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I have watered thy flowers. Hast thou seen our glasses and
our bottles? Our friends are always jolly. Where are your
pocket-handkerchiefs and ours? My (female) cousin has lost
our pens and hers. I have given (to) this poor child my pens
and thine. My father has sold his dogs and mine. Have you
also sold yours? Thy wife has bought ten glasses and four
bottles for her daughter. I have given a lead-pencil to thy
sister; she has lost hers. Our neighbour has sold all his
houses. We have sold all our meadows. I have lost all my
pocket-handkerchiefs. All these bottles belong to our uncle.
The whole house is beautiful. I love all these beautiful
flowers. I think every day (i.e. all days) of Henry and of
Charles. Where have you bought these six glasses? I have
seen the whole town. All your letters have (i. e. are) arrived.
Louisa has (ie. is) departed with all her (female) friends.
Have you found all these apples in the garden of your father?
Our neighbour has been shedding tears (i. e. has shed tears)
the whole week; she has lost all her children.

EXERCISE.-ENGLISH-ITALIAN,

The adversities are for the soul what a tempest is for the air. The count has bought the livery for the waiting-man and for the huntsman. The merchant was reputed to be (i. e. passed for) an honest man. The new singer made yesterday her appearance for the first time in the part of the prima donna. The indignant servant seized some by the hair (i. e. hairs), others by the neck and by the arm. He went through the wood and not through the village. He put the dress on the chair, the watch, however, and the money on the table. God has punished him for his sins. The bird was now on the roof, now on the tree. Let us climb this hill together. We bave taken to-day a long walk upon the bastion. The suspicion fell upon him and upon her. On similar matters books have been written by a great many excellent men. There is a great difference between him and his brother. Between longing and fear he knew not which resolution to take. excited mortal enmity between these (men) and those. VOCABULARY.

Adversity, av-ver-si-tà, f.
Soul, d-ni-ma, f,
What, ciò che

Tempest, tem-po-rú-le, m.
Air, d-ria, f.

Has bought, ha com-prá-to
Waiting-man, ca-me-rié-re, m.
Huntsman, cac-cia-tó-re, m.
Livery, livré-a, f.

Merchant, mer-can-te, m.

Was reputed to be, pas-sú-va

His sins, i sub-i pec-ca-ti.
Bird, uc-cel-lo, m.

Was now

ó-ra

He

now, ê-ra ó-ra

Roof, tét to, m.
Tree, al-be-ro, m.
Let us climb together, sa-liá-
mo in-sie-mo (with su).
This hill, questa col-li-na, f.
We have taken, ab-biá-mo
fát-to

MATHEMATICAL ILLUSTRATIONS.-No. II.

(Continued from p. 150, Vol. IV.)

ARITHMETICAL LOGARITHMS.

1. The term Logarithms is derived from two Greek words (arithmoi and logon), and literally signifies the numbers of ratios; but we ought to invert the words ace rding to their present use and meaning (logoi and arithmon), and then it will literally signify the ratios or indices of numbers. Hence logarithms are defined, a system of artificial numbers which indicate, or represent, the natural numbers.

2. By the use of logarithms instead of the natural numbers, the tedious and difficult calculations of arithmetic may be superseded; the irksome and laborious operations of multiplication and division may be effected by simple additions and subtractions; and the operose and intricate processes of involution and evolution, or the raising of powers and the extraction of roots, may be reduced to simple multiplications and divisions.

NATURE OF POWERS AND ROOTS.

3. If a number be multiplied by itself, the product is called the square or second power; and the number itself, the root or first power. If the second power be multiplied by the root, the product is called the cube or third power. If the third power be multiplied by the root, the product is called the fourth power; and so on. Thus, the multiplication of a number by itself, and by each successive product, produces the powers of that number. This process is called involution, or raising of

powers.

4. The numbers which indicate how often the root enters as a factor into each of the powers, are called the indices of those powers. Thus, the index of the first power is 1; of the second power, 2; of the third power 3; of the fourth power, 4; of the fifth power, 5; of the sixth power, 6: and so on.

5. If a number be considered as a power of another, the latter is called such a root of the former as is denoted by the name of that power. Thus, if one number be considered as the square of another, the latter is called the square root of the former. If one number be considered as the cube of another, the latter is called the cube root of the former. If one number be considered as the fourth power of another, the latter is called the fourth root of the former; if one number be considered as the fifth power of another, the latter is called the fifth root of the former; and so on. The process by which the roots are found is called erolution, or the extraction of roots, and is explained in the common treatises on arithmetic.

6. The reciprocal of a number is a fraction whose numerator is unity, and whose denominator is the number itself. Thus,

Honest man, ga-lant-uo-mo, m. Long walk, lun-ga pas-seg-gid- the reciprocal of 2 is; of 3 is; of 4 is ; and so on.

New singer, mủ và can la

tri ce, f.

ance,

Made yesterday her appear-
re-ci-to je-ri
First time, pri-ma ról-ta, f.
Part, par-te, f.
Servant, ser-vo, m.
Indignant, sde-gnu-to
Seized some, pré-se al-cu-ni
Hair, ca pél-lo, m.

Other, al-tro, m.

Neck, col-lo, m.

Arm, brac-cio, m.

He went through, é-gli è pas

sá-to per
Wood, bo-sco, m.
And not, e non

Village, vil-lug-gio, m.

He put, é-gli pó-se

Dress, á-bi-to, m.
On, su

Chair, sé-dia, f.

WVatch, o-ri-cô lo, m.

However, all' in-cón-tro

Money, da-na-ro, m.

Table, ta-vo-li-no, m.

ta, f.

Bastion, ba-stió-ne, m.
Suspicion, so-spét-to, m.
Fell, & ca-du-to
Him, lui
Her, le-i.

Similar, sí-mi-le
Matter, ma-te-ria, f.
Books have been written (.e.
was written) fu scrit-to
A great many, mol-tis-si-mo
(i. e. very much).
Excellent man, va-lent-uô-m

m.

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8. If the number 2 be assumed as the root or first power, then the successive powers of 2 are calculated by simple multiplication in the following manner :

Table of Products.

2 First Power
4 Second Power
8 Third Power

2=

2 X 2=

Longing, brá-ma, f.

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Fear, ti-mo-re, m.

8 X

16 Fourth Power

He knew not, non sa-pé-a

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Which resolution to take, che

32 X

64 Sixth Power

ri-sôl-ve-re.

64 X2

128 Seventh Power

He excited, spár-se

128 × 2

256 Eighth Power

This, qué-sto, m.

Enmity, ni-mi-cí zia, f.

That, quel-lo, m.

Has punished him for, lo ha Mortal, mor-tá-le.

pu-ni-to per

256 X 2

512 Ninth Power

512 × 2=1024 Tenth Power 1024 X 22048 Eleventh Power 2048 X 24096 Twelfth Power

etc.

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2, Ninth 100t of 512

2, Tenth root of 1024

2, Eleventh root of 2048 2, Twelfth root of 4096 etc.

10. The multiplication of the powers or roots of the same number, is indicated by the addition of the indices of the factors. Thus, if 16, the fourth power of 2, be multiplied by 6, its third power, the product 128 is its seventh power. This process is thus indicated: if 4 and 3, the indices of the factors, 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 root, the product 8 is its square root. This process is thus indicated if and, the indices of the factors, be added together, the sum is the index of the product. See Prob. VI. Addition of Fractions, p. 329, vol. II.

11. The division of powers or roots of the same number is indicated by the subtraction of the index of the divisor from that of the dividend. Thus, if 512, the ninth power of 2, be divided by 128, its seventh power, the quotient 4 is its square. This process is thus indicated: if 7, the index of the divisor, be subtracted from 9, the index of the dividend, the remainder 2 is the index of the quotient. Again, if 8, the square root of 64, be divided by 4, its cube root, the quotient 2, is This process is thus indicated: if the index of the divisor, be subtracted from, the index of the dividend, the remainder is the index of the quotient. See Prob. VII. Subtraction of Fractions, p. 357, vol. ÎI.

its sixth root.

12. The involution of powers or roots of the same number is indicated by the multiplication of the indices of the given powers or roots of those of the required powers. Thus, if 16, the fourth power of 2, be squared, the square 256 is the eighth power of 2. This process is thus indicated: if 4, the index of the fourth power of 2, be multiplied by 2, the index of the square, the product 8 is the index of the required power of 2. Again, if 2, the sixth root of 64, be cubed, the cube 8 is the square root of 64. This process is thus indicated: if, the index of the sixth root, be multiplied by 3, the index of the cube, the product is the index of the required power of 64. See Prob. VIII. Multiplication of Fractions, p. 25, vol. III. 13. The evolution of powers or roots of the same number is indicated by the multiplication of the indices of the given powers or roots, by the indices of the required roots; or, by the division of those indices by the denominators of the indices of the required roots. Thus, if the cube root of 512, the ninth

power of 2, be extracted, the cube root 8 is the third power of 2. This process is thus indicated: if'9, the index of the ninth power, be divided by 3, the denominator of the index of the cubo root, the quotient 3 is the index of the required power of 2. Again, if the square root of 4, which is the same root of 16, be extracted, the square root 2 is the fourth root of 16. This process is thus indicated: if, the index of the given root, be multiplied by, the index of the required root of 4, the product is the required root of 16. See Prob. VIII. above cited.

14. If any power or root of a number be divided by itself, the quotient is unity. On the principle that the division of powers and roots of the same number is indicated by subtracting the index of the divisor from that of the dividend, the index of the quotient unity, in this case, is 0. Hence, that power of any number, whose index is 0, is unity. Consequently, in the expression 2o 1, unity may be denominated the zero power of 2.

CORRESPONDENCE.

THE FOUR-BALL QUESTION, EUCLID, ETC.

SIR, AS I was informed by the last page of the last monthly part of the P. E. that the Ball Question remains unanswered, was induced to give it a trial, with the humble assurance (though I have not studied Euclid) that the application of a little common sense would furnish a correct answer.*

I should like to know, by your favour, whether mathematics or algebra teaches usto find answers to similar questions, and with the same nicety, without the aid of paper and compasses.;

As to the study of Euclid, which all the world esteems and recommends as an important item of education, excuse my (ignorance, perhaps,) want of veneration in saying, that I felt somewhat amused and deceived, on finding, after an hour's perusal of Euclid [your edition] for the first time, that it consisted in definitions that seemed to be a mere play upon words, and demonstrations shrouded in dark sentences, and mystified by letter repetitions to set forth so many apparently simple and seif evident facts. I therefore placed it aside for an undisturbed repose, till I could better understand the qualities which render it an essential portion of education, and the merits that have gained for it a world-wide fame. These, perhaps, you will direct me to discover. add, that, as I have the happiness of being a young man in this Pardon my having trespassed at this length; whilst I briefly golden age for the advantages of mental culture, and better opportunities for study than those who lived before me, I may have the pleasure of witnessing the consequent good effects on myself and society in general. To this end I hope that whilst the good aim is being made to bring knowledge within the reach of the many, that its genuineness will not lose in the attempt to make it cheap (rather let us have quilty than quantity); that this may ever be the case with that boon to the world, the P. E., is the sincere desire of yours respectfully, R. F. T.

[We have inserted this let'er because it contains a strange mixture of good and evil-of the phrenological bumps marked Nos. 15 and 17, with those marked Nos. 5 and 10. We have so often recommended the study of Euclid and algebra, that our correspondent must excuse us this time; but if any of our readers only let them remember the advice that David gave to his capwill kindly take up the cudgels on our behalf, we shall be thankful; tains concerning Absalom-2 Samuel xviii. 5.]

MANUSCRIPT MAGAZINES.

SIR,-Your correspo dent W. B., in No. 100, gives a plan for the formation of Mutual Instruction Societies." This plan is applicable where there are several young men in a town who pursue the same course of study; but there are, no doubt, many students of the P. E. who reside in small places, where it is impossible to form such classes, and many who cannot make it convenient to attend them when formed.

To such students the following plan may be of some use; it is ene which I have tried, in conjunction with some friends, for the last eighteen months, and I have found it successful. The idea is not mine. Four or five young men agree to circulate a MS. Magazine on the subject which they wish to study. The first thing is to choose one of their number as conductor, who regulates the postage, the time of circulation, etc. He commences by writing a

Our correspondent's answer is 4 inches 3-16ths; and nearly the same answer has been sent by hundreds: but it is entirely wrong.

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 quota to the work, asks any questions which he wishes to be answered, and passes it on to the next in rotation. Thus it circulates among all the members. When it comes into the conductor's hands the second time, he withdraws his first contribution and replaces it by a second one, prepared, for the most part, whilst No. 1 was in the possession of the other members. He again forwards the magazine, when each member does the same, circulating

it as before.

It adds much to the value of such a plan, if the members mutually agree to correct each other's spelling.

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know.

great help to him. He is right in the solution of the algebraic question.
A. A. A. (Sherborne): Read "Cassell's Emigrants' Hand-book; or a Guide
to the various Fields of Emigration in all parts of the Globe."-JOHN FORD
to the most eminent physician he can find. Good men are generous in pro
(Derby): Hutton's Mensuration.-ASPIRANT (Lower House) should apply
portion to their greatness.-T. THOMAS (Derby): Still to be procured; we
presume the Society of Arts has agents or connexions in all great towns.-
ELLEN SMITH (Westbury): We should be only too happy to answer her
question to her satisfaction, but we cannot.-R. S. (Bradford): We forget.
-R. THOMPSON (Upleatham): Yes.-M. SCHOFIELD (Waterhead): The
female elephants have tusks as well as the males, but they are not so large;
and when the animals are young, the tusks are not visible.-JUVENIS (Car-
lisle): His 45 solutions are received. The impropriety of his concluding
remark is excused, because he is Juvenis.-T. WELLS (Linfield): We don't
T. WATKINSON (Stanstead): If your answer to the Four-ball Question be
2 inches, It is very near the truth-very near, indeed, to be found by expe-
root as the word excel.-W. CHEETHAM (Stockport): Study chemistry, and
you will see at once how the earth is neither lighter nor heavier since the
creation, by the immense consumption of coal.-AN ASPIRING APPRENTICE
is entitled 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
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
we can't fill our pages. He may learn either Latin or French; both are
useful: but we think his labour in learning French will be lost, if it be for
the sole purpose of reading Voltaire, Volney, Mirabeau, Rousseau, or any of
that motley infidel crew. His poetry is pretty fair, and indicates a better
state of mind than his former letter. It is essential to blank verse that
every line should contain ten syllables; and these syllables must, if possible,
be made iambic, that is, consisting of a short and a long syllable alternately,
as in Milton, thus:-

I should be glad to meet with four or five earnest young men
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-
dence" columns, in studying the theoretical part of their profes-riment alone. The word Excelsior means higher; it is from the same Latin
sion. Address, "T. J., Post-office, Bedminster, Bristol." Yours
respectfully,
T. J.

ALGEBRA, ETC.

SIR,-In compliance with your request, appended in a note to the "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 never knew anything of algebra until last Christmas, when I was induced to commence the study from seeing the easy lessons in your truly excellent publication, I hope you will give me credit for my statement.

I am really a self-taught man; but as languages and other subjects have occupied so much of my time and attention, I was surprised at the improvement I made within the short space of a fortnight in the study of algebra. Although I began in real earnest, 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 in the cause of self improvement and of education and literature generally, that I have formed a slight acqua atance with a science that always appeared to me to be involved in the most inexplicable

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ANSWERS TO CORRESPONDENTS.

A. C. H. (Lambeth) is recommended to make his own solutions. The exact strength for most purposes of testing is of no consequence. The great consideration is to have them pure. When a solution of any particular strength is necessary, we shall take care to intimate this. The occasion will not occur so long as we are occupied with qualitative not quantitative chemistry.

J. G. CUNNINGHAM (Sunderland): Right, and thanks.-A. B. (Hoxton): The subject on which he makes inquiry will soon appear.-H. HOLBROOK (Ipswich): Cassell's Arithmetic, Algebra, and Geometry, price 18. each, are the best for him; and they are indiensable.-PETER (Ripponden) should apply to the Bishop's chaplain, as he says.-T. R. (Durham): We can't give the required information.-J. G. THORNLEY (Ballyclare): It is entirely out of our power to oblige him and his friends with the subject requested at present.-PEREGRINE PICKLE (Aberystwith): We would strongly advise him to persevere with the Lessons on Music and Drawing in the P. E.; he will be sure ultimately to succeed. If he could by any means

"Before the heav'ns thou wērt, ănd at the võice
Ŏf God, ǎs with a mantle didst Invest."

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ON PHYSICS, OR NATURAL PHILOSOPHY.

No. XXVIII.

(Continued from page 8.)

VIBRATION OF THE AIR IN SONOROUS PIPES.

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 instruments 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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body, as well as in others formerly investigated, there are portions which vibrate and portions which do not vibrate, that is, swells and nodes. When a pipe is closed at the extremity opposite to that where the passage is, it is almost evident that the stratum of air in contact with the bottom

Fig. 148.

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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Sound 5,

2 nodes and 2 swells.

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 is called the passage; by this aperture the air is introduced into the pipe. The opening bo is called the mouth, and its upper lip, b, is made with a feathered edge. The upper part of both figures represents the pipe, which may be either open or closed according to the nature of the instrument. In fig. 146 the foot, or extremity, P, is employed to fix the pipe on a blowing machine, as shown in fig. 134, p. 364, vol. iv. When a rapid current of air rushes through the passage, it strike against the upper lip, and produces a concussion which prevents the air from issuing in a continuous manner from the mouth, and causes it to proceed intermittingly. From this cause the pulsations which are transmitted to the air in the pipe make it vibrate and emit a sound. In order that the emitted sound may be clear and distinct, a certain relation must be established among the lips, the aperture of the mouth, and the size of the passage; and the pipe must have a great length in comparison with its diameter. The number of vibrations depends, in general, on the dimensions of the pipe and the velocity of the current of air. In the German fute the mouth is a simple lateral and circular opening. It is owing to the arrangement of the lips that the current of air is made to strike against the edges of this aperture; this is seen in the Pandian reed, and in the perforated key which is made to whistle. The following representation (fig. 148) of an organ-pipe or flute, exhibits a side section and front view, in which shows the extremity of the passage, and the mouth. Since the vibration of air contained in the pipes is really the cause of former. the production of the sound, we shall find that in this sonorous pipes open at both ends, a series of sounds which arise from

After the first or fundamental sound, that which is produced the most naturally allows the node and the swell at the extremities of the pipe to remain; but a new node appears at a third part of the column of air reckoning from the mouth, and a new swell at a third part reckoning from the bottom. The higher sound which immediately follows forms an additional node and an additional swell. correspond to these different states of the pipe are higher and The sounds which higher. The manner in which the column of air is divided in a closed pipe, shows that the different sounds emitted are to each other as the series of odd numbers 1, 3, 5, 7, etc.

VOL. V.

In a pipe open at both ends the arrangement of the nodes and swells is different. The bottom, which in the former case determined at once a node of vibration, no longer exists; consequently, a swell takes its place. Thus, there is at each extremity of an open pipe, a swell; it is necessary, therefore, that there should be at least one node in the interior. Expe riment shows that this node is in the middle, and that it divides the column of air into two parts which vibrate in an inverted direction. Each of these parts is, of course, one-half of that which would vibrate if the pipe were closed at one end, and this explains the reason why the fundamental sound of the pipe open at both ends is the higher octave of the After this fundamental sound, there exist also, in

the subdivision of the column of air into aliquot parts, and
which produce new swells and new nodes, as exhibited in
fig. 150.
Fig. 150.

Fundamental sound,

or sound I.

called the tongue, and which, when it is fixed, fits the edges of the channel so as nearly to close it--this tongue is fixed only at its upper part; and 4th, an iron wire, bent at its lower part, which presses upon the tongue. This iron wire, which is called the catch, can be raised or lowered so as to regulate 1 node and 2 swells. 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

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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 of air. This tongue acts in a manner easily comprehended, see figs. 151 and 152, which show a front and side view of it. Fig. 151

Fig. 152.
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L

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M

N

pipe м N, 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 waves producing a sound whose height increases with the velocity of the current.

Membraneous Tongues.-The effects of membraneous tongues are not yet well ascertained; but they are interesting, as connected with the theory of the human voice. The following are the principal results of the researches of M. Muller, the physiologist, on this subject:

Lis the tongue fixed at one extremity, in front of a rectangular
opening, o, made in the side of the pipe, whether square or
round. This tongue is constructed in such a manner that it
is bent in a state of rest, and leaves the opening free to the
admission of air. But if a current of air be produced in any
manner so as to have a tendency to rush through the aperture
o, in the direction indicated by the arrow, the tongue is forced
against the side of the pipe, and for a moment it stops the flow
of the current; then its elasticity restores it to its first posi-
tion: after this it is again impelled against the side of the a
pipe, and so on alternately, as long as the air exerts a tendency
to rush through the aperture o. Thus we see that the impulse
of the air must be made intermittingly, and when these inter-
mittent impulses are sufficiently rapid, a sound arises which is
stronger than that proceeding from the vibrations of the tongue

or lamina alone.

This kind of tongue is used in hautboys, bassoons, clarionets, children's trumpets, and other simple instruments of this kind. Some organ-pipes have a mouth-piece like that already described, fig. 146; others have rigid tongues. Fig. 153 is a representation of one of the latter, with the arrangement best adapted for demonstration. It is mounted on the box of a blowing machine, and glass, inserted at E in the sides of the pipe, admits of the vibrations of the tongue being seen. A wooden apparatus, like a trumpet, is used for strengthening the sound. Fig. 154 represents the tongue outside of the pipe. It is composed of four pieces:-1st, a rectangular wooden pipe closed at the bottom, and open at the top, at o; 2nd, a brass plate, ce, pierced by a longitudinal aperture, called the channel, which is intended to give a passage to the air of the pipe N to the or fice o; 3rd, an elastic lamina, i, which is

1st. If a narrow strip or ribbon of caoutchouc be stretched over a ring about of an inch in diameter, and if by means of pipe of small diameter we blow on one of the edges of the ribbon in a direction oblique to its surface, the ribbon will be made to vibrate and a sound will be emitted.

2nd. If a strip of caoutchouc, of about of an inch in breadth, be stretched over a ring or a wooden frame, and if on each side of this elastic strip we fix a rigid plate made of card or of wood, so as to admit of a very straight and narrow slit between the plates and the caoutchouc, this strip will be made to vibrate by surrounding the ring or frame with the lips and blowing through it, or by blowing through a porte-vent or pipe, at one end of which the apparatus is placed.

3rd. If one-half of the orifice of a very short pipe be covered with an elastic membrane, and the other half with a rigid plate, leaving a narrow slit between them for the passage of the air, the membrane will be put into a state of vibration by the same means as in the preceding case.

4th. If the orifice of a very short pipe be covered with two elastic membranes instead of one elastic membrane and a rigid plate, leaving the narrow slit between them as before, the same effect will be produced by employing the same means. This arrangement is most like that of the glottis, or chink of the larynx, which is the upper part of the windpipe.

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