Optical reflection and mechanical rebound: the shift from analogy to axiomatization in the seventeenth century. Part 2

RUSSELL SMITH

doi:10.1017/S0007087407000374

Optical reflection and mechanical rebound: the shift from analogy to axiomatization in the seventeenth century. Part 2

Published online by Cambridge University Press: 25 September 2007

RUSSELL SMITH

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RUSSELL SMITH: Affiliation:
Studley View, 113 Lower End, Leafield, Oxfordshire, OX29 9QG, UK. Email: citizensmithx@yahoo.co.uk.

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Type: Research Article
Information: The British Journal for the History of Science , Volume 41 , Issue 2 , June 2008, pp. 187 - 207
DOI: https://doi.org/10.1017/S0007087407000374 [Opens in a new window]
Copyright: Copyright © 2007 British Society for the History of Science

Descartes: principles of light and ideal motion

Thomas Harriot, whose work was discussed in the first part of this paper, generalized the kinematics and dynamics of optical reflection to form the foundations of an atomistic theory of impact.Footnote ¹ What of Descartes and his abhorrence of the vacuum? What insights, if any, does he obtain from his early studies in optics? There are two striking differences between Harriot's conception of principles of motion, through the grounding of theories of impact on generalized optical principles, and that of Descartes.Footnote ² Firstly, Descartes presents his principles as universal laws of motion, applicable not only to how all things actually move but also to how all things would move though their paths are impeded – that is, to their tendency to move. Secondly, his conception of these laws of motion does not arise from his (much later) formulation of a theory of impact.Footnote ³ It rather stems from cosmological considerations that involved a mechanical explanation for the propagation of sunlight and analyses of light–matter interactions.Footnote ⁴

Descartes is the first explicitly to apply the geometry of the parallelogram rule found in the Aristotelian Mechanica to the reflection of light. Its provenance is established by Descartes's identical kinematical explication of this geometry.Footnote ⁵ He gives a new physical interpretation of the dynamics of reflection in line with the tradition of light mechanics. However, this interpretation follows his novel conception of the nature of light as a tendency to move, not motion itself. Hence the mechanical explanation of the equality of light's angles of incidence and reflection in terms of an unaffected parallel component motion and a resisted perpendicular component must apply to a tendency to motion. He then resolves the ball's motion (AB, Figure 1) into a determination (a directed tendency to move) and its force of motion, which are physically independent. One can be changed without affecting the other.Footnote ⁶ Using this distinction, Descartes deviates from the light-mechanical explanation of reflection by ignoring the perpendicular component's reversal upon impact.Footnote ⁷ He favours, instead, an analysis in terms of the conservation of motion.Footnote ⁸

Figure 1.

Dioptrics: Descartes on reflection.

Descartes's explanation for the path of rebound/reflection incorporates the geometry of image location, the parallelogram rule of the Mechanica and a framing circle taken from long-established diagrammatic representations of refraction. The incident line forms the circle's radius. It represents both the direction of the ‘movable’ (such as a ray or tennis ball), the measure of its force and the distance travelled, in a unit of time. In the same time as it takes to travel from A to B, if unhindered at B it would travel to D (Figure 1). Yet since it is prevented from doing so by the surface, and with its quantity of motion AB conserved, its rebound path must also be of radius length. After combining this conserved quantity of motion with the conserved parallel component determination (HF), the only physically possible path possessing both these determinations is that which results in rebound at equal angles, BF.Footnote ⁹ In this way, Descartes sidesteps the key dynamical element of mechanical impact – the resistance/repulsion acting along the line perpendicular to the tangent plane at the point of impact – focusing instead on the conservation of determinations and their quantities of motion.

Descartes's selection from the kinematical and dynamical principles of light mechanics creates a bridge between optics and mechanics different to that of Harriot and Marci. Though Harriot's resultant ‘doctrine of reflexions’ is superior to Marci's theory of impact, they are built upon the same light-mechanical foundations. In disregarding the dynamical reversal of the perpendicular component of oblique motion, Descartes's bridge between optics and mechanics could not therefore lead to a theory of impact grounded in the law of reflection. Rather, this Cartesian bridge allows the formulation of dynamical principles not limited to impact phenomena. From the kinematics and dynamics of reflection he draws the following general principles: (i) the conservation of both quantity of motion and component determinations, and (ii) the tendency to uniform rectilinear motion in the absence of obstacles, or interfering forces (in his appeal to the mechanical interpretation of the geometry of image location and in the light-mechanical analysis of rebound, respectively).

Of course, Descartes's natural philosophy necessitates a law of motion specifically governing mechanical collisions.Footnote ¹⁰ Yet it, and the later rules of impact derived from it, concern direct collisions. The law is largely premised upon the physics of reflection.Footnote ¹¹ However, the subsequent rules also appeal to statical concepts and to the scholastic doctrine that motion results from an excess of force over resistance. This approach is very similar to that of Marci.Footnote ¹² It is possible that Descartes knew of Marci's work before publishing his Principia.Footnote ¹³ Indeed, Marci's failure to consider oblique collisions between unequal balls is reflected in Descartes's solitary and flawed analysis of just such a case, in which he failed to incorporate the light-mechanical principle of action and reaction. Descartes therefore failed to apply the kinematics and dynamics of oblique reflection to collision between unequal bodies (unlike Harriot), let alone equal bodies (unlike Marci). However, the generalization of the dynamics of reflection to mechanical collisions facilitated the generalizability of the parallelogram rule in mechanics. This was something that Descartes therefore signally failed to achieve, unlike Harriot and, especially, Newton.Footnote ¹⁴

In conclusion, Descartes's introduction of a hydrostatically derived model of light's propagation and interaction with matter entailed the generalization of physical principles employed in interpreting the geometry of reflection from a particle's local motion to its tendency to motion.Footnote ¹⁵ This extension of the dynamical principles employed in the mechanical analogy of optical reflection to light propagation allowed Descartes to construct a single theory of dynamics for both motion (through light metaphysics' focus on light propagation) and resistance to motion (through light mechanics' focus on light–matter interaction). Light's actual propagation as a tendency to motion constitutes a physical demonstration of a material body's tendency to motion, which does not determine its actual motion in a plenum. The mechanical analysis of a tendency's interaction with matter (such as incidence and reflection) demonstrates that the dynamical principles employed apply equally to a body's local motion (such as incidence and rebound) and its component tendencies. However, this dynamical synthesis of the twin optical traditions of light mechanics and light metaphysics was weighted toward the latter. His hydrostatic mechanization of the multiplication of species equally asserted the infinite velocity of light, contrary to the kinematical requirements of light mechanics. This process of pressure-pulse transmission was also entirely dissimilar to that of mechanical rebound. His treatment of optical reflection as mechanical rebound thus remained purely analogical.Footnote ¹⁶ Yet his dynamical analysis of mechanical rebound via this analogy, when combined with the atomistic mechanization of light propagation, was what allowed the formulation of a unified dynamics of mechanical motion.

Axiomatization

The paper has thus far charted the historical continuity and conceptual concretization of the mechanical analogy of reflection. It has shown how the widespread yet implicit belief in the generalizability of the geometrical law of reflection to material collision led key kinematical and dynamical features of the physical interpretation of that law to be applied to oblique mechanical collisions in the early to mid-seventeenth century. There is of course more to mechanics than can be generalized from optics, since there are variables such as differing weights and speeds involved in analysing collisions between physical bodies. These are not applicable to light motion/reflection. There exists, therefore, a traditional ‘non-optical’ strand of mechanical thought originating in ancient Greek mechanics and statics that developed further via the medieval science of weights through to Galileo's more immediate predecessors in the sixteenth-century Italian school of Tartaglia, Cardano, Benedetti, Guido Ubaldo and Baldi.Footnote ¹⁷ Yet as the seventeenth-century search for laws of motion gathered pace after Descartes, atomism's particulate theory of light offered a conceptualization of light motion as ideal physical motion, a species of material motion completely unaffected by gravity or air friction. In this way, the kinematics and dynamics of light motion become the kinematics and dynamics of mechanical motion, when idealized to the degree necessary for the construction of laws of motion at a higher level of abstraction than those achieved by Galileo for motion affected by gravity (such as projectile and free fall).

Newton: reflection and material collisions

In Newton's early dynamical writings he applies optical principles both to the propagation of forces such as gravity and magnetismFootnote ¹⁸ and to material impact/rebound in light mechanics.Footnote ¹⁹ These two traditions allow the conceptualization of phenomena ordinarily treated in distinct disciplines (such as astronomy and mechanics) via a single theoretical framework, physical optics. Descartes had previously similarly conjoined the two through his mechanization of the multiplication of (light) species in the celestial medium and the compound analysis of light reflection/mechanical rebound. However, he offered a confused account of circular motion by applying to the tangential component of circular motion the above compound analysis of light as a rectilinear tendency to motion, which failed to grasp the dynamics involved.Footnote ²⁰ With Newton's atomistic conception of light as local (projectile) motion,Footnote ²¹ his application of the compound analysis of reflection/rebound to circular motion entailed the direct generalization of the dynamics of reflection from rectilinear (optical and mechanical) motion to circular (geometrical and mechanical) motion. Newton was able to identify the dynamical principles requiring axiomatization in the construction of a general theory of mechanics.

Newton's early application of optical principles to rectilinear mechanical motion begins with mechanical definitions of reflection and refraction.Footnote ²² We shall focus on the former: ‘A quantity [body] is reflected when meeting another quantity it looseth the determination of its motion by rebounding from it’, where their parting is due as if to ‘some springing motion in themselves or in the matter crouded betwixt them’.Footnote ²³ Newton then clarifies the Cartesian distinction between determination and motion employed in this definition of reflection in a marginal note on circular motion: ‘noe motion is lost in reflection. For then circular motion being made by continuall reflection would decay’.Footnote ²⁴ A body loses its determination to travel in a certain direction, but its quantity of motion is simply redirected. He reiterates this element of conservation in the definition a little later by proposing that motion before reflection is the same as motion after reflection.Footnote ²⁵ This matches the mechanical generalization of optical principles as conducted by, for example, Ibn al-Haytham and Harriot.

Newton continues to apply the dynamical principles developed within light mechanics to the analysis of material collisions of a body both against a surface and against another body. Regarding the former type of collision (Figure 2), he emphasizes the preserved parallel component of a body's obliquely incident motion and suggests that it is entirely independent of the perpendicular component. It is preserved even when the latter is lost, implicitly raising Harriot's first motive action (see Figure 2 in Part 1 of this paper).Footnote ²⁶ In examining two-body collisions, Newton later explicitly applies to each body this compound analysis of a body's reflection from a surface, where the ‘surface’ is reconstructed as the plane of contact tangent to the point of collision.Footnote ²⁷ Regarding the latter type of collision (Figure 3 Footnote ²⁸), under the marginal heading ‘Of the mutuall force in reflected bodys’, he sets out several axioms which duplicate the generalization of the dynamics of reflection found in the light mechanics of writers such as al-Haytham and Harriot. These include the identification of the line of action in impact with that joining the two bodies' centres, the perpendicular component of oblique motion, Kepler's equality of action and effect in opposite directions and the change of motion in rebound via alteration of this perpendicular component.Footnote ²⁹ The concluding axiom (122) states: ‘Therefore if the body p come from c and the body r from d soe much as p's motion is changed towards w, so much the motion of r will be changed towards v’.Footnote ³⁰ Hence here, in Newton's early axioms governing the ‘reflection’ of bodies in rectilinear motion employing generalized optical principles, is the first conceptualization of what later became his third law of motion: ‘To every action there is an equal and opposite reaction: or the mutual action of two bodies upon each other are always equal, and directed to contrary parts.’ This idea of action (impact) and reaction (rebound) being equal and opposite has long been evident in the ideal case of optical reflection and its mechanical analogy and was expressed in Kepler's Optics.Footnote ³¹ Newton's analysis of two bodies' mutual ‘reflections’ generalized the simple case of equal and opposite reflection/rebound from a surface to the reflections between two incident movables.Footnote ³²

Figure 2.

Early Newton: reflection from a surface.

Figure 3.

Early Newton: two-body collision.

The equality of action and reaction in opposite directions is evident when the incident and rebound angles are equal, in cases such as equal bodies colliding with equal motion, whether oblique or direct. However, where a body's angle of incidence differs from its rebound angle on colliding with another incident body (see Figure 7 in Part 1 of this paper), the equality of action and reaction is difficult to determine. Newton overcomes this problem by applying this principle to the motion of both bodies relative to their common point of motion, from which their rebound motions are equal and opposite (Figure 4).Footnote ³³

Figure 4.

Newton: rebound – equal and opposite.

Newton's creative introduction of statical concepts into the dynamical analysis of rectilinear collisions enabled the application of the dynamics of reflection to be extended beyond the simple cases of mechanical reflection mirroring the analogies of light mechanics to those cases in which the geometry of reflection does not hold.Footnote ³⁴ This in turn allowed Newton to appreciate their utility with respect to the formulation of general laws of motion, such as the equality of action and reaction evident in the pure case of rebound: optical reflection.

Newton's mechanical laws of reflection

Around 1666 Newton wrote a paper on the motion of bodies through space and their reflections from other bodies, entitled ‘The Lawes of Motion’, most probably drafted as his virtually identical paper entitled ‘On the Laws of Reflection’.Footnote ³⁵ Newton's notion that his own Harriot-like ‘doctrine of reflexions’ constituted general laws of motion is doubtless linked to his statement elsewhere that the ability to reflect light or other bodies, or to be reflected, in collisions, is the defining property of a corporeal, impenetrable, body, where such (reflective/rebound) motion occurs ‘in accord with certain laws’.Footnote ³⁶ This linking of optical and material incidence upon corporeal surfaces allows Newton to refine the geometrical content of principles of motion, as is evident in the conceptual shift occurring between the geometrical treatment of compound rectilinear motion in his ‘To resolve problems by Motion’ and ‘The Lawes of Motion’.

In the former paper, he applies a ‘mechanical’ method to the resolution of mathematical problems via the introduction of motion into geometrical constructions. His statement and demonstration of the basic kinematical principles of component and compound motion are themselves geometrical. Hence Newton (Figure 5a) represents the velocity of a body moving along diameter ac relative to any point (d) on the circumference as a chord (ad),Footnote ³⁷ and his subsequent geometrical demonstration of this proposition on the relative velocities of a moving body rests on Pythagoras' theorem.Footnote ³⁸ Having introduced the notion of direct motion possessing various component velocities, Newton moves on to show that such direct motion can result from a compounding of these motions (Figure 5b), by identifying two components of body a's motion arranged in a parallelogram with the direct motion of two other bodies.Footnote ³⁹ His demonstration appeals to the geometrical argument of the previous proposition (compare adf in Figure 5a and 5b),Footnote ⁴⁰ in determining the component motions of the first and second bodies towards c.

Figure 5a.

Newton: component chords.

Figure 5b.

Newton: summed motion.

On the other hand, in ‘The Lawes of Motion’, Newton is examining bodies reflecting from others, so their motion towards impact can be conceived in terms of optical incidence and its mechanical generalization, through the compound analysis of obliquely incident motion. The paper has three chapters: ‘How solitary bodyes are moved’, ‘How bodys are Reflected’ and ‘Some Observations about Motion’. At the beginning of the first chapter Newton restates the two principles above, but in a way which now requires no separate geometrical demonstration. This is because he employs the well-established compound analysis of optical incidence/reflection to state general principles of motion, making them seem virtually self-evident. Regarding the first principle, the motion of a body A along the diameter AB (Figure 6a) is no longer considered relative to a point on the circumference (C), but as being oblique to a line (BC). What was once a chord (AC) is thus transformed into the perpendicular component of A's oblique motion AB towards ‘surface’ CB. Moreover, Newton introduces here a body's several tendencies to motion, as presented in Descartes's dynamical analysis of optical reflection and its mechanical analogy. In this way the component velocities of A's velocity towards B, relative to other lines/surfaces, are, as its component of impact motion upon each of these lines/surfaces, perpendicular to these incident ‘surfaces’.Footnote ⁴¹ Newton's law of a solitary body's various tendencies to motion and their relative measures follows uncontroversially once Figure 6a is viewed as representing a light ray or tennis ball moving from A to B, obliquely incident upon various surfaces (BC, BD, BE); for the component of their motion directed towards these surfaces will be the perpendicular component of such oblique incidence.

Figure 6a.

Newton: oblique components.

With respect to the second proposition (Figure 6b), Newton demonstrates that these components of direct motion introduced in the first proposition can produce the direct motion, by representing the direct motion possessing component tendencies to motion (AB – Figure 6a), as motion resulting from the composition of two such tendencies (AD – Figure 6b).Footnote ⁴² The continued appeal to the compound analysis of reflection is evident in Newton's identical representation of a line/surface (BD), upon which the actual motion of the body is obliquely incident (AD), as found in the first proposition (DB and AB respectively – Figure 6a). This implies that the component of velocity AB (Figure 6b) corresponds to the component of velocity AD as the perpendicular component of oblique motion AB (Figure 6a). Newton's appeal to bodies in collision as a means of compounding motions constitutes a mechanical generalization of the dynamics of impact derived from optical reflection: A's physical tendency towards B and the impressed force of impact towards C, composing its (oblique) motion AD. He thus provides a mechanical substitution for the perpendicularly related components of oblique optical incidence constructed within the tradition of light mechanics, as is evident in his subsequent clarification of the above parallelogram principle.

Figure 6b.

Newton: compound motion.

In the paper's last section, ‘Some Observations about Motion’, Newton demonstrates (through Figure 7) both that the actual motions of two bodies are convertible into two component tendencies to motion of one body moving obliquely to a plane of incidence (diagonally),Footnote ⁴³ and that the two component tendencies to motion of one body moving obliquely to a plane of incidence (diagonally) are convertible into the actual motions of two bodies via mechanical collision.Footnote ⁴⁴ The latter case matches Harriot's first motive act (see Figures 2 and 3 in Part 1 of this paper), and shows that his second principle's composition of tendencies to motion via the parallelogram rule constitutes a mechanical generalization of the dynamical analysis of reflection. Taking Cα as the oblique line of impact of particle α (of light or matter) upon ‘immovable surface’ δβ, its parallel component CB would be preserved unchanged while its perpendicular component Cβ would be reversed, resulting in the usual reflection at an angle equal to BαC. However, taken as a body hitting instead an equal but stationary movable body, its parallel component (CB) is still preserved unchanged, but its perpendicular component Cβ is entirely transferred to this other body, resulting in the incident body's rebound entirely in the direction αδ of its sole remaining (parallel) component motion, and the struck body's motion in the direction AD of its sole, newly acquired and perpendicular, component motion. Running the collision forwards, as two moving bodies meeting at right angles, it becomes clear that α's motion along δβ is perpendicular to the line of body A's impact (DB). This is dynamically identical to motion parallel to a surface of impact vis-à-vis oblique reflection, and so it will be preserved unchanged upon its impact with body A. But since the components of reflection are independent of each other, body α's initial motion is preserved and combined with the motion obtained from the perpendicular impact of body A. We thus arrive at body α possessing perpendicularly related components of motion, just like a light ray αC obliquely incident upon surface βC (or, running the collision backwards, Cα upon βα).

Figure 7.

Newton: compounded collisions.

It should now be clear why Newton conceives of his ‘Lawes of Motion’ as ‘Laws of Reflection’. The kinematical and dynamical principles of reflection developed within the optical tradition of light mechanics become directly applicable to colliding bodies upon the atomistic conception of light. Shifting from the simple case of a body rebounding at equal angles from an immovable surface, to that of the mutual collision of two moving or movable bodies disobeying the geometrical law of reflection, rests on the recognition that it is these principles of reflection which require mechanical generalization, not the geometrical law, as previously assumed in the long-espoused mechanical analogy of reflection. Harriot's ‘doctrine of reflexions’ achieved this, but he was unable to generalize Kepler's dynamical analysis of impact/rebound from a surface, equal in power and opposite in direction, to mechanical collisions where the geometrical law of reflection does not hold. Newton, on the other hand, did so by analysing the mutual rebound of bodies from each other, irrespective of a body's incident and rebound angles relative to the mutual ‘plane of contact’. However, the crucial advance made by Newton was his use of the kinematics and dynamics of reflection to analyse the dynamics of circular motion, through their reasonable extension to impulse as well as to impact forces.

Newton: reflection and circular motion

Newton's extension of his analysis of impact to circular motion in his early writings is well known.Footnote ⁴⁵ But no one has taken his constant appeals to ‘reflection’ seriously enough, so no one has identified or examined the conceptual consequences of Newton's mechanical generalization of the dynamics of optical reflection for the construction of a general theory of mechanics. This is best introduced through Descartes's confused analysis of circular motion, in which he analyses the tangential, rectilinear tendency to motion of a stone whirled around in a sling into two component determinations to establish the body's centrifugal determination (Figure 8):Footnote ⁴⁶

imagine this inclination that the stone has to move from A to C as if it were composed of two others, that is, one to turn around the circle AB, and the other to go straight along the line VXY, in such proportion that, finding itself at the point of the sling marked V when the sling is at the point of the circle marked A, it ought to find itself later at the point marked X when the sling would be at B, and at the point marked Y when it would be at F, and thus it ought always to remain on the straight line ACG. Then, since we know that one of the parts of its inclination, namely that which carries it along the circle AB, is in no way impeded by this sling, it is easily seen that the stone meets resistance only in its other part, namely that which would cause it to move along the line DVXY …Footnote ⁴⁷

This confusion arises because Descartes applies the compound analysis of an oblique light ray as a rectilinear tendency to motion to the whirled stone's (tangential) rectilinear tendency to motion. The propagation of light manifests a God-given rectilinear tendency to motion of all bodies and is, when obliquely incident, resolvable into perpendicularly related determinations. Thus this tendency, made manifest upon a body's release from circular motion, must also be so resolvable. Such a compound analysis of the tangential component of circular motion and the conceptualization of a simple circular tendency to motion both rest upon Descartes's metaphysical rejection of the possible existence of centripetal attraction. He can therefore only consider the erroneous centrifugal component motion AE of a falsely compounded tangential motion ACG, rather than the correct centripetal component motion DA of genuinely compounded circular motion AB.

Figure 8.

Descartes: circular motion, 1644.

Newton begins to break away from this analysis of circular motion by attempting to analyse the centrifugal force involved in circular motion via his optically derived analysis of impact. He conceives of such motion as resulting from the continual resistance of an external concave surface to the body's tangential tendency to motion, which is continually reflected by the resisting surface (Figure 9). This inwardly directed force/pressure upon the ball from the surface must be equal and opposite to the ball's centrifugal force.Footnote ⁴⁸ The consequence of such ‘continued reflection’ upon the ball's tangential tendency cg in half a revolution is the equal and opposite tangential tendency bk.Footnote ⁴⁹ This is in direct agreement with the dynamics of optical reflection and its mechanical generalization to rebound from a surface. Around this time, Newton knew of Descartes's and Hooke's geometrical representations of a raindrop's internal reflections of light.Footnote ⁵⁰ He was also aware of Hobbes's representation of chords of a circle as obliquely incident lines subject to reflection and his use of the law of reflection as a principle of geometrical construction.Footnote ⁵¹ He may also have known of Marci's work on the squaring of the circle via motion.Footnote ⁵² Whether Newton was influenced by one or all of these geometrical arguments, the result is the same: the analysis of circular motion via reflection from four tangent planes (Figure 10). Newton immediately applied to this problem Archimedes' geometrical solution to the squaring of the circle by increasing the number of sides (reflections) to infinity until it approximates the circle itself.Footnote ⁵³

Figure 9.

Newton: circular motion, 1664.

Figure 10.

Newton: circular motion by reflection.

Newton here gives physical meaning to a purely geometrical problem and its Archimedean solution by introducing the kinematics and dynamics used in the physical interpretation of geometrical optics.Footnote ⁵⁴ In this way, the incident path of motion ab (of light or matter) upon surface fbg evidently has perpendicular component af and parallel component fb, while the reflected path bc has perpendicular component gc and parallel component bg. This provides Newton with an insight into the dynamics of circular motion by enabling him to relate the force of the body moving away from the centre n via path ab to the force of one entire revolution (abcd).Footnote ⁵⁵

As with optical reflection, the impact force of the ball's motion ab upon fg is its perpendicular component fa, reversed by the equal and opposite force of resistance directed towards the centre n of the circular orbit. Hence this centripetal impulse force at b is twice the body's perpendicular component of motion ab, i.e. 2fa. This is shown most clearly in Erlichson's illustration (Figure 11), in which he unknowingly yet appropriately replicates the geometry of image location, providing a dynamical demonstration of the law of reflection. Had the ball encountered no resistance at b it would have continued with uniform motion to x, in the same time that it had moved from a to b. So the impulse force at b produces the displacement xc=2fa.Footnote ⁵⁶ Therefore the relation between b's motion (ab) and its force (fa) is proportional to the relation between b's ‘force or pression upon fg at its reflecting’ and its force of motion (fa), and the perpendicular component (fa) of obliquely incident motion ab determines the force at fg, due to the reversal of this perpendicular force fa according to the dynamics of reflection. Newton could then apply this analysis to all four reflections (ab, bc, cd, da), making a complete circuit (4ab), and see that the force of reflection in a circuit is proportional to the force of b's motion (fa), where the latter constitutes the required circle's radius.

Figure 11.

Erlichson: modern illustration.

Newton immediately goes on to apply the purely geometrical method of exhaustion employed in the classical problem of ‘squaring the circle’ to this square circuit (4ab) crudely approximating circular motion. He conceives of the polygon's many sides as a body's ever-decreasing paths of reflected motion occurring in ever-shorter periods of time. This is entirely kinematical. Yet the number of sides equals the number of the body's multiple reflections, hence the forces involved in its complete circuit. Consequently, Newton reasons that the relation between the total force of all these reflections and the force of the body's motion (perpendicular to the reflecting surface) is proportional to the relation between the sum of all the polygon's sides and the radius of the circle to which the polygon approximates. He then considers a polygon of infinite sides.Footnote ⁵⁷ He draws the conclusion that the relation between the total force involved in a body's motion around centre n and the force of its motion from centre n (that is, fa) is proportional to the relation between the circle and its radius, i.e, 2πr:r (or 2πr/r).Footnote ⁵⁸ He obtains this insight by providing a novel dynamical interpretation of the geometrical method of exhaustion, applied not to the area of a circle but to its circumference via infinitesimal line segments representing the continually reflected motions of a body under centripetal impulses.

Newton's synthesis of the physico-mathematics of reflection and the geometrical method of exhaustion extends the dynamics of reflection from rectilinear to circular motion through its employment of the parallelogram rule for rectilinear motion. Such a physical interpretation of the geometry of exhaustion in terms of material motion and impulse forces, grounded in a creative appeal to geometrical optics, was a breakthrough in the conceptualization of compound circular motion.Footnote ⁵⁹ Its enduring conceptual utility is most evident in the Principia's further development (Figure 12) of the original polygonal approach to circular motion (Figure 10). Its first two propositions employ this optically derived approach to show that a body in uniform (inertial) motion subject to a centripetal force will move according to Kepler's law of areas.Footnote ⁶⁰ The geometrical application of the parallelogram rule via a reflecting ‘surface’ to the classical ‘squaring of the circle’ (as in Figure 10) is here abandoned by Newton in favour of the rule's direct application to the inertial rectilinear motion of a particle and its ‘collision’ with a deviating force (as in Figure 7). Since the ‘collision’ is no longer referred to a reflecting ‘surface’, the resulting component motions of ‘rebound’ at B are no longer conceived as being perpendicular and parallel to a plane. Instead, the body's rectilinear inertial tendency forms one component (Bc) while the centripetal impulse directed in a straight line forms the other (BB′), producing the rhomboid parallelogram BcCB′ and the body's compound motion (diagonal BC) (Figure 12). In this way Newton is able to resolve circular motion into two independent and rectilinear component motions: one towards the centre of motion, the other ‘obliquely’ away from it, becoming tangent to the curve as these inertial line segments (chords of the orbit's arcs) become infinitely small. He reconstructs the ancient Greek parallelogram rule for circular motion, similarly employing a tangential inertial motion but substituting centripetally directed impulse forces for a centrally attractive force.Footnote ⁶¹

Figure 12.

Newton: polygonal method.

Between 1664 and 1666 Newton developed his own ‘doctrine of reflexions’ for the mutual collisions of bodies, grounded in the physico-mathematical principles of optical reflection. This was more advanced than Harriot's, because it incorporated the mechanical generalization of the law of reflection's equality of action and reaction in opposite directions (as stated by Kepler) to the mutual collisions of bodies having unequal angles of incidence and reflection. Newton also applied these physico-mathematical principles to circular motion through their employment in the physical interpretation of the purely geometrical method of squaring the circle. By 1666 these generalized principles of reflection became for Newton principles of rectilinear and circular motion. He could subsequently construe them as general laws of motion.

We can now identify these generalized principles of reflection and their Newtonian axiomatization. Firstly, all applications of the dynamics of reflection to moving bodies under impact (and, later, impulse) forces assume the mechanical generalization of the principle of the rectilinearity of light propagation: (Descartes's) principle of rectilinear inertia.Footnote ⁶² Hence everything, whether light corpuscles, material particles or planets, uniformly moves in straight lines before encountering obstacles or deviating forces. This constitutes the kinematic element of Newton's first law of motion. Secondly, all applications of the dynamics of reflection to moving bodies under impact and impulse forces employ the parallelogram rule in the construction and combination of component motions and forces, incorporating their dynamical independence, the conservation of the body's initial inertial motion and the perpendicular action of any deviating force. These constitute key dynamical elements of Newton's second law of motion. Thirdly, having successfully generalized the equality of action and ‘effect’ in opposite directions evident in optical reflection (as stated by Kepler), Newton was the first to incorporate this in the application of the dynamics of reflection to moving bodies under impact and impulse forces. This constitutes his third law of motion.

Conclusion

The specifically seventeenth-century emergence of a general mechanics based upon universal laws of motion cannot be understood without reference to the history of optics. The universality of such laws lies in their equal applicability to light motion and material motion. In principle, the most practical ways of obtaining these laws would be either the generalization of the laws of light to cover material motion or the generalization of the laws of material motion to cover light motion. Historically, the geometrical law of light reflection and its physical interpretation were developed first. From the ancient Greeks, this process involved the (initially) analogical application of the geometrically and experientially demonstrated law of reflection to material motion, in the direct and oblique rebound of bodies from hard surfaces. For centuries the dominance of Aristotelian natural philosophy and the Aristotelian view of light propagation ensured its analogical status. Within optics the physico-mathematical development of this analogical interpretation of reflection constitutes the tradition of light mechanics. At the beginning of the seventeenth century the conceptual hold of Aristotelianism was weakened by competing anti-Aristotelian natural philosophers. In the hands of some of these competitors to Scholastic Aristotelianism, the process of light propagation became mechanized and the physico-mathematical principles of reflection were extended beyond optics into natural philosophy.

The renaissance of atomism prompted a preference for generalizing the physico-mathematics of reflection (light mechanics) across the entire geometrical content of optics as developed within the medieval tradition of light metaphysics. Natural philosophers with mechanist leanings, articulating their view of nature as simply matter in motion, were able to transform the light-metaphysical notion of emanationist force into one of particles in collision. The light-mechanical analysis of reflection/rebound could thus be generalized to form the foundations of a new physics. Yet before this could be achieved there were several conceptual hurdles to be overcome: first, the realization that what requires generalization is the physico-mathematical principles of reflection, not the geometrical law of reflection, for these principles apply even when the geometrical law of reflection does not; second, the realization that one of these generalized principles is the equality of action and reaction in opposite directions; and third, the realization that these principles can be extended to circular motion via their application to impulse, as well as impact, forces.

The tradition of light mechanics provided a dynamical interpretation of the kinematical description of the parallelogram rule for rectilinear geometrical point motion, stated in the ancient Greek Mechanica, through the compound analysis of oblique incidence and reflection. Newton extended the application of this rule to (traditionally simple) circular motion through the creative employment of the emergent conception of the law of reflection as a principle of geometrical demonstration to interpret the purely geometrical resolution of the problem of squaring the circle. He successfully completed the partial transference of the physico-mathematical character of the ‘middle science’ of optics (specifically on light motion) to a science of general motion. This was achieved by the early seventeenth-century construction of theories of impact through the further generalization of its domain of application from impact to impulse forces. Newton's systematic and novel use of the physico-mathematics of reflection provided the key dynamical and mathematical principles upon which his Principia was built, thus founding the new ‘middle science’ of motion – Newtonian mechanics.

References

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, 153–64CrossRef Google Scholar; Sorensen, K. E., ‘A study of the De proportione motus by Marcus Marci’, Centaurus (1976), 20, 50–76CrossRef Google 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, 535–72CrossRef Google 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, 645–64CrossRef Google 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, 3–32CrossRef Google 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, 303–21CrossRef Google 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, 46–63CrossRef Google Scholar, 47; Erlichson, H., ‘Huygens and Newton on the problem of circular motion’, Centaurus (1994), 37, 210–29CrossRef Google 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.