14 MATHEMATICS. 
three pianes, then 1s the sum of any two of the plane angles greater than 
the third. In any case, however, the sum of any number of plane angles, 
forming a solid angle, is .ess than four right angles. 
2. OF ANGULAR SOLIDS. 
A solid may be inclosed either by plane surfaces alone, in which case it 
is called a polyhedron, or by curved surfaces alone, or by both plane and 
curved at the same time. Bodies of the first and third kind have a base, 
that is, a plane surface upon which the solid is supposed to rest. If such a 
body should have another plane bounding surface parallel to this base (in 
which case this plane may be considered another base), or a vertex opposite 
to the base, the distance between the two surfaces, or between the vertex 
and the base (in both cases measured by a perpendicular let fall), is called 
the altitude of the solid. The planes bounding a polyhedron are called its 
faces; their intersections, its edges. No polyhedron can have less than four 
faces, four solid angles, or six edges. Furthermore, no polyhedron can be 
inclosed by figures of six or more sides, or have equal solid angles formed 
by six or more plane angles. 
Two solids are said to be equivalent when the spaces inclosed between 
their bounding surfaces are equivalent; they are equal when they agree 
exactly in shape and size, so that the one may be taken for the other. 
A polyhedron is called regular when it is inclosed by perfectly regular 
and equal figures, and has all its angles equal. There are only five 
regular solids; 1, tetrahedrons, bounded by four triangles (pl. 2, fig. 56); 2, 
octohedrons, by eight (fig. 58) ; 3, tcosahedrons, by twenty (fig. 60); 4, 
hexahedrons, bounded by six squares (fig. 57); 5, dodecahedrons, by 
twelve pentagons (fig. 59). The expansion of some of these solids, or the 
representation of their surfaces as spread out in a plane, may be found in 
pl. 4, where jig. 49 is the expansion of the tetrahedron, fig. 50 that of the 
hexahedron, fig. 51 of the dodecahedron. A solid, bounded by regular 
figures of two kinds, and which has, at the same time, all the solid angles 
equal, is called an Archimedean solid. If we limit ourselves to polyhedrons 
having triangles and squares for faces, such a solid may be contained, 1, 
by two triangles and three squares (a special case of the three-sided prism) ; 
2, by eight triangles and six squares; 3, by eight triangles and eighteen 
squares (pl. 2, fig. 73”) ; and, 4, by thirty-two triangles and six squares. 
The most important angular solids are the prisms and pyramids. A 
prism is a solid bounded by two equal and parallel rectilineal figures 
(forming the bases) and as many parallelograms as each base has sides. 
It is called three, four, five-sided, as the bases are triangles, quadrilaterals, 
pentagons, &c. (pl. 2, figs. 61, 62, 63). The prism is called a right prism if 
the lateral faces are perpendicular to the bases, otherwise it is oblique. A 
fowr-sided prism whose bases are parallelograms, is called a parallelopipedon ; 
when all the faces are squares, it is a cube or hexahedron. If a prism be 
intersected by a plane parallel to the base, the section formed will be equal 
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