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 met in a point at the top, called the vertex of the pyra

mid.

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

9. The axis of a solid, is a line drawn from the middle 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 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 cen tre 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 a 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 has 12 pentagonal faces; fifth, the icosaedron, which has 20 triangular faces.

NOTE. 1. 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.

Cube one of its sides for the content; that is, multiply the side by itself, and that product by the side again.

* DEMONSTRATION.

Conceive the base of the cube to be di vided into a number of little squares, each equal to the superfi cial measuring unit.

Then will those squares be the bases of a like number of small cubes, which are each equal to the solid measuring unit.

But the number of little squares, contained in the base of the cube, are equal to the square of the side of that base, as has been shown already.