This having been done, and a small grap nel, with a rope attached, hung to the bottom of the magazine or box, by an eyebolt fixed there for the purpose, the fuse is to be set on fire, and the machine liberated. The direction of the wind, as before observed, on a lee-shore, will always drive it towards the land; the distance from which will, of course, regulate the size of the rope that should be applied to the grapnel, being smaller in proportion to the length it has to go. It is scarcely necessary to observe, that if sufficient rope is supplied, the balloon cannot fail in conveying the grapnel over the land; or that, when the fire of the fuse reaches the powder, it will explode, and thereby detach the grapnel and rope from it. This, however, is not the only purpose the fuse is intended to answer; as, should the calamity occur at night, the light, whilst in the air, together with the explosion, will direct the inhabitants on the beach to the place where the grapnel will fall. The advantages of a communication thus opened with the land, it is needless to dwell upon; personal experience having shown me, could such have been effected, a number of valuable lives might have been saved. Should, however, even one unfortunate be rescued in the hour of his distress, the knowledge would constitute in itself more than a reward for the anxiety and trouble the perfecting this machine has entailed on the inventor. smoke into light, and light into glory." This early idea of gas-light, to which it seems plainly to refer, was received as an idle dreamn, and is only preserved to us, like straws in amber, by the wit and satire of Addison or Steele. We are to learn, therefore, not too hastily to reject even those hints which are not immediately clear to us. Under protection of this remark, I venture to propose to you, that a telegraphic communication may be held, at whatever distance, without a moment's loss of time in transmission, and equally applicable by day or night, by means of the electric shock. An experiment of this kind has been tried on a chain of conductors of three miles in extent, and the shock returned without any perceptible time spent in its going round; and may not the same principle be applicable for 100 or 10,000 miles? Let the conductors be laid down under the centre of the post-roads, imbedded in rosin, or any other, the best nonconductor, in pipes of stone ware. The electric shock may be so disposed as to ignite gunpowder; but if this is not sufficient to rouse up a drowsy officer on the night-watch, let the first shock pass through his elbows, then he will be quite awake to attend to the second; and by a series of gradations in the strength and number of shocks, and the interval between each, every variety of signal may be made quite intelligible, without exposure to the public eye, as in the usual telegraph, and without any obstruction from darkness, fogs, &c. It was mentioned F, the rope communicating with the before, that electricity will fire gunship. G, a grapnel hung from the eye-bolt. powder-that is known; we may imagine, therefore, that on any wor Fig. 2 represents the scene of a ship- thy occasion, preparations having wreck on a lee-shore, and the instrument in operation. ELECTRIC TELEGRAPHS. SIR,-There is, I think, in one of the numbers of the Spectator, dated about a hundred years ago, a passage tending to ridicule some projector of that day, who had proposed to "turn been made for the expected event, as the birth of a royal heir, a monarch might at one moment, with his own hand, discharge the guns of all the batteries of the land in which he reigns, and receive the congratulations of a whole people by the like return. I am, Sir, Very respectfully yours, MODERATOR. SHORT METHOD OF CALCULATING PROFIT AND DISCOUNT. INFLUENCE OF COPPER, ETC. ON MAGNETIC NEEDLES. 149 valuable miscellany, a short Table, which I formed some years ago for my own use, and which, I think, will be found to be according to "Cocker," and upon the universally received and correct principles of our best_commercial writers-e. g. Evans's Tables of Profit and Discount, a 4to. volume, appeared, price only one guinea. Now, the seventh edition of which has recently M. Arago lately communicated to the Academy of Sciences some interesting experiments relative to the oscillations of a magnetic needle, surrounded by different substances. He had ascertained that the copper rings, with which dipping needles are generally surrounded, exerted on the needles a very singular action, the effect of which was rapidly to diminish the amplitude of the oscillations without sensibly altering their duration; thus, when a horizontal needle, suspended in a ring of wood by a thread, without tension, was moved 45° from its natural position, and left to itself, it made 145 oscillations before the amplitude was reduced to 10o. In a ring of copper the amplitude diminished so rapidly Windsor-street, Islington, as I believe that each item of this exsubjoined method, be as correctly and pensive work may, by the aid of the very expeditiously obtained, I need not make any other apology for troubling you. Should this be deemed worthy insertion, I shall feel a pleasure in appearing again upon your pages, in that the same needle moved 45° from its natural position, and only oscillated 33 times before the arc was reduced to 10o. In another ring of copper, of less weight, the number of oscillations between the arcs of 45° and 10° were 66. The time of the oscillations appeared to be the same in all the rings. In the ring of wood, 145 oscillations, 45 deg. to 10 deg. In the ring of copper, 33 oscillations, 45 deg. to 10 deg. In a lighter ring, 66 oscillations, 45 deg. to 10 deg. further elucidation of this and other I am, Sir, May 15, 1825. WL. A TABLE, Exhibiting to immediate view the additional rate per cent. required to sustain the reaction of a given discount, without affecting the principal; also, show-* ing the gain or profit upon goods purchased under a discount, and resold at the gross price. Rate per Cent. Profit or Discount. 150 A HINT TO MECHANICS' INSTITUTIONS. Example. To be able to allow 20 per cent. it is required to put on 25 per cent. for 25 +100=125-20 per cent. (or) =100; and goods bought at 20 per cent. discount off, clears 25 per cent. profit; for 100-20-80, nett cost, and 80+ 4 (or 25 per cent.) = 100. 123 much would he gain if he sold them at 1421. 17s. 1ąd. 120. 12 per cent. (or }) And, again,-Suppose (a common case) a manufacturer produces a new article, which he sells at a profit of 40 per cent. but, competitors starting, he is compelled to lower his price; and, from time to time, as opposition increases, he comes down until he offers the article at 30 per cent. under his original charge, he will (it is very probable) tell you that he is still gaining 10 per cent., but, in fact, he is losing -he is selling at 981. what cost him 1007. 0:10-10-10 A HINT TO MECHANICS' INSTITUTIONS. At the end of one of the gratuitous lectures which Dr. Gregory has lately been delivering to the Mechanics' Institution at Crayford, he suggested to them a plan by which they might become their own lecturers, which was this:-Select some instructive popular work, as Ferguson's Select Lectures, Millington's Lectures, the second volume of Gregory's Mechanics, Bonnycastle's or Brinkley's Astronomy, Tilloch's Mechanics' Oracle, the London and Glasgow Mechanics' Magazines: let these, or appropriate portions of them, selected by the Committee, be read in order, and made the subject of conversation at alternate meetings. To render the subjects equally plain to readers and auditors, let three or four boards be provided, each presenting a face of about four feet by three, and let them be painted black. Let some member or members of ON THE QUADRATURE OF THE CIRCLE. the Committee draw such figures and diagrams of reference, with chalk, on a large scale, upon one or more of these boards, as may be required for the evening's reading. These boards, placed in suitable positions, and with a good light thrown upon them, will render the respective subjects as intelligible to the auditors as to the individual who 151 reads, and thus remove the great obstruction to the communication of knowledge, where diagrams are concerned, to a large auditory by a single reader. Moulds of circles, ellipses, squares, parallelograms, &c. along the contours of which chalk may be drawn, will much shorten the labour of sketching the diagrams. AN ESSAY ON THE QUADRATURE OF THE CIRCLE. [TO THE EDITOR OF THE MECHANICS' MAGAZINE.] SIR, Although the celebrated problem of finding the quadrature, or the ratio, which the periphery bears to the diameter of the circle, has passed through the hands of many eminent mathematicians since its solution was first attempted by Apollonius Pergæus, and afterwards by Archimedes of Syracuse, the greatest geometricians of antiquity, it is still susceptible of various novel solutions, quite different from those previously given to it by any of the famous authors who have treated on the subject; and as I conceive the grand object of the Mechanics' Magazine is to give publicity to new improvements in the Arts and Sciences-I will, with the Editor's permission, present its readers with one of these new methods of solution, intending to publish the whole myself in a separate pamplet at some future period. This solution assigns the length of a given circular arc in an elegant and novel scries, which is afterwards summed or converted into another series, for the purpose of numerical computation. Subsequently a new and general theorem is investigated, and applied to finding various serieses of very rapid convergency; among these are deduced, from our theorem, the best of those given by the celebrated authors, Euler, Machin, Hutton, and Bonnycastle. From the same source is also derived a new and quickly converging series in terms of the arc of 45o, and consequently its root is the eighth part of the whole circumference of the circle. The same fertile theorem enables us to divide the whole periphery into 8 N equal parts, and having just glanced at the method of doing this, I shall conclude the Essay, not, however (I hope), without exciting some interest among the geometrical readers of your valuable miscellany, should you, Mr. Editor, deem it worthy of insertion. 152 ON THE QUADRATURE OF THE CIRCLE, PROPOSITION I.-PROBLEM. INVESTIGATION AND SUMMATION OF A NEW SERIES, EXPRESSING THE LENGTH OF A CIRCULAR ARC. H G F Let BL, the tangent of the circular arc BN, be divided into n equal parts at the points C, D, E, F, &c., the radius AB being unity; then, assuming the whole tangent BL equal, each of the parts, BC, CD, DE, EF, &c. will be 1 equal the tangent of the angle BAC. Now, by means of the well-known n nt &c. Where the law of the conti nuation is obvious, the general expression for the nth tangent being n2 t2 + n (n+1)' in which n represents any integer whatever, finite or infinite. If we suppose n to be indefinitely great, then the equal parts into which the given tangent BL is divided, and likewise the angles BAC, CAD, DAE, &c. become indefinitely small. But the tangents and the arcs of such indefinitely small angles being ultimately equal to one another, it thus appears that the length of the arc BN is equal to, or expressed by, the sum of the series + nt + &c. in n t n2 12 + 2 n2 12+ 6 n2 12 + 12 n2 t2 + 20 n° to + 30' infinitum); or,, since each term of this series is, bona fide, a mere point, we are permitted to reject the first term, as being indefinitely small, which' will rather simplify the series for the purpose of summation. Thus the rectification of an arc of the circle, on which the quadrature depends, is reduced to 1 1 + 1 + 1 1 nt + |