CASES OF PROP. V.

It will often happen that, under particular conditions of the data, the practical construction becomes much simplified; and the simplified cases which occur in usual practice require to be specially noticed. 1. Let the plane be parallel to the elevation, and P, be its trace, a, a, being the given point, and R, R, the direction of the ray.

Draw the parallels through a, and a, to the projections of the ray; and let a, a, cut P, in a,: then a, is the horizontal projection of the shadow point on P1, and drawing, as usual, we get a for the projection on the vertical.

R

4. Let the plane be perpendicular to both planes: that is profile. Draw a a1, and a, a, to meet the traces (parallel to the ray as before): then a, a, is the shadow point required.

For all these operations are only finding the intersection of the ray with the given plane, by the methods of Descriptive Geometry, as already explained and studied.

PROPOSITION VI.

To find the projections of the shadow of a given line on a given plane. Let P be the given plane, a, ß, the given line, and R the direction of the rays.

1. Through any point c of the given line, draw a parallel to the ray R: then the plane of shadow of the given line will pass through this line and the given line; and its intersection with P will be the shadow on P.

Find one of the traces of the line through e parallel to the ray, as Y1; then trace the plane 2 Y2 α, which will be the shadow plane. Find its intersection with P: then the lines Q. q, and Q2 2 will be the projections of the shadow cast upon the plane P by the line a1 ẞ2.

2. If the light emanate from a point, the difference of construction

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 penumbra 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, develop

* It might perhaps be as well to call the lines of separation of light and neutral lines; or in polyhedrons, neutral edges.

able 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.

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, Y, & 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. VII. 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. Our 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, Y1 be the trace of the cylinder on the plan; draw a, a, parallel to R, to meet the trace in a1, and a, a, parallel to Re, to meet the perpendicular from a, in a,: then a, a, is the projected point of section.

If a, a, meet the trace again in B1 and we find B, as we found a2, 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, any convenient distances from each other; either equal or otherwise.

...n taken at

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