57. Thus, for example, if the co-ordinate plane be assumed perpen dicular to the diagonal of a cube, the projections of the three edges meeting in either end of that diagonal will form angles of 120° with each other, and the three projections of the edges at one extremity will, respectively, bisect the equal angles formed by those of the edges at the other extremity ; while the lines joining the ends of these six equal radii, which lines must obviously form a regular hexagon, will be the projections of the remaining edges of the solid. Each face of the cube is projected into an equilateral rhombus, as Acne, BCDO, ACM, DFCG, &c., the sides of which form angles respectively of 120° and of GO° each. If the Bide of the cube be unity, the equal projections of those sides will be -3165, which is equal to the cosine of the angle at which the originals are inclined to the co-ordinate plane. The original diagonals of the three faces, An, nu, DA, are obviously, from the symmetry and position of the solid, parallel to the co-ordinate plane ; their projections are therefore equal to those originals, or arc each equal to ... if an original solid be made up of rectangular parallelopipeds, having their faces mutually parallel, and the co ordinate, or plane of projection, be assumed as equally inclined to the three faces forming any of the solid angles, the projections of all its edges, and of all lines parallel to thorn, would be in the constant ratio to the originals of •8164: 1 ; the dimensions, consequently, of those originals, as measured in the directions of lines which would be isometrically projected, may be set off from any scale along the isometric projections of any lines parallel to the edges of the original solids, and a figure or image of the original constructed which would show the three principal series of planes of which that original was composed.
58. The projections of all circles equally inclined to the co-ordinate plane will be similar ellipses ; the axes of these ellipses, when repre senting circles lying in planes parallel to the faces of a cube equally inclined to the co-ordinato plane, will be to each other in the ratio of the diagonals of the rhombus representing the inscribed or circum scribing square isometrically projected. The following simple method of constructing a scale for determining the lengths of the axes of the isometric projection of a circle will be of service to the practical draughtsman. Construct a right-angled triangle the base and perpen dicular of which are iu the ratio of the aide to the diagonal of a square, or as 1 : 1.4142. Set off the length of the isometric projection of the circumscribing square of any original circle along the side of this triangle, from the acute angle, and draw a line parallel to the other 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 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 a/3 : WI; a/2.
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, &c., 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 regarl 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.