Descartes, Newton, and Snell’s law
JOSA, Vol. 66, Issue 1, pp. 1-8 (1976)
http://dx.doi.org/10.1364/JOSA.66.000001
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Abstract
Descartes’ insightful derivation of Snell’s law is translated into English and seen to be largely equivalent to the mechanical-particle or corpuscular derivation often attributed to Newton (who was seven years old at Descartes’ death). Such mechanical theories of refraction are now widely held to yield unphysical results for light, and thus to constitute clear evidence that light cannot consist of classical particles, and must therefore be classical waves or else a quantum-mechanical phenomenon. We show that this evidence is illusory and derives only from confusion of momentum with velocity. An alternate operational formulation of the law of refraction, which applies to any type of reversibly propagating energy, including light and material particles, is summarized. It thus follows that the principal modern reason for rejecting a classical-particle or corpuscular description of light (speed change upon refraction) is unfounded and that such a description is actually possible and useful.
© 1976 Optical Society of America
Citation
W. B. Joyce and Alice Joyce, "Descartes, Newton, and Snell’s law," J. Opt. Soc. Am. 66, 1-8 (1976)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-66-1-1
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References
- I. Newton, Opticks (4th ed. of 1730, reprinted by Dover, N.Y., N.Y., 1952); Book One, Proposition VI, Theorem V; Book Two, Proposition X. Newton (1642–1727) cited Descartes (1596–1650) when criticizing his theory of colors (p. 169) but ignored Descartes when reproducing the essential elements of his mechanical analysis of light refraction (pp. 79–82). See also p. xlvii (I. B. Cohen) and p. lxxiv (E. T. Whittaker). Newton was quite familiar with Descartes<s analysis of refraction and discussed it in a lecture about 1670 and elsewhere. See I. B. Cohen, in The Annus Mirabilis of Sir Issac Newton, edited by R. Palter (MIT Press, Cambridge, Mass., 1970), p. 151, Ref. 19, and The Mathematical Papers of Isaac Newton, edited by D. T. Whiteside (Cambridge University Press, Cambridge, U. K., 1967), Vol. I., p. 559.
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- R. W. Ditchburn, Light (Interscience, New York, 1963), Secs.3. 35 and 4.41.
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- Project Physics, G. Holton et al., Directors (Holt, Rinehart, and Winston, New York, 1975), Sec. 13.3
- R. Descartes, La Dioptrique (Leyden, 1637). Reprinted in Oeuvres de Descartes, edited by C. Adam and P. Tannery (Léopold Cerf, Paris, 1902), Vol. VI, Discours II.
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- A. I. Sabra, Theories of Light from Descartes to Newton (Old-bourne, London, 1967), Chaps. IV and XII. Many assessments of Descartes< analysis of refraction, which have appeared from its publication to present day, are summarized here.
- W. B. Joyce, Phys. Rev. D 9, 3234 (1974).
- Thus υ= ∂ E / ∂ p = ∂ω/∂k is the physical quantity known as the group or particle speed; e. g., υ is not the phase speed = E/p = ω/k.
- W. B. Joyce, Am. J. Phys. 43, 245 (1975).
- Thus p is the physical quantity known for various particles as the generalized, crystal, Minknowski, virtual, or pseudo-momentum; e.g., p is not υ = υE′ /C^{2}, the Abraham, true, mass-carrying, or energy-carrying momentum, where C is the vacuum phase and group speeds of light and E ′ is the total relativistic energy (see Refs. 22 and 20). A familiar one-dimensional example is a springless toy car propelled by kinetic energy stored primarily in a flywheel geared directly to the car's wheels. The car's true momentum (υ times its weight E ′/C^{2} in mass units) is significantly less than its p momentum which equals the impulse applied in bringing the car to speed υ or in stopping it. The momenta differ because of the presence of a medium (the floor) which acts to set the flywheel in rotation; if either the floor or flywheel were absent, the two momenta would be identical (with the usual neglect of the wheels in the latter case). Similarly the car's mass m defined as p/υ exceeds its weight-mass p/υ = E ′/C^{2}. It is the p momentum which the child feels, whereas the so-called true momentum P cannot even be sensed without removing the car from the level floor. Thus in the description of the car regarded as a point particle on a level or nonlevel stationary floor, it is the p momentum, not the true momentum, which is conserved in car-car scattering, which appears in Hamilton's classical equations, and which equals hk and becomes - ih ∂ / ∂x in Schrödinger's equation or the Feynman path integral for the car.
- G. Green, Proc. R. Soc. Edinburgh 13, 54 (1836).
- H. Lamb, Hydrodynamics (Cambridge Univ. Press, Cambridge, England, 1932; reprinted by Dover, New York, 1945), Secs. 92 and 93.
- Inclusion of the ball's potential energy Φ = (m_{0} - ρV)gy is straightforward (e. g., nCp / E = [(1 - Φ / E)2m/E]^{½}C) but distracting since, in the geometrical optics of material media, we ignore the potential energy of light in a gravitational acceleration g. [For example, if g> 0 and 0 < m_{o} < ρV, then in the point-particle representation the ball, launched as in Fig. 1, follows a trajectory given everywhere by the conservation of E and n sinθ i. e., parabolas on alternate sides of the interface joined by (6)]. As another example, a photon of momentum p = hk at radius r in an optical fiber of refractive index n(r,p) has the same trajectory, velocity, and momentum as a classical particle with Hamiltonian E = Cp/n, and conversely for any Hamiltonian E(r, p) there is a corresponding fiber with n = Cp/E (r, p) (although the resulting refractive index is not necessarily found in nature). Clearly then a photon's energy E, angular momentum rp_{θ}, and axial component of momentum P_{x} (but not necessarily υ_{x}) are constants of motion in a fiber (whether n is a continuous or discontinuous function of r). The étendue or throughput U, or equivalently the number of modes N = 2U / λ^{2}^{vacuum}, of a multimode fiber has been calculated on this basis [Ref. 20, Eqs. (83) and (101) plus Ref. 43, Eqs. (61) and (62); see also C. Pask, A. W. Snyder, and D. J. Mitchell, J. Opt. Soc. Am. 65, 356 (1975)].
- For other particle-medium systems, m may increase, stay constant, or decrease, as when an electron passes from the vacuum into a semiconductor. We emphasize that this dependence of mass on position (medium) is quite distinct from the familiar relativistic dependence on speed.
- It has long been recognized that the corpuscular description of light could be made to yield the correct velocity change upon refraction by simply postulating a correspondingly chosen mass increase at the interface. [See T. Preston, in The Theory of Light, 3rd ed., edited by C. Jolly (MacMillan, London, 1901), p. 18; F. R. Tangherlini, Am. J. Phys. 36, 1001 (1968); NuovoCimentoB4, 13 (1971), and other sources cited therein. ] Apart from questions about the physical origin of the additional mass, this still seemed to distinguish light from material particles because of the resulting unfamiliar and somewhat ad hoc dynamics of corpuscles. In the present article, as in Ref. 20, the group velocity concept, the distinction among various masses, the operational classical-particle definition of p for light, and the immersed-sphere example show that the m mass of even a classical material particle is changed by the medium (without medium accompaniment) and that corpuscles and material particles share a common classical-particle dynamics (in which neither Planck's constant nor wave concepts need appear).
- We introduce Plancks quantum of action h because it provides a unique conversion, rather than analogy, between our particle and wave representations, but we. cannot emphasize too strongly that h appears in neither of those self-consistently classical descriptions of light.
- In terms of the superfluous but valid force concept we say that the short-range force on the (clothed) bail equals dp→/dt and has no x component with the result that px is conserved. The same cannot, of course, be said of the bare sphere.
- D. J. Struik, in Dictionary of Scientific Biography, edited by C. C. Gillispie (Scribner's, New York, 1975), Vol. 12, pp. 499–502. As Struik also points out, there is little justification for the spelling Snell which has become conventional in the technical literature while remaining less common in the history-of-science literature. The name was actually Snel, embellished into Snel van Royen and, following the practice of the period, Latinized into Snellius.
- Newton was able to compute the final speed of a (nonrelativistic) material particle which passes between two locally indistitiguishable regions (i. e., m_{0} = m_{l}) that differ only by a constant amount of potential energy. However, this special case, although embraced by (4), is sufficiently remote from the refraction of either a material particle or light between locally distinguishable media as to be more misleading than enlightening in the present context. Newton's example is sometimes modeled as a ball rolling between flat surfaces at different heights joined by a ramp.
- W. R. Hamilton, Proc. R. Irish Acad, 1, 267 and 341 (1841).
- In taking the photon speed v as the measurable transit or group velocity ∂ω/∂k of a light pulse we are inconsistent with Ref. 9 (p. 13) where photons are held to travel in all media at the vacuum speed of light, and with Ref. 10 (pp. 111 and 260) and J. J. Muray, F. E. Nicodemus and I. Wunderman, Appl. Opt., 1465 (1971) (final table) where they are held to travel at the phase speed c = ω/k = C/n. In addition to (15), a reason for the definition n = Cp/E [rather than n = P or n = pƒƒ(E)] appears in the theory of adiabatic invariants where n and E change while p is conserved in time-varying homogeneous media (Ref. 20).
- O. Klemperer and M. E. Barnett, Electron Optics (Cambridge Univ. Press, Cambridge, England, 1971), Sec. 1.2,
- p. Grivet, Electron Optics (Pergamon, New York, 1972), Chap. 6.
- P. A. Sturrock, Static and Dynamic Electron Optics (Cambridge Univ. Press, Cambridge, England, 1955), p. 9.
- Mathematics Dictionary, edited by G. James and R. C. James (Van Nostrand, New York, 1968), p. 306.
- F. G. Smith and J. H. Thomson, Optics (Wiley, New York, 1971), p. 92.
- Reference 20, Eqs. (48), (50), (60), (71), and following (72).
- Reference 19, p. 313.
- J. B. Keller, Am. J. Phys. 40, 1035 (1972).
- W. B. Joyce, R. Z. Bachrach, R. W. Dixon, and D. A. Sealer, J. Appl. Phys. 45, 2229 (1974).
- W. B. Joyce, J. Acoust. Soc. Am. 58, 643 (1975).
- J. W. Shirley, Amer. J. Phys. 19, 507 (1951); and J. W. Shirley (private communication based on his forthcoming publication of Harriot's papers).
- J. A. Lohne, in Dictionary of Scientific Biography, edited by C. C. Gillispie (Scribner's, New York, 1972), Vol. 6, pp. 124–129; and private communication.
- M. L. Schagrin, Am. J. Phys. 42, 927 (1974). In an early-17th-century remark that Schagrin also discusses, Kepler came fairly close to the realization that solar-radiation force orients a comet's tail.
- Although Descartes cites a motive force for philosophical reasons, from an operational standpoint he follows Galileo and employs what has since become known as Newton<s first law of motion.
- Descartes repeatedly emphasized a distinction between determination to move and actual motion or velocity, although he analyzed the former as if it were the latter. It is sometimes held that determination to move should be interpreted as the more recent and precise term momentum, but we feel that his determination concept is too vague operationally for such a translation. Note also that the concept of mass is not explicit in Descartes' derivation.
- Descartes clearly understands the decomposition of the velocity vector into independent Cartesian components—a procedure used earlier by Galileo and others (Ref. 19).
- Instead of, for example, incorrectly halving the vertical component of velocity at every incidence angle (with fixed υ_{0}), Descartes correctly makes the major assumption that the magnitude of the velocity is halved independently of the incidence angle. The conservation-of-energy principle was not yet recognized (see Refs. 48 and 58), and if Descartes knew of the Harriot-Snell law (11), it would have provided guidance in formulating this assumption. For such reasons, including his presence in Leyden at the time of Snell's lectures, we tend to believe Descartes knew of it or of some equivalent statement.
- Figure 2 is deficient here in obvious ways that were corrected in the 1824 Victor Cousin edition and then restored to the original form in the more faithful 1902 Adam and Tannery edition.
- This is Descartes' basic error-conserving the horizontal component of velocity υ_{x}. rather than the component of momentum P_{x}. Actually the ball, like light, refracts toward the normal (Fig. 1). (Deflection toward the normal is immediately apparent if one regards the ball as a composite of two hemispherical airfoils and notes that the airfoil on the higher density side of the trajectory has the greater lift.) It is not entirely clear that the x component of Descartes's determination is assumed conserved upon immersion. On the one hand he appears to say that his analysis of the cloth case (where conservation is asserted) applies here. Since velocity and determination presumably remain collinear this implies that their ratio (mass ?) is independent of the medium. On the other hand, Descartes may have meant to say elsewhere that the ratio can change (Ref. 19, p. 120).
- Total internal reflection (previously observed with light by Kepler) is clearly derived here, although because of the conservation of υ_{x} rather than p_{x} it occurs on the wrong side of the interface.
- Descartes' analysis makes no mention of partial reflection, although he must certainly have been aware of it—say, from reflections from a pond. Can the probability of reflection of a particle of light (given by the Fresnel coefficients in the case of an abrupt interface) be deduced within the framework of classical particle mechanics? With our definition (Ref. 20) of a classical particle as one whose local properties (e.g., momentum, kinetic energy, velocity, etc.) are unvarying as it passes through a homogeneous time-independent medium the answer is no, It would be necessary to borrow from wave theory and attach some measure of distance traveled to the particle, such as phase or fits of easy reflection, in order to determine the reflection probability from a thin film or, equivalently, from a region of high density gradient. An exception might, conceivably, seem to be the case of an abrupt interface, but even this would not meet the physical requirement we chose to impose in Ref. 20—that the relection probability from an abrupt interface be obtainable as the infinite steepness limit of the reflection deduced in the presence of a gradient.
- Having erroneously deduced that light gains speed when refracted toward the normal, Descartes seeks a plausible mechanism.
- Neither Descartes nor Newton was entirely consistent or confident throughout his writings on the physical conceptuaiization of light, and we make no attempt to summarize the evolving physical models they gave nor the extent of their allegiance to these models. For examples, Descartes often appeared to believe the speed of light infinite, and Newton's corpuscles became increasingly wavelike.
- Reversibility is recognized here although it is hardly present when a ball rolls on a rug. Thus as Leibniz later noted, Descartes might better have tried to explain his unphysical speedup of light in media by emphasizing an analogy with, for example, flow through the narrowing of a channel.
- J. A. Lohne, Arch. Hist. Exact Sci. 1, 389 (1961).
- Although we have emphasized the deficiencies and, particularly, the nonuniqueness of traditional optico-mechanical analogies which yield unphysical speeds for particles of light, we do not, of course, question the value of these analogies as computational aids for determining trajectories of either light or material particles. Similarly, the elimination of time from the equations of motion is often a convenient procedure when trajectories are sought, but equally so^{20} for light and material particles.
- It is true for both material particles and light that the group (= particle = energy) speed may increase or decrease upon reversible refraction toward the normal whereas the phase speed c always decreases.
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