3. Quelles paroles énergiques la jeune fille adressa-t-elle à Bonaparte?

4. Que dit l'Empereur et que

fit-il ?

5. Pourquoi Mlle de Lajolais ne se laissa-t-elle pas intimider? 6. Qu'ajouta-t-elle en se traînant sur les dalles de marbre? 7. Que fit alors l'Empereur?

9. Où étaient les mains de

l'enfant ?

10. Que faisait-elle aux pieds de
Napoléon ?

11. Que lui demanda-t-il alors?
12 Quelle réponse lui fit

Maria?

13. Que lui dit Napoléon, rela-
tivement à son père?

14. Que répondit-elle ?
15. Que repliqua Bonaparte?
16. L'Empereur semblait il
ému ?

17. Que dit-il ?

innumerable scientific truths for the benefit and to the astonishment of man; and in the midst of our wonder, we are forced to acknowledge and admire the omnipotence of study in exploring the secret bosom of Nature, and snatching therefrom the hidden treasures she would willingly conceal; but in the child, Zarah Colburn, we find the young and powerful mental giant, with amazing alacrity, performing his incredible feats of intellectual gymnastics over the rugged play-ground of mathematical calculation.

Zarah Colburn, the subject of the present short memoir, was an American boy belonging to the state of Vermont; according to our authority he was born in 1807. When only six years of age, his knowledge of arithmetic began to be discovered, his father having, to his astonishment, accidentally heard him tell the product of two numbers; and on asking him the multiplication table, and a series of similar questions in the rule of multiplication, he found that the little prodigy answered them 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 his way from Dublin to Belfast, whence he intended to proceed to Glasgow College. During his short stay in Newry, he had scarcely time to take his hurried refreshment; for the people of that intelligent town were too eager to see and to hear the young philosopher, of whom they had previously heard so much. Here, many intricate questions were proposed to him, and, as if by instinct, he solved them all with the greatest rapidity and accuracy; and "all by the mere operation of the mind, without the assistance of any visible symbol or contrivance." Zarah Colburn, be it remembered, never made use of pen or pencil in solving the most difficult problems, no matter how long or how abstruse the process might be that was required. I have been often told that an humble house in Water-street, Newry, was for many years afterwards pointed out to the inquisitive tourist as the place where "the calculating boy" once stopped.

In his progress on his journey, before Zarah Colburn reached Belfast, his extraordinary intellectual capacity was a favourite topic; and many a juvenile arithmetician was ransacking his own brains in order that he might find "a few puzzlers, questions, to propose to the much talked-of youth.

as

Zarah, shortly after his arrival in Belfast, was, on the 16th of November, 1813, introduced to a meeting of the members of the Royal Academical Institution; and the young readers of the POPULAR EDUCATOR may rest assured that his capabilities were sufficiently tested under a high standard by those 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
Maria ?

gner?

NOTES AND REFERENCES.-a. passer outre, to go on, to proeeed.—b. aura, has without doubt, probably; the future tense, in French, is often used to express probability.-c. assez rudement, with some abruptness.— d. il y allait, etc., so precious a life was in danger, at stake.-e. dalles, floor; literally, flat stones.f. L. part ii., § 61-2.—g. déchirant, heart-rending-h. avec tant d'instance, so earnestly.-i. tenait, resembled that.-j. L. part ii., § 49. R. (4).-k. fini par, mechanically, unconsciously; literally, at last.-7. from reprendre; L. part ii., p. 100.-m. from savoir; L. part ii., p. 104.—n. l', so.—o. from mourir; L. part i., p. 96. -p. L. part ii., § 49, R. (1).—q. tenues, held; from tenir; L. part ii., p. 108.

BIOGRAPHY.-No. XIII.

ZARAH COLBURN.

WHEN Nature, as if to show her own dignity, bestows on a mere child, extraordinary mental powers, exceeding in magnitude what experience and the most wonderful development can scarcely approach, our pride is so completely humbled, that the pleasure we feel in contemplating the sublime phenomenon is almost lost in the disappointment we meet in being unable to reach its superlative grandeur.

In men such as Newton and La Place, we find genius, by the force of culture and indefatigable application, calling forth

Here the amazing "calculating boy," of only nine years of age, stood to be questioned; and here it was that he called without a parallel in history. He was first asked the product forth such a strength of intellectual energy as to be almost of 365 and 13, and his answer, on the moment, was simply 4,745. But it was soon seen that such questions as this were too easy for such a pet of Nature's choosing. He was next told to extract the cube root of 307,546,875, and with the greatest readiness he answered 675. Other questions of a similar nature were proposed, and their solutions were effected by him with the same readiness and accuracy. A writer in one of the journals of the day says, "in short, there appeared to be no limits to the powers of his mind in calculation.

After exhibiting such rare proficiency at the Institution, and 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 coffee-room of the inn where he resided for the time, in order to prove by their own experience what the most flexible credulity could scarcely believe.

Of the complex nature of the numerous questions proposed to Zarah at this exhibition, our readers may form some remote idea from the few following examples. He was requested to immediextract the cube root of 51,230,158,344; his answer, ately given, was 3,714.

He, in an instant, multiplied 349,621 by 5, and gave the correct product, 1,748,105.

Again, he divided 2,608,732 by 4, and gave for answer 652,183. Here, it is to be expected that the astonishment and

20

10

11

are

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 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 of an inch; | from 2 to the same point, are contained 20 of such equal parts, 10, 11, 12, 133, 15, 16, 18, 20, 22, 25; or 2 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 arethe bottom of the space it occupies, and into 12 equal parts at 28, 32, 36, 40, 45, 50, 60, 70, S5, 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 as these subdivisions are of great use in drawing the parts of a 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 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 as a line of equal parts, and prefer the duodecimal to the decimal subdivision of the unit.

on.

12

12

In our next Lesson we shall explain the Logarithmic Lines on the Engineer's Rule and on Gunter's Scale.

ANSWERS TO CORRESPONDENTS.

IACO: The employment of the words "either ""or" certainly are am"and" in their biguous; let our correspondent supply the words "both" What we meant to state was, that iron, manganese, cobalt, and nickel, are not precipitated from their solutions by hydrosulphuric acid alone, but are precipitated from their solutions by hydrosulphate of ammonia.-R. G. SMITH: The precipitate furnished by sulphuretted hydrogen in a pure solution of silver, is black.

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, 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 of an inch; hence, from 1 to the this purpose, must be constructed on peculiar principles; ordinary lamps 12 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 g of an inch; from 2 to the same point, for this description of lamp by Mr. Holloway? He has a depot in Holborn are contained 20 of such equal parts or of an inch; and so inconvenience, and have given rise to numerous accidents, some of them (No. 117, we believe). Wickless lamps are attended with considerable 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 occupies, and into 12 equal parts at the top of this space. Of the former subdivisions, each is a tenth part of a unit of this line, or a tenth part of ten-twelfths of an inch, that is, onetwelfth of an inch; hence any number of twelfths of an inch may be obtained from this line as far as the scale will permit, as 7, 14, 27, etc. twelfths of an inch. Of the top subdivisions, each one is a twelfth part of a unit of the scale. Next, the fourth line on the scale, reckoning upwards, has reckoning upwards, has 133 marked at the left-hand side, and the units 1, 2, 3, 4, 5, 6, marked along the line from left to right; this means that an inch contains 133 equal parts or subdivisions of this line, each 10 20 unit on the line containing of an inch, or of an inch; 13/2 27 hence, from 1 to the beginning of the subdivisions on the right, are contained 10 of such equal parts, or of an inch; from to the same point, are contained 20 of such equal parts, or of an inch; and so on. A unit of this line, the first adjacent to the number 13, is subdivided into 10 equal parts at the bottom of the space it occupies, and into 12 equal parts at the top of this space. Of the former subdivisions, each one is a tenth 10 part of a unit of this line, or a tenth part of of an inch, or 13/

40

27

2

a tenth part of twenty twenty-sevenths of an inch, that is, two
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,
each one is a twelfth part of a unit of the scale. In the latter
case, the part of an inch thus obtained is a compound and
10
1 20
complex fraction denoted by of
of
› or
that is,
131 12 27

5

of an inch.

81

1

12

In the same way we might proceed to explain the remaining lines of this scale; but we presume that we have sufficiently explained the first four lines on the scale, to render the remaining ten lines equally easy of comprehension. Besides, we must leave a little to the ingenuity of our students, otherwise there would be no excitement for them to study. Moreover, all these different lines may be used as scales of equal parts differing from one another in the magnitude of the unit by very small and gradual differences, so that a student from amongst them may get almost any scale that will answer his drawings. The relation of the units of these scales to an inch is a matter

in general of small importance to a vast variety of mechanical drawings; still, if such relation be wanted, it can be fully obtained on the principles which we have already explained. Some plane scales of the kind which we have described may be had with twenty lines on them adapted to different scales of measurement, namely, the one half or ten on the one side,

W. B. HODSON (Harby): It is not necessary to study Logic before Euclid; the geometry of Euclid is the finest specimen of Logic the world ever saw. Whateley's Logic is reckoned among our best modern treatises.H. K. W. (Brixton): Won't do; must try again; but before doing so, read much good poetry, as Milton, Cowper, etc.-DOUBTFUL (Bishop-Auckland): Moody is quite correct as far as Latin is concerned. The English has no ablative.

E. H. B. (Birkenhead) should study Cassell's Arithmetic, Algebra, and Geometry in their order, that is, Arithmetic first; then Algebra; and then Geometry; or if he prefers it, the lessons on these subjects in the P. E. will do as well. The French Dictionary will soon be finished, if not so already. The suggestions made will be kept in view.-H. GEORGE (Bristol): It is most likely that the Lessons in Geology will assume a separate form. Any book on navigation is a guide to the compass. WILTON (Bebbington): The Geography will be issued with an Atlas; the Lessons in English are published at 3s.-J. CHADWICK (Royton) should make the experiment referred to; we have never seen the statement. It is generally considered that the best glass for an electrical machine is the 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 thick.

ERRATA.

Vol. iv. page 2, col. 2, line 23 from bottom, for 400th. read 40,000th.
page 126, col. 2, line 15 from top, for £170 read £180.
page 126, col. 2, line 10 from bottom, for Liverpool read Manchester.

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ON PHYSICS, OR NATURAL PHILOSOPHY.

No. XXVI.

(Continued from page 364.)

PHYSICAL THEORY OF MUSIC

Quality of Musical Sound.-The result of continued, rapid, and isochronous vibrations, which produce on the organ of 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 high. Those sounds, therefore, which are at the extremities of the scale of perceptible sounds, are properly called low or high. All the intermediate sounds are called low or high only in a relative manner. Yet we speak of a low sound or a high sound, as we speak of a low temperature or a high temperature, by comparing the sound with those which most commonly fall upon the ear. The relative depth or height of two sounds is called tone; that is, this word expresses the degree of the height of a given sound; and in a musical point of view, it expresses the degree of the height of the scale to which it belongs.

means of fragments of lines which indicate their relative position. The other consists in the use of keys or clefs, which are signs employed to raise or lower the gamut by several tones, and even several octaves. The clefs are commonly at the beginning of a piece of music, and the normal intonation is determined by them. They are accompanied with a variety of other signs which regulate the different conditions of the performance; these will be afterwards explained.

It has been already shown that the intensity or the force of the sound depends on the amplitude of the oscillations, and not on their number. degree of height or depth, and yet assume a greater or less intensity, according to the amplitude of the oscillations which produce it. This is seen in a tense cord, as it is made to depart more or less from its position of equilibrium.

(A) { The same sound may preserve the same

Distinctness, or timbre, is observed in the case of two different instruments which yield each a sound of the same height or intensity; and yet these two sounds can be perfectly distinguished from each other. Thus, the sound of the hautboy is very different from that of the flute; or the sound of the horn from that of the bassoon. In the same way, the human voice varies much; that is, it presents a very different timbre, according to the individual, the age, or the sex. The cause of this quality is unknown. It appears to depend not only on the matter of which the instruments are composed, but also on their form, and on the mode in which they are put in action. Thus, the sound of a brass trumpet is completely changed by being strongly heated in an oven, and a straight trumpet has a louder sound than a curved one.

Unison. When two sounds are produced by the same number of vibrations per second, they are said to be in unison; that is, they are equally low or equally high. Thus, the wheel of Savart and the siren are in unison when their counters indicate the same number of vibrations in the same time. The unison of a musical sound can always be determined; but not that of a noise. The number of vibrations of any sonorous body

3

8

15

The notes of the garnut can be represented by numbers. For this purpose, we take for ut, the fundamental sound of the produced by a cord vibrating throughout its whole length. sonometer explained in a former Lesson; that is, the sound By varying the position of the moveable bridge B, fig. 129, No. 1, Page 362, an experimenter, who has a practised ear, can easily find the length which must be successively given to the Thus, by representing by unity or 1, the length of the cord vibrating part A B, in order to produce the six other notes. which gives ut, we find that the lengths of the cords which give the other notes will be represented by the following scale of numbers containing fractions of unity:Names of the Notes ut, ré, mi, fa, sol, la, si; Relative lengths of the Cords 1, §, §, 2, 3, 3, f. Thus, the length of the cord which gives the note ré, is only of the length of that which gives the note ut; the length of the cord which produces the note mi, is only of that which produces the note ut; and so on. Such are the numbers which are employed to represent the notes of the gamut, according to the relative length of the cords which produce them. By continuing to advance the place of the bridge on the sonometer, we find that the eighth sound produced by the half of the length of the cord is the same as the fundamental sound. The same series of ratios already given recommences at this sound, and we obtain a new gamut, perfectly corresponding to the first; the length of the cord corresponding to each note of this second gamut, being the half of that which answers to the note of the same name in the preceding gamut; and so on, for a third and a fourth gamut.

In order to ascertain the relative number of vibrations in the same time corresponding to each note, we have only to take the reciprocals of the fractions in the preceding table; for, according to the first law of the vibrations of cords, formerly stated, the number of the vibrations of a cord is in the inverse ratio of its length. Representing, therefore, the number of the vibrations of a cord which give the fundamental sound ut by unity or 1, we have the following table (B) of the reciprocals of the preceding table (A) :—

thaly is, in fact, determined by putting it in unison with the (B) Notes of the Gram Vibrations ut, re, mi, fa, sol, la, si

siren.

The Musical Scale, or Gamut.-We give the name of the Musical Scale to a series of sounds separated from one another by intervals, which appear to have their origin in the nature of our organization. In this series, the sounds are reproduced in the same order by periods of seven sounds, each period being denominated a Gamut, and the seven sounds, or notes of each gamut, are known by the names, ut, ré, mi, fa, sol, la, si.

The gamut, of which the ratios of the vibrations of the notes have now been given, is called the Diatonic Scale; the gamut which proceeds by semitones, and which contains thirteen sounds, is called the Chromatic Scale.

Absolute Number of Vibrations to each Note.-The siren affords a simple method of deducing from the preceding table the 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 staff, 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 music 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, two methods are used. The one consists in giving to the notes which exceed the normal staff, a supplementary staff, by

VOL. IV.

Now, as the fundamental sound which is taken for the note ut, varies with the length of the cord of the sonometer, with its tension and with its nature, so the number of vibrations cor 104

(C)

si.

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 gamuts. 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. notes of this gamut are indicated by giving the index 1, as ut1; 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 ut2, 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 ut t_1, ut_2, ré 1, 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 have only to multiply this number by the ratios given in table the numbers of their corresponding vibrations are to each other (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 each note; whence, we have the following table :— ut, mi, sol; or sol, si, re', form three perfect concords. These (Notes of the Gamut rst, ré, mi, filles salg lag are the harmonies which produce on the ear the most agreeable Abs. No. of Sim. Vib. 128, 144, 160, 170, 192, 213, 240. musical sensation. The absolute numbers of vibrations for the notes of the bigher scales, are obtained by multiplying the numbers of this table (C), by 2, by 4, by 8, etc.; and for those of the lower scales, by dividing the same numbers by 2, by 4, by 8, etc. Thus, the number of simple vibrations corresponding to sals, is equal to 192 × 4, or 768 per second; and the number corres- If the pulsations are sufficiently near each other to produce ponding to si is equal to 2404, or sixty per second. 3, 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. We know that sound passes over about 1,120 feet per second, at a mean state of the atmosphere, or about 60° Fahrenheit. If, therefore, a body made only one simple vibration per second, the length of the wave would be 1,120 feet; if it made two such vibrations, the length of the wave would be half of 1,120 feet; and so on. Now we have seen that 128 simple vibrations per second correspond to the note ut; the length 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 gamuts or scales:

Pulsation. When two sounds, which are not in unison, are produced simultaneously, there is heard at equal intervals a strengthening of sound which is called a pulsation. Thus, if the numbers of the vibrations of two sounds are 30 and 31, after 30 vibrations of the first, or 31 of the second, there will be a coincidence, and consequently a pulsation.

The Tuning Fork.-The tuning-fork is a small instrument by the aid of which we can reproduce at pleasure an invariable note; it thus becomes a suitable apparatus for regulating musical instruments as well as the human voice. It consists of a steel rod bent into the form of a pair of sugar tongs; fig. 136. It is made to vibrate by drawing a bow across its

Fig. 136.

Intervals, Sharps and Flats.-The ratio of one sound to another 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

(D) Relative No. of vibrations

81

10

9,

10

1

ut, re, mi, fa, sol, la, si, ut. edges, or by suddenly separating its two branches by means of 1, 8, 1, 3, 4, 1, 1, 3, 5, 2.a cylindrical piece of iron which is forcibly drawn between Intervals %, 6o, 18, 4, 4o, §, 15. 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, and 18. The first of these, certain number of vibrations, and produce a constant sound 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 semitone major. The interval between the tone major and the or both of its extremities. It may also be put into vibration tone minor is §. This is the smallest interval which is taken by holding it in the hand by the piece attached to the bend in into consideration; it is only a practised 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 distinguished by the names of sharps and fats. To sharpen a note, is to increase the number of its vibrations in the ratio of 24 to 25; to flatten it, is to diminish the same number in the ratio of 25 to 24. In music, the sharp is denoted by the sign, and the flat by the sign b.

Harmony, Discord.- The co-existence of several sounds

VIBRATION OF RODS, PLATES, AND MEMBRANES. Vibration of Rods and Lamina.-Rods and thin laminæ in wood, glass, and especially in tempèred steel, vibrate in consequence of their elasticity, and exhibit, like cords, two kinds of vibrations; the one transversal, and the other longitudinal. The transversal are produced by fixing the rods or lamine at one extremity, and passing a bow over the part which is free