417 



PERSPECTIVE. 



PERSPECTIVE. 





side from the point thus marked off; this parallel, and the segment of 

 the hypothenuse cut off by it, will be the minor and major axes of the 

 ellipse. Since the major axis of the elliptic projection of a circle is 

 always equal to the diameter of that circle (49), the major axis of the 

 isometrical projection of a 'circle is equal to the side of the circum- 

 scribing square. Hence the axes of the ellipse and the side of the 

 circumscribing square, when isometrically projected, are as V3 : 

 <S\\ V2. 



59. The projecting lines and planes are assumed perpendicular to 

 the rectangular co-ordinate planes, simply to facilitate the construction ; 

 but it is obvious that lines and figures may be projected on a plane by 

 parallel projecting lines, making any angle with the plane of projec- 

 tion ; such projections are termed oblique, but as they are but seldom 

 employed, we shall only here give two general theorems relating to 

 them ; since we shall have occasion to recur to this subject in a 

 subsequent part of this article. The oblique projection of a straight 

 line, figure, or curve, lying wholly in a plane parallel to the co- 

 ordinate plane, will be similar and equal to the original; and the 

 oblique projections of parallel right lines will be parallels. 



60. The oblique projection of a sphere must be an ellipse, for the 

 parallel projecting lines which are tangential to the spherical surface 

 must always form a right cylinder, the oblique section of which must 

 be an ellipse. The major axis of this ellipse will be the intersection 

 with the co-ordinate plane of one perpendicular to it, and passing 

 through the oblique projecting line of the centre of the sphere. This 

 major axis will consequently pass through the perpendicular, or 

 ordinary projection of the centre of the sphere. The conjugate axis 

 must clearly be equal to the diameter of the sphere. 



61. We now proceed to show how, by a modification of the 

 principles of projection, an image of an object, or a pictorial outline of 

 it, may be obtained. It is however only to buildings, engines, 

 machines, 4c., consisting of strictly geometrical forms, that this 

 modified projection can be applied ; since the constructions by which 

 these projections are obtained are as strictly geometrical as those by 

 which we obtain the projections of such objects on co-ordinate 

 planes. 



62. Each point on the surface of an object is seen in the direction of 

 a straight line,* supposed to be drawn from that point to the eye, and 

 representing the reflected ray of light by which that point is rendered 

 visible. The rays from every point of that surface will obviously form 

 a geometrical solid pyramid, the surface of which will be composed 

 of those rays which, touching the object, might be supposed prolonged 

 in the same straight direction beyond it, without penetrating its 

 surface. But when considering the subject of outline alone, we need 

 only regard such of the internal rays of the pyramid as proceed from 

 lines on the surface of the object, produced by the intersections of 

 portions of that surface not continuous : and from our limitation of the 

 class of objects, such lines must be either straight, or else geometrical 

 curves ; resulting from the mutual intersection of planes and curved 

 surfaces with each other. 



63. The general pyramid of rays will therefore be made up of a 

 series of others, having one common vertex, and for their several bases 

 the perimeter of a portion of continuous surface. 



64. If we imagine these pyramids of rays cut by a plane, the 

 common section will obviously be an outline of the object as it would 

 present itself to an eye placed at the vertex, each line and point of the 

 section coinciding with the corresponding lino and point of the 

 original. 



65. As long as the object and the spectator's eye retain the same 

 relative position, it is immaterial in what direction or at what 

 distance the plane cute the pyramids of rays ; for the lines and points 

 produced at each position of the plane must necessarily coincide with 

 the originals when viewed from the vertex ; although the outlines on 

 the plane would vary for each of its positions. But each of these 



nt outlines would suggest to the mind the same original com- 

 bination of forms, provided it be viewed from the true vertex, and 

 cannot be a correct representation or image of the object, if viewed 

 from any other point, t 



66. When wo revert to the connection between this subject and 

 drawing, in the common acceptation of the word, we shall point out 

 the precautions that must be taken by the draughtsman, when applying 

 the principles of projection to the pictorial delineation of object*, to 

 prevent his drawing from appearing distorted when viewed indifferently 

 from other than the correct point, which it must inevitably be on most 

 occasions. But at present, dismissing all considerations of light, 

 vision, and art, we shall proceed to treat this branch of the subject of 

 projections, termed perspective, in a purely geometrical manner. 



67. Instead of the simple elements alone, which entered into the 



* We need hardly allude to tho modification of this a-ertioii rendered 

 necenwiry by the effect. 1 * of atmospheric refraction. It U evident Unit the forms 

 of objects which can be seen by the eye at one time are in no way influenced by 

 these effect 1 *, which may therefore be neglected in treating this subject. 



,' re in a common toy which well illustrates, this subject ; we allude to a 

 distorted image of a building, which, when viewed from a certain fixed point, 

 present* a natural appearance. This distorted image is a correct section of a 

 pyramid of rays supposed to proceed from the building to the point in question, 

 and therefore when viewed from that true vertex, conveys the same impression 

 that the building itielf would do. 



ARTS ASD SCI. DIV. VOL. VI. 



constructions for determining lines and points, referred to a co-ordinate 

 plane, by parallel lines perpendicular to that plane, we have in per 

 spective projection the additional elements of the convergence of the 

 projecting lines, or rays, intersecting the plane at different angles, 

 depending conjointly on the distance of their point of convergence 

 from the original lilies, and from the plane. This variation in the 

 conditions necessitates a different course of proceeding : in the former 

 kind of projection the object of our constructions was to determine 

 magnitudes ; in that we are about to consider, our object is to delineate 

 apparent and not real form. 



68. The following definitions are here given to avoid unnecessary 

 repetitions. The plane, on which the projection is supposed to be 

 formed, and which is represented by the drawing board or paper on 

 which the constructions are made, will always be termed the plane of 

 the picture. 



69. The point of convergence of the rays, or projecting lines, or the 

 vertex of the pyramids of rays, will be designated as the vertex. 



70. The centre of the picture is the point in which a line, through the 

 vertex, perpendicular to the plane of the picture, meets that plane ; 

 and the length of this perpendicular, from the vertex to the centre of 

 the picture, is the distance of the picture or vertex : this term will also 

 be applied to the line itself when we have occasion to refer to it. 



71. The vertical plane is one passing through the vertex, parallel to 

 the plane of the picture. 



72. Let x Y z . . . and u T r. s, in the figure, be the plane of the picture 

 and the vertical plane; v in the latter being the vertex. Let BB" be 

 any straight line taken as an elementary original object : the rays from 

 every point in B B" will lie in one plane, the intersection, 6' 6", of which 

 with the plane of the picture will be the indefinite perspective pro- 

 jection or image of B B" : the projecting plane (9) passing through any 

 original line B B" and the vertex will also intersect the vertical plane in 

 a line V D, parallel tob'b". V D is the director of the original line. 



73. If the original line B B" were parallel to the plane of the picture, 

 and therefore also to the vertical plane, its indefinite image and director 

 would be parallel to the original line. But if B B" be not parallel to 

 the plane of the picture and vertical plane, it must intersect them 

 both. 



74. The point A, in which any original line cuts the plane of the 

 picture, in termed its intersecting pnint ; and n, in which it cuts the 

 vertical plane, is termed the station point of that original line. 



75. If a line V P be supposed to pass through the vertex, parallel to 

 any original line B B", it will cut the plane of the picture, if the original 

 line iteelf be not parallel to that plane. This line V P is termed the 

 radial of B B", and the point p, in which| this radial cuts the plane of 

 the picture, is the ranithivg point of the original line. 



76. If any original line pass through the vertex, its radial will 

 coincide with it, and the point in which such a line cuts the plane of 

 the picture will not only be its common intersecting and vanishing 

 points, but also the common image of all points in the original line, acd 

 consequently of the entire line itself. 



77. It follows from these theorems that the original line, its director, 

 its radial, and its image, all lie in its projecting plane, and therefore its 

 image must pass through its intersecting and vanishing points ; while 

 its director must pass through its station point : and that these four 

 lines must form a parallelogram, unless the original line be parallel to 

 the plane of the picture ; in which case the director and radial will 

 coincide in one line, lying in the vertical plane, parallel both to tho 

 original and to its image. 



78. Let us now consider the projection or image of any point B in 

 an original line B B", and the situation of that image in the indefinite 

 one of the line, according to the position of the point u. 



I E 



