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2017-01-23
Huygens' Light Theory: A Text Analysis
https://ia800307.us.archive.org/13/items/treatiseonlight031310mbp/treatiseonlight031310mbp.pdf

[pagination is given as page number in pdf file/page number in original text]
Chapter One
The first lines sound very strange coming from the founder of wave theory since they consecrate the standing of the opposite view, Geometric Optics:
"As happens in all the sciences in which Geometry is applied to matter, the demonstrations concerning Optics are founded on truths drawn from experience." Huygens sees therefore no conflict between his new approach and Optics which are based on the behavior of particles of light. Einstein's wave-particle duality was certainly nothing new!
Huygens therefore does not doubt the validity of optical laws that say that light travels in straight lines, that the angles of incidence and reflection are equal or that refraction obeys the law of sines.
That is all and well, but what is the relation between those particles that make up light, and those waves that the same particles are said to be creating?
His next affirmation is certainly significant: "It is inconceivable to doubt that light consists in the motion of some sort of matter." It might seem obvious, but that still does not tell us much on what kind of matter we are talking about. Also interesting for historians is his allegiance  to "the true Philosophy, in which one conceives the causes of all natural effects in terms of mechanical motions."
Next follows a theory of vision which we will find held by Newton and many others: the movement of light particles create in us visual sensations, which makes him say: "yet one reason more for believing that light consists in a movement of the matter which exists between us and the luminous body." A very nice circular argument, but we will not hold it against him.
The following paragraphs are very interesting in their attempt to differentiate light from other matter, and in particular "it cannot be by any transport of matter coming to us from this object, in the way in which a shot or an arrow traverses the air". This is a declaration of war pure and simple. Light is made of particles, but these cannot be understood as projectiles. How are we then supposed to understand them?
Let us skip the details to arrive at the punch line: light should be understood the way we understand sound, a phenomenon that propagates in circular waves around a point of origin: "for I call them waves from their resemblance to those which are seen to be formed in water when a stone is thrown into it..." Follows a long argumentation aimed at proving that light is not instantaneous, as Descartes believed, but needs time to travel from one point to the other, however incredibly fast its speed might be.
But light is not sound, even though both are propagated in circular waves. Sound needs air to propagate while light is dependent on a much more mysterious entity, the ether:
"all those of the luminous bodies which are liquid, such as flames, and apparently the sun and the stars, are composed of particles which float in a much more subtle medium which agitates them with great rapidity, and makes them strike against the particles of the ether which surrounds them, and which are much smaller than they."

The ether is completely different from the familiar medium air, the vibrations that cause light being much more rapid that those needed to produce sound. More than the differences between the medium air and the ether, the properties of the latter medium itself are something that need our undivided attention. One very important distinction is, surprisingly, the hardness of the ether particles which, miraculously do not seem to form an obstacle for matter to go through them.
The way Huygens explains the behavior of this ether particles is certainly edifying. They are compared to hard spheres, one would say billiard balls, that collide with each other and move in straight lines. This is a rather familiar picture and I will move on to more important details like this one: "But by supposing springiness in the ethereal matter, its particles will have the property of equally rapid restitution whether they are pushed strongly or feebly; and thus the propagation of Light will always go on with an equal velocity." Let us just take his word for it, won't we? We have after all to make allowance for the fact that this text is more than 300 years old and stopping at every detail wouldn't do us any good. let us leave that to historians who are much better qualified to appreciate the value of each point.
What we shall retain from all this is that motion in the ether is propagated in straight lines for reasons that seemed cogent to the author. What interests us also is the little drawing p.29/15 which shows a group of spheres and, isolated from them, a single sphere which will collide with them and send them flying in different directions, all straight lines. Furthermore, and that will prove to be essential, we must not forget a very important effect of the motions of these spheres. They do not really move from their place but transmit their vibrations to their neighbors! It is Newton Cradle, but then without the first and final motions! Everything happens between the balls in the middle which never leave their place.
He rounds up his analysis with the following declamation: "it is not beyond the limits of probability that the particles of the ether have been made equal for a purpose so important as that of light, at least in that vast space which is beyond the region of atmosphere and which seems to serve only to transmit the light of the Sun and the Stars."
Notice that nowhere in the text does he yet justify the use of the concept of wave to explain the behavior of light. Only particles, of the ether, seem to play any role, and if the propagation somehow looks like waves in a pond after a stone has been thrown in, such an image can hardly be understood as more than an allegory: particles vibrate and transmit their motions to their neighbors in ever widening circles reminiscent of water waves.

That does not stop him from affirming: "I have then shown in what manner one may conceive Light to spread successively, by spherical waves..." The following precision is certainly worth noting: "although the particles are supposed to be in continual movement [...] the successive propagation of the waves cannot be hindered by this; because the propagation consists nowise in the transport of those particles but merely in a small agitation which they cannot help communicating to those surrounding, notwithstanding any movement which may act on them causing them to be changing positions amongst themselves." The propagation of light is independent of the motions of the particles forming the ether. How are we to understand that? How could light still be propagated in circles if the motions of the ether can be so easily discounted? Hadn't Huygens declared just a few lines before that:
"if one wishes to seek for any other way in which the movement of Light is successively communicated, one will find none which agrees better, with uniform progression, as seems to be necessary, than the property of springiness..." The fact that ether particles are hard and "springy" is the perfect explanation for the fact that light can be propagates at such long distances. At the same time, the smallness of these particles explains why in a collision, a particle moves not only another particle which is directly in its path, but also the surround particles, thereby creating a wave.
There is an abrupt change of perspective in which Huygens goes from a metaphorical use of the concept of wave while presenting strictly corpuscular, optical, explanations for the propagation of light, to a literal use of the concept. Light waves become somehow something real. While until now waves were simply a by-product of the vibrations of particles in the ether, they are now the main players and the particles are only there to make the role of waves even more prominent.
"each little region of a luminous body, such as the Sun, a candle, or a burning coal, generates its own waves of which that region is the centre. Thus in the flame of a candle, having distinguished the points A, B, C, concentric circles described about each of these points represent the waves which come from them. And one must imagine the same about every point of the surface and of the part within the flame." Notice how in the second part of this quote particles have been reduced to geometrical points in space. Only waves seem to be real.
This dematerialization process becomes more and more pronounced. It affects not only waves:
"waves which traverse one another without confusion and without effacing one another..." It manifests itself with particles as well: "If against [a] row there are pushed from two opposite sides at the same time two similar spheres A and D, one will see each of them rebound with the same velocity which it had in striking, yet the whole row will remain in its place, although the movement has passed along its whole length twice over." This myth is still reaffirmed in all textbooks without any grain of criticism. Two waves on a string, starting from the opposing ends, will go through each other to the other end, and back. As I said before, the possibility that the waves bounce off each other and retrace their path back is never even considered, while it is the most plausible explanation. But then, that would turn waves into something tangible which follows the known mechanical laws that everybody is so eager to abandon, starting with Huygens, despite his statements of loyalty to the "true philosophy".

Before we go any further, we should ask ourselves what Huygens means with particles of light. The ether is obviously the medium, and it consists in very small but hard spheres that hardly move from their place, but which can transmit their rapid vibrations to neighboring ether particles all around them. At the same time, those emanations propagate in straight lines in all directions just like water waves. But is light only an effect of these vibrations? And how do these vibrations start?
Well yes. Apparently light is nothing else but those vibrations which somehow reach the eye. It can be considered as an effect of heat which itself is nothing but motion. There is no room in Huygens theory for photons and neither for caloric, a heat substance that would somehow be transfered from one element to the other. But since he is building a theory of light and not of heat, what is important is the emphasis he puts on the motions of the ethereal particles.
The danger of course is the identification of light with the ether itself. If there are no specific light particles that somehow interact with the ether particles, then how is light produced? 
This is how Huygens approaches the problem of "the production of light". First we have to remember that what travels between the object and our eyes cannot be compared with "a shot or an arrow". Also, let us not forget that sound is propagated through air "by a movement which is passed on successively from one part of the air to another". Here again, the air "parts" are themselves not changing place but transmitting their vibrations to their neighbors. The same can be said of light, and since no matter is transported from one place to the other, from an object to our eyes, we are obliged to say that whatever matter there is between us and the objects, only its vibrations will be propagated from one part to the other.
This is a powerful proof by analogy because of its suggestive power: it sounds very plausible and is therefore easy to accept.
There are therefore only the particles of the ether, they are the ones that transmit their motions to each other and excite our retina so that we see light and objects.

What Huygens did not make clear is where those waves are coming from if they are more than the vibrations of ether particles.
What does the term "region" mean in the quote mentioned above: "each little region of a luminous body, such as the Sun, a candle, or a burning coal, generates its own waves of which that region is the centre. Thus in the flame of a candle, having distinguished the points A, B, C, concentric circles described about each of these points represent the waves which come from them."?
Suddenly there is no direct link anymore between particles and waves; particles have been drowned in regions which in turn give rise to waves. The explanation sounds very plausible: "one and the same particle of matter can serve for many waves coming from different sides or even from contrary directions, not only if it is struck by blows which follow one another closely but even for those which act on it at the same instant." As we have seen, a single sphere, or ball, can affect many neighboring spheres, which means indeed that any one of them will be the origin of many waves.
it sounds so logical, doesn't it? But try and imagine to picture it. A single sphere giving birth to a multitude of waves, and reacting to different hits, even when coming from opposite directions. We are far from a mechanical view of physical reactions. Of course, successive causes with different effects are perfectly understandable in any world view, but you wonder how simultaneous opposite causes do not simply cancel out, instead of waiting until the waves are full-grown and big enough to interfere destructively with each other.
More importantly, the material support of waves has completely disappeared and been reduced to isolated particles that only serve as anchors to waves. The corpuscular approach, vibrating particles of the ether, is abruptly abandoned in favor of a concept of wave that rises suddenly to power in what amounts to a philosophical coup d'etat.
Compare the drawing p.33/19 which is almost entirely based on the idea of waves with all the preceding drawings where only particles are involved (with the exception of the candle p.31/17).
We have until now the following picture in our mind: a heat source [the end of the 19th century will add electricity as a possible cause of light] that makes its molecules vibrate. These vibrations are transmitted by the ether particles. This transmission takes the form of waves. The question now is, what is the exact relation between those particles and the waves? Are waves an optical effect, a mathematical construct, or both? Or are waves maybe real physical phenomena in their own right?
The last view is what transpires through the rest of the text, even though it is nowhere explicitly stated. It is built on the suggestion of real waves, as we know water waves to be, but no effort is made to analyze their nature. It all seems like the word itself is explanation enough. You do not need to explain how it comes that a wave can be active at many places at once, it is what Hegel will later call "immediate experience". We have, by uttering a single word, unified the antinomic ideas of the one and the many!
That does not of course so much solve the antinomy as it hides it behind an illusion of clarity. Take the following quote:
"After all, this prodigious quantity of waves which traverse one another without confusion and without effacing one another must not be deemed inconceivable; it being certain that one and the same particle of matter can serve for many waves coming from different sides or even from contrary directions, not only if it is struck by blows which follow one another closely but even for those which act on it at the same instant. It can do so because the spreading of the movement is successive. This may be proved by the row of equal spheres of hard matter, spoken of above."
Huygens does not see any contradiction between the last sentence, which will be followed by the drawing of particles interacting with each other, and the idea of an infinity of immaterial waves meeting at the same location. Where are the vibrations of the ether in all this,  aren't they supposed to be transmitted from particle to particle? How can a particle transmit vibrations of group A to one particle, and vibrations of group B to another particle, and this at the same time? The problem ceases magically to exist if you stop speaking of particles and switch to waves. Then it becomes completely evident that we need another logic altogether. A single particle, like a stone falling in a pond, creates multiple waves. Isn't that something anyone can witness with his own eyes?
But is it what is really happening? Are we allowed to consider the stone as a single particle while we differentiate its effects in a multitude of waves?
Logically, Huygens should have continued pursuing his idea of vibrations transmitted from particle to particle, and asked himself what happens exactly when a stone falls into a pond. Can we really claim that all the waves we see have the same cause, or should we consider the stone as a convenient container of multiple causes, each linked to a single wave? What right did Huygens have to leave the main path of his analysis, the vibrations of individual particles?


2017-01-24
Huygens' Light Theory: A Text Analysis
Chapter One (2)
Huygens' central idea is "that one and the same particle of matter can serve for many waves coming from different sides or even from contrary directions". That is also what, in his mind, explains the fact that light can travel such immense distances. It is therefore crucial that we understand what make such a feat possible. Unluckily, Huygens does not give us much to go on. His explanations are rather sketchy and I fear, quite meaningless.
We have already met his first argument, represented by the drawing p.32/19 of a Newton Cradle, a sphere which collides with a row of other spheres and makes the last one be ejected.
We can understand the first collision as the emission of light by a heat source, and that light is then propagated along the spheres until it reaches its final destination, the last jump, the ejection of the last sphere being understood metaphorically, something like a spark we see at the end of a lightening.
Huygens realizes that it is quite plausible that the waves created will be too weak to be sustained over long distances, but since that is evidently not  the case. we can after all see very distant stars in the sky, light somehow is able to bridge those incredible distances. Still, all he has to offer is the strong belief that it it exactly the infinite number of waves that explains their strength and stamina. For that there is, he thinks, only one sensible explanation, they must all cooperate in creating and sustaining the same forward motion through space.
"Thus this infinite number of waves which originate at the same instant from all points of a fixed star, big it may be as the Sun, make practically only one single wave which may well have force enough to produce an impression on our eyes. Moreover from each luminous point there may come many thousands of waves in the smallest imaginable time, by the frequent percussion of the corpuscles which strike the Ether at these points: which further contributes to rendering their action more sensible."
It sounds like Huygens is begging us to take him at his word and put all our doubts aside. But then, how do we explain the fact that this infinite number of waves are propagated in an infinite number of directions? More importantly, how could we ever accept the idea that an infinite number of waves all move in the same direction and aim at the same location? The image of "an infinitude of waves, though they have issued from different points of this body, unite together in such a way that they sensibly compose one single wave" (p.32/18) calls for more questions than it can answer. It explains how we can burn things by concentrating a great number of rays at a single point, but it certainly cannot explain how light can travel such great distances, or more simply, how we can see the sun as a large sphere and not as a single luminous point.
What is most surprising is that this argument has withstood the attacks of time and is still found in all physic textbooks. Nobody finds it strange that secondary waves magically unite to create a big wave that can survive immense distances. But where should these secondary waves come from? As we have seen, light is produced by a source of heat and then propagated from particle to particle in straight lines. How could those straight lines suddenly change direction and converge unless an external force makes them do so? But such a convergence of rays, or waves, can never be the rule, otherwise the light of suns and stars would shine in only one or a limited number of directions.

The idea of secondary waves is in fact a temporal and causal aberration.
In the drawing p.33/19 Huygens presents one particle A that somehow creates a cone of light with sides AC and AE. He places also other particles which create secondary waves. One wonders where those secondary particles, B and G, come from, and whether D, C, E and F are also particles or simply abstract geometric locations. The same way, one can ask how the particles bb and dd could ever come to existence. Huygens draws a static scene where all participants are fixed at their position for eternity, but light is a dynamic process and if we are speaking of a particle A that creates a cone of light then no other particles could ever get ahead of it and somehow create secondary waves to reinforce the main wave. The drawing makes it seem like the main wave is waiting for the secondary waves to come reinforce it, but how could these secondary waves ever catch up with the main wave, and what would trigger them? Other light sources situated at the points B and G? But we have only one source of light, and that is A. That can only mean that the wave created by A passes successively by all the other points, but since it is one and the same wave, there can be no external reinforcement present, no secondary waves that would magically appear under way and keep the main wave going.
Huygens drawing is the best proof that his whole argumentation is built on a fallacy. The fallacy of a main wave that can only survive its long journey if it is helped by an infinite number of secondary waves. The problem is that Huygens can offer no plausible explanation for the existence of those secondary waves. There is no place for them in his model of light produced by heat and transmitted from particle to particle.

2017-01-29
Huygens' Light Theory: A Text Analysis
[For the French version:
http://gallica.bnf.fr/ark:/12148/bpt6k5659616j/f1.image#
]

Chapter Two: On Reflexion
revolves around the drawing p.37/23. The drawing, just like those in the first chapter, shows the discrepancy between the verbal affirmations of the author, and what he really can prove. You could look for waves for a very long time, you will not find any. I do not mean of course real waves, but a (mathematical) definition of abstract waves. There is nothing in the drawing that could not be explained with particles. I will try and show what I mean.

1) We have a distant light source that somehow creates a wave AC. While C is continuing towards B, A has already arrived at the boundary between the two media, and is reflected at a certain angle from that surface towards N. Since A is going as fast as C, it will arrive at N having traveled as far as C had to travel to B. AN=BC (or CB).
2) Between A and C are "pieces", or even better, locations, ("endroits" in the French original) which are moving in the same direction along the parallel lines HK.
This is where the magic, or rather the sleigh of hands, happens. How are we supposed to understand these elements which are between the extremities A and C? Huygens tells us that they are themselves waves. Okay, but what is there then between two of these H's? Are there even smaller waves between them? You can see how delighted Zeno would have been if a contemporary had written such a treatise!
3) The wave AC has become the smaller wave KL, and what happened with AC happens with KL. And this is where the troubles begin. While A is moving towards N, and H has arrived at K, C is now at L. We now need K to be reflected in a parallel direction to AN, which was the path A had been reflected on. After all, HK is parallel to the line on which A is moving, and we can therefore safely assume that it will be reflected on a parallel trajectory to AN. That holds of course for all H's and K's.This is Huygens' first error.
4) How far will the waves starting from K travel? We already know the direction, parallel to AN, and have the distance and time of arrival at B. Which means that by the time C has arrived at B, all the other K's will have indeed arrived at a point located on a line BS, passing by N.
5) An other analytical error, besides the reflection path of the K's, is the distance to the M's. Huygens assumes that they are equal to the relative distances to the points on BS, KM being promoted to semi-diameter or radius of all the circles created by the different parts of AC. But then, we are dealing with two different media, in each of which light has a different speed. The time it takes light to get to N can certainly be considered as equal to the time it takes to get to B, since both waves remain in the same medium. But that is not the case with the waves that go beyond AB, into the second medium. The distances AG and all the KM's cannot be equal to their counterparts on BS, and KM cannot therefore be considered as a common semi-diameter, nor can BG be parallel to AC.
6) Here is a third issue in this analysis. where do the O's come from? 
According to Huygens:
"And if one wishes to see how the wave AC has come successively to BN, one has only to draw in the same figure the straight lines KO parallel to BN, and the straight lines KL parallel to AC. Thus one will see that the straight wave AC has become broken up into all the OKL parts successively, and that it has become straight again at NB." (p.38-39/24-25)
Another magical trick? It certainly looks like it. By the time A arrives at the first O, the first H will be at the first K, and so on. The problem has already been sketched at (3): the OK's are not parallel to each other, and neither are the KL's. In fact, it is the lines themselves drawn from K to BS that can be said to be parallel to each other and to AN. We will see that the second drawing in this chapter in fact confirms this analysis.
7) Huygens' analytical errors do not prevent him from arriving at a correct result. The triangles  ACB and BNA are identical, and one can rightly say that the incident ray CB (or BC) is identical to the reflecting ray AN.
8) The second part of this chapter is at least as interesting as the first one. Huygens starts by creating problem for himself and his view. The line BN can be considered as the tangent lines of all the circles formed by the waves created by the reflection of AC, but how do we know it is the only tangent that matters? After all, as spheres, all those little waves arriving on BN can be said to have an infinitude of tangents.
Huygens thinks he has the logical solution: put all those tangents in the same bag! "I say then that the wave AC, being regarded only as a line, produces no light." It is much better to consider it as a circle, in the second figure indicated by HC.
It is this circle which, while contained between the sides HA and CB, is reflected from AB between N and B. This tidies up any loose ends created by all the waves and their circles. One ring to rule them all!
9) We can see now how the second drawing in fact neutralizes the errors of the first one. We do not have to worry over the exact place of the K's, M's and O's anymore, and even the question which one was parallel to which has become irrelevant. They have all been integrated in one single wave, carried by a single circle that encompasses everything.
10) Having done that, Huygens can then reconcile what was considered as irreconcilable: "But the thing to be above all remarked in our demonstration is that it does not require that the reflecting surface should be considered as a uniform plane... but only an evenness such as may be attained by the particles of the matter of the reflecting body being set near to one another..." (p.41/27)
See, the problem with waves is that they hit at multiple places at the same time, and if the surface is not smooth, each part of the wave will be reflected in another direction. By eliminating the need of "uniformity" and replacing it by that, less stringent, of "evenness", Huygens can keep the best of both worlds and explain how waves can be reflected off a surface the way a ball is reflected off a wall.
Huygens therefore creates a super-particle which he then calls wave. It is in fact the mother of all waves.



2017-01-30
Huygens' Light Theory: A Text Analysis
Chapter Three: On Refraction

revolves around the drawing p.49/35. The drawing, just like those in the first and the second chapter, shows the discrepancy between the verbal affirmations of the author, and what he really can prove. You could look for waves for a very long time, you will not find any. I do not mean of course real waves, but (a mathematical definition of) abstract waves. There is nothing in the drawing that could not be explained with particles. I will try and show what I mean.

1) We have a distant light source that somehow creates a wave AC. While C is continuing towards B, A has already arrived at the boundary between the two media, and is refracted at a certain angle from that surface towards N. Since A is going as fast as C, it will arrive at N having traveled as far as C had to travel to B. AN=BC (or CB).
2) Between A and C are "pieces", or even better, locations, ("endroits" in the French original) which are moving in the same direction along the parallel lines HK.
This is where the magic, or rather the sleigh of hands, happens. How are we supposed to understand these elements which are between the extremities A and C? Huygens tells us that they are themselves waves. Okay, but what is there then between two of these H's? Are there even smaller waves between them? You can see how delighted Zeno would have been if a contemporary had written such a treatise!
3) The wave AC has become the smaller wave KL, and what happened with AC happens with KL. And this is where the troubles continue. While A is moving towards N, and H has arrived at K, C is now at L. We now need K to be refracted in a parallel direction to AN, which was the path A had been refracted on. After all, HK is parallel to the line on which A is moving, and we can therefore safely assume that it will be refracted on a parallel trajectory to AN. That holds of course for all H's and K's.
4) How far will the waves starting from K travel? We already know the direction, parallel to AN, and have the distance and time of arrival at B. Which means that by the time C has arrived at B, all the other K's will have indeed arrived at a point located on a line BN.
5) An other analytical error, besides the refraction path of the K's, is the distance to the M's. Huygens assumes that they are equal to the relative distances to the points on BS, KM being promoted to semi-diameter or radius of all the circles created by the different parts of AC. But then, we are dealing with two different media, in each of which light has a different speed. The time it takes light to get to N can certainly be considered as equal to the time it takes to get to B, since both waves remain in the same medium. But that is not the case with the waves that go beyond AB, into the second medium. The distances AG and all the KM's cannot be equal to their counterparts on BS, and KM cannot therefore be considered as a common semi-diameter, nor can BG be parallel to AC.
6) The most surprising is that Huygens is more than aware of the difference in speed, but that does not stop him from presenting a mathematical model that simply ignores reality. The line BG passing through the points M is a counterfactual line. It is the line that would be drawn if light kept the same speed in both media. In other words, Huygens asks us to make abstraction of an empirical fact and concentrate on pure geometrical relations in an ideal situation. He needs CAGB to be a rectangle and all triangles to be equivalent. This is the only way he knows to give a semblance of rationality to his argumentation.
7) Here is a third issue in this analysis. where do the O's come from? 
According to Huygens:
"And if one would know how the wave AC has come progressively to BN, it is necessary only to draw in the same figure the straight lines KO parallel to BN, and all the lines KL parallel to AC. Thus one will see that the wave CA, from being a straight line, has become broken in all the positions LKO successively, and that it has again become a straight line at BN." (p.51/37)
Another magical trick? It certainly looks like it. By the time A arrives at the first O, the first H will be at the first K, and so on. The problem has already been sketched at (3): the OK's are not parallel to each other, since the distance between two O's will always be a fraction of the distance between two K's.. In fact, it is the lines themselves drawn from K to BS that can be said to be parallel to each other and to AN. We will see that the first drawing in this chapter in fact confirms this analysis.
8) Huygens' analytical errors cannot be as easily forgotten as with reflection. There the misrepresentation of the ratios between the different parts in the second medium had no direct consequence on how light was reflected within the same medium. With refraction we are dealing with border-crossing phenomena and a wrong representation of the geometrical relations lead to even more errors. The fact that Huygens is able to arrive at the correct conclusion (Snell's Law) shows how irrelevant his reasoning is.
9) Just like in Chapter Two, the line BG cannot be parallel to AC except in a counterfactual sense. KM's are parallel to AN and not to AG.
10) We can see now how the first drawing (p.48/34) in fact neutralizes the errors of the second one. We do not have to worry over the exact place of the K's, M's and O's anymore, and even the question which one is parallel to which becomes irrelevant. It is all about the sines of the incident and the refracted rays. in fact, you could erase all K's, L's, M's, O's, and the whole triangle ABG, and it would not change anything to the argumentation since it is incorrect anyway. These geometrical elements, as counterfactual drawings, cannot play any role in the actual behavior of light which is refracted along lines parallel to AN. Huygens does not add anything to Snell's Law, and he certainly does not explain refraction. Just like he did not explain reflection in the first chapter.


2017-01-30
Huygens' Light Theory: A Text Analysis
Chapter Three: On Refraction (2)
What is the empirical status of the line BN? Remember that it is formed by drawing the line between the point N, reached by the light coming from D and passing through A, after it has been refracted on exiting the first medium and entering the second one, and the point B. The time it takes light to reach B from C, is equal to the time it takes light to reach N from A.
It is, in other words, not a physical boundary but an abstract, mathematical line with no empirical significance. Light rays, or waves, do not suddenly stop when they have reached this line, nor do they change direction. As far as Nature is concerned, the line BN does not exist.
Therefore, it does not really matter whether this line goes through the triangle ABG or not. That means very simply that such an abstract line cannot in any way explain why light is not refracted beyond a critical angle.
It also shows that the idea of waves KM falling, or not, on BN is a (geometrical) fiction.

Another point of importance is that light, after it has reached B, is also refracted after entering the second medium, and that, not in the direction of N, but in a path parallel to AN. Another reason why BN has no physical significance whatsoever.

2017-02-01
Huygens' Light Theory: A Text Analysis
Fermat's Illusion: the Rule of Least Possible Time
This rule has made a great impression not only on illustrious thinkers of the 17th century, but also on renown scientists up to modern times, as Richard Feynman can testify. Allow me to use Huygens version which is, as he himself says "simpler and easier." (p.57/43).
The rule is very simple, We have a starting point A in a fast medium like air, a boundary point B, and a refracted arrival at C in a slower medium like water. The rule says that the trajectory light takes from A to C, passing through B, is the fastest light could follow, taking the speed of light in each medium into consideration. It is therefore not only a matter of distance, because in this case, a straight line from A to C would be the shortest. But then, that would mean, as Feynman would say, running very fast for a short time, and swimming very slowly for a much longer time.
All other combinations turn out to take more time than the refraction trajectory of light. Isn't that something magical? It looks like light knows which path to follow to get there as fast as possible! Just like it knows not to go through two slits at the same time when somebody is watching!
This mysticism is present in all of physics. Scientists somehow seem to nurture the conviction that Nature is something beyond human comprehension. It has been said that they are all atheists in week days, and believers in the weekend. I believe that most of them never stop believing, which makes them very sensitive to theories that go beyond the boring mechanical Newtonian view. Such a view was considered revolutionary as long as ignorant priests tried to control the scientific minds, but now that this menace has been eliminated, our scientists suddenly feel the need of something transcending their smothering labs. Is that all there is to science? Tedious experiments and calculations?
I am afraid so.
Let us now look at Fermat's special brand of mysticism. Could light really somehow know which path is the fastest in all situations?
Well, of course! The fastest trajectory is always the one the light takes. The question is, could it take another route to its objective? And what does it really mean that it always takes the fastest path?
Take three points A, B and C. There is a simple rule: you have to get to C by passing through B. For the rest, you are free to go any way you want.
I will bet you one whole kilogram of chocolate (make it 10 K's, I never seem to have enough of it in house) that wherever you place B, the shortest path will always be AB + BC.
There is one condition though: C should be placed relative to B the way a ray coming from A would be refracted to C.
What you are not allowed to do is what Feynman does so nonchalantly: change the rules to fit your own conclusions.
You cannot decide that you have only A, a life-guard on the beach, and C, someone drowning in the water, and then ask A to choose the fastest path to C.
That would mean that light is free to choose where it will shine and refract, and that is simply not true. Any point B you choose has a specific refracting destination depending on the position of A. All other points will have their specific A and C positions, and each combination A, B, C will be the fastest simply because it is the only possible one.
You could then say that Fermat is right, that light always chooses the fastest path, but you would be terribly wrong. Light does not choose, because it has never more than one possibility.
But that possibility is always the fastest, right? 
Wherever you put C, the straight line BC will be the shortest path. And whatever point B you point the light at, AB will the shortest path. But not every BC will correspond to the trajectory of a refracted ray coming from a specific A. Move A and you will move C at the same time. 
The fallacy in Fermat's (and Feynman's) approach is to attribute free choice to the life-guard, and therefore to light, while it is in fact like pointing the life-guard, who has to keep his eyes closed, in a certain direction, and tell him to keep running until he reaches the water, and once in the water, to let himself get carried away by the stream. Would you then still say that the life-guard always chooses the fastest path?
Feynman needs the imprecision inherent to the concept of point source of light. It sounds like one and the same ray of light (or even wave if you prefer) can take different directions randomly. This is a petitio principii. It is like shooting a gun fastened on a tripod and being surprised that the bullet does not hit at the same spot each time. It is funny how the Uncertainty Principle only gets to be used when it is convenient. How can one assume that the position of the gun, and therefore the trajectory of each bullet (it is never the same one and they are never absolutely identical at the subatomic level), does not change from one shot to the other?
If you absolutely need to leave the choice to light, then start at the beginning: why does light always have to travel in straight lines? Could it choose not to?
The trouble with Fermat's Rule of Least Time is that it claims to be an explanation for the behavior of physical processes, while it is at most an epiphenomenon. It gives the false impression that we understand why light behaves as it does, and why the Law of Sines is as it is, while in fact we are simply stating the existence of empirical regularities.
Yes, however light gets refracted, its path will always be the fastest of all other possible trajectories... if the world were not as it is, and if light could behave differently.
Fermat was really wrong for believing he had discovered a law of nature that explained the behavior of light, and Huygens, Feynman and others were misguided when they took him at his word. I know these are harsh words, not generally used in academic essays, but there is no excuse for you when you are so smart. Scientists have had more than 300 years to ponder this fallacy, but apparently all it did is give credibility to the likes of Bohr.