So lang er glaubt an die goldene Zeit, Und erstickst du ihn nicht in den Lüften frei, 9. Translate into German (a) When it was known that Cæsar was approaching, Cato caused all the gates to be closed, except that which led to the sea. He urged all those who would to betake themselves to the ships, and dismissed his friends, of whom a few only, and among them his own son, insisted on remaining with him. With these cherished associates he sat down to a supper, and discoursed with unusual warmth on the great questions of philosophy. The principal matter of the discourse was the famous teaching of the Stoics that the good man alone is free, and all the bad are slaves. His companions easily guessed what his secret purpose was, and betrayed their unhappiness in their features. Cato observing that they were sad tried to encourage them, and to divert their thoughts by conversation on the affairs of the day. (b) The Marquis Tseng has just made a profound remark in France which may be taken to heart in this country, where the depressed condition of agriculture forms the gloomy background of all our thoughts. Asked what he thought of France, the Chinese statesman replied:-" He thought it a great country. The railways, the manufactures, and the application of steam and mechanics to industry filled him with envy. But as regarded agriculture, France, and indeed the rest of Europe, was in a state of infancy when compared with China. France might find it worth her while to import Chinese agriculturists to teach her how to make the most of her fertile soil." Here is a prospect indeed! LOWER MATHEMATICS. Professor Nanson. (Candidates must answer satisfactorily in each of the three divisions of this paper.) I.—1. The sum of the squares on two sides of a triangle is double the sum of the squares on half the base and on the line joining the vertex to the middle point of the base. The sum of the squares on the sides of a parallelogram is equal to the sum of the squares on the diagonals. 2. Angles in the same segment of a circle are equal to one another. Two chords AB, CD of a circle intersect in E. Shew that the triangles AEC, DEB are equiangular. 3. About a given circle circumscribe a triangle equiangular to a given triangle. About a given circle circumscribe a quadrilateral equiangular to a given quadrilateral. 4. If two triangles have their angles respectively equal, they are similar, and those sides which are opposite to the equal angles are homologous. Two equal triangles are drawn upon the same base and upon the same side of it; a straight line is drawn through them parallel to the base; shew that the parts of it intercepted between the sides of each triangle are equal to one another. II.—1. Prove that a quadratic equation cannot have more than two roots. then each of these ratios is equal to the ratio Prove also that on the same supposition (a2 + b2 + c2) (a'2 + b2 + c'2) = (aa' + bb' + cc')2. 4. Define a geometrical progression, and find the sum of any number of terms of such a progression. If a, b, c, d, e be in geometrical progression prove that c (a + 2c + e) = (b + d)2. III.-1. Define the sine of an angle, and trace the changes in its sign and magnitude as the angle increases from zero to four right angles. Find all the values of x between zero and four right angles which satisfy the equation 2. If A and B be positive angles whose sum is less than a right angle prove that cos (A + B) = cos A cos B sin A sin B. The sides of a triangle are 7, 8, 13; find the greatest angle. 4. Shew how to solve a triangle having given two sides and an angle opposite to one of them. = If a √6, b = 3, A = 45°, solve the triangle. UPPER MATHEMATICS. Professor Nanson. (Candidates must answer satisfactorily in each of the three divisions of this paper.) I.-1. Prove that in the parabola PN24AS. AN. 2. Prove that the sum of the focal distances of any point on an ellipse is constant, 3. Prove that in any conic the tangents at the extremities of any chord meet on the diameter which bisects the chord. 4. Prove that in a central conic supplemental chords are parallel to conjugate diameters. II.-1. Prove that an equation of the nth degree cannot have more than n roots. 2. Prove that an infinite series is convergent if from and after some fixed term the ratio of each term to the preceding term is numerically less than some quantity which is itself numerically less than unity. 3. If the relation f(m) × f(n) = f(m + n) be true for all values of m and n, prove that ƒ(m) = {ƒ(1)}". D |