SENIOR FRESHMEN. Mathematics. A. DR. HART. 1. Find the area of the quadrilateral, the co-ordinates of whose vertices are (1.2) (3.4) (5.3) (6.2). 2. Find the angle contained by the lines x2 - 5xy + 6y2 = 0; and find the equation of the bisectors of this angle. 3. Two vertices of a triangle move on given right lines, and its sides pass through three given points in a right line; find the locus of the third vertex. 4. If three chords of a circle are drawn from the same point in the circumference, prove that semicircles described on these chords will intersect in three points which are on one right line. MR. SALMON. 5. Given one side of a spherical triangle and the two adjacent angles, how are the other sides found? and prove the formulæ. 6. Determine B so that the following equation may represent two right lines : x2 + Bxy + 3y2 — 5x — 3y +6. 7. What is the equation of the pair of tangents drawn from the point 1, 2 to the circle x2 + y2 — 4x — 7y — I = 0. 8. A dairyman, after selling a certain quantity of milk at 3d. a quart, fills the vessel up with water, and, selling the mixture at the same price, finds that he has gained the same profit as if he had added no water, and charged 3 d. at first: what proportion of water did the mixture contain? MR. TOWNSEND. 9. Prove the formula for tan ne in powers of tan 0; viz. : 10. Prove the formula for the area of a spherical triangle, viz. :— If r = area = r2. (A + B + C − π). 4000 miles, and A + B + C = 181°, find the area. 11. If x be a large number, and & any small increase or diminution to it, prove that M log (x+8)= log x ± .d. 12. Prove the rule by which the number corresponding to a given logarithm not found in the tables is calculated from the tables. MR. CARMICHAEL. 13. Prove, by the method of abridged notation, that the three perpendiculars on the sides of a triangle meet in a point, and that the three bisectors of the sides meet in a point. 14. If a chord of a constant length be inscribed in a given circle, prove analytically that it will always touch another circle. 15. Express the side of the square inscribed in a triangle in terms of the base and perpendicular height. 16. Expand sin x in terms of x. B. DR. HART. 1. Find the area of the triangle contained by the right lines Ax+By+C=0, A'x + By + C = 0, A′′x + B′′y + C" = 0. 2. Find the angles of a triangle such that if perpendiculars be let fall on its sides from any point on the circumference of a circle, three times the sum of the squares of two of these perpendiculars will be equal to the square of the third. 3. Given base and area of a triangle; find the equation of the locus of the centre of inscribed circle. 4. Write the equation whose roots are harmonic means between the roots of the equation x3 +5x2+7x=11. MR. SALMON. 5. The equations of the sides of a triangle being a=0, B = 0, y = o, find the equations of the lines joining the feet of the three perpendiculars. 6. Prove that the conditions that the following equation shall represent a circle are a sin2B + b sin2A – 2ƒ sin A sin B =b sin2C+csin2B - 2e sin C sin B 7. Write down the equation of the locus of a point such that its perpendicular distance from a given line is equal to the intercept made from the origin on the axis of a by a perpendicular through the point to the radius vector from the origin. 8. Given two sides of a spherical triangle and the included angle; give a logarithmic formula for finding the third side, and prove it. MR. TOWNSEND. 9. Taking two of the three diagonals of a quadrilateral as axes of coordinates, form the equation of the line joining their middle points, and show that it passes through the middle point of the third diagonal. 10. Given the equations of any four lines in the abridged form, a=o, ẞ=o, y=0, d=0; form the equation of the line joining the intersections of the perpendiculars of two of the four triangles they determine; and show from its form that it passes through the intersections of the perpendiculars of the remaining two. 11. Denoting by a, B, y the three altitudes of an inaccessible object observed from three points a, b, c in the same right line in a horizontal plane; show that the height h of the object above the plane is given by the formula I 12. If the function (r+ 1) be expanded in an infinite series of the form (Ak.xk), show that every three consecutive coefficients Ak-1, Ak, Ak+1 are connected by the relation, (2 k − 1) Ak-1+ (2 k +1) Ak+1 = 2kAk. MR. CARMICHAEL. 13. Deduce, by the method of abridged notation, the equation of the line connecting the point of intersection of perpendiculars with the point of intersection of bisectors of sides, of any triangle. 14. If the perpendiculars from the vertices of one triangle on the sides of another meet in a point, prove that, vice versa, the perpendiculars from the vertices of the second on the sides of the first meet in a point. 15. For a given circle investigate the general expressions for the sides of the inscribed and circumscribed regular pentagons. 1. What is the meaning of the phrase "Reductio ad Impossibile"? What circumstance renders it necessary? Show that there can be only two cases of this; and prove the conclusiveness of the new mode in each case. D 2. Show the conclusiveness of ostensive reduction in all other cases; and investigate if "Reductio ad Impossibile" be equally applicable to them. 3. If the substitution of Conclusion for minor premiss renders the new premises legitimate, investigate, on this supposition, the mode and figure. MR. BARLOW. 1. Distinguish accurately between the Matter and the Form of Thought. What is the Form of the Concept, the Judgment, and the Syllogism, respectively? 2. State Aristotle's definition of iv0úμnua. What does he mean here by ɛikòç and onμetov? Point out the error of the derivation ordinarily assigned for ἐνθύμημα. 3. What is the difference between Induction and Deduction? State, and illustrate by examples, the principal successive steps in the Baconian method of Induction. 4. Write down in Sir William Hamilton's notation the modes ordinarily received in the second and third Figures. 5. Which of the following modes are legitimate : IYI-NY-UAU-UAA-wYw-wEw-IUn-AwE-YEE-EYE. nUn-UOO-UOn-Uww-UwO-YYY. 6. Apply the process of "Reductio ad Impossibile" to the following modes:-Cesare, Felapton, Dimaris, Camenes, Fresison. 7. Show that it is impossible to verify an illegitimate mode by Reduction. 8. "The angles at the base of an isosceles triangle are equal." Exhibit the proof of this proposition in a series of rigorous syllogisms. MR. CONNER. 1. Explain the ambiguities in the use of the following words :Experience; species; same; reason; tendency; impossibility; indiffe rence. 2. Apply the syllogistic test to the following arguments: a. A proof of a future life would not be a proof of religion: for that we are to live hereafter is just as reconcileable with the scheme of Atheism as that we are now alive; and therefore nothing can be more absurd than to argue from that scheme that there can be no future state. But as religion implies a future state, any presumption against such a state is a presumption against religion. b. The real friend is not the man who overlooks one's faults. Therefore, men are better friends to others than to themselves, since they see the faults of others better than their own. c. If this world were the last stage of man's existence, wickedness would not pass unpunished here; but there is evidence of a future life. Therefore affliction is not evidence of wickedness. Classics. PLATO-SELECTED DIALOGUES. MR. RUTLEDGE. Translate into English Prose one of each pair of the following passages: ̓Αλλ ̓, ὦ Σώκρατες, πειθόμενος ἡμῖν τοῖς σοῖς τροφεῦσι μήτε παῖδας περι πλείονος ποιοῦ μήτε τὸ ζῆν μήτε ἄλλο μηδὲν πρὸ τοῦ δικαίου, ἵνα εἰς ̔́Αιδου ἐλθὼν ἔχῃς ταῦτα πάντα ἀπολογήσασθαι τοῖς ἐκεῖ ἄρχουσιν· οὔτε γὰρ ἐνθάδε σοι φαίνεται ταῦτα πράττοντι ἄμεινον εἶναι οὐδὲ δικαιότερον οὐδὲ ὁσιώτερον, οὐδὲ ἄλλῳ τῶν σῶν οὐδενί, οὔτε ἐκεῖσε ἀφικομένῳ ἄμεινον ἔσται. ἀλλὰ νῦν μὲν ἠδικημένος ἄπει, ἐὰν ἀπίῃς, οὐχ ὑφ ̓ ἡμῶν τῶν νόμων ἀλλ' ὑπ ̓ ἀνθρώπων· ἐὰν δὲ ἐξέλθῃς οὕτως αἰσχρῶς ἀνταδικήσας τε καὶ ἀντικακουργήσας, τὰς σαυτοῦ ὁμολογίας τε καὶ ξυνθήκας τὰς πρὸς ἡμᾶς παραβὰς καὶ κακὰ ἐργασάμενος τούτους, οὓς ἥκιστα ἔδεί, σαυτόν τε καὶ φίλους καὶ πατρίδα καὶ ἡμᾶς, ἡμεῖς τέ σοι χαλεπανοῦμεν ζῶντι, καὶ ἐκεῖ οἱ ἡμέτεροι ἀδελφοὶ οἱ ἐν Αιδου νόμοι οὐκ εὐμενῶς σε ὑποδέξονται, εἰδότες, ὅτι καὶ ἡμᾶς ἐπεχείρησας ἀπολέσαι τὸ σὸν μέρος. ἀλλὰ μή σε πείσῃ Κρίτων ποιεῖν ἄ λέγει μᾶλλον ἢ ἡμεῖς.—CRITO. ̓Αλλὰ πρῶτον εὐλαβηθῶμέν τι πάθος μὴ πάθωμεν. Τὸ ποῖον, ἣν δ ̓ ἐγώ. Μὴ γενώμεθα, ἡ δ ̓ ὅς, μισολόγοι, ὥσπερ οἱ μισάνθρωποι για γνόμενοι· ὡς οὐκ ἔστιν, ἔφη, ὅ τι ἄν τις μεῖζον τούτου κακὸν πάθοι ἢ λόγους μισήσας. γίγνεται δὲ ἐκ τοῦ αὐτοῦ τρόπου μισολογία τε καὶ μισανθρωπία. τε γὰρ μισανθρωπία ἐνδύεται ἐκ τοῦ σφόδρα τινὶ πιστεῦσαι ἄνευ τέχνης, καὶ ἡγήσασθαι παντάπασί τε ἀληθῆ εἶναι καὶ ὑγιῆ καὶ πιστὸν τὸν ἄνθρωπον, ἔπειτα ὀλίγον ὕστερον εὑρεῖν τοῦτον πονηρόν τε καὶ ἄπιστον, καὶ αὖθις ἕτερον. καὶ ὅταν τοῦτο πολλάκις πάθῃ τις, καὶ ὑπὸ τούτων μάλιστα, οὓς ἂν ἡγήσαιτο οἰκειοτάτους τε καὶ ἑταιροτάτους, τελευτῶν δὴ θαμὰ προσκρούων μισεῖ τε πάντας καὶ ἡγεῖται οὐδενὸς οὐδὲν ὑγιὲς εἶναι τὸ παράπαν. ἢ οὐκ ᾔσθησαι οὕτω πως τοῦτο γιγνόμενον; Πάνυ γε, ἦν δ ̓ ἐγώ. Οὐκοῦν, ἡ δ ̓ ὅς, αἰσχρόν, καὶ δῆλον, ὅτι ἄνευ τέχνης τῆς περὶ τἀνθρωπεια ὁ τοιοῦτος χρῆσθαι ἐπιχειρεῖ τοῖς ἀνθρώποις; εἰ γάρ που μετὰ τέχνης ἐχρῆτο, ὥσπερ ἔχει, οὕτως ἂν ἡγήσατο, τοὺς μὲν χρηστοὺς καὶ πονηροὺς σφόδρα ὀλιγους εἶναι ἑκατέρους, τοὺς δὲ μεταξὺ πλείστους.-PHÆDO. *Ακουε δὴ ὄναρ ἀντὶ ὀνείρατος. ἐγὼ γὰρ αὖ ἐδόκουν ἀκούειν τινῶν, ὅτι τὰ μὲν πρῶτα οἷονπερεὶ στοιχεῖα, ἐξ ὧν ἡμεῖς τε συγκείμεθα καὶ τἆλλα, λόγον οὐκ ἔχοι. αὐτὸ γὰρ καθ ̓ αὑτὸ ἕκαστον ὀνομάσαι μόνον εἴη, προσειπεῖν δὲ οὐδὲν ἄλλο δυνατόν, οὔθ ̓ ὡς ἔστιν, οὔθ ̓ ὡς οὐκ ἔστιν· ἤδη γὰρ ἂν οὐσίαν ἢ μὴ οὐσίαν αὐτῷ προστιθεσθαι, δεῖν δὲ οὐδὲν προσφέρειν, εἴπερ αὐτὸ ἐκεῖνο μόνον τις ἐρεῖ. ἐπεὶ οὐδὲ τὸ αὐτὸ οὐδὲ τὸ ἐκεῖνο οὐδὲ τὸ ἕκαστον οὐδὲ τὸ μόνον οὐδὲ τὸ τοῦτο προσοιστέον, οὐδ ̓ ἄλλα πολλὰ τοιαῦτα, ταῦτα μὲν γὰρ περιτρέχοντα πᾶσι προσφέρεσθαι ἕτερα ὄντα ἐκείνων, οἷς προστίθεται, δεῖν δέ, εἴπερ ἦν δυνατὸν αὐτὸ λέγεσθαι καὶ εἶχεν οἰκεῖον αὑτοῦ λόγον, ἄνευ τῶν ἄλλων ἁπάντων λέγεσθαι. νῦν δὲ ἀδύνατον εἶναι ὁτιοῦν τῶν πρώτων ῥηθῆναι λόγῳ· οὐ γὰρ εἶναι αὐτῷ ἀλλ ̓ ἢ ὀνομάζεσθαι μόνον· ὄνομα γὰρ μόνον ἔχειν, τὰ δὲ ἐκ τούτων |