5. Draw a right line from a given point either without or in the circumference, which shall touch a given circle. If three circles touch each other externally and the three common tangents be drawn, these tangents shall intersect in a point equidistant from the points of contact of the circles. 6. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles. If perpendiculars be drawn from any point in the circumference of a given circle to the sides of an inscribed triangle, the points where those perpendiculars cut the sides shall lie in the same right line. 7. Upon a given right line describe a segment of a circle which shall contain an angle equal to a given rectilineal angle. On the same base and on the same side of it two segments of circles are described, and a point is taken in the one from which chords are drawn to the extremities of the base, find the point in the other from which chords being drawn in like manner, the sum of the chords from the point in the one segment shall be equal to the sum of the chords from the point in the other segment. 8. Describe a circle about a given triangle. If four right lines intersect each other forming four triangles, the circles circumscribing them shall all pass through one point. 9. Give Euclid's definition of proportional magnitudes. To what objection has this definition been considered liable? 10. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it shall be similar to the whole triangle and to one another. Produce a given right line so that the rectangle under it. and the produced part shall be equal to a given square. 11. Cut a given finite right line in extreme and mean ratio. On a given right line construct a right-angled triangle whose three sides shall be in continued proportion. 12. If two right lines meeting one another be paralled to two other right lines which meet one another, but are not in the same plane as the first two, the plane which passes through these is parallel to the plane which passes through the others. VI.-ALGEBRA. 1. Multiply a b by cd, where a is greater than b, and e greater than d; and hence deduce the rule of signs in multiplication. 2. Prove the truth of the equation a" m and n are positive integers. m Assuming that the above equation continues true when m and n are negative or fracțional, attach a consistent meaning Р to the expression a2. 3. Extract the square root of the following quantities: 1 + x1 + 6x3 + 11x2 + 6x; ·001, 1, 01 to 3 decimal places. 4. Prove a rule for the addition of fractions with different lenominators. Simplify the expression 5. Shew that the greatest common measure of two compound algebraical quantities is the least common multiple of all the common measures. 6. Solve the equations 8. A number consisting of three digits, the absolute value of each digit being the same, is 37 times the square of any digit. Find the number. 9. A and B run a mile. At the first heat A gives B a start of 20 yards, and beats him by 30 seconds; at the second heat A gives B a start of 32 seconds, and beats him by 9 yards. At what rate an hour does A run. 10. Find the sum of n terms of an arithmetic series. Given the first term and the common difference find n, so that the sum of 2n terms may be equal to p times the sum of n terms. Explain the result when p=4, and when p=2. 11. Find the number of combinations of n things taken together, without assuming the number of variations. In how many of the combinations do any 3 particular things occur? 12. Prove the Binomial Theorem for a positive integral index. Find the coefficient of x" in the expansion of (1 + x)2”. 13. Find the number of different ways in which a substance of a ton weight may be weighed by weights of 9 lbs. and 14 lbs. If the right angle be centesimally divided, and the measure of an angle according to that division be 5.5, where the measure of the right angle is 100, express its magnitude in the common circular measure. 2. Prove the formula sin (A + B) = sin A cos B + cos A sin B. Hence obtain sin 2.4 and cos 24. 4. Prove that in any plane triangle of which a, b, c are the lengths of the sides, and A, B, C the magnitudes of the angles, 2 /sin A + sin B + sin C a cos A + b cos B + c cos C u + b + c sincy= 2abc 5. Given in any triangle two sides and the included angle, solve the triangle; and adapt your result to logarithmic computation. 6. Express the area of a triangle in terms of its sides. In the "ambiguous case" find the difference between the areas of the two triangles. 7. A statue 10 feet high, standing on a column 100 feet high, subtends at the eye of an observer in the horizontal plane, from which the column springs, the same angle as a man 6 feet high standing at the foot of the column; find the distance of the observer from the column. 8. Find the radius of a circle inscribed in a triangle. Find also the radius of the circle which touches one side and the other two produced. 9. A polygon of n sides is inscribed in a circle; find its area; and the area to which it continually approximates as n is increased without limit. 10. If two circles, the radii of which are a and b, touch each other externally; and if O be the angle contained by the two common tangents to the circles; shew that 11. Prove Demoivre's Theorem; and apply it to find the 3 values of (– 1). 12. Expand sin✪ in terms of 0; and prove that 13. Prove Machin's series for the value of π. 14. Find the sum of the series EXAMINATION FOR MINOR SCHOLARSHIPS. DOWNING COLLEGE. November, 1861. Ι. Translate : From "Ο ΔΕ Κῦρος, ἅτε παῖς ὤν, καὶ φιλόκαλος.... ....ἀφικνείσθε ὅποι ἡμεῖς πάλαι ἥκομεν. to From ΔΥΝΑΤΩΤΕΡΑΣ δὲ γενομένης τῆς Ἑλλάδος.... to .... From ΤΙ δέ; ἂν αὖ γυμναστικῇ πολλὰ.... to .... καὶ γὰρ ἔοικεν ἔφη. THUCYDIDES. PLATO. From ΕΤΙ ἐκ τῶν αὐτῶν καὶ διὰ.... From ΠΑΙΕ, παῖε τὸν πανούργον..... to .... ὡς ὑπ ̓ ἀνδρῶν τύπτομαι ξυνωμοτών. ARISTOPHANES. 1. Ποθοίη, παροψίδας, λεκάνια, ελιγμούς, ἄγει, ἐξηρτύετο, ἀφνειὸν, ταραξιππόστρατους, Ἡλιασταὶ, φράτορες τριωβόλου. Εxplain and derive. 2. Δύναμις, ἐνέργεια, ἐντελέχεια. With what distinction of meaning does Aristotle use these words? What is the importance of the last in his scheme of philosophy, and how does Cicero explain it? 3. Give the names and laws, according to Porson or Buttmann, of the common Iambic and Trochaic metres in the passages from Euripides and Aristophanes. II.` Translate the following passages, and answer the questions appended: I. From Ceterum nemini omnium major justiorque.... to ....suæ ditionis fecisse? LIVY. Explain the use of the moods and tenses in the "oratio obliqua." Account for the genitive "suæ ditionis." Give the date of the second Punic War, the principal battles fought during its course, and the ultimate fate of Hannibal. What was the origin, geographical position, and fate of Carthage? II. From A. d. III. Kal. Maias quum essem.... to ....si, quid maxime expediat, obscurum. CICERO. Explain the use of the subjunctive mood in the first sentence. Who formed the first and second triumvirates? What was the battle which decided the fate of the Roman republic? When and for what reason was Cicero put to death? III. From Lauti, cibum capiunt:... to ....constituunt, dum errare non possunt. TACITUS. During the reign of what Roman Emperors did Tacitus live? What other works of his have come down to our time? Derive Lauti, conviciis, simplices. |