GEOMETRY. 



BSA equal to the face BSC, 

 then the dihedral angle SC is 

 equal to the dihedral angle SA. 

 Let SA'B'C' be the correspond- 

 ing trihedral formed by pro- 

 ducing the edges SA, SB, SC. 

 Then since the trihedral SABC 

 has two equal faces, SABC, 

 SA'B'C' are superposable, and 

 therefore the dihedral SC = 

 dihedral SA'; but SA'= SA, therefore dihedral 

 SC = dihedral SA. 



The converse of the proposition is evidently true. 



Cor. i. If the three faces are equal, the three 

 dihedral angles are also equal, and conversely. 



Equality of Trihedral Angles. 



It may be shewn that two trihedral angles are 

 equal : 



(i.) If they have two faces in the one equal to 

 two in the other, and also the contained dihedral 

 angles equal. 



(2.) If they have two dihedral angles in the one 

 equal to two in the other, and a face in the one 

 equal to the corresponding face in the other. 



(3.) If they have three faces in the one equal to 

 three faces in the other, and similarly placed. 



(4.) If they have the three dihedral angles in 

 the one equal to the three in the other, and simi- 

 larly placed. 



Scholium. It is important to notice the analogy 

 which exists between the properties of the tri- 

 hedral angle and those of the triangle ; thus the 

 sides and angles of the triangle correspond with 

 the plane angles and dihedral angles respectively 

 of the trihedral. 



PROP. XIX. The sum of the faces of a tri- 

 hedral angle is less than four 

 right angles. 



Let SABC be the trihedral 

 angle ; to prove that the sum of 

 its faces is less than four right 

 angles. Produce AS through 

 the vertex S to A'. Then in the 

 trihedral SA'BC, we have 

 BSC^BSA'+CSA'. 

 Add to each ASB + ASC ; 



/. BSC + ASB + ASC 

 ^ BSA'+ BSA + CSA'+ CSA 

 ^ 2 right angles + 2 right angles 

 *. 4 right angles. 



PROP. XX. The sum of the faces of any con- 

 vex polyhedral angle is less than four right angles. 



Let SABCDE be a polyhedral ; to prove that 

 the sum of its faces is less 

 than four right angles. 

 Let a plane be drawn 

 cutting the edges on the 

 same side of the vertex 

 in A, B, C, D, E. Take 

 any point O within the 

 figure. Join it with S, A, 

 B, C, D, E. Then in 

 the trihedral angle A we 

 have 





SAE + SAB ^ EAB 



^ OAE + OAB. 



Similarly for the other angular points of the poly- 

 gon ; therefore, by adding, we have the sum of 

 the angles at the base of the triangles SAB, SBC, 

 &c. greater than the sum of the angles at the base 

 of the triangles OAB, OBC, &c. ; but we have as 

 many triangles in the one set as in the other, and 

 therefore the sums of all these angles are equal ; 

 and therefore the sum of the angles at the vertex 

 S is less than the sum of the angles at the 

 vertex O. But all the angles round O make four 

 right angles, therefore the angles at vertex S are 

 together less than four right angles. 



POLYHEDRA. 



Definitions. A polyhedron is a geometrical 

 solid bounded by planes. The bounding planes, 

 by their mutual intersections, limit each other, 

 and determine the faces (which are polygons), the 

 edges, and the vertices of the polyhedron. A 

 diagonal of a polyhedron is a straight line joining 

 any two of its vertices not in the same face. The 

 least number of planes that can form a polyhedral 

 angle is three ; but the space within the angle is 

 indefinite in extent, and it requires a fourth plane 

 to inclose a finite portion of space, or to form a 

 solid ; hence the least number of planes that can 

 form a polyhedron is four. 



A polyhedron of four faces is called a tetra- 

 hedron ; one of six faces, a hexahedron ; one of 

 eight faces, an octahedron ; one of twelve faces, 

 a dodecahedron ; one of twenty faces, an icosa- 

 hedron. 



A polyhedron is convex, when the section 

 formed by any plane intersecting it is a convex 

 polygon. 



A prism is a polyhedron, two of whose faces are 

 equal polygons lying in parallel 

 planes, and having their homolo- 

 gous sides parallel ; the other faces 

 being parallelograms formed by 

 the intersections of planes passed 

 through the homologous sides of 

 the equal polygons. The parallel 

 faces are called the bases of the 

 prism ; the parallelograms taken 

 together constitute its lateral or 

 convex surface ; the intersections of 

 its lateral faces are its lateral edges. The altitude of 

 a prism is the perpendicular distance between the 

 planes of its bases. A triangular prism 

 is one whose base is a triangle; a //\^ 

 quadrangular prism, one whose base is 

 a quadrilateral, &c. 



A right prism is one whose lateral 

 edges are perpendicular to the planes 

 of its bases. In a right prism, any 

 lateral edge is equal to the altitude. 

 An oblique prism is one whose lateral 

 edges are oblique to the planes of its 

 bases. 



A parallelepiped is a prism whose bases are 

 parallelograms. It is therefore a 

 polyhedron all of whose faces are 

 parallelograms. From this defini- 

 tion, it is evident that any two 

 opposite faces of a parallelepiped 

 are equal parallelograms. 



A right parallelepiped is a paral- 

 lelepiped whose lateral edges are 

 perpendicular to the planes of its 



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