MENSURATION OF SOLIDS.

By a solid body we understand whatever is described under the three dimensions of length, breadth, and depth or thickness. Solids are bounded by plane surfaces, as a cube, a prism, &c. or by curved surfaces, as a globe, a cone, &c.

All solid figures, whose extremities are similar, equal and parallel planes, and all whose other sides are parallelograms, are in general termed prisms, such as are represented in Plate 6, Fig. 5, 6, 7, and 8; and the prism may be produced by the motion of one of the planes at its extremity along a right line, in a position always parallel to that which it had at the beginning of the motion.

The two parallel extremities are called the bases of the prisin, and a perpendicular let fall from one base to the other, gives the altitude of the prism.

If a prism be cut across in any place by a plane paralle! to the base, this section will be perfectly similar, and equal to the base.

A prism is said to be right when the sides are perpendicular to the base; but it is an oblique prism when the sides are not perpendicular to the base.

Prisms are denominated according to the number of their sides; thus, one of three sides, as in Fig. 5, is a triangular prism; Fig. 6 and 7 are quadrangular prisms, their bases consisting of four sides; and Fig. 8 a hexagonal prism, the base consisting of six sides.

Amongst quadrangular prisms are particularly distin guished the parallelopipedon and the cube; the parallelopipedon is a prism whose bases, as well as sides, are

parallel

parallelograms, and, if all the angles are right, the figure is said to be a right-angled parallelopipedon: but if all the angles be right, and all the sides equal to the base, then the figure is termed a cube, which is, therefore, a solid comprehended under six equal squares.

The cube is the common measure of the solidity of all other solid bodies, in the same way that the square is the common measure of all surfaces (see page 441). Thus in Fig. 7, Plate 6, let A be a regular cube, each of whose sides is equal to 6 inches; to find the solid contents of this body, we must adopt some standard or unit to which the solidity must be referred, as, for instance, the small figure B, being a cube of 1 inch every side; when it is evident that the whole question is to discover how often the cube B can be contained in the cube A. As the side of A is 6 inches, it may be divided into 6 equal parts, each equal to B, of which the side is but 1 inch. Let then the cube B be cut or sawn through at each 6th division of the side, when it will be divided into 6 equal figures, each 6 inches long, 6 inches broad, and 1 inch thick; and containing 36 solids of 1 inch a side again, let each of these sections be divided into 6 equal parts by lines drawn through the divisions on one side, when each will be 6 inches long, 1 inch broad, and 1 inch thick, and consequently contain 6 solids of 1 inch a side: lastly, let this section be cut through into 6 equal parts, when each part will be a solid of 1 inch every way, or 1 cubic inch. If now we trace back the operation from this cube, we have in the 3d section a cube of 1 inch; in the 2d a solid figure or parallelopiped of 6 inches; in the 1st section a parallelopiped of 36 inches, and in the whole cube A, a figure containing 216 solid or cubical inches. Hence, in order to find the solid contents of a cube, we multiply its length into its breadth, and the product into its thickness, when the last product will give the contents in

:

solid measures corresponding to those in which the dimensions are taken.

Further, let us suppose a right-angled box A, (Fig. 7, Plate 6,) whose dimensions on the inside are 6 inches in depth, breadth, and length, or that it is a cube of 6 inches a side: let there be a small cube or die B, of 1 inch every way, and let it be required to know how many of these small cubes or dice can be placed within the box A. If the bottom of the box be divided into squares by lines drawn crossing one another at right angles, at the distance of I each asunder, it is evident that on each of these squares 1 die may be placed, and that the bottom will contain 6 rows of 6 dice each, or in all 6x6 36 dice.

But as the box is 6 inches in depth, while the die B is only 1 inch deep, each square in the bottom will support a pile or column of 6 dice, and consequently the 36 squares will support each a pile of 6 dice, making in all 36×6=216 dice contained within the given box A: but each die being

cubic inch, the box will, of course, contain 216 cubic inches, being the cube of the side of the box, 6×6×6

= 216.

From what has been said it follows, that in order to have the solid contents of any prism, we must multiply the superficial area of the base by the altitude of the prism, which is always measured at right angles to the base.

EXAMPLE I.

To find the solid contents of the triangular prism represented in Fig. 5, of which the base is an equilateral triangle 6 inches a side, and the perpendicular altitude is 10 inches.

The area of the base will be found to be 15,588 square inches, (Mensuration of Surfaces, Example 5th,) and this quantity, multiplied by the perpendicular altitude 10 inches, will give 155,88 solid or cubic inches for the solid contents of the given triangular prism.

EXAMPLE

EXAMPLE II.

To find the solid contents of the rectangular parallelopiped shown in Fig. 6, of which the length is 24 inches, the breadth is 18 inches, and the altitude 12 inches.

In the first place, by the 2d Example of Mensuration of Surfaces, we find the superficial contents of the base, which is 24 inches long by 18 broad, to be 432 square inches ; and this multiplied by the altitude 12 inches, will give 5184 solid or cubic inches, for the solid contents of the giveri parallelopiped.

EXAMPLE III.

To find the solid contents of a cube, which is only a paralellopiped, of which the length, breadth, and altitude, are all equal, we have only to cube the side, and the product will be the solid contents. Thus, in the cube represented, Fig. 7, the sides are all equal, each being given 6 inches, the cube of which, or 6×6×6=216 cubic inches, are the solid contents of the given figure.

EXAMPLE IV.

To find the solid contents of a regular hexagonal prism, such as in Fig. 8, whose base is 6 inches a side, and whose altitude is 10 inches.

Agreeably to the rule laid down in Example 9th of Mensuration of Surfaces, the area of the hexagonal base of the prism will be found to be 93,528 square inches, and this quantity multiplied by the altitude, 10 inches, will give 935,28 cubic inches for the solid contents of the given prism.

EXAMPLE V.

In the same manner the solidity of any prism, whatever be the form of its base, or the number of its sides, may be found ?

3 a

found and hence we discover the solidity of a cylinder, which is only a prism on a circular base, that is, on a base formed by a polygon of an infinite number of sides: so that, to discover the contents of the cylinder shown in Fig. 9, we first find the area of the circular base, (Mensuration of Surfaces, Example 13,) and then multiply that quantity by the perpendicular altitude, to obtain the solidity.

If, therefore, the base of the cylinder were a circle of 10 inches diameter, the circumference of course being 31,4159 inches, the area would be 78,53975 square inches, which quantity multiplied by the perpendicular altitude, say 25 inches, the solid contents of the cylinder would be 1953,49375 cubic inches.

EXAMPLE VI.

To find the solid contents of a pyramid, such as are represented in Fig. 10, 11, and 12, of Plate 6.

A pyramid is a solid figure of which the base may be any polygon, but not a circle, and the sides are all triangles, whose bases are the sides of this polygon, and whose vertices unite in one point above the base of the figure, which is called its summit or vertex. Thus Fig. 10 represents a pyramid whose base is a triangle, on each side of which is erected a triangle, and the three unite in the vertex V. Fig. 11 represents a pyramid whose base is a square, and consequently the faces of the figure are four triangles meeting together at the vertex. pyramid whose base is a hexagon, and its sides are of course formed by six triangles all meeting at the vertex of the pyramid V.

Fig. 12 is a

A perpendicular let fall from the vertex to the base of a pyramid will give the measure of its altitude, whether it fall within the base, as in Fig. 11, or without it, which will be the case when the pyramid is much inclined, as in