THEOREM XVIII. If two arcs of great circles intersect on the surface of a hemisphere, the sum of the opposite spherical triangles which they form is equivalent to a lune whose angle is the angle between the arcs in question, THEOREM XIX. A lune is to the surface of the sphere as the angle of the lune is to four right angles. THEOREM XX. The area of a spherical triangle is equal to its spherical excess. THEOREM XXI. The shortest line that can be drawn on the surface of a sphere between two points is the arc of a great circle, not greater than a semi-circumference, joining the two points. THE MEASUREMENT OF THE SPHERE. THEOREM XXII. The area of the surface generated by a straight line revolving about an axis in its plane is equal to the length of the projection of the line on the axis multiplied by the circumference the circle the radius of which is the perpendicular erected at the middle of the line and terminated by the axis. THEOREM XXIII. The area of a zone is equal to the product of its altitude and the circumference of a great circle. S=2wrh, THEOREM XXIV. The area of the surface of a sphere is equal to the product of its diameter by the circumference of a great circle. S = 27r X 2 r = 4712. Hence the surface of a sphere is equivalent to four great circles. Corollary. The surfaces of two spheres are to each other as the squares of the diameters, or as the squares of the radii. THEOREM XXV. The volume of a sphere is equal to the area of the surface multiplied by one third of the radius. V=8773 Corollary. The volumes of two spheres are to each other as the cubes of the diameters, or as the cubes of the radii. THEOREM XXVI. The volume of a spherical sector is equal to the area of the zone which forms the base multiplied by one third the radius of the sphere. PROBLEMS IN CONSTRUCTION, The pupil is expected to be able to make the following constructions by the aid of straight lines and circles (ruler and compasses) and to prove their correctness : 1. To bisect a given straight line. 2. To bisect a given angle. 3. At a given point in a straight line to erect a perpendicular to that line. 4. From a given point to let fall a perpendicular upon a given straight line. 5. At a given point in a straight line to construct an angle equal to a given angle. 6. Through a given point to draw a parallel to a given straight line. 7. To find the centre of a given circular arc. 8. At a given point in a given circumference to draw a tangent. 9. Through a given point without a given circumference to draw a tangent to the circumference. 10. To inscribe a circle in a given triangle. 11. To draw a circumference through three given points not lying in the same straight line. 12. On a given straight line as a base to construct a circular segment in which a given angle can be inscribed. 13. To divide a given straight line into any given number of equal parts. 14. To divide a given straight line into parts proportional to two given straight lines. 15. To find a fourth proportional to tbree given straight lines. 16. To find a mean proportional between two given straight lines. 17. To divide a given straight line in extreme and mean ratio. 18. To construct a triangle equivalent to a given polygon. RATIO AND PROPORTION. The pupil is assumed to be familiar with the following propositions : 1. If a, b, c, d are numbers and a: b = cid, then (1) ad bc, 2. If the numerical measures of four quantities A, B, C, D form a proportion, and if A and B are of the same kind and C and D of the same kind, then A:B = C:D. |