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douleur donnait à ses traits un caractère énergique et pas- innumerable scientific truths for the benefit and to the astosionné ;s ses beaux cheveux blonds ruisselaientj sur ses nishment of man; and in the midst of our wonder, we are épaules; ses petites mains, crispées par la fièvre, avaient forced to acknowledge and admire the omnipotence of study fini park saisir une des mains de l'Empereur, et lui com- in exploring the secret bosom of Nature, and snatching theremuniquaient leur chaleur brûlante..... Agenouillée, le from the hidden treasures she would willingly conceal; but in visage baigné de larmes, levant ses grands yeux bleus vers mental giant, with amazing alacrity, performing his incredible

the child, Zarah Colburn, we find the young and powerful celui duquel clle semblait attendre la vie ou la mort,20 elle feats of intellectual gymnastics over the rugged play-ground ne pouvait plus ni parler, ni pleurer, ni respirer.

of mathematical calculation. -N'êtes-vous pas Mlle de Lajolais ? 11 lui demanda l'Em

Zarah Colburn, the subject of the present short memoir, was pereur.

an American boy belonging to the state of Vermont; according Sans répondre, Maria pressa la main de l'Empereur avec to our authority he was born in 1907. When only six years plus de force.12

of age, his knowledge of arithmetic began to be discovered, his Il reprit avec sévérité : Savez-vous que c'est la seconde father having, to his astonishment, accidentally heard him tell fois que votre père se rend coupable d'un crime envers the product of two numbers ; and on asking him the multil'Etat, Mademoiselle ? 13

plication table, and a series of similar questions in the rule of -Je le sais m répondit Mlle de Lajolais, avec la plus multiplication, he found that the little prodigy answered them grande ingénuité; mais la première fois il était innocent,

all with the greatest possible ease.

In November, 1813, this astonishing youth, accompanied by

his father, happened to be in Newry for a few hours, on bis - Mais, cette fois, il ne l'est" pas, répliqua Bonaparte.15

way from Dublin to Belfast, whence he intended to proceed to Aussi c'est sa grâce que je vous demande, sire, reprit Glasgow College. During his short stay in Newry, he had Maria, grâce .... ou je mourraio devant vous.

scarcely time to take his hurried refreshment; for the people L'Empereur, ne pouvant plus maîtriser 16 son émotion, se of that intelligent town were too eager to see and to hear the baissa vers elle en lui disant :

young philosopher, of whom they had previously heard so -Eh! bien, oui, Mademoiselle, oui je vous l'accorde. much. Here, many intricate questions were proposed to him, Mais, relevez-vous. 17

and, as if by instinct, he solved them all with the greatest Et, lui jetant un sourire d'encouragement et de bonté, il rapidity and accuracy; and “ all by the mere operation of the dégageap ses mains tenues a toujours avec forces et s'éloigna mind, without the assistance of any visible symbol or convirement.

trivance." Zarah Colburn, be it remembered, never made

use of pen or pencil in solving the most difficult problems, no COLLOQUIAL EXERCISE.

matter how long or how abstruse the process might be that 1. Que fit l'Empereur en enten- 9. Où étaient les mains de was required. I have been often told that an humble house dant ces cris ?

in Water-street, Newry, was for many years afterwards 2. Que dit-il d'un ton d’impa- 10. Que faisait-elle aux pieds de pointed out to the inquisitive tourist as the place where " the

Napoléon ?

calculating boyonce stopped. 3. Quelles paroles énergiques la 11. Que lui demanda-t-il alors? Belfast, his extraordinary intellectual capacity was a favourite

In his progress on his journey, before Zarah Colburn reached jeune fille adressa-t-elle à Bo- 12 Quelle réponse lui fit topic; and many a juvenile arithmetician was ransacking his naparte?

own brains in order that he might find “a few puzzlers," as 4. Que dit l'Empereur et que 13. Que lui dit Napoléon, rela. I questions, to propose to the much talked-of youth. fit-il ? tivement à son père ?

Zarah, shortly after his arrival in Belfast, was, on the 16th 5. Pourquoi Mlle de Lajolais ne 14. Que répondit-elle ? of November, 1813, introduced to a meeting of the members

se laissa-t-elle pas intimider? 15. Que repliqua Bonaparte ? of the Royal Academical Institution; and the young readers 6. Qu'ajouta-t-elle en se traî- 16. L'Empereur semblait - il of the POPULAR EDUCATOR may rest assured that his capanant sur les dalles de marbre?

bilities were sufficiently tested under a high standard by those 7. Que fit alors l'Empereur ? 17. Que dit-il ?

literary gentlemen, and that to a degree that raised wonder 8. Quel caractère la douleur 18. Que fit-il avant de s'eloi- and delight to their very climax. donnait-elle aux traits de

Here the amazing calculating boy,” of only nine years of Maria ?

age, stood to be questioned; and here it was that he called NOTES AND REFERENCES.-----. passer outre, to go on, -2. passer outre, to go on, to pro without a parallel in history. He was first asked the product

forth such a strength of intellectual energy as to be almost ceed.--b. aura, has without doubt, probably; the future tense, in French, is often used to express probability. +0. assez rudement,

of 365 and 13, and his answer, on the moment, was simply with some abruptness.-d. il y allait

, etc.
, so precious a life was in 4,745. But it was soon seen that such questions as this

were danger, at stake.-e. dalles, floor; literally, flat stones.-. L. too easy for such a pet of Nature's choosing. He was next told part ii., $ 61–2.-9. déchirant, heart-rending h. avec tant d'in: to extract the cube root of 307,546,875, and with the greatest stance, so earnestly.-i. tenait, resembled that.-). L. part ii., $nature were proposed, and their solutions were effected by

readiness he answered 675. Other questions of a similar 49. R. (4):--K. fini par, mechanically, unconsciously; literally, at him with the same readiness and accuracy. A writer in one Zast.2. from reprendre; L. part ii., p. 100.-m. from savoir ; of the journals of the day says, “in short, there appeared to L. part ii., p. 104.—n. l, so.—7. from mourir; L. part ii., p. 96. be no limits to the powers of his mind in calculation. -p. L. part ü., $ 49, R. (1).--. tenues, held; from tenir ; L.

After exhibiting such rare proficiency at the Institution, and part ii., p. 108.

before such a learned body, the result of his examination naturally spread through all parts of the town, to garret, cellar, and drawing-room alike, as if carried on the wings of

electric agency; so that crowds of the literati thronged to the BIOGRAPHY --No. XIII.

coffee-room of the inn where he resided for the time, in order

to prove by their own experience what the most flexible crea ZARAH COLBURN.

dulity could scarcely believe. WHEN Nature, as if to show her own dignity, bestows on a Of the complex nature of the numerous questions proposed mere child, extraordinary mental powers, exceeding in magni- to Zarah at this exhibition, our readers may form some remote tude what experience and the most wonderful development idea from the few following examples. He was requested to can scarcely approach, our pride is so completely humbled, extract the cube root of 51,230,158,344 ; his answer, immedithat the pleasure we feel in contemplating the sublime pheno- ately given, was 3,714. menon is almost lost in the disappointment we meet in being He, in an instant, multiplied 349,621 by 5, and gave the corunable to reach its superlative grandeur.

rect product, 1,748,105. In men such as Newton and La Place, we find genius, by Again, he divided 2,608,732 by 4, and gave for answer the force of culture and indefatigable application, calling forth | 652,183.' Here, it is to be expected that the astonishment and

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pleasure of his auditors were great indeed; and it may be obtained by this means he should be enabled to get a college Bafely inferred that the problems proposed were still becoming education. An eminent literary gentleman even undertook to more and more abstruse.

write his life; but whether the laudable proposal was ever Now, it was proposed to him, given the sum and difference executed or not, I cannot tell, or whether the “ calculating of two numbers, 728, and 16, to find the numbers themselves; boy” ever got to college I never was able to ascertain. he answered 372 and 356. On being asked what factors would Perhaps, indeed, the vigorous flame of his intellect, so early produce 765,621, his answer was 85,069, multiplied by 9. kindled by “nature's touch," and so often called on to act Again, 877 was given as one of the factors of the same number mechanically, was neglected, suffered to run to waste, flicker, to find the other; and he instantaneously gave 873 as the and die, without ever knowing the blessings of a proper

development! For the sake of humanity and science, I hope Again, he was required tell the fourth root of 3,701,506; but pot-but I cannot banish my doubts on the subject, as I never he immediately said there was no root, which was indeed true, heard of him figuring in the mathematical world after he left the proposer having intentionally read the number wrong, for Belfast. the purpose, if possible, of " flooring" the young genius.

In disposition, Zarah was modest and playful, and in But the active powers of his mind seemed by far too great appearance presented nothing singular beyond other children to be taken by surprise on the broad arena of culculation, of his age, not even in the formation of his forehead, that Shortly after the proposal of the preceding question, he was

portion of the human fabric to which critics so eagerly direct asked the fourth root of 37,015,056 (the right number), and attention. Yet there is no doubt that, in after years, the gradual the modest little arithmetician, with his usual expertness, 'ease, development of such a mind would have acted on the counteand accuracy, answered 78, to the great surprise and delight nance of such an extraordinary person. of the whole auditory. Thus it was that the “ American Cal.

As some of the correspondents of the POPULAR EDUCATOR (of culating Boy,” Zarah Colburn, spent some time in Belfast, which I have been a constant reader from the first) expressed experiencing kindness wherever he went, and exciting the

a desire to know something of this wonderful boy, I have admiration of all by his truly wonderful facililty of managing endeavoured thus hurriedly to place before them this rather numbers, through the unassisted instrumentality of mental scanty memoir, collected from what I had heard of him, and operation.

from my old papers of the year 1813. Like those of many an eminent genius in humble life, his

My fellow-readers, keep good heart; my promise concernparents were poor, his father was struggling to send him to ing the memoir of our celebrated Dr. Thomson will be fulfilled. the University, but he had no money; it was therefore sug.

The POPULAR EDUCATOR shall have it when finished, before gested that a memoir of his life should be published, that it the ink is quite dry; but some time must yet elapse. should cost a guinea and a half, and that with the money Katesbridge, February 20th, 1854.

H. H. ULIDIA,

are contained 10 of such equal parts, or 18 of an inch; from 2 INSTRUMENTAL ARITHMETIC.-Vo. IV. to the same point, are contained 20 of such equal parts or

iof an inch; from 3 to the same point, 30 equal parts, or to SCALES OF VARIOUS EQUAL PARTS TO AN INCH.

of an inch; and so on. A unit of this line, the first one adjaIN Lesson No. III. on Instrumental Arithmetic, we gave a cent to the number 10, which should have been marked with drawing and description of an instrument called a Plane Scale zcro or 0, at its extremity on the right hand, is subdivided into and Protractor; we then omitted the drawing of the other side 10 equal parts at the bottom of the space which it occupies, and of the instrument for want of room; but we now insert it below, into 12 equal parts at the top of this space. Of the former subfig. 1, with a short description of its nature and use.

divisions, each one is a tenth part of an inch; hence this line

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In this scale there are fourteen lines of equal parts, all of with the bottom subdivisions is a decimal scale of inches, and which contain a certain number of equal parts to an inch. from it we can take off or measure any number of inches and The numbers placed at the left-hand side of the scale show tenths of an inch as far as the scale will permit, as, 1.7, 2.5, how many equal parts of the line on which it stands, one inch 3.8, etc. inches. Of the top subdivisions, each one is a titelfth contains, each unit of the line containing ten of these equal part of an inch; hence this line with the top subdivisions is : parts. Thus, the first line on the scale at the bottom has 10 duodecimal scale of inches, and from it we can take off or meamarked at the left-hand side, and the units 1, 2, 3, and 4, sure any number of inches and primes or twelfths of an inch as marked along the line from left to right; this means that an far as the scale will allow, as lin. 7', 2 in. 5', 3 in. S, etc. inch contains ten equal parts or subdivisions of this line, each Again, the second line on the scale, reckoning from the unit on this line containing 19 of an inch; hence, from 1 to bottom upwards, has 11 marked at the left-hand side, and the the beginning of the smaller divisions on the line on the right, l units 1, 2, 3, 4, and 5, marked along the line from left to right:

ablative.

this means that an inch contains 11 equal parts or subdivisions and the other half or ten on the other side. The range of of this line, each unit on this line containing 11 of an inch; these scales is as follows:hence, from 1 to the beginning of the subdivisions on the On the one side, the numbers of the subdivisions to an inch right, are contained 10 of such equal parts, or 11 of an inch; arefrom 2 to the same point, are contained 20 of such equal parts,

10, 11, 12, 133, 15, 163, 18, 20, 22, 25; or å of an inch; and so on. A unit of this line, the first one and on the other side, the numbers of the subdivisions to an adjacent to the number 11, is subdivided into 10 equal parts at inch are the bottom of the space it occupies, and into 12 equal parts at

28, 32, 36, 40, 45, 50, 60, 70, 85, 100. . the top of this space. Of the former subdivisions, each one is the left-hand primary division or unit of the lines on these a tenth part of a unit of this line, or a tenth part of ten-elevenths scales is sometimes subdivided into 10, 12, and 8 equal parts; of an inch, that is, one-cleventh of an inch; hence any number of elevenths of an inch may be obtained from this line as far as fortress, a piece of cannon, an engine, or of the different orders

as these subdivisions are of great use in drawing the parts of a the scale will permit, as 7, 12, 25, etc. elevenths of an inch. of architecture. Of the top subdivisions, each one is a twelfth part of a unit of the scale; this is intended for those who use this line merely on the Engineer's Rule and on Gunter's Scale.

In our next Lesson we shall explain the Logarithmic Lines as a line of equal parts, and prefer the duodecimal to the decimal subdivision of the unit.

Again, the third line on the scale, reckoning upwards, has 12 marked at the left-hand side, and the units 1, 2, 3, 4, 5,

ANSWERS TO CORRESPONDENTS. marked along the line from left to right; this means that an YORICK: The only fluids adapted for burning in lamps without wicks inch contains 12 equal parts or subdivisions of this line, each are some of the highly volatile hydrocarbons, of which those are best which unit on the line containing 19 of an inch; hence, from 1 to the this purpose, must be constructed on peculiar principles; ordinary lamps

approach nearest to the composition of Benzole. The lamp, however, for beginning of the subdivisions on the right, are contained 10 will not do. Is our correspondent aware that a patent has been taken out of such equal parts, or 12 of an inch; from 2 to the same point, for this description of lamp by Mr. Holloway? Ảe has a depot in Holborn are contained 20 of such equal parts or îi of an inch; and so inconvenience, and have given rise to numerous accidents, some of them

Wickless lamps are attended with considerable on. A unit of this line, the first adjacent to the number 12, is fatal. subdivided into 10 equal parts at the bottom of the space it IACO: The employment of the words “either"“or” certainly are amoccupies, and into 12 equal parts at the top of this space.

Of biguous; let our correspondent supply the words “both” “and” in their

place. the former subdivisions, each is a tenth part of a unit of this nickel, are not precipitated from their solutions by hydrosulphuric acid

What we meant to state was, that iron, manganese, cobalt, and line, ur a tenth part of ten-twelfths of an inch, that is, one- alone, but are precipitated from their solutions by hydrosulphate of twelfth of an inch; hence any number of twelfths of an inch ammonia.-R. G. SMITH: The precipitate furnished by sulphuretted may be obtained from this line as far as the scale will permit, hydrogen in a pure solution of silver, is black.

W. B. HODSON (Harby): It is not necessary to study Logic before as 7, 14, 27, etc. twelfths of an inch. Of the top sutdivisions, Euclid; the geometry of Euclid is the finest specimen of Logic the world each one is a twelfth part of a unit of the scale.

ever saw. Whateley's Logic is reckoned among our best modern treatises.Next, the fourth line on the scale, reckoning upwards, has H. K. W. (Brixton): Won't do; must try again ; but before doing so, read 13} marked at the left-hand side, and the units 1, 2, 3, 4, 5, 6, Moody is quite correct as far as Latin is concerned. The English has no marked along the line from left to right; this means that an inch contains 133 equal parts or subdivisions of this line, each

E. H. B. (Birkenhead) should study Cassell's Arithmetic, Algebra, and

Geometry in their order, that is, Arithmetic first; then Algebra; and 10

20 unit on the line containing of an inch, or of an inch; will do as well. The French Dictionary will soon be finished, if not so

then Geometry; or if he prefers it, the lessons on these subjects in the P. E. 131

27

already. The suggestions made will be kept in view.-H. GEORGE (Bristol): hence, from 1 to the beginning of the subdivisions on the right, it is most likely that the Lessons in Geology will assume a separate form. are contained 10 of such equal parts, or in of an inch ; from 2 Any book on navigation is a guide to the compass. to the same point, are contained 20 of such equal parts, or of Lessons in English are published at 35.-J. CHADWICK (Royton) should

WILTON (Bebbington): The Geography will be issued with an Atlas; the an inch; and so on. A unit of this line, the first adjacent to make the experiment referred to; we have never seen the statement. It is the number 133, is subdivided into 10 equal parts at the boto generally considered that the best glass for an electrical machine is the tom of the space it occupies, and into 12 equal parts at the top that which contains a large proportion of silex; and it should not be too

whitest, the most transparent, the hardest, and freest from bubbles, and of this space. Of the former subdivisions, each one is a tenth thick.

10 part of a unit of this line, or a tenth part of of an inch, or Vol. iv. page 2, col.2, line 23 from bottom, for 400th. read 40,009th.

132

page 126, col. 2, line 15 from top, for £170 read £180. a tenth part of twenty twenty-sevenths of an inch, that is, two

page 126, col, 2, line 10 from bottom, for Liverpool read Manchester. twenty-sevenths of an inch; hence any number of parts of which 13] make an inch may be obtained from this line as before, only as far as the scale will permit. Of the top subdivisions,

LITERARY NOTICES. each one is a twelfth part of a unit of the scale. In the latter

CASSELL'S DATIN DICTIONARY, BY J. R. BEARD, D.D. - The publicacase, the part of an inch thus obtained is a compound and tion of this Dictionary has commenced, and will be completed in about

1 10
1 20

Twenty-six Nuinbers, ,THREEPENCE each, or in Monthly Parts, ONE complex fraction denoted by of

of that is, SHILLING each. Part the First is now ready; Part the Second will be ready 12 132 12 27'

with the Magazines for April. 5

CASSELL'S FRENCH AND ENGLISH DICTIONARY.--The FRENCH and of an inch.

ENGLISH portion of this important Dictionary is now completed, and may be 81

had, price 4s., or strongly bound, 5s. The ENGLISH and FRENCH portion In the same way we might proceed to explain the remaining is in the course of publication, and will be completed in about Twelve lines of this scale; but we presume that we have sufficiently volume, will be ready with the Magazines for April, price 98.6d.

Numbers, THREEPEXCE each. The entire Dictionary, forming one handsome explained the first four lines on the scale, to render the remain

DICTIONARY.The GERMANing ten lines equally easy of comprehension. Besides, we ENGLISH Portion of this Dictionary is now ready, price 5s. in stiff covers, must leave a little to the ingenuity of our students, otherwise or 5s.6d, strong cloth.--The ENGLISH-German Portion will be completed there would be no excitement for them to study. Moreover, all Volume, strongly bound,

at 9s., will shortly be issued.

as quickly as possible, in Numbers, THRLEPENCE each; and the entire these different lines may be used as scales of equal parts differ- CASSELL'S SHILLING EDITION or FIRST LESSONS IN LATIN. By Proing from one another in the magnitude of the unit by very fessors E, A. ANDREWS and S. Stoddard. Revised and Corrected. Price small and gradual differences, so that a student from amongst

ls. paper covers, or ls.6d. neat cloth.

CASSELL'S LATIN GRAMMAR. For the use of Schools and Colleges. By them may get almost any scale that will answer his drawings. Professors E. A. Andrews and S. STODDARD. Revised and Corrected. The relation of the units of these scales to an inch is a matter Price 3s. 6d. in cloth boards. in general of small importance to a vast variety of mechanical unrivalled by thousands of students. Dany who have studied Latin from

CASSELL'S LESSONS IN LATIN. These Lessons have been pronounced drawings; still, if such relation be wanted, it can be fully other grammars and on other systems, and have completely failed, havo obtained on the principles which we have already explained. acquired more real knowledge of the Latin Tongue from these Lessons in Some plane scales of the kind which we have described may six months, than they have acquired in as many years by the means hitherto be had with twenty lines on them adapted to different scales adopted. Price 28.6d. in paper covers, or 3s. in cloth.

A KEY TO CASSELL'S LESSONS IN LATIN. Containing Trarslations of of measurement, namely, the one half or ton on the one side, I all the Exercises. Price ls. in paper covers, or 1s. 6d. cloth.

ERRATA.

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means of fragments of lines which indicate their relative ON PHYSICS, OR NATURAL PHILOSOPHY, position. The other consists in the use of keys or clefs, which

are signs employed to raise or lower the gamut by several No. XXVI.

tones, and even several octaves. The clefs are commonly at

the beginning of a piece of music, and the normal intonation (Continued from page 364.)

is determined by them. They are accompanied with a variety

of other signs which regulate the different conditions of the PHYSICAL THEORY OF MUSIC

performance ; these will be afterwards explained. Quality of Musical Sound. The result of continued, rapid, and isochronous vibrations, which produce on the organ of

Fig. 135. hearing a prolonged sensation, is called a musical sound. Such a sound can always be compared with others of the same kind as to their unison or discord; and in this respect it is to be wholly distinguished from mere noise. The ear can perceive in musical sounds three particular qualities--height, intensity, and distinctness; the last of which the French call timbre.

The impression made upon the organ of hearing by the greater or less number of vibrations made in a given time, is called the height of a musical sound. Sounds which are produced by a small number of vibrations, are called low; and those which arise from a great number of vibrations, are called

The notes of the garnut can be represented by numberto of the scale of perceptible sounds, are properly called low For this purpose, we take for it, the fundamental sound of the or high. All the intermediate sounds are called low or high produced by a cord vibrating throughout its whole length. only in a relative manner. high sound, as we speak of a low temperature or a high By varying the position of the moveable bridge B, fig. 129, No. 1, temperature, by comparing the sound with those which most page 362, an experimenter, who has a practised ear, can easily commonly fall upon the ear. The relative depth or height of find the length which must be successively given to the two sounds is called tone ; that is, this word expresses the vibrating part a B, in order to produce the six other notes. degree of the height of a given sound; and in a musical point which gives ut, we find that the lengths of the cords which

, the of view, it expresses the degree of the height of the scale to give the other notes will be represented by the following which it belongs.

It has been already shown that the intensity or the force of scale of numbers containing fractions of unity :the sound depends on the amplitude of the oscillations, and not

(A) Relatise lengths of the Cords 1, LTE, A

ut, mi, fa, sol, la, si; The same sound may preserve the same on their number. degree of height or depth, and yet assume a greater or less

Thus, the length of the cord which gives the note ré, is only intensity, according to the amplitude of the oscillations which of the length of that which gives the note ut; the length of produce it. This is seen in a tense cord, as it is made to the cord which produces the note mi, is only of that which depart more or less from its position of equilibrium.

produces the note ut; and so on. Such are the numbers Distinctness, or timbre, is observed in the case of two dif- which are employed to represent the notes of the gamut, ferent instruments which yield each a sound of the same according to the relative length of the cords which produce height or intensity; and yet these two sounds can be per- them. By continuing to advance the place of the bridge on fectly distinguished from each other. Thus, the sound of the the sonometer, we find that the eighth sound produced by the hautboy is very different from that of the flute; or the sound half of the length of the cord is the same as the fundamental of the horn from that of the bassoon. In the same way, the sound. The same series of ratios already given recommences human voice varies much; that is, it presents a very different at this sound, and we obtain a new gamut, perfectly cortimbre, according to the individual, the age, or the sex. The responding to the first; the length of the cord corresponding cause of this quality is unknown. It appears to depend not to each note of this second gamut, being the half of that only on the matter of which the instruments are composed, which answers to the note of the same name in the preceding but also on their form, and on the mode in which they are pụt gamut; and so on, for a third and a fourth gamut. in action. Thus, the sound of a brass trumpet is completely In order to ascertain the relative number of vibrations in changed by being strongly heated in an oven, and a straight the same time corresponding to each note, we have only to take trumpet has a louder sound than a curved one.

the reciprocals of the fractions in the preceding table; for, Unison.- When two sounds are produced by the same according to the first law of the vibrations of cords, formerly number of vibrations per second, they are said to be in unison; stated, the number of the vibrations of a cord is in the inverse that is, they are equally low or equally high. Thus, the ratio of its length. Representing, therefore, the number of wheel of Savart and the siren are in unison when their counters the vibrations of a cord which give the fundamental sound ut indicate the same number of vibrations in the same time. The by unity or 1, we have the following table (B) of the reci. unison of a musical sound can always be determined; but not procals of the preceding table (A) :that of a noise. The number of vibrations of any sonorous body is, in fact, determined by putting it in unison with the

Notes of the Gamut

ut, , mi, fa, sol, la, ;

(B) Relative Num. of Vibrations siren.

1, š, , , , , 20 The Musical Scale, or Gainut.-We give the name of the The gamut, of which the ratios of the vibrations of the notes Musical Scale to a series of sounds separated from one another have now been given, is called the Diatonic Scale; the gamut by intervals, which appear to have their origin in the nature of which proceeds by semitones, and which contains thirteed our organization. In this series, the sounds are reproduced sounds, is called the Chromatic Scale. in the same order by periods of seven sounds, each period being Abrolute Number of Vibrations to each Note.--The siren affords denominated a Gamut, and the seven sounds, or notes of each a simple method of deducing from the preceding table the gamut, are known by the names, ut, , ani, fa, sol, la, si. real number of vibrations which are produced by each of the

The notes of the Gamut are represented to the eye by notes of the musical scale. Thus, if we put this apparatus in placing them on what is called the staf; which consists of five unison with the fundamental note ut, it will point out to us parallel straight lines and four intervening spaces. The the exact number of vibrations which correspond to this note. double staff used for the niusic of the pianoforte represents, We have then only to multiply this number by the ratios , within its extent, a series of three octaves, as exhibited in the , etc., of the preceding table, in order to find the exact following table, fig. 135.

number of the vibrations of the other notes. When it is necessary to go beyond the extent of this staff, Now, as the fundamental sound which is taken for the note two methods are used. The one consists in giving to the ut, varies with the length of the cord of the sonometer, with its notes which exceed the normal staff, a supplementary staff, by / tension and with its nature, so the number of vibrations cos

104

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responding to this note will also vary. The real number of which produce on the ear an agreeable sensation, is generally vibrations, calculated as we have now shown, may be repre- denominated harmony, concord, or accord. There is harmony sented by an infinity of numbers, to which will correspond as only' when the numbers of the vibrations of the simultaneous many different gamüts. Among all the scales which may be sounds are connected with each other by a simple ratio; if the thus represented, that has been selected of which the note ut ratio is complex, the ear is affected in a disagreeable manner, corresponds to the lowest sound of the bass, and in physics the and the simultaneous sounds are called a discord or dissonance. 110tes of this gamut are indicated by giving the index 1, as uti; The simplest concord is unison ; then follow the octave, the whilst to the notes of the higher gamuts are given the indices fifth, the third, the fourth, and the sixth. A perfect harmony 2, 3, etc., as uth, uts, etc.; and to the notes of the lower gamuts, or concord is the name given to three simultaneous sounds, the indices --1,--2, etc., as aut

ut 2, ré ré_2, etc.

such as the first and the second forming a third major, the Having ascertained by experiment, that the number of vibra- 'second and the third forming a third minor, and the first and tions corresponding to the lowest sound of the bass is 128, we the third forming a fifth ; that is, to three sounds, such that (B), in order to obtain the absolute number of vibrations for as the numbers 4, 5, and 6. Thus, the three notes fa, la, ut; or (B), in order to obtain 38 number by the ratios giren in table the numbers of their corresponding vibrations are to each other each note; whence, we have the following table :

ut, mi, sol; or sol, si, r'e', form three perfect concords. These

si. ist, re, mi, fllo sol, la,

are thé harmonies which produce on the ear the most agreeable (C) Notes

of the Gamut
Abs. No. of Sim. Vib. 128, 146, 1.60, 170$, 192, 213,240.

musical sensation.

Pulsation. When two sounds, which are not in unison, are The absolute numbers of vibrations for the notes of the produced simultaneously, there is heard at equal intervals a bigher scales, are obtained by.maltiplying the numbers of this strengthening of sound which is called a pulsation. Thus, if table (C), by 2, by 4, by 8, etc.; and for those of the lower the numbers of the vibrations of two sounds are 30 and 31, scales, by dividing the same numbers by 2, by 4, by 8, etc. after 30 vibrations of the first, or 31 of the second, there will Thus, the number of simple vibrations corresponding to sals, is be a coincidence, and consequently a pulsation. equal to 192 X 4, or 768 per second; and the number corres- If the pulsations are sufficiently near each other to produce ponding to si_3, is equal to 240 - 4, or sixty per second. a continued sound, it will be evidently lower than those from

Length of the Waves. When we have ascertained the num- which it is derived, since it proceeds from a single vibration, ber of simple vibrations which a sonorous body makes per when the other sounds make 30 and 31 vibrations. second, it is easy to deduce from them the length of the waves. The Tuning Fork.--The tuning-fork is a small instrument by We know that sound passes over about 1,120 feet per second, the aid of which we can reproduce at pleasure an invariable at a mean state of the atmosphere, or about 60° Fahrenheit. note ; it thus becomes a suitable apparatus for regulating If, therefore, a body made only one simple vibration per musical instruments as well as the human voice. It consists second, the length of the wave would be 1,120 feet; if it made of a steel rod bent into the form of a pair of sugar tongs; two such vibrations, the length of the wave would be half of fig. 136. It is made to vibrate by drawing a bow across its 1,120 feet; and so on. Now we have seen that 128 simple vibrations per second correspond to the note uti; the length

Fig. 136. of the waves, therefore, is the quotient of 1,120 feet divided by 128, that is, 8.75 feet. The following table shows the length of the wave corresponding to the first note of the successive garuts or scales :

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First Notes.

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_13 aut zat

1 uta

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Number of
Vibrations,

16
32
64
128
256
512
1024

:

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1

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Lengths of Waves.

70.00 feet
35.00
17:50
8.75

4.375
it 3

2.1:875

1.09375, Intervals, Sharps and Flats.- The ratio of one sound to abrother in music is called an interval, that is, the quantity which indicates by how.much one sound is higher than another. The interval from ut to ré is called a second; from ut to mi, a third; from ut to fa, a fourth; from ut to sol, a fifth; from.ut to la, a sixth; from ut to si, a seventh ; and from ut to ut, an octave. The following table shows the intervals of the consecutive notes, which are obtained by dividing the number of the vibrations of any note by that of the vibrations of the note immediately below it :Notes of the scale

act, re, mi, fa, sol, la, si, ut. ledges, or by suddenly separating its two branches by means of PD) Relative No. of vibrations 1, 8, , , , , , 2. a cylindrical piece of iron which is forcibly drawn between Intervals

$, 14, 19, , , , $, 14. them, as shown in the figure. The two branches being thus From this table we perceive that the different intervals are forced out of their state of equilibrium, return to it after a reduced to three, which are $, 10 and 15. The first of these, certain number of vibrations, and produce a .constant spund which is the greatest, is called the tone major; the second, the for each instrument of this kind. The sound of this apparatus tone minor; and the third, which is the least, is called the is greatly increased by fixing it on a wooden box, open at one semitune major. The interval between the tone major and the l'or both of its extremities. It may also be put into vibration tone minor is i. This is the smallest interval which is taken by holding it in the hand by the piece attached to the bend, in into oonsideration; it is only a praotised ear which can appre- the instrument, and striking either end of it against a wooden ciate this interval, a quantity known in music by the name of 'board, comma. Composers have been led to intercalate: between the notes of the Gamut certain intermediate notes, which are VIBRATION OF RODS, PLATES, AND MEMBRANES. distinguished by the names of sharps and fats. To sharpen a Vibration of Rods and Lamina.-Rods and thin laminæ in note, is to increase the numbep of its vibrations in the ratio of wood, glass, and especially in tempered steel, vibrate in con24 to 25 ; to flatten it, is to diminish the same number in the sequence of their elasticity, and exhibit, like cords, two kinds ratio of 25 to 24. In music, the sharp is denoted by the sign of vibrations; the one transversal, and the other longitudinal. and the flat by the sign D.

The transversal are produced by fixing the rods or laminæ at Harmony, Discord. -- The co-existence of several sounds one extremity, and passing a bow over the part which is free

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