be formed on the hemisphere ADEFG, 25 triangles, all equal to each other, being mutually equilateral. The entire sphere will contain 50 of these small triangles, and the lune ADBE 8 of them. Hence the area of the lune is to the surface of the sphere, as 8 to 50, or as 4 to 25; that is, as the arc DE to the circumference.

Secondly. When the ratio of the arc to the circumference can not be expressed in whole numbers, it may be proved, as in Prop. XIV., B. III., that the lune is still to the surface of the sphere, as the angle of the lune to four right angles.

Cor. 1. On equal spheres, two lunes are to each other as the angles included between their planes.

Cor. 2. We have seen that the entire surface of the sphere is equal to eight quadrantal triangles (Prop. X., Cor.). If the area of the quadrantal triangle be represented by T, the surface of the sphere will be represented by 8T. Also, if we take the right angle for unity, and represent the angle of the lune by A, we shall have the proportion

area of the lune: 8T:: A : 4.

Hence the area of the lune is equal to

8AXT

or 2A×T.

9 4

Cor. 3. The spherical ungula, comprehended by the planes ADB, AEB, is to the entire sphere, as the angle DCE is to four right angles. For the lunes being equal, the spherical ungulas will also be equal; hence, in equal spheres, two ungulas are to each other as the angles included between their planes.

PROPOSITION XIX. THEOREM.

If two great circles intersect each other on the surface of a hemisphere, the sum of the opposite triangles thus formed, is equivalent to a lune, whose angle is equal to the inclination of the two circles.

Let the great circles ABC, DBE intersect each other on the surface of the hemisphere BADCE; then will the sum of the opposite triangles ABD, CBE be equivalent to a lune whose A angle is CBE.

For, produce the arcs BC, BE till they meet in F; then will BCF be a semicircumference, .as also ABC. Sub

D

T

B

E

C

tracting BC from each, we shall have CF equal to AB. For the same reason EF is equal to DB, and CE is equal to AD.

Hence the two triangles ABD, CFE are mutua ly equilateral; they are, therefore, equivalent (Prop. XV.). But the two triangles CBE, CFE compose the lune BCFE, whose angle is CBE; hence the sum of the triangles ABD, CBE is equivalent to the lune whose angle is CBE. Therefore, if two great circles, &c.

PROPOSITION XX, THEOREM.

The surface of a spherical triangle is measured by the excess of the sum of its angles above two right angles, multiplied by the quadrantal triangle.

Let ABC be any spherical triangle; its surface is measured by the sum of its angles A, B, C diminished by two right angles, and multiplied by the quadrantal triangle.

I

Produce the sides of the triangle ABC, until they meet the great circle DEG, drawn without the triangle. The two triangles ADE, AGH are together equal to the lune whose angle is A (Prop. XIX.); measured by 2A ×T (Prop. XVIII., Cor. 2).

ADE+AGH=2A×T.

For the same reason,

also,

BFG+BDI =2B×T;

CHI+CEF=2C×T.

H

D

B

F

E

and this lune is Hence we have

But the sum of these six triangles exceeds the surface of the hemisphere, by twice the triangle ABC; and the hemisphere is represented by 4T; hence we have

4T+2ABC-2A ×T+2BxT+2CxT;

or, dividing by 2, and then subtracting 2T from each of these equals, we have

ABC

AXT÷B×T+C×T—2T, =(A+B+C-2) ×T.

Hence every spherical triangle is measured by the sum of its angles diminished by two right angles, and multiplied by the quadrantal triangle.

Cor. If the sum of the three angles of a triangle is equal to three right angles, its surface will be equal to the quadrantal triangle; if the sum is equal to four right angles, the surface of the triangle will be equal to two quadrantal triangles; if the sum is equal to five right angles, the surface will be equal to three quadrantal triangles, etc.

PROPOSITION XXI. THEOREM.

The surface of a spherical polygon is measured by the sum of its angles, diminished by as many times two right angles as it has sides less two, multiplied by the quadrantal triangle.

E

Let ABCDE be any spherical polygon. From the vertex B draw the arcs BD, BE to the opposite angles; the polygon will be divided into as many triangles as it has sides, minus two. But the surface of each triangle is measured by the sum of its angles minus two right angles, mul- A tiplied by the quadrantal triangle. Also,

D

B

the sum of all the angles of the triangles, is equal to the sum of all the angles of the polygon; hence the surface of the polygon is measured by the sum of its angles, diminished by as many times two right angles as it has sides less two, multiplied by the quadrantal triangle.

Cor. If the polygon has five sides, and the sum of its an gles is equal to seven right angles, its surface will be equal to the quadrantal triangle; if the sum is equal to eight right angles, its surface will be equal to two quadrantal triangles; if the sum is equal to nine right angles, the surface will be equal to three quadrantal triangles, etc.

BOOK X.

THE THREE ROUND BODIES.

Definitions.

1. A cylinder is a solid described by the revolu tion of a rectangle about one of its sides, which remains fixed. The bases of the cylinder are the circles described by the two revolving opposite sides of the rectangle.

2. The axis of a cylinder is the fixed straight line about which the rectangle revolves. The opposite side of the rectangle describes the convex surface.

3. A cone is a solid described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed. The base of the cone is the circle described by that side containing the right angle, which revolves.

4. The axis of a cone is the fixed straight line about which the triangle revolves. The

hypothenuse of the triangle describes the convex surface. The side of the cone is the distance from the vertex to the circumference of the base.

5. A frustum of a cone is the part of a cone next the base, cut off by a plane parallel to the base.

6. Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals.

PROPOSITION I. THEOREM.

The convex surface of a cylinder is equal to the product of its altitude by the circumference of its base.

Let ACE-G be a cylinder whose base is the circle ACE and altitude AG; then will its convex surface be equal to the product of AG by the circumference ACE.

H

B

In the circle ACE inscribe the regular polygon ABCDEF; and upon this polygon G et a right prism be constructed of the same altitude with the cylinder. The edges AG, BH, CK, &c., of the prism, being perpendicular to the plane of the base, will be contained in the convex surface of the cylinder. The convex surface of this prism is equal to the product of its altitude by the perimeter of its base (Prop. I., B. VIII.). Let, now, the arcs subtended by the sides AB, BC, &c., be bisected, and the number of sides of the polygon be indefinitely increased; its perimeter will approach the circumference of the circle, and will be ultimately equal to it (Prop. XI., B. VI.); and the convex surface of the prism will become equal to the convex surface of the cylinder. But whatever be the number of sides of the prism, its convex surface is equal to the product of its altitude by the perimeter of its base; hence the convex surface of the cylinder is equal to the product of its altitude by the circumference of its base.

Cor. If A represent the altitude of a cylinder, and R the radius of its base, the circumference of the base will be represented by 24R (Prop. XIII., Cor. 2, B. VI.); and the convex surface of the cylinder by 2πRA.

The solidity of a cylinder is equal to the product of its base by its altitude.

Let ACE-G be a cylinder whose base is the circle ACE and altitude AG; its solidity G is equal to the product of its base by its altitude.

14

H

K

E

B

In the circle ACE inscribe the regular polygon ABCDEF; and upon this polygon let a right prism be constructed of the same altitude with the cylinder. The solidity of this prism is equal to the product of its base by its altitude (Prop. XI., B. VIII.). Let, now, the number of sides of the polygon be indefinitely in creased; its area will become equal to that of the circle, and the solidity of the prism becomes equal to that of the cylinder. But whatever be the number of sides of the prism, its solidity is equal to the product of its base by its altitude; hence the solidity of a cylinder is equal to the product of its base by its altitude