Hostname: page-component-5db58dd55d-jhf8m Total loading time: 0 Render date: 2026-05-28T14:13:51.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Robert Grosseteste on the Subalternate Sciences

Published online by Cambridge University Press:  29 July 2016

W. R. Laird
W. R. Laird*
Affiliation:
Carleton University
Get access

Extract

It is well known that in the Middle Ages mathematics had little part in the study of nature. Natural philosophy, which had in its purview all of nature and natural things, was considered fundamentally distinct from mathematics, both in subject matter and in method. Yet there was a handful of sciences in which mathematics and natural philosophy came together, sciences that were to have a very significant role in later scientific thought. These were what Thomas Aquinas, in the thirteenth century, called the ‘intermediate sciences’ (scientiae mediae), since they were thought of as in some way intermediate between the natural and the mathematical; they included astronomy, optics, harmonics, and sometimes mechanics. They were also known as the ‘subalternate sciences,’ since they were considered under, or subalternate to, pure mathematics, and sometimes to natural philosophy as well.

Information

Type
Articles
Information
Traditio , Volume 43 , 1987 , pp. 147 - 169
Copyright
Copyright © 1987 by Fordham University 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1 Pedersen, O., ‘Du quadrivium à la physique,’ in Artes liberales, von der antiken Bildung zur Wissenschaft des Mittelalters (ed. Koch, Josef; Leiden 1959) 107–23, esp. 120.Google Scholar

2 Gagné, J., ‘Du quadrivum aux scientiae mediae,’ Arts libéraux et philosophie au moyen âge (Montreal 1969) 975–86.Google Scholar

3 Weisheipl, J. A., ‘Classification of the Sciences in Medieval Thought,’ Mediaeval Studies 27 (1965) 54–90; repr. in Weisheipl, J. A., Nature and Motion in the Middle Ages (ed. W. E. Carroll, Studies in Philosophy and the History of Philosophy 11; Washington, D.C. 1985) 203–37; Wallace, W. A., Causality and Scientific Explanation (2 vols.; Ann Arbor 1972–74).Google Scholar

4 Ribeiro do Nascimento, C. A., ‘Le statut épistémologique des “sciences intermédiaires” selon s. Thomas d'Aquin,’ La Science de la nature: théories et pratiques (Cahiers d'Études Médiévales 2; Montreal–Paris 1974) 33–95.Google Scholar

5 Livesey, S. J., ‘Metabasis: The Interrelationship of the Sciences in Antiquity and the Middle Ages’ (diss. Univ. of California, Los Angeles 1982; for Grosseteste see pp. 264–88); ‘William of Ockham, the Subalternate Sciences, and Aristotle's Theory of Metabasis,’ British Journal for the History of Science 18 (1985) 127–45.Google Scholar

6 Crombie, A. C., Robert Grosseteste and the Origins of Experimental Science 1100–1700 (Oxford 1953; repr. 1962), esp. 91–98.Google Scholar

7 Crombie, A. C., ‘Grosseteste's Position in the History of Science,’ in Robert Grosseteste, Scholar and Bishop (ed. Callus, D. A.; Oxford 1955) 98–120, esp. 109–11; Eastwood, B. S., ‘The Geometrical Optics of Robert Grosseteste’ (diss. Univ. of Wisconsin 1964) esp. 10–15; ‘Medieval Empiricism: The Case of Robert Grosseteste's Optics,’ Speculum 43 (1968) 306–21; ‘Robert Grosseteste's Theory of the Rainbow: A Chapter in the History of Non-Experimental Science,’ Archives Internationales d'Histoire des Sciences 19 (1966) 313–32; McEvoy, J., The Philosophy of Robert Grosseteste (Oxford 1982) 206–11; Weisheipl, J. A., ‘Science in the Thirteenth Century,’ in The History of the University of Oxford (ed. Catto, J. I.; Oxford 1984) I 435–69, esp. 440–53.Google Scholar

8 Boethius, , De Trinitate 2 (PL 64.1250–51); see Weisheipl, , ‘Classification of the Sciences’ (above, n. 3) 60–62.Google Scholar

9 Weisheipl, , ‘Classification of the Sciences’ 68–72; see also Eastwood, , Geometrical Optics (above, n. 7) 12–15.Google Scholar

10 Grosseteste, Robert, Commentarius in Posteriorum Analyticorum libros 1.2 (ed. Rossi, P. [Florence 1981], p. 99 lines 9–22).Google Scholar

11 Comm. Post. Anal. 1.17 (ed. Rossi, , pp. 256. 340–257.363); for a translation of this passage, see Crombie, , Robert Grosseteste 129.Google Scholar

12 Comm. Post. Anal. 1.11 (ed. Rossi, , pp. 178.123–181.189).Google Scholar

13 ‘Mathematici magnitudines abstrahunt a motu et a materia et subiciunt magnitudines abstractas et de hiis demonstrant accidencia per se magnitudinibus. 'Physicus vero non demonstrat per se accidencia magnitudinibus [= magnitudinum ?] de magnitudinibus inquantum accidunt simpliciter magnitudinibus, sed de corporibus physicis demonstrat magnitudines figuratas secundum quod accidunt corporibus physicis ex parte ea qua physica sunt’ (Roberti Grosseteste episcopi Lincolniensis Commentarius in viii libros Physicorum Aristotelis [= Comm. Phys.] 2 (ed. Dales, R. C. [Boulder, Colorado 1963], pp. 36–37). Dales paraphrases and quotes extensively from this section of the Commentarius in ‘Robert Grosseteste's Commentarius in octo libros Physicorum Aristotelis,’ Medievalia et Humanistica 11 (1957) 19–20; the translations quoted here are, with slight changes, from Crombie, , Robert Grosseteste (above, n. 6) 94.Google Scholar

14 Comm. Phys. 2 (ed. Dales, , p. 37).Google Scholar

15 Aristotle, , De Caelo 2.14, 297a24; quoted by Heath, T. L., The Thirteen Books of Euclid's Elements (2nd ed. Cambridge 1926; repr. New York 1956) III 269.Google Scholar

16 Comm. Phys. 2 (ed. Dales, , p. 37).Google Scholar

17 ‘Quod itaque est physico predicatum hoc abstractum est pure mathematico subiectum, astrologo vero et physico idem subiectum et predicatum’ ( Comm. Phys. 2; ed. Dales, , p. 37).Google Scholar

18 Comm. Post. Anal. 1.11 (ed. Rossi, [above, n. 10], pp. 174. 38–175.43); cf. Grosseteste, Robert, De sphaera, ed. Baur, L., Die philosophischen Werke des Robert Grosseteste, Bischofs von Lincoln (Beiträge zur Geschichte der Philosophie des Mittelalters 9; Münster 1912), 12–13.Google Scholar

19 ‘Propterea subiectis pure mathematicis superadduntur accidencia naturalia et fit subiectum compositum ex mathematico et naturali, et demonstratur accidens mathematicum de tale subiecto composito secundum quod accidit ei propter accidens naturale quod est in subiecto, utpote ex linea et radiositate componitur linea radiosa et demonstrantur ex ea accidencia et figuraciones linee que accidunt ei ex parte radiositatis, et propter hoc magis physicum quam mathematicum est hoc. Et forte astrologia in quibusdam conclusionibus suis est huic simile’ ( Comm. Phys. 2; ed. Dales, , p. 37; trans. Crombie, , Robert Grosseteste, p. 94 [slightly altered]).Google Scholar

20 ‘Magis physica quam mathematica’ in the translation by James of Venice; the phrase in Aristotle is (Aristotle, , Physics 2.2, 194a7–8). For evidence that Grosseteste used this translation, known as the translatio vetus, see Lacombe, G., Aristoteles Latinus: Codices 1 (Rome 1939; 2nd ed. Bruges–Paris 1957) 51–52; A. Mansion (ed.), Physica: Translatio Vaticana. Aristoteles Latinus 7.2 (Bruges–Paris 1957) viii–ix; Minio-Paluello, L., ‘James of Venice,’ Dictionary of Scientific Biography 7.66.Google Scholar

21 Comm. Post. Anal. 1.18 (ed. Rossi, [above, n. 10], p. 260.31–37).Google Scholar

22 Comm. Post. Anal. 1.7 (ed. Rossi, , p. 137.55–62).Google Scholar

23 ‘Ex XI conclusione proximo ostensa et explanatione eiusdem conclusionis sequitur hec XII conclusio quod omnem demonstrationem necesse est esse ex principiis appropriatis conclusioni’ (Comm. Post. Anal. 1.8; ed. Rossi, , p. 150. 102–104).Google Scholar

24 Comm. Post. Anal. 1.8 (ed. Rossi, , pp. 154.191–155.201); in line 198, I read demonstratur with MSS O, Va, and W rather than Rossi's reading demonstrantur.Google Scholar

25 Comm. Post. Anal. 1.7 (ed. Rossi, , p. 137.51–55); see also 1.10 (pp. 170.1–172.54).Google Scholar

26 Comm. Post. Anal. 1.8 (ed. Rossi, , pp. 155. 202–156.217).Google Scholar

27 Elements 10 deals with incommensurable magnitudes; see Heath, , Euclid's Elements III 1–10.Google Scholar

28 Comm. Post. Anal. 1.7 (ed. Rossi, , pp. 137. 63–138.79).Google Scholar

29 The phrase alterum sub altero occurs at Posterior Analytics 1.7, 75b15, in James of Venice's translation; subalterne scientiarum occurs at 79a14 of the ‘Ioannes’ translation, and subalterna at 81a27 of that version (Analytica posteriora, translationes Iacobi, anonymi sive ‘Iaonnis’, Gerardi et recensio Guillelmi de Moerbeke: edd. Minio-Paluello, L. and Dod, B. G., Aristoteles Latinus 4.1–4 [2nd and 3rd eds.; Leiden 1968], pp. 20, 131, 136). For evidence that Grosseteste used James of Venice's translation of the Posterior Analytics see Rossi, , Commentarius (above, n. 10) 80; Rossi also observes here that Grosseteste, in one passage, mentions littera aliarum translationum, and in his note to this passage Rossi gives parallels both to the ‘Ioannes’ translation and to Gerard's (Comm. Post. Anal. 1.9 [ed. Rossi, p. 166. 107] and n.).Google Scholar

30 ‘Scientia autem est subalternata alii cuius subiectum addit conditionem super subiectum subalternantis, que conditio non est totaliter exiens a natura subiecti subalternantis, sed extra sumitur, velut radiositas non est aliqua natura totaliter exiens a natura magnitudinis, sed extra assumpta est’ (Comm. Post. Anal. 1.18 [ed. Rossi, , p. 261.41–45]).Google Scholar

31 Comm. Post. Anal. 1.8 (ed. Rossi, , p. 147.33–39).Google Scholar

32 ‘Verumtamen, sicut subiectum scientie subalternate habet in se subiectum scientie subalternantis cum conditione superadiecta que appropriat ipsum scientie subalternate, sic medium sumptum de scientia subalternante, cum venit in sillogismum demonstrantem conclusionem scientie subalternate, recipit supra se conditiones per quas appropriatur scientie subalternate, et ipsum medium tale quale est in sillogismo demonstrante conclusionem scientie subalternate est in predicta proximitate cum extremis illius scientie, et dicitur de tertio secundum quod ipsum est et primum de ipso medio similiter secundum quod ipsum est’ ( Comm. Post. Anal. 1.8 [ed. Rossi, , pp. 148.54–149.63]).Google Scholar

33 ‘Demonstratur in perspectiva quod omnes duo anguli quorum alterum constituit radius incidens cum speculo et reliquum radius reflexus sunt duo anguli radiosi equales …’ (Comm. Post. Anal. 1.8 [ed. Rossi, , p. 149.66–69]).Google Scholar

34 Wallace, , Causality (above, n. 3) 1.38; cf. Euclid, , Catoptrica (ed. Heiberg, J. L., Euclidis opera omnia 7 [Leipzig 1895] 286–89).Google Scholar

35 ‘… et hec conclusio probatur per istam geometrie: omnium duorum triangulorum quorum unus angulus unius est equalis uni angulo alterius et latera equales angulos continentia sunt proportionalia, reliqui anguli prout se respiciunt sunt equales’ (Comm. Post. Anal. 1.8 [ed. Rossi, , p. 149.69–72]). The proof is translated in part by Crombie, , Robert Grosseteste, pp. 95–96. Rossi identifies the geometrical proposition as Euclid, , Elements 6, Prop. 6 (Commentarius [above, n. 10] 149n.).Google Scholar

36 ‘Et hec propositio secundum quod est simpliciter geometrica abstrahit a triangulis et ab angulis et a lateribus radiosis et non radiosis, sed secundum quod venit in sillogismum demonstrantem conclusionem predictam speculative appropriatur ad triangulos et angulos et latera radiosa hoc modo: omnium duorum triangulorum radiosorum, quorum unus angulus radiosus unius est equalis uni angulo radioso alterius et latera radiosa equos angulos radiosos continentia proportionalia, reliqui anguli radiosi prout se respiciunt sunt equales’ (Comm. Post. Anal. 1.8 [ed. Rossi, , p. 149.72–80]).Google Scholar

37 ‘Sed omnes duo anguli, quorum alterum constituit radius incidens cum speculo et reliquum radius reflexus, sunt duo anguli radiosi sese respicientes duorum triangulorum radiosorum quorum unus angulus radiosus unius est equalis uni angulo radioso alterius et latera radiosa equos angulos radiosos continentia proportionalia…’ ( Comm. Post. Anal. 1.8 [ed. Rossi, , p. 149.80–84]).Google Scholar

38 Ptolemy, , Optics 3: trans. in Cohen, M. R. and Drabkin, I. E. (edd.), A Source Book in Greek Science (Cambridge, Mass. 1948) 1268–71; a complete translation of Ptolemy's Optics is being prepared by A. Mark Smith.Google Scholar

39 ‘…ergo omnes duo anguli quorum alterum constituit radius incidens cum speculo et reliquum radius reflexus sunt duo anguli radiosi equales’ ( Comm. Post. Anal. 1.8 [ed. Rossi, , pp. 149. 85–150.86]).Google Scholar

40 Comm. Post. Anal. 1.8 (ed. Rossi, , p. 150.87–93).Google Scholar

41 ‘Causa namque equalitatis duorum angulorum factorum super speculum ex radio incidente et reflexo non est medium sumptum ex geometria, sed eius causa est natura radiositatis sese generantis secundum incessum rectum, que, cum congregatur super obstaculum habens in se naturam humidi spiritualis, fit ibi sicut principium regenerans se secundum similem viam ei per quam generatur. Cum enim operatio nature sit finita et regularis, necesse est ut via regenerationis sit similis vie sue generationis et ita regeneratur in angulo equali angulo incidenti’ (Comm. Post. Anal. 1.8 [ed. Rossi, , p. 150.93–101]; trans. Wallace, , Causality [above, n. 3] 1.39 [my insertion]). Cf. ‘Aut transitus eius [i.e., radii] est secundum rectum ad corpus habens naturam huius modi spiritualis, per quam ipsum est speculum, et ab ipso reflectitur ad rem visam’ (De iride, ed. Baur, [above, n. 18], p. 73.22–25); Eastwood suggests that huius modi spiritualis here means ‘a straight line, the path of a light ray, and is here used to characterize the surface of a plane mirror’ (Geometrical Optics [above n. 7] 187n.). But since the law of reflection holds for curved mirrors as well as plane mirrors, this phrase and its equivalent in the passage just quoted more likely refers to the nature of surfaces that makes them shiny and reflective, which is how Weisheipl has taken it (‘Science in the Thirteenth Century’ [above, n. 7] 448).Google Scholar

42 ‘Sed natura operatur breviori et meliori modo, quo potest; quare melius operatur super lineam rectam’ (De lineis, ed. Baur, [above, n. 18], p. 61.5–6).Google Scholar

43 De lineis (ed. Baur, , p. 62.8–15).Google Scholar

44 Euclid, , Catoptrica (ed. Heiberg, [above, n. 34] 290–91).Google Scholar

45 De lineis (ed. Baur, , p. 62.15–21).Google Scholar

46 Wallace (above, n. 3) discusses this part of Grosseteste's Commentary in some detail: 1.33–37.Google Scholar

47 Comm. Post. Anal. 1.12 (ed. Rossi, , pp. 188.1–189.22).Google Scholar

48 Comm. Post. Anal. 1.12 (ed. Rossi, , p. 189.23–35).Google Scholar

49 Comm. Post. Anal. 1.12 (ed. Rossi, , p. 189.36–38).Google Scholar

50 Comm. Post. Anal. 1.12 (ed. Rossi, , p. 193.106–112).Google Scholar

51 Comm. Post. Anal. 1.12 (ed. Rossi, , pp. 189.38–190.41).Google Scholar

52 Comm. Post. Anal. 1.12 (ed. Rossi, , p. 190. 41–53).Google Scholar

53 Comm. Post. Anal. 1.12 (ed. Rossi, , pp. 190.54–192.105); see also De sphaera (ed. Baur [above, n. 18], p. 29. 17–30.7).Google Scholar

54 Comm. Post. Anal. 1.12 (ed. Rossi, , p. 193.113–15); such a syllogism would be in the second figure camestres (all P is M; no S is M; no S is P) or cesare (no P is M; all S is M; no S is P).Google Scholar

55 Comm. Post. Anal. 1.12 (ed. Rossi, , pp. 193.115–194.125).Google Scholar

56 Comm. Post. Anal. 1.12 (ed. Rossi, , p. 194.126–136); beginning here, much of the rest of this chapter is translated in Crombie, , Robert Grosseteste 91–93.Google Scholar

57 Comm. Post. Anal. 1.12 (ed. Rossi, , pp. 194.136–195.153).Google Scholar

58 Comm. Post. Anal. 1.12 (ed. Rossi, , p. 195.153–163).Google Scholar

59 Comm. Post. Anal. 1.12 (ed. Rossi, , pp. 195.163–196.167).Google Scholar

60 Comm. Post. Anal. 1.12 (ed. Rossi, , p. 196.167–171).Google Scholar

61 Comm. Post. Anal. 1.12 (ed. Rossi, , pp. 196.172–197.190).Google Scholar

62 Comm. Post. Anal. 1.12 (ed. Rossi, , p. 197. 190–198).Google Scholar

63 De iride (ed. Baur, [above, n. 18], pp. 72–75); for the combined theory of vision, see also Comm. Post. Anal. 2.4 (ed. Rossi, , p. 386. 464–467); Crombie, , Robert Grosseteste 114; and Lindberg, D. C., Theories of Vision from Al-kindi to Kepler (Chicago 1976) 101.Google Scholar

64 Comm. Post. Anal. 1.12 (ed. Rossi, , pp. 197. 198–198.214).Google Scholar

65 Laird, , ‘The Scientiae Mediae’ (above, n. ∗) 96–131.Google Scholar