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I have watered thy flowers. Hast thou seen our glasses and MATHEMATICAL ILLUSTRATIONS.-No. II. our bottles ? Our friends are always jolly. Where are your pocket-handkerchiefs and ours? My (female) cousin has lost

(Continued from p. 150, Vol. IV.) our pens and hers. I have given (to) this poor child my pens

ARITHMETICAL LOGARITH SIS, and thine. My father has sold his dogs and mine. Have you also sold yours? Thy wife has bought ten glasses and four 1. The term Logarithens is derived from two Greek words bottles for her daughter, I have given a lead-pencil to thy (arithmoi and logon), and literally signifies the numbers af ratios; sister; she has lost hers. Our neighbour has sold all his but we ought to invert the words accrding to their present use houses. We have sold all our meadows. I have lost all my and meaning (logoi and arithmon), and then it will literally pocket-handkerchiefs. All these bottles belong to our uncle. signify the ratios or indices of numbers. Hence logarithms are The whole house is beautiful. I love all these beautiful defined, a system of artificial numbers which indicate, or repreHowers. I think every day (i.e. all days) of Henry and of sent, the natural numbers, Charles, Where have you bought these six glasses ?' I have 2. By the use of logarithms instead of the natural numbers, seen the whole town. All your letters have (i. e. are) arrived. the tedious and difficult calculations of arithmetic may be Louisa lias (i.e. is) deparied with all her (female) friends. superseded; the irksome and laborious operations of multipliHave you found all these apples in the garden of your father cation and division may be effected by simple additions and Our neighbour has been shedding tears (i.c, has shed tears) subtractions; and the operose and intricate processes of involu. the whole week; she has lost all her children.

tion and cvolution, or the raising of powers and the extraction

of roots, may be reduced to simple multiplications and Exercise. -Exclisi-ITALIAN.

divisions. The adversities are for the soul what a tempest is for the

NATURE OF POWERS AND ROOTS. air. The count has bought the livery for the waiting-man and for the huntsman. The merchant was reputed to be (i.c. 3. If a number be multiplied by itself, the product is called passed for) an honest inan. The new singer made yesterday the square or second power; and the number itself, the root or her appearance for the first time in the part of the prima donna. first pover. If the second power be multiplied by the root, the The indignant servant seized some by the hair (i.e. hairs), product is called the cube or third power. 'If the third power others by the neck and by the arm. He went through the be multiplied by the root, the product is called the fourth wood and not through the village. He put the dress on the power ; and so on. Thus, the multiplication of a number by chair, the watch, however, and the money on the table. itself, and by each successive product, produces the powers of God has punished him for his sins. The bird was now on the that number. This process is called involution, or raising of roof, now on the tree. Let us climb this hill together. We

powers. bave taken to-day a long walk upon the bastion. The suspi- 4. The numbers which indicate how often the root enters as ciun fell upon him and upon her. On similar matters books a factor into each of the powers, are called the indices of those have been written by a great many excellent men. There is powers. Thus, the index of the first power is 1; of the second a great difference between him and his brother. Between power, 2; of the third power 3; of the fourth power, 4; of the lunging and fear he knew not which resolution to take. He fifth power, 6; of the sixth power, 6 : and so on. excited mortal enmity between these (men) and those.

5. If a number be considered as a power of another, the VOCABULARY,

latter is called such a root of the former as is denoted by the Adversity, av.ver-si-, f. His sins, i suó-i pec-ca-ti.

name of that power. Thus, if one number be considered as the Soul, á-ni-ina, f. Bird, uc-cello, m.

8qtare of another, the latter is called the square root of the for.

mer. If one number be considered as the cube of another, the What, ciò che

Was now –
now, é.ra 6.ra

latter is called the cube root of the former. If one number be
Tempest, tem. po-rii-le, m.
Air, d-ria, f.
Roof, tét. to, m.

considered as the fourth power of another, the latter is called Has bought, ha com-pru-to Tree, al-be-ro, m.

the fourth root of the former; if one number be considered as Waiting-man, ca-me-riê-re, m. Let us climb'together, sa-liá- the fifth power of another, the latter is called the fifth root of Huntsman, cac-cia--re, m. mo in-sie-mo (with su).

the former ; and so on. The process by which the roots are Livery, li-vré-a, f. This hill, qui-sta col-li-na, f.

found is called ero!ution, or the extraction of roots, and is Merchant, mer-con-le, m. We have taken, ab-bid - mo

explained in the common treatises on arithmetic. Was reputed to be, pas-su-va fut-to

6. The reciprocal of a number is a fraction whose numerator Honest man, ga-lant-ui-nuo, m. Long walk, lin-ga pas-seg-gid- | the reciprocal of 2 is }; of 3 is } ; of 4 is }; and so on. If the

is unity, and whose denominator is the number itself. Thus, New singer, 110.va cau latrí ce, f. Bastion, ba-stió-ne, m.

number' be a fraction, its reciprocal is the fraction inverted.

Thus, the reciprocal of : is $; of į is t; of } is 1 or 4; and so Made yesterday her appear- Suspicion, 30-spit-to, m. ance, re-ci.to je-ri

Fell, é ca-du-to First time, pri-va ról-la, f. Him, lui

7. The reciprocals of the numbers which indicate how often Part, pir-te, f. Her, le-i.

the root of a number enters as a factor into that number, are Servant, sér.10, m. Similar, -ini-le

called the indices of its roots. Thus, the index of the square Indignant, sde-gnu-10 Matter, ma-te-ria, f.

root is }; of the cube root, }; of the fourth root, 1 ; of the Seized some, pré-se al-cu-ni Boks have been written (i.e. being fractions, the roots themselves are called fractional powers

fifth root, }; and so on. The indices of the roots of a number Hair, ca pel.lo, m.

was written) fu scrit-to

of that number. Ocher, al-tro, m.

A great many, mol-tis-si-mo Neck, cól-lo, m.

(i.e. very much).

8. If the number 2 be assumed as the root or first power, Arm, bruc-cio, m, Excellent man, va-lent-uô-mo, multiplication in the following manner:

then the successive powers of 2 are calculated by simple He went throughi, é.gli è pasThere is, c'è

Table of Products.
Wood, bi-sco, m.
Great difference, mol-ta dif-fe-

2= 2 First Power
And not, e non
tên ca, f.

2 X 2= 4 Second Power Village, vil-lug-gio, m. Longing, brá-ma, f.

4 x

8 Third Power He put, éogli -se Fear, ti-mo-re, m.

8 x 16 Fourth Power Dress, a.bi-to, m. He knew not, non sa--a

16

32 Fifth Pover On, sit Which resolution to take, che

32 x 64 Sixth Power Chair, -dia, f. ri-sól-ve-re.

128 Seventh Power Watch, o-ri-urô lo, m. He excited, spur-se

128 X 2=

256 Eighth Power However, all' in.con tro This, quié-sto, m.

256 X 2= 512 Ninth Power Money, da-nu-ro, m. That, quel-lo, m.

512 X 2= 1024 Tenth Power Table, ta-vo-li-no, ir. Enmity, ni-mi-zia, f.

1024 X 2= 2048 Eleventh Power Has punished him for, lo ha Mortal, mor-td-le.

2048 X 2=4096 Twelfth Power pu-ni-lo per

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ta, f.

on.

m.

sd-to per

61 X

etc.

28

etc.

4

8

256

2018 i4

These powers of the number 2 are denoted by placing their power of 2, be extracted, the cube root 8 is the third power indices on the right of the root, as in the following table : of 2. This process is thus indicated : it'9, the index of the Table of Powers,

ninih power, be divided by 3, the denominator of the index of

the cubo root, the quotient 3 is the index of the required 2, Root or First Power

power of 2.

Again, if the square root of 4, which is the same 4, Square or Second Power,

root of 16, be extracted, the square root 2 is the fourth root of 8, Cube or Third lower

16. This process is thus indicated: if }, the index of the given 24 16, Fourth Power

root, be multiplied by }, the index of the required root of 4, 25 32, Fifth Power

the product is the required root of 16. See Prob. VIII. 26 64, Sixth Power

above cited. 27 128, Seventh Power

14. If any power or root of a number be divided by itself, = 256, Eighth Power

the quotient is unity: On the principle that the division of 29 = 512, Ninth Power

powers and routs of the same number is indicated by subtract210— 1024, Tenih Power

ing the index of the divisor from that of the dividend, the 211= 2048, Elerenth Power

index of the quotient unity, in this case, is 0. Hence, that 21 = 4096, Twelfth lower

power of any number, whose index is 0, is unity. Conse

quently, in the expression 2 =l, unity may be denominated 9. The roots of these powers are, in like manner, denoted the cero power of 2. by placing their indices on the right of the powers, each being considered a separate number, as in the following table : Table of Roots.

CORRESPONDENCE.
}
= 2, Square root of 4

TIJE FOUR-BALL QUESTION, EUCLID, ETC,
= 2, Cube root of 8

S11,--As I was informed by the last page of the last monthly 16 * = 2, Fourth root of 10

part of the P. E. that the Ball Question remains unanswered, I 32 = 2, Fifth root of 32

was induced 10 give it a trial, with the humble assurance (though

I have not studied Euclıd) that the application of a little common 64 – 2, Sixth root of 6

sense would furnish a correct answer."

I should like to know, by your favour, whether mathematics or 128 = 2, Seventh root of 128

algebra teaches usto find answers to similar questions, and with the = 2, Eighth root of 256

same nicety, without the aid of paper and compasses.;

As to the study of Euclid, which all the world esteems and 512 = 2, Ninth root of 512

recommends as an important item of education, excuse my (igno. 1024

to
= 2, Tenih root of 1024

rance, perhaps,) want of veneration in saying, that I felt some

what amusei and deceived, on finding, after an hour's perusal of = 2, Eleventh root of 2048

Euclid (your cditiun) for the first time, that it consisted in definis 21

cions that seemed to be a jnere play upon words, and demonstra4096 = 2, Twelfth root of 4096

tions shrouded in dark sentences, and mystified by letter repetitions etc.

to set forth so many apparenily simple and seit evident iacts. I 10. The multiplication of the powers or roots of the same herefore placed it aside for all uudisturbed repose, till I could number, is indicated by the addition of the indices of the better understand the qualities which render it an essential factors. Thus, if 16, the fourth power of 2, be multiplied by portion of education, and the merits that have gained for it a 8, its third power, the product 128 is its seventh power. This rorld-wide fame. These, perhaps, you will direct me to discover. process is thus indicated : if 4 and 3, the indices of the factors, adu, that, as I liave the happiness of being a young man in this be added together, the sum 7 is the index of the product. Again, if 4, the cube root of 64, be multiplied by 2, its sixth golden age for the advantages of mental culture, and betier oppor

tunities jor study iban those who lived before me, I may have the root, the product 8 is its square root. This process is thus indicated: if } and d, the indices of the factors, be added society in general. To this end I hope that whilst the good aim is

pleasure of witnessing the consequent good effects on inyself and together, the sumn } is the index of the product. See Prob. VI. being made to bring knowledge within the reach of the many, that Addition of Fractions, p. 329, vol. II.

its genuinene:s will nut lose in the attempt to make it cheap 11. The division of powers or roots of the same number is (rather let us have qu...cy than quantity); that this may ever be indicated by the subtraction of the index of the divisor from that ine case with that boon to the world, the P. E., is the sincere of the dividend. Thus, if 512, the ninth power of 2, be divided desire of yours respectfully,

R. F. T. by 128, its seventh power, the quotient 4 is its square. This

[We have inserted this let'er because it contains a strange process is thus indicated: if 7, the index of the divisor, be mixture of good and evil-of the phrenological bumps marked subtracted from 9, the index of the dividend, the remainder 2 Nos. 15 and 17. with chore marked

Nos. 5 and 10. We have so is the index of the quotient. Again, if 8, the square root

otten recommended the study of Euclid and algebra, that our of 64, be divided by 4, its cube root, the quotient 2, is correspondent must excuse us this time ; Lut if any of uur readers its sixth root.

will kinuly take up the cudgels on our behalf, we shall be thankful; This process is thus indicated : if }, the index of the divisor, be subtracted from 1, the index of the city leonthemo remember the advice that David gave to his cap

tains concerning . dividend, the remainder ) is the index of the quotient. See Prob. VII . Subtraction of Fractions, p. 357, vol. II.

MANUSCRIPT MAGAZINES. 12. The involution of powers or roots of the same number is indicated by the multiplication of the indices of the given powers

SIR,– Your correspoi dept W. B., in No. 100, gires a plan for Our roots of those of the required powers. Thus, if 16, the the formation of Mutual Instruction Societies. fourth power of 2, be squared, the square 256 is the eighth applicable where there are several young men in a 10wn who power of 2. This process is thus indicated : if 4, the index of students of the P. E. who reside in small places, where it is inthe fourth power of 2, be multiplied by 2, the index of the possible to form such classes, and many who cannot make it consquare, the product 8'is the index of the required power of icnient to attend them when formed. 2. Again, if 2, the sixth root of 64, be cubed, the cube 8 is the To such studenis the following plan may be of some use; it is square root of 61. This process is thus indicated: if }, the one which I have tried, in conjunction will some friends, for the index of the sixth root, be multiplied by 3, the index of the last eighteen months, and I have found it successful. The idea is cube, the product is the index of the required power of 64. not mine. Four or tive young men agree

to circulate a MS. Magas See Prob. VIII. Multiplication of Fractions, p. 25, vol. IIf. 13. The evolution of powers or roots of the same number is to choose one of their number as conductor, who regulates the indicated by the multiplication of the indices of the given postage

, the time of circulation, etc. He commences by waiting a powers or roots, by the indices of the required roots; or, by the division of those indices by the denominators of the indices • Our correspondent's answer is 4 inches 3-10ths; and nearly the same of the required roots. Thus, if the cube root of 512, the ninth answer has been sent by hundreds: but it is entirely wrong.

know.

ܙܙ

the

paper on the particular subject of study, and then forwards it by join a Mutual Instruction or other Class on these subjects, it would be a post to the next member in rotation ; this member contributes his great help to him. He is right in the solution of the algebraic question.quota to the work, asks any questions which he wishes to be .. A. A. (Sberborne): Read “ Cassell's Emigrants' Hand-book; or a Guide answered, and passes it on to the next in rotation. Thus it circus to the various Fields of Emigration in all parts of the Globe." -'JOHN FORD lates among all ihe members. When it comes into the conductor's | to rlie most eminent physician he can find.

(Derby): Hutton's Mensuration.-ASPIRANT (Lower House) should apply

Good men are generous in prohands the second time, he withdraws his first contribution and portion to their greatness.-T. THOMA8 (Derby): Still to be procured; we replaces it by a second one, prepared, for the most part, whilst presume the Society of Arts bas agents or connexions in all great towns.No. I was in the possession of the other members. He again for-Eller SMITH (Westbury): We should be only too happy to answer her wards the magazine, when each member does the same, circulating question to her satisfaction, but we cannot.--R. 8. (Bradford): We forget. it as before.

-R. THOMPSON (U pleatham): Yes.-M. SCHOFIELD (Waterhead); The

female elephants have tusks as well as the males, but they are not so large; It adds much to the value of such a plan, if the members mutu- and when the animals are young, the lusks are not visible.-JUVENIS (Carally agree to correct each other's spelling.

lisle): His 45 solutions are received. The impropriety of his concluding I should be glad to meet with four or five earnest young meni remark is excused, because he is Juvenis.-T, WELLS (Linfield): We don't who desire to study physiology in this way; or I would be glad to unite with any of the would-be artists” of your “ Correspon- 24 inches, it is very near the truth-very near, indeed, to be found by expe

T. WATKINSON (Stanstead): If your answer to the Four-ball Question be dence” columns, in studying the theoretical part of their profes-riment alone. The word Exceisior means higher; it is from the same Latin sion. Address, “ T. J., Post-office, Bedminster, Bristol." Yours root as the word excel.-W. CHEETHAM (stockport): Study chemistry, and respectfully,

T. J. you will see at once how the earth is neither lighter nor heavier since the

creation, by the iminense consumption of coal.-AN ASPIRING APPRENTICE ALGEBRA, ETC.

is entitied to know why we wished him to call on us; we saw that he was

enveloped in a mist of error, and we thought that a few words spoken to SIR--In compliance with your request, appended in a note to him in a kind and proper manner, in a free and easy conversation with him,

would do him more good than a bushel of letters or arguments; with these centenary of problems," which you gave in No. 101 of the P. E., I venture to send you 'two solutions. If I tell you that I useful: but we think his labour in learning French will be lost, if it be for

we can't fill our pages. He may learn either Latin or French ; both are never knew anything of algebra until last Christmas, when I was

the sole purpose of reading Voltaire, Volney, Mirabeau, Rousseau, or any of induced to commence the study from seeing the easy lessons in that motiey intidel crew, "His poetry is pretty fair, and indicates a beiter your truly excellent publication, I hope you will give me credit for state of mind than his former letter. It is essential to blank verse that my statement.

every line should contain ten sillables; and these syllables must, if possible, I am really a self-taught man; but as languages and other sub-be made iambic, that is, consisting of a short and a long syllable alternately,

As in Milton, thus :jects have occupied so much of my time and attention, I was surprised at the improvement I made within the short space of a

“ Before the heav'ns thou wort, and at the voice fortnight in the study of algebra. Although I begna in real

of God, ås with ă mântlē didst învēst." earoest, yet I am not insensible to the fact that my determination may be ascribed to the very simple and easy method laid down in the first lessons you published. For this acquisition, moderate as it is, I beg you will accept my most sincere thanks. It is, indeed, entirely owing to your useful and indefatigable exertions

LITERARY NOTICES. in the cause of self improvement and of education and literature generally, that I have formed a slight acqua'atance with a science THE HISTORICAL EDUCATOR.-The First Volume of this important that al vays appeared to me to be involved in the most inexplicable work, containing Ancient Voyages and Travels, with Popular Histories of mystery.

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ALEPU.

Twenty-six Numbers, TOREEPENCB each, or in Monthly Parts, ONE
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CASSELL'S FRENCH AND ENGLISH DICTIONARY.-The FRENCH and Algebra,” which I have found very acceptable. I should recom- had, price 48., or strongly bound, 58.-The ENGLISH aud FRENCH portion mend all students in algebra to procure it.

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Origin of the Sound in Wind Instruments.--In the different apparatus described in former lessons, the sound arose from the vibrations of solid bodies; the air was only the medium of conveyance. In wind instruments, constructed of pipes whose sides are sufficiently strong to resist vibration, it is the column of the air inclosed in the pipes which is alone the sonorous body. It is, in fact, demonstrable that the matter of the pipes has no influence on the sound, which is the same under equal dimensions, whether they are made of wood, glass, or metal, with the exception of the timbre or distinct quality of the sound. As to the manner of putting the air into vibration in pipes, wind instruments are divided into instruinents with a mouth-piece, instruments with a rigid tongue, and instruments with a membraneous tongue.

Instruments with a Mouth-piece.--In instruments having a mouth-piece, all the parts of the latter are fixed. Fig. 146

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belongs to a node of vibration; and it is equally manifest that the portion of air in the neighbourhood of the mouth belongs to a swell, for it is at this point that the disturbing cause exists. This node and this swell occur only when the pipe emits the lowest or fundamental sound. A change in the diameter of the pipe, in the dimensions of the mouth, or in the velocity of the current of air, will produce a series of sounds rising gradually higher and higher, and affecting the ratios of the different portions of the column of air in the pipe, as shown in fig. 149.

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3 nodes and 3 swells.

represents the mouth-piece of an organ-pipe ; and fig. 147
represents that of a whistle or flageolet. In these two figures,
the opening i is called the passage; by this aperture the air
is introduced into the pipe. The opening 60 is called the
mouth, and its upper lip, 5, is made with a feathered edge. Souud 3,

C The upper part of both figures represents the pipe, which may be either open or closed according to the nature of the instru- After the first or fundamental sound, that which is proment. In fig. 146 the foot, or extremity, P, is employed to duced the most naturally allows the node and the swell at fix the pipe on a blowing machine, as shown in fig. 134, p. 364, the extremities of the pipe to remain ; but a new node appears vol. iv. When a rapid current of air rushes through the at a third part of the column of air reckoning from the mouth, passage, it strike against the upper lip, and produces a con- and a new swell at a third part reckoning from the bottom. cussion which prevents the air from issuing in a continuous The higher sound which immediately follows forms an addi. manner from the mouth, and causes it to proceed intermit- tional node and an additional swell. The sounds which tingly. From this cause the pulsations which are transmitted correspond to these different states of the pipe are higher and to the air in the pipe make it vibrate and emit a sound. In higher. The manner in which the column of air is divided in order that the emitted sound may be clear and distinct, a a closed pipe, shows that the different sounds emitted are to certain relation must be established among the lips, the aper- each other as the series of odd numbers 1, 3, 5, 7, etc. ture of the mouth, and the size of the passage ; and the pipe In a pipe open at both ends the arrangement of the nodes must have a great length in comparison with its diameter. and swells is different. The bottom, which in the former case The number of vibrations depends, in general, on the dimen- determined at once a node of vibration, no longer exists ; conbions of the pipe and the velocity of the current of air. In sequently, a swell takes its place. Thus, there is at each the German Aute the mouth is a simple lateral and circular exiremity of an open pipe, a swell; it is necessary, therefore, opening. It is owing to the arrangement of the lips that the that there should be at least one node in the interior. Espe current of air is made to strike against the edges of this riment shows that this node is in the middle, and that it aperture ; this is seen in the Pandian reed, and in the per. divides the column of air into two parts which 'vibrate in an forated key which is made to whistle.

inverted direction. Each of these parts is, of course, one-half The following representation (fig. 148) of an organ-pipe or of that which would vibrate if the pipe were closed at one Aute, exhibits a side section and front view, in which i shows end, and this explains

the reason why the fundamental sound the extremity of the passage, and the mouth. Since the of the pipe open at both ends is the higher octave of the vibration of air contained in the pipes is really the cause of former.

After this fundamental sound, there exist also, in the production of the sound, we shall find that in this sonorous pipes open at both ends, a series of sounds which arise from

TOL. V.

the subdivision of the column of air into aliquot parts, and called the tongue, and which, when it is fixed, fits the edges which produce new swells and new nodes, as exhibited in of the channel so as nearly to close it—this tongue is fixed fig. 150.

only at its upper part ; and 4th, an iron wire, bent at its lower Fig. 150.

part, which presses upon the tongue. This iron wire, which Fundamental und

is called the catch, can be raised or lowered so as to regulate 1 node and 2 swells. J all the motions of the tongue, and fix the height of the sound;

and it admits of the complete adjustment of the pipes to the tongues. Suppose, now, that the tongue is replaced in the

Fig. 153. Fig. 154. 2 nodes and 3 swells.

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Sound 2.

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The existence of these nodes and swells is proved by introducing a piston into the pipe at each of the nodal points, and making holes in its side at each of the swell points; and when they are accurately adjusted, these different operations do not alter the height of the sound. The length of the column of air comprised between two nodes is called a concameration. The length of a concameration has been found by experiment nearly equal to the length of the wave of the sound produced by propagation in free air.

Instruments with a Rigid Tongue, -The air is put into vibration in instruments having a rigid tongue by a simple elastic piece of metal or of wood, which is put in motion by a current bf air. This tongue acts in a manner easily comprehended, see figs. 151 and 152, which show a front and side view of it.

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pipe mn, and that a current of air is admitted into the latter by the foot p; the tongue will then be acted on by the air, and by this action, being bent inwardly, will give a passage to the air, which will escape by the orifice o. But the tongue, in consequence of its elasticity, will return immediately to its first position ; it will thus form a series of oscillations which will alternately open and close the channel, and cause the

current of air to act intermittingly; hence arise the sonorous Lis the tongue fixed at one extremity, in front of a rectangular waves producing a sound whose height increases with the opening, o, made in the side of the pipe, whether square or velocity of the current. round. This tongue is constructed in such a manner that it Membraneous Tongues.-The effects of membraneous tongues is bent in a state of rest, and leaves the opening free to the are not yet well ascertained; but they are interesting, as admission of air. But if a current of air be produced in any connected with the theory of the human voice. The followmanner so as to have a tendency to rush through the aperture ing are the principal results of the researches of M. Muller, o, in the direction indicated by the arrow, the tongue is forced the physiologist, on this subject :against the side of the pipe, and for a moment it stops the flow 1st. If a narrow strip or ribbon of caoutchouc be stretched of the current; then its elasticity restores it to its first posi- over a ring about of an inch in diameter, and if by means of tion : after this it is again impelled against the side of the a pipe of small diameter we blow on one of the edges of the pipe, and so on alternately, as long as the air exerts a tendency ribbon in a direction oblique to its surface, the ribbon will be to rush through the aperture o. Thus we see that the impulse made to vibrate and a sound will be emitted. of the air must be made intermittingly, and when these inter- 2nd. If a strip of caoutchouc, of about * of an inch in mittent impulses are sufficiently rapid, a sound arises which is breadth, be stretched over a ring or a wooden frame, and if on stronger than that proceeding from the vibrations of the tongue each side of this elastic strip we fix a rigid plate made of card or lamina alone.

or of wood, so as to admit of a very straight and narrow slit This kind of tongue is used in hautboys, bassoons, clarionets, between the plates and the caoutchouc, this strip will be made children's trumpets, and other simple instruinents of this kind to vibrate by surrounding the ring or frame with the lips and Some organ-pipes have a mouth-piece like that already blowing through it, or by blowing through a porte-vent or pipe, described, fig. 146; others have rigid tongues. Fig. 153 is a at one end of which the apparatus is placed. representation of one of the latter, with the arrangement best adapted for demonstration. It is mounted on the box of a with an elastic membrane, and the other half with a rigid

3rd. If one-half of the orifice of a very short pipe be covered blowing machine, and glass, inserted at e in the sides of the plate, leaving a narrow slit between them for the passage of pipe, admits of the vibrations of the tongue being seen. A the air, the membrane will be put into a state of vibration by wooden apparatus, like a trumpet, is used for strengthening the same means as in the preceding case. the sound. Fig. 154 represents the tongue outside of the pipe. 4th. If the orifice of a very short pipe be covered with two It is composed of four pieces :-Ist, a rectangular wooden elastic membranes instead of one elastic membrane and a rigid pipe closed at the bottom, and open at the top, at 0; 2nd, a plate, leaving the narrow slit between them as before, the brass pla:e, ce, pierced by a longitudinal aperture, called the same effect will be produced by employing the same means. channel, which is intended to give a passage to the air of the This arrangement is most like that of the glottis, or chink of pipe un to the or fice 0; 3rd, an elastic lamina, i, which is the larynx, which is the upper part of the windpipe.

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