The research on which this paper is based was supported by the National Endowment for the Humanities and the Graduate School of the University of Wisconsin. I am grateful to A. G. Molland, James A. Weisheipl, O.P., and Sabetai Unguru for their very helpful comments.

¹ See, for example, Charles, Emile, Roger Bacon, sa vie, ses ouvrages, ses doctrines (Paris, 1861), p. 135Google Scholar; The Opus Majus of Roger Bacon, ed. Bridges, J. H. (London, 1900), I. IxxixGoogle Scholar; Adamson, Robert, as quoted by Little, A. G., ‘Introduction: On Roger Bacon's Life and Works,’ in Roger Bacon Essays, ed. Little, (Oxford, 1914), pp. 16–17Google Scholar; Frankowska, Malgorzata, Scientia as Interpreted by Roger Bacon, trans. Zienkiewicz, Ziemislaw (Warsaw, 1971), p. 126Google Scholar. I do not deny the possibility of close similarities between Bacon and the seventeenth century; I merely question whether they shed light on Bacon's achievement (see the argument immediately following). On Bacon's attitude toward mathematics, the best sources are Fisher, N. W. and Unguru, Sabetai, ‘Experimental Science and Mathematics in Roger Bacon's Thought,’ Traditio, 27 (1971), 353–78CrossRefGoogle Scholar; Wallace, William A., Causality and Scientific Explanation, I (Ann Arbor, 1972), 47–50Google Scholar; Crombie, A. C. and North, J. D., ‘Roger Bacon,’ Dictionary of Scientific Biography, i, 379–81Google Scholar; Smith, David Eugene, ‘The Place of Roger Bacon in the History of Mathematics,’ in Roger Bacon Essays, ed. Little, pp. 153–83.Google Scholar

² On Bacon's mathematical knowledge, see the works of Fisher, Unguru, and Smith cited in the previous note.

³ Timaeus, 28b–29d, 48eGoogle Scholar. As all students of Plato, I am much indebted to the perceptive commentary of Cornford, Francis M., Plato's Cosmology: The Timaeus afflato (London, 1937).Google Scholar

⁴ Ibid., 53c–55c; Cornford, , Plato's Cosmology, pp. 210–39.Google Scholar

⁵ Ibid., 36b–d.

⁶ Ibid., 33b, trans. Cornford, , in Plato's Cosmology, p. 54.Google Scholar

⁷ Ibid., 35b–36b.

⁸ Timaeus., 53d, trans. Cornford, , Plato's Cosmology, p. 212.Google Scholar

⁹ Ibid., 56b, trans. Cornford, , Plato's Cosmology, p. 223.Google Scholar

¹⁰ Ibid., 53d, trans. Cornford, , p. 212Google Scholar. Cornford suggests that the more remote principles are still mathematical in character—namely, lines and numbers—and that those favoured by heaven to know them are mathematicians.

¹¹ Ibid., 56c–57c.

¹² Ibid., 32c, trans. Cornford, , Plato's Cosmology, pp. 44–5.Google Scholar

¹³ For this interpretation, see ibid., pp. 51–2.

¹⁴ Solmsen, Friedrich, Aristotle's System of the Physical World (Ithaca, 1960), p. 47Google Scholar. Innumerable scholars have commented on these matters; two who have influenced me are Weisheipl, James A., O.P., The Development of Physical Theory in the Middle Ages (New York, 1959), p. 16Google Scholar; Wallace, , Causality and Scientific Explanation, i. 19–21.Google Scholar

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¹⁷ Posterior Analytics, i. 13. 78^b34–79^a3Google Scholar. This is G. R. Mure's translation (The Basic Works of Aristotle, ed. Ross, W. D., vol. 1 [Oxford, 1908]Google Scholar), except for the phrases ‘reason’ and ‘reason for the fact,’ which I substitute for Mure's ‘reasoned fact’ (which, though it may be excellent Greek, is virtually unintelligible English). For discussion of this passage, see Crombie, A. C., Robert Grosseteste and the Origins of Experimental Sciente 1100–1700 (Oxford, 1953), pp. 25–7Google Scholar. See also Aristotle, , Physics, ii. 2. 194^a7–8Google Scholar; and the discussion of this passage in Weisheipl, James A., ‘The Relationship of Medieval Natural Philosophy to Modern Science: The Contribution of Thomas Aquinas to its Understanding,’ Manuscripta, 20 (1976), 185–6.CrossRefGoogle Scholar

¹⁸ Meteorology, iii. 5Google Scholar. For a modern analysis, see Boyer, Carl B., The Rainbow: From Myth to Mathematics (New York, 1959), pp. 38–55.Google Scholar

¹⁹ Physics, iv. 8Google Scholar; viii. 5. De caelo, I. 6.Google Scholar

²⁰ Solmsen, , Aristotle's System, pp. 259–64.Google Scholar

²¹ De caelo, iii. 1. 299^a25 ff.Google Scholar; iii. 7. 306^a1 ff.

²² Ibid., iii. 7. 306^a9–12, trans. Stocks, J. L. (Works, ed. Ross, , vol. 2 [Oxford, 1930]).Google Scholar

²³ Lloyd, , Aristotle, p. 167.Google Scholar

²⁴ Aristotle takes his tripartite division of speculative knowledge from Plato, but reinterprets it. See Weisheipl, James A., O.P., ‘The Nature, Scope, and Classification of the Sciences,’ in Science in the Middle Ages, ed. Lindberg, David C. (Chicago, 1978), pp. 464–8.Google Scholar

²⁵ Metaphysics, xi. 3Google Scholar. 1061^a30 5, trans. Tredennick, Hugh (London, 1935), pp. 67–9Google Scholar. See also Metaphysics, vi. 1.Google Scholar

²⁶ De anima, iii. 7. 431^b16–17.Google Scholar

²⁷ Gilson, Etienne, The Christian Philosophy of Saint Augustine, trans. Lynch, L. E. M. (New York, 1960), pp. 127–30Google Scholar. The quotation (which Augustine takes from the Book of Wisdom) is given by Bourke, Vernon J., ed., The Essential Augustine (New York, 1964), p. 104Google Scholar (from De Genesi ad litteram, iv. 3. 7Google Scholar); it is also found elsewhere in Augustine's works, as in De trinitate, xi. 11Google Scholar. For stress on number, see also De trinitate, iv. 4–6.Google Scholar

²⁸ Migne, J. P., Patrologia Latina, lxiii, col. 1083bGoogle Scholar; quoted by Taylor, Jerome, The Didascalicon of Hugh of St. Victor: A Medieval Guide to the Aits (New York, 1961), p. 202, n. 41.Google Scholar

²⁹ Weisheipl, , ‘Classification of the Sciences,’ p. 61Google Scholar. De aritmetica, in Migne, , Patrologia Latina, lxiii, col. 1081d.Google Scholar

³⁰ Weisheipl, James A., O.P. ‘Classification of the Sciences in Medieval Thought,’ Mediaeval Studies, 27 (1965), 58–62CrossRefGoogle Scholar; Weisheipl, , ‘Nature, Scope, and Classification,’ pp. 470–1.Google Scholar

³¹ Taylor, Didascalicon of Hugh of St. Victor, pp. 61–73Google Scholar; Commentaries on Boethius by Thierry of Chartres and His School, ed. Häring, Nikolaus M. (Toronto, 1971), pp. 70–3, 160–4, 273–4Google Scholar; The Commentaries on Boetkius by Gilbert of Poitiers, ed. Häring, Nikolaus M. (Toronto, 1966), pp. 79–86Google Scholar; de Conches, Guillaume, Glosae super Platonem, ed. Jeauneau, Edouard (Paris, 1965), pp. 61–2Google Scholar; Häring, Nikolaus M., The Life and Works of Clarembaud of Arras (Toronto, 1965), pp. 66–9.Google Scholar

³² Taylor, , Didascalicon of Hugh of St. Victor, p. 72Google Scholar. The bracketed expression is mine.

³³ ‘One, two, three—but where, my dear Timaeus, is the fourth of those guests of yesterday who were to entertain me today?’ Quoted from Cornford's translation, in Plato's Cosmology, p. 9.Google Scholar

³⁴ Glosae, ed. Jeauneau, , p. 71; cf. p. 305.Google Scholar

³⁵ Ibid, p. 154.

³⁶ Taylor, , Didascalicon of Hugh of St. Victor, pp. 64–5.Google Scholar

³⁷ De sex dierum operibus, quoted by Taylor, , Didascalicon of Hugh of St. Victor, p. 202, n. 41.Google Scholar

³⁸ For a brief biographical sketch, see Callus, D. A., ‘Robert Grosseteste as a Scholar,’ in Robert Grosseteste, Scholar and Bishop, ed. Callus, D. A. (Oxford, 1955), pp. 3–11.Google Scholar

³⁹ i, 12. Quoted by Crombie, , Grosseteste, pp. 912Google Scholar. For discussion of Grosseteste's methodology, see ibid., pp. 91–104; Crombie, , ‘Grosseteste's Position in the History of Science,’Google Scholar in Grosseteste, Scholar and Bishop, ed. Callus, , pp. 101–12Google Scholar; Wallace, , Causality and Scientific Explanation, i, 28–47Google Scholar; Weisheipl, , ‘Classification of the Sciences,’ pp. 72–5Google Scholar; and Leff, Gordon, Paris and Oxford Universities in the Thirteenth and Fourteenth Centuries (New York, 1968), pp. 277–86.Google Scholar

⁴⁰ Commentary on Posterior Analytics, i, 12Google Scholar. Quoted by Weisheipl, , ‘Classification of the Sciences,’ p. 74Google Scholar, with alteration of a single word (‘subalternated’ to ‘subordinate’).

⁴¹ Commentarius in VIII libros physicorum Aristotelis, ed. Dales, Richard C. (Boulder, Colo., 1963), pp. 35–6Google Scholar. See also the translation in Crombie, , Grosseteste, p. 94.Google Scholar

⁴² See especially Crombie, , Grosseteste, pp. 104–16Google Scholar; Baur, Ludwig, Die Philosophie des Robert Grosseteste (Beiträge zur Geschichte der Philosophie des Mittelalters, xviii, 4–6) (Münster, 1917), pp. 76–84Google Scholar; Hedwig, Klaus, Sphaera Lucis: Studien zur Intelligibilität des Seienden im Kontext der mittelalterlichen Lichtspekulation (Beiträge zur Geschichte der Philosophie und Theologie des Mittelalters, N.S., xviii) (Münster, 1980), pp. 119–50Google Scholar; Lindberg, David C., Theories of Vision from d-Kindi to Kepler (Chicago, 1976), pp. 94–102Google Scholar. See the latter, p. 95, on reasons for preferring the expression ‘philosophy of light’ to Baur's and Crombie's ‘metaphysics of light.’

⁴³ Grosseteste, Robert, On Light, trans. Riedl, Clare C. (Milwaukee, 1942).Google Scholar

⁴⁴ Concerning Lines, Angles, and Figures, trans. Lindberg, David C.Google Scholar, in Source Book in Medieval Science, ed. Grant, Edward (Cambridge, Mass., 1974), p. 385Google Scholar. The Latin text is given in Die philosphischen Werke des Robert Grosseteste, ed. Baur, Ludwig (Beiträge zur Geschichte der Philosophie des Mittelalters, ix) (Münster, 1912), pp. 59–60.Google Scholar

⁴⁵ Ibid., pp. 65–6. This passage is from De natura locorum, which is frequently joined to De lineis in manuscripts and which can be considered a continuation of it. There is another passage relevant to this same issue in Grosseteste, 's De irideGoogle Scholar; Grant, , Source Book, pp. 388–9.Google Scholar

⁴⁶ Such an apparent discrepancy causes one to look immediately to the possibility of a change of mind or a process of development on Grosseteste's part. However, the dating of the various works seems to preclude such an approach, for, as Dales has argued, the commentary on the Posterior Analytics predates De lineis, while the commentary on the Physics was written about the same time as De lineis and presupposes much of Grosseteste's philosophy of light. See Dales, Richard C., ‘Robert Grosseteste's Scientific Works,’ Isis, 52 (1961), 401–2CrossRefGoogle Scholar; Grosseteste, , Commentarius in VIII libros physicorum, ed. Dales, p. xv.Google Scholar

⁴⁷ Weisheipl, , ‘Classification of the Sciences,’ pp. 73–4Google Scholar, and Development of Physical Theory, pp. 51–2Google Scholar, main tains that only through a mathematical middle term are propter quid demonstrations obtainable in physics. Wallace, , Causality and Scientific Explanation, i, 37–46Google Scholar, argues that propter quid demonstrations are sometimes obtainable in physics without the use of mathematics. There is no need to settle the question here.

⁴⁸ For an example of a physical argument that Grosseteste apparently did not regard as a propter quid demonstration (though the passage is confusing and has been variously interpreted), but which was a useful argument nonetheless, see the discussion of the equality of angles of incidence and reflection in Weisheipl, , ‘Classification of the Sciences,’ pp. 73–4Google Scholar; Wallace, , Causality and Scientific Explanation, i, 39Google Scholar; Crombie, , Grosseteste, pp. 95–6.Google Scholar

⁴⁹ If Grosseteste was prepared to reduce physics to the behaviour of light (as I think he may have been), he was clearly not willing to reduce the behaviour of light of geometry alone. For a similar conclusion, see Crombie, , Grosseteste, pp. 96, 104Google Scholar; Wallace, , Causality and Scientific Explanation, i, 37–9, 46Google Scholar. On the place of qualitative considerations in Grosseteste's mathematical law of refraction, see Eastwood, Bruce S., ‘Grosseteste's “Quantitative” Law of Refraction: A Chapter in the History of Non-Experimental Science,’ Journal of the History of Ideas, 28 (1967), 403–14.CrossRefGoogle Scholar

⁵⁰ See Wallace, , Causality and Scientific Explanation, i, 31–40Google Scholar; Crombie, , Grosseteste, pp. 92–103.Google Scholar

⁵¹ See the opening phrase in the quotation from Grosseteste's commentary on the Physics, above, at n. 41.

⁵² But note the possibility of exceptions to this rule, above, n. 47.

⁵³ Weisheipl, James A., O.P., ‘The Life and Works of St. Albert the Great,’ in Albertus Magnus and the Sciences, ed. Weisheipl, (Toronto, 1980), pp. 21–8.Google Scholar

⁵⁴ The best work on the relationship between Bacon and Albert is Jeremiah Hackett, M. G., ‘The Attitude of Roger Bacon to the Scientia of Albertus Magnus,’ in Albertus Magnus and the Sciences, ed. Weisheipl, pp. 53–72Google Scholar; see also Easton, Stewart, Roger Bacon and His Search for a Universal Science (Oxford, 1952), pp. 210–31Google Scholar. Which of Albert's writing Bacon knew cannot be easily established because of the inspecific nature of Bacon's criticisms. The work in which Albert most forcefully dealt with the relationship between mathematics and physics was his Metaphysics, not written until 1264–7, probably too late to have influenced any but Bacon's very late works. However, Albert's position had been outlined earlier in his Physics, written in the late 1240s, and his Posterior Analytics, probably written between 1261 and 1264; these, and especially the former, may have been known to Bacon earlier. On the dating of Albert's works, see Weisheipl, , ‘Life and Works,’ in Albertus Magnus and the Sciences, pp. 39–40Google Scholar; Appendix I to the same volume, pp. 565, 576; and Weisheipl, , ‘Albert the Great,’ New Catholic Encyclopedia, i, 257.Google Scholar

⁵⁵ Albert the Great, Opera omnia, ed. Geyer, Bernhard, xvi, 1 (Cologne, 1960), p. 2Google Scholar, lines 31–5. For discussions of Albert's position, see Molland, A. G., ‘Mathematics in the Thought of Albertus Magnus,’ in Albertus Magnus and the Sciences, ed. Weisheipl, pp. 463–78Google Scholar (the fullest account); Ashley, Benedict M., O.P., ‘St. Albert and the Nature of Natural Science,’Google Scholaribid., pp. 94–102; Weisheipl, James A., O.P., ‘Albertus Magnus and the Oxford Platonists,’ Proceedings of the American Catholic Philosophical Association, 32 (1958), 124–39CrossRefGoogle Scholar; Weisheipl, , Development of Physical Theory, pp. 57–62Google Scholar; Wallace, , Causality and Scientific Explanation, i, 66–71.Google Scholar

⁵⁶ Ultimately, both mathematics and physics are rooted in metaphysics. See Albert, , Opera omnia, ed. Geyer, xvi, 1, p. 2Google Scholar; Molland, , ‘Mathematics in the Thought of Albert,’ p. 467.Google Scholar

⁵⁷ Except for a phrase or two, this translation is from Ashley, , ‘Albert and the Nature of Natural Science,’ p. 96Google Scholar. For the Latin text, see Albert, , Opera Omnia, ed. Borgnet, A., iii (Paris, 1890), 109–10Google Scholar. For another illustration of the same point, see Molland, , ‘Mathematics in Albertus’, p. 468.Google Scholar

⁵⁸ I am counting pages in modern editions. Bacon commits more than 200 pages to the subject in the Opus tertium, more than 300 in the Opus maius, and a handful in the Communia mathematica and other works.

⁵⁹ No doubt the difference in character between Bacon's treatment of the problem and those of his predecessors can be explained in functional terms: they were commenting on Aristode or some other philosophical authority, or perhaps writing a handbook of the sciences, while he was appealing for patronage. That Bacon was capable of more traditional and sober philosophical analysis is evident from his Aristotelian commentaries. Indeed, these Aristotelian commentaries occasionally touch upon issues relevant to our topic, but in none of them is the relationship of mathematics to nature met head on. See for example, the discussion of number in Bacon, 's Questions on the Metaphysics, in Opera hactenus inedita, ed. Steele, Robert and Delorme, Ferdinand M., xi (Oxford, 1932), 75–83Google Scholar; and the discussion of causal knowledge and abstraction in the Questions on the Physics, in Opera, ed. Steele, and Delorme, , xiii (Oxford, 1935), 4–5, 93–7Google Scholar. In the latter passage, Bacon acknowledges that physics and mathematics are distinguished by the kind of abstraction of which they partake.

⁶⁰ Opus maius, ed. Bridges, i, 97–8Google Scholar. Burke's translation (The Opus Majus of Roger Bacon, trans. Burke, Robert B. [Philadelphia, 1928]Google Scholar is so generally defective that I will not quote from it, though I have, of course, continually benefitted from its existence).

⁶¹ Ibid., i, 108.

⁶² Fr. Rogeri Baconi Opera quaedam hactenus inedita, i, ed. Brewer, J. S. (London, 1859), p. 111Google Scholar; cf. ibid., pp. 105, 268–9.

⁶³ Opera, ed. Steele, and Delorme, , xvi (Oxford, 1940), 7.Google Scholar

⁶⁴ Opus maius, ed. Bridges, , i, 66; ii, 223.Google Scholar

⁶⁵ Opus maius, ed. Bridges, , i, 145.Google Scholar

⁶⁶ Opera, ed. Brewer, , pp. 43–4Google Scholar; Opus maius, ed. Bridges, , ii, 172–3.Google Scholar

⁶⁷ Ibid., i, 168–9.

⁶⁸ This claim is made in both Communia mathematica, in Opera, ed. Steele, and Delorme, , xvi, 8Google Scholar; and Opus maius, ed. Bridges, , i, 99Google Scholar. The same claim appears also in what seem to be student notes taken during Bacon's Parisian lectures; see Steele, Robert, ‘Roger Bacon as a Professor: A Student's Notes,’ Isis, 20 (1933), 62.CrossRefGoogle Scholar

⁶⁹ Opus maius, ed. Bridges, , i, 98–9.Google Scholar

⁷⁰ Opus maius, ed. Bridges, , i, 102.Google Scholar

⁷¹ Ibid., i, 103.

⁷² Ibid., i, 103–5, 175–6. This and the previous conclusion make clear that Bacon's claims about the indispensability of mathematics are motivated by pedagogical, as well as metaphysical and epistemological, concerns. This may seem to reflect confusion on Bacon's part, but I suspect that it bespeaks simply his breadth of purpose and his helter-skelter style. Fisher, and Unguru, , ‘Experimental Science and Mathematics,’ p. 361Google Scholar, argue that Bacon's references to mathematics as the ‘gate and key’ to the other sciences simply reflect his conviction that mathematics was the first discovered and should be the first taught of the sciences.

⁷³ Ibid., i, 107.

⁷⁴ Opus maius, ed. Bridges, .Google Scholar

⁷⁵ Ibid., i, 108. Bacon's reference in the middle sentence of this quotation to the subjects dealt with in the writings of those who have employed mathematics appears to be, in large part, a list of Grosseteste's works.

⁷⁶ The distinction is obscure, since presumably the sciences treat things.

⁷⁷ Cf. the discussion in Communia mathematica, Opera, ed. Steele, and Delorme, , xvi, 50.Google Scholar

⁷⁸ Opus maius, ed. Bridges, , i, 109–11Google Scholar. Bacon returns to material causation in ibid., i, 143–65.

⁷⁹ Opus maius, ed. Bridges, , i, 111Google Scholar. This translation is reprinted from Grant, , Source Book, p. 393Google Scholar. See also Opus tertium, in Opera, ed. Brewer, , pp. 107–9.Google Scholar

⁸⁰ Opus maius, ed. Bridges, , i, 111 27.Google Scholar

⁸¹ See my forthcoming edition and translation of De multiplicatione specierum and De speculis comburentibus, to be published as Roger Bacon's Philosophy of Nature (Oxford University Press).

⁸² Opus maius, ed. Bridges, , i, 127–37.Google Scholar

⁸³ Ibid., i, 138 44.

⁸⁴ Ibid., i, 143 8. For a full analysis of Bacon's position on the possible unity of matter, see Crowley, Theodore, Roger Bacon: The Problem of the Soul in His Philosophical Commentaries (Louvain/Dublin, 1950), pp. 91–100.Google Scholar

⁸⁵ Opus maius, ed. Bridges, , i, 148–52.Google Scholar

⁸⁶ Opus maius, ed. Bridges, , i, 152–64.Google Scholar

⁸⁷ Ibid., i, 164–5.

⁸⁸ Grant, , Source Book, pp. 382–3Google Scholar; Clark, David L., ‘Optics for Preachers: The De oculo morali by Peter of Limoges,’ Michigan Academician, 9 (1977), 329–43Google Scholar; Lindberg, , Theories of Vision, p. 99Google Scholar. I am assured by Mr. Peter Brown, of the University of Kent at Canterbury, that Bacon knew and used Peter of Limoges' De oculo morali.

⁸⁹ In addition to the section of the Opus maius analysed below, see the Opus tertium, ed. Brewer, , pp. 199–232.Google Scholar

⁹⁰ Opus maius, ed. Bridges, , i, 175–80.Google Scholar

⁹¹ Ibid., i, 175, 182–238.

⁹² Opus maius, ed. Bridges, , i, 175.Google Scholar

⁹³ Ibid., i, 185.

⁹⁴ Ibid., i, 211.

⁹⁵ Ibid., i, 180.

⁹⁶ Ibid., i, 269–85.

⁹⁷ On the importance of mathematics in directing the commonwealth, see ibid., i, 253–403.

⁹⁸ Astrology is treated repeatedly by Bacon; see especially ibid., i, 109–10, 138–9, 238–69, 286–8, 376–98; Opus tertium, in Opera, ed. Brewer, , pp. 106–7, 269–70Google Scholar. See also Molland, A. G., ‘Roger Bacon: Magic and the Multiplication of Species,’ forthcoming in Paideia.Google Scholar

⁹⁹ That is, utterances having physical effects; see Opus maius, ed. Bridges, , i, 395–6.Google Scholar

¹⁰⁰ Fascination, as Bacon uses the term, is the ability of some people to contaminate things in their vicinity through sight and the spoken word; see ibid., i, 398–9.

¹⁰¹ Ibid., i, 399, 402. On the likelihood that this section belonged originally to the Opus minus, but was inserted in the Opus maius by a later editor, see Little, A. G., ed., Part of the Opus tertium of Roger Bacon, Including a Fragment Now Printed for the First Time (Aberdeen, 1912), pp. xviixviii, 18Google Scholar. I thank A. G. Molland for calling this to my attention.