lines, which, as you will see, when many forces have to be compounded, would cause much confusion in your figures.

R

Fig. 6.

Let us apply this principle now to compound any number of forces acting on a point. Let there be five, and that will illustrate the rule as well as a thousand could. Suppose forces, o A, O B, O C, O D, O E, applied to the point, o. By the triangular rule, if I draw A R equal and parallel to o B, the line joining o with R is the resultant of the first two forces. I shall not actually draw this line, o R; let us suppose it drawn. Now, if I compound this resultant with o c, I have the resultant of three of the forces. But that, by the same rule, is got by drawing from R a line R R, equal and parallel to o c. The line o R is this resultant of three. Again we shall not draw it. The resultant of this and o D, for the same reason, would be o R, got by drawing R, R2 parallel and equal to o D, and, lastly, the resultant of this and o E would be o E, the line, R, R., being equal and parallel to o E. We have thus exhausted all the forces, and evidently o R, is the resultant of the whole five. There was here no confusing ourselves with parallelograms; all we had to do was to draw line after line, one attached to the other, carefully observing to keep their magnitudes and directions aright. A kind of unfinished polygon was thus formed, and the line o B,, which closes up the polygon, joining the last point E with the point of application, is the resultant in magnitude and in direction. Thus you have made another step in advance, and arrived at the Polygon of Forces. You have learned how, by the mere careful drawing of lines, to determine the resultant of any number of forces. All you require is paper and pencil, a rule, compasses, a scale, and a pair of parallel rulers.

Now, there is one point about this polygon I wish you carefully to note. You will observe that the arrows on its sides, representing the directions of the forces you have compounded, all point from left to right, as you go round the figure, turning it with you so as to bring each side in succession to the top. The resultant, however, points in the opposite direction, from right to left, when that side is uppermost. This is as it should be; the direction of the resultant, as you go round the figure, must be opposite to those of the components. The use of this you will see in the next lesson.

It

Now, let us suppose that, in determining the resultant after this method, as we come to the end of the operation, the end, E of the last line, R, R, chanced to coincide with, or fall upon the point of application, o. The polygon would close of itself without any joining line; what is the meaning of this? means that there is no resultant; the line, o R,, is nothing, and therefore the resultant is nothing, and the forces produce equilibrium. What a valuable result we have arrived at! A method by which we can, by rule and compass, tell at once whether any number of forces make equilibrium at a point or not. All we have to do is to describe the polygon of forces, and if it closes up of itself, there is equilibrium; if it does not, there cannot be equilibrium, and the resultant is in magnitude the side which is necessary to close the figure.

Deferring the further expansion of this subject to the next lesson, I shall now turn back and apply these principles to a few elementary examples.

First Example.-Three equai forces act at a point in different directions what condition should they fulfil in order to be in equilibrium? Get your ruler and compass, and commence constructing the figure by which their resultant may be found. From the end of one of the forces you are to draw a line equal and parallel to the second equal force, and from the end of that | another line, equal and parallel to the third. You will thus have three lines strung together, all equal to each other. But if the forces are in equilibrium, the end of the last line must fall on the point of application, that is to say, the polygon of forces must close up, and form a triangle. Your construction will then give you a triangle of three equal sides, commonly called an equilateral triangle. But such a triangle must have

all its angles equal; also the angles between the sides of the triangle, or of the polygon of forces, are the angles between the forces themselves, since they are parallel to these forces. This is evident from the properties, 1 and 2, of the parallelogram referred to above; therefore, in the case we are considering, the three equal forces must act at equal angles, as I showed other. wise must be the case at the close of the last lesson.

B

R

P

Second Example.-Let a weight hang from the ceiling by means of two cords of unequal length, as in Fig. 7. It is evidently at rest. Whatever be the forces called into action, they produce equilibrium. Is there nothing further to ascertain ? There is; it may be desirable to know by how much each cord is strained. Our assurance that the cords will support the weight depends on this knowledge. Let p and q be the two points of support of the strings which meet at o. Now, whatever be the strains on the cords, o P, o Q, they make equilibrium with w, the weight. Therefore, if we suppose a length, o A, of O P to represent the strain on o P, and from a draw a line, A R, parallel to o Q, equal to the strain, o B, on o Q, then, since the three forces are in equilibrium, the line, R o, closing up the triangle must be equal to, and be in the direction as, the third force, or weight, w. This, then, tells us what to do. Measure on o R upward as many inches as there are pounds in w; and from R then draw RA parallel to the cord o q to meet the cord o A. The number of inches in o A will represent in pounds the strain cn o P, and those on RA the strain on o o. All, therefore, that we desire to know is determined.

W

Fig. 7.

Third Example.-A horse pulls a roller up a smooth inclined plane or slope; what is the force he must exert when he just keeps the roller at rest? And by how much does the roller press on the plane?

B

Let the horse pull in any direction, o A. Then there will be three forces acting on the roller; namely, its own weight right downwards, the horse's pull, and the resistance of the plane or slope, perpendicular to itself. There must be this third force, for the other two, not being opposite to each other, cannot make equilibrium. The roller is somehow supported by the plane; and that it cannot be unless by its resistance; and plane cannot resist except perpendicularly to itself. This third force, you thus see, must be

a

C

Fig. 8.

taken perpendicular to the plane. It is represented in the figure by o B. Now apply the polygon of forces. Let o c represent the weight of the roller, and from c suppose a line, c R, drawn equal and parallel to o A, the horse's pull. Then, since there is equilibrium, the polygon of forces should close up and become a triangle-that is, the line joining R with o should be the pressure, and therefore should be perpendicular to the plane. What then are we to do? Take o c, equal in inches to the number of pounds in the roller, draw then from c a line c R parallel to the horse's pull, to meet the line drawn from the centre o of the roller perpendicularly to the plane; C R so determined will in inches tell the pounds in the horse's pull, and O R the amount by which the roller presses the plane. You can easily see from this that as the slope increases the pull will increase and the pressure diminish. This is what naturally we should expect. The plane I have supposed to be smooth; for, where there is friction against the roller caused by roughness in itself or in the plane, of in both, the question is much altered, as in due time you will see.

thus accented.

This vowel receives but one kind of accent, and that is the circumflex, viz.:-1, ; though it is comparatively seldom found This vowel has two sounds, viz., long and short; long, as ee in the English word see, and short, like i in the English word pin, or nearly like it. It becomes nasal in combination with the letters m and n, in which case the character of its own sound is completely changed, which is indeed true of all the vowels.

In these Lessons, the vowel I, i, will be represented by the two letters ee, when long or under the circumflex accent, and by e when it has the short sound.

euses ?

Elles ne sont pas très-studieuses.
Ces règles sont-elles générales?
Leurs habillements sont superbes.
Ces principes sont généraux.
Avez-vous peur de ces chevaux

rétifs ?

Vos montres d'or sont excellentes. Les miennes sont-elles meilleures que les vôtres ?

Les vôtres sont meilleures que les

miennes.

Agréable, agreeable. Ainé, e, elder. Allemande, f., German. Jamais, never. Indulgent, -e, indulgert Laine, f., wool; woollen. Maroquin, m., morocco.

They are not very studious.
Are those rules general?
Their clothes are superb.
Those principles are general.
Are you afraid of those restive

horses ?

Your gold watches are excellent. Are mine better than yours?

Yours are better than mine.

1. Les chevaux de notre ami sont-ils rétifs ? 2. Ses chevaux ne sont pas rétifs, mais ses mules sont très-rétives. 3. Les chevaux et les mules de votre frère sont excellents. 4. Vos sœurs sont-elles très-vives? 5. Mes frères et mes sœurs sont très-vifs. 6. Sont-ils souvent oisifs ? 7. Non, Monsieur, mes sœurs ne sont jamais oisives. 8. Avez-vous peur de votre frère ? 9. Non, Monsieur, je n'ai peur de personne. 10. Ne sommes-nous pas indulgents ? 11. Vous êtes indulgents, et vous avez raison. 12. Ai-je vos livres ? 13. Vous ne les avez pas, vous avez ceux de mon frère aîné. 14. Ne les avez-vous pas ? 15. Je ne les ai pas. 16. Avez-vous une bonne paire de bas de laine? 17. J'ai une belle paire de bas de soie. 18. Avez-vous les bonnes maisons ou les mauvaises ? 19. Je n'ai ni les bonnes ni les mauvaises, j'ai celles de ma cousine. 20. Le travail est-il agréable? 21. Le travail est utile et agréable. 22. Avez-vous mes beaux souliers de maroquin ? 23. Je n'ai pas vos beaux souliers de maroquin, j'ai vos belles pantoufles de velours.

EXERCISE 22.

brothers are quick, but my sisters are not quick. 3. Have you 1. Are your brothers and sisters very (bien) quick? 2. My not two restive horses ? 4. No, but I havo a restive mule. 5. Have you not two good pairs of silk gloves? 6. I have a good pair of cotton gloves, and two pairs of silk gloves. 7. Are you not afraid of your friends? 8. No, Sir, I am never afraid of my friends. 9. I am afraid of nobody. 10. Are you right or wrong? 11. I am right. 12. Have you my beautiful leather slippers, or my old satin slippers? 13. I have your old leather shoes and your velvet slippers. 14. Are those ladies pleased? 15. Those ladies are pleased, and they are right. 16. Has the German lady your father's shoes or mine? 17. She has neither his nor yours, she has my sister's. 18. Has your elder brother good houses or bad? 19. His houses are better than yours and than mine.* 20. Are his houses old? 21. His houses are old, but they are good. 22. Have you them? 23. No, Sir, I have them not, I have no houses. 24. Have you my brother's or my sister's? 25. Your sister has hers and my mother's. 26. Are your scholars attentive? 27. My scholars are very attentive and very studious. 28. Are those German ladies studious? 29. They are very studious and very attentive. 30. Are you

Part II.

9. PRESENT OF THE INDICATIVE OF ETRE, TO BE.

Negatively.

Je ne suis pas,

Tu n'es pas,

I am not. Thou art not.

He is not.

She is not.

SECTION XIII.-PLACE OF THE ADJECTIVES.-RELATIVE

PRONOUN EN.

Is he not? N'est-elle pas ? Is she not? sommes-nous Are we not?

Ne

of life, to bring them to the knowledge of God and the belief of the holy Church; for if they had continued still to chastise them according to the rigour of the said Inquisition, they would have caused them all to die by fire."

Such then were the causes of the deep hatred already spoken of as existing among Englishmen during the reign of Elizabeth. The Spanish political power and the Spanish ecclesiastical power, each lusted after dominion, and allowed no considerations nor scruples to stand in their way. Each helped the other; the priests taught the "right divine" of the Spanish king "to govern wrong," and the Spanish king in return upheld, with brutal obstinacy, the priests' Inquisition-an institution of which more will be said in another paper; but of which it will be enough here to say that it was a spiritual tribunal, irresponsible and acting in secret, which punished men and women with all punishments, including death, for not acting in strict accordance with the rules of the Roman-Catholic Church.

ENGLISH SHIP OF WAR.

Englishmen, after the Reformation especially, hated both these powers. The one cramped their action and their enterprise, forbidding them under pain of being treated as pirates to trade to places where the Spaniards claimed to have a monopoly, as in America; the other oppressed their souls with burdens too heavy to be borne, and then killed them for stumbling. Generous sympathy also for those who suffered wrong at the oppressor's hands, and were unable to help themselves, glowed in the English breast; and that sympathy, in an age of adventure and of chivalrous feeling, was not slow to express itself in action. It had received a fillip, too, in a point which nearly concerned the best interests of the nation. An attempt had been made after the death of Edward VI., in 1553, to introduce both the detested powers into England. Philip II. of Spain, was actually married to Queen Mary of England, and though the nation was, to a man, hostile to the introduction of the Inquisition, and swore it would not have it at any price, the energy and watchfulness of the best men were required to prevent the planting of the Spanish political power. In 1558 Elizabeth came to the throne, and not only roused the wrath of disappointment and jealousy by her prompt rejection of Spanish advances, directly and indirectly she challenged the Spaniards by the uncompromising Protestantism of her policy.

but

Her subjects were imbued with the same spirit as the Queen. The Spaniards were looked upon as public enemies, whom to destroy was to do God service; and many was the private adventure made by persons of good name and reputation, to make war upon them. In a time when the two governments were at peace, cruisers were fitted out in England- notably in West-country ports-to prey upon the enemy's commerce on the Spanish

could be nothing but perpetual war between the nations, and a fresh declaration of an old fact would have been useless as well, as tiresome. So whenever a Spanish treasure fleet was coming home, or a Spanish squadron of merchantmen was known to be on the seas, the English royal vessels slipped out of port, and smote the Philistines wherever they found them.

One of the most courageous and indomitable of the English rovers was Sir Richard Grenville, of Stowe, in Cornwall, a gentleman of ancient family and large fortune, an enthusiastic admirer of all that was generous and manly. He hated the Spaniards with an exceeding bitter hatred, and again and again left his pleasant home in Cornwall to roam the seas after the enemies of God and man, as he considered them to be. He had been eminently successful, both in distant expeditions and in repelling the attack of the Armada on the English coast itself, and his name was a terror to many & Spanish sailor. It happened, in the year 1591, that a Government expedition of the kind above-mentioned was about to sail under orders of Lord Thomas Howard, to intercept the Spanish treasure ships on their way from the West Indies. Sir Richard was appointed second in command, and hoisted his flag on board the Revenge; the rest of the squadron inIcluding eight fighting ships, with tenders and victuallers. The account of the action in which the Revenge fought single-handed for England is given here as best showing the kind of spirit it was which animated Englishmen at the time when their enemies were the detested upholders of Absolutism in Church and State.

TIME OF ELIZABETH.

Lord Thomas Howard sailed with his ships in August, 1591, and after cruising about for some time, put into the Western Islands, to recruit his men, ill with scurvy, and to wait there for the treasure ships. On the 31st of August, 1591, the lookout men reported a fleet in sight, and great was the joy and

greedy, perhaps, the expectation of the English warriors. But a nearer view disclosed, not the Spanish treasure ships, but a fleet of fifty-three ships of war, which had been equipped and sent out for the very pur pose of pouncing on the pouncers. Half the English crews were on shore, ill, and the rest were busy watering and victualling the ships. Lord Thomas looked at his vessels and sickly crews, and then at the enemy's ships, concerning which the cry was still, "They come." Eight against fifty-three-the disproportion was too great. He determined not to try conclusions with them, and having recalled his crews by signal, stood out of the Bay of Flores, and succeeded in getting away.

There was one ship, however, which did not follow. Sir Richard Grenville felt it to be almost an immoral act to retreat before a Spaniard, and though he was too good an officer wilfully to

advantage of some unavoidable delay which occurred in getting his men from the shore, to stay behind. The other English ships gained the offing, and thither, too, was sent the master of one of the victuallers, who, seeing Sir Richard's danger, offered to stay and share it with him.

On came the Spanish fleet, on the weather bow of the Revenge. Some of the officers remonstrated with the admiral, and advised him to crowd all sail and try to outsail the enemy; but Sir Richard declared "he would much rather die than leave such a mark of dishonour ou himself, his country, and the Queen."