8. What does Aristotle mean by saying that men can become just and temperate only by doing what is just and temperate'? Surely, if men's acts are just and temperate, they themselves are already just and temperate ? SECTION B. [For Candidates who take Course I. in Calendar.] 4. (a) Necesse est quod homo sit liberi arbitrii ex hoc ipso, quod rationalis est.' What is meant by liberi arbitrii' in its ethical import, and how do reason and will respectively contribute to the production of a moral act? 5. Positive law supposes the existence of Natural law.' Discuss this statement, and determine the essential conditions for the validity of a positive law. 6. What bearing on Ethics have the following Psychological facts: (a) The Association of Ideas; (b) The tendency of Sense-images to realization? SECTION C. [For Candidates who take Course II. in Calendar.] 7. Discuss-In genuine cases of perplexity of Conscience, Green's theory of the Moral Ideal may give us help, but Utilitarianism will afford none.' 8. Explain and examine Kant's attempt to deduce all particular duties from one Categorical Imperative. 9. (a) Can any theory of Ethics account for moral progress? (b) Can the acceptance of Evolution be reconciled with an Idealistic system of Ethics? HISTORY OF PHILOSOPHY. PROFESSOR MAGENNIS; REV. DR. MAGILL; PROFESSOR PARK. [Candidates will answer on two sections only, ONE of which must be section A.] SECTION A. 1. How does Kant think that his critical view of knowledge explains the antinomy or dilemma which arises from the Kantian view of Free Will? How was the similar difficulty met in Ancient or in Medieval Philosophy? 2, What metaphysical questions most interested ancient thought? With what modifications do these remain the problems of later philosophy? 3. Write historical notes on the following: 'Pythagoras proclaimed numbers as the truth for all. The Eleatics took their stand upon being. Heraclitus contended for becoming. Plato advanced his theory of ideas. Plotinus shifted the ground of the absolutely true from the thing thought to the thought itself of the thing. He was led to it by the inconsequences of which Scepticism had convicted all antecedent systems.' SECTION B. 4. Give a brief historical survey of the development of empirical psychology in the Scholastic period. 5. How does Duns Scotus combine the ontological, cosmological, and teleological Theistic arguments? 6. Write a short account of the Thomistic Aesthetic or Theory of Beauty. SECTION C. 7. Descartes, Locke, Hume, Kant, and, with all his shortcomings, even Reid, were among the great original thinkers who have carried philosophy into one of its indispensable phases.' Examine this historically in detail. 8. AnnotateThe post-Kantian philosophy is perhaps more rife in metaphysical speculation than that of any other period. We shall look for Metaphysics, however, less in the works of the idealists, Fichte, Schelling, Hegel, than in Kant's realistic followers, Herbart and Schopenhauer.' 9. Scepticism in its absolute form is self-destructive; on the other hand-and that is the great advantage that it has over Dogmatism-its method possesses a very high degree of value in all departments of scientific investigation.' Illustrate this with special reference to French and to British speculation. IV.-MATHEMATICAL SCIENCE. MATHEMATICS. FIRST PAPER. PLANE GEOMETRY AND THEORY OF EQUATIONS. PROFESSOR BROMWICH. [Full credit will be given for answering TWO-THIRDS of this Paper.] 1. Prove that the six anharmonic ratios of the roots of the quartic (abcde(x 1)* are the values of a determined by 4 (λ2 − λ + 1)3 ̄ ̄ (λ + 1)2(λ − 2)2(2λ − 1)2 ace + 2bcd ad2 — eb2 — c3. 2. Two real conics S, S' are given: show that two real parabolas can be drawn through the points of intersection of S, S', unless S, S' are both hyperbolas; and if S, S' are hyperbolas, prove that the two parabolas are real, unless the directions of the asymptotes of S are interlaced with those of S'. For example, consider the case when S, S' are rectangular hyperbolas. Prove further that if the parabolas coincide, they must degenerate into straight lines, unless S, S' are hyperbolas with one asymptote of S parallel to one of S'. 3. Explain how to find the common self-polar triangle of two conics whose equations are given. Determine the triangle for the conics 4. In any real linear transformation of a plane, show that at least one real point and one real line are unchanged. Taking the special case find the points and lines unaltered by the transformation. Show that the group of linear transformations for which the conic zxy20 is invariant, is given by = x' = a2x + 2aẞßy + ẞ2z, y' = ayx + (ad + ẞy) y + Bdz, == y2x + 2ydy + S2%, where a, B, y, 8 are arbitrary, except that ad - By must not vanish. With regard to the last transformation, prove that the three points (p2, p, 1), (q2, q, 1), (2ẞ, 8-a, - 2y), are unaltered by the transformation, provided that p, q are the roots of the equation in t yt2 + (8-a) t- B = 0. Consider what happens in the case p = q, i.e. (a − 8)2 + 4ßy = 0. 5. Prove that every cubic has at least one real inflexion, and may be transformed by a real projection to the form Draw roughly the 5 distinct types which are included in this equation. The line y-bt (x − a) is drawn from (a, b) a point of the cubic to touch the curve elsewhere: prove that t satisfies the equation Applying question 1 to this quartic, show that the anharmonic ratios of the pencil of four tangents are given by that equation (though now, of course, 92, 93 have different meanings). 6. Explain how to express the coordinates of points on a unicursal cubic as cubic polynomials of a parameter t. Taking the cubic (a‚b‚¢‚d‚X† 1)3 = (a,b,c,d‚¤† 1 )3 = (ab ̧ ̧d ̧Xƒ 1 )31 show that three points t1, t2, t, are collinear, provided that where = P } (t1 + tz + ts), 9 = } (t2t3 + tзtı + t1t2), r = − tâtats. Deduce a quadratic for the parameters of the double point, and find the condition that it should be a cusp. 7. A curve of the fourth class touches the line at infinity at each of the circular points: show that the curve has one focus and six asymptotes, that the six asymptotes touch conic whose centre is the focus, and that the asymptotes of the conic touch the original curve of the fourth class. η 8. Let έ, be ordinary rectangular Cartesian coordinates, and let x = & + in, y = - in, while t is a variable complex parameter of absolute value unity. Show that the equations x = tm, y=t"", where m, n are conjugate complex indices, determine an equiangular spiral. Hence, or otherwise, prove that an equiangular spiral is its own polar reciprocal with respect to any concentric rectangular hyperbols which touches the spiral. |