base, and whose vertex is at the centre of the lens O, falls on a surface beyond the lens, and produces an inverted picture A' B' of the object A B. The picture and the object form thus parallel bases of opposite cones. (446.) The camera obscura is one of the feeble attempts of art to imitate nature. The eye is a camera obscura of exquisite perfection and sensibility. In front of the sphere which forms the eyeball is the circular opening called the pupil, which produces the black circular spot seen in the centre of the iris, or coloured membrane of the eye. Immediately behind this opening is suspended a double convex lens, formed of a perfectly transparent fluid called the crystalline humour. This lens, in the phenomena of vision, plays the part of the lens of glass O in the camera obscura. The cones of rays coming from visible objects to the eye, having their vertex in this lens, are continued to the posterior surface of the inner chamber of the eyeball, on which is depicted, with its proper form and colours, but in an inverted position, a luminous representation of all the objects of vision; and it is such luminous pictures acting on the optic nerve that produce the effect on the brain which is the immediate cause of vision. (447.) The whole art of perspective, and therefore a considerable part of the art of the painter, depends upon the properties of conical surfaces. A picture delineated on a plane surface, being intended to produce upon the eye the same effect as visible objects seen at certain distances behind that surface, the relative positions, forms, and magnitudes of the objects on the canvass must be determined by the intersection of the plane of the canvass with the conical surfaces formed by visual rays drawn from the eye of the spectator to the real positions which the objects represented on the canvass are supposed to have. Thus, if we suppose a distant landscape viewed through a rectangular frame placed at a certain distance from the eye of the spectator, a cone, or rather a pyramid, having a rectangular base, must be imagined, the vertex of which shall be at the eye of the spectator. The frame bounding the landscape, and through which it is viewed, is a section or generatrix of this pyramid; and the diverging faces of the pyramid being continued indefinitely in the direction of the landscape, the actual objects comprehended in it will be included within the four triangular surfaces extending from the eye of the spectator, passing through the four sides of the rectangular frame, and continued indefinitely beyond them. If a line be drawn from the vertex of the pyramid to any point within the limits of the landscape, the place where that line would penetrate the canvass, if canvass were extended in the frame, would be the place of such a point in the painting. If the surface of any object in the view be parallel to the canvass, the section of the cone of which the object is the base made by the canvass will be similar to the object; but if the plane of the object be not parallel to the canvass, then the form of the section of the cone by the canvass will be different from that of the object, and nothing but the application of exact geometrical principles can determine the form of such section. This effect, which, in particular applications of it, is called fore-shortening, is one, therefore, which an artist cannot expect to produce with correctness if he be not conversant with the principles of geometry which are required in the solution of such problems. There is, accordingly, no department of the arts of design in which errors so glaring are committed even by the most eminent artists. The collection of general theorems relating to the intersection of conical and pyramidal surfaces by a plane, which is necessary for the solution of such problems, constitutes the theory of perspective. As an example of such theorems, the following, which are of very universal application and general utility, may be given. (448.) Parallel lines which are parallel to the plane of the picture will be represented by parallel lines upon the canvass; for if a plane be drawn through any one of these parallels, and through the point of sight the intersection of such plane with the plane of the canvass will be a line parallel to that through which the plane is drawn, and this line will be that which represents the parallel upon the canvass. Since, therefore, the representations of the parallel lines on the canvass are parallel to the lines themselves, and since the latter are parallel to each other, the lines on the canvass representing them will also be parallel to each other. (449.) If a system of parallel lines be not parallel to the plane of the drawing, then the lines which represent them on the drawing will be lines which all converge to a point, so placed on the plane of the drawing that a straight line drawn from it to the point of sight will be parallel to the lines thus delineated. For, take any two of the parallels to be delineated, and suppose planes drawn through them, and through the point of sight; these planes will intersect in a certain line parallel to the lines to be delineated, and this line will therefore not be parallel to the plane of the drawing, and will therefore meet it at some determinate point. The intersections of the two planes drawn through the point of sight, and through the two parallels, with the plane of the drawing must meet at the same point, that being in fact the point where all the three planes inter sect. That point will therefore be the point to which the representations of the two parallel lines on the canvass must converge, and it may in like manner be shown that all the lines representing the parallels will converge to that point. This, in fact, amounts to little more than the statement that all planes which are drawn through a number of parallel lines must have a common line of intersection. For their line of intersection must be parallel to the parallels; and since only one such parallel can pass through the given point, that one must be their common line of intersection. (450.) These general principles are brought into frequent application in architectural and mechanical drawing, where the forms of the objects represented are generally determined by systems of parallel lines, as in the case of a building which is composed principally of vertical lines and of horizontal lines at right angles to each other. (451.) The eye is an organ incapable of estimating actual magnitude. All visible objects appear to the eye of equal magnitudes, provided the angle of the cone formed by the visual rays which bound them is the same. Let E (fig. 173.) be the eye, and let A B be an object placed at any distance from it, and A' B' be another object at a greater distance; if the visual ray from the upper extremity A' coincide with the visual ray from the upper extremity of the other, and the visual rays from the lower extremities B, B' also coincide, then the objects will have the same apparent magnitude. In fact the one will entirely cover and intercept the other. In this case, the real magnitudes of the objects will be proportional to their distances from the eye; for they are the bases of similar triangles of which those distances are the sides. (452.) In general, similar objects will have the same apparent magnitude when their linear dimensions are proportional to their distances from the eye; for in that case their sections are the bases of similar cones of which the altitudes are the distances of the objects from the eye. (453.) A remarkable example of this is presented by the sun and moon, whose apparent magnitudes are very nearly the same, although the actual diameter of the sun is about 400 times greater than that of the moon. The reason of the equality of their apparent magnitudes is, that while the distance of the moon from the earth is only £40,000 miles, that of the sun is 96,000,000 miles, the one distance being 400 times greater than the other. P (455.) Since the centre O is equally distant from every point in the revolving circle, and since that circle as it moves coincides successively with every part of the spherical surface, it follows that the point O is equally distant from every part of the surface of the sphere. This point is therefore called the centre of the sphere. (456.) The circle by the revolution of which on its diameter P P' the spherical surface is produced is called a meridian of the sphere. (457.) All sections of the sphere made by planes passing through P P' are circles equal to the meridian by the revolution of which the sphere is produced; for the meridian as it revolves coincides successively with all such circles. All such circles are therefore called meridians. (458.) The diameter PP' on which the generating circle turns is called the axis of the sphere, and its extremities P P' are called the poles of the sphere. (459.) The axis of the sphere is therefore the common line of intersection of the planes of all the meridians, and the poles are the common points of intersection of the circumferences of such meridians. (460.) As the meridian revolves, all points, such as L, upon it describe circles whose planes are at right angles |