will merely be that which results from drawing the line through c from the luminary instead of parallel to R. The cases of this proposition are correspondent to those of the preceding one, the operation being performed for each extremity of the given line. LIGHT AND SHADE. We shall omit, for the moment, all consideration of the circumstances arising out of the supposed magnitude of the luminous body, and therefore all that relates to penumbræ and penumbrals. The light will, therefore, be considered to either move in parallel straight lines, or in lines meeting in a single point. The influence of magnitude will be considered hereafter. Only the actual effect of direct light as distinguished from its absence will be considered in this stage of the inquiry : all that relates to reflected and refracted light upon the parts of the body not in direct light, and all that relates to the different intensities of light upon the enlightened as well as upon the shaded parts, involving other and separate considerations, will not be discussed at present. We shall discriminate the shade as here considered by the name of geometrical shade, and the shade modified by the other circumstances referred to, by the name of natural shade. All figures bounded by plane surfaces have these surfaces bounded by straight lines meeting, three or more, in points. The boundary of light and shade must, evidently, be in some of these lines, and hence must also pass through some of these points. It will hence follow that the portion of space from which the light is intercepted by a polyhedron is a prism or a pyramid, according as the rays are parallel or emanate from a point. It is moreover clear that the extreme plane boundaries of the prism or pyramid will be the boundaries of light and shade; and that the parts (or faces) which lie on the side of the polyhedron on the side of the light will be enlightened, whilst the remaining ones are in geometrical shade. When a plane face coincides with one of the planes of light and shade, we shall call it neutral :* the slightest variation of its position in one way placing it in light, and in the opposite way shade. If the surface be curved, the boundary of light and shade will be a cylinder or cone, according as the rays are parallel or emanate from a point; and in all cases the cylinder or cone will be tangential to the surface, and may be considered as generated by the consecutive intersections of tangent planes to the surface drawn parallel to the given ray, or emanating from the given point, as the case may be. To find the shaded parts of a polyhedron, which are always bounded by straight lines, we have only to find, therefore, the extreme shadows ; and mark the lines whose shadows these are, which will be the lines of separation sought. In the case of the cone and cylinder, the determination of the shadow also suffices for the determination of the shade, inasmuch as the generatrices which limit the shadow also limit the shade. For other curve surfaces, as the surfaces of the second order, developable surfaces, and surfaces of revolution 'generally, a different mode of actual construction must be employed adapted to the known properties of each of these classes of surfaces ; but the methods themselves are both rigorous and strictly geometrical. * It might perhaps be as well to call the lines of separation of light and neutral lines ; or in polyhedrons, neutral edges. PROPOSITION VII. Given the plan and elevation of a polyhedron, to find the shadows of the figure on the plan and elevation, the light being parallel to a given line. Find the shadows a, b, 7, 8 of the angular points on whichever plane they may lie; and draw lines to join them if on the same plane; or if on different planes, find the broken line of shadow as before explained. These will be the boundaries of the shadow. In the tetrahedron in the figure the shadow of ad is wholly on the horizontal plane, and of cb wholly on the vertical; whilst those of all the other lines are partly on each plane. PROPOSITION VIII. To find the shadow of a given point on a given right cylinder. This is, in fact, a particular case of the intersection of a given line and a given cylinder, which has been constructed in the “Descriptive Geometry," (Prop. vir. Sec. Iv.): but in this case, like many others in the application of general principles, the actual problem is almost invariably to the right circular cylinder, perpendicular to the plan. solution merely requires that the cylinder shall have its axis perpendicular to the plan. Let a, a, be the given point, and lett he curve a, B. Y be the trace of the cylinder on the plan ; draw a, a, parallel to R, to meet the trace in Qy, and a, a, parallel to Ro, to meet the perpendicular from a in : then a, a, is the projected point of section. If a, a, meet the trace again in B, and we find B, as we found dz, we get the shadow upon the internal face of the cylinder, the half next the light being supposed to be removed. In this we have in fact constructed the point on the tangent plane of the cylinder, by the preceding propositions; and it follows from that without further reasonings. CASES AND APPLICATIONS OF PROPOSITION VIII. 1. To find the shadow of a line anyhow situated on the cylinder. This is effected by the construction of the points 1, 2, ...n taken at any convenient distances from each other; either equal or otherwise. In the left figure, the line is anyhow situated: the construction for the point 7 being exhibited. The vertical edges of the cylinder 1 and 9 are the boundaries of light and shade, and 1, 9 on the plane is the trace of the neutral diametral plane. In the right-hand figure, we have a cylinder surmounted by a square cap; &, €, is the line of shade; Oy Ey is the visible shadow of ae ; and ach is the shadow of ac, which, being perpendicular to the elevation, is a straight line. 2. To construct the shadow of a circular cap surmounting a cylin drical column upon the shaft of the column. The process is here of pre ot cisely the same character ; the construction being given for the point 6 of the cap. And in precisely the same way will the construction be performed for any shadow whatever, the projections of the object being given. 3. To find the shadow of a given circle on a given plane. (a.) Let the circle be parallel to the plane on which it is to be projected : then the shadow will be equal to the circle itself, and the homologous lines of the circle, and its projection will be equal and parallel, both in fact and in their respective plans aud sections. (6.) When the plane of projection is one of the coordinate planes, we have only to find the projection of the centre, and with the radius of the given circle, describe the contour of the shadow. (c.) When the plane has any given position, and the projection of the centre upon one of the coordinate planes is given, as well as that of any one point in the circumference, the other projections can be found as taught in the Descriptive Geometry, and the diameters parallel to and perpendicular to one of the traces of the containing plane can be projected on both coordinate planes. These diameters are perpendicular to each other in the given circle ; and their projections are conjugate diameters of the elliptic shadow. From these the principal axis can be found by several methods, and the ellipse traced. (d.) When the plane which contains the circle is perpendicular to one of the planes, the actual work is slightly reduced, as far as relates to finding the vertical projections of the two rectangular diameters of the circle. PROPOSITION IX. To find the shadow of a point on a right cone standing on the plane. Let a, a, be the given point, R, R, the given ray, and v, v, the vertex of the cone. Through a, draw a, c parallel to Ry, and through a draw the parallel to R, and eT, perpendicular to the axis, giving the point T. Find also the horizontal trace t, of a v: then t, T, is the horizontal trace of the plane through av parallel to R. Whence it will cut the trace of the cone in e, i, which are the horizontal traces of the edges of the cone on which the shadow falls externally and internally. The intersection of the vertical and horizontal projections of these edges with those of the ray give e, è, and is i, for the positions of the shadows externally and internally. A moment's reflection will show that the process is precisely the same whether the cone be right or oblique. |