Marching inexorably towards the final goal. The decisive prismatic experiments that will settle the matter. Part 1

There are two kinds of prismatic experiments: objective and subjective. Our entire understanding of light and colours is fully and unconditionally dependent and defined by those two kinds of experiments. This year marks exactly 350 years since Newton’s “...letter containing his new theory about light and colours” was published in the Philosophical Transactions, which is the event that introduced the Newtonian view of optics to the world, and which has remained ever since the mainstream theoretical understanding of all related phenomena.

In spite of all the opposition that Newton’s theory has faced in its 350-year reign there is not a single thing that has had any significant effect upon the mainstream establishment, and that basically means that, conventionally, we believe that Newton’s theory is as valid in 2022 as it was in 1672. Now, this 350-year-old status quo may be deemed unjust by many today (especially by those who believe that Goethe’s body of work is categorically worthy of an objective assessment and consideration). But the reality is that no decisive and irrefutable evidence has been brought forth as a truly viable alternative to the Newtonian view. Until today. 

We have good reasons to believe that there is a decisive, conclusive, direct and simple way to find out once and for all if the Newtonian theory of light and colours is indeed the definitive descriptor of optical phenomena. This is the main objective of this post. In that quest we will present a number of simple prismatic experiments that will comprehensively test both Newton’s and Goethe’s theories, and ultimately unambiguously determine which, if either, is correct in their respective claims. The experiments that we'll present will address both objective and subjective observations, leaving in the process no room for any doubt about which (if either) of the two theories is a better descriptor of the optical phenomena.

The first experiment we’ll present consists of a subjective prismatic observation of a white rectangle placed upon a black background. (Figure 1)

What the observer will see upon conducting a subjective observation of the white rectangle in Figure 1 is thought to be well known and fully accounted by Newton’s theory. The reality, however, is totally different to the conventional belief, as you will see in a few moments. To that end, let us first present you with the image that the observer will see upon looking at the white rectangle through a triangular prism (prism oriented with its apex pointing to the left) from a distance of about 0.5m. (See Figure 2)

The two figures above contain everything we need to comprehensively determine which (if any) of the two competing theories can correctly account for the result of our experiment.

A Newtonian account of the image above would begin with-and from-the argument that the rectangle under observation appears white to the naked eye because the named rectangle is actually formed by an infinity of superposed rectangles of all spectral colours. This argument will then be followed thus. When that apparently white rectangle is passing through a prism in its way to the observer’s eye, all those infinite superposed rectangles of different colours will be refracted (in the same direction, toward the prism’s apex) by precise and unique amounts for each individual colour. Thus, if we take as an example only those four coloured bands that are flanking the white rectangle, the R rectangle is refracted the least (toward the apex of the prism), the Y one a little more than the R, the C rectangle a little more than the Y one, with the B rectangle being refracted the most. Finally, the Newtonian account would say that the reason that only four colours are visible through the prism is due to the fact that only those colours are not superposed with any of the other myriad of rectangular spectral hues, with the reason for the white display in the middle being the result of all that infinity of colours being once again superposed upon each other over that particular area of the image.

A Goethean account, on the other hand, would argue that the reason for the image depicted in Figure 2, is simply due to the fact that an observation through a prism of the white rectangle will basically create a double image of the original rectangle. Specifically, Goethe says that a prism refracts (or deflects) the entire image of the object under observation in a direction toward the apex of the prism, and the reason for those four visible colours is due to the fact that in one instance we look at the original white rectangle through a turbid medium (the prism), whilst in the other we look at the dark background that surrounds the original white rectangle, through a turbid medium (the prism) that contains a white copy of the original rectangle.

In examining the process of the experiment just given, we find that in the one case we have, to appearance, extended the white edge upon the dark surface; in the other we have extended the dark edge upon the white surface, supplanting one by the other, pushing one over the other. (Paragraph 203 from Goethe's Theory of colour.)

The time has come to answer now whose theory can better account for the results of our experiment. First, though, let us state in advance that the answer to that question is both definitive and categorical. Now, the answer is… neither. Let us show you next why that is the case.

Incredibly, the reality is that neither Newton nor Goethe (nor anybody else, for that matter) seem to have realised that in order to make sure that one’s theory could correctly account for all prismatic observations, one must, before anything else, ensure that the position of any object under observation is precisely determined and known at all times and from all relevant perspectives. Otherwise, how could the experimenter, or the observer, determine to what extent and in which directions does a prism refracts (or deflects) any component of whatever the object of observation may be?

There are many ways of doing that. For instance, in our case the rectangle is placed right at the centre of the picture, and therefore any refraction would change those coordinates. We choose to employ another way, however, in which we place a green line along each side of the white rectangle. (See Figure 3) Green because objects of that colour are not deflected at all by a prism.

Next, we’ll look through our prism at the marked rectangle and thus find that the green lines are right on the borders between the R band and the Y one, and the C band and the B one, respectively. (See Figure 4) Thus we know exactly where the white rectangle is, which in turn means that we can now determine which-if any-of the two theories is correct.

First, Goethe’s theory fails in that quest because if it were correct the G line on the right side would have been laying on the border between the R band and the black background, and the other one would have marked the border between the C band and the white rectangular area at the centre of the image. (See Figure 5)

Second, the Newtonian understanding fails because in that case one of the G lines should be seen laying a little to the right of the R band and the other one a little to the right of the C band. (See Figure 6)

Now, if we examine carefully the image in Figure 4 we have no choice but to accept that the R rectangle has been refracted (deflected) in a direction toward the base of the prism. Needless to say, this is an extremely inconvenient fact, for it flies straight in the face of Newton’s theory. (And Goethe’s, as we know.) Nonetheless, the reality is that (at least) in subjective experiments R and B refract in opposite directions—with R towards the base of the prism and B towards its apex. And that’s not all, either. To that most inconvenient fact we also have to add the reality that G does not refract at all in a prism. (We’ll show that in a moment.) And if all those things weren’t enough, let us finally close this topic by stating that it is not only G that defies the conventional theory by refusing to refract in a prism: neither C nor Y do it, either.

It is easy to prove everything stated in the paragraph above, as you will see from the demonstrations below. First, we shall prove that R and B are refracted by a prism exactly as we have said. At the same time, we shall also prove the G is not refracted at all by a prism. 

In Figure 7 we have an image of the white rectangle we have used in all our experiments thus far. Right under that white rectangle there are three other rectangles of the same width, but of different height (for convenience) coloured R, G, and B. Those three rectangles are all the superposed colours that are (in our understanding) needed, in order to account for all prismatic observations. If we now conduct a subjective observation of the image in Figure 7 (with the prism oriented as before—apex to the left—from a distance of about 0.5 m) we shall be confronted by an image like that shown in Figure 8. (Some typical prismatic artefacts are not shown.)

As you can see, the results of this experiment are categorical. The R rectangle has been deflected towards the base of the prism by an amount equal to the width of the C band. At the other end of the spectrum the B rectangle has been deflected in the opposite direction, towards the apex, by an amount equal to the width of the Y band. Finally, in the middle of the spectrum, the G rectangle has not been deflected at all from its original position. Just as I said. 

There is no doubt that the refraction of the spectral colours in subjective observations does not happen as Newton and Goethe had imagined. The subjective refraction in a prism is not seen to happen in the same direction for all spectral colours, as both Newton and Goethe believed. It happens, instead, in opposite directions for the boundary colours and not at all for the central hue. 

It had become manifest the fact that in subjective experiments a prism refracts the spectral colours in the same fashion as when one opens a Fan with one's hand: from the centre towards the sides, in both directions at the same time. To my mind that made perfect sense. It was a far more efficient process. It was more beautiful, simpler, faster. 

In spite of all those revelations one thing continued, nevertheless, to haunt me long after completing my work in that particular matter. The thing that haunted me came in the form of one simple question: Why should those wonderful facts that I had discovered be factors of manifestation only in subjective experiments? Isn’t God the true epitome of parsimoniousness? Why would He then have such simplicity in one aspect of reality, yet so convoluted a story in its mirror counterpart? It was time to look deeper into the nature of objective experiments.

Subjective and objective spectra created in the same experiment

On rainbows. Part 4.

There is one particularly interesting little story that occurred in my former correspondence with Dr. Markus Selmke I don't think I ever mentioned on these pages before, so I'll do it today because it's certainly worth knowing (by the discerning mind). 

After a few intense weeks of being involved in a torrid exchange of emails with Dr. Selmke I was beginning to get a pretty confident picture of what the man in question might be really like and about, so one day I decided to put it to a certain test, for good measure. Specifically, in a short and seemingly unrelated email to any other since then I asked the good Dr. to put a percentage number on how confident he truly was that the conventional understanding of rainbows was correct. A short time later his answer was crying out from my inbox and under the cheeky gaze of my Greek I opened the email and wryly grinned aloud the (rather sadly, and quite disappointingly) unsurprising figure. The test was over and in an instant I was completely freed of any lurking sense of either guilt, regret, remorse or wonder.

I have been taking my time with this post. There's no need to hurry. I remember reading somewhere, a long time ago now, someone's answer to a question whether one who might harbour a belief in possessing some great scientific insight should be wary about how to expose it to the world at large, in order to prevent/avoid/insure that it won't be stolen or claimed by some one else, later. One, the answer went, should not worry at all about that, for if one's unspecified scientific insight should happen to really be "great" then one shall find that no one will even think about doing anything of the kind. 

Clever. More importantly, though, true 😎!😂 

Almost a year ago to the day I received the email below from Markus Selmke. Today I decided to answer it.

Dear Remus,

Your answer does not address the points I have raised. I take it that you have no meaningful response to offer.

Why don't you do all those things yourself

I did indeed, indirectly many times. The direct measurement is too simple to warrant a dedicated section in a lab course, but you will find it in the curricula of many schools with dedicated experimental setups. In my physics training I have for instance done many of the following experiments myself:

http://home.uni-leipzig.de/physfp/fprakt.html

You will find spectroscopy among them, which for dispersive spectrometers is based precisely on snells law of refraction and dispersion.

You may also remember that I have published research. I have used Snells law in combination with dispersion many times and found good agreement with experiments I have done myself. My latest contribution in this direction being the fairly simple deomstration experiment and its quantification here: https://128.84.21.199/abs/1608.08664 (to appear in the next issue of Am. J. Phys., after a vetting and partial rewriting within the due course of a proper and lengthy critical review process; in fact you will be able to notice the difference between the final version and the submitted manuscript according to the ArXiv version). I found the agreement with experiments only WITH the inclusion of dispersion, and inferior results without. Unsurprisingly, I should say, because dispersion is, contrary to your baseless and evidently uninformed claim, a century-old well-supported, understood, measured and quantified fact. It may even be derived and understood from more fundamental laws like the Maxwell equations together with an appropriate atomistic model for matter.

BTW: You again demonstrate that you lack a fundamental understanding of the concept of light altogether. There is simply no way to individually send a single wavelength through a prism in an experiment. You will always be concerned with spectral distributions, even for narrow bandwidth laser sources! But this is, admittedly, the smallest of errors you make, with the more grave one being that you lack an understanding of the scientific method.

Also, I am curious: Did you test yourself whether gravity works as Newton described it? Did you ever jump off a cliff to find your velicity to increase linearly with time? I’m afraid not, but other people tested it (well, not by jumping) and found precise agreement. They even made quantitativly testable predictions of experimental outcomes. Your car will use the mechanical principles so eloquently framed by Newton, yet you didn’t test them all, did you? And so does your camera use the different refrangibility of colors. Or binoculars for that matter which truly use prisms directly. They are color-corrected in fact to counteract the measurable color dispersion effect. It should be unneccessary to state it again, but of course there is abundant evidence in the scientific literature to support this (including my miniscule contributions) contrary to your laughable claim (again, abstracting from the fact that you can only send spectral distributions of colors through a prism).

You may now apologize if you understand the above and the previous email in full.

Frustratedly,

Markus

Dear Markus,

Last year when I received your email, cited in full above, I had a good look at your  usual array of childish assumptions and their commensurately stupid and even more infantile extensions, I smiled and then decided to leave you continuing to bask in your pompous ignorance for the time being. Today, however, when I happened to stumble across it once again I suddenly realised that it in fact would be the best starting cue for this particular post. So here I am, ready to address more points than you can even imagine--let alone raise.

Leaving aside your incoherent ranting about dispersion, Snell's law, spectroscopy, etc. let me first tell you why I asked you conduct a proper experiment in order to confirm once and for all that Snell's law is indubitably valid. 

If you google the most relevant terms concerned with the subject of colour dispersion you will find that basically all sites that cover the issue use one and the same setup in every case. Specifically, starting with a version of Newton's experimentum crucis they separate colours from a dispersed beam of white light and thereof proceed to pass them through prisms placed at minimum deviation and finally compare the results according with Snell's refractive equations. Now, to my mind that particular method was fraught with danger, especially when used exclusively, as it seemed to be to the best of my research. (Why fraught with danger I'll tell you in a moment.)

And there was an additional reason for my request. I knew you simply had not only the knowledge and expertise to address the issue properly: You also had far superior means than my Kmart laser pointers and prisms.

Now, regarding my suspicious outlook towards Snell you'd be wise to treat it carefully, for I not only have a much firmer grasp of the subject than you'd be willing to consider, but that I have also found such a beautiful and healthy reason for dispersion that will make your heart skip a few beats when you will hear it.

Being aware that Snell developed his law a long time before Newton himself came unto the scene made me wonder how the man could have possibly managed to conceive of such an apparently accurate tool of assessment without having a real understanding of how white light was dispersed by a prism in a full spectrum of colours. Had Snell developed his law after Newton I certainly wouldn't have had a problem with the idea. But before?! (These days Snell's law is stated to be derivable from Fermat's principle of least time, but we should not forget that Snell's law was created some 30 years before Fermat laid down that principle.) Nonetheless, in time I came to realise that one could rather easily begin with compiling empirical results from prismatic observations, combine then those with certain mathematical and geometrical principles, derive symmetries by relating observational results and variables, unite all of those via some binding coefficients and finally use the recipe to practically enforce, define and in the end 'standardise' the whole industry, so to speak. In fact that has been the mainstream practice since those very times. And that is a most perilous methodology. So perilous that by now it has contaminated your mob so chronically that you seem to have all become decidedly impotent to see, or even imagine, how blatantly wrong your celebrated optical doctrines are. 

...dispersion is, contrary to your baseless and evidently uninformed claim, a century-old well-supported, understood, measured and quantified fact. 

There is only one thing I can say about the above statement: that when all current facts are considered you have to be either a genuine prophet or a fair dinkum fool to make it. Time will undoubtedly reveal the truth of that matter. (And at this point I should also tell you now the percentage figure I got from Markus Selmke in regard to the little story I shared with you at the beginning of this post, even though I suspect that most of you guessed it already. The figure was, yes, exactly 100%.)

If Markus Selmke understands perfectly well what dispersion is he certainly knows more about it than at least one of the creators of QED ever did. I'm referring here to none other than Richard Feynman, who candidly admitted in these videotaped lectures that he did not understand what exactly happens in the interaction of light with different media. The only thing the great man admitted that he could say on the subject was what effects appeared to unfold under the eye of a careful observer. As my friend Michael Heffron pointed out to me not long ago, when it comes to the subject in question what a genuine physicist of these times can only say is what Sheldon Cooper said in one episode of The Big Bang theory: "I can explain it to you, but I can't make you understand".

Now that is one thing I have always known myself. Furthermore, that is exactly the reason why I have embarked all these years ago on my personal foray into the subject. As most laymen out there, I simply did not understand what our crop of physicists were talking about so I set out (on my own) to change all that. Yes, I know, it was rather typically arrogant from someone in my position to even imagine that I could do that, but that is the truth and that's what I did. Moreover, that's not all either, for these days I have become even more arrogant, if you will, for these days I am here claiming that I do understand what happens when light interacts with different media, creating the chromatic dispersion that is observed in prismatic experimentation and the atmospheric optics phenomena in the process. Moreover still, in contrast to what Sheldon Cooper said (and to what a genuine physicist should say on the subject of light-matter interaction) I am asserting here that I can not only explain the subject to you, but that I can also make you understand. Make of this what you will, just don't make it in haste. Stay with me.

-----------------------

Since I have always been a layman who dared to preach in the conventional temple you can imagine that my 'sermons' are vastly different to those that have pharisaically been preached to the world for a number of centuries now. For instance mine contain very little mathematics, while theirs are by and largely chanted in that language. Furthermore, my own presentations are routinely embedded in familiar pictures, while theirs are most of the time encrypted in complicated graphics, charts and other similar concoctions of arcane data. But perhaps the greatest difference between our two forms of expression is the fact that my own language is very simple while theirs is convoluted and alien to the majority of those listening. Then I must also mention that their expositions can often put a dizzying spell on their audience, while mine are most of the time noticeably un-impressing even the best of those genuinely trying to feign some  degree of interest. What can I say, the truth is that we have always come to the same temple, but always from the exactly opposite ends of the town: when I come from the South they come from the North, and when I happen to arrive from the East they invariably choose to do so from the West👍👎👉👈.

But let me give you next a number of other discrepancies between us that are more relevant to our current bone of contention.

As a concrete example, take Newton's famous experimentum crucis and compare it to mine. (Yes, I do have one of those myself--see below.)

In spoken language: I laid down on a flat surface a short message written on a piece of paper, I then positioned my eye on a line of sight running perpendicularly along the same plane and finally I slid a prism in between them. Hello.

That was my experimentum crucis,  in a nutshell. It was crucis because it was due to it that in a swift period of time I managed to learn the most important things about what happens (about what really happens) in all prismatic interactions between light and matter. See the pictures I will drop below and think a little, if you really want to see where I'm coming from and where I'm heading to. Alternatively, make an effort and read all my previous posts. Otherwise consider leaving this site (right now, please) for I'm surely not interested in your company either😎.

Let me now show you the main reason that had led and driven me to the entire process. See the picture below and remember one particular prismatic observation, which Newton had been well aware of but left  alone (without any explanation).

The observational fact in question is that when a beam of white light is looked at directly with the naked eye, which is a so-called subjective observation, the order of colours in the spectrum is reversed--VBGYOR instead of ROYGBV. As we know, Newton was well aware of that fact but he never left anything more  to us on that subject beside casually mentioning it in his Opticks. Now to my mind that showed that he did not have any plausible explanation for it and that he sure as hell wasn't going to admit it. And indeed that was to eventually become poignantly evident when he categorically refused to discuss that subject when Lucas mentioned it in one of his letters. 

Collaboration Day 5

Ethos: To live in, by, and only, for truth

From VBGYOR to ROYGBV to VBGYOR/ROYGBV, and then to their implicative extensions Part 1.

My analysis of Newton's theory of light and colours

My analysis of Goethe's work on light and colours

Goethe's edge spectra

If the slit of the spectral apparatus is extremely widened or if a broad white strip is observed against black paper through a prism, as described by Goethe in the Didactic Part of his Theory of Color, the edge spectrum shown in Fig. 7a will be perceived. Goethe explains these edge spectra as being the shift of the objects from their real position caused by the effect of the prism. According to Goethe, the image is not shifted completely as if it in fact resisted the shift. As a result, a ”secondary image” is produced which slightly precedes the actual image. If the bright rectangle is viewed through a prism, it is shifted to the left by refraction, and the bright secondary image is superposed on the dark paper. Goethe propounds that bright on dark produces blue which changes into violet if the effect of the dark increases. On the right edge, the image of the dark surface shifts over the remaining bright ”principal image”. Dark on bright produces yellow which, according to Goethe, accounts for the yellow seam. Where the effect of the dark increases, yellow changes into red.

From the physico-optical viewpoint, it is an untenable interpretation that edge spectra should be caused by principal and secondary images and their resistance to displacement. In the case of a wide slit, the edge spectra can be demonstrated to result from the overlap of monochrome slit images, as illustrated in Fig. 7b. For greater clarity, the slit images of the individual colors are shown in a vertical arrangement. On the right, (starting from 1), the red edge spectrum is very obvious because both red and yellow are fully represented here. On the left in the illustration, the blue edge spectrum is visible (at 1’ and 2’). At the position marked with 4, all colors are present and produce white. 

Extraordinary observations were made by Goethe on the ”negative slit” (Fig. 8a): Unlike the experiment described above, a broad black strip is viewed against a white background through the prism. An unusual ”reversed spectrum” is observed here, displaying the respective complementary colors of the previously described edge spectrum. The formation of this ”reversed spectrum” can be demonstrated in Fig. 8b. Starting at the top, a dark field should be drawn in the middle between the strips of the same color. The background at 0 and 0’ – previously black – is now white because all colors are present here. The previously white center at 4 is now black due to the lack of any color. On the left, the sequence of colors towards the edge is red (3’), reddish yellow (2’) and yellow (1’), and on the right violet (3), blue (2) and bluish green (1). Goethe lists the following ”elements” between white and white, from right to left: blue, bluish red, black, reddish yellow, yellow (Theory of Colors; Didactic Part § 246), corresponding to the positions marked here with 2, 3, 4, 2’, 1’. If the normal slit or the white strip becomes increasingly narrow, the standard prismatic spectrum is gradually obtained, with green instead of white in the middle. If the ”negative slit” or the black strip becomes increasingly narrow, the red and violet spectral ends overlap at position 4 to form purple, the complementary color of green, as can be seen in the illustration. As a result, the following color sequence is obtained with a thin black strip or negative slit: white, yellow, orange, red, purple, violet, blue, bluish green, white.

This excerpt is from an article written for a magazine published for the famous Carl Zeiss Company. Its authors are: Prof. Lutz Wenke (Dean of the Faculty of Physics and Astronomy at the Friedrich Schiller University in Jena), Dr. Friedrich Zollner, Manfred Tettweiler (both from the Institute of Applied Optics) and Hans-Joachim Teske (Manager of the Astronomical Instruments business unit at Carl Zeiss). There are a few interesting points you must have noticed in the “demonstration” above. Firstly, the orientation of the prism is not mentioned, and the sentence which was probably meant to reveal that orientation (“If the bright rectangle is viewed through a prism, it is shifted to the left by refraction...”) is still not clear enough. In any event, we know the orientation necessary to produce the colours observed: The prism has to be oriented with its refractive angle (vertex) pointing to the observer’s left. Secondly, the spectral colours do not extend for the whole width of the white strip. This is rather odd and, in any case, it’s an ad hoc decision. Thirdly, I’m sure you have noticed how convoluted the ‘explanation’ is (especially for the so-called “negative spectrum”), considering how simply it could have been shown where the observed colours originate. Fourthly, the “demonstrations” illustrated in the figures make no sense, when the orientation of the prism is taken into consideration—for in figures 7b and 8b the spectral colours are depicted to run at a 90 degree angle to the normal way of refraction!

This is truly a very strange “demonstration” and I wonder how many physicists, apart from these authors, are accepting it. There could be, however, a possibility that I may have misunderstood something in the “demonstrations” above, and in that case I would love to hear from those who could clarify the situation. On the other hand, I (and I have reasons to believe that you, too) can explain the colours observed much, much easier.

The second example I want to give you is from a paper written by David Seamon, titled “Goethe’s way of science as a phenomenology of nature”.

To understand Goethe’s style of looking and seeing, I want to focus on the prism experiments in part two of Theory of Color. These easy-to-do exercises are a helpful way to introduce students to phenomenological looking because a phenomenon is present—the appearance of color in a prism—which, on one hand, most people are unfamiliar with yet which, on the other hand, can be readily examined, described, and verified through sustained work with the prisms. Table 1 indicates the kind of questions one should keep in mind in doing these experiments and, for that matter, all Goethean science. 

Participants are asked to begin by simply looking through the prism, seeking to become more and more familiar with what is seen. They record their observations in words and colored drawings. Ideally, the experiments are done by a group of four or five, so that participants can report their observations to each other and bring forth descriptive claims—e.g., “I see a halo of color around all objects” or “I notice that there only seem to be colors along edges of objects.” Other participants can then confirm or reject these observations in their own looking and seeing. Gradually, the group moves toward a consensus as to exactly how, where, and in what manner colors appear.

This process of looking is slow and requires continual presentation, corroboration, recognition of error, and correction. Eventually, however, group members can establish a thorough picture of what their experience of color through the prism is and end with a set of descriptive generalizations like those in table 2.

SEEING AND UNDERSTANDING BROADER PATTERNS

The general exercise of looking through the prism just described is excellent for introducing students to the effort, care, and persistence required to produce accurate phenomenological description, but Goethe’s aim is considerably larger: to discover a theory of color that arises from the colors themselves through our growing awareness and understanding of them.

Here, we move into a stage of looking and seeing that explores the wholeness of color by describing in what ways the colors arrange themselves in relationship to each other and to the edge of light and darkness that, as discovered in the experiment just described, seems to be a prerequisite for any color to arise at all.

To identify such patterns and relationships, Goethe presents a series of experiments using a set of cards with black and white patterns that are to be viewed carefully through the prism and results accurately recorded. Examples of these cards are illustrated in table 3 and instructions for the use of three of these cards is provided in tables 4 and 5.

The value of the cards in these experiments is that they provide a simple way to direct the appearance of color and, thereby, provide a more manageable and dependable context for looking and describing. Rather than seeing color along any edge, participants are now all looking at the same edge displaced in the same way so they can be certain that they will see the same appearance of colors.

In regard to card A, for example, we begin with the white area above the black and, through the prism, look at the white-black horizontal edge in the middle of the card. If the image that we see is displaced by the prism below the actual card, then at the edge we see the darker colors of blue above violet (see drawing 1). If we turn the card upside down so that black is above white, we now see something quite different—a set of lighter edge colors that, from top down, are red-orange and yellow (see drawing 2).

As drawings 3 and 4 indicate, the experiments with cards B and C are perhaps the most intriguing because they generate two colors not as regularly seen as in the dominant spectra of yellow-orange-red and blue-indigo-violet. As one moves card B farther away toward arm’s length, there is a point at which the yellow and blue edges merge, and a vivid green appears horizontally so that the original white rectangle is now a band of rainbow (drawing 3). For card C, a similar point is reached where the red and violet edges merge to create a brilliant magenta (drawing 4)

The first thing you might have realised is that from what we have discussed thus far you can explain why the colours in the drawings 1 and 2 are observed. You might have also established—without looking through a prism or reading the instructions—that the colours in those drawings are observed only when the prism is oriented with the vertex pointing down. In fact, you might have realised that, in principle, you could predict what colours would be observed by looking through a prism at all six cards in table 3—even though we haven’t yet discussed the colours observed in experiments like those depicted in drawings 3 and 4. The new colours seen in those drawings (green and magenta) are the products of a mixture of certain spectral colours, as indeed it is explained. But the most important thing you have probably realised is the simplicity with which you can predict and explain the origins of the colours observed in Goethean experiments. You might thing that this explanation is accommodated by common sense and that—therefore—it is rather conspicuous. You might also think that it follows directly from Newton’s reply to Lucas and, on that basis alone, that physicists would have embraced it a long time ago. You might indeed think about all this, but you would be wrong. As you have seen in the first example I gave you earlier, and as it will become evident from the next two examples written by physicists, the explanation that can best predict and describe the observations of Goethean prismatic experiments is not part of the defensive arsenal of the scientific community. 

In "Beiträge zur Optik" Goethe advises us to look through a glass prism and observe the colour phenomena that appear. It soon becomes evident to the observer that colours appear at distinct borders between dark and bright areas in the field of view. If you vary the geometrical conditions you find that all of the various configurations can be boiled down to four principal spectra: The two border-spectra [red-yellow] and [violet-blue] and the two aperture-spectra [cyan-magenta-yellow] and [red-green-violet].

An essential feature of the world of prismatic colours is a basic symmetry: whenever white and black are interchanged in the pattern, the other colours are interchanged specifically, i.e. yellow is interchanged with violet, purple with green, and cyan with red.

Thus, if the upper half of the picture (in the illustration above) should be additively superimposed upon the lower half, the result would ideally be a full white rectangle. If they were instead superimposed subtractively, i.e. as colour slides, laid upon each other, then the result would be a wholly black rectangle. The two halves are perfectly complementary: They have not a single wavelength in common and together cover the whole range.

Goethe was enthusiastic over the discovery he had made, namely that the complementary relationship among colours, since long well known to the painters, had such an evident foundation in the physics of colour. For that reason he was anxious to stress that all four spectra had to be considered as basis for a true theory of colour –not only the particular one, obtained in case of a narrow aperture, studied by Newton. The physicists of Goethe's time told him that all these phenomena could very well be explained by help of Newton's concept of rays of light, differently refrangible. But Goethe stubbornly maintained that it was not just a question of explanation but of basic principles.

Pondering things over during the years, I think I have come to an understanding of what Goethe was after. He was pointing out a lack, or shall we say imperfection, in Newton's theory, especially as this theory was propagated by Newton's followers and late disciples.

The above was written by the physicist Pehr Sällström. Let us compare our (unconventional) explanation of the colours observed in the picture given with Goethe’s explanation (as described by the named physicist). Before doing that, however, notice that the orientation of the prism is not mentioned. Nevertheless, we don’t need to look through a prism in order to establish the orientation of the prism that will result in the specific colours displayed—we can firmly deduce that the only orientation which will result in the depicted colours is an orientation of the prism pointing with its refractive angle (vertex) to the right. I should also mention here that we’ll ignore some of the spectra that will be generated by the picture above, concentrating only on the spectra discussed. To make our task easier I have numbered the spectra of interest in the figure below, and I have also numbered the sources responsible for those spectra (according to our understanding, of course).

On the right picture I have numbered the four spectra observed, in the order in which they were listed in the article. On the left picture I have correspondingly numbered the sources generating the spectra, as I said before. Now, in the article there isn’t an explanation per se of where the four spectra come from—there is only a rather observational (phenomenological) comment that accompanies the picture. Apart from that comment the author says: “An essential feature of the world of prismatic colours is a basic symmetry: whenever white and black are interchanged in the pattern, the other colours are interchanged specifically, i.e. yellow is interchanged with violet, purple with green, and cyan with red”. These observations are quite useless in understanding the phenomenon—they are similar to learning the multiplication table by heart, where knowing the answer does not mean understanding the principle. (Besides that, Goethe’s enthusiasm for finding a physical basis for “colour-complementarity” is rather mystifying, in my opinion. But that’s another story.)

From our perspective, the spectra observed can be easily explained as being generated by an observation through a prism of four ‘independent’ sources.

Thus, spectrum number 1 in the picture on the right (the red-yellow) is one half of the spectrum generated by the white rectangle 1 in the left picture. The other half of the spectrum generated by the rectangle 1 is formed by a violet-blue combination, which is contributing to the creation of spectrum 3. The orientation of the colours in spectrum 1 points to where the other half is. Spectrum 1 is observed to appear towards the base of the prism, while the other half (the violet-blue combination) is observed to appear towards the vertex of the prism—just like we’ve already established.

Spectrum 2 (the violet-blue) is one half of the full spectrum generated by the white rectangle 2. The other half of that full spectrum is the red-yellow combination, and it can be seen (albeit, less vividly than its counterpart) at the border between the white rectangle 2 and the grey background of the page.

Spectrum 3 (the blue-magenta-yellow) is formed by the violet-blue half of the spectrum generated by the rectangle 1 and the red-yellow half spectrum generated by the white rectangle 3. In effect, the magenta component of spectrum 3 is formed by the mixing of red and violet—the blue and yellow components remaining unaffected. The violet-blue combination of the full spectrum generated by rectangle 3 can be seen at its border with the page itself.

Spectrum 4 (the red-green-violet) is the full spectrum generated by the narrow white rectangle 4. The green component of that spectrum is the result of the mix of its yellow and blue components. Spectrum 4 displays three colours (red-green-violet) only if the observation is conducted from a distance greater than approx. 20cm. If you look at spectrum 4 from a smaller distance you will see the yellow and blue components instead.

This is my explanation for those so-called four spectra. Compare it with Goethe’s, or with the one offered by the physicists from whom I gave you the first example, and judge for yourself. I know that my explanation can account for all possible subjective prismatic experiments, and that it can also predict what colours will be seen in all circumstances. This explanation is so accurate and comprehensive that I will therefore call it, henceforth, the law of colour-display in subjective prismatic experiments. In the last example I want to show you we will apply the law of colour-display to some more complicated shapes. Then in the next chapter I’ll continue to test the law and I will also showwhere the spectral colours originate and how they come into observation.

The final example I want to show you comes from an article titled “Exploratory Experimentation: Goethe, Land, and Color Theory” which appeared in Physics Today in July 2002.

Goethe's experimental procedure comprised two stages: an analytic one that moved from complex appearances through simpler ones to a first principle, and a synthetic stage that moved in reverse order, showing how more complex appearances are related to the first principle. The analytic stage is illustrated by a set of experiments with black-and-white images. Figure 2 shows how a few of the images Goethe used look when viewed through a prism with its refracting angle held downward. The general law determined by Goethe was that colored fringes arose at black-white borders parallel to the prism's axis: yellow and red when the white was below the black, blue and violet when it was above, as shown in the prism view of Figure 2e. For Goethe, these fringes constituted an elementary appearance of prismatic color from which all others could be derived. For example, Goethe's experiments with black and white rectangles showed that the Newtonian and complementary spectra (see the prism views of Figures 2c and d) were generated when the colored fringes from two closely spaced black-white boundaries encountered each other: The yellow and blue fringes mixed to produce green; the red and violet produced magenta. For Goethe, therefore, the Newtonian and complementary spectra were compound phenomena that could be derived from the law of colored fringes.

The synthetic stage of Goethe's investigation is illustrated by his experiments on the colored fringes that appear when gray and colored images on various backgrounds are viewed through a prism. Figure 3 shows how part of one of Goethe's diagrams (see the cover of this issue), from Theory of Colors, looks through a prism with its refracting angle held downward. Experiments with squares in different shades of gray against white and black backgrounds showed that the intensity of the colored fringes increased with the lightness contrast at the boundary. More complex phenomena were seen using colored squares, which exhibited fringes with new colors not seen in the previous experiments. Goethe argued, quite plausibly, that those new colors were due to the mixing of the elementary fringe colors with the colors of the squares themselves. Goethe regarded that mixing the true explanation of Newton's observation that a red square, viewed through a prism against a black background, appears displaced slightly higher than a blue one, as seen in the upper right of Figure 3. Whereas Newton had adduced this observation to prove that different colors of light have different refrangibilities—the first proposition of his Opticks—Goethe saw it as merely a special case of the more general law of colored fringes.

Goethe's analytic investigations proceeded from the complex to the simple. Shown are five black-and-white images selected from a series studied by Goethe, viewed with the naked eye (top, adapted from Contributions to Optics, ref. 1) and through a prism with its refracting angle held downward (bottom). The up-down sequence of all the colors is reversed if the refracting angle is held upward. (a) An irregular arrangement of black and white exhibited colored fringes with no apparent order. (b) The colors generated by a simpler checkerboard pattern were periodic and exhibited regular changes as the checkerboard was rotated, but were still too complicated to be expressed in a law. (c) The colored fringes generated by a white rectangle depended on the width of the rectangle and its distance from the prism. A very narrow rectangle, or one at a great distance, exhibited a spectrum with just three colors. Wider rectangles, such as the one shown, displayed fringes whose colors--red, yellow, green, blue, and violet--were consistent with those of the Newtonian spectrum. (d) A black rectangle on a white background exhibited a spectrum—blue, violet, magenta, red, and yellow—complementary to that of (c). The complementary spectrum's central magenta, called "pure red" by Goethe, is not in the Newtonian spectrum. (e) The boundaries of wider rectangles acted as isolated black-white contrasts, displaying red and yellow fringes when the black was above, blue and violet when it was below. No colors appeared at vertical black-white borders.

The experiments just described are only a small fraction of those that Goethe performed during his career. Others included novel experiments with refracted sunlight that displayed at a glance the evolution of both the Newtonian and complementary spectra as a function of distance from the prism, and careful replications and variations of many of the experiments in book 1 of Newton's Opticks.

We shall pay close attention to this description of Goethe’s work, for it is a good summary and it mentions the most important aspects of Goethe’s theory of colours. In the first paragraph cited we encounter again Goethe’s explanation of the colours observed in subjective prismatic experiments like those depicted in figure 2. Notice that Goethe’s mechanistic observation is called “general law”, although it falls well short of accounting for all subjective experiments—as it will become evident soon. The stipulations of GMore complex phenomena were seen using colored squares, which exhibited fringes with new colors not seen in the previous experiments. Goethe argued, quite plausibly, that those new colors were due to the mixing of the elementary fringe colors with the colors of the squares themselves. Goethe regarded that mixing the true explanation of Newton's observation that a red square, viewed through a prism against a black background, appears displaced slightly higher than a blue one, as seen in the upper right of Figure 3. Whereas Newton had adduced this observation to prove that different colors of light have different refrangibilities—the first proposition of his Opticks—Goethe saw it as merely a special case of the more general law of colored fringes. Goethe’s “general law” we already discussed. A more interesting observation is mentioned in the second paragraph.

More complex phenomena were seen using colored squares, which exhibited fringes with new colors not seen in the previous experiments. Goethe argued, quite plausibly, that those new colors were due to the mixing of the elementary fringe colors with the colors of the squares themselves. Goethe regarded that mixing the true explanation of Newton's observation that a red square, viewed through a prism against a black background, appears displaced slightly higher than a blue one, as seen in the upper right of Figure 3. Whereas Newton had adduced this observation to prove that different colors of light have different refrangibilities—the first proposition of his Opticks—Goethe saw it as merely a special case of the more general law of colored fringes.

This is the most important contribution Goethe made to the research into the nature of colour. It is also the only observation that truly shows deficiency in Newton’s theory—although, alas, it failed to attract the attention it genuinely deserves. In fact, as you will see, Goethe’s argument on this issue is not only quite plausible—it is undoubtedly true. We shall analyse that argument in detail, and then you can assess my claim.

More on Newton's theory of light and colours

Newton’s theory of light and colours