since the triangles ABC, abc are similar (Prop. 8, ante).

Substituting now the first ratio of (3) for its equivalent, the second ratio of (2), and then the first ratio, of (2) for its equivalent (since AD = ab), the second ratio of (1), we have

▲ ABC: A ADC:: A ADC: & abc. Q. E. D. Corol. The same proposition is true of the frustum of any pyramid or of a cone (Prop. 9 and 10) which is equivalent to three cones having the upper base, the lower base and a mean proportional between the two for bases, and for a common altitude the altitude of the frustum. In symbols r and r', being the radii of the bases, and h the altitude of the frustum, its volume would be expressed by (th. 73, schol.)

If a sphere be cut by a plane, the section will be a circle.

Because the radii of the sphere are all equal, each of them being equal to the radius of the describing semicircle, it is evident that if the section pass through the center of the sphere, then the distance from the center to every point in the periphery of that section. will be equal to the radius of the sphere, and the section will, therefore, be a circle of the same radius as the sphere. But if the plane do not pass through the center, draw a perpendicular to it from the center, and draw any number of radii of the sphere to the intersection of its surface with the plane; then these radii are evidently the hypothenuses of a corresponding number of right-angled triangles, which have the perpendicular from the center on the plane of the section, as a common side; consequently, their other sides are all equal, and, therefore, the section of the sphere by the plane is a circle, whose center is the point in which the perpendicular cuts the plane,,

Scholium. All the sections through the center are

equal to one another, and are greater than any other section which does not pass through the center. Sections through the center are called great circles, and the other sections small or less circles.

PROP. XIV.

Every sphere is two thirds of its circumscribing cylinder.

Let ABCD be a section of the cylinder, and EFGH a section of the sphere through the center I, and join AI, BI. Let FIH be parallel to AD or BC, and EIG and KL parallel to AB or DC, the base of the cylindric section; the latter line KL meeting BI in M, and the circular section of the sphere in N.

E

A

D

/P

F

B

L

Q

K

H

Then, if the whole plane HFBC be conceived to revolve about the line HF as an axis, the square FG will describe a cylinder AG, and the quadrant IFG will describe a hemisphere EFG, and the triangle IFB will describe a cone IAB. Also, in the rotation, the three lines, or parts, KL, KN, KM, as radii, will describe corresponding circular sections of these solids, viz., KL a section of the cylinder, KN a section of the sphere, and KM a section of the cone.

Now, FB being equal to FI or IG, and KM parallel to FB, then, by similar triangles, IK = KM (Geom. Theor., 63), and IKN is a right-angled triangle; hence IN is equal to IK+KN2 (theor. 26). But KL is equal to the radius IG or IN, and KM=IK; therefore KL' is equal to KM2+KN', or the square of the longest radius of the above-mentioned circular sections is equal to the sum of the squares of the two others. Now circles are to each other as the squares of their diameters, or of their radii, therefore the circle described by KL is equal to both the circles described by KM and KN; or the section of the cylinder is equal to both the corresponding sections of the

sphere and cone. And as this is always the case in every parallel position of KL, it follows that the cylinder EB, which is composed of all the former sections, is equal to the hemisphere EFG and cone IAB, which are composed of all the latter sections, the number of the sections being the same, because the three solids have the same altitude.

But the cone IAB is a third part of the cylinder EB (Prop. 11, cor. 3); consequently, the hemisphere EFG is equal to the remaining two thirds, or the whole sphere EFGH is equal to two thirds of the whole cylinder ABCD.

Corol. 1. A cone, hemisphere, and cylinder of the same base and altitude are to each other as the numbers 1, 2, 3.*

Corol. 2. All spheres are to each other as the cubes of their diameters, all these being like parts of their circumscribing cylinders.

Corol. 3. From the foregoing demonstration it appears that the spherical zone or frustum EGNP is equal to the difference between the cylinder EGLO and the cone IMQ, all of the same common height IK. And that the spherical segment PFN is equal to the difference between the cylinder ABLO and the conic frustum AQMB, all of the same common altitude FK.

Scholium. By the scholium to Prop. 11, we have cone AIB: cone QIM :: IF: IK':: FH3: (FH-2FK)' ...cone AIB: frust. ABMQ:: FH': FH-(FH-2FK)" ::†FH3:6FH2.FK-12FH.FK2+8FK3; but cone AIB = one third of the cylinder ABGE; hence cyl. AG: frust. ABMQ:: 3FH":6FH'.FK-12FH.FK2

Now cyl. AL: cyl. AG:: FK: FI.

+8FK'.

Multiplying the last two proportions, and striking out the common factors from the ratios, observing, also, that FIFH, we have

*The surfaces of the sphere and circumscribing cylinder are in the same ratio as their solidities. For the demonstration, see Mensuration. + Raising the binomial to the third power.

cyl. AL: frust. ABMQ::6FH: 6FH-12FH.FK + 8FK2 .. (dividendo, and by corol. 3 of this Prop., 14), cyl. AL: segment PFN :: 6FH2: 12FH.FK-8FK' ::FH: FK(3FH—2FK). But cylinder AL = circular base whose diameter is AB or FH, multiplied by the height FK; hence cylinder AL circle EFGH× FK.

=

..segment PFN=

2 circle EFGH
FH

3

(3FH—2FK)FK2.

SPHERICAL GEOMETRY.

DEFINITIONS.

1. A SPHERE is a solid terminated by a curve surface, and is such that all the points of the surface are equally distant from an interior point, which is called the center of the sphere.

We may conceive a sphere to be generated by the revolution of a semicircle APB about its diameter AB; for the surface described by the motion of the curve APB will have all its points equally dis- o tant from the center O.

The sector of a circle AOC at the same time generates a spherical sector.

A

2. The radius of a sphere is a straight B

line drawn from the center to any point on the surface. The diameter or axis of a sphere is a straight line drawn through the center, and terminated both ways by the surface.

It appears from Def. 1 that all the radii of the same sphere are equal, and that all the diameters are equal, and each double of the radius.

3. It will be demonstrated (Prop. 1) that every section of a sphere made by a plane is a circle; this being assumed,

A great circle of a sphere is the section made by a plane passing through the center of the sphere.

A small circle of a sphere is the section made by a plane which does not pass through the center of the sphere.

4. The pole of a circle of a sphere is a point on the surface of the sphere equally distant from all the points in the circumference of that circle.

It will be seen (Prop. 2) that all circles, whether great or small, have two poles.