Vertex

“ A right line drawn from the vertex and turning about it in the plane of the angle, from the position of coincidence with one leg to that of coincidence with the other, is said to turn through the angle, and the angle is the greater as the quantity of turning is the greater. Again, since the line may turn from one position to the other in either of two ways, two angles are formed by two lines drawn from a point. ”

“ With them, the eye is situated so far from their vertex that they can scarcely take cognizance of what goes on at that extremity of their frame. They are moreover of a rough coarse nature, not sensitive to the delicate touch of the highly organized Polygon. ”

“ The axis which contains the foci is called the principal axis, and the intersection of the axis with the curve is called the vertex of the curve. The directrix is at right angles to the principal axis. In a parabola the vertex is equally distant from the focus and the directrix, these three points and the point at infinity on the axis being four harmonic points. In the ellipse the vertex is nearer to the focus than it is to the directrix, for the same reason, and in the hyperbola it is farther from the focus than it is from the directrix. ”

“ An elliptical field may be obtained, by giving the bounding line the shape of one quarter of an ellipse, and placing it in such a manner that the vertex of the conjugate axis falls upon one of the reflectors, and the vertex of the transverse axis upon the other. ”

“ This last theorem is the basis for the construction of the parabolic reflector. A ray of light from the focus is reflected from such a reflector in a direction parallel to the axis of the reflector.
157. Directrix. Principal axis. Vertex. The polar of the focus with respect to the conic is called the directrix. ”

“ You have heard of the fallacy 'A dicto secundum Quid ad dictum Simpliciter'?
Nie. (hastily) We are not going to commit ourselves to that. You may assume that we mean, by 'angle,' a rigid angle, which cannot change its magnitude.
Min. In that case you assert that, when a pair of Lines, terminated at a point, is transferred so that its vertex has a new position, these three conditions can be simultaneously fulfilled:—
(1) the right arm has 'the same direction' as before; (2) the left arm has 'the same direction' as before; (3) the magnitude of the angle is unchanged. Nie. ”

“ The different positions of the several parts of the body with regard to the eye, when put together, make a real figure, which is truly extended in the length and breadth, and which represents a figure that is extended in length, breadth, and thickness. In like manner, a projection of the sphere is a real figure, and hath length and breadth, but represents the sphere, which hath three dimensions. ”

“ This surface which thou attributest to bodies, thou canst indeed neither see nor feel, nor perceive by any organ; but it may be said, in a certain sense, that thou canst see the red colour upon it, or feel the smoothness. But thou addest something more even to this surface: thou extendest it to a solid mathematical figure; as by thy previous admission thou hast extended the line to a surface. Thou assumest a substantial interior existence of the body behind its surface. ”

“ The Sphere would willingly have continued his lessons by indoctrinating me in the conformation of all regular Solids. Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I ventured to interrupt him. Not that I was wearied of knowledge. On the contrary, I thirsted for yet deeper and fuller draughts than he was offering to me.
“Pardon me,” said I, “O Thou Whom I must no longer address as the Perfection of all Beauty; but let me beg thee to vouchsafe thy servant a sight of thine interior.”
Sphere. My what?
I. Thine interior: thy stomach, thy intestines. ”

“ V. The Angle of Reflexion or refraction, is the Angle which the line described by the reflected or refracted Ray containeth with the Perpendicular to the reflecting or refracting Surface at the Point of Incidence.
Defin. VI. The Sines of Incidence, Reflexion, and Refraction, are the Sines of the Angles of Incidence, Reflexion, and Refraction.
Defin. VII. The Light whose Rays are all alike Refrangible, I call Simple, Homogeneal and Similar; and that whose Rays are some more Refrangible than others, I call Compound, Heterogeneal and Dissimilar. ”

“ And whenever you get long vertical lines in a composition, no matter whether it be a cathedral interior, a pine forest, or a row of scaffold poles, you will always have the particular feeling associated with rows of vertical lines in the abstract. And further, whenever you get the swinging lines of the volute, an impression of energy will be conveyed, no matter whether it be a breaking wave, rolling clouds, whirling dust, or only a mass of tangled hoop iron in a wheelwright's yard. ”

“ And a line flowing out into breadth begets a surface, and the limits of the surface are lines; but a surface flowing out into breadth becomes body. And when what is solid has in this manner derived existence from, altogether, the smallest point, the nature of a huge body is constituted ”

“ To sum up, the eye, owing to the laws of its mechanism, follows a line much more easily in the central than in the peripheral parts of the field, and in the central parts it follows a vertical line more easily than an oblique, and an horizontal more easily than a vertical. ”

“ This being so, the glands of the investing fleshy tissue secrete lime from the sea-water, which holds it in solution, and constantly deposits it, after a determinate and orderly pattern, on every part of the surface of each shelly piece; the inner face, the outer face, and each of the sides and angles of the polyhedron grow together, and all so evenly, that while the dimensions [Pg 179] increase, both of thickness and superficies, the form characteristic of that individual piece is maintained with immutable mathematical precision. ”

“ The whole mass being nearly white, we want a contrasting shadow somewhere; and get it, under our piece of decrepitude. This shade, with the tiles of the wall below, forms another pointed mass, necessary to the first by the law of repetition. Hide this inferior angle with your finger, and see how ugly the other looks. A sense of the law of symmetry, though you might hardly suppose it, has some share in the feeling with which you look at the battlements; there is a certain pleasure in the opposed slopes of their top, on one side down to the left, on the other to the right. ”

“ The superficies of any opake body participates of the colour of the transparent medium interposed between the eye and such body, in a greater or less degree, in proportion to the density of such medium and the space it occupies. ”

“ That every visible figure will be represented by that part of the surface of the sphere, on which it might be projected, the eye being in the centre. And every such visible figure will bear the same ratio to the whole of visible space, as the part of the spherical surface which represents it, bears to the whole spherical surface. ”

“ Simultaneity to these painters meant the combined presentation of a number of aspects of the same object from many different angles. In the visualisation of an object in nature during the absence of that object, we conceive it, not only as a silhouette or as a form with three dimensions, but as a congeries of silhouettes which, when imagined simultaneously, constitute the appearance of the object from every known angle. ”

“ A space that is arched looks wider than it actually is, for the eye unconsciously follows the lines of the arch, and a distance or width effect is the result. The same space treated with a straight line is quickly bridged. The same space treated with lines that come to an angle looks narrower, for the reason that the eye becomes focused by the apex of the angle, and a height effect, not a breadth effect, is the result. ”

“ And is not the density thereof greater in free and open Spaces void of Air and other grosser Bodies, than within the Pores of Water, Glass, Crystal, Gems, and other compact [Pg 350] Bodies? For when Light passes through Glass or Crystal, and falling very obliquely upon the farther Surface thereof is totally reflected, the total Reflexion ought to proceed rather from the density and vigour of the Medium without and beyond the Glass, than from the rarity and weakness thereof. ”

“ So seen, then, all objects appear to the human eye simply as masses of colour of variable depth, texture, and outline. The outline of any object is the limit of its mass, as relieved against another mass. ”

“ If, however, the eye, or point of vision, is regarded as being at an infinite distance from the object all the rays will be parallel, and edges which are parallel on the crystal will be represented by parallel lines in the drawing. The plane of the drawing, in which the parallel rays joining the corners of the crystals and the eye intersect, may be either perpendicular or oblique to the rays ”

“ In proof of the fact that the half balance, or that on either side of the vertical is sufficient in many subjects, see such portraits in which the head alone is attractive, the rest being suppressed in detail and light, for the sake of this attraction.
It is rarely that figure art deals with balance over the horizontal central line in conjunction with balance over the vertical. ”

“ Thus the most perfect idea we can possibly acquire of a sphere, is by conceiving an infinite number of straight rays of equal lengths, issuing from the center, as from the eye, spreading every way alike; and circumscribed or wound about at their other extremities with close connected circular threads, or lines, forming a true spherical shell. ”

“ We have seen that a film tends to set itself at equal angles to the surface which it meets, and therefore, when that surface is a solid, to meet it (or its tangent if it be a curved surface) at right angles. Our vertical partition is, therefore, everywhere normal to the original cell-walls, and constitutes a plane surface. ”

“ But it does not appear that such a person could arrive at any conception of geometrical solidity. For that which does not come in contact with the sensory surface is non-existent for the sense of touch; and a solid body, impressed upon the palm of the hand, gives rise only to the notion of the extension of that particular part of the solid which is in contact with the skin. ”

“ We should also aim for the expression of contrasting elements of forms, such as curves with straight lines, sharp curves with others of less curvature, horizontal lines to counteract vertical tendencies, or mouldings and lines of varying widths arranged to fit in such positions that will give, or suggest, constructive strength; all of which are simply architectural principles, which, if applied to the design of common objects, would give them a definite claim to be considered as works of art. ”

“ We may avoid all such confusion by using and to represent a strain. Let us consider an element of the body strained which in the state is a cube with its edges parallel to the axes of , and call the edges according to the axes to which they are parallel, and consider the ends of the edges as positive for which the values of or are the greater. ”

“ There must be horizontal divisions of colour which attract the attention and make one oblivious of what is above them.
Every one knows the effect of a paper with perpendicular stripes in apparently heightening a ceiling which is too low, but not every one is equally aware of the contrary effect of horizontal lines of varied surface. ”

“ Shadows Cast by Moldings.—Always bear in mind that a curved surface makes a blended shadow. A straight, flat or plain surface does not, and it is for that reason the concaved and the convexed surfaces, brought out by moldings, become so important. ”

“ The devices which aid the figure painter in disposing of one or many verticals have been briefly viewed. A consideration of the horizontal will necessarily take us out of doors to earth and sky, where nature constructs on surfaces which follow the horizon.
The problem in composition which each of these lines presents is the same and the principle governing the solution of each identical; balance by equalization of forces. [pg 117] Given a line which coincides with but one side of the picture it becomes necessary for the poise of the quadrilateral to cross it with an opposing line. ”

“ You might, at first, imagine that, given what we may call our scale of solidity, or scale of depth, the diminution from nature would be in regular proportion, as, for instance, if the real depth of your subject be, suppose, a foot, and the depth of your bas-relief an inch, then the parts of the real subject which were six inches round the side of it would be carved, you might imagine, at the depth of half an inch, and so the whole thing mechanically reduced to scale. ”

“ Let’s see from what you would define a mirror—from a surface per se, in quantum est superficies, or from a substance that forms the surface, or from the substance upon which the surface rests, the raw material, modified by the attribute ‘surface,’ since it is clear that, surface being an accidental property of bodies, it cannot exist without substance. ”

“ Now the first peculiarity distinguishing things from shapes is that they can occupy more or less cubic space: we can hit up against them, displace them or be displaced by them, and in such process of displacing or resisting displacement, we become aware of two other peculiarities distinguishing things from shapes: they have weight in varying degrees and texture of various sorts. Otherwise expressed, things have body, they exist in three dimensional space ”