metic, the one treating of quantity continued, and the other of quantity dissevered. Mixed mathematics hath for subject some axioms or parts of natural philosophy, and considers quantity determined, as it is auxiliary and incident unto them; for many parts of nature can neither be discovered with sufficient subtilty, nor explained with sufficient perspicuity, nor accommodated unto practice with sufficient dexterity, without the aid and intervention of the mathematics; of which sort are perspective, music, astronomy, cosmography, architecture, engineering, and divers others. In the mathematics, I can discover no deficiency, except it be that men do not sufficiently understand the excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be dull they sharpen it, if too wandering they fix it, if too inherent in the sense they abstract it; so that as tennis is a game of no use in itself, but of great use in respect that it maketh a quick eye, and a body ready to put itself into all postures, so in the mathematics that use which is collateral and intervenient, is no less worthy than that which is principal and intended. And as for the mixed mathematics, I may only make this prediction, that there cannot fail to be more kinds of them as nature grows farther disclosed." The admirable distinction between pure and mixed mathematics, which is drawn by the celebrated Lord Chancellor, in the preceding extract, remains correct to the present day; but the wonderful verification of his own prediction, has added in an extraordinary degree to both kinds. Hence, under pure mathematics are now included, arithmetic, algebra, called by Newton universal arithmetic, logarithms, or exponential arithmetic, and the theory of equations, probabilities, &c. Also geometry, plane, solid, and analytical; trigonometry, or the application of arithmetic to geometry, plane, spherical, and analytical; and the new geometry, or the differential and integral calculus, including the whole theory of curves and curved surfaces. Under mixed mathematics, are included mechanics, which comprehends statics and dynamics, hydrostatics, and hydrodynamics or hydraulics; also pneumatics, optics, heat, electricity, and magnetism; with astronomy, plane, nautical, and physical; the latter being sometimes denominated celestial mechanics. To these may be properly added many other mixed sciences, which perpetually call in the aid of pure mathematics; as geology, geography, geodesy, land-surveying, navigation, civil, practical, and military engineering, life assurances, steam-locomotion by sea and land, &c. It will be the object of the "Popular Educator" to make its readers acquainted more or less with the subjects just enumerated, and their application to the present state of society; and to convey instruction in such a manner, as that any one willing to learn, may do so without the aid of a master, provided he can only read and write. That this can be done, has been often proved by the history of those who have followed" the pursuit of knowledge under difficulties." The example of Edmund Stone may be cited for the encouragement of all. He was the son of the gardener of the Duke of Argyle. At the age of eight years he was taught to read; and, at that of eighteen, he had, without assistance, made such progress in mathematical knowledge, that he could read the works of Sir Isaac Newton. As the duke was one day walking in his garden, he saw a copy of Newton's Principia lying on the grass, and called some one near him to take it back to his library. Young Stone modestly observed that the book was his own. "Yours!" replied the duke; "do you understand geometry, Latin, Newton ?" "I know a little of them," said the young man, with an air of simplicity. The duke was surprised, and having himself a taste for the sciences, he entered into conversation with the young mathematician. He asked him several questions, and was astonished at the force, the accuracy, and the candour of "But how," said the duke, "did you come by the knowledge of all these things?" Stone replied: "A servant taught me ten years ago to read. Does any one need to know more than the letters of the alphabet, in order to learn any thing else he wisnes?" The duke's curiosity was redoubled: he sat down on a bank, and requested a detail of all his proceedings. "I first learned to read," said Stone; "the masons were then at work upon your house; I went near them one day and saw that the architect used a rule and his answers. compasses, and that he made calculations. I inquired what was the meaning and use of these things, and was informed that there is a science called arithmetic. I purchased a book of arithmetic, and learned it. I was told there was another science called geometry; I bought the books, and learned geometry. By reading, I found that there were good books on these two sciences in Latin; I bought a dictionary and learned Latin. I understood also, that there were good books of the same kind in French; I bought a dictionary and learned French; and this, my lord, is what I have done. It seems to me, that we may learn everything when we know the letters of the alphabet." The duke highly pleased with this account, brought the young man out of obscurity, and provided him with an employment, which left him leisure to apply himself to the sciences. Edmund Stone, F.R.S., was the author of a new Mathematical Dictionary, published in 1726; a treatise on Fluxions and Fluents, 1730; the Elements of Euclid, 1731; and a variety of other useful works, besides papers in the Philosophical Transactions. With the example of Stone before us, we propose to begin our Mathematical Lessons with one on Arithmetic, which will be followed by one on geometry; and so on alternately, until we come to the applied sciences. This will leave a fortnight's interval between each lesson; and we sincerely hope that many of our readers will avail themselves of this interval to master each lesson in succession. In order to keep up the spirit of inquiry, and to bring out the latent abilities of those who nobly and generously aspire to be learned, we shall insert in each number "Queries and Problems," relating more or less to the subjects in hand. In general, these shall be proposed one week and solved the next. It is hoped that these also will afford both pleasure and instruction to our readers in general. LESSONS IN ARITHMETIC.-No. I. arithmeo, to count, is properly applied to the science of numThe term Arithmetic, which is derived from the Greek verb bers. To a certain extent, this science must have been coeval with the history of man. As an art, it is indispensable in daily business; and the man who is best acquainted with its practical details has always the preference in every mercantile fold-to develop its principles as a science, and to show the establishment. Our object in these lessons shall be twoapplication of its rules as an art. For this purpose, it will be necessary to begin with the first principles of Numeration and Notation, and to give such rules as will enable any one to read and write a given number correctly. After the tys, or multiples of ten, by the first nine names, which are again combined by addition with the same names, so as to reach from twenty-one up to ninety-nine, a new series commences by the adoption of a new name for ten times ten, or tenty-viz., a hundred. This enables us, by the help of previous combinations, to, reach from one hundred and one up to nine hundred and ninety-nine, when a new series commences by the adoption of the new name, a thousand, for ten hundred. After this, no new name occurs till we reach a million, or a thousand times a thousand. It is true, that we have adopted from the Greek the term myriad, which signifies ten thousand, and which might properly commence a new series; but it has not been admitted into the nomenclature of our system of numeration. If the same process of analogy had been followed out, a new name ought also to have been adopted for a hundred thousand; but this has not been done, evidently for the simple reason that such high numbers were seldom in use, either in speaking or in writing. Names, indeed, would have increased so fast, and their combinations would have become so laborious to remember and to apply, in any language, that the adoption of a conventional system of signs to denote numbers, was absolutely necessary to supply the wants of mankind. Accordingly, we find that at a very early period a variety of signs or characters were invented and employed to denote numbers, and to enable men not only to express very large numbers by a few of these characters, but to make calculations of various kinds, essential both for the purposes of commerce and science. The system of numeration adopted in our language, and explained above, proceeds on the decimal scale of numbers, in which every new name or rank is tenfold, or ten times, greater than the preceding-a system which is evidently founded on the digital structure of the human hand. So natural, indeed, is the practice of counting by the fingers, that both learned and unlearned adopt it, whenever any calculation is to be made which does not require the pen. The names of the successive ranks of numbers in their decimal order, to a certain extent, are the following: Decimal System of Numeration. Tens of units. Hundreds of units. Thousands of units. Tens of thousands of units. Hundreds of thousands of units. MILLIONS (thousands of thousands). Tens of millions. Hundreds of millions. Thousands of millions. Tens of thousands of millions. Hundreds of thousands of millions. BILLIONS (millions of millions). Tens of billions. Hundreds of billions. Thousands of billions. Tens of thousands of billions. Hundreds of thousands of billions, &c. In the preceding table, the first six ranks of names are called the first period of numbers; the next six, the second period; the six after this, the third period ; and so on. It will be observed that after the first period of units, the same names are applied, in the same order, to the second period of millions; and then, to the third period of billions. This process is continued, in our system of numeration, and the table may be extended to any length required, by applying the same names to each successive period in order, the names of these periods being as follows: The Higher Periods. QUINTILLIONS (millions of quadrillions). SEXTILLIONS (millions of quintillions). SEPTILLIONS (millions of sextillions). OCTILLIONS (millions of septillions). NONILLIONS (millions of octillions), &c. The preceding system of numeration is that adopted by all the oldest and best English writers on arithmetic; and, up to the last name in the higher periods above mentioned, is capa ble of expressing a number containing sixty figures in the common system of notation. In many of our recent works on arithmetic, the French system of numeration is adopted, which differs very considerably from ours, and which has the merit of greater simplicity to recommend it; but what it gains in simplicity, it loses in power. Instead of dividing numbers into periods of six ranks each, they divide them into periods of three ranks each, the first period being called units; the second, thousands; the third millions; and so on, as in the following table: French System of Numeration. UNITS (ones). This table, when compared with the table of the English system of numeration, will clearly show the difference between the two systems. For example, trillions in the preceding table signifies only millions of millions; whereas in the English table it signifies millions of millions of millions. This comes of the French using the word mille for a thousand, and the word million for a thousand thousand; hence, also, the confusion arising from the similarity of the names. In consequence of the French division of the numeration table into periods of three ranks instead of six, it is plain that with the exception of one period, viz., the thousands, their system is capable of expressing a number containing only half the quantity of figures which the English system can express; and is therefore so much inferior in power. QUESTIONS ON THE PRECEDING LESSON. What is the origin of the term, Arithmetic, and to what science is it properly applied? How many primitive words, as names of numbers, are to be found in almost all languages? How are names obtained for the numbers beyond ten? State the combinations by addition of the primitive names of numbers in English. State the combinations by multiplication of the same primitives. Give an idea of the manner in which the names of numbers, up to one hundred, are filled up. What is the meaning of the name myriad ? What is the name of the next rank after myriads, which, accord ing to analogy, would have required a new name? What method was necessary to denote numbers to prevent the increase of names? What are the advantages of denoting numbers by characters or signs? What scale of numbers, or system of numeration, is adopted in English? What is the origin of this system? State the different ranks of numbers in our system of numeration up to trillions. Mention the names of the higher periods, and states how many ranks each period consists of. Give some account of the French system of numeration. State the difference between the French and English systems and the advantages of each. LESSONS IN FRENCH.-No. I. By Professor LOUIS FASQUELLE, LL.D. IN commencing these French Lessons, we have thought it best to begin with a chapter exclusively devoted to the pronunciation of words, and the variations which are caused in the sounds of vowels and consonants by changes in their relative position. We shall then enter into the construction of the language, and endeavour in as plain a manner as persible to make our readers familiar with its various idiom.s 14 and peculiarities. It will greatly facilitate their progress to have made themselves familiar with the instructions contained in the French Lessons, reprinted from the Working Man's Friend.* We also recommend careful attention in applying the rules and exceptions to the writing and reading of exercises. This is little more than a work of the memory, and an ordinary amount of diligence at the outset will soon enable any one not only to translate but to speak the French language with ease and comfort. First, then, as to the letters and their sounds. THE LETTERS. 9. o nearly like o in rob. Ex. robe, robe; globe, globe; cachot, dungeon; haricot, bean. Ex. dépôt, deposit; prévôt, provost; bientôt, soon; suppot, supporter. 10. ô like o in bone. 11. u. The exact French sound of this letter is not found in English. The position of the lips in whistling, is very nearly the position which they should have in emitting the French u. Ex. urne, urn; lune, moon; but, aim; tribu, tribe; tribut, tribute; élu, elected. 12. à is the u with a prolonged sound. Ex. mare, mulberry ; dú, due; cra, growth; brûler, to burn. 13. y. See 28, y. THE DIPHTHONGS, A vowel surmounted by an accent cannot form a dipththong with another vowel, it must be pronounced separately. Ex. obeir, to obey; deité, deity; réussite, success. Exceptions, où, where-pronounced so; i and e accented (if e follow i) form a diphthong. A vowel surmounted by a diæresis (") follows the above rule. Ex. haï, hated; païen, pagan ; maïs, maize. Exceptions; u followed by ë at the end of a few words, as in ciguë, hemlock, is pronounced like û alone. 16. ai is like a in fate. Ex. j'ai, I have; je ferai, I will make; baie, bay; mai, May; balai, broom. Old names: New names. Examples. ah ah amas. 14. bay say be § ke T barre. cas, cil. day de ¶ dard. a a effet. X x Y y Z z tay u vay eeks e grec zed urne. vase. rixe. yeux. zèle. W, called in French double V, might be added, as many foreign words which have that letter, have been adopted into the French language. THE VOWELS. Vowels are rendered long or short by certain marks placed These marks, which are three in number, are over them. called accents. The acute accent () is laced over the e to give it a sharp or close sound. (See 4, 6.) When the diphthong ai is followed by s, d, or t, it assumes a broader sound, resembling the French è, or ai in the English word, pair. Ex. j'avais, I had; je ferais, I should make; lait, milk; laid, ugly. 17. au nearly like oh in English. Ex. taux, rate; chaud, warm. e preceding au is blended with that diphthong without changing its sound. Ex. beau, handsome : château, castle, tableau, picture; eau, water. 18. ei nearly like a in fate. Ex. beige, serge; neige, snow; seigle, rye; reine, queen; peigne, comb. 19. eu approaches the sound of u in tub. Ex. jeu, play; lieu, place; peu, little; peur, fear; chaleur, heat. Exceptions, in eu, had: j'eus, &c., I had ; j'eusse, &c., I might have; eu is pronounced like u alone. 20. ia nearly like ia in medial. Ex. il lia, he bound; il cria, he cried; dialogue, dialogue. 21. ie like ee in bee. Ex. il lie, he binds; il étudie, he studies; harpie, harpy; mie, the soft part of bread. 22. oi nearly like wa in was. Ex. croix, cross; il boit, he drinks; roi, king. 23. ou like oo in cool. Ex. doux, soft; coup, blow; nous, we; 24. ua The grave accent () is placed over a, e, u, to give to these vowels a grave or open sound. (See 5, è.) 25. ue The circumflex accent (") is placed over a, e, i, o, u, to give to these letters a long and broad sound.‡‡ 26. ui 1. a like a in mass. Ex. face, face; bateau, boat; tableau, picture; patte, paw; malade, sick. 27. uo vous, you; cou, neck. 2. a like a in bar, far. Ex. âge, age; château, castle; pâte, 28. y when initial, when coming between two consonants, or paste; blame, blame; crâne, cranium. 3. e nearly like u in bud, and frequently silent at the end of polysyllables. Ex. le, the; me, me; te, thee; que, that; meuble, piece of furniture; peuple, people; rime, rime. 4. é like a in fate. Ex. été, summer; amitié, friendship; élevé, raised; épée, sword. 5. è like ai in pair. Ex. père, father; frère, brother; mère, mother; elève, pupil. 6. ê nearly like e in there. Ex. rêve, dream; extrême, extreme; crême, cream; crêpe, crape; forêt, forest. 7. i nearly like i in pin. Ex. midi, mid-day; ici, here; fini, finished; credit, credit. 8. f like ee in bee. Ex. ile, island; gîte, lodging; épître, epistle; dîme, tithe; abime, abyss. Published at the office of the "Popular Educator," price 6d. when forming a syllable of itself, has the sound of the French . Ex. style, style; type, type; yeux, eyes; Ypres, Ypres; y, there; between two vowels y has the power of two i's, one of which forms a diphthong with the preceding, and the other with the following vowel ; the syllabic division taking place between the i's. Ex. moyen, means; essayer, to try; nettoyer, to clean; citoyen, citizen; abbaye, abbey; these words are pronounced as if they were written mo-ien, essai-ier, nettoi-ier, citoi-ien, abbai-ie. The words, pays, country, paysage, landscape; paysan, peasant, are pronounced pé-is, pé-isage, péi-san. THE NASAL SOUNDS. The combination of the vowels with the consonant m or n, produces what the French call le son nasal, the nasal sound. The e of the other letters has 30. When the consonant m or n is doubled, or is immedi tse in rose. This accent indicates the suppression of the letter s after the vowel on which it is placed, thus: fete, tête, bête, were formerly written, feste, teste, beste, the's was not sounded, but gave to the preceding vow that prolonged sound now represented by the circumflex accers. ately followed by a vowel the nasal sound does not The words ennui, ennuyer, emmener, enivrer, enorgueillir, form excep tions to this rule. The first syllable of ennui, ennuyer, emmener is nasal enivrer, enorgueillir are pronounced en-nivirer, en-orgueillir. 17 LESSONS IN MUSIC.-No. I. By JOHN CURWEN. helpful, perhaps just then only confusing to you ;-the second that when something is set before you to be done, you will really do it, instead of supposing it to be done, and going on, for only "by doing we truly understand" FIRST PRINCIPLES OF MUSIC. You must allow us to lay before you certain fundamental principles of music itself-of music considered apart from any method of teaching or of writing it-principles which would be true of music, if Guido had never invented the "staff," and if "crotchet," and "quaver," "flat," and "sharp," had never been heard of. We have a friend, who was long persuaded by his relatives, who were all "musical," that he had "no voice." Any innocent attempt of his to unite in the vocal pleasures of the family circle was instantly checked by some compassionate expression or imploring look. He humbly acquiesced in this judgment of his friends, but found it often difficult to resist the sympathy of song, and sometimes startled the singers by adding his honest voice to the closing strain. In public worship, too, no frowns or dissuasives could hinder him from "doing his best" to join in the praises of God. He often wondered how it was that he came to be born with "no voice," especially when he observed that the infants of the present day are so much more highly endowed, every one of them who attends an infantYou know what is the difference between "high" and school apparently taking for granted that he "has a voice," "low" in music. The "squeak" is high, the "growl" is and using it accordingly. As a religious man, also, he could low. Recognise this difference to yourself now by singing not help noticing that one whole book of the scriptures was first a high and then a low note. Between the highest and written for the promotion of public vocal praise, and that it the lowest sounds which the human ear can appreciate, an in abounded in such expressions as this: "Let the People praise definite number of other sounds may be produced. But how, thee, O God, let ALL the people praise thee." The example out of this vast chaos of possible sounds, are the distinct and of Christ and the precepts of his apostles seemed also to set choice notes of a TUNE to start into life and power? The quesforth the same duty. "It cannot be," he sometimes reflected, tion is thus answered. Before a TUNE can be created, a certain "that the Father of all should command us to sing,' in addi- sound, whether high or low in pitch, must be chosen and fixed tion to making melody with our hearts,' and yet give to so as the KEY note (sometimes called the governing note, and in many of his children no voice!" Such thoughts as these led books of science the tonic) of the coming tune. Immediately. him to the conclusion, that it is "no practice" and "no culti-according to those laws of nature by which God has tuned our ears vation," rather than "no voice" and "no ear," with which and souls, six other notes spring forth, at measured distances the majority of men are afflicted. In consequence of this, to from the key-note, claiming the sole right of attendance upon the no small amusement of his musical acquaintance, our it. Let this be clearly understood. Any sound may be taken friend was soon found to have become an attentive and pains- for the KEY-NOTE; and that being fixed, the places of the six taking member of a singing class. He was soon deep in other notes are known. "thirds" and "fifths" and "sevenths," toiling at a series of the most unmusical exercises that could well be invented. But hope sweetens toil,-and the expectation of conquering at last gave to our friend courage and long patience. When sixty laborious lessons, relieved by an occasional song, were over, he made the discovery that he had learnt "a system," that he had gained also some confidence and much command of the organs of voice. But what did he know of music? Could he take the plainest Psalm tune (not in the key of C), unseen before, and sing it? Alas! no. His labour had not been lost, but it had produced small fruit. He could follow the "leader" more promptly and easily, but he could not go without him. There was still an indecision and uncertainty about his endeavours. He could seldom be sure whether he was right or not by half a tone. And many a choice song, and not a few tunebooks, which he had purchased in his hopeful days, lay on his table unenjoyed because of this musical uncertainty in which he was left. Once more, however, our friend has "taken heart," and has promised to follow the course of effort which we shall prescribe,-we, on our part, undertaking that he shall in that case be enabled to sing at first sight by himself, and to make good use of the books on his table. We shall begin at the beginning, however, for your sake, gentle reader,-if you will join him in his efforts. We have no "Royal Road" to music. No worthy attainment is won without labour. But we have a straight and clear road, and that is a great advantage when the common road is very circuitous, and abounding with needless hindrances. We have only two things to ask of you: the first, that you will be content to learn one thing at a time, instead of being impatient for knowledge, not, at the moment. VOL. I. The common human ear throughout the world is pleased when these sounds attend that key-note, and is displeased when other sounds, not holding the same relation to the key-note, and not standing at precisely the same relative pitch, are used in their stead; for even an uncultivated ear would promptly mark the difference between the accurate singer and the inaccurate, between the singer in tune and the singer out of tune. This distinct arrangement of six sounds around a key-note, is called the musical "scale." It may be high in pitch in one tune, and low in another, but the relative position of its notes remains unchanged. These notes may be reproduced in replicates or "octaves" of higher or lower pitch, but they still retain the same relation. Transition, or "modulation" (which will be afterwards explained) may change the key-note in the course of a tune, but the new key-note governs its dependents exactly as the old one did. Every apparent exception only proves the rule. This one scale is the foundation of all music. Some speak of this scale as though it were of human invention; but if so, how is it that every newly-discovered nation is found either using it (if they are musical at all), or possessed of ears which readily approve it? How is it that the Chinese or Indians have not "invented" some other scale? The truth is, some of these nations do omit a note or two, but they do not alter the rest; and when the question is fairly examined, it is found the omissions were caused by their rude and incomplete instruments, rather than by defective ears. Again, let me ask, going back to the time of the ancient Greeks, of whose musical notation there is not a remnant from which we could have copied, how is it that we learn, from their philosophical 2 18 66 treatises, that the scale which the people used was the same as ours? Could not that refined people have invented" something better? Are we not right, then, in calling it the scale of all nations and of all times, the scale to which the ear and soul of man are tuned by the all-wise Creator. DOH' 5 TE 9 When we examine its structure more closely, we find other proofs that it comes from the hand of God. Like many of his works-the rainbow, for instance-it seems to the careless bserver irregular, but discloses a beautiful harmony and The distances in purpose to him who is more thoughtful. pitch (that is, highness or lowness of sound); or, in other words, the intervals between the notes of this scale, are very delicately arranged. In another lesson we shall be able to describe its structure more minutely; but let it suffice for the present to say, that the simplest measurement of the scale in plain figures is that which divides it into Such a division is only inacfifty-three degrees. curate to the extent of being a third of one of these degrees too large. If you will make use of the sol-fa syllables to represent the notes of this scale, DOH standing for THE KEY NOTE OF A TUNE, at whatever pitch it is taken, then the number of such degrees between each couple of notes may be set forth by the figures at the side. Why the scale of music, found most acceptble to human ears, should be thus curiously and delicately formed, and why it does not exhibit greater apparent uniformity, we cannot tell. It is an "ultimate fact" of philosophy, like the structure of the rainbow. We must take it as it is, and reverently Sir Isaac Newstudy the laws of its structure. ton's division of the spectrum into seven colours, bore some analogy to these seven notes; and Mr. Hay, of Edinburgh, has established, in a large work recently published, a clear relationship be LAH 8 SUH 9 FAR 5 ME 8 RAY '9 DOH tween the principles of beauty in the human form, and certain angles founded on the proportions of the musical scale. Doubtless there are in the various departments of nature which shall prove them all to have sprung from one creating certain uniting principles, certain secret affinities of things hand. It may, however, be noticed here, that every note of the scale sounds pleasantly, when heard at the same time with most difficult notes of the scale, more will be said when our the key note, excepting only RAY and Te, and of these, the lessons are further advanced. For the present we wish your attention confined to the three notes, DоH, ME, and Soн, the first, the third, and the fifth. We may call them "the They are the strong notes of the scale, on which, as you will afterwards learn, the others lean. framework of the building." When sounded together they are commonly called the "chord of the tonic," tonic being the Take, then, some low sound of your scientific name for key-note. Chiefly by these notes your voice must be tuned. voice for the key note, or DоH, and try to sing the following exercises, pointing to the notes on the scale given above, as you sing. This pointing on the scale is more important than you would at first imagine. In no other way can you obtain so clear a notion of the relative position of notes. If previously uninstructed, you must ask some musical friend to sing these notes to you, or play them on an instrument for a pattern. Do not, on any account, however, sing with him or let him sing with you. Remember that you are learning to sing alone. Your friend will know what notes to play when you tell him D; F sharp; A; and upper D'; or, if he prefers it, C; E; G; and upper C. You will notice that when a note is repeated in a higher pitch, we put a mark thus-DOH' above it. You need not trouble yourself with the "staff" of five lines at present except to notice that Doн is printed as a square note. EXERCISE 1. DOH ME SOH pattern" NOTE.-Sing these notes first slowly, then quickly, and again with a sound "long drawn out." Do not be disappointed if your friend pronounces you inaccurate in the first and second notes, though they are the easiest. Let him patiently set the " of those two notes again, and, if need be, many times again. Master one note at a time. Some pupils require several lessons, with much patient patterning of the teacher, and much careful listening, followed by vocal effort of the learner, before this exercise is perfectly done. NOTE. You observe the upright bars. Sing the note immediately after them with a stronger accent or force of voice than the others. You notice that two of the notes on the "staff" of five lines are open, and that the names beneath are followed by a stroke of continuance." Sing those notes twice as long as the rest. Sing all these exercises again, while some one else repeats the note Dox for every note you sing. This we call "tolling the ball." |