=

Frustum of a Pyramid or Cone.

= } H'' × V + } H′′ × V' + } H" × √V V', and we are to prove that these solidities are equal, or that

V× H— V1× H = V × H' + V'× H' + √ VV' × H". Now

and, by art. 379,

whence

H" = H— H',

and, multiplying extremes and means

NV X H = NVX H.

If we multiply this equation successively by ✅V and V we obtain, by transposing the members of the second product,

√ VV X H = V × H',

NV V X H = V X H;

the difference between which is

√VV × (H— H') = V X H-VX H, or

X

VV X H = VX H-VX H.

And if we add to this last equation the equations

VX HVXH-VX H

VX H = VXH-V × H',

we get, by cancelling the terms which destroy each other,

NVV × H+V × H"'+V1× H" =V× H-V × H,

:

which is the equation to be proved, and the solidity of the frustum is therefore equal to

{ H' × (V + V' +√VV).

401. Corollary. If R is the radius of the lower base of

Faces of Similar Solids.

the frustum of a cone, and R' the radius of the upper base,

VV = √2 × R2 × R12 =1× R × R',

and the solidity of the frustum is

2

3 π × H' × (R2 + R12 + R × R').

402. Scholium. The solidity of any polyedron may be found by dividing it into pyramids.

CHAPTER XVII.

SIMILAR SOLIDS.

403. Definition. Similar polyedrons are those in which the homologous solid angles are equal, and the homologous faces are similar polygons.

404. Corollary. Hence, from art. 169, the sides of similar polyedrons are proportional to each other.

405. Corollary. From art. 268, the faces of similar polyedrons are to each other as the square of their homologous sides; and, from the theory of proportions, the sums of the faces, or the entire surfaces of

Ratios of Similar Solids.

the polyedrons are also to each other as the squares. of the homologous sides.

406. Corollary. The bases of similar prisms or pyramids are to each other as the squares of their altitudes; and the perimeters of their bases are to each other as their altitudes.

407. Corollary. The bases of similar cylinders or cones are to each other as the squares of their altitudes; and their altitudes are to each other as the circumferences of the bases, or as the radii of the

bases.

408. Corollary. The convex surfaces of similar prisms, pyramids, cylinders, or cones are to each other as their bases, or as the squares of their altitudes.

409. Corollary. The convex surfaces of similar prisms or pyramids are to each other as the squares of their homologous sides.

410. Corollary. The convex surfaces of similar cylinders or cones are to each other as the squares of the radii of their bases.

411. Theorem. Similar prisms, pyramids, cylinders or cones are to each other as the cubes of their altitudes.

Demonstration. Prisms, pyramids, cylinders, or cones are to each other, by arts. 366 and 386, as the products of their bases by their altitudes. But where these solids are similar,

Ratios of Similar Solids.

their bases are to each other, by arts. 406 and 407, as the squares of their altitudes; and the products of the bases by their altitudes, or their solidities are to each other, as the products of the squares of their altitudes by their altitudes, or as the cubes of their altitudes.

412. Corollary. Similar prisms or pyramids are to each other as the cubes of their homologous sides.

413 Corollary. Similar cylinders or cones are to each other as the cubes of the radii of their bases.

414. Theorem. Similar polyedrons are to each other as the cubes of their homologous sides.

Demonstration. Let a polyedron be divided into pyramids by drawing lines from one of its vertices to all its other vertices; any similar polyedron may be divided into similar pyramids by lines similarly drawn from the homologous ver

tex.

Now these similar pyramids are to each other, by art. 412, as the cubes of their homologous sides, or as the cubes of any two homologous sides of the polyedrons; and, from the theory of proportions, their sums, that is, the polyedrons themselves, are to each other in the same ratio, or as the cubes of their homologous sides.

Radii and Section of a Sphere.

CHAPTER XVIII.

THE SPHERE.

415. Definition. A sphere is a solid terminated by a curved surface, all the points of which are equally distant from a point within called the centre.

416. Corollary. The sphere may be conceived to be generated by the revolution of a semicircle, DAE (fig. 177) about its diameter DE.

417. Definitions. The radius of a sphere is a straight line drawn from the centre to a point in the surface; the diameter or axis is a line passing through the centre, and terminated each way by the surface.

418. Corollary. All the radii of a sphere are equal ; and all its diameters are also equal, and double of the radius.

419. Theorem. Every section of a sphere made by a plane is a circle.

Demonstration. From the centre C (fig. 178) of the sphere draw the perpendicular CO to the section AMB and the radii CA, CM, CB, &c. Since these radii are equal, they must, by art. 319, be at equal distances from the perpendicular CO; that is, OA, OM, OB, &c. are equal, or AMB is a circle.