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Carry 4; then, 6 times 0 being 0, put down the carried 4; again, the product of 6 and 0 being 0, put down 0, as nothing was arried, and proceed. Next say 6 times 6 are 36, put down 6 and carry 3; and the next product b ing 6 times 0, or 0, put down 3 in the next place of figures; last.y, say 6 times 6 are 36, and put down the complete product being the last in the operation. On the abridged process, above recommended, this operation would be very short; thus you say 0, and put down 0; 48, and put down 3: 0, 4, and put down 4; 0, and put down 0; 36, and put down ; 0, and put down 3; 42, and put down 42.

EXERCISES.-1. Find the products of the number 142857 by the nine digits.

2. Find the products of the number 98998, the smallest number contained in the table in Exercise 4, page 55, No. 4, by the nine digits, and you will find these products in the same table.

3. 857142X9; 76876898×2; 101040060 0x7; 79806090×8; and 999999999999 × 5.

DEFINITION. As the cipher or 0, denotes nothing by itself, and is only employed to keep the combinations of the other nine figures in any number, in their proper places according to their rank or value, it is called an insignificant (not-meaning) figure; but the nine digits are called significant (meaning) figures; yet when the cipher, or 0, occurs between any of the other figures, as in the numbers 206 and 3007, it is called, and reckoned by arithmeticians, as significant; but when it occurs at the end of any of the other figures, that is, on their right, it is called insignificant, as in the numbers, 90, 100, and 25000. Ciphers are never put at the beginning of other figures, except in decimal fractions, as shall be explained in future numbers of this work.

If the multiplier to any proposed multiplicand, happens to consist of any one of the nine digits, followed by one cipher or any number of ciphers, then multiply according to the following rule :Rule 2.-Place the multiplier under the multiplicand, so that the significant figure of the multiplier shall stand under the units' figure of the multiplicand; and multiply the successive figures of the multiplicand by the significant figure of the multiplier according to Rule 1, explained above. Then place the same number of ciphers in a line at the end of the product thus obtained, (that is, on the right,) as are contained in the multiplier. The reason of placing these ciphers at the end of the product, is in order that the significant figures of the product may be preserved in their proper places, according to the decimal scale of notation.

EXAMPLES: Multiply 123456789 by 90, and by 90000.

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In the first of these examples, you multiply by 9, according to Rule 1; you then annex to the product four ciphers, because the multiplier contains two ciphers, and the multiplicand two ciphers, thousands; for the multiplicand having two ciphers, expresses so in order to preserve the product in its proper place, as tens of many hundreds as the significant figures denote, and the multiplier produce tens of thousands; and this is manifest from the consideraexpresses 9 hundreds; now, hundreds multiplied by hundreds tion, that ten times a hundred are ten hundreds or a thousand; and ten times a thousand, are ten thousands.

1.

This operation might be further illustrated, by saying that 78912345600 789123456×100; and 900=9x100; therefore X100 x 100 = 7102111104 x 1000071021111040000. This 78912345600 x 900 = 789123456 × 100×9×100=789123456X9 illustration proceeds on two important principles: That, (that is the successive) product of their factors; and 2. That the the product of any two or more numbers is equal to the continued continued product of the factors of any number will be the same in whatever order they may be taken. For the sake of younger readers, we shall give some simpler examples of the application of are 2, 2, and 3; now 12 is produced from these factors by multiplicathese principles. Thus, we know that the simplest factors of 12 tion in any of the following ways: 2x2x3=12; or 2×3×2=12; or 3X2X2=12; here, 12 is the continued product of the three factors, taken either way. In like manner, if we had to multiply 18 by 10, we might separate 18 into its factors 3, 3, and 2 ; and 10 order, thus 3X3X2X2X5=180; or, 3×2×3×2×5=180; or, into its factors 2 and 5; and then multiply these factors in any 2×2×3×3×5=180; or, 5×2×2×3×3=180; and so on, as often as the numbers can be varied as to their position: in all these cases, the product of the factors will be equal to the product of the two numbers,-viz., 18 x 10-180.

ELUCIDATION OF THE PRECEDING REMARKS.

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In the first of these examples, you multiply first by 9, according to Rule 1; you then annex (i.e., join to) to the product one cipher, because the multiplier contains one cipher, in order to preserve the product in its proper place, as the product of 9 tens. In the second example, you follow the same rule, but you annex to the product four ciphers, because the multiplier contains four ciphers, in order to preserve the product in its proper place, as the product of 9 tens of thousands.

EXERCISES: 1. Multiply 76854 by 80; 90763 by 700; and 142857 by 70000.

2. Find the following products: 98998X7000; 692986×80000; and 395992X7000000.

When the multiplicand consists of figures, both significant and insignificant, and the multiplier of one digit followed by ciphers, then multiply according to the following rule :

Rule 3.-Place the multiplier under the multiplicand so that the significant figure of the multiplier shall stand under the first significant of the multiplicand, that is, the one nearest to the right hand, and multiply the successive figures of the multiplicand by the significant figure of the multiplier, according to Rule 1. Then place the same number of ciphers in a line at the end of the product thus obtained (that is, on the right), as are contained both in the multiplier and the multiplicand. The reason of this is, in

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lesson, in order to see if she can answer them; if so, she may then safely proceed to the second lesson, and do the same thing in reference to it; then to a third, a fourth, &c.

R. H.-Bailey's Dictionary was a very good one in its day, and even yet it possesses some claims to the notice of the student. Dr. Samuel as an authority for the meaning of words, especially hard words Johnson's Dictionary, however, is that which is universally referred to

derived from the Latin, of which the author was remarkably fond. For instance, instead of saying "a chain of events," the Dr. would say “a concatenation of events," and yet mean the same thing.

A. Z. B. Y.-The students of the POPULAR EDUCATOR are expected to answer the Problems and Queries in it if they can. If new and ingenious problems, either in Geometry or Algebra, or new and ingenious solutions of old problems be sent, they will be admitted.

J. S. E.-We thank him for his offer, but really we have so much in hand, that we must decline it at present.

PERKINS-We are much pleased with his communication on the descent of nations, and we would insert it at once, but we should like to see his authorities before giving it our imprimatur.

W. R.-We think very highly of his proposal to engage a teacher of French for the purpose of assisting himself and his friends in the study of the lessons given in the POPULAR EDUCATOR and in the WORKING MAN'S FRIEND. If any properly qualified teacher should

tercommunication.

The major being equal to the sum of the minor and remainder, there-apply to us for this purpose, we shall be happy to be the medium of infore the excess of the nines in the one must be equal to that in the other. 1. Place the excess of the nines in the major at the top of a cross, and the excess of the minor and remainder (as if they were items in addition) at the bottom; consequently, the figure at the top and bottom will be the same, if the work is right. 2. Begin with the highest denomination, reducing always the excess to the next inferior Thus, 7+3= 10; the excess 1 reduced to shillings is 20; the excess is 2; then say 2 + 1 + 5 = 8, which reduced to pence is 96; and the excess 6 + 1 = 7, which is the excess of the major.

denomination.

Again, 4+ 8 = 12, excess 3; then 3+2+5 10, excess 1 = 208.: then 2+1+2+3 = 8, and 8 X 12 96, the excess is 6; and 6+612, excess 3; and 3+4=7, which is the excess of the CHARLES CURRIE.

minor and remainder.

Glasgow, April, 1852.

LITERARY NOTICES.

J. E. SMYTH.-The meaning of the exercise in section 2 of the French lessons, which he calls a vocabulary, is to give the student some good practice in learning to pronounce French words aright, or as they ought to be pronounced according to the rules laid down in the words to memory; but they should be read over and over several times first section. There is no necessity for committing such a mass of in order, if possible, to acquire a correct method of pronouncing vowels, this is practised, the student will insensibly acquire a knowledge of a diphthongs and consonants, as well as the peculiar nasal sounds. While great many words, and that almost without effort; but if it be read as a task, it is to be feared that very little knowledge either of words or sounds will be gained. Our reason for placing the sons of Noah in the ethnographic table in the order in which they stand, is simply this, that in Scripture the same order is preserved; and we consider the reason of the order to be such as an attentive reader of the Bible might fairly discover. Shem was the head of the sacred line of descent in which the Saviour was born, and therefore ought to stand first; Ham was guity of a grievous offence towards his father the aged patriarch, and second head of the human race, and therefore ought to stand last; of course Japheth, under such circumstances, must naturally stand N. A. G.-Jos. HOLLINGWORTH-A plan is in contemplation in order to meet their wishes, and those of their friends.

F. T. RAYNER.—He is right; it was a misprint, and has been corW. J-N BELPER.-We are obliged by your courteous note, and hope the defect alluded to will not again occur.

The ILLUSTRATED EXHIBITOR and MAGAZINE OF ART, is in course of issue in Weekly Numbers, price 2d.; Monthly parts, price 9d.; or if containing five weekly numbers, price 11d. The charac-second. teristics of this splendid publication are, that it presents specimens of the various departments of Fine Art, and the highest attainments in Wood Engraving, with a perfection of printing hitherto unattempted in works of so moderate a price. The new Printing Esta-rected. blishment of JOHN CASSELL is now in full operation, so that he can present to the public a work unique in character, and certainly never surpassed in grandeur of design, in execution of details, or in magnificence of illustration, though issued at double or treble the cost. The fourth Monthly Part is just published, price 9d., and contains a Portrait of the late J. M. W. Turner, R.A, drawn by J. Gilbert, and engraved by W. J. Linton; together with upwards of forty other choice illustrations. The first half-yearly volume of this truly National Illustrated Work will be published July 1st, in a style of binding in strict accordance with the character of the work, price 68. 6d. It mays with safety be asserted that this volume will contain a larger number of first-class engravings, executed in the most finished manner, than has ever previously appeared in works of three times the cost.

SCRIPTURE LIBRARY FOR THE YOUNG, in Shilling Volumes.-The first two volumes of this instructive series of works, "The LIFE of JOSEPH," illustrated with sixteen choice engravings and maps, and" The TABERNACLE, its PRIESTS, and SERVICES," with twelve engravings, are now ready.

ANSWERS TO CORRESPONDENTS.

H. A. W. C.-ELIZABETH-ASTRONOMER-MENES-T. J. S.G. S. T.-All we can say is, that everything mentioned by our correspondents will come in its turn.

W. M.-His suggestion will meet with attention. W. JAMES.—In some of the latest works on Chronology, it is shown, by a careful and searching examination of Holy Scripture itself, that the deluge took place 3215 years before the birth of Christ; according to this mode of reckoning, the revolution of Menes took place 433 years after the deluge. For the demonstration of the truth of this chronology, see Professor Wallace's work, entitled "The True Age of the Worli, &c.," published by Smith, Elder, and Co., Cornhill.

J. G. A. S. Professor of Music.-The syllable TE is used instead of SI (pronounced SEE) that we may have a different initial letter from SOL (or SOH) for the simple "Interpreting Notation" which we afterwards use. The great variety of syllables used by different teachers (see prefaces to Hickson's " Part Music" and to Turle and Taylor's "Singing Book") will surely be sufficient excuse for this very slight innovation which Miss Glover has introduced. You have only to stop the sibilant with the tongue at the root of your teeth, and you make it a T. The clefs and flats, and sharps and time signatures, are removed from the staff in the earlier exercises, to prevent the pupil attempting to learn too many things at once. At present he has to do only with the key. note, the chord, and the scale. The key-note is marked for him by a peculiar form. The pitch of the key-note is given him either in the title, thus, "Key D"" Key G," or by the general directions of the note below. The places of the semi-tones are already known to him in all keys. They are between ME FAII and TE DOH. If our correspondent will be pa tient, for a few weeks, he shall be gratified by seeing clefs and time signatures, and flats and sharps gathering in all their ominous dignity at the beginning of the staff. The key of D is preferred to that of c, for the earlier scale exercises, because its compass is more in the middle of the various voices. See Crivelli. Several subscribers to the POPULAR EDUCATOR have expressed a wish to have an edition printed on finer paper, and declared their willingness to pay a higher price for it. To this there can be no pos sible objection should the demand so greatly increase as to warrant the trial of the experiment. Those who wish to have covers can most easily obtain them by purchasing the monthly parts, for which an extra penny is charged, in order to pay the expense of the cover and stitching. Portfolios for enclosing and preserving the numbers contained in a volume are now ready, and may be procured at our office, price 1s. 64. each.

M. L.-She should read the first lesson very carefully over a good many times, and try to remember as much of it as she can; she should then get an intelligent friend to ask her the questions at the end of theinted and Published by JOHN CASSELL, 355, Strand, and Ludgate-bill,

London.-May 8, 1852.

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THE success which has already attended the publication | appear from time to time in our pages. The utility of a of the POPULAR EDUCATOR, demonstrates the readiness sound and practical acquaintance with the harmony of with which the people do appreciate any well-directed sweet sounds, and with the principles of vocal and instrueffort for their improvement and their elevation. It cannot mental music, will not be questioned in our day. It is but be very pleasing to us to reflect that each successive well known also what great advantages would be gained week nearly a hundred thousand families are undergoing a by our handicraftsmen and mechanics, if they were incourse of useful instruction by means of this periodical. structed in the principles of Geometry-a science so necesThe adult as well as the youthful members of each of these sary to the perfection of the practical arts of carpentry, families have something in its pages from which they can masonry, building, and engineering of all kinds; to navilearn; and it is not only among the humbler classes that our gation, nautical astronomy, and architecture, and, in short, work is read and appreciated, but many among the affluent to all the elegant and refined arts of common life. Living welcome its appearance. Sitting in the House of Commons instructions in these branches, added to those given by us, the other evening, a member for one of our large boroughs would create an interest and a desire to learn among our came up to thank us for the publication of the POPULAR working population which would have the happiest effects EDUCATOR, and exclaimed, "I find it invaluable; indeed, on their social and industrial habits, and on the welfare of I have commenced my education over again." all with whom they are connected by the nearest and the dearest ties. To speak of the utility of a knowledge of Arithmetic and English Grammar, and even of the Latin under certain points of view, must appear superfluous, as every one acknowledges the fact; yet how few possess that acquaintance with these branches which would fit them for that position both in business and in society to which, by little patient industry, they might naturally aspire. The barriers to their upward progress are now being rapidly removed; and the efforts of the POPULAR EDUCATOR Will not be wanting in the hour of need.

Encouraged, therefore, by what has already been done, and believing that much more may be accomplished in the great work of education, by building on the simple but solid foundation of having learned to read, we are anxious to develop our views on this important subject. We feel our own deficiencies in many respects, and particularly in not being able to convey all the instruction that is required by means of type only; but we ask, why should not the living voice of the teacher be made to bear upon the subject, and convey at once to the ear of the student of the POPULAR EDUCATOR, any necessary information which we fall short of imparting to him through its pages? Take, for instance, the knowledge of the French language; we cannot, even in the best lesson that could be written, convey all that is required concerning its sounds and pronunciation; but a teacher could do this at once, and with the greatest ease.

We sincerely believe that we shall do more towards cementing the friendship of England and France, and preventing the horrors and atrocities of war, by making the mass of the people able to read and to speak the French language, than could be done by forming battalions of militia-men from our industrial population. We say to the young men of the United Kingdom, follow our advice; only devote as much time to the facilities for the acquisition of this or any other kind of knowledge which we are placing before you, as you do to the pleasures and frivolities of youth, and you will not only realise the truth of what we have said, but you will elevate and improve your condition in life, and secure for yourselves a green and happy old age-loved, honoured, and respected by all.

What we wish to see organised for the instruction of the people, in connexion with the POPULAR EDUCATOR, is the establishment of Public Instructors or lecturers on the various lessons it contains; and, first of all, on the lessons in French, as being at present the most required to render our teaching perfect. Instructions on the other lessons would of course follow if success attended the first experiment. Such instructions might next be given in Music, in Mathematics, including Geometry and Arithmetic, in Latin and English Grammar, and in other subjects as they

VOL. I.

We propose that a room properly adapted for the purpose be selected in any town or district where these instructions are most wanted-that it be engaged for one evening or so in the week-and that the students of the POPULAR EDUCATOR be admitted to the lectures on the payment of one penny each evening, if it is possible to arrange it at so low a figure; that the funds thus raised be appropriated to the payment of the lecturer and other necessary expenses— that in general the lecturer should occupy the first half-hour in explaining the peculiarities of the French language, such as its sounds, its pronunciation, its idioms, &c.; and the next half-hour in answering queries on the subject, put to him by any of the students, for the benefit of all-that none but fully qualified gentlemen be appointed in each department-and that the names of the public teachers, with the appointments for each evening, be announced in the POPULAR EDUCATOR.

In large towns there can be no doubt that this plan would not be the case, and where a great desire for the lectures be found self-supporting; but in places where this would existed, assistance would be given from a fund to be raised by subscription. We have the means of knowing that our publication is read by many wealthy and liberal-minded individuals, who are warm-hearted friends to the cause of POPULAR EDUCATION; and we trust that the mere announcement of our plan will secure their co-operation and assistance. We shall cheerfully subscribe £50 to this fund. A list will be opened, and subscriptions will be received, by JOHN CASSELL, Belle Sauvage Yard, Ludgatehill, London. All subscriptions will be acknowledged, and EDUCATOR; and the amount placed to an account opened the names of the subscribers inserted in the POPULAR with a Bank in London. The fund thus raised will be expended under the direction of a Committee to be specially appointed.

7

LESSONS IN MUSIC.-No. III.

By JOHN CURWEN.

In order to fix in the memory yet more distinctly, the real structure of the scale, it may be well to notice that it is divisible into two similar sets of four notes, each set including two tones, crowned by a tonule. These sets of four notes have been called TETRACHORDS. If the replicato of the key-note is included, you will have Don RAY ME FAH for the first tetrachord, and SoH LAH TE DOH' for the second. These are called disjunct tetrachords, because the tone, between FAH and Soн separates them. If the key-note is made the highest note of the one and the lowest of the other tetrachord, SOH, LAH, TE, DOH, will form the first, and Don (repeated), RAY, ME, FAH, the second. These are called the conjunct tetrachords, because they are joined in the key-note. Take coins or counters to represent the notes, and arrange them on the table, first with the disjunct, and then with the conjunct tetrachords

thus:

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Learn to do this from memory, and, having done it, to name the notes you have arranged. Some scale-makers, beginning upon RAY and ascending to the upper RAY', suppose they have got hold of a different scale, because the tonules (semi-tones) are between the second and third, and sixth and seventh notes from RAY, while they were between the third and fourth, and seventh and eighth from Dou! In the same way, they begin upon ME, and ascending to the upper ME', suppose they have discovered another new scare with its tonules differently placed! And so on, making every note of the scale the beginning of what they strangely fancy to be a new scale! You will be saved from this delusion by simply observing that, whatever note you begin on, the tonules are divided by two tones on the one hand, and by three on the other. This will best appear by your placing the notes in the form of a circle, thus:

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Our pupil will not blame us for having so long drawn his attention to this foundation scale of all music, when he comes to see the importance in his after progress of thus thoroughly comprehending its structure. He will now be prepared to understand the "modulator, or pointing board for teaching tunes." (See the margin.) The middle column represents the seven notes of the scale in their proper order and at their proper distances. The replicates (octave notes) are added, both above and below, with the figures attached to them as already described. It will be seen that, with the exception of the middle octave (eight notes), the initial letters of the solfa syllables alone are used. The side columns (which are but repetitions of the same thing at different heights in pitch) and the additional notes ("accidentals" as they are called) Tv, FI, NE, NI, and Nu, need not be attended to at present. They are only printed here for the sake of completeness. The scale is sometimes called the "common mode" (the common mode in which notes are arranged for a tune), and the word modulate means properly to sing "in mode," or, in other words, to sing correctly in tune." The uses of the modulator are the following:-1. It supplies the learner with a perfect pictorial representation of the notes he is singing, and thus enables him, as he sings and "points," to measure to the eye the exact intervals which the voice is taking. This cannot be done on the staff of five lines, for there is nothing there to

OR, POINTING BOARD FOR TEACHING TUNES.

indicate pictorially the place of the tonules (semi-tones) and it is not easy for the learner to know at all times, from that staff, what part of the scale he is in; a knowledge which every true singer should carry with him, and which the learner cannot escape possessing if he faithfully and constantly uses the modulator. Let him steadily do this for the next twenty of thirty lessons, and he will find that the modulator has become a ready interpreter of the "staff," and a clear, sure light, guiding him through all the maze of flats and sharps, and clefs and keys, and whatever other difficulties may be crowded upon it. 2. It gives to the learner a simple and uniform "language of interval," for DOH being always the key-note, the intervals remain always the same, to THE MODULATOR, whatever pitch the scale may be raised or lowered. Thus the tonules are always between ME FAH, and TEDổн, and the pupil is so accustomed to sing those syllables to that interval, that he would find it difficult to sing them wrongly. This constant use of the syllables in connexion always with the same intervals, helps the mind to re-. call those intervals with great ease. We all acknowledge the power of this mental association of syllable and interval. When we wish to remember some favourite tune, for inask ourselves in aid of memory, stance, how frequently do we

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How is this? It is plain that the first syllables of the hymn or song had so often coexisted in our minds along with the first intervals of the tune, that the one had gained a power to suggest the other. This power of association," proved to be occasionally so useful, we systematise and make of constant use. Several persons, recently made acquainted with this method of teaching to sing, have written to us in this manner: "I was reckoned a very fair sight-singer before I became acquainted with this method, but I frequently, in preparing for our choral meetings, met with passages which I could not conquer without the help of an instrument. I now, however, simply trace out such passages on the modulator, translate them into this accurate and unchanging language of interval, and then it becomes really difficult to sing them wrongly." 3. It facilitates the practice of teaching by patters. This is of great importance. The teacher sings, scitly and distinctly, a short phrase of the tune to be taught. To this vocal pattern the pupils so listen that they may be able to imitate immediately afterwards. There are two mental processes in learning to sing a note. The first is an effort (if we may so speak) of perception in seeking to appreciate clearly the note to be imitated. The second is an effort of will, commanding the organs of voice to reproduce the note thus clearly perceived. The "pattern" cultivates each of these distinctly. It stimulates the pupil to a strong mental effort in endeavouring to bring the ear and the voice to do the mind's bidding. In this mental effort alone consists the real work of learning to sing. That method is the best, therefore, which requires the most of it. One hour's training of this kind a far more effective than five spent in singing with a leader. The teacher also, not singing with his pupils, is better able

LESSONS IN MUSIC.

criticise and patiently correct their mistakes. The pupil who has to teach himself, with only an occasional pattern from some voice or instrument, must make himself thoroughly perfect in pointing on the modulator, and singing those pieces in which he has had the advantage of a pattern, and they will The first sign of intelligence in a help him to the rest. learner is that he knows when he sings wrong. Let him always, in that case, go back to the key-note and chord, and "try again." Many persons have taught themselves to sing in this way, often making mistakes of which they were ignorant for a while, but discovering their error and the means of correcting it, in their efforts to sing some following exercise. A teacher always by our side will, doubtless, save us from many misunderstandings and blunders; but he who cannot enjoy this advantage, may work on sturdily and hopefully Let him remember that his first business is to without one. use the modulator so constantly that it shall become "printed" in the eye of memory.

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This introduces as our next topic that simple way of writing vocal music which we intend to use as the companion and inIt is the interpreter of the more difficult and complex "old notation," of which we hope finally to make you master. vention of an excellent and intelligent lady-Miss Glover, of Norwich-and has been modified and adapted to popular purposes by Mr. Curwen, in his "Grammar of Vocal Music,' Tonic-Solfa Edition of the People's Service of Song," and It consists of the first letters of the solfa other works. syllables, which you have used in learning a tune from the modulator, written down. And, if you have used the modulator till you are able to carry one "in your mind's eye," this simple notation answers the purpose of pointing out the notes on that mental scale. But, let it be remembered that this notation should never be used apart from a perfect modulator either on paper before the learner, or clearly seen in his mind's

eye.

It may be easily noticed that, at certain distances throughThis combination of distinctness and force is and force. out a tune, the voice is delivered with increased distinctness Close observation will enable you to discalled "accent.' tinguish three degrees of accent thus produced-the louder (or stronger), the softer (or weaker), and the medium. Listen to a well sung tune more closely still, and you will find that the accents recur in regular order, and at equal distances of time. Take care to verify all these assertions by singing some (It is somtimes inwell-known tune yourself, or by listening to another. Then remember that-the distance of time from one of the louder accents to the next, is called a MEASURE. of the voice. accurately called a BAR.) The distance of time between any It may also be called a "pulse accent and the next is called an ALIQUOT, or equal part, of the measure. We There are four sorts of MEASURE in common use. The BINARY or Two-PULSE MEASURE contains two aliquots, one having the louder and the other the softer accent. use an upright bar to represent the louder accent, and two dots to represent the softer. The binary measure may, there: | &c. : &c., or fore, be represented thus :| : | : The TRINARY or THREE-PULSE MEASURE contains three aliquots, one of which has the louder and the other two the t may be represented thus:: &c. : &c., or softer accent. : | : : The QUATERNARY or FOUR-PULSE MEASURE is formed from the Binary by changing every alternate louder accent into one of medium force. We represent the medium accent by a shorter bar than that used for the louder accent. then, be thus represented:

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When we remember that to secure this mental modu-:::: lator it is only necessary to learn the proper position of seven notes, the effort does not appear a difficult one; yet, so inrooted is laziness in some people, that we have found many who go on using the solfa syllables to no advantage for years, without taking the trouble to learn this little ten minutes' lesson, which would make those syllables, in connexion with the power of association just described, clear interpreters of music to them. You will perceive, then, that these notes of the new notation do not appear to our own pupils as they would to others, only on one horizontal line, but seem, as they sing them, to rise or fall to their proper places in the scale. Some persons have objected to this marking of the notes by the solfa syllables, saying, "If the old notation must be learnt at last, however difficult it is, because it contains all the stores of classical music, then why not begin with that at once? why teach two notations?" First, because there is really no trouble in teaching the solfa notation; we have seen children in an infants' school use it before they had learnt to read. It was to them, as we have described it, simply the letters from the modulator "written down." Secondly, because the old notation presents such difficulties to the learner as to make it impossible to teach music in any short time by its means alone. Many of the best systems make use of some simpler Mr. Gall, of Edinburgh, Mr. notation to interpret the old. Waite, and some others, make use of a notation by figures. Dr. Bryce, of Belfast, uses both the figures and the solfa syllables. And we have lately learnt that a sort of solfa notation was printed under the notes with some of the very earliest English psalm-tunes. It consisted of the initial letters of the solfa syllables placed under the notes much as we shall use them. Thirdly, because the use of some such new notation is the quickest and most perfect means of gaining a real com mand of the old. Already, by the method which we are now developing, many children in day-schools, in addition to a large number of adults, have learnt to sing "at sight" from

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It is of small consequence what syllables are used for this purpose. A great variety have been used at different times. We have chosen those given above because they are best known, only changing SE into Te for the sake of having a different initial letter from Soн. We have given the English spelling of the syllables instead of the Italian, as we have nothing to do with the Italian language in these lessons.

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THE SENARY OR SIX-PULSE MEASURE is formed from the trinary measure by changing every alternate louder accent : :|: &c. : 1 : 1 : : or | : into a medium accent, and may be represented thus:: 1: : You perceive that these measures often begin on the softer or medium accents, but the imperfect measure is always completed at the end of a tune. Much of the delicacy and expressiveness of music depends on this proper recurrence of accent, sometimes called Rhythm. By neglect of this a properly beautiful tune is often made dull, heavy, and unmeaning, while careful attention to it will give beauty to some of the plainest melodies. Many of our most popular tunes owe their effect almost entirely to Rhythm, and it forms nearly the sole power of such instruments as the drum and the tambourine. It makes even the regulated step of the soldier and the dancer akin to music. The philosophy of the origin of our sense of Rhythm is treated very admirably in the appendix to Dr. Bryce's "Rational Introduction to Music." He shows its connexion with the pulsations of the heart, which are multiples of the respirations of the lungs. "About the commencement of each expiration of the breath, there is one moment at which the effort, whether muscular or elastic, is stronger than at any other time in the whole breathing. This is most * * Between the expira apparent in a person sleeping soundly when the mechanism of the body, not being controlled by the mind follows unceremoniously its own laws. tion and inspiration there seems to intervene a pause during which the lungs are at rest, but during or immediately after great bodily exertion-running for example-this pause diswith a very brief period of rest. The same happens when appears, and exspiration succeeds inspiration immediately or the breathing is impeded by disease. Hence a respiration may be divided into two (Binary) or into three (Trinary) parts. 3rd inspiration. If into two, 1st expiration, 2nd inspiration.' If into three parts they will be 1st expiration, 2nd pause, Rhythm in its fullest sense has a wider range and more Colonel Thompson ("Westminster delicate expression than can be given within the boundaries of a single measure. Review," Oct. 1832) very beautifully describes it thus:"Whoever has been rocked in a boat upon what, in plain prose that besides the petty furrow which lifted its head and stern may be called "the ocean waves" will have been conscious

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