are 40, 36, 30, 24 and 20; and let CB, Fe, Gd, Kb and L a be the lesser series of factors, originating at the points C, F, G, K and L, and having their numerical values respectively 9, 10, 12, 15 and 18; then by bisecting the sum of each pair of factors, andwith their halves describing the several semicircles A D B, AE e, AHd, A Ib and A M a, it will be found that these semicircles intersect the perpendiculars CD, FE, G H, KI and LM, in the points D, E, H, I and M, all of which are equally distant from the straight line AB, or all subsisting in O P, which is parallel to A B, so that any one of the perpendiculars may be taken as the root of the proposed number 360; hence it appears, that any pair of factors into which any number can be separated, will give the square root of that number, but since some of the pairs are found more convenient for the purpose than others, it remains for the operator to select that pair which is likely to succeed the best. The two methods by which we have resolved the proposed numerical example, have only been carried out to show the generality of the process; but in practice one operation will always be sufficient, and

in that case, the steps of the construction will be the same as those that we have described in delineating the first of the foregoing figures, where the mode of procedure is sufficiently obvious and the quantity of labour very trifling.

We have stated that there are several propositions in Euclid's Elements from which the process may be inferred; it is however unnecessary to specify them, as the one which we have quoted is the most directly applicable, and the only one in which the assimilation of figures is implied, this being the very circumstance that suggests the idea of quadrate evolution. The process itself is nothing more than finding a mean proportional between two given lines, a problem which is given in every book on practical geometry, and therefore well known; but the identification of the solution with the extraction of the square root, has not, to our knowledge, been pointed out, at least in such a way as to render it available to practical men, who may not be ready in applying the arithmetical process; and having done this is the only merit to which the present paper lays claim.

best clocks for railway stations, &c., I did however expect would have been more of an original and instructive character than it is, coming, as it does, from so long established a vender of clocks. The whole of the information is, indeed, no more than is usually given to a customer on making a purchase. I am anx

ious to impress upon persons interested in the improvement of clocks, that something more is required of them than an enumeration of well-known facts; and it is for this reason, that I propose (with your permission), to enter a little into the remarks contained in Mr. Vulliamy's paper.

Clocks, we are informed, must either have a weight or spring for the maintaining power, and the weight clock, of course, is recommended, with a seconds pendulum. Then we are informed, that the spring or gut-line may break, and that the new spring may be stronger or weaker, which will materially alter the time. These remarks only show that the art of clock making is still in its infancy, for there can be no doubt that the suspension spring may be so managed as to prevent any such variation. I have seen marine chronometers keep the same time with a spring one-third weaker than the proper spring. We are informed, that if the spring be a little stronger or weaker, it will cause a considerable alteration in the rate, and that the pendulum is thought by workmen to possess superior qualities to the balance: surely, if that is the case, it can be made subservient to the isochronal law. I have had silk lines in clocks that have lasted six or eight years, which, I think, is as long as clocks in general will keep going without cleaning. It would be absurd to think of employing half-second pendulum clocks, instead of weight clocks, with an idea of saving, as there is more work in the former, which the difference in the case of the weight can scarcely compensate. Now for the escapement: Graham's dead beat is the one most strongly recommended with all its thirst for oil, which alone sometimes causes it to fail in less than a year. The oil, I think, is sooner driven away from the acting part of the pallet in the above escapement than any other; it is likewise an escapement that possesses no control over a pendulum that may be deficient in adjustment, in long and short vibrations, which is a great consideration, as the cold thickens the oil, and the construction of the escapement drives it from the part that requires it. Is there not a recoil escapement that might be recommended? It has several advantages over the dead beat, such as the three following (or more perhaps): the oil cannot be driven on the repose part, and there remain of no use; the

71

wheels are always in motion, which prevents some of that loss of power in the impulse from the oil after the clock has been going three or four years; and the escapement can be varied in construction to suit errors in the pendulum, which is of consequence. When we are told, for example, that a springclock varies considerably, say ten minutes per week after cleaning, this must be from a disregard of adjustment in the suspension spring; it is proper to have adjustment for beat, and it is of consequence that it should be well made. Clocks ought likewise to have going ratchets. With respect to the dial, whether it is engraved or painted, silvered or gilt, or of any of the following dimensions, -6, 12, or 18 in. diameter,-it matters not, except to the parties who use it. Pinions are not very expensive, and I should think twenty shillings additional expense not worth considering in a concern like a railway.

The pendulum, we are further told, is a part of the clock on which its good performance mainly depends, and to which too much attention cannot be paid. Let me ask your readers what attention it has received from Mr. Vulliamy? Does he give us any fresh information? Does he hint at anything that might improve it? I should say he does not, when the antiquated wooden rod is all he can recommend. He makes no mention of any laborious endeavours made to perfect compensation-pendulums, or to bring them within the limit of what he terms "moderate expense." The suspension spring ought to be, we are informed,-0·5, 0·6 of an inch in length. Is that rule given in consequence of its being more uniform in different arcs of vibration? Is there not considerable variation between the 0.5 and the 0·6 of an inch in isochronism? Is not the metal of the spring sometimes thicker at one part, or in other words, is it not used taper? And is it always tapered alike? These questions must be answered before any length for pendulum-suspension springs can be given to the world as of universal application. It is, in fact, deceiving the working clock-maker, who confides in the proposed length on the supposition that no person would issue to the world dimensions for a suspension spring, unless he could answer for the properties contained in the given length.

ing slides are recommended for the purpose of keeping the arc of vibration up to a given point. This is another proof that clocks are not scientifically made; otherwise diminution in the vibration would not alter the time. It is true, a continual decrease of vibration must eventually stop a clock, and this, I am afraid, will require considerable addition of weight to prevent, which again would produce another evil-increased friction, which oiling the pallets might obviate, instead of increasing the weight.

There is such a gradual connection or intimate relation between all the numerous curves of the very easiest description, and the conic section, that it is not unreasonable to suppose that the whole are sections, or projections of intersections of some other solids, gradually varying from the cylinder or cone, or in some other way related to them.

The cardioid, by different modes of generation, is the reciprocal of the ellipse; the ellipse is a conic section, the section of a cylinder, and the projection, and perspective representation of a circle. Then, in any of these, or any other respect, what is the cardioid ?

But the cardioid by another mode of generation is the reciprocal of the cardioid.

Again, the cardioid is to the ellipse, what the conchoid is to the reciprocal of the conchoid.

In the other case, the cardioid is to its

its reciprocal cissoid.

I am, Sir, yours, &c.

JOSEPH JOPLING.

29, Wimpole-street, July 22, 1845.

MR. NASMYTH'S STEAM HAMMER, PILE DRIVER, ETC., NOT NEW. INVENTED AND PATENTED BY MR. WM. DEVERELL FORTY YEARS AGO.

Sir,-On turning over an old volume of the Repertory of Arts (vol. ix., second series, p. 387) a few days ago, I was somewhat amazed at finding the specification for a patent, of which I send you a copy below, for a steam hammer, identical with Mr. Nasmyth's, patented so long ago as the year 1806, by a person named Deverell, who seems to have been fully alive to the importance of this principle of motion, and who describes or alludes to almost every conceivable variety of form or construction, and nearly every purpose or application possible for the steam hammer or blow to be used for. If I am right in my reading of this specification, it fully anticipates all that is contained in each of Mr. Nasmyth's subsequent patents for the steam hammer and its applications; and if so, leaves the making, vending, and using these important machines henceforth free to the public.

If this be so, Mr. Nasmyth will have nothing to complain of in hereafter reaping a limited harvest from inventions to which he can have no further right than as a second inventor, and thence without any claim to monopoly. And, indeed, any man who now-a-days conceives something which he thinks is new, and patents it without a rigid search as to prior patents or inventions identical with his own, deserves (we might almost say) to learn at his own cost that there are very few new things, if any, under the

sun.

I am, Sir, your obedient servant, LECTOR.

(Copy.)

"Specification of the patent granted to William Deverell, of Charles-street, Blackfriars'-road, in the parish of Christ Church, and county of Surrey, engineer, for certain improvements in the mode of giving motion to hammers, stampers, knives, shears, and other things, without the application of wheel, pinion, or any ro

two series of pillars it appears that there is a falling off in strength in all columns from the shortest to the longest; but that the diminution is so small, when the height of the column is not greater than about twelve times the side of its square, that the strength may be considered as uniform, the mean being 10,000lbs. per square inch, or upwards. From the experiments on the columns one inch square, it appears that when the height is 15 times the side of the square the strength is slightly reduced; when the height is 24 times the base, the falling off is from 138 to 96 nearly; when it is 30 times the base, the strength is reduced from 138 to 75; and when it is 40 times the base, the strength is reduced to 52, or to little more than onethird. These numbers will be modified to some extent by the experiments in progress. In all columns shorter than 30 times the side of the square, fracture took place by one of the ends failing; showing the ends to be the weakest parts; and the increased weakness of the longer columns over that of the shorter ones seemed to arise from the former being deflected more than the latter, and therefore exposing a smaller part of the ends to the crushing force. The cause of failure is the tendency of rigid materials to form wedges with sharp ends, these wedges splitting the body up in a manner which is always pretty nearly the same; some attempts to explain this matter theoretically were made by Coulomb. As long columns always give way first at the ends-showing that part to be the weakest-the material might be economized by making the areas of the ends larger than that of the middle, increasing the strength from the middle both ways towards the ends. If the area of the ends be to the area in the middle as the strength of a short column is to that of a long one, we should have for a column whose height was 24 times the breadth, the area of the ends and middle as 13.766 to 9.595 nearly. This, however, would make the ends somewhat too strong; since the weakness of long columns arises from their flexure, and increasing the ends would diminish that flexure. Another mode of increasing the strength of the ends would be that of preventing flexure by increasing the dimensions of the middle. From the experiments it would appear that the Grecian columns, which seldom had their lengths more than about ten times the diameter, were nearly of the form capable of bearing the greatest weight when their shafts were uniform; and that columns tapering from the bottom to the top were only capable of bearing weights due to the smallest part of their section, though the larger end might serve to prevent lateral thrusts. This last remark applies, too, to the Egyptian columns, the strength of the column being only that of

the smallest part of the section. From the two series of experiments, it appeared that the strength of a short column is nearly in proportion to the area of the section, though the strength of the larger one is somewhat less than in that proportion.

Prof. CHALLIS inquired whether Mr. Hodgkinson had found the columns to give way chiefly in the direction of the cleavages of the stone? Mr. Hodgkinson replied that he had; and that hence two pieces of the same size and shape of stone cut out of the same block, required very different forces to crush them across the grain from what they did with it.

Improved Method of taking Positive Talbotypes (Calotypes).

In the method now in use the face of the negative Talbotype is placed directly upon the side of the paper which has been brushed over with a solution of nitrate or ammoniacal nitrate of silver, and which is to receive the positive picture. In strong sunlight the picture is thus taken very quickly; but there is a roughness in the shades, owing to the formation of black specks, which destroys the softness of the picture, and in portraits gives a disagreeable harshness to the human face. In order to remove this defect, Sir David Brewster first interposed thin plates of glass, with their surfaces sometimes ground and sometimes polished; but, though the divergency or diffusion of the light, passing through the negative picture, produced great softness in the positive, yet the outlines were too indistinct, though the Talbotypes looked very well, when placed at a distance. Sir David now informed the meeting that he had tried the effect of interposing a sheet of writing-paper, without the water-mark and of uniform texture. The result of this experiment fully answered his expectations. The diffusion of the light thus occasioned shaded off, as it were, all the sharp lines and points, and gave a high degree of softness to the picture. The effect was even improved by interposing two sheets of clean paper; and with a very bright meridian sun, he found that three sheets may be used with advantage. A similar effect may be obtained, in a smaller degree, by placing the back of the negative upon the positive paper, so as to cause the light to traverse the thickness of the negative, and this may be combined with one or more sheets of clean paper. This, of course, will be appropriate only with portraits; and it has the advantage (sometimes required) of making the figure look another way. To those who see the experiment above described for the first time, the effect is almost magical; when the negative is removed, we see only a blank sheet of white paper, and