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 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 deci. sion), 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, teeth; and his lips should have the position most easily ex plained by referring to that of a gentle smile, but really expressing no smile, and giving no emotional expression. Some 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 inspira tion 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 difficulty 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 steadily 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 expanded. 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 EXERCISE 16.-LEYBURN. KEY B. M. 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. (An old English Ballad Tune. Words by M. A. Stodart, from "Poetry" by the Home and Colonial School Society.) Then hurrah for merry England, And may we all be seen True to our well-loved country, And faithful to our Queen. Then hurrah, etc. 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 (with five sharps), whichever he prefers; one is as easy as the other to you. Take care to point on the modulator without book, and to "figure" the tune (one, two, three, fo-ur, five, si-x, seven; one, two, three, fo-ur, five, six, etc.) before you sing it to words are learned "by heart." You will observe the various signs of repetition which are explained in the preceding lesson. A second line of words is given, in each case, for the repetition of the music. The tune is harmonised with a bass in "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. heure vous éveillez-vous le matin? 12. Je m'éveille ordinaire13. Vous levez-vous aussitôt que vous vous éveillez? 14. Je me lève aussitôt que je m'éveille. 15. De quels livres vous servez-vous ? 16. Je me sera des miens et des vôtres. 17. Ne vous servez-vous pas de ceux de votre frère? 18. Je m'en sers aussi. 19. Les plumes dont [Sect. XXX. 8] vous vous servez sont-elles bonnes? 20. Pourquoi votre ami s'éloigne-t-il du feu ? 21. Il s'en éloigne parcequ'il a 2. Se servir [2, ir.; see § 62], to use, also requires the prepo- 23. Il s'en approche pour se chauffer. trop chaud. 22. Pourquoi votre domestique s'en approche-t-il? 24. Vous ennuyez-vous sition de before its object. 25. Je ne m'ennuie pas. Vous passez-vous de ce livre? Je ne puis m'en passer, Je me sers de votre canif, Do you do without that book? I use your penknife. I do not use it. ici? 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)]. I apply to you and to her. 5. S'endormir [2, ir.; see § 62], to fall asleep, and s'éveiller, leave the window? to awake, are also reflective. 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 quelle * The rule does not apply to the reflective pronoun, which is sometimes an indirect object. 11. warm enough? [Sect. XXXIII. 3.] 10. She is very cold. 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 many of them are taken. 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 of, or otherwise expressed which is the same as or otherwise expressed, 5: 3. Hence we see that the inverse ratio of two numbers is erpressed by inverting the order of the terms when the ratio is * The reciprocal of a number or fraction is the number or fraction obtained by inverting it. Thus, the reciprocals of 5, . à, etc., are respectively, 1, 6. expressed by points, or by inverting the fraction which expresses the direct ratio. A ratio is said to be compounded of two other ratios when it is equal to the product of the two ratios. Thus, is a ratio compounded of the ratios and J. 3. Proportion. Any set of numbers are said to be respectively proportional to any other set containing the same number when the one set can be obtained from the other by multiplying or dividing all the numbers of that set by the same number. Thus, 3, 4, 5 are proportional respectively to 9, 12, 15, or to 3, 7. To divide a given number into parts which shall be propor Different pairs of numbers may have the same ratio. Thus, tional to any given numbers. the ratios,,, are all equal. When two pairs of numbers have the same ratio, the four numbers involved are said to form a proportion; and they themselves, in reference to this relation subsisting among them, are called proportionals. Thus, 3, 4, 12, 16, are proportionals, because the ratio 2, or 3 : 4 the ratio, or 12: 16. A proportion is expressed either by writing the sign of equality (=) between the two equal ratios, or by placing four dots in the form of a square, thus, :: between them. = Thus we see that either product may be separated to form the extremes, and that, the order of either the means or the extremes being interchanged, the numbers still form a proportion. 5. If three numbers be given, a fourth can always be found which will form a proportion with them. Add the given numbers together, and then, dividing the given number into a number of parts equal to this sum, take as many of these parts as are equal to the given numbers respectively. EXAMPLE.-Divide 420 in proportion to the numbers 7, 5, 7+ 5 + 3 = 15; and 3. And therefore the respective parts are- × 420 196. × 420 == 84. Find in their simplest form: 1. The ratio of 14 to 7, 36 to 9, 8 to 32, 54 to 6. 3. The inverse ratio of 4 to 12, and of 42 to 6. 4 Find the fourth term of the propertions, 3:5::6; 4:8::9: -; 1 5. Insert the third term in the following proportions-3:5::-:6; 4:8:9; } } : 6. Insert the second term in the following proportions-3: -::5:6; This is the same thing as saying that if three terms of a pro- 48:9; 18. portion be given, the fourth can be found. Take any three numbers-3, 4, 5, for instance. Then we have 7. Insert the first term in the following proportions- -:35:6; -:48:9; 8. Find a fourth proportional to 2:13, 579, and 3·14159, correct to 5 places of decimals. 9. Divide 100 in the ratio of 3 to 7. 10. Two numbers are in the ratio of 15 to 34, and the smaller is 75; find the other. 11. What two numbers are to each other as 5 to 6, the greater of them being 210? As tests by which the correctness of the processes of addition, subtraction, multiplication, and division may be has not been thought requisite to give answers to the Exercises ascertained, were given in Lessons in Arithmetic, II. to V., it already given in abstract Arithmetic. The answers will, however, be supplied to future examples in concrete Arithmetic. MECHANICS.—IX. THE STEELYARD. ANOTHER weighing instrument is the steelyard, which (Fig. 54) is a lever of the first order, to the short arm of which is attached at b a hook from which the substance, w, to be weighed is suspended, while on the long arm slides the movable counterpoise P. The object aimed at in this instrument being that a small weight, P, should balance a large one, w, on the hook, it is clear that there must be a corresponding disproportion in the arms-the fulcrum, a, must be near one of the ends of the beam. Further, since it is necessary that the steelyard should take an horizontal position, both when loaded and unloaded at its hook, it is essential that its own centre of gravity should lie somewhere on the short arm; for then the counterpoise can balance it when placed in some position on the other arm, such as that marked o, in the figure. For this reason steelyards are made heavy at one end. 13 12 11 10 9 86 54321 To Graduate a Steelyard.-The centre of gravity of the beam being on the hook side of the fulcrum, let it be brought into an horizontal position, no weight being on the hook. Then, as proved in Lessons VII. and VIII., the moment of P is equal to the moment of the beam, that is, the weight of the beam multiplied into the distance of its centre of gravity from a vertical line through the fulcrum, is equal to P multiplied into the distance of o from that line. At the point o so found draw a line across the beam; that line represents the zero division of the long arm, or the division at which P produces equilibrium, the weight on the hook being nothing, cipher, or zero. Р Fig. 54. W Now, supposing that any number of pounds, w, of any substance are hung on the hook, while P is shifted to the left until, as in the figure, the arm is again horizontal, we have P multiplied by the distance of its ring from the fulcrum a equal to w multiplied by ab (this line ab being supposed horizontal), together with the moment of the beam. But P multiplied by the distance of the zero division from a, is equal to the moment of the beam, as already proved; therefore it follows that P multiplied by its distance from the zero division is equal to w multiplied by a b. Now, in order to graduate, let us suppose P one pound and w seven. Then we have in numbers seven times a b equal once the distance of the counterpoise from o, which tells us the exact position of P for 7 pounds on the hook, namely, that you find it by measuring from o to the left seven pieces each equal to a b. Let w be 13 pounds or 3 pounds, then in like manner you measure 13 or 3 pieces equal to a b. It thus appears that the subdivisions for the successive pounds are equal to each other; and we may therefore lay down the following rule for graduating a steelyard: Find first the zero subdivision by bringing the unloaded instrument into an horizontal position by the counterpoise. Put then on the hook, or in the pan, such a number of even pounds as will push the counterpoise to the greatest distance it can go on its arm for even pounds, and divide the distance between this last position and the zero point into as many equal parts as there are then pounds on the hook. The points of division so obtained are the positions of the counterpoise for the several pounds up to that number. must be shifted to the point in which FR is to F P in the proportion of 16 to 1, there being 16 ounces in the pound. This comes to dividing the distance R P (which is known) into seventeen equal parts, as proved in Lesson IV., and taking the first point of division next to P for the fulcrum. If there be 2 ounces in the pan, RF must be to FP as 16 to 2; that is, you divide RP into 18 parts, and take the fulcrum 2 from P. If there be 7 ounces, you divide into 23 parts, and take 7 next to P; and so on for all the ounces from 1 to 16 you may determine the several positions of the fulcrum, marking them as you proceed. If the beam be of any other weight, you follow a similar course, dividing RP into as many equal parts as there are ounces in the sum of the weights of the beam and substance, and counting off as many divisions from P as there are ounces in the latter. From all this it is evident, first, that the subdivisions are not equal to each other, as in the steelyard; secondly, that the operation of graduation is more troublesome than in that instrument. The Danish balance, however, has the advantage of not being encumbered with a movable counterpoise; it carries its own imperial standard weight within itself. M THE BENT LEVER BALANCE. The principle of this instrument, a species of which is largely sold for weighing letters, may be understood by the aid of the accompanying Fig. 56. On an upright stand is placed a quadrant arc, м o, of which c is the centre. Round c as a fulcrum revolves a lever, usually bent, but in the figure represented as formed of two arms at right angles to each other. The arm C B is generally of small weight, being lightly constructed, while the other, CG, called the "index arm," is heavily weighted at its lower end, the centre of gravity of the whole lever thus being nearly at some point, G, on that arm. On some substance, w, to be weighed, being suspended from B, the index moves from its zero point, o, up the quadrant until the weight of the lever acting at G balances w at B, that is, until the moments of these forces are equal, which will be when w multiplied by B H is equal to the weight of the lever multiplied by G I. The divisions of the quadrant corresponding Fig. 56. to the several weights 1, 2, 3, 4, etc., suspended from B are, however, best determined by experiment for each weight. THE LEVER WHEN THE FORCES ARE NOT PARALLEL. In all the cases of levers and weighing instruments we have so far considered, the forces were supposed parallel-in weighing instruments necessarily so. The treatment of the subject is, however, not complete until the condition of equilibrium is de This is the most general case that can occur, and indeed it includes all the others. To clearly understand it, let a lever be defined a mass of matter of any shape which has one fixed point in it. It may be a bar straight, or simply bent, or bent and twisted, or it may be a solid block. So long as there is one point fixed, we may treat it as a lever, that point being the fulcrum. For half and quarter pounds these divisions must be sub-termined for levers the forces acting on which are not parallel. divided; and for greater weights than one pound will balance on the long arm, the counterpoise must be doubled or trebled, etc. If the steelyard be intended for weighing small objects, such as letters, the counterpoise may be ounces, or tenths of an ounce, or even smaller weights, as occasion requires. It thus appears that the construction of a steelyard is very simple, and that any handy person of a mechanical turn may make one of steel or iron, or even of a piece of hard wood, without much trouble. Fig. 55, R THE DANISH BALANCE. This is a species of steelyard, in which (Fig. 55) the fulcrum is movable, and the counterpoise is the weight of the beam acting at its centre of gravity, P, the substance to be weighed being suspended from a hook or placed in a pan, at the extremity, R, on the other side of the fulcrum. The question is, how may you graduate such an instrument? To do this, let us suppose the beam to weigh 1 pound, and that 1 ounce of some substance is placed in the scale; then it is evident that the fulcrum, F, Moreover, the two forces which act on it are supposed to be such that their directions when produced meet, and that their plane passes through the fulcrum. In cases where the two forces do not meet, or their plane does not pass through the fulcrum, there cannot be equilibrium. For example, the outstretched right arm of a man is a lever, of which the fulcrum is in the right shoulder. Suppose, as he stretches it before him in a horizontal position, one force is applied to the hand obliquely from him towards the left to the ground, while another acts horizontally at his elbow towards the right and at right angles to the arm; these forces cannot meet, and therefore would not under any circumstances keep the arm in equilibrium; further. even were they to meet, they would not so keep it unless their plane passed through the fulcrum in the shoulder socket. Sup posing the forces, therefore, to be as described, namely, that their directions meet and their plane passes through the fulcrum. |