c, will similarly cast their shadow, b h and d k: h l' and l'k drawn parallel to b c and c d will complete it.

On the elevation, determine likewise the shadow of the cube by making eg equal to ef, making thereby the angle eg f equal to 45°.

The rays of light cast on an object standing on a horizontal plane will produce direct rays of shadow; but if that plane is cut by another plane forming any angle with it or with the horizon, the projection of the ray of shadow will follow the direction of the intervening plane. Thus, in Fig. 94, to determine the projection of the shadow cast by a cube on a vertical plane at any given distance behind it. Let a b c d represent its plan, e f g h its elevation, and determine the ray of light g k, forming an angle of 45° with the horizon. Draw bl, cm, dn, equal to the diagonal a c, and forming likewise angles of 45° with x y. Join b l m n d for the projection of the shadow cast on the horizontal plane.

To show that portion of shadow which is cast on the vertical plane, at o, where c m intersects x y, erect the perpendicular o p, equal to k m, and join k p. At s draw st perpendicular to ry and equal to o p, join p t s, and the whole shadow will be represented under the required conditions by the figure blk ptsd.

Under similar conditions, determine the shadow cast by a square block, say 2 inches side, 1.5 inches high, surmounted by a square pyramid 3 inches high, whose base coincides with the upper surface of the block.

After having obtained in the usual manner the plan and elevation of the solid, determine the direction of the vertical and horizontal rays of light, as already explained, and, at the junction of the horizontal rays with x y, erect the perpendiculars, whose heights are determined by the same process as in the preceding example; from which the

present question is only a slight variation. The inspection of Diagram 95 should prove a sufficient explanation for its complete solution.

The mode of projecting the shadows described in the two last diagrams being well understood, no difficulty whatever need be experienced in treating any kind of prisms in a similar manner. Fig. 96, which represents the shadow cast under similar conditions by a given cone, may serve as a typical example for all like questions, whether relating to cones or pyramids.

The plan a, elevation b, and directions of shadow c and d, being determined, draw e ƒ parallel to cd and equal to it. Join gf and hf for the vertical portion of the cast shadow. Draw a k and a l at right angles to kg and h to determine the shade of the cone.

To project the shadow of a cylinder cast upon a wall under the same conditions. (Fig. 97.)

The plan and clevation of the cylinder being drawn according to the usual method, determine the angles of shadow. From centre A draw A lg produced, forming with x y an angle of 45o. From its vertical projection a' draw likewise the line a' t, forming the same angle with xy. Draw t a" perpendicular to xy: a" will be the projection of the point A, from which describe a circle equal to its projector. At right angles with A a' draw the diameter b c. Draw b s s' and c t t' tangent to both circles, in order to complete the horizontal projection of the shadow, that portion of which, however, lying in the horizontal plane is only required. That portion of the shadow which is projected on the vertical plane is determined thus:

Parallel to k A1 draw d n, and parallel to d d' draw k b b', s a, n d", el c', e e′ and g g'. Parallel to a' t, from the points d' b' a' c', draw lines intersecting these in o d" and g", also in d" and in g'. The point u is determined by drawing t'u parallel to ay.

The visible part

The ellipse

of the curve is drawn through o d" g' and u.
is completed by continuing it through d"' and g".

bsod" g'ut and c will determine the shape of the shadow required.

Determine the boundary of light on a sphere in horizontal and vertical projection; also the shadow projected by that sphere on both planes of projection, according to the same conditions.

Let A represent the plan, and B the elevation, of the sphere. Through the centre a and its vertical projection a' draw the line b d, b' d', forming angles of 45°, and intersecting in d'. At right angles with these lines draw the diameters ef, e'f', and draw the tangents ep, fq, e' p', f' q', parallel to the ray of light: join e c and transfer the distance e c from a to g, g will become the centre of the elliptical shadow of the sphere: through g draw ogn parallel to e a f, for its minor axis.

Parallel to eg draw sr tangent to the plan of the sphere, and from the point s, through the centre a, draw sk, make kt parallel to e c, and produce q f in l, intersecting k t in u. At u, the intersection of these two lines, draw u m parallel to ef; the distanced g m will equal half the major axis or the ellipse. Make g v equal to g m, and through the points m o vn describe the ellipse. At 8 draw si parallel to ef, the semi-ellipse e if will determine the limits of light cast on the sphere. Make a z equal to a i, and the point z will determine the culminating point of light.

To represent the elevation, perpendicular to x y draw z z', m m' i i', gg', intersecting b' d'. These points will correspond with those similarly lettered in the plan: thus f'i' e' will determine the boundary of light in the elevation, z' its culminating point; the distance g' m', produced in g v', will complete the major axis, and n'o', drawn through g, perpendicular to it, will determine the minor

axis of the ellipse formed by the shadow whose projections will be completed as shown in Diagram 98.

The following examples are mostly founded on constructions obtained from the "Traité de Dessin Industriel, par Messrs. Armangaud Jeune et Amouroux;" and a careful inspection of the diagrams will show the analogy between each and the method of construction adopted in similar cases.

Fig. 99 represents a cylinder 1.25 inches in diameter, surmounted by a cylindrical cap 2 inches in diameter and .25 inch high. In this, as well as in the following diagrams, the direction of the rays of light form the usual angle of 45° with both planes of projection; and, for the sake of convenience, only one half of the surface of each horizontal cap surmounting the solids is shown.

From centre A draw the radii A c c' and A ef, at angles. of 45° with the horizontal plane, and draw A d d' at right angles to p q. Draw b b', d d', e e', parallel to A c c', and from each point thus obtained on both circumferences project the perpendiculars b'-b 3, c'-c3, d'-d3, e'-e 3, and from the centre 62, with radius b 2-63, determine the distance b2 64; from centre c 2, with radius c 2-c 3, determine the point c 4; from d 2, with radius d 2-d 3, determine the point d 4; and from the centre ce, with radius e 2-e 3, find the point e 4. A curve drawn by the hand through the points 64, c 4, d 4, e 4 will show the shadow required. A careful inspection of the following diagrams will demonstrate that the mode of obtaining their shadows is precisely similar to that just explained, and therefore requires no further observations.

Fig. 100 represents an octagonal prism, surmounted by an octagonal cap of larger dimensions, and whose sides are parallel to the faces of the prism.

Fig. 101. A cylinder surmounted by a hexagonal cap.

Fig. 102. An octagonal prism, surmounted by a cylindrical cap.

Fig. 103. An octagonal prism, surmounted by a square

cap.

Fig. 104. A square prism, surmounted by a cylindrical

cap.

In these few examples on shadows I have purposely avoided entering into details which belong more to the province of the artist-the influence of reflected lights, of position, of distance, of the atmosphere, &c. The art of keeping or graduating lights or shades, under different circumstances, will be found in "Hints on Freehand Drawing," now preparing for publication.

The foregoing questions being understood, the solution of the following questions will be found easy.

EXAMPLES AND EXERCISES ON SHADOWS.

A pentagonal pyramid, 2 inches side and 3 inches high, rests on the plane of its base. Show its shadow when cast on the horizontal plane, the ray of light forming an angle of 40° parallel to the vertical plane.

Find the shadow of a frustum of a square pyramid, 3 inches high, and whose sides at the base are 3 inches long, and at top 2 inches. One side forms an angle of 30° with the vertical plane, and the ray of light falls at an angle of 60° with the horizon, and 45° with the vertical plane.

A cylinder, 3 inches long and 2 inches in diameter, lies on its side, which forms an angle of 30° with the vertical plane.

Project its shadow, the ray of light forming an angle of 50° with the horizon, and of 60° with the vertical plane.

A tetrahedron of 3 inches edge has neither of its sides parallel to the vertical plane, determine its shadow, the