Proposition XIV.—Planes to which the same straight line is perpendicular, are parallel to each other.
Proposition XV.—If two straight lines, meeting each other, be parallel to two other straight lines, which also meet, but are not in the same plane with the first two ; the plane which passes through the hitter is parallel to the plane which passes through the former two lines.
Proposit on XVI.—If two parallel planes be cut by another plane, their common sections with it are parallels. Proposition XV 11.-1f two straight lines be cut by parallel planes, they will be cut in the same ratio.
Proposition XVIII.—If a straight line be at right angles to a plane, every plane which passes through it will be at right angles to that plane.
Proposition two planes cutting each other be each of them perpendicular to a third plane, their common section will also be perpendicular to it.
Proposition 11.—If a solid angle be contained by three plane angles, any two of them are greater than the third.
Proposition 11I.—Every solid angle is contained by plane angles, which together are less than four right angles.
Proposition every two of three plane angles be greater than the third, and if the straight lines which con tain them be all equal, a triangle may be made of the straight lines that join the extremities of those equal straight lines.
Properties of Solids.
In a prism, all parallel sections which cut the sides, are similar and equal figures ; • or, all parallel sections which would cut the plane of the base, if produced, are similar and equal figures.
In a pyramid, all the parallel sections which are not parallel to the plane of the base, are unequal similar figures.
The properties of a cone are numerous and interesting. If cut parallel to the plane of the base, the section is a circle ; if in any direction through the apex, the section is a plane right-lined triangle; if the cone be cut by a plane inclined to the plane of the base, at any given angle, the section is an ellipsis ; if cut by a plane parallel to any straight line within the solid passing through the apex, the section is denominated an hyperbola ; and if cut by a plane parallel to another plane which touches the curved surface, the section formed by this position of the cutting plane, is called a parabola.
For the purposes of stereotomy, we shall suppose the cone a right one ; and consequently the abscissa of the curves, or sections, will bisect all the double ordinates at right angles.
Definition 1.-1f any semi-conic section be supposed to revolve upon its abscissa, so as to perform an entire revolution, the surface generated by the curve-line is called a conoid, and the abscissa the axis.
Definition 2.—If the semi-conic section be a semi-ellipsis, the solid generated is called an ellipsoid.
Definition 3.—If the generating figure be a semi-parabola, the solid is called a paraboloid.
Definition 4.-1f the generating figure be a semi-hyperbola, the solid is called an hyberboloid.
Definition solids whatever, generated by revolving plane figures upon an axis, are called solids of revolution. Definition parallel sections of conoids are similar figures.
General Principles of Construction. Definition.—Solid angles, which consist of three plane angles, are called trihedrals.
In the construction of trihedrals, besides the three plane angles which form the boundaries of the solid, there are the three inclinations. These inclinations are, by way of dis tinction, called the angles ; the three boundaries are called the sides ; and the sides and angles are indifferently called parts ; any three of which, excepting the three angles, may be found by the following constructions :