If we denote the volume by V, convex surface of a cone by S, its radius of its base by R, its altitude by H, and its slant height by H', we have (P. III., V.), If we denote the convex surface of a frustum of a cone by S, its volume by V, the radius of its lower base by R, the radius of its upper base by R', its altitude by H, and its slant height by II', we have (P. IV., VI.), S = T(R+R') × H' V = } «(R2 + R22 + √√R × R') × H (5.) (6.) If we denote the surface of a sphere by S, its volume If we denote the radius of a sphere by R, the area of any zone of the sphere by S, its altitude by H, and the volume of the corresponding spherical sector by V, we shall have (P. X., C. 2), If we denote the volume of the corresponding spherical segment by V, the radius of its lower base by R', the radius of its upper base by R", the distance of its lower base from the centre by H', and the distance of its upper base from the centre by II", we have (P. XIV., S.), V = {(2R2 × H+ R'2 × H' = R''2 × H') BOOK IX. SPHERICAL GEOMETRY. DEFINITIONS. 1 A SPHERICAL ANGLE is an angle included between the arcs of two great circles of a sphere meeting at a point. The arcs are called sides of the angle, and the point at which they meet is called the vertex of the angle. The measure of a spherical angle is the same as that of the diedral angle included between the planes of its sides. Spherical angles may be acute, right, or obtuse. 2. A SPHERICAL POLYGON is a portion of the surface of a sphere bounded by arcs of three or more great circles. The bounding arcs are called sides of the polygon, and the points in which the sides meet are called vertices of the polygon. Each side is supposed to be less than a semi-cir cumference. Spherical polygons are classified in the same manner as plane polygons. 3. A SPHERICAL TRIANGLE is a spherical polygon of three sides. Spherical triangles are classified in the same manner as plane triangles. 4. A LUNE is a portion of the surface of a sphere bounded by semi-circumferences of two great circles. 5. A SPHERICAL WEDGE is a portion of a sphere bounded by a lune and two semicircles meeting in a diameter of the sphere. 6. A SPHERICAL PYRAMID is a portion of of a sphere bounded by a spherical polygon and sectors of circles whose common centre is the centre of the sphere. The spherical polygon is called the base of the pyramid, and the centre of the sphere is called the vertex of the pyramid. 7. A POLE OF A CIRCLE is a point on the surface of the sphere, equally distant from all the points of the circumference of the circle. 8. A DIAGONAL of a spherical polygon is an arc of a great circle joining the vertices of any two angles which are not consecutive. PROPOSITION I. THEOREM. Any side of a spherical triangle is less than the sum of the other two. Let ABC be a spherical triangle situated on a sphere whose centre is : then will any side, as AB, be less than the sum of the sides AC and BC. For, draw the radii OA, OB, and OC: these radii form the edges of a triedral angle whose vertex is 0, and the plane angles included between them are measured by the arcs AB, AC, and BC (B. III., P. XVII., Sch.). But any plane angle, as AOB, is less than the sum of the plane angles AOC and BOC (B. VI., P. XIX.): hence, A B the arc AB is less than the sum of the arcs AC and BC; which was to be proved. Cor. 1. Any side AB, of a spherical polygon ABUDE, is less than the sum of all the other sides. For, draw the diagonals AC and AD, dividing the polygon into triangles. The arc AB is less than the sum of AC and BC, the arc AC is less than the sum of AD and DC, and the arc AD is less than the sum of DE and EA; hence, AB is less than the sum of BC, CD, DE, and EA. D E B A Cor. 2. The arc of a great circle joining any two points on the surface of a sphere, is less than the arc of a small circle joining the same points. For, divide the arc of the small circle into equal parts, and through the extremities of each part pass the arc of a great circle. The arc of the great circle joining the given points will be less than the ever may be their number. sum of these arcs (C. 1), whatBut when this number is infinite, the arcs of the great circle coincide with the corresponding ares of the small circle, and their sum is equal to the entire arc of the small circle. Cor. 3. The shortest distance between two points on the surface of a sphere, is measured on the arc of a great circle joining them. PROPOSITION II. THEOREM. The sum of the sides of a spherical polygon is less than the circumference of a great circle. Let ABCDE be a spherical polygon situated on a sphere whose centre is 0: then will the sum of its sides be less than the circumference of a great circle. E A D B For, draw the radii OA, OB, OC, OD, and OE: these radii form the edges of a polyedral angle whose vertex is at O, and the angles included between them are measured by the arcs AB, BC, But the sum of CD, DE, and EA. these angles is less than four right angles (B. VI., P. XX.): hence, the sum of the arcs which measure them is less than the circumference of a great circle; which was to be proved. PROPOSITION III. THEOREM. If a diameter of a sphere be drawn perpendicular to the plane of any circle of the sphere, its extremities will be poles of that circle. Let C C be the centre of a sphere, FNG any circle of the sphere, and DE a diameter of the sphere perpendicular to the plane of FNG: then will the extremities D and E, be poles of the circle FNG. P. V.), consequently, the arcs themselves will be equal. But these arcs are the shortest lines that can be drawn from the |