Chap. VIII.

GEOMETRY.

787

cessary to fix its position on the plane of the shadow, and this can only be effected by theordinary rules of projection : if we regard the cylinder of rays which produces the shadowof a circle as a solid capable of being cut by a plane in anydirection, we shall find no difficulty in fixing the points ofeach ray upon the plane so cutting it, and through themtracing the form of the shadow.

A pyramid may be projected on a plane or upright wall,by continuing the lines of its base, and then drawing per-pendiculars to them.

The figure more clearly shows the orthographical pro-jection on two planes at right angles with each other, one ofwhich is termed the horizontal, the other vertical: it simplydepends upon representing a point on any space by drawinga perpendicular from it to both the vertical and horizontalplanes, that on which the perpendicular falls being termedthe projection of the proposed point; if we then imaginelines made up of points, we shall have no difficulty whatever in projecting the entire outlineof any figure.

We may suppose the figure to represent a building, and if it be required to show theposition on the plan of the tiles or slates with which it is covered, w r e have only to dropperpendiculars from them to the plane below, and find where their lines intersect: the apexof the pyramid, which may be the point of a hipped roof, w’ould fall where the diagonalsof the square cross on the plan below ;it is therefore not difficult to transferthe seat of a point from one planeto another.

Fig. 897,

Fig. 898.

In the cylinder it is necessary,before we can project it, toimagine its surface covered by asystem of lines or a series ofpoints, after which each may beprojected to a vertical or hori-zontal plane, and when these areunited by lines, the figure will beprojected. Cylinders may be re-garded as prisms whose bases areeither circular or elliptical, andthese may be resolved into poly-gons which have straight lines forsides, uniting in as many points.

Solids, which have a doublecurvature, require that we shouldconsider them as inclosed withina single surface, and as suchbodies present neither anglesnor lines, they must be re-presented by some apparentcurve which will bound theirsuperficies *. this may accuratelybe shown by a series of tan-gents made parallel to a linedrawn from the centre of thesolid, perpendicular to the plane of projection. When such solids are cutwe must by points and lines find their true figure upon them.

3 e 2

,-4:H pEEfEl

Fig. 899

by other planes,