Physics

Light

DESCARTES

REFRACTION OF Light

Since we shall need hereafter to know exactly the quantity of this refraction, and that it may be conveniently understood by the comparison which I am going to use, I believe that it is proper that I attempt here a complete explanation and that I speak first therefore of reflection, so as to make it easier to understand the explanation of refraction. Let us suppose therefore that a ball driven from A to B (Fig. 45) encounters at the point B the surface of the earth C B E; which prevents its going on and is the reason that it changes its direction. Let us see toward what side. But first, in order that we shall not embarrass ourselves with new difficulties, let us suppose that the earth is perfectly flat and hard, and that the ball proceeds always with constant velocity, both as it descends and as it rises again, not inquiring in any way about the force which maintains its motion after it is no longer in contact with the racquet, and not considering any effect of its weight or its size or its shape. For there is no question here of looking so closely into the matter, and none of these things come into the action of the light, to which this motion is to be compared. It only needs to be noticed that the force, whatever it may be, which keeps up the motion of the ball, is different from that which makes it move in one direction rather than in another, as it is very easy to see from this, that it is the force by which it has been driven by the racquet on which its motion depends, and that this same force would have been able to make it move in any other direction as easily as toward B, while in fact it is the position of the racquet which makes it move toward B, and which would have been able to make it move in the same way, even if another force had moved it. This shows already that it is not impossible that the ball may be turned in its path by its encounter with the earth, and that the tendency which it had to go to B may be changed without anything being changed in the force of its motion, since they are two different things; and consequently that we should not imagine that it is necessary that it should stop for a moment at the point B, before turning toward F, as several of our philosophers would have it do; for if its motion were once interrupted by this check, there would be no cause which would thereafter make it start off again. Furthermore, it must be remarked that the tendency to move itself in any direction, just as well as the motion itself, and generally as any other sort of quantity, may be divided into all the parts of which we may imagine that it is compounded, and that we may easily imagine that this motion of the ball which moves it from A to B is compounded of two others, one of which would make it descend from the line AF toward the line CE, and the other would, at the same time, make it go from the left-hand line AC toward the right-hand line FE, so that these two motions together carry it to B along the line AB. And further it is easy to understand that the encounter with the earth can only prevent one of these two motions, and cannot affect the other in any way. For it certainly ought to prevent that motion which would make the ball descend from AF toward CE, because it occupies all the space which is below CE; but why should it prevent the other motion, which would make it advance toward the right, seeing that it is not opposed in any way in that sense? Therefore, to find out correctly in what direction the ball ought to rebound, we describe a circle with the center B, which passes through the point A, and we say that, in the same time that it would take to move from A to B, it infallibly should return from B to some point of the circumference of this circle, in as much as all the points which are as distant from the center B as A is, are found in this circumference, and we suppose the motion of the ball to be always equally swift. Then finally, to find out precisely to which one of all the points of this circumference it ought to return, we draw three straight lines AC, HB, and FE, perpendicular to CE, and in such a way that there is neither more nor less distance between AC and HB than between HB and FE: and we say that in the same time that the ball has advanced toward the right from A, one of the points of the line AC, to B, one of the points of the line HB, it should also move from the line HB as far as some point of the line FE; for all the points of this line FE are as far away from the line HB in this sense on the one side as those of the line AC are on the other, and it is also as ready to move in this direction as it was before. Now it cannot in the same time reach some point in the line FE, and also some point in the circumference of the circle AFD, except at the point D, or at the point F, since there are only these two points where these lines cut each other; so that since the earth prevents its passing toward D, we must conclude that it must go infallibly toward F, and so you see easily how reflection occurs, that is, with an angle equal to that which we call the angle of incidence. Thus if a ray coming from the point A falls at the point B on the surface of a plane mirror CBE, it is reflected toward F in such a way that the angle of reflection FBE is neither greater nor less than the angle of incidence ABC.

FIG. 45.

We now come to refraction. And, first, we suppose that a ball, driven from A to B, (Fig. 46) encounters at the point B, not now the surface of the earth, but a cloth CBE, which is so weak and so thin that the ball can break it and pass entirely through it, losing only a part of its velocity, for example, half of it. Now, this being supposed, in order to determine what path it should follow, we notice first that its motion differs entirely from its tendency to move in one direction rather than in another; from which it follows that the quantities of these motions should be considered separately. And we notice also that of the two parts of which we may imagine that this tendency is compounded, only that part which would make the ball move from above downward can be changed in any way by encountering the cloth; and that the tendency which made it move toward the right should always remain the same as it has been, because the cloth is in no way opposed to it in that sense. Then, having described from the center B the circle AFD and drawn at right angles to CBE the three straight lines AC, HB, FE, in such a way that there is twice as much distance between FE and HB as between HB and AC, we see that the ball ought to move toward the point I. For since it loses half of its velocity when it passes through the cloth CDE, it ought to take twice as long to move downward from B to some point of the circumference of the circle AFD as it has taken above it to pass from A to B. And, since it loses none of the tendency which it had previously to move toward the right, in twice the time that it has taken to pass from the line AC to HB it ought to travel twice as far in this same direction, and consequently should reach some point of the straight line FE at the same instant that it reaches also some point of the circumference of the circle AFD. This would be impossible if it did not go to I, since that is the only point below the cloth CBE where the circle AFD and the straight line FE cut each other.

FIG. 46.

Let us now think of the ball which moves from A toward D, as encountering at the point B, no longer a cloth, but water, of which the surface CBE deprives it of half of its velocity just as the cloth did. And supposing everything else to be the same as before, I say that the ball ought to pass from B in a straight line, not toward D but toward I. For, first, it is certain that the surface of the water ought to turn it toward that point in the same way that the cloth did, seeing that it deprives it of just as much of its force and is opposed to it in the same sense. Considering the body of water which fills all the space between B and I, while it may resist its motion more or less than the air did which we supposed before, we cannot say nevertheless that it ought to change its path; for it may open up to give it passage as easily in one direction as in another. At least that is so, if we suppose always, as we have done, that neither the weight nor the lightness of the ball, nor its size, nor its shape, nor any other such cause changes its course. And we may here remark that it is so much more changed in direction by the surface of the water or by the cloth as it encounters it more obliquely; so that if it encounters the surface at right-angles, as when it is driven from H toward B, (Fig. 47) it ought to go on in a straight line toward G without turning out of it all But if it is driven along a line such as AB, which is so much inclined to the surface of the water or to the cloth CBE, that when the line FE is drawn as before it does not cut the circle AD, then the ball does not penetrate the surface at all but rebounds from the surface B toward the air L, just as if it had encountered the earth. This effect has sometimes produced the regrettable result that when cannon have been shot for fun toward the surface of a river, men have been wounded who were on the bank on the other side.

FIG. 47.

But now let us make here another supposition, and assume that the ball which has been first driven from A to B is driven just when it is at the point B by the racquet CBE which increases the force of its motion, for example, by a third, so that it can afterwards move over as great a distance in two moments as it did in three before. This will have the same effect as if the ball had encountered at the point B a body of such a nature that it passes through the surface CBE a third more easily than through air. It follows manifestly from that which has already been demonstrated, that if we describe the circle AD as before (Fig. 48) and the lines AC, HB, FE, in such a way that there is a third less distance between FE and HB than between HB and AC, the point I, in which the straight line FE and the circular line AD cut each other, will determine the place toward which the ball which is at the point B should turn.

Now we may also take the reverse of this conclusion, and say that, since the ball which comes from A moves in a straight line as far as B, and at the point B turns and proceeds to the point I, this means that the force of facility with which it enters the body CBEI is to that with which it leaves the body ACBE, as the distance between AC and HB is to that which is between HB and FI, that is to say, as the line CB is to BE.

FIG. 48.

To conclude, inasmuch as the action of light follows in this respect the same laws as the motion of the ball, we must say that when its rays pass obliquely from one transparent body into another, which receives them more or less easily than the first body, they turn in such a way that they are always less inclined to the surface separating these bodies on the side where that body is which receives them more easily than on the side where the other body is, and this just in proportion to that which receives them more easily than the other does. Only we must take notice that this inclination should be measured by the magnitudes of the straight lines, like CB or AH, and EB or IG, and others like them, compared one to the other; not by the magnitudes of the angles, such as ABH or GBI; and much less by the magnitudes of the angles, such as DBI, which are called angles of refraction. For the ratio or proportion between these angles changes for all the different inclinations of the rays, while that which holds between the lines AH and IG, or the like, remains the same for all the refractions which are caused by the same bodies. So, for example, (Fig. 49) if a ray passes in air from A to B and encounters at the point B the surface of glass CBR, so that it turns toward I in the glass; and if another one comes from K to B which turns toward L; and another from P toward R which turns toward S, there ought to be the same proportion between the lines KM and LN, or PQ and ST, as between AH and IG, but not the same proportion between the angles KBM and LBN, or PRQ and SRT, as between ABH and IBG.

Now that you see in what way refraction should be measured and further that to determine the quantity of refraction, in so far as it depends on the particular nature of the bodies in which it occurs, there is need of proceeding by experiment, there is found to be no difficulty in doing this with sufficient certainty and facility, since all refractions are thus reduced to the same measure; for it is only necessary to determine them for a single ray to determine all those which occur at the same surface, and we can avoid all error if we examine in addition some others. Thus if we wish to know the measure of the refractions which occur in the surface CBR, which separates air AKP from glass LIS, we have only to test the refraction of the ray ABI, by finding the ratio between the lines AH and IG. Then if we fear we have made some mistake in this experiment, we can test our result by using other rays, such as KBL or PRS, and when we find the same ratio of KL to LM and of PQ to ST as that of AH to IG, we shall have no further reason to question the accuracy of our experiment.

FIG. 49.

But perhaps you will be astonished when you make these experiments to find that the rays of light are more inclined in air than in water to the surface where the refractions occur; and still more in water than in glass, exactly the opposite from the course of a ball, which is more inclined to the surface in water than in air and can not enter glass at all. For example, if a ball, which is driven in air from A to B (Fig. 50), encounters the surface of water CBE at the point B, it will be deflected from B toward V; and if it is a ray of light it will go on the contrary from B toward I. You will cease, however, to find this a strange effect, if you recall the nature that I have attributed to light, when I said that it is nothing other than a certain motion or an action conceived in a very subtle matter, which fills the pores of all other bodies; and when you consider that as a ball loses more of its motion when it strikes against a soft body than against a hard one, and that it rolls less easily on a table-cloth than on a bare table; so the action of this subtle matter may be much more restrained by the parts of the air which, being as they are soft and loosely joined together, do not offer much resistance to it than by the parts of the water, which offer more resistance, and still more by the parts of the water than by those of glass or crystal. Thus it happens that so much as the small parts of a transparent body are harder and firmer so much the more do they allow the light to pass more easily; for the light should not drive any of them out of their places, as a ball ought to drive out the parts of the water to find passage among them.

FIG. 50.

Further, as we now know the cause of the refractions which occur in water and in glass, and generally in all other transparent bodies which exist about us, we may remark that they should be in all respects similar when the rays come out from the bodies and when they enter them. Thus, if the ray which passes from A toward B is bent at B toward I in passing from air into glass, the ray which will come back from I toward B should also bend at B toward A. It may possibly be that other bodies may be found, principally in the skies, where refractions proceed from other causes and are not thus reciprocal. And there may also be other certain cases in which the rays ought to bend even though they pass only through a single transparent body; just as the motion of a ball is often a curved motion because it is turned in one direction by its weight, and in another by the action which has set it going; or for divers other reasons. In fact I dare to say that the three comparisons which I have just employed are so suitable that all the particularities which can be noticed are comparable with others which are found just like them in light; but I have not tried to explain those which are not of the most importance to my subject. And I shall not ask you to consider anything further except this, that the surfaces of transparent bodies which are curved bend the rays which pass at each of their points in the same way that plane surfaces would that we may imagine touching these bodies at the same points. Thus for example, the refraction of the rays AB, AC, AD, (Fig. 51) which come from the flame A and fall on the curved surface of the crystal ball BCD ought to be treated as if AB fell on the plane surface EBF and AC on GCH and AD on IDK, and so for the others. Thus you may see that these rays may be brought together or may separate in different ways, according as they fall on surfaces which are differently curved. It is time that I begin the description of the structure of the eye, that I may make you understand how the rays which enter within it conduct themselves so as to cause the sensation of sight.

FIG. 51.