ones are covered by the nearest. Moving further along, this changes. At II we shall not have to look round so far to see the remotest tree 3 as to see the nearer tree 2, nor so far to see this as to see 1. Hence, as we move onward, objects that are near to us seem to lag behind as compared with objects that are remote from us, the lagging increasing with the proximity of the objects. Very remote objects, towards which we must always look in the same direction as we proceed, appear to travel along with us.
If we should see, therefore, jutting above the brow of yonder hill the tops of two trees whose distance from us we were in doubt about, we should have in our hands a very easy means of deciding the question. We should take a few steps forward, say to the right, and the tree-top which receded most to the left would be the one nearer to us. In truth, from the amount of the recession a geometer could actually determine the distance of the trees from us without ever going near them. It is simply the scientific development of this perception that enables us to measure the distances of the stars.
Hence, from change of view in forward motion the distances of objects in our field of vision can be measured.
Rigorously, however, even forward motion is not necessary. For every observer is composed really of two observers. Man has two eyes. The right eye is a short step ahead of the left eye in the right-hand direction. Hence, the two eyes receive different pictures of the same woods. The right eye will see the near trees displaced to the left, and the left eye will see them displaced to the right, the displacement being greater, the greater the proximity. This difference is sufficient for forming ideas of distance.
We may now readily convince ourselves of the following facts:
1. With one eye, the other being shut, you have a very uncertain judgment of distances. You will find it, for example, no easy task, with one eye shut, to thrust a stick through a ring hung up before you; you will miss the ring in almost every instance.
2. You see the same object differently with the right eye from what you do with the left.
Place a lamp-shade on the table in front of you with its broad opening turned downwards, and look at it from above. (Fig. 21.) You will see with your right eye the image 2, with your left eye the image 1. Again, place the shade with its wide opening turned upwards; you will receive with your right eye the image 4, with your left eye the image 3. Euclid mentions phenomena of this character.
3. Finally, you know that it is easy to judge of distances with both eyes. Accordingly your judgment must spring in some way from a co-operation of the two eyes. In the preceding example the openings in the different images received by the two eyes seem displaced with respect to one another, and this displacement is sufficient for the inference that the one opening is nearer than the other.
I have no doubt that you, ladies, have frequently received delicate compliments upon your eyes, but I feel sure that no one has ever told you, and I know not whether it will flatter you, that you have in your eyes, be they blue or black, little geometricians. You say you know nothing of them? Well, for that matter, neither do I. But the facts are as I tell you.
You understand little of geometry? I shall accept that confession. Yet with the help of your two eyes you judge of distances? Surely that is a geometrical problem. And what is more, you know the solution of this problem: for you estimate distances correctly. If, then, you do not solve the problem, the little geometricians in your eyes must do it clandestinely and whisper the solution to you. I doubt not they are fleet little fellows.
What amazes me most here is, that you know nothing about these little geometricians. But perhaps they also know nothing about you. Perhaps they are models of punctuality, routine clerks who bother about nothing but their fixed work. In that case we may be able to deceive the gentlemen.
If we present to our right eye an image which looks exactly like the lamp-shade for the right eye, and to our left eye an image which looks exactly like a lamp-shade for the left eye, we shall imagine that we see the whole lamp-shade bodily before us.
You know the experiment. If you are practised in squinting, you can perform it directly with the figure, looking with your right eye at the right image, and with your left eye at the left image. In this way the experiment was first performed by Elliott. Improved and perfected, its form is Wheatstone's stereoscope, made so popular and useful by Brewster.
By taking two photographs of the same object from two different points, corresponding to the two eyes, a very clear three-dimensional picture of distant places or buildings can be produced by the stereoscope.
But the stereoscope accomplishes still more than this. It can visualise things for us which we never see with equal clearness in real objects. You know that if you move much while your photograph is being taken, your picture will come out like that of a Hindu deity, with several heads or several arms, which, at the spaces where they overlap, show forth with equal distinctness, so that we seem to see the one picture through the other. If a person moves quickly away from the camera before the impression is completed, the objects behind him will also be imprinted upon the photograph; the person will look transparent. Photographic ghosts are made in this way.
Some very useful applications may be made of this discovery. For example, if we photograph a machine stereoscopically, successively removing during the operation the single parts (where of course the impression suffers interruptions), we obtain a transparent view, endowed with all the marks of spatial solidity, in which is distinctly visualised the interaction of parts normally concealed. I have employed this method for obtaining transparent stereoscopic views of anatomical structures.
You see, photography is making stupendous advances, and there is great danger that in time some malicious artist will photograph his innocent patrons with solid views of their most secret thoughts and emotions. How tranquil politics will then be! What rich harvests our detective force will reap!
By the joint action of the two eyes, therefore, we arrive at our judgments of distances, as also of the forms of bodies.
Permit me to mention here a few additional facts connected with this subject, which will assist us in the comprehension of certain phenomena in the history of civilisation.
You have often heard, and know from personal experience, that remote objects appear perspectively dwarfed. In fact, it is easy to satisfy yourself that you can cover the image of a man a few feet away from you simply by holding up your finger a short distance in front of your eye. Still, as a general rule, you do not notice this shrinkage of objects. On the contrary, you imagine you see a man at the end of a large hall, as large as you see him near by you. For your eye, in its measurement of the distances, makes remote objects correspondingly larger. The eye, so to speak, is aware of this perspective contraction and is not deceived by it, although its possessor is unconscious of the fact. All persons who have attempted to draw from nature have vividly felt the difficulty which this superior dexterity of the eye causes the perspective conception. Not until one's judgment of distances is made uncertain, by their size, or from lack of points of reference, or from being too quickly changed, is the perspective rendered very prominent.
On sweeping round a curve on a rapidly moving railway train, where a wide prospect is suddenly opened up, the men upon distant hills appear like dolls.[17] You have at the moment, here, no known references for the measurement of distances. The stones at the entrance of a tunnel grow visibly larger as we ride towards it; they shrink visibly in size as we ride from it.
Usually
