of accent in the French language, I defire to refer to example within the English language.' p. 431. And fhortly after, The difficulties of the paffage, for turning into French, &c.'

We are far from requiring high-wrought expreffion, bold imagery, or fonorous periods, in a work of this nature; but we think that he, who propofes fchemes for improving the euphony of the English language, would be heard with more deference, if he were ftudious of writing that language, as it now exifts, with propriety and elegance. The work has, however, intrin fic merit, which will compenfate the faults of style; and we think it our duty to recommend it to the attention of all who take any interelt in fuch fpeculations.

ART. XI. Principi di Statici per i Tetti, per i Ponti, e per le Volti. Di Paolo de Langes.

From Memorie de Mat. e Fis. de la Societa Italiana. 1803. Vol. X. Part I. p. 183.

T was remarked by an eminent mathematician, that while we give ourselves infinite trouble to pursue investigations relating to the motions and maffes of bodies which move at immeasurable distances from our planet, we have never thought of determining the forces neceffary to prevent the roofs of our houfes from falling on our heads. To accomplish this investigation, various methods have been employed fince his time; and the author of the very elegant tract now before us begins by defcribing two of thefe, and fhewing their defects.

The firft was that of the ordinary compofition and refolution of forces. By very easy steps, this method leads us to an equation between the fine and cofine of a given angle of inclination and the preffure exerted by a beam refting diagonally between two given planes. If is the angle of its inclination to the horizontal plane, a its length, b the distance of its centre of gravity from the upper extremity, and Pits whole weight, then the force of the other extremity in an horizontal direction is equal to P(ab). col.

There were other values of this force deduced

a. fin. from different principles. It was given either as

€ (a — b). fin. q. cof. 4),

or as (P. fin. . col. 4.)

Upon these refults, our author obferves, that when the beam is horizontal, the first formula gives infinity for the value of its horizontal force; and when the beam is vertical, we have, by the

fame

fame folution, nothing for the value of the horizontal force. But the other two formulas, he adds, give nothing for the value of that force in both thefe cafes; and one of them gives the folution independent of the pofition of the centre of gravity. This mode of folving the problem takes into view, as it ought to do, the distance of the centre of gravity from the extremity of the diagonal beam, and is fuch as to give the value of the horizontal preffure equal to nothing in each of the extreme cafes, both when the beam is placed at right angles to the horizontal plane, and when it coincides with that plane.

The form of the question here alluded to, is the fimpleft cafe of this problem. It is the only cafe, as we fhall afterwards fee, which our author difcuffes; but it is certainly the moft general and fundamental, as well as the eafieft to be inveftigated. It is treated with great neatness and elegance, and with geometrical rigour, in this paper, which we recommend as an excellent intro. duction to the great fubject of arches.

The first cafe of the problem in queftion is thus enunciated. Two planes being given, at right angles to each other, and a rod or beam of any shape, but of a given weight and length, being placed between the planes at any inclination, it is required to find the preffure exerted by the inferior extremity of the beam in an horizontal direction. Of this problem, our author gives a geometrical folution of great elegance by means of the ellipfe, having previously demonftrated the following property of that curve; that if, from a point in its tranfverfe axis a straight line be inclined fo, that the curve and the point intercept a portion equal to half the conjugate, the portion intercepted between the conjugate and the curve is equal to the femitranfverfe. He then adds an analytical folution of the problem. If is the angle of the beam's inclination to the horizon, P its weight, m and n the two portions of its length on each fide of its centre of gravity, then the horizontal preffure of the lower extremity is equal to m n. fin. 4. cof.

P.

(m2. fin. 4+n2. cof. q); an expression certainly of very great neatness and fymmetry, and which is found to agree exactly with folutions of lefs general cafes drawn from other methods. He adds two investigations to difcover the pofition of the beam's centre of gravity which gives the greatest poffible preffure at a given angle of inclination, and to discover the angle of inclination which gives the fame maximum at a given pofition of the beam's centre of gravity. The ordinary method of maxima and minima, applied to the above formula, gives, for the firft cafe, an expreffion from whence we may deduce the following fimple folution-that the distance between the centre and the vertical plane must be a

Bb 2

fourth,

fourth, proportional to the length of the beam, the cofine of its inclination, and the fum of the fine and cofine of inclination. For the folution of the fecond cafe, let a = the beam's length, and b its given fuperior fegment, the height of the fuperior extre

=

mity above the horizontal plane must be a

(a - b)3

-

(a - b)3 + b2 (a - b)' in order that the preffure may be the greatest poffible. This expreffion, tranflated into geometrical language (as we tranflated the former), fhews that the fquare of the height must be a fourth proportional to the fquare of the beam's length, the cube of its inferior fegment, and the fum of that cube, and the parallelopiped, whose bafe is the fquare of the fuperior fegment, and altitude the inferior fegment. From thefe proportions, it is easy to conclude, that if the beam is inclined at an angle of 45°, the horizontal preffure is a maximum when the centre of gravity falls in the middle of the beam, and converfely.

Having condered the fimple. cafe, of one beam pressing on two planes at right angles to each other, our author proceeds to confider, how his refults are modified by the combination of different beams, which, it may be remarked, is the ordinary cafe in practice. He fhews by confiderations which muft immediately prefent themfelves to those who attend to the foregoing analysis, that when two equal beams lean against each other, they are exactly to each other as to the verticle plane in the cafes above folved. In like manner, the folution of the cafe in which three beams are connected together, the one lying over the other horizontally, is reduced to a variety of the first cafe, and when four beams are combined, two meeting in a point above and refting upon the other two, inclined at any acute angle to the horizon, he hews that when the two latter are attached by a chain paling horizontally across the interval, the cafe is reducible to the ori ginal and general problem, being a variety of the last hypothe fis. But when four beams are thus combined, and no connexion is made between the two inferior ones at their upper ex tremities, the folution of the problem becomes much more intricate. This will be apparent if we confider that all the former folutions depend upon the difcovered nature of the trajectory, which the centre of gravity of the beam defcribes, when it de fcends freely in the angle of two planes vertical to each other. This is well known to be an elliptical arch; and, confequently, the folution of the problem is effected by the application of that curve's obvious properties. But in the cafe of four beams, the two higheft of which meet and reft on the two lowest with their other extremities, feveral curves must be discovered, in which the

centres

centres of gravity of each pair of beams and the juncture of the two feverally move. One curve must be found for the trajectory of the centres of the two upper beams, another for the trajectory of the centres of the lower beams, and a third for the trajectory of the mutual joinings of the two beams on each fide. Our author has not thought proper to make any attempt to difcover these three curves. But he limits the problem in different ways in order to fimplify it, and to facilitate an approximation to the folution. He first fuppofes the lower beam to be ftopped on the horizontal plane, by an obstacle at a given point; then it is clear that its centre of gravity gyrates in a circle, whofe radius is given: ftill the trajectory of the upper beam's centre remains to be found. He fuppofes it to be elliptical, which it is nearly, when the angie of the upper beam's inclination to the horizon is very small. This cafe, therefore, is refolved into the cafes formerly inveftigated. He then takes the cafe of the lower beam ftanding fixed in the perpendicular; and in this hypothefis, alfo, the centre of the upper beam gyrates in an elliptical arch when the angle of its horizontal inclination is fmall. These are obviously the fim. pleft cafes; the latter is that of a roof fupported by two pillars or pilaftres, but very flat, and of the kind known by the name of Tetti alla Manfarda. Such roofs are by far the most eles gant; they are univerfally employed all over Italy, both for the coverings of ufeful and of ornamental buildings. They are the roofs found in all ancient temples, and other structures of the Greek and Roman ages. But they are obviously adapted only to a climate where little or no fnow falls. In the northern countries of Europe, the roofs affume a very different appearances they are built very nearly in the vertical line: inftead of two beams refting on the upright pillars or walls at fmall horizontal angles, we there find four, fix, and eight beams, joined or little feparated at the top, and inclined to one another and to the co lumns or walls, in directions which deviate but little from the perpendicular. To fuch cafes, the solution, or rather the rough approximation of the Italian mathematician, does not at all ap ply; and as fuch cases were evidently within the scope of his general queftion, and prefented themfelves to his obfervation in the course of his inveftigations, it may be thought that he was bound to furnish a folution of them. Inftead of this, he does not even state how their examination is to be carried on. He contents himself with remarking, that they are extremely intricate; and immediately leaves the fubject. We fhall offer a few Strictures upon thofe cafes, and point out the general method of refolving them.

Our author has certainly been too eafily alarmed by the ap

Bb 3

pearance

pearance of their difficulties at a diftance; and, by not attending to the steps which lead to the introduction of the elliptical trajectory, in the branches preliminary to his own investigations of the fimple cafes. He feems to have thought that the ellipfe was difcovered to be the trajectory of the beam's centre of gravity, when it moves in the angle of two perpendiculars, in confequence of fome property peculiar to the centre of gravity. On the contrary, this is a propofition derived from the investigation of a very general problem of inclinations. If it is required to incline a given ftraight line in a given angle, fo that another straight line given in pofition, fhall cut the line given in magnitude in a given point; we know that this problem can be folved by the ancient geometrical analyfis. But when there is no straight line given in pofition, the problem becomes indeterminate or local; and the given point in the straight line given in magnitude, is always in an ellipfe which may be found. Let a = the length of the given line, b its fuperior fegment, confequently a — b = its other part, y the perpendicular drawn from the given point to one of the given lines, and x = the perpendicular drawn from the fame point to the other line. By fimilar triangles, we have babx to the fegment intercepted between the perpendi cular y, and the line to be inclined. This fegment is therefore equal to (a - b); and (by the property of the rightangled

triangle) (a — b)2 = ~ (a—b)2 + y2, or y2 =

(a - b)
ba

Which

× (ba— x2), an equation to the ellipfe, whofe tranfverfe axis is equal to twice (ab), and whose conjugate axis is equal to twice b. is the very propofition, including the lemma, by whofe affiftance the author's folutions are accomplished. And therefore it is evident that this method of folving the problem depends on no property of the beam's centre of gravity, but folely on the datum of any point in the length of the beam.

We fhall now fuppofe that it is required to find the preffure of the flanting fides of the roof of a temple, supported by upright pillars or pilaftres. The problem is reduced to this. To find the curve line in which the centre of gravity of one of the flanting beams moves, while its lower extremity gyrates with the pillar's upper end in a circle, and the upper extremity moves along a vertical plane; or, which is the fame thing, to find the locus of a given point in a ftraight line given in magnitude, one end of which is carried along a ftraight line given in pofition, and the other along the circumference of a given circle. This is one of the cafes fuppofed by our author to be of too intricate a nature to

justify