}"×EF×(2BE2+2DF2+2BE×DF+EF2+DF®—2DF×BE+RÈ”), ¿«×EF×(3BE2+3DF2+EF2);

equal to

an expression which may be written in two parts, viz.,

EFX ("XEE°+=XDF") and ↓ «XEF";

2

and these parts correspond with the enunciation.

Cor. If the radius of either base is nothing, the seg ment becomes a spherical segment with a single base; hence, any spherical segment, with a single base, is equivalent to half the cylinder having the same base and the same altitude, plus the sphere of which this altitude is the diameter.

GENERAL SCHOLIUMS.

1. Let R be the radius of a cylinder's base, H its altıtude: the solidity of the cylinder is

XR2XH.

2. Let R be the radius of a cone's base, H its altitude: the solidity of the cone is

«>R2×}H=}«×R3×H.

3. Let A and B be the radii of the bases of a frustum of a cone, H its altitude: the solidity of the frustum is

4. Let R be the radius of a sphere; its solidity is

5. Let R be the radius of a spherical sector, H the altitude of a zone, which forms its base: the solidity of the sector is

6. Let P and Q be the two bases of a spherical segment, H its altitude: the solidity of the segment is

P+Q
2

7. If the spherical segment has but one base, its solidity is

BOOK IX.

SPHERICAL GEOMETRY.

DEFINITIONS.

1. A SPHERICAL TRIANGLE is a portion of the surface of a sphere, bounded by three arcs of great circles.

These arcs are named the sides of the triangle, and each is less than a semicircumference. The angles which the planes of the circles make with each other, are the angles of the triangle.

2. A spherical triangle takes the name of right-angled, isosceles, equilateral, in the same cases as a rectilineal triangle.

3. A SPHERICAL POLYGON is a portion of the surface of a sphere bounded by three or more arcs of great circles.

4. A LUNE is a portion of the surface of a sphere included between two semi-circles intersecting in a common diameter of the sphere.

5. A SPHERICAL WEDGE, or UNGULA, is that portion of a solid sphere, included between two planes passing through the centre, and the lune which forms its base.

6. A SPHERICAL PYRAMID is a portion of the solid sphere, included between three or more planes. The base of the pyramid is the spherical polygon intercepted by the same planes. These planes bound a polyedral angle, whose vertex is at the centre of the sphere.

7. The POLE OF A CIRCLE is a point on the surface of the sphere. equally distant from every point in the circum ference.

PROPOSITION I. THEOREM.

In every spherical triangle, any side is less than the sum of the

two other sides.

Let O be the centre of the sphere, and ACB a spheri cal triangle: then will any side be less than the sum of the two other sides.

For, draw the radii OA, OB, OC. Conceive the planes AOB, AOC, COB, to be drawn; these planes bound a polyedral angle whose vertex is at the centre 0; and the plane angles AOB, AOC, COB, are measured by AB, AC, BC, the sides of the spherical triangle. But each of the three plane angles which bound a polyedral

B

C

angle is less than the sum of the two other angles (B. VI., P. 19); hence, any side of a spherical triangle is less than the sum of the two other sides.

PROPOSITION II. THEOREM.

The sum of all the sides of any spherical polygon is less than the circumference of a great circle.

Let ABCDE be any spherical polygon, and O the cen tre of the sphere.

Conceive O to be the vertex of a polyedral angle bounded by the plane angles AOB, BOC, COD, &c. Now, the sum of the plane angles which bound a polyedral angle is less than four right angles (B. VI., P. 20); hence, the sum of the sides of any

E

D

spherical polygon is less than the circumference.

Cor. The sum of the three sides of any spherical tri angle is less than the circumference; for, the triangle is a polygon of three sides.

PROPOSITION III. THEOREM.

The poles of a great circle of a sphere are the extremities of that diameter of the sphere which is perpendicular to the circle; and these extremities are also the poles of all small circles parallel to it.

Let ED be perpendicular to the great circle AMB; then will E and D be its poles; and they will also be the poles of every parallel small circle FNG.

For, DO being perpendicular to the plane AMB, is perpendicular to all the straight lines CA, CM, CB, &c., drawn through its foot in this plane (B. VI., D. 1); hence, all the arcs DA, DM, DB, &c., are quarters of the circumference. So likewise are all the arcs EA, EM, EB, &c.; therefore, the points D and E

F

B

M

E

are each equally distant from all the points of the circumference AMB; hence, they are the poles of that circumference (D. 7).

Again, the radius DC, perpendicular to the plane AMB, is perpendicular to the parallel FNG; hence, it passes through O, the centre of the circle FNG (B. VIII., P. 7, c. 4); hence, if the chords DF, DN, DG, be drawn, these oblique lines will cut off equal distances measured from 0; hence, they will be equal (B. VI., P. 5). But, the chords being equal, the arcs are equal; hence, the point D is the pole of the small circle FNG; and for like reasons, the point E is the other pole.

Cor. If through the pole D and any point M, in the are of a great circle AMB, an arc of another great circle MD be drawn, the arc MD is a quarter of the circumference, and is called a quadrant. This quadrant makes a right angle with the arc AM. For, the line DC being perpendicular to the plane AMC, every plane DME, passing through the line DC is

perpendicular to the plane AMC (B VI., P. 16); hence, the angle of these planes, or the angle AMD is a right angle.

Cor. 2. Conversely: If the distance of the point D from each of the points A and M, in the circumference of a great circle, is equal to a quadrant, the point D is the pole of the arc AM.

F

/N

M

M

For, let be the centre of the sphere, and draw the radii CD, CA, CM. Since the angles ACD, MCD, are right angles, the line CD is perpendicular to the two straight lines CA, CM; hence, it is perpendicular to their plane (B. VI., P. 4): hence, the point D is the pole of the arc AM.

Scholium. The properties of these poles enable us to describe arcs of a circle on the surface of a sphere, with the same facility as on a plane surface. It is evident, for instance, that by turning the arc DF, or any other line extending to the same distance, round the point D, the extremity F will describe the small circle FNG; and by turning the quadrant DFA round the point D, its extremity A will describe the arc of a great circle AMB.

PROPOSITION IV. THEOREM.

The angle formed by two arcs of great circles, is equal to the angle formed by the tangents of these arcs at their point of intersection. The angle is measured by the arc of a great circle described from the vertex as a pole, and limited by the sides, produced if necessary.

Let the angle BAC be formed by the two arcs AB, AC; then will it be equal to the angle FAG formed by the tangents AF, AG, and be measured by the arc DE of a great circle, described about A as a pole.