786

THEORY AND PRACTICE OF ENGINEERING.

Rook II.

It is not possible thoroughly to compre-hend the seat of a shadow upon either a ver-tical or horizontal plane, without using themethods already described for projectingthem: by the various positions in which thecube is placed, we can perceive that itsshadow is dependent upon the same prin-ciples as those already described for the for-mation of a solid.

The geometrical form of the cube, shownin the figure, would be as here represented.

The diagonal lines dotted on the plan showthe position of the upper edges of the cube;the perpendicular dotted line on side pro-jection shows its diagonal : by a variety ofinclinations the projections of the cube may-be made to exhibit the parts of machinery',for on every side may be drawn a wheelor other figure, and its projection found eitherupon the vertical or horizontal plane.

Fig. 891.

A sphere or ball, whether its projection be on a ver-tical or horizontal plane, is always found to be circular.

To project a curved line, when the surface in which itlies is curved, and it is not perpendicular to the plane ofthe projection, it is better to form a polygon, and thenfrom each of its angles to drop a perpendicular, fromwhich parallel lines may be drawn to the chords whichsubtend the arcs. But as the curved line has a doubleflexure, it is necessary to inscribe within it another poly-gon, which shall represent the surface in which the curvedline is situated. If the plane on which the shadow of theglobe is represented was not in perspective, the form of itwould be circular; its diameters in every direction beingthe same, so must be that of its shadow.

The cone, when projected upon a perpendicular face, assumes the character of the pyramid,and on the horizontal plane the circle. The projection of any of these figures is exceedinglysimple, and needs but little further observation.

In the shadow of the cone the same rules must be adopted,the seat having been found on the horizontal can be readily,by means of parallel lines, transferred to the vertical; thedirection of the vertical plane does not alter the height of theprojected shadow, though its breadth depends upon its position.

The shadows of curved lines being projections of thosecurves, they may be treated as such : that of a circle, in anyposition where the luminary is not in its plane, will he a conicsection if the shadow be received on a plane, and the form ofthe curve, which will represent it, will depend on the relativeposition of the circle, the luminary, and the plane of theshadow ; therefore the shadow of a circle can only be ob-tained by establishing a sufficient number of points in itscurve, and then drawing lines through them. To find the shadow of any point it is ne-

Fig. 896.