EXERCISES. 1. Prove that every spherical triangle may be inscribed in a circle. 2. Through a given point on the arc of a great circle to draw an arc of a great circle perpendicular to the former. 3. The same through a point without the given arc. 4. Prove that the rectangles of the parts of all lines passing through the same point within a sphere, and terminating at the surface, are equal. 5. Prove that circles whose planes are equidistant from the center of the sphere are equal. 6. Prove that every plane passing through the point of contact of a tangent plane to a sphere cuts this plane in a line tangent to the circle cut from the sphere. 7. That the line of centers of two spheres which cut each other is perpendicular to the plane of the circle of section of the two spheres. 8. Prove that their intersection is a circle. 9. Show how to construct a spherical triangle with any three parts given. 10. Prove that the sum of all the sides of a spherical polygon is less than the circumference of a great circle. 11. Make a sphere pass through four given points, or prove that every tetrahedron may be circumscribed by a sphere. 12. Also, inscribed. 13. Prove that the measure of the surface of a spherical polygon is equal to the excess of the sum of its angles over as many times two right angles as the figure has sides less two. 14. Make a great circle tangent* to a small circle on the surface of a sphere. 15. Change a spherical quadrangle into an equivalent spherical triangle. 16. Upon the base of a spherical triangle to construct an isosceles spherical triangle of equal surface. 17. To construct on the base of a given spherical triangle another of equal surface, 10, having a given base angle; 2°, having a given side. 18. Prove that the sums of the opposite angles of a spherical quadrilateral inscribed in a circle of the sphere are equal.t * One circle is said to be tangent to another on the surface of a sphere when the two circles have a common tangent line at a common point. + This is done by connecting the four vertices of the quadrilateral 19. Prove that if two spherical triangles, having a cominon base, be inscribed in the same circle of a sphere, the difference between the sum of the base angles and the vertical angle will be equal in the two triangles. Corol. Spherical triangles having the same base, and the sums of their base angles equal, and also their vertical angles equal, have their vertices lying in the same circumference on the sphere. 20. Prove that if the base of a spherical triangle be prolonged to become a complete circumference, and the other two sides prolonged beyond the vertex till they meet this; then, if through the points of meeting and the vertex a small circle of the sphere be made to pass, every triangle having its vertex in this, and its base the same with the given triangle, will have an equal surface. Corol. If one of the other sides of the triangle falls in the prolongation of the base, and the vertex coincides with one of the abovementioned points of meeting, the small circle passing through the three points vanishes or reduces to a point, viz., the point in which these three points coalesce; the triangle then degenerates into a lune, which is still, however, equal to the given triangle in surface. 21. Upon the base of a given spherical triangle to construct another of equal surface of which the vertex shall lie in a given great circumference. 22. To change a spherical triangle into another of equal surface with a given side and given angle adjacent. 23. To construct a spherical triangle with two given sides and of surface equal to a given triangle. 24. Prove that if P denote the number of polyhedral angles of a polyhedron, F the number of its faces, and E the number of its edges, P+F=E+2. 25. Also, that the sum of the plane angles of a polyhedron is equal to P-2 times four right-angles. 26. To construct the length of the radius of a sphere when confined to its exterior. 27. To describe the circumference of a great circle through two given points. with the pole of the circle in which it is inscribed, thus forming four isosceles spherical triangles. PROVE that if m n APPENDIX III. expresses the ratio of an arc to a quadrant, will express the ratio of the arc to the radius. Knowing the ratio of an arc to the radius, show how to find the number of degrees which it contains. Two angles subtended by arcs of different radii are to each other as the ratios of the arcs to their respective radii (th. 71, corol. 3). In symbols, if V and V' be two angles, A and A' the arcs subtending them, described with the radii R and R', Taking the right angle as the unit of angles, supposing for a moment V' to be this unit, A' the unit of arc, and R' the unit of length, the above proportion becomes A that is, an angle at the center has for its measure the quotient of the arc which subtends it, divided by the radius. It must, however, be understood that the quantities V, A, and R are referred to their respective units. THEOREM. Of two arcs, each less than a semicircumference, subtended by the same chord, the shortest is that whose center is furthest from the middle of the chord. Let AB be the common chord, AMB, AM'B the two arcs, O the center of the former, O' of the latter. Then, if OP>O'P, A AMBAM'B. For (by th. 17) OA>O'A; and, if the arc AM'B be turned over round AB as a hinge, it will evidently contain the arc AMB within it;* and it may be easily *This may be seen more distinctly by observing that an indefinitely small portion of the arc of a circle may be regarded as a M O B proved, that of two lines, the one enveloping the other, and terminating at the same points, the enveloped line is the least. This may be shown, supposing them to be polygonal lines at first, by repeated application of the principle that a straight line is the shortest distance between two points, and then supposing the straight portions of the polygonal lines to become infinitely small, or the polygonal lines to become curves. Prove that every small circle of a sphere has a less radius than the sphere. THEOREM. The arc of a great circle comprehended between two given points on the surface of a sphere is less than any arc of any small circle comprehended between the same two points. This follows from the last theorems. THEOREM. The shortest path from one point to another on the surface of a sphere is the arc of a great circle. To prove this, let it be observed that the sphere is perfectly round in all directions, so that every section of it made by a plane is a circle. This being premised, suppose an irregular line upon its surface between the two given points; this may be considered either an arc of a small circle, or made up of small portions of such circles. In the first case, it has already been proved that the arc of a great circle between the points is shorter than this. In the second case, arcs of great circles between the extremities of the portions are less than these portions, and, by the repetition of the principle that one side of a spherical triangle is less than the sum of the other two, it may be shown that the arc of a great circle between the two given points is less than the polygonal combination of arcs of great circles between the same points, so that in both cases the theorem is demonstrated. straight line, which, prolonged both ways, becomes a tangent; the tangent, therefore, shows the direction of the curve at the point of contact. If, now, after the arc AM'B is turned over, it be observed that the direction of this arc at the point A is perpendicular to AO", while the direction of AMB is perpendicular to AO, it is evident that the latter arc will run within the former. By joining the point O" with any point of the inverted arc AM'B, and the point in which this line intersects the arc AMB with the point O, it may be shown that the arc AM'B is every where diverging in direction from the arc AMB, except at M, M'. APPENDIX IV. ISOPERIMETRY ON THE SPHERE. 1. PROVE that of all spherical triangles formed with two given sides, the greatest is that in which the angle formed by the given sides is equal to the sum of the other two angles. 2. That of all spherical triangles formed with one side, and the perimeter given, the greatest is that in which the undetermined sides are equal. 3. That of all isoperimetrical spherical polygons, the greatest is an equilateral polygon. 4. That of all spherical polygons formed with given sides, and one side taken at pleasure, the greatest is that which can be inscribed in a circle, of which the chord of the undetermined side is the diameter. 5. The greatest of spherical polygons formed with given sides is that which can be inscribed in a circle of the sphere. 6. The greatest of spherical polygons having the same perimeter and same number of sides is that in which the sides and angles are equal. Note. All the above apply, also, to polyhedral angles, of which the spherical triangles are the measures. |