Tue Notes of the SCALE.

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shortening the results of previous experiments by a third, and The following are the results of such experiments as those just lengthening them by an eighth, * * and here was the be- referred to. Arithmeticians may notice that the proportion of ginning of sorrows. * * The attempt (beyond these three the vibrations is inversely as the length of the strings given above. steps) at the division of the “Canon"-in other words, at the But we here print the fractions with a common denominator to division of a string into the lengths which produce the sounds make the relations more obvious. that make music in a single key—was a failure." The experiments of modern philosophers have been rewarded

THE NOTES OF THE SCALE. with the discovery that a musical string divided in the proportions given underneath will produce the notes of the scale as

The proportion of vibrations there described. Let it be noticed that the figure 1 stands for given by each note. the whole length of the string, whether a foot, a yard, or any other measure, and whatever sound (in pitch) it gives—that

While the key note gives the | 21 21 21 24 24 24 24 24

following number. sound being taken for the key note-Don. It may also be mentioned that the same numbers denote the comparative

If our arithmetical friend will now work a few sums in prolengths of organ pipes capable of sounding the corresponding

portion, he will be able to show the value of the intervals be. notes.

tween the several notes of this scale. Thus the vibrations of

Doh differ from those of RAY, in being three less, and (three Name of Notes. Don Ray Me Fau Sou Lan | Te Don'

being one-ninth of twenty-seven) Doh has therefore only eight

ninths of Ray's vibrations. The same proportion will be found Length of string. I il S L 131 2131811

between Fah SOH, and LAH TE. These intervals are called the “great tones." The proportion of RAY ME, and of Sou Lah is

nine-tenths. These are the “small tones." The proportion of Perhaps these proportions will be better understood by the ME FAH, and of TE Doh, is fifteen-sixteenths. These are called annexed diagram. A single string thus stretched and used

semitones, or, more properly, Tonules. If you calculate from the for these experiments is called a monochord. If the student

length of the string given above you will find still the same prois of a mechanical turn, let him make one and verify the

portions existing.
measurements here given. Let him suspend a board Let our arithmetical friend reduce these "ratios,” or pro-
of four or five feet in length against a wall. To the portions, of the three intervals in the scale to fractions having a
upper part of this board fasten the end of a piano-

common denominator. They will then stand thus:--
forte-wire or other musical string which is of the
same thickness throughout. Let the wire pass down THE GREAT 1280 THE SMALL 1296 TнE

1350 the face of the board, over a firm wooden bridge,

the board, over a firm wooden bridge, I TONE 1440 TONE 1440 TONULE 1940 an inch or so high, and close to the top, and over a movable bridge at the bottom; and let it be kept Now this evidently means that the lower note of the great stretched by a heavy weight. Set your movable tone” has 1,280 vibrations, while the higher note bas 1,440, and bridge (which the weight will keep in its place) at (as the lengths of string are in inverse proportion to the vibrathe bottom, marking the spot, and take the sound of tions) that it takes 1,440 degrees of the string, while the higher the whole string, by the help of a fiddle bow, for takes only 1,280 such degrees. Therefore the proportional your Dor, or key-note. Then (having properly difference between them, whichsoover way you look at it, is one measured and marked the board) move the bridge hundred and sixty degrees. In the same way you will find that to the other divisions, sounding each note as before. the difference between tho two notes of this “small tone" is one It may be well to mention that Colonel Thompson hundred and forty-four degrees, and that the interval of the maintains, and with good show of reason, what he “tonule" is ninety degrees. The degrees in each case are of calls the “ duplicity” of Ray and TE. They are similar value, all measured on the same scale (common de

sometimes sounded by good singers and violin- nominator) of 1,440 degrees. We may therefore treat them as DOH players a very small degree lower than their usual belonging to one scale, and adding three “great tones,” two E position given above. These experiments will fix “small toncs," and two “tonules” together, we shall obtain a in your mind a clear notion of the scale.

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 com

tions 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 “ chordæ vocales"-the vocal chords of that the scale in plain figures. But if the pupil will go one step 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 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 aërial 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 "mono

trated by an instrument called the syren, the in-chord,” and try for himself whether this is not in truth an 1,4-Dol vention of Cagniard de la Tour. A blast of air is accurate description of that scale of related notes which God has

a forced through a narrow aperture in a pipe; and a made most suitable to human ears and souls. All the books of I IVI910XS

flat circular disk, perforated near its circumference science are agreed that it is; and experience bears the same STUIEG OF with a number of small holes equidistant, and in a testimony. It is the more important that you should underAXGSa circle 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 “cqual 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 arial 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 394 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 mon. other methods." The more rapid the vibrations of the sonorous strous distortion. You will urderstand now why it is so body, the more “ acute” (shriller, or higher) the note produced. pleasanter to sing “ without the piani."



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L ! exercises, are to be sung twice as 12. 1 2.

Dotes; and the open notes without a

the 17 t his exercise, are to be twice as long

e 1 5 a Tuis relative length does not, however, Le mere 2 de Tome An open note in one tune may e >> Res t note in another, and a black note in C D I re g open note in another. Let it, how.

be much better for the learner not ca present to the old " notation ” (way of . I N e s ti as placed between brackets. He SE : La Tith the potation of music, when he

de p u bove attention to music itself. Sing S

e s sad never leave an exercise until -I * s :xcel a memory, pointing on the

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L :bes all a rect. thy paths -e

me and he st every mistake or uncertainty striking the .478 m deginning again with great patience. When you

as the cast from the modulator by memory, then learn w p o stal of the syllables, the words “ One, two, three,

on a three, four, five, six," still pointing to the right . V a the medalstor. It may be well for you now to learn he

me of notes (to be sung by another voice along with Et ve Imel as you learnt the first, and not, for the present, N

the unds But if you wish to use the words, then first chem Nam e the words “ Trust in the Lord with " on the single

elere ve To do this with distinct utterance, you should divide

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the note (in your mind) into two equal beats or “pulses” of notes. Take care not to sing the syllables " standing" quickly time (you can mark them by beating on the table with your and sharply. Let them take as much time as tho syllables hand), and then the words “ Trust in tho” will go to the first “under" in the same word. The second verse of words, printed beat, and “ Lord with” to the second. The largo dot above the underneath, is divided on the same plan as the first. The line shows this division. The words “all thine heart” fall double bar, you will observe, separates the words of the “reeasily to their right notes. To the note Son you will sing the citing note" from those of the "cadence" (as it is called) of tho words " and lean not unto thine.” “And” is scarcely heard. chant. On the upper “staff” you miss the square note Dividing Son, like the other “reciting note" ME, into two for Don, because the tune begins on ME, and Don does not beats (and reciting notes of a chant like this may be divided occur in the “air.” The place of Don, however, is in the first into as many beats as you please), you will have the words space, reckoning from the bottom of the staff. On the lower " lean not unto" to the first beat, and “thine” alone filling the staff it was necessary to make an additional line to carry Son. second. The word "own" you perceive is "slurred” on to two This is called a " ledger line.”

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In preparing paper for copying all our elementary copy-slipg

in large text, the learner must not omit to insert the diagonal Ix our last lesson wo showed our learners how to make the lines as they appear in Copy-slips Nos. 1 to 6 inclusive, as long letters m and n by combinations of the top-turn and the top as he finds them absolutely necessary for the regulation of the and-bottom-turn. In the present lesson we set before them five slope of his letters, and to enable him to preserve a proper Copy-slips for practice, comprising various combinations of the distance between them. To save trouble in ruling sheet after letters m and n with the letters i, u, t, and l, which they sheet of paper with horizontal and diagonal lines at the proper learned to form from the simple bottom-turn. Doubtless many distances from each other, and to save expense as well, the of those who have been endeavouring to teach themselves the learner might rule with a sharp-pointed steel instrument, such art of Penmanship by the aid of our lessons, are now beginning as a bradawl, one side of a cheap slate with sets of lines similar to see that, after all, it is not such a difficult matter to learn to those in our copy-slips, taking care, however, to leave a space to write, since by learning to form only three strokes of a very of one inch and a quarter between each set, to enable him to add simple kind, they have found that they have acquired the power the extra horizontal lines that will be required when he begins di writing six out of the twenty-six letters that form the alphabet. to make looped letters, and letters such as p and q, that'extend They will soon see that this is by no means the utmost limit of below the lower of the two horizontal lines that contain what we their progress, when in the course of future lessons they dis. have called the body of the letter. A slate thus prepared will be cover how many letters there are into whose composition these found useful for acquiring facility in forming the various letters, three elementary strokes also enter.

i but the learner must by no means omit to write on paper as well.


D. Gut-er, to or for good;

roth-er, to or for red;
. Out-e, good;

roth-e, red. 35619 -97,LT10 OP ADJECTIVES DENOTING ATZRIAL.

When preceded by either of the articles, or by any one of the

adjective pronouns (see lists Sect. IX. and X.), the adjective 2.14" *** tang til material of which & thing is made

terminates in the nominative and accusative as in the old an wat je ming tonouns the letters 11, en, or ern. Ex.,

declension, but in the genitive and dative in the letters en. .8.5 171 an athero); Gult, golten (gold, golden); Blei,

Thus: 91567


. Il the root vowol boa, o, or u, it is STILII. magos to its corresponding Umlaut, as : - (Wlas, N. tie gut-e, the good;

meine alt-e, my old; : to su A glamu); Hols, hölzern (wood, wooden). (See

G. der gut-en, of the good ; meiner alt-en, of my old ;

D. der gut-en, to or for the good; meiner alt-en, to my old; VOCABULARY.

A. rie gut-e, the good;

meine alt-e, my old. Dostupcu v muy tenker. Koch, m. cook. Reif, ripe.

I. The personal pronoun Sie (you) is always written with a 'T M A · Kupfern, ooppor. Silbern, silver.

capital initial, while sie (she or her) is only thus written at the pow . til, i Marmorni, marble. | Tin'tenfaß,n.inkstand. beginning of a sentence. Hence in writing, no ambiguity can Her, , Mein, my Tisd, m, table.

arise. Ex., Ich sehe Sie, I see you ; ich sehe sie, I see her. When Be sur barrel, cank. Derfer, m. mortar. Tischler, m. joiner. Sie is used in the nominative, the form of the verb determines 4 . gent. Obst, n, fruit.

Better, m, cousin.

the person. Ex., Sie sehen ihn, you see him; Sie sicht ihn, she %,h, winden, Duftmesser, n. fruit. Weder-noch, neither sees him. Whether, however, Sie (when in the accusative) stands Awidin kettl, boilor. knifo.


for you or her, can only be determined by the context. The RÉSUMÉ OF EXAMPLES.

orthography of the possessive pronouns Ihr (your) and ihr (her)

is also identical, and, in speaking, is liable to equal ambiguity. Riberner toffet in fchön, und Your silver spoon is beautiful,

Thus, Ihr Buch ist groß, may signify, your book is large, or her mais ofernes Meffer ist schwer and my iron knife is heavy.

book is large; and, Ich habe ihr Buch, may mean, I have their ulex Themerne i fower This stone table is heavy

book, or I have her book. The significations of sie in 31 Mebeu tee Kindes ist ein goltner The life of the child is a golden

the accusative, and of ihr in all the cases, must of course, when dream.

spoken, be determined by the connection. (See Declension, aft midt en eljerned Schiff dauerhaft? Is not an iron ship durable

Sect. XVIII.)

1. Haben Sie mein reifed Doft! 2. Nein, ich habe Ihr silbernes Obit.
meffer, und 36r alter Freund, der febrer, bat tas reife Chit. 3. babch

| America, n. America. - Fretent in,j.friend, 510. C'pernglas, n. operain meinen filbernen Bleistift? 4. Nein, ter gute Rebrer bat iba. 5. Bibliotbet", f. hbrary. Olajern, glass.

glass. bat ter alte Kod meinen balzernen Tisd! 6. Nein, ter Tudler bat ibn. Brille, f. spectacles. ! (Bolten, golden. Schecre, f. scissors. aber ter sodo bat cinen marmornen Tiid 7. Dat er au cin bölacrncs Dame, J. lady. Ibr, her (see above). Schwester, f. sister. any 8. 3a, und dieser Fleistge Schüler bat ein idence, bleiernes Tinteni

Fetet, f. pen.

Sein, no, not any. Scite, f. silk. fah 9. Dat er and cinen ilbetten Beter? 10. Ja, und er hat auc Bein, fine.

Kette, f. chain. Tarte, j. aunt. einen hausfernen Seifel and ciren cuirmen Mitier 11, baben Sie das neue i Branfreich, n. France. Lampe, j. lamp. lhr, f. watch, clock. Meffer meinod parou dreunto 12 Nena s lake cin neues Meiler pon! Bräulcin, 1. miss, Peintvand, f. linen. lþrótasche, f. watchbem futen Kaufintaan 13 ut tort tacir Sauler das gute Buc! young lady. Mutter, f. mother. pocket. tied alten Freunde, der det fleres et inn guten Wetters? 14. Gr Mat verder en gutt Be e mer cent Binti- hat nur

RÉSUMÉ OF EXAMPLES. einen Malpermen Kletzt 15 Se der Perrier Section LX. 2) Keid Der Bruder fat ta8 Buch der The brother has the book of het sota! 16. Det arme Winket ret CTE Qvinnen said


the sister. EXES 18

Der Sater giebt der Tattor cu The father gives the daughter i Buch

a book. 1. Has she tu my rige fret! : Thc o cook has my Der ut meiner Mutter ut idin. The hat of my mother is beausilver falbernet pencil Fas be also a new en knife

tiful. 4. The good merchant aiman has un nen, old mazbleme uit die Uhr ibrer gräulein Where is your cousin's watch? tabla 3. He has neatha a golden riches fruit-knife mor a' Geuliner silver tubarnen oup Summer Sa Sommar is a ane golden Sie uit in ter bant ihrer Mutter. It is in her mother's hand. time Zeci The dizagent joiner has the iron ciernen kettle


1. Jit tie funge Sorefter treica jungen Dame in Tcuticlaxi? 2. Nein, FoXS ADJECTIVES, Etc.

i fic it in granftrach, aber ihr Bruter ut in America. 3. Wo ift meine The series n she fede siagals are declined thus - acne, goldene Fetet? 4. Ihre junge Freundin friulein , hat sie (Sect. t her. The cinc a;


XVIII. 3.) 5. Hat Ihre Mutter die schöne Seite Ihrer Tante? 6. Ja, (meint)

i und and tic ibène, frine Leinwand. 7. Be it Ihre gettone Brille? 8. de toer ise sipe: aria conec. to or for a : imeiner) 34 babe tane gelten Brille 9. Paben Sie eine filberne, der eine


geltene Uhr ? 10. 10 babe cine fülhetne ufr. 11. li fie cine gute lihr?

12. Ja afer Tie it must ichr itin 13. Be it the litt? 14. Sie ist The papil having now had in dne course all the forms of the in miner Uhitajde 15. 641 Jhce Sct meter eine geltene Ihr? 16. 3a, article in the singular, may note, that like ticier (which differs wat je bat and an itine, Jaltene Sate. 17. Be it meine neue from the definite article only in having es instead of as in the Scherre ? 18. 30 babe mic, she fic macht ist itari. 19. Bo ist nom, and a neuter, Seot. VII.), are doolined all the words in Ihre Schweiter? 2: Sre bei bet (Sect. XTIL. 3.) Mutter in ter Hi , Sect. IX. and that like ein, are inflected all those in the Pillrethef. 21. Be u mane glaverne Samner? 22. So habe sie. Nu, ein, mein, fein, eto, Seet X.

i 23. Ber bat mrin arnes Opernglas? 24. hate ed und Site nete winine nouns are in the surgular indeclinable; as nom. sic Brille exice (the will) gen der Seite; dat der beide; ace. die Seite. ! Who wjective in the feminine singular has two forms. When

EXERCISE 13 i sunt alone, or thaitected ly a proceding word ($ 29), the: 1. The mother of this lady is in France. 2. Has the beautiful nominative and necusative end in the genitive and dative in er danghter of the good der guten annt a golden edtene watch? 16 is then wall to be of

3. My diligent brother has neither s golden wateh nor a THE OLD DROSION.

i pood gutes opera-glass. 4. My good sister has no (trime fine

linen, but abct she has a DET net glass lomp 5. My cousin with itthe ret golden spectacles is with bei? my (meinen beautiful brother in the horary.



product of all the divisors except the last two, and so on. Add LESSONS IN ARITHMETIC.–VI.

all these results and the first remainder together; the sum will ABRIDGED METHODS OF MULTIPLICATION AND DIVISION. be the whole remainder.

EXAMPLE.—To divide 17285 by 84. 1. The methods of multiplication and division explained in the previous lessons are those ordinarily employed; and the learner

81 = 7 * 4 * 3 must make himself perfectly familiar with them before pro

7) 17285 ceeding further.

4) 2469—2 These processes, however, in particular cases, can often be materially facilitated by various artifices. Some of these shorter

3) 617-1 methods we subjoin, not only because they are useful in themselves, but because they are valuable as exercises, in explaining

205—2 the fundamental principles of arithmetic.

And the whole remainder is 2 * 4 * 7 + 11 x 7 + 2, that is, 2. Any number which is formed by multiplying two or more

56 + 7 + 2, or 65. numbers or factors together is called a composite number. It It does not follow that this process is in all cases simpler has already been explained in a former lesson that the same than the method of Long Division; sometimes, however, it is numbers multiplied together will give the same product, in more convenient whatever order the multiplication is effected.

EXERCISE 11. Hence, to multiply any number by one which is composite i.e., which is composed of several factors—we

(1.) Work the following examples in division : have only to multiply the number first by one

1. 16128 • 24.

4. 91080 = 72. 2. 17220 + 84.

5. 142857 + 112. factor, the result by another factor, and so on.

3. 25760 • 56.

6. 123456 • 168. Thus, to multiply 352 by 28, since 28 = 7 X 4,

(2.) How many acres of land, at 35 crowns an acre, can you we can perform the operation as indicated in the margin.

buy for 4650 crowns. 9858 = 23 352

. (3.) A man divided 837 crowns equally among 27 persons, EXERCISE 10.

who belonged to three families, each family containing nine per

sons: how many crowns did each person receive ? (1.) Resolve the following sets of numbers into their factors :

(4.) A man bought a quantity of clover seed amounting to 1, 9, 10, 14, 22.

4, 8, 16, 18, 20, 24.

507 pints, which he wished to divide into parcels containing 64 2, 35, 54, 56, 63.

5, 27, 32, 36, 40, 48. 3, 45, 72, 64, 81, 96. 6, 12, 28, 54, 72, 84.

pints each: how many parcels can ho make ? (2.) What will 24 horses cost at 74 crowns apiece.

5. Multiplying and dividing by powers of 10, and by numbers (3.) What will 45 hogsheads of tobacco cost, at 128 crowns ending in any number of ciphers. a hogshead?

The products of two tens, three tens, four tens, etc., are called (4.) What will 54 acres of land cost, at 150 crowns per acre ? respectively the second, third, fourth, etc., powers of 10. They

(5.) At 118 shillings per week, how much will it cost a family are 100, 1000, 10000, etc. Thus, the second power is 1 followed to board 49 weeks?

by two ciphers, the third 1 followed by three ciphers, and so (6.) If a man travel at the rate of 72 miles a day, how far on; the number of the ciphers in each case being the same as will he travel in 64 days?

that of the power. (7.) At 163 crowns per ton, how much will 72 tons of lead It has been already explained that to multiply by 10, or any

power of 10, we have only to annex to the multiplicand the (8.) What will 81 pieces of broadcloth cost, at 245 shillings number of ciphers corresponding to the power. Thus, 345 apiece ?

| multiplied by 1000 is 345000. (9.) What will 84 carriages cost, at 384 crowns apiece ? | If any number of the right-hand figures in the multiplier be

(10.) What will a railway 132 miles in length cost, at the ciphers—as, for instance, in 75000—then, as we have already rate of £1,960 a mile ?

Reen in Lesson IV., Art. 5, we need only multiply the multiplicand (11.) If I can walk a mile in 16 minutes, how long will it take by 75, and annex to the product the same number of ciphers, in me to walk 374 miles ?

this case three.

EXERCISE 12 3. Similarly, it will be seen that to divide by any composite number, we have only to divide by one

(1.) Work the following examples in multiplication : factor, then divide the quotient by another

1. 153186 X 10000.

11. 2370000 x 52. 148 1408 factor, and so on.

2. 3120 167 * 100000.

12. 48120000 x 48.

3. 52690078 X 1000000. Thus, to divide 9856 by 28, arrange the

13. 356300000 x 74. 4. 689063157 X 10000000.

14. 1623000000 x 89. 352 Answer. process as indicated in the margin.

5. 4946030506 x 100000000.

15. 540000 x 700.
In this case there is no remainder. But suppose it be re 6. 87831206507 X 1000000000. 16. 1563800 x 20000.
quired to divide 9873 by 28.

7. 67856005109 x 10000000000. 17. 31230000 x 120000.
8. 14376 X 25000.

18. 5310200 X 3400000. (7) 6373 Proceeding as before, we get a remainder 3 9. 350634 X 410000.

19. 82065000 x 8100000. 23 - after the division by 7, and a remainder 2 after 10, 4630425 * 6200000.

20. 210909000 x 5100000. (4) 1410-3 the division by 4. The first remainder 3 means (2.) What will 10 boxes of lemons cost, at 63 shillings per

3 units; and the 2 which remains after box ? 359

- dividing 1410 by 4, means two sevens of the 1410 (3.) How many bushels of corn will 465 acres of land produce, sevens which are contained in 9873. Hence the whole remainder at 100 bushels per acre ? will be 2 sevens + 3 units—i.e., 17.

(4.) Allowing 365 days for a year, how many days are there The process may be exhibited analytically thus :

in 1000 years ? 9873 = 1410 * 7 + 3

(5.) How much will 50 hogs weigh, at 375 pounds apiece ? = 352 X 4 X 7 + 2 * 7 + 3

(6.) If i barrel of flour weighs 192 pounds, how much will 352 x 28 + 14 + 3

500 barrels weigh? = 352 x 28 + 17 Therefore 9873 divided by 28 has a quotient 352, and remainder

LESSONS IN GEOMETRY.-III. 17. Hence, when there are two factors, to find the whole remainder, multiply the second remainder into the first divisor,

INSTRUMENTS USED IN PRACTICAL GEOMETRY. and add the first remainder.

In the operations of practical geometry, a case of mathematical 4. If there are more than two factors, similar considerations instruments must be considered as an essential requisite. will show that the following rule will give the whole remainder : | These instruments vary in number and quality, according to

-Multiply the last remainder into the continued product of all their price. Some are made of wood, bone, and ivory-as the divisors but the last, the last but one remainder into the l rulers and Ecales; cthers are made of brass and steel, German

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