Let H represent the altitude of the frustum of a pyramid, having parallel bases A and B; VAB will be the mean protional between those bases; and the solidity of the frustum will be H×(A+B+√AB).

Let B represent the base of the frustum of a triangular prism; H, H, H' the altitudes of its three upper vertices: the solidity of the truncated prism will be B×(H+H+H′′).

Ja fine, let P and p represent the solidities of two similar polyedrons; A and a two homologous edges or diagonals of these polyedrons: we shall have P : p : : A3 : a3.

BOOK VII.

THE SPHERE.

Definitions.

437. The sphere is a solid terminated by a curve surface, all the points of which are equally distant from a point within, called the centre.

438. The radius of a sphere is a straight line drawn from the centre to any point of the surface; the diameter or axis is a line passing through this centre, and terminated on both sides by the surface.

All the radii of a sphere are equal; all the diameters are equal, and each double of the radius.

439. It will be shewn (452.) that every section of the sphere, made by a plane, is a circle: this granted, a great circle is a section which passes through the centre; a small circle one which does not pass through the centre.

440. A plane is tangent to a sphere, when their surfaces have but one point in common.

441. The pole of a circle of a sphere is a point in the surface equally distant from all the points in the circumference of this circle. It will be shown (464.) that every circle, great or small, has always two poles.

442. A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.

Those arcs, named the sides of the triangle, are always supposed to be each less than a semicircumference. The angles, which their planes form with each other, are the angles of the triangle.

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

444. A spherical polygon is a portion of the surface of a sphere terminated by several arcs of great circles.

445. A lune is that portion of the surface of a sphere, which is included between two great semicircles meeting in a common diameter.

446. A spherical wedge or ungula is that portion of the solid sphere, which is included between the same great semicircles, and has the lune for its base.

447. A spherical pyramid is a portion of the solid sphere, included between the planes of a solid angle whose vertex is the centre. The base of the pyramid is the spherical polygon intercepted by the same planes.

448. A zone is the portion of the surface of the sphere, included between two parallel planes, which form its buses. One of those planes may be tangent to the sphere; in which case, the zone has only a single base.

1

449. A spherical segment is the portion of the solid sphere, included between two parallel planes which form its bases.

One of these planes may be tangent to the sphere; in which case, the segment has only a single base.

450. The altitude of a zone or of a segment is the distance between the two parallel planes, which form the bases of the zone or segment.

451. Whilst the semicircle DAE (437.) revolving round its diameter DE, describes the sphere; any circular sector, as DCF or FCH, describes a solid, which is named a spherical sector.

THEOREM.

452. Every section of a sphere, made by a plane, is a circle,

Let AMB be a section, made by a plane, in the sphere whose centre is C. From the point C, draw CO perpendicular to the plane AMB ; and different lines CM, CM to different points of the curve AMB, which terminates the section.

The oblique lines CM, CM, CA

are equal, being radii of the sphere;

Ꭰ

M

B

M

hence (329.) they are equally distant from the perpendicular CO; therefore all the lines OM, MO, OB are equal; consequently the section AMB is a circle, whose centre is O.

453. Cor. 1. If the section passes through the centre of the sphere, its radius will be the radius of the sphere; hence all great circles are equal.

454. Cor. 2. Two great circles always bisect each other; for their common intersection, passing through the centre, is a diameter.

455. Cor. 3. Every great circle divides the sphere and its surface into two equal parts: for, if the two hemispheres were separated, and afterwards placed on the common base, with their convexities turned the same way, the two surfaces would exactly coincide, no point of the one being nearer the centre than any point of the other.

456.

Cor. 4. 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.

457. Cor. 5. Small circles are the less the further they lie from the centre of the sphere; for the greater CO is, the less is the chord AB, the diameter of the small circle AMB.

458. Cor. 6. An arc of a great circle may always be made to pass through any two given points of the surface of the sphere; for the two given points, and the centre of the sphere make three points which determine the position of a plane. But if the two given points were at the extremities of a diameter, these two points and the centre would then lie in one straight line, and an infinite number of great circles might be made to pass through the two given points.

THEOREM.

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

the other two.

Let O be the centre of the sphere, and A ACB the triangle; draw the radii OA, OB, OC. Imagine the planes AOB, AOC, COB to be drawn; those planes will form a solid angle at the centre O; and the angles AOB, AOC, COB will be measured by AB, AC, BC, the sides of the spherical triangle. But (356.) each of the three plane angles forming a solid angle is less than the sum of the other O

B

two (356.); hence any side of the triangle ABC is less than the sum of the other two.

460.

THEOREM.

The shortest path from one point to another, on the surface of a sphere, is the arc of the great circle which joins the two given points.

Let ANB be the arc of the great circle which joins the points A and B; and without this line, if possible, let M be a point of the shortest path between A and B. Through the point M, draw MA, MB, arcs of great circles; and take BN=MB.

B

By the last theorem, the arc ANB is shorter than AM+MB; take BN=BM respectively from M both; there will remain AN AM. Now, the distance of B from M, whether it be the same with the arc BM or with any other line, is equal to the distance of B from N; for by making the plane of the great circle BM to revolve about the diameter which passes through B, the point M may be brought into the position of the point N; and the shortest line between M and B, whatever it may be, will then be identical with that between N and B: hence the two paths from A to B, one passing through M, the other through N, have an equal part in each, the part from M to B equal to the part from N to B. The first path is the shorter, by hypothesis; hence the distance from A to M must be shorter than the distance from A to N; which is absurd, the arc AM being proved greater than AN; hence no point of the shortest line from A to B can lie out of the arc ANB; hence this arc is itself the shortest distance between its two extremities.

THEOREM.

461.

The sum of the three sides of a spherical triangle is less than the circumference of a great circle.

Let ABC be any spherical triangle; produce the sides AB, AC till they meet again in D. The arcs ABD, ACD will be semicircumferences, since (454.) two great circles always bisect each A other. But in the triangle BCD, we have (459.) the side BC BD +CD; add AB+AC to both; we shall have AB+AC+BCZ ABD+ACD, that is to say, less than a circumference.

B

D

E