TRIGONOMETRY. MR. H. THRIFT. 1. Assuming that the Sun subtends an angle of 30' at an observer's eye, what is the greatest distance from his eye at which he can hold a coin one inch in diameter so as to completely obscure the sun's disc? 2. Prove the formula for cos (A + B), where A and B are acute angles, but A + B is an obtuse angle. 3. Express the sine and cosine of an angle in terms of the tangent of half the angle. 7. The lengths of the sides of a triangle are 39, 41, 50: calculate the radii of its inscribed and circumscribed circles. 8. Given in a triangle that two sides are in the ratio 3: 2, and the included angle is 77° 18', find the base angles 9. A castle stands on the top of a cliff near the sea. A man in a boat observes that the angles of elevation of the top and bottom of the castle are 75° and 30° respectively. He then rows 100 feet directly away from the castle, and finds that the angle of elevation of the top of the castle is 45°. Find the height of the castle. 10. Given that log 2 = 3010, find the number of ciphers between the decimal point and the first significant figure in (2) 1000 LOGIC. MR. ROGERS. I. Express each of the following in one of the four categorical forms, stating in each case the subject and predicate used: 'Any two sides of a triangle are greater than the third.' 'Not all triangles are scalene.' Tares and wheat are mixed together in every field." 2. Express in categorical form, and give the contradictories of— 'All that shivers is not cold.' 'Only the strong deserve the meat.' 2. If A is true, which of the other three propositions with the same subject (a) must be true, (b) may be true, (c) must be false, (d) may be false? 4. What are the five principal general rules of a categorical syllogism? Prove that two particulars give no conclusion. 5. Draw up a scheme to show the positions of the terms in the four figures of a syllogism. Prove from the general rules that only the first figure can give an A conclusion. 6. Quote the mnemonic lines for the four figures, explaining the meaning of the vowels and consonants used. What general syllogistic rules and what rules of immediate inference would be violated by Bemasop in any figure, the letters having their usual meaning? 7. Explain and illustrate the terms: Conditional Proposition,' 'Disjunctive Proposition,' 'Conditional Syllogism,' 'Disjunctive Syllogism.' 8. What is an argument from Analogy? Give an example. Has such reasoning any value? MECHANICS. MR. FRY. 1. Prove the method for getting the magnitude and position of the resultant of any number of concurrent forces, when their magnitudes are known, and the angles their lines of action make with a given line. 2. A weight of 15 lbs. on a smooth inclined plane, 5 feet long and 4 feet high, is acted on by a force of 3 lbs. in a downward direction parallel to the plane. What force must act parallel to the base of the plane to keep it at rest, and what will be the pressure on the plane? 3. Place a light bar, 24 inches in length, on two props, 10 inches apart in the same horizontal line, so that when weights of 15 and 25 lbs. are attached to the ends of the bar, the pressures on the props shall be equal. 4. The sides AB, BC of a light rectangular plate are 4 and 8 inches in length respectively. Find the point at which the plate will balance horizontally, when weights of 2, 3, 4, 5, and 6 lbs. are attached to the points A, B, C, D, and the centre of the plate respectively. 5. A slider weighing 10 lbs., resting on a rough horizontal plane, can just be moved by a force of 4 lbs. acting parallel to the plane. What is the smallest weight which must be placed on it to prevent motion when it is pulled by a force of 14 lbs. acting parallel to the plane? 6. A mass of 30 tons, moving with a velocity of 15 miles an hour, is brought to rest by a force in 100 yards. How many pounds can the force support? 7. Prove the formulæ for the acceleration and the tension of the string, when a mass of P lbs. on a smooth horizontal table, is connected by a light string passing over a smooth light pulley at the edge of the table, to a mass of Q lbs. hanging freely. 8. Find the initial horizontal and vertical components of the velocity of a projectile, so that it may rise to a height of 81 feet, and have a horizontal range of 405 feet. 9. With what velocity must a mass of 12 lbs. be projected against one of 56 lbs., moving with a velocity of 60 feet per second, so that the mass of 56 lbs. shall be brought to rest? Find also the energy lost in footpounds. 10. When a particle is moving on a smooth inclined plane, prove that the increase of its kinetic energy is equal to the work done on it by gravity. For Supplemental Senior Freshmen Trinity Examination. The first FIVE questions set for the Final Freshman Examination, and the following: 6. A weight of 40 ozs. is suspended by a string tied at C to two strings CA, CB, of lengths 3 and 4 feet respectively, which are tied to two points A, B, 5 feet apart in the same horizontal line. Find the tensions in CA, CB when another string is tied to C and pulled parallel to AB with a force of 20 ozs. 7. Prove the method for getting the resultant of two unlike parallel forces. 8. A uniform beam AB, 26 feet long, weighing 60 lbs., rests on a smooth plane AC, and against a smooth vertical wall CB. If AC is 10 feet, what force must be applied horizontally at a point P in the beam, I foot from A, in order to keep the beam at rest? ὡς ἔφατ ̓, ὦρτο δὲ Ιρις ἀελλόπος ἀγγελέουσα, τῶν μιμνησκόμεναι οἳ δὴ πολέες τε καὶ ἐσθλοί 66 θάρσει, Δαρδανίδη Πρίαμε, φρεσί, μηδέ τι τάρβει 1. Parse—κίχεν, καταμήσατο, λύσασθαι, κέατο, ἔλλαβε. HOMER. 2. Mark the quantities in—ἔφυρον, ἱκάνω, λύσασθαι, δάκρυσιν, ἔκτανε. 3. Translate : : (α) ἁρπάξων ἢ ἄρν' ἀμαλὴν ἤ πτῶκα λαγωόν. (6) οὐδ ̓ ἄρα οἵ τις ἀνουτητί γε παρέστη. (c) τῆς δ' ἐλελίχθη γυῖα, χάμαι δέ οἱ ἔκπεσε κερκίς. (α) ήριπε δ' ἐξοπίσω, ἀπὸ δὲ ψυχὴν ἐκάπυσσε. Quae contra breviter fata est Amphrysia vates: At ramum hunc (aperit ramum, qui veste latebat) VIRGIL. 1. Parse and give the principal parts of-fata est, absiste, gemuit, ferunt, aperit. 2. Translate: (α) Sidoniam picto chlamydem circumdata limbo. (c) Maeonia mentum mitra crinem que madentem (α) Litora litoribus contraria, fuctibus undas Imprecor, arma armis; pugnent ipsique nepotesque. 3. Mark the quantities in-limen, fatalis, tempore, incolumis, genitorem. φίλον πρὸς ἄνδρα χρὴ λέγειν ἐλευθέρως, EURIPIDES. 1. Parse-ἐλειψάμην, κἄστεψα, τύχοιμι, ἔλθω, παρεστώς. 2. Translate : (a) ἔχω λόγον δὴ καὶ προθυμίαν σέθεν. (6) κοὔτις ἦν οὕτω κακός ὅν οὐ προσεῖπε, καὶ προσερρήθη πάλιν. 3. Give the meanings of—-νεολαία, ἄωρος, ἀμπλακών, περιπτύσσων, σύγκασις. Translate : [11.] Ph. Verum ego quando te et amicitiam et gratiam in nostram domum Video adlicere, etsi advorsatus tibi fui, istac iudico: Tibi permitto, posce duce. Lu. Di te servassint mihi. Sed ad istam adde gratiam unum. Ph. Quid id est autem unum? Lu. Eloquar. Tute ad eum adeas ut concilies, tute poscas. Ph. Eccere. Lu. Nimio citius transiges: firmum omne erit quod tu egeris. Gravius tuom erit unum verbum ad eam rem quam centum mea. |