MENSURATION.

OF

SOLID S.

I.

DEFINITIONS.

SOLIDS, or BODIES, are figures, having length, breadth

and thickness.

2. A prism is a solid, or body, whose ends are any plane figures, which are equal and similar; and its sides are parallelograms.

A prism is called a triangular prism, when its ends are triangles; a square prism, when its ends are squares; a pentagonal prism, when its ends are pentagons; and so on.

3. A cube is a square prism, having six sides, which are all squares. It is like a die, having its sides perpendicular to one another.

4. A parallelopipedon is a solid, having six rectangular sides, every opposite pair of which are equal and parallel.

5. A cylinder is a round prism, having circles for its ends.

6. A pyramid is a solid, having any plane figure for a base, and its sides are triangles, whose vertices meet in a point at the top, called the vertex of the pyramid.

The pyramid takes names according to the figure of its base, like the prism; being triangular, or square, or hexagonal, &c.

7. A cone is a round pyramid, having a circular base.

9. The axis of a solid is a line, drawn from the midde of one end to the middle of the opposite end; as between the opposite ends of a prism. Hence the axis of a pyramid is the line from the vertex to the middle of the base, or the end, on which it is supposed to stand. And the axis of a sphere is the same as a diameter, or a line passing through the centre, and terminated by the surface on both sides.

10. When the axis is perpendicular to the base, it is a right prism, or pyramid otherwise it is oblique.

11. The height or altitude of a solid is a line, drawn from its vertex, or top, perpendicular to its base. This is equal to the axis in a right prism or pyramid; but in an oblique one, the height is the perpendicular side of a right-angled triangle, whose hypotenuse is the axis.

12. Also a prism or pyramid is regular or irregular, as its base is a regular or an irregular plane figure.

13. The segment of a pyramid, sphere, or any other solid, is a part, cut off the top by a plane parallel to the base of that figure.

14. A frustum, is the part, that remains at the bottom after the segment is cut off.

15. A zone of a sphere is a part, intercepted between two parallel planes; and is the difference between two segments. When the ends, or planes, are equally distant from the centre on both sides, the figure is called the middle zone.

16. The sector of a sphere is composed of a segment less than a hemisphere or half sphere, and of a cone, having the same base with the segment, and its vertex in the centre of the sphere.

17. A circular spindle is a solid, generated by the revolution of a segment of a circle about its chord, which remains fixed.

18. A spheroid, or ellipsoid, is a solid, generated by the revolution of an ellipse about one of its axes. It is prolate, when the revolution is about the transverse axis; and oblate, when about the conjugate.

19. A conoid is a solid, formed by the revolution of a parabola, or hyperbola, about the axis; and is accordingly called parabolic, or hyperbolic. The parabolic conoid is also called á paraboloid; and the • hyperbolic conoid, a hyperboloid.

D

20. A spindle is formed by any of the three conic sections, revolving about a double ordinate, like the circular spindle.

21. A regular body is a solid, contained under a certain number of equal and regular plane figures of the same

sort.

22. The faces of the solid are the plane figures, under which it is contained. And the linear sides, or edges, of the solid are the sides of the plane faces.

23. There are only five regular bodies; namely, first, the tetraedron, which is a regular pyramid, having four triangular faces; second, the hexaedron, or cube, which has 6 equal square faces; third, the octaedron, which has 8 triangular faces; fourth, the dodecaedron, which bas 12 pentagonal faces; fifth, the icosaedron, which has 20 triangular faces.

NOTE I. If the following figures be exactly drawn on pasteboard, and the lines cut half through, so that the parts may be turned up and glued together, they will represent the five regular bodies; namely, figure 1 the tetraedron, figure 2 the hexaedron, figure 3 the octaedron, figure 4 the dodecaedron, and figure 5 the icosaedron.

Figure 1.

Figure 2.

Figure