Let ABC be an isosceles triangle, base BC, and DGE an arc struck with centre A, and cutting the base or base produced in D and E.

From A drop a perpendicular AF to the base.

Then AB AC; angle ABF=ACB (I. 5); angle BFA=AFC; therefore BF=FC (I. 26).

But because AF is at right angles to the chord DE, DF=FE (III. 3).

DF FE and BF=FC. Take away the equal parts from the whole, and the remainders DB and CE are equal.

Draw any other chord FG. From A drop AD perpendicular to FG.

In the triangle ADC the angle ADC is a right angle; therefore ACD is less than ADC, and therefore AĎ is less than AC (I. 19).

Therefore FG is greater than BE (III. 15). In the same way it can be proved that all chords through C are greater than BE.

To divide a straight line PQ into two parts, so that the rect angle contained by PQ, QR, may be equal to the square on PR.

Prove that rect. PQ, PR+ sq. on PR=sq. on PQ, and that rect. PQ, RQ+rect. PR, RQ=rect. PQ, PR.

Let PQ be the given line.

On PQ describe a square (I. 46).

Bisect PS in V. Join VQ.

Produce PV to W, making_VW=VQ (I. 3).

On WP describe a square PWKR, then R divides the line as required.

Let ABCD be a circle, diameter AD, FG and BC other chords.

From E drop perpendiculars to FG and BC.

Join FE, GE, BE, CE.

Then FE=AE, GE=ED. But FE, EG are greater than FG (I. 20). Therefore AD is greater than FG. In the same manner it can be proved that AD is greater than any other line in the circle, except it be a diameter.

Secondly, let BC be nearer to the centre than FG. Then BC shall be greater than FG. FE=BE (def. 15). Therefore sq. on FE=sq. on BE. Therefore sqs. on FK + sq. on KE=sq. on BH+sq. on EH (I. 47).

But KE is greater than EH, and therefore sq. on EK is greater than sq. on EH ; and therefore the sq. on KF is less than the sq. on EH.

Therefore FK is less than BH. BC double of BH (III. 3). FG.

But FG is double of FK, and Therefore BC is greater than

Thirdly, let BC be greater than FG, then EH shall be less than KE. For, as before, sq. on FK+sq. on KE-sq. on BH + sq. on EH. But FK is half of FG, and BH half of BC (III. 3), and

But sq. on VQ=sq. on PV+sq. on PQ.

Therefore rect. SW, WP=sq. on PQ.

But rect. SW, WP is the figure WL, since WPWK. And the sq. on PQ is the figure PT.

Take away the common part PL.

And WR=RT. That is the square on PR = rectangle made up of QT or PQ, and the part RQ.

The rectangle PL is the rectangle contained by PR, PQ, since PSPQ, and RT=rect. PQ, QR=sq. on PR.

But PL + RT make up the figure PT, which is the sq. on PQ. Therefore sq. on PQ=rect. PQ, PR + sq. on PR.

Since PQ is divided in R, rect. PQ, PR=sq. on PR +rect. PR, RQ (II. 3).

But sq. on PR=rect. PQ, RQ (cons.).

Therefore rect. PQ, PR=rect. PQ, RQ+rect. PR, RQ.

12. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another. State and prove the corollary.

Let ABC be a right-angled triangle with perpendicular dropped from B to base AC.

B

In the triangle PVB, the angle VPB is a right angle (III. 31). Therefore the angles PVB, PBV together make a right angle.

But PBV is half a right angle because it is half of POQ (III. 20). Therefore PVB is half a right angle, and is equal to PBV, and therefore also PB=PV.

In the same way we may prove ARB to be half a right angle.

Again, because RPA is a right-angled triangle, RPA being the right angle (III. 31).

And PRA being half a right angle (part 1).
Therefore PARPRA, and therefore RP = PA.

In the two right-angled triangles RPV BPV we have PV= PB (Part 1), RP=PA (part 2). every respect (I. 4), and

Therefore the triangles are equal in

therefore RVP=PBA.

But POA is twice PBA (III. 18). twice PVR

Therefore also POA is

14. On OA the radius of one circle as diameter another is described, and a chord PQ is drawn in it parallel to OA. If through PQ lines be drawn perpendicular to OA, and meeting the larger circle in D, E and F, G respectively, then the sum of the rectangles contained by PD, PE and QE, QG is equal to the square on the radius of the larger circle.

F

Angle ABC BDA because both are right angles. Angle BAD is common to the two triangles ABD, ACB. Therefore the remaining angles ABD, BCA are equal. Therefore the triangle ABD is similar to ACB (VI. 4).

In the same way we may prove that BCD is similar to ACB. And therefore, since ABD, BCD are both similar to ABC, they are similar to each other.

Corollary. The perpendicular BD is a mean proportional to AD, DC. And also AB is mean prop. to AC and AD, and BC is mean prop. to BC, DC.

For in the triangles ADB, DBC, AD is to BD as BD to DC (VI. 4).

In the triangles ABC, ABD, AC is to AB as AB to AD.

ABC, DCB, AC is to BC as BC to DC.

Sq. on FL rect. FQ QG+sq. on QL (II. 5).

Sq. on FL=rect. AĽ·ĽK (III. 35). Rect. AL·LK+sq. on LO=sq. on AO (II. 5).

. sq. on AO=rect. FQ QG+sq. on QL+sq. on LO. But sq. on LO+sq. on QL=sq. on QO (I. 47).

Therefore sq. on AO is greater than rect. FQ QG by sq. on Qo.

By an exactly similar process with DP, ME we can get that the sq. on AO is greater than rect. PD, PE by sq. on PO. Therefore twice the sq. on AO is greater than rect. PD, PE +rect. FQ, QG by the sum of the sqs. on PO+QO.

But because PMQL is a rectangle + PM=QL, MS =SL (III. 14).

Therefore also AL-OM, and PM=QL, and the included angles are right angles, whence AQ=PO (I. 4).

But AQO is a right angle (III. 31).

Wherefore sq. on AQ (that is, sq. on PO) + sq. on QO=sq.

on AO.

Therefore twice the sq. on AO is greater than rect. PD, PE +rect. FQ. QG by the sq. on AO.

Therefore the sq. on AO=rect. PD, PE+rect. FQ, QG.

To describe an isosceles triangle having each of the angles at the base double of the third angle.

Divide a right angle into 5 equal parts.

Draw any line AB and divide it at C so that sq. on AC=rect. AB, BC (II. II).

Describe a circle with centre A radius AB.

Cut off BD in this circle AC (IV. 1).

Describe a circle through ACD (IV. 5).

Join CD, AD. ABD is the required triangle sq. on ED (=AC)=rect. AB, BC.

Therefore BDA=BCD (ax. 1). But BDA=ABD (I. 5).
Wherefore BCD=CBD, and therefore CD=BD (I. 6).
But BD=CA (cons.). Therefore angle CAD=CDA..
But CAD BDC. Therefore CAD is half BDA.

Let ABC be the given right angle, and on BC describe an isosceles triangle with the angles at the base double of the vertical angle (IV. 10). Bisect DBC by EB, and then bisect DBE and EBC by FB, GB (I. 9).

D

Let ABCD be a quadrilateral having ABC+ADC=Two right angles.

Bisect AB in F, and draw FE at right angles.
Bisect BC in G and draw GE at right angles.

For AF FB.

So that FE, GE meet in E. Then a circle described with radius BE, centre E, will pass through AC. FE is common; included angle is a right angle. = BE, similarly BE=EC.

H

D

F

E

Therefore AE

A

B

G

G

Because DBC, BCD are each double of BDC, and the three angles are together equal to two right angles, DBC is two-fifths of two right angles, whence DBE, EBC are a fifth part of two right angles.

Therefore GBC, EBG, FBE, DBF are each one-fifth of a right angle, and therefore ABD is the remaining fifth.

The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

If the opposite angles of any quadrilateral be together equal to two right angles, show that a circle may be described round it, and find its centre.

Let ABCD be a quadrilateral inscribed in a circle.

Join BD, AC.

Angle ADB ABC (III. 21).

Angle DCA=DBA (III. 21).

Therefore angle DCB-ADB+DBA.

To each add DAB. Then DAB+DCB=ADB +DBA+ DAB.

But ADB+DBA+DAB=two right angles.
Therefore DAB+DCB=two right angles.

1. Recite the chief terms of the treaty between France and England at the close of the Seven Years' War.

England was to retain the following possessions :-Tobago, Dominica, St. Vincent, Grenada, Nova Scotia, and Canada. She was to give up St. Lucia, Cuba and Havannah, Pondicherry.

England was to receive Florida, Senegal, Minorca. Certain boundaries were fixed, and rights of fishing, etc., arranged.

Write a brief account of Burgoyne's disastrous campaign and capitulation.

In 1777 General Burgoyne set out from Canada with a body of British troops to push across country and form a connection with the Hudson River, and so cut off the New England States from the others in rebellion. After some early successes he struck boldly out and engaged the Americans, under Generals Gage and

Arnold, at Stillwater, and, though victorious, suffered severely; the enemy, however, succeeded in cutting him off in the rear, and he soon found himself surrounded by a hostile and ever-increasing force. He entrenched himself at Saratoga, and vainly waited for help and relief. So closely did the enemy hem him in that he was unable to employ foraging parties, and at last, supplies failing, he capitulated, and, by the Convention, his troops were allowed to return to England, on condition of not taking any further part in the war.

Give some account of the critical condition of England financially, and as to foreign alliances in the years 1796-97.

The expense of carrying on the war with France was very great, and the Government had to draw very heavily on the Bank. At the same time, the fear of a French invasion led to money being held, and not invested, by persons who were fortunate enough to have it; the consequence of this was that the Government had to resort to paper-money, and it was some time before they were able again to pay in bullion

England was deserted by Prussia and Spain, and then Spain soon after went over to the other side, while the French had also the Dutch as allies, so that England, single-handed, was opposed to all these nations.

Recite the chief provisions of the Act of Union, passed by the Irish Parliament in the year 1800.

(1) That Ireland should have ICO members in the House of Commons.

(2) Twenty-eight peers elected by their fellows to represent them in the House of Lords.

(3) That the Churches of England and Ireland should be united.

(4) And that both countries should have the same privileges of trade and justice.

Give some account of Lord Chatham's Ministry, 17661770.

On the failure of the Rockingham ministry, Chatham was induced to take office and form a Cabinet, including among others Grafton, Conway, and Townshend. But Pitt soon became seriously ill, or, at any rate, this was the cause stated for his withdrawal from public life, and the Ministry had to float on without him. Some of their Acts were the duties imposed on articles sent to the colonies, and the persecution of Wilkes. After two years' silence, Chatham reappeared, but by this time the Ministry had changed, Townshend had died, and the Grenville and Rockingham party were linked against him, Lord North was Chancellor of the Exchequer, and the opinions of the Government were no longer Chatham's opinions, and he consequently resigned office.

Who were the chief members of the Ministry of Rockingham ? What were the chief difficulties they encountered at the outSet?

The chief members of this ministry were the Duke of Newcastle, Grafton, Conway, and Edmund Burke.

They found themselves sadly puzzled how to act towards America; the Colony was in arms against the Stamp Act; Grenville and his party clamoured to have it enforced; Pitt called for its repeal, unconditionally, as being illegal.

The king supported Grenville's view, and a faction called the 'king's friends,' and whom Macaulay describes as a 'reptile body of politicians,' did their best to thwart the Ministry and support the monarch, and they thus had the full force of the opposition; and another body, who, holding Government offices and sinecure places, were counted as supporters, also against them. Besides this they had taken office with the support of the Duke of Cumberland, who had great influence; but soon afterwards he died, and they had among their number only one orator (who had his name to make), viz., Edmund Burke.

Detail the principal causes of the unexampled distress of the years 1816-17 in England.

There had been a very heavy strain on the country by the French war, which had drained men and money and spoilt trade. The Government had consequently had to increase taxes con siderably.

To add to this there was a succession of bad harvests, and the importation of foreign corn was restricted by the Corn Bill of 1815. Thus bread was dear, money scarce, and trade bad, and the lower classes would suffer most by all this.

Prepare the outline of a lesson for an advanced class on one of the great measures of reform passed between 1828 and 1837.

REFORM BILL.

Ask who is member for the boys' town or county. How he becomes member, and who elects him, viz., householders or lodgers in boroughs, etc. Select some well-known town in their neighbourhood which has members and another has not; by naming several, draw out the general principle that the number of members depends on the population. Sketch the state of affairs before 1831, describing such places as Old Sarum and Gatton, and the consequent farce of election, also the rioting at elections and the small number of voters, and show the need of reform.

The Passing of the Bill.-Describe the course of the agitation; Lord Grey's Bill; its rejection; the riots at Nottingham, Bristol, aad Derby; the obstruction by peers; and the threat to create new peers; the final passing of the Bill.

Provisions. Forty-four new boroughs, two members each. Let the boys try to name some of these by recalling the names of towns that have grown rapidly, as Birmingham, Manchester, London, etc. Twenty-one boroughs one member, more county members, several places disfranchised.

Describe the qualifications necessary for voters, and those added by this Bill, such as £10 householders in boroughs. Ask why the time of election should have been reduced, and why counties had two days and boroughs one day allowed. Compare with present time.

Recapitulate, and make complete blackboard sketch.

(b) A voyager embarking at a port would often find that though everything seemed in readiness for an immediate start, still the vessel waits and waits, and the captain watches the water until at last he seems satisfied. and gives the order to let go. If the traveller, impatient of the delay, inquired of a sailor the reason for the apparent waste of time, the answer would be, 'We're waiting for the tide, sir,' and further questioning would elicit the explanation that the vessel could only cross the harbour bar at the flood tide. This is evidently the kind of incident that was in Shakespeare's mind in writing the words that head this paper. In the scene where the words occur, Brutus and Cassius are consulting as to the future movements of their forces. Brutus urges immediate action and attack, Cassius suggests waiting for the enemy's onslaught, when the former replies that their cause is ripe, their friends are all with them, and that future delay will only be running the risk of desertion, and this view he urges with the speech, There is a tide, etc.'

Most men in reviewing their life can look back on some period when their mode of action has led, either to continued after success, or, on the other hand, to years of failure or mediocrity.

Who in reading the life of Lord Beaconsfield can help admiring the tact of the man in seizing the opportunity of Sir R. Peel's acceptance of the Anti-Corn Law League, to combine his fortunes with those of the disappointed Tories, and abuse their once-admired leader; and many other noted men have attained their eminent position mainly by means of the clear foresight that has led them to embark on their enterprises at the critical period which would command success.

Grammar.

Paraphrase the following passages, and comment briefly upon the words italicised:

Vexed I am

Of late with passions of some difference,
Conception only proper to myself.
Which give some soil perhaps to my behaviours.

'Tis a common proof,

That lowliness is young ambition's ladder, Whereto the climber-upwards turns his face, But, when he once attains the upmost round, He then unto the ladder turns his back,

DD

Looks in the clouds, scorning the base degrees By which he did ascend.

Swear priests and cowards and men cautelous
Old feeble carrions and such suffering souls
That welcome wrongs; unto bad causes swear
Such creatures as men doubt; but do not stain
The even virtue of our enterprise,

Nor the insuppressive mettle of our spirits,
To think that or our cause or our performance
Did need an oath.

These couchings and these lowly courtesies
Might fire the blood of ordinary men
And turn pre-ordinance and first decree
Into the law of children. Be not fond

To think that Cæsar bears such rebel blood
That will be thaw'd from the true quality
Like that which melteth fools.

I have lately been sorely perplexed by contending emotions and with thoughts which concern me alone, and which somewhat sully and spoil my demeanour towards others.

It has very often occurred to me that humility is like a ladder which assists a man to rise. While climbing he turns his face towards it, but when once he has attained the desired eminence he turns round, no longer caring for those steps that helped his ascent, but viewing the wide expanse of heaven.

Priests and cravens, deceitful men, and faint-minded old persons are those from whom you should require an oath, and from those who delight in wrong-doing. Oaths are needed when the cause is bad, and when the performers of the act are untrustworthy; but our cause is not of this class, and we should only tarnish the moral excellence of our undertaking, and our strength of purpose by supposing that either need an oath to bind them.

All this stooping and prostration might puff up some men with pride, and cause them to alter their previous decisions with the captiousness of children. Do not be so foolish as to think that Cæsar's blood is so easily inflamed as to lead him astray in his actions.

Difference-that are in conflict one with another.
degrees-the steps or rungs of the ladder.

cautelous-crafty or cautious, first meant cautious, then
suspicious, then crafty or deceitful.
carrions-carcases, used contemptuously.

couchings-stoopings, prostrations, to couch 'is to stoop or
bend low.

pre-ordinance-things decreed from the beginning.

fond-foolish, still used in the north part of England with this meaning.

Explain the ides of March—I had as lief not be—the cross blue lightning-no fleering tell-tale-richest alchemy-the genius and the mortal instruments-the eighth hour--wilt thou lift up Olympus? - signed in thy spoil-cry 'Havoc -Cæsar's angel-abjects, orts, and imitations.

The ides of March.-The 13th of some months and the 15th of others was called 'the ides.' The ides of March would be the 15th.

I had as lief not be-I would as willingly not exist. The cross blue lightning-the zigzag path of the lightning. No fleering tell-tale-no grinning or sneering sneak. Richest alchemy.-Alchemy professed to change base metal into gold.

Genius and the mortal instruments-the soul and the bodily powers.

Wilt thou lift up Olympus?— Olympus, a mountain in Greece. The question is asked to imply, Wilt thou try to do an impossible thing?

Signed in thy spoil-linked together at thy capture.

Cry Havoc-raise a cry of slaughter without giving any quarter.

Caesar's angel-was like his guardian angel, so inseparable was he from him.

Abjects, orts, and imitations-castaway pieces, odds and ends, and mocking others.

I am constant as the northern star.

Hollow men, like horses hot at hand,

Make gallant show and promise of their mettle.

In watching a sunrise, the even dulness of the sky would be observed to change, and a few lines break the clouds, then gradually spreading and changing colours.

The northern or pole star is said to be the centre of the heavenly systems, and our axis points to it, hence it never changes its position in the heavens.

Those horses that prance much and are restless are often thought to be splendid animals, and seem to be only too eager to perform great feats. So with hypocritical persons; they promise much, but perform but little of it.

Point out any grammatical peculiarity in the following :— I'll about.-A verb of motion (as turn) omitted before an adverb or preposition, common in Shakespeare.

Be not jealous on me.-'On' used for 'of.'

What need we any spur.- What' for 'why.'

And that were much he shall.—' Much ' means 'hard,' and that is put before 'were.'

Let no man abide this deed but we the doers.-'Abide' is used transitively, as in 'I cannot abide it,' and 'we' is subject of 'will abide it,' which is understood.

There is tears for his love.-'Tears' plural and 'is' singular. 'Tears is there.'

Ay, and truly you were best-i.c., it were best for you.' The objective pronoun put before the verb.

As the flint bears fire who shows a hasty spark.—'Who, which we use only for person, here refers back to a neuter thing, 'fire.' 'We should say that' or 'which.'

Only I yield to die.-The place of the adverb 'only' changed from behind the verb to before the subject.

Write an analysis of the Essay on Study, explain and illustrate from your own reading, books to be 'read by deputy,' to be 'swallowed,' and the best cure for 'wandering wits.'

Studies are for delight, ornament, and improvement of mind or intellect.

Too much study is idleness and laziness.

Used too much for ornament is affectation.

To judge solely by, and depend entirely on, weakness.
They must be linked with and controlled by experience.
Read to weigh and consider.

Some books read

(1) Only in part.

(2) Others scanned through.
(3) Others wholly and carefully.
(4) Others by deputy.

Reading makes a full man.
Writing an exact man.
Conversation a ready man.

Different studies serve different purposes.

As Mathematics-to give power of concentration. History--a knowledge of the ways of the world. Logic and Rhetoric-argumentative power. Books to be swallowed,' such as novels, descriptions of his torical characters, or adventures, give one a general knowledge, but would be waste of time to study carefully every detail.

'Read by deputy.' Large volumes, such as Mill's Political Economy,' or Hamilton's Mental Philosophy,' which are too compendious for students in college to wade through, but if read by the tutors, and abstracts given of the parts most useful to them, the abstracts would be of great value.

Best cure for wandering wits.' Mathematics. It is no use trying in a half-hearted style to study Euclid, for instance; for if we let our wits wander, and once drop the thread of the demonstra tion it is necessary to begin all over again.

Name some of the 'false points of praise'; and point out how praise may be judiciously employed.

False points of Praise-When it is only flattery or praising to one's hurt, maliciously, either to him, and so cause him to be conceited, or to others to excite jealousy. Over-praising one. Judicious use of Praise-If a boy has done a difficult task, praise will be a reward to him, and stimulate him to further effort. If the praise be public, others will also be excited to emulation.

State briefly the antithesis between the conduct of the young and the old, suggested in Bacon's essay on Youth and Age.'