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the inferior creation. The Supreme Being himself is most pleased with praise and thanksgiving: the other part of our duty is but an acknowledgment of our faults, whilst this is the immediate adoration of His perfections. 'Twas an excellent observation, that we then only despise commendation when we cease to deserve it; and we have still extant two orations of Tully and Pliny, spoken to the greatest and best princes of all the Roman emperors, who, no doubt, heard with the greatest satisfaction what even the most disinterested persons, and at so large a distance of time, cannot read without admiration. Cæsar thought his life consisted in the breath of praise, when he professed he had lived long enough for himself, when he had for his glory. Others have sacrificed themselves for a name which was not to begin till they were dead, giving away themselves to purchase a sound which was not to commence till they were out of hearing. But by merit and superior excellence, not only to gain, but, whilst living, to enjoy a great and universal reputation, is the last degree of happiness which we can hope for here.-SPECTATOR.
Translate the following passage into Latin Alcaic Verse :
in this manner.
1. Find the equation of the tangents from a given point to a given conic.
2. If two lines be drawn through any point cutting a curve of the second order, and if the points where they meet the curve be joined directly and transversely, prove analytically that the joining lines will intersect on the polar of that point.
3. Find the eccentricity of a conic given by its general equation.
4 Prove the formulæ in Spherical Trigonometry which give the sine and cosine of half the sum of the base angles of a triangle in terms of the sides.
5. Prove the formula for the spherical excess, cos2a+cos2b+cos2c - I
cos Σ =
6. What relation exists between the coefficients of an equation in order that it may be possible to remove the second and third terms by the same transformation? Solve the equation
x2 + 4x3 + 6x2 + 3x=0
7. Prove the logarithmic formula for the sine of any angle of a spherical triangle in terms of the sides.
8. Prove the logarithmic formula for determining any two angles of a spherical triangle in terms of the opposite sides and third angle.
9. Investigate the algebraic solution of the cubic equation
and explain why it fails to give the numerical values of the roots when all real.
10. Transform the same equation into another whose roots shall be the sums of the roots of the original taken in pairs.
II. Required the locus of the intersection of tangents to a parabola which intersect at right angles.
12. Required the locus of the intersection of tangents to an ellipse or hyperbola which intersect at right angles.
13. Find the locus of the centre of a conic which passes through four given points.
14. Find the locus of the intersection of the perpendicular from the centre on any tangent to a conic, with the radius vector from a focus to the point of contact.
15. Determine the radius of curvature, and the co-ordinates of the centre of curvature, at any point of a parabola.
16. If r and R be the radii of the circles inscribed in and circumscribed about a spherical triangle, prove that
2 sina sin sinc
17. Solve the equation
tan r.tan R =
18. Eliminate 0 from the equations
4 cot 20 cot 20 - tan 20.
sin 0 + sin 20a, cos 0+ cos 20 = b.
1. Prove that
A cos 20+ 2H cos 0 sin 0+ B sin 20
has equal values for any two values of 0 which correspond to the directions of lines equally inclined to the two represented by
Ax2+2Hxy+ By2 = 0.
2. Two angles of a triangle which circumscribes a given circle move on parallel lines; find the equation of the locus of the third angular point.
3. Reduce the equation
x3 +px2 + qx+r=0
to the form y3-3y+m=0, by assuming x = ay+b, and solve this equation by making
find the relation between the coefficients which is required in order that the original equation may have equal roots.
4. If tank tan o, prove that
tan (0-4) =
49 + 3 ( 1 − 1 ) sin 64 + &c.
5. The perimeter of a spherical figure being less than the circumference of a great circle, determine the form under which a given perimeter will enclose the maximum area.
6. For the complete cubic,
Ax3 +3Вx2+3Cx + D=0,
investigate the absolute term in the equation of the squares of the differences of the roots.
7. Two tangents to a parabola intersect at a constant angle; investigate, geometrically, the locus of their point of intersection.
8. Given the centre of an ellipse or hyperbola, and the directions of any two pairs of conjugate diameters; determine, geometrically, the directions of the axes.
9. Find the locus of the intersection of normals drawn to a conic at the extremities of a focal chord.
10. Prove that normals drawn to an ellipse at three points whose eccentric angles are a, ß, y, will intersect in a point, if
sin (B+ y) + sin (y + a) + sin (a + ß) = o.
1. Define Logic, and explain the terms of your definition.
2. What are the different kinds of Wholes, and in which is reasoning possible?
3. What is the difference between Intuitive and Symbolical Thinking?
4. How are exceptive and exclusive Propositions to be analyzed? 5. Explain the canon, "Nota notæ est nota rei ipsius.”
6. Point out the fallacy in the following argument:-There are new discoveries made every day: therefore some things are discovered many times over.
7. State the "Achilles and Tortoise" fallacy, and show where the error lies.
8. Distinguish the different fallacies which arise when the falsehood of the conclusion is inferred from the falsehood of a premiss, or the truth of a premiss from the truth of the conclusion.
1. State the various objections to Logic enumerated by Whately, and give his answers.
2. According to Whately, there are no grounds for believing in the existence of Abstract Ideas in the ordinary sense of the term?
3. Give some account of the controversy as to whether Induction can be resolved into Syllogism or not.
4. State Whately's answer to the question, "whether the Discoveries which have been made in Natural Philosophy were accomplished by Reasoning?"
5. Distinguish between the Organon of Aristotle and the Organon of Bacon.
6. Whence arises the difficulty of distinguishing between Real and Verbal Questions? Illustrate the positions which Whately takes.
7. What are the causes to which Whately attributes the rise of Realism? The Realistic doctrine is repudiated by Aristotle ?
8. Give some account of the controversy between King and Berkeley on the subject of the Divine Attributes.
9. Give some account of the different senses of the word "Impossibility."
10. Point out the fallacies which are to be detected in Hume's argument against Miracles.
1. Show that while the first figure is equally adapted for either inductive or deductive reasoning, the second figure is more suitable for the deductive, and the third for the inductive form.
2. Taking into account those propositions in which the subject and predicate are reciprocal terms, assign the modes which are common to the first, second, and third figures.
3. Representing the extension of a concept by the area of a circle, an affirmative proposition may be symbolized by one circle wholly or partly contained within another, and a negative by two separate circles. Draw such diagrams of the six modes in the third figure.
4. Resolve a sorites into syllogisms in such a manner that the first syllogism shall be in the first figure, the second in the fourth, and all the rest in the third.
5. Investigate the question, whether the conclusion of the sorites and that of the last syllogism be necessarily identical?