Abstract
This chapter reconstructs the genealogy of Descartes’ discovery of the law of refraction; initial development of a theory of lenses; and first attempts, in the years 1626–1628, to explain the law through a mechanistic theory of light. These events of the mid to late 1620s constitute the greatest of Descartes’ achievements in mixed- and physico-mathematics. They were also of the utmost importance for his emergence, from the late 1620s, as a systematic corpuscular-mechanical natural philosopher. He would use the discovery of the law of refraction as a putative example of his supposedly all conquering method. More importantly, the optical work led him to the mature formulation of the central concepts of his dynamics—the causal register of his emerging system of corpuscular-mechanism, when he later came to write Le Monde. Thus, his optical triumph of the 1620s was both the climax of his early physico-mathematical agenda, as well as the exemplar for important parts of his mature, systematic natural philosophical work to come.
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Notes
- 1.
The centrality of light and its action in the system of corpuscular-mechanical natural philosophy, as a set of phenomena and as an exemplar of action and explanation, will be discussed in Chap. 10.
- 2.
- 3.
- 4.
- 5.
It should also be noted that Le Monde itself contains a reference to the text of the Dioptrique, attributing the distinction between force of motion and directional force of motion to that text. AT X. 9. cf Alquié (1963) t. 1, 321 note 2. The importance of the priority of optics in the elaboration of the dynamics will emerge clearly from our reconstruction.
- 6.
AT xi. 38; SG 25–6; MSM 61.
- 7.
AT xi. 43–44: SG 29, ‘I shall add as a third rule that, when a body is moving, even if its motion most often takes place along a curved line and, as we said above, it can never make any movement that is not in some way circular, nevertheless each of its parts individually tends always to continue moving along a straight line. And so the action of these parts, that is the inclination they have to move, is different from their motion (…leur action, c’est à dire l’inclination qu’elles ont à se mouvoir, est different de leur mouvement).’ And,
‘This rule rests on the same foundation as the other two, and depends solely on God’s conserving everything by a continuous action, and consequently on His conserving it not as it may have been some time earlier, but precisely as it is at the very instant He conserves it. So, of all motions, only motion in a straight line is entirely simple and has a nature which may be grasped wholly in an instant. For in order to conceive of such motion it is enough to think that a body is in the process of moving in a certain direction (en action pour se mouvoir ver un certain coté), and that this is the case at each determinable instant during the time it is moving.’ (pp.29–30)
- 8.
In the passages discussing the third law, cited above, Descartes defines ‘action’ as ‘l’inclination à se mouvoir’. He then says that God conserves the body at each instant ‘en action pour se mouvoir ver un certain coté’. This would seem to mean that at each instant God conserves both a unique direction of motion and a quantity of ‘action’ or force of motion. In other words, the first law certifies God’s instantaneous conservation of the absolute quantity of tendency to motion, the ‘force of motion’. The third law specifies that as a matter of fact in conserving ‘force of motion’ or ‘action’, God always does this in an associated unique direction. The first law asserts what today one would call the scalar aspect of motion, the third law its necessarily conjoined vector manifestation. Just because he recognizes that some rectilinear direction is in fact always annexed to a quantity of force of motion at each instant, Descartes often slips into abbreviating ‘directional force of motion’ by the terms ‘action’, ‘tendency to motion’ or ‘inclination to motion’, all now seen in context as synonyms for ‘determination’.
- 9.
Le Monde, AT xi. 45–6, 85. SG 30, 54–55; MSM 73–75, 147–151.
- 10.
Le Monde, AT xi. 85. For the sake of Whiggish edification it can be noted that had Descartes dealt with the centripetal constraint on the ball, offered by the sling, instead of the ‘circular’ tendency (which violates the first law in any case), he might have moved closer to Newton’s subsequent analysis of circular motion. For an analysis of Newton’s success and Descartes’ pitfalls in dealing with circular motion, as a function of their respective theories of dynamics see Smith (2008a)
- 11.
Le Monde, AT xi. 85.
- 12.
AT vi. 94.
- 13.
On this interpretation of ‘determination’ in the Dioptrique see Sabra (1967) 118–21.
- 14.
AT vi. 95–6.
- 15.
AT vi. 95.
- 16.
AT vi. 96.
- 17.
- 18.
This crucial point was first noted by Mahoney (1973) 378–9 in the course of his path breaking reinterpretation of Descartes’ optical proofs in terms of relations amongst quantities and directional quantities of forces.
- 19.
See below note 25, and the argument in Sect. 4.4 below.
- 20.
AT vi. 97.
- 21.
AT vi. 97–8.
- 22.
AT vi. 97.
- 23.
AT vi. 97–8. Descartes later supplies arguments concerning the mechanical structure of optical media to explain why light bends toward the normal when passing into a denser medium. AT vi. 103.
- 24.
Mahoney (1973) 379, was the first to suggest how the tennis ball model could be referred back to an imputed Cartesian dynamics in order to explicate Descartes’ proof.
- 25.
It is noteworthy that Descartes himself thought about his tennis ball model proof in precisely the manner we have just used to render it in terms of his dynamics and apply it to light rays. He later wrote to Mydorge for Fermat to explain the manipulation of the speeds (forces of motion) and determinations in the tennis ball proof: (To Mydorge for Fermat, 1 Mar. 1638, AT ii. 20): ‘The (principal) determination is forced to change in various ways, in accordance with the requirement that it accommodate itself to the speed (force of motion). And the force of my demonstration consists in the fact that I infer what the (principal refracted) determination must be, on the basis that it cannot be otherwise than I explain in order to correspond to the speed, or rather the force which comes into play at B.’ Here Descartes views his proof in dynamical terms, as a deduction of the new refracted principal determination induced at the instant of impact with the surface, rather than in kinematical terms, as a deduction of the position of the tennis ball at a certain time after impact with the surface.
- 26.
This derivation merely reworks Sabra’s well known analysis of Descartes’ demonstration. (Sabra 1967, 97–100, 105–6, 116.) The only difference is that here we deal with quantities of forces and their directional components (determinations), rather than with quantities of speed and their directional components, as Sabra did. The reason is that we have insisted upon the centrality of the former concepts for Descartes, and we have argued that Descartes could reduce phenomenal speeds to instantaneously exerted quantities of force of motion, so that speeds and tendencies to motion could be treated under the same conceptual and geometrical framework. We shall return to Sabra’s analysis below in Sect. 4.6, concentrating on his contentions about the timing of Descartes’ discovery and its possible relation to the optical fragment of 1620, discussed in, Sect. 3.6.
- 27.
We take it that in the spirit of Bachelard’s epistemological and historiographical conception of récurrence, such analytical Whiggism is not at all a thing to be avoided. Cf. Gaukroger (1976) 229–34.
- 28.
AT vi. 98.
- 29.
AT vi. 98–9.
- 30.
to Mersenne, for Bourdin, 3 December 1640 AT iii. 250. Bourdin (1595–1653) an almost exact contemporary of Descartes, was a Jesuit, lecturing, since 1635, on natural philosophy and mathematics at the College of Clermont. He had most likely attended La Flèche during the time Descartes had been there, and had taught there from 1618 (Clarke 2006, 194).
- 31.
Descartes is tacitly appealing on the empirical level to an indubitable fact: when dealing with a pair of homogenous media, refraction is an interface phenomenon. His dynamical premises are consistent with this fact, but they cannot be consistently articulated so as to allow the deduction of this fact, and this fact only.
- 32.
AT vi. 99–100.
- 33.
ibid.
- 34.
Lohne (1963, 1959), Vollgraff (1913, 1936), deWaard (1935–6). Here and throughout, ‘exact form’ of the law of refraction means not allowing for chromatic dispersion, they are working with and articulating assumption that all light rays are refracted in exactly same manner at a given interface. For the sequel, see Dijksterhuis (2004).
- 35.
Mersenne (1932–88) I 404–415.
- 36.
loc cit. p.404.
- 37.
loc cit. p.405.
- 38.
loc. cit. p.406
- 39.
AT x. pp.336ff; also Beeckman (1939–1953) fol. 333v ff.
- 40.
Descartes repeatedly mentioned that during this period he recruited Mydorge and the master artisan Ferrier in an attempt to confirm the law and construct a plano-hyperbolic lens. Eg. Descartes to Golius, 2 February 1632, AT i. 239; Descartes to C. Huygens, December 1635, AT i. 335–6.
- 41.
In addition to the material cited in previous note, see Descartes to Ferrier, 8 October 1629, AT i. 32; 13 November 1629, AT i. 53ff; Ferrier to Descartes, 26 October 1629, AT i. 38 ff. In the mid 1620s Mydorge annotated Leurechon’s Récréations mathématiques, a popular work dealing with mathematical tricks and fancies of a natural magical character. Leurechon’s work was first published anonymously in 1624 and reprinted several times thereafter with additional notes, including those by Mydorge. I have consulted (Jacques Ozanam) Les Récréations Mathématiques…Premierement revu par D. Henrion depuis par M. Mydorge (Rouen 1669). Mydorge notes concerning the nature of refraction ‘Ce noble sujet de refractions dont la nature n’est point esté cogneue n’y aux anciens, n’y aux modernes Philosophes et Mathematiciens iusque à present, doit maintenant l’honneur de sa découverte à un brave Gentilhomme de nos amis, autant admirable en scavoir et subilité d’esprit.’ p.157.
- 42.
DeWaard admits that the copy he examined dated from 1631 at the earliest, Mersenne (1932–88) I. 404.
- 43.
Shea (1991) 243 note 38.
- 44.
Mersenne (1932–88) I. 411–413. The anaclastic problem is to define the refracting surface that will focus all parallel incident rays to one point.
- 45.
Mersenne (1932–88) I. 408–11. The textual and mathematical claims made in this and the next paragraph are documented in Appendix 1, ‘Descartes, Mydorge and Beeckman—The Evolution of Cartesian Lens Theory 1627–1637’. The sceptical reader should examine this Appendix immediately after finishing the present Sect. 4.5.2.
- 46.
Mersenne (1932–1988) I 408–9.
- 47.
cf. Figure 4.3 above.
- 48.
AT x. 341–2; Beeckman (1939–1953) fol. 338r.
- 49.
ibid.
- 50.
This principle appears in Alhazen, Pecham, Witello, Roger Bacon and Maurolico; cf Robert Smith, A Compleat System of Optics (Cambridge, 1738) para 212, cited in Turbayne (1959) 467.
- 51.
Lohne (1959) pp.116–7, (1963) 160. Gerd Buchdahl (1972) 284 provides a particularly clear statement of the methodological role played by the image principle in Harriot’s discovery of the law. Willebrord Snel’s initial construction of the law of refraction also followed the type of path indicated by the Lohne analysis. See Vollgraff (1913, 1936), deWaard (1935–6), and Schuster (1977) pp. 313–5.
- 52.
- 53.
Sabra (1967) 97–100, 105–6,116.
- 54.
cf Note 26 above. Sabra, of course, spoke in terms of the ‘speed’ of light in the two media. The reader should note both here and in Sect. 4.4, we correct Sabra, speaking of the ratio of the ‘force of light’ in the two media. (For Descartes, the speed of the propagation of light being instantaneous, but with variable, finite, degrees of ‘force’, as explained above.)
- 55.
AT x. 242–3
- 56.
Sabra (1967) 106, 111.
- 57.
As we commented in Chap. 3, Notes 106 and 111 on this point: Had Descartes assumed that the parallel component varies either directly or inversely with the density, he would have again deduced ‘tangent laws’ with slightly differing indices of refraction. There seems no way to proceed directly from the assumptions of 1620 to the sine law of refraction, unless one is prepared to introduce Newtonian complications about the variation in components as functions of the angle of incidence, a way of conceiving the problem foreign to Descartes in 1620, 1626, as well as 1637. Sabra, of course, assumed that penetration varied with density regardless of the angle of incidence, an assumption that does indeed yield the sine law when conjoined with the assumption that the parallel component of the motion, force or penetration of the incident ray is unaffected by refraction. Sabra’s error consisted in his construal of the first premise: Descartes was envisioning that the normal component of penetration varied with density.
- 58.
The discerning reader will note a difficulty in this reconstruction. It has been argued that Descartes and Mydorge (as well as Snel) used the traditional image finding rule in their path of research leading to the law of refraction. But, unlike Harriot, the three later discoverers presumably were well aware of Kepler’s new theory of vision, which cast grave doubt on the use of the traditional rule. Descartes, after all, was working on a mechanistic version of Kepler’s theory of vision around the same time he and Mydorge discovered the law, and his 1620 optical fragment already indicates familiarity with Kepler’s new work on vision. This fascinating issue cannot be addressed in full here. Suffice it to say that the problem is more Descartes’ than our own. That is, there is evidence that Descartes suppressed discussion of his actual path of discovery for several reasons, one of which was the embarrassing point that his work depended upon an optical principle he could no longer accept. For example, his odd methodological story about how the law might be discovered, offered in rule 8 of the Regulae ad directionem ingenii, seems intended to occlude this fact, and to mythologize several of his other theoretical quandaries, under a cloak of persuasive, but necessarily vacuous ‘method talk’. See below Sect. 4.9 and Chap. 6 where the issue of the efficacy of Descartes’ method is discussed. These matters are also discussed Schuster (1993).
- 59.
One can also imagine slightly lesser degrees of articulation, involving, for example, merely a corpuscular-mechanical explanation of optical sources and media, but lacking cosmological articulation, and possibly lacking a highly articulated theory of dynamics.
- 60.
For Descartes’ similar reaction to Beeckman’s celestial mechanical speculations see Schuster (2005) 70–2 and below, Sect. 10.3.
- 61.
On the larger functions and uses of the tennis ball model and Descartes’ difficulties with it, see below Sect. 4.8.2.
- 62.
- 63.
- 64.
Schuster (1980) 59–64, and, Sect. 4.7.3: In rule 12 Descartes claims that the external senses ‘perceive in virtue of passivity alone, just in the way that wax receives an impression /figuram/ from a seal.’ He intends no mere analogy: just as the wax is impressed with the image of the seal, ‘the exterior figure of the sentient body is really modified by the object’. All of our sensations, whether of light, color, odor, savor, sound or touch, are ultimately caused by the mechanical disturbance of the external sense organs. From the sense organs the impressed ‘figures’ are transmitted instantaneously to the common sense via the nerves, by means of the passing of a pattern of mechanical disturbance. ‘No real entity travels from one organ to the other’, just as the motions of the tip of a pen are instantaneously communicated to its other end, for ‘who could suppose that the parts of the human body have less interconnection than those of the pen’. Patterns in the common sense can then be imprinted in the imagination, either to be stored in memory for the future ‘attention’ of the vis cognoscens, or to be immediately attended to in sense perception. AT x. 412–4
- 65.
Schuster (1980) 61–2 and, Sect. 4.7.3 below: Although Descartes focuses upon the mechanical causation of sensation and perception, it is clear that a mechanical theory of light underpins the entire discussion. Whatever the essential nature of external objects may be, Descartes implies, they act upon the perceiving subject in a mechanical manner. In the case of visual perception, therefore, light (or the optical media through which it acts) mechanically impresses the ‘figures’. Presumably light is an instantaneously transmitted mechanical impulse: Descartes’ mention of instantaneous mechanical nervous action, and his analogy of it to the instantaneous transmission of motion from one end of a pen to the other, suggest that light is considered to act in the same fashion. Note also that although the pen analogy is applied to nervous action (see previous Note), it is similar to the analogy of the blind man’s staff, used later in Partie 1 the Dioptrique to illustrate the instantaneous mechanical transmission of light. AT VI 85–6.
- 66.
AT x 336; Beeckman (1939–53) fol. 333v.
- 67.
Milhaud (1921) 110.
- 68.
It would also illustrate the case of a ‘tennis’ or cannon ball whose motion is refracted away from the normal in water, as discussed later in the Dioptrique (AT vi. 97–8). Beeckman and Descartes might perhaps also have discussed this phenomenon in 1628.
- 69.
The only problem with Descartes’ analogy of course is that greater force (effective weight) depends upon placement in a rarer medium and vice versa, thus implying a disanalogy between specific gravity and refractive ‘density’ of an optical medium
- 70.
As Stevin, the stimulus for the hydrostatic manuscript of 1619, had taught with his near approach to the parallelogram of forces, mainly applied to the non-vertical components of weight. Stevin (1955–66) Vol. 1. 183–5.
- 71.
- 72.
Beeckman (1939–53) i. 24–5.
- 73.
To Mersenne December 18 1629, AT I 90
- 74.
To Mersenne, 8 October 1629, AT i. 23. His work at this time is discussed in more detail below, Sections 8.4.3, 8.4.4 and 8.4.5.
- 75.
Météores, AT vi. 331–32
- 76.
First, he uses the analogy of the blind man’s staff to illustrate the instantaneous propagation of light without the passage of any material (or immaterial) entity. The analogy clearly derives from the pen analogy used earlier in the Regulae. As the blind man receives from the far end of his staff only instantaneously conveyed tendencies or resistances to motion, so light rays are only lines of tendency to motion propagated instantaneously through the contiguous particles of optical media. (AT vi. 84–6) The second analogy deals with the rectilinear propagation of light rays, their propagation in infinitely many directions from a luminous point, and their ability to cross without impeding each other. Descartes’ model is a vat filled with half crushed grapes and new wine. The analogy is carried out by manipulating putative lines of tendency-to-descend running from wine particles on the surface of the vat to hypothetically voided points on its bottom, a procedure clearly borrowed from the hydrostatics manuscript of 1619. (AT vi. 86–8). On a closely related set of observations, regarding Descartes’ theory of light in its cosmic setting in Le Monde, see, Sect. 10.7.4 below.
- 77.
Although he will later deal with the production of colors through refraction of light, Descartes introduces the ‘spin/speed’ articulation of the tennis ball model in the case of reflection (AT vi. 90–1), because it is much more easily grasped in common sense terms, and because, he has not yet even shown how the simple tennis ball model can be applied to the law of reflection and then extended to the law of refraction.
- 78.
Météores, AT vi. 331.
- 79.
loc. cit. p. 332.
- 80.
loc. cit. pp.331–4. This piece of explanation in turn is fundamental to Descartes’ groundbreaking work on the rainbow. The best modern explication of Descartes’ research on this classic problem is Buchwald (2008), which also brilliantly demonstrates how within this work Descartes achieved the only instance in his corpus where a corpuscular-mechanical model is applied and further articulated with relation to novel experiments which have quantitative implications.
- 81.
At times Descartes speaks of a part of the speed of translation of a ball being converted into spin. (eg. AT vi. 90) He was no doubt thinking of everyday macroscopic analogies, such as a tennis ball appearing to lose some its incident speed upon acquiring a spin after bouncing obliquely on the ground.
- 82.
Descartes uses this infelicitous locution at AT vi. 333.
- 83.
For, as we have established above, at the moment of impact, the tennis ball (reduced to a weightless, frictionless point) behaves exactly the way a light impulse would—indeed dynamically speaking the two are identical—and the superficially kinematical aspects of the model ‘momentarily’ drop from view.
- 84.
When presenting his real theory of light in Chap.14 of Le Monde, he lists 12 properties of light and explains them as arising from tendencies to motion transmitted through the spherical boules of his ‘second element’. Color is not mentioned explicitly as one of these properties; but, it is implicitly contained in the last two properties, described in terms of capacity of the ‘force’ of a light ray to be increased or decreased ‘by the diverse dispositions or qualities of the matter that receives them’. Descartes’ ‘explanation’ of these properties makes no mention of color and seems intended more to elaborate the explanation of the tenth property, refraction. As for refraction and reflection themselves, Descartes passes up the opportunity to introduce the tennis ball model (or moving boules), and simply refers the reader to the Dioptrique. (AT x. 97-103)
- 85.
The exception occurs in an obscure corner of the final part of the French version of the treatise (Principia IV 131, AT IXB. 270; MM 241), where Descartes explains the properties of colored glass. Leaving aside this limited and late passage, which is Descartes’ and/or Picot’s afterthought, we see that Descartes steadfastly refused to introduce the spin/speed model into his systematic work. And the likely reason for this is that the model cannot be made to agree with his real theory of light as a tendency to motion. Further evidence of Descartes’ awareness of the problem, and its intractability, may be found in the Météores. In the passages discussed above (Note 82 above), Descartes twice writes of the boules ‘tendency’ to move and ‘tendency’ to spin. Evidently he was caught between the content and the grammar of his real theory, on the one hand, and the mechanical rationale of his spin/speed model, on the other. At this point of tension his discourse falters and wavers, despite the fact that here in the published text of 1637 he could (for the foreseeable future) have gotten away with the pretence that light consists in the translation (and spin) of boules.
- 86.
The little we know about the course of composition of the Dioptrique tends to confirm this picture of a Descartes reluctantly satisfied, for the time being, with the tennis ball model in the publications of 1637. The Dioptrique is first mentioned in a letter to Mersenne of 25 November 1630 (AT i. 179), over a year after the problems of parhelia and the rainbow had first stimulated his work on a system of corpuscular-mechanical natural philosophy. Descartes writes that he wishes to insert into the Dioptrique an explanation of ‘the nature of light and colors’, a task which has held him up for six months. This will virtually turn the Dioptrique into a ‘system of physics’, an ‘abridgment of Le Monde’, and so acquit him of his promise to Mersenne, made in April 1630, to finish the system within three years. He adds that if the reception of the Dioptrique shows he can persuade people of the truth, then he will proceed to complete his treatise on metaphysics begun earlier in 1629.
Two main difficulties seem to have been haunting Descartes. First, the explanation of the nature of color had proven a most difficult proposition. One suspects this was not only due to the intricacies of his articulated tennis ball model, but also because of the dawning realization that it bore no convincing analogy in the real theory of the ‘nature of light’. Second, Descartes was clearly still undecided about how much material from his emerging system of corpuscular-mechanism should or could appear in the Dioptrique. In the letter he toys with the idea of adding a section on the true nature of light and color, and thus implying that he already possessed some version of the model-based presentation he later published. Again, part of his hesitation and indecision may have related to the difficulty of linking the spin/speed articulation to his real theory of light. In January 1632 he sent to Golius what he termed ‘the first portion of the Dioptrique’, dealing with ‘refractions without touching upon the rest of philosophy’. (AT i. 235) This, too, tends to indicate that Descartes still contemplated publishing in the Dioptrique more of his dynamics and real theory of light than we find in the publication of 1637. If so, he was probably then still facing the problem of the relevance of the spin/speed articulation to the real theory.
In the end Descartes’ problems were solved on a pragmatic basis, motivated by external events. When he learned of the condemnation of Galileo and decided to withhold Le Monde from publication, he reorganized his publication program, producing within three years the Discours and three Essais in the form with which we are now familiar. The reorganization allowed him to design the Dioptrique and the optical portions allotted to the Météores around the tennis ball model, without having to face up to the problem of whether the model in its articulated form could represent aspects of the real theory of light. In this respect, perhaps, he came to see the demise of Le Monde as something less than a complete disaster, since it allowed him to resolve the problem of presenting and justifying his optical achievements. Again, from this perspective, he may well have viewed the tennis ball model as a qualified success.
- 87.
It can also be shown that it is the first of the passages added to the Regulae in Paris and leads directly to the core of the third stratum of the text. See below, Sects. 7.2 and 7.3. Cf Schuster (1980) 58–9.
- 88.
AT x 393–5.
- 89.
We shall learn more about Descartes’ methodological terms, ‘absolute’ and ‘relative natures’ in Chap. 5, where we examine his dream of a universal method and the opening portions of the text of the Regulae.
- 90.
Perhaps he also had in mind other analogies for the action and refraction of light, for example, a rudimentary and unarticulated kinematic model, a tennis ball model; we simply do not know.
- 91.
- 92.
Like a myth viewed in a Lévi-Straussian perspective, the method discourse provides a structure which imposes order on this jumble of biographical and in part contradictory conceptual meaning-tokens, by means of a narrative of particular events and actions which is, at bottom, yet another instance of his core myth of method. Lévi-Strauss (1972), 216, 224. Alternatively, if one prefers Roland Barthes’ view of myth, we might say Descartes’ account amounts to a none too convincing rational reconstruction, motivated by a host of personal, philosophical and ideological concerns, and posing as a true story of the discovery. Barthes, ‘Myth Today’ in (1973), 109–59. We return to these theoretical reflections on ‘method-talk’ as akin to mythopoeic talk in Chap. 6.
- 93.
This conceit of ‘seeing (natural philosophical) causes inside well grounded mixed mathematical results’ emerged in discussion of ‘Baroque Optics’ with my colleagues, Dr. Ofer Gal (Unit for HPS, University of Sydney) and Dr. Sven Dupré (then of the Department of History of Science, University of Ghent). We have put this notion to work in research on the physico-mathematization of optics in the work of Kepler and Descartes, brought together in a dedicated issue of Synthèse.See Schuster (2012).
- 94.
See below Chap. 6. On the specific issue of the necessary vacuity of the rules of grand methods see Schuster (1984, 1986) and, as noted therein the very important, and little noted paper of Paul Feyerabend (1970) on exactly this issue, which is to be preferred to his wider ranging and better known works on method in relation to this critically important point.
- 95.
And where the arrows in the figure, of course, do not, and cannot, represent strictly valid logical movements.
References
Works of Descartes and Their Abbreviations
AT = Oeuvres de Descartes (revised edition, 12 vols.), edited by C. Adam and P. Tannery (Paris, 1964–76). References are by volume number (in roman) and page number (in Arabic).
SG = The World and Other Writings, edited and translated by Stephen Gaukroger (Cambridge,1998).
MM = René Descartes, The Principles of Philosophy, translated by V. R. Miller and R. P. Miller (Dordrecht, 1991)
MSM = Rene Descartes, Le Monde, ou Traité de la lumière, translated by Michael S. Mahoney (New York, 1979).
CSM(K) = The Philosophical Writings Of Descartes, 3 vols., translated by John Cottingham, Robert Stoothoff, and Dugald Murdoch, and (for vol. 3) Anthony Kenny, (Cambridge, 1988) References are by volume number (in roman) and page number (in arabic).
HR = The Philosophical Works of Descartes, vol I translated by E.S. Haldane and G.R.T. Ross (Cambridge, 1968 [1st ed. 1911])
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Alquié, F. (ed.). 1963. Oeuvres philosophiques de Descartes, t.1. Paris, Garnier Frères.
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Bossha, J. 1908. ‘Annexe note’, Archives Neerlandaises des Sciences Exactes et Naturelles, ser 2 t. 13, pp.xii–xiv.
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Schuster, J. (2012). Descartes Opticien: The Optical Triumph of the 1620s. In: Descartes-Agonistes. Studies in History and Philosophy of Science, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4746-3_4
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