SECTION XV.-POLYEDRALS. DEFINITIONS. 1. The divergence of three or more planes from a single point forms a polyedral, or polyedral angle. If the divergence is of three planes, the angle is called a triedral. The corner of the room, where two walls and the ceiling have a common point, is a triedral. 2. When, as in the example just given, the three diedrals of a triedral are each right, the angle is called a trirectangular triedral. 3. An isosceles triedral is one which has two equal face angles. 4. An equilateral triedral is one which has three equal face angles. 5. Two triedrals are equal when, being applied one to the other, they coincide in all their parts. 6. Two triedrals are symmetrical if they are composed of parts which, taken one by one, are respectively equal, but are not similarly arranged. Suppose we have three lines passing through a common point, but not all in one plane; and planes passing through these lines, two and two, as in this figure. The lines AOE and COH are in the plane of the paper. Conceive the point B as behind that plane, and the point S in front of it. O-ABC and O-HSE are symmetrical triedrals. Pupils should analyze the figure, and show that its two parts correspond to the definition of symmetrical. H B A C S E THEOREM XI. The sum of any two face angles of a triedral is greater than the third face angle. Let BAD be the greatest of the face angles of the triedral at A. Then, BAC+ CAD > BAD. B Make the angle BAC' BAC, and AC' = AC. = Then will the triangles BAC and BAC' be equal (?). In the triangles CAD and C'AD the angle CAD > C'AD (Book I, Theo. XVII, Cor.). Add the equal angles BAC and BAC', and D we have BAC + CAD > BAD. THEOREM XII. The sum of the three face angles of any triedral is less than four right angles. Draw the figure used in the last theorem, omitting AC'. The sum of the angles of the triangles having a common point at A is six right angles (?). Prove, by Theorem XI, that the sum of the angles ABD + ABC + ACB + ACD + ADC + ADB is greater than DBC + BCD + CDB; that is, greater than two right angles. Complete the demonstration. THEOREM XIII. Two triedrals having the face angles of the one equal to the face angles of the other, each to each, and similarly arranged, are equal. OS and VS are drawn perpendicular to AC in their respective planes. NX and PX are drawn under similar conditions. Hence, the angle OSV measures the inclination of the two planes whose intersection is AC; that is, the diedral D-AC-B; and NXP measures the diedral G-EH-F (Definitions 5 and 6, Sec. XIV). = But the triangle OSV the triangle NXP; for, OS NX, and VS = PX (Theo. XVIII, Book I), and OV NP (Theo. XII, Book I). Hence, the triangles are mutually equilateral and (Theo. XIX, Book I) equiangular. = = Therefore the angle OSV the angle NXP; that is, the diedrals are equal. In the same way the diedrals whose edges are AB and EF, and those whose edges are AD and EG, may be proved equal. Ques.- Will the equality of the plane angles OVS and NPX prove the equality of the diedrals B-AD-C and F-EG-H? Sch.-Notice that the corresponding faces, the triangles ADC and EGH, etc,, are not necessarily equal; the planes cutting BCD and FHG being used simply to aid the eye to comprehend the riedral angles. A THEOREM XIV. An isosceles triedral and its symmetrical are equal. E F By definition, the triedral O-DEF is the symmetrical of O-ABC. By hypothesis, the angle AOB BOC. If the upper part of the figure rotate about the point O till DOF coincides with AOC, prove that the triedrals coincide throughout, and are equal. Cor. An isosceles triedral has two equal diedrals. SECTION XVI.-POLYEDRONS AND THE CYLINDER. Three planes may be conceived which have no intersection. They are parallel each to the others. Three planes may have one intersection. They pass through a common line, as lines in a plane may pass through a common point. Three planes may have two intersections. Two of them are parallel, like two opposite walls of the room, and the third cuts them. B Three planes may have three intersections. Two of C them may be to each other as the adjacent walls of a room, and the third may cut them like a partition passed through the diagonals of the floor and ceiling; or the third may cut them as the ceiling cuts them, and form a trirectangular triedral. The face angles may not be angles. In the triedral A, each ri of the triangles BAD, BAC, and DAC is a portion of a plane, and three planes have three intersections each passing through A, and each of the face angles may be greater or less than a right angle. It is evident that three planes in no one of these positions enclose a space on all sides; but that, in one case, a fourth plane may complete the enclosure. For example, a plane cutting the intersections AB, AD, and AC, at B, D, and C. Here we have the smallest number of planes which can bound a portion of space, viz., four. DEFINITIONS. 1. A polyedron is a geometrical solid, or portion of a space bounded by planes, called faces. The intersections of the faces are called edges; and of the edges, vertices. 2. The volume of a polyedron is the number of times it contains some unit of measure. What kind of a unit? 3. A diagonal of a polyedron joins two vertices not in the same face. A diagonal plane divides a polyedron by passing through two of its edges. 4. A polyedron of four faces is called a tetraedron; of five, a pentaedron; of six, a hexaedron, etc. Evidently, a polyedron may have any number of sides more than three. 5. A regular polyedron has equal polygons for its faces, and its polyedral angles all equal. There are but five regular polyedrons. 6. Similar polyedrons are identical in form. They have the same number of faces respectively similar and similarly placed, and their respective polyedrals are equal. |