Ratio of Similar Solids.

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.

Proof. 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

vertex.

Now these similar pyramids are to each other, by § 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.

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.

The

417. Definitions. The

radius of a

radius of a sphere is a

straight line drawn from the centre to a point in the

Great and Small Circles. Poles.

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.

Proof. 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 § 321, terminate in a circumference AMB, of which O is the centre.

420. Definitions. The section made by a plane which passes through the centre of the sphere is called a great circle. Any other section is called a small circle.

421. Corollary. The radius of a great circle is the same as that of the sphere, and therefore all the great circles of a sphere are equal to each other.

422. Corollary. The centre of a small circle and that of the sphere are in the same straight line perpendicular to the plane of the small circle.

423. Definition. The points, in which a radius of the sphere, perpendicular to the plane of a circle, meets the surface of the sphere, are called the poles of the circle; thus P, P' are the poles of AMB.

424. Corollary. Since the oblique lines PA, PM, &c. are equally distant from the perpendicular PO, they are equal; and also the arcs of great circles PA, PM,

Arcs traced upon a Sphere.

&c. are, by § 113, equal; that is, the pole of a circle is equally distant from all the points in the circumference of the circle.

425. Corollary. Since the distance DM (fig. 177) of a point, in the circumference of a great circle from the pole, is measured by the right angle DCM, it is a quadrant.

426. Scholium. By means of poles, arcs may be traced upon the surface of a sphere as easily as upon a plane surface.

We see, for example, that by turning the arc DF (fig. 177) about the point D, the extremity F describes the small circle FNG; and by turning the quadrant DFA about the point D, the extremity A describes the arc of a great circle AM.

427. Theorem. A point upon the surface of a sphere which is at the distance of a quadrant from each of two other points, is one of the poles of the great circle which passes through these two points.

Proof. Thus, if the distances DA, DM (fig. 177) are quadrants, the angles DCA and DCM are right angles, and, therefore, by § 318, DC is perpendicular to the circle AMB, and its extremity D is, by § 423, a pole of the circle ABM.

428. Corollary. Since the common intersection of two great circles is, by § 420, a diameter, they bisect each other.

429. Theorem. Every great circle bisects the sphere. Proof. For if, having separated the two hemispheres from each other, we apply the base of one to that of the other,

Spherical Triangle, Polygon, Wedge, Pyramid.

turning the convexities the same way, the two surfaces must coincide; otherwise, there would be points in these surfaces unequally distant from the centre.

430. Definitions. A spherical triangle is a part of the surface of a sphere comprehended by three arcs of great circles.

These arcs, which are called the sides of the triangle, are always supposed to be smaller each than a semicircumference. The angles, which their planes make with each other, are the angles of the triangle. Since the sides are arcs, they may be expressed in degrees and minutes, as well as the angles.

431. Definitions. A spherical triangle takes the name of right, isosceles, and equilateral, like a plane triangle, and under the same circumstances.

432. Definition. A spherical polygon is a part of the surface of a sphere terminated by several arcs of great circles.

433. Definitions. The portion of a sphere comprehended between the halves of two great circles is called a spherical wedge, and the portion of the surface of the sphere comprehended between them is called a lunary surface, and is the base of the wedge.

434. Definitions. A spherical pyramid is the part of a sphere comprehended between the planes of a solid angle whose vertex is at the centre.

The base of the pyramid is the spherical polygon intercepted by these planes.

435. Definition. A plane is tangent to a sphere, when it has only one point in common with the surface of the sphere.

Spherical Segment, Sector.

436. Definitions. When two parallel planes cut a sphere, the portion of the sphere comprehended between them is called a spherical segment, and the portion of the surface of the sphere comprehended between them is called a zone.

The bases of the segment are the sections of the sphere, and the bases of the zone are the circumferences of the sections.

The altitude of the segment or zone is the distance between the sections.

One of the cutting planes may be tangent to the sphere, in which case the zone or segment has but one base.

437. Definition. While the semicircle DAE (fig. 177) turning about the diameter DE describes a sphere, every circular sector, as DCF or FCH, describes a solid, which is called a spherical sector. The base of the sector is the zone generated by the arc DF, or FH. 438. Theorem. Either side of a spherical triangle is less than the sum of the other two.

Proof. From the centre 0 (fig. 179) of the sphere draw the radii OA, OB, OC to the vertices A, B, C of the spherical triangle ABC. The three plane angles AOB, AOC, BOC form a solid angle at O; and each of these angles is, by § 338, less than the sum of the other two. But they are measured by the arcs AB, AC, BC; and, therefore, each of these arcs is less than the sum of the other two.

439.

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