This is my fourth post on the Traité de la lumière [Treatise on Light1], published in 1690 by Cristiaan Huygens (1629-1695). The previous posts were
Christiaan Huygens Proposes an Infinite Series of Levels to the Universe
Cristiaan Huygens Presents Ole Rømer's Proof that the Speed of Light is Finite
In this post, I will present Huygens’s demonstration that the wave theory of light explains reflection.
Huygens initially presents the reflection by a perfectly smooth plane surface of a light wave emanating from a light source so distant that the wave front can be considered to be a straight line. This is done through the use of the following diagram:
Huygens explains the diagram.
Let there be a surface AB, plane and polished, of some metal, glass, or other body, which at first I will consider as perfectly uniform (reserving to myself to deal at the end of this demonstration with the inequalities from which it cannot be exempt), and let a line AC, inclined to [AB], represent a portion of a wave of light, the centre of which is so distant that this portion AC may be considered as a straight line; for I consider all this as in one plane, imagining to myself that the plane in which this figure is, cuts the sphere of the wave through its centre and intersects the plane AB at right angles. This explanation will suffice once for all. [p.23]
So AB represents the perfectly smooth and plane (flat) surface. The line AC represents the wave whose light source is very distant. The image, which is two-dimensional, must be understood as a slice of a three-dimensional situation, in which the wave AC is at right angles to the surface AB in the third dimension. By making this assumption, the third dimension can be ignored, at least for now.
Now Huygens considers what happens to the different parts of the wave AC as they reach the surface AB at different moments.
The piece C of the wave AC, will in a certain space of time advance as far as the plane AB at B, following the straight line CB, which may be supposed to come from the luminous centre, and which in consequence is perpendicular to AC. Now in this same space of time the portion A of the same wave, which has been hindered from communicating its movement beyond the plane AB, or at least partly so, ought to have continued its movement in the matter which is above this plane, and this along a distance equal to CB, making its own partial spherical wave, according to what has been said above. Which wave is here represented by the circumference SNR, the centre of which is A, and its semidiameter AN equal to CB. [pp.23-24]
He considers first the two extremities A and C of the wave AC. Piece A gets reflected at point A, and a wave propagates out from that point. Meanwhile, piece C heads towards point B. When piece C reaches point B, the wave emanating from point A will correspond to arc SNR and will have a semidiameter [radius] exactly equal to the distance CB.
So the question remains about what happens to all of the pieces of the wave between A and C, which he labels H.
If one considers further the other pieces H of the wave AC, it appears that they will not only have reached the surface AB by straight lines HK parallel to CB, but that in addition they will have generated in the transparent air, from the centres K, K, K, particular spherical waves, represented here by circumferences the semi-diameters of which are equal to KM, that is to say to the continuations of HK as far as the line BG parallel to AC. But all these circumferences have as a common tangent the straight line BN, namely the same which is drawn from B as a tangent to the first of the circles, of which A is the centre, and AN the semi-diameter equal to BC, as is easy to see. [p.24]
For a given piece H, it will reach a corresponding point K on the surface AB, and a wave will emanate from that point K. When piece C reaches point B, the wave emanating from that point K will have semidiameter KM, where HKM forms a straight line.
And now all this can be brought together:
It is then the line BN (comprised between B and the point N where the perpendicular from the point A falls) which is as it were formed by all these circumferences, and which terminates the movement which is made by the reflexion of the wave AC; and it is also the place where the movement occurs in much greater quantity than anywhere else. Wherefore, according to that which has been explained, BN is the propagation of the wave AC at the moment when the piece C of it has arrived at B. For there is no other line which like BN is a common tangent to all the aforesaid circles, except BG below the plane AB; which line BG would be the propagation of the wave if the movement could have spread in a medium homogeneous with that which is above the plane. 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. [pp.24-25]
There is exactly one line which passes through B and which is tangent to the arc SNR (at point N) and also tangent to all of the circular waves centered on the points labeled K. This line is BN. The only other such line is the line BG, which is where the wave AC would reach were it not reflected by the surface AB. The implication is that when wave piece C has reached point B, the wave AC has been completely reflected into the wave BN.
The diagram also shows what happens to the wave AC when a given point K on surface AB is reached. Part of that wave, labeled OK, will be reflected, while the other part, labeled KL, will still be unreflected.
The passage “it is also the place where the movement occurs in much greater quantity than anywhere else” refers to the fact that Huygens has previously stated that if a wave propagates from a given point, then every point in which that wave propagates to will in turn propagate its own wave, although of a lesser intensity. This means that there might be some light beyond the point N on the line BN, but much more feeble than along BN. See the previous post Cristiaan Huygens Explains How Light Waves Advance, along with the associated comments therein.
Finally, given this diagram, it is straightforward to demonstrate that the angle of incidence, i.e., the angle at which the wave AC strikes the surface AB, is equal to the angle of reflection, i.e., the angle at which the wave NB leaves the surface AB.
Now it is apparent here that the angle of reflexion is made equal to the angle of incidence. For the triangles ACB, BNA being rectangular and having the side AB common, and the side CB equal to NA, it follows that the angles opposite to these sides will be equal, and therefore also the angles CBA, NAB. But as CB, perpendicular to CA, marks the direction of the incident ray, so AN, perpendicular to the wave BN, marks the direction of the reflected ray; hence these rays are equally inclined to the plane AB. [p.25]
Now, what was demonstrated above was what happens when a circular wave, of two dimensions, is reflected. But real waves are spherical, of three dimensions. What happens when these are reflected? This question is addressed using the next diagram.
Huygens writes:
But in considering the preceding demonstration, one might aver that it is indeed true that BN is the common tangent of the circular waves in the plane of this figure, but that these waves, being in truth spherical, have still an infinitude of similar tangents, namely all the straight lines which are drawn from the point B in the surface generated by the straight line BN about the axis BA. It remains, therefore, to demonstrate that there is no difficulty herein: and by the same argument one will see why the incident ray and the reflected ray are always in one and the same plane perpendicular to the reflecting plane. I say then that the wave AC, being regarded only as a line, produces no light. For a visible ray of light, however narrow it may be, has always some width, and consequently it is necessary, in representing the wave whose progression constitutes the ray, to put instead of a line AC some plane figure such as the circle HC in the following figure, by supposing, as we have done, the luminous point to be infinitely distant. Now it is easy to see, following the preceding demonstration, that each small piece of this wave HC having arrived at the plane AB, and there generating each one its particular wave, these will all have, when C arrives at B, a common plane which will touch them, namely a circle BN similar to CH; and this will be intersected at its middle and at right angles by the same plane which likewise intersects the circle CH and the ellipse AB. [pp.25-26]
Huygens considers here the reflection of a circle HC, once again supposing the source of light to be infinitely distant, being reflected by a surface AB. He then breaks up HC into infinitely many slices, each of which gets reflected as in the discussion around the first diagram. It then follows that the whole of HC gets reflected to BN.
One sees also that the said spheres of the partial waves cannot have any common tangent plane other than the circle BN; so that it will be this plane where there will be more reflected movement than anywhere else, and which will therefore carry on the light in continuance from the wave CH. [p.26]
The passage “where there will be more reflected movement that anywhere else” should be understood in the same manner as the passage “it is also the place where the movement occurs in much greater quantity than anywhere else” in the discussion of the first diagram.
Huygens then addresses the initial supposition of a perfectly smooth plane surface:
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, as has been supposed by all those who have tried to explain the effects of reflexion; 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; which particles are larger than those of the ethereal matter, as will appear by what we shall say in treating of the transparency and opacity of bodies. For the surface consisting thus of particles put together, and the ethereal particles being above, and smaller, it is evident that one could not demonstrate the equality of the angles of incidence and reflexion by similitude to that which happens to a ball thrown against a wall, of which writers have always made use. [p.27]
What Huygens is saying is that his model of reflection supposes that the particles of ponderable matter (what we can see and touch), which are an order of magnitude larger than the particles of the aether, be “set near to another”. He says that reflection of a light wave cannot be explained, say, as the bouncing of a simple ball against a wall, but rather of the reflection of multiple wave pieces on these particles of ponderable matter.
In our way, on the other hand, the thing is explained without difficulty. For the smallness of the particles of quicksilver [mercury], for example, being such that one must conceive millions of them, in the smallest visible surface proposed, arranged like a heap of grains of sand which has been flattened as much as it is capable of being, this surface then becomes for our purpose as even as a polished glass is: and, although it always remains rough with respect to the particles of the Ether it is evident that the centres of all the particular spheres of reflexion, of which we have spoken, are almost in one uniform plane, and that thus the common tangent can fit to them as perfectly as is requisite for the production of light. And this alone is requisite, in our method of demonstration, to cause equality of the said angles without the remainder of the movement reflected from all parts being able to produce any contrary effect. [pp.27-28]
Therefore, using the wave theory of light, the huge numbers of particles of ponderable matter in a surface, all minute, although of an order of magnitude greater than aether particles, collectively act in practice as would a completely smooth planar surface. It follows that the angle of incidence equals the angle of reflection, not just for the perfect surfaces of theory, but also for real, actual surfaces.
When Huygens wrote his Traité de la lumière, it had long been known, at least as far back as Ibn al-Haytham (965-1040), that the angle of incidence equals the angle of reflection. What Huygens’s aether model provides is a simple mechanical understanding of how the reflection of light could place.
Christiaan Huygens. Treatise on Light. In which are explained the causes of that which occurs in Reflexion, & in Refraction. And particularly in the strange Refraction of Iceland Crystal. Rendered into English By Silvanus P. Thompson. New York: Dover, 1962. First published by Macmillan and Company, Limited, in 1912.
Thanks! Where you write "while the other part, labeled KM, will still be unreflected.", do you mean KL instead of KM?
The spread of a tiny bit of light in all directions from A on the arc SNR is interesting. Light from Jupiter reaches the earth with such precision that we see an image of the planet. The bits diffused in all directions by the ether particles in between must be so small that they do not blur the overall image.