Refraction

by Rene Descartes

Especially since we shall need hereafter to know exactly the quantity of this refraction, and since it can quite conveniently be understood by the comparison which I have just used, I believe it is apropos that I mark here to explain it all at once, and that I speak first of reflection, in order to make it all the easier to understand. Let us therefore think that a ball being pushed from A towards B[6] encounters at point B the surface of the earth CBE, which, preventing it from passing further, causes it to turn away; and let’s see which way.

But in order not to encumber ourselves with new difficulties, let us suppose that the earth is perfectly flat and hard, and that the ball always goes of equal speed, as much in descending as in ascending, without inquiring in any way about the power which continues to move it, after it is no longer touched by the racket, nor consider any effect of its weight, nor of its size, nor of its shape;

For there is no question here of looking at it so closely, and there is none of these things which take place in the action of light, to which this must be referred. Only it should be noted that the power, such as it is, which causes the movement of this ball to continue is different from that which determines it to move rather towards one side than towards another, just as it is very easy to to know that it is the force on which it has been pushed by the racket, on which its movement depends, and that this same force could have made it move towards any other side as easily as towards B; whereas it is the situation of this racket which determines it to tend towards B, and which could have determined it there in the same way, even if another force would have moved it; which already shows that it is not impossible for this ball to be deflected by the encounter with the earth, and so that the determination it had to tend towards B is changed, without there being anything for that changed to the force of its motion, since these are two different things, and therefore one must not imagine that it is necessary for it to stop for any moment at the point B before returning to F, as well as do many of our philosophers: for, if its movement were once interrupted by this arrest, there would be found no cause which would make it begin again afterwards. Further, it must be remarked that the determination to move towards some side can as well as the movement, and generally as any other kind of quantity, be divided into all the parts of which it can be imagined to be composed, and that one can easily imagine that that of the ball which moves from A towards B is composed of two others, one of which makes it descend from the line AF towards the line CE, and the other at the same time makes it go from the left AC to the right FE, so that these two joined together lead it to B along the straight line AB, And then it is easy to understand that the meeting of the earth can only prevent one of these two determinations, and not the other in any way: for it must indeed prevent the one which caused the ball to descend from AF towards CE, because it occupies all the space which is below CE; but why should she prevent the other who made her advance towards the right hand, seeing that she is in no way opposed to him in that sense? In order to find therefore precisely towards which side this ball must return, let us describe a circle from the center B, which passes through the point A, and let us say that in as much time as it will have taken to move from A to B, it must infallibly return from B to some point on the circumference of this circle, especially since all the points which are as distant from this one B, as is A, are in this circumference, and that we suppose the movement of this ball always be equally fast. Then, in order to know precisely to which of all the points of this circumference it must return, let us 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 say that in as much time as the ball took to advance towards the right side, from A one of the points of the line AC, to B one of those of the line HB, it must also to advance from the line HB to some point on the line FE: for all the points of this line FE are as far from HB in this direction as each other, and as much as those of the line AC , and she is as determined to move towards that side as she was before. But is it that it cannot arrive at the same time at some point on the line FE and together at some point on the circumference of the circle AFD, if not at the point D or at the point F, especially since there are only these two where they intersect each other, so that the earth preventing it from passing towards D it must be concluded that it must go infallibly towards F. And so you easily see how the reflection takes place, namely according to an angle always equal to that which is called the angle of incidence; as if a ray coming from point A falls at point B on the surface of the flat mirror CBE, it is reflected towards F, so that the angle of the reflection FBE is no longer less than that of the incidence ABC.

Let us now come to refraction; and first suppose that a ball pushed from A[7] towards B encounters at point B, no longer the surface of the earth, but a web CBE, which is so weak and loose that this ball has the force to break it and pass through it, losing only part of its speed, namely, for example, half. Now, this having been established, in order to know what path it must follow, let us again consider that its movement differs entirely from its determination to move rather towards one side than towards another, whence it follows that their quantity must be examined separately; and let us also consider that of the two parts of which one can imagine this determination to be composed, it is only that which caused the ball to stretch from top to bottom which can be changed in some way by the meeting of the web, and that for the one that made it tend towards the right hand, it must always remain the same as it was, because this canvas is in no way opposed to it in this sense. Then having described the circle AFD from the center B, and drawn

at right angles on CBE the three straight lines AC, HB, FE, such that there is twice as much distance between FE and HB as between HB and AC, we will see that this ball must tend towards point I ; for, since it loses half its speed in crossing the web CBE, it must take twice as much time to pass under it from B to some point on the circumference of the circle AFD which it has made above to come from A to B: and, since it loses none of the determination it had to advance to the right in twice as much time as it took to pass from the line AC up to HB, it must make twice as much distance towards this same side, and consequently arrive at some point on the straight line FE, at the same instant as it also arrives at some point on the circumference of the circle AFD; which would be impossible if it did not go towards I, especially since it is the only point below the canvas CBE where the circle AFD and the straight line FE intersect.

Let us now think that the ball which comes from A towards D encounters at point B, no longer a canvas, but water, the surface of which CBE deprives it of precisely half of its speed as this canvas did; and the rest posed as before, I say that this ball must pass from B in a straight line, not towards D, but towards I: for first it is certain that the surface of the water must divert it towards there in the same way as the canvas, since it deprives him just as much of his strength, and is opposed to him in the same sense. Then for the rest of the body of water which fills all the space which is from B to I, although it resists it more or less than did the air which we previously supposed there, it is not to say for that that he must more or less deflect it: for he can open himself to make it pass just as easily towards one side as towards another; at least if we always suppose as we do that neither the weight or lightness of this ball, nor its size, nor its shape, nor any other such extraneous cause, changes its course;

and we can here notice that it is all the more deflected by the surface of the water or the canvas, the more obliquely it meets it; so that if it meets it at right angles, as when it is pushed from H[8] towards B, it must pass in a straight line towards G, without turning away at all; but if it is pushed along a line like AB, which is so steeply inclined on the surface of the water or the canvas CBE, that the line FE being drawn as before does not intersect the circle AD, this ball must in no way to penetrate, but to rebound from its surface B towards the air L, just as if it had encountered earth there. What we have sometimes experienced with regret, when, for pleasure, firing artillery pieces towards the bottom of a river, we have injured those who were on the other side on the shore.

But here let us make yet another supposition and suppose that the ball, having been pushed from A[9] towards B, is pushed again, being at point B, by the racket CBE, which increases the force of its movement, for example, d a third, so that it can afterwards make as much distance in two moments as it did in three before, which will have the same effect as if it encountered at point B a body of such nature that it passed through its surface area CBE by one-third more easily than by air. And it obviously follows from what has already been demonstrated that, if we describe the circle AD as in front, and the lines AG, HB, FE, in such a way that there is a third less distance without FE and HB that between HB and AC, the point I, or the straight line FE and the circular AD intersect, will designate the place towards which this ball, being at point B, must be diverted.

Now we can also take the reverse of this conclusion, and say that, since the ball which comes from A in a straight line as far as B turns away being at point B and takes its course from there towards I, this means that the force or the ease with which it enters the body CBEI is with that with which it leaves the body ACBE, as the distance which is between AC and HB with that which is between HB and FI, that is to say as the line CB is with BE .

Finally, especially since the action of light follows in this 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, which receives them more or less easily than the first, they are diverted there in such a way that they are always less inclined on the surface of these bodies on the side where is that which receives them most easily than on the side where is the other; and it is precisely in proportion to the fact that he receives them more easily than the other does. Only it is necessary to take care that this inclination must be measured by the quantity of the straight lines, like CB or AH, and EB or IG, and similar, compared ones with the others; not by that of the angles, such as ABH or GBI, nor much less by that of the similar to DBI, which are called the angles of refraction; for the ratio or proportion which is between these angles varies at all the different inclinations of the rays, whereas that which is between the lines AH and IG, or similar, remains the same in all the refractions which are caused by the same bodies. As, for example, if a ray passes through the air from A[10] to B,

which, meeting the surface of the CBR glass at point B, turns towards I in this glass, and there comes another from K to B which turns away towards L, and another from P towards R which turns away towards S , there must be the same proportion between the lines KM and LN, as between AH and IG; but not the same between the angles RBM and LBN, as between ABH and IBG.

So that you now see in what way refractions must be measured; and even that, to determine their quantity as it depends on the particular nature of the bodies in which they are made, it is necessary to come to experience, one cannot fail to be able to do it quite certainly and easily from that they are thus all reduced to the same measure; for it suffices to examine them in a single ray to know all those which are made in the same area, and we can avoid all error, if we examine them in addition to that in a few others. As if we want to know the quantity of those which are made in the surface CBR, which separates the air AK from the glass LI, we have only to test it in that of the radius ABI, by seeking the proportion which is between lines AH and IG. Then, if we are afraid of having failed in this experiment, we must still test it in some other rays, like KBL; and, finding the same proportion of KM to LN, as of AH to IG, we shall no longer have any occasion to doubt the truth.

But perhaps you will be surprised, in making these experiments, to find that the rays of light bend more in air than in water, on the surfaces where their refraction takes place; and even more in water than in glass, quite the contrary of a ball, which tilts more in water than in air, and can in no way pass through glass: for, for example, if it is a ball which, being pushed in the air from A[11] towards B, meets at point B the surface of the water CBE, it will turn away from B towards V; and if it is a ray, it will go quite the contrary from B towards I. What you will however cease to find strange, if you remember the nature which I attributed to light, when I said that it was nothing else than a certain movement or an action received in a very subtle matter, which fills the pores of other bodies; and that you consider that, as a ball loses more of its agitation by hitting a soft body than against a hard one, and that it rolls less easily on a carpet than on a bare table, so the action of this subtle matter can be hindered much more by the parts of the air, which being soft and badly joined together, do not give it much resistance, than by those of the water, which give it more; and even more by those of water than by those of glass or crystal: so that as much as the small parts of a transparent body are harder and firmer, so much the more do they allow light to pass through. easily, for this light must not drive any out of their places, just as a bullet must drive out those of water to find passage among them.

Moreover, knowing thus the cause of the refractions which take place in water and in glass, and commonly in all the other transparent bodies which are around us, we can remark that they must all be similar when the rays come out. of these bodies and when they enter them: as if the ray which comes from A towards B is diverted from B towards I while passing from the air in the glass, that which will return from I towards B must also be diverted from B towards A However, other bodies may well be found, chiefly in the sky, where the refractions, proceeding from other causes, are not so reciprocal. And there may also be certain cases in which the rays must bend even though they only pass through a single transparent body; as the motion of a ball often bends, because it is deflected to one side by its weight, and to another by the action by which it has been pushed, or for various other reasons; for, finally, I dare to say that the three comparisons which I have just used are so specific that all the particularities which can be observed in them relate to a few others which are all similar in light; but I only tried to explain those which were most relevant to me, and I do not want you to consider anything else here, except that the surfaces of transparent bodies which are curved divert the rays which pass through each of their points, in the same way as would make the flat surfaces which one can imagine touching these bodies at the same points: as, for example, the refraction of the rays AB[12], AC, AD, which, coming from the torch A, fall on the curved surface of the crystal ball BCD, must be considered in the same way as if AB fell on the flat surface EBF, and AC on GCH, and AD on IDK, and so on the others; whence you see that these rays can be assembled or separated differently, according as they fall on surfaces which are curved differently.

And it is time that I begin to describe to you what is the structure of the eye, in order to be able to make you understand how the rays which enter it arrange themselves there to cause the feeling of sight.