steam was made of wood, but what contained the fire and smoke was of iron. Two boilers were thus constructed for the two dredging engines: a wooden box of 4 inch fir planks bolted together like a wooden cistern, about 9 feet long, 6 feet wide, and 6 feet 6 inches high, with iron tubes running through it for the fireplace, flues, and chimney. The first wooden case had the flues renewed two or three times, and itself had a new top once; but the principal part of it lasted twenty years."

It will be perceived, that the digging engine was in some particulars superior to other machines afterwards introduced; and this fact is proved by the following note of expense found amongst Sir Samuel's papers:

"Details of Expense of Digging Engine per day.

0

"Total per day...... 3 18 "Or less than 1d. per ton, as the engine dug out 2 tons per minute."

Mr. Goodrich, under other circumstances, states that it costs less than 2d. per ton to raise and deliver shingle from a depth of 27 feet, allowing a per centage of 50l. per annum for wear and tear; and to show what immense savings were, or might have been effected by the introduction of the dredging machine worked by steam, I shall conclude by the following extract from one of Sir Samuel Bentham's papers in the United Service Journal*:

"The profits upon the contract for digging mud off Woolwich Dock-yard, when the contractors were raising 700 tons per day, amounted to about 407. per day, comparing the price paid to them with the expense at which it would have been raised, had Government raised it as at Portsmouth, on their own account, the engine employed by

• Year 1830, Part I. page 198. See also "Naval Papers," No. VIII., page 66,

the contractors been similar to the one I contrived and brought first into use off Portsmouth Dock-yard."

I am, Sir, your's respectfully,
T. G. CHESNEL.
Hampstead, August 4, 1845.

GEOMETRICAL CONVERSION OF CONVEX SURFACES.

(Continued from page 100.)

PROBLEM III.-Having given the diameters of the two parallel ends of a conic frustrum, and the perpendicular distance between them, to determine, by a geometrical construction, the diameter of a circle whose area shall be equal to the convex surface of the frustrum.

The solution of this problem is rather more difficult than either of the preceding ones, yet it is sufficiently easy when the principles of construction have once been investigated. In order to this,

Put D=the diameter of the greater end of the frustrum.

d the diameter of the lesser end. s=the slant distance between the ends or length of the side.

And r=the radius of the equivalent circle.

Then, because the convex superficies of the frustrum is expressed by half the sum of the circumferences, drawn into the slant distance between them, we have, by comparing this with the area of the equivalent circle,

3·1416r2=3·1416 × (D+d)s;

and this, by expunging the common factor 31416, becomes (D+ d)s ; that is, the radius of the equivalent circle is a mean proportional between the slant distance of the ends and half the sum of the two diameters; hence this construction.

Let A B C D, fig. 3, be the conic frustrum; A B, D C, the diameters of its ends, and E F the axis or perpendicular distance between them, AD being the slant height, or length of the side between the parallel ends of the frustrum. From the point C, the extremity of the lesser diameter D C, let fall the perpendicular CH, meeting the greater diameter A B in the point H; then is A H equal to A F+D E, the sum of the semi-diameters. Produce the side D A directly forward, until A Q becomes

side and two semi-diameters. Bisect DQ in P, and on the point P as a centre, with the distance P D or P Q as radius, describe the semicircle D m R n Q; then, at the point A, in the line D Q, erect the perpendicular A S, intersecting the semicircle DR Q in the point R, so that AR shall be a mean proportional between the slant side A D and the sum of the semidiameters A F and D E. For, if from the points D and Q, the extremities of the diameter D Q, we draw the straight lines DR and QR; then, by the nature of the figure, the right-angled triangles QRA and R D A are similar, so that AQ AR :: AR; AD; hence it is manifest that AR is a mean proportional between A D and A _Q. But it has been proved that the radius of the circle whose area is equal to the convex surface of the frustrum, is also a mean proportional between A D and AQ; hence AR is equal to the radius of the equivalent circle. Upon the point R as a centre, with RA as radius, describe the circle Am Sn; then shall the area of the circle thus described be equal to the convex superficies of the conic frustrum A B CD, contained between the parallel planes or ends whose diameters are AB and D.C.

By developing the superficies of the three foregoing figures, an equivalent area could also be readily found, but in neither case would the construction be purely geometrical, and the results would be only approximative. Not so however with the constructions here given, for they depend on the principles of pure geometry, and the results, geometrically considered, are rigorously correct with

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regard to the principles employed in obtaining them; viz., the simple determination of a mean proportional between quantities whose absolute magnitudes are known.

What has been said above with respect to the development will be rendered manifest by what follows. If the convex superficies of a right cylinder, fig. 1, be developed on a plane, the development as we have already stated, will be a rectangular parallelogram, having one side equal to the length of the cylinder, and the other equal to the circumference of its base. Here, then, it becomes necessary to lay down a straight line equal in length to the circumference of a circle of which the diameter is given. Now, it is a well-known fact, that there is no method of doing this by principles which are purely geometrical, although there are methods of approximating to it, and which methods are more than correct for every practical purpose. Again, if the convex superficies of a cone, fig. 2, be developed on a plane, the development will be a sector of a circle whose radius is equal to the slant side of the cone, and the bounding curve equal to the circumference of its base. Here again the principles of pure geometry are at fault, for there is no method, purely geometrical, by which we can describe a portion of one circle that shall be exactly equal to the circumference of another, although the thing may be done approximatively to any practical degree of exactness required.

Lastly, if the convex superficies of the conic frustrum, fig. 3, be developed on a plane, the development will be a portion of a circular annulus, of which the breadth is equal to the length of the slant side of the conic frustrum, and the bounding curves equal to the circumferences of its ends, the radii of these bounding curves being respectively the slant side of the entire cone from which the frustrum is cut, and that of the cone cut off above the frustrum, so that the principle of construction in this case fails, for the same reason that we stated for the cone; for the developed area is manifestly nothing more than the difference between two circular sectors, whose radii differ from each other by the breadth of the development.

PROBLEM IV. Having given the diameter of a sphere or globe, to find geo

metrically the diameter of a circle whose area shall be equal to the convex superficies.

This problem may be constructed on the same principle as the first in respect of the right cylinder, being only a particular case of the general construction there given, viz., that in which the length of the cylinder and the diameter of the base are equal to one another; for it is a well-known fact in geometry, that the surface of a sphere is equal to the convex surface of its circumscribing cylinder; hence the mode of construction is manifest; for we have only to describe a circle with a radius equal to the diameter of the sphere, and the circle thus described will be equal in area to the spheric surface, and it will contain an area equal to four great circles of the sphere, or four times the area of a plane that passes through its centre. A separate construction for this case is unnecessary, the principle being so obvious as to require no illustration by a diagram.

The surface of a sphere can be projected in several ways upon a plane, but it cannot be developed; this arises from the circumstance of its being a surface of double curvature, which does not admit of being expanded upon a plane; and what is said of the whole surface is equally true of any portion of it; but a circle can always be found that shall contain an area, equal to any portion of the spheric surface, by the method just described; for the superficies of any segment of a sphere cut off by a plane, parallel to the base of the circumscribing cylinder, is equal to the corresponding part of the convex superficies of that cylinder.

THE HISTORY AND RESUSCITATION OF THE CLAVIOLE, OR FINGER-KEYED VIOL. BY JOHN ISAAC HAWKINS, ESQ., C.E.

Sir, The present communication was commenced immediately after the publication of your Number for December 17, 1842, but I was at the time so pressingly engaged, that when the paper was finished several weeks afterwards, I thought it too late to be sent in as an answer to Mr. Savage's remarks in that Number of your Journal. It has therefore lain on the shelf until now.

The republication, however, of the Report of a Committee of the Franklin

Institute, in the last Number of your very useful Journal, has stimulated me to send the paper for your consideration and insertion in your pages, should that meet your approval, and I will now add some further remarks.

It is curious that the Franklin Institute should call attention to the Claviole at this particular time, since I have had several men at work for a few weeks past on the instrument in putting it in thorough repair, intending again to bring it before the public, and I expect in the course of the next month it will astonish and delight the philosophical and musical world as it did forty years ago, in Philadelphia and in London.

The communication of my worthy friend, Mr. Alfred Savage, at page 561, vol. 37, December 17th, 1842, of your valuable Magazine, has awakened in me the desire of giving to the public some account of the Claviole, which Mr. Savage in his last paragraph hints "has for ever gone to sleep."

The pressing avocations of superintending my everlasting pen manufactory, and in giving professional advice as consulting engineer and patent agent, will prevent my racking my brain much on the subject; but I have ransacked my papers and found documents enough to form an interesting history of that extraordinary instrument, copies of which I hand over to you, to be culled and inserted according to your judgment, as the whole might be tiresome to your unmusical readers, who, I have no doubt, bear but a small proportion to those who delight in the concord of sweet sounds; for it is well known that the lovers of science are almost always lovers of music.

After the history of the Claviole, I purpose giving an outline of some of the principal points of its construction, referring the reader to Rees's Cyclopædia, under the term "finger-keyed viol" for a more particular description, and an elaborate engraving by the celebrated Wilson Lowry, from a drawing made upwards of thirty years ago, by the accurate and indefatigable John Farey, who, no doubt will rejoice at an opportunity of hearing the instrument again.

I also send you herewith a drawing exhibiting the manner in which the horsehair is affixed to the bows, or rather rings, to produce the perfect effect of the violin bow, without the breaks occasioned by

its reciprocating motion, the revolution of the claviole bow producing a continuous tone, and in that respect the instrument is a perfect organ, of a much purer tene than any other organ, there being but one unison, and no discord-gene. rating stops, such as the twelfth, tierce, cornet, &c.

I have not yet found any memoranda to determine the exact date of the first idea of the instrument, but I made the first bow before the year 1800, at Bordenton, in the State of New Jersey, United States. The invention therefore belongs to the last century and to the United States of America.

The earliest document I find, is a handbill of a concert which I gave in Philadelphia, on the 21st of June, 1802; the following is a copy :

"Grand Concert of Vocal and Instrumental Music. John I. Hawkins proposes having a Concert on the evening of the 21st instant, at the Hall of the University, in Fourth-street, when he will perform on the

The concert was given at the time appointed, and from the warm manner in which the audience expressed their unqualified approbation of the varied powers of the instrument, and the pressing requests that I would afford an opportunity for their friends to hear it, I issued the following hand-bill :

:

"The Claviole having excited great curiosity, and many persons being desirous to see it, J. I. Hawkins has set apart every evening this week to exhibit its powers and construction, at Mr. Peale's Museum, between the hours of 8 and 10, and as it cannot be expected he should devote his time gratis, he will demand for admittance 25 cents. Philadelphia, June 23, 1802."

The Claviole was exhibited according to the hand-bill to crowds of visitors;

This instrument was the grandfather of the present Cabinet, Piccolo, and other pianos with short upright strings.

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but engagements in my profession precluded my continuing the exhibition. It was soon known, however, that I usually played on the instrument in the evening, and hundreds of persons assembled in the street opposite the house, on all the fine evenings of the month of July. Ladies of the first respectability had chairs and stools brought by their servants to sit in the street, opposite the windows, which of course I gallantly threw open, that the company might hear the better.

A long journey kept me some months out of Philadelphia, and when I returned I was too much occupied in preparing for a trip to Europe, to spare much time for the Claviole.

The following three letters were received from Mr. Joseph Leacock, a philosophical character residing in Philadelphia.

LETTER I.

"Philadelphia, August 4th, 1802. "Sir,-Having a great desire (though far advanced in years) of seeing and hearing the new musical instrument of your invention, in America, ere your embarking with it for England, I intimated that longing desire to Mr. Raphael Peale, who said you had declined exhibiting it any longer in this country, notwithstanding which he hoped I might be gratified, and that he would endeavour to prevail on Mr. Hawkins to grant this favour to me and a few friends of genius, and lovers of good music, on some evening when most convenient to you; some who have heard it regret my omitting the availing myself of the gratification in due time. Notwithstanding the great desire I have to be indulged, I must waive the gratification rather than give unnecessary trouble to you, Sir. "With good wishes for your success, "I am, yours, &c.

"JOSEPH LEACOCK."

"P.S.-I have heard that when the Viol shall have been completed in London two are to be sent to America, one for the President, and one for Mr. Peale. This will be as it ought to be, and I hope for the credit of the country where the Viol originated, that the first two, when completed, may be forwarded regardless of all other considerations. What delight will not this music afford our good President, (Jefferson,) whose soul will be in unison with its harmony, and what an acquisition will it be to the Museum. It will, like the attraction of the needle to the pole, attract and draw all harmonious souls to that repository of the wondrous productions of the God of nature, the contem

plation of which surpasses all that can be uttered by the tongues of men."

LETTER II.

"Philadelphia, August 6th, 1802. "Joseph Leacock's respects to Mr. Hawkins, thanks him for his indulgent kindness in exhibiting his lovely Claviole, and delighting him with its harmony. His calling at the residence of Mr. Peale on the 5th inst. was merely to learn when it might be most convenient for Mr. Hawkins to gratify not only himself, (as not being of a niggard or selfish disposition,) but a few friends of a congenial turn, having previously anticipated the favour to be granted.

"Should it not be deemed too troublesome and too presuming on good nature, he wishes to accompany his friends to that pleasing apartment at six o'clock on Saturday evening next, being the 7th of August, 1802, if convenient to the ingenious inventor of that first of all musical instruments, exceeding even the organ.

"Mr. Peale informs me you will leave the city this day. I imagine the Viol will be left where it is; in that case, (provided you have no objection,) I am persuaded Mr. Peale will have none to show it to a few friends."

LETTER III.

"Philadelphia, February 14th, 1803. "Mr. Peale,-Your well-wisher, Joseph Leacock, is strongly of opinion that nothing can possibly be more conducive to attract company to your Museum than one of the Clavioles invented by the ingenious Mr. Hawkins; such music as that is irresistible, and cannot fail drawing harmonious souls there.

"I felt the force of it powerfully, although the instrument was not in good order; you expect to receive one from London, but it will be too long coming.

"I wish to see one set about immediately, ere that gentleman's departure. I have been told he intends having another completed ere his departure.

"Cannot you set your ingenious workman, now fixed snugly under your immediate eye in the Steeple, (and a good warm place it is,) to imitate progressively every part of the wood-work? By observing the operator occasionally he can undoubtedly imitate it exactly.

"Do but get one of these rapturous Clavioles, and a chandelier for lighting the apartment, and, without doubt, many will make that place their evening resort. I have no objection to your showing this to Mr. Hawkins, being persuaded his sentiments will be in unison with mine, for the gratifi

cation of the public and the promotion of your merited interests.

"Let us have the Claviole, say I; such music will charm and draw me every night to that pleasing retreat, and although my visit may not afford pecuniary benefit, it matters not, I know your generosity thinks little of that.

"The Museum is brilliant at present, but when the Claviole is in operation you will have leisure time to embellish it as much as you please, and the more so the better, till it vies with and even eclipses that of London."

I left Philadelphia in June, 1803, after having dismantled the Claviole, and packed it up to remain safely till my return. I arrived in London in August 1803, where, under my patent, dated Nov. 13, 1800, I commenced the manufacture of the Claviole, in the year 1805, and finished one, to which I called the public attention, in September, 1806.

Some of the opinions of the musical world were recorded in a prospectus which I drew up six years afterwards, which is as follows:

"PROPOSALS for building, by subscription, a superlatively grand musical instrument, to be called The Millechord Claviole, or thousand-stringed finger-keyed viol; to be constructed with one thousand gut strings, the tones produced by rosined horsehair bows, and artificial fingers, acted on by four sets of finger-keys and eighty pedals, together with barrels of large dimensions, which, with the bows, will be turned by machinery, giving the full power and variety of a band of two hundred performers.

"The instrument will yield the perfect sounds of the violin, viola, violoncello, double bass, harp, and organ; will closely imitate the flute, clarionet, oboe, bassoon, fife, flageolet, union pipes, horn, trumpet, buglehorn, musical glasses, celestina, eolian harp, &c., as well as produce sounds entirely new and peculiarly delightful; and will scarcely ever be out of tune.

"The expense of the Millechord Claviole, tastefully ornamented, and designed for constant exhibition, is estimated at 4,2007. It is therefore proposed,

"1st. To raise the sum of five thousand pounds, in 100 shares of 50l. each, payable by instalments.

"2nd. As soon as the whole number of shares are subscribed for, a meeting of the subscribers shall be called, for the purpose of choosing a Treasurer, to receive the instalments; and a committee to manage all the money concerns, to order the commencement of the work, to inspect its progress,