plane <$, BAC, CAD, DAE, EAF, FAB; these together shall be less than four right 28. Let the planes in which the 1s are, be cut by a plane, and let BC, CD, DE, EF, FB, be its common sections with the planes. Then : the solid B is contained by three plane < $, CBA, ABF, FBC, of which any two are > the third, <CBA + LABF are> FBC; for the same reason, the two plane Zs at each of the points C, D, E, F, viz., those <s which are at the bases of the As having a common vertex A, are greater than the third Z at the same point, which is one of the angles of the polygon, BCDEF; :, all the 2$, at the bases of the As, are together greater than all the Zs of the polygon; and .: all the 4s of the As = twice as many right's as there are As, i.e., as there are sides in the polygon BCDEF; and : all the <$ of the polygon + four right 28 = twice as many right _$ as there are sides in the polygon; :: all the 28 of the A$ are = all the 4s of the polygon + four right 28; but the _s at the bases of the As are > _$ of the polygon; :: the remaining <s of the As, which are those at the vertex, and contain the solid at A, are < four right Zs. CoR. Hence may we find how many of the plane <s of an equilateral and equiangular polygon be taken to form a solid angle. 1°. Let the plane < be that of an equilateral A. * Then : each <= 60° ; and :: 3 x 60 = 180, 4 x 60 = 240, 5 x 60 = 300: a solid < may be formed of 3, 4, 5 equal _s of an equilateral A ; but not more, : 6 x 60 = 360 = four right Zs. The solids so formed are respectively called the Tetrahedron, the Octahedron, the Icosahedron. 1 2°. Let the equilateral polygon be a square; then each plane <=90°; and :: 3 x 90 = 270 < 360, a solid angle with three right 28 may be formed, but not more, :: 4 x 90 is not <360. This solid is the cube. 3o. Let the polygon be the pentagon; the interior 2 of which = 108°; and : 3 x 108 = 324 < 360. A solid Z, containing three of these 25, may be formed. A regular solid, which has this kind of Z, is called the Dodecahedron. As the interior _ of a hexagon = 120°, no solid < can be formed of the plane _s of a regular hexagon; and the same may be said of the <s of a polygon containing more than six sides. DESCRIPTIVE GEOMETRY. 1. The object of Descriptive Geometry is the invention of methods by which we may represent upon a plane, having only two dimensions, length and breadth, the form and position of a body which possesses three dimensions, length, breadth, and height. This definition of the chief object of descriptive geometry includes its application to points and lines situated above or below the plane, to which they are to be referred ; and thus gives us means of drawing such lines and points with an accuracy unnecessary and unknown to common geometry. For the results obtained in plane geometry do not depend upon the faithfulness with which the figures are drawn, but upon the accuracy with which the reasoning is conducted. But even in these problems we often make assumptions with regard to the mechanical drawing of lines, which we should find it extremely difficult to execute. Thus in Proposition XI. of the Introduction we say, draw AD + BC; but the practical difficulty of drawing ad is as great as that of drawing an, the object of the problem. The means by which descriptive geometry attains its object is the method of Projections, which we now proceed to explain. 2. The projection of a point upon a plane is the foot of the perpendicular let fall from the point upon the plane. Thus, if pp be drawn from P, perdicular to the plane mn, p is the projection of p upon the plane mn. And mn is called the plane of projection. It is also evident that p is the projection of every point of the line pp. 3. The projection of a line upon a plane is the line formed by the projections of every point of the former line. From the different points of the curve a B, let fall perpendiculars on the plane mn; the curve line ab, formed by the feet of these perpendiculars, is the projection of the curve AB. The perpendiculars so drawn will form a surface, called, from the parallelism of the lines which may be supposed to generate it, a cylindrical surface; and for this reason it is often known by the name of the projecting cylinder. The projection ab may be considered as the intersection of the projecting cylinder with the plane mn; and it is evident that every curve, as A'B' traced on this cylinder, will have ab for its projection. 4. When AB is a plane curve, and in a plane perpen dicular to the plane of projection, the projecting cylinder coincides with the plane of the curve; and the projection of the curve AB is the straight line ab: but when the curve is in a plane parallel to the plane of projection, it is manifest that the projection is an equal curve. |