Mirrors and Telescopes

What mathematics gives for the mysteries of vision, and why those reasons are not sufficient.

The rays that some power — unknown until our own day — makes rebound to your eyes from the surface of a mirror, without touching that surface, and from the pores of that mirror without touching the solid parts — these rays, I say, return to your eyes in the same direction in which they arrived at that mirror.

If it’s your own face you are looking at, the rays leaving your face, parallel and perpendicular to the mirror, return just like a ball that bounces perpendicularly off the floor.

If you look in this mirror M (figure 11) at an object next to you, like A, the rays leaving that object behave exactly like a ball that would bounce off point B, where your eye is. This is what we call the angle of incidence being equal to the angle of reflection.

The line A C is the line of incidence, and the line C B is the line of reflection. It is well known — and the simple statement demonstrates it — that these lines form equal angles on the surface of the mirror.

But why do I not see the object in A, where it really is, nor at C, from which the rays actually come to my eyes, but rather at D, behind the mirror itself?

Geometry will tell you (figure 12): It’s because the angle of incidence equals the angle of reflection; because your eye at B traces the object to D; because objects can only act on you in a straight line, and because the straight line continuing from your eye B to behind the mirror at D is as long as the lines A C and C B taken together.

It will also tell you: You only ever see objects from the point where the rays begin to diverge.

Take this mirror MI.

The bundles of rays leaving each point of the object A begin to diverge the instant they leave the object; they strike the surface of the mirror. There each ray hits, deflects, and reflects toward the eye. The eye traces them back to the points D D, at the end of the straight lines, where those same rays would have met. But, in meeting at D D, those rays would do the same thing they did at A A: they would begin to diverge again. Therefore you see the object A A at the points D D.

These angles and lines certainly help you understand the mechanism nature uses; but they are far from telling you the real physical reason why your soul instinctively places the object beyond the mirror at the same distance it is on this side. These lines show you what happens — but they do not tell you why it happens.

If you want to know how a convex mirror makes objects appear smaller, and how a concave mirror makes them appear larger, these same lines of incidence and reflection will give you the explanation.

You’re told: This cone of rays diverging from points A (figure 13), falling on this convex mirror, makes angles of incidence equal to angles of reflection, whose lines go to our eye. But these angles are smaller than they would be on a flat surface; therefore, if they are imagined to pass on to B, they will converge much sooner; therefore the object that would be in B B will be smaller.

Your eye traces the object in B B back to the points where the rays would begin to diverge: therefore the object must appear smaller — as it indeed does in this figure. And for the same reason that it appears smaller, it also appears closer, since the points where the rays B B would end are nearer the mirror than the rays A A.

By the opposite reasoning, you must see objects larger and farther away in a concave mirror, if the object is placed close enough to the mirror (figure 14).

The cones of rays A A, coming to diverge at the points where they hit the mirror — if they were to reflect through the mirror, they would only meet at B B: therefore it’s at B B that you see them. But B B is larger and farther from the mirror than A A: therefore you see the object bigger and farther away.

This, in general, is what happens with the rays reflected into your eyes; and this single principle — that the angle of incidence is always equal to the angle of reflection — is the foundation of all the mysteries of catoptrics (the science of mirrors).

Now we come to how lenses enlarge sizes and bring distances closer; and finally, why objects are painted upside down in your eyes, yet you still see them upright.

Regarding sizes and distances, here is what mathematics will tell us:

The larger an angle an object makes in your eye, the larger the object will appear: nothing could be simpler.

This line H K, which you see from a hundred paces, makes an angle in eye A (figure 15); from two hundred paces, it makes an angle half as big in eye B (figure 16).

Now, the angle formed on your retina, of which your retina is the base, is equal to the angle whose base is the object itself. These are vertically opposite angles: therefore, by the first elements of geometry, they are equal.

So if the angle formed in eye A is double the angle formed in eye B, the object must appear twice as large to eye A as it does to eye B.

Now, for the eye at B to see the object as large as the eye at A does, we must somehow make this eye B receive an angle as large as that of the eye A, which is twice as close. Telescope lenses accomplish this effect (figure 17).

Let us put only one lens here for simplicity, ignoring the other effects of multiple lenses.

The object H K sends its rays to this lens. They meet at some distance from the glass. Imagine a lens cut so that these rays cross to form in the eye at C an angle as large as that in the eye at A: then, we’re told, the eye “judges” by this angle.

It therefore sees the object the same size as the eye at A does. But at A, the object is seen from a hundred paces away: therefore at C, receiving the same angle, the eye will again see it as if it were a hundred paces away.

The entire effect of multiple spectacle lenses, of various telescopes, and of microscopes that enlarge objects, therefore consists solely in making things appear under a larger angle.

The object A B (figure 18) is seen, by means of this lens, under the angle D C D, which is much larger than the angle A C B.

You still ask optics why you see objects in their true position, even though they are painted upside down on our retina.

The ray leaving the man’s head A (figure 19) strikes the lower point A of your retina; his feet B are seen by the rays B B at the upper point of your retina B.

Thus this man is literally painted head down and feet up at the back of your eyes.

Why, then, do you not see him upside down — but upright, as he really is?

To answer this question, people use the comparison of a blind man holding crossed canes, with which he very accurately determines the positions of objects.

For the point to his left, being touched by his right hand via the cane, he immediately judges it to be on the left; and the point his left hand senses via the other cane, he judges to be on the right, without error.

All the masters of optics therefore tell us that the lower part of the eye instantly reports its sensation to the upper part of the object, and that the upper part of the retina just as naturally reports its sensation to the lower part; thus we see the object in its true orientation.

But even if you perfectly understood all these angles and all these mathematical lines, by which one traces the path of light to the back of the eye, do not think you therefore know how you perceive the sizes, distances, and positions of things.

The geometric proportions of these angles and lines are correct, yes — but there is no more connection between them and our sensations than there is between the sound we hear and the size, distance, or position of the thing we hear.

By sound, my ear is struck: I hear tones, and nothing more. By sight, my eye is struck: I see colors, and nothing more.

Not only can the proportions of these angles and lines not possibly be the immediate cause of the judgment I make about objects, but in many cases, these proportions do not match at all the way we actually see objects.

For example: A man seen at four paces, and at eight paces, appears the same size.

Yet the image of this man, at four paces, is — to within very little — double the size in your eye of the one he traces at eight paces. The angles are different, but you see the object always equally large; therefore it is obvious from this single example, chosen from many, that these angles and lines are not at all the immediate cause of how we see.

Therefore, before continuing the research we have begun on light and the mechanical laws of nature, you command me to explain here how the ideas of distance, size, and position of objects are received in our soul.

This investigation will provide something new and true — the only excuse for writing a book.