Linear perspective

“ When, on account of some particular quality of the air, you can no longer distinguish the difference between the lights and shadows of objects, you may reject the perspective of shadows, and make use only of the linear perspective, and the diminution of colours, to lessen the knowledge of the objects opposed to the eye; and this, that is to say, the loss of the knowledge of the figure of each object, will make the same object appear more remote. ”

“ The young student is the spectator on the other side of it, the dotted lines drawn from the corners of the cube to the eye of the spectator are the visual rays, and the points on the transparent picture plane where these visual rays pass through it indicate the perspective position 8 of those points on the picture. To find these points is the main object or duty of linear perspective.
Fig. 3.
Perspective up to a certain point is a pure science, not depending upon the accidents of vision, but upon the exact laws of reasoning. ”

“ The knowledge of perspective is no small help to the seeing objects truly, for which purpose Dr. Brook Taylor's Linear perspective made easy to those who are unacquainted with geometry, proposed to be publish'd soon by Mr. ”

“ If we saw everything as depicted by plane geometry, that is, as a map, we should have no difference of view, no variety of ideas, and we should live in a world of unbearable monotony; but as we see everything in perspective, which is infinite in its variety of aspect, our minds are subjected to countless phases of thought, making the world around us constantly interesting, so it is devised that we shall see the infinite wherever we turn, and marvel at it, and delight in it, although perhaps in many cases unconsciously. ”

“ In an extended view or landscape seen from a height, we have to consider the perspective plane as in a great measure lying above it, reaching from the base of the picture to the horizon; but of course pierced here and there by trees, mountains, buildings, &c. As a rule in such cases, we copy our perspective from nature, and do not trouble ourselves much about mathematical rules. ”

“ You know how on a flat surface, which has only two dimensions, we can represent a figure of a three-dimensional solid, and similarly they think that by models of three dimensions they could represent one of four—if they could master the perspective of the thing. ”

“ And we may further notice, how, in this first step of rational mechanics, we see illustrated that truth awhile since referred to, that as magnitudes of linear extension are the only ones of which the equality is exactly ascertainable, the equalities of other magnitudes have at the outset to be determined by means of them. ”

“ The sort of cases with which we shall be exclusively occupied are those comparatively simple ones in which only linear magnitude, or some quality which can be adequately represented by linear magnitude, is the object under consideration. ”

“ But the mistakes we occasionally make, for instance taking a road seen from above for a church-tower projecting out of the plain, or the perspective of a mountain range for its cubic shape, occasionally reveal that we do not really see three-dimensional objects, but merely infer them by connecting visual data with the result of locomotor experience. ”

“ The test of validity of any theory, as you know, is the accuracy of the predictions which are made possible by its use, and our greatest thinkers have shown that the completeness and fidelity of any visualization of the Cosmic All are linear functions of the clarity of definition of the components of that interface. Full knowledge of that indeterminate zone would mean infinite power and a statistically perfect visualization. ”

“ But if, for example, the sounds which I hear are to belong to the same perspective with the patches of colour which I see, there must be particulars which have no direct spatial relation and yet belong to the same perspective. We cannot define a perspective as all the data of one percipient at one time, because we wish to allow the possibility of perspectives which are not perceived by any one. There will be need, therefore, in defining a perspective, of some principle derived neither from psychology nor from space. ”

“ The world which we have so far constructed is a world of six dimensions, since it is a three-dimensional series of perspectives, each of which is itself three-dimensional. We have now to explain the correlation between the perspective space and the various private spaces contained within the various perspectives severally. It is by means of this correlation that the one three-dimensional space of physics is constructed ”

“ We spoke earlier of two ways of classifying particulars. One way collects together the appearances commonly regarded as a given object from different places; this is, broadly speaking, the way of physics, leading to the construction of physical objects as sets of such appearances. The other way collects together the appearances of different objects from a given place, the result being what we call a perspective. In the particular case where the place concerned is a human brain, the perspective belonging to the place consists of all the perceptions of a certain man at a given time. ”

“ All lines situated in a plane that is parallel to the picture plane diminish in proportion as they become more distant, but do not undergo any perspective deformation; and remain in the same relation and proportion each to each as the original lines. ”

“ You think yourself the perfection of existence, while you are in reality the most imperfect and imbecile. You profess to see, whereas you can see nothing but a Point! You plume yourself on inferring the existence of a Straight Line; but I can see Straight Lines and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. ”

“ As we approach nearer to the object, the effect of the intervening medium grows less. In a generalized sense, all the intrinsic laws of change of appearance may be called "laws of perspective." Given any appearance of an object, we can construct hypothetically a certain system of appearances to which the appearance in question would belong if the laws of perspective alone were concerned. ”

“ To get rid of this difficulty, and at the same time to save the credit of the supposed system of necessary truth, it is customary to say that the points, lines, circles, and squares which are the subject of geometry, exist in our conceptions merely, and are part of our minds; which minds, by working on their own materials, construct an à priori science, the evidence of which is purely mental, and has nothing whatever to do with outward experience. ”

“ With the great principle of linear composition in mind, namely, that the vision travels in the path of least resistance, no rule need be formulated and no further examples produced to prove that the various items of a composition are taken at their required value to the extent to which they adhere to and partake of the established plan of observation. ”

“ For the introduction into a picture of a suggested third dimension is what prevents our thinking away from a merely two-dimensional aspect by supplying subsidiary imaginary aspects susceptible of being co-ordinated to it. So perspective and modelling in light and shade satisfy our habit of locomotion by allowing us, as the phrase is, to go into a picture; and going into, we remain there and establish on its imaginary planes schemes of horizontals and verticals besides those already existing on the real two-dimensional surface. ”

“ Our idea of a point, I apprehend to be simply our idea of the minimum visibile, the smallest portion of surface which we can see. A line, as defined by geometers, is wholly inconceivable. We can reason about a line as if it had no breadth; because we have a power, which is the foundation of all the control we can exercise over the operations of our minds; the power, when a perception is present to our senses, or a conception to our intellects, of attending to a part only of that perception or conception, instead of the whole. ”

“ Plane diagrams also can be conceived as solids, and that in more than one way. Figs. 74, 75, 76, for example, are ambiguous perspective projections, and may each of them remind us of two different natural objects. Whichever of these [Pg 257] objects we conceive clearly at the moment of looking at the figure, we seem to see in all its solidity before us. A little practice will enable us to flap the figures, so to speak, backwards and forwards from one object to the other at will. ”

“ As we already mentioned, reality is the one generated by the model associated to a given external reality. Each time we state a truth, we have to specify first the model.
Example: There are an A and a B person. A is taller than B, as it results from measurement. The term 'length' is generated of the model 'space', as Euclid's Geometry and Newton's Mechanics understand it. These fundamental models characterize this truth as objective. If we say that "A is more attractive than B", this is a subjective truth. ”

“ We draw with a continuous line, after having looked at, or thought of, our model. How shall we explain such a faculty, except by our habit of discovering at once the organization of the outlines of common objects, that is to say, by a motor tendency to draft their diagram in one continuous line? ”

“ On the contrary, the larger the collection of seemingly heterogeneous facts we can bring under the rule of a single formula, the nearer we feel that we have reached towards the source of knowledge, and the more perfectly we trace the little arc of the immeasurable circle which comes within the range of our hasty observations, at first like the broken fragments of a many-sided polygon, but at last as a simple curve which encloses all we know, or can know, of nature. ”

“ We draw a continuous line as regularly as possible between the points given by observation. Why do we avoid angular points and inflexions that are too sharp? Why do we not make our curve describe the most capricious zigzags? It is because we know beforehand, or think we know, that the law we have to express cannot be so complicated as all that. ”

“ Why should that absent and imagined slant-legged image displace the present and felt square image from our mind? Why, when the objective cross gives us so many shapes, as it varies its position, should we think we feel the true shape only when the cross is directly in front? And when that question is answered, how can the absent and represented feeling of a slant-legged figure so successfully intrude itself into the place of a presented square one? ”

“ But when we come to unformulated nature—to the vast world of complex forms, ever changing their aspect, full of life and movement, trees, flowers, woods and waters, birds, beasts, fishes, the human form—the problem how to represent any of these forms, to express and characterize them by means of so abstract a method as line-drawing, seems at first difficult enough. ”

“ The perception of distance by the eye, which seems so like intuition, is thus, in reality, an inference grounded on experience; an inference, too, which we learn to make; and which we make with more and more correctness as our experience increases; though in familiar cases it takes place so rapidly as to appear exactly on a par with those perceptions of sight which are really intuitive, our perceptions of color. ”

“ These linear phenomena direct his attention to the linear properties of surrounding objects. Now the head becomes a circle, and now the circular line represents the head, the elliptical curve connected with it represents the body; arms and legs appear as straight or broken lines, and these again represent arms and legs; the fingers he sees as straight lines meeting in a common point, and lines so connected are, for the busy child, again hands and fingers; the eyes he sees as dots, and these again represent eyes; and thus a new world opens within and without. ”

“ It is not necessary to live and experience everything, or all the intermediate links; exactly as we need not, supported by a scientific geometry, actually measure all the lines which we require to use, but can discover some through mere calculation. ”

“ Whenever such signs of changes in the distance are impressed upon our mind, we immediately supplement them by ideas of the distances themselves. Thus our original two-dimensional perception of space is expanded into a three-dimensional perception.
All knowledge of things, of their properties, their names, their uses, their meanings, consists in supplementing our consciousness of those qualities which they present to our senses, by images previously obtained through any senses. The force of this supplementing can be understood from the drawings of children and primitive peoples. ”

“ A mathematical line and its direction are not two things, like the feet of a pair of compasses—that while the line is moving one way, the direction may be going astray in another—but mutually implied and inseparably connected together in nature or the understanding—let the realists or idealists determine which they please. ”

“ If, within the world of our perceptions, we try to order parts of our perceptual data under any ordinal characteristic arbitrarily chosen from among those which we immediately apprehend, we find that we can order (that is, correlate with one or more series of natural numbers) under this characteristic a determinate part of the totality of events into which the mind divides its Experience; but we can as a general rule expect that in a given Experience we shall always be able to find events which remain outside the ordering, that is, outside the system of co-ordinates which we have chosen. ”