shortening the results of previous experiments by a third, and lengthening them by an eighth, * and here was the beginning of sorrows. * # The attempt (beyond these three steps) at the division of the "Canon"-in other words, at the division of a string into the lengths which produce the sounds that make music in a single key-was a failure." The experiments of modern philosophers have been rewarded with the discovery that a musical string divided in the proportions given underneath will produce the notes of the scale as Let it be noticed that the figure 1 stands for there described. the whole length of the string, whether a foot, a yard, or any other measure, and whatever sound (in pitch) it gives-that sound being taken for the key note-Doн. It may also be mentioned that the same numbers denote the comparative lengths of organ pipes capable of sounding the corresponding Perhaps these proportions will be better understood by the annexed diagram. A single string thus stretched and used for these experiments is called a monochord. If the student is of a mechanical turn, let him make one and verify the measurements here given. Let him suspend a board of four or five feet in length against a wall. To the upper part of this board fasten the end of a pianoforte-wire or other musical string which is of the same thickness throughout. Let the wire pass down the face of the board, over a firm wooden bridge, an inch or so high, and close to the top, and over a movable bridge at the bottom; and let it be kept stretched by a heavy weight. Set your movable bridge (which the weight will keep in its place) at the bottom, marking the spot, and take the sound of the whole string, by the help of a fiddle bow, for your Doн, or key-note. Then (having properly measured and marked the board) move the bridge to the other divisions, sounding each note as before. It may be well to mention that Colonel Thompson maintains, and with good show of reason, what he calls the "duplicity" of RAY and TE. They are sometimes sounded by good singers and violinDO players a very small degree lower than their usual position given above. These experiments will fix in your mind a clear notion of the scale. TE LAH If our arithmetical friend will now work a few sums in proportion, he will be able to show the value of the intervals between the several notes of this scale. Thus the vibrations of Dон differ from those of RAY, in being three less, and (three being one-ninth of twenty-seven) Doн has therefore only eightninths of RAY's vibrations. The same proportion will be found between FAH SоH, and LAH TE. These intervals are called the "great tones." The proportion of RAY ME, and of SOH LAH is nine-tenths. These are the "small tones." The proportion of ME FAH, and of TE DOH, is fifteen-sixteenths. These are called semitones, or, more properly, Tonules. If you calculate from the length of the string given above you will find still the same proportions existing. Let our arithmetical friend reduce these "ratios," or proportions, of the three intervals in the scale to fractions having a common denominator. They will then stand thus:--THE GREAT 1280 TONE = 1440 THE SMALL 1296 1440 THE 1350 = 1440 Now this evidently means that the lower note of the "great tone" has 1,280 vibrations, while the higher note has 1,440, and (as the lengths of string are in inverse proportion to the vibrations) that it takes 1,440 degrees of the string, while the higher takes only 1,280 such degrees. Therefore the proportional difference between them, whichsoever way you look at it, is one hundred and sixty degrees. In the same way you will find that the difference between the two notes of this small tone" is one hundred and forty-four degrees, and that the interval of the "tonulo" is ninety degrees. The degrees in cach case are of similar value, all measured on the same scale (common denominator) of 1,440 degrees. We may therefore treat them as belonging to one scale, and adding three "great tones," two "small tones," and two "tonules" together, we shall obtain a perfectly measured scale of 948 degrees. As all these numbers, It will be well for you to understand the con- however, will divide by 2, retaining, of course, the same pronection between these musical notes and the vibra- portion to one another, it is better to regard the scale as comtions of the sonorous body which produces them-posed of 474 degrees, containing three "great tones" of 80 Son whether that body be the string of a violin, the air degrees, two "small tones" of 72 degrees, and two "tonules" in an organ pipe, a small plate of glass or metal, or of 45 degrees, and this is the smallest perfect measurement of the "chorda vocales"-the vocal chords of that the scale in plain figures. But if the pupil will go one step FAR Wonderful little box instrument, called the " larynx," further, and divide each of these intervals by nine, he will see which you can feel in your own throat. Sounds how we obtain the proximate scale of fifty-three degrees. The ME produced by irregular vibrations are not musical. tonule will be exactly 5 degrees, the small tone exactly 8 They form the " roar, rattle, hiss, buzz, crash," or degrees, and the great tone only one-ninth of a degree less than some other noise. But sounds produced by equal 9 degrees. Adding these together, as before, you will have the and regular vibrations are musical. "That musical "Index scale," as Colonel Thompson calls it, "of fifty-three," notes are produced by a rapid succession of aerial and you will see that it is three-ninths or one third of a degree impulses at equal intervals, is very clearly illus. too large. We strongly advise the pupil to construct a monotrated by an instrument called the syren, the in- chord," and try for himself whether this is not in truth an DOH vention of Cagniard de la Tour. A blast of air is accurate description of that scale of related notes which God has forced through a narrow aperture in a pipe; and a made most suitable to human ears and souls. All the books of flat circular disk, perforated near its circumference science are agreed that it is; and experience bears the same STRING OF with a number of small holes equidistant, and in a testimony. It is the more important that you should undercircle concentric with the disk, is so applied to the stand these points, because the true scale is dreadfully abused pipe, that the blast is interrupted by it, excepting by the common keyed-instruments. Many of these are tuned by when one of the holes in the disk is opposite to that of the pipe; what is called "equal temperament;" that is, the scale is and when the former is made to revolve rapidly, the resulting divided into twelve equal semi-tones, and it follows that the aerial impulses cause a series of isochronous vibrations that pro- tones are all 79 degrees (of the perfect scale of 474), while they duce a musical note, and the corresponding number of its ought to be sometimes 80 and sometimes 72 degrees! and the vibrations can very easily be computed, from knowing the tonules (semitones) are both 39 instead of 45!! They might as number of holes and of revolutions of the plate. The results well cut down the fingers of a statue to "equal temperament!"' obtained by this instrument agree exactly with those found by Human ingenuity will surely deliver us soon from this me other methods." The more rapid the vibrations of the sonorous strous distortion. You will understand now why it is so body, the more "acute" (shriller, or higher) the note produced. pleasanter to sing "without the pians." 1 RAY I IVISIONS A MONO- 66 4 S = FAH ME RAY Don Dou LE NE RAT DоH TE1 fers from the present only in this and ret bow great the difference! Learn the upper and the lower "parts." FAR ME RAY DOH RAY DOп TE, DOH we them at every mistake or uncertainty striking the A and beginning again with great patience. When you the chant from the modulator by memory, then learn sal of the syllables, the words "One, two, three, v far, three, four, five, six," still pointing to the right www.on the modulator. It may be well for you now to learn the second be of notes (to be sung by another voice along with 1: re he' as you learnt the first, and not, for the present, Sapt the words But if you wish to use the words, then first mx iu to sing the words "Trust in the Lord with" on the single To do this with distinct utterance, you should divide the note (in your mind) into two equal beats or "pulses" of time (you can mark them by beating on the table with your hand); and then the words "Trust in the" will go to the first beat, and "Lord with" to the second. The large dot above the line shows this division. The words "all thine heart" fall easily to their right notes. To the note Soн you will sing the words" and lean not unto thine." "And" is scarcely heard. Dividing Soн, like the other "reciting note" ME, into two beats (and reciting notes of a chant like this may be divided into as many beats as you please), you will have the words "lean not unto" to the first beat, and "thine" alone filling the second. The word “own” you perceive is "slurred" on to two un notes. Take care not to sing the syllables “standing" quickly and sharply. Let them take as much time as the syllables "under" in the same word. The second verse of words, printed underneath, is divided on the same plan as the first. The double bar, you will observe, separates the words of the ". reciting note" from those of the "cadence" (as it is called) of the chant. On the upper "staff" you miss the square note for Doн, because the tune begins on ME, and Doн does not occur in the "air." The place of Doн, however, is in the first space, reckoning from the bottom of the staff. On the lower staff it was necessary to make an additional line to carry Soн. This is called a "ledger line." nut COPY-SLIP, NO. 17.-COMBINATION OF THE LETTERS n, u, t. nit COPY-SLIP, NO. 18.--COMBINATION OF THE LETTERS n, i, 1. tun COPY-SLIP, NO. 19.-COMBINATION OF THE LETTERS t, u, n. LESSONS IN PENMANSHIP.-VI. In our last lesson we showed our learners how to make the letters m and n by combinations of the top-turn and the topand-bottom-turn. In the present lesson we set before them five Copy-slips for practice, comprising various combinations of the letters m and n with the letters i, u, t, and 1, which they learned to form from the simple bottom-turn. Doubtless many of those who have been endeavouring to teach themselves the art of Penmanship by the aid of our lessons, are now beginning to see that, after all, it is not such a difficult matter to learn to write, since by learning to form only three strokes of a very simple kind, they have found that they have acquired the power of writing six out of the twenty-six letters that form the alphabet. They will soon see that this is by no means the utmost limit of their progress, when in the course of future lessons they discover how many letters there are into whose composition these three elementary strokes also enter. In preparing paper for copying all our elementary copy-slips in large text, the learner must not omit to insert the diagonal lines as they appear in Copy-slips Nos. 1 to 6 inclusive, as long as he finds them absolutely necessary for the regulation of the slope of his letters, and to enable him to preserve a proper distance between them. To save trouble in ruling sheet after sheet of paper with horizontal and diagonal lines at the proper distances from each other, and to save expense as well, the learner might rule with a sharp-pointed steel instrument, such as a bradawl, one side of a cheap slate with sets of lines similar to those in our copy-slips, taking care, however, to leave a space of one inch and a quarter between each set, to enable him to add the extra horizontal lines that will be required when he begins to make looped letters, and letters such as p and q, that extend below the lower of the two horizontal lines that contain what we have called the body of the letter. A slate thus prepared will be found useful for acquiring facility in forming the various letters, but the learner must by no means omit to write on paper as well. (las, Reif, ripe. --nor. Ex Alberner Puffel ist schen, und Your silver spoon is beautiful, micin eiterned Deffer ist schwer. Suter Beinerne Tisch ist schwer. 248 Velen kes Sintes ist ein gettner 18 nicht ein eisernes Schiff dauerhaft? dream. Is not an iron ship durable? 1. Haben Sie mein reifes Obst? 2. Nein, ich babe Zbr silbernes Obi meffer, und Ihr alter Freund, der Lehrer, bat das reife Obf. 3. Haben ie meinen filbernen Bleistift? 4. Nein, ter gute Sebrer bat ibn. 5. Dat ter alte Koch meinen bölzernen Tisch? 6. Nein, der Tischler bat ihn, aber ter Koch bat einen marmornen Tisch. 7. Hat er auch ein bölzernes' #aß 8. Ja, und dieser Aleisige Schüler bat ein schines, bleiernes Tinten fasi 9. Hat er auch einen üldernen Becher? 10. Ja, und er bat auch einen fupreinen Kefel und einen ersernen Mörser 11. Haben Sie das neue Meffer meines jungen Freundes? 12. Nam is babe ein neues Messer von bem auten Kaumanne 13. Hut ve Anšize Shüler das gute Buch des alten Freundet, eder den übernen Büntet setnes guten Betters? 14. Gx bat weder ein extes Das rod ever. Übernen Sieht—er bat nur roth-er, to or for red; When preceded by either of the articles, or by any one of the adjective pronouns (see lists Sect. IX. and X.), the adjective terminates in the nominative and accusative as in the old declension, but in the genitive and dative in the letters cn. Thus : N. tie gut-e, the good; . der gut-en, of the good; meine alt-e, my old; meiner alt-en, of my old; meiner alt-en, to my old ; meine alt-e, my old. I. The personal pronoun Sie (you) is always written with a capital initial, while fte (she or her) is only thus written at the beginning of a sentence. Hence in writing, no ambiguity can arise. Ex., 3ch sebe Sic, I see you; ich sehe sie, I see her. When Sic is used in the nominative, the form of the verb determines the person. Ex., Sie sehen ihn, you see him; Sie sicht ihn, she sees him. Whether, however, ic (when in the accusative) stands for you or her, can only be determined by the context. The orthography of the possessive pronouns 3hr (your) and ihr (her) is also identical, and, in speaking, is liable to equal ambiguity. Thus, 3br Buch ist groß, may signify, your book is large, or her book is large; and, 3ch habe ihr Buch, may mean, I have their accusative, and of ibr in all the cases, must of course, when book, or I have her book. The significations of fie in the spoken, be determined by the connection. (See Declension, Sect. XVIII.) einen Fölzernen Miecht 13. Bet var ferie (Section IX. 2) Kennel, Der Bruter hat das Buch ter The brother has the book of EXERCISE 13. Schwester Der Vater giebt der Tochter en Der Hut meiner Mutter ist schön. 1. Has she my npe fruit 2 The old book has my SPOTION -THE FEMININE GENDER OF ARTICLES, The acquies in the framine singular are declined thus :--- Gouine? the sister. The father gives the daughter a book. The hat of my mother is beautiful. Where is your cousin's watch? It is in her mother's hand. EXERCISE 14. 1. 34 vie lunge Schwester dieser jungen. Tame in Teutickland? 2. Nein, se it in Frankreich aber ihr Bruter it in America 3. We ist meine neue, goldene Herer? 4. Ihre junge Freundin, Fräulein S. bat ñie (Sect. XVIII. 3.) 5. Hat Ihre Mutter die ichine Seite Ihrer Tante? 6. Ja, und auch die schine, feine Seinwand 7. Be in Ihre geldene Brille ? 8. 3 babe keine geltene Brille 9. Haben Sie eine filderne, eder eine goldene Uhr ? 10. It babe eine saberne libr. 11. If fie eine gute Übe? 12. Ja aber e vit mucht icht ichin. 13. Be in Ihre Ube? 14. Sie ist in meiner Übrtaiche 15. Hat Ihre Schweiter eine geltene Ubr? 16. Ja, wat nie hat auch eine schöne, goldene Kerte. 17. Be it meine neue Serre 18. 3d babe sie aber sie in nicht sehr scharf. 19. Be it Ihre Swenter? 2). Sie in bei der Sect. XVII. 3.) Nutter in ter The pup having now had in due course all the forms of the article in the singular, may note, that like tiefer (which differs from the definite article only in having es instead of as in the nom, and ace, neuter, Sect. VII), are declined all the words in list 2, Sect, IX. ; and that likə tin, are inflected all those in the bactbef. 21. Seit meine alaserne Samze * 22. Ich babe nie. list, ein, mein, sein, etc., Seet. X. 23. Wer hat mein neues Coerngias 24. 3d habe es und Ihre neue Powinine nouns are in the singular indeclinable; as, nom. Die Brille Serre (the silk); gen der Seite; dat, der Seite, ace, die Seite The adjective in the limizëse s'agular has two forms. When it stands alone, or unaffected by a preceding word (§ 29), the nominative and accusative end in e, the genitive and dative in er It is then and to be of LESSONS IN ARITHMETIC.—VI. ABRIDGED METHODS OF MULTIPLICATION AND DIVISION. 1. THE methods of multiplication and division explained in the previous lessons are those ordinarily employed; and the learner must make himself perfectly familiar with them before proceeding further. These processes, however, in particular cases, can often be materially facilitated by various artifices. Some of these shorter methods we subjoin, not only because they are useful in themselves, but because they are valuable as exercises, in explaining the fundamental principles of arithmetic. 2. Any number which is formed by multiplying two or more numbers or factors together is called a composite number. 352 It has already been explained in a former lesson that the same numbers multiplied together will give the same product, in whatever order the multiplication is effected. Hence, to multiply any number by one which is compositei.e., which is composed of several factors-we have only to multiply the number first by one factor, the result by another factor, and so on. Thus, to multiply 352 by 28, since 287 X 4, we can perform the operation as indicated in the margin. 7 2134 4 9858 = 23 × 352 4, 8, 16, 18, 20, 24. 5, 27, 32, 36, 40, 48. 6, 12, 28, 54, 72, 84, (2.) What will 24 horses cost at 74 crowns apiece. (3.) What will 45 hogsheads of tobacco cost, at 128 crowns a hogshead? (4.) What will 54 acres of land cost, at 150 crowns per acre? (5.) At 118 shillings per week, how much will it cost a family to board 49 weeks? (6.) If a man travel at the rate of 72 miles a day, how far will he travel in 64 days? (7.) At 163 crowns per ton, how much will 72 tons of lead cost? (8.) What will 81 pieces of broadcloth cost, at 245 shillings apiece? (9.) What will 84 carriages cost, at 384 crowns apiece? (10.) What will a railway 132 miles in length cost, at the rate of £1,960 a mile ? (11.) If I can walk a mile in 16 minutes, how long will it take me to walk 374 miles ? 3. Similarly, it will be seen that to divide by any composite number, we have only to divide by one factor, then divide the quotient by another factor, and so on. 179856 41408 Thus, to divide 9856 by 28, arrange the 352 Answer. process as indicated in the margin. In this case there is no remainder. But suppose it be required to divide 9873 by 28. (2.) How many acres of land, at 35 crowns an acre, can you buy for 4650 crowns. (3.) A man divided 837 crowns equally among 27 persons, who belonged to three families, each family containing nine persons: how many crowns did each person receive? (4.) A man bought a quantity of clover seed amounting to 507 pints, which he wished to divide into parcels containing 64 pints each: how many parcels can he make ? 5. Multiplying and dividing by powers of 10, and by numbers ending in any number of ciphers. The products of two tens, three tens, four tens, etc., are called respectively the second, third, fourth, etc., powers of 10. They are 100, 1000, 10000, etc. Thus, the second power is 1 followed by two ciphers, the third 1 followed by three ciphers, and so on; the number of the ciphers in each case being the same as that of the power. It has been already explained that to multiply by 10, or any power of 10, we have only to annex to the multiplicand the number of ciphers corresponding to the power. Thus, 345 multiplied by 1000 is 345000. If any number of the right-hand figures in the multiplier be ciphers-as, for instance, in 75000-then, as we have already Seen in Lesson IV., Art. 5, we need only multiply the multiplicand by 75, and annex to the product the same number of ciphers, in this case three. EXERCISE 12 (1.) Work the following examples in multiplication: 1. 153486 × 10000. 2. 3120467 x 100000. 3. 52690078 x 1000000. 4. 689063157 x 10000000. 5. 4946030506 × 100000000. 6. 87831206507 × 1000000000. 7. 67856005109 x 10000000000. 8. 14376 x 25000. 9. 350634 x 410000. 10. 4630425 × 6200000. 11. 2370000 × 52. 12. 48120000 × 48. 13. 356300000 × 74. 14. 1623000000 × 89. 15. 540000 × 700. 16. 1563800 × 20000. 17. 31230000 x 120000. 18. 5310200 × 3400000. 19. 82065000 x 8100000. 20. 210909000 x 5100000. (2.) What will 10 boxes of lemons cost, at 63 shillings per box? at 100 bushels per acre? (3.) How many bushels of corn will 465 acres of land produce, (4.) Allowing 365 days for a year, how many days are there in 1000 years? (5.) How much will 50 hogs weigh, at 375 pounds apiece? (6.) If 1 barrel of flour weighs 192 pounds, how much will 500 barrels weigh? LESSONS IN GEOMETRY.—III. INSTRUMENTS USED IN PRACTICAL GEOMETRY. IN the operations of practical geometry, a case of mathematical instruments must be considered as an essential requisite. These instruments vary in number and quality, according to their price. Some are made of wood, bone, and ivory-as rulers and scales; cthers are made of brass and steel, German |