That bears the name of life? Yet in this life SHAKESPEARE. EXAMINATION FOR JUNIOR AND SCHOOL EXHIBITIONS. MR. WILLIAMSON. 1. If from any point a tangent and a secant be drawn to a circle, prove that the rectangle under the whole secant and its external segment is equal to the square of the tangent. Describe a circle passing through a given point and touching two given right lines. 2. Find the locus of a point from which the tangents to two given circles are of equal length. 3. Construct a figure which shall be similar to a given rectilinear figure, and double its area. 4. Prove the expression for the area of a plane triangle in terms of its sides. The sides of a triangle are 114, 101, and 25, respectively; calculate its area. 5. Given of a triangle, two sides and its area, construct it; and point out when a solution is impossible. 6. Find the locus of a point such that the sum of its distances from three right lines, given in position, shall be equal to a given length. 7. If a quadrilateral be inscribed in a circle, prove that the rectangle under its diagonals is equal to the sum of the rectangles under its opposite sides. 8. Inscribe a square in a given triangle. If the base of a triangle be 5 feet and its area be 20 square feet, find the length of the side of its corresponding inscribed square. 3. Find the values of x and y that satisfy the equations √ 2x2 − 13x + 15 + √ 3x2 – 19x + 20 + √ 4x2. the three quadratic expressions having a common factor. 7. Write down the first seven terms of the expansion of 8. Find an expression for the amount of a given sum for a given time at compound interest. 9. What are eggs selling at, when, if they be raised three pence the dozen, one would get four fewer in a shilling's worth? 10. A and B can do a piece of work in 8 days, B and C can do it in 12 days, and A, B, and C can do it in 6 days. In how many days can A and C do it? ARITHMETIC. MR. GALBRAITH. 1. Find the greatest common measure of the numbers 667 and 2059, also of 1957 and 6137. 2. Find the sum in the lowest terms of the mixed numbers 13, 5, 210, 421, 122. 3. Find by Practice the price of 29 cwt. 3 qrs. 17 lbs., at £3 68. 9d. per cwt. 4. Find by Practice the rent of a farm of 113 acres, 2 roods, 37 perches, at £3 108. 8d. per acre. 5. If the circumference of the earth be 40,000,000 metres, the length of the metre being 39.37079 inches, calculate, by the chain rule, the diameter of the earth in miles, the ratio of circumference to diameter being 355: 113. 6. Compute the interest on £348 168. 7d. at 4 per cent. per annum, for 213 days. 7. Define true Discount, and find its amount for 259 days on £100, at 32 per cer cent. per annum. 8. Define Compound Interest, and compute how much it is on £175 for 5 years, the rate being 4 per cent. per annum. 9. Explain the rule for Contracted Multiplication, and by means of it calculate to five places the square, and also the cube, of the number 3.14159. 10. If the weight of a cubic inch of water be 252.5 grains, find, by the chain rule, the number of cubic inches in a ton of water, and find, by Horner's rule for extracting the cube root, the side of a cube to seven places of decimals which will just hold this weight of water. TRIGONOMETRY. DR. TRAILL. 1. Find the area of a plane triangle, whose three sides are 131, 246, and 327 feet respectively. 2. Explain what is called "the ambiguous case" in the solution of oblique-angled triangles; show in what cases there is but one solution; and find c from the equation 3. In any triangle prove 2bc sin A = √2a2b2 + 2b2c2 + 2c2a2 — aa — ba — c4. 4. Solve the simultaneous equations 7. In any plane triangle whose angles are a, B, y, prove 1 - cos 2a cos 2B cos 2 2 cos a cos ẞ cos y = 0. 8. If a cos + b cos (0+ a) be thrown into the form A cos (0 + B), determine the values of A and B. 9. Find the values of the circular measures of 30° and 45° to four places of decimals. 1. Beginning, Τὴν δὲ κρίσιν αὐτὴν τοῦ βίου πέρι ὧν λέγομεν, κ. τ. λ. Ending, ὁπότερος αὐτοῖν εὐδαιμονέστερος. PLATO, De Repub., lib. ii. 4. 2. Beginning, Consul interim omnium spe longiorem .... Ending, trepidationem insanam superstantibus armatis praebuerit. LIVY, lib. xxxi. cap. 17. GREEK AND LATIN PROSE COMPOSITION. Translate into Greek : PROFESSOR BRADY. Observing the enemy in possession of this lofty ground, Cheirisophus halted until all the army came up; in order that the generals might take counsel. Here Kleanor began by advising that they should storm the pass with no greater delay than was necessary to refresh the soldiers. But Xenophon suggested that it was far better to avoid the loss of life which must thus be incurred, and to amuse the enemy by feigned attack, while a detachment should be sent by stealth at night to ascend the mountain at another point and turn the position." However (continued he, turning to Cheirisophus), stealing a march upon the enemy is more your trade than mine. For I understand that you the full citizens and peers at Sparta practise stealing from your boyhood upward; and that it is held no way base, but even honourable, to steal such things as the law does not distinctly forbid. And to the end that you may steal with the greatest effect, and take pains to do it in secret, the custom is, to flog you if you are found out." Translate into Latin : Thus entangled in a situation nearly similar to that of Flaminius at Thrasymenus, the Romans were completely defeated. Night, however, saved them from total destruction, but to retreat to the plains was impossible. The pass in their rear, by which they had entered the valley, was secured by the enemy, so that they had no other resource but to encamp in the valley, not far from the scene of their defeat, and there hopelessly to abide the issue. The Samnites having thus got them in their power, waited quietly till famine should do their work for them. Occupying the road both in front and on the rear of the Romans, and guarding every possible track by which the enemy might try to escape over the hills on either side of the valley, they easily repulsed some desperate attempts made by the Romans to break out, and a large army surprised on its march, and hemmed in within a single narrow valley, could not possibly have the means of subsistence beyond a very short period. Accordingly the Romans soon threw themselves on the mercy of the conqueror. HISTORY AND GEOGRAPHY. MR. MAHAFFY. 1. Sketch the life and policy of Phidon of Argos. stances show the usual date assigned to him to be false? What circum 2. What Athenian politicians of the Periclean age might be designated as Conservatives, Whigs, and Radicals? Discuss the main points at issue between them? 3. What sides did the various literary men of the day take? 4. Describe the policy of Epaminondas in the Morea. How far was his work permanent ? 5. What is known of the early relations of Rome and Carthage ? State the causes of the first Punic War? 6. Explain the various political interests combined by the First Triumvirate. 7. Give some account of the reign of the Emperor Claudius. 8. Describe the principal mountain chains in Greece. What are the six highest mountains in these chains? 9. In what features may the countries of Italy and Sicily be contrasted with Greece? 10. Give the modern names of the following:-Patavium, Lugdunum Batavorum, Stabiæ, Euboea, Orchomenos (in Boeotia), Delphi, Kos, Nauplia. What cities of Italy and Greece do not occupy the sites of ancient towns? II. Write a note on the history of Corinth. 12. Sketch the Roman conquest of the Greek settlements in Sicily. PROFESSOR DOWDEN. ENGLISH LANGUAGE AND LITERATURE. 1. Trench enumerates a number of the dissimilating forces by the action of which languages spoken by two sections of a divided people become unlike one another? |