11. There is a number such that if 63 be subtracted from it, its digits will be inverted; and if it be multiplied by the sum of its digits, the product will be 1012; find the number.

12. Let r and R be the radii of the inscribed and circumscribed circles, and p the perpendicular let fall from the vertical angle C of a triangle, show that the base angles A and B are determined by the equations

13. If arcs drawn through the vertices of one spherical triangle perpendicular to the sides of another meet in a point, then arcs drawn through the vertices of the second perpendicular to the sides of the first meet also?

14. Prove that in a spherical triangle

15. If an algebraical equation be deprived of its second term, show that the coefficients of the transformed equation are functions of the differences of the roots of the original one.

16. Solve the recurring equation

x511x4+17x3 + 17x2 - 11x+1=0

17. Find the locus of the pole of a given line with regard to a series of confocal conics.

18. Find the equation of a chord of an ellipse in terms of the eccentric angles of the two points of intersection.

2. Let AB be the diameter of a circle, C'the centre, and P a point on

the radius CA such that CP =

n

CA; let Q, R be the points in which

any line passing through P meets the circumference, and draw the lines

BR, QC; let QC produced meet BR in S. Denoting the angle BSQ by 4, and CBS by 4, it is required to prove that

3. Let two equilateral hyperbolas having a common centre 0 intersect each other in a point Q, and let the same right line touch one at the point P, and the other at R; it is required to prove that the three distances OP, OQ, OR are in continued proportion.

4. From a fixed point in the minor axis produced, let tangents be drawn to the ellipses of a confocal system; find the locus of the points of contact.

DR. SHAW.

5. Show, from the general equations of a curve of the second degree, that any point on the curve may be regarded as the intersection of two consecutive tangents.

6. The length of the chord of an ellipse which touches a confocal ellipse whose semiaxes are a2 - k2, √ b2 - k2, is

9. If from any point 0 two lines be drawn to the foci of an ellipse meeting the conic in R, R, S, S', prove that

10. Find the locus of the intersection of tangents to a conic which intercept a given length on a given fixed tangent.

11. Find the locus of the intersection of normals to an ellipse which are at right angles to each other.

12. If a1 a1⁄2 . . . a, be the roots of ☀ (x), show that

...

1. Describe the processes of Abstraction and Generalization. How are they related to one another?

2. Give the substance of his Remarks on Definition.

3. Whately gives two different rules for computing probabilities in arguments; state these, and show the reason of their difference.

4. Give examples of petitio principii and ignoratio elenchi from Whately's own arguments.

5. Whately's analysis of Induction and Example? Criticize his arguments to prove that it must be reducible to syllogism.

6. Is syllogism competent to discover new truths? Discuss this question, giving examples, and notice the errors of Plato and Clarke on the point.

7. πᾶσα δὲ οὐσία δοκεῖ τόδε τι σημαίνειν. Ἐπὶ μὲν οὖν τῶν πρώτ των οὐσιῶν ἀναμφισβήτητόν ἐστι καὶ ἀληθές. ὅτι τόδε τι σημαίνει. ἀτομον γὰρ καὶ ἕν ἀριθμῷ το δηλούμενόν ἐστι. Επὶ δὲ τῶν δευτέρων οὐσιῶν φαίνεται μὲν ὁμοίως τῷ σχματι τῆς προσηγορίας τόδε τι σημαίνειν, ὅταν εἴπη, ἄνθρωπος, ἢ ζῶν· οὐ μήν γε ἀληθές· ἀλλὰ μᾶλλον Tоióv TI onμaive. Translate, and write a comment on this passage.

8. Assuming the necessary axioms of equality, and the construction, prove in syllogisms that the angles below the base of an isosceles triangle are equal.

9. Whately remarks that it is more difficult to prove a negative than an affirmative; why so? and explain the exception which seems to exist to this rule.

10. Construct, on the model of the "dictum de omni," &c., dicta which will apply directly to the second, third, and fourth figures.

MR. BARLOW.

1. Show that the process of reduction ad impossibile, as defined by Murray, does not apply to the modes Darapti and Felapton.

2. Prove that IEO is irreducible—(a) ostensively, (b) ad impossibile. 3. (a). Prove that the order of the three propositions in a legitimate syllogism is determinate.

(b). Assign the mode and figure of a syllogism in which the three propositions are-no Y is X, all X is Z, no Z is Y.

4. Substitute for one of the premises of a legitimate syllogism its contradictory-(a) Is the new syllogism legitimate? (b) If illegitimate, why? (c) If legitimate, how are the two conclusions related?

5. In reduction ad impossibile what difficulty is peculiar to the fourth figure, and why?

6. Prove the laws of the Sorites by the rules de omni and de nullo.

7. Various reasons have been assigned by different writers for the rejection of the fourth figure; mention any of these.

8. What is the meaning of the "ultratotal distribution of the middle term"? Give an example of a syllogism in which this occurs.

9. A copulative proposition has n subjects, and m predicates; determine the total number of simpler propositions into which it may be resolved.

10. Investigate of what proposition should the quantity and quality be assigned in the several figures, in order to determine the mode.

Classics.

PLATO.

MR. GRAY.

Translate the following passages into English :

1. Beginning, "Ην γάρ ποτε χρόνος, ὅτε θεοὶ μὲν ἦσαν, κ. τ. λ. Ending, σωτηρίαν τῷ γένει πορίζων.

Protag., xi.

2. Beginning, ἡμᾶς οὖν αἰσχρὸν τὴν μὲν φύσιν τῶν, κ. τ.λ.
Ending, ὃς ὑμῖν φυλάξει τὸ μέτριον μῆκος τῶν λόγων ἑκατέρου.

Ibid., xxiv.

3. Beginning, "Ω Σώκρατες, δοκεῖς νεανιεύεσθαι ἐν, κ. τ. λ. Ending, μηδὲ ἄλλῳ, οὗ ἂν κήδεται.

Gorgias, xxxviii.

4. Beginning, 'Αλλ ̓ οὔτ ̓ ἀπολέσθαι τὰ κακὰ δυνατόν, κ. τ. λ. Ending, ἐν δὲ τέχναις βάναυσοι.

Theatetus, xxv.

1. Compare the character of Socrates presented in the Apology with that presented in the Crito; and hence point out the purpose of the Crito.

2. What is the difference between Rhetoric and Dialectic? Under which head does the Crito come?

3. What is the difference in use between riμãv and ruãoba? With what cases are they constructed?

4. Αὐτίκα, πολλάκις and τὸ δέ have peculiar uses in Plato ? Explain how rò dé comes to have its meaning.

5. Contrast imperial Sparta with imperial Athens.

6. Compare Sparta in 432 B. C., with Sparta after 404 B. C., so as to show how badly the Spartan empire worked for the character of the Spartans themselves.

7. State accurately the real position of the kings at Sparta in 400 B. C. What was the change attempted by Lysander?

8. Give a full account of the conspiracy of Cinadon, and the causes of it.

9. Trace the steps by which the maritime war against Sparta was revived.

10. How long did the empire of Sparta last? Name the principal events which led to its overthrow.

CICERO.

MR. ABBOTT.

Translate the following passages into English:

(N. B.-Special credit will be given for elegance of translation.) 1. Beginning, Jam, quum se ille septimo die venisse a freto,...... Ending, At ego, inquit, novus patronus instituam.

Ep. ad Att., lib. ii. ep. 1.

......

2. Beginning, Quod de Quinti fratris epistola scribis,.. Ending, quoniam maritimum hoc contempsisti.

3. Beginning, Quum aliquem apud te laudaro tuorum Ending, Mirabiliter enim moratus est, sicut nosti.

4. Beginning, Hæc pactio, non verbis,.... Ending, commodum, ut ad me scribit.

5. Beginning, Hæc cum ad me frater pertulisset,. Ending, quam tibi, si potero, breviter exponam.

Ibid., lib. ii. ep. 16. familiarium, . . . ..

Ibid., lib. ii. ep. 25.

Ibid., lib. iv. ep. 18.

Epist. Famil., lib. i. ep. 9.

1. Write notes on the preceding passages where required by the construction or by historical allusions.

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