In his letter to Vatier of 22 February 1638, Descartes writes:
I have, however, given a brief sample of it [my method] in my account of the rainbow, and if you take the trouble to reread it, I hope that it will satisfy you more than it did the first time; the matter is, after all, quite difficult in itself.
(AT 1: 559, CSM 3: 85) This is one of the few places in which Descartes unequivocally claims to provide a sample of his method; hence, we briefly examine that work, and find that his procedural moves comply with our account of the method.1
Descartes’s discussion of the rainbow in Discourse Eight of the Meteorology differs in certain ways from the method we have discussed and which we will discuss in detail with respect to the Meditations. At first blush, one might think that he conducts no searches for general principles, no exercises in conceptual elucidation, and fails to appeal to the natural light. Since we contend that these characteristics are hallmarks of the Cartesian method, does their absence expose a fundamental flaw in our account of the method?
Not at all. First, being a fairly complex phenomenon, the rainbow is far from fundamental. His discussion presupposes the conclusions reached in the previous seven discourses of the Meteorology and the Optics. We should not be surprised that he discovers no new laws. Second, the discourse on the rainbow is explanatory. As such, it exemplifies the fundamentality of coherence to the Cartesian method. Third, it is only at first blush that one finds no exercises in conceptual elucidation. As we show, Descartes is concerned with the essential elements necessary for the occurrence of a rainbow. While his interests are straightforwardly causal, the search for
essential elements may be construed as an exercise in ideational clarification.
Finally, as one would expect when examining a topic that is several levels removed from the metaphysical foundations of the Cartesian system, it is only reasonable to contend that he tacitly justifies his conclusions by means of the natural light; to explicitly introduce the natural light at this level of discourse would be inappropriate.
Descartes describes the phenomenon to be explained in the first paragraph of Discourse Eight. He begins by observing the occasions on which one perceives a rainbow. He writes:
First, I considered that this arc can appear not only in the sky, but also in the air near us, whenever there are many drops of water in the air illuminated by the sun, as experience shows us in certain fountains;
thus it was easy for me to judge that it came merely from the way that the rays of light act against those drops, and from there tend toward our eyes. Then, knowing that these drops are round, as has been proven above, and seeing that their being larger or smaller does not change the appearance of the arc, I then took it into my head to make a very large one, the better to examine it.
(AT 6:325, O 332) Descartes sets the initial elements of the problem as follows. Experience indicates that it is only on occasions in which droplets of water are in the air that one perceives a rainbow. This occurs not only in cases when a rainbow appears at a distance after a rainstorm, but also when rays of sunlight pass through droplets of water close at hand. Thus, the phenomenon to explain is how the interaction of water droplets and light result in an observable rainbow. Given his conclusion earlier in the Meteorology that all drops of water are round – “those of fresh water are round like strings, and those of salt like cylinders or rods; for all bodies that move in different ways over a long period of time usually become rounded” (AT 6: 203, O 285; cf. AT 6:
280–1, O 298–9, where water droplets are said to be perfectly round) – Descartes proposes to examine the phenomenon by using a large round flask filled with water to simulate a water droplet. Let us call this round flask of water a “super-droplet.”
Descartes observes the effects of sunlight passing through the super-droplet. (See Figure 2.) There are two rainbows. In the primary rainbow, the color red is perceived at point D. In the secondary rainbow, a fainter red is
perceived at point K. In the case of D, the angle formed by point D, the observer E and the point M at the base of the rainbow is about 42°. The angle KEM is approximately 52°. So long as the angle DEM remains at approximately 42°, the brilliant red remains. If the angle becomes bigger, the color disappears; if it becomes smaller, it divides into other colors. The fainter rainbow is inverted. If the angle KEM is made slightly larger than 52°, weaker colors (yellow, blue) appear; if it is much larger or somewhat smaller than 52°, the colors disappear. This is the phenomenon to be explained.
Descartes now turns to explanation. He writes:
After this, examining in more detail what caused the part D of the ball BCD to appear red, I found that it was the rays of the sun which, coming from A toward B, were curved as they entered the water at point B, and went toward C, whence they were reflected toward D; and there, being curved again as they left the water, they tended toward E. For as Figure 2 The rainbow
soon as I put an opaque or dark body in some place on the lines AB, BC, CD, or DE, this red color would disappear. And even if I covered the whole ball except for the two points B and D, and put dark bodies everywhere else, provided nothing hindered the action of the rays ABCDE, the red color nevertheless appeared. Then I was also searching for the cause of the red which appeared at K; and I discovered that it was the rays which came from F toward G, where they curved toward H, and in H reflected toward I, and in I reflected again toward K, and then finally they curved at point K and tended toward E. Therefore the primary rainbow is caused by the rays which reach the eye after two refractions and one reflection, and the secondary by other rays which reach it only after two refractions and two reflections; which is what prevents the second from appearing as clearly as the first.
(AT 6: 328–9, O 334) To understand how this passage agrees with our account of the Cartesian method, one should take the initial observation – that wherever rainbows appear water droplets and sunlight must also appear – as placing certain limits on the explanans: any explanation of the rainbow must appeal only to properties of water and light. The discussions of reflection and refraction in the Optics (see AT 6: 93–105, CSM 1: 156–64) provide the guiding thread:
namely, water bends rays of light. Given this, Descartes may formulate a hypothesis regarding the roles of reflection and refraction in explaining the physical side of color perception: in the case of the primary rainbow, one perceives red as a result of two refractions and one reflection, and in the case of the secondary rainbows, one perceives red as a result of two refractions and two reflections. The hypothesis, in turn, guides the subsequent experimental tests and observations: by blocking the light following any of the lines represented on the diagram, the phenomenon (red) ceases to occur. These observations confirm the supposition that to see red requires light. They also show that to perceive redness requires two refractions.
If Descartes’s method complies with the model we have suggested, he should seek the minimum number of characteristics of light interacting with a medium to explain the phenomenon (the rainbow) sufficiently. If one were to enumerate the characteristics he has considered to this point, these would include: one, curved water droplets, two, light traveling in straight lines, three, two refractions of light, four, one or two reflections of the light, and five, refractions resulting in angles of very specific degrees. Are all of these
characteristics necessary for the phenomenon? No. His discussion of the prism reduces the number of characteristics necessary for explaining the phenomenon.
A prism is a triangle of crystal which causes a rainbow phenomenon. As he describes the case, he assumes a prism with an angle of 30° to 40°, such that the sun’s rays that enter the prism are perpendicular to its angled surface and bend to form a rainbow. (See Figure 3.) The phenomenon is the same as it was in the case of the rainbow, but there are significant differences in the circumstances that result in the rainbow-like phenomenon. Descartes lists these:
From this I learned, first, that the surfaces of the drops of water need not be curved in order to produce these colors, for those of this crystal are completely flat; nor does the angle under which they appear need to be of any particular size, for it can be changed there without their changing. And although we can cause the rays going toward F to curve sometimes more and sometimes less than those going toward H, they nevertheless always paint red, and those going toward H always paint blue; neither is reflection necessary, for there is none of it here;
nor finally do we need a plurality of refractions, for there is only one of them here. But I judged that there must be at least one refraction, and even one such that its effect was not destroyed by another, for experiment shows that if the surfaces MN and NP were parallel, the rays, being straightened as much in the one as they were curved in the other, would not produce these colors.
(AT 6: 330–1, O 335) Notice that Descartes eliminates a certain number of characteristics. The rainbow phenomenon occurs when light passes through a prism. But the prism has only straight edges, so the curvature of a water droplet is not necessary for the formation of a rainbow. The size of the angle appears to be irrelevant to the phenomenon. In the case of the prism, there is only one refraction and no reflection of the light, so it appears that only one refraction is necessary for the phenomenon.2 So he leaves us with the refraction of light as the sole necessary condition for explaining the rainbow phenomenon.
Before going on, we should notice what Descartes has done here and how it complies with our general account of the Cartesian method. Notice, first, that a rainbow as a phenomenon that occurs after a thunderstorm is
sensibly indistinguishable either from a rainbow that occurs when light passes through water in a fountain or from the result of light passing through a prism. Second, insofar as the visual effect is the same, prima facie reasons arise for assuming a common cause. In examining the prism, Descartes has clarified the idea of the cause of a rainbow; it is in many ways analogous to the clarification of the idea of the self in Meditation Two (see AT 7: 25–8, CSM 2:17–19).3 The meteorological rainbow is phenomenally identical with the rainbow caused by light passing through a prism. In showing the differences between the two cases, he enables himself to hone the notion of the cause of a rainbow to its single fundamental element, namely, the refraction of light.
But if the refraction of light alone is essential to the rainbow phenomenon, it remains an open question why such refraction results in the apparent change from white light to colored light. In turning to this problem, Descartes again allows the conclusions reached in the Optics to guide his hypotheses.
Light is construed as the movement of fine particles. The movement of these particles is, in principle, no different from the movement of other bodies. So to explain refraction is nothing more than to explain changes in the motion of the moving particles constituting light. Descartes explains this (see Figure 4) by means of the interaction of a particle V (whose spin is represented by the numbers 1, 2, 3 and 4) with four other particles, Q,R, S, Figure 3 The prism
and T. If Q and R are moving more rapidly toward some point X than V, and S and T are moving more slowly toward X than V, then Q increases the spin of V without hindrance from R, since R is already moving faster. The result is that V has a stronger tendency to rotate than to move in a straight line, which causes the color red. If the opposite situation obtains – if Q and R are moving more slowly than V, which is moving more slowly than T and S – then the rotation of V is decreased and it veers in such a way that it is visible as the colors green to blue.4
Notice what has happened. Descartes focuses on the rainbow phenomenon. He makes certain observations of a meteorological rainbow.
He clarifies the concept of the cause of a rainbow by considering rainbows caused by a prism. This ideational analysis clarifies the idea of the cause of a rainbow by reducing it to a fundamental common element, namely the refraction of light. He then explains refraction in terms of his optical theory.
All that remains for him to do is explain why, in the case of the meteorological rainbow, the primary and secondary rainbows are perceived only at about 42° and 52°. This he explains in terms of shadows, the blockage of light (see AT 6: 336–7, O 339).
We need not go into the details of that explanation. Nor need we go into his explanations of inverted rainbows (AT 6: 341–2, O 343–4) and tertiary rainbows (AT 6: 342–3, O 344). Suffice it to say his pattern throughout Figure 4 Movement of particles
Discourse Eight follows the model we set forth in the first three chapters.
He analyzes the phenomenon (he looks for clear and distinct ideas). He seeks natural laws which will allow him to explain it. In this case, unlike those we have considered or shall consider below, the laws are found in the more general work he has already completed, namely, the Optics.
Having found the laws and a “clear and distinct idea” of the cause of the rainbow, he can explain the meteorological phenomenon on the basis of his laws. This is precisely what one would expect given our account of the Cartesian method.