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John of Tynemouth alias John of London: emerging portrait of a singular medieval mathematician

Published online by Cambridge University Press:  05 January 2009

Wilbur R. Knorr
Wilbur R. Knorr
Affiliation:
Program in the History of Science, Stanford University, Stanford CA 94305, U.S.A.
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Extract

In 1953 Marshall Clagett presented a preliminary scheme of the medieval Latin versions of Euclid's Elements. Since then a considerable body of these texts has become available in critical editions, thanks to Clagett's labours on the Archimedean tradition and H. L. L. Busard's work on the Euclidean versions. Further, Busard, M. Folkerts, R. Lorch and C. Burnett have scrutinized the pivotal ‘second’ version of Adelard of Bath, and have thereby exposed a diversity of text forms that spells real complications for the effort to establish its provenance and use. In his recent overview of the medieval Euclidean tradition, Folkerts displays not only how these studies have filled out and expanded upon Clagett's initial framework, but also how they have compelled rethinking of some basic issues, such as on the source relations and authorship of the various versions.

Type
Research Article
Information
The British Journal for the History of Science , Volume 23 , Issue 3 , September 1990 , pp. 293 - 330
Copyright
Copyright © British Society for the History of Science 1990

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References

1 ‘The Medieval Latin Translations from the Arabic of the Elements of Euclid’, Isis, (1953), 44, pp. 16–42Google Scholar. For a synopsis in the light of later findings, see Murdoch, J. E., ‘Euclid: Transmission of the Elements’, Dictionary of Scientific Biography, vol. iv, (1971), pp. 437–459.Google Scholar

2 Clagett, M., Archimedes in the Middle Ages, 5 vols, vol. i (Madison, 1964), ii–v (Philadelphia, 1976–84).Google Scholar

3 See his editions of the versions of Hermann of Carinthia (in two parts: Leiden, 1968 and Amsterdam, 1977); Adelard of Bath, the ‘first’ version (Toronto, 1983); Gerard of Cremona (Leiden, 1984); and the anonymous Graeco-Latin version (Stuttgart, 1987).

4 See Busard, H. L. L., ‘Some Early Adaptations of Euclid's Elements,’ in Mathemata: Festschrift für Helmut Gericke (eds Folkerts, M. and Lindgren, U.), Stuttgart, 1984, pp. 135–142Google Scholar, Folkerts, M., ‘Adelard's Versions of Euclid's Elements’, in Adelard of Bath: An English Scientist and Arabist of the Early Twelfth Century (ed. Burnett, C. S. F.), London (Warburg Institute Surveys and Texts, 14), 1987, pp. 55–68Google Scholar; Lorch, R., ‘Some Remarks on the Arabic-Latin Euclid’Google Scholar, ibid., pp. 45–54. Busard and Folkerts are currently preparing a critical edition of ‘Adelard II’.

5 An overview of the general characteristics of the medieval Euclids, including the Adelardian versions, together with an inventory of the known manuscripts of all versions, has been compiled by Folkerts and his as sociates; see his ‘Euclid in Medieval Europe’ (Questio II de rerum natura), Winnipeg, 1989Google Scholar. For an account of the pedagogical and metamathematical aspects, see Murdoch, J. E., ‘The Medieval Euclid: Salient Aspects of the Translations of the Elements by Adelard of Bath’, Revue de Synthèse, (1968), 89, pp. 67–94.Google Scholar

6 See the critical edition by Clagett, , op. cit. (2), vol. i, Chapter 6.2Google Scholar. For recent studies, see Knorr, W. R., ‘The Medieval Tradition of Archimedes' Sphere and Cylinder’, in Mathematics and its Applications to Science and Natural Philosophy in the Middle Ages (eds Grant, E. and Murdoch, J. E.), Cambridge, 1987, pp. 3–42Google Scholar; and Sato, T., ‘Quadrature of the Surface Area of the Sphere in the Early Middle Ages’, Historia Scientiarum, (1985), 28, pp. 61–90.Google Scholar

7 On the early citations, see Clagett, , op. cit. (2), vol. iii, pp. 1253–1254Google Scholar (cf. also note 45 below); for the later influence of De curvis, see ibid., passim (especially pp. 1235–1246, for a summary of citations by Maurolico, Giovanni Fontana, Luca Pacioli, Leonardo da Vinci and others).

8 See the entry in Emden, A. B., A Biographical Register of the University of Oxford to A.D. 1500, 3 vols, Oxford, 1957–1959, iii, p. 1923Google Scholar; cited in Clagett, , op. cit. (2), vol. i, p. 720; v, p. 11.Google Scholar

9 Clagett, , op. cit. (2), vol. iii, p. 1253Google Scholar. To similar effect, Busard calls this author ‘mysteriös’ in ‘Der Traktat De isoperimetris’, Mediaeval Studies, (1980), 42, p. 64.Google Scholar

10 For my analysis of the versions of De quadratura circuli, see Textual Studies in Ancient and Medieval Geometry, Boston/Basel/Berlin, 1989, Part III, Chapter IXGoogle Scholar. For a similar breakdown of the versions of De ysoperimetris see my ‘Paraphrase Editions of Latin Mathematical Texts: De figuris ysoperimetris’, Mediaeval Studies, (1990), 52, (in press).Google Scholar

11 For a summary of the general philological basis of the scheme for analysing Plato's style, see Ross, D., Plato's Theory of Ideas, Oxford, 1951, Chapter IGoogle Scholar, and Guthrie, W. K. C., A History of Greek Philosophy, 6 vols, Cambridge, 1962–1981, iv, Chapter IIIGoogle Scholar. On the Federalist Papers, see Mosteller, F. and Wallace, D. L., Applied Bayesian and Classical Inference: The Case of The Federalist Papers, 2nd edn, New York, 1984CrossRefGoogle Scholar; in Chapter X the authors review several related studies that have used statistical methods for authorship determination.

12 Even in such a difficult case, where the editor has consulted several sources, if his use of each is systematic it may be possible to distinguish their relative influence; see, for instance, my analysis of the ‘Corpus Christi Version’ of De quadratura in Textual Studies, op. cit. (10), pp. 655–663.Google Scholar

13 For example, among historians of Greek art, the principal schemes for the identification of the painters of the ancient Athenian vases depend critically on the analysis of style; on the stylistic analyses by Sir John Beazley, which are considered a paradigm by many, see Kurtz, D. C., The Berlin Painter, Oxford, 1983Google Scholar. From the study of medieval art, I mention as a specific example the efforts to identify the craftsman of the so-called ‘Bury St Edmunds Cross’, for which the focus of debate lies in the interpretation of stylistic correspondences with the illuminations in the Bury Bible; for a summary, see Scarfe, N., ‘The Walrus-Ivory Cross in the MetropolitanMuseum’, in his Suffolk in the Middle Ages, Bury St Edmunds, 1986, Chapter VIIGoogle Scholar. Ina similar vein, the discipline of palaeography relies primarily on close analyses of handwriting style to determine features of the provenance and dating of manuscripts; for discussions, see the works of Thompson, Jenkinson and Denholm-Young cited in notes 114–117 below.

14 On the stylistic analyses of Plato's works, see the discussions cited in note 11. In New Testament studies source analyses are critical in the investigation of the relative placement of the synoptic gospels, as are stylistic analyses for arguments on the authenticity of works in the Pauline group; these matters are summarized in all New Testament introductions, e.g. Kümmel, W. G., Introduction to the New Testament, Nashville, 1975 (trans of the 17th German edn, 1973).Google Scholar

15 For listings of other passages of this type, see Knorr, , op. cit. (6), pp. 5–8, 33–34Google Scholar. I include in this group a few reminiscences of Biblical passages, e.g. the line cited in item (2), imitating ‘lest thou dash thy foot…’ from Psalms 90.12 (Vulgate), Matthew 4.6, and parallels.

16 See Busard's edition of ‘Adelard I’, passim.

17 This phrase, imitating a Biblical passage (Ecclesiasticus 37.23–24), thus also exemplifies item (1); cf. note 15.

18 For a more detailed survey, see my Textual Studies, op. cit. (10), pp. 621–623Google Scholar and ‘Paraphrase Editions’, op. cit. (10), Section II.Google Scholar

19 For the critical edition of the text, see Clagett, , op. cit. (2), vol. i, pp. 91–142Google Scholar; for a detailed analysis of its style in comparison with De curvis, see my Textual Studies, op. cit. (10), pp. 620–623Google Scholar. Clagett speaks of the ‘Florence Versions’ (in the plural), because the extant copy provides two different proofs of Proposition 1, while it would seem most reasonable that a given editor had produced only one proof. But the basic elements of style are consistent throughout, indicating the hand of a single editor. I cannot say whether the duplicate version was the editor's original intent, or whether the extant copy is a combination of separate versions.

20 The critical edition of the original translation is given by Busard, , op. cit. (9), pp. 61–88Google Scholar. I set out a detailed comparison of CS1 with the Digby version (FY1) in my ‘Paraphrase Editions’, op. cit. (10), Section II.

21 For the analysis of these source relations, see my ‘Medieval Tradition’, op. cit. (6); Textual Studies, op. cit. (10), Part III, Chapters VIII and IX; and ‘Paraphrase Editions’, op. cit. (10), Section I.

22 For a complete listing of manuscripts of CS1, see Clagett, , op. cit. (2), vol. i, pp. 446–449Google Scholar. No one manuscript holds all three works. The Digby manuscript holds a second copy of the isoperimetric tract (in the literal Graeco–Latin form) and De quadratura circuli (in the Gerardian translation, rather than the Florence adaptation). In fact, most of the codices that hold De curvis also have the Gerardian De quadratura (see the listings in ibid., pp. 37–38, 446–449). This would be expected if John composed De curvis first (so that Gerard's version would still be his source for the circle quadrature), and only later produced his adaptation of De quadratura.

23 See Clagett, , op. cit. (1), pp. 23–25Google Scholar; for discussion, listing of manuscripts, and specimens of text, see ibid., pp. 33–38. The same Digby manuscript also holds excerpts from ‘Adelard II’on fols 154r–l59v (not so noted by either Clagett or Folkerts) and from Gerard of Cremona's Euclid on fols 160r–173v. On recent findings bearing on Versions I and II, see the articles by Folkerts, and Lorch, , op. cit. (4)Google Scholar, and the inventory of Folkerts, , op. cit. (5)Google Scholar. On the alternative Version IIIB, see Section IV below.

24 For the passage, with the definitions of exemplum, dispositio and the other terms, see Clagett, , op. cit. (1), p. 34Google Scholar; and ‘King Alfred and the Elements of Euclid’, Isis, (1954), 45, p. 274Google Scholar; cited also in Clagett, , op. cit. (2), vol. i, p. 444n.Google Scholar

25 For the discussion of ‘Quelibet media proportionalia’ see my ‘On a Medieval Circle Quadrature: De circulo quadrando’, Historia mathematica, (1990), 17 (in press).Google Scholar

26 See Clagett, , op. cit. (1), p. 25Google Scholar; note that this copy of ‘Euclid's stereometry’ includes only Books X–XV, and so lacks the initial heading.

27 Monoculuslum ought literally to denote a person or thing that has one eye. Although the invention of eye-glasses only came early in the fourteenth century, the magnifying property of a lens (such as a transparent sphere of crystal or glass) was recognized earlier. For instance, Robert Grosseteste speaks of the use of optical principles to ‘enable us to see the smallest letters at incredible distance, or to count sand, grain, grass, or whatever minute objects’; cf. ‘De iride et speculo’, Die philosophischen Werke des Robert Grosseteste (ed. Baur, L.), Münster, 1912, p. 74Google Scholar; the passage is paraphrased in Bacon, 's Communia mathematica (ed. Steele, R.), Opera hactenus inedita Rogeri Baconi, fasc. 16, Oxford, 1940, p. 46.5–8Google Scholar. Bacon also mentions the utility of such a spherical lens for the elderly and those with weak eyes; cf. Opus Maius, Part V, Chapter IV (ed. Bridges, J. H.), 2 vols, Oxford, 1897, vol. ii, p. 15Google Scholar. Among physicians, the use of an oculus berillinus for improving vision is recorded with Bernard of Gordon in 1303; the plural ocularia, thus unambiguously denoting ‘spectacles’, appears only in a later edition of his work. For surveys, see Sarton, G., Introduction to the History of Science, 3 vols in 5, Washington, 1927–1947, ii, pp. 1024–1027Google Scholar; and Rosen, E., ‘The Invention of Eyeglasses’, Journal of the History of Medicine, (1956), 11, pp. 13–46, 183–218Google ScholarPubMed. Conceivably, then, the term monoculus in the heading of the Digby copy of AIIIa could denote an eyepiece worn by a vision-impaired master.

28 On the life and work of Roger Bacon, I have consulted principally the following: Lindberg, D., Roger Bacon's Philosophy of Nature, Oxford, 1983, pp. xv–xxviGoogle Scholar; Crombie, A. C. and North, J. D., ‘Bacon, Roger’, D.S.B., vol. i, (1970), pp. 377–385Google Scholar; Easton, S. C., Roger Bacon and his Search for a Universal Science, Oxford, 1952Google Scholar; and Little, A. G. (ed.), Roger Bacon: Essays, Oxford, 1914.Google Scholar

29 Opus Tertium (ed. Brewer, J. S.), Rolls Series, xv, London, 1859, pp. 34–35Google Scholar; for Brewer's translation see p. lxxv. I have preserved the spelling of the names in the Latin text.

30 For background, see Grant, E., ‘Peter Peregrinus’, D.S.B., vol. x (1974), pp. 532–540Google Scholar; Toomer, G. J., ‘Campanus of Novara’, D.S.B., vol. iii (1971), pp. 23–29Google Scholar; Benjamin, F. S. Jr, ‘Campanus of Novara’, in Benjamin, and Toomer, , Campanus of Novara and Medieval Planetary Theory, Madison, 1971, pp. 3–24.Google Scholar

31 Cf. Bémont, C., Simon de Montfort (trans by Jacob, E. F.), Oxford, 1930, p. 283Google Scholar; Kingsford, C. L., ‘Montfort, Almeric of (d. 1292?)’, Dictionary of National Biography, vol. xxxviii (1894), p. 282Google Scholar. The identification of Roger's ‘Almaricus’ as de Montfort's son is made by Brewer, , op. cit. (29), p. lxxvGoogle Scholar, as also by Crombie, and North, , op. cit. (28), p. 381Google Scholar. But others have identified this figure instead as the older brother of Simon, Amaury de Montfort, count of the French estate of the family at Montfort-l'Amaury; cf. Bémont, , op. cit., p. 2nGoogle Scholar, and Prothero, G. W., Life of Simon de Montfort, London, 1877, p. 39Google Scholar. Both Bémont and Prothero rely on a remark by D. E. D. Smith, who, however, identifies the tutee only as Amaury de Montfort, without specifying which Amaury; cf. ‘The Place of Roger Bacon in the History of Mathematics’, in Little, , op. cit. (28), p. 159Google Scholar. But the older Amaury was lord of the estate from 1218 until his death in 1241, whence any tutor of his would have been active very early in the century (say, c. 1210), far too early to be relevant to Roger's remark in Opus Tertium. Moreover, no Amaury, other than these two, figures in the history of the de Montfort family in this period; see the Dictionnaire historique de la France (ed. Lalanne, L.), 2nd edn, 1877 (reprinted New York, 1968), vol. ii, p. 1306Google Scholar. Thus, Bacon must indeed intend by ‘Almaricus’ the son of de Montfort.

32 I would have assumed that Amaury, namesake of his uncle, count Amaury, would be Simon's first son born after the count's death in 1241. The biographers propose a somewhat later date, but the difference affects our estimates by only a few years.

33 On the dating of Campanus' Euclid to the period 1255–1261, see Toomer, , op. cit. (30), pp. 23–24Google Scholar, and Benjamin, , op. cit. (30), pp. 4–5, 12–13Google Scholar. Roger quotes a passage from Campanus' edition in extenso (though without explicit attribution) in his Communia mathematica, op. cit. (27), pp. 125–129Google Scholar; see also note 51 below.

34 For Roger's citations of these works, see note 40. In his Compendium studii philosophie, Roger describes refractory students, ‘ut ait Archimenides, sapere non audent, sed laborant ut nesciant, et a palude [!] vultum suum gratis detegente avertunt oculos imprudenter’; ed. Brewer, , Rolls Series, xv (cf. op. cit. 29), p. 417Google Scholar. The passage is being taken not from Archimedes as such, however, but from De curvis: that rebellious students are not driven to learning, ‘cum sapere non audeant, cum etiam laborent ut nesciant, cum et a Pallade ultro vultus detegente oculos avenant’ (ed. Clagett, , Proposition 1, lines 18–20; op. cit. (2), vol. i, p. 452Google Scholar). The paraphrase is rather close, save that Roger (or a careless scribe) has transformed Athena (Pallas) into, a swamp (palus)!. On this passage of De curvis, see also Knorr, , op. cit. (6), p. 6.Google Scholar

35 For summaries of data, see Emden, , op. cit. (8), ii, p. 1157Google Scholar; Russell, D. S., Dictionary of Writers of 13th Century England, London, 1936, pp. 68–69Google Scholar; Kingsford, C. L., ‘John (fl. 1267), called of London’, Dictionary of National Biography, vol. xxix (1892), p. 448 (cf. p. 449)Google Scholar; Sarton, G., op. cit. (27), ii (ii), p. 582Google Scholar; Paetow, L. J., Morale scolarium of John of Garland, Berkeley (California University Memoirs, 4), 1927, pp. 83–85 (especially 84n)Google Scholar; Powicke, F. M. and Emden, A. B. (eds), The Universities of Europe in the Middle Ages (new edition of the work by H. Rashdall), 3 vols, Oxford, 1936, vol. ii, p. 164n, and iii, p. 50nGoogle Scholar (noting and criticizing the discussion by Paetow); Thorndike, L., A History of Magic and Experimental Science, 8 vols, New York, 1923–1958, vol. ii, pp. 95–96Google Scholar. Recognizing that Rogery's assistant John could not be the ‘master John of London’ praised by him in Opus Tertium, M. R. James has made the interesting proposal that a certain other John of London, known as a major donor of books to St Augustine's Abbey in Canterbury in the fourteenth century, might in fact be Roger's young friend; see his Ancient Libraries of Canterbury and Dover, Cambridge, 1903, pp. lxxiv–viiGoogle Scholar. (Interestingly, our Oxford ms Digby 174, containing ‘Adelard III’, is one of the books in this donation; ibid., pp. 302, 519.) The suggestion has since been rejected on chronological grounds, but I think a view along the lines of James' position is defensible. I intend a separate discussion.

36 See Clagett, , op. cit. (1), p. 23nGoogle Scholar, identifying four passages of Communia mathematica that transcribe portions of the preface to ‘Adelard III’; two of these are explicitly attributed to ‘Alardus Batoniensis in sua editione speciali’: [1] a definition of conceptiones (ed. Steele, , op. cit. (27), p. 65.34–66.1Google Scholar; cf. Clagett, , ‘King Alfred’, op. cit. (24), p. 275.105–107Google Scholar); and [2] a long passage on the units of practical measurement (Steele, 78.10–79.10; cf. ‘King Alfred’, 274.52–72); another [3] merely cites ‘Alardus Batoniensis’ on the grammatical difference of propositiones and proposita (Steele, 69.34–35, 70.3–5; cf. ‘King Alfred’, 275.100–102). Clagett notes further, [4] Roger's remark to the effect that Adelard links the terms proportionalitas and mediatas, ‘non in Commento sed in editione speciali’ (Steele, 83.25–29), relates to a passage of ‘Adelard III’ (Book V, Definition 4), not found in the other versions; one may observe that later in the same section, when Roger continues, ‘sed Euclides in quinto Elementorum non prosecutus est conclusiones arismetice et musice medietatis, sed tantum geometrice, ut dicit Alardus’ (Steele, 84.8–11), he seems to have in mind the same remark from ‘Adelard III’: ‘nota, cum in arismetica x medietates distinguantur, hic tamen una illarum diffinitur, unde et ibi geometrica appelatur’ (ms Balliol 257, fol. 22r). Several other passages of like kind can be cited. For instance, [5] a comment on the terms for axiom, ‘hee dicuntur concepciones vel dignitates vel maxime proposiciones vel anxionata [!] secundum Alardum’ (Steele, 65.30–32), seems to paraphrase the remark ‘hec autem distinguuntur per anxiomata, petitiones, et conceptiones; anxioma dignitas interpretatur’ (‘King Alfred’, 275.103–104); [6] a citation of ‘Alardus Batonensis in editione speciali’, on unequal ratios: ‘nam sexqualtera major est quam sesquitercia’ (Steele, 91.30–32), and continuing later, ‘nam dicit quod universaliter major est proporcio quando dux pluries continet comitem, vel plus de comite’, can be associated with a remark to V, Definition 8 of ‘Adelard III’ (Balliol, fol. 22v): ‘unde et maior est proportio AB, s. sexqualtera, quam CD, s. sexquitertia,… generaliter ubi dux pluries continet comitem, vel plus de comite, maior est proportio’; [7] the definition of proportion according to the ‘intencionem, Alardi Batoniensis in edicione speciali’ (Steele, 86.5–6) agrees verbatim with V, Definition 5 of ‘Adelard III’ (Balliol, fol. 22r), although this happens also to agree with ‘Adelard I’ (cf. Busard, edn, p. 145Google Scholar) and ‘Adelard III’ (ms Gonville, and Caius, 504, fol. 39rGoogle Scholar). Other passages, where Adelard is not explicitly named, include [8] a remark on the etymology of ‘geometria’ (Steele, 39.11–13; cf. ‘King Alfred’, 274.24–26); [9] a passage on plane and solid units (Steele, 107.11–17; cf. ‘King Alfred’, 275.73–76); [10] the statement ofan axiom, ‘quicquid est equale majori, majus est equali minori’ (Steele, 114.20–21); cf. ‘ut quodlibet maius equali maiori, maius est equali minori’ (‘King Alfred’, 275.108–109); [11] the statement and proof of a theorem, ‘quelibet media proporcionalia’ (Steele, 129.35–130.5), in verbatim agreement with part of the text appended to the Paris copy of ‘Adelard III’ (see remarks in Section II above); [12] the statements of selected propositions from Books II, VI, X, VII and VIII (Steele, , pp. 140–143Google Scholar), which all conform to ‘Adelard III’, even where the wording is different from the other Adelardian versions. Certain other passages of Communia mathematica appear to be reminiscences of ‘Adelard III’, but their precise identification has not yet been established.

37 See note 33 above.

38 See note 26 above. All Roger's explicit citations of Adelard have been covered in the survey in note 36 above. Clagett has taken these explicit attributions as evidence that Adelard himself produced ‘Version IIIA’. This view is now questioned, however; cf. Folkerts, , op. cit. (4), p. 65Google Scholar, maintaining that Version IIIA is ‘an anonymous commentary on the second’; and Folkerts, , op. cit. (5), p. 15Google Scholar. Note that Roger sometimes interchanges Adelard and Euclid (e.g. ‘intentio Alardi Batoniensis in edicione speciali. Dicit enim Euclides…’, ed. Steele, 86.5–6). Similarly, when Roger elsewhere expressly quotes Archimedes (‘as Archimedes says…’), the lines are actually taken directly from Proposition 1 of De curvis; see the passage cited in note 34 above. It is thus clear that Roger can sometimes speak loosely when he makes attributions of this sort. Just as he can cite CS) as if by Archimedes, when he knows it is a commentum of John (for the name is in the colophon), so also, presumably, he could substitute the names of the prior sources, Euclid or Adelard, even when he knew he was working from the Euclid commentum of John.

39 It is noteworthy that in the recopied sequence of propositions of ‘Adelard III’, II Definition 1 to V 2 (Digby 174, fols 146r–153v), many of the proofs are labelled ‘commentum’. On the sense of this term, with particular reference to the style of ‘Adelard II’, see Murdoch, , op. cit. (5), pp. 70–72.Google Scholar

40 See, for instance, Communia mathematica (ed. Steele, ), op. cit. (27), p. 44Google Scholar; cf. also Opus Maius (ed. Bridges, ), op. cit. (27), i, pp. 154–155Google Scholar (on isoperimetric figures). For the implicit citation of De curvis, see note 34 above.

41 See Clagett, , op. cit. (1), pp. 29–30Google Scholar; Murdoch, , op. cit. (5), pp. 73–74Google Scholar; and Folkerts, , op. cit. (4), p. 59.Google Scholar

42 In my analysis of the medieval adaptions of De quadratura, for instance, I have found that the Florence Version is based on Gerard of Cremona's translation, but all other versions derive, either directly or indirectly, from the Florence Version; see Textual Studies, op. cit. (10), Part III, Chapter IX.Google Scholar

43 Etudes d'histoire des sciences et de la philosophie du Moyen Age, Warsaw, 1970, pp. 117–210 (especially p. 121)Google Scholar; cited by Clagett, , op. cit. (2), vol. v, p. 6.Google Scholar

44 Birkenmajer, , op. cit. (43), pp. 162–163.Google Scholar

45 The work is cited (but miswritten as De eternis superficiebus) in Robert's Commentary on Aristotle's Physics (ed. Dales, R. C., Boulder, Colo., 1963, p. 128)Google Scholar; the work appears to derive from Robert's lectures 1228'1232 (ibid., pp. xvii–xviii). For a citation of the passage, see also Clagert, , op. cit. (2), vol. iii, pp. 1253–1254.Google Scholar

46 Communia mathematica (ed. Steele, ), op. cit. (27), pp. 117.34–118.5.Google Scholar

47 Cf. Steele, ibid., p. vii; Crombie, and North, , op. cit. (28), p. 378Google Scholar; Lindberg, , op. cit. (28), p. xxv.Google Scholar

48 Cf. Emden, , op. cit. (8), i, p. 103Google Scholar. Citing this very passage, Smith assumes that ‘Bandoun’ is ‘London’ and claims that the line is ‘difficult to read’ in the manuscript; cf. Smith, , op. cit. (31)Google Scholar, in Little, , op. cit. (28), p. 164nGoogle Scholar. He could be right; but Steele indicates no such uncertainty in his edition.

49 Kurath, H. and Kuhn, S. M., Middle English Dictionary, Ann Arbor, MI, 1956, vol. i, pp. 630–631Google Scholar (cf. also ‘abandoun’, ibid., p. 7). In later usage, the word takes on senses as in the modern phrase, ‘with abandon’, that is, with freedom of action to the point of recklessness.

50 I have considered alternatives, based on Old French bandel (fem., ‘headband’), bandel (mase., ‘scar’), or medieval Latin bandum, bandonum, etc. (‘headband’; ‘banner’). These all have a certain suggestiveness, in view of our one-eyed mathematician, but none commands any strong credibility.

51 In the Communia mathematica, a similar observation—that the ‘studiosi in mathematica et qui floruerunt in ea’ require around thirty or forty years—is restated (ed. Steele, , p. 121Google Scholar). Roger then proposes that the theory of proportions would be easy to transmit, if treated via specific numerical examples. As a specimen of a notoriously ‘cruel and horrible’ manner of proof, he cites in extenso demonstrations of V 1, 3, 5, 8, 11 (ibid., pp. 125–129), where the text runs in agreement with Campanus' edition; cf. note 33 above. Note that Steele incorrectly inverts the order of textual dependence, hence inferring Campanus' dependence on Roger (ibid., pp. x–xi).

52 Ed. Brewer, , op. cit. (29), p. 139.Google Scholar

53 Communia mathematica (ed. Steele, ), op. cit. (27), p. 44.Google Scholar

54 See notes 8 and 35 above.

55 For the text of the star table, see Kunitzsch, , Typen von Sternverzeichnissen in astronomischen Handschriften des zehnten bis vierzehnten Jahrjunderts, Wiesbaden, 1966, pp. 39–46Google Scholar. For discussion of issues bearing on the catalogue and its editor, see the same author's ‘John of London’, Journal for the History of Astronomy, (1986), 17, pp. 51–57.Google Scholar

56 The letter has been edited by Fontes, , ‘Le manuscrit de Jean de Londres’, Bulletin de l'Académie des sciences, inscriptions et belles-lettres de Toulouse, (1897–1898), 1, pp. 146–160.Google Scholar

57 See Fontès, ibid., pp. 158, 159n; his account of the issue is in ‘Deux mathématiciens peu connus du xiiie siècle’, Mémoires de l'Académie des sciences, inscriptions et belles-lettres de Toulouse, (1897), IX, 9, pp. 382–386Google Scholar. One may note that Fontès assumes this John of London also to be Roger's young Franciscan associate (see note 35 above). He seems not to have noticed that the young John, being no older than age twenty at the time of Roger's writing in 1267, would barely have been born at the time of the composition of the letter in 1246.

58 The astrological letter of John of London does not betray any of the striking features of the pedagogical style I have delineated in the geometric works. This does not render the identification impossible, however, since the same aspects of style are also absent from the preface to ‘Adelard III’. For the latter, the form of writing is within the genre of ‘accessus’: introductions providing students with useful information preliminary to the study of a given text; see, for instance, Evans, G. R., ‘Introductions to Boethius' “Arithmetica” of the Tenth to the Fourteenth Century’, History of Science, (1978), 16, pp. 22–41CrossRefGoogle Scholar. Thus, John appears to write in a manner appropriate to the context. The pedagogical flourishes that so brighten his editions would be out of place in a scholarly letter to his own mentor.

59 ‘Tabula stellarum fixarum que in astrolabio poni soient verificata per instrumentum considerationis anno domini .1246. deinde post annos .4. examinata ad concordiam instrumenti quod fecit Rogerus Linconus secundum doctrinam magistri Jo. de London’ famosi astronomi cuius nempe R. fuit discipulus', in Kunitzsch, , Typen, op. cit. (55), p. 44Google Scholar. The gloss thus assigns the instrument (an astrolabe) to Roger, its design to John, but is unclear as to which of them made the later astronomical observations. It seems to me more likely that they were made by the author of the table (that is, John, although he is not named as such in the colophons of the table), with the assistance of his disciple Roger.

60 Three manuscripts of ‘Adelard II’ have the annotation ‘Lincol' Zeob' Rog' Hel'’ by a long note to the end of Book X; cf. Folkerts, , op. cit. (4), p. 64Google Scholar. M. Gibson suggests filling out this cryptic abbreviation as ‘Lincol(niensis), Zeob', Rog(erius), Hel(iensis)’; that is, ‘Zeob' of Lincoln, Roger of Ely’; ‘Adelard of Bath’, in Burnett, (ed.), op. cit. (4), p. 15Google Scholar. I offer another conjecture: ‘Lincol(nienses:) Zech(arias) [or possibly, Jacob(us)], Rog(erus), Heli(as)’, that is, denoting three Lincolnian annotators. The confusions of ‘e’ with ‘c’ or ‘o’, and ‘h’ with ‘b’ are orthographically easy in many medieval Latin scripts. Moreover, we know from the star table that there did indeed exist a technical adept, ‘Rogerus Linconus’. It seems to me that this Roger may also be the man who became Robert Grosseteste's steward around 1251; see Major, K., ‘The Familia of Robert Grosseteste’, in Callus, D. A. (ed.), Robert Grosseteste, Scholar and Bishop, Oxford, 1955, p. 237.Google Scholar

61 Clagett has edited the ‘Cambridge Version’ (principal copy in Cambridge ms, Gonville and Caius 504, thirteenth century, fols 108v–109v) in op. cit. (2), vol. i, pp. 63–79Google Scholar, but he has edited only a portion of the paraphrase version of De curvis, namely its three supplementary propositions (ibid., pp. 530–547). As for the corresponding version of De ysoperimetris, it is listed by Busard, , op. cit. (9), p. 65Google Scholar, but has not been edited, save for specimens that I provide in the Appendix to my ‘Paraphrase Editions’, op. cit. (10). The prospect of providing complete editions of these versions of De curvis and De ysoperimetris is daunting, however, in view of the abundance of scribal errors in the Florence codex (bibl. naz. conv. soppr. J.V. 18) which preserves unique witness to them, whence Clagett himself chose not to attempt a full edition of this alternative version of De curvis (cf. Clagett, , op. cit. (2), vol. i, pp. 65, 530).Google Scholar

62 For details of the comparative stylistic analysis of these works, see my Textual Studies, op. cit. (10), pp. 628–631Google Scholar, and ‘Paraphrase Editions’, op. cit. (10), Section III.

63 For details of the source analyses of these works, displaying their dependence on the versions of John, see my Textual Studies, op. cit. (10), pp. 625–628Google Scholar, and ‘Paraphrase Editions’, op. cit. (10), Section I. Use of the term adversarius in place of falsigraphus may be influenced by the Euclid version ‘Adelard II’, where the term adversarius (but not falsigraphus) appears.

64 For comments and textual specimens, see Clagett, , op. cit. (1), pp. 25, 36–38Google Scholar. The unique extant witness of this version is in the Oxford Bodleian, Savile 19, fols 1r–37r. This copy is incomplete, since the text abruptly switches to the version of Campanus at the end of the definitions in Book VI; cf. Folkerts, , op. cit. (5), pp. 36, 41.Google Scholar

65 Clagett does not expressly state the direct dependence of ‘IIIB’ on ‘IIIA’, however. Although he includes it within his section on ‘III’, he advises that it ‘needs further study before its position is clarified’ (op. cit. 1, p. 25). Folkerts' list of mss of ‘Adelard III’ includes the Savile ms among them (see preceding note), with the notation ‘the so-called Version IIIB’ (op. cit. 5, p. 36).

66 But the situation may be more complex than this suggests. In the beginning ot book 1, ‘IIIB’ indeed reveals awareness of special readings from ‘IIIA’ (as the specimens given by Clagett show); but elsewhere the relation is less clear, as the editor appears to have incorporated readings from other versions into his text; for instance, his term adversarius most probably indicates a connection with version ‘II’ (see note 63 above). I intend to examine this question more systematically in a separate study.

67 See Clagett, , op. cit. (2), vol. v, pp. 3–11Google Scholar for a synopsis of what is known of the life and work of Gerard and the history and significance of the De motu.

68 For the critical edition of the text, with commentary, see Clagett, , op. cit. (2) vol. i, pp. 80–81Google Scholar. Clagett associates the ‘Naples Version’ with Gerard, ibid., pp. 80–81.

69 I set out the stylistic similarities between Gerard's De motu and the ‘Naples Version’ in my Textual Studies, op. cit. (10), pp. 636–637.Google Scholar

70 See De motu, Part II, Proposition 3, line 66 (Clagett, , op. cit. (2), vol. v, p. 95).Google Scholar

71 Ibid. vol. v, p. 39.

72 Ibid., Part II, Proposition 3, line 103, op. cit. (2), vol. v, p. 97; cf. also p. 41. A similar citation occurs at Part I, Proposition 2, line 11: ‘per primam de piramidibus’ (ibid., p. 76). Two marginal notes also cite Proposition 4: ‘in fine commenti super iiiiam de piramidibus’ (ibid., p. 66n) and ‘istud probat quarta de piramidibus’ (ibid., p. 76n), in agreement with De curvis. Presumably, these were written by a later scribe who knew the cited work in its adaptation by Gerard.

73 Ibid., p. 25 (text, pp. 76–77); cf. a similar lemma in Proposition 1, ibid., p. 21 (text, pp. 64–66) on the difference of circles. It seems odd that Gerard proves in full the lemma to Proposition 2 (on the difference of the surfaces of two similar cones), when it appears in De curvis Proposition 4, and could not have been dispensed with in his own adaptation De piramidibus. But his proof of the lemma is markedly abridged, in comparison with the form in De curvis, and he replaces the closing section of the proof in De curvis (Proposition 4, lines 30–37) with an appeal to his own lemma on circles in Proposition 1 (cf. De motu, Part I, Proposition 2, lines 24–25).

74 For detailed descriptions of all six manuscripts employed by Clagett for his edition of De motu, see op. cit. (2), vol. v, pp. 60–62. Four of these also hold copies of De curvis: Oxford (Bodl.) Auct. F.5.28; Berlin (Staatsbibl., Preuss. Kulturbesitz) lat. Q. 510; Naples (Bibl. naz.) lat. VIII.C.22; Vienna (Nationalbibl.) lat. 5303. In two copies of De curvis the name ‘Gervasius de Essexta’ (or ‘de Assassia’) appears in place of Johannes de Tinemue in the colophon (ibid., i, p. 449). I think this can be explained through a scribal error, influenced by the proximity of Gerard's name; in the Naples codex De motu appears at fols 60v–65v, immediately following De curvis (fols 57r–60r), so that the name ‘Gerardi de Brussel’ in the heading of De motu occurs immediately after ‘Gervasii de Essexta’ in the colophon of De curvis. Clagett has suggested a possible connection with one John Gervais of Exeter, bishop of Winchester, who died at Viterbo in 1268 (see the article by Kingsford, C. L. in Dictionary of National Biography, vol. xxix [1892], pp. 448–449Google Scholar). But there are at best only a few scribal differences between the Johannine and Gervasian forms of De curvis, and no reason at all to assign mathematical interests to this bishop. Moreover, to get the name ‘Gervasius de Essexta’ from ‘Johannes Gervasius de Exonia’ requires no smaller scribal error than to get it from ‘Gerardus de Brussel’.

75 Birkenmajer, , op. cit. (43), p. 166Google Scholar. The codex of entry No. 43 is extant in Edinburgh, Crawford Library of the Royal Observatory, Cr. 1.27; cf. Clagett, , op. cit. (2), vol. v, pp. 61–62.Google Scholar

76 None of the seven or so different versions of De ysoperimetris that are listed by Busard, , op. cit. (9), pp. 62, 65Google Scholar, and analysed by me in ‘Paraphrase Editions’, op. cit. (10), conforms to Gerard's style.

77 Clagett notes these entries, and draws the inference on dating for De motu (op. cit. (2), vol. v, p. 6).Google Scholar

78 For the source analysis of the ‘Naples’ and ‘Florence Versions’, see my Textual Studies, op. cit. (10), pp. 631–635.Google Scholar

79 Victor presents a critical edition, with translation and commentary, in his Practical Geometry in the High Middle Ages, Philadelphia Memoirs of the American Philosophical Society, 134, 1979.Google Scholar

80 For details of the stylistic analysis of Artis in comparison with Gerard's De motu, see my Textual Studies, op. cit. (10), pp. 672–675.Google Scholar

81 Literally so: the author describes the study of theory without practice as like ‘picking a spring flower without waiting for its fruit’; cf. Victor, , op. cit. (79), pp. 108–109.Google Scholar

82 Ibid., pp. 108–109; Victor points out the wordplay on ‘taste’ (sapor) and ‘wise’ (sapientes).

83 Proposition 1, lines 18–22; Clagett, , op. cit. (2), vol. i, pp. 452–453Google Scholar. The same passage is also cited by Roger Bacon; cf. note 34 above. I note its classical allusions in my op. cit. (6), p. 33. By ‘philosophus’ I suppose the author means Archimedes; for the heading identifies him as author of De curvis, but John as author of its commentum (cf. Clagett, , op. cit. (2), vol. i, p. 450).Google Scholar

84 Victor, , op. cit. (79), pp. 108–111Google Scholar; for emphasis, I have changed his translation of rimari from ‘discover’ to ‘delve into’.

85 Although I have long known this passage of De curvis (cf. note 83 above), it never dawned on me that it alluded to the quadrivium until I encountered the parallel in Artis. Neither does Clagett, in his edition of De curvis, make any comment on the possible nuance; cf. Clagett, , op. cit. (2), vol. i, p. 507.Google Scholar

86 Victor notes the quadrivial interest in the passage of Artis (op. cit. 79, p. 111n) and discusses in detail medieval views on the place of practical geometry within the university curriculum (ibid., pp. 31–42). He also surveys views on the relation of theoretical and practical geometry (ibid., pp. 42–53), e.g. as in the prologue of the Adelardian Euclid, ‘Version IIIA’. This last is of particular interest for my purposes, since I have set the author of Artis in specific dependence on the author of ‘Adelard IIIA’. In the preface to ‘Adelard IIIA’, geometric practice is firmly subordinated to theory, whereas Artis takes practice to be the fruition of theory. The views are complementary, of course, but differ in emphasis. Yet we would hardly insist that the author of Artis, even if a disciple of John, could not manifest an opinion of his own on such an issue.

87 Describing the rigours of mathematical study, Boethius asks, ‘whether the speed of a trained mind can take in the flights of subtle things’ (‘an rerum subtilium fugas exercitatae mentis velocitas conprehendat’); cf. Friedlein, G. (ed.), Boetii De institutione arithmetica, Leipzig, 1867, p. 4, lines 21–22.Google Scholar

88 In the preface of Artis the puzzling term pape appears. Victor follows a precedent from classical Latin where the interjection papae (‘excellent’) is known as a transliteration from Greek. But if the work is in the form of a dedication, as we have seen in the preceding, one might take the term as a vocative form, referring to the addressee. In medieval Latin (with precedents from Greek and classical Latin), the word papas (sometimes declined as if papa) is an affectionate term for ‘teacher’. (Cf. Niermeyer, J.F., Mediae latinitatis lexicon minus, Leiden, 1976, p. 758Google Scholar, s. v. papa and papas; and Du Cange, , Glossarium mediae et infimae latinitatis, Paris, 1840, vol. v, pp. 66–67.Google Scholar) Strictly speaking, the vocative should be papas (or possibly, papa); I would suppose the form pape has been influenced by the usual masculine form, or may simply be a minor grammatical error. According to Victor, (op. cit. 79, p. 105)Google Scholar, L. Thorndike, taking pape in its more familiar sense to denote the pope or other high prelate, had identified the author of Artis as Campanus, addressing the work to his patron, the Pope. Victor rightly rejects this interpretation of pape, as being entirely too disrespectful a form of address in such a context.

89 Clagett, , op. cit. (2), vol. iii, p. 211Google Scholar; Victor, , op. cit. (79), p. 25Google Scholar; cf. Victor's text, ibid., pp. 232, 234, 238, 240, 258.

90 Victor, , op. cit. (79), pp. 24, 282Google Scholar (cf. 283n); Clagett, , op. cit. (2), vol. iii, p. 211.Google Scholar

91 Victor, , op. cit. (79), pp. 282–284Google Scholar; I have modified his translation to be in more literal agreement with the text. See also Clagett, , op. cit (2), vol. iii, p. 211.Google Scholar

92 Victor notes the good agreement with figures from Ibn Ezra, 1154 (op. cit. 79, p. 285n). Both sets entail a shift of about ten degrees from those of the Toledan Tables, which take the Hegira as epoch (A.D. 622); cf. Toomer, , ‘A survey of the Toledan Tables’, Osiris, (1968), 15, pp. 44–45.CrossRefGoogle Scholar See also note 102 below.

93 I suspect the value ‘15 degrees of Gemini’ has suffered scribal error, for in Almagest III, 7 Ptolemy finds the solar apogee to lie at 5° 30′ of Gemini; see Toomer, , Ptolemy's Almagest, New York, 1984, pp. 166–169Google Scholar; Pedersen, O., A Survey of the Almagest, Odense, 1974, pp. 146–147.Google Scholar

94 Victor, , op. cit. (79), pp. 286–288.Google Scholar The discussion here concerns the different values proposed for the length of the year and the times and positions of the equinoxes.

95 See Vernet, J., ‘al-Zarqâlî’, D.S.B. xiv (1976), pp. 592–595.Google Scholar

96 For citation, see Duhem, P., Le système du monde, 10 vols, Paris, 1913–1959, vol. iii, p. 234Google Scholar; cf. also Wright, J.K., ‘Notes on the knowledge of latitudes and longitudes in the middle ages’, Isis, (1923) 5, p. 85Google Scholar, and Vallicrosa, J.M. Millás, Estudios sobre Azarquiel, Madrid–Granada, 1943–1950, p. 373.Google Scholar These Paris Tables are not extant, and are undatable from the London reference. For notes on comparable tables, for Marseilles (c. 1140), Hereford (1178), and others, see Wright, (op. cit., p. 84)Google Scholar, Millás Vallicrosa (vide supra, Chapter VII), and Sarton, G., op. cit. (27), vol. ii, pp. 210, 621.Google Scholar

97 The Toulouse Tables are extant in Paris ms lat. 16658, according to Wright, (op. cit. 96, p. 84n)Google Scholar and Vallicrosa, Millás (op. cit. 96, p. 394n)Google Scholar, but neither indicates a date.

98 Actually, Paris is about two degrees east; but medieval manuscripts displace it considerably to the west, so that their longitudes become about the same; indeed, Toulouse ends up slightly more eastward; cf. Wright, , op. cit. (96), pp. 91–95.Google Scholar

99 See the articles by Poulle, E. on ‘William of Saint-Cloud’, D.S.B., vol. xiv (1977), pp. 389–391Google Scholar and on ‘John of Lignères’, D.S.B., vol. vii, (1973), pp. 122–128.Google Scholar

100 On the medieval values, see Dreyer, J.L.E., A History of Astronomy from Thales to Kepler, 2nd edn, Cambridge, 1906 (reprinted, New York, 1953), pp. 276–277.Google Scholar The principal values (as of Ibn Ezra) are also cited by Victor, , op. cit. (79), p. 285nGoogle Scholar (cf. note 92 above).

101 On Thâbit's trepidation theory, see Mercier, R., ‘Studies in the Medieval Conception of Precession’ (Part I), Archives internationales d'histoire des sciences, (1976), 26, pp. 209–220.Google Scholar On its place within the Toledan Tables, see ibid., pp. 201–206, and Toomer, , op. cit. (92), pp. 118–122.Google Scholar For a brief account of ancient and medieval precession theories, see Swerdlow, N., ‘The Commentariolus of Copernicus’, Proceedings of the American Philosophical Society, (1973), 117, pp. 445–450.Google Scholar

102 Compared with the Toledan Tables, the apogees in Artis have advanced by at least 8° 28′ (for Jupiter) and by no more than 10° 44′ (for Saturn), where the solar apogee has advanced 9° 50′. (For the Toledan figures, see Toomer, op. cit. (92), pp. 44–45.) Assigning the discrepancy to precession alone, we derive values of between 53″ 23 and 67″ 40 per annum (and 62″ 0 for the sun), since the epoch of the Tables is A.D. 622 (that is, 571 years before the date stated in Artis, 1193). It is clear, then, that the apogees in Artis, if based on the Toledan Tables, are not compatible with the stated annual precession of 51″, for the difference would then only have amounted to 8° 5′. The discrepancy might be accountable via suitable assumptions on the motion of the apogee; for instance, al-Battânî and other Islamic astronomers did propose, in disagreement with Ptolemy, that the solar apogee shifted in the ecliptic; cf. Dreyer, , op. cit. (100), pp. 250–251Google Scholar, and Pederson, , op. cit. (93), p. 147n.Google Scholar

103 See Fontès' edition of the letter, op. cit. (56), pp. 148–149. The critique of Thâbit's theory reads as if the author is merely quoting directly from Albategni. But the discrepancy of sixteen degrees that, he says, ‘I have frequently observed’ (quod ego frequenter consideravi) is in full agreement with the co-ordinates in the star catalogue associated with the letter (cf. Kunitzsch, , Typen von Sternverzeichnissen, op. cit. 55, pp. 42–43)Google Scholar; e.g. No. 18 ‘edubh .i. ursa’ with a longitude of four degrees in Leo corresponds to Ptolemy's ursa major No. 16, with longitude Cancer seventeen and two-thirds degrees; cf. Toomer, , op. cit. (93), p. 342Google Scholar (cf. the figures for Regulus, scilicet ‘calbelezed .i. cor leonis’, No. 21, in note 104).

104 Taking the author's stated nineteen-degree deviation between his own observations (made in A.D. 1246) and Hipparchus' data, and setting limits of 140 B.C. and 125 B.C. for Hipparchus, Fontès derives the implied values of between 49″ 36 and 49″ 40 per annum for the precession (op. cit. 56, p. 156n). On the other hand, the observations Ptolemy reports as his own lie within the interval A.D. 125 to A.D.141 (cf. Pedersen, , op. cit. 93, p. 12)Google Scholar, as John could infer from his study of the Almagest, so that the 16° difference he states would imply a precession of between 51″ 23 and 52″ 8 per annum. If, alternatively, we adopt the figures from Almagest VI, 2, actually cited by John, Ptolemy observes Regulus at 2° 30′ Leo in A.D. 139, whereas John finds it tobe 18° Leo in A.D. 1246; the difference of 15° 30′ in 1107 years amounts to a precession of 50″ 24 per annum. But John expressly frames the parameter in terms of how many years result in a shift of one degree; these data yield the value 71.42 years, or 71 approximately, in contrast to Ptolemy's 100 and Albategni's 66. In turn, the value of one degree per seventy-one years becomes 50″ 42 per annum, or approximately 51″ per annum, the value stated in Artis. Adjusting these computations to Hipparchus' data yields a significantly lower rate: for Ptolemy reports that 265 years earlier (129/8 B.C.) Hipparchus found the same star 2° 40′ farther to the rear; thus, in comparison with John's data, the difference of 18° 10′ in 1374 years yields a precession rate of 47′ 36 per annum (or one degree in 75.6 years). Thus, the 51″ figure of Artis conforms with what John's letter implies, but only in connection with the Ptolemaic figures. (On Ptolemy's data for the precession, see Toomer, , op. cit. (93), pp. 327–329Google Scholar; Pedersen, , op. cit. (93), pp. 240–245.)Google Scholar

105 For comparison, the modern value (scilicet, A.D. 1900) for the precession is 50″ 25, while the mean value over the interval from the sixth through the nineteenth century is 50″ 6 per annum (cf. Mercier, op. cit. (101), p. 205). Note that the rather impressive agreement of the figure inferred from John's data (50″ 24) results only from comparison with Ptolemy's data. However, as the Ptolemaic stellar longitudes have been exposed as dependent on a faulty precession figure (scilicet, 36″ per annum), one would expect that any alternative derivation made relative to John's figures would produce a quite different value; on Ptolemy's data and an effort to rebut his modern critics, see Pedersen, , op. cit. (93), pp. 253–258Google Scholar, and Evans, J., ‘On the Origin of the Ptolemaic Star Catalogue’, Journal for the History of Astronomy, (1987), 18, pp. 155–172, 233–278.CrossRefGoogle Scholar

106 Note that a confusion of this kind has affected the dating of Robert of Hereford. His Computus was issued in 1176, as the colophon asserts; but an alternative date of 1124, employed in the course of a computation, has sometimes wrongly been cited as the date of composition. See Haskins, C.H., Studies in the History of Mediaeval Science, 2nd edn, Cambridge, Mass, 1927 (reprinted, New York, 1960), pp. 124–125.Google Scholar By a similar argument, one could date the present article to 1900 on the basis of the ‘modern value’ cited in the preceding note.

107 Indirect indicators for the dating of Artis come from the existence of certain derivative versions: Artis is the source for a French translation, Pratike de geometrie, made before 1276, and if it is one of the sources consulted by the editor Johannes Anglicus of the Quadrata vetus, Artis would have to antedate the interval 1277–1284. On the dating and sources of Quadrans vetus, see Hahn, N.L., Medieval Mensuration: Quadrans vetus and Geometrie due sunt partes principales, Philadelphia (Transactions of the American Philosophical Society, 72, Part 8), 1982, pp. xxxv–xxxvi, xxxviii–xli.Google Scholar Paragraphs 69, 70, 79, 81–82 of Quadrans (also cited by Clagett, , op. cit. (2), vol. iii, p. 217n)Google Scholar display strong verbal affinities with the corresponding discussions in Artis, I 22 (cf. I 31), I 21, II 35, III 7, respectively (Hahn, however, appears to doubt the use of Artis by Quadrans p. xxxix). On the date of the Pratike, see Victor, , op. cit. (79), p. 27Google Scholar, reporting the finding of C. Henry (1882) in his examination of the Pratike; Victor includes a critical edition of the Pratike as an appendix to his edition of Artis. The Pratike is rendered into the dialect of Picardy, a commercially vital region in this period, and the native place of such distinguished literary and scientific figures as Petrus de Maricourt and Richard de Fournival (see notes 30 and 43 above), and practitioners like Villard de Honnecourt. Tracing the connections between the formal geometries and the practical geometries in Latin (such as Artis), thence to the French (as with the Pratike), we can discern one of the routes of the vernacularization of learning in this period and the sort of scholars responsible for it. For discussion, see Victor, , op. cit. (79), pp. 68–73.Google Scholar

108 For the passages, see Victor, 's index, s. v. ‘Euclides’, ‘Euclidens’, ‘Ptholomeus’, ‘Almagesti’, op. cit. (79), pp. 615, 617, 620.Google Scholar To be sure, the provision of such citations does not necessitate the author's familiarity with the source treatises, since he could know of them through secondary treatments, e.g. tracts on the astrolabe, tables, and the like.

109 See the text of the letter, as edited by Fontès, op. cit. (56). Kunitzsch describes John of London ‘as a learned and well-read man’ and lists seven of the authorities he cites (‘John of London’, op. cit. (55), pp. 51–52).Google Scholar

110 On the date of Robert's commentary on the Posterior Analytics, see the summary by Rossi, P. in his edition of the work (Florence, 1981), pp. 18–19.Google Scholar Whereas Rossi leaves the issue open, not insisting on any specific date within the 1220s, the view earlier advocated by Crombie, assigning it to the first half of that decade, has recently been supported anew by Southern, R.W., Robert Grosseteste: The Growth of an English Mind in Medieval Europe, Oxford, 1986, pp. 131–133.Google Scholar I intend a separate discussion of the connections between Robert Grosseteste and ‘Adelard III’.

111 On the dates of these two manuscripts, see Folkerts, , op. cit. (5), pp. 15, 36.Google Scholar (There is a discrepancy, however, in that Folkerts writes ‘s. xiii’ for the Digby ms on p. 31.)

112 For Clagett's critique of Macray's dating, see op. cit. (2), vol. i, p. xx. Note that in Clagett's later discussion of the dating of Jordanus de Nemore and his works (ibid., vol. v, pp. 145–146), despite uncertainties, he sees no evidence for dating Jordanus' activity as early as the twelfth century. Similarly, R.B. Thomson, who has produced a critical edition of Jordanus' De spera and compiled an exhaustive inventory of the mss of all Jordanus' known works, is committed to dating him to the early-thirteenth century; cf. ‘Jordanus de Nemore: Opera’, Mediaeval Studies, (1976), 38, p. 97Google Scholar (he assigns a thirteenth-century date to the Digby ms, without noting the discrepancy with Macray, ibid., p. 100). Further, in articulating the nature of Jordanus' scholarly program, this later dating has been central for Høyrup, J., as his very title makes clear: ‘Jordanus de Nemore, 13th Century Mathematical Innovator’, Archive for History of Exact Sciences, (1988), 38, pp. 307–363.CrossRefGoogle Scholar

113 Catalogue of the Manuscripts of Balliol College, Oxford, Oxford, 1963, pp. 278–279Google Scholar: ‘Late 12th century. … One or more skilled English hands of chancery type’.

114 Much resembling the style of the Balliol hand is the specimen from 1206 reproduced by Thompson, E.M., A Handbook of Greek and Latin Palaeography, London, 1901 (reprinted, Chicago, 1980), p. 304Google Scholar; cf. the similar specimen from 1204 (ibid., p. 303). It resembles also the style of a specimen in Johnson, C. and Jenkinson, H., English Court Hand, Oxford, 1915Google Scholar, Part II, Plate xii (b), an official document written in 1225. By these examples, I do not wish to deny that the same stylistic features can be amply documented in twelfthcentury handwriting.

115 Jenkinson observes of scribal practice in the thirteenth century: ‘it has to be remembered that of any two clerks writing at one time similar, or parts of the same, documents one might be 60 years of age, the other 20;… instances are not unknown where two contiguous pages in a Register are written in styles so distinct as to belong in appearance to dates a generation apart’; Palaeography and the Practical Study of Court Hand, Cambridge, 1915, p. 13.Google Scholar

116 Displaying many of the features of the Balliol hand is a charter of Henry II that Jenkinson reproduces (ibid., Plate I). Ostensibly from the twelfth century, and accepted as such by the editors of the Facsimiles of National Manuscripts, nevertheless, this document is a forgery, made in the time of Edward III, two centuries later. Jenkinson observes: ‘the handwriting of the time of Henry II—a period in which anyone would be inclined to allow the science of Palaeography full scope—may be imitated so well as to deceive … even the most skilled palaeographer’ (ibid., p. 16).

117 Among the other documents selected by Jenkinson to illustrate the stylistic changes typifying hands over the period 1150–1250, a few of those described as falling within the twelfth-century styles offer parallels to the Balliol hand (cf. ibid., Plate VI especially, and also Plates III, IV and V); but the entire set was taken from a single roll assembled in the same year, 1225. Jenkinson observes: ‘Court Hand documents … can nearly always be dated with accuracy [sc. through knowledge of Administrative History], but not by their handwriting’ (ibid., p. 37). His account has been noted as ‘a fascinating paper and a salutary warning to researchers’ by another palaeographical expert, Denholm-Young, N., Handwriting in England and Wales, Cardiff, 1954, p. 81.Google Scholar

118 I develop the argument in a separate article, ‘On the Birthplace of Robert Grosseteste’ (available in preprint).

119 The figure ‘Nicholas Grecus’ may be one of these translators; see Russell, , op. cit. (35), p. 89Google Scholar; Major, in Callus, (ed.), op. cit. (60), p. 229Google Scholar; Stevenson, F.S., Robert Grosseteste: Bishop of Lincoln, London, 1899, pp. 224–227.Google Scholar Robert's use of translators is explicitly stated by Bacon, Roger, Compendium studii philosophiaeGoogle Scholar(ed. Brewer, J.S., Rolls Series, xv, no. 3), pp. 434, 472Google Scholar; as also in Opus Tertium, op. cit. (29), p. 91Google Scholar; cf Hirsch, S.A., ‘Roger Bacon and Philology’Google Scholar, in Little, (ed.), op. cit. (28), pp. 101–102.Google Scholar

120 The adaptor, one ‘Joannes’, of the Latin translation of the Posterior Analytics, explains how the obscurity of the extant translation had kept the ‘French masters’ from making due use of Aristotle's work, whence the need for his own retranslation; cf. the preface, quoted by Minio-Paluello, L., Aristoteles Latinus, iv (Analytica Posteriora), Leiden, 1968, p. xliv.Google Scholar Despite that claim, this revised translation is only a light reworking of the version by James of Venice, as a comparison of passages makes evident (ibid., pp. xlvi–xlviii).

121 Cf. note 39 above.

122 Victor suggests to me that such characteristics may reflect the oral delivery of lectures. The role of oral tradition, e.g. in the reportage of lectures, is emphasized also by Høyrup, , op. cit. (112), pp. 349–351.Google Scholar

123 This point depends on what conclusions are finally established about the nature of ‘Adelard II’ and how ‘Adelard III’ is related to it.

124 Compendium studii philosophiae (ed. Brewer, ), op. cit. (119), pp. 471–472Google Scholar (cf. the translation by Brewer, ibid., pp. lix–Ix); the passage is similar to Opus Tertium, ibid., pp. 91–92.

125 See Roger's remark on mathematicians in Opus Tertium, op. cit. (29), p. 34Google Scholar, quoted in Section III above.

126 On Campanus, see Toomer, , op. cit. (30)Google Scholar, and Benjamin, , op. cit. (30)Google Scholar; see note 30 above. For a description of the Viterbo group, with special reference to William of Moerbeke, see Clagett, , op. cit. (2), vol. ii, pp. 3–13.Google Scholar

127 See my Textual Studies and ‘Paraphrase Editions’, op. cit. (10).

128 Dialogue concerning the two chief world systems, ed. Drake, S., Berkeley, Los Angeles and London, 1967, p. 464.Google Scholar