Now consider the case of two objects of the same dimensions situated at different distances from the viewpoint, in the same direction. If the nearer of the two objects is at a distance from the viewpoint equal to n, times the distance between the objects, the respective scales of their images will be in the ratio + 1). Thus the images will become less different as n gets greater, as is shown in the following table, where the values of n/ (n 1) are given for different values of n.
Thus it can be seen that if the distance of two equal objects is equal to the distance of the nearer of them from the viewpoint, one of the objects will be represented twice the size of the other. If we multiply by ten the distance to the first of the objects, and compensate for this increase of distance by extending the principal distance until an image of the nearer object is obtained which is the same as previ ously, the more distant object will not differ from it more than io per cent.
Returning now to the case of the front view of a portrait, and bearing in mind that the point of the nose is about 4 in. or 51. in. in front of the back outline of the ears, it can be calculated that in a portrait taken at about 4 ft. from the sitter, a rigorous application of the laws of perspective would result in the nose being represented on a scale greater by io per cent than the scale of the ears. A painter, when sketching a portrait, is always at least io or 15 ft. from his model.
Let us take the case of a house, and consider its perspective at a distance of about 300 yd. At this distance the house is in correct relation with the distant landscape. If now we approach to within 20 yd. of it, whilst keeping the same principal distance, the image of the house will be magnified 15 times, but the distance will he practically the same size as before, and will thus be on a much smaller scale.
Similarly, a painter, when prevented from going back far enough to see properly, would design the background on a magnified scale in order to correct this effect, which, though it would be scarcely noticeable in the examination of a landscape itself, because our brain corrects the sensations which our eyes transmit to it, might be displeasing in the case of a plane image.
30. Binocular Vision. Only a rough idea of the relative distances of objects can be obtained by monocular vision (using one eye only). One
knows how difficult it is to place a finger in the neck of a bottle placed by someone else at the height of the observer's eyes, when one eye is shut. The factors to be appreciated are the varia tion of the apparent dimensions of an object of known size, the changes in the relative position of the objects when the observer moves transversely, the aerial perspective (§ 15), and the variations in the effort necessary to accom modate the eye (focussing the eye) according to the distance of the object.
The causes which give rise to the sense of relief in binocular vision (using two eyes) are, on one hand, the dissimilarity of the two retinal images, each eye seeing a single near point projected on two different points of the back ground, and on the other hand, the effort of convergence of the ocular axes towards the fixed point, this effort becoming greater as the point becomes nearer. These two circumstances only play a part for not very distant objects. Aviators and balloonists verify daily that at a few hundred yards above the earth all sensation of relief disappears, even for the highest buildings.
Consider two perspectives of a single subject, each perspective having the same principal distance, on two parts of the same plane, from two viewpoints the separation between which is equal to the mean separation of the eyes (about 65 mm.). If the centres of rotation of one's eyes be placed at the viewpoints, each eye only seeing the perspective of its own viewpoint, the same sensation of relief will be experienced as in direct observation of the object with the two eyes (the variations of the accommodation no longer come in in this case). This relief may be so striking that an observer who did not already know would scarcely believe that the solid image which he could see was actually the result of two plane images.
This fact forms the basis of stereoscopy. Stereoscopic vision implicitly assumes that the observer has two equal and symmetrical eyes.
31. Perspective on a Non-vertical Plane. If from the viewpoint 0 (Fig. 8) the perspective of a solid body S be drawn on the non-vertical plane T, the images of all the vertical lines of the solid will converge to the vanishing point V where V is the intersection of the plane T with the vertical dropped from the viewpoint.