DR. SHAW. 7. The radii of two circles which do not intersect are r and r; and the distances between their centres is a. Find an expression for the entire length of a string wrapped round both circles a. In case the string crosses itself between the circles; and b. In case it does not. 8. Find the locus of the foot of a perpendicular let fall from a fixed point on a chord of a circle; the chord so moving that the product of the perpendiculars let fall on it from the fixed point and the centre = r2 — d2 d being distance of point from centre. 9. Find by Sturm's theorem the number and situation of the real roots of the equation 23-112- 10. Prove by analytic geometry that the three diagonals connecting opposite vertices of a hexagon circumscribed to a circle pass through a point. 11. Join the middle points of the sides a, b, c of a triangle. In the new triangle thus formed inscribe a circle, and from its centre draw right lines to the angles A, B, C of original triangle. Show that these lines divide A, B, Cinto parts a, a'; ß, ß'; y, y'; such that 12. Prove that a circle which cuts two fixed circles at constant angles will also touch two fixed circles. 13. Given base and vertical angle of a triangle; find the locus of the intersection of the lines drawn from the extremities of the base bisecting the opposite sides; taking the base for one of the axes, and its middle point for origin of co-ordinates. 14. Find the equation of a circle which touches two right lines, and passes through a given point; the given lines being the axes. 15. If Sm be the sum of the mth powers of the roots of a given equation, and om that of the roots of the equation of differences; prove that and hence form the equation of differences for a cubic. Logics. DR. MALET. 1. Without resolving a Sorites into syllogisms, prove that in such a series of propositions the chain of reasoning stops when we come to a negative proposition; and is broken when we come to any particular proposition after the first. 2. In substituting conclusion for a premiss, prove that (in order to have the new premisses legitimate) the suppressed premiss cannot be either I or 0. 3. The mutual substitution of the conclusion and a premiss is never valid; show this on general principles, without inquiring into the quantity of the terms. The same remark applies to Sorites and other forms? 4. The assertion that from falsehood truth may follow, does not hold on all occasions; what are the circumstances under which it is true? and give instances of logical deduction, in which, from falsehood, falsehood must follow. 5. Without examining particular cases, prove generally that, when the conclusion is substituted for a premiss, the resulting premisses cannot be in the fourth figure. 6. Prove that IEO cannot be brought to a legitimate mode in any of the four figures by the application of Reductio ad impossibile. 7. Give instances of IEO in which the premisses are true, and the conclusion true; also, the conclusion false from true premisses; and, in the latter case, how are you guided in the selection of the terms? The figure is in all cases indifferent; why? 8. Given the three propositions (not in their syllogistic order); how do you determine the middle term? MR. LONGFIELD. 1. The following arguments are put by Cicero ("De Nat. Deorum") into the mouth of Velleius, to prove that the gods have the human form : "Quod si omnium animantium formam vincit hominis figura, Deus autem animans est; ea figura profecto est, quæ pulcherrima sit omnium : quoniamque Deos beatissimos esse constat, beatus autem esse sine virtute nemo potest, nec virtus sine ratione constare, nec ratio usquam inesse, nisi in hominis figura; hominis esse specie Deos, confitendum est." Exhibit the first of these arguments in the form of a syllogism in the first figure, and the second in the form of a series of syllogisms in the first figure. 2. Put the following arguments into syllogistic form, and show where the fallacy, if any, in each lies: "None but maritime countries are prosperous. Portugal is a maritime country, and therefore is prosperous." "None but maritime countries are prosperous. Bavaria is not a maritime country, and therefore is not prosperous." "Things which are identical are the same, but things may be nearly the same; therefore identity admits of degrees." "All mines that produce gold are not worked with profit. The mines in a certain district produce gold, and therefore cannot be worked with profit." 3. Show that an inductive argument, when reduced to the syllogistic form, is apparently a syllogism in the third figure; and explain how the reasoning is valid, though contrary to the rules of the third figure, the conclusion is A. 4. Reasoning is compared by D. Stewart to the piling together blocks of stone; on each of which, as on a pedestal, a man can raise himself a small, and but a small, height above the plain; but which, when skilfully built up, will form a flight of steps which will raise him to a great elevation. How does Archbishop Whately extend this illustration? 5. Mention the various ways in which words come to be of ambiguous signification. WHATELY'S LOGIC. MR. CARMICHAEL. 1. a. Point out the ambiguity in the derivation of the term "Logic;" and hence account for the twofold aspect in which the science of Logic has been viewed. b. Dr. Whately is charged with representing Logic at one time as a formal, at another, as a real, science. Quote the passages on which this charge is founded, and consider its justice. z. State the strict logical sense of the term "Argument," according to Dr. Whately, and the objection which has been made to his view and enumerate the other senses in which this term has been employed. 3. Write a note upon the term "Enthymeme," = Evdúμnμa. 4. Four distinct objections have been raised against the leading principles of the science of Logic: enumerate these; name their respective advocates; and give an abstract of Dr. Whately's observations in reply. 5. Give an abstract of Dr. Whately's observations upon Definition and Division. Classics. PLATO. MR. DICKSON. Translate the following passages into English Prose : Τί δ ̓ αὖ ; ἔστι τι καὶ τοιόνδε ἐν αὐτοῖς, οἷον μεταξὺ ἀμφοτέρων πάν των τῶν ἐναντίων δυοῖν ὄντοιν δύο γενέσεις, ἀπὸ μὲν τοῦ ἑτέρου ἐπὶ τὸ ἕτερον, ἀπὸ δ ̓ αὖ τοῦ ἑτέρου πάλιν ἐπὶ τὸ ἕτερον; μείζονος μὲν γὰρ πράγματος καὶ ἐλάττονος μεταξὺ αὔξησὶς καὶ φθίσις, καὶ καλοῦμεν οὕτω τὸ μὲν αὐξάνεσθαι, τὸ δὲ φθίνειν ; Ναί, ἔφη. Οὔκουν καὶ διακρίνεσθαι καὶ συγκρίνεσθαι, καὶ ψύχεσθαι καὶ θερμαίνεσθαι, καὶ πάντα οὕτω, κἂν εἰ μὴ χρώμεθα τοῖς ὀνόμασιν ἐνιαχοῦ, ἀλλ ̓ ἔργῳ γοῦν πανταχοῦ οὕτως ἔχειν ἀναγκαῖον, γίγνεσθαί τε αὐτὰ ἐξ ἀλλήλων γένεσίν τε εἶναι ἐξ ἑκατέρου εἰς ἄλληλα ; Πάνυ γ', ή δ' ὅς.—PHADO. ὡς μέντοι διὰ ταῦτα ποιῶ ἃ ποιῶ, καὶ ταύτῃ νῷ πράττω, ἀλλ ̓ οὐ τῇ τοῦ βελτίστου αἱρέσει, πολλὴ ἀν καὶ μακρὰ ῥᾳθυμία εἴη τοῦ λόγου. τὸ γὰρ μὴ διελέσθαι οἷόν τ ̓ εἶναι, ὅτι ἄλλο μὲν τί ἐστι τὸ αἴτιον τῷ ὄντι, ἄλλο δ' ἐκεῖνο, ἄνευ οὗ τὸ αἴτιον οὐκ ἄν ποτ ̓ εἴη αἴτιον. ὃ δή μοι φαίνονται ψηλαφῶντες οἱ πολλοὶ ὥσπερ ἐν σκότει, ἀλλοτρίῳ ὀνόματι προσχρώμενοι, ὡς αἴτιον αὐτὸ προσαγορεύειν. διὸ δὴ καὶ ὁ μέν τις τίνην περιτιθεὶς τῇ γῇ ὑπὸ τοῦ οὐρανοῦ μένειν δὴ ποιεῖ τὴν γὴν, ὁ δὲ ὥσπερ καρδόπῳ πλατείᾳ βάθρον τὸν ἀέρα ὑπερείδει.—PHADO. Πότερον δὲ τὸν αὐτὸν βελτίω καλεῖς σὺ καὶ κρείττω; οὐδὲ γάρ τοι τότε οἷός τ ̓ ἦ μαθεῖν σου, τί ποτε λέγεις. πότερον τοὺς ἰσχυροτέρους κρείττους καλεῖς, καὶ δεῖ ἀκροᾶσθαι τοῦ ἰσχυροτέρου τοὺς ἀσθενεστέ ρους, οἷόν μοι δοκεῖς καὶ τότε ἐνδείκνυσθαι, ὡς αἱ μεγάλαι πόλεις ἐπὶ τὰς σμικρὰς κατὰ τὸ φύσει δίκαιον ἔρχονται, ὅτι κρείττους εἰσὶ καὶ ἰσχυρότεραι, ὡς τὸ κρεῖττον καὶ ἰσχυρότερον καὶ βέλτιον ταὐτὸν ὄν, ἢ ἔστι βελτίω μὲν εἶναι, ἥττω δὲ καὶ ἀσθενέστερον, καὶ κρείττω μὲν εἶναι, μοχθηρότερον δέ· ἢ ὁ αὐτὸς ὅρος ἐστὶ τοῦ βελτίονος καὶ τοῦ κρείττονος; τοῦτό μοι αὐτὸ σαφῶς διόρισον, ταὐτὸν ἢ ἕτερόν ἐστι τὸ κρείττον καὶ τὸ βέλτιον καὶ τὸ ἰσχυρότερον.--GORG. Τί οὖν, ὦ Πρωταγόρα, ἀποκρινούμεθα αὐτῷ ταῦτα ὁμολογήσαντες, ἐὰν ἡμᾶς ἐπανέρηται, Οὐκ ἄρα ἐστὶν ὁσιότης οἷον δίκαιον εἶναι πρᾶγ μα, οὐδὲ δικαιοσύνη οἷον ὅσιον, ἀλλ ̓ οἷον μὴ ὅσιον· ἡ δ ̓ ὁσιότης οἷον μὴ δίκαιον, ἀλλ ̓ ἄδικον ἄρα, τὸ δὲ ἀνόσιον ; τί αὐτῷ ἀποκρινούμεθα ; ἐγὼ μὲν γὰρ αὐτὸς ὑπέρ γε ἐμαυτοῦ φαίην ἂν καὶ τὴν δικαιοσύνην ὅσιον εἶναι καὶ τὴν ὁσιότητα δίκαιον· καὶ ὑπὲρ σοῦ δέ, εἴ με ἐψης, ταὐτὰ ἂν ταῦτα ἀποκρινοίμην, ὅτι ἤτοι ταὐτὸν γέ ἐστι δικαιότης ὁσιότητι ἢ ὅτι ὁμοιότατον, καὶ μάλιστα πάντων ἥ τε δικαιοσύνη οἷον ὁσιότης καὶ ἡ ὁσιότης οἷον δικαιοσύνη.—PROT. Καὶ ὅταν τοίνυν τῷ μὲν παρᾷ αἴσθησις τῶν σημείων, τῷ δὲ μή, τὸ δὲ τῆς ἀπούσης αἰσθήσεως τῇ παρούσῃ προσαρμόσῃ, πάντη ταύτῃ ψεύδεται ἡ διάνοια. καὶ ἑνὶ λόγῳ, περὶ ὧν μὲν μὴ οἶδέ τις μηδὲ ᾔσθετο πώποτε, οὐκ ἔστιν, ὡς ἔοικεν, οὔτε ψεύδεσθαι οὔτε ψευδὴς δόξα, εἴ τι νῦν ἡμεῖς ὑγιὲς λέγομεν· περὶ δὲ ὧν ἴσμεν τε καὶ αἰσθανόμεθα, ἐν αὐτ τοῖς τούτοις στρέφεται καὶ ἑλίττεται ἡ δόξα ψευδὴς καὶ ἀληθὴς γιγνομένη, καταντικρὺ μὲν καὶ κατὰ τὸ εὐθὺ τὰ οἰκεῖα συνάγουσα ἀποτυ τώματα καὶ τύπους, ἀληθής, εἰς πλάγια δὲ καὶ σκολιὰ ψευδής.— THEET. 1. Write short notes explanatory of the following words:-αὐτοσχε διαζειν, σκιαμαχία, κολούειν. 2. What were the employments of the τραπεζίται at Athens? In what estimation were they held? 3. What officers were called οἱ ἑνδεκα? How were they appointed, and what were their duties ? 4. What was the true character and calling of the Sophist, according to Grote? 5. Give some account of the personal history of Socrates, and his leading traits of character. 6. What were the particular circumstances which brought on the trial of Socrates? 7. Draw a coast line of the Ægean Sea and the principal islands. CICERO. MR. ABBOTT. Translate into English the first two of the following passages, and two of the remaining group, adding brief notes where necessary : I. Ita male instituta ratio exitum reperire non potest. Sic enim dicebas, speciem Dei percipi cogitatione, non sensu: nec esse in ea ullam soliditatem, neque eamdem ad numerum permanere, eamque esse ejus visionem, ut similitudine et transitione cernatur, neque deficiat umquam ex infinitis corporibus similium accessio: ex eoque fieri, ut in hæc intenta mens nostra, beatam illam naturam et sempiternam putet. Hoc, per ipsos Deos, de quibus loquimur ! quale tandem est? Nam si tantummodo ad cogitationem valent, nec ullam habent soliditatem nec eminentiam quid interest, utrum de hippocentauro, an de Deo cogitemus? omnem enim talem conformationem animi ceteri philosophi motum inanem vocant: vos autem adventum in animos et introitum imaginum dicitis. Ut igitur, Ti. Gracchum quum videor concionantem in Capitolio videre, de M. Octavio deferentem sitellam, tum eum motum animi dico esse inanem: tu autem, et Gracchi et Octavii imagines remanere, quæ, in Capitolium quum pervenerim, tum ad animum meum referantur: hoc idem fieri in Deo, cujus crebra facie pellantur animi; ex quo esse beati atque æterni intelligantur. Fac imagines esse, quibus pulsentur animi. Species dumtaxat objicitur quædam. Num etiam cur ea beata sit, cur æterna? Quæ autem istæ imagines vestræ, aut unde? A Democrito omnino hæc lieentia. Sed et ille reprehensus a multis est, nec vos exitum reperitis: totaque res vacillat et claudicat.-De Nat. Deor. Munus autem animi est, ratione bene uti et sapientis animus ita semper affectus est, ut ratione optime utatur. Numquam igitur est perturbatus. At ægritudo perturbatio est animi. Semper igitur ea sapiens vacabit. Veri etiam simile illud est, qui sit temperans, quem Græci σúφρονα appellant eamque virtutem σωφροσύνην vocant, quam soleo equidem tum temperantiam, tum moderationem appellare, nonnumquam etiam modestiam: sed haud scio an recte ea virtus frugalitas appellari possit, quod angustius apud Græcos valet, qui frugi homines xpnoiμovs appellant, id est, tantummodo utiles: at illud est latius. Est enim omnis abstinentia, omnis innocentia (quæ apud Græcos usitatum nomen nullum habet, sed habere potest aßλáßetav. Nam est innocentia, affectio talis animi, quæ noceat nemini) reliquas etiam virtutes frugalitas continet. Quæ nisi tanta esset, et si iis angustiis, quibus plerique putant, teneretur, numquam esset L. Pisonis cognomen tantopere laudatum. Sed quia |