ex humanis legibus nascencium. Et hec vocata est compotus ab auctoribus. Roger Bacon , Compotus fratris Rogeri Baconis , ed. by Steele R. ( Opera Hactenus inedita Rogeri Baconi , Fasc. 6; Oxford, 1926), 2, 11. 23–30. Compotus est scientia considerans tempora ex Solis et Lunæ motibus & eorum ad invicem coæquatione distincta…. Hanc scientiam ecclesiæ: Subalternatam ab astronomia distare constat…. Haec vero temporum discretionem secundum cursum Solis & Lunæ metitur. Quicquid etiam Astronomus de temporis fractionibus determinat…. Computistæ vero, temporum discretiones. John Sacrobosco , De anni ratione, seu ut vocatur vulgò, Compotus ecclesiasticus (Paris, 1550). Lynn Thorndike Pearl Kibre , A catalogue of incipits of mediaeval scientific writings in Latin (Cambridge, Mass., 1963), cols 242–3, list thirteen texts (mainly from the thirteenth and fourteenth centuries) beginning with the phrase Computus est scientia….

A valuable introduction to the application of this critical attitude to a wide range of scholastic disciplines is Edward Grant , God and reason in the Middle Ages (Cambridge, 2001).

Grosseteste , Compotus (ref. 50), 215.

Bacon , Compotus (ref. 50), 32–33.

Sacrobosco , De anni ratione (ref. 50), ff. 29r–30v; Pedersen O. , “In quest of Sacrobosco”, Journal for the history of astronomy , xvi (1985), 175–221, pp. 209–10.

Grosseteste , Compotus (ref. 50), 232–6; Bacon , Compotus (ref. 50), 32–33; Ludwig Baur , Die Philosophie des Robert Grosseteste, Bischofs von Lincoln (Beiträge zur Geschichte der Philosophie des Mittelalters, 18, 4–6; Münster , 1917), 53–55; North John D. , “The Western calendar — ‘Intolerabilis, Horribilies, et Derisibilis’: Four centuries of discontent”, in Coyne G. V. Hoskin M. A. Pedersen O. (eds), Gregorian Reform of the calendar: Proceedings of the Vatican Conference to Commemorate its 400th Anniversary, 1582–1982 (Rome, 1983), 75–113.

Grosseteste , Compotus (ref. 50), 235.

The classic critique of the tendency to imagine that the path of progress “represents something like a line of causation” is Herbert Butterfield , The Whig interpretation of history (New York, 1965), 12.

Coyne Hoskin Pedersen (eds), Gregorian Reform of the calendar (ref. 55).