Cylinder. Parallelopiped.

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 centre of its bases, is called the axis of the cylinder.

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

349. Corollary. The right cylinder (fig. 166) may be considered as generated by the revolution of the 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.

Equality of Prisms and Cylinders.

353. Theorem. The area of the convex surface of a right prism or cylinder is the perimeter or circumference of its base multiplied by its altitude.

Demonstration. a. The area of each of the parallelograms ABFG, BCGH, &c., which compose the convex surface of the right prism (fig. 164) is, by art. 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.

Demonstration. a. Let LMNO &c. (fig. 163) be a section of the prism made by a plane parallel to the bases. It follows, from art. 333, that LM is parallel to AB, MN to BC, &c.; and, consequently, the angle LMN is equal to ABC, by art. 29, the angle MNO to BCD, &c. Moreover, in the parallelograms ABLM, BCMN, &c., AB is equal to LM, BC to MN &c., and the polygons ABCD &c., LMNO &c. are equiangular and equilateral with respect to each other, and would coincide by superposition.

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

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

Solidity of Parallelopipeds.

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

357. Theorem. Two right parallelopipeds are to each other as the products of their bases by their altitudes.

Demonstration. Let the two right parallelopipeds be ABCD EFGH, AKLM NOPQ (fig. 168) which we will denote by AG and AP.

Then, if the sides of the rectangles ABCD and AKLM are commensurable, the rectangles can, by art. 241, be divided into equal rectangles; and, if through each of the vertices of these small rectangles perpendiculars are erected to the plane AL, the parallelopipeds AG and AP are divided into smaller right parallelopipeds. All the parallelopipeds of AG are equivalent, by art. 355, as well as all those of AP; and the number of parallelopipeds in AG is equal to the number of rectangles in ABCD; and the number of parallelopipeds in AP is equal to the number of rectangles in AKLM.

If now the altitudes AE and AN are commensurate, AN can be divided into equal parts, of which AE contains a certain number; and if, through the points of division of AN, planes are drawn parallel to the base AL, each of the partial parallelopipeds of AG and AP are divided into smaller equal parallelopipeds, and all these smallest parallelopipeds are equal to each other.

Now, the whole number of the smallest parallelopipeds contained in AG is the product of the number of rectangles in its base ABCD by the number of divisions of its altitude AE, and the number contained in AP is the product of the

Solidity of Parallelopipeds.

number of rectangles in its base AKLMG by the number of divisions in its altitude AN. Hence

AG : AP = ABCD × AE : AKLM × AN.

358. Corollary. The solidity of any parallelopiped or its ratio to the unit of solidity is, by art. 351, the product of its base by its altitude, that is,

AG ABCD × AE.

359. Corollary. Since, by art. 242,

we have

ABCD = AB × AD,

AG = AB × AD × AE;

or the solidity of a parallelopided is the product of its three dimensions.

360. Corollary. The solidity of a cube is the cube of one of its sides.

361. 'Corollary. Since, by art. 355, any parallelopiped of a rectangular base is equivalent to a right parallelopiped of the same base and altitude, the solidity of any parallelopiped of a rectangular base is the product of its base by its altitude.

362. Theorem. The solidity of any parallelopiped is the product of its base by its altitude.

Demonstration. Any parallelopiped which has ABCD (fig. 169) for its base is, by art. 355, equivalent to the parallelopiped AF, which has the same base, and its sides AH, BE, CF, DG, perpendicular to the base ABCD.

But any other face may as well be assumed for the base of AF as ABCD; taking, then, the rectangle ABEH for

Solidity of Triangular Prism, Prism, Cylinder.

the base, the parallelopiped AF is, by art. 361, equal to the right parallelopiped of the same base and altitude, that is, by drawing DK perpendicular to AB,

AF = DK × ABEH = DK × AB × AH.

But,

hence

ABCD DK × AB;

AF ABCD × AH.

363. Corollary. Any two parallelopipeds of equivalent bases and the same altitude are equivalent.

364. Corollary. Parallelopipeds of the same base are to each other as their altitudes, and parallelopipeds of the same altitude are to each other as their bases.

365. Theorem. The solidity of a triangular prism is the product of its base by its altitude.

Demonstration. Let ABC DEF (fig. 170) be a triangular

prism.

Draw BG parallel to AC, CG parallel to AB, GH parallel to AD, meeting the plane ADF in H. Join ÈH; FH; AH is, evidently, a parallelopiped; and BCG EFH is a triangular prism.

The triangular prisms ABC DEF and BCG EFH are equivalent, by art. 355; since their altitude is the same and their bases ABC and BCG are equal, by art. 76. Hence each of the prisms is half of the parallelopiped AH, and has half its measure, or the product of ABCD by the altitude, that is, the product of its own base by its altitude.

366. Theorem. The solidity of any prism or cylinder whatever is the product of its base by its altitude.

Demonstration. a. The prism ABC &c. FGH &c. (fig. 163) may be divided into the triangular prisms ABC