1 Independently of Harriot, Marcus Marci (1595–1667), a university professor of medicine in Prague, also attempted to construct a dynamical theory of impact using generalized optical principles in his De proportione motus … (1639). As with Harriot, the physico-mathematics of oblique reflection gave Marci the principle of composition of motions, the perpendicular line of impact and resistance, the equality and opposition of action and effect and the preserved parallel-component motion. His approach differs radically from Harriot's by simultaneously treating the oblique incidence of a rebounding body as one in perpendicular fall and confusing weight with impulse, which allows his appeal to the law of the lever in his analysis of rectilinear rebound. See Aiton, E. J., ‘Ioannes Marcus Marci (1595–1667)’, Annals of Science (1970), 26, 153–64CrossRefGoogle Scholar; Sorensen, K. E., ‘A study of the De proportione motus by Marcus Marci’, Centaurus (1976), 20, 50–76CrossRefGoogle Scholar and (1977), 21, 246–77; and P. Svobodny (ed.), Johannes Marcus Marci: A Seventeenth-Century Bohemian Polymath, Prague, 1998.

2 Descartes's principles of motion are found in Treatise on Light, drafted 1630–2 but abandoned in 1633 and published as The World (1664), for which see Descartes, Descartes: The World and Other Writings (ed. and tr. S. Gaukroger), Cambridge, 1998, 3–75; Dioptrics, drafted at the same time as the Treatise, with a final draft written 1634–6, published in 1637, for Discourse 2 of which see the above Descartes: The World and Other Writings, 76–84, whereas the entire Dioptrics can be found in idem, Descartes: Discourse on Method, Optics, Geometry and Metereology (tr. P. J. Olschamp), New York, 1961, 65–173; and Principles of Philosophy, written 1641–4, for which see idem, Principles of Philosophy (tr. V. R. Miller and R. P. Miller), London, 1983.

3 His laws of motion are found in his Treatise on Light, yet his rules of impact first appear in his Principles of Philosophy.

4 Descartes's research into optics was crucial to the subsequent formulation of his laws of nature. His insight into the conservation of motion before and after impact arose from the long-established dynamics of optical reflection. I agree with Schuster: ‘How better to base the laws of nature than to use as an exemplar the dynamical principles revealed by successful optical research: Light, after all is just an impulse, so its behaviour clearly reveals the basic dynamics of forces and determinations.’ J. A. Schuster, ‘Descartes opticien: the construction of the law of refraction and the manufacture of its physical rationale, 1618–29’, in Descartes' Natural Philosophy (ed. S. Gaukroger, J. Schuster and J. Sutton), London, 2000, 258–312, 303. See also Gaukroger, S. and Schuster, J., ‘The hydrostatic paradox and the origins of Cartesian dynamics’, Studies in History and Philosophy of Science (2002), 33, 535–72CrossRefGoogle Scholar, 558. But this methodological move was not original to him. In the early seventeenth century his work formed part of a larger pattern of individuals constructing theories of impact or motion based upon optical principles.

5 Unlike al-Haytham, Descartes does not talk of a body's component motion simply towards the perpendicular or the surface plane; rather he reiterates the Mechanica's purely geometrical description of the compounded motion of a point (A) along two moving lines (AB to CD, and AC to BD, Figure 9) in his Principles of Philosophy. See Descartes, Principles of Philosophy, op. cit. (2), 55.

6 Dioptrics, Discourse 2, in Descartes, Descartes: The World and Other Writings, op. cit. (2), 77.

7 As noted by A. I. Sabra in his Optics, Astronomy and Logic: Studies in Arabic Science and Philosophy, Aldershot, 1994, 552.

8 Descartes claims the source of conserved motion is to be found in the immutability of God. But the fact that forces are conserved in Nature was evident to Descartes, and others, within the optical tradition of light mechanics and its analysis of reflection, a subject with which Descartes was very familiar, from at least 1620.

9 Descartes, Descartes: The World and Other Writings, op. cit. (2), 78.

10 ‘When one [body] … pushes against another it cannot give the other any motion except by losing as much of its own motion at the same time; nor can it take any of the other's motion unless its own is increased by the same amount.’, Law 2, Treatise on Light, in Descartes, Descartes: The World and Other Writings, op. cit. (2), 27.

11 See Principles of Philosophy, II, 40–1 in Descartes, Principles of Philosophy, op. cit. (2), 62. This point is also made in D. Des Chene, Physiologia: Natural Philosophy in Late Aristotelian and Cartesian Thought, Ithaca, 1996, 288–9.

12 Marci has eight rules of impact to Descartes's seven. Descartes's rules 2 and 4 describe the same cases as Marci's rules 2 and 4. Where Descartes's famously erroneous rule 4 states that a smaller body will (always) rebound off a larger stationary body without moving it, so does Marci's. One significant difference between the two is to be found in Marci's rule 1 and Descartes's rule 6 on the collision between equal bodies where one is at rest (as Harriot's motive action one, Figure 2, but in direct collision). Marci correctly states that the moving body will stop on impact and the other will move off. Descartes claims that they will both rebound. Compare the lists presented in Svobodny, op. cit. (1), 143; and D. Garber, Descartes' Metaphysical Physics, Chicago, 1992, 256–60.

13 Marci had given a copy of his work to Galileo in 1640; see Aiton, op. cit. (1), 154. He is also mentioned in Mersenne's correspondence; see A. Solcova, ‘Johannes Marcus Marci of Kronland and mathematics’, in Svobodny, op. cit. (1), 173–202, 193.

14 See P. Damerow, G. Freudenthal, P. McLaughlin and J. Renn, Exploring the Limits of Preclassical Mechanics. A Study of Conceptual Development in Early Modern Science: Free Fall and Compounded Motion in the Work of Descartes, Galileo, and Beeckman, New York, 1992, 122. Their interpretation of Descartes's argument for the relation between the speeds of colliding balls in terms of the mechanical analogy of oblique reflection through a tangent plane and components of oblique motion is untenable, given his failure to take account of the basic dynamical feature of reflection: the reciprocity of impact forces. So their claim that Descartes's system provided the basis for the generalized application of the parallelogram rule in mechanics is also untenable.

15 For Descartes's hydrostatic conception of light propagation see Schuster, op. cit. (4); Gaukroger and Schuster, op. cit. (4). This mechanization of light propagation shared defining characteristics of the light-metaphysical process of multiplication of species – the instantaneous, rectilinear, contiguous-contact transmission of force (agency) through a medium without local motion.

16 No doubt the unsuitability of his model of light with respect to reflection informed his decision to ignore the dynamical reversal of an obliquely incident movable's perpendicular component of motion in terms of action and reaction, which led to the overwhelming failure of his impact rules to address oblique collisions.

17 M. Clagett, The Science of Mechanics in the Middle Ages, Madison, 1959; M. Clagett and E. A. Moody, The Medieval Science of Weights, Madison, 1952; S. Drake and I. E. Drabkin, Mechanics in Sixteenth-Century Italy, Madison, 1969, 5–16; R. Dugas, A History of Mechanics, New York, 1988, Chapters 1–6; De Groot, J., ‘Aspects of Aristotelian statics in Galileo's dynamics’, Studies in History and Philosophy of Science (2000), 31, 645–64CrossRefGoogle Scholar; C. Lewis, The Merton Tradition and Kinematics in Late Sixteenth and Early Seventeenth Century Italy, Padova, 1980; Bertoloni Meli, D., ‘Guidobaldo dal Monte and the Archimedean revival’, Nuncius (1992), 7, 3–32CrossRefGoogle Scholar; W. Wallace, Prelude to Galileo: Essays on Medieval and Sixteenth-Century Sources of Galileo's Thought, Dordrecht, 1981, Chapters 1–5.

18 In his student notebook Newton conceives of both gravity and magnetism in terms of light metaphysics' conception of causal agencies as radiative phenomena operating according to optical principles such as reflection and refraction. See J. E. McGuire and M. Tamny, Certain Philosophical Questions: Newton's Trinity Notebook, Cambridge, 1983, 431 and 379.

19 Just as Harriot constructed his ‘doctrine of reflexions’ in constructing a theoretical treatment of mechanical collisions, so Newton conceives of laws of motion as ‘laws of reflection’.

20 Just as Descartes characterized light as a determination, or tendency to motion, which could itself be a compound of two other (orthogonal) determinations, so he characterized the tangential tendency of a body in circular motion as a compound of two other determinations, circular and centrifugal.

21 Unlike Descartes's hydrostatic view of light, Newton's atomistic view guarantees a unified conception of light's propagation and its interaction with matter as one of physical projection and mechanical impact.

22 J. Herivel, The Background to Newton's Principia: A Study of Newton's Dynamical Researches in the Years 1664–1684, Oxford, 1965, 132 ff. Dated ‘20th January 1664’, these owe much to the Cartesian analogies of reflection and refraction in the Dioptrics.

23 Herivel, op. cit. (22), 137–8.

24 Herivel, op. cit. (22), 139, note f.

25 ‘Of the separation of bodies: If two equall and equally swift bodys … meet one another they shall bee reflected, soe as to move as swiftly frome one another after the reflection as they did to one another before it.’ Herivel, op. cit. (22), 142.

26 ‘If the body g reflect on the immoveable surface (dv) at its corner o [Figure 2] its parallel motion (viz, from d to v) shall not be hindered by the surface dv, (viz. if the center of g's motion were distant from the perpendicular dn an inch at one minute before reflection it shall bee so farr distant from it at one minute after reflection) ffor dv is noe ways opposed to motion parallel to it, and a body might slide/move upon it without looseing any motion, and if at the first moment of contact the body g should loose its perpendicular and only keep its parallel motion it would (perhaps) continue to slide upon it and not reflect.' Herivel, op. cit. (22), 179.

27 Newton's Principia, Law III, Corollary III. This tangent plane of mechanical contact originates from the identical construction in ancient optics for the reflection of light from curved mirrors, a geometrical feature declared essential by Cavalieri for the application of the law of reflection to the rebound of anything moving in straight lines which is incident upon a curved surface, in his Lo specchio ustoria overo Trattato delle settioni coniche, Bologna 1632. See Ariotti, P. E., ‘Bonaventura Cavalieri, Marin Mersenne, and the reflecting telescope’, Isis (1975), 66, 303–21CrossRefGoogle Scholar, 307–8.

28 Herivel, op. cit. (22), 159.

29 ‘Of the mutuall force in reflected bodys. Ax.121. If 2 bodys p and r meet the one the other, the resistance in both is the same for soe much as p presseth upon r so much r presseth on p. And therefore they must both suffer an equall mutation in their motion; 119. If r presse p towards w then p presseth r towards v. Tis evident without explication; 120. A body must move that way which it is pressed.’ Herivel, op. cit. (22), 159.

30 Herivel, op. cit. (22), 159.

31 Taking together Propositions 18 and 19, Shapiro rejects the claim that Newton read Kepler's Optics as an undergraduate, arguing that he paid scant attention to the Optics of al-Haytham and Witelo. The Optical Papers of Isaac Newton, Vol. 1. The Optical Lectures 1670–1672 (ed. A. E. Shapiro), Cambridge 1982, 9. However, Newton had certainly been influenced by light metaphysics – and he had access to Isaac Barrow's library during 1664 and 1665, around the time of his writings in the Notebook and Waste Book. This library contained, inter alia, Risner's 1572 edition of both al-Haytham's and Witelo's Optics, Cavalieri's Lo specchio ustoria, the Aristotelian Mechanica and Problemata, Blancanus's Aristotelis loca mathematica, Hooke's Micrographia and Kepler's Ad Vitellionem Paralipomena. See M. Feingold, ‘Isaac Barrow's library’, in Before Newton: The Life and Times of Isaac Barrow (ed. M. Feingold), Cambridge, 1990, 333–72, 341 ff.

32 Westfall has argued that Newton arrived at this idea by considering the mutual rebound of bodies relative to their mutual centre of motion or gravity (R. Westfall, Never at Rest, Cambridge, 1980, 147). But its optical origins are evident: Newton states this fundamental concept within Axioms 119–122 prior to their subsequent application to mutual rebound relative to a common centre of motion – as will be shown below. Herivel also takes Newton's Axiom 121 and Axiom 119 as expressing the third law of motion, in which there is no appeal to a common centre of motion. Herivel, op. cit. (22), 161.

33 ‘If two bodies (b and c) meet at q their center of motion shall bee in the same line (kp) after reflection in which it was before it. For the motion of b towards d the center of their motion is equall to the motion of c towards d … therefore the bodys b and c have equal motion towards the points k and m, that is towards the line kp … And at their reflection so much (c) presseth (b) from the line kp; so much (b) presseth (c) from it (ax. 121). Wherefore they must have equal motion from the line kp after reflection … Therefore e and g have equall motion from the point o which … must therefore be the center of motion of the bodys (b and c) when they are in the places g and e, and it is in the line kp. The demonstration is same in all cases.’ Herivel, op. cit. (22), 168; emphasis added.

34 This is a fascinating story which, unfortunately, cannot be pursued here. But see his extended analyses combining motion, impact and statics in his Waste Book – in Herivel, op. cit. (22), 161–79. Notice how Figure 9's representation of the line joining the two bodies with their common centre of motion and their respective parallel and perpendicular components of motions (Herivel, op. cit. (22), 169) is geometrically identical to Guidobaldo del Monte's geometrical analysis of the balance in his Mechanicorum liber (1577) – a book Newton could also find in Barrow's library. See Feingold, op. cit. (31), 359; also Drake and Drabkin, op. cit. (17), 279 ff.

35 This latter (equally undated) paper does not include the section ‘Observations’ which will be discussed later. As such, ‘On the Laws of Reflection’ is not as complete as ‘The Lawes of Motion’, and it is reasonable to assume that the latter indicates development in Newton's conceptualization of his theoretical foundations via the generalization of laws of impact and rebound to motion per se. See Unpublished Scientific Papers of Isaac Newton (ed. A. R. Hall and M. Hall), Cambridge, 1978, 157.

36 Newton's De gravitatione et aequipondio fluidorum, in Unpublished Scientific Papers of Isaac Newton, op. cit. (35), 89 ff.

37 ‘If the body a in the Perimeter of ye cirkle or sphaere adce moveth towards its centre … its velocity to each point … of yt circumference is as ye chords … drawne from that body to those points are.’ Unpublished Scientific Papers of Isaac Newton, op. cit. (35), 15.

38 Unpublished Scientific Papers of Isaac Newton, op. cit. (35), 33.

39 ‘If ye >s adc, aec are alike viz. ad=ec & co … & 3 bodys move from the point a uniformely & in equall times ye first to d, the 2^nd to e ye 3d to c; Then is the thirds motion compounded of ye motions of the first & second.’ Unpublished Scientific Papers of Isaac Newton, op. cit. (35), 16.

40 Unpublished Scientific Papers of Isaac Newton, op. cit. (35), 33–4

41 ‘The motion of a body tends one way directly & severall other ways obliqly. As if ye body A move directly towards ye point B it also moves obliquely towards all ye lines BC, BD, BE & c wch passe through yt point B; & shall arrive to ym all at ye same time. Whence its velocity towards ym is in such proportion as its distance from them yt is, as AB, AC, AD, AE & c.’ Unpublished Scientific Papers of Isaac Newton, op. cit. (35), 157; emphasis added.

42 ‘If a body A moveth towards B wth the velocity R, & by ye way hath some new force done to it wch had ye body rested would have propell'd it towards C wth ye velocity S. Then making AB:AC::R:S, & Completing ye Parallelogram BC ye body shall move in ye Diagonall AD & arrive at ye point D wth this compound motion in ye same time it would have arrived at ye point B with its single motion.’ Unpublished Scientific Papers of Isaac Newton, op. cit. (35), 158.

43 ‘Motion may be lost by reflection. As if two equall Globes A & α wth equall motions from D & δ done in the perpendicular lines DA & δα, hit one another when the center of ye body α is in ye line DA. Then ye body A shall loose all its motion & yet ye motion of α is not doubled. For completing ye square Bβ, ye body α shall move in ye Diagonall α, & arrive at C but at the same time it would have arrived at β wthout reflection.’ Unpublished Scientific Papers of Isaac Newton, op. cit. (35), 164. That is, body A loses its motion by transferring it to body α as a component motion.

44 ‘Motion may be gained by reflection. For if the body α return wth ye same motion back again from C to α. The two bodyes A & α after reflection shall regain ye same equall motions in ye lines AD & αδ (though backwards) wch they had at first.’ Unpublished Scientific Papers of Isaac Newton, op. cit. (35), 164.

45 R. Westfall, Force in Newton's Physics: The Science of Dynamics in the Seventeenth Century, London, 1971, 351 ff; and idem, op. cit. (32), 148; Erlichson, H., ‘Newton and Hooke on centripetal force motion’, Centaurus (1992), 35, 46–63CrossRefGoogle Scholar, 47; Erlichson, H., ‘Huygens and Newton on the problem of circular motion’, Centaurus (1994), 37, 210–29CrossRefGoogle Scholar, 222; J. B. Brackenbridge, ‘Newton's mature dynamics: a crooked path made straight’, in Isaac Newton's Natural Philosophy (ed. J. Z. Buchwald and I. B. Cohen), Cambridge, 2001, 105–37, 106.

46 Found both in Descartes's Treatise on Light, posthumously published in 1664, and in his Principles of Philosophy, 1644.

47 Treatise on Light, in Descartes, Descartes: The World and Other Writings, op. cit. (2), 54–5.

48 Herivel, op. cit. (22), 146–7.

49 ‘ … supposing [body oc] have moved from (c) by (h) to (b). Then the resistance of the body def (which is equall to its pressure upon def) is able to destroy its force of moveing from c to g and to generate in it as much force from (b) to [k] the quite contrary way.’ Herivel, op. cit. (22), 146–7; emphasis added.

50 Descartes's Meteorology, Discourse 8 in Descartes, Descartes: The World and Other Writings, op. cit. (2), 92; Hooke's Micrographia, in R. Gunther, Early Science in Oxford, Vol 13: The Life and Work of Robert Hooke (Part V), Oxford, 1938, 61, Figure 3.

51 See The English Works of Thomas Hobbes of Malmesbury (ed. W. Molesworth), 11 vols., London, 1839, i, 273 ff.

52 See I. Stoll, ‘Johannes Marcus Marci and mechanics’, in Svobodny, op. cit. (1), 147 ff.

53 Herivel, op. cit. (22), 130.

54 Recent discussions of this development in the dynamical analysis of circular motion have failed to appreciate its use of optical principles. Westfall asserts that Newton's ability to ‘compare the force of one impact, in which the component of the body's motion perpendicular to the side it strikes is reversed, to the force of the body's motion’ arises simply from the ‘geometry of the square’ (Westfall, op. cit. (32), 149–50), rather than from a direct application of the geometrical law of reflection and its dynamical interpretation. Erlichson asserts that Newton here applies Galileo's idea of the composition of motions from his Two New Sciences (Erlichson, op. cit. (45), 47–8), without appreciating that the compound analysis of such oblique rectilinear impact and reflection pre-dates Galileo by centuries, and was explicitly geometrized as the parallelogram rule for rectilinear motion in Descartes's Dioptrics (1637) – published before Galileo's work. The knowledge of optics common to sixteenth- and early seventeenth-century mathematicians (including Galileo in the 1590s) ensured their exposure to the dynamics of independent and perpendicularly related components of rectilinear motion. On the other hand, Cohen addresses Newton's explicit use here both of the geometrical law and of the dynamics of reflection. See Isaac Newton: The Principia (tr. I. B. Cohen and A. Whitman), Berkeley, 1999, 70.

55 Herivel, op. cit. (22), 129–30.

56 Indeed, if the impulse was sufficient only to prevent its motion along bx without repulsing it along bc, then it would move along bg – just as when a billiard ball obliquely strikes an equal but stationary ball.

57 Herivel, op. cit. (22), 130.

58 Herivel, op. cit. (22), 129. From this result we can deduce the modern formula for centripetal force. See Westfall, op. cit. (45), 355.

59 It was a crucial creative leap because Newton then accepted the Cartesian rejection of any centripetally attractive force responsible for circular motion – a metaphysical conviction which conflicted with the parallelogram rule's employment of two rectilinear component motions in the theoretical analysis of circular motion found in the ancient Greek Mechanica, i.e. one tangential and the other centripetally attractive. Without this creative use of light mechanics, he could not have applied the parallelogram rule for compound rectilinear motion to circular motion.

60 Principa, Book 1, Section 2, To find centripetal forces, Propositions 1 and 2. See Isaac Newton: The Principia, op. cit. (54), 444 ff.

61 See M. Cohen and I. E. Drabkin (eds.), A Source Book in Greek Science, Cambridge, MA, 1948, 190; Aristotle: Minor Works (tr. W. S. Hett), London, 1963, 341.

62 Ibn al-Haytham had stated that a light ray and a body ‘share the same nature of reflecting’. It was only the body's weight which pulled it away from the ray's straight-line motion. In the absence of gravity a body would behave precisely as light, travelling in uniform motion in straight lines and being reflected in straight lines. Ever since Risner's edition of al-Haytham's work in 1572, mathematicians and natural philosophers reading this would be aware of such a mechanical generalization of the optical principle of rectilinearity.