CHAMBERS'S INFORMATION FOR THE PEOPLE. 



bases. Hence, its lateral faces are rectangles 

 but its bases may be either rhom- 

 boids or rectangles. A rectangular 

 parallelepiped is a right parallel- 

 epiped whose bases are rectangles. 

 Hence it is a parallelepiped all of 

 whose faces are rectangles. Since 

 the perspective of figures in space 



distorts the angles, the diagram may [/' \/ 



represent either a right or a rect- 

 angular parallelepiped. 



A cube is a rectangular paral- 

 lelepiped whose six faces are all 

 squares. 



PROP. XXI. The four diagonals of a paral- 

 lelopiped bisect each other. 



Let ABCD G be a parallelepiped ; its four 

 diagonals AG, EC, BH, DF 

 bisect each other. Through 

 the opposite and parallel 

 edges AE, CG pass a plane 

 which intersects the parallel 

 faces ABCD, EFGH in the 

 parallel lines AC and EG. 

 The figure ACGE is a paral- 

 lelogram, and its diagonals 

 AG and EC bisect each other 

 in the point O. In the same 

 manner it is shewn that AG and BH, AG and 

 DF bisect each other, therefore the four diagonals 

 bisect each other in the point O. 



Scholium. The point O, in which the four 

 diagonals intersect, is called the centre of the 

 parallelepiped ; and it is easily proved that any 

 straight line drawn through O, and terminated by 

 two opposite faces of the parallelepiped, is bisected 

 in that point. 



The Regular Polyhedra. 



PROP. XXII. There cannot be more than five 

 regular polyhedra. 



The faces of a regular polyhedron must be 

 regular polygons, and at least three faces are 

 necessary to form a polyhedral angle. Since 

 (Prop. XX., p. 631) the sum of the plane 

 angles at each vertex is less than four right 

 angles, therefore the faces of a regular poly- 

 hedron must be equilateral triangles, squares, 

 or regular pentagons ; for three hexagons placed 

 with an angular point in common, and having 

 edges common, would make the adjacent angles 

 equal to four right angles. The only possible 

 polyhedra therefore will have their polyhedral 

 angles composed respectively: 



(i.) By three equilateral triangles. 



(2.) By three squares. 



(3.) By four equilateral triangles. 



(4.) By three pentagons. 



(5.) By five equilateral triangles. 

 These figures are named respectively the tetra- 

 hedron, cube, octahedron, dodecahedron, and icosa- 

 hedron. 



Remark. The five regular polyhedra may be 



632 



readily constructed by drawing on card-board the 

 following diagrams. Let them be cut out entire, 

 and at the lines separating adjacent polygons cut 

 the card-board half through. The figure will 

 then readily bend into the form of the respective 

 surfaces. 



Tetrahedron. 



Cube. 



Octahedron. 



Dodecahedron. 



Icosahedron. 



MENSURATION. 



Mensuration is that branch of mathematics 

 which enables us, by means of the principles 

 established in the preceding pages, to measure the 

 length of lines, the areas of surfaces, and the 

 volumes of different bodies. 



OF LINES. 



PROP. I. Having given any two sides of a 

 right-angled triangle, to find the remaining side. 



Thus, for example, in the 

 right - angled triangle A C B, 

 of which ACB is a right 

 angle, let the side AC = 30 ft., g^/ 20 



BC = 40 ft. ; to find the 

 length of the hypotenuse AB. 

 It has been proved that the B 

 square upon the hypotenuse 

 is equal to the sum of the squares on the other 

 two sides, 

 or AB 2 = AC 2 + BC 2 



.'. AB 



: a + BC 2 = V9o + 1600 = 50 



that is, the hypotenuse is found by taking the 

 square root of the sum of the squares of each of 

 the sides containing the right angle. 



Again, let the two sides which are given be AB 

 and BC ; to find AC. 



