(115.) The centrobaryc method is also applicable to surfaces generated by the motion of a line, by multiplying the length of the line by the distance moved through by its centre of gravity.

Thus, referring to the last figure, supose we wish the surface of the ring generated by the revolution of the circle AB about the axis DE. While the circle generates the solid ring, its circumference generates the surface of the ring. The centre of gravity of the circumference is evidently at A, and moves through a distance denoted by 2 R; this, multiplied by the circumference, which is denoted by 2πr, gives 4 2 R r for the surface of the ring.

While the right-angled triangle SAB, (see figure on the preceding page,) by its revolution about the side SA, generates the cone, the hypothenuse generates the cone's convex surface. The centre of gravity of SB is evidently at the middle point of SB; it therefore describes the circumference of a circle half as great as that of the base, and denoted by AB: this, multiplied by the length SB, gives π ABXSB for the convex surface of the cone.

PROPOSITION XIII.

THEOREM. Every spherical sector is measured by the zone which forms its base, multiplied by a third of the radius; and the whole sphere has for its measure a third of the radius, multiplied by its surface.

Let ABC be the circular sector, which, by its revolution about AC, describes the spherical sector; the zone described by AB being AD × circ. AC, or 2 π AC . AD: and it is to be shown that this zone multiplied by of AC, or that. AC. AD, will measure the sector.

First, suppose, if possible, that T. AC. AD is the measure of a greater spherical sector, say of the spherical sector described by the circular sector ECF similar to ACB.

In the arc EF, inscribe ECF a portion of a regular polygon, such that its sides shall not meet the arc AB; then imagine the polygonal sector ENFC to turn about EC, at the same time with the circular sector ECF. Let CI be a radius of the circle inscribed in the polygon, and let FG be drawn perpendicular to EC. The solid described by the polygonal sector will have for its measure

T. CI2. EG ; but CI is greater than AC by construction, and EG is greater than AD; for, joining AB, EF, the similar triangles EFG, ABD give the proportion

EG AD FG : BD :: CF : CB;

7. CI2. EG

hence EG >AD. For this double reason, is greater than . CA2. AD. The first is the measure of the solid described by the polygonal sector; the second, by hypothesis, is that of the spherical sector described by the circular sector ECF: hence the solid described by the polygonal sector must be greater than the spherical sector; whereas, in reality, it is less, being contained in the latter. Hence our hypothesis was false: therefore, in the first place, the zone or base of a spherical sector multiplied by a third of the radius, cannot measure a greater spherical sector.

Secondly, it is to be shown that it cannot measure a less spherical sector. Let CEF be the circular sector, which, by its revolution, generates the given spherical sector; and suppose, if possible, that .CE. EG is the measure of some smaller spherical sector, say of that produced by the circular sector ACB.

The construction remaining as above, the solid described by the polygonal sector will still have for its measure. CI. EG. But CI is less than CE; hence the solid is less than . CE. EG, which, according to the supposition, is the measure of the spherical sector described by the circular sector ACB: hence the solid described. by the polygonal sector must be less than the spherical sector described by ACB; whereas, in reality, it is greater, the latter being contained in the former. Therefore, in the second place, it is impossible that

the zone of a spherical sector, multiplied by a third of the radius, can be the measure of a smaller spherical sector. Hence every spherical sector is measured by the zone which forms its base, multiplied by a third of the radius.

A circular sector ABC may increase till it becomes equal to a semicircle; in which case, the spherical sector described by its revolution is the whole sphere. Hence the solidity of a sphere is equal to its surface multiplied by a third of the radius.

Cor. The surfaces of spheres being as the squares of their radii, these surfaces multiplied by their radii must be as the cubes of the latter. Hence the solidity of two spheres are as the cubes of their radii, or as the cubes of their diameters.

Schol. Let R be the radius of a sphere: its surface will be 4R; its solidity, 4 R2x R, or T. R3. If the diameter is named D, we shall have RD, and R3= D3: hence the solidity may likewise be expressed by. D3, or D3.

PROPOSITION XIV.

THEOREM. The surface of a sphere is to the whole surface of the circumscribed cylinder, (including its bases,) as 2 is to 3; and the solidities of these two bodies are to each other in the same ratio.

Let MNPQ be a great circle of the sphere, and ABCD the circumscribed square. If the semicircle PMQ and the half square PADQ are at the same time made to re

volve about the diameter PQ, the semicircle will generate the sphere, while the half-square will generate the cylinder circumscribed about that sphere.

The altitude AD of that cylinder is equal to the diameter PQ; the base of the cylinder is equal to the great circle, its di

N

B

M

ameter AB being equal to MN: hence (B. VIII, Prop. Iv,) the convex surface of the cylinder is equal to the circumference of the great circle multiplied by its diameter. This measure (B. VIII, Prop. 1x,) is the same as that of the surface of the sphere: hence the surface of the sphere is equal to the convex surface of the circumscribed cylinder.

But the surface of the sphere is equal to four great circles; hence the convex surface of the cylinder is also equal to four great circles; and adding the two bases, each equal to a great circle, the total surface of the circumscribed cylinder will be equal to six great circles : hence the surface of the sphere is to the total surface of the circumscribed cylinder as 4 is to 6, or as 2 is to 3; which is the first branch of the proposition.

In the next place, since the base of the circumscribed cylinder is equal to a great circle, and its altitude to the diameter, the solidity of the cylinder (B. VIII, Prop. 1,) will be equal to a great circle multiplied by its diameter. But (B. VIII, Prop. xIII,) the solidity of the sphere is equal to four great circles multiplied by a third of the radius; in other terms, to one great circle multiplied by of the radius, or by of the diameter: hence the sphere