them then, and paid him. 13. Have you not seen your eldest sister
during your stay in Lyons? 14. I have not seen her.
15. Did you
not go to bed too soon last night? 16. I went to bed late. 17. At
what hour did you rise this morning? 18. I rose at five o'clock; I

write the terms of the subtrahend after those of the minuend. (Art. 54).

and write it after the quantity from which it is to be subtracted, Otherwise.-Put the quantity to be subtracted in brackets, generally rise early. 19. Did you not seek to escape from your prison last year? 20. I have never tried to escape. 21. Have with the sign you sold your between them; then apply the Rules of property? 22. I have not sold it. 23. What have you given to the Addition. EXAMPLES. soldier? 24. I have given him nothing. 25. During his stay at B., we gave him all that he wished.

EXERCISE 96 (Vol. II., page 42).

1. Que reçûtes-vous la semaine dernière? 2. Nous reçûmes cinquante francs de votre ami, et vingt-cinq de votre frère. 3. Menâtes-vous votre fils à l'église hier? 4. Je ne l'y menai pas. 5. Que perdites. vous l'année dernière? 6. Nous perdimes notre argent, nos habillements et nos chevaux. 7. Les avez-vous cherchés? 8. Je les ai cherchés, mais je ne les ai pas trouvés. 9. Parla-t-on de votre frère, hier? 10. On parla de lui et de vous. 11. Qu'est-ce que le médecin vous a donné? 12. Il ne m'a rien donné. 13. À quelle heure votre sœur se leva-t-elle hier? 14. Elle se leva à cinq heures. 15. Vous êtes-vous levé de bonne heure ce matin? 16. Nous nous sommes levés à six heures et demie. 17. Votre cousin a-t-il vendu toutes ses propriétés? 18. Il ne les a pas vendues, il les a données à sa sœur aînée. 19. Le voyageur vous a-t-il raconté ses aventures? 20. Il me les a racontées. 21. Cet homme a-t-il cherché à parler à votre père? 22. Il a cherché à lui parler. 23. Le professeur a-t-il parlé de votre frère, 25. Votre ami pendant son séjour chez vous? 24. Il a parlé de lui. a-t-il porté son habit neuf? Il ne l'a pas encore porté. 27. Avezvous remercié votre frère? 28. Je l'ai remercié. 29. Qu'avez-vous donné à votre sœur aînée? 30. Je ne lui ai rien donné, je n'ai rien à lui donner. 31. Quand M. votre frère vous donna un livre, l'année dernière, le remerciâtes-vous? 32. Je ne le remerciai pas. 33. Est-il tard? 34. Il n'est pas tard, il n'est que six heures. 35. Fait-il beau temps ou mauvais temps? 36. Il fait très-beau temps.

26.

LESSONS IN ALGEBRA.—IV.

SUBTRACTION.

56. SUBTRACTION is the finding of the difference between any two quantities or collections of quantities.

EXAMPLES.-(1.) Charles has 5a pears, and James has 3a pears. How many more pears has Charles than James? In this example, we wish to take 3a pears from 5a pears. But subtraction is denoted by the sign sion 5a 3a pears represents the answer. pears; which is the answer.

Hence the

expres

But 5a3a2a

£4,500, but he is in

=

£3,700. Ans.

(2.) A gentleman owns a house valued at debt £800. How much is he worth? Here we have £4,500 £800 57. Let us now attend to the principle upon which these operations are performed. Let us suppose that you open a book account with your neighbour, and that when cast up, the debtor side, which is considered positive, is £500, and the credit side, which is considered negative, is £300. On balancing the account, you find that he owes you £500 £300200. Now, if you take £50 from the positive or debtor side, it will have the same effect on the balance, as if you added £50 to the negative or credit side; and on the other hand, if you take £50 from the negative or credit side, it will have the same effect on the balance, as if you added £50 to the positive or debtor side.

(1.) From 6a + 9b, take 3a + 4b.

Here, change the signs of the subtrahend, but not those of the minuend, thus :

6a9b За 4b. Next reduce these terms, by Art. 52, and you have the answer, 3a + 5b.

(2.) From 166 (3.) 14da (4.)-28 (5.)-16b (6.) — 14da
Take 126
6da
- 126

- 16

12

6da (10.) — 16 (11.)

14da

8da

- 28

+12

6da

(14.) +14da (15.) — 28 (16.) — 166 (17.) — 14da +126

(13.) + 16b
- 126
6da
+ 286
+ 20da
(18.) From 8ab, take 6ry.
(19.) From 6aay
Take 17ay

+16

+ 6da

61. From these examples, it will be seen that the difference between a positive and a negative quantity may be greater than either of the two quantities. In a thermometer, the difference between 28 degrees above zero, and 16 degrees below, is 44 degrees. The difference between gaining 1,000 pounds in trade, and losing 500 pounds, is equivalent to 1,500 pounds.

62. Proof.-Subtraction may be proved, as in arithmetic, by adding the remainder to the subtrahend. The sum ought to be equal to the minuend, upon the obvious principle, that the dif ference of two quantities added to one of them, is equal to the other.

EXAMPLES. (1.) From 2xy-1, subtract — xy + 7.

Operation.
Here, Minuend 2xy-1
Subtrahendxy + 7

Remainder 3xy - 8

58. In like manner, if, in the expression 12a-5a, you take 3a from 12a, it will have the same effect on the expression, as if you added 3a to 5a, and retained the negative sign in the sum; thus, 9a 5a is the same as 12a-8a. Again, if in the expression 12a-5a, you take 3a from 5a, and retain the negative sign in the difference, it will have the same effect on the expression,+2am + 6am. as if you added 3a to 12a; thus, 12a-2a, is the same as 15a 5a.

59. Hence universally, taking away a positive quantity from an algebraic expression is the same in effect as adding an equal negative quantity; and taking away a negative quantity is the same as adding an equal positive one.

60. Upon this principle is founded the following

GENERAL RULE FOR SUBTRACTION.

3am + am + 7am

(2.) From y, subtract a + a +a+a.
(3.) From ax-bc+3ax + 7bc, subtract 4bc-2ax+be+4ax.
Answer. 2ax + bc.

(4.) From ad+3dc-bæ, subtract 3ad +7bxde + ad.
Answer. 4dc-8bx 3ad.

64. The sign placed before the marks of parenthesis which include a number of quantities, requires that, when these marks Change the signs of all the quantities to be subtracted, i.e., of are removed, the signs of all the quantities thus included should the subtrahend, or suppose them to be changed from + to- and be changed. Thus a (b-cd) signifies that the quantities from to; then if the quantities are ALIKE, unite the terms b-c and d are to be subtracted from a. Remove the as in addition (Arts. 49, 50); but if the quantities are UNLIKE, parenthesis, and the expression will then become a — —b+c―d,

an expression which has exactly the same meaning as the land if not King of France," had died away, and there was former.

EXAMPLE. From ry+d, take 7ad-xy + d + hm. Here, ay+d(7ad — xy + d + hm) = — 7ad + 2xy - hm. Answer. 65. On the other hand, when a number of quantities are to be introduced within the marks of parenthesis, with immediately preceding it, their signs must be changed. Thus, -m+bdomination was stamped with such indelible plainness that all -de + 3h = - (m—b + dx-3h).

26. From a + 4x3y + 6x2y2 + 4x73 + y*, take a+ — 4x3y + 6xoy2 — 4xy3 heresy which were preferred against her; and she had been told + y*.

true cause of their unholy zeal in setting the church's law in motion against her, that cause being the crushing defeat Jeanne Darc had inflicted upon the English political influence in the country. But how came the English in the country at all? Was it by way of revenge for the conquest by William the Norman, or did it spring out of some after-born political entanglements?

having been treated badly by Philip the Fair, took an ignoble revenge by giving his services to the foes of his country. Edward looked about for allies before launching forth on a great war with France, and he secured the friendship of the Counts of Flanders, Brabant, Namur, Gueldres, and Hainault, and the powerful assistance of the rich citizens of Ghent, represented by the brewer, Jacob van Artevelde. Having gained these allies, and coaxed Parliament to give a large supply in aid of the war, Edward proceeded to pick a quarrel. He complained that Philip had helped the Scots in the late war between Scotland and England, and that he still protected the Scotch king, a personal enemy of his. Finally, he renounced his homage, and defied the French king, who, knowing that the contest must come, buckled to with a will, determined to suffer anything rather than admit Edward's claim to the French crown.

The claim of the English kings to be kings also of France began to be seriously mooted when Edward III. was Prince of Wales, and when he came to the throne, the question was taken up with ardour when once he was aroused from the lethargy which in the earlier days of his reign seemed to be the forerunner of an inglorious era. In 1337, ten years after he had been on the throne, Edward lacked occupation, and manifesting a desire to let his energies find vent in true Plantagenet fashion, listened to the advice and remonstrance of some of those about him, who urged him to assert his right to the crown of France. At first matters did not go happily; the English king, who The way in which he claimed was almost too barefaced to be elected to attack from the side of Flanders, had great difficulty written down; and while it is certain that few of those who in keeping his allies together; and though he did advance with fought on his side so valiantly and well, knew the real merits 50,000 men into French territory, he did not fight, and returnof the case, it is likely that he himself was not very expert in ing into Flanders, disbanded many of his troops. Charges to tracing his genealogy. Those who had motives of their own for the extent of £300,000 had been incurred; the money given by the war, and who hoped to win fortune and rank for themselves Parliament, and that raised by pawning the crown jewels and s། out of it, told him he had a righteous cause, and he, gladly con- the personal effects of the king, was all gone, and not a foot vinced, believed them. It was the custom in France, borrowed of French land had been won. The Parliament, in the king's from the Salian Franks, who had become absorbed in the nation, absence, refused supply except on the condition of redress of to exclude women from the succession to the throne, and when a grievances, and it seemed as if the royal expedition after the woman came in the direct line of succession her place was taken French crown must end in an inglorious fiasco. Suddenly by the male heir nearest related to the late king. This custom Edward appeared in London, wrung a heavy grant out of the had been sanctioned by the approval of several hundred years, Parliament, and proceeded to fit out a fresh force against during which time no one who founded his title through a female Philip, notwithstanding that the Parliament told him it owed had mounted the throne. When Philip the Fair died in 1314, him no allegiance as King of France, and that if won, France he was succeeded by his son, Louis the Stubborn, who died, must ever remain a separate part of the kingdom. without male issue, and Louis's brother, Philip the Long, succeeded him, being himself followed in 1322 by his younger brother, Charles the Fair.

Philip the Long had died without issue, and on the birth of a daughter to Louis the Stubborn, the estates of France decreed, her exclusion, and the exclusion of all females in future. Charles the Fair's only child was a daughter, and with Charles was extinct the direct male line of Philip the Fair. Philip's fourth child was a daughter, Isabella, married to Edward II. of England, and it was taken for granted that the law of exclusion which applied to Louis the Stubborn's daughter, married to Charles, King of Navarre, and which applied to the daughter of Charles the Fair, applied also to Isabella, their aunt. So, thoroughly did this opinion prevail, that when Philip of Valois, nephew of Philip the Fair, claimed the throne on the death of his last male cousin, his claim was allowed as reasonable and unquestionable by the whole nation, and no one so much as thought of a claim being made on behalf of Isabella by her son. The exact position of affairs may be best seen from the annexed Ciagram.

to Edward II.
of England.

On June 24, 1340, Edward's fleet, well manned and sound, came up, off Sluys, with the French fleet of four hundred sail which Philip had prepared to intercept Edward's army in its descent on the coast. A bloody battle ensued. The English were the better sailors, and manoeuvred so as to take every advantage of the enemy, who lost the greater part of thei ships and upwards of 25,000 men. This crushing victory of which Edward was not prepared at the moment to tak advantage, fixed an unbridgeable gulf between the good will of the two nations. National prejudice, national hatre had their birth in it, and from the battle of Sluys dates th dreadful animus which existed down to quite recent time between the English and French. From the same event, how ever, dates the welding of the English nation into one hom geneous whole; the lords ceased to affect French ways and th French language-which, historically speaking, was theirs-an identified themselves with the country which was their ne home. After the battle of Sluys the word "Englishman" cease to be a term of reproach.

The battle of Sluys, the first brilliant victory of the Engli navy, was barren of immediate result so far as Edward's cla to the French crown was concerned. As usual, when a Fren war broke out, the King of Scotland broke the peace by way diversion on his side, and Edward had to turn the whole of Isabella, married strength against his northern enemy, who was, necessarily, be crushed before a foreign war could be carried on. In 13 however, Edward, with the English nation at his back, set on the campaign which ended on the field of Crecy, and wh was followed some years afterwards by the rout of Poicti (1356), where the French king, John, was captured by the Bla Prince, and brought prisoner to London. The exhausting effe made during the campaign were such as to prevent Edward fr following up his splendid successes, and he was glad to arran by the Treaty of Bretigni, for a long truce. Various reas conspired to prevent the resumption of hostilities on a gr scale during the rest of Edward's long reign. The English mained masters of large portions of French territory, the claim of the English king to the crown was not abander but kept as a sword in the scabbard, for use at a nient season. The son of the Black Prince, Richard of deaux, who succeeded to his grandfather's crown, did not ceed to his energy or his ability, and the English claim virtually dormant during the whole of his reign, while French were employing the time in recovering from the eff of Edward's blows, and from the disastrous results of the

Not only did Philip take undisturbed possession of the throne, but, as a matter of course, he summoned King Edward to do homage to him, as his liege lord, for the province of Guienne. which belonged to Edward as feudal tenant of the French king. Edward obeyed, rendered homage, and thus virtually acknowledged Philip's right to be king. But he did so only because it was not convenient to have a quarrel on his hands at the time. He had a Scotch war to fight, troublesome subjects at home to, curb, and there was a plentiful lack of that sinew of warmoney-without which it is useless to back even the strongest claim. When these troubles were over he listened to the suggestions of Robert of Artois, a renegade French nobleman, who,

CO

regency which continued all the years King John was in captivity. Henry IV. had not leisure from home troubles to pursue the war, though he seems to have been desirous of doing so, not only as King of England, but by way of paying out the French king for his something more than neglect of him at the time he was in exile as Henry of Bolingbroke. When Henry V. succeeded, he had a large stock of energy to expend, a quiet kingdom, and a fairly stocked treasury; he had plenty of brave spirits about him, and within him was an ambition which would have taken him to Constantinople or to the capital of the Great Mogul. He determined to assert his claim to the crown of France.

To a king in his frame of mind an occasion of declaring war could not long be wanting, and there were several causes which allowed of his choosing his own time and opportunity. He set about his work deliberately, sent a special embassy to France to demand his right, and when that embassy returned from its bootless errand, he prepared with diligence and the utmost circumstance to enforce his claim with the sword. In the month of August, 1415, he sailed from Southampton with one of the fest armies ever mustered in England, landed at Harfleur, which he besieged and captured, and then prepared to advance o Paris. An enemy worse than all the French armies put together came into his camp. Dysentery smote down hundreds f his men, including some of the bravest and wisest, and so weakened the remainder that they could scarcely walk. Henry obliged to abandon the idea of going to Paris, and gave orders for a march to Calais, whence he proposed to embark his enfeebled army for England. At Agincourt, the French army, which had been hanging about him, barred his advance. It consisted of full three times the number of the English, and was xzmanded by the Dauphin, the French king's eldest son, and | by the flower of the French nobility. The French were confident of victory, the English were in a desperate case, and the battle a joined with an amount of fury seldom witnessed even in those days. The French were utterly routed (October 25, 1415), ast numbers of them were slain, and the shattered remains the English army pursued its march unmolested to Calais. In the next campaign, which was not undertaken till two Jears afterwards, Henry met with but little resistance in the country of Normandy, though Rouen was stoutly defended. H- reduced Rouen and other towns, and marched to Paris, ich he mastered, and dictated terms in the capital of his my. The French king, Charles VI., was imbecile, and the Thaty of Troyes, to which the Dauphin refused to be a py, provided that Charles should be called King of France ng his lifetime, but that Henry should really administer the 20vernment, and that after Charles's death he and his sucessors should be acknowledged as kings of France. Henry trengthened the band by marrying Catherine, daughter of the Frach king, and during the rest of his life he did actually rule France and receive the homage of her vassals. In the height of his power Henry was struck down with fistula, ich the doctors could not cure. He died, still a young man, and eft his son, an infant of nine months old, to the guardianship of Dake of Bedford and the Earl of Warwick. This was in 1422. For five years Bedford, who managed with singular tact i prudence, succeeded in keeping things pretty straight, in pe of numerous causes of trouble and disturbance, including, ourse, the efforts of the Dauphin, who in the meantime bad me Charles VII., to regain his father's throne. Charles had *arge following, especially in the south-east of France, and he able to possess himself of a few towns of strength and Orleans was of the number, but it was closely ged by the English under the best of their generals, and arles despaired of relieving it, and thought of going to Ledoc, there to make a final stand. Iben arose Jeanne Darc, a peasant girl, who saw, or believed Aw, vi-ions of the saints, especially of St. Catherine, who to her and told her she must deliver France from the ence of the English. Her “voices," as she called them, her don man's attire, and directed her to fetch a certain word from a neighbouring church dedicated to St. Catherine., permitted by the authorities to follow her bent, and was rushed with armour and a horse. At first the regular soldiers ghed at her, but soon they got to regard her as a prophetess, ent for the deliverance of France. Under her guidance in strictly military operations she was assisted by Dunois,

portance.

governor of Orleans-the men fought with a courage which increased in proportion as her fame as a prophetess grew, and struck fear into the ranks of the English. Orleans was relieved by "the Maid" in person, and the garrison, now strong enough to attack its besiegers, sallied forth and drove the English from several of their positions. Subsequently another sally was made, a bloody battle was fought, the English lost 2,000 men, and Lord Talbot, afterwards Earl of Shrewsbury, was made prisoner. The Duke of Suffolk raised the siege, retiring to Paris, and Charles was crowned King of France with great solemnity at Rheims.

With these signs of returning prosperity many wavering nobles and towns declared for Charles, and the Duke of Bedford had enough to do to hold Paris and the strictly English parts of France. Jeanne, believing her mission to be over, was anxious to return to her former home in Lorraine, but was over-persuaded by Dunois to remain with the army till the English should be driven out of France. She remained, and in a sortie made by the garrison of Compiègne, was captured and given over to the English authorities. The English, partly from superstition, partly to excuse the disgrace of their defeats, said that "the Maid" had a devil, and that she had done her work through magic. They hoped also by punishing their prisoner not only to take revenge, but to show the French that their prophetess was a woman after all.

ness.

The Duke of Bedford handed her over to the Church, with what effect we have seen already; and from the moment of her death the English power seemed to be stricken with mortal sickPlace after place was wrested from them, Paris drove them out, the Duke of Burgundy forsook their alliance, and when in 1435 the Duke of Bedford died, their influence in France was at a very low ebb. A war of reprisals was carried on till 1443, and then a truce was agreed upon which either side broke or kept as it suited their convenience.

Then came the English Wars of the Roses, during which disastrous period the claims to France were not thought of, and it never happened to any prince after Henry VI. to have power or opportunity to pursue the right which was never formally renounced. Kings of England continued, nevertheless, to write themselves down kings of France, even after the loss, in Mary's reign, of their last remaining possession, Calais. Indeed, it was not, as stated at the beginning of this paper, until George III. ascended the throne that the title appeared to those interested so ridiculous that it was ordered to be expunged from the style and description of his Majesty of Great Britain.

LESSONS IN GEOGRAPHY.-XXIX.

ASIA.

Position on the Earth's Surface.-Asia, the cradle of the human race, and the original seat of the Garden of Eden, lies within the northern and eastern hemispheres, and to the east and south-east of Europe. The greater part of this continent lies within the north temperate zone; the exceptions being part of Siberia, which lies within the north frigid zone, and parts of Arabia, Hindostan, Further India, and China, which lie within the torrid zone.

Boundaries.-Asia is bounded on the north by the Arctic Ocean; on the south by the Indian Ocean and the Chinese Sea; on the east by the Pacific Ocean; and on the west by the Ural Mountains, Ural River, Caspian Sea, Mount Caucasus, Black Sea, Sea of Marmora, the Mediterranean Sea, the Isthmus of Suez, which connects it with Africa, and the Red Sea or Arabian Gulf.

Extent, Length, Breadth, etc.-This continent extends from lat. 78° 25′ N., to lat. 1° 20' S.; and from long. 26° 4′ E., to long. 170° W. This shows that a very small portion of this continent lies in the western half of the northern hemisphere. Its length, from the Strait of Bab-el-Mandeb, at the entrance to the Red Sea, to Behring Strait, between Asia and North America, is about 7,000 miles, measured across the continent in a straight line, as the crow flies. Its breadth, from Cape Severo, also called Cape Tcheliuskin, and North East Cape, in Siberia, in a straight line from north to south, almost identical with the 102nd meridian of east longitude, to Cape Romania, at the extremity of the Malay Peninsula, is about 5,250 miles. This continent is connected with that of Africa, at the Isthmus of Suez, which is about 80 miles wide. The surface of Asia, in