782

THEORY AND PRACTICE OF ENGINEERING.

Book II.

To draw an Oblique Prism, we must commence bytracing the profile of the prism parallel to the degree ofits inclination : having defined the inclined axis of theprism in the direction of its length, and lines to showthe surfaces by which it is terminated, upon the axis sodrawn, the polygon which forms the plane of the prismis to be drawn perpendicular to the axis: thus the fouredges of the prism will be obtained.

Right and Oblique Cones may be formed in a similarmanner: thus the base of a cone is the sector of a circlewhose radius is equal to the side, and the arc equal tothe circumference of the circle which is its base.

If we regard the cone as a polygon with an indefinitenumber of sides, we shall have little difficulty in de-veloping it; in the example we have shown twelve sides,and the lines may be imagined to show the edges orarrises.

The Cone may be formed by setting out its superficies, and bending it round until itunites at the edges.

The Base is a regular polygon, of an in-finite number of sides, and consequently itsdevelopment is the sector of a circle, whoseradius is equal to the side of the cone, andthe arc equal to the circumference of thecircle which forms its base.

The ancients made considerable progressin the discovery of the properties of conicsections : Archimedes incidentally refers tothe subject, but his writings do not explainthe whole of the theory relating to it; hetreats, however, of the areas of the sections,and the solids formed by their revolutionabout an axis; he also shows that the area ofa parabola is two-thirds that of its circum-scribing parallelogram, and this for a longtime was the only true quadrature of acurvilineal space known. This great geo-metrician also pointed out what was the Fig. 881.

proportion of elliptic areas to their circum-scribing circles, and of solids formed by the revolution of the different sections to their cir-cumscribing cylinders.

Apollonius , the Greek geometer , cultivated the science of conic sections, and made con-siderable advances on the subject: before his time the different curves were defined by sup-posing right cones to be cut by planes perpendicular to their sides, by which method it wasnecessary to have three different cones to produce the three sections, as a right-angled conefor the parabola, an acute-angled cone for an ellipse, and an obtuse-angled cone for the hy-perbola. Apollonius showed that the three sections might be obtained from any one cone,whether right or oblique.

In setting out a series of courses or zones on the cone, it is necessary to define all thedivisions intended on the base, as well as on the slant lines, and then form each portionseparately; where a cone is to be cased with masonry the thickness of the stone, whenapplied, forms an outer cone, consequently two developments are required.

Fig. 880.

Fig. 882.

Upon a cone so set out may be traced the varieties of curves or sections which produce