Sacrobosco's Quadrans : Date and Sources

Pedersen Olaf , “In quest of Sacrobosco”, Journal for the history of astronomy, xvi (1985), 175–221.

Ibid., 185. Pedersen actually lists five manuscripts, but one of them (Munich CLM 353, f. 35–38), holds Quadrans vetus, rather than the tract by Sacrobosco.

Of the remaining copies, at least two date from the fifteenth century and one from the sixteenth; I have no data on the others, but assume they fall in the earlier range, namely, thirteenth or early fourteenth century.

The Quadrans vetus was edited by Tannery P. in 1897 in “Le traité du quadrant de maître Robert Anglès” (reprinted in Mémoires scientifiques, v (Toulouse and Paris, 1922), 118–97). A critical edition was prepared by Hahn N. L. , Medieval mensuration (Transactions of the American Philosophical Society, lxxii (1984)). For a discussion of issues relating to this work, see my “Latin sources of Quadrans vetus” (forthcoming).

Other tables included in some sets of the Tabule solis list the length of rising time of the signs; the length of daylight in the seventh climate (sc. the region of Paris); and the shadow table.

Knorr Wilbur , “Solar tables in the tradition of 13th century tracts on the quadrant” (forthcoming). A Latin text of the canon of the Tabule Humeniz has been edited by Millás J. M. y Vallicrosa, Estudios sobre Azarquiel (Madrid, 1943–50), 379–82. On the questions about the origins of these tables, see also North J. D. , Richard of Wallingford (3 vols, Oxford, 1976), iii, 248–9.

Tabule solis (Paris, Bibl. nat., lat. 7475, f. 65) and Canon Humeniz (ed. Millás , 389): In civitate Parisiensi sol ascendit in Ariete et in Libra descendit 42 graduum. An alternative text of the Tabule solis canon (New York, Publ. Lib., MS 69, f. 12v): Et nota quod elevatio capitis Arietis … est Parisius 42 graduum.

I disregard purely syntactic variants, or minor lexical variants, e.g. quo v. quoto.

Variants will be cited from Paris, Bibl. nat., MS lat. 7475, f. 64v, Oxford, Bodl. Lib., MS Laud misc. 644, f. 11; London, Brit. Lib., MS Add. 30380, f. 5v; Vatican, Bibl. Apost., MS urb. lat. 1428, f. 98v.

The Oxford copy presents this text in the lower margin of f. 11, below the first of the bissextile tables; below the fourth of the tables, f. 12v, it presents an alternative, somewhat abbreviated text of the same canon. For the present lines, this second version agrees with the Paris copy in its omission of que declinatio … a predicta elevatione, but it agrees with the London copy on the rest of the text.

When Sacrobosco later describes how one determines the solar altitude, he produces the correct statement (New York, Publ. Lib., MS 69, f. 76v–77: Si sol fuit in signis septemtrionalibus, declinatio septemtrionalis erit; eam igitur de altitudine quama accepisti minue. Et si in australibus, australis erit eius declinatio, sive meridiana; eam itaque altitudini quam accepisti adde). This Statement does not correspond to the wording in the Tabule solis or Humeniz, as we have given it in [2]. Presumably, then, Sacrobosco is following a different source in this later passage.

In the case of the longitude tables, the only change in the Tabule solis is to begin the year with 1 March, rather than with 1 September as in Humeniz.

Pedersen , “In quest of Sacrobosco” (ref. 1), 190–1.

The only such source is Boethius's Arithmetica, whose preface is cited implicitly by paraphrase in the opening line of the Algorismus.

Pedersen , “In quest of Sacrobosco” (ref. 1), 192.

Ibid., 188.

Ibid., 183, 185. The Copenhagen manuscript in its extant state is defective, now missing its original folios 1–85, 118–34, 144, 147, 160, 168 (as one infers from its pagination); see the description by Jørgensen E. , Catalogus codicum latinorum medii aevi Bibliothecae Regiae Hafniensis (Copenhagen, 1926), 417–18. Since it includes at its beginning several works on arithmetic, it may well also have once included Sacrobosco's Algorismus. Near its close it holds the astrolabe tract of Hermannus Contractus, but no indication of any other comparable work, such as the Quadrans of Sacrobosco.

Pedersen discusses the passage; cf. “In quest of Sacrobosco” (ref. 1), 189. In comparing the solar and lunar cycles (which coincide, with a small difference, every 19 years), Sacrobosco states the current year to be 1235 (sed ab incamatione Domini elapsi sunt 1235 anni), and as constituting 65 decemnovennial cycles, whence the difference between 19 Julian years and 235 synodic months, at the rate of 1 hours per cycle, amounts to 65 horis et totidem terciis, or 3 dies et 14 horas. But in the Copenhagen MS GKS 277, 2°, f. 95, the date is given instead as 1232. Since this is the earliest datable copy of the Compotus (see ref. 17 above), Pedersen is inclined to accept its variant 1232 over the virtually unanimous reading of 1235 in the other manuscripts. The difference can readily be assigned to scribal confusion, since the figures for 2 and 5 are somewhat similar in early thirteenth-century script (cf. specimens in Figure 2). But I am not persuaded that a scribe would misconstrue 1232 as 1235, since the appearance of the first 2 in 1232 would only reinforce the second 2; conversely, however, with 1235 the 2 might instigate a miscopying of the 5. One might argue, however, that 1235 is a correction for original 1232, since the former is exactly 65 cycles of 19 years, while the latter is only approximately so. But this would leave unexplained why the tradition of the correct figure, if 1232, is so poorly represented in the manuscripts; for there would be an interval of at least three years (that is, between 1232 and 1235) during which copies with the original figure were made. I am thus inclined to accept the consensus figure of 1235 and view the variant 1232 as a scribal error.

As observed by Pedersen (op. cit. (ref. 1), 192), Michael Scot's commentary on De spera must have been composed before 1235 (the date of Michael's death) and after 1231 (when the ban on the teaching of Aristotle's works was lifted at Paris). It follows that De spera itself appeared before 1235; Pedersen is inclined also to date it after 1231, by reason of its citations of Aristotle. But one must distinguish between Michael's references to Aristotle, which are frequent and specific (including the controversial Metaphysics and natural treatises), and Sacrobosco's, which are sparse (mostly to the noncontroversial logical corpus). Indeed, Sacrobosco's hesitancy in this regard could well indicate that he still felt the pressure of the ban. I thus favour assigning the De spera a somewhat earlier date, around 1230 or even a few years before that. As for the Algorismus, there is no clear evidence for assigning a date. Its elementary character, however, might recommend placing it before the De spera and Compotus, that is, to the 1220s.

Cf. Pedersen , op. cit. (ref. 1), 188 and 218 n89. Miraeus's Rerum toto orbe gestarum chronicon ab anno Christi M.CC. ad nostra usque tempora of 1608 is clearly a work of the most general kind, based on standard sources, rather than primary research. One can hardly suppose that such a compiler as Miraeus consulted original documents to establish dates for events like the death of Sacrobosco. As Pedersen admits, moreover, Miraeus's reasons for the date 1236 are entirely unknown, so that “it is impossible to verify it”. The testimony of Thomas Dempster (1627), who likewise reports the date 1236, simply follows Miraeus, and so it is rightly discounted by Pedersen.

Cf. Pedersen (op. cit. (ref. 1), 187), who quotes from a slightly variant text in the somewhat later Basel MS O. II. 7, f. 38v.

Namely, the Scottish family name Halybush, or the place name Haliwood or Holy wood in Scotland or Ireland, or Halifax in Yorkshire, etc.; cf. Pedersen (op. cit. (ref. 1)), 176–81.

I overlook the alternative proposal mentioned by Pedersen, as suggested by Taisbak C. M. , that would construe deno quater as “in the fourth decade”, in order to make the date 1234.

In the scansion of quinquagesimo sexto anno, the antepenultimate syllable will be elided because of the juxtaposition of vowels, thus to be read … sext’ anno.

As in the preceding note, by elision the phrase would be read… quart’ anno.

While insisting on 1256 as the correct reading, I do not mean to dismiss the alternative 1244 as unreasonable. I have indicated a medieval annotation (in the New York copy) that supports 1256, but another precedent can be cited in support of 1244: The initial for the Algorismus on f. 1 of the lost Tournai MS 87, according to the cataloguer Faider P. , shows “Un personnage traçant sur une table le nombre 1244” (Catalogue des manuscrits conservés à Tournai = Catalogue général des manuscrits des bibliothèques de Belgique, vi (Gembloux, 1950), 86); and this, Faider surmises, alludes to the date in the epigram ending the Compotus , on f. 96, which he takes to be its date of composition: “La date 1244 (cfr. fol. 1 r° lettrine) se refère à l'année ou fut terminé l'ouvrage” (ibid., 87), that is, the Compotus.

See ref. 18 above.

Cf. Pedersen , op. cit. (ref. 1), 187, 218 nn71 and 73.

The annotation on f. 10 conforms to the rubric title on the same page (in a hand less compact than the text hand), and agrees also with the text hand for the verse algorism beginning on f. 72. Pedersen (op. cit. (ref. 1), 189, 218 n74) cites this same Paris manuscript, at f. 55v, where the epigram in its usual place at the end closes with its fourth line (“Tu stabilire … evum”), while the five additional lines are inserted in a later hand.

Pedersen , op. cit. (ref. 1), 181. Note that the switch in word order results only from metrical considerations.

Ibid., 188.

The passage is quoted by Thorndike L. (The Sphere of Sacrobosco and its commentators (Chicago, 1949), 2), and cited therefrom by Pedersen (op. cit. (ref. 1), 183).

Compare the Quadrans vetus of Johannes Anglicus, for instance, who names the latitude of Paris (as 48°) from his Paris-based source, the Artis cuiuslibet consummate, but then adds the latitude of Montpellier (44°), his own location; cf. my “Latin sources of Quadrans vetus” (ref. 4).

New York, Publ. Lib., MS 69, f. 74v: Ponatur enim regula supra centrum quadrantis et finem 24 gradus a dextris fiat linea intersecans cursorem. Similiter in sinistra posita regula supra centrum quadrantis et finem 18 gradus in limbo fiat linea intersecans cursorem. Inter istas igitur duas lineas extremas remanebit longitudo cursoris essentialis, s. 48 graduum.

The Latin text has been edited by Bubnov N. as Geometria incerti auctoris in Gerberti Opera mathematica (Berlin, 1899), 317–36. Note that in the pseudo-Gerbertian geometry, the phrase Si fuerit altitudo… commences what Bubnov labels its “Liber III”, a collection of geometrical problems; and Geometrieales traetanti… commences “Liber IV”, a miscellany of rules in practical geometry. But in one manuscript not cited by Bubnov, Copenhagen GKS 277, 2° (c. 1240; see refs 17–18 above), the latter incipit prefaces a collection like the former (with, however, many changes of order).

A usable, although not critical, edition of the Latin text is presented by Gunther R. T. in Chaucer and Messahalla on the astrolabe (= Early science in Oxford, v (Oxford, 1929)).

Kunitzsch P. , “On the authenticity of the treatise on the composition and use of the astrolabe ascribed to Messahalla”, Archives internationales d'histoire des sciences, fasc. 106 (1981), 42–62. A Latin text of Johannes Hispalensis's version is given by Millás in Las traducciones orientales en los manuscritos de la Biblioteca Catedral de Toledo (Madrid, 1942), Appendix Kunitzsch I. (op. cit., 49) maintains this Latin version to be “totally identical with Ibn al-Saffár's Arabic treatise” (emphasis his), which is extant and has been edited by Millás in Arabic and modern Catalan (for references, see Kunitzsch , op. cit., 48).

Kunitzsch , op. cit., 43–48.

Poulle E. , “L'astrolabe médiéval d'après les manuscrits de la Bibliothèque Nationale”, Bibliothèque de l'École de Chartes, cxii (1954), 86–87.

Kunitzsch , op. cit., 53–56.

Ibid., 54 n55.

The Geometria does not cover any astronomical applications.

I have made a few trivial changes of spelling to conform to medieval conventions.

In parentheses I indicate variants in London, MS Royal 12 c ix, f. 41.

Poulle , “L'astrolabe médiévale” (ref. 39), 86–87.

I indicate by “L” a few of the variants in London, MS Royal 12 c ix, f. 41.

Note that the word quadrans where it occurs here in Maslama and Messehallah (or the quadratum in the Geometria) refers to the shadow “square”, not the quadrant.

Specifically, in Messehallah [45, a] the proportion is given correctly: 12 is to umbra extensa as the altitude is to the distance. Although Sacrobosco [a] makes the identical claim, he should instead have written: 12 is to umbra extensa as distance is to altitude. Likewise, the rule in [b] (which follows from [a] in accordance with the relation 12: extensa = versa: 12) is correct in Messehallah, but inverted in Sacrobosco.

The passage goes on to consider certain other cases without parallels in Messehallah or Sacrobosco.

Poulle (“Astrolabe médiévale”, op. cit. (ref. 39), 86–87) observes that Messehallah's section [46] (as well as part of the preceding sect. 45) “se retrouve inchangé” in a certain anonymous tract on the quadrant (inc.: Cum quadrantem volueris componere), in the Paris manuscript, Bibl. nat., lat. 7416B (f. 62–63v, specifically f. 63). But, in fact, the verbal correspondence here is quite loose, especially in the latter half, and certainly far less than the agreement between Messehallah and the Geometria. One notes, further, that the anonymous tract sets this passage immediately after paraphrases of Messehallah's sections [42] and [45], and follows it with a paraphrase of [47]. The anonymous tract also presents paraphrases of Messehallah's [2] (on finding the altitude of the Sun), and [21–22] (on determining one's latitude by measuring the altitude of the Sun, or of a circumpolar star, respectively), together with alternative versions of the latter, resembling other sources. For instance, a variant of the stellar method of latitude determination agrees with that given in Quadrans vetus. One infers, then, that Messehallah's astrolabe tract was a source exploited (extensively so, it appears) for the account of applications in Cum quadrantem volueris componere. From a note attached immediately after its end, and written in the same scribal hand, one infers some English connection for the anonymous tract; for it states the latitude of Oxford along with that of Paris (f. 63v: polus elevatur in partibus Oxonie per 52 gradus, Parisius vero per 48). Most of Paris MS lat. 7416B was copied in the first quarter of the fourteenth century; for it contains a copy of the Quadrans novus of Petrus de Sancto Audomaro (St Omer) from 1293 (cf. F. S. Pedersen), and a table of eclipses for the years 1312 to 1330 (f. 114v) in a different hand from the rest, as well as a calendaric annotation (f. 44) interpretable as relating to 1330. But one part of the manuscript (comprising f. 59–65v, 120–125v) is in a hand different from the rest, and may be earlier; it consists of the anonymous quadrant tract, along with a commentary on Sacrobosco's Algorismus, tables of declination and longitude (f. 64v, for the second year after the bissextile only, in the Humeniz-based series), a tract on the cylindrical sundial, an account of the use of the shadow table, and a copy of the Grosseteste Kalendarium.

The corresponding accounts in other works, such as the Quadrans vetus of Anglicus Johannes (sect. [57–58]) and the Practica quadrantis of Campanus, specify the side correctly (as umbra recta in their variant terminology); cf. the parallel citations in my “Latin sources of Quadrans vetus” (ref. 4). If in either station the sightline happens to cut the umbra versa instead, Campanus and Johannes instruct the measurer to convert to the umbra recta (as equal to 144/umbra versa); cf. ref. 48 above.

The passage next explains in full the same operation, if the sightline cuts the other side (latus umbre verse) of the shadow square. It continues with an account of how to determine which of two places is higher, and what is the gradient between them. I find no analogue to this account in any of the three other tracts.

Kunitzsch , claiming to find no parallel to Messehallah [47], appears to have missed the connection with Maslama [34]; cf. “On the authenticity” (ref. 37), 54 n55.

All three state the proportion: umbra is to 12 as the measurer's height is to the width of the plane. For the Geometria and Messehallah, in the context of the astrolabe, the vertical side (or umbra versa) would be intended; for Sacrobosco's quadrant, it would be the base (or umbra extensa) of the shadow square.

It is clear, in this case, that the Messehallah editor has both books on his desk, as it were. Perhaps the manuscript of Maslama he consulted was unusable for the three consecutive sections where the shift occurs.

Sacrobosco's citations of the Geometria agree specifically with the variant readings in the group of manuscripts Bubnov designates “E”. The same is true of the passages of the Geometria incorporated into Messehallah.

Among the instances: Sacrobosco usque ad radicem rei, Geometria [1] usque ad radicem altitudinis (Messehallah [45] in radicem rei); S & G [4] vis (Mssh [47] queris). But in comparable passages Messehallah is sustained by the source, e.g. Mssh [45] summitatem, G [1] summitas (S cacumen); Mssh [47] & G [4] post hec (Spostea); Mssh [47] steterit regula, G [4] mediclinium stet (S ceciderit perpendiculum).

Kunitzsch , “Authenticity of the treatise” (ref. 37), 56 n60a.

A table of solar longitudes on f. 59v is explicitly dated for this year; another table (f. 60) of the very same form has slightly lower values, suggesting a date perhaps four years earlier; for discussion see my “Solar tables” (ref. 6). A marginal note to the calendar (at the bottom of f. 70), in an annotator's hand, states the “5th year of the 4th cycle”, which would correspond to 1278, or 62 years after the base of the calendar, 1216.

The passages are reviewed in my “Latin sources” (ref. 4).

The date follows from textual parallels to Campanus's Practica quadrantis, as I explain in “Latin sources” (ref. 4).

Ibid. The tract is briefly discussed by Benjamin F. S. Jr , in Campanus of Novara and medieval planetary theory, edited with Toomer G. J. (Madison, Milwaukee and London, 1971), 4–5.

The sole exception known to me is the quadrant diagram appearing with Campanus's tract in Vatican, MS urb. lat. 1428, f. 95v (see Figure 1).

Millás J. M. , “La introduction del cuadrante con cursor en Europa”, Isis, xvii (1932), 218–58 (esp. pp. 251–3).

Poulle E. , “Two medieval texts”, Journal for the history of astronomy, xv (1984), 135–137).

Poulle E. , “Les instruments astronomiques de l'Occident latin aux xie et xiie siècles”. Cahiers de civilisation médiévale, no. 25 (1972), 27–40 (esp. pp. 38–39). Similarly, King D. views the quadrans vetus in the thirteenth century as a trivial consolidation of features well known in older Arabic devices: “The neglected astrolabe”, in Mathematische Probleme im Mittelalter, ed. by Folkerts M. (Wolfenbütteler Mittelalter Studien, 10; Wiesbaden, 1996), 45–55 (esp. pp. 51–52).

The passage (from Paris MS lat. 7195) is quoted by Millás (“Introduction del cuadrante” (ref. 64), 251), with references to prior discussions by Sédillot and Tannery.

Cf. Plates 2 and 3 in “Introductión del cuadrante” (ref. 64), from London, Brit. Lib. MS Royal 15 B ix; and Vatican, MS regin. lat. 1661 (both manuscripts from the eleventh century, as dated by Millás).

Ibid., 232. The attribution to Hermannus is proposed by Pez in the Migne edition, but Bubnov assigns it instead to Gerbert. Millás (ibid., 230–5) seems inclined to accept Hermannus as author, as does Poulle, although with reservations (cf. “Instruments astronomiques” (ref. 66), 38: “qui n'est sûrement pas de lui”).

Sumpto astrolabio … per utrumque mediclinii foramen polo perspecto …; cf. Thorndike L. (ed.), op. cit. (ref. 32), 85.

Cf. the edition of Thorndike (ibid.), 297: per demonstrationem astrolabii vel quadrantis; 300: probet per quadrantem, sicut dicitur in littera. The latter passage is striking, in that it seems to imply that Sacrobosco's text also names the quadrant here. But only one of Thorndike's manuscripts of Sacrobosco holds the variant reading astrolabio vel quadrante (cf. ibid., 85n, citing a variant in Princeton, MS Garrett 99). It may be, then, that Michael is simply imputing to Sacrobosco his own knowledge of the quadrant.