1 The Pseudo-Aristotelian Problemata, Chap. XVI.13; emphasis added. See The Works of Aristotle, Volume VII: Problemata (tr. E. S. Forster), Oxford, 1971. For discussions on its Peripatetic provenance between the second and sixth centuries AD see A. Blair, ‘The Problemata as a natural philosophical genre’; and J. Monfasani, ‘The pseudo-Aristotelian Problemata and Aristotle's De Animalibus in the Renaissance’, both in Natural Particulars: Nature and the Disciplines in Renaissance Europe (ed. A. Grafton and N. Siraisi), Cambridge, MA, 1999, 171–204.

2 See Aristotle, On the Soul, II.8 419b2 ff., in The Complete Works of Aristotle, Vol. 1 (ed. J. Barnes), Princeton, 1994; and idem, Posterior Analytics, II. 15. 98a27, in The Works of Aristotle, Vol. I (ed. W. D. Ross), Oxford, 1968.

3 Either from Euclid's Catoptrics or Hero of Alexandria's Catoptrics. See G. Irby-Masssie and P. Keyser (eds.), Greek Science of the Hellenistic Era: A Sourcebook, London, 2002, 195.

4 Taken from Euclid's Catoptrics. See K. Takahashi, The Medieval Latin Traditions of Euclid's Catoptrica: A Critical Edition of De Speculis with an Introduction, English Translation and Commentary, Kyushu, 1992, 155.

5 I therefore disagree with Sabra: ‘the author of this passage seems to be concerned to bring out a contrast, rather than an analogy, between the appearance of images in mirrors and the rebound of objects … We are told that the object returns at an angle equal to the angle of incidence, but we are not told why this must be so’. A. I. Sabra, Theories of Light from Descartes to Newton, London, 1967, 69; original emphasis. This very appeal to the geometry of reflection and image location is the reason why the object rebounds in this way. Just as the visual ray rebounds on being prevented from following its straight-line path beyond the mirror, so the object rebounds on being prevented from following its straight-line path beyond the obstacle.

6 Aristotle, Sense and Sensibilia, 6. 446a25 and 446b28, in The Complete Works of Aristotle, Vol. 1 (ed. J. Barnes), Princeton, 1994; idem, On the Soul, II.7.418b14, in ibid.

7 See M. Cohen and I. E. Drabkin (eds.), A Source Book in Greek Science, New York, 1948, 261–5; and Irby-Masssie and Keyser, op. cit. (3), 193–6.

8 Cohen and Drabkin, op. cit. (7), 263.

9 Cohen and Drabkin, op. cit. (7), 263–4.

10 Cohen and Drabkin, op. cit. (7), 263–4.

11 Ptolemy's Optics, c.160 AD. See A. M. Smith, Ptolemy's Theory of Visual Perception: An English Translation of the Optics with Introduction and Commentary, in Transactions of the American Philosophical Society (1996), 86, 2. Smith argues (15) against the influence of Hero's Catoptrics upon Ptolemy because (a) whatever he might have found in Hero's Catoptrics could also be found in Euclid's, and (b) he failed to mention Hero's ‘minimum-line’ proof of the law of reflection. However, (a) the mechanical analogies of reflection and rectilinearity are not to be found in Euclid's Catoptrics, though they are in Hero's, and (b) Ptolemy sought a physical explanation of the geometrical law of reflection applicable to both the rebound of bodies and the reflection of rays, whereas Hero's geometrical proof of minimum lines offers no such physical understanding of the phenomena of rebound and reflection, or of the law itself, and sharply contrasts with his use of mechanical analogies. Of course, Ptolemy was under no obligation to introduce Hero's metaphysical explanation (a principle of Nature's economy) for what he sought to explain via general physical principles. For Hero on this principle see Irby-Masssie and Keyser, op. cit. (3), 195. It is reasonable to assume, with Smith, Ptolemy's knowledge among the Aristotelian corpus of De anima and Problemata; see Smith, op. cit., 11.

12 Smith, op. cit. (11), 76–7.

13 Smith, op. cit. (11), 139; emphasis added.

14 Smith mentions that ‘the likening of optical reflection to physical rebound is found not only in Hero's Catoptrics, but also in the Pseudo-Aristotelian Problemata’, but makes no mention of the conceptual difference between Ptolemy's development of the analogy and the Problemata's simple analogical appeal to the geometry of image formation; see Smith, op. cit. (11), 139.

15 Though there is doubt about the exact date at which Ptolemy's work was translated into Arabic, it was certainly available to Ibn al-Haytham. See A. I. Sabra, The Optics of Ibn al-Haytham, Books I–III on Direct Vision, 2 vols., London, 1989, ii, 58–9, 74.

16 See D. C. Lindberg, Theories of Vision from Alkindi to Kepler, Chicago, 1976, 65.

17 F. Risner (ed.), Opticae Thesaurus …, Basel, 1572, 112–13, quoted in E. Grant (ed.), A Source Book in Medieval Science, Cambridge, MA, 1974, 418.

18 Risner, op. cit. (17), 112–13, quoted in Grant, op. cit. (17), 418.

19 Risner, op. cit. (17), 112–13, quoted in Grant, op. cit. (17), 418; emphasis added.

20 While the key section of the Pseudo-Aristotelian Mechanica (Problem 1) was transmitted to Arabic culture via al-Khazini's early twelfth-century Book on the Balance of Wisdom (Kitab mizan al-hikma), this was too late for al-Haytham. On the other hand, Hero's Mechanica became accessible to Arab mathematicians in the tenth century via Qusta ibn Luqa's On Lifting Heavy Objects (Fi raf al ashya al-thaqila). See Abbatouy, M., ‘Greek mechanics in Arabic context: Thabit ibn Qurra, al-Isfizari and the Arabic traditions of Aristotelian and Euclidean mechanics’, Science in Context (2001), 14, 179–248Google Scholar, 183–7.

21 This rule is also in the Pseudo-Aristotelian Mechanica: ‘Now whenever a body is moved in two directions in a fixed ratio it necessarily travels in a straight line, which is the diagonal of the figure which the lines arranged in this ratio describe’, 848b10, in Aristotle, Aristotle: Minor Works (tr. W. S. Hett), London, 1963, 337.

22 It can be argued that if the conceptual source of al-Haytham's analysis of rebound/reflection had been the parallelogram rule of Hero's Mechanica, he would have employed it directly. However, such a move would have brought the kinematics of geometrical point motion explicitly into the domain of Aristotelian physics, contradicting its classification of rectilinear motion as simple, not compound.

23 Risner, op. cit. (17), 112–13, in Grant, op. cit. (17), 419.

24 ‘But in the rebound of a heavy body, when the motion of repulsion ceases, the body descends because of its nature and tends toward the centre [of the world]. However, light, which has the same nature of reflecting [as a heavy body], does not by nature ascend or descend; therefore, in reflection it is moved along its initial line until it meets an obstacle which terminates its motion.’ Risner, op. cit. (17), 112–13, in Grant, op. cit. (17), 419; emphasis added.

25 Ibn al-Haytham's On Light, in Ibn Al-Haytham. Proceedings of the celebrations of 1000th Anniversary Held under the Auspices of Hamdard National Foundation (ed. M. H. Said), Karachi, 1969, 215–16.

26 Bruno's three basic principles are humidity (spherical, indestructible atoms), dryness (an infinite substratum in which atoms exist and move, usually identified as ‘ether’) and light (which unites the dry and humid and functions as a principle of motion within atoms related to ‘soul’). Patrizi's four basic principles are space (a three-dimensional, infinite container of matter lacking resistance), light (a three-dimensional, infinite filler of space lacking resistance), heat (produced by a special kind of light) and ‘fluor’ (a matter principle of rarefaction and condensation via expansion and contraction, which is also light). In his 1591 book Patrizi explains that ‘just as Aristotle discovered the Prime Mover by way of motion, so … I find it by way of “lumen” and “lux” and then … by way of a Platonic Method I descend to the products of light’, quoted in E. E. Maechling, ‘Light metaphysics in the natural philosophy of Francesco Patrizi da Cherso’, unpublished M.Phil. thesis, University of London (Warburg Institute), 1977. For Bruno see H. Gatti, ‘Giordano Bruno's soul-powered atoms from ancient sources towards modern science’, in Late Medieval and Early Modern Corpuscular Matter Theories (ed. C. Luthy, J. Murdoch and W. Newman), Leiden, 2001, 163–80. For Patrizi see also B. Brickman, An Introduction to Franceso Patrizi's ‘Nova de universis philosophia’, New York, 1941; L. Deitz, ‘Space, light, and soul in Francesco Patrizi's Nova De Universis Philosophia (1591)’, in Natural Particulars: Nature and the Disciplines in Renaissance Europe (ed. A. Grafton and N. Siraisi), Cambridge, MA, 1999, 139–69; Schrenk, L. P., ‘Proclus on space as light’, Ancient Philosophy (1989), 9, 87–94CrossRefGoogle Scholar.

27 This was a group of gentlemen scholars such as Nathaniel Torporley, Robert Hues, Walter Warner, Nicholas Hill and Thomas Harriot, centred around their patron the ninth Earl of Northumberland Henry Percy, who also knew John Dee. See J. Jacquot, ‘Harriot, Hill, Warner and the New Philosophy’, in Thomas Harriot: Renaissance Scientist (ed. J. W. Shirley), Oxford, 1974, 107–28; G. R. Batho, ‘Thomas Harriot and the Northumberland household’, in Thomas Harriot: An Elizabethan Man of Science (ed. R. Fox), Aldershot, 2000, 28–47; Clucas, S., ‘The atomism of the Cavendish circle: a reappraisal’, The Seventeenth Century (1994), 9, 247–73Google Scholar.

28 For instance, in a manuscript written between 1610 and 1620 Walter Warner defines vis as ‘an efficient power or virtue which may be called light whether sensible or insensible’. He and Nicholas Hill attempted to marry the new atomistic outlook with the medieval light metaphysics of Robert Grosseteste and Roger Bacon: ‘Warner needed a source of local motion, generation and alteration in his atomistic system – medieval theories of vis radiativa provided him with a solution’. S. Clucas, ‘Corpuscular matter theory in the Northumberland circle’, in Late Medieval and Early Modern Corpuscular Matter Theories (ed. C. Luthy, J. Murdoch and W. Newman), Leiden, 2001, 181–207, 187 and 196. Clucas also makes a strong case for Nicholas Hill adopting this medieval light metaphysical approach to a corpuscular philosophy in his Philosophia Epicurea …, Paris, 1601 (reprinted Geneva, 1619), quoting Hill describing light as one of the ‘efficient principles of the universe’ which ‘insinuates itself into the material parts of the world forming everything’, in ibid., 198 ff. See also S. Clucas, ‘Mediaeval concepts of force in the atomism of the Northumberland circle’, in Science and Cultural Diversity (ed. J. J. Saldana), Proceedings of the XXIst International Congress of History of Science, CD-ROM (Sociedad Mexicana de Historia de la Ciencia y de la Tecnologia – Universidad Nacional Autonoma de Mexico), Mexico City, 2005, 3060–73.

29 There are reliable records establishing its popularity among the aristocracy in, for example, France between 1574 and 1579 and England around 1587. See C. Everton, The Story of Billiards and Snooker, London, 1986, 9–11, and N. Clare, Billiards and Snooker Bygones, Princes Risborough, 1996, 5–9.

30 F. Risner, Optica thesaurus. Alhazani Arabis libri septem … Vitellonis Thuringopolini libri X … (Basel, 1572), reprinted with an introduction by D. C. Lindberg, New York, 1972. While Alhazen's work was only printed once, the works of medieval optics were regularly reprinted, even into the seventeenth century. See D. C. Lindberg, A Catalogue of Medieval and Renaissance Optical Manuscripts, Toronto, 1975.

31 It is worth considering why Harriot chose to tackle this problem. Lohne suggests that Harriot sought to analyse and refute the ‘common belief that in all collisions the angles of incidence and reflexion are equal’, something which he would have seen contradicted playing billiards. Lohne, J. A., ‘Essays on Thomas Harriot: I. Billiard balls and laws of vollision; II. Ballistic parabolas; III. A survey of Harriot's scientific writings’, Archive for History of Exact Sciences (1979), 20, 189–312CrossRefGoogle Scholar, 193. This belief undoubtedly grew because of the increasing popularity of the Problemata as evidenced by its repeated printing in the sixteenth century. See Blair, op. cit. (1), 189; and Monfasani, op. cit. (1), 232 ff. We should note that Harriot had attended Oxford University, where the 1549 Statutes of Edward VI called for the Problemata to be taught (Blair, op. cit. (1), 179). Another suggestion about Harriot's choice of topic is that ‘he may have connected it with a corpuscular theory of light and his solution of the refraction problem. It may be that it was for this reason that he made a special point of drawing attention to the general inequality of the angles of reflection and refraction’. Pepper, J. V., ‘Harriot's manuscript on the theory of impacts’, Annals of Science (1975), 33, 131–47CrossRefGoogle Scholar, 141. M. Kalmar suggests a connection between Harriot's experience of billiards and the rise of atomism in his ‘Thomas Hariot's “De Reflexione Corporum Rotundorum”: an early solution to the problem of impact’, Archive for History of Exact Sciences (1976), 16, 201–27, 216.

32 DSB, Harriot, Thomas; Lohne, op. cit. (31); Batho, op. cit. (27).

33 Quoted in H. Gatti, ‘The natural philosophy of Thomas Harriot’, in Thomas Harriot: An Elizabethan Man of Science (ed. R. Fox), Aldershot, 2000, 177.

34 Smith, op. cit. (11), 139.

35 Ibn al-Haytham's Optics, Kitab al-Manzir, MS Ayasofia 2448, 353, quoted in Rashed, R., ‘Optique géometrique et doctrine optique chez Ibn Al Haytham’, Archive for History of Exact Sciences (1970), 6, 271–98CrossRefGoogle Scholar, 284.

36 Risner, op. cit. (17), 112, quoted in Rashed, op. cit. (35), 286. Sabra has pointed out the problem for al-Haytham in arguing that the force of rebound is related to the force of incidence, when the latter is abolished on impact, which could be resolved by introducing elasticity. See Sabra, op. cit. (5), 76; and A. I. Sabra, Optics, Astronomy and Logic: Studies in Arabic Science and Philosophy, Aldershot, 1994, 551.

37 Harriot's research in optics from 1597 to 1601–2 centred on refraction, leading to his discovery of the correct sine law, later propounded by Descartes and Snell. See Lohne, op. cit. (31), 279 and 283; Batho, op. cit. (27), 38. After reading Kepler's Optics, Harriot corresponded with him between 1606 and 1609, propounding a novel conception of refraction as the multiple reflection of light rays from atoms throughout the medium. See A. R. Alexander, Geometrical Landscapes: The Voyages of Discovery and the Transformation of Mathematical Practice, Stanford, 2002, 115–25. In 1605 Harriot's patron Henry Percy was imprisoned in the Tower of London, where he installed a billiard table (Lohne, op. cit. (31), 193). Harriot's fusion of optics with atomistic matter theory whilst observing billiard ball collisions may have led him to consider the latter via the former or vice versa.

38 ‘For demonstrations of … [reflection and refraction] … all philosophers and optical writers establish a certain comparison between physical bodies (and their motions) and light.’ Optics, 1.6: see W. Donahue (tr.), Johannes Kepler, Optics: Paralipomena to Witelo and Optical Part of Astronomy, Santa Fe, 2000, 18.

39 Proposition 18; Donahue, op. cit. (38), 26.

40 Proposition 19; Donahue, op. cit. (38), 27.

41 Propositions 18 and 19; Donahue, op. cit. (38), 26 and 27.

42 Kepler refers to the Problemata a little later, in his introduction to Chapter 2; Donahue, op. cit. (38), 54.

43 Where the obstacle can impact upon a body at rest or where both bodies are in motion and yet differ in size or weight and relative speeds. Harriot determined that the mechanical rebound of an incident body does not obey the law of reflection only when the obstacle is immovable. See his motive actions 1 and 2, here Figure 5.

44 See individual papers of Lohne, Pepper and Kalmar, opera cit. (31). Some writers lament Harriot's failure to set out his first principles or assumptions, with Kalmar speculating that ‘much or even most of the medieval use of the word “reflexion” of Harriot's title refers particularly to optics. There may be nothing more than an analogy here, but it could indicate a useful question for further study’. Kalmar, op. cit. (31), 142. Yet if we treat Harriot's use of such terms seriously, these first principles can be discovered by looking at what Harriot actually learned about the physics of impact from the physics of ‘reflexion’.

45 It differs from Descartes's later rectangular version, undoubtedly taken from the Aristotelian Mechanica due to its virtually identical kinematical explication. This is because Descartes's mechanical analogy of oblique reflection only involved the oblique incidence of one body (ball) upon an extended surface, not the oblique collision of two moving bodies (balls). The addition of material variability via size (and speed) in such collisions entailed a level of dynamical complexity which the parallelogram of reflection/rebound from an unmoving surface (mirror, ground, wall) could not accommodate.

46 The meaning of this term is a little problematic and is usually left untranslated. Kalmar interprets it as referring to ‘the tendency that a body has to move in a given direction with a given speed, whether that tendency is manifested by an actual motion or not’, explaining that the ‘primary meaning … is a nodding or downward inclining of the head. It can also refer to the downward movement or tendency of a heavy body to move due to its gravity. Harriot generalised the term to include tendencies to move in any direction and exclusive of any cause. This use of nutus is, as far as I know, unique to him’. See Kalmar, op. cit. (31), 204. See also Pepper, op. cit. (31), 138; Lohne op. cit. (31), 276 ff.

47 Lohne, op. cit. (31), 202.

48 Lohne, op. cit. (31), 202.

49 Hence the appeal to the geometry of image location and its mechanical application.

50 See C. B. Daish, The Physics of Ball Games, London, 1972, 193. Lohne makes the interesting point that Harriot was most likely the first person to state that colliding bodies sometimes move off at right angles to each other (a common feature in billiards) and provide a dynamical analysis of this situation. Lohne, op. cit. (31), 193.

51 Lohne, op. cit. (31), 202–3.

52 As implicitly employed by Pseudo-Aristotle and al-Haytham, and explicitly employed by Kepler. Indeed, the same mechanical argument and geometrical demonstration, taken from the Problemata, was further popularized via Josephus Blancanus's Aristotelis loca mathematica ex universis ipsius operibus collecta et explicata, Bologna, 1615, 243–4.

53 This is one of three cases of rebound, in which the law of reflection is obeyed, the other two being where the obstacle body is infinitely heavy, and its equivalent case, where the obstacle body is taken to be fixed (see Figure 6). Kalmar, op. cit. (31), 205; Lohne, op. cit. (31), 225.

54 Lohne, op. cit. (31), 203.

55 Parity of reasoning and the lesser's diminution in size and active power enables the construction of a shorter diagonal (BF), its parallelogram (ABCF), and B's path of rebound BC.

56 Lohne, op. cit. (31), 204.

57 This diverges from the optical case because it covers both equal and unequal angles of incidence and reflection, and employs rhomboid parallelograms whose adjacent sides do not correspond to the perpendicularly related components of oblique motion. Descartes's later use of the rectangular parallelogram for compound motion in oblique impact and rebound considers only the oblique impact of light or a ball against a plane surface, which therefore rebounds in line with the law of reflection.

58 Lohne, op. cit. (31), 202.

59 Lohne, op. cit. (31), 203. Compare with Figure 4 above, of Kepler's Proposition 19. Harriot's line fc here corresponds to Kepler's line DE.

60 ‘Hence it follows that at the same time as the body b continues its motion from time f to c and the whole line fc is translated as described above, the body's centre will describe a certain third line which when completed will be the line bc, drawn from the point b to c’. Lohne, op. cit. (31), 203.

61 Lohne, op. cit. (31), 203.

62 Kalmar, op. cit. (31), 214; Lohne, op. cit. (31), 196.

63 Harriot's letter to the Duke of Northumberland, dated 13 June 1619, which accompanied a copy of his De reflexione and wherein he talks of perfecting his ‘auntient notes of the doctrine of reflections of bodies’ and imparting them to Warner. See J. O. Halliwell, A Collection of Letters Illustrative of the Progress of Science in England, London, 1841, 45. There is renewed interest in the likelihood of earlier collaboration between Harriot and Warner on collisions (around 1603), due to the discovery of previously unknown manuscripts of Warner's on collisions and motion; see S. Clucas, ‘Thomas Harriot and Walter Warner on collisions: English mechanics in the early seventeenth century’, paper read at the XXII International Congress of the History of Science, Beijing, China, 24–30 July 2005. Yet the recognition of the applicability of optical principles to material motion, which grounds Harriot's De reflexione, is implicit in Henry Percy's listing of ‘The doctrine of motion of the optics’, written in 1595. See G. B. Harrison (ed.) Advice to His Son, by Henry Percy Ninth Earl of Northumberland, London, 1930, 67. Given that Harriot had thoroughly studied al-Haytham's Optics in the early 1590s, his ‘auntient notes’ may well date as far back as then. However, this phrase of Percy's has been ignored by many, including Clucas. See Clucas's 2005 conference paper, cited above, and his ‘Thomas Harriot and the field of knowledge in the English Renaissance’, in Thomas Harriot: An Elizabethan Man of Science (ed. R. Fox), Aldershot, 2000, 93–136, 107. In the latter, Clucas employs the misquotation of ‘the doctrine of motions, optics’ found in Christopher Hill's The Intellectual Foundations of the English Revolution, Oxford, 1965, 142.

64 Mainly William and Charles Cavendish, Thomas Hobbes, John Pell and Robert Payne. Kenelm Digby became associated with the Circle via correspondence with Hobbes. See Jacquot, J., ‘Sir Charles Cavendish and his learned friends’, Annals of Science (1952), 8, 13–27CrossRefGoogle Scholar and 175–91, and Clucas, op. cit. (27).

65 Lohne, op. cit. (31), 215.

66 Jacquot, op. cit. (64); Clucas, op. cit. (27), 252–9. This explains, for instance, why Kenelm Digby's Two Treatises of 1644, containing his arguments for the material nature of light (as a body) and the notion of reflection as violent motion, was first published in Paris, where there was a receptive intellectual environment and where he lived from 1641 to 1645. See Clucas op. cit. (27), 256. Implicit in this mechanization of light is not simply the idea that light should therefore obey the same laws of motion as bodies, but the more fundamental notion that the laws governing material bodies must be based upon laws of light because light motion represents the ideal case of material motion/impact, free from the effects of gravity and air friction.

67 Collins had been described by Isaac Barrow as the ‘English Mersenne’ (see his DSB entry), corresponding with scholars such as Barrow, Newton, Wallis, Borelli, Huygens and Leibniz.

68 Lohne, op. cit. (31), 190 and 215.

69 Hall, A. R., ‘Mechanics and the Royal Society, 1668–70’, BJHS (1966), 3, 24–38CrossRefGoogle Scholar, 28.

70 We must include the conceptual spurs provided in the late sixteenth century and the earlier seventeenth by Aristotle's presentation of the mechanical analogy of reflection in his Problemata, the exposition of the parallelogram rule in the Pseudo-Aristotelian Mechanica, Kepler's explicit linking of light and material motion in his Optics and, of course, Descartes's introductory illumination of his mechanical philosophy via his Dioptrics, in which he explicitly applies the said parallelogram rule to reflection and, via the above analogy, to rebound.

71 In 1670 Pardies sought to correct Descartes's erroneous laws of impact by explicitly introducing the dynamics and kinematics of optical reflection, shifting from the analysis of direct to oblique collisions between bodies. See Ignace-Gaston Pardies, Discours du movement local, Paris, 1670, translated into English by Henry Oldenburg as A Discourse of Local Motion, London, 1670, in Descartes in Seventeenth-Century England, Vol. 7: Critiques of Descartes (ed. R. Ariew and D. Garber), Bristol, 2002, 38 ff. There are striking similarities between his, Harriot's and Newton's work of 1666 on mutual rectilinear collisions in their use of the physics of reflection, which deserve a fuller examination. But whilst, unlike Harriot, he includes circular motion in his analysis, Pardies signally fails to extend the parallelogram rule to circular motion, unlike Newton a few years earlier. This is because he does not make Newton's creative conceptual leap of analysing such motion via the geometry, and dynamics, of reflection. It is unlikely that Pardies had access to Harriot's writings, yet he was familiar with the optico-mechanical writings of Maignon and Hobbes.

72 This common view no doubt arose from the widespread dissemination of Aristotle's Problemata in the sixteenth century (see Blair and Monfasani, opera cit. (1)) and its specific application of the geometry of image location to mechanical rebound, which was popularized in 1615 by Josephus Blancanus (op. cit. (52)). Moreover, the Latin publication of Ibn al-Haytham's light-mechanical treatment of reflection and refraction in 1572 would have reinforced interest in exploring the dynamics of reflection and mechanical collision, given the imminent rise of an atomistic conception of light. In 1624 Jean Leurechon noted that ‘hence by mathematical principles, the games of Tennis may be assisted, for all the moving in it is by right lines and reflections’, and that ‘the maximes of reflections cannot be exactly observed by locall motion, as in the beames of light and other qualities, whereof it is necessary to supply it by industry or by strength otherwise one may be frustrated in that respect’. Leurechon, Récréations mathématiques …, Paris, 1624, Problem 71 (there were twelve editions before 1630 and more after); see also the 1654 English edition, W. Oughtred, Mathematical Recreations …, London, 1654, 122–4, and also 1633 and 1674. Interestingly, Descartes refers to it in a letter to Mersenne, in April/May 1634, and this Problem 71 seems to inform Discourse 2 of his later Dioptrics, while Mydorge's first major writing in 1630 (reprinted 1643) was a review of Récréations mathématiques (see DSB, Mydorge). Seventeenth-century appeals to the law of reflection in analysing (especially oblique) mechanical impact can also be found in Marcus Marci, De proportione motus …, Prague, 1639, N-3; Kenelm Digby, Two Treatises: One Concerning Bodies …, Paris, 1644, 45; E. Torricelli, De motu gravium …, Florence, 1644 (see Opere di Evangelista Torricelli (ed. G. Loria and G. Vassura), 4 vols., Faenza, 1919, ii, 230); G. B. Baliani, De motu naturali gravium …, Genoa, 1646, 105; E. Maignan, Perspectiva Horaria …, Paris, 1648, 292 ff. (including Lemma III on the equality of a rebounding ball's impulse and repulse, 297); W. Charleton, Physiologia Epicuro-Gassendo-Charletona …, London, 1654, 471–4 (compare P. Gassendi, Syntagma philosophicum, in idem, Opera omnia, 6 vols., Lyons, 1658, i, 354, 360–1); T. Hobbes, De Corpore …, London, 1655 (see the English translation of 1656 in The English Works of Thomas Hobbes of Malmesbury (ed. W. Molesworth), 11 vols., London, 1839–45, i, 384); and idem, Problemata physica, London, 1662, in which he provides an update of al-Haytham's claim that a heavy body and a light one share the same nature of reflecting until gravity alters the body's motion via discussion of a bullet shot against a wall (see English translation of 1682, Seven Philosophical Problems, in English Works (ed. Molesworth), vii, 50 ff.); and Pardies, op. cit. (71), 38 and 46 ff.

73 This analysis could be broadened by considering the convergence of geometrical treatments of oblique motion in both traditional mechanics and optics. The analysis of bodies moving on an inclined plane serves to idealize physical motion of a falling body in a similar way to the horizontal plane of a billiard table idealizes the motion of colliding bodies, where the geometrical representation of perpendicularly related components of such oblique fall is identical to that of oblique optical incidence. See Galileo Galilei, Dialogues Concerning Two Chief World Systems (tr. S. Drake), Berkeley, 1967, 23; and idem, Two New Sciences (tr. H. Crew and A. de Salvio), New York, 1954, 181 ff. As a foundational mathematical discipline optics was studied by all late sixteenth-century mathematicians. After Risner's Latin edition of Ibn al-Haytham's Optics in 1572, they would have been very familiar with its physical-projection interpretation of the geometry of optical reflection and dynamical analysis in terms of perpendicularly related and independent components of motion/force. In 1588, well before his work in mechanics and kinematics, Galileo had applied to the Academia del Disegno in Florence for the post of professional geometer to teach artists linear perspective, among other disciplines, probably using Daniel Barbaro's Practica della Perspectitiva, published in Venice several times in the 1560s and consulted by members of the Academia. See S. Edgerton, The Heritage of Giotto's Geometry: Art and Science on the Eve of the Scientific Revolution, Ithaca, 1991, 224 ff. In 1594 Galileo had also visited Guidobaldo del Monte to view his as yet unpublished Perspectiva libri sex, Pesaro, 1600. By 1601 Galileo was teaching optics; see S. Drake, Galileo at Work, Chicago, 1978, 35 and 51; and Dupre, S., ‘Mathematical instruments and the theory of the concave spherical mirror: Galileo's optics beyond art and science’, Nuncius (2000), 15, 551–88Google Scholar. The relevance of optical principles for the conceptualization of circular and projectile motion as resulting from two independent component motions was explicitly raised in the seventeenth century by Stefano degli Angeli. Arguing against Riccioli's erroneous analysis of projectile motion he said, ‘As for the mixture of two or more motions, nothing is easier. They always mix or, if we prefer, nature is perfectly able to hold them apart, so that the effects of the one do not impair or hinder those of the other, as Kepler and Descartes have shown it occurring in the case of the reflection of light’ (refering to the Optics, and Dioptrics, respectively, both published before Galileo's Two New Sciences). S. degli Angeli, Considerationi sopra la forza di alcune ragioni fisico mattematiche …, Venetia, 1667, Quartae Considerationi, 20, quoted in A. Koyré, A Documentary History of the Problem of Fall from Kepler to Newton, Philadelphia, 1955, 395.

74 The common feature of these three analyses of mutual collision, conducted to determine ‘laws of motion’, is that they are all concerned with the dynamically simpler case of direct impact. This allows Huygens, Wren and Wallis to conceptualize such phenomena via appeal to statical principles, the balance or lever. See Hall, op. cit. (69), 24–38; and R. Dugas, A History of Mechanics, New York, 1988, 172 ff.