From a (Fig. 442) in the plan, and from c in the elevation, draw the rays ab and ce; then ao will show the extent of the shadow on the first riser, as at A; fg will determine the shadow on the second riser, as at B; cd gives the amount of shadow on the first tread, as at C, and h i that on the second tread, as at D; which completes the shadow of the left-hand pedestal, both on the plan and elevation. A mere inspection of the figure will be sufficient to show how the shadow of the right-hand pedestal is obtained.

Fig. 442.

Fig. 443.

Fig. 444.

## 581. - To Find The Shadow Thrown On A Column By A Square Abaeus

From a and b (Fig. 443) draw the rays ac and be, and from c erect the perpendicular ce; tangical to the curve at d draw the ray df, and from k, corresponding to f in the plan, draw the ray ho; take any point between a and f, as i, and from this, as also from a corresponding point n, draw the rays ir and ns; from r and from d erect the perpendiculars rs and do; through the points e, s, and o trace the curve as shown in the figure; then the extent of the shadow will be defined.

Fig. 445.

## 582. - To Find The Shadow Thrown On A Column By A Circular Abacus

This is so nearly like the last example that no explanation will be necessary, farther than a reference to the preceding article.

## 583. - To Find The Shadows On The Capital Of A Column

This may be done according to the principles explained in the examples already given; a quicker way of doing it, however, is as follows: if we take into consideration one ray of light in connection with all those perpendicularly under and over it, it is evident that these several rays would form a vertical plane, standing at an angle of 45 degrees with the face of the elevation. Now we may suppose the column to be sliced, so to speak, with planes of this nature - cutting it in the lines a b, c d, etc. (Fig. 445), and, in the elevation, find by squaring up from the plan, the lines of section which these planes would make thereupon. For instance: in finding upon the elevation the line of section ab, the plane cuts the ovolo at e, and therefore f will be the corresponding point upon the elevation; h corresponds with g, i withy, 0 with s, and l with b. Now, to find the shadows upon this line of section, cast from m the ray m n, from h the ray h o, etc.; then that part of the section indicated by the letters m f i n, and that part also between h and o will be under shadow. By an inspection of the figure, it will be seen that the same process is applied to each line of section, and in that way the points p, r, t, u, v, w, x, as also 1, 2, 3, etc., are successively found, and the lines of shadow traced through them.

Fig. 446.

Fig. 447.

Fig. 446 is an example of the same capital with all the shadows finished in accordance with the lines obtained on Fig. 445.