SEPTEMBER 1896. SOLID. 1. Define a pyramid, a regular pyramid. Show that the lateral area of a regular pyramid is equal to one-half the product of its slant height and the perimeter of its base. 2. The volumes of two similar tetrahedrons are to each other as the cubes of their homologous edges. 3. Every section of a circular cone made by a plane parallel to its base is a circle. 4. The angle of two arcs of great circles on a sphere is equal to the angle of their planes, and is measured by the arc of a great circle described from its vertex as a pole and included between its sides (produced if necessary). 5. Find the volume of the earth's atmosphere, if it extends 50 miles from the surface, assuming the earth to be a sphere with a radius of 4000 miles. JUNE 1897. PLANE. 1. When two parallel lines are cut by a third straight line, the alternate interior angles are equal and conversely. 2. What is meant by measuring a magnitude? When are two magnitudes of the same kind commensurable, and when incommensurable? Explain what is meant by the limit of a varying magnitude, and give illustrations drawn from geometry. 3. The product of two sides of a triangle is equal to the product of the perpendicular on the third side let fall from their intersection and the diameter of the circumscribed circle. 4. Show how to construct a square equivalent to the sum of two given squares. 5. Show how to inscribe a regular hexagon in a circle of radius R, and prove that the area of the hexagon is equal to 12 R13, SEPTEMBER 1897. PLANE. 1. Enumerate and define the different kinds of quadrilaterals. Show that the diagonals of a rhombus are perpendicular. 2. Upon a given line to describe a segment of a circle to contain a given angle. 3. What is meant by the locus of a point? If through a point within a circle the chord of a circle is drawn, find the locus of the middle point of that chord. 4. (a) The perimeters of regular polygons of the same number of sides have the same ratio as any two homologous sides. (b) The areas of regular polygons of the same number of sides have the same ratio as the squares of two homologous sides. 5. Find the ratio of the areas of a square and the inscribed circle. Is the ratio commensurable? JUNE 1897. SOLID. 1. Define a plane, dihedral angle, polyhedral angle, polyhedron, parallelopiped, similar polyhedrons, regular polyhedrons. How many regular polyhedrons are possible? 2. (a) Two planes perpendicular to the same straight line are parallel. (b) If one of two parallel planes is perpendicular to a straight line, the other is also. 3. If a pyramid is cut by a plane parallel to the base, the edges and altitude are divided proportionally, and the section made is a polygon similar to the base. 4. A sphere of radius 10 inches is cut by a plane at a distance of 6 inches from the center; (a) prove that the section made is a circle; (b) find the area and volume of the cone whose base is this circle and whose vertex is the center of the sphere; (c) find the ratio of the areas of the two portions into which the spherical surface is divided by the plane. SEPTEMBER 1897. SOLID. 1. Two angles not in the same plane whose sides are parallel lie in parallel planes and are either equal or supplementary. 2. When two planes are perpendicular to a third plane their intersection is also perpendicular to that plane. 3. (a) The volume of a triangular prism equals the product of the base and altitude. (b) The volume of any prism equals the product of the base and altitude. 4. (a) What is the pole of a circle on a sphere? Show how to locate the pole of a great circle. (b) What is meant by the distance between two points on a spherical surface? Prove that all points of a circle are equally distant from the poles. 5. Prove that the surface of a sphere is two-thirds of that of the circumscribed right cylinder; show that the same theorem holds for the volumes. |