Equal Solid Angles.

CDS+ SDE > CDO + ODE, &c.

Hence ABS + SBC+BCS+ SCD+&c., or the sum of the angle at the bases of the triangles, which have their vertices at S, is greater than ABO + OBC +BCO + OCD+ &c., or the sum of the angles at the bases of the triangles which have their vertices at 0.

If, then, these two sums of the angles at the bases of the triangles are subtracted from the common sum of all the angles of each set of triangles, the remaining sum of the angles which have their vertices at S must be less than the sum of the angles which have their vertices at 0, or, by 26, than four right angles.

340. Theorem. If two solid angles are respectively contained by three plane angles which are equal, each to each, the planes of any two of these angles in the one have the same inclination to each other as the planes of the homologous angles in the other.

Proof. Let the solid angles S, S' (fig. 161) be included by the plane angles ASBA'S'B', ASC-A'S'C', BSC B'S'C'.

Take SASA' of any length at pleasure. Draw AB, AC, perpendicular to SA, in the planes ASB and ASC; and draw A'B', A C', perpendicular to S'A' in the planes A S'B' and A'S'C'.

In the triangles ASB, A'S'B, the side ASA'S, the angle ASBA'S'B'; and the right angle SAB = S'A'B' ; hence, by § 54, ᎯᏴ - A'B' and SB = S'B'.

In the same way, it may be shown that AC: SC S'C'.

A'C',

Join BC, B'C', and, in the triangles SBC, S'B'C', the angle BSC B'S'C', the side SB S'B', and the side SC S'C'; hence, by § 52, BC — B'C'.

In the triangles ABC, A'B'C' the three sides are respec

Solids of equal Heights and equivalent Sections are Equal.

tively equal, and, therefore, by § 61, the angle BAC, which, by § 315, measures that of the planes ASB, ASC is equal to B'A'C', which measures the angle of the planes A'S'B', A'S'C'.

In the same way, it may be shown that the angles of the other planes are equal; some changes, easily made, are, however, required in the demonstration when either of the angles ASB, ASC is obtuse.

CHAPTER XVI.

SURFACE AND SOLIDITY OF SOLIDS.

341. Definitions.

Equivalent solids are those which

have the same bulk or magnitude.

A lamina or slice is a thin portion of a solid included between two parallel planes.

342. Theorem. If two solids have equal bases and heights, and if their sections, made by any plane parallel to the common plane of their bases, are equal, they are equivalent.

Proof. Let ABCDEF, A'B'C'D'EF' (fig. 162) be the two solids. Let MNO, MNO be two equal sections made by a plane parallel to the base, and let PQR, P'Q'R' be two other equal sections made by a plane infinitely near the former plane, and parallel to it.

The infinitely thin laminæ MNOPQR, M'NO' P'Q'R' are equal; for if M'NO' be applied to its equal MNO, PQ'R' must be infinitely near coincidence with its equal PQR; and the lamina themselves can differ from coincidence only by a quantity infinitely smaller than either of them, and which may, by § 99 and 205, be neglected.

Polyedron, Prism.

But by drawing a series of parallel planes, infinitely near each other, the given solids are divided into laminæ, which are respectively equal to each other; and, therefore, their sums or the entire solids must be equivalent.

343. Definitions. Every solid bounded by planes is called a polyedron.

The bounding planes are called the faces; whereas the sides or edges are the lines of intersection of the faces.

344. Definitions. A polyedron of four faces is a tetraedron, one of six is a hexaedron, one of eight is an octaedron, one of twelve a dodecaedron, one of twenty an icosaedron, &c.

The tetraedron is the most simple of polyedrons; for it requires at least three planes to form a solid angle, and these three planes leave an opening, which is to be closed by a fourth plane.

345. Definitions. A prism is a solid comprehended under several parallelograms, terminated by two equal and parallel polygons, as ABC &c. FGH &c. (fig. 163).

The bases of the prism are the equal and parallel polygons, as ABC &c., and FGH &c.

The convex surface of the prism is the sum of its parallelograms, as ABFG + BCGH+ &c.

The altitude of a prism is the distance between its bases, as PQ.

346. Definitions. A right prism is one whose lateral faces or parallelograms are perpendicular to the bases, as ABC &c. FGH &c. (fig. 164).

In this case each of the sides AF, BG &c. is equal to the altitude.

Cylinder, Parallelopiped, Cube, Unit of Solidity, Volume.

347. Definitions. A prism is triangular, quadrangular, pentagonal, hexagonal, &c., according as its base is a triangle, a quadrilateral, a pentagon, a hexagon, &c.

348. Definitions. The prism, whose bases are regular polygons of an infinite number of sides, that is, circles, is called a cylinder (fig. 165).

The line OP, which joins the centres of its bases, is called the axis of the cylinder.

In the right cylinder (fig. 166) the axis is perpendicular to the bases, and equal to the altitude.

349. Corollary. The right cylinder (fig. 166) may be considered as generated by the revolution of the right parallelogram OABP about the axis OP.

The sides OA and PB generate, in this case, the bases of the cylinder, and the side AB generates its convex surface.

350. Definitions. A prism whose base is a parallelogram (fig. 167) has all its faces parallelograms, and is called a parallelopiped.

When all the faces of a parallelopiped are rectangles, it is called a right parallelopiped.

351. Definitions. The cube is a right parallelopiped, comprehended under six equal squares.

The cube, each of whose faces is the unit of surface, is assumed as the unit of solidity.

352. Definition. The volume, solidity, or solid contents of a solid, is the measure of its bulk, or is its ratio to the unit of solidity.

353. Theorem.

The area of the convex surface of

Convex Surface of right Prism or Cylinder.

a right prism or cylinder is the perimeter or circumference of its base multiplied by its altitude.

Proof. a. The area of each of the parallelograms ABFG, BCGH, &c., which compose the convex surface of the right prism (fig. 164) is, by § 247, the product of its base AB, BC &c., by the common altitude AF; and the sum of their areas, or the convex surface of the prism, is the sum of these bases, or the perimeter ABCD &c., by the altitude AF.

b. This demonstration is extended to the right cylinder by increasing the number of sides to infinity.

354. Theorem.

The section of a prism or cylinder made by a plane parallel to the bases is equal to either base.

Proof. a. Let LMNO, &c. (fig. 163) be a section of the prism made by a plane parallel to the bases. It follows, from § 334, that LM is parallel to AB, MN to BC, &c.; and, consequently, the angle LMN is equal to ABC, by 29, the angle NMO to BCD, &c. Moreover, in the parallelograms ABLM, BCMN, &c., AB is equal to LM, BC to MN, &c., and the polygons ABCD &c., LM NO, &c. are equiangular and equilateral with respect to each other, and are, therefore equal, by § 195.

b. The demonstration is extended to the cylinder by increasing the number of sides to infinity.

355. Corollary. Hence, from § 342, two prisms or cylinders of equal bases and altitudes are equivalent.

356. Corollary. Any prism or cylinder is equivalent to a right prism or cylinder of the same base and altitude.