Descartes's Experimental Journey Past the Prism and Through the Invisible World to the Rainbow

Summary

Descartes's model for the invisible world has long seemed confined to explanations of known phenomena, with little if anything to offer concerning the empirical investigation of novel processes. Although he did perform experiments, the links between them and the Cartesian model remain difficult to pin down, not least because there are so very few. Indeed, the only account that Descartes ever developed which invokes his model in relation to both quantitative implications and to experiments is the one that he provided for the rainbow. There he described in considerable detail the appearances of colours generated by means of prisms in specific circumstances. We have reproduced these experiments with careful attention to Descartes's requirements. The results provide considerable insight into the otherwise fractured character of his printed discovery narrative. By combining reproduction with attention to the rhetorical structure of Descartes's presentation, we can show that he worked his model in conjunction with experiments to reach a fully quantitative account of the rainbow, including its colours as well as its geometry. In this one instance at least, Descartes produced just the sort of explanatory novelties that the young Newton later did in optics. That Descartes's results in respect to colour are in hindsight specious is of course irrelevant.

Acknowledgements

I thank an anonymous referee and John Schuster for their insightful and helpful comments on a draft of this paper. Most of all, I thank Moti Fengold and Diana Kormos-Buchwald.

Notes

¹The Cartesian rainbow has been discussed often, notably by Carl B. Boyer, The Rainbow. From Myth to Mathematics (New York, 1959), Stephen Gaukroger, Descartes. An Intellectual Biography (Oxford, 1995), William R. Shea, The Magic of Numbers and Motion. The Scientific Career of René Descartes (Canton, MA, 1991), A.I. Sabra, Theories of Light from Descartes to Newton (Cambridge, 1981), Stephen Gaukroger, Descartes’ System of Natural Philosophy (Cambridge, 2002), and Richard S. Westfall, ‘The Development of Newton's Theory of Color’, Isis 53 (1962) 339–580, Boyer's now half-century-old discussion remains an insightful introduction not only to Descartes but to the long history of speculations concerning the ‘iris’ and to the post-Cartesian developments, culminating in George Biddell Airy's nineteenth-century analysis based on interference. Shea and Gaukroger rely primarily on Boyer. A recent book by two optical scientists nicely supplements Boyer (Raymond L. Lee and Alistair B. Fraser, The Rainbow Bridge. Rainbows in Art, Myth, and Science (University Park, PA, 2001)). See also M. Minnaert, The Nature of Light & Color in the Open Air (New York, 1954), 174–76. Jean-Robert Armogathe, ‘The Rainbow: A Privileged Epistemological Model’, Descartes’ Natural Philosophy, edited by Stephen Gaukroger, John Schuster and John Sutton (London, 2000), 249–57, ‘L'arc-En-Ciel Dans Les Météores’, Le Discours Et Sa Méthode, edited by Nicolas Grimaldi and Jean-Luc Marion (Paris, 1987) 145–62, offers another perspective on Descartes's account.

²John Schuster, ‘‘Waterworld’: Descartes’ Vortical Celestial Mechanics’, The Science of Nature in the Seventeenth Century. Patterns of Change in Early Modern Natural Philosophy, edited by Peter R. Anstey and John A. Schuster, Studies in History and Philosophy of Science (Dordrecht, 2005), 37.

³The original of René Descartes, Discours De La Méthode Pour Bien Conduire Sa Raison & Chercher La Verité Dans Les Sciences Plus La Dioptrique, Les Méteores, Et La Géometrie, Qui Sont Des Essais De Cette Methode (Leyden, 1637), with minor orthographic corrections, was published in full in vol. 6 of the first edition of his Oeuvres, edited by Charles Adam and Paul Tannery, Oeuvres De Descartes, 12 vols. (Paris, 1897–1910) (hereafter AT; Les Météores are on pp. 231–366). The whole minus the Géometrie was translated into Latin by the Protestant minister and theologian Etienne de Courcelles (1586–1659) and published in 1644, with Descartes approving the overall sense but not assisting in the translation, which is infelicitous (also included in vol. 6 of AT). ‘Of the rainbow’ occupies only thirteen and a half pages in translation. The Olscamp translation (René Descartes, Discourse on Method, Optics, Geometry, and Meteorology, trans. Paul J. Olscamp, The Library of Liberal Arts (New York, 1965): hereafter PO), where used, has been checked against AT; the corresponding location in AT has been provided in all cases.

⁴PO, 332; AT, v6, 325.

⁵Daniel Garber, ‘Descartes and Experiment in the Discourse and Essays’, Essays on the Philosophy and Science of René Descartes, edited by Stephen Voss (Oxford, 1993), 298.

⁶Garber, ‘Descartes and Experiment in the Discourse and Essays’, 293–94.

⁷Garber bravely constructs a diagrammatic tree to represent Descartes's account of the rainbow. Though plausible, the path exhibits yawning gaps and requires parallel branchings that inevitably result from the ‘confused mass of experiment and reasoning’ that Descartes's readers encounter (Garber, ‘Descartes and Experiment in the Discourse and Essays’, 299).

⁹Franciscus Aguilonius, Opticorum Libri Sex, Philosophis Iuxta Ac Mathematicis Utiles (Antwerp, 1613), 215–16, cited and translated in Peter Dear, ‘The Meanings of Experience’, The Cambridge History of Science. Early Modern Science, edited by Katherine Park and Lorraine Daston, vol. 3 (2006), 122.

⁸On which see William R. Newman, Atoms and Alchemy. Chymistry & the Experimental Origins of the Scientific Revolution (Chicago, 2006).

¹⁰Franciscus Maurolycus, Abbatis Francisci Mavrolyci Messanensis Photismi De Lvmine, & Vmbra Ad Perspectiuam, & Radiorum Incidentiam Facientes … (Naples, 1611).

¹¹Libert Froidmont, Liberti Fromondi Meteorologicorum (Antwerp, 1627).

¹²Under the assumption that size does not make the colours and their disposition appear ‘in any other way’ than they would in the natural phenomenon: AT, v6, 325.

¹³In the Opticks, Newton implied that Descartes drew nearly everything worthwhile from de Dominis, who taught ‘how the interior Bow is made in round Drops of Rain by two refractions of the Sun's Light, and one reflexion between them, and the exterior by two refractions and two sorts of reflexions between them in Each Drop of Water’. Further, Newton wrote that de Dominis ‘proves his Explications by Experiments made with a Phial full of water, and with Globes of Glass filled with Water, and placed in the Sun to make the Colours of the two Bows appear in them. The same Explication Des Cartes hath pursued in his Meteors’ (Isaac Newton, Opticks: Or, a Treatise of the Reflexions, Refractions, Inflexions and Colours of Light. Also Two Treatises of the Species and Magnitude of Curvilinear Figures (London: Sam Smith and Benjamin Walford, Printers to the Royal Society, 1704), 127). Although the De Dominis book was in his library, Newton's characterization of the explication is wide of the mark, since de Dominis never mentions an emergent refraction for either the primary or the secondary bow, or two reflections for the secondary: see Marci Antonii de Dominis, De Radiis Visus Et Lucis in Vitris Perspectivis Et Iride (Venetiis, 1611); de Dominis’ account is discussed in Boyer, The Rainbow. From Myth to Mathematics, 187–92, R.E. Ockenden, ‘Marco Antonio De Dominis and His Explanation of the Rainbow’, Isis 26 (1936), and on Newton see Alan E. Shapiro, ed., The Optical Papers of Isaac Newton. The Optical Lectures. 1670–1672, vol. 1 (Cambridge, 1984), 593, note 1.

¹⁴AT, v6, 325.

¹⁵Froidmont, Liberti Fromondi Meteorologicorum, 358, noted in Armogathe, ‘The Rainbow: A Privileged Epistemological Model’, 252. Froidmont remarks ‘Sed alterum Iridis artificialis genus est, merae διακλασιoζ refractionis filia. Talem prismata, & vitra triangularia efficiunt. Item urinale, aut vitru etiam vulgare vinarium … ’. That is, an artificial kind of rainbow can be produced with a ‘triangular’ glass (i.e. a prism), a urinal, or a wine flask.

¹⁶In the felicitous phrase of Lee and Fraser, The Rainbow Bridge. Rainbows in Art, Myth, and Science .

¹⁷This is, in modern parlance, the scattering angle for a refracting sphere.

¹⁸AT, v6, 326–27.

¹⁹PO, 334: AT, v6, 328.

²⁰PO, 334; AT, v6, 329.

²¹PO, 334; AT, v6, 329.

²²Garber, ‘Descartes and Experiment in the Discourse and Essays’, 298 notes the point.

²³PO, 333; AT, v6, 327.

²⁴PO, 335; AT, v6, 330.

²⁵Although we do not know whether Descartes closely controlled the dimensions of Figure 3 as printed, measurement of the figure itself yields angles , respectively, of 38° 40′, 51° 20′.

²⁶For an index n of refraction, total internal reflection sets in for counter-clockwise rays when their angle of incidence on NM reaches .

³⁰PO, 335; AT, v6, 330.

²⁷See immediately below for a discussion of what seems at first to be a strange claim in respect to apertures.

²⁸The placing of colours as intermediates on a chromatic scale ranging from white to black had been abandoned by artists for pigment mixing by the end of the sixteenth century, many of whom considered neither white nor black to be colours or generators of colours. Their views had become increasingly common by mid-century, on which see Alan E. Shapiro, ‘Artists’ Colors and Newton's Colors’, Isis 85 (1994), 627, who remarks that the ‘widespread use of color mixing culminated in the early seventeenth century in the disclosure of the painters’ primaries, and henceforth the painters’ trinity would play a fundamental role in color theory … A transformation had also occurred in the very conception of what a color is, with the shift from a tonal classification based on black and white to a chromatic one based on hue and a separate black–gray–white scale’. Descartes did not apparently adopt the latter scheme, because, we shall see below, he used a mechanical structure to locate white between red and blue, though black was necessarily the simple absence of light.

²⁹For Descartes, bodily and emphatic colours could not be different in kind: both were produced in the final instance by the actions of the very same mechanical configurations on the visual apparatus. The configurations could, however, be generated originally in alternative ways, so that the difference between the bodily and the emphatic now became one between their originating mechanical causes and not one of form.

³¹In terms of Descartes's mechanics, which associates increased perturbation of the normal state (white) inversely with aperture size, even a wide aperture will produce coloration, but its mechanical effect on the microspheres (see below) will not become visible except at the edges until the beam has sufficiently diverged—before that the microspheres interfere with one another, leaving the overall state (white) visibly unchanged.

³³PO, 336; AT, v6, 331.

³²This right-angled prism is isosceles, and so its exceeded Descartes's by 5°.

³⁴PO, 335; AT, v6, 330.

³⁵PO, 335; AT, v6, 330.

³⁶In Descartes's experimental configuration, the beam within the prism is effectively undispersed. Furthermore, since he used a right-angle prism with the sunlight entering its long face NM, the beam will always emerge from the surface NP refracted towards the other side of the prism, MP. Because blue light has a higher index than red, the border colour at the edge EH must then show blue, and conversely, red always shows at the border DF. Where, for Newton, the boundary coloration would exemplify the unequal refrangibility of red and blue light, for Descartes it meant that the order of colours could not be linked to the relative angles of emergence of the beam's edges.Confusion about what Descartes had in mind with respect to the prism's colour orders remains to this day. Lee and Fraser, The Rainbow Bridge. Rainbows in Art, Myth, and Science, 355, note 195, for example, look to Descartes's diagram to conclude that ‘he must mean that the red ray emerges from the prism at a less oblique angle than the blue, as his accompanying diagram shows’. On the other hand Sabra, Theories of Light from Descartes to Newton, 64 correctly notes Descartes's claim.

³⁷The original reads ‘Mais j'ay jugé qu'il y en falloit pour le moins une, & mesme une don't l'effect ne fust point destruit par une contraire’: AT, v6, 330. We will return below to the question of what Descartes meant by a refraction whose ‘effect’ is not destroyed by a subsequent one, particularly since, in the rainbow, the second refraction produces a ray that emerges at the same angle with respect to the drop normal at which the incoming ray had entered.

³⁸PO, 336; AT, v6, 331.

³⁹PO, 332; AT, v6, 325.

⁴⁰PO, 336; AT, v6, 331.

⁴¹ Dioptrique, First Discourse: AT, v6, 89.

⁴²Shea, The Magic of Numbers and Motion. The Scientific Career of René Descartes, 212–18 discusses Ciermans and Morin. We will return below to these critiques and Descartes's response to them.

⁴³PO, 336 (translation altered); AT, v6, 331. The original reads ‘il faut imaginer les parties ainsi que de petites boules qui roullent dans les pores des cors terrestres’.

⁴⁴In the ninth Discourse of the Météores, Descartes remarked that ‘the normal movement of the small particles of this material—of those in the air around us, at least—is to roll in the same way that a ball rolls on the ground, when it is propelled only in a straight line. And it is the bodies that make them roll in this way which we properly call white’ (PO, pp. 346–47; AT, v6.). Westfall, ‘The Development of Newton's Theory of Color’, 341 notes the point concerning white.

⁴⁵Descartes explained to Ciermans that ‘all these little globes contained within the pores of glass, air, and other bodies always, or at least most often, have an inclination or propensity to turn in some direction, and even to turn with a speed equal to that with which they are moved in a straight line, as long as they do not encounter a specific cause which augments or diminishes this speed … most of them have different inclinations, according to their diverse encounters with the confines of the pores where they are located; so that if some among them incline to turn to one side, others incline to turn at the same time to another’ (Clerselier, ed., Lettres De Mr Descartes, 3 vols. (Paris, 1724), 298–99). He follows this with an example that insists on the random directional effects of encounters with large particles. Descartes seems not to have thought that these encounters would alter the rotational tendencies per se but only their directions, the motion proper being conserved except in the special cases of coloured surfaces and at the edges of shadowed regions in refraction. For Descartes, a rotating sphere would necessarily continue to rotate unless forced not to for the same reasons that undergird his famous analysis of a stone whirled ‘round in a sling: the glue that holds the parts of the sphere together pulls them back from the innate force that, as he understood the dynamics, tends to move them outwards in order to remain along the path's tangent.

⁴⁶This establishes a chromatic scale running from red through white to blue, but the Cartesian scale applies only to prismatic colours and furthermore has no clear implications even for the mixing of lights.

⁴⁷As noted in Westfall, ‘The Development of Newton's Theory of Color’, 343.

⁴⁸There are, however, potential difficulties in forming a consistent understanding of optical intensity in Descartes's scheme if ray counts are to be used: see below, note 58.

⁴⁹Descartes accordingly had to perform the following computations. Given the refractive index, n, the drop radius R, and the distance FH, the corresponding arcs , are, respectively, . If—like Harriot before him—Descartes had worked with angles of incidence and refraction, then he would instead have computed r as followed by simple additions and subtractions to obtain ,. Descartes's method requires two trigonometric look-ups together with squarings and square roots.A treatise probably written by Benedict de Spinoza (1632–1677), but printed posthumously without an author's name in 1687, provided the tiresome details that Descartes left to the reader. ‘As is his wont’, Spinoza wrote, ‘he [Descartes] simply presents his table, without revealing to those interested in algebra how he discovered the two laws of refraction by means of which he worked it out’ (M.J. Petry, ed., Spinoza's Algebraic Calculation of the Rainbow & Calculation of Chances, vol. 108 (Dordrecht, 1985), 39). Spinoza proceeded to introduce x, y to represent , in order to ‘reproduce it in algebraic terms’ (p. 53).

⁵⁰At 20°C, sodium light (589.3 nm) has an index of 1.33299, which reaches 1.33348 at 14°C. Descartes's value, 1.3369 to four decimal places, is too high for the central spectrum, which perhaps indicates (assuming he did measure) that he observed a dispersed beam and took the index towards its most refracted terminus. Presumably, Descartes would have used the apparatus he described in the Dioptrique (PO, 162; AT, v6, 212), though without reconstructing the device it is difficult to tell how accurately an index can be measured with it.

⁵¹All of the first-refracted rays are internally reflected from a comparatively small arc at the top of the sphere from G to the point Z that lies on a diameter along line CZ which is parallel to the rays of the incoming set. Isaac Barrow (1630–1677) determined that (in our Figure 9) the counterclockwise limit (G) of the region marks the point at which the intersection of a pair of indefinitely close refractions lies on the surface of the sphere itself. From this, it followed without the computation of an extremum that the viewing angle must be a maximum: see Alan E. Shapiro, ‘The Optical Lectures and the Foundations of the Theory of Optical Imagery’, Before Newton. The Life and Times of Isaac Barrow, edited by M. Feingold (Cambridge, 1990), 144–47 for Barrow on the rainbow. At this angle, neighbouring rays among an initially parallel set accordingly remain parallel to one another on emergence; at other viewing angles, they diverge. In Figure 9, rays 8 and 9 remain nearly parallel, whereas all other ray pairs are divergent.

The loci of emergence for both singly and doubly reflected rays move counterclockwise with increasing incidence until certain angles are reached, at which point the loci reverse direction. The two angles differ, since the respective positions of N and Q are and . Consequently, the angles for reversal are, again respectively, and or, for Descartes's index of 250/187, about 76° 45′, 81° 23′; the corresponding radial distances FH are (in Descartes's units) 9734, 9887. Although the last entry in each of Descartes's two tables could be used to show the reversal, he did not list the actual position of the points of emergence N and Q because he did not tabulate the angles of incidence (Figure 11). Below the reversal angle, the singly reflected emergent rays project back to intersect at the point Z in Figure 9. No such point exists for doubly reflected rays.

⁵²Their loci can be computed using Descartes's tabulated arcs (see below, Figure 11) from a corresponding table of the (though Descartes did not provide one), since these latter are the angles of incidence, . Then, moving clockwise, Q is located at , while N is located at .

⁵³‘ … its part D appeared to me completely red and incomparably more brilliant than the rest … this part K would appear red too, but not as brilliant as at D’.

⁵⁴There are three effects here: extreme brightness, coloration, and cutoff. Only the first in and of itself suggests ray clustering.

⁵⁵Edmund Halley, ‘A Discourse of the Rule of the Decrease of the Height of the Mercury in the Barometer, According as Places Are Elevated above the Surface of the Earth, with an Attempt to Discover the True Reason of the Rising and Falling of the Mercury, Upon Change of Weather’, Philosophical Transactions 16 (1686) 104–15. For Huygens, see Anders Hald, History of Probability and Statistics and Their Applications before 1750 (Hoboken, NJ, 2003), 108–109; the graph appears in Christiaan Huygens, Oeuvres Complètes De Christiaan Huygens, edited by Société_Hollandaise_des_Sciences (La Haye, 1888–1950), 526–31, vol. 6.

⁵⁶Although the angle which the extreme ray forms with a line perpendicular to the horizon depends only on the index of refraction, the Sun's angular extension does affect the viewing angle (figure 10). The tabular computation assumes rays from the Sun's centre and must accordingly be adjusted.

⁵⁷The singly and doubly reflected extremal angles , are, respectively, .

⁵⁸Another possible difficulty with Descartes's scheme, though not one that seems to have occasioned much if any comment, concerns optical intensity proper. Descartes here associated it with the number of rays in a given region, implicitly assuming that each ray has the same, or effectively the same, visual effect. Optical intensity in the Cartesian mechanical model can presumably be changed in only two ways: either the originating luminous pressure is altered, or the number of spheres in a given region that are subject to the pressure is itself changed. Descartes's world is always completely full, and so to alter the ball numbers requires that they displace the larger particles of air, for example. However, the number of rays can be increased without doing anything directly physical to a region and without changing the originating pressure at the luminous source. Indeed, that is just what occurs in any reflection or refraction, which means that the notion of a light ‘ray’, in so far as optical intensity is concerned, is somewhat problematic here even though Descartes effectively counts rays in his tables. Light rays might be treated as geometrical entities that have a dual character: on the one hand, a ray traces a path of pressure through the balls, while, on the other hand, the number of rays in a given region specifies the degree of pressure sustained by the spheres within it, with the further implication that the pressure acts laterally among the rays as well as linearly along each of them (which follows from Descartes's understanding of pressure).

⁵⁹Despite Descartes's silence on the point, in his Nova Stereometira Kepler and explicitly recognized that a property which changes with another will alter the least when the latter is at a maximum (Ch. Frisch, ed., Joannis Kepleri, Astronomi Opera Omnia (Frankfurt, 1858–1871), 612, vol. IV), where he writes that ‘circa maximam vero utrinque circumstantes decrementa habent initio insensibilia’, viz. ‘near a maximum the decrements on both sides are in the beginning only imperceptible’: translated in Dirk Struik, ed., A Source Book in Mathematics, 1200–1800 (Cambridge, MA, 1969), 222. I thank Jesper Lützen for the reference. A simple method for determining the extrema of polynomials was developed by Johann Hudde in the 1650s (see Victor J. Katz, A History of Mathematics. An Introduction, 2nd edition (Reading, MA, 1998), 473–74).

⁶⁰Johannes Lohne, ‘Regenbogen Und Brechzahl’, Sudhoffs Archiv für Geschichte der Medizin und der Naturwissenschaften 49 (1965), 408–409. Since the viewing angle for the primary bow is 4r – 2i, at an extremum we must have 4dr = 2di. The law of refraction yields cos(i)di = ncos(r)dr, with n the index, and so the two relations together entail tan(i) = 2tan(r). Barrow (see above, note 51) obtained an equivalent relation, since he had found the conditions which place the intersection of neighbouring refractions on the sphere itself. His version had the form cos(r)/cos(i) = 2/n: Shapiro, ‘The Optical Lectures and the Foundations of the Theory of Optical Imagery’, 146, in A.G. Bennett, ed., Isaac Barrow's Optical Lectures (Lectiones Xviii) (London, 1987), 146–52.

⁶¹Johannes Lohne, ‘Thomas Harriot Als Mathematiker’, Centaurus 11 (1965), 35–38.

⁶²Descartes tabulated arcs and , which are (180°–) 2i and 2r. He accordingly could read off twice the differences between successive angles. The differences in the incidences proper in the vicinity of the primary maximum from distances of 8300 through 8800 (the tabular extremum occurring at 8500 and 8600) are 1° 1′, 1° 2′, 1° 5′, 1° 6′, 1° 9′, 1° 10′; the corresponding differences in the refractions are 1° 3′, 1° 2.5′, 1° 3.5′, 1° 3′, 1° 8′, 1° 8′. The problem is not that Descartes's differences in the refractions are any larger than Harriot's in the same vicinity, but that his incidences are not fixed at arithmetic differences. Consequently, what jumps to the eye in Harriot's table—that the refraction differences are very nearly half the fixed differences in the incidences—cannot be divined from Descartes's.

⁶³Alternatively, Harriot may have been thinking of the concentration of light near extrema, since Newton, without having seen anything of Harriot's, made the terminological connection based on the slow motion of the Sun at the solstices: ‘Now it is to be observed, that as when the Sun comes to his Tropicks, days increase and decrease but a very little for a great while together; so when by increasing the distance CD [which measures the sine of the angle of incidence], these Angles come to their limits, they vary their quantity but very little for some time together, and therefore a far greater number of the rays … ’ (Newton, Opticks: Or, a Treatise of the Reflexions, Refractions, Inflexions and Colours of Light. Also Two Treatises of the Species and Magnitude of Curvilinear Figures, 128: Prop. IX, Prob. IV).

⁶⁴Johannes Lohne, ‘Thomas Harriott (1560–1621). The Tycho Brahe of Optics. Preliminary Notice’, Centaurus 6 (1959), 120. Harriot used both a hollow glass prism filled with turpentine, saltwater, wine or just water, as well as one of solid glass. He found, for water, for example, that what he termed the ‘chief primary’ light, which must be just below the coloured border, had an index of 1.3351921, while the ‘secondary’, or red ray, had an index of 1.3415993. Though Harriot as usual left no remarks in MS, his diagram (ibid.) shows that he observed the upper boundary of the paper through the prism apex, where the colour will be red. He did not it seems observe a lower border, which would be coloured blue. Since Harriot gave his measured incidences and refractions to the minute, his computed values for the indexes are accurate to about .002 for angles above 20°.

⁶⁵Descartes remarked ‘how much the ancients were deceived in their Catoptrics, when they tried to determine the locations of images in concave and convex mirrors’ (AT, v6, 144; PO, 110).

⁶⁶For Kepler, see Shapiro, ‘The Optical Lectures and the Foundations of the Theory of Optical Imagery’, 119–27, whom I thank for discussions concerning Kepler, Descartes and perceived images. All of the situations with which Descartes was concerned involve sets of divergent rays, and so the ‘image’ in question is the one produced by the eye or by an equivalent lenticular system. The images are all, in later parlance, virtual—they cannot be received directly on a screen without lenticular transformation. Kepler did not consider the perception of an image by a single eye, which is just the situation that Descartes had in mind for the prism and, subsequently, for the rainbow: viz. (referring to Figure 3) ‘when the eye is in the location of the white screen FGH’ (AT, v6, 341; PO, 342).John Schuster has argued that Descartes may have discovered the law of refraction by reasoning from the cathetus method despite Kepler (John Schuster, ‘Descartes and the Scientific Revolution, 1618–1634: An Interpretation’, Ph.D., Princeton University, 1977, John Schuster, ‘Descartes Opticien: The Construction of the Law of Refraction and the Manufacture of Its Physical Rationales, 1618–1629’, Descartes’ Natural Philosophy, edited by Stephen Gaukroger, John Schuster and John Sutton (London, 2000), 299–329). Schuster's argument is inferential, based in substantial part on similarities between Claude Mydorge's (1585–1647) diagrammatic formulation of the method in a letter to Marin Mersenne (1588–1648) and the way in which the law would be expressed if it did originate in the cathetus. There is substantial evidence, Schuster notes, that both Harriot and Willebrord Snel (1580–1626), both of whom produced the law independently, did reach it in just that way, or at least made the association between image locus and the refraction law. Schuster's essential point is that if an object is placed sequentially on the circumference of a circle within water, for example, then the locus of its image points drawn according to the cathetus will appear to lie on a concentric circle of smaller radius, within observational limits. If the emergent rays are assumed accurately to entail the smaller circle, then the law of refraction follows at once, albeit expressed in a different manner than the ratio of sines.

⁶⁷AT, v6, 144; PO, 110. See also Descartes's remarks in his posthumously published Traité de l'Homme, in which he associates the judgement of distance with the displacement of the point in the pineal gland that is moved by the effect of the animal spirits impelled by the visual motions produced, in this case, in the retinas of both eyes (AT, v11, 183). Also see Gary Hatfield, ‘Descartes’ Physiology and Its Relation to His Psychology’, The Cambridge Companion to Descartes, edited by John Cottingham (Cambridge, 1992), 357, who remarks Descartes's notion that ‘the idea of distance is caused by a brain state without judgmental mediation’.

⁶⁸The sequential application of the cathetus rule to multiple reflections was applied by Hero in his Catoptrics to mirrors (Morris R. Cohen and I.E. Drabkin, eds., A Source Book in Greek Science (Cambridge, MA, 1975), 267: sec. 18), which (though attributed to Ptolemy) was translated into Latin by the Flemish Dominican William of Moerbeke (1215–1286). In Hero's construction, the locus of the penultimate reflection constitutes the object point for applying the cathetus. In the application that, I suggest, Descartes made to the raindrop, the final action is a refraction, but the penultimate effect is, as in Hero's Catoptrics, a reflection, so that the generalization of Hero's procedure to this case would consider the locus of the final reflection to constitute the object point for the emergent refraction.Medieval opticians do not seem to have concerned themselves with images produced by multiple reflections (much less by a refraction preceded by a reflection) which is hardly surprising since, unlike Hero, they were not interested in temple illusions. Giambattista della Porta (1535–1615) did, however, apply the cathetus to the case of two refractions by a sphere or by a convex lens. In doing so, he did not consider the locus of the first refraction to be the object point for the image produced by the second; he instead retained the locus of the original object for the construction. In an altogether unusual departure from the customary uses of the cathetus, he did not even employ the tangent to the surface at the point of emergence. Instead, he passed a line from the object through the centre of the sphere, placing the image at the point of the latter's intersection with the ray within the sphere (and not the extension of the ray from the eye to the sphere). See Giambattista della Porta, De Refractione Optices Parte (Naples, 1593); I thank Sven Dupré for the reference and for discussion about the issue.

⁶⁹The following relations determine the locus of the image via the ‘cathetus'construction:

⁷⁰Specifically, at the single-reflection extremum (corresponding to an incident ray that strikes at .85 or .86 radii), the image is located at .99 radii from the drop's centre, while at the double-reflection extremum (for an incident ray striking at .95 radii) it is at 1.06 radii, using Descartes's tabulated angles. The image distance from the centre increases very rapidly as the incident ray moves closer in towards the centre from, so that it is entirely reasonable to argue that red should occur near the extrema for both bows.

⁷¹Boyer, The Rainbow. From Myth to Mathematics, 217. Shea, The Magic of Numbers and Motion. The Scientific Career of René Descartes, 224 follows Boyer on this point, though he notes Descartes's remark about prism thickness, while Gaukroger, Descartes. An Intellectual Biography, 269 writes that Descartes could not explain the colour inversion.

⁷²See above, note 67.

⁷³Recall that the concentration of rays is one of only two necessary conditions for the production of colour: the other is either the appropriate physical character of a reflecting surface, or else a refraction. Polished mirrors may concentrate rays, but their surfaces are not of the right sort to produce coloration.

⁷⁴The criticisms are also discussed in Shea, The Magic of Numbers and Motion. The Scientific Career of René Descartes, 212–18.

⁷⁵The Latin correspondence with Ciermans is printed in AT, v2, 55–62 and 69–81. The letters are translated into French in Clerselier, ed., Lettres De Mr Descartes, 262–72 and 86–303, vol. 1 and (on different pages) in earlier editions.

⁷⁶On which, see Shea, The Magic of Numbers and Motion. The Scientific Career of René Descartes, 213–16.

⁷⁷We will not pursue the point, but Descartes insisted on identifying red with great agitation because he linked greater agitation or tendency also to heat, and hot things are often red or turned red by heat.

⁷⁸Descartes of course used neither word in his explanation for Ciermans. Nevertheless, the discussion quite clearly proceeds by compensating one action by another in an unrefracted beam, and by invoking the absence of local differences in an unshadowed one: see Clerselier, ed., Lettres De Mr Descartes, 298–303. According to this way of thinking, shadow combined with a sufficient degree of refraction resets an unrefracted beam so that the spheres all now rotate in the same direction (ibid., pp. 300–301) but at different rates. The rate increases from the ‘usual’ at the centre up to a maximum at the first edge, and decreases from the centre to a minimum at the opposite edge. Without a general ordering of the spheres into the same rotational direction, it would be difficult to understand why the rate is a maximum at one edge and a minimum at the other, since if a random distribution of rotational directions continued to obtain on refraction, then presumably the rate-changing actions would themselves have random effects, leaving the beam overall untinted. However, the direction of rotation still remains irrelevant to colour—only the magnitude of the rotation rate's difference from the ‘usual’ counts.

⁷⁹AT, v1, 536–57.

⁸⁰‘I would willingly attack the essence or nature of light, which you say is action, or motion, or the inclination to motion, or like an action and a motion, &c. of subtle matter, &c’. (ibid.,547).

⁸¹I would willingly attack the essence or nature of light, which you say is action, or motion, or the inclination to motion, or like an action and a motion, &c. of subtle matter, &c’. (ibid.,547).

⁸³AT, v2, 208. Cited and discussed, with a slightly different translation, in Shea, The Magic of Numbers and Motion. The Scientific Career of René Descartes, 217.

⁸²AT, v2, 196–221.

⁸⁷AT, v6., 368.

⁸⁴AT, v2, 288–305.

⁸⁵AT, v6, 331. In fact, Descartes had written ‘il faut imaginer’, which has an aura of deliberate ambiguity.

⁸⁶AT, v6, 362–73.

⁸⁸See Peter Galison, ‘Descartes's Comparison: From the Invisible to the Visible’, Isis 75 (1984), 323–24 on comparison and the Cartesian imagination.

⁸⁹AT, v6, 408–19.

⁹⁰AT, v6., 411.

⁹¹AT, v6, 330: ‘il y en falloit pour le moins une, & mesme une don't l'effect ne fust point destruit par une contraire’.

⁹²Robert Hooke, Micrographia: Or Some Physiological Descriptions of Minute Bodies Made by Magnifying Glasses with Observations and Inquiries Thereupon (London, 1665), 59: ‘quidem talent ut ejus effectus alia contraria (refractione) non destruatur’—‘refractione’ is absent from the original, cf. AT, v6, 702.

⁹³Hooke, Micrographia: Or Some Physiological Descriptions of Minute Bodies Made by Magnifying Glasses with Observations and Inquiries Thereupon, 54.

⁹⁴There is an extensive body of literature on More. See Alan Gabbey, ‘Philosophia Cartesiana Triumphata: Henry More (1646–1671)’, Problems of Cartesianism, edited by Thomas M. Lennon, John M. Nicholas and John W. Davis (Kingston, 1982), A Rupert Hall, Henry More and the Scientific Revolution (Cambridge, 1990) and especially the illuminating introductory discussion in A. Jacob, ed., Henry More's Manual of Metaphysics. A Translation of the Enchiridium Metaphysicum (1679) with an Introduction and Notes (Hildesheim, 1995 (1679)).

⁹⁵Jacob, ed., Henry More's Manual of Metaphysics. A Translation of the Enchiridium Metaphysicum (1679) with an Introduction and Notes, v.

⁹⁶Jacob, ed., Henry More's Manual of Metaphysics. A Translation of the Enchiridium Metaphysicum (1679) with an Introduction and Notes, v, 148.

⁹⁷Jacob, ed., Henry More's Manual of Metaphysics. A Translation of the Enchiridium Metaphysicum (1679) with an Introduction and Notes, v., 164–67. More even revived the argument from authority: ‘Thus Plotinus, Whose opinion approaches very closely this of ours, and differs in almost no way, except that he is seen to expressly invoke a certain World-Soul and we are concerned to detect in the present case nothing apart from a certain Hylarchic Principle or World-Spirit in general. On which, however, we acknowledge all those sympathies and harmonies of life to be based, and apart from which we divine the phenomena of light and colours would be either not at all or very weak and fading, and not perceptible from almost any distances’ (p. 167)—‘hylarchic’ meaning that More's World-Spirit rules over matter.

⁹⁸Robert Hooke, ‘Lampas: Or, Descriptions of Some Mechanical Improvements or Lamps & Waterpoises. Together with Some Other Physical and Mechanical Discoveries’, Lectiones Cutlerianae, or a Collection of Lectures Physical, Mechanical, Geographical, & Astronomical. Made before the Royal Society on Several Occasions at Gresham Colledge. To Which Are Added Divers Miscellaneous Discourses. (London, 1679 (1677)), 33–34 in Lampas.

⁹⁹Hooke might have noted a correlated criticism for the rainbow: namely, that if colours do not occur when light emerges from thick plates after an internal reflection, then why should they do so in the case of a sphere, which apparently satisfies precisely the same conditions? More's argument based on deviation cannot answer this objection, which is moreover one that Descartes might have known since he was well aware of internal reflection—after all, it underlies his theory of the rainbow. The only way to solve the problem, and one which Descartes would likely have used, involves taking account of the shape of the emergent beam, for there is a significant difference in this respect between unshadowed light directly from the Sun that emerges after internal reflection from a plate and the same light that emerges from a sphere: the light from the plate remains effectively without shadow, whereas a marked shadow has been produced in the case of the sphere, thereby providing a rationale for colour. Suppose, however, that a wide beam of shadowed light enters both plate and sphere: it too will emerge uncoloured from the plate but not from the sphere, even though the shadow exists in both cases before entry. To handle this situation in Descartes's scheme would require specifying that the shadow must be significantly altered in the exit beam from its structure on entry, thereby (in effect) generating a new and possibly colour-producing situation. There is indeed a difference available here: in the case of the plate, the shape of a beam is unchanged at emergence; the sphere, on the other hand, markedly alters it, splaying the beam out and redistributing the rays within it. In fact, the rays that had been shadowed at entry are generally not shadowed at exit, and the rays that on exit are now shadowed were not so on entry. All of which offers ample opportunities for exploitation. Hooke's thin films certainly counter this, but the point is that in their absence, Descartes's scheme, at least in this respect, could be reasonably well defended.

¹⁰⁰Alan E. Shapiro, ‘The Gradual Acceptance of Newton's Theory of Light and Color, 1672–1727’, Perspectives on Science 4 (1996) 59–140.

¹⁰¹For another example of imperfect rhetoric, but one which in this case has the virtue of being backed by manuscript evidence, see Jed Z. Buchwald, ‘The Scholar's Seeing Eye’, Reworking the Bench: Research Notebooks in the History of Science, edited by Larry Holmes, Jürgen Renn and Hans-Jörg Rheinberger, Archimedes (Dordrecht, 2003) 309–25. And for an account of how to recover past practice by a master of science history, see Frederic L. Holmes, Investigative Pathways. Patterns and Stages in the Careers of Experimental Scientists (New Haven, CT, 2004).