Acoustic Instrument, or auricular tube. See ACOUSTICS, n° 26.

Acoustic Vessels, in the ancient theatres, were a kind of vessels, made of braids, shaped in the bell fashion, which being of all tones within the pitch of the voice or even of instruments, rendered the sounds more audible, so that the actors could be heard through all parts of theatres, which were even 400 feet in diameter.

Acoustic Disciples, among the ancient Pythagoreans, those more commonly called Acousmatici.

The Science of ACOUSTICS ACOUSTICS

INSTRUCTS us in the nature of sound. It is divided by some writers into Diacoustics, which explains the properties of those sounds that come directly from the sonorous body to the ear; and Catacoustics, which treats of reflected sounds: but such distinction does not appear to be of any real utility.

CHAP. I. Different Theories of Sound.

Most sounds, we all know, are conveyed to us on the bosom of the air. In whatever manner they either float upon it, or are propelled forward in it, certain it is, that, without the vehicle of this or some other fluid, we should have no sounds at all. Let the air be exhausted from a receiver, and a bell shall emit no sound when rung in the void; for, as the air continues to grow less dense, the sound dies away in proportion, so that at last its strongest vibrations are almost totally silent.

Thus air is a vehicle for sound. However, we must not, with some philosophers, assert, that it is the only vehicle; that, if there were no air, we should have no sounds whatsoever: for it is found by trial, that sounds are conveyed through water almost with the same facility with which they move through air. A bell rung in water returns a tone as distinct as if rung in air. This was observed by Derham, who also remarked that the tone came a quarter deeper. Some naturalists assure us also, that fishes have a strong perception of sounds, even at the bottom of deep rivers (a). From hence, it would seem not to be very material in the propagation of sounds, whether the fluid which conveys them be elastic or otherwise. Water, which, of all substances that we know, has the least elasticity, yet serves to carry them forward; and if we make allowance for the difference of its density, perhaps the sounds move in it with a proportional rapidity to what they are found to do in the elastic fluid of air.

One thing however is certain, that whether the fluid which conveys the note be elastic or non-elastic, whatever sound we hear is produced by a stroke, which the sounding body makes against the fluid, whether air or water. The fluid being struck upon, carries the impression forward to the ear, and there produces its sensation. Philosophers are so far agreed, that they all allow that sound is nothing more than the impression made by an elastic body upon the air or water (b), and propagated, this impression carried along by either fluid to the organ of hearing. But the manner in which this conveyance is made, is still disputed: Whether the sound is diffused into the air, in circle beyond circle, like the waves of water when we disturb the smoothness of its surface by dropping in a stone; or whether it travels along, like rays diffused from a centre, somewhat in the swift manner that electricity runs along a rod of iron; these are the questions which have divided the learned.

Newton was of the first opinion. He has explained the progression of sound by an undulatory, or rather a Newton's vermicular, motion in the parts of the air. If we have theory, an exact idea of the crawling of some insects, we shall have a tolerable notion of the progression of sound upon this hypothesis. The insect, for instance, in its motion, first carries its contractions from the hinder part, in order to throw its fore-part to the proper distance, then it carries its contractions from the fore-part to the hinder to bring that forward. Something similar to this is

(a) Dr Hunter has proved this, and demonstrated the auricular organ in these animals. See Fish, and Comparative Anatomy.

(b) Though air and water are both vehicles of sound, yet neither of them seems to be so by itself, but only as it contains an exceedingly subtle fluid capable of penetrating the most solid bodies. Hence, by the medium of that fluid, sounds can be propagated through wood, or metals, even more readily than through the open air. By the same means, deaf people may be made sensible of sounds, if they hold a piece of metal in their mouth, one end of which is applied to the sounding body. As it is certain, therefore, that air cannot penetrate metals, we must acknowledge the medium of sound to be of a more subtle nature; and thus the electrical fluid will naturally occur as the proper one. But why then is sound no longer heard in an exhausted receiver, if the air is not the fluid by which it is conveyed, seeing the electrical matter cannot be excluded? The reply to this is obvious: The electrical fluid is so exceedingly subtle, and pervades solid bodies with so much ease, that any motion of a solid body in a quantity of electric matter by itself, can never excite a degree of agitation in it sufficient for producing a sound; but if the electric fluid is entangled among the particles of air, water, wood, metal, &c. whatever affects their particles will also affect this fluid, and produce an audible noise. In the experiment of the air-pump, however, there may be an ambiguity, as the gradual exhausting of the air creates an increasing difference of pressure on the outside, and may occasion in the glass a difficulty of vibrating, so as to render it less fit to communicate to the air without the vibrations that strike it from within. From this cause the diminution of sound in an exhausted receiver may be supposed to proceed, as well as from the diminution of the air. But if any internal agitation of its parts should happen to the electrical fluid, exceeding loud noises might be propagated through it, as has been the case when large meteors have kindled at a great distance from the earth. It is also difficult to account for the exceeding great swiftness of sound, upon the supposition that it is propagated by means of air alone; for nothing is more certain, than that the strongest and most violent gale is, in its course, inert and sluggish, compared with the motion of sound. Different is the motion of the air when struck upon by a founding body. To be a little more precise, suppose ABC, the string of an harpsichord screwed to a proper pitch, and drawn out of the right line by the finger at B. We shall have occasion elsewhere to observe, that such a string would, if let go, vibrate to E; and from E to D, and back again; that it would continue thus to vibrate like a pendulum for ever, if not externally retarded, and, like a pendulum, all its little vibrations would be performed in equal times, the last and the first being equally long in performing; also, that, like a pendulum, its greatest swiftness would always be when it arrived at E, the middle part of its motion. Now then, if this string be supposed to fly from the finger at B, it is obvious, that whatever be its own motion, such also will be the motion of the parts of air that fly before it. Its motion, as is obvious, is first uniformly accelerated forward from B to E, then retarded as it goes from E to D, accelerated back again as it returns from D to E, and retarded from E to B. This motion being therefore sent in succession through a range of elastic air, it must happen, that the parts of one range of air must be sent forward with accelerated motion, and then with a retarded motion. This accelerated motion reaching the remotest end of the first range will be communicated to a second range, while the nearest parts of the first range being retarded in their motion, and falling back with the recession of the string, retire first with an accelerated, then with a retarded motion, and the remotest parts will soon follow. In the mean time, while the parts of the first range are thus falling back, the parts of the second range are going forward with an accelerated motion. Thus there will be an alternate condensation and relaxation of the air, during the time of one vibration; and as the air going forward strikes any opposing body with greater force than upon retiring, so each of these accelerated progressions have been called by Newton a pulse of sound.

Thus will the air be driven forward in the direction of the string. But now we must observe, that these pulses will move every way; for all motion impressed upon fluids in any direction whatsoever, operates all around in a sphere: so that sounds will be driven in all directions, backwards, forwards, upwards, downwards, and on every side. They will go on succeeding each other, one on the outside of the other, like circles in disturbed water; or rather, they will lie one without the other, in concentric shells, shell above shell, as we see in the coats of an onion.

All who have remarked the tone of a bell, while its sounds are decaying away, must have an idea of the pulses of sound, which, according to Newton, are formed by the air's alternate progression and recession. And it must be observed, that as each of these pulses is formed by a single vibration of the string, they must be equal to each other; for the vibrations of the string are known to be so.

Again, as to the velocity with which sounds travel, this Newton determines, by the most difficult calculation that can be imagined, to be in proportion to the thickness of the parts of the air, and the distance of these parts from each other. From hence he goes on to prove, that each little part moves backward and forward like a pendulum; and from thence he proceeds to demonstrate, that if the atmosphere were of the same density everywhere as at the surface of the earth, in such a case, a pendulum, that reached from its highest surface down to the surface of the earth, would by its vibrations discover to us the proportion of the velocity with which sounds travel. The velocity with which each pulse would move, he shows, would be as much greater than the velocity of such a pendulum swinging with one complete vibration, as the circumference of a circle is greater than the diameter. From hence he calculates, that the motion of sound will be 979 feet in one second. But this not being consonant to experience, he takes in another consideration, which destroys entirely the rigour of his former demonstration, namely, vapours in the air; and then finds the motion of sound to be 1142 feet in one second, or near 13 miles in a minute: a proportion which experience had established nearly before.

Thus much will serve to give an obscure idea of a theory which has met with numbers of opposers. Even John Bernoulli, Newton's greatest disciple, modestly professed that he did not pretend to understand this part of the Principia. He attempted therefore to give a more perspicuous demonstration of his own, that might confirm and illustrate the Newtonian theory. The subject seemed to reject elucidation: his theory is obviously wrong, as D'Alembert has proved in his Theory of Fluids.

Various have been the objections that have been made to the Newtonian system of sounds. It is urged, that this theory can only agree with the motion of sound in an elastic fluid, whereas sounds are known to move forward through water that is not elastic. To explain their progress therefore through water, a second theory must be formed: so that two theories must be made to explain a similar effect; which is contrary to the simplicity of true philosophy, for it is contrary to the simplicity of nature. It is farther urged, that this slow vermicular motion but ill represents the velocity with which sounds travel, as we know by experience that it is almost 13 miles in a minute. In short, it is urged, that such undulations as have been described, when coming from several sonorous bodies at once, would cross, obstruct, and confound each other; so that, if they were conveyed to the ear by this means, we should hear nothing but a medley of discord and broken articulations. But this is equally with the rest contradictory to experience, since we hear the fullest concert, not only without confusion, but with the highest pleasure. These objections, whether well founded or not, have given rise to another theory: which we shall likewise lay before the reader; though it too appears liable to objections, which shall be afterwards mentioned.

Every sound may be considered as driven off from the founding body in straight lines, and impressed upon the air in one direction only: but whatever impression is made upon a fluid in one direction, is diffused upon its surface into all directions; so that the sound first driven directly forward soon fills up a wide sphere, and is heard on every side. Thus, as it is impressed, it instantaneously travels forward with a very swift motion, resembling the velocity with which we know electricity flies from one end of a line to another.

Now, as to the pulses, or close shakes as the musicians express it, which a founding body is known to make make, each pulse (say the supporters of this theory) is itself a distinct and perfect sound, and the interval between every two pulses is profoundly silent. Continuity of sound from the same body is only a deception of the hearing; for as each distinct sound succeeds at very small intervals, the organ has no time to transmit its images with equal swiftness to the mind, and the interval is thus lost to sense: just as in seeing a flaming torch, if flared round in a circle, it appears as a ring of fire. In this manner a beaten drum, at some small distance, presents us with the idea of continuous sound. When children run with their sticks along a rail, a continuous sound is thus represented, though it need scarce be observed that the stroke against each rail is perfectly distinct and insulated.

According to this theory, therefore, the pulses are nothing more than distinct sounds repeated by the same body, the first stroke or vibration being ever the loudest, and travelling farther than those that follow; while each succeeding vibration gives a new sound, but with diminished force, till at last the pulses decay away totally, as the force decays that gives them existence.

All bodies whatsoever that are struck return more or less a sound; but some, wanting elasticity, give back no repetition of the sound; the noise is at once begotten and dies: while other bodies, however, there are, which being more elastic and capable of vibration, give back a sound, and repeat the same several times successively. These last are said to have a tone; the others are not allowed to have any.

This tone of the elastic string, or bell, is notwithstanding nothing more than a similar sound of what the former bodies produced, but with the difference of being many times repeated while their note is but single. So that, if we would give the former bodies a tone, it will be necessary to make them repeat their sound, by repeating our blows swiftly upon them. This will effectually give them a tone; and even an unmusical instrument has often had a fine effect by its tone in our concerts.

Let us now go on then to suppose, that by swift and equably continued strokes we give any non-elastic body its tone: it is very obvious, that no alterations will be made in this tone by the quickness of the strokes, though repeated ever so fast. These will only render the tone more equal and continuous, but make no alteration in the tone it gives. On the contrary, if we make an alteration in the force of each blow, a different tone will then undoubtedly be excited. The difference will be small, it must be confessed; for the tones of these inflexible bodies are capable but of small variation; however, there will certainly be a difference. The table on which we write, for instance, will return a different sound when struck with a club, from what it did when struck only with a switch. Thus non-elastic bodies return a difference of tone, not in proportion to the swiftness with which their sound is repeated, but in proportion to the greatness of the blow which produced it; for in two equal non-elastic bodies, that body produced the deepest tone which was struck by the greatest blow.

We now then come to a critical question, What is it that produces the difference of tone in two elastic founding bells or strings? Or what makes one deep and the other shrill? This question has always been hitherto answered by saying, that the depth or height of the note proceeded from the swiftness or slowness of the times of the vibrations. The slowest vibrations, it has been said, are qualified for producing the deepest tones, while the swiftest vibrations produce the highest tones. In this case, an effect has been given for a cause. It is in fact the force with which the sounding string strikes the air when struck upon, that makes the true distinction in the tones of sounds. It is this force, with greater or less impressions, resembling the greater or less force of the blows upon a non-elastic body, which produces correspondent affections of sound. The greatest forces produce the deepest sounds: the high notes are the effect of small efforts. In the same manner a bell, wide at the mouth, gives a grave sound; but if it be very maffly withal, that will render it still graver; but if maffly, wide, and long or high, that will make the tone deepest of all.

Thus, then, will elastic bodies give the deepest sound, in proportion to the force with which they strike the air: but if we should attempt to increase their force by giving them a stronger blow, this will be in vain; they will still return the same tone; for such is their formation, that they are famous only because they are elastic, and the force of this elasticity is not increased by our strength, as the greatness of a pendulum's vibration will not be increased by falling from a greater height.

Thus far of the length of chords. Now as to the frequency with which they vibrate the deepest tones, it has been found, from the nature of elastic strings, that the longest strings have the widest vibrations, and consequently go backward and forward slowest; while, on the contrary, the shortest strings vibrate the quickest, or come and go in the shortest intervals. From hence those who have treated of sounds, have asserted, as was said before, that the tone of the string depended upon the length or the shortness of the vibrations. This, however, is not the case. One and the same string, when struck, must always, like the same pendulum, return precisely similar vibrations; but it is well known, that one and the same string, when struck upon, does not always return precisely the same tone: so that in this case the vibrations follow one rule, and the tone another. The vibrations must be invariably the same in the same string, which does not return the same tone invariably, as is well known to musicians in general. In the violin, for instance, they can easily alter the tone of the string an octave or eight notes higher, by a softer method of drawing the bow; and some are known thus to bring out the most charming airs imaginable. These peculiar tones are by the English fiddlers called flute-notes. The only reason, it has been alleged, that can be assigned for the same string thus returning different tones, must certainly be the different force of its strokes upon the air. In one case, it has double the tone of the other; because upon the soft touches of the bow, only half its elasticity is put into vibration.

This being understood (continue the authors of this theory), we shall be able clearly to account for many things relating to sounds that have hitherto been inexplicable. Thus, for instance, if it be asked, When two strings are stretched together of equal lengths, tensions, and thicknesses, how does it happen, that one of them being struck, and made to vibrate throughout, Different throughout, the other shall vibrate throughout also? Theories of Sound, the answer is obvious: The force that the string struck receives is communicated to the air, and the air communicates the same to the similar string; which therefore receives all the force of the former; and the force being equal, the vibrations must be so too. Again, put the question, If one string be but half the length of the other, and be struck, how will the vibrations be? The answer is, The longest string will receive all the force of the string half as long as itself, and therefore it will vibrate in proportion, that is, through half its length. In the same manner, if the longest string were three times as long as the other, it would only vibrate in a third of its length; or if four times, in a fourth of its length. In short, whatever force the smaller string impinges upon the air, the air will impress a similar force upon the longer string, and partially excite its vibrations.

From hence also we may account for the cause of those charming, melancholy gradations of sound in the Eolian lyre; an instrument (says Sir John Hawkins) lately introduced upon the public as a new invention, *Vide Kircher Musurgia, lib. ix.*

This instrument is easily made, being nothing more than a long narrow box of thin dale, about 30 inches long, 5 inches broad, and 1½ inches deep, with a circle in the middle of the upper side or belly about 1½ inch diameter, pierced with small holes. On this side are seven, ten, or (according to Kircher) fifteen or more strings of very fine gut, stretched over bridges at each end, like the bridge of a fiddle, and screwed up or relaxed with screw-pins (b). The strings are all tuned to one and the same note; and the instrument is placed in some current of air, where the wind can buffet over its strings with freedom. A window with the glass just raised to give the air admission, will answer this purpose exactly. Now when the entering air blows upon these strings with different degrees of force, there will be excited different tones of sound; sometimes the blast brings out all the tones in full concert; sometimes it links them to the softest murmurs; it feels for every tone, and by its gradations of strength solicits those gradations of sound which art has taken different methods to produce.

It remains, in the last place, to consider (by this theory) the loudness and softness, or, as the musicians speak, the strength and softness of sound. In vibrating elastic strings, the loudness of the tone is in proportion to the deepness of the note; that is, in two strings, all things in other circumstances alike, the deepest tone will be loudest. In musical instruments upon a different principle, as in the violin, it is otherwise; the tones are made in such instruments, by a number of small vibrations crowded into one stroke. The refined bow, for instance, being drawn along a string, its roughnesses catch the string at very small intervals, and excite its vibrations. In this instrument, therefore, to excite loud tones, the bow must be drawn quick, and this will produce the greatest number of vibrations. But it must be observed, that the more quick the bow passes over the string, the less apt will the roughness of its surface be to touch the string at every instant; to remedy this, therefore, the bow must be pressed the harder as it is drawn quicker, and thus its fullest sound will be brought from the instrument. If the swiftness of the vibrations in an instrument thus rubbed upon, exceed the force of the deeper sound in another, then the swift vibrations will be heard at a greater distance, and as much farther off as the swiftness in them exceeds the force in the other.

By the same theory (it is alleged) may all the phenomena of musical sounds be easily explained.—The fables of the ancients pretend, that music was first found out by the beating of different hammers upon the anvil of smithy's anvil. Without pursuing the fable, let us endeavour to explain the nature of musical sounds by a similar method. Let us suppose an anvil, or several similar anvils, to be struck upon by several hammers of different weights or forces. The hammer, which is double that of another, upon striking the anvil will produce a sound double that of the other: this double sound musicians have agreed to call an Octave. The ear can judge of the difference or resemblance of these sounds with great ease, the numbers being as one and two, and therefore very readily compared. Suppose that an hammer, three times less than the first, strikes the anvil, the sound produced by this will be three times less than the first: so that the ear, in judging the similarity of these sounds, will find somewhat more difficulty; because it is not so easy to tell how often one is contained in three, as it is to tell how often it is contained in two. Again, suppose that an hammer four times less than the first strikes the anvil, the ear will find greater difficulty still in judging precisely the difference of the sounds; for the difference of the numbers four and one cannot so soon be determined with precision as three and one. If the hammer be five times less, the difficulty of judging will be still greater. If the hammer be six times less, the difficulty still increases, and so also of the seventh, inasmuch that the ear cannot always readily and at once determine the precise gradation. Now, of all comparisons, those which the mind makes most easily, and with least labour, are the most pleasing. There is a certain regularity in the human soul, by which it finds happiness in exact and striking, and easily-made comparisons. As the ear is but an instrument of the mind, it is therefore most pleased with the combination of any two sounds, the differences of which it can most readily distinguish. It is more pleased with the concord of two sounds which are to each other as one and two, than of two sounds which are as one and three, or one and four, or one and five, or one and six or seven. Upon this pleasure, which the mind takes in comparison, all harmony depends. The variety of sounds is infinite; but because the ear cannot compare two sounds so readily as to distinguish their discriminations when they exceed the proportion of one and seven, musicians have been content to confine all harmony within that compass, and allowed but seven notes in musical composition.

Let us now then suppose a stringed instrument fitted up

(b) The figure represents the instrument with ten chords; of which some direct only eight to be tuned unisons, and the two outermost octaves below them. But this seems not to be material. Of Musical up in the order mentioned above. For instance: Let Sounds, the first string be twice as long as the second; let the third string be three times shorter than the first; let the fourth be four times, the fifth string five times, and the sixth five times as short as the first. Such an instrument would probably give us a representation of the lyre as it came first from the hand of the inventor. This instrument will give us all the seven notes following each other, in the order in which any two of them will accord together most pleasingly; but yet it will be a very inconvenient and a very disagreeable instrument: inconvenient, for in a compass of seven strings only, the first must be seven times as long as the last; and disagreeable, because this first string will be seven times as loud also; so that when the tones are to be played in a different order, loud and soft sounds would be intermixed with most disgusting alternations.

In order to improve the first instrument, therefore, succeeding musicians very judiciously threw in all the other strings between the two first, or, in other words, between the two Octaves, giving to each, however, the same proportion to what it would have had in the first natural instrument. This made the instrument more portable, and the sounds more even and pleasing. They therefore disposed the sounds between the Octave in their natural order, and gave each its own proportional dimensions. Of these sounds, where the proportion between any two of them is most obvious, the concord between them will be most pleasing. Thus Octaves, which are as two to one, have a most harmonious effect; the fourth and fifth also sound sweetly together, and they will be found, upon calculation, to bear the same proportion to each other that Octaves do.

Let it not be supposed (says Mr Saver), that the musical scale is merely an arbitrary combination of sounds; it is made up from the consonance and differences of the parts which compose it. Those who have often heard a fourth and fifth accord together, will be naturally led to discover their difference at once; and the mind unites itself to their beauties. Let us then cease to assign the coincidences of vibrations as the cause of harmony, since these coincidences in two strings vibrating at different intervals, must at best be but fortuitous; whereas concord is always pleasing. The true cause why concord is pleasing, must arise from our power, in such a case, of measuring more easily the differences of the tones. In proportion as the note can be measured with its fundamental tone by large and obvious distinctions, then the concord is most pleasing; on the contrary, when the ear measures the discriminations of two tones by very small parts, or cannot measure them at all, it loses the beauty of their resemblance: the whole is discord and pain (c).

But there is another property in the vibration of a musical string not yet taken notice of, and which is alleged to confirm the foregoing theory. If we strike the string of an harpsichord, or any other elastic sounding chord whatever, it returns a continuing sound. This till of late was considered as one simple uniform tone; but all musicians now confess, that instead of one tone it actually returns four tones, and that constantly. The notes are, beside the fundamental tone, an octave above, a twelfth above, and a seventeenth. One of the bass-notes of an harpsichord has been dissected in this manner by Rameau, and the actual existence of these tones proved beyond a possibility of being controverted. In fact, the experiment is easily tried; for if we smartly strike one of the lower keys of an harpsichord, and then take the finger briskly away, a tolerable ear will be able to distinguish, that, after the fundamental tone has ceased, three other shriller tones will be distinctly heard; first the octave above, then the twelfth, and lastly the seventeenth: the octave above is in general almost mixed with the fundamental tone, so as not to be easily perceived, except by an ear long habituated to the minute discriminations of sounds. So that we may observe, that the smallest tone is heard last, and the deepest and largest one first: the two others in order.

In the whole theory of sounds, nothing has given greater room for speculation, conjecture, and disappointment, than this amazing property in elastic strings. The whole string is universally acknowledged to be in vibration in all its parts, yet this single vibration returns no less than four different sounds. They who account for the tones of strings by the number of their vibrations, are here at the greatest loss. Daniel Bernoulli supposes, that a vibrating string divides itself into a number of curves, each of which has a peculiar vibration; and though they all swing together in the common vibration, yet each vibrates within itself. This opinion, which was supported, as most geometrical speculations are, with the parade of demonstration, was only born soon after to die. Others have ascribed this to an elastic difference in the parts of the air, each of which, at different intervals, thus received different impressions from the string, in proportion to their elasticity. This is absurd. If we allow the difference of tone to proceed from the force, and not the frequency of the vibrations, this difficulty will admit of an easy solution. These sounds, though they seem to exist together in the string, actually follow each other in succession: while the vibration has greatest force, the fundamental tone is brought forward; the force of the vibration decaying, the octave is produced, but almost only instantaneously; to this succeeds, with diminished force, the twelfth; and, lastly, the seventeenth is heard to vibrate with great distinctness, while the three other tones are always silent. These sounds, thus excited, are all of them the harmonic tones, whose differences from the fundamental tone are, as was said, strong, and distinct. On the other hand, the discordant tones cannot be heard. Their differences being but very small, they are overpowered, and in a manner drowned in the tones of superior difference: yet not always neither; for Daniel Bernouilli has been able, from the same stroke, to make the same string bring out its harmonic and its discordant tones also (d). So that from hence we may justly infer, that every note whatsoever

(c) It is certain, that in proportion to the simplicity of relations in sound, the ear is pleased with its combinations; but this is not to be admitted as the cause why musicians have confined all harmony to an octave. Discriminated sounds, whose vibrations either never coincide, or at least very rarely, do not only cease to please, but violently grate, the ear. Harmony and discord, therefore, are neither discriminated by the judgment of hearers, nor the imitation of musicians, but by their own essential and immutable nature.

(d) Vid. Memoires de l'Academie de Berlin, 1753, p. 153. Of musical sounds is only a succession of tones; and that those are most distinctly heard, whose differences are most easily perceivable.

To this theory, however, though it has a plausible appearance, there are strong and indeed insuperable objections. The very fundamental principle of it is false. No body whatever, whether elastic or non-elastic, yields a graver sound by being struck with a larger instrument, unless either the sounding body, or that part of it which emits the sound, is enlarged. In this case, the largest bodies always return the gravest sounds.

In speaking of elastic and non-elastic bodies in a musical sense, we are not to push the distinction so far as when we speak of them philosophically. A body is musically elastic, all of whose parts are thrown into vibrations so as to emit a sound when only part of their surface is struck. Of this kind are bells, musical strings, and all bodies whatever that are considerably hollow.—Musical non-elasticities are such bodies as emit a sound only from that particular place which is struck: thus, a table, a plate of iron nailed on wood, a bell sunk in the earth, are all of them non-elasticities in a musical sense, though not philosophically so. When a solid body, such as a log of wood, is struck with a switch, only that part of it emits a sound which comes in contact with the switch; the note is acute and loud, but would be no less so though the adjacent parts of the log were removed. If, instead of the switch, a heavier or larger instrument is made use of, a larger portion of its surface then returns a sound, and the note is consequently more grave; but it would not be so, if the large instrument struck with a sharp edge, or a surface only equal to that of the small one.

In sounds of this kind, where there is only a single thwack, without any repetition, the immediate cause of the gravity or acuteness seems to be the quantity of air displaced by the sounding body; a large quantity of air displaced, produces a grave sound, and a smaller quantity a more acute one, the force wherewith the air is displaced signifying very little.—What we hear advance is confirmed by some experiments made by Dr Priestley, concerning the musical tone of electrical discharges. The passage being curious, and not very long, we shall here transcribe it:

"As the course of my experiments has required a great variety of electrical explosions, I could not help observing a great variety in the musical tone made by the reports. This excited my curiosity to attempt to reduce this variation to some measure. Accordingly, by the help of a couple of spinets, and two persons who had good ears for music, I endeavoured to ascertain the tone of some electrical discharges; and observed, that every discharge made several strings, particularly those that were chords to one another, to vibrate; but one note was always predominant, and founded after the rest. As every explosion was repeated several times, and three of us separately took the same note, there remained no doubt but that the tone we fixed upon was at least very near the true one. The result was as follows:

"A jar containing half a square foot of coated glass founded F sharp, concert pitch. Another jar of a different form, but equal surfaces, founded the same.

"A jar of three square feet founded C below F sharp. A battery consisting of sixty-four jars, each containing half a square foot, founded F below the C.

"The same battery, in conjunction with another of thirty-one jars, founded C sharp. So that a greater quantity of coated glass always gave a deeper note.

"Differences in the degree of a charge in the same jar made little or no difference in the tone of the explosion: if any, a higher charge gave rather a deeper note."

These experiments show us how much the gravity or acuteness of sounds depend on the quantity of air put in agitation by the sounding body. We know that the noise of the electric explosion arises from the return of the air into the vacuum produced by the electric spark. The larger the vacuum, the deeper was the note: for the same reason, the discharge of a musket produces a more acute note than that of a cannon; and thunder is deeper than either.

Besides this, however, other circumstances concur to produce different degrees of gravity or acuteness in sounds. The sound of a table struck upon with a piece of wood, will not be the same with that produced from a plate of iron struck by the same piece of wood, even if the blows should be exactly equal, and the iron perfectly kept from vibrating.—Here the sounds are generally said to differ in their degrees of acuteness, according to the specific gravities or densities of the substances which emit them. Thus gold, which is the most dense of all metals, returns a much graver sound than silver; and metallic wires, which are more dense than others, return a proportionally greater sound.—But neither does this appear to be a general rule in which we can put confidence. Bell-metal is denser than copper, but it by no means appears to yield a graver sound; on the contrary, it seems very probable, that copper will give a graver sound than bell-metal, if both are struck upon in their non-elastic state; and we can by no means think that a bell of pure tin, the least dense of all the metals, will give a more acute sound than one of bell-metal, which is greatly more dense.—In some bodies hardness seems to have a considerable effect. Glass, which is considerably harder than any metal, gives a more acute sound; bell-metal is harder than gold, lead, or tin, and therefore sounds much more acutely; though how far this holds with regard to different substances, there are not a sufficient number of experiments for us to judge.

In bodies musically elastic, the whole substance vibrates with the slightest stroke, and therefore they always give the same note whether they are struck with a large or with a small instrument; so that striking a part of the surface of any body musically elastic is equivalent, in it, to striking the whole surface of a non-elastic one. If the whole surface of a table was struck with another table, the note produced would be neither more nor less acute whatever force was employed; because the whole surface would then yield a sound, and no force could increase the surface; the sound would indeed be louder in proportion to the force employed, but the gravity would remain the same. In like manner, when a bell, or musical string, is struck, the whole substance vibrates, and a greater stroke cannot increase the substance.—Hence we see the fallacy of what is said concerning the Pythagorean anvils. An anvil is a body musically elastic, and no difference in the tone can Of musical can be perceived whether it is struck with a large, or with a small hammer; because either of them are sufficient to make the whole substance vibrate, provided nothing but the anvil is struck upon: smiths, however, do not strike their anvils, but red-hot iron laid upon their anvils; and thus the vibrations of the anvil are stopped, so that it becomes a non-elastic body, and the differences of tone in the strokes of different hammers proceed only from the surface of the large hammers covering the whole surface of the iron, or at least a greater part of it than the small ones. If the small hammer is sufficient to cover the whole surface of the iron as well as the large one, the note produced will be the same, whether the large or the small hammer is used.

Lastly, the argument for the preceding theory, grounded on the production of what are called flute-notes on the violin, is built on a false foundation; for the bow being lightly drawn on an open string, produces no flute-notes, but only the harmonics of the note to which the string is tuned. The flute-notes are produced by a particular motion of the bow, quick and near the bridge, and by fingering very gently. By this management, the same sounds are produced, tho' at certain intervals only, as if the vibrations were transferred to the space between the end of the finger-board and the finger, instead of that between the finger and the bridge. Why this small part of the string should vibrate in such a case, and not that which is under the immediate action of the bow, we must own ourselves ignorant; nor dare we affirm that the vibrations really are transferred in this manner, only the same sounds are produced as if they were.

Though these objections seem sufficiently to overturn the foregoing theory, with regard to acute sounds being the effects of weak strokes, and grave ones of stronger impulses, we cannot admit that longer or shorter vibrations are the occasion of gravity or acuteness in sound. A musical sound, however lengthened, either by string or bell, is only a repetition of a single one, whose duration by itself is but for a moment, and is therefore termed inappreciable, like the smack of a whip, or the explosion of an electrical battery. The continuation of the sound is nothing more than a repetition of this instantaneous inappreciable noise after the manner of an echo, and it is only this echo that makes the sound agreeable. For this reason, music is much more agreeable when played in a large hall where the sound is reverberated, than in a small room where there is no such reverberation. For the same reason, the sound of a string is more agreeable when put on a hollow violin than when fastened to a plain board, &c.—In the sound of a bell, we cannot avoid observing this echo very distinctly. The sound appears to be made up of distinct pulses, or repetitions of the same note produced by the stroke of the hammer. It can by no means be allowed, that the note would be more acute though these pulses were to succeed one another more rapidly; the sound would indeed become more simple, but would still preserve the same tone.—In musical strings the reverberations are vastly more quick than in bells; and therefore their sound is more uniform or simple, and consequently more agreeable than that of bells. In musical glasses*, the vibrations must be inconceivably quicker than in any bell, or stringed instrument: and hence they are of all others the most simple and the most agreeable, though neither the most acute nor the loudest.—As far as we can judge, quickness of vibration contributes to the uniformity, or simplicity, but not to the acuteness, nor to the loudness, of a musical note.

It may here be objected, that each of the different pulses, of which we observe the sound of a bell to be composed, is of a very perceptible length, and far from being instantaneous; so that it is not fair to infer that the sound of a bell is only a repetition of a single instantaneous stroke, seeing it is evidently the repetition of a lengthened note.—To this it may be replied, that the inappreciable sound which is produced by striking a bell in a non-elastic state, is the very same which, being first propagated round the bell, forms one of these short pulses that is afterwards re-echoed as long as the vibrations of the metal continue, and it is impossible that the quickness of repetition of any sound can either increase or diminish its gravity.

CHAP. II. Of the propagation of Sound. Newton's Doctrine explained and vindicated.

The writers on sound have been betrayed into these difficulties and obscurities, by rejecting the 47th proposition, B. ii. of Newton, as inconclusive reasoning. Of this proposition, however, the ingenious Mr Young of Trinity college, Dublin, has lately given a clear, explanatory, and able defence. He candidly owns that the demonstration is obscurely stated, and takes the liberty of varying, in some degree, from the method of Newton.

"1. The parts of all sounding bodies, (he observes), vibrate according to the law of a cycloidal pendulum: for they may be considered as composed of an indefinite number of elastic fibres; but these fibres vibrate according to that law. Vide Huyghen, p. 270.

"2. Sounding bodies propagate their motions on all sides in direction, by successive condensations and rarefactions, and successive goings forward and returnings backward of the particles. Vide prop. 43, B. 2. Newton.

"3. The pulses are those parts of the air which vibrate backwards and forwards; and which, by going forward, strike (pulsant) against obstacles. The latitude of a pulse is the rectilineal space through which the motion of the air is propagated during one vibration of the sounding body.

"4. All pulses move equally fast. This is proved by experiment; and it is found that they describe 1070 Paris feet, or 1142 London feet in a second, whether the sound be loud or low, grave or acute.

"5. Prob. To determine the latitude of a pulse. Divide the space which the pulse describes in a given time (4) by the number of vibrations performed in the same time by the sounding body, (cor. 1. prop. 24. Smith's Harmonics), the quotient is the latitude.

"M. Sauveur, by some experiments on organ-pipes, found that a body, which gives the gravest harmonic sound, vibrates 12 times and a half in a second, and that the shrillest sounding body vibrates 51,100 times in a second. At a medium, let us take the body which gives what Sauveur calls his fixed sound: it performs 100 vibrations in a second, and in the same time the pulses describe 1070 Parisian feet; therefore the space described by the pulses whilst the body vibrates once, that that is, the latitude or interval of the pulse, will be 10.7 feet.

6. Prob. To find the proportion which the greatest space, through which the particles of the air vibrate, bears to the radius of a circle, whose perimeter is equal to the latitude of the pulse.

During the first half of the progress of the elastic fibre, or sounding body, it is continually getting nearer to the next particle; and during the latter half of its progress, that particle is getting farther from the fibre, and these portions of time are equal (Helmholtz); therefore we may conclude, that at the end of the progress of the fibre, the first particle of air will be nearly as far distant from the fibre as when it began to move; and in the same manner we may infer, that all the particles vibrate through spaces nearly equal to that run over by the fibre.

Now, M. Sauveur (Acad. Science, an. 1700, p. 141) has found by experiment, that the middle point of a chord which produces his fixed sound, and whose diameter is \( \frac{1}{8} \)th of a line, runs over in its smallest sensible vibrations \( \frac{1}{8} \)th of a line, and in its greatest vibrations 72 times that space; that is \( 72 \times \frac{1}{8} \)th of a line, or 4 lines, that is, \( \frac{1}{8} \)d of an inch.

The latitude of the pulses of this fixed sound is 10.7 feet (5); and since the circumference of a circle is to its radius as 710 is to 113, the greatest space described by the particles will be to the radius of a circle, whose periphery is equal to the latitude of the pulse as \( \frac{1}{8} \)d of an inch is to 1.7029 feet, or 20.4348 inches, that is, as 1 to 61.3044.

If the length of the string be increased or diminished in any proportion, ceteris paribus, the greatest space described by its middle point will vary in the same proportion. For the inflicting force is to the tending force as the distance of the string from the middle point of vibration to half the length of the string (see Helmholz and Martin); and therefore the inflicting and tending forces being given, the string will vibrate through spaces proportional to its length; but the latitude of the pulse is inversely as the number of vibrations performed by the string in a given time, (5) that is, directly as the time of one vibration, or directly as the length of the string (prop. 24, cor. 7, Smith's Harmonics); therefore the greatest space through which the middle point of the string vibrates, will vary in the direct ratio of the latitude of the pulse, or of the radius of a circle whose circumference is equal to the latitude; that is, it will be to that radius as 1 to 61.3044.

7. If the particles of the aerial pulses, during any part of their vibration, be successively agitated, according to the law of a cycloidal pendulum, the comparative elastic forces arising from their mutual action, by which they will afterwards be agitated, will be such as will cause the particles to continue that motion, according to the same law, to the end of their vibration.

Let AB, BC, CD, &c. denote the equal distances of the successive pulses; ABC the direction of the motion of the pulses propagated from A towards B; Ee, Ff, Gg, three physical points of the quiescent medium, situated in the right line AC at equal distances from each other; Ee, Ef, Gg the very small equal spaces through which these particles vibrate; e, f, g any intermediate places of these points.

Draw the right line PS equal to Ee, bisect it in O, and from the centre O with the radius OP describe the circle SIPb. Let the whole time of the vibration of a particle and its parts be denoted by the circumference of this circle and its proportional parts. And since the particles are supposed to be at first agitated according to the law of a cycloidal pendulum, if at any time PH, or PHSb, the perpendicular HL or Hb be let fall on PS, and if Ee be taken equal to PL or P4, the particle E shall be found in e. Thus will the particle E perform its vibrations according to the law of a cycloidal pendulum. Prop. 52. B. i. Principia.

Let us suppose now, that the particles have been successively agitated, according to this law, for a certain time, by any cause whatsoever, and let us examine what will be the comparative elastic forces arising from their mutual action, by which they will afterwards continue to be agitated.

In the circumference PHSb take the equal arches HI, IK in the same ratio to the whole circumference which the equal right lines EF, FG have to BC the whole interval of the pulses; and let fall the perpendiculars HL, IM, KN. Since the points E, F, G are successively agitated in the same manner, and perform their entire vibrations of progress and regress while the pulse is propagated from B to C, if PH be the time from the beginning of the motion of E, PI will be the time from the beginning of the motion of F, and PK the time from the beginning of the motion of G; and therefore Ee, Ff, Gg will be respectively equal to PL, PM, PN in the progress of the particles. Whence ef or EF + Ff - Ee is equal to EF - LM. But ef is the expansion of EF in the place ef, and therefore this expansion is to its mean expansion as EF - LM to EF. But LM is to IH as IM is to OP, and IH is to EF as the circumference PHSb is to BC; that is, as OP is to V, if V be the radius of a circle whose circumference is BC; therefore, ex aequo, LM is to EF as IM is to V; and therefore the expansion of EF in the place ef is to its mean expansion as V - IM is to V; and the elastic force existing between the physical points E and F is to the mean elastic force as

\[ \frac{1}{V-IM} \text{ is to } \frac{1}{V} \quad (\text{Cotes Pneum. Let. 9.}) \]

By the same argument, the elastic force existing between the physical points F and G is to the mean elastic force as

\[ \frac{1}{V-KN} \text{ is to } \frac{1}{V} \]

and the difference between these forces is to the mean elastic force as

\[ \frac{1}{V^2} = \frac{1}{V} \quad \text{or as } IM - KN \text{ is to } V; \text{ if only (upon account of the very narrow limits of the vibration) we suppose IM and KN to be indefinitely less than } V. \text{ Wherefore, since } V \text{ is given, the difference of the forces is as } IM - KN, \text{ or as } HL - IM \text{ (because } KH \text{ is bisected in I); that is, (because } HL - IM \text{ is to IH as OM is to OI or OP, and IH and OP are given quantities) as OM; that is, if } EF \text{ be bisected in } \Omega \text{ as } \Omega_2. \]

In the same manner it may be shown, that if PHSb be the time from the beginning of the motion of E, PHSi will be the time from the beginning of the motion of F, and PHSk the time from the beginning of the the motion of G; and that the expansion of EF in the place \( \varphi \) is to its mean expansion as \( EF + F - E \), or as \( EF + h' \) is to \( EF \), or as \( V + h' \) is to \( V \) in its regres; and its elastic force to the mean elastic force as

\[ \frac{1}{V + h'} \text{ is to } \frac{1}{V}; \]

and that the difference of the elastic forces existing between E and F, and between F and G is to the mean elastic force as \( h' - im \) is to \( V \); that is, directly as \( \varphi \).

"But this difference of the elastic forces, existing between E and F, and between F and G, is the comparative elastic force by which the physical point \( \varphi \) is agitated; and therefore the comparative accelerating force, by which every physical point in the medium will continue to be agitated both in progress and regression, will be directly as its distance from the middle point of its vibration; and consequently, will be such as will cause the particles to continue their motion, undisturbed, according to the law of a cycloidal pendulum."

Prop. 38. L. 1. Newton.

"Newton rejects the quantity \( V \times IM + KN + IM \times KN \), on supposition that IM and KN are indefinitely less than \( V \). Now, although this may be a reasonable hypothesis, yet, that this quantity may be safely rejected, will, I think, appear in a more satisfactory manner from the following considerations derived from experiment: PS, in its greatest possible state, is to \( V \) as 1 is to 61.3044 (6); and therefore IM or KN, in its greatest possible state, (that is, when the vibrations of the body are as great as possible, and the particle in the middle point of its vibration) is to \( V \) as 1 is to 122.5. Hence \( V^2 = 15030.76 \), \( V \times IM + KN = 245.2 \) and \( IM \times KN = 1 \); therefore \( V^2 \) is to \( V^2 - V \times IM + KN + IM \times KN \) as 15.03076 is to 14786.56; that is, as 61 is to 60 nearly.

"Hence it appears, that the greatest possible error in the accelerating force, in the middle point, is the \( \frac{1}{10} \) part of the whole. In other points it is much less; and in the extreme points the error entirely vanishes.

"We should also observe, that the ordinary sounds we hear are not produced by the greatest possible vibrations of which the founding body is capable; and that in general IM and KN are nearly evanescent with respect to \( V \). And very probably the disagreeable sensations we feel in very loud sounds, arise not only from IM or KN bearing a sensible proportion to \( V \), by which means the cycloidal law of the pulses may be in some measure disturbed, but also from the very law of the motion of the founding body itself being disturbed. For, the proof of this law's being observed by an elastic fibre is founded on the hypothesis that the space, through which it vibrates, is indefinitely little with respect to the length of the string. See Smith's Harmonics, p. 237, Helmam, p. 270.

"8. If a particle of the medium be agitated, according to the law of a cycloidal pendulum, the comparative elastic force, acting on the adjacent particle, from the instant in which it begins to move, will be such as will cause it to continue its motion according to the same law.

"For let us suppose, that three particles of the medium had continued to move for times denoted by the arches PK, PI, PH, the comparative elastic force, acting on the second during the time of its motion, would have been denoted by HL—IM, that is, would have been directly as MO (7). And if this time be diminished till it becomes coincident with P, that is, if you take the particles in that state when the second is just beginning to move, and before the third particle has yet been set in motion; then the point M will fall on P, and MO become PO; that is, the comparative elastic force of the second particle, at the instant in which it begins to move, will be to the force with which it is agitated in any other moment of time, before the subsequent particle has yet been set in motion, directly as its distance from the middle point of vibration. Now this comparative elastic force, with which the second particle is agitated in the very moment in which it begins to move, arises from the preceding particle's approaching it according to the law of a pendulum; and therefore, if the preceding particle approaches it in this manner, the force by which it will be agitated, in the very moment it begins to move, will be exactly such as should take place in order to move it according to the law of a pendulum. It therefore acts out according to that law, and consequently the subsequent elastic forces, generated in every successive moment, will also continue to be of the just magnitude which should take place, in order to produce such a motion.

"9. The pulses of the air are propagated from founding bodies, according to the law of a cycloidal pendulum. The point E of any elastic fibre producing a sound, may be considered as a particle of air vibrating according to the law of a pendulum (1). This point E will therefore move according to this law for a certain time, denoted by the arch IH, before the second particle begins to move; for sound is propagated in time through the successive particles of air (4). Now from that instant, the comparative elastic force which agitates F, is (8) directly as its distance from the middle point of vibration. F therefore sets out with a motion according to the law of a pendulum; and therefore the comparative elastic force by which it will be agitated until G begins to move, will continue that law (8). Consequently F will approach G in the same manner as E approached F, and the comparative elastic force of G, from the instant in which it begins to move, will be directly as its distance from the middle point of vibration; and so on in succession. Therefore all the particles of air in the pulses successively set out from their proper places according to the law of a pendulum, and therefore (7) will finish their entire vibrations according to the same law.

"Cor. 1. The number of pulses propagated is the same with the number of vibrations of the tremulous body, nor is it multiplied in their progress: because the little physical line \( \gamma \), (fig. 7.) as soon as it returns to its proper place, will there quiete; for its velocity, which is denoted by the fine IM, then vanishes, and its density becomes the same with that of the ambient medium. This line, therefore, will no longer move, unless it be again driven forwards by the impulse of the founding body, or of the pulses propagated from it.

"Cor. 2. In the extreme points of the little space through which the particle vibrates, the expansion of the air is in its natural state; for the expansion of the physical line is to its natural expansion as \( V - IM \) is to V; but IM is then equal to nothing. In the middle point of the progress the condensation is greatest; for IM is then greatest, and consequently the expansion V—IM least. In the middle of the regres, the rarefaction is greatest; for im, and consequently V+im, is then greatest.

"10. To find the velocity of the pulses, the density and elastic force of the medium being given.

"This is the 49th prop. B. 2. Newton, in which he shows, that whilst a pendulum, whose length is equal to the height of the homogeneous atmosphere, vibrates once forwards and backwards, the pulses will describe a space equal to the periphery of a circle described with that altitude as its radius.

"Cor. 1. He thence shows, that the velocity of the pulses is equal to that which a heavy body would acquire in falling down half the altitude of that homogeneous atmosphere; and therefore, that all pulses move equally fast, whatever be the magnitude of PS, or the time of its being described; that is, whether the tone be loud or low, grave or acute. See Hales de Sonis, § 49.

"Cor. 2. And also, that the velocity of the pulses is in a ratio compounded of the direct subduplicate ratio of the elastic force of the medium, and the inverse subduplicate of its density. Hence sounds move somewhat faster in summer than in winter. See Hales de Sonis, p. 141.

"11. The strength of a tone is as the moment of the particles of air. The moment of these particles, (the medium being given) is as their velocity; and the velocity of these particles is as the velocity of the string which sets them in motion (9). The velocities of two different strings are equal when the spaces which they describe in their vibrations are to each other as the times of these vibrations; therefore, two different tones are of equal strength, when the spaces, through which the strings producing them vibrate, are directly as the times of their vibration.

"12. Let the strength of the tones of the two strings AB, CD, which differ in tension only (fig. 1, 2.) be equal. Quere the ratio of the inflecting forces F and f. From the hypothesis of the equality of the strength of the tones, it follows (11), that the space GE must be to the space HF as \( \frac{F}{f} \) to \( \frac{F}{f} \), (Smith's Harm. Prop. 24, Cor. 4.) Now the forces inflecting AB, CD through the equal spaces GE, HP are to each other as the tending forces, that is, as \( \frac{F}{f} \) to \( \frac{F}{f} \), (Malcolm's Treatise on Music, p. 52.) But the force inflecting CD through HP is to the force inflecting it through HF as HP or GE to HF, (ib. p. 47.) that is, by the hyp. as \( \frac{F}{f} \) to \( \frac{F}{f} \). Therefore, ex aequo, the forces inflecting AB and CD, when the tones are equally strong, are to each other as \( \frac{F}{f} \times \frac{F}{f} \) to \( \frac{F}{f} \times \frac{F}{f} \), or as \( \frac{F}{f} \) to \( \frac{F}{f} \). That is, the forces necessary to produce tones of equal strength in various strings which differ only in tension, are to each other in the subduplicate ratio of the tending forces, that is, inversely as the time of one vibration, or directly as the number of vibrations performed in a given time. Thus, if CD be the acute octave to AB, its tending force will be quadruple that of AB, (Malcolm's Treatise on Music, p. 53); and therefore to produce tones of equal strength in these strings, the force impelling CD must be double that impelling AB; and so in other cases.

N° 3.

"Suppose, now, that the strings AB, CD, (fig. 2, 3.) differ in length only. The force inflecting AB through GE is to the tending force, which is given, as GE to AG; and this tending force is to the force inflecting CD through the space HP equal to GE, as HD to HP. Therefore, ex aequo, the forces inflecting AB and CD through the equal spaces GE and HP, are to each other as HD to AG, or as CD to AB. But the force inflecting CD through HP is to the force inflecting it through HF, as HP or GE to HF, that is, because these spaces are as the times (11), as AB to CD. Therefore, ex aequo, the forces inflecting AB and CD, when the tones are equally strong, are to each other in a ratio of equality. Hence we should suppose, that in this case, an equal number of equal impulses would generate equally powerful tones in these strings. But we are to observe, that the longer the strings, the greater, ceteris paribus, is the space through which a given force inflicts it (Malcolm); and therefore whatever diminution is produced in the spaces through which the strings move in their successive vibrations, arising either from the want of perfect elasticity in the strings, or from the resistance of the air, this diminution will bear a greater proportion to the less space, through which the shorter string vibrates. And this is confirmed by experience; for we find that the duration of the tone and motion of the whole string exceeds that of any of its subordinate parts. Therefore, after a given interval of time, a greater quantity of motion will remain in the longer string; and consequently, after the successive equal impulses have been made, a greater degree of motion will still subsist in it. That is, a given number of equal impulses being made on various strings differing in length only, a stronger sound will be produced in that which is the longer."

Chap. III. Of the Velocity, &c. of Sound. Axioms.

Experience has taught us, that sound travels about velocity of the rate of 1142 feet in a second, or near 13 miles in a found minute; nor do any obstacles hinder its progress, a contrary wind only a small matter diminishing its velocity. The method of calculating its progress is easily made known. When a gun is discharged at a distance, we see the fire long before we hear the sound. If then we know the distance of the place, and know the time of hearing the report, this will show us exactly the time the sound has been travelling to us. For instance, if the gun is discharged a mile off, the moment the flash is seen, you take a watch and count the seconds till you hear the sound; the number of seconds is the time the sound has been travelling a mile.—Again, by the above axiom, we are enabled to find the distance between objects that would be otherwise immeasurable. For example, suppose you see the flash of a gun in the night at calculated sea, and tell seven seconds before you hear the report, by means of it follows therefore, that the distance is seven times 1142 feet, that is, 24 yards more than a mile and a half. In like manner, if you observe the number of seconds between the lightning and the report of the thunder, you know the distance of the cloud from whence it proceeds.

Derham has proved by experience, that all sounds travel at whatever rate, the sound of a gun, same rate, and and the striking of a hammer, are equally swift in their motions; the softest whisper flies as swiftly, as far as it goes, as the loudest thunder.

To these axioms we may add the following.

Smooth and clear sounds proceed from bodies that are homogeneous, and of an uniform figure; and harsh or obtuse sounds, from such as are of a mixed matter and irregular figure.

The velocity of sound is to that of a brisk wind as fifty to one.

The strength of sounds is greatest in cold and dense air, and least in that which is warm and rarefied.

Every point against which the pulses of sound strike, becomes a centre from which a new series of pulses are propagated in every direction.

Sound describes equal spaces in equal times.

**Chap. IV. Of Reverberated Sounds.**

Sound, like light, after it has been reflected from several places, may be collected in one point, as into a focus; and it will be there more audible than in any other part, even than at the place from whence it proceeded. On this principle it is that a whispering gallery is constructed.

The form of this gallery must be that of a concave hemisphere (ε), as ABC; and if a low sound or whisper be uttered at A, the vibrations expanding themselves every way will impinge on the points DDD, &c., and from thence be reflected to EEE, and from thence to the points F and G, till at last they all meet in C, where, as we have said, the sound will be the most distinctly heard.

The augmentation of sound by means of speaking-trumpets, is usually illustrated in the following manner:

Let ABC be the tube, BD the axis, and B the mouthpiece for conveying the voice to the tube. Then it is evident, when a person speaks at B in the trumpet, the whole force of his voice is spent upon the air contained in the tube, which will be agitated through the whole length of the tube; and, by various reflections from the side of the tube to the axis, the air along the middle part of the tube will be greatly condensed, and its momentum proportionally increased, so that when it comes to agitate the air at the orifice of the tube AC, its force will be as much greater than what it would have been without the tube, as the surface of a sphere whose radius is equal to the length of the tube, is greater than the surface of the segment of such a sphere whose base is the orifice of the tube. For a person speaking at B, without the tube, will have the force of his voice spent in exciting concentric surfaces of air all around the point B; and when those surfaces or pulses of air are diffused as far as D every way, it is plain the force of the voice will there be diffused through the whole surfaces of a sphere whose radius is BD; but in the trumpet it will be so confined, that at its exit it will be diffused through so much of that spherical surface of air as corresponds to the orifice of the tube. But since the force is given, its intensity will be always inversely as the number of particles it has to move; and therefore in the tube it will be to that without, as the superficies of such a sphere to the area of the large end of the tube nearly.

But it is obvious, Mr Young observes, that the confinement of the voice can have little effect in increasing the strength of the sound, as this strength depends on the velocity with which the particles move. Were this reasoning conclusive, the voice should issue through the smallest possible orifice; cylindrical tubes would be preferable to any that increased in diameter; and the less the diameter, the greater would be the effect of the instrument; because the place or mass of air to be moved, would, in that case, be less, and consequently the effect of the voice the greater; all which is contradicted by experience.

The cause of the increase of sound in these tubes must therefore be derived from some other principles; and amongst these we shall probably find, that what the ingenious Kircher has suggested in his Phonurgia is the most deserving of our attention. He tells us, that "the augmentation of the sound depends on its reflection from the tremulous sides of the tube; which reflections, conspiring in propagating the pulses in the same direction, must increase its intensity." Newton also seems to have considered this as the principal cause, in the scholium of prop. 50. B. 2. Princip. when he says, "we hence see why sounds are so much increased in tentorophonic tubes, for every reciprocal motion is, in each return, increased by the generating cause.

Farther, when we speak in the open air, the effect on the tympanum of a distant auditor is produced merely by a single pulse. But when we use a tube, all the pulses propagated from the mouth, except those in the direction of the axis, strike against the sides of the tube, and every point of impulse becoming a new centre, from whence the pulses are propagated in all directions, a pulse will arrive at the ear from each of those points; thus, by the use of a tube, a greater number of pulses are propagated to the ear, and consequently the sound increased. The confinement too of the voice may have some effect, though not such as is ascribed to it by some; for the condensed pulses produced by the naked voice, freely expand every way; but in tubes, the lateral expansion being diminished, the direct expansion will be increased, and consequently the velocity of the particles, and the intensity of the sound. The substance also of the tube has its effect; for it is found by experiment, that the more elastic the substance of the tube, and consequently the more susceptible it is of these tremulous motions, the stronger is the sound.

If the tube be laid on any non-elastic substance, it deadens the sound, because it prevents the vibratory motion of the parts. The sound is increased in speaking-trumpets, if the tube be suspended in the air; because the agitations are then carried on without interruption. These tubes should increase in diameter from the mouthpiece, because the parts, vibrating in directions perpendicular to the surface, will conspire in impelling forward the particles of air, and consequently, by increasing their velocity, will increase the intensity of the sound; and the surface also increasing, the number of points of impulse and of new propagations will increase.

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(ε) A cylindric or elliptic arch will answer still better than one that is circular. proportionally. The several causes, therefore, of the increase of sound in these tubes, Mr Young concludes to be, 1. The diminution of the lateral, and consequently the increase of the direct, expansion and velocity of the included air. 2. The increase of the number of pulses, by increasing the points of new propagation. 3. The reflections of the pulses from the tremulous sides of the tube, which impel the particles of air forward, and thus increase their velocity.

An echo is a reflection of sound striking against some object, as an image is reflected in a glass; but it has been disputed what are the proper qualities in a body for thus reflecting sounds. It is in general known, that caverns, grottoes, mountains, and ruined buildings, return this image of sound. We have heard of a very extraordinary echo, at a ruined fortress near Louvain, in Flanders. If a person sung, he only heard his own voice, without any repetition; on the contrary, those who stood at some distance, heard the echo but not the voice; but then they heard it with surprising variations, sometimes louder, sometimes softer, now more near, then more distant. There is an account in the memoirs of the French academy, of a similar echo near Rouen.

As (by n° 21 and 22) every point against which the pulses of sound strike becomes the centre of a new series of pulses, and sound describes equal distances in equal times; therefore, when any sound is propagated from a centre, and its pulses strike against a variety of obstacles, if the sum of the right lines drawn from that point to each of the obstacles, and from each obstacle to a second point, be equal, then will the latter be a point in which an echo will be heard. Thus let A be the point from which the sound is propagated in all directions, and let the pulses strike against the obstacles C, D, E, F, G, H, I, &c., each of these points becomes a new centre of pulses by the first principle, and therefore from each of them one series of pulses will pass through the point B. Now if the several fums of the right lines AC+CB, AD+DB, AE+EB, AG+GB, AH+HB, AI+IB, &c., be all equal to each other, it is obvious that the pulses propagated from A to these points, and again from these points to B, will all arrive at B at the same instant, according to the second principle; and therefore, if the hearer be in that point, his ear will at the same instant be struck by all these pulses. Now it appears from experiment (see Mauchtenbroek, V. ii. p. 210), that the ear of an exercised musician can only distinguish such sounds as follow one another at the rate of 9 or 10 in a second, or any slower rate; and therefore, for a distinct perception of the direct and reflected sound, there should intervene the interval of \( \frac{1}{12} \)th of a second; but in this time sound describes \( \frac{1}{12} \times 1142 \) feet nearly. And therefore, unless the sum of the lines drawn from each of the obstacles to the points A and B exceeds the interval A.B by 127 feet, no echo will be heard at B. Since the several fums of the lines drawn from the obstacles to the points A and B are of the same magnitude, it appears that the curve passing through all the points C, D, E, F, G, H, I, &c., will be an ellipse, (prop. 14. B. 2. Ham. Con.) Hence all the points of the obstacles which produce an echo, must lie in the surface of the oblong spheroid, generated by the revolution of this ellipse round its major axis.

As there may be several spheroids of different magnitudes, so there may be several different echoes of the same original sound. And as there may happen to be a greater number of reflecting points in the surface of an exterior spheroid than in that of an interior, a second or a third echo may be much more powerful than the first, provided that the superior number of reflecting points, that is, the superior number of reflected pulses propagated to the ear, be more than sufficient to compensate for the decay of sound which arises from its being propagated through a greater space. This is finely illustrated in the celebrated echoes at the lake of Killarney in Kerry, where the first return of the sound is much inferior in strength to those which immediately succeed it.

From what has been laid down it appears, that for the most powerful echo, the founding body should be in one focus of the ellipse which is the section of the echoing spheroid, and the hearer in the other. However, an echo may be heard in other situations, though not so favourably; as such a number of reflected pulses may arrive at the same time at the ear as may be sufficient to excite a distinct perception. Thus a person often hears the echo of his own voice; but for this purpose he should stand at least 63 or 64 feet from the reflecting obstacle, according to what has been said before. At the common rate of speaking, we pronounce not above three syllables and an half, that is, seven half syllables in a second; therefore, that the echo may return just as soon as three syllables are expressed, twice the distance of the speaker from the reflecting object must be equal to 1000 feet; for, as sound describes 1142 feet in a second, \( \frac{1}{2} \)ths of that space, that is, 1000 feet nearly, will be described while six half or three whole syllables are pronounced: that is, the speaker must stand near 500 feet from the obstacle. And in general, the distance of the speaker from the echoing surface, for any number of syllables, must be equal to the seventh part of the product of 1142 feet multiplied by that number.

In churches we never hear a distinct echo of the voice, but a confused sound when the speaker utters his words too rapidly; because the greatest difference of distance between the direct and reflected courses of such a number of pulses as would produce a distinct sound, is never in any church equal to 127 feet, the limit of echoes.

But though the first reflected pulses may produce no echo, both on account of their being too few in number, and too rapid in their return to the ear; yet it is evident, that the reflecting surface may be formed, as that the pulses which come to the ear after two reflections or more may, after having described 127 feet or more, arrive at the ear in sufficient numbers, and also so nearly at the same instant, as to produce an echo, though the distance of the reflecting surface from the ear be less than the limit of echoes. This is confirmed by a singular echo in a grotto on the bank of the little brook called the Dinan, about two miles from Cattlecomber, in the county of Kilkenny. As you enter the cave, and continue speaking loud, no return of the voice is perceived; but on your arriving at Entertain a certain point, which is not above 14 or 15 feet from the reflecting surface, a very distinct echo is heard.

Now this echo cannot arise from the first course of pulses that are reflected to the ear, because the breadth of the cave is so small, that they would return too quickly to produce a distinct sensation from that of the original sound; it therefore is produced by those pulses, which, after having been reflected several times from one side of the grotto to the other, and having run over a greater space than 127 feet, arrive at the ear in considerable numbers, and not more distant from each other, in point of time, than the ninth part of a second.

This article shall be dismissed with a few inventions founded on some of the preceding principles, which may amuse a number of our readers.

**Entertaining Experiments and Contrivances.**

I. Place a concave mirror of about two feet diameter, as A B (c), in a perpendicular direction. The focus of this mirror may be at 15 or 18 inches distance from its surface. At the distance of about five or six feet let there be a partition, in which there is an opening E F, equal to the size of the mirror; against this opening must be placed a picture, painted in watercolours, on a thin cloth, that the sound may easily pass through it (H).

Behind the partition, at the distance of two or three feet, place another mirror G H, of the same size as the former, and let it be diametrically opposite to it.

At the point C let there be placed the figure of a man seated on a pedestal, and let his ear be placed exactly in the focus of the first mirror: his lower jaw must be made to open by a wire, and shut by a spring; and there may be another wire to move the eyes: these wires must pass through the figure, go under the floor, and come up behind the partition.

Let a person, properly instructed, be placed behind the partition near the mirror. You then propose to any one to speak softly to the statue, by putting his mouth to the ear of it, assuring him that it will answer instantly. You then give the signal to the person behind the partition, who, by placing his ear to the focus I, of the mirror G H, will hear distinctly what the other said; and, moving the jaw and eyes of the statue by the wires, will return an answer directly, which will in like manner be distinctly heard by the first speaker.

This experiment appears to be taken from the Century of Inventions of the Marquis of Worcester; whose designs, at the time they were published, were treated with ridicule and neglect as being impracticable, but are now known to be generally, if not universally practicable. The words of the Marquis are these: "How to make a brazen or stone head in the midst of a great field or garden, so artificial and natural, that though a man speak ever so softly, and even whisper into the ear thereof, it will presently open its mouth, and resolve the question in French, Latin, Welsh, Irish, or English, in good terms, uttering it out of its mouth, and then shut it until the next question be asked."—The two following, of a similar nature, appear to have been inventions of Kircher, by means of which (as he informs us *) he used to "utter foreigned and ludicrous confutations, with a view to show the fallacy and imposture of ancient oracles."

II. Let there be two heads of plaster of Paris, placed on pedestals, on the opposite sides of a room. There must be a tin tube of an inch diameter, that must pass from the ear of one head, through the pedestal, under the floor, and go up to the mouth of the other. Observe, that the end of the tube which is next the ear of the one head, should be considerably larger than that end which comes to the mouth of the other. Let the whole be so disposed that there may not be the least suspicion of a communication.

Now, when a person speaks, quite low, into the ear of one butt, the sound is reverberated thro' the length of the tube, and will be distinctly heard by any one who shall place his ear to the mouth of the other. It is not necessary that the tube should come to the lips of the butt.—If there be two tubes, one going to the ear, and the other to the mouth, of each head, two persons may converse together, by applying their mouth and ear reciprocally to the mouth and ear of the butts; and at the same time other persons that stand in the middle of the chamber, between the heads, will not hear any part of their conversation.

III. Place a butt on a pedestal in the corner of a room, and let there be two tubes, as in the foregoing amusement, one of which must go from the mouth and the other from the ear of the butt, through the pedestal, and the floor, to an under apartment. There may be likewise wires that go from the under jaw and the eyes of the butt, by which they may be easily moved.

A person being placed in the under room, and at a signal given applying his ear to one of the tubes, will hear any question that is asked, and immediately reply; moving at the same time, by means of the wires, the mouth and the eyes of the butt, as if the reply came from it.

IV. In a large cage, such as is used for dials and spring-clocks, the front of which, or at least the lower part of it, must be of glass, covered on the inside with gauze; let there be placed a barrel-organ, which, when wound up, is prevented from playing, by a catch that takes a toothed wheel at the end of the barrel. To one end of this catch there must be joined a wire, at the end of which there is a flat circle of cork, of the same dimension with the inside of a glass tube, in which it is to rise and fall. This tube must communicate with a reservoir that goes across the front part of the bottom of the cage, which is to be filled with spirits, such as is used in solar systems.

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(c) Both the mirrors here used may be of tin or gilt pasteboard, this experiment not requiring such as are very accurate.

(h) The more effectually to conceal the cause of this illusion, the mirror AB may be fixed in the wainscot, and a gauze or any other thin covering thrown over it, as that will not in the least prevent the sound from being reflected. An experiment of this kind may be performed in a field or garden, between two hedges, in one of which the mirror AB may be placed, and in the other an opening artfully contrived. thermometers, but not coloured, that it may be the better concealed by the glaze.

This case being placed in the sun, the spirits will be rarefied by the heat; and rising in the tube, will lift up the catch or trigger, and set the organ in play: which it will continue to do as long as it is kept in the sun; for the spirits cannot run out of the tube, that part of the catch to which the circle is fixed being prevented from rising beyond a certain point by a check placed over it.

When the machine is placed against the side of a room on which the sun shines strong, it may constantly remain in the same place, if you inclose it in a second case, made of thick wood, and placed at a little distance from the other. When you want it to perform, it will be only necessary to throw open the door of the outer case, and expose it to the sun.

But if the machine be moveable, it will perform in all seasons by being placed before the fire; and in the winter it will more readily stop when removed into the cold.

A machine of this sort is said to have been invented by Cornelius Drebel, in the last century. What the construction of that was, we know not; it might very likely be more complex, but could scarce answer the intention more readily.

V. Under the keys of a common harpsichord let there be fixed a barrel, something like that in a chamber organ, with stops or pins corresponding to the tunes you would have it play. These stops must be moveable, so that the tunes may be varied at pleasure. From each of the keys let there go a wire perpendicular down: the ends of these wires must be turned up for about one-fourth of an inch. Behind these wires let there be an iron bar, to prevent them from going too far back. Now, as the barrel turns round, its pins take the ends of the wires, which pull down the keys, and play the harpsichord. The barrel and wires are to be all inclosed in a case.

In the chimney of the same room where the harpsichord stands, or at least in one adjacent, there must be a smoke jack, from whence comes down a wire, or cord, that, passing behind the wainscot adjoining the chimney, goes under the floor, and up one of the legs of the harpsichord, into the case; and round a small entertainer wheel fixed on the axis of that first mentioned. There should be pulleys at different distances, behind the wainscot and under the floor, to facilitate the motion of the chord.

This machinery may be applied to any other keyed instrument as well as to chimes, and to many other purposes where a regular continued motion is required.

An instrument of this sort may be considered as a perpetual motion, according to the vulgar acceptation of the term; for it will never cease going till the fire be extinguished, or some parts of the machinery be worn out.

VI. At the top of a summer-house, or other building, let there be fixed a vane AB, on which is the pinion C, A Ventofal that takes the toothed wheel D, fixed on the axis EF, Plate I., which at its other end carries the wheel G, that takes fig. 6. the pinion H. All these wheels and pinions are to be between the roof and the ceiling of the building. The pinion H is fixed to the perpendicular axis IK, which goes down very near the wall of the room, and may be covered after the same manner as are bell-wires. At the lower end of the axis IK there is a small pinion L, that takes the wheel M, fixed on the axis of the great wheel NO. In this wheel there must be placed a number of stops, corresponding to the tunes it is to play. These stops are to be moveable, that the tunes may be altered at pleasure. Against this wheel there must hang 12 small bells, answering to the notes of the gamut. Therefore, as the wheel turns round, the stops striking against the bells, play the several tunes. There should be a fly to the great wheel, to regulate its motion when the wind is strong. The wheel NO, and the bells, are to be inclosed in a case.

There may be several sets of bells, one of which may answer to the tenor, another to the treble, and a third to the bass; or they may play different tunes, according to the size of the wheel. As the bells are small, if they are of silver, their tone will be the more pleasing.

Instead of bells, glasses may be here used, so disposed as to move freely at the stroke of the stops. This machinery may likewise be applied to a barrel-organ; and to many other uses.