## Friday, July 20, 2018

### Huygens on Descartes Awareness of Snell's Law

In his Dioptrica, 1703, Huygens indicates that Descartes had come into contact with Snell's manuscript containing the ratio of sines for the angles involved in diffraction. Snell is most likely to have used the particle impulse model to arrive at his results. Huygens states the following,

"Hæc autem omnia, quæ de refractionis inquisitione volumine integro Snellius exposuerat‚ inedita mansere; quae & nos vidimus aliquando, & Cartesium quoque vidisse accepimus, ut hinc fortasse mensuram illam, quae in sinibús consistit, elicuerit; quá in explicanda iride & vitrorum figuris investigandis felicissimè est usus.

Cujusmodi vero sit illa Refractionis in sinubus proportio, cum radius ex aere in aquam, vitrumve, aut alia corpora diaphana desertur, id vel prismate, ut Cartesius praecipit, inquiri potest, vel aliis modis; quos, qui praecedentia intellexerit, non difficulter inveniet."

"All these things, about the refraction question in the work Snell completely explained, but it remained unpublished; what both we've seen for some time and Descartes had seen and heard so here possibly is the measure for it; which he concluded depends on the sines; which in explaining the rainbow of glass figures investigated is most successfully employed.

However with the preceding having been understood, it will not be difficult to discover that the Refraction is in the proportion of sines when a ray has been conveyed from air into water; glass; or any other transparent body; thus or for a prism or in other ways as Descartes has informed us, it is possible to inquire into."

### The Law of Refraction From Wave Vectors

The impulse acting on a particle analogy gives an incorrect value for the index of refraction so this theory has to be rejected. The direction unit vectors were not the source of the error. If we assume that ê₂ can be decompose in terms of ê₁ and n̂ it is not too difficult to solve for their coefficients.

We should have used the photon's momentum which is proportional to the wave vector, the gradient of the phase. ħ=h/(2π) is the reduced Plank's constant.

Thus we deduce the speed of light will be lower in denser mediums. The ratio of the angles of refraction and reflection equal a constant since the speed of light in the various media is due to the homogeneous nature of the media and so the ratio of the speeds is therefore independent of the angle of incidence. This turned out to be true for the particle impulse theory but we get the inverse of the index of refraction.

## Wednesday, July 18, 2018

### Regarding Descartes' Description of the Law of Refraction

One finds in Descartes' La Dioptrique (1637) a description of the law of refraction with the proportion between the sines of the angles of incidence and those of refraction being the same for all angles of incidence. His discussion is very general and focuses on a ball struck by a tennis racket and bouncing off of the ground for reflection and penetrating it for refraction. For refraction he speculates (see figure on p. 20 in the link above),

"...In the end, as long as the action of light follows the same laws as the movement of this ball, it must be said that when its rays pass obliquely from one transparent body into another, as the receiption is more or less easier than the first, they turn away in such a manner, that they are all less inclined to the surface of these bodies, from the side where the reception is easier, than that where it's the contrary: and this adjustment is in proportion to that which receives them more easily than the other. But we should take care that this inclination is measured by the quantity of straight lines, such as CB or AH, EB or IG, and the like, compared to each other; not by that of the angles, such as are ABH, or GBI, much less by those similar to DBI, which are called the angles of Refraction. For the ratio or proportion between these angles varies by all the various inclinations of the rays, whereas that which is between the lines AH & IG or the like, remains the same in all the refractions which are caused by the same bodies..."

One can clean this argument up some by assuming a particle of light receives an impulse as it enters a transparent body in the direction of the surface normal altering its trajectory. Let's assume that it is drawn into a denser body increasing its speed in the direction of the normal and so turning it in that direction as is observed.

Before and after the interaction with the surface we find uniform rectilinear motion. The interaction with the surface can be treated as an impulse similar to that of a collision by adding the velocity change to the initial motion.

One finds the proportion between the ratio of the sines is in inverse proportion to the speed as Huygens pointed out. If Δv is positive, the impulse being attractive, the speed increases and the angle of refraction is reduced. If it is negative, as with repulsion, the speed after entering the body will decrease and the angle of refraction increases. One gets a better match with observations if one assumes higher speeds in denser bodies. A complication is that the speed within the body isn't always the same but depends on the angle incidence. Later observations showed that the speed of light decreases in denser bodies.

## Monday, July 16, 2018

### Maupertuis' Three Laws of Motion for Light and Action as a Universal Principle

In his Accord de différentes lois de la nature of 1744 Maupertuis gives Three Laws for the motion of light.

"Here are the laws that light follows.

The first is that, in a uniform medium, it moves in a straight line.

The second, that when light meets a body which it cannot penetrate, it is reflected; and the angle of its reflection is equal to the angle of its incidence; that is to say, after its reflection, it makes an angle with the surface of the body equal to that under which it had met it.

The third is, that when the light passes from one diaphanous medium to another, its path after the meeting of the new medium, makes an angle with that which it held in the first; & the sine of the angle of refraction is always in the same ratio to the sine of the angle of incidence. If, for example, a ray of light passing from the air into the water is broken so that the sine of the angle of its refraction is three quarters of the sine of its angle of incidence; under some other obliquity that it meets the surface of the water, the sine of its refraction will always be three quarters of the sine of its new incidence."

He also contrasts the term, quantity of action, with Descartes' quantity of motion in Letter X found in his Works and describes action as a universal principle governing all motion.

"There is a truly universal principle, from which these Laws follow, in regards to the movement of hard bodies, elastic bodies, light, and all bodily substances: That is, in all the changes which occur in the Universe, the sum of the products of each body multiplied by the space which it traverses, and by the speed with which it traverses it, what is called the quantity of action, is always the smallest that is possible."

## Sunday, July 15, 2018

### Maupertuis On The Quantity Of Action (1744)

Here's a rough translation of a 1744 passage by Maupertuis containing his statement of the quantity of action.

"It was by this principle that Fermat solved the problem, by this principle so plausible, that light which, in its propagation & in its reflection always goes in the shortest possible time, still follows that same law in its refraction; & he did not hesitate to consider that light does not move with greater ease and more swiftly in rarer mediums than in those where, for the same space, there is found a greater quantity of matter: in fact, couldn't one assume at the onset that light traverses with greater ease & more swiftly in crystal and water than in air and the void?
Also many of the most famous mathematicians are known to embrace the feelings of Fermat; Leibnitz is the one who has made the most use of it, and by his name and by a more elegant analysis which he has given of this problem: he was so charmed by the metaphysical principle, & here to find its final causes to which he was considerably attached, that he regarded, as an unmistakable fact, that light moves faster in air than in water or glass.
It is however the opposite. Descartes had advanced the first, that the light moves most rapidly in the densest mediums, and although the explanation of the refraction which he had deduced from it was insufficient, his fault did not come from the supposition that he had made. All systems that give some plausible explanation for the phenomena of refection, assume the paradox, or confirm it.
If one now supposes, that light moves most rapidly in the densest mediums, the whole edifice which Fermat & Leibnitz had built, is destroyed: the light, when it crosses different mediums, neither by the shortest way, or by the quickest way; the ray which passes from the air into the water making the greater part of its path in the air, arrives later than if it did not make the slightest difference. We can see in the Memoir that Mr. de Mairan gave on Reflection and Refraction, the history of the dispute between Fermat & Descartes, & the difficulty & impotence to which we have so far been able to grant the law of refection with metaphysical principle.
While meditating deeply on this matter, I thought that the light, when it passes from one medium to another, already abandoning the shortest path, which is that of the straight line, could well also not follow that at the most rapid time: indeed, what preference should there be here of time over space? the light being unable to go at once by the shortest way, and by that of the quickest time, why should it go by one of these paths rather than by the other? so it does not follow either of the two, it takes a path that has a more real assertion: the path it takes is that by which the amount of action is a minimum.
Now I have to explain what I mean by the quantity of action. When a body is carried from one point to another, it requires a certain action, this action depends on the speed of the body and the space it travels, but it is neither the speed nor the space taken separately. The quantity of action is all the greater as the speed of the body is greater, and the path which it traverses is longer, it is proportional to the sum of the spaces multiplied each by the speed with which the body travels. It is this, that quantity of ation which is here the true expenditure of Nature, and which it spares as much as possible in the movement of light.
Let two different media, separated by a common surface be represented by the line CD, such that the speed of light in the medium which is above, set = V, & the speed in the medium which is below, set = W. Let a ray of light AR, which from a given point A must reach the given point B.

To find the point R where it must break, I search for the point where the ray breaks, the amount of action is the least, & I have V.AR + W.RB which must be a minimum, or V.√(AC²+CR²) + W.√(BD²+CD²-2CDxCR+CR²) = min. So AC, BD & CD being constant, I have V.CR.dCR/√(AC²+CR²)-W.(CD-CR).dCR/√(BD²+DR²)=0, or V.CR/AR=W.DR/BR. CR/AR:DR/BR::W:V, that is, the sine of incidence to the sine of refraction is in inverse ratio to the speed that light has in each medium.
All the phenomena of refraction now agree with the great principle, that Nature in the production of its effects always acts in the simplest ways. From this principle follows that when the light passes from one medium to another, the sine of its angle of refraction is to the sine of its angle of incidence in inverse ratio to the velocities of light in each medium."

## Friday, July 13, 2018

### Ptolemy, Kepler, Newton & Huygens on Refraction

Some references on the history of refraction:

160s -
Ptolemy - Optics  Lejeune  Smith

1611 -
Kepler - Dioptrice - refringuntur & refracti

1672 -
Newton - Philos. Trans., concave mirror vs convex lens (aberration)
Newton - use of the word aberration in a response

1687 -
Newton - Principia Mathematica, 1st Law, uniform rectilinear motion

1690 -
Huygens - Traité de la lumière, refraction, proportion of sines

1703 -
Huygens - Dioptrica, Snell, refraction, proportion of sines

1704 -
Newton - Opticks, ratio of sines, refrangible rays (aberration)

1727-8² -
Bradley - Philos. Trans. stellar aberration

1747¹ -
Bradley - Philos. Trans., stellar aberration
_____________
¹ edit (Jul 13): Bradley 2nd ltr
² edit (Jul 13): Bradley 1st ltr

## Wednesday, July 11, 2018

### References for Hamilton's Theory of Systems of Rays, 1st Part

Here are some references that may give some insight into Hamilton's terminology and sources regarding the presentation in his theory of optics and rays.

Malus - Treaté de l'Optique 1807

Least Action

Maupertuis - quantité d'action 1744

Laplace - Mechanics: Maupertuis least action footnote 1814

pencil

Hamilton -  pencils 1824

Coddington - A System of Optics, pencil 1829

Characteristic

cnrtl - caractéristique

Laplace - Mécanique céleste, caractéristiques 1799

Monge - Application de l'analyse à la géometrie, caractéristique 1807

Lagrange - caractéristique (symbol δ for variation) 1811

Lagrange (Eng) caractéristique -> symbol trans.

caustic*

Coddington - An Elementary Treatise on Optics, caustics 1825

arétes de rebroussement (cusp edges)

Hamilton - arétes de rebroussement

cnrtl - aréte de rebroussement

Monge - Géométrie descriptive, arétes de rebroussement (see also pages 18, 208)

Aberration

Hamilton - aberrations

Optical aberration wiki

*edit (Jul 11): Found an earlier Coddington book in my notes. Hamilton was associated with Trinity College, Dublin while Coddington was associated with Trinity College, Cambridge. They were contemporaries.