Proof of the errors of our sight concerning motion

General proof of the errors of our sight concerning motion

Let:

• A be the spectator’s eye
• C the object which is far from A

Although the object remains immobile at C, one may believe it to be moving away to D or approaching to B. That although the object is moving away toward D, one may believe it to be immobile at C and even approaching toward B; and conversely, although it is approaching toward B, one may believe it to be immobile at C and even moving away toward D. That although the object has advanced from C to E or H, or to G or K, one may believe that it has moved only from C to F or I; and conversely, that although the object has moved from C to F or I, one may believe that it has moved to E, or H, or even to G or K. That if the object moves along a line equally distant from the spectator, that is to say along a circumference of which the spectator is the center: even though this object moves from C to P, one may believe that it moves only from B to O; and conversely, although it moves only from B to O, one may believe it to be moving from C to P. If beyond object C there is another object M, which one believes to be immobile, and which however moves toward N; although object C remains immobile, or moves much more slowly toward F, than M toward N, it will appear to move toward Y, and conversely, if, etc.

it is necessary to know the distance of objects in order to judge the magnitude of their motion. — III. Examination of the means for recognizing distances.**

It is evident that the proof of all these propositions, except the last, where there is no difficulty, depends only on one thing, which is, that we cannot ordinarily judge with assurance of the distance of objects. For if it is true that we cannot judge it with certainty, it follows that we cannot know whether C has advanced toward D, or whether it has approached toward B, and so on for the other propositions.

Now to see whether the judgments that we form about the distance of objects are assured, we have only to examine the means we use to judge them; and if these means are uncertain, it cannot be that the judgments are infallible. There are several of them, and they must be explained.

III. The first, most universal, and sometimes most sure means that we have for judging the distance of objects is the angle made by the rays of our eyes, of which the object is the vertex, that is to say, of which the object is the point where these rays meet. When this angle is very large, we see the object very close, and conversely when it is very small, we see it very far away. And the change that occurs in the situation of our eyes according to the changes of this angle is the means by which our soul judges the distance or proximity of objects. For just as a blind man who had two straight sticks in his hands, of which he did not know the length, could, by a kind of natural geometry, judge approximately the distance of some body by touching it with the ends of these two sticks, because of the disposition and distance of his hands; so one can say that the soul judges the distance of an object by the disposition of its eyes, which is not the same when the angle by which it sees it is large as when it is small, that is to say, when the object is close as when it is distant.

One will easily persuade oneself of what I say, if one takes the trouble to make this very easy experiment. Let one suspend at the end of a thread a ring whose opening does not face us, or let one drive a stick into the ground, and take another in one’s hand that is curved at the end; let one withdraw three or four paces from the ring or the stick; let one close one eye with one hand and with the other try to thread the ring, or to touch sideways and at about the height of one’s eyes the stick with the one held in one’s hand; and one will be surprised at not being able perhaps in a hundred tries to do what one thought very easy. If one even puts down the stick and still wants to thread the ring sideways with one of one’s fingers, one will find some difficulty, although one is quite close to it.

But it must be carefully noted that I said one should try to thread the ring or touch the stick sideways, and not in a straight line from our eye to the ring; for then there would be no difficulty, and it would even be easier to succeed with one eye closed than with both eyes open, because that would guide us.

Now one can say that the difficulty one finds in threading a ring sideways with only one eye open comes from the fact that the other being closed, the angle of which I have just spoken is not known. For it is not enough to know the magnitude of an angle to know the length of the base and that of an angle made by one of its sides on this base; which is known by the preceding experiment. But it is still necessary to know the other angle made by the other side on the base, or the length of one of the sides, which cannot be known exactly except by opening the other eye. Thus the soul cannot use its natural geometry to judge the distance of the ring.

The disposition of the eyes that accompanies the angle formed by the visual rays that cross and meet in the object is therefore one of the best and most universal means that the soul uses to judge the distance of things. If therefore this angle does not change sensibly, when the object is a little distant, whether it approaches or recedes from us, it will follow that this means will be false, and that the soul will not be able to use it to judge the distance of this object.

Now it is very easy to recognize that this angle changes notably when an object that is one foot from our sight is moved to four: but if it is only moved from four to eight, the change is much less sensible; if from eight to twelve, still less; if from a thousand to a hundred thousand, almost none; finally this change will no longer be sensible, even if one were to carry it into imaginary spaces. So that if there is a sufficiently considerable space between A and C, the soul will not be able by this means to know whether the object is close to B or to D.

It is for this reason that we see the sun and the moon as if they were enveloped in the clouds, although they are extraordinarily far from them; that we naturally believe that all the stars are at an equal distance, and that comets are stable and almost without any motion at the end of their course. We even imagine that comets dissipate entirely after a few months, because they move away from us along a line almost straight or direct to our eyes, and so go to lose themselves in those great spaces, from which they return only after several years, or even after several centuries; for it is very likely that they do not dissipate as soon as we cease to see them.

To explain the second means by which the soul judges the distance of objects, one must know that it is absolutely necessary that the shape of the eye be different, according to the different distance of the objects we see; for when a man sees an object close to himself, it is necessary that his eyes be longer than if the object were farther away: because in order that the rays of this object may be gathered together on the optic nerve, which is necessary in order to see it distinctly, principally because the object is poorly lit, the distance between this nerve and the crystalline lens must be greater.

It is true that if the crystalline lens became more convex when the object is close, that would produce the same effect as if the eye lengthened; but it is not believable that the crystalline lens can easily change its convexity; and on the other hand, we have a fairly probable proof that the eye lengthens; for anatomy teaches that there are muscles that surround the eye in the middle, and one feels the effort of these muscles that press it, and apparently lengthen it, when one wants to see something very close.

But it is not necessary to know here in what manner this happens; it suffices that a change occurs in the eye, either because the muscles that surround it press it, or because the small nerves that respond to the ciliary ligaments, which hold the crystalline lens suspended among the other humors of the eye, slacken to increase the convexity of the crystalline lens, or stiffen to diminish it, or finally because the pupil dilates or contracts, for there are many people whose eyes receive no other change.

In short, the change that occurs, whatever it may be, is only to make the rays of objects gather exactly on the optic nerve. Now it is certain that when the object is five hundred paces or ten thousand leagues away, one looks at it with the same disposition of the eyes, without any sensible change in the muscles that surround the eye, nor in the nerves that respond to the ciliary ligaments of the crystalline lens, nor finally in the opening of the pupil, and the rays of objects gather very exactly on the retina or optic nerve. Thus the soul would judge that objects ten thousand or a hundred thousand leagues away are only at five or six hundred paces, if it judged their distance only by the disposition of the eyes of which I have just spoken.

However, it is certain that this means could serve the soul when the object is close. If, for example, an object is only half a foot from us, we can distinguish its distance fairly well by the disposition of the muscles that press our eyes, in order to make them a little longer, and even this disposition is painful. If this object is at two feet, we still distinguish it, because the disposition of the muscles is somewhat sensible, although it is no longer painful. But if one moves the object away by a few more feet, this disposition of our muscles becomes so little sensible that it is entirely useless to us for judging the distance of the object.

These then are two means that the soul can use to judge the distance of the object, which are very useless when this object is five to six hundred paces away, and which are not even assured although the object is closer.

The third means consists in the size of the image that is painted at the back of the eye and that represents the objects we see. It is admitted that this image diminishes in proportion as the object moves away, but this diminution is all the less sensible the more distant the object that changes distance is. For when an object is already at a reasonable distance, such as five to six hundred paces, more or less in proportion to its size, very considerable changes occur in its distance without any sensible change occurring in the image that represents it, as is easy to demonstrate. Thus this third means has the same defect as the other two of which we have just spoken.

There is moreover to be noted that the soul does not judge those objects to be the most distant whose image, painted on the retina, is smaller. When I see, for example, a man and a tree at a hundred paces, or several stars in the sky, I do not judge that the man is farther away than the tree, and the small stars farther away than the larger ones, although the images of the man and the small stars that are painted on the retina are smaller than those of the tree and the large stars; one must still know by sensory experience the size of the object in order to judge approximately its distance; and because I know or have seen several times that a house is larger than a man, although the image of a house is larger than that of a man, I nevertheless do not judge it or see it as closer.

It is the same with the stars. Our eyes represent them all at the same distance, although it is very reasonable to believe some of them much farther from us than others. Thus there is an infinity of objects whose distance we cannot know at all, since there is an infinity whose size we do not know.

We also judge the distance of the object by the force with which it acts on our eyes, because a distant object acts much more weakly than another; and by the distinctness and clarity of the image that forms in the eye; because when the object is distant, the opening of the eye must open wider and consequently the rays gather somewhat confusedly. This is why objects that are poorly lit, or that we see confusedly, appear to us farther away than they are, and, conversely, that luminous bodies, which we see distinctly, appear closer to us. It is quite clear that these latter means are not assured for judging with any certainty the distance of objects; and we do not wish to dwell on them in order to come finally to the last of all, which is the one that most helps the imagination and more easily leads the soul to judge that objects are very far away.

The sixth and principal means consists in the fact that the eye does not report to the soul a single object separated from others, but that it also shows it all those that are found between us and the principal object we are considering. When, for example, we look at a bell tower quite far away, we ordinarily see at the same time several stretches of land and several houses between us and it; and because we judge the distance of these lands and these houses, and yet see that the bell tower is beyond them, we also judge that it is much farther away and even larger and greater than if we saw it alone. However, the image traced at the back of the eye is always of equal size, whether there are lands and houses between us and it, or whether there are none, provided we see it from an equally distant place, as is supposed. Thus we judge the size of objects by the distance at which we believe them to be; and the bodies we see between us and the objects greatly aid our imagination in judging their distance, just as we judge the length of our duration or the time that has passed since we performed some action by the confused memory of the things we have done or the thoughts we have had successively since that action. For it is all these thoughts and all these actions that have succeeded one another that help our mind to judge the length of some time or some part of our duration; or rather, the confused memory of all these successive thoughts is the same thing as the judgment of our duration, just as the confused sight of the lands that are between us and a bell tower is the same thing as the natural judgment of the distance of the bell tower, for these judgments are only composite sensations.

From this it is easy to recognize the true reason why the moon appears larger to us when it rises than when it is very high on the horizon; for when it rises it appears to us to be several leagues away and even beyond the sensible horizon or the lands that terminate our view, whereas we judge it to be only about half a league from us or seven or eight times higher than our houses when it has risen above our horizon. Thus we judge it much larger when it is near the horizon than when it is far from it, because we judge it much farther from us when it rises than when it is very high on our horizon.

It is true that a very great number of philosophers attribute what we have just said to the vapors that rise from the earth. They claim that the vapors, breaking the rays of objects, make them appear larger. But it is certain that they are mistaken, for refractions only increase their elevation above the horizon and they diminish on the contrary somewhat the visual angle under which they are seen. They do not prevent the image traced at the back of our eyes, when we see the moon rising, from being smaller than that which forms there when it has long been risen.

Astronomers who measure the diameter of planets note that that of the moon increases in proportion as it moves away from the horizon, and consequently in proportion as it appears smaller to us; thus the diameter of the image we have of it at the back of our eyes is smaller when we see it larger. In effect, when the moon rises, it is farther from us by the diameter of the earth than when it is perpendicularly above our head; and that is the reason why its diameter increases when it rises on the horizon, because then it approaches us.

What makes us see it larger when it rises is therefore not the refraction its rays undergo in the vapors that come from the earth, since the image formed by these rays is then smaller; but it is the natural judgment that forms in us of its distance, because it appears to us beyond the lands that we see very far from us, as explained earlier; and one is astonished that philosophers hold that the reason for this appearance and this deception of our senses is more difficult to find than the greatest algebraic equations.

This means that we have for judging the distance of some object by the knowledge of the distance of things that are between us and it is often quite useful to us when the other means of which I have spoken can be of no use to us; for we can judge by this last means that certain objects are several leagues away from us, which we cannot do by the others. However, if one examines it, one will find several defects.

For first, this means serves us only for objects that are on the earth, since one can make use of it only very rarely and even very uselessly for those that are in the air or in the heavens. Secondly, on the earth one can use it only for things a few leagues distant. Thirdly, one must be assured that there are found between us and the object neither valleys, nor mountains, nor anything else similar that prevents us from using this means. Finally, I believe that there is no one who has not made enough experiments on this subject to be persuaded that it is extremely difficult to judge with any certainty the distance of objects by the visible sight of the things that are found between them and us, and one has perhaps dwelt on this too long.

These are all the means we have for judging the distance of objects; we have pointed out considerable defects in them, and we must conclude that judgments based on such uncertain means must also be very uncertain. From this it is easy to show the truth of the propositions I have put forward. Object C has been supposed sufficiently far from A, and it may in many cases advance toward D or approach toward B without one recognizing it, since one has no assured means for judging its distance. It may even retreat toward D when one will believe it to be approaching toward B, because the image of the object sometimes increases and enlarges on the optic nerve, either because the transparent matter between the object and the eye can produce a greater refraction at one time than at another, or because small tremblings of this nerve sometimes occur, or finally because the impression made by the union of rays not exactly on this same nerve spreads and communicates to parts that should not have been agitated, which can come from several different causes. Thus the image of the same objects being found larger on these occasions, it gives the soul reason to believe that the object is approaching. The same must be said of the other propositions.

Before finishing this chapter, it must be noted that it greatly concerns us, for the preservation of our life, to know better the motion or rest of bodies in proportion as they are closer to us, and that it is quite useless to us to know with exactitude the truth of these things when they occur in places very far away. For this shows evidently that what I have advanced generally of all the senses, that they make us know things only with relation to the preservation of our body and not according to what they are in themselves, is found exactly true in this encounter, since we know the motion or rest of objects better in proportion as they approach us, and that we cannot judge by the senses when they are so far away that they seem to have no longer or almost no longer any relation to our bodies; as when they are five or six hundred paces from us, if they are of medium size; or even closer than that, if they are smaller, or finally farther away, if they are larger.