83. A Solid has length, breadth and thickness.

84. A Prism is a solid of which the ends are equal, similar and parallel straight-sided figures, and of which the other faces are parallelograms.

85. A Triangular Prism is one whose bases or ends are triangles. (Fig. 48.)

86. A Quadrangular Prism is one whose bases or ends are quadrilaterals. (Fig. 49.)

87. A Pentagonal Prism is one whose bases or ends are pentagons. ( Fig. 50.)

88. A Hexagonal Prism is one whose bases or ends are hexagons. (Fig. 51.)

89. A Cube is a prism of which all the faces are squares. ( Fig. 52.)

90. A Cylinder, or properly speaking a Circular Cylinder, is a round solid of uniform diameter, of which the ends or bases arc equal and parallel circles. (Fig.53.)

Fig. 40.   A Quadrangular Prism.

Fig. 40. - A Quadrangular Prism.

Fig. 50.   A Pentagonal Prism.

Fig. 50. - A Pentagonal Prism.

Fig. 51.   A Hexagonal Prism.

Fig. 51. - A Hexagonal Prism.

Fig. 52.   A Cube.

Fig. 52. - A Cube.

Fig. 53.   A Cylinder.

Fig. 53. - A Cylinder.

Fig 54.   A Cone.

Fig 54. - A Cone.

Fig 55.   A Right Cone.

Fig 55. - A Right Cone.

Fig. 56   An Oblique or Scalene Cone.

Fig. 56 - An Oblique or Scalene Cone.

Fig. 57.   A Tiun cated Cone.

Fig. 57. - A Tiun-cated Cone.

91. An Elliptical Cylinder is one whose bases are ellipses.

92. A Right Cylinder is one whose curved surface is perpendicular to its bases.

93. An Oblique Cylinder is one whose curved surface is inclined to its base.

94. A Cone is a round solid with a circle for its base, and tapering uniformly to a point at the top called the apex. (Fig .54.)

95. A Right Cone is one in which the perpendicular let fall from the vertex upon the base passes through the center of the base. This perpendicular is then called the axis of the cone. (Fig. 55.)

96. An Obliqe Cone or Scalene Cone is one in which the axis is inclined to the plane of its base. (Fig. 56)

97. A Truncated Cone is one whose apex is cut off by a plane parallel to its base. (Fig. 57.) This figure is also called a frustum of a cone. A pyramid may also be truncated. (See Figs. 69 and 70 and definition 112.)

98. A Pyramid is a solid having a straight-sided base and triangular sides terminating in one point or apex. Pyramids are distinguished as triangular, quad-rongular, pentagonal, hexagonal, etc., according as the base has three sides, four sides, live sides, six sides, etc. (Figs. 58, 59 and 60.)

Fig 58.   A Trian gular Pyramid.

Fig 58. - A Trian-gular Pyramid.

Fig. 59.   A Quadrangular Pyramid.

Fig. 59. - A Quadrangular Pyramid.

Fig. 60.   An Octagonal Pyramid.

Fig. 60. - An Octagonal Pyramid.

Fig. 61   A Right Pyramid.

Fig. 61 - A Right Pyramid.

Fig. 62.   Altitude of a Cone.

Fig. 62. - Altitude of a Cone.

Fig 63   Altitude of a Pyramid.

Fig 63 - Altitude of a Pyramid.

Fig. 64.   Altitude of a Prism.

Fig. 64. - Altitude of a Prism.

Fig. 65   Altitude of a Cylinder.

Fig. 65 - Altitude of a Cylinder.

Fig. 66.   A Sphere, or Globe.

Fig. 66. - A Sphere, or Globe.

99. A Right Pyramid is one whose base is a regular polygon, and in which the perpendicular let fall from the apex upon the base passes through the center of the base. This perpendicular is then called the axis of the pyramid. (Fig. 61.)

100. The Altitude of a pyramid or cone is the length of the perpendicular let fall from the apex to the plane of the base. The altitude of a prism or cylinder is the distance between its two liases or ends, and is measured by a line drawn from a point in one base perpendicular to the plane of the other. (Figs. 56, 62, 68, 64 and 65.)

101; The Slant hight of a pyramid is the distance from its apex to the middle of one of its sides at the base. The slant hight of a cone is the distance from its apex to any point in the circumference of its base.

102. A Sphere or Globe is a solid bounded by a uniformly curved surface, any point of which is equally distant from a point within the sphere called the center. (Fig. 66.)

103. A Polyhedron is a solid bounded by plane figures. There are live regular polyhedrons, viz.:

104. A Tetrahedron is a solid bounded by four equilateral triangles. It is one form of triangular pyramid. (Fig. 67.)

105. A Hexahedron is a solid bounded by six squares. The common mime for this solid is cube, which see. (Fig. 52.)

Fig. 67.   A Tetrahedron.

Fig. 67. - A Tetrahedron.

Fig. 68.   An Octahedron.

Fig. 68. - An Octahedron.

Fig. 69.   Frustum of a Scalene Cone.

Fig. 69. - Frustum of a Scalene Cone.

Fig. 72.   A Cone Cut by a Plane Parallel to One of Its Sides.

Fig. 72. - A Cone Cut by a Plane Parallel to One of Its Sides.

Fig. 73   A Cone Cut by a Plane Which Makes an Angle with the Base Greater than the Angle Formed by the Side.

Fig. 73 - A Cone Cut by a Plane Which Makes an Angle with the Base Greater than the Angle Formed by the Side.

Fig. 70.   Frustum of a Pyramid.

Fig. 70. - Frustum of a Pyramid.

Fig. 71.   A Cone Cut by a Plane Obliquely through Its Opposite Sides.

Fig. 71. - A Cone Cut by a Plane Obliquely through Its Opposite Sides.

106. The Octahedron is a solid bounded by eight equilateral triangles. (Fig. 68.)

107. The Dodecahedron is a solid bounded by twelve pentagons.

108. The Icosahedron is a solid bounded by twenty equilateral triangles.

109. An Axis is a straight line, passing through a body on which it revolves, or may be supposed to revolve. (Figs. 55 and 61.)

110. By the Envelope of a solid is meant the surface which encases or surrounds it, as the envelope of a cone.

111. Intersection Of Solids is a term used to describe the condition of solids which arc so joined and fitted to each other as to appear as though one passes through the other. The intersection of their surfaces forms the basis of the greater part of the problems of Chap. VI.

112. The Frustum of a Cone or Frustum of a Pyramid is that portion of the original solid which remains after the apex has been cut away upon a plane parallel to the base.. (Figs. 57. 69 and 70.) When the cutting plane is oblique to the base of the solid they are spoken of as oblique frustums.

113. A Conic Section is a curved line formed by the intersection of a cone and a plane. The different conic sections are the triangle, the circle, the ellipse, the parabola and the hyperbola. When the cutting plane passes obliquely through its opposite sides the resulting figure is called an ellipse. (Fig. 71.) (An ellipse is also an oblique section through a cylinder.) When a cone is cut by a plane parallel to one of its sides, the resulting figure is a parabola. Thus in Fig. 72 the cutting plane A B is parallel to the side of the cone C D. See definition 79. When the cutting plane makes a greater angle with the base than the side of the cone makes, or when it passes vertically through the cone to one side of the axis, the resulting figure is a hyperbola, Thus in Fig. 73 the angle A B C is greater than the angle ADE. See definition 80. The parabola and hyperbola resemble each other, both being incomplete figures, with arms extending indefinitely.

The ellipse is a complete figure, but of varying proportions, as the cutting plane is inclined more or less. 114. Concave means hollowed or curved inward, said of the interior of an arched surface or curved line in opposition to convex. (Fig. 74.)

Solids 74Fig. 74.   Sections of Curved Surfaces.

Fig. 74. - Sections of Curved Surfaces.

115. A Convex surface is one that is curved outward, that is regularly protuberant or bulging, when viewed from without. The opposite of convex is concave. (Fig. 74.)