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nöthig als schwer: ob das, was uns von Alters her überliefert und von unsern Vorfahren für gültig geachtet worden, auch wirklich zuverlässig sey, in dem Grade, dass man darauf fernerhin sicher fortbauen möge? oder ob ein herkömmliches Bekenntniss nur stationär geworden, und desshalb mehr einen Stillstand als einen Fortschritt veranlasse? Ein Kennzeichen fördert diese Untersuchung, wenn nämlich das Angenommene lebendig und in das thätige Bestreben einwirkend und fördernd gewesen und geblieben.

Im Gegensatze steht die Prüfung des Neuen, wo man zu fragen hat: ob das Angenommene wirklicher Gewinn, oder nur modische Uebereinstimmung sey ? denn eine Meinung, von energischen Männern ausgehend, verbreitet sich contagios über die Menge, und dann heiszt sie herrschend-eine Anmassung, die für den treuen Forscher gar keinen Sinn ausspricht. Staat und Kirche mögen allenfalls Ursache finden, sich für herrschend zu erklären: denn die haben es mit der widerspenstigen Masse zu thun, und wenn nur Ordnung gehalten wird, so ist es ganz einerlei, durch welche Mittel ; aber in den Wissenschaften ist die absoluteste Freiheit nöthig : denn da wirkt man nicht für heut und morgen, sondern für eine undenklich vorschreitende Zeitenreihe.

Gewinnt aber auch in der Wissenschaft das Falsche die Oberhand, so wird doch immer eine Minorität für das Wabre übrig bleiben, und wenn sie sich in einen einzigen Geist zurückzöge, so hätte das nichts zu sagen. Er wird im Stillen, im Verborgenen fortwaltend wirken, und eine Zeit wird kommen, wo man nach ihm und seinen Ueberzeugungen fragt, oder wo diese sich, bei verbreitetem allgemeinem Licht, auch wieder hervorwagen dürfen.

Die Zweifelnden.
(3) Ihr liebt und schreibt Sonette! Web der Grille!

Die Kraft des Herzens, sich zu offenbaren,
Soll Reime suchen, sie zusammenpaaren ;
Ihr Kinder, glaubt, ohnmächtig bleibt der Wille.
Ganz ungebunden spricht des Herzens Fülle
Sich kaum noch aus: sie mag sich gern bewahren;
Dann Stürmen gleich durch alle Saiten fahren;
Dann wieder senken sich zu Nacht und Stille.
Was quält ihr euch und uns, auf jähem Stege
Nur Schritt vor Schritt den läst'gen Stein zu wälzen,
Der rückwärts lastet, immer neu zu mühen ?

Die Liebenden.
Im Gegentheil, wir sind auf rechtem Wege!
Das Allerstarrste freudig aufzuschmelzen,
Musz Liebesfeuer allgewaltig glühen.

FRIDAY, JUNE 8, from 2.30 to 5.30 P.M.
SECTION III. French and German.

IX. 1. What determines the gender of compound nouns? Give examples. Mention any exceptions to the rule. Give the gender of Gewähr, Scheu, Abscheu, Lust, Verlust, Rath, Heirath, Petschaft, Befugniss.

2. In what cases is the definite article used in German where it is not used in English ? Illustrate your answer by examples.

3. Illustrate the force of be, ent, ver, er and ge, when prefixed to verbs.

4. Explain and illustrate the different senses of mögen and sollen used as auxiliaries.

5. Translate in as many different ways as possible :-there were ten of us, a quarter past eleven, they went two and two, they got five shillings apiece, to drive four in hand, three and

a half.

6. Give the past participle of liebkosen, lossprechen, frohlocken, mahlen; the past subj. ist person singular of treffen, weisen, schwimmem, schwären, glimmen ; and the imperative of löschen, brechen, scheren, erbleichen.

7. Distinguish between sondern and aber, wenn and wann, als and da, während and indem.

8. Explain Umlaut and Ablaut, giving instances of each in German and English.

9. Sketch shortly the relationship between the German and English languages.

10. Compare French, German and English, as regards the different modes in which the adjectival conception can be expressed.

SATURDAY, JUNE 9, from 9.30 A, M. to 12.30.

SECTION III. French and German.

X.

Translate into German :

(1) Aurelia, though a woman of great quality, delights in the privacy of a country life, and passes away a great part of her time in her own walks and gardens. Her husband, who is her bosom friend and companion in her solitudes, has been in love with her ever since he knew her. They both abound with good sense, consummate virtue, and a mutual esteem; and are a perpetual entertainment to one another. Their family is under so regular an economy, in its hours of devotion and repast, employment and diversion, that it looks like a little commonwealth within itself. They often go into company, that they may return with the greater delight to one another; and sometimes live in town, not to enjoy it so properly as to grow weary of it, that they may renew in themselves the relish of a country life. By this means they are happy in each other, beloved by their children, adored by their servants, and are become the envy, or rather the delight, of all who know them.

(2) There is a natural affinity between goodness and the cultivation of the Beautiful, when it is real cultivation, and not a mere unguided instinct. He who has learnt what beauty is, if he be of a virtuous character, will desire to realize it in his own life, will keep before himself a type of perfect beauty in human character, to light his attempts at self-culture. There is a true meaning in the saying of Goethe, though liable to be misunderstood and perverted, that the Beautiful is greater than the Good; for it includes the Good, and adds something to it: it is the Good made perfect, and fitted with all the collateral perfections which make it a finished and completed thing. (3) In Anna's wars, a soldier poor and old

Had dearly earned a little purse of gold :
Tired with a tedious march, one luckless night,
He slept, poor dog! and lost it, to a doit.
This put the man in such a desperate mind,
Between revenge, and grief, and hunger joined
Against the foe, himself, and all mankind,

He leaped the trenches, scaled a castle-wall,
Tore down a standard, took the fort and all.
* Prodigious well;' his great commander cried,
Gave him much praise, and some reward beside.
Next pleased his Excellence a town to batter;
(Its name I know not, and it's no great matter)
'Go on, my friend, he cried, see yonder walls !
Advance and conquer ! go where glory calls !
More honours, more rewards, attend the brave.'
Don't you remember what reply he gave ?
'D'ye think me, noble general, such a sot?
Let him take castles who has ne'er a groat.'

SATURDAY, JUNE 2, from 9.30 A.m. to 12.30.

SECTION IV. Mathematics. 1. Elementary Algebra and Trigonometry. 1. A sum of £P accumulates at compound interest for n years at kr per unit per annum, k > 1. It is then the present value, interest being at the rate of r per unit per annum, of a perpetual annuity of £krP to begin at the end of another n years. Find the value of n. 2. Solve the equations :

(1) V(x-1) (~—2) + N(2-3) (0-4)= V2;
(2) 202 + y2 = 3xy, 205 + y = 2;
3

y
= I,
6

72 c2
(3)

y
+ + a

a3 63 c3 3. Prove the Binomial Theorem for fractional indices, assuming its truth when the index is a positive integer.

Find the sum of the products of the coefficients in the expansion of (1+x)", taken 2 and 2 together, n being a positive integer. 4. If az, az, az ... an are n positive numbers, prove ay taat

> (a, ag... am)".

[ocr errors]
[ocr errors]

+ a

+

+

+

=l,

[ocr errors]

1

tan

n

5. Sum the series

I (1)

+ +

2.5

+... to n terms;

1.4

3.6

I

[ocr errors]

I

+

1+2

I + x4

I

(2) C+C cos 0 + c, cos 20+etc., where Co, C1, C2 ... are

the coefficients in the expansion of (1 +X)" ; and (3) shew that if <

4.23 8.07
+
+

+ ... ad inf. is equal to I+

I+x8 6. A real root of f'(x) = o lies between every adjacent two of the real roots of f(0) = 0. Prove this, and shew that if a negative root of f'(= o is numerically greater than a negative root of f (cc) = 0, then f (x) = o must have two negative roots at least.

Hence shew that the equation given in Question 7 has one positive and three negative roots.

7. Solve the equation 2+ + 2 203 — 1022 – 20x–8 = 0, in which the sum of one pair of roots is equal to the product of the other pair, and the sum of the latter pair is equal to the product of the former pair.

8. Prove Demoivre's Theorem, and expand sin 0 in a series of powers of 0.

9. (1) Find the radii, r and R of the inscribed and circumscribed circles of a triangle ABC, and prove

(2) a cos A + 6 cos B+c cos C = 4R sin A sin B sin C;

(3) 2 aber = (a+b+c) d, d, dg, where , d,, dg are the distances of the vertices of the triangle from the centre of the inscribed circle. 10. In any spherical triangle ABC, prove that

(1) cos A sin b sin c = cosa — and, if the triangle be right angled at C,

(2) tana = cos B tanc. 11. An event has happened which must have arisen from some one of a given number of causes, determine the probability of the existence of each of the causes.

There are three coins in a bag, each one of which is à priori equally likely to be a sovereign, shilling, or sixpence. The offer is made to a man that he may either draw out one coin and keep it, or draw out one coin, which he is not to keep, and in case this is a sovereign, but not otherwise, draw out a second coin and keep it. Which is the better offer?

- cos 6

COSC;

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