PRACTICAL SQUARING OF THE CIRCLE.

Sir,-Understanding that contributions connected with mechanics are acceptable for the columns of your Magazine, I send you the accompanying demonstration of the square of the circle, as applicable to the measurement of trees and other cylindrical bodies.

The mathematician will not object to a subject of scientific study since the days of Euclid and Archimedes, as it affords, I believe, the readiest and nearest practical data, as a relative proportion between the square and circle, ever yet offered to the public in integers, and the young arithmetician, or humble artisian, will be able by it to make his calculations free of decimals and fractional parts.

To the private gentleman wishing to ascertain the dimensions of the beam that he can get from a tree; to the merchant in purchasing timber; to the builder, the mechanic, and in short, for numerous purposes, it may be found useful in simplifying and shortening calculations.

The author, Captain Cortlandt Taylor, of the Madras artillery, when on a mission in 1837, from the Madras government to the Malabar coast, to procure teak timber for the government guncarriage manufactory, forwarded it with some calculations in the reduction of logs, to the military board of that presidency.

I am, Sir, your obedient servant,
EXODUS.

London, July 23, 1838.

meters (A C, and BD perpendicular to each other; then are the triangles AEB, BEC, CED, and DEA (formed by the radii of the circle and the points A, B, C, and D being joined), right angle triangles and similar; having the common centre E a right angle in each; the legs A E, BE, CE, and D E (or circles radii,) equal; and the lines or hypothenuses A B, BC, CD, and A D, also equal, and together forming an inscribed square to the circle A B C D.

Then, as the circumferences of all circles are to their diameters, as 3.1416 is to unit. So 3.1416 is to 1, as the circumference of any given circle A BCD, is to its diameter, DB: but D B the diameter is the double of EB,-the radius; and EB is one leg or side of the right angle triangle A E B, of which AE is the other leg, and A B, is the third side, or hypothenuse.

Then, as in right angle triangles, the square of the hypothenuse is equal to the sum of the squares of the other two sides, in the right angle triangle AEB, BE+AEA B2: but as B E, and A E are equal, their squares are equal also, therefore 2 E B2= A B2, or √2 EB2 = AB, the one-fourth of the perimeter of the required square.

Thus, knowing the circumference of any circle, the perimeter, or measurement of the inscribed square can be easily ascertained; and vice versa, knowing the square the circle can be obtained, and the difference, (or loss on the latter being squared,) is known.

Practical Illustration.-To illustrate this demonstration by numbers: In the above figure, assume the measurement of the inscribed square* A B C D at 100, (one hundred, say inches,) of which the side A B is one-fourth or (twenty-five,) 25. Then in the right angle triangle A B E, (as before), the square of the hypothenuse (or side A B), or 252, is equal to A E+E B2, that is 625 = A E2+ E B2; but as AE and EB are radii of of the same circle and equal, their squares are equal also, and 6252 E B2, or A E2, or √/ 25 = E B, or again √312.5 =EB. (Extract root of 312.5-) • 312,5)17,6776 the root.

Let A B C D be a circle, and the dia

This is working the reverse way, but as it enabled me to come, by a single process, to the required information, it was adopted.

SCIENCE.

Mr. Russell, of Edinburgh, brought up the "Report of the Committee (consisting of Sir John Robison and himself) on Waves."

This report was a continuation of that of last year, published in the volume of the Transactions just issued. The following were the duties of the Committee:-1. To examine the phenomena of a certain kind of wave generated in a fluid, with the view of enabling us to understand the mechanism of its propagation, and so advance this department of hydrodynamical science. 2. To examine the nature of the connexion which exists between the generation of these waves in a fluid, and the resistance of the fluid to the motion of a floating body moved through it, as in the instance of a ship. And, 3. To investigate the nature of the connexion which exists between this wave, which Mr. Russell has called the "Great Wave of Translation,' and the tidal wave, remarkable analogies having been already ascertained to exist between them; and to determine the effect of the wind on the propagation of the tide wave. In regard to the first department, the phenomena of the great wave of translation, the report of the preceeding year contained a great portion of what had been done this year; but the observations were such that they would probably not be completed in several years.

What they had done this year, had been to complete, as far as they possibly could, the second of these departments of investigation-that which related to the physical investigation of waves; and he was happy to think that all that could be done, had, he believed, been done, towards an understanding of the mechanism of the wave. He should, first of all, show what had been done in regard to the physical mechanism. They had previously determined the formal law of the wave; that was to say, they had determined that the velocity of the wave was totally independent of the manner in which it was generated, and depended only upon one circumstance-namely, the depth of the wave. But this was a merely formal result, and his present object was to investigate the physical constitution of the wave. In the formal investigation, they experienced considerable difficulty in ascertaining its form. The nature of a wave was such, that it was exceedingly difficult to detect its form, because it required an instantaneous observation, made with great precision, by a number of observers, in a small space, and with a minuteness which exceeded any means of

ordinary observation. The method proposed by Prof. Stevelly had been satisfactorily applied, to determine the length of the wave, which they found little more than six times the depth of the fluid. But they also found that this did not hold in all cases; for when the wave was very high, the quantity was much greater; but when it was low, it was exactly the same. So that just when the wave was about to disappear, it was found that its length was six times the depth of the fluid. Now they could not define a form, unless by fitting it into some form already known; and from all that he could learn, the only class of curve to which there was any liklihood of fitting this wave, were the trochoidal curves, of which the cycloid was the limit. Now the form, as observed by him (Mr. Russell,) was almost precisely cycloidal; but of double the length which the circumstance of its being the cycloid would have given, because it was manifest that taking the limits, and supposing the wave to be a cycloidal wave, it would only be the circumference of a circle, of which its height was the diameter, or little more than three times the depth of the fluid. But the true length was six times that quantity. Thus another set of curves became necessary. They found that the curve which the wave would fit was not a cycloid, but an analogous curve, which he (Mr. Russell) should be disposed to call the semi-circular cycloid. He should now state the results which they had come to. For the purpose of ascertaining the physical constitution of the wave, and how it was propagated, the reservoir which they had previously constructed was formed in a certain part of plate glass, the object being to float particles in water, and to observe the phenomena which took place among these particles while the wave was passing over them. He had previously examined many observations which had been made in Germany and elsewhere. Weber had described the oscillating wave and the ocean wave in all their phenomena; but neither the Webers nor any other investigator appeared to have recognized the existence of this great solitary wave of translation; they seemed to have limited their observations to the oscillatory and gregarious waves. He (Mr. Russell) called his the primary wave, or great wave of translation of the fluid. The glass side of the vessel was carefully divided, so as to enable the eye to determine the results;

and the following phenomena took place among the particles so invariably, that on the slightest observation he could calculate the results. Suppose that the particles were in a particular plane, at right angles to the direction of the motion of the wave, when first the wave came to that place, the particles would begin to move in the direction of that motion. They would move with accelerating velocity; they were at their maximum when the top of the wave was immediately over them; and from that moment the particles began to move forward with retarded velocity, and, at the instant when the wave left the place, they were at rest in precisely the same position to one another as they occupied previously to the translation. They were put forward without the slightest displacement. The next question was the path of translation, which was a curious and yet simple matter. While these particles were in their progress forward, they were also raised. They were transferred forward horizontally to a distance equal to twice the height of the wave; and the curves, which the uppermost particles described, were as exactly as possible semi-circles, described on a radius equal to the height of the wave; and of the other particles, at greater depths, each of them described a semi-ellipse, whose major axis was equal to the diameter of the semi-circle, and whose minor axis was to the radius of the semi-circle in the same ratio as the height of the particle above the bottom to the whole depth of the fluid; the path of the lowest being a straight line. Considering, next, the vertical motion of the particles, it appeared that during the transit of the great wave, each particle was lifted upwards from a state of rest with an accelerated motion, left at its highest point for an instant at rest, from which it descended with a motion first of all accelerated, and then retarded, so as to be left perfectly at rest at its original height at the instant when the wave

had passed away. Supposing, then, an

elevation of fluid of this kind to have been once generated, the manner in which it propagated the wave might be adequately conceived to take place thus: any given series of particles, in the same vertical plane, would be more pressed forwards than a similar series behind them, under the anterior part of the elevation, and the difference of pressures would be the differential of the vertical ordinates in those planes; and this excess of pressure necessarily produced two effects, it forced the particles in one plane nearer to those in the other, and thus caused progression or horizontal translation of the particles; then the same excess of pressure diminishing the distance

between the particles, elevated the intervening column of fluid to a height inversely proportional to the distance of these planes from each other. The planes situated in the latter part of the elevation of fluid, were in a situation the reverse of this, and the difference of pressure permitted the particles to descend, and diminished the velocity of translation; and the sum of these increments and decrements being equal and in opposite directions, the particles were successively accelerated and elevated, retarded and depressed, by the same law. The curve which the wave described, was a very remarkable one. It was very nearly related to the cycloid, yet differed essentially from it. It was also related to the curve of sines. The ordinates of the cycloid consisted of the ordinates of a curve of sines, added to those of a circle. The ordinates of this wave-curve consisted of the ordinates of the curve of sines, added to those of a semi-circle, whose diameter was double that of the circle. He, therefore, should designate this curve as the semi-circular cycloid. It was the only one which appeared to represent the observed phenomena. The next part of the subject to which he directed attention, was the relation which the translation wave bore to the phenomena of resistance of fluids. He had previously ascertained that the displacement of a fluid by a vessel took place, not in the body of the current, but solely by the generation of waves. Now, the manner in which they were generated, appeared to throw light upon the subject of the resistance of fluids; because they wished to have exactly the same transference for particles of matter which was required for transference of waves. They wished to remove the particles of fluid from a state of rest, and admit the vessel to pass through, and then allow them to return to their former places, just as in the wave the particles were first elevated above the surface, and then permitted to subside. Now they found that whenever the displacement took place, as in the wave, they had the phenomena of least resistance. So that, in forming a floating vessel with this wave-line disposed on alternate sides of the keel, so as to give such motion to the particles as to displace nothing more than was necessary, nor for a greater distance than was necessary to allow the vessel to pass, they obtained the solid of least resistance. Since that time, a variety of experiments on large vessels had been performed; steam vessels were now constructing on this form; and it was a remarkable fact, that the fastest vessel on the Thames was one to which this form had been given. Several other vessels had since

been built on a large scale on this construction, and he was happy to find that a great number of ship-builders had adopted and were building vessels with great success on this line. It was scarcely credible, that a vessel should move at the rate of fifteen miles an hour, and not raise a spray-not raise anything like that high mass of water which was always found at the bows of vessels going at speed, but enter the water perfectly smooth, and leave it smooth, and as much at rest in the direction of the displacement as it was before the floating solid passed. This phenomenon had invariably accompanied all the vessels formed on this line. Now this appeared to him (Mr. Russell) to show the correctness of a remark made by D'Alembert, who, in his Opuscules, had given a demonstration, from the mathematical theory of fluids, that the resistance to a solid body, if properly formed, was nothing. He challenged the mathematicians of the day to disprove his assertion, which was never done; though what the proper form necessary for this purpose was, had not been assigned. Now he (Mr. Russell) thought he had quite manifested the possibility of a vessel moving through the water with little or no resistance. On making allowance for adhesion to the sides of the vessel, (which they knew might be done correctly, from experiments made by others,) they found that the resistance of the vessel was not one-twentieth part of the mere adhesion of the water to the sides of the vessel; so that the resistance from displacement of transference was nearly nothing. A large vessel having been made in this form, the following experiment was performed. Two oranges were placed in the direction in which the vessel moved; the person steering, after many attempts, at last succeeded in insinuating the prow of the vessel between the oranges; they rolled along the side of the vessel, remained in contact therewith, and returned at the wake, and when the vessel passed they remained at rest; they had been transferred horizontally, in the manner of a wave, and remained at rest in precisely the same position as they were when the transference commenced. This appeared to him to be the strongest test; and if this vessel was not a solid of least resistance, it was closely allied to it. [The Chairman: I should say it was a vesɛel of no resistance.] There was another thing which he might mention-namely, that as steam vessels built on this line did not produce the waves which were at present so injurious to the banks of rivers, &c. perhaps its introduction would be attended with great advantages in this respect. He felt certain, indeed, that this was a form to

which ship-builders must ere long be driven. It was the theoretical form of least resistance, which he (Mr. Russell) gave at Dublin three years ago; but it was not until he discovered the law of transference of the wave, that he found he had hit upon the very form of displacement of the wave. Ship-builders had been in the habit of saying, Whatever you do, let us have no hollow lines. The inaxim now would be, at least of those ship-builders who had carefully examined the subject,-Let us have the hollow lines where we want them, and then we shall have plenty of scope for mak ing fuller lines where they will not injure the progress of the vessel. He (Mr. Russell) should now have entered upon the connexion of the subject with the theory of tides, because he thought he had identified the theory of this wave with that of the tidal wave; and whatever influence the celestial mechanism might have upon the tides, they must yet depend upon terrestrial mechanism for bringing it to their doors: he thought they could get a great deal more knowledge about the theory of waves by tidal observations, because there they had a large, long, slow wave, which could be examined with great minuteness. Time, however, would not then permit him to enter upon the subject.

Sir W. R. Hamilton congratulated the section on the increasing interest excited by these valuable researches of Mr. Russell on waves. It was now evident that they had a most important and direct relation to the doctrine of the tides; and he had no doubt, from the many imperfect glimpses which he was enabled to catch of their relation to the undulatory propagation of light, that they were in progress towards elucidating some of the mysteries of that mysterious physical process. Mr. Whewell felt peculiar interest in that part of Mr. Russell's communication in which he described the internal motion of the molecules of the fluid during the progress of the wave, and traced each from its first disturbance of position to its return to rest after the passage of the wave; and he felt confident this was only the first step towards the solution of a most important problem in general hydrodynamics, by which we should at length be led to know the manner in which motions were propagated through fluids as perfectly as we at present know how forces communicate motions to cohering masses of matter. In describing the form of the curve of the wave Mr. Russell had proposed to call it the semi-circular cycloid. He would be inclined to suggest the propriety of the term hemi-cycloid, in which both parts of the term are borrowed from the Greek, and the term semi-cycloid had already another accepta