58. Let AB, CD, EF be three diameters of a sphere each at right angles to the other two, and intersecting each other in O the center of the sphere, the extremities of the lines meeting the surface of the sphere. Join AC, CB, BD, DA, then these four edges of the figure may be proved equal to one another by the right-angled triangles. In the same way the other edges may be proved equal. Having proved all the edges equal, the faces of the figure are equilateral triangles. Lastly prove the inclinations of every two faces to be equal. It may also easily be shewn that if lines be drawn joining the centers of the faces of a cube; these will be the edges and diagonals of a regular octahedron. APPENDIX. EXAMINATION PAPERS IN EUCLID SET TO CANDIDATES FOR First and Second Class Provincial Certificates, AND TO STUDENTS MATRICULATING IN THE UNIVERSITY OF TORONTO. SECOND CLASS PROVINCIAL CERTIFICATES, 1871. TIME-TWO HOURS AND A HALF. 1. If two triangles have two sides of the one equal to two sides of the 2. other, each to each, and have likewise their bases equal, the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides, equal to them, of the other. Triangles upon the same base, and between the same parallels, are equal to one another. 3. If the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it, the angle contained by these two sides is a right angle. 4. If a straight line be divided into two equal, and also into two unequal, parts, the squares on the two unequal parts are together double of the square on half the line, and of the square on the line between the points of section. 5. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts are together equal to the square on the whole line. 6. 7. Bisect a parallelogram by a straight line drawn from a point in one Let A B C be a triangle, and let B D be a straight line drawn to D, 8. In a triangle A B C, A D being drawn perpendicular to the straight line B D which bisects the angle B, show that a line drawn from D parallel to B C will bisect A C. NOTE. The percentage of marks requisite, in order that a candidate may be ranked of a particular grade, will be taken on the value of the above paper, omitting question 8. SECOND CLASS PROVINCIAL CERTIFICATES, 1872. TIME-2 HOURS. 1. Define a straight line, a plane rectilineal angle, a right angle, a Gnomon. Enunciate Euclid's Postulates. 2. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle. 2. If two triangles have two angles of the one equal to angles of the other, each to each, and one side equal to one side, namely, either the sides adjacent to the equal angles, or sides which are opposite to equal angles in each; then shall the other sides be equal, each to each; and also the third angle of the one equal to the third angle of the other. (Take the case in which the assumed equal sides are those opposite to equal angles.) In every triangle, the square on the side subtending an acute angle is less than the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle, and acute angle. (Take the case where the perpendicular falls within the triangle.) 5. If a straight line be divided into any two parts, the squares on the whole line, and one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square on the other part. 6. Prove that, if a straight line AD be drawn from A, one of the angles of a triangle ABC, to D, the middle point of the opposite side BC, BA × AC is greater than 2 AD. . Let the equilateral triangle ABC, and triangle ADB, in which the angle ABD is a right angle, be on the same base AB, and between the same parallels AB and CD. Prove that 4 AD2 =7 AB2. E. From D, a point in AB, a side of the triangle ABC, it is required to draw a straight line DE, cutting BC in E, and AC produced in F, so that DE may be equal to EF. SECOND CLASS PROVINCIAL CERTIFICATES, 1873. TIME-TWO HOURS AND A HALF. NOTE.-Candidates who take only Book I, will confine themselves to the first eight questions; those who take Books I and II, will omit the first two questions. 1. If two angles of a triangle be equal to one another, the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another. 2. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles. 3. The opposite side, and angles of a parallelogram, are equal to one another. 4. The complements of the parallelograms,which are about the diameter of any parallelogram, are equal to one another. 5. To describe a square on a given straight line. 6. Let A B C D be a quadrilateral figure whose opposite angles A B C and ADC are right angles. Prove that, if A B be equal to A D, C B and C D shall also be equal to one another. 7. If A B C D be a quadrilateral figure, having the side A B parallel to the side C D, the straight line which joins the middle points of A B and D C shall divide the quadrilateral into two equal parts. The straight line, which joins the middle points of two sides of a triangle, is parallel to the base. 8. 9. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the parts. 10. In an obtuse angled triangle, is the sum of the sides containing the obtuse angle greater or less than the square of the side opposite to the obtuse angle? And, by how much? Prove the proposition. SECOND CLASS PROVINCIAL CERTIFICATES, 1874, TIME-TWO HOURS AND THREE QUARTERS. NOTE.-Candidates who take only Book I. will confine themselves to the first 7 questions. Those who take Books I. and II., will omit questions 1, 2, and 3. 1. When is one straight line said to be perpendicular to another? To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it. 2. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles. 3. If two triangles have two angles of the one equal to two angles of the other, each to each; and one side equal to one side, namely, sides which are opposite to equal angles in each; then shall the other sides be equal, each to each. 4. What are parallel straight lines? If a straight line, falling on two other straight lines, make the alternate angles equal to one another, the two straight lines shall be parallel to one another. 5. What is a parallelogram? Parallelograms on equal bases, and between the same parallels, are equal to one another. 6. If two isosceles triangles be on the same base, and on the same side of it, the straight line which joins their vertices, will, if produced, cut the base at right angles. 7. Let ABC be a triangle, in which the angle ABC is a right angle. From AC cut off AD equal to AB, and join BD. Prove that the angle BAC is equal to twice the angle CBD. S. If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to &c. (5, II.) 9. In every triangle, the square on the side subtending an acute angle is less than the squares on the sides containing that angle, by &c. (13. II). (It will be sufficient to take the case in which the perpendicular falls within the triangle.) 10. To describe a square that shall be equal to a given rectilineal figure. 11. The square on any straight line drawn from the vertex of an isosceles |