Abstract
Some time ago during a course I was teaching at Caltech several students were puzzled by the differences between Descartes and Newton over prismatic colors. One afternoon a week or two later Moti and I were discussing a project we had in mind about Newton, when our conversation turned to the questions my students had asked. Dan Garber and John Schuster, whom we had known as friends and superb scholars for years, had greatly illuminated Descartes’ work, but something kept nagging us as we turned back to the elaborate account that he had written on prismatic colors and the rainbow. Hidden within Descartes’ seemingly fractured rhetorical construction, was something that hinted at a uniquely ‘Cartesian’ way with experiment. So having obtained a suitable prism and attached pieces of paper to it in Descartes’ fashion, I went with Moti outside to play with colors as he had. It soon became apparent that the configuration and size of this recreated ‘Cartesian’ device markedly controlled the band of colors that could be seen. This suggested ways to understand several otherwise puzzling aspects of Descartes’ observations and claims. To go further, I went home, filled a bowl with water and with the assistance of Diana Kormos Buchwald worked to reproduce what Descartes had seen in that type of configuration as well. These experiments led me to write a long paper on Descartes and the rainbow. What follows here conveys the essence of what we had found.
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Notes
- 1.
Buchwald 2007, 1–46.
- 2.
- 3.
Schuster 2005, 37.
- 4.
The original of René Descartes’s Discourse 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 Method (1637) with minor orthographic corrections, was published in full in vol. 6 of his Oeuvres (1897–1910) (Hereafter AT). Olscamp’s translation of Descartes (1965), (hereafter PO) where used, has been checked against AT; the corresponding location in AT has been provided in all cases.
- 5.
PO, 332; AT, v6, 325.
- 6.
Garber 1993, 298.
- 7.
Maurolycus 1611.
- 8.
Froidmont 1627.
- 9.
De Dominis 1611.
- 10.
- 11.
In the felicitous phrase of Lee and Fraser 2001.
- 12.
This is, in modern parlance, the scattering angle for a refracting sphere.
- 13.
AT, v6, 326–7.
- 14.
PO, 334: AT, v6, 328.
- 15.
PO, 334; AT, v6, 329.
- 16.
PO, 334; AT, v6, 329.
- 17.
Garber 1993, 298 notes the point.
- 18.
PO, 333; AT, v6, 327.
- 19.
PO, 335; AT, v6, 330.
- 20.
Although we do not know whether Descartes closely controlled the dimensions of Figure 3 as printed, measurement of the figure itself yields angles PNM, NMP respectively of 38°40′, 51°20′.
- 21.
For an index n of refraction, total internal reflection sets in for counter-clockwise rays when their angle of incidence on NM reaches sin−1(1/n)-PNM).
- 22.
See immediately below for a discussion of what seems at first to be a strange claim in respect to apertures.
- 23.
The placing of colors 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 colors or generators of colors. Their views had become increasingly common by mid-century, on which see Shapiro 1994, 627. 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.
- 24.
For Descartes bodily and emphatic colors 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.
- 25.
PO, 335; AT, v6, 330.
- 26.
In terms of Descartes’ mechanics, even a wide aperture will produce coloration but its mechanical effect on the microspheres (see below) won’t 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.
- 27.
This right-angled prism is isosceles, and so its angle PNM exceeded Descartes’ by 5°.
- 28.
PO, 336; AT, v6, 331.
- 29.
PO, 335; AT, v6, 330.
- 30.
PO, 335; AT, v6, 330.
- 31.
In Descartes’ 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 color 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 colors could not be linked to the relative angles of emergence of the beam’s edges.
- 32.
The 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.
- 33.
PO, 336; AT, v6, 331.
- 34.
PO, 332; AT, v6, 325.
- 35.
PO, 336; AT, v6, 331.
- 36.
Dioptrique, First Discourse: AT, v6, 89.
- 37.
Shea 1993, 212–18 discusses Ciermans and Morin. We will return below to these critiques and Descartes’ response to them.
- 38.
PO, 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”.
- 39.
In 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. 346–7; AT, v6.). Westfall 1962, 341 notes the point concerning white.
- 40.
Clerselier 1724, 298–99.
- 41.
This establishes a chromatic scale running from red through white to blue, but the Cartesian scale applies only to prismatic colors and furthermore has no clear implications even for the mixing of lights.
- 42.
As noted in Westfall 1962, 343.
- 43.
There are however potential difficulties in forming a consistent understanding of optical intensity in Descartes’ scheme if ray counts are to be used: see below, note 59.
- 44.
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” Petry 1985, 39.
- 45.
Presumably Descartes would have used the apparatus he described in the Dioptrique (PO, 162; AT, v6, 212).
- 46.
Their loci can be computed using Descartes’ tabulated arcs (see below,) from a corresponding table of the \( \hat{FA} \) (though Descartes did not provide one) since these latter are the angles of incidence, sin−1FH/R. Then, moving clockwise, Q is located at \( \hat{FA} \)+3\( \hat{FK} \), while N is located at \( \hat{FA} \)+2\( \hat{FK} \).
- 47.
There are three effects here: extreme brightness, coloration, and cutoff. Only the first in and of itself suggests ray clustering.
- 48.
Although 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. The tabular computation assumes rays from the sun’s center and must accordingly be adjusted.
- 49.
Descartes 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).
- 50.
- 51.
AT, v6, 144; PO, 110. See Hatfield 1992, 357, who remarks Descartes’ notion that “the idea of distance is caused by a brain state without judgmental mediation”.
- 52.
The sequential application of the cathetus rule to multiple reflections was applied by Hero in his Catoptrics to mirrors (Cohen and Drabkin 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.
- 53.
The following relations determine the locus of the image via the ‘cathetus’ construction:
$$ CI=\frac{FH}{n}, FK=2\sqrt{R^2-C{T}^2} $$$$ N{L}_N= FKcos\left(\hat{\frac{FK}{2}}\right),Q{L}_N= FKsi{n}^{`}\left(\hat{\frac{FK}{2}}\right) $$$$ {I}_N{L}_N=\frac{Q{L}_N}{\tan \left(\frac{\hat{FG}}{2}\right)},{I}_NN=N{L}_N-{I}_N{L}_N $$$$ C{I}_N\sqrt{R^2+{I}_N{N}^2+2(R)\left({I}_NN\right)\cos \left(\hat{FG}\right)} $$ - 54.
Specifically, 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 center, while at the double-reflection extremum (for an incident ray striking at .95 radii) it is at 1.06 radii, using Descartes’ tabulated angles. The image distance from the center increases very rapidly as the incident ray moves closer in towards the center from, so that it’s entirely reasonable to argue that red should occur near the extrema for both bows.
- 55.
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Buchwald, J.Z. (2023). Descartes’ Experimental Journey Past the Prism and Through the Invisible World to the Rainbow. In: Roos, A.M., Manning, G. (eds) Collected Wisdom of the Early Modern Scholar. Archimedes, vol 64. Springer, Cham. https://doi.org/10.1007/978-3-031-09722-5_10
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