TRUMPET, Articulate, comprehends both the speaking and the hearing trumpet, is by much the most valuable instrument, and has, in one of its forms, been used by people among whom we should hardly have expected to find such improvements. That the speaking trumpet, of which the object is to increase the force of articulate sounds, should' have been known to the ancient Greeks, can excite no wonder; and therefore we easily admit the accounts which we read of the horn or trumpet, with which Alexander addressed his army, as well as of the whispering caverns of the Syracusan tyrant. But that the natives of Peru were acquainted with this instrument, will probably surprise many of our readers. The fact, however, seems incontrovertible.

In the History of the Order of Jesuits, published at Naples in 1601 by Beritaria, it is said, that in the year 1595 a small convent of that order in Peru, situated in a remote corner, was in danger of immediate destruction by famine. One evening the superior, Father Samaniac, implored the help of the cacique. Next morning, on opening [7:21:391]the gate of the monastery, he found it surrounded by a number of women, each “f whom carried a small basket of provisions. He returned thanks to heaven for having miraculously interposed, by inspiring the good people with pity for the distress of his friars. But when he expressed to them his wonder how they came all to be moved as if by mutual agreement with these benevolent sentiments, they told him it was no such thing; that they looked upon him and his countrymen as a pack of infernal magicians, who by their sorceries had enslaved the country, and had bewitched their good cacique, who hitherto had treated them with kindness and attention, as became a true worshipper of the sun; but that the preceding evening at sunset he had ordered the inhabitants of such and such villages, about six miles off, to come that morning with provisions to this nest of wizards. The superior asked them in what manner the governor had warned so many of them in so short a time, at such a distance from his own residence. They told him that it was by the trumpet; and that every person heard at his own door the distinct terms of the order. The father had heard nothing; but they told him that none heard the trumpet but the inhabitants of the villages to which it was directed. This is a piece of very curious information; but, after allowing a good deal to the exaggeration of the reverend Jesuits, it cannot, we think, be doubted that the Peruvians actually possessed this stentorophonic art. For we may observe that the effect described in this narration resembles what toe now know to be the effect of speaking trumpets, while it is unlike what the inventor of such a tale would naturally and ignorantly say. Till speaking trumpets were really known, we would expect the sound to be equally diffused on all sides, which is not the case; for it is much stronger in the line of the trumpet than in any direction very oblique to it.

About the middle of the seventeenth century, Athanasius Kircher turned his attention to the philosophy of sound, and in different works threw out many useful and scientific hints on the construction of speaking trumpets; but his mathematical illustrations were so vague, and his own character of inattention and credulity so notorious, that for some time these works did not attract the notice to which they were well entitled.

About the year 1670, Sir Samuel Morland, a gentleman of great ingenuity and science, directed his attention to this subject, and proposed as a question to the Royal Society of London, What is the best form for a speaking trumpet? which he called a stentorophonic horn. He accompanied his demand with an account of his own notions on the subject (which he acknowledged to be very vague and conjecturaD, and an exhibition of some instruments constructed according to his views. They were in general very large conical tubes, suddenly spreading at the very mouth to a greater width. Their effect was really wonderful. They were tried in St James’s Park; and his majesty, King Charles IL, speaking in his ordinary colloquial pitch of voice through a trumpet only five and a half feet long, was clearly and most distinctly heard at the distance of 1000 yards. Another person, selected we suppose for the loudness and distinctness of his voice, was perfectly understood at the distance of four miles and a half. The fame of this instrument soon spread. Morland’s principles were refined, considering the novelty of the inquiry, and differed considerably from Father Kircher’s. The aerial undulations (for he speaks very accurately concerning the nature of sound) endeavour to diffuse themselves in spheres, but are stopped by the tube, and therefore reundulate towards the axis like waves from a bank; and meeting in the axis, they form a strong undulation a little farther advanced along the tube, which again spreads, is again reflected, and so on, till it arrives at the mouth of the tube greatly magnified, and then it is diffused through the open air in the same manner as if all proceeded from a very sonorous point in the centre of the wide end of the trumpet. The author distinguishes with great judgment between the prodigious reinforcement of sound in a speaking trumpet and that in the musical trumpet, bugle-horn, conch-shell, &c.; and shows that the difference consists in the violence of the first sonorous agitation, which can be produced by us only on a very small extent of surface. The mouth-piece diameter, therefore, of the musical trumpet must be very small, and the force of blast very considerable. Thus one strong but simple undulation will he excited, which must be subjected to the modifications of harmony, and will be augmented by using a conical tube.^[1. Accordingly the sound of the bugle-horn, of the musical trumpet, or the French horn, is prodigiously loud, when we consider the small passage through which a moderate blast is sent by the trumpeter. ] But a speaking trumpet must make no change on the nature of the first undulations; and each point of the mouth-piece must be equally considered as the centre of sonorous undulations, all of which must be reinforced in the same degree, otherwise all distinctness of articulation will be lost. The mouthpiece must therefore take in the whole of the mouth of the speaker.

When Sir Samuel Morland’s trumpet came to be generally known on the continent, it was soon discovered that the speaker could be heard at a great distance only in the line of the trumpet; and this circumstance was by a Mr Cassegrain (Journ. des Sçavans, 1G72, p. 131) attributed to a defect in the principle of its construction, which he said was not according to the laws of sonorous undulations. He proposed a conoid formed by the revolution of a hyperbola round its assymptote as the best form. A Mr Hase of Wittemberg, on the other hand, proposed a parabolic conoid, having the mouth of the speaker placed in the focus. In this construction he plainly went on the principle of a reflection similar to that of the rays of light; but this is by no means the case. The effect of the parabola will be to give one reflection, and in this all the circular undulations will be converted into plane waves, which are at right angles to the axis of the trumpet. But nothing hinders their subsequent diffusion; for it does not appear that the sound will be enforced, because the agitation of the particles on each wave is not augmented.

The subject is exceedingly difficult. We do not fully comprehend on what circumstance the affection or agitation of our organ, or simply of the membrana tympani, depends. A more violent agitation of the same air, that is, a wider oscillation of its particles, cannot fail to increase the impulse on this membrane. The point therefore is to find what concourse of feeble undulations will produce or be equivalent to a great one. The reasonings of all these restorers of the speaking trumpet are almost equally specious, and each point out some phenomenon which should characterise the principle of construction, and thus enable us to say which is most agreeable to the procedure of nature. Yet there is hardly any difference in the performance of trumpets of equal dimensions made after these different methods.

The propagation of light and that of elastic undulations seem to require very different methods of management. Yet the ordinary phenomena of echoes are perfectly explicable by the acknowledged laws either of optics or acoustics; still however there are some phenomena of sound which are very unlike the genuine results of elastic undulations. If sounds are propagated spherically, then what conies into a room through a small hole should diffuse itself from that hole as round a centre, and it should be heard equally well [7:21:392]at twelve feet distance from the hole in every direction. Yet it is very sensibly louder when the hearer is in the straight line drawn from the sonorous body through the hole. A person can judge of the direction of the sounding body with tolerable exactness. Cannon discharged from the different sides of a ship are very easily distinguished, which should not he the case by the Newtonian theory; for in this the two pulses on the ear should have no sensible difference.

The most important fact for our purpose is this: an echo from a small plane surface in the midst of an open field is not heard, unless we stand in such a situation that the angle of reflected sound may be equal to that of incidence. But by the usual theory of undulations, this small surface should become the centre of a new undulation, which should spread in all directions. If we may make an analogous experiment on watery undulations, by placing a small flat surface so as to project a little above the water, and then drop in a small pebble at a distance, so as to raise one circular wave, we shall observe, that when this wave arrives at the projecting plane, it is disturbed by it, and this disturbance spreads from it on all sides. It is indeed sensibly stronger in that line which is drawn from it at equal angles with the line drawn to the place where the pebble was dropped. But in the case of sound, it is a fact, that if we go to a very small distance on either side of the line of reflection, we shall hear nothing.

Here then is a fact, that whatever may be the nature of the elastic undulations, sounds are reflected from a small plane in the same manner as light. We may avail ourselves of this fact as a mean for enforcing sound, though we cannot explain it in a satisfactory manner. We should expect from it an effect similar to the hearing of the original sound along with another original sound coming from the place from which this reflected sound diverges. If therefore the reflected sound or echo arrives at the car in the same instant with the original sound, the effect will be doubled; or at least it will be the same with two simultaneous original sounds. Now we know that this is in some sense equivalent to a stronger sound; for it is a fact, that a number of voices uttering the same or equal sounds is heard at a much greater distance than a single voice. How this happens, we cannot perhaps explain by mechanical laws, or assign the exact proportion in which ten voices exceed the effect of one voice; or the proportion of the distances at which they seem equally loud. We may therefore, for the present, suppose that two equal voices at the same distance are twice as loud, three voices three times as loud, &c. Therefore, if, by means of a speaking trumpet, we can make ten equal echoes arrive at the ear at the same moment, we may suppose its effect to be to increase the audibility ten times; and we may express this shortly by calling the sound ten times louder or more intense.

But we cannot do this precisely. We cannot by any contrivance make the sound of a momentary snap, and those of its echoes, arrive at the ear in the same moment, because they come from different distances. But if the original noise be a continued sound, a man’s voice, for example, uttering a continued uniform tone, the first echo may reach the ear at the same moment with the second vibration of the larynx, the second echo along with the third vibration, and so on. It is evident, that this will produce the same effect. the only difference will be, that the articulations of the voice will be made indistinct, if the echoes come from very different distances. Thus, if a man pronounce the syllable taw, and ten successive echoes are made from places which are ten feet farther off, the tenth part of a second (nearly) will intervene between hearing the first and the last. This will give it the sound of the syllable thaw, or perhaps raw, because r is the repetition of t. Something like this occurs when, standing at one end of a long line of soldiers, we hear the muskets of the whole line discharged in one instant. It seems to us the sound of a running the aim therefore in the construction of a speaking trumpet may be, to cause as many echoes as possible to reach a distant ear without any perceptible interval of time. This will give distinctness, and something equivalent to loudness. Pure loudness arises from the violence of the single aerial undulation. To increase this may be the aim in the construction of a trumpet; but we are not sufficiently acquainted with the mechanism of these undulations to bring this about with certainty and precision; whereas we can procure this accumulation of echoes without much trouble, since we know that echoes are, in fact, reflected like light. We can form a trumpet so that many of these lines of reflected sound shall pass through the place of the hearer. We arc indebted to Mr Lambert of Berlin for this simple and popular view of the subject; and shall here give an abstract of his most ingenious Dissertation on Acoustic Instruments, published in the Berlin Memoirs for 1763.

Sound naturally spreads in all directions; but we know that echoes or reflected sounds proceed almost strictly in certain limited directions. If therefore we contrive a trumpet in such a way that the lines of echo shall be confined within a certain space, it is reasonable to suppose that the sound will become more audible in proportion as this diffusion is prevented. Therefore, if we can oblige a sound which, in the open air, would have diffused itself over a hemisphere, to keep within a cone of 120°, we should expect it to be twice as audible within this cone. This will be accomplished by making the reflections such that the lines of reflected sound shall be confined within this cone. We here suppose that nothing is lost in the reflection. Let us examine the effect of a cylindrical trumpet.

Let the trumpet be a cylinder ABED (fig. 1), and let C be a sounding point in the axis. It is evident that all the sound in the cone BCE will go forward without any reflection. Let CM be any other line of sound, which we may, for brevity’s sake, call a sonorous or phonic line. Being reflected in the points M, N, O, P, it is evident that it will at last escape from the trumpet in a direction PQ, equally diverging from the axis with the line CM. The same must be true of every other sonorous line. Therefore the echoes w ill all diverge from the mouth of the trumpet, in the same manner as they would have proceeded from C without any trumpet. Even supposing, therefore, that the echoes are as strong as the original sound, no advantage is gained by such a trumpet, but that of bringing the sound forward from C to c . This is quite trifling when the hearer is at a distance. Yet we find that sounds may be heard at a very great distance, at the end of long, narrow, cylindrical, or prismatical galleries. It is known that a voice may be distinctly heard at the distance of several hundred feet in the Roman aqueducts, whose sides are perfectly straight and smooth, being plastered with stucco. The smooth surface of the still water greatly contributes to this effect. Cylindrical or prismatical trumpets must therefore be rejected.

Let the trumpet be a cone BCA (fig. 2), of which CN is the axis, DK a line perpendicular to the axis, and DFHI [7:21:393]the path of a reflected sound in the plane of the axis. The last angle of reflection IHA is equal to the last angle of incidence FHC. The angle BFH, or its equal CFD, is equal to the angles FHD and FCH; that is, the angle of incidence CFD exceeds the next angle of incidence FHC by the angle FCD, that is, by the angle of the cone. In like manner, FDH exceeds CFD by the same angle FCD. Thus every succeeding angle, either of incidence or reflection, exceeds the next by the angle of the cone. Call the angle of the cone a , and let b be the first angle of incidence PDC. The second, or DFC, is b—a. The third, or FHC, is b — 2 a, &c.: and the nth angle of incidence or reflection is b — n a, after n reflections. Since the angle diminishes by equal quantities at each subsequent reflection, it is plain, that whatever be the first angle of incidence, it may be exhausted by this diminution; namely, when n times a exceeds or is equal to b. Therefore, to know how many reflections of a sound, whose first incidence has the inclination b, can be made in an infinitely extended cone, whose angle is a, divide b by a ; the quotient will give the number n of reflections, and the remainder, if any, will be the last angle of incidence or reflection less than a . It is very plain, that when an angle of reflection IHA is equal to or less than the angle BCA of the cone, the reflected line HI will no more meet with the other side CB of the cone.

We may here observe, that the greatest angle of incidence is a right angle, or 90°. This sound would be reflected back in the same line, and would be incident on the opposite side in an angle = 90° — a , &c.

Thus we see that a conical trumpet is well suited for confining the sound; for, by prolonging it sufficiently, we can keep the lines of reflected sound wholly within the cone. And when it is not carried to such a length as to do this, when it allows the sounding line GH, for example, to escape without farther reflection, the divergency from the axis is less than the last angle of reflection BFH by half the angle BCA of the cone. Let us see what is the connection between the length and the angle of ultimate reflection.

*****We have sin. (b — a) : sin. b = CD: CF, and CF = sin. b CD * sin. (b —a)' aπd sin. (0 — 2 a): sin. (6—α) = CF: CH, and CH = CF × ⅞ ’-⅛°> = CD × -≠,"∕ ∖ × ειn.(o—2<∕) sin.(6— a) oo l tS b - a ) = CDx . ζ' n ∙* ι , &C.

sin (b —2α) sm.(o— ia)

Therefore if we suppose X to be the length which will give us n reflections, we shall have X = CD × sin. b / sin. ( b - na ) . Hence we see that the length increases as the angle (b—na) diminishes; but is not infinite, unless n a is equal to b. In this case, the immediately preceding angle of reflection must be a, because these angles have the common difference a. Therefore the last reflected sound was moving parallel to the opposite side of the cone, and cannot again meet it. But though we cannot assign the length which w ill give the nth reflection, we can give the length which will give the one immediately preceding, whose angle with the side of the cone is a . Let Y be this length. We have Y = CD × sin. b / sin. a . This length will allow every line of sound to be reflected as often, saving once, as if the tube were infinitely long. For suppose a sonorous line to be traced backwards, as if a sound entered the tube in the direction i h, and were reflected in the points h,f, d, D, the angles will be continually augmented by the constant angle a. But this augmentation can never go farther than 90° + ½ a. For if it reaches that value at D, for instance, the reflected line DK will be perpendicular to the axis CN; and the angle ADK will be equal to the angle DKB, and the sound will come out again. This remark is of importance on another account.

Now, suppose the cone to be cut off at D by a plane perpendicular to the axis, KD will be the diameter of its mouth-piece; and if we suppose a mouth completely occupying this circle, and every point of the circle to be sonorous, the reflected sounds will proceed from it in the same manner as light would from a flame which completely occupies its area, and is reflected by the inside of the cone. The angle FDA will have the greatest possible sine when it is a right angle, and it never can be greater than ADK, which is = 90° + ½ a. And since between 90° + ½ a, and 90° — ½ a , there must fall some multiple of a , call this multiple b. Then, in order that every sound may be reflected as often as possible, saving once, we must make the length of it X = CD × sin. b / sin. a .

Now, since the angle of the cone is never made very great, never exceeding ten or twelve degrees, b can never differ from 90° above a degree or two, and its sine cannot differ much from unity. Therefore X will be very nearly equal to , which is also very nearly equal to ****—— i —; 1 sin. a j 2sm.⅜α

because a is small, and the sines of small arches are nearly equal and proportional to the arches themselves. There is even a small compensation of errors in this formula. For as the sine of 90° is somewhat too large, which would give X too great, 2 sin. ½ a is also larger than the sine of a. Thus let a be 12°: then the nearest multiple of a is 84° or 96°, both of which are as far removed as possible from 90°, and the error is as great as possible, and is nearly 1 / 160 th of the whole.

This approximation gives us a very simple construction. Let CM be the required length of the trumpet, and draw ML perpendicular to the axis in O. It is evident that sin. MCO: rad. = MO: CM, and CM or X = ****-—r— = sin. £ a

——; but X =— CD , and therefore LM equal to CD. 2sin.∣β 2sm.⅞<ι

If therefore the cone be of such a length, that its diameter at the month is equal to the length of the part cut off, every line of sound will have at least as many reflections, save one, as if the cone were infinitely long; and the last reflected line will either be parallel to the opposite side of the cone, or he nearer the axis than this parallel; consequently such a cone will confine all the reflected sounds within a cone whose angle is 2 a, and will augment the sound in the proportion of the spherical base of this cone to a complete hemispherical surface. Describe the circle DKT round C, and making DT an arch of 90°, draw the chord DT. Then, since the circles described with the [7:21:394]radii DK, DT, are equal to the spherical surfaces generated by the revolution of the arches DK and DKT round the axis CD, the sound will be condensed in the proportion of DK 2 to DT 2 .

This appears to be the best general rule for constructing the instrument; for, to procure another reflection, the tube must be prodigiously lengthened, and we cannot suppose that one reflection more will add greatly to its power.

It appears, too, that the length depends chiefly on the angle of the cone, for the mouth-piece may be considered as nearly a fixed quantity. It must be of a size to admit the mouth when speaking with force and without constraint. About an inch and a half may be fixed on for its diameter. When therefore we propose to confine the sound to a cone of twice the angle of the trumpet, the whole is determined by that angle. For since in this case LM is equal to CD, CD 2 we have DK: CD = LM (or CD): CM and CM = *** iyκ ~.

***But 2 sin. Ie: 1 = DK: CD, and 2 sin. À a : 1 = CD: CM; therefore 4 sin. 2 ⅛ a: I = DK: CM,

And CM = 7 = very nearly. And

4sin∕⅛α sin. a since DK is an inch and a half, we get the length in inches counted from the apex of the cone = -r-⅛-, or 1 sm. a a

3 ———. From this we must cut oft’ the part CD, which 2sm.-α 1

DK , DK 3

is ;— i -, or very nearly - ; , or - , —-, measured in

2sm.⅜o sin. α 2 sin. a inches, and we must make the mouth of the same width 3

2 sin. d

On the other hand, if the length of the trumpet is fixed on, we can determine the angle of the cone. For let the 3 length (reckoned from C) be L; we have 2 sin. 9 α = γ, or sin. 9 a =.,; “bod sin. α =

Thus let six feet or 72 inches be chosen for the length of the cone, we have sin. a — ∣J =0’144-34

= sin. 8° 17' for the angle of the cone; and the width at 3

the mouth is ^g s ∙ n tf = 10’4 inches. This being taken from 72, leaves 61-6 inches for the length of the trumpet. ’

And since this trumpet confines the reflected sounds to DT “a cone of 16° 34 ’, we have its magnifying power = ∣jj-⅛, (⅜DT)' sin. 2 45° ne , 1 t r = (t^DK)≈ = sh‰ r 4≡8'∣ zz 96 near ' y ∙ lt therefure condenses the sound about 96 times; and if the distribution were uniform, it would be heard √7j⅛, or nearly 10 times farther off; for the loudness of sounds is supposed to be inversely as the square of the distance from the centre of undulation.

But before we can pronounce with precision on the performance of a speaking trumpet, we must examine into the manner in which the reflected sounds are distributed over the space in which they are all confined.

Let BKDA (fig. 3) be the section of a conical trumpet by a plane through the axis; let C be the vertex of the cone, and CW its axis; let TKV be the section of a sphere, having its centre in the vertex of the cone; and let P be a sonorous point on the surface of the sphere, and P a f e l the path of a line of sound lying in the plane of the section. In the great circle of the sphere take KQ — KP, DR = DQ, and KS = KR. Draw QB h ; also draw Q d n parallel to DA; and draw PB, P d, P A.

1. Then it is evident that all the lines drawn from P, within the cone APB, proceed without reflection, and are diffused as if no trumpet had been used.

2. All the sonorous lines which fall from P on KB are reflected from it as if they had come from Q.

3. All the sonorous lines between BP and d P have suffered but one reflection; for d n will no more meet DAA' so as to be reflected again.

4. All the lines which have been reflected from KB, and afterwards from DA, proceed as if they had come from R. For the lines reflected from KB proceed as if they had come from Q; and lines coming from Q and reflected by DA, proceed as if they had come from R. Therefore draw R A o , and also draw R g m parallel to KB, and draw Q c A q, Q b g , P c , and P b . Then,

5. All the lines between b P and c P have been twice reflected.

Again, draw SB p , B r R, r u q, S x A , R y x , Q z y .

6. All the lines between u P and z P have suffered three reflections.

Draw the tangents T A t, VBr, crossing the axis in W.

7. The whole sounds will be propagated within the cone v W t. For to every sonorous point in the line KD there corresponds a point similar to Q, regulating the first reflection from KB; and a point similar to R, regulating the second reflection from DA; and a point S regulating the third reflection from KB, &c. Ami similar points will be found regulating the first reflection from DA, the second from KB, and the third from DA, &c.; and lines drawn from all these through A and B must lie within the tangents TA and VB.

8. Thus the centres of reflection of all the sonorous lines which lie in planes passing through the axis, will be found in the surface of this sphere; and it may be considered as a sonorous sphere, whose sounds first concentrate in W, and are then diffused in the cone v W t .

It may be demonstrated nearly in the same manner, that the sonorous lines which proceed from P, but not in the plane passing through the axis, also proceed, after various reflections, as if they bad come from points in the surface of the same sphere. The only difference in the demonstration is, that the centres Q, R, S of the successive reflections are not in one plane, but in a spiral line winding round the surface of the sphere according to fixed laws. The foregoing conclusions are therefore general for all the sounds which come in all directions from every point in the area of the mouth-piece.

Thus it appears, that a conical trumpet is well fitted for increasing the force of sounds by diminishing their final divergence. For had the speaker’s mouth been in the open air, the sounds which are now confined within the cone v W t would have been diffused over a hemisphere; and we [7:21:395]see that prolonging the trumpet must confine the sounds still more, because this will make the angle BWA still smaller; a longer tube must also occasion more reflections, and consequently send more sonorous undulations to the ear at a distance placed within the cone v W t .

We have now obtained a very connected view of the whole effect of a conical trumpet. It is the same as if the whole segment TKDV were sounding, every part of it with an intensity proportional to the density of the points Q, R, S, &c. corresponding to the different points P of the mouthpiece. It is easy to see that this cannot be uniform, but must be much rarer towards the margin of the segment. It would require a good deal of discussion to show the density of these fictitious sounding points; and we shall content ourselves with giving a very palpable view of the distribution of the sonorous rays, or the density (so to speak) of the echoes, in the different situations in which a hearer may be placed.

We may observe, in the mean time, that this substitution of a sounding sphere for the sounding mouth-piece has an exact parallel in optics, by which it will be greatly illustrated. Suppose the cone BKDA (fig. 3) to be a tube polished in the inside, fixed in a wall B α , perforated in BA, and that the mouth-piece DK is occupied completely by a flat flame. The effect of this on a spectator will be the same, if he is properly placed in the axis, as if he were looking at a flame as big as the whole sphere. This is very evident.

It is easy to see that the line l e S is equal to the Vine l e f a P ; therefore the reflected sounds also come to the ear in the same moments as if they had come from their respective points on the surface of the substituted sphere. Unless, therefore, this sphere be enormously large, the distinctness of articulation will not be sensibly affected, because the interval between the arrival of the different echoes of the same snap will be insensible.

Our limits oblige us to content ourselves with exhibiting this evident similarity of the progress of echo from the surface of this phonic sphere, to the progress of light from the same luminous sphere shining through a hole of which the diameter is A B. The direct investigation of the intensity of the sound in different directions and distances would take up much room, and give no clearer conception of the subject. The intensity of the sound in any point is precisely similar to the intensity of the illumination of the same point; and this is proportional to the portion of the luminous surface seen from this point through the hole directly, and to the square of the distance inversely. The intelligent reader will acquire a distinct conception of this matter from fig. 4, which represents the distribution of the sonorous lines, and by consequence the degree of loudness which may be expected in the different situations of the hearer.

As we have already observed, the effect of the cone of the trumpet is perfectly analogous to the reflection of light from a polished concave conical mirror. Such an instrument would be equally fitted for illuminating a distant object. We imagine that these would be much more powerful than the spherical or even parabolic mirrors commonly used for this purpose. These last, having the candle in the focus, also send forward a cylinder of light of equal width with the mirror. But it is well known, that oblique reflections are prodigiously more vivid than those made at greater angles. Where the inclination of the reflected light to the plane of the mirror does not exceed eight or ten degrees, it reflects about three fourths of the light which falls on it. But when the inclination is 80°, it does not reflect one fourth part.

We may also observe, that the density of the reflected sounds by the conical trumpet ABC (fig. 4) is precisely similar to that of the illumination produced by a luminous sphere TDV shining through a hole AB. There will be a space circumscribed by the cone formed by the lines TB t and VA v, which is uniformly illuminated by the whole sphere (or rather by the segment TDV), and on each side there is a space illuminated by a part of it only, and the illumination gradually decreases towards the borders. A spectator placed much out of the axis, and looking through the hole AB, may not see the whole sphere. In like manner, he will not hear the whole sounding sphere: he may be so far from the axis as neither to see nor hear any part of it.

Assisting our imagination by this comparison, we perceive that beyond the point w' there is no place where all the reflected sounds are heard. Therefore, in order to preserve the magnifying power of the trumpet at any distance, it is necessary to make the mouth as wide as the sonorous sphere. Nay, even this would be an imperfect instrument, because its power would be confined to a very narrow space; and if it be not accurately pointed to the person listening, its power will be greatly diminished. And we may observe, by the way, that we derive from this circumstance a strong confirmation of the justness of Mr Lambert’s principles; for the effects of speaking trumpets are really observed to be limited in the way here described. Parabolic trumpets have been made, and they fortify the sound not only in the cylindrical space in the direction of the axis, but also on each side of it; which should not have been the case, had their effect depended only on the undulations formed by the parabola in planes perpendicular to the axis. But to proceed.

Let BCA (fig. 5) be the cone, ED the mouth-piece, TEDV the equivalent sonorous sphere, and TBAV the circumscribed cylinder. Then CA or CB is the length of cone that is necessary for maintaining the magnifying power at all distances. We have two conditions to be fulfilled. The diameter ED of the mouth-piece must be of a certain fixed magnitude, and the diameter A B of the outer end must be equal to that of the equivalent sonorous sphere. These conditions determine all the dimensions of the trumpet and its magnifying power. And, first, with respect to the dimensions of the trumpet.

The similarity of the triangles ECG and BCF gives CG: ED = CF: AB; but CG = BF = ½AB, and CF = CG + GF = GF + ½AB; therefore ½AB: ED — GF + ½AB; AB, and AB: ED = 2 GF + AB: AB; therefore 2 GF × ED + AB × ED = AB 2 , and 2 GF × ED = AB 2 — AB × ED = AB × (AB — ED) and GF [7:21:396]*****AB×(AB-ED) . , , . , , , = - ⅛ fD ' 0n t * le 0t * ler hand, because

AB* — AB × ED = 2 GF × ED, we have AB≈ — AB X ED +; ED = 2 GF × ED + ί ED≡, or (AB — ⅛ ED) 2 = 2 GF × ED + ( ED 2 , and AB = √ (2 GF × ED 4- ⅛ ED i ) + i ED.

Let x represent the length of the trumpet, y the diameter at the great end, and m the diameter of the mouth-piece.

Then x =. ——-, and y — √(2 xm -f-1 m 2 ) -∣- ⅛∕λ. Thus the length and the great diameter may be had reciprocally. The useful case in practice is to find the diameter for a proposed length, which is obtained by the last equation.

Now if we take all the dimensions in inches, and fix m at an inch and a half, we have 2 x m ~ 3 x, and ⅛ ι∣r = 0∙5625, and m = 0∙75; so that our equation becomes y = √ ’ (3 x -∣- 0’5625) -f- 0∙75. The following table gives the dimensions of a sufficient variety of trumpets. The first column is the length of the trumpet in feet; the second column is the diameter of the mouth in inches; the third column is the number of times that it magnifies the sound; and the fourth column is the number of times that it increases the distance at which a man may be distinctly heard by its means; the fifth contains the angle of the cone.
[table]

The last two columns are constructed on the following considerations. We conceive the hearer placed within the cylindrical space whose diameter is BΛ. In this situation he receives an echo coming apparently from the whole surface TGV; and we account the effect of the trumpet as equivalent to the united voices of as many mouths as would cover this surface. Therefore the quotient obtained by dividing the surface of the hemisphere by that of the mouthpiece will express the magnifying power of the trumpet. If the chords g E, g T, be drawn, we know that the spherical surfaces T g V, E g D, are respectively equal to the circles described with the radii T g , E g , and are therefore as T g 2 and E g 2 . Therefore the audibility of the trumpet, when compared with a single voice, may be expressed by T g 2

*****E~s∙ Now the ratio of T g 2 to E g 2 is easily obtained. For if E f be drawn parallel to the axis, it is plain that BAED

B, ∕= g , and that E/is to B∕ as radius to the tangent of BCF; which angle we may call a. Therefore tan. α = , and thus we obtain the angle a. But if the radius CE be
accounted
1, Tÿ is = √ 2, and Eÿ is — 2 sin. -. Therefore ∣, — ~~~ u ' an ^ the magnifying

∙ ’ 2 sin. “

2 ]

power of the trumpet is = — . The num-

4 sin. 9 2sin.*^∙

'∙> *2

bers, therefore, in the third column of the table are each = .

2sin,≡l

But the more usual way of conceiving the power of the trumpet is, by considering how much farther it will enable us to hear a voice equally well. Now we suppose that the audibility of sounds varies in the inverse duplicate ratio of the distance. Therefore if the distance d, at which a man may be distinctly heard, be increased to z, in the proportion of EG to T g, the sound will be less audible, in the proportion of T g 2 to E G 2 . Therefore the trumpet will be as well heard at the distance z as the simple voice is heard at the distance d. *****Therefore ~ will express the ex- V 2 tending power of the trumpet, which is therefore = .

2 sin A 2 ln this manner were the numbers computed for the fourth column of the table.

When the angle BCA is small, which is always the case in speaking trumpets, we may, without any sensible error, consider Eg> as = = And Tg = TC × √ 2 =

ABAB y v 2 = —— = ——∙. 1 his gives a very easy com-

ä 4 V

putation of the extending and magnifying powers of the trumpet.

The extending power is = √ 2 —.

The magnifying power is = 2

We may also easily deduce from the premises, that if the mouth-piece be an inch and a half in diameter, and the length x be measured in inches, the extending power is very nearly ***= ⅛ x ∣ and the magnifying power — ⅛ x.

An inconvenience still attends the trumpet of this construction. Its complete audibility is confined to the cylindrical space in the direction of the axis, and it is more faintly heard on each side of it. We are therefore obliged to direct the trumpet very exactly to the spot where we wish it to be heard. This is confirmed by all the accounts we have of the performance of great speaking trumpets. It is evident, that by lengthening the trumpet, and therefore enlarging its mouth, we make the lines TB t and VA v expand (fig. 4); and therefore it will not be so difficult to direct the trumpet.

But even this is confined within the limits of a few degrees. Even if the trumpet were continued without end, the sounds cannot be reinforced in a wider space than the cone of the trumpet. But it is always advantageous to increase its length; for this makes the extreme tangents embrace a greater portion of the sonorous sphere, and thus increase the sound in the space where it is all reflected. And the limiting tangents TB, VA, expand still more, and thus the space of full effect is increased. But either of these augmentations is very small in comparison of the augmentation of size. If the trumpet of fig. 5 were made an [7:21:397]hundred times longer, its power would not be increased one half.

We need not therefore aim at much more than to produce a cylindrical space of full effect; and this will always be done by the preceding rules, or table of constructions. We may give the trumpet a third or a fourth part more length, in order to spread a little the space of its full effect, and thereby make it more easily directed to the intended object. But in doing this we must be careful to increase the diameter of the mouth as much as we increase the length; otherwise we produce the very opposite effect, and make the trumpet greatly inferior to a shorter one, at all distances beyond a certain point. For by increasing the length while the part CG remains the same, we cause the tangents TB and VA to meet on some distant point, beyond which the sound diffuses prodigiously. The construction of a speaking trumpet is therefore a problem of some nicety; and as the trials are always made at some considerable distance, it may frequently happen that a trumpet which is not heard at a mile’s distance, may be made very audible two miles off, by cutting off a piece at its wide end.

After this minute consideration of the conical trumpet, we might proceed to consider those of other forms. In particular the hyperbolic, proposed by Cassegrain, and the parabolic, proposed by Hase, seem to merit consideration. But if we examine them merely as reflectors of echoes, we shall find them inferior to the conical.

With respect to the hyperbolic trumpet, its inaptitude is evident at first sight; for it must dissipate the echoes more than a conical trumpet. Indeed Mr Cassegrain proceeds on quite different principles, depending on the mechanism of the aerial undulations: his aim was to increase the agitation in each pulse, so that it may make a more forcible impulse on the car. But we are too imperfectly acquainted with this subject to decide a priori; and experience shows that the hyperbola is not a good form.

With respect to the parabolic trumpet, it is certain that if the mouth-piece were but a point, it would produce the most favourable reflection of all the sounds; for they would all proceed parallel to the axis. But every point of an open mouth must be considered as a centre of sound, and none of it must be kept out of the trumpet. If this be all admitted, it will be found that a conical trumpet, made by the preceding rules, will dissipate the reflected sounds much less than the parabolic.

Those who have endeavoured to improve the speaking trumpet on mechanical principles, have generally aimed at increasing the violence of the elastic undulations, that they may make a more forcible impulse on the ear. This is the object in view in the parabolic trumpet. All the undulations arc converted into others which are in planes perpendicular to the axis of the instrument; so that the same little mass of air is agitated again and again in the same direction. From this it is obvious to conclude, that the total agitation will be more violent. But, in the first place, these violent agitations must diffuse themselves laterally as soon as they get out of the trumpet, and thus be weakened, in a proportion that is perhaps impossible for the most expert analyst to determine. But, moreover, we are not sufficiently acquainted with the mechanism of the very first agitations, to be able to perceive what conformation of the trumpet will cause the reflected undulations to increase the first undulations, or to check them. For during the production of a continued sound in a trumpet, it must happen that a parcel of air, which is in a state of progressive agitation as it makes a pulse of one sound, may be in a state of retrograde agitation as it is part of a pulse of air producing another sound. We cannot (at least no mathematician has yet done it) discriminate, and then combine these agitations, with the intelligence and precision that are necessary for enabling us to say what is the ultimate accumulated effect. Mr

Lambert, therefore, did wisely in abstaining from this intricate investigation; and we are highly obliged to him for deducing such a body of demonstrable doctrine from the acknowledged, but ill understood, laut of the reflection of echoes.

We know that two sounds actually cross each other without any mutual disturbance; for we can hear either of them distinctly, provided the other is not so loud as to stun our ears, in the same manner as the glare of the sun dazzles our eyes. We may therefore depend on all the consequences which are legitimately deduced from this fact, in the same manner as we depend on the science of catoptrics, which is all deduced from a fact perfectly similar, and as little understood.

But the preceding propositions by no means explain or comprehend all the reinforcement of sound which is really obtained by means of a speaking trumpet. In the first place, although we cannot tell in what degree the aerial undulations are increased, we cannot doubt that the reflections which are made in directions which do not greatly deviate from the axis, do really increase the agitation of the particles of air. We see a thing perfectly similar to this in the waves on water. Take a long slip of lead, about two inches broad, and having bent it into the form of a parabola, set it into a large flat trough, in which the water is about an inch deep. Let a quick succession of small drops of water fall precisely on the focus of the parabola. We shall see the circular waves, proceeding from the focus, all converted into waves perpendicular to the axis; and we shall frequently see these straight waves considerably augmented in their height and force. We say generally, for we have sometimes observed that these reflected waves were not sensibly stronger than the circular or original waves. We do not exactly know to what this difference must be ascribed; we arc disposed to attribute it to the frequency of the drops. This may be such, that the interval of time between each drop is precisely equal, or at least commensurable, to the time in which the waves run over their own breadth. This is a pretty experiment; and the ingenious mechanician may make others of the same kind which will greatly illustrate several difficult points in the science of sounds. We may conclude, in general, that the reflection of sounds, in a trumpet of the usual shapes, is accompanied by a real increase of the aerial agitations; and in some particular cases we find the sounds prodigiously increased. Thus, when we blow through a musical trumpet, and allow the air to take that uniform undulation which can be best maintained in it, namely, that which produces its musical tone, where the whole tube contains but one or two undulations, the agitation of a particle must then be very great; and it must describe a very considerable line in its oscillations. When we suit our blast in such a manner as to continue this note, that is, this undulation, we are certain that the subsequent agitations conspire with the preceding agitation, and augment it. And accordingly we find that the sound is increased to a prodigious degree. A cor de chasse, or a bugle horn, when properly winded, will almost deafen the ear; and yet the exertion is a mere nothing in comparison with what we make when bellowing with all our force, but with not the tenth part of the noise. We also know, that if we speak through a speaking trumpet in the key which corresponds with its dimensions, it is much more audible than when we speak in a different pitch. These observations show, that the loudness of a speaking trumpet arises from something more than the sole reflection of echoes considered by Mr Lambert. The very echoes are rendered louder.

In the next place, the sounds are increased by the vibrations of the trumpet itself. The elastic matter of the trumpet is thrown into tremors by the undulations which proceed from the mouth-piece. These tremors produce [7:21:398] pulses in the contiguous air, both in the inside of the trumpet and on that which surrounds it. These undulations within the trumpet produce original sounds, which are added to the reflected sounds; for the tremor continues for some little time, perhaps the time of three, or four, or more pulses. This must increase the loudness of the subsequent pulses; we cannot say to what degree, because we do not know the force of the tremor which the part of the trumpet requires; but we know that these sounds will not be magnified by the trumpet to the same degree as if they had come from the mouth-piece; for they are reflected as if they had come from the surface of a sphere which passes through the agitated point of the trumpet. In short, they are magnified only by that part of the trumpet which lies without them. The whole sounds of this kind, therefore, proceed as if they came from a number of concentric spherical surfaces, or from a solid sphere whose diameter is twice the length of the trumpet cone.

All these agitations arising from the tremors of the trumpet tend greatly to hurt the distinctness of articulation; because, coming from different points of a large sphere, they arrive at the ear in a sensible succession, and thus change a momentary articulation to a lengthened sound, and give the appearance of a number of voices uttering the same words in succession. It is in this way that, when we clap our bands together near a long rail, we raise an echo from each post, which produces a chirping sound of some continuance. For these reasons it is found advantageous to check all tremors of the trumpet by wrapping it up in woollen lists. This is also necessary in the musical trumpet. With respect to the undulations produced by the tremors of the trumpet in the air contiguous to its outside, they also hurt the articulation. At any rate, this is so much of the sonorous momentum uselessly employed, because they are diffused like common sounds, and receive no augmentation from the trumpet.

It is evident that this instrument may be used for aiding the hearing; for the sonorous lines are reflected in either direction. We know that all tapering cavities greatly increase external noises; and we observe the brutes prick up their ears when they want to hear uncertain or faint sounds. They turn them in such directions as are best suited for the reflection of the sound from the quarter whence the animal imagines that it comes. Let us apply Mr Lambert’s principle to this very interesting case, and examine whether it be possible to assist dull hearing in like manner as the optician has assisted imperfect sight. The subject is greatly simplified by the circumstances of the case; for the sounds to which we listen generally come in nearly one direction, and all that we have to do is to produce a constipation of them. And we may conclude, that the audibility will be proportional to this constipation.

Therefore let ABC, fig. 6, be the cone, and CD its axis.

The sound may be conceived as coming in the direction RA, parallel to the axis, and to be reflected in the points A, b, c, d, e, till the angle of incidence increases to 90°; after which the subsequent reflections send the sound out again. We must therefore cut off a part of the cone; and, because the lines increase their angle of incidence at each reflection, it will be proper to make the angle of the cone an aliquot part of 90°, that the least incidence may amount precisely to that quantity. What part of the cone should be cut off, may be determined by the former principles. Call the angle ACD, a. We have C e = CA·sin. a / sin.(2 n + 1) a , when the sound gets the last useful reflection. Then we have the diameter of the mouth AB = 2 CA ∙ sin. α, and that of the other end er = Ce ∙ 2 sin. a. Therefore the sounds w ill be constipated in the ratio of CA 2 to Ce 2 , and the trumpet will bring the speaker nearer in the ratio of CA to Ce.

When the lines of reflected sound are thus brought together, they may be received into a small pipe perfectly cylindrical, which may be inserted into the external ear. This will not change their angles of inclination to the axis, nor their density. It may be convenient to make the internal diameter of this pipe 1 / 3 d of an inch. Therefore Ce ∙ sin. a is = 1 / 6 th of an inch. This circumstance, in conjunction with the magnifying power proposed, determines the other dimensions of the hearing trumpet. ***For Ce = ,rJ—

’ b sin. α

CA sin. a , _ , sin. (2n J- 1 a) = -—τπ rv > ≡ CA = —-J—√- -.

sin. (in -f- 1 )a b sin? a

Thus the relation of the angle of the cone and the length of the instrument is ascertained, and the sound is brought nearer in the ratio of CA to Ce, or of sin. (2n -∣- 1) a to sin. e. And seeing that we found it proper to make (2n -J- 1) a — 90°, we obtain this very simple analogy, 1: sin. a — CA: Ce. And the sine of the angle of the cone is to radius as I to the approximating power of the instrument.

Thus let it be required that the sound may be as audible CA

as if the voice were 12 times nearer. This gives — = 12; and sin. o = γ^, and a — 4° 47 ’, and the angle of the cone = 9° 34 ’. Then CA = —!— = —— = —

6 sin.*« 6χ-,-[ r 6

≡ 24. Therefore the length of the cone is 24 inches.

CA

From this take Ce as — = 2, and the length of the trumpet is 22 inches. The diameter at the mouth is 2 Ce = 4 inches. With this instrument one voice should be as loud as 144.

If it were required to approximate the sound only four times, making it 16 times stronger than the natural voice at the same distance, the angle ACB must be 29°, A e must be 2 inches, AB must be 1 1 / 3 d inch, and er must be 1 / 3 d of an inch.

It is easy to see that when the size of the ear-end is the same in all, the diameters at the outer end are proportional to the approximating powers, and the lengths of the cones are proportional to the magnifying powers.

We shall find the parabolic conoid the preferable shape for an acoustic trumpet; because, as the sounds come into the instrument in a direction parallel to the axis, they are reflected so as to pass through the focus. The parabolic conoid must therefore be cut off through the focus, that the sounds may not go out again by the subsequent reflections; and they must be received into a cylindrical pipe of one third of an inch in diameter. Therefore the parameter of this parabola is one sixth of an inch, and the focus is one twelfth of an inch from the vertex. This determines the whole instrument; for they are all portions of one parabolic conoid. Suppose that the instrument is required to approximate the sound 12 times, as in the example of the conical instrument. The ordinate at the mouth must be 12 times the 6th of an inch, or 2 inches; and the mouth diameter is four inches, as in the conical instrument. Then, [7:21:399]for the length, observe that DC in fig. 7 is 1 / 6 th of an inch, and MP is 2 inches, and AC is 1 / 12 th of an inch, and DC 2 : MP 2 = AC: AP. This will give AP = 12 inches, and CP = 11 11 / 12 ths; whereas in the conical tube it was 22. In like manner, an instrument which approximates the sounds four times, is only 1¼th inch long, and 1¼th inch diameter at the larger end. Such small instruments may be very exactly made in the parabolic form, and are certainly preferable to the conical. But since even these are of a very moderate size when intended to approximate the sound only a few times, and as they can be accurately made by any tinman, they may be of more general use. One of 12 inches long, and 3 inches wide at the larger end, should approximate the sound at least 9 times.

A general rule for making them.— Let m express the approximating power intended for the instrument. The length of the instrument in inches is m ( m -1) / 6 , and the diameter at the mouth is m / 3 . The diameter at the small end is always one third of an inch.

In trumpets for assisting the hearing, all reverberation of the trumpet must be avoided. It must be made thick, of the least elastic materials, and covered with cloth externally. For all reverberation lasts for a short time, and produces new sounds, which mix with those that are coming in. We must also observe, that no acoustic trumpet can separate those sounds to which we listen from others that are made in the same direction. All are received by it, and magnified in the same proportion. This is frequently a very great inconvenience.

There is also another imperfection, which we imagine cannot be removed, namely, an odd confusion, which cannot be called indistinctness, but a feeling as if we were in the midst of an echoing room. The cause seems to be this: Hearing gives us some perception of the direction of the sounding object, not indeed very precise, but sufficiently so for most purposes. In all instruments which we have described for constipating sounds, the last reflections are made in directions very much inclined to the axis, and inclined in many different degrees. Therefore they have the appearance of coming from different quarters, and instead of the perception of a single speaker, we have that of a sounding surface of great extent. We do not know any method of preventing this, and at the same time increasing the sound.

There is an observation which it is of importance to make on this theory of acoustic instruments. Their performance does not seem to correspond to the computations founded on the theory. When they are tried, we cannot think that they magnify so much. Indeed it is not easy to find a measure by which we can estimate the degree of audibility. When a man speaks to us at the distance of a yard, and then at the distance of two vards, we can hardly think that there is any difference in the loudness; though theory says that it is four times less in the last of the two experiments; and we cannot but adhere to the theory in this very simple case, and must attribute the difference to the impossibility of measuring the loudness of sounds with precision. And because we are familiarly-acquainted with the sound, we can no more think it four times less at twice the distance, than we can think the visible appearance of a man four times less when he is at a quadruple distance. Yet we can completely convince ourselves of this, by observing that he covers the appearance of four men at that distance. We cannot easily make the same experiment with voices.

But, besides this, we have compared two hearing trumpets, one of which should have made a sound as audible at the distance of 40 feet as the other did at 10 feet distance; but we thought them equal at the distance of 40 and 18. The result was the same in many trials made by different persons, and in different circumstances. This leads us to suspect some mistake in Mr Lambert’s principle of calculation; and we think him mistaken in the manner of estimating the intensity of the reflected sounds. He conceives the proportion of intensity of the simple voice and of the trumpet to be the same with that of the surface of the mouth-piece to the surface of the sonorous hemisphere, which he has so ingeniously substituted for the trumpet. But this seems to suppose that the whole surface, generated by the revolution of the quadrantal arch TEG round the axis CG (fig.∙4), is equally sonorous. We are assured that it is not; for even if we should suppose that each of the points Q, R, and S (fig. 8), are equally sonorous with the point P, these points of reflection do not stand so dense on the surface of the sphere as on the surface of the mouthpiece. Suppose them arranged at equal distances all over the mouth-piece, they will be at equal distances also on the sphere, only in the direction of the arches of great circles which pass through the centre of the mouth-piece. But in the direction perpendicular to this, in the circumference of small circles, having the centre of the mouth-piece for their pole, they must be rarer in the proportion of the sine of their distance from this pole. This is certainly the case with respect to all such sounds as have been reflected in the planes which pass through the axis of the trumpet; and we do not see (for we have not examined this point) that any compensation is made by the reflection which is not in planes passing through the axis. We therefore imagine that the trumpet does not increase the sound in the pro- ***f/E' oT 2 portion of gE 2 to g T 2 (fig. 5), but in that of ***jjg t0 (J τ p∙

Mr Lambert seems aware of some error in his calculation, and proposes another, which leads nearly to this conclusion, but is founded on a principle which we do not think in the least applicable to the case of sounds. (b. b. b.)

Trumpet, Murine, an old musical stringed instrument, said to derive its name from its inventor Marino or Marigni. Mersenne describes it as a monochord and as a dichord. See his Harmonicoum libri xii. Paris, 1652, proposition 37 of book 2d, pp. 56, 57, 58. It was played with a bow, and the sounds were stopped by the fingers gently touching the string, so as to produce the harmonics of the string, in the same manner as is practised on the violin, &c. The loud, harsh, and peculiar tone of the trumpet-marine was increased by the bridge being fixed by one end only to the sound-board, while the other end was free, and allowed to strike against the sound-board, according to the vibration of the string, &c. The sounds above the fundamental of the open string, followed those of the aliquots mentioned in article Music. The reader may consult an ingenious paper upon the marine trumpet in the fourth volume of the late Professor Robison’s Mechanical Philosophy, edited by Dr Brewster in 1822, pp. 486-500.