OF THE

MENSURATION

OF

SOLIDS.

DEFINITIONS.

1. The measure of any solid body, is the whole capacity or content of that body, when considered under the triple dimensions of length, breadth, and thickness.

2. A cube whose side is one inch, one foot, or one yard, &c. is called the measuring unit; and the content or solidity of any figure is computed by the number of those cubes contained in that figure.

3. A cube is a solid contained by six equal square sides.

4. A parallelopipedon is a solid contained by six quadrilateral planes, every opposite two of which are equal and parallel.

5. A prism is a solid whose ends are two equal, parallel, and similar plane figures, and whose sides are parallelo

grams.

Note. When the ends are triangles it is called a triangular prism; when they are squares, a square prism; when they are pentagons, a pentagonal prism, &c.

6. A cylinder is a solid described by the revolution of a right angled parallelogram about one of its sides, which remains fixed.

7, A * pyramid is a solid whose sides are all triangles meeting in a point at the vertex, and the base any plane figure whatever.

Note.-When the base is a triangle, it is called a triangular pyramid; when a square, it is called a square or quadrangular pyramid; when a pentagon, it is called a pentagonal pyramid, &c.

8. A sphere is a solid described by the revolution of a semicircle about its diameter, which remains fixed.

9. The centre of a sphere is a point within the figure, every where equally distant from the convex surface of it. 10. The diameter of the sphere is a straight line passing

* The definition of a cone has been given already.

through the centre, and terminated both ways by the convex superficies.

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

12. A spheroid is a solid generated by the revolution of a semi-ellipsis about one of its diameters, which is considered as quiescent.

The spheroid is called prolate, when the revolution is made about the transverse diameter, and oblate when it is made about the conjugate diameter.

13. Elliptic, parabolic, and hyperbolic spindles, are generated in the same manner as the circular spindle, the double ordinate of the section being always fixed or quiescent.

14. Parabolic and hyperbolic conoids, are solids formed by the revolution of a semi-parabola or semi-hyperbola about its transverse axis, which is considered as quiescent.

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

16. A frustum or trunk, is the part that remains at the bottom, after the segment is cut off.

17. The zone of a sphere, is that part which is inter

cepted between two parallel planes; and when those planes are equally distant from the centre, it is called the middle zone of the sphere.

18. The height of a solid is a perpendicular, drawn from its vertex to the base or plane on which it is supposed to stand.

PROBLEM I.

To find the solidity of a cube, the height of one of its sides

being given.

RULE.*

Multiply the side of the cube by itself, and that product again by the side, and it will give the solidity required.

EXAMPLES.

1. The side AB, or BC, of the cube ABCDFGHE, is 25.5: what is the solidity?

* Demon. Conceive the base of the cube to be divided into a number of little squares, each equal to the superficial 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.

And consequently, the number of small cubes contained in the whole figure, must be equal to the square of the side of the base multiplied by the height of that figure; or, which is the same thing,

L

Here AB3= 25.53=25.5× 25.5×25.5=25.5 × 650.25 =16581.375 the answer, or content of the cube.

2. The side of a cube is 15 inches: what is the solidity? Ans. ft. 11in. 5pa.

3. What is the solidity of a cube whose side is 17.5 inches ? Ans. 3.1015 feet.

PROBLEM II.

To find the solidity of a parallelopipedon.

RULE.*

Multiply the length by the breadth, and that product again by the depth or altitude, and it will give the solidity required.

EXAMPLES.

1. Required the solidity of a parallelopipedon ABCD FEHG, whose length AB is 8 feet, its breadth FD 4 feet, and the depth or altitude AD 63 feet?

the square of the side of the base multiplied by the base, is equal to the solidity. Q. E. D.

Note. The surface of the cube is equal to six times the square of its side.

*The reason of this rule, as well as the following ones for the prism and cylinder, is the same as that for the cube.

Note. The surface of the parallelopipedon is equal to the sum of the areas of each of its sides or ends.