This fourth operation of the Understanding consists in acquiring knowledge of the distance of objects from us: it is this precisely which constitutes that third dimension of which we have been speaking. Visual sensation, as we have said, gives us the direction in which objects lie, but not their distance from us: that is, not their position. It is for the Understanding therefore to find out this distance; or, in other words, the distance must be inferred from purely causal determinations. Now the most important of these is the visual angle, which objects subtend; yet even this is quite ambiguous and unable to decide anything by itself. It is like a word of double meaning: the sense, in which it is to be understood, can only be gathered from its connection with the rest. An object subtending the same visual angle may in fact be small and near, or large and far off; and it is only when we have previously ascertained its size, that the visual angle enables us to recognise its distance: and conversely, its size, when its distance is known to us. Linear perspective is based upon the fact that the visual angle diminishes as the distance increases, and its principles may here be easily deduced. As our sight ranges equally in all directions, we see everything in reality as from the interior of a hollow sphere, of which our eye occupies the centre. Now in the first place, an infinite number of intersecting circles pass through the centre of this sphere in all directions, and the angles measured by the divisions of these circles are the possible angles of vision. In the second place, the sphere itself modifies its size according to the length of radius we give to it; therefore we may also imagine it as consisting of an infinity of concentric, transparent spheres. As all radii diverge, these concentric spheres augment in size in proportion to their distance from us, and the degrees of their sectional circles increase correspondingly: therefore the true size of the objects which occupy them likewise increases. Thus objects are larger or smaller according to the size of the spheres of which they occupy similar portions—say 10°—while their visual angle remains unchanged in both cases, leaving it therefore undecided, whether the 10° occupied by a given object belong to a sphere of 2 miles, or of 10 feet diameter. Conversely, if the size of the object has been ascertained, the number of degrees occupied by it will diminish in proportion to the distance and the size of the sphere to which we refer it, and all its outlines will contract in similar proportion. From this ensues the fundamental law of all perspective; for, as objects and the intervals between them must necessarily diminish in constant proportion to their distance from us, all their outlines thereby contracting, the result will be, that with increasing distance, what is above us will descend, what is below us will ascend, and all that lies at our sides will come nearer together. This progressive convergence, this linear perspective, no doubt enables us to estimate distances, so far as we have before us an uninterrupted succession of visibly connected objects; but we are not able to do this by means of the visual angle alone, for here the help of another datum is required by the Understanding, to act, in a sense, as commentary to the visual angle, by indicating more precisely the share we are to attribute to distance in that angle. Now there are four principal data of this kind, which I am about to specify. Thanks to these data, even where there is no linear perspective to guide us, if a man standing at a distance of 200 feet appears to me subtending a visual angle twenty-four times smaller than if he were only 2 feet off, I can nevertheless in most cases estimate his size correctly. All this proves once more that perception is not only a thing of the senses, but of the intellect also.—I will here add the following special and interesting fact in corroboration of what I have said about the basis of linear perspective as well as about the intellectual nature of all perception. When I have looked steadily at a coloured object with sharply defined outlines—say a red cross—long enough for the physiological image to form in my eye as a green cross, the further the surface on to which I project it, the larger it will appear to me: and vice versa. For the image itself occupies an unvarying portion of my retina, i.e. the portion originally affected by the red cross; therefore when referred outwards, or, in other words, recognised as the effect of an external object, it forms an unchanging visual angle, say of 2°. Now if, in this case, where all commentary to the visual angle is wanting, I remove it to a distant surface, with which I necessarily identify it as belonging to its effect, the cross will occupy 2° of a distant and therefore larger sphere, and is consequently large. If, on the other hand, I project the image on to a nearer object, it will occupy 2° of a smaller sphere, and is therefore small. The resulting perception is in both cases completely objective, quite like that of an external object; and as it proceeds from an entirely subjective reason (from the image having been excited in quite a different way), it thus confirms the intellectual character of all objective perception.—This phenomenon (which I distinctly remember to have been the first to notice, in 1815) forms the theme of an essay by Séguin, published in the "Comptes rendus" of the 2nd August, 1858, where it is served up as a new discovery, all sorts of absurd and distorted explanations of it being given. Messieurs les illustres confrères let pass no opportunity for heaping experiment upon experiment, the more complicated the better. Expérience! is their watchword; yet how rarely do we meet with any sound, genuine reflection upon the phenomena observed! Expérience! expérience! followed by twaddle.
To return to the subsidiary data which act as commentaries to a given visual angle, we find foremost among them the mutationes oculi internæ, by means of which the eye adapts its refractory apparatus to various distances by increasing and diminishing the refraction. In what these modifications consist, has not yet been clearly ascertained. They have been sought in the increased convexity, now of the cornea, now of the crystalline lens; but the latest theory seems to me the most probable one, according to which the lens is moved backwards for distant vision and forwards for near vision, lateral pressure, in the latter case, giving it increased protuberance; so that the process would exactly resemble the mechanism of an opera-glass. Kepler, however, had, in the main, already expressed this theory, which may be found explained in A. Hueck's pamphlet, "Die Bewegung der Krystallinse," 1841. If we are not clearly conscious of these inner modifications of the eye, we have at any rate a certain feeling of them, and of this we immediately avail ourselves to estimate distances. As however these modifications are not available for the purposes of clear sight beyond the range of from about 7 inches to 16 feet, the Understanding is only able to apply this datum within those limits.
Beyond them, however, the second datum becomes available: that is to say, the optic angle, formed by the two optic axes, which we had occasion to explain when speaking of single vision. It is obvious that this optic angle becomes smaller, the further the object is removed: and vice versa. This different direction of the eyes, with respect to each other, does not take place without producing a slight sensation, of which we are nevertheless only in so far conscious as the Understanding makes use of it, as a datum, in estimating distances intuitively. By this datum we are not only enabled to cognize the distance, but the precise position of the object viewed, by means of the parallax of the eyes, which consists in each eye seeing the object in a slightly different direction; so that if we close one eye, the object seems to move. Thus it is not easy to snuff a candle with one eye shut, because this datum is then wanting. But as the direction of the eyes becomes parallel as soon as the distance of the object reaches or exceeds 200 feet, and as the optic angle consequently then ceases to exist, this datum only holds good within the said distance.
Beyond it, the Understanding has recourse to atmospheric perspective, which indicates a greater distance by means of the increasing dimness of all colours, of the appearance of physical blue in front of all dark objects (according to Göthe's perfectly correct and true theory of colours), and also of the growing indistinctness of all outlines. In Italy, where the atmosphere is very transparent, this datum loses its power and is apt to mislead: Tivoli, for instance, seems to be very near when seen from Frascati. On the other hand, all objects appear larger in a mist, which is an abnormal exaggeration of the datum; because our Understanding assumes them to be further from us.
Finally, there remains the estimation of distance by means of the size (known to us intuitively) of intervening objects, such as fields, woods, rivers, &c. &c. This mode of estimation is only applicable where there is uninterrupted succession: in other words, it can only be applied to terrestrial, not to celestial objects. Moreover, we have in general more practice in using it horizontally than vertically: a ball on the top of a tower 200 feet high appears much smaller to us than when lying on the ground 200 feet from us; because, in the latter case, we estimate the distance more accurately. When we see human beings in such a way, that what lies between them and ourselves is in a great measure hidden from our sight, they always appear strikingly small.
The fact that our Understanding assumes everything it perceives in a horizontal direction to be farther off, therefore larger, than what is seen in a vertical direction, must partly be attributed to this last mode of estimating distances, inasmuch as it only holds good when applied horizontally and to terrestrial objects; but partly also to our estimation of distances by atmospheric perspective, which is subject to similar conditions. This is why the moon seems so much larger on the horizon than at its zenith, although its visual angle accurately measured—that is, the image projected by it on to the eye—is not at all larger in one case than in the other; and this also accounts for the flattened appearance of the vault of the sky: that is to say, for its appearing to have greater horizontal than vertical extension. Both phenomena therefore are purely intellectual or cerebral, not optical. If it be objected, that even when at its zenith, the moon occasionally has a hazy appearance without seeming to be larger, we answer, that neither does it in that case appear red; for its haziness proceeds from a greater density of vapours, and is therefore of a different kind from that which proceeds from atmospheric perspective. To this may be added what I have already said: that we only apply this mode of estimating distances in a horizontal, not in a perpendicular, direction; besides, in this case, other correctives come into play. It is related of Saussure that, when on the Mont Blanc, he saw so enormous a moon rise, that, not recognising what it was, he fainted with terror.
The properties of the telescope and magnifying glass, on the other hand, depend upon a separate estimate according to the visual angle alone: i.e., that of size by distance, and of distance by size; because here the four other supplementary means of estimating distances are excluded. The telescope in reality magnifies objects, while it only seems to bring them nearer; because their size being known to us empirically, we here account for its apparent increase by a diminution of their distance from us. A house seen through a telescope, for instance, seems to be ten times nearer, not ten times larger, than seen with the naked eye. The magnifying glass, on the contrary, does not really magnify, but merely enables us to bring the object nearer to our eyes than would otherwise be possible; so that it only appears as large as it would at that distance even without the magnifying glass. In fact, we are prevented from seeing objects distinctly at less than from eight to ten inches' distance from our eyes, by the insufficient convexity of the ocular lens and cornea; but if we increase the refraction by substituting the convexity of the magnifying glass for that of the lens and cornea, we then obtain a clear image of objects even when they are as near as half an inch from our eyes. Objects thus seen in close proximity to us and in the size corresponding to that proximity, are transferred by our Understanding to the distance at which we naturally see distinctly, i.e. to about eight or ten inches from our eyes, and we then estimate their magnitude according to this distance and to the given visual angle.
I have entered thus fully into detail concerning all the different processes by which seeing is accomplished, in order to show clearly and irrefragably that the predominant factor in them is the Understanding, which, by conceiving each change as an effect and referring that effect to its cause, produces the cerebral phenomenon of the objective world on the basis of the à priori fundamental intuitions of Space and Time, for which it receives merely a few data from the senses. And moreover the Understanding effects this exclusively by means of its own peculiar form, the law of Causality; therefore quite directly and intuitively, without any assistance whatever from reflection—that is, from abstract knowledge by means of conceptions and of language, which are the materials of secondary knowledge, i.e. of thought, therefore of Reason.
That this knowledge through the Understanding is independent of Reason's assistance, is shown even by the fact, that when, at any time, the Understanding attributes a given effect to a wrong cause, actually perceiving that cause, whereby illusion arises, our Reason, however clearly it may recognise in abstracto the true state of the matter, is nevertheless unable to assist the Understanding, and the illusion persists undisturbed in spite of that better knowledge. The above-mentioned phenomena of seeing and feeling double, which result from an abnormal position of the organs of touch and sight, are instances of such illusions; likewise the apparently increased size of the rising moon; the image which forms in the focus of a concave mirror and exactly resembles a solid body floating in space; the painted relievo which we take for real; the apparent motion of a shore or bridge on which we are standing, if a ship happens to pass along or beneath it; the seeming proximity of very lofty mountains, owing to the absence of atmospheric perspective, which is the result of the purity of the air round their summits. In these and in a multitude of similar cases, our Understanding takes for granted the existence of the usual cause with which it is conversant and forthwith perceives it, though our Reason has arrived at the truth by a different road; for, the knowledge of the Understanding being anterior to that of the Reason, the intellect remains inaccessible to the teaching of the Reason, and thus the illusion—that is, the deception of the Understanding—remains immovable; albeit error—that is, the deception of the Reason—is obviated.—That which is correctly known by the Understanding is reality: that which is correctly known by the Reason is truth, or in other terms, a judgment having a sufficient reason; illusion (that which is wrongly perceived) we oppose to reality: error (that which is wrongly thought) to truth.
The purely formal part of empirical perception—that is, Space, Time, and the law of Causality—is contained à priori in the intellect; but this is not the case with the application of this formal part to empirical data, which has to be acquired by the Understanding through practice and experience. Therefore new-born infants, though they no doubt receive impressions of light and of colour, still do not apprehend or indeed, strictly speaking, see objects. The first weeks of their existence are rather passed in a kind of stupor, from which they awaken by degrees when their Understanding begins to apply its function to the data supplied by the senses, especially those of touch and of sight, whereby they gradually gain consciousness of the objective world. This newly-arising consciousness may be clearly recognised by the look of growing intelligence in their eyes and a degree of intention in their movements, especially in the smile with which they show for the first time recognition of those who take care of them. They may even be observed to make experiments for a time with their sight and touch, in order to complete their apprehension of objects by different lights, in different directions and at different distances: thus pursuing a silent, but serious course of study, till they have succeeded in mastering all the intellectual operations in seeing which have been described. The fact of this schooling can be ascertained still more clearly through those who, being born blind, have been operated upon late in life, since they are able to give an account of their impressions. Cheselden's blind man[82] was not an isolated instance, and we find in all similar cases the fact corroborated, that those who obtain their sight late in life, no doubt, see light, outlines, and colours, as soon as the operation is over, but that they have no objective perception of objects until their Understanding has learnt to apply its causal law to data and to changes which are new to it. On first beholding his room and the various objects in it, Cheselden's blind man did not distinguish one thing from another; he simply received the general impression of a totality all in one piece, which he took for a smooth, variegated surface. It never occurred to him to recognise a number of detached objects, lying one behind the other at different distances. With blind people of this sort, it is by the sense of touch, to which objects are already known, that they have to be introduced to the sense of sight. In the beginning, the patient has no appreciation whatever of distances and tries to lay hold of everything. One, when he first saw his own house from outside, could not conceive how so small a thing could contain so many rooms. Another was highly delighted to find, some weeks after the operation, that the engravings hanging on the walls of his room represented a variety of objects. The "Morgenblatt" of October 23rd, 1817, contains an account of a youth who was born blind, and obtained his sight at the age of seventeen. He had to learn intelligent perception, for at first sight he did not even recognise objects previously known to him through the sense of touch. Every object had to be introduced to the sense of sight by means of the sense of touch. As for the distances of the objects he saw, he had no appreciation whatever of them, and tried to lay hold indiscriminately of everything, far or near.—Franz expresses himself as follows:[83]—