lines which have been thus put together at right angles. The triangle formed in this manner, as the triangle C E G in Fig. 40, is a right-angled triangle; and as in the case of this triangle it has been shown practically that the square G E L M described on G E, the side which subtends the right angle E C G, is equal in superficial area to the squares C F K G, C D H E, described on the sides & c, CE, which contain the right angle E C G, so it is true that in any and every right-angled triangle the square described on the side which subtends the right angle is equal to the squares of the sides by which the right angle is contained. But there are yet other facts that may be gathered from an examination of Fig. 40, and a consideration of the dotted lines that are drawn in the figure. First, through the point c the straight line C N is drawn parallel to G M or E L, intersecting the straight line E G in the point T, and dividing the square GEL M into the unequal rectangles (see Definition 27, page 53) TELN, TGM N. Of these the rectangle T E L N is equal to the square C D H E, and the rectangle T N M G equal to the square C G K F, as we will proceed to show. The reader will remember that in Problem XXIV (page 308) it was shown that triangles on the same base and between the same parallels are equal to one another, and that triangles on equal bases and between the same parallels are also equal to one another, Now in the trapezoid (see Definition 31, page 53) GDHE, of which the sides G D, H E are parallel, there are two triangles, DHE, G H E. These triangles stand upon the same base HE, and between the same parallels G D, H E, and are therefore equal to one another. But the dotted line D E is a diagonal of the square C D H E, and divides it into two equal parts; therefore the triangle D H E is equal to the triangle C D E, or, in other words, the square c DHE is double of the triangle D H E, and as the triangle D H E is equal to the triangle G H E, the square C D H E is also double of the triangle GH E; and this brings us to the fact, that when a square and a triangle happen to be on the same base, and between the same parallels, the area of the square is double the area of the triangle. Now let us turn to the trapezoid CEL N, of which the sides CN, E L are parallel, and which contains the rectangle, or rectangular parallelogram E L N T within its limits. In this there are also two triangles, C E L, E L N, standing on the same base, E L, and between the same parallels, the parallel sides E L, C N, of the trapezoid CEL N, and these triangles are consequently equal to one another. Now the rectangle E L N T is divided into two equal parts by the diagonal E N, and the triangle E L N is therefore equal to the triangle E T N, or in other words, the rectangle E L N T is double of the triangle E L N, and as the triangle E L N is equal to the triangle C E L, the rectangle E LNT is double of the triangle c E L. And this teaches us that when a rectangle or right-angled parallelogram and a triangle are upon the same base, and between the same parallels, the area of the rectangle is double the area of the triangle. And as it is true that when a square and a triangle, or a rectangle and a triangle, are upon the same base and between the same parallels, the area of the square or rectangle, as the case may be, is double that of the triangle, so it is equally true that when a square and a parallelogram, or a rectangle and a parallogram, are upon the same base and between the same parallels, the areas of the square and rectangle, or the areas of the rectangle and the parallelogram, thus situated, are equal to one another, as may be seen by drawing the straight line Ho through H, parallel to E G, when we have the square C D H E, and the parallelogram O HEG on the same base E H, and between the same parallels H E, G D, equal to one another; and by drawing the straight line L V through L, parallel to E C, when we get the rectangle ELNT and the parallelogram CELV equal to one another. Parallelograms also on the same base and between the same parallels are equal to one another, and when a parallelogram and a triangle are on the same base, the area of the parallelogram is double the area of the triangle; and more than this, as triangles on equal bases and between the same parallels are equal to one another, so also rectangles and parallelograms on equal bases and between the same parallels are equal to one another. But to proceed to show that the rectangle E L N T is equal to the square C D H E, let us look at the triangle G H E, which was proved to be equal to half the square C D H E, and the triangle CEL, which was proved to be equal to half of the rectangle ELN T, and compare their sides and angles. On inspecting them we find that the side E L of the triangle c E L is equal to the side E G of the triangle G H E, each being also a side of the square G E L M, and that the side c E of the triangle C E L is equal to the side E H of the triangle G H E, each of them being also a side of the square C D H E; and the angle C E L, contained by the sides C E, E L of the triangle C E L, is equal to the angle G E H contained by the sides G E, E H of the triangle G H E, for each of these angles is composed of a right angle and the angle c E G, which is common to both, the angle c E L being composed of the right angle L E G and the angle G E C, while the angle GE H is composed of the angle GE c and the right angle c E H. Here, then, we have two triangles, each having two sides of the one, namely, G E, E H, equal to two sides of the other, L E, E C, and the angles contained by these sides equal, namely, the angle G E H to the angle C EL; and this being true, it is plain that their bases or third sides are also equal, namely, HG to CL; and the areas of the triangles are equal, as we may prove practically by cutting out the triangle C E L, and turning it, as on a pivot, round the point E, until it rests on the triangle G H E. But the square C D H E has been shown to be double of the triangle G H E, and the rectangle E L N T has been shown to be double of the triangle c E L, and as things which are double of equal things must be equal to one another, the rectangle E L N T must be equal to the square c D H E. In the same way it may be shown that the rectangle G T N M is equal to the square C F K G, and the learner is recommended to work out the proof of this as a useful exercise. READING AND ELOCUTION.-XI. EMPHASIS distinguishes the most significant or expressive words of a sentence. It properly includes several functions of voice, in addition to the element of force. An emphatic word is not unfrequently distinguished by the peculiar "time," "pitch," "stress," and "inflection" of its accented sound. But all these properties are partially merged, to the ear, in the great comparative force of the sound. Hence it is customary to regard emphasis as merely special force. This view of the subject would not be practically incorrect, if it were understood as conveying the idea of a special force superadded to all the other characteristics of tone and emotion, in the word to which it applies. Emphasis is either "absolute" or "relative." The former occurs in the utterance of a single thought or feeling, of great energy; the latter, in the correspondence or contrast of two or more ideas. "Absolute" emphasis is either "impassioned" or tive." The former expresses strong emotion, as :False wizard, AVAUNT! 66 distinc But the latter designates objects to the attention, or distinguishes them to the understanding, as : The fall of man is the main subject of Milton's great poem. "Relative" emphasis occurs in words which express comparison, correspondence, or contrast, as : Cowards die many times; the brave but once. Rule 1.-Exclamations and interjections usually require "impassioned" emphasis, or the strongest force of utterance, as in the following examples : Woe to the traitor, WOE! UP! comrades, UP! AWAKE! ARISE! or be for EVER FALLEN! Motionless torrents! silent cataracts! GOD! GOD! the torrents, like a shout of nations, * Three degrees of emphasis are usually thus denoted in type: the first by Italic letters; the second, by small capitals; and the third, by large capitals. Thus, "You shall DIE, BASE DOG! and that before yon cloud has passed over the sun!"-Sometimes a fourth, by Italis capitals, thus :-NEVER, NEVER, NEVER! Rule 2.-Every new incident in a narration, every new object in a description, and every new subject in a didactic passage, requires "distinctive" emphasis, or a force of utterance sufficient to render it striking or prominent. Examples. Their frail bark was, in a moment, overset, and a watery grave seemed to be the inevitable doom of the whole party. The eye rested with delight on the long, low range of beautifully tinted clouds, which skirted the horizon. The power of faith was the subject of the preacher's discourse. Rule 3.-All correspondent, and all antithetic, or contrasted words, require a force sufficient to distinguish them from all the other words in a sentence, and to make them stand out prominently. When the comparison or contrast is of equal force in its constituent parts, the emphasis is exactly balanced, in the words to which it is applied: when one of the objects compared or contrasted is meant to preponderate over the other, the emphasis is stronger on the word by which the preponderance is expressed. Examples. The gospel is preached equally to the rich and to the poor. Exercises in "Relative" Emphasis. VIRTUE is better than riches. Study not so much to show knowledge, as to acquire it. He that cannot bear a jest, should not make one. It is not so easy to hide one's faults, as to mend them. I that denied thee gold, will give my heart. You have done that you should be sorry for. Why beholdest thou the mote that is in thy brother's eye, but considerest not the beam that is in thine own eye? As it is the part of justice || never to do violence, so it is the part of modesty never to commit offence. A friend cannot be known | in prosperity, and an enemy | cannot be hidden in adversity. Emphatic clauses (those in which every word is emphatic) are sometimes pronounced on a lower, sometimes on a higher key, but always with an intense force. Examples. Heaven and earth will witness IF ROME MUST FALL-that we are innocent. This state had then not one ship-NO, NOT ONE 1 WALL. religion. He must avoid everything which may look like moroseness and gloom. He must cultivate a cheerfulness of spirit. He must endeavour to show, in his whole deportment, the contentment and tranquillity which naturally flow from heavenly affections, from a mind at peace with GoD, and from a hope full of IMMORTALITY. The spirit which Christianity enjoins and produces is so widely different from the spirit of the world, and so immensely superior to it, that, as it cannot fail of being noticed, so it cannot fail of being admired, even by those who are strangers to its power. Do you ask in what particulars this spirit shows itself? I answer, in the exercises of humility, of meekness, of gentleness; in a patient bearing of injuries; in a readiness to forgive offences; in a uniform endeavour to overcome evil with good; in self-denial and disinterestedness; in universal kindness and courtesy; in slowness to wrath; in an unwillingness to hear or to speak evil of others; in a forwardness to defend, to advise, and to assist them; in loving our enemies; in blessing them that curse us; in doing good to them that hate us. These are genuine fruits of true Christianity. The Christian must "let his light shine before men, by discharging in a faithful, a diligent, and a consistent manner, the personal and particular duties of his station. As a member of society, he must be distinguished by a blameless and an inoffensive conduct; by a simplicity and an ingenuousness of character, free from every degree of guile; by uprightness and fidelity in all his engagements. As a neighbour, he must be kind, friendly, and accommodating. His discourse must be mild and instructive. He must labour to prevent quarrels, to reconcile those who differ, to comfort the afflicted. In short, he must be "ready for every good work;" and all his dealings with others must show the HEAVENLY PRINCIPLE which dwells and works in his HEART. Exercise.-The Benefits of a Popular Government. The real glory and prosperity of a nation does not consist in the hereditary rank or titled privileges of a very small class in the community; in the great wealth of the few, and in the great poverty of the many; in the splendid palaces of nobles, and the wretched huts of a numerous and halffamished peasantry. No such a state of things may give pleasure to proud, ambitious, and selfish minds, but there is nothing here on which the eye of a patriot can rest with unmingled satisfaction. In his deliberate judgment "Il fares the land, to hastening ills a prey, It is an intelligent, virtuous, free, and extensive population, able by their talents and industry to obtain a competent support, which constitutes the strength and prosperity of a nation. But youth, it seems, is not my only crime: I have been accused of into operation a greater amount of talent than any other. ecting a THEATRICAL part. As to the present ministry, I cannot give them my confidence. Pardon me, gentlemen: Confidence is a plant of SLOW growth. General Remark.-Young readers are commonly deficient in emphasis, and hence feeble and unimpressive, in their style of reading. Students should exert much vigilance on this point. At the same time, an overdone emphasis is one of the surest indications of defective judgment and bad taste. Faults which result from study are always the most offensive. Exercise.-The Duty of a True Christian. The true Christian must show that he is in earnest about religion. In the management of his worldly affairs, he must let it clearly be seen, that he is not influenced by a worldly mind; that his heart is not upon earth; that he pursues his worldly calling from a principle of DUTY, not from a sordid love of gain; and that, in truth, his treasures are in HEAVEN, He must, therefore, not only "provide things honest in the sight of all men;" not only avoid everything which is fraudulent and unjust in his dealings with others; not only openly protest against those iniquitous practices which the custom of trade too frequently coun tenances and approves ;-but, also, he must "let his moderation be known unto all men." He must not push his gains with seeming eagerness, even to the utmost LAWFUL extent. He must exercise forbearance. He must be content with moderate profits. He must sometimes even forego advantages, which, in themselves, he might innocently take, lest he should seem to give any ground for suspecting that his heart is secretly set upon these things. Thus, also, with respect to worldly pleasures: he must endeavour to convince men that the pleasures which RELIGION furnishes, are far greater than those which the world can yield. While, therefore, he conscientiously keeps from joining in those trifling, and, too often, profane amusements, in which ungodly men profess to seek their happiA, he must yet labour to show, that, in keeping from those things, he is, in respect to real happiness, no loser, but even a GAINER by It is not the least advantage of a popular government, that it brings It is acknowledged by every one, that the occurrence of great events awakens the dormant energies of the human mind, and calls forth the most splendid and powerful abilities. It was the momentous question, whether your country should be free and independent, and the declaration that it was so, which gave to you orators, statesmen, aud generals, whose names all future ages will delight to honour. The characters of men are generally moulded by the circumstances in which they are placed. They seldom put forth their strength, without some powerfully exciting motives. But what motives can they have to qualify themselves for stations, from which they are for ever excluded on account of PLEBEIAN EXTRACTION ? How can they be expected to prepare themselves for the service of their country, when they know that their services would be REJECTED, because, unfortunately, they dissent from the established religion, and have the honesty to avow it! But in a country like OURS, where the most obscure individuals in society may, by their talents, virtues, and public services, rise to the most honourable distinctions, and attain to the highest offices which the people can give, the most effectual inducements are presented. It is indeed true, that only a few who run in the race for political honour, can obtain the prize. But, although many come short, yet the exertions and the progress which they make, are not lost either on themselves or society. The suitableness of their talents and characters for some other important station may have been perceived; at least the cultivation of their minds, and the effort to acquire an honorable reputation, may render them active and useful members of the community. These are some of the benefits peculiar to a POPULAR government; benefits which we have long enjoyed. LESSONS IN MUSIC.-VI. FOR Exercise 15, in the following page, the pupil will pitch his own key-note as indicated in the title. If, however, he has not got a tuning-fork, let him take DOH at a rather low pitch. A stroke beneath two or more notes shows that they are to be sung to one syllable of the words, or "slurred." The comma after a note gives it a quarter of an aliquot; the dot and comma, three quarters. Be careful in singing this correctly. Exercise yourself in singing the two notes, first with a dot only, and then with a dot and comma between them. The tune is Mr. Burnet's copyright. It may be found harmonised for four voices in "People's Service of Song." All the early exercises in this course are given in two-part harmony, because we are persuaded that, by two-part harmony, the ear is best taught to understand that which is more complex. These exercises should be sung by "equal voices;" that is, by two male voices, or by two female or children's voices. It will not sound quite so well if the air (or upper part) being sung by a female voice, the lower part is sung by a male, for the male and female voice are naturally an octave apart, and the intervals cannot be so "close" and sweet. When you have traced and sol-faed this tune from the modulator perfectly, your next step will be to "figure it; that is, sing EXERCISE 15.-GRIFFIN. it to the words "one, two, three, four, five, six, seven, eight; one, two, three, four, five, six, seven," etc. As you know these words very familiarly, your attention will not be distracted by them (as it might be by other words), while you try to strike the intervals correctly, without that help to the memory which the sol-fa syllables give. You may afterwards sing the words; but remember that this tune must be sung with spirit (abrupt decision), or not at all. A curve over or under two or more notes, indicates a slur. In previous exercises we have had a black note (crotchet) to correspond with an aliquot or pulse of the measure. In this tune we have used an open note (a minim for the aliquot. We prefer using the crotchet as the standard aliquot; but, as it is not always so used, we have made this change to indicate that fact. It makes no difference to the music. There are still four pulses to the measure, and they move at the rate indicated by the metronome. The proper management of the voice in singing is of great importance, and will require a few suggestions from us. First, notice that a sound of the voice in singing is distinctly held and continues the same from the beginning to the end. It is thus distinguished from the speaking voice, each sound of which has a change in it called an "inflection." A sound of the singing voice is commonly called a "note"-though the word note is more properly limited to the mark upon paper-the sign of a sound. With a violin you can produce either a "note" or an "inflection." Press your finger steadily on the upper part of a string, while you draw the bow, and that will give you a clear and beautiful note. But if, instead of that, you move your finger up or down the string, while you draw the bow, that will give you an inflection. You perceive, therefore, that a note ought to have nothing of the inflection about it-no "scraping" up or down as some sing—but it should be clear, steady, and distinct. To produce a good note, the singer should be in an easy posture, with his head upright and his shoulders back, so as to allow the muscles of the chest and the larynx (that little box in the throat which we can feel with our fingers) to have free movement. His mouth should be moderately open. His tongue should lie down, just touching the roots of the lower 4 Trust no future, howe'er pleasant; 5 Let us, then, be up and doing, Some teeth; and his lips should have the position most easily explained by referring to that of a gentle smile, but really expressing no smile, and giving no emotional expression. teachers require a small cork of the thickness of a little finger, or the little finger itself, to be placed between the back teeth during the earlier exercises. We have a friend who, to improve his voice for speaking, used to read aloud for half an hour before breakfast every morning with a large cork between his front teeth. Of course this did not cultivate his enunciation-his words were curiously pronounced-but it strengthened the larynx and lungs, and prevented his over-exertion of the throat, so that he could speak in public with the greatest ease, and without the slightest fatigue of voice, as we have had ample proof, nearly a whole day long. The pupil who would learn to sing without fatigue, should practise, for a few minutes every day, the taking a full inspiration into the lungs, and then giving out the air very slowly and steadily. This will give him command of the muscles of the chest. He will be surprised, at first, to discover the diffi culty of a slow and steady expiration. But let him persevere, making this the first of his exercises for the improvement of his voice, every morning. The next of his morning exercises should be in singing the chord and scale, holding the notes as long and stezdily as possible, and ascending as high as his voice will allow (with the cork, if necessary, to keep his mouth open), and with the most careful observance of the following directions. Expand the ribs, so that they press against the dress at the sides, and, by drawing in the muscles of the lower belly, keep the ribs thus erpanded. This will allow free and easy play to the lungs. For courses of exercises on these subjects, see the two small books named in Lesson V. The sounds of the voice, in singing, should be delivered promptly and easily. If the voice is given out carelessly, it comes roughly through the throat, and is called guttural; and if produced in a forced manner, it is driven through the nose, and so becomes nasal. Correctness in singing depends upon mental effort, for it is the mind which commands the delicate muscles of the larynx and throat. Lazy singing is always flat and miserable; hence we always sing musically better when our hearts are most engaged in the song. A note may be loud or soft. The loudness or softness of the voice is called its force. It is very important to cultivate the habit of using a medium force of voice, so that it may be always easy to sing a note or strain more loudly or more softly than the rest. This habit is important to comfort and pleasure in singing, and absolutely necessary for expression and refinement. The medium voice of one person is, of course, different from that of another, according to the size of the larynx and the strength of the lungs. The suggestions given above must be kept constantly in mind in every daily practice. If you enjoyed the advantage of a private teacher, such points as these would be constantly in his mind, and he would see to it that you observed them. Indeed, one of the chief uses of a private teacher is to keep us to our work. The self-educator, however, must summon to his aid sturdy determination and steady perseverance. A lady went to a distinguished teacher of singing, to receive a course of costly lessons in the art. For a large proportion of these lessons, in the early part of the course, he did not permit her to sing a single note, but made her simply pace the room, expanding her lungs, and taking breath in every way which was required to give her command of the material of which voice is made. We have heard that even the great public singers do not think of omitting the daily practice of the scale and chord in long "holding" notes, as we have recommended. Crotchet = 66, beating only twice in a measure. EXERCISE 16.-LEYBURN. KEY B. M. (An old English Ballad Tune. Words by M. A. Stodart, from "Poetry" by the Home and Colonial School Society.) If your friend gives you "pattern" with an instrument, tell | words. Indeed, no song is rightly learned till both tune and him to play in the key of B flat (with two flats), or in that of B words are learned " by heart." You will observe the various (with five sharps), whichever he prefers; one is as easy as the "signs of repetition" which are explained in the preceding other to you. Take care to point on the modulator without lesson. A second line of words is given, in each case, for the book, and to "figure "the tune (one, two, three, fo-ur, five, si-x, repetition of the music. The tune is harmonised with a bass in seven; one, two, three, fo-ur, five, six, etc.) before you sing it to "School Music." LESSONS IN FRENCH.-XXII. SECTION XXXVIII.-USES OF REFLECTIVE VERBS (continued). 1. THE reflective verb se passer is used idiomatically in the sense of to do without. It is followed by the preposition de, when it comes before a noun or a verb. Vous passez-vous de ce livre ? Do you do without that book? 12. Je m'éveille ordinaire13. Vous levez-vous aussitôt heure vous éveillez-vous le matin ? 2. Se servir [2, ir.; see § 62], to use, also requires the prepo- 23. Il s'en approche pour se chauffer. sition de before its object. Je me sers de votre canif, I use your penknife. EXERCISE 72. 1. Will you lend me your penknife? 2. I cannot do without it, I want it to mend my pen. 3. Do you want to use my book? 4. I want to use it, will you lend it to me? 5. What knife does your brother use ? 6. He uses my father's knife, and my brother's fork. 7. Will you not draw near the fire? 8. We 4. The pronoun* used as indirect object of a reflective verb, are much obliged to you, we are warm. 9. Is that young lady if representing a person, follows the verb [§ 100 (4)]. Je puis me passer de lui, Je m'adresse à vous et à elle, I can do without him. I apply to you and to her. warm enough? [Sect. XXXIII. 3.] 10. She is very cold. 11. Tell her (dites-lui) to come near the fire. 12. Why do you go from the fire ? 13. We are too warm. 14. Does your brother 5. S'endormir [2, ir.; see § 62], to fall asleep, and s'éveiller, leave the window? 15. He leaves the window because he is to awake, are also reflective. Vous servez-vous de ce couteau ? Does your son draw near the fire? Do you use that knife? I do not use it, it does not cut. Nous nous servons de couteaux We use steel knives. d'acier. Pouvez-vous vous passer d'argent? vous. Vous vous endormez facilement. Je m'en approche parceque j'ai Nous nous éloignons du feu. Nous nous en éloignons. Can you do without money? Do you do without your teacher? Do you apply to those gentlemen? You go to sleep easily. I come near it because I am cold. cold. 16. To whom does that gentleman apply? 17. He applies to me and to my brother. 18. Why does he not apply to me? 19. Because he is ashamed to speak to you. 20. Do you awake early every morning? 21. I awake early when I go to bed early. 22. Why do you go to sleep? 23. I go to sleep because I am tired. 24. Are you afraid to go near your father? 25. I am not afraid to approach him. 26. Can you do without us? 27. We cannot do without you, but we can do without your brother. 28. Do you want my brother's horse? 29. No, Sir, we can do without it. 30. Do you intend to do without money? 31. You know very well that we cannot do without it. 32. Is your brother weary of being here? 33. He is not weary of being here. 34. Come near the fire, my child. LESSONS IN ARITHMETIC.-XX. 1. In comparing two numbers or magnitudes with each other, This latter relation-namely, that which is expressed by the quotient of the one number or magnitude divided by the otheris called their Ratio. Thus the ratio of 6 to 2 is 62, or 3. The ratio of 7 to 5 is 75, or, as it would be written, the fraction . The two numbers thus compared are called the terms of the ratio. The first term is called the antecedent, the last the consequent. It will be seen that any ratio may be expressed as a fraction, the antecedent being the numerator, and the consequent the denominator. A ratio is, in fact, the same thing as a fraction. When we talk of a ratio, we regard the fraction from rather a different point of view, namely, as a means of comparing the magnitude of the two numbers which represent the numerator and the denominator, rather than as an expression indicating that a unit is divided into a number of equal parts, and that so Ordinairement, gene- many of them are taken. We go from the fire. 1. Pouvez-vous vous passer d'encre ? 2. Nous pouvons nous en passer, nous n'avons rien à écrire. 3. Vous servez-vous de votre plume? 4. Je ne m'en sers pas; en avez-vous besoin? 5. Ne voulez-vous pas vous approcher du feu? 6. Je vous suis bien obligé, je n'ai pas froid. 7. Pourquoi ces demoiselles s'éloignent-elles de la fenêtre ? 8. Elles s'en éloignent parcequ'il y fait trop froid. 9. Ces enfants ne s'adressent-ils pas à vous? 10. Ils s'adressent à moi et à mon frère. 11. À quello The rule does not apply to the reflective pronoun, which is sometimes an indirect object. 2. The ratio of two numbers is often expressed by writing two dots, as for a colon, between them. Thus the ratio of 6 to 3 is written 6: 3; that of 3 to 5, 3: 5, etc. The expressions and 3: 5, it must be borne in mind, mean exactly the same thing. A direct ratio is that which arises from dividing the antecedent by the consequent. An inverse or reciprocal ratio is the ratio of the reciprocals of the two numbers. Thus, the inverse ratio of 3: 5 is the ratio which is the same as or |