or CENTENARIO, in the middle ages, an officer who had the government or command, with the administration of justice, in a village. The centenarii as well as vicarii were under the jurisdiction and command of the court. We find them among the Franks, Germans, Lombards, Goths, and other nations.

CENTENARIUS was also used to signify an officer who had the command of one hundred men, most frequently called a centurion.

in monasteries, was an officer who had the command of one hundred monks.

CENTRE.

CENTRE, or CENTER, a word borrowed from the French name cintre or cintre, given to the frame of timber by which the brick or stone of arched vaulting is supported during its erection, and from which it receives its form and curvature.

It is not our intention to describe the variety of constructions which may be adopted in easy situations, where the arches are of small extent, and where sufficient foundation can be had in every part of it for supporting the frame. In such cases the frequency of the props which we can set up dispenses with much care; and a frame of very slight timbers, connected together in an ordinary way, will suffice for carrying the weight, and for keeping it in exact shape. But when the arches have a wide span, and consequently a very great weight, and when we cannot set up intermediate pillars, either for want of a foundation in the soft bottom of a river, or because the arch is turned between two lofty piers, as in the dome of a stately cathedral, we are then obliged to rest everything on the piers themselves; and the framing which is to support our arch before the keystone is set must itself be an arch, depending on the mutual abutment of its beams. One should think that this view of the construction of a centre would offer itself at the first, naturally derived from the erection it was to assist; but it has not been so. When intermediate pillars were not employed, it was usual to frame the mould for the arch with little attention to anything but its shape, and then to cross it and recross it in all directions with other pieces of timber, till it was thought so bound together that it could be lifted in any position, and, when loaded with any weight, could not change its shape. The frame was then raised in a lump, like any solid body of the same shape, and set in its place. This is the way still practised by many country artists, who, having no clear principles to guide them, do not stop till they have made a load of timber almost equal to the weight which it is to carry.

But this artless method, besides leading the employer into great expense, is frequently fatal to the undertaker, from the unskilfulness of the construction. The beams which connect its extremities are made also to support the middle by means of posts which rest on them. They are therefore exposed to a transverse or cross strain, which they are not able to bear. Their number must therefore be increased, and this increases the load. Some of these cross strains are derived from beams which are pressed very obliquely, and therefore exert a prodigious thrust on their supports. The beams are also greatly weakened by the mortises which are cut in them to receive the tenons of the crossing beams; and thus the whole is exceedingly weak, in proportion to what the same quantity of timber may be made by a proper disposition of its parts.

The principles from which we are to derive this disposition are the general mechanical principles of carpentry, of which an account has already been given under the proper head. See CARPENTRY. These furnish one general rule: When we would give the utmost strength possible to a frame of carpentry, every piece should be so disposed that it is subject to no strain but what either pushes or draws it in the direction of its length; and, if we would be indebted to timber alone for the force or strength of the centre, we must rest all on the first of these strains; for when the straining force tends to draw a beam out of its place, it must be held there by a mortise and tenon, which possesses but a very trifling force, or by iron straps and bolts. Cases occur where it may be very difficult to make every strain a thrust, and the best artists admit of ties; and indeed where we can admit a tie-beam connecting the two feet of our frame, we need seek no better security. But this may sometimes be very inconvenient. When it is the arch of a bridge that we are to support, such a tie-beam would totally stop the passage of small craft up and down the river. It would often be in the water, and thus exposed to the most fatal accidents by freshes, &c. Interrupted ties, therefore, must be employed, whose joint or meetings must be supported by something analogous to the king-posts of roofs. When this is judiciously done, the security is abundantly good. But great judgment is necessary, and a very scrupulous attention to the disposition of the pieces. It is by no means an easy matter to discern whether a beam, which makes a part of our centre, is in a state of compression or in a state of extension. In some works of the most eminent carpenters even of this day, we see pieces considered as struts, and considerable dependence had on them in this capacity, while they are certainly performing the office of tie-beams, and should be secured accordingly. This was the case in the boldest centre, we think, that has been executed in Europe, that of the bridge of Orleans, by M. Hupeau. Yet it is evidently of great consequence not to be mistaken in this point; for when we are mistaken, and the piece is stretched which we imagine to be compressed, we not only are deprived of some support that we expected, but the expected support has become an additional load.

To ascertain this point, we may suppose the piers to yield a little to the pressure of the arch-stones on the centre frames. The feet, therefore, fly outwards, and the shape is altered by the sinking of the crown. We must draw our frame anew for this new state of things, and must notice what pieces must be made longer than before. All such pieces have been acting the part of tie-beams.

But a centre has still another office to sustain; it must keep the arch in its form; that is, while the load on the centre is continually increasing as the masons lay on more courses of arch-stones, the frame must not yield and go out of shape, sinking under the weight on the haunches, and rising in the crown, which is not yet carrying any load. The frame must not be supple, and must derive its stiffness, not from the closeness and strength of its joints, which are quite insignificant when set in competition with such immense strains, but from struts or ties, properly disposed, which hinder any of the angles from changing its amplitude.

It is obvious, from all that has been said, that the strength and stiffness of the whole must be found in the triangles into which this frame of carpentry may be resolved. We have seen that the strains which one piece produces on two others with which it meets in one point, depend on the angles of their intersection; and that it is greater as an obtuse angle is more obtuse, or an acute angle more acute. And this suggests to us the general maxim, "to avoid as much as possible all very obtuse angles." Acute angles, which are not necessarily accompanied by obtuse ones, are not so hurtful, because the strain here can never exceed the straining force; whereas, in the case of an obtuse angle, it may surpass it in any degree.

Such are the general rules on this subject. Although something of the mutual abutment of timbers, and the support derived from it, has been long perceived, and employed by the carpenters in roofing, and also, doubtless, in the forming of centres, yet it is a matter of historical fact, that no general and distinct views had been taken of it till about the beginning of last century, or a little earlier. Fontana has preserved the figure of the frames on which the arches of St Peter's at Rome were turned. The one employed for the dome is constructed with very little skill; and those for the arches of the nave and transepts, though incomparably superior, and of considerable simplicity and strength, are yet far inferior to others which have been employed in later times. It is much to be regretted that no trace remains of the forms employed by the great architect and consummate mechanic Sir Christopher Wren. We should doubtless have seen in them every thing that science and great sagacity could suggest. We are told, indeed, that his centreing for the dome of St Paul's was a wonder of its kind; begun in the air at the height of 160 feet from the ground, and without making use of even a projecting cornice whereon to rest.

The earliest theory of the kind that we have met with, that is proposed on scientific principles, and with the express purpose of serving as a lesson, are two centres by M. Pitot of the Academy of Sciences, about the beginning of last century. As they have considerable merit, greatly resembling those employed by Michael Angelo in the nave of St Peter's, and afford some good maxims, we shall give a short account of them. We crave the excuse of the artists if we should employ their terms of art somewhat awkwardly, not being very familiarly acquainted with them. Indeed, we observe very great differences, and even ambiguity, in the terms employed.

What we shall describe under the name of a centre is, properly speaking, only one frame, truss, or rib, of a centre. They are set up in vertical planes, parallel to each other, at the distance of five, six, seven, or eight feet, like the trusses or main couples of a roof. Bridging joists are laid across them. In smaller works these are laid sparingly, but of considerable scantling, and are boarded over; but for great arches, a bridging joist is laid for every course of arch-stones, with blockings between to keep them at their proper distances. The stones are not laid immediately on these joists, but beams of soft wood are laid along each joist, on which the stone is laid. These beams are afterwards cut out with the chisel, in order to separate the centre from the ring of stones, which must now support each other by their mutual abutment.

The centre is distinguishable into two parts, ALLB (fig. 1) and LDL, which are pretty independent of each other, or at least act separately. The horizontal stretcher LL cuts the semicircle ADB half way between the spring and the crown of the arch; the arches AL, LD, being 45° each. This stretcher is divided in the same proportion in the points G and H; that is, GH is one half of LL, and LG, HL, are each one fourth of LL nearly. Each end is supported by two struts EI, GI, which rest below on a sole or bed properly supported. The interval between the heads of the struts GI, HK, is filled up by the straining beam GH, abutting in a proper manner on the struts (see Carpentry). The extremities L, L, are united in like manner by butting joints, with the heads of the outer struts. The arch moulds AP, BP, are connected with the struts by cross pieces PQ, which we shall call bridles, which come inwards on each side of the struts, being double, and are bolted to them. This may be called the lower part of the frame. The upper part consists of the king-post DR, supported on each side by the two struts or braces ML, ON, mortised into the post; and also mortised into the stretcher, at the points L, N, where it is supported by the struts below. The arches LD, LD, are connect- There is an evident propriety in the manner in which he has distributed the supports of the upper part. The struts which carry the king-post spring from those points of the stretcher where it rests on the struts below. Thus the stretcher, on which all depends, bears no transverse strains. It is stretched by the strut above it, and it is compressed in a small degree between the struts below it, at least by the outer ones. M. Pitot proposes the straining beam GH as a lateral support to the stretcher, which may therefore be of two pieces; but although it does augment its strength, it does not seem necessary for it. The stretcher is abundantly carried by the strap, which may and should suspend it from the king-post. The great use of the straining piece is to give a firm abutment to the inner struts, without allowing any lateral strain on the stretcher.—N.B. Great care must be taken to make the hold sufficiently firm and extensive between the stretcher and the upper struts, so that its cohesion to resist the thrusts from these struts may be much employed.

The only imperfection that we find in this frame is the lateral strains which are brought upon the upper struts by the bridles, which certainly transmit to them part of the weight of the arch-stones on the curves. The space between the curves and ML should also have been trussed. M. Pitot's form is, however, extremely stiff; and the clasping of the middle bridle to reach down to the stretcher seems to secure the upper struts from all risk of bending.

This centre gives a very distinct view of the offices of all the parts, and makes therefore a proper introduction to the general subject. It is the simplest that can be in its principle, because all the essential parts are subjected to one kind of strain. The stretcher LL is the only exception, and its extension is rather a collateral circumstance than a step in the general support.

The examination of the strength of the frame is extremely easy. M. Pitot gives it for an arch of sixty feet in span, and supposes the arch-stones seven feet long, which is a monstrous thickness for so small an arch; four feet is an abundant allowance; but we shall abide by his construction. He gives the following scantlings of the parts:

- The ring or circumference consists of pieces of oak twelve inches broad and six thick. - The stretcher LL is twelve inches square. - The straining piece GH is also twelve by twelve. - The lower struts ten by eight. - The king-post twelve by twelve. - The upper struts ten by six. - The bridles twenty by eight.

These dimensions are French, which is about one seventeenth larger than ours, and the superficial dimensions (by which the section and the absolute strength is measured) are almost one eighth larger than ours. The cubic foot, by which the stones are measured, exceeds ours by nearly one fifth. The pound is deficient about one thirteenth. But since very nice calculation is neither easy nor necessary on this subject, it is needless to depart from the French measures, which would occasion many fractional parts, and a troublesome reduction.

The arch is supposed to be built of stone which weighed a hundred and sixty pounds per foot. M. Pitot, by a computation, in which he has committed a mistake, says that only eleven fourteenths of this weight is carried by the frame. We believe, however, that this is nearer the truth than M. Couplet's assumption of four ninths, already mentioned.

M. Pitot further assumes, that a square inch of sound oak will carry 8640 pounds. By his language we should imagine that it will not carry much more; but this is very far below the strength of any British oak that we have tried; so far, indeed, that we rather imagine that he means that this load may be laid on it with perfect security for any time. But to compensate for knots and other accidental imperfections, he assumes 7200 as the measure of its absolute force.

He computes the load on each frame to be 707,520 pounds, which he reduces to this, or 555,908 pounds.

The absolute force of each of the lower struts is 576,000 (at 7200 per inch), and that of the curves 518,400. M. Pitot, considering that the curves are kept from bending outwards by the arch-stones which press on them, thinks that they may be considered as acting precisely as the outer struts EI. We have no objection to this supposition.

With these data we may compute the load which the lower truss can safely bear, by the rule delivered in the article CARPENTRY. We therefore proceed as follows:

Measure off by a scale of equal parts \(a\), \(a'\), each 576,000, and add \(t\) to 518,400. Complete the parallelogram \(aexs\), and draw the vertical \(xc\), meeting the horizontal line \(ac\) in \(c\). Make \(ab\) equal to \(ca\). Join \(xb\), and complete the parallelogram \(axby\).

It is evident that the diagonal \(xy\) will represent the load which these pieces can carry; for the line \(a\) is the united force of the curve AP and the strut IE, and \(as\) is the strength of IG. These two are equivalent to \(ax\); \(wb\) is, in like manner, equivalent to the support on the other side, and \(xy\) is the load which will just balance the two supports \(ax\) and \(bx\).

When \(xy\) is measured on the same scale, it will be found =2,850,000 pounds. This is more than five times the load which actually lies on the frame. It is therefore vastly stronger than is necessary. Half of each of the linear dimensions would have been quite sufficient, and the struts needed only to be five inches by four. Even this would have carried twice the weight, and would have borne the load really laid on it with perfect safety.

We proceed to measure the strength of the upper part. The force of each strut is 432,000, and that of the curve is 518,400; therefore, having drawn \(mv\) parallel to the strut ON, make \(me = 432,000\), and \(ma = 432,000 + 518,400\). Complete the parallelogram \(msrv\). Draw the horizontal line \(rk\), cutting the vertical \(mc\) in \(k\), and make \(ky = mk\). It is plain, from what was done for the lower part, that \(my\) will measure the load which can be carried by the upper part. This will be found =1,160,000. This is also greatly superior to the load, but not in so great a proportion as the other part. The chief part of the load lies on the upper part, but the chief reason of the difference is the greater obliquity of the upper struts. This shortens the diagonal \(my\) of the parallelogram of forces. M. Pitot should have adverted to this, and instead of making the upper struts more slender than the lower, he should have made them stouter.

The strain on the stretcher LL is not calculated. It is measured by \(rk\), when \(my\) is the load actually lying on the upper part. Less than the sixth part of the cohesion of the stretcher is more than sufficient for the horizontal thrust, and there is no difficulty in making the foot joints of the struts abundantly strong for the purpose.

The reader will perceive that the computation just now given does not state the proportions of the strains actually exerted on the different pieces, but the load on the whole, on the supposition that each piece is subjected to a strain proportioned to its strength. The other calculation is much more complicated, but is not necessary here.

This centre has a very palpable defect. If the piers should yield to the load, and the feet of the centre fly out, the lower part will exert a very considerable strain on the stretcher, tending to break it across between N and L, and on the other side. HKF of the lower part is firmly bound together, and cannot change its shape, and will therefore act like a lever, turning round the point E. It will draw the strut HK away from its abutment with GH, and the stretcher will be strained across at the place between H and F, where it is bolted with the bridle. This may be resisted in some degree by an iron strap uniting ON and HK, but there will still be a want of proportional strength. Indeed, in an arch of such height, a semicircle, there is but little risk of this yielding of the piers, but it is an imperfection.

The centre (fig. 2, No. 1) is constructed on the same principles for an elliptical arch. The calculation of its strength is nearly the same also; only the two upper struts of a side being parallel, the parallelogram \(msrv\) (of fig. 1) is not needed, and in its stead we measure off on ON a line to represent twice its strength. This comes in place of \(mr\) of fig. 1.—N.B. The calculation proceeds on the supposition that the shortening piece MM makes but one firm body with the king-post. Mr Pitot employed this piece, we presume, to separate the heads of the struts, that their obliquity might be lessened thereby; and this is a good thought, for when the angle formed by the struts on each side is very open, the strain on them becomes very great.

The stretcher of this frame is scarfed in the middle. Suppose this joint to yield a little, there is a danger of the lower strut ON losing its hold, and ceasing to join in the support; for when the crown sinks by the lengthening of the stretcher, the triangle ORN of fig. 2 will be more distorted than the space above it, and ON will be loosened. But this will not be the case when the sinking of the crown arises from the mere compression of the struts. Nor will it happen at all in the centre, fig. 1. On the contrary, the strut ON will abut more firmly by the yielding of the foot of ML.

The figure of this arch of M. Pitot's consists of three arches of circles, each of 60 degrees. As it is elegant, it will not be unacceptable to the artist to have a construction for this purpose.

Make \(BY = CD\), and \(CZ' = \frac{1}{2} CY\). Describe the semi-circle \(Z'AEY\), and make \(ZS' = ZE\). S is the centre of the construct side arches, each of 60 degrees. The centre T of the arch, such an arch which unites these two, is at the angle of an equilateral triangle STS'.

This construction of M. Pitot's makes a handsome oval, and very nearly an ellipsis, but lies a little without it. We shall add another of our own, which coincides with the ellipse in eight points, and furnishes the artist, by the way, with a rule for drawing an infinite variety of ovals.

Let \(AB, DE\) (fig. 2, No. 2), be the axes of an ellipse, C the centre, and \(f, f'\) the two foci. Make \(Cb = CD\), and describe a circle \(ADbe\) passing through the three given points A, D, and b. It may be demonstrated, that if from any point P of the arch AD be drawn a cord PD, and if a line PR be drawn, making the angle DPR = PDC; and meeting the two axes in the points R and r, then R and r will be the centres of circles, which will form a quarter APD of an oval, which has AB and DE for its two axes.

We want an oval which shall coincide as much as possible with an ellipsis. The most likely method for this is to find the very point P where the ellipsis cuts the circle ADbe. The easiest way for the artist is to describe an arch of a circle am, having AB for its radius, and the remote focus f for its centre. Then set one foot of the com-

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1 It is the middle arch of the bridge at Lille Adam, of which M. Pitot had the direction. It is of eighty feet span, and rises thirty-one feet. passes on any point P, and try whether the distance PF from the nearest focus F is exactly equal to its distance P m from that circle. Shifting the foot of the compasses from one point of the arch to another will soon discover the point. This being found, draw PD, make the angle DP r = PD r, and R and r are the centres wanted. Then make CS = CR, and we get the centres for the other side.

The geometer will not relish this mechanical construction. He may therefore proceed as follows: "Draw ed the diameter of the circle, cutting AB in N; join Dd, and produce it so that dH may be equal to CD, and join eH, meeting AB in Q. On B and Q, as centres with any radius, describe arcs cutting each other in X, and on C and N with the same radius describe arcs cutting each other in Y. Take the distance XN in the compasses, and on Y as a centre describe an arc cutting AB in M and M', and draw MP, MP' perpendicular to AB. These ordinates will cut the circle in the points P and P', where it is cut by the ellipse." We leave the demonstration as an exercise for the dilettante.

We said that this centring of M. Pitot's resembled in principle the one employed by Michael Angelo for the nave and transepts of St Peter's church at Rome. Fontana, who has preserved this, ascribes the construction of it to one of the name of San Gallo. A sketch of it is given in fig. 3. It is, however, so much superior, and so different in principle, from that employed for the cupola, that we cannot think it the invention of the same person. It is, like Pitot's, not only divisible, but really divided into two parts, of which the upper carries by much the greatest part of the load. The pieces are judiciously disposed, and every important beam is amply secured against all transverse strains. Its only fault is a great profusion of strength. The innermost polygon aghb is quite superfluous, because no strain can force in the struts which rest on the angles. Should the piers yield outwards, this polygon will be loose, and can do no service. Nor is the triangle gah of any use, if the king-post above it be strapped to the tie-beam and straining sill. Perhaps the inventor considered the king-post as a pillar, and wished to secure the tie-beam against its cross strain. This centring, however, must be allowed to be very well composed; and we expect that the well-informed reader will join us in preferring it to M. Pitot's, both for simplicity of principle and scientific propriety, as well as for strength.

There is one considerable advantage which may be derived from the actual division of the truss into two parts. If the tie-beam LL, instead of resting on the stretcher EF, had rested on a row of chocks formed like double wedges, placed above each other head to point, the upper part of the centring might be struck independent of the lower, and this might be done gradually, beginning at the outer ends of the stretcher. By this procedure the joints of the arch-stones will close on the haunches, and will almost relieve the lower centring, so that all can be pulled out together. Thus may the arch settle and consolidate in perfect safety, without any chance of breaking the bond of the mortar in any part; an accident which frequently happens in great arches. This procedure is peculiarly advisable for low pitched or elliptical arches. But this will be more clearly seen afterwards, when we treat of the internal movements of an arch of masonry.

This may suffice for an account of the more simple construction of trussed centres; and we proceed to such as have a much greater complication of principle. We shall take for examples some constructed by M. Perronet, a very celebrated French architect.

M. Perronet's general maxim of construction is to make the truss consist of several courses of separate trusses, independent, as he thinks, of each other, and thus to employ the joint support of them all. In this construction it is not intended to make use of one truss, or part of one truss, to support another, as in the former set, and as is practised in the roofs of St Paul's church, Covent Garden, and in Drury Lane theatre. Each truss spans over the whole distance of the piers, and would stand alone (having, however, a tottering equilibrium). It consists of a number of struts, set end to end, and forming a polygon. These trusses are so arranged that the angles of one are in the middle of the sides of the next, as when a polygon is inscribed in a circle, and another (of the same number of sides) is circumscribed by lines which touch the circle in the angles of the inscribed polygon. By this construction the angles of the alternate trusses lie in lines pointing towards the centre of the curve. King-posts are therefore placed in this direction, between the adjoining beams of the trusses. These king-posts consist of two beams, one on each side of the truss, and embrace the truss-beams between them, meeting in the middle of their thickness. The abutting beams are mortised, half into each half of the post. The other beam, which makes the base of the triangle, passes through the post, and a strong bolt is driven through the joint, and secured by a key or a nut. In this manner is the whole united; and it is expected, that when the load is laid on the uppermost truss, it will all butt together, forcing down the king-posts, and therefore pressing them on the beams of all the inferior trusses, causing them also to abut on each other, and thus bear a share of the load. M. Perronet does not assume the invention to himself, but says that it was invented and practised by M. Mansard de Sagonne at the great bridge of Moulins. It is much more ancient, and is the work of the celebrated physician and architect Perrault, as may be seen in the collection of machines and inventions of that gentleman, published after his death, and also in the great collection of inventions approved of by the Academy of Sciences. It is this which we propose to examine.

Fig. 4 represents the centring employed for the bridge of Cravant. The arches are elliptical, of sixty feet span and twenty feet rise. The arch-stones are four feet thick, and weigh 176 pounds per foot. The truss-beams were from fifteen to eighteen feet long, and their section was nine inches by eight. Each half of the king-posts was about seven feet long, and its section nine inches by eight. The whole was of oak. The five trusses were five and a half feet asunder. The whole weight of the arch was 1,350,000 lbs., which we may call 600 tons (it is 558). This is about 112 tons for each truss. We must allow nearly ninety tons of this really to press the truss. A great part of this pressure is borne by the four beams which make the feet of the truss, coupled in pairs on each side. The diagonal of the parallelogram of forces drawn for these beams is, to one of the sides, in the proportion of 360 to 285. Therefore say, as 360 to 285, so is 90 to 71.

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1 The construction given in the Supplement to the Third Edition, by the writer of this article (the late Professor Robison), is here left out, and another more simple (that marked by inverted commas) put in its stead. If we put \( s \) for the semi-transverse cA, \( b \) for the semi-conjugate cD, and denote the distance of the ordinates CM, CM', from the centre by \( x \), then CM and CM' will be the roots of this quadratic equation:

\[ x^2 - \frac{a-b}{a+b}x = \frac{2ab}{(a+b)^2} \]

The above construction has been derived from this equation. (n.) tons, the thrust on each foot. The section of each is 144 inches. We may with the utmost safety lay three tons on every inch for ever. This amounts to 432 tons, which is more than six times the strain really pressing the foot-beams in the direction of their length; nay, the upper truss alone is able to carry much more than its load. The absolute strength of its foot-beam is 216 tons. It is much more advantageously placed, for the diagonal of the parallelogram of forces corresponding to its position is to the side as 438 to 285. This gives $\frac{58}{10}$ tons for the strain on each foot, which is not much above the fourth part of what it is able to carry for ever. No doubt can therefore be entertained of the superabundant strength of this centring. We see that the upper row of struts is quite sufficient, and all that is wanted is to procure stiffness for it; for it must be carefully kept in mind that this upper row is not like an equilibrated arch. It will be very unequally loaded as the work advances. The haunches of the frame will be pressed down, and the joints at the crown raised up. This must be resisted.

Here then we may gather by the way a useful lesson. Let the outer row of struts be appropriated to the carriage of the load, and let the rest be employed for giving stiffness. For this purpose let the outer row have abundant strength. The advantages of this method are considerable. The position of the beams of the exterior row is more advantageous when, as in this example, the whole is made to rest on a narrow foot; for this obliges us to make the last angle, at least of the lower row, more open, which increases the strain on the strut; besides, it is next to impossible to distribute the compressing thrusts among the different rows of the truss-beams; and a beam which, during one period of the mason work, is acting the part of a strut, in another period is bearing no strain but its own weight, and in another it is stretched as a tie. A third advantage is, that, in a case like this, where all rests on a narrow foot, and the lower row of beams are bearing a great part of the thrust, the horizontal thrust on the pier is very great, and may push it aside. This is the most ruinous accident that can happen. An inch or two of yielding will cause the crown of the arch to sink prodigiously, and will instantly derange all the bearings of the abutting beams; but when the lower beams already act as ties, and are quite adequate to their office, we render the frame perfectly stiff or unchangeable in its form, and take away the horizontal thrust from the piers entirely. This advantage is the more valuable, because the very circumstance which obliges us to rest all on a narrow foot, places this foot on the very top of the pier, and makes the horizontal thrust the more dangerous.

But, to proceed in our examination of the centring of Cravant Bridge, let us suppose that the king-posts are removed, and that the beams are joined by compass joints. If the pier shall yield in the smallest degree, both rows of struts must sink; and since the angles, at least the outermost, of the lower rows are more open than those of the upper row, the crown of the lower row will sink more than that of the upper.

The angles of the alternate rows must therefore separate a little. Now restore the king-posts, they prevent this separation. Therefore they are stretched; therefore the beams of the lower row are also stretched; consequently they no longer butt on their mortises, and must be held in their places by bolts. Thus it appears that, in this kind of sagging, the original distribution of the load among the different rows of beams is changed, and the upper row becomes loaded beyond our expectation.

If the sagging of the whole truss proceed only from the compression of the timbers, the case is different, and we may preserve the original distribution of mutual abutment more accurately. But in this case the stiffness of the frame arises chiefly from cross strains. Suppose that the frame is loaded with arch-stones on each side up to the posts HC, h; the angles E and e are pressed down, and the beams EOF, eo F, push up the point F. This cannot rise without bending the beams EOF, eo F; because O and o are held down by the double king-posts, which grasp the beams between them. There is therefore a cross strain on the beams. Observe also that the triangle EHF does not preserve its shape by the connection of its joints; for although the strut-beams are mortised into the king-post, they are in very shallow mortises, rather for steadying them than for holding them together. M. Perronet did not even pin them, thinking that their abutment was very great. The triangle is kept in shape by the base EF, which is firmly bolted into the middle post at O. Had these intersections not been strongly bolted, we imagine that the centres of some of M. Perronet's bridges would have yielded much more than they did; yet some of them yielded to a degree that our artists would have thought very dangerous. M. Perronet was obliged to load the crown of the centring with very great weights, increasing them as the work advanced, to prevent the frames from going out of shape; in one arch of 120 feet he laid on forty-five tons. Notwithstanding this imperfection, which is perhaps unavoidable, this mode of framing is undoubtedly very judicious, and perhaps the best which can be employed without depending on iron work.

Fig. 5 represents another, constructed by Perronet, for the arch of ninety feet span and twenty-eight feet rise, bridge of Nogent. The trusses were seven feet apart, and the arch was four feet high; so that the unreduced load on each frame was very nearly 235 tons. The scantling of the struts was fifteen by twelve inches. The principle is the same as that of the former. The chief difference is, that in this centre the outer truss-beam of the lower row is not coupled with the middle row, but kept nearly parallel to the outer beam of the upper row. This adds greatly to the strength of the foot, and takes off much of the horizontal thrust from the pier.

M. Perronet has shown great judgment in causing the polygon of the inner row of truss-beams gradually to approach the polygon of the outer row. By this disposition, the angles of the inner polygon are more acute than those of the outer. A little attention will show that the general sagging of all the polygons will keep the abutments of the lower one nearer, or exactly, to their original quantity. We must indeed except the foot-beam. It is still too oblique; and, instead of converging to the foot of the upper row, it should have diverged from it. Had this been done, this centre is almost perfect in its kind. As it is, it is at least six times stronger than was absolutely necessary. We shall have occasion to refer to this figure on another occasion.

This maxim is better exemplified by M. Perronet in St Maxence's centring of the bridge of St Maxence, exhibited in fig. 5, No. 2, than in that of Nogent, fig. 5, No. 1. But we think that a horizontal truss-beam ab should have been inserted, in a subordinate manner, between the king-posts next the crown on each side. This would prevent the crown from rising while the haunches only are loaded, without impairing the fine abutments of cd, ed, when the arch is nearly completed. This is an excellent centring, but is not likely to be of much use in these kingdoms, because the arch itself will be considered as ungraceful and ugly, looking like a huge lintel. Perronet says that he preferred it to the ellipse, because it was lighter on the piers, which were thin. But the failure of one arch must be immediately followed by the ruin of all. We know much better methods of lightening the piers. Fig. 6 represents the centring of the bridge of Neuilly, near Paris, also by Perronet. The arch has 120 feet span and thirty feet rise, and is five feet thick. The frames are six feet apart, and each carries an absolute (that is, not reduced to 1/3 or to 1/4) load of 350 tons. The strut-beams are seventeen by fourteen inches in scantling. The king-posts are fifteen by nine each half; and the horizontal bridles, which bind the different frames together in five places, are also fifteen by nine each half. There are eight other horizontal binders of nine inches square.

This is one of the most remarkable arches in the world; not altogether on account of its width, for there are several much wider, but for the flatness at the crown; for about twenty-six feet on each side of the middle it was intended to be a portion of a circle of 150 feet radius. An arch (semicircular) of 300 feet span might therefore be easily constructed, and would be much stronger than this, because its horizontal thrust at the crown would be vastly greater, and would keep it more firmly united.

The bolts of this centre are differently placed from those of the former, and the change is judicious. M. Perronet had doubtless found by this time that the stiffness of his framing depended on the transverse strength of the beams, and therefore he was careful not to weaken them by the bolts. But notwithstanding all his care, the framing sunk upwards of thirteen inches before the keystones were laid; and during the progress of the work, the crown rose and sunk, by various steps, as the loading was extended along it. When twenty courses were laid on each side, and about sixteen tons laid on the crown of each frame, it sunk about an inch. When forty-six courses were laid, and the crown loaded with fifty tons, it sunk about half an inch more. It continued sinking as the work advanced; and when the keystone was set it had sunk thirteen inches and a quarter. But this sinking was not general; on the contrary, the frame had risen greatly at the very haunches, so as to open the upper part of the joints, many of which gaped an inch; and this opening of the joints gradually extended from the haunches towards the crown, in the neighbourhood of which they opened on the under side. This evidently arose from a want of stiffness in the frame. But these joints closed again when the centres were struck, as will be mentioned afterwards.

We have taken particular notice of the movements and twisting of this centre, because we think that they indicate a deficiency, not only of stiffness, but of abutment among the truss-beams. The whole has been too flexible, because the angles are too obtuse; this arises from their multiplicity. When the intercepted arches have so little curvature, the power of the load to press it inward increases very fast. When the intercepted arch is reduced to one half, this power is more than doubled; and it is also doubled when the radius of curvature is doubled. The king-posts should have been farther apart near the crown, so that the quantity of arch between them should compensate for its diminished curvature.

The power of withstanding any given inequality of load would therefore have been greater had the centre consisted of fewer pieces, and their angles of meeting been proportionally more acute. The greatest improvement would have been to place the foot of the lower tier of truss-beams on the very foot of the pier, and to have also separated it at the head from the rest with a longer king-post, and thus to have made the distances of the beams on the king-posts increase gradually from the crown to the spring. This would have made all the angles of abutment more acute, and would have produced a greater pressure on all the lower tiers when the frames sagged.

Fig. 7 represents the centring of the bridge of Orleans. The arch has 100 feet span, and rises thirty, and the arch-stones are six feet long. It is the construction of M. Hupeau, the first architect of the bridge. It is the boldest work of the kind that we have seen, and is constructed on clear principles. The main abutments are few in number. Because the beams of the outer polygon are long, they are very well supported by straining beams in the middle; and the struts or braces which support and butt on them are made to rest on points carried entirely by ties. The inventor, however, seems to have thought that the angles of the inner polygon were supported by mutual compression, as in the outer polygon. But it is plain that the whole inner polygon may be formed of iron rods. Not but that both polygons may be in a state of compression; this is very possible; but the smallest sagging of the frame will change the proportions of the pressures at the angles of the two polygons. The pressures on the exterior angles will increase, and those on the lower or interior angles will diminish most rapidly; so that the abutments in the lower polygons will be next to nothing. Such points could bear very little pressure from the braces which support the middle of the long bearings of the upper beams, and their pressures must be borne chiefly by the joints supported by the king-posts. The king-posts would then be in a state of extension. It is difficult, however, to decide what is the precise state of the pressure at these interior angles.

The history of the erection of this bridge will throw much light on this point, and is very instructive. M. Hupeau died before any of the arches were carried farther than a very few of the first courses. M. Perronet succeeded to the charge, and finished the bridge. As the work advanced, the crown of the frame rose very much. It was loaded; and it sunk as remarkably. This showed that the lower polygon was giving very little aid. M. Perronet then thought the frame too weak, and inserted the long beam DE, making the diagonal of the quadrangle, and very nearly in the direction of the lower beam ab, but falling rather below this line. He now found the frame abundantly strong. It is evident that the truss is now changed exceedingly, and consists of only the two long sides, and the short straining beam lying horizontally between their heads. The whole centring consists now of one great truss, a E e b, and its long sides, a E, e b, are trussed up at B and f. Had this simple idea been made the principle of the construction, it would have been excellent. The angle a DE might have been about 176°, and the polygon D c g h employed only for giving a slight support to this great angle, so as not to allow it to exceed 180°. But M. Perronet found that the joint c, at the foot of the post E e, was about to draw loose, and he was obliged to bolt long pieces of timber on each side of the joint, embracing both beams. These were evidently acting the same part as iron straps would have done; a complete proof that, whatever may have been the original pressures, there was no abutment now at the point c, and that the beams that met there were not in a state of compression, but were on the stretch. M. Perronet says that he put these cheeks to the joints to stiffen them. But this was not their office, because the adjoining beams were not struts, but ties, as we have now proved.

We may therefore conclude that the outer polygon, with the assistance of the pieces a b, DE, were carrying the whole load. We do not know the distance between the frames; but supposing them seven feet apart, and the arch six feet thick, and weighing 170 pounds per foot, we learn the load. The beams were sixteen inches square. If we now calculate what they would bear at the same very moderate rate allowed to the other centres, we find that the beams AB and a b are not loaded to one sixth of their strength. We have given this centre as a fine example of what carpentry is able to perform, and because, by its simplicity, it is a sort of text on which the intelligent artist may make many comments. We may see plainly that, if the lower polygon had been formed of iron rods, firmly bolted into the feet of the king-posts, it would have maintained its shape completely. The service done by the beam DE was not so much an increase of abutment as a discharge of the weight and of the pull at the joint c. Therefore, in cases where the feet of the truss are necessarily confined to a very narrow space, we should be careful to make the upper polygon sufficient to carry the whole load, say by doubling its beams, and we may then make the lower polygon of slender dimensions, provided we secure the joints on the king-posts by iron straps which embrace a considerable portion of the tie on each side of the joint.

We are far from thinking that these centres are of the best kind that could be employed in their situation; but they are excellent in their kind, and a careful study of them will teach the artist much of his profession. When we have a clear conception of the state of strain in which the parts of a frame really are, we know what should be done in order to draw all the advantages possible from our materials. We have said in another place, that where we can give our joints sufficient connection, as by straps and bolts, or by cheeks or fishes, it is better to use ties than struts, because ties never bend.

We do not approve of M. Perronet's practice of giving his trusses such narrow feet. By bringing the foot of the lower polygon farther down, we greatly diminish all the strains, and throw more load on the lower polygon; and we do not see any of M. Perronet's centres where this might not have been done. He seems to affect a great span, to show the wonders of his art; but our object is to teach how to make the best centre of a given quantity of materials, and how to make the most perfect centre, when we are not limited in this respect; nor in the extent of our fixed points.

We shall conclude this series of examples with one where no such affectation takes place. This is the centring of the bridge at Blackfriars, London. The span of the arch is one hundred feet, and its height from the spring is about forty-three. The drawing, fig. 8, is sufficiently minute to convey a distinct notion of the whole construction. We need not be very particular in our observations, after what has been said on the general principles of construction. The leading maxim, in the present example, seems to be, that every part of the arch shall be supported by a simple truss of two legs, resting, one on each pier. H, H, &c., are called apron-pieces, to strengthen the exterior joints, and to make the ring as stiff in itself as possible. From the ends of this apron-piece proceed the two legs of each truss. These legs are twelve inches square; they are not of an entire piece, but of several, meeting in firm abutment. Some of their meetings are secured by the double king-posts, which grasp them firmly between them, and are held together by bolts. At other intersections, the beams appear halved into each other, a practice which cannot but weaken them much, and would endanger their breaking by cross strains, if it were possible for the frame to change its shape. But the great breadth of this frame is an effectual stop to any such change. The fact was, that no sinking or twisting whatever was observed during the progress of the mason-work. Three points in a straight line were marked on purpose for this observation, and were observed every day. The arch was more than six feet thick, and yet the sinking of the crown, before setting the keystones, did not amount to one inch.

The centre employs about one third more timber than Perronet's great centre in proportion to the span of the arch; but the circumference increases in a greater proportion than this, because it is more elevated. In every way of making a comparison of the dimensions, Mr Mylne's arch employs more timber; but it is beyond all comparison stronger. The great elevation is partly the reason of this. But the disposition of the timbers is also much more advantageous, and may be copied even in the low pitched arches of Neuilly. The simple truss, reaching from pier to pier for the middle point of the arch, gives the strong support where it is most of all wanted; and in the lateral points H, although one leg of the truss is very oblique, the other compensates for it by its upright position.

The chief peculiarity of this centre is to be seen in its base. This demands a more particular attention; but we must first make some observations on the condition of an arch, as it rests on the centring after the keystones are set, and on the gradual transference of the pressure from the boards of the centring to the joints of the arch-stones.

While all the arch-stones lie on the centring, the lower courses are also leaning pretty strongly on each other. But the mortar is hardly compressed in the joints, and the least of all in the joints near the crown. Suppose the arch to be Catenarean, or of any other shape that is perfectly equilibrated: when the centring is gradually withdrawn, all the arch-stones follow it. Their wedge-like form makes this impossible, without the middle ones squeezing the lateral ones aside. This compresses the mortar between them. As the stones thus come nearer to each other, those near the crown must descend more than those near the haunches, before every stone has lessened its distance from the next by the same quantity, for example, by the hundredth part of an inch. This circumstance alone must cause a sinking in the crown, and a change of shape. But the joints near the crown are already more open than those near the haunches. This produces a still greater change of form before all is settled. Some masons endeavour to remedy, or at least to diminish this, by using no mortar in the joints near the crown. They lay the stones dry, and even force them together by wedges and blocks laid between the stones on opposite sides of the crown; they afterwards pour in fine cement. This appears a good practice. Perronet rejects it, because the wedging sometimes breaks the stones. We should not think this any great harm, because the fracture will make them close where they would otherwise lie hollow. But, after all our care, there is still a sinking of the crown of the arch. By gradually withdrawing the centring, the joints close, the arch-stones begin to butt on each other, and to force aside the lateral courses. This abutment gradually increasing, the pressure on the haunches of the centring is gradually diminished by the mutual abutment, and ceases entirely in that course, which is the lowest that formerly pressed it; it then ceases in the course above, and then in the third, and so on. And in this manner not only the centring quits the arch gradually, from the bottom to the top, by its own retiring from it, but the arch also quits the centring by changing its shape. If the centring were now pushed up again, it would touch the arch first at the crown; and it must lift up that part gradually, before it come again in contact with the haunches. It is evident, therefore, that an arch built on a centre of a shape perfectly suited to equilibration, will not be in equilibrium when the centring is removed. It is therefore necessary to form the centring in such a manner, by raising the crown, that it shall leave the arch of a proper form. This is a very delicate task, requiring a previous knowledge of the ensuing change of form. This cannot be ascertained by the help of any theory we are acquainted with. But suppose this attained, there is another difficulty. While the work advances, the centring is warped by the load laid on it, and continually increasing on each side. The first pressure on the centring forces down the haunches, and raises the crown. The arch is therefore less curved at the haunches than is intended; the joints, however, accommodate themselves to this form, and are close, and filled with mortar. When the masons approach the middle of the arch, the frame sinks there; and rises up at the haunches. This opens all the joints in that place on the upper side. By the time that the keystones are set, this warping has gone farther, and joints are opened on the under side near the crown. It is true, we are here speaking rather of an extreme case, when the centring is very flexible; but this occurred to M. Perronet in the two great bridges of Neuilly and of Mantz. In this last one the crown sunk above a foot before the key was set, and the joints at the haunches opened above an inch above, while some nearer the crown opened near a quarter of an inch below.

In this condition of things, it is a delicate business to strike the centring. Were it removed in an instant, all would probably come down; for the arch-stones are not yet abutting on each other, and the joints in the middle are open below. M. Perronet's method appears to us to be very judicious. He began to detach the centring at the very bottom, on each side equally, where the pressure on the centring is very slight. He cut away the blocks which were immediately under each arch-stone. He proceeded gradually upwards in this way with some speed, till all was detached that had been put out of shape by the bending of the centring. This, being no longer supported, sunk inward, till it was stopped by the abutment which it found on the arch-stones near the crown, which were still resting on their blocks. During part of this process, the open joints opened still more, and looked alarming. This was owing to the removal of the load from the haunches of the centring. This allowed the crown to sink still more, by forcing out the arch-stones at the haunches. He now paused some days, and during this time the two haunches, now hanging in the air, gradually pressed in toward the centring, their outer joints closing in the meanwhile. The haunches were now pressing pretty hard on the arch-stones nearer the crown. He then proceeded more slowly, destroying the blocks and bridgings of these upper arch-stones. As soon as he destroyed the support of one, it immediately yielded to the pressure of the haunch; and if the joint between it and the one adjoining toward the crown happened to be open, whether on the under or the upper side, it immediately closed on it. But in proceeding thus, he found every stone sink a little while it closed on its neighbour; and this was like to produce a ragged soffit, which is a deformity. He therefore did not allow them to sink so much. In the places of the blocks and bridgings which he had cut away, he set small billets, standing on their ends, between the centring and the arch-stones. These allowed the pendulous arch to push toward the crown without sensibly descending; for the billets were pushed out of the perpendicular, and some of them tumbled down. Proceeding in this way, he advanced to the very next course to the keystone on each side, the joints closing all the way as he advanced. The last job was very troublesome; we mean the detaching of the three uppermost courses from the centring; for the whole elasticity of the centring was now trying to unbend, and pressing hard against them. He found that they were lifted up; for the joints beyond them, which had closed completely, now opened again below; but this job was finished in one day, and the centre sprung up two or three inches, and the whole arch sunk about six inches.

This was an anxious time; for he dreaded the great momentum of such a vast mass of matter. It was hard to say where it would stop. He had the pleasure to see that it stopped very soon, settling slowly as the mortar was compressed, and after one or two days settling no more. This settling was very considerable, both in the bridge at Neuilly and in that at Mantz. In the former, the sinking during the work amounted to thirteen inches. It sunk six inches more when the blocks and bridgings were taken out, and one and a half when the little standards were destroyed, and one and a fourth more next day; so that the whole sinking of the pendulous arch was nine inches and a half, besides what it had sunk by the bending and compression of the centring.

The crown of the centring was an arch of a circle described with a radius of 150 feet; but by the sinking of the arch its shape was considerably changed, and about sixty feet of it formed an arch of a circle whose radius was 244 feet. Hence M. Perronet infers that a semicircle of 500 feet span may be erected. It would no doubt be stronger than this arch, because its greater horizontal thrust would keep the stones firmer together. The sinking of the arches at Mantz was not quite so great, but everything proceeded in the same way. It amounted in all to twenty and a half inches, of which twelve inches were owing to the compression and bending of the centring.

In fig. 5, No. 1, may be observed an indication of this procedure of the masonry. There may be noticed a horizontal line ac, and a diagonal ab. These are supposed to be drawn on the masonry as it would have stood had the frames not yielded during the building. The dotted line A'bc shows the shape which it took by the sinking of the centring. The dotted line on the other side was actually drawn on the masonry when the keystone was set; and the wavy black line on the same side shows the form which the dotted line took by the striking of the centring. The undulated part of this line cuts its former position a little below the middle, going without it below, and falling within it above. This shows very distinctly the movement of the whole masonry, distinguishing the parts that were forced out and the parts which sunk inward.

We presume that the practical reader will think this account of the internal movements of a stupendous arch very instructive and useful. As M. Perronet observed it to be uniformly the same in several very large arches which he erected, we may conclude that it is the general process of nature. We by no means have the confidence in the durability or solidity of his arches which he prudently professes to have. We have conversed with some very experienced masons, who have also erected very great arches, and in very difficult situations, which have given universal satisfaction; and we have found them uniformly of opinion, that an arch which has settled to such a proportion of its curvature as to change the radius from 150 to 244 feet, is in a very hazardous situation. They think the hazard the greater because the span of the arch is so great in proportion to its weight (as they express it very emphatically) or its height. The weight, say they, of the haunches is too small for forcing together the keystones, which have scarcely any wedgelike form to keep them from sliding down. This is very good reasoning, and expresses very familiar notions. The mechanician would say that the horizontal thrust at the crown is too small. When we questioned them about the propriety of M. Perronet's method of removing the centring, they unanimously approved of its general principle, but said that it was very ticklish indeed in the execution. The cases which he narrates were new to them. They should have almost despaired of success with arches which had gone so much out of shape by the bending of the centres; because, said they, the slope of the centring, to a great distance from the crown, was so little, that the arch-stones could not slide outwards along it; to close even the under side of the joints which had opened above the haunches; so that all the arch-stones were at too great a distance from each other; and a great and general subsiding of the whole was necessary for bringing them even to touch each other. They had never observed such bendings of the centrings which they had employed, having never allowed themselves to contract the feet of their trusses into such narrow spaces. They observed that nothing but lighters with their masts down can pass under the trusses, and that the sides must be so protected by advanced works from the accidental shock of a loaded boat, that there cannot be left room for more than one. They added, that the bridges of communication necessary for the expeditious conducting of the work made all this supposed roominess useless; besides, the business can hardly be so urgent and crowded anywhere as to make the passage through every arch indispensably necessary. Nor was the inconvenience of this obstruction greatly complained of during the erection of Westminster or Blackfriars Bridges. Nothing should come in competition with the undoubted solidity of the centring and the future arch; and all boasting display of talent and ingenuity by an engineer, in the exhibition of the wonders of his art, is misplaced here.

These appeared to us good reasons for preferring the more cautious, and incomparably more secure, construction of Mr Mylne, in which the breadth given to each base of the trusses permitted a much more effective disposition of the abutting timbers, and also enabled the engineer to make it incomparably stiffer; so that no change need be apprehended in the joints which have already closed, and in which the mortar has already taken its set, and commenced an union that never can be restored if it be once broken in the smallest degree, nor not even by greater compression.

Here we beg leave to mention our notions of the connection that is formed by mortar composed of lime or gypsum. We consider it as consisting chiefly, if not solely, in a crystallization of the lime or gypsum and water. As much water is taken up as is necessary for the formation of the crystals during their gradual conversion into mild calcareous earth or alabaster, and the rest evaporates. When the free access of air is absolutely prevented, the crystallization never proceeds to that state, even although the mortar becomes extremely dry and hard. We had an opportunity of observing this accidentally, when passing through Maestricht in 1770, while they were cutting up a massy revetment of a part of the fortifications more than three hundred years old. The mortar between the bricks was harder than the bricks (which were Dutch clinkers, such as are now used only for the greatest loads); but when mixed with water it made it lime-water, seemingly as strong as if fresh lime had been used. We observed the same thing in one small part of a huge mass of ancient Roman work near Romney, in Kent; but the rest, as well as all the very old mortar that we have seen, was in a mild state, and was generally much harder than what produced any lime-water. Now when the mortar in the joints has begun its first crystallization, and is allowed to remain in perfect rest, we are confident that the subsequent crystals, whether of lime, or of calcareous earth, or of gypsum, will be much larger and stronger than can ever be produced if they are once broken; and the farther that this crystallization has been carried, that is, the harder that the mortar has become, less of it remains to take any new crystallization. Why should it be otherwise here than in every other crystallization that we are acquainted with?

We think therefore that it is of great consequence to keep the joints in their first state if possible; and that the strength, in as far as it depends on the mortar, is greatly diminished by their opening; especially when the mortar has acquired considerable hardness, which it will do in a month or six weeks, if it be good. The cohesion given first state by mortar is indeed a mere trifle, when opposed to a force which tends to open the joints, acting, as it generally does, with the transverse force of a lever. But in situations where the overload on any particular arch-stones tends to push them down through between their neighbours, like wedges, the cohesion of the mortar is then of very great consequence.

We must make another observation. M. Perronet's ingenious process tended very effectually to close the joints. In doing this the forces which he brought into action had little to oppose them; but as soon as they were closed the contact of the parts formerly open opposed an obstruction incomparably greater, and immediately balanced a force which was but just able to turn the stone gently about the two edges in which it touched the adjoining stones. This is an important remark, though seemingly very trifling; and we wish the practitioner to have a very clear conception of it; but it would take a multitude of words to explain it. It is worth an experiment. Form a little arch of wooden blocks; and form one of these so, that when they are all resting on the centring, it may be open at the outer joint; remove the centring; then press on the arch at some distance from the open joint; you will find that a very small pressure will make the arch bend till that joint closes. Press a little harder, and the arch will bend more, and the next joint will open. Thus you will find, that by pressing alternately on each side of the open joint, that stone can easily be made to flap over to either side; and that immediately after this is done the resistance increases greatly. This shows clearly that a very moderate force, judiciously employed, will close the joints, but will not press the parts strongly together. The joints therefore are closed, but no more than closed, and are hanging only by the edges by which they were hanging while the joints were open. The arch, therefore, though apparently close and firm, is but loose and tottering. M. Perronet says that his arches were firm, because hardly a stone was observed to chip or splinter off at the edges by the settlement. But he had done everything to prevent this, by digging out the mortar from between the headers, to the depth of two inches, with saws made on purpose. But we are well informed, that before the year 1791 (twenty years after the erection) the arches at Neuilly had sunk very sensibly, and that very large splinters had flown off in several places. It could not be otherwise. The original construction was too bold; we may say needlessly and ostentatiously bold. A very gentle slope of the roadway, which would not have slackened the gallop of a ducal carriage, nor sensibly checked the laborious pull of a loaded waggon, and a proper difference in the size of the arches, would have made this wonderful bridge incomparably stronger, and also much more elegant and pleasing to the eye. Indeed it is far from being as handsome as it might have been. The ellipse is a most pleasing figure to every beholder; but this is concealed as much as possible, and it is attempted to give the whole the appearance of a tremendous lintel. It has the oppressive look of danger. It will not be of long duration. The bridge at Mantz is still more exceptionable, because its piers are tall and slender. If any one of the arches fails, the rest must fall in a moment. An arch of Blackfriars Bridge might be blown up without disturbing its neighbours.

M. Perronet mentions another mode of striking the centring, which he says is very usual in France. Every second bridging is cut out. Some time after every second of the remainder; after this every second of the remainder; and so on till all are removed. This is never practised in this country, and is certainly a very bad method. It leaves the arch hanging by a number of distant points; and it is wonderful that any arch can bear this treatment.

Our architects have generally proceeded with extreme caution. Wherever they could they supported the centring by intermediate pillars, even when it was a trussed centre, having a tie-beam reaching from side to side. The centre was made to rest, not immediately on these pillars, but on pieces of timber formed like acute wedges, placed in pairs, one above the other, and having the point of the one on the thick end of the other. These wedges were well soaked and rubbed with black lead, to make them slippery. When the centres are to be struck, men are stationed at each pair of the wedges with heavy mauls. They are directed to strike together on the opposite wedges. By this operation the whole centring descends together; or when any part of the arch is observed to have opened its joints on the upper side, the wedges below that part are slackened. The framing may perhaps bend a little, and allow that part to subside. If any part of the arch is observed to open its joints on the under side, the wedges below that part are allowed to stand after the rest have been slackened. By this process the whole comes down gradually, and as slowly as we please, and the defects of every part of the arch may be attended to. Indeed the caution and moderation of our builders have commonly been such, that few defects have been allowed to show themselves. We are but little acquainted with joints opening to the extent of two inches, and in such a case would probably lift every stone of the arch again. We have not employed trussed centring so much perhaps as we should have done; nor do we see their advantage (speaking as mere builders) over centres supported all over, and unchangeable in their form. Such centres must bend a little, and require loading on the middle to keep them in shape. Their compression and their elasticity are very troublesome in the striking of the centres in M. Perronet's manner. The elasticity is indeed of use when the centres are struck in the way now described.

These observations on the management of the internal movements of a great arch will enable the reader to appreciate all the merit of Mr Mylne's very ingenious construction. We proceed therefore to complete our description.

The gradual enlargement of the base of the piers of Mr Blackfriars Bridge enabled the architect to place a series of five posts C, C, C, C, C, one on each step of the pier, the ingenious contexture of which made it like one solid block of stone. (See Arch.) These struts were gradually more and more oblique, till the outer one formed an obtuse angle with the lowest side of the interior polygon of the truss. On the top of these posts was laid a sloping seat or beam D of stout oak, the upper part of which was formed like a zig-zag scarfing. The posts were not perpendicular to the under side of the seat. The angles next the pier were somewhat obtuse. Short pieces of wood were placed between the heads of the posts, but not mortised into them, to prevent them from slipping back. Each face of the scarf was covered with a thick and smooth plate of copper.

The feet of the truss were mortised into a similar piece F, which may be called the sole of the truss, having its lower side notched in the same manner with the upper side of D, and like it covered with copper. Between these two lay the striking wedge E, the faces of which corresponded exactly with the slant faces of the seat and the sole. The wedge was so placed that the corresponding faces touched each other for about half of their length. A block of wood was put in at the broad end or base of this wedge, to keep it from slipping back during the laying of the arch-stones. Its outer end E was bound with iron, and had an iron bolt several inches long driven into it. The head of this bolt was broad enough to cover the whole wood of the wedge within the iron ferrule.

We presume that the reader by this time foresees the use of this wedge. It is to be driven in between the sole and the seat, having first taken out the block at the base of the wedge. As it advances into the wider spaces, the whole truss must descend, and be freed from the arch; but it will require prodigious blows to drive it back. Mr Mylne did not think so, founding his expectation on what he saw in the launching of great ships, which slide very easily on a slope of ten or twelve degrees. He rather feared, that taking out the block behind would allow the wedge to be pushed back at once, so that the descent of the truss would be too rapid. However, to be certain of the operation, he had prepared an abundant force in a very ingenious manner. A heavy beam of oak, armed at the end with iron, was suspended from two points of the centre like a battering ram, to be used in the same manner. Nothing could be more simple in its structure, more powerful in its operation, or more easy in its management. Accordingly the success was to his wish. The wedge did not slip back of itself; and very moderate blows of the ram drove it back with the greatest ease. The whole operation was over in a very few minutes. The spectators had suspected that the space allowed for the recess of the wedge was not sufficient for the settlement of the arch, but the architect trusted to the precautions he had taken in its construction. The reader, by turning to the article Arch, will see that there was only the arch LY (Plate LIII., fig. 1) which could be expected to settle; accordingly, the recess of the wedge was found to be much more than was necessary. However, had this not been the case, it was only necessary to take out the pieces between the posts below the seat, and then to drive back the heads of the struts; but this was not needed, we believe, in any of the arches. We are well assured that none of the arches sunk an inch and a half. The great arch of one hundred feet span did not sink one inch at the crown. It could hardly be perceived whether the arch quitted the centring gradually or not, so small had been the changes of shape. We have no hesitation in saying, that, if we except some waste of great timber by uncommon joggling, the whole of this performance is the most perfect of any that has come to our knowledge.

The subject which we have been considering is very closely connected with the construction of wooden bridges. These are not always constructed on the sole principles of equilibrium by means of mutual abutment. They are stiff frames of carpentry, where, by a proper disposition, beams are put into a state of extension, as well as of compression, so as to stand in place of solid bodies as big as the bridge.

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1 The writer of this article can only say, that, after much inquiry, he has no information of any arch being received from the builder as sufficient that had suffered half the change of shape mentioned by M. Perronet. The arch of Dublin Bridge, built by an excellent, but a very private mason, Mr Steven, is 105 feet wide, with only twenty-two feet rise. It was erected, but not on a trussed centring, without changing one full inch in its elevation; and when the centring was removed, it sunk only 1/16th inch, and about half an inch more when the parapets were added and the bridge completely finished.

2 The reader will find a representation of the centring of Waterloo Bridge in Plate CXXXIV., art. Bridge. spaces which the beams inclose; and thus we are enabled to couple two, three, or four of these together, and set them in abutment with each other like mighty arch-stones. We shall close this article, therefore, with two or three specimens of wooden bridges, disposed in a series of progressive composition, so as to serve as a sort of introduction to the art in general, and furnish a principle which will enable the intelligent and cautious artist to push it with confidence as far as it can go.

The general problem is this. Suppose that a bridge is to be thrown over the space AB (fig. 9), and that this is too wide for the strength of the size of timber which is at our command; how may this beam AB be supported with sufficient effect? There are but two ways in which the middle point C, where the greatest strain is, can be supported: 1. It may be suspended by two ropes, iron rods, or wooden ties, DC, EC, made fast to two firm points D, E above it; or it may rest on the ridge of two rafters dC, eC, which rest on two firm points d, e, below it. 2. It may be supported by connecting it with a point so supported; and this connection may be formed, either by suspending it from this point, or by a post resting on it. Thus it may hang, by means of a rod or a king-post FC, from the ridge F of two rafters AF, BF; or it may rest on the strut Cf, whose lower extremity f is carried by the ropes, rods, or wooden ties Af, Bf.

Whichever of these methods we employ, it follows, from the principles of carpentry, that the support given to the point C is so much the more powerful, as we make the angle DCE, or dCe, or the equivalent angles AFB, or AfB, more acute.

Each of these methods may be supposed equally strong. Our choice will depend chiefly on the facility of finding the proper points of support D, E, d, e; except in the second case, where we require no fixed points but A and B. The simple forms of the first case require a great extent of figure. Very rarely can we suspend it from points situated as D and E. It is even seldom that we have depth enough of bank to allow the support of the rafters dC, eC, but we can always find room for the simple truss AFB. This, therefore, is the most usually practised.

In the construction, we must follow the maxims and directions prescribed in the article Carpentry and the article Roof. The beams FA, FB, must be mortised into AB in the firmest manner, and there secured with straps and bolts; and the middle must hang by a strap attached to the king-post FC, or to the iron rod that is used for a king-post. No mortising in the point C must be employed; it is unnecessary, and it is hurtful, because it weakens the beam, and because it lodges water, and soon decays by rot. The best practice is not to suspend the beam immediately by this strap, but to let it rest, as in fig. 10, on a beam C, which crosses the bridge below, and has its other end supported in the same manner by the other truss.

It is evident that the length of the king-post has no effect on the support of C. We may therefore construct everything, and preserve the same strength of support, by finding two points a and b (fig. 11) in the banks, at a moderate distance below A and B, and setting up the rafters aF, bF, and suspending C from the shortened king-post. In this construction, when the beam AB rests on a cross bearer, as is drawn here, the struts aF, bF, are kept clear of it. No connection between them is necessary, and it may be hurtful, by inducing cross strains on both. It will, however, greatly increase the stiffness of the whole. This construction may safely be loaded with ten times the weight that AB can carry alone.

Suppose this done, and that the scantling of AB is too weak for carrying the weight which may be brought on the parts AC, CB. We may now truss up each half; as Centre in fig. 12, and then the whole will form a handsome bridge, of the simplest construction possible. The intersections of the secondary braces with those of the main truss will form a hand-rail of agreeable figure.

We are not confined to the employment of an entire piece AB, nor to a rectilineal form. We may frame the bridge as in fig. 13, and in this form we dissuade from allowing any connection with the middle points of the main braces. This construction also may be followed till each beam AC and CB is loaded to ten times what it can safely bear without the secondary trussing.

There is another way by which a bridge of one beam Another may be supported beyond the power of the first and simplest construction. This is represented in fig. 14 and fig. 15. The truss-beam FG should occupy one third of AB. The advantage of this construction is very considerable. The great elevation of the braces, which is a principal element of the strength, is preserved, and the braces are greatly shortened.

This method may be pushed still farther, as in fig. 16. And all these methods may be combined, by joining the constructions of fig. 14 and fig. 15 with that of fig. 16, thus combining skill in the proper adjustment of the scantling of the timber, and the obliquity of the braces to the lengths of the different bearings. A very oblique strut, or a slender one, will suffice for a small load, and may often give an opportunity to increase the general strength; while the great timbers and upright supports are reserved for the main pressures. Nothing will improve the composition so much as reflecting progressively, and in the order of these examples, on the whole. This alone can preserve the great principle in its simplicity and full energy.

These constructions are the elements of all that can be done in the art of building wooden bridges, and are to be found more or less obviously and distinctly in all attempts that can be made of this kind. We may assert, that the more obviously they appear, the more perfect the bridge will be. It is astonishing to what extent the principle may be carried. We have seen a bridge of forty-two feet span formed of two oak trusses, the biggest timber of which did not exceed six inches square, bearing with perfect steadiness and safety a waggon loaded with more than two tons, drawn by four stout horses. It was framed as fig. 16 nearly, with the addition of the dotted lines, and was near thirty years old, protected, however, from the weather by a wooden roof, as many bridges in Germany are.

We recollect another in the neighbourhood of Stettin, which seemed constructed with great judgment and spirit. It had a carriage-road in the middle, about twenty feet, we think, wide, and on each side a footway about five feet wide. The span was not less than sixty feet, and the greatest scantling did not appear to exceed ten inches by six.

This bridge consisted of four trusses, two of which formed the outside of the bridge, and the other two made the separation between the carriage-road and the two footways. We noticed the construction of the trusses very particularly, and found it similar to the last, except in the middle division of the upper truss, which, being very long, was double trussed, as in fig. 17.

The reader will find in that volume of Leupold's Theatrum Machinarum, which he calls Theatrum Pontificum, many specimens of wooden bridges, which are very frequent in the champaign parts of Germany. They are not, in general, models of mechanic art; but the reflecting reader, who considers them carefully, will pick up here and there subordinate hints, which are ingenious, and may sometimes be useful. What we have now exhibited are not to be considered as models of construction, but as elementary examples and lessons, for leading the reader systematically into a thorough conception of the subject.

We cannot quit the subject without taking notice of a very wonderful bridge at Wittingen in Switzerland, slightly described by Mr Coxe (Travels, vol. i. 132). It is of a construction more simple still than the bridges we have been describing. The span is 230 feet, and it rises only twenty-five. The sketch (fig. 18) will make it sufficiently intelligible. ABC is one of two great arches, approaching to a Catenarian shape, built up of seven courses of solid logs of oak, in lengths of twelve or fourteen feet, and sixteen inches or more in thickness. These are all picked of a natural shape, suited to the intended curve; so that the wood is nowhere cut across the grain to trim it into shape. These logs are laid above each other, so that their abutting joints are alternate like those of a brick wall; and it is indeed a wooden wall simply built up, by laying the pieces upon each other, taking care to make the abutting joints as close as possible. They are not fastened together by pins or bolts, or by scarfings of any kind. They are, however, held together by iron straps, which surround them, at the distance of five feet from each other, where they are fastened by bolts and keys.

These two arches having been erected (by the help, we presume, of pillars, or a centring of some kind), and well butted against the rock on each side, were freed from their supports, and allowed to settle. They are so placed that the intended road abc intersects them about the middle of their height. The roadway is supported by cross joists, which rest on a long horizontal summer beam. This is connected with the arches on each side by uprights bolted into them. The whole is covered with a roof, which projects over the arches on each side to defend them from the weather. Three of the spaces between these uprights have struts or braces, which give the upper work a sort of trussing in that part.

This construction is simple and artless; and appears, by the attempt to truss the ends, to be the performance of a person ignorant of principle, who has taken the whole notion from a stone arch. It is, however, of a strength much more than adequate to any load that can be laid on it. Mr Coxe says, but does not explain how, that it is so contrived that any part of it can be repaired independently of the rest. It was the last work of one Ulrich Grubenhamm of Tuffen, in the canton of Appenzell, a carpenter without education, but celebrated for several works of the same kind; particularly the bridge over the Rhine at Schafhausen, consisting of two arches, one of 172 and the other of 193 feet span, both resting on a small rock near the middle of the river.

While writing this article, we got an account of a wooden bridge erected in North America, in which this simple notion of Grubenhamm's is mightily improved. The span of the arch was said to exceed 250 feet, and its rise exceedingly small. The description we got is very general, but sufficient, we think, to make it perfectly intelligible.

In fig. 19, DD, EE, FF, are supposed to be three beams of the arch. They consist of logs of timber of small lengths, suppose of ten or twelve feet, such as can be found of a curvature suited to its place in the arch without trimming it across the grain. Each beam is double, consisting of two logs applied to each other side to side, and breaking joint, as the workmen term it. They are kept together by wedges and keys driven through them at short intervals, as at K, L, &c.

The manner of joining and strongly binding the two side pieces of each beam is shown in fig. 20. The mortise ai eb and de io, which is cut in each half beam, is considerably longer on the outside than on the inside, where the two mortises meet. The two keys, BB and CC, are formed, each with a notch b cd, or ai o, on its side; which notch fits one end of the mortise. The inner side of the key is straight, but so formed that when both keys are in their places, they leave a space between them wider at one end than at the other. A wedge AA, having the same taper as the space just mentioned, is put into it and driven hard. It is evident that this must hold the two logs firmly together.

This is a way of uniting timber not mentioned in the article Carpentry, and it has some peculiarities worthy of notice. In the first place, it may be employed so as to produce a very strong lateral connection, and would then co-operate finely with the other artificial methods of scarfing and tabling that we described in the article referred to. But it requires nice attention to some circumstances of construction to secure this effect. If the joints are accurately formed to each other, as if the whole had been one piece divided by an infinitely thin saw, this manner of joining will keep them all in their places. But no driving of the wedge AA will make them the firmer, or cause one piece to press hard on the other. If the abutment of two parts of the half beam is already close, it will remain so; but if open in the smallest degree, driving of the wedge will not make it tighter. In this respect, therefore, it is not so proper as the forms described in Carpentry.

In order that the method now described may have the effect of drawing the halves of the beams together, and of keeping them hard squeezed on each other, the joints must be made so as not to correspond exactly. The prominent angle ai o (fig. 21), formed by the ends of the two half mortises, must be made a little more obtuse than the angle af o of the notch of the key which this prominence is intended to fill up. Moreover, the opposite side et of this key should not be quite straight, but a little convex. With these precautions, it is easy to see that, by driving the wedge AA, we cause the notch af o to take hold, first at the two points a and o, and then, by continuing to drive the wedge, the sides af, af, of the notch gradually compress the wood of the half beams, and press them on each other. By continuing to drive the wedge, the mutual compression of the key and the beam squeezes all together, and the space af o is completely filled up. We may see, from this process, that the mutual compression and drawing together of the timber will be greater in proportion as we make the angle ai o more prominent, and its corresponding angle af o more deep; always taking care that the key shall be thick enough not to break in the narrow part.

This adjustment of the keys to the mortise is necessary on another account. Supposing the joints to fit each other exactly before driving the wedge, and that the whole shrinks a little by drying—by this the angle ai o will become more prominent, and the angle af o will become more shallow; the joint will open at a and o, and the mutual compressure will be at an end.

We may also observe that this method will not give any additional firmness to the abutments of the different lengths employed to piece out the arch-beam; in which respect it differs materially from the other modes of joining timber.

Having shown how each beam is pieced together, we must now show how a number of them are united, so as to compose an arch of any thickness. This is done in the very same way. The beams have other mortises worked out of their inner sides, half out of each half of the beam. The ends of the mortises are formed in the same way with those already described. Long keys, BB, CC (fig. 19), are made to fit them properly, the notches being placed so as to keep the beams at a proper distance from each other. It is now plain that driving in a long wedge AA will bind all together.

In this manner may an arch be extended to any span, and made of any thickness of arching. The bridge over Portsmouth river in North America was more than 250 feet in length, and consisted of several parallel arches of beams. The inventor (we think his name is Bladet) said that he found the strength so great that he could with perfect confidence make one of four times the span.

We admire the ingenuity of this construction, and think it very effectual for bringing the timbers into firm and uniform abutment; but we imagine that it requires equilibration, because it is extremely flexible. There is nothing to keep it from bending, by an inequality of load, but the transverse strength of the beams. The keys and wedges can have very little power to prevent this bending. The distance between the beams will also contribute little or nothing to the stiffness; nay, we may imagine that a great distance between them will make the frame more flexible. Could the beams be placed so near each other that they could be somehow juggled on each other, the whole would be stiffer; but at present they will bend like the plates of a coach-spring. But nothing hinders us from adding diagonal pieces to this construction, which will give it any degree of stiffness, and will enable it to bear any inequality of loading. When completed in this manner, we imagine that it will be at least equal to any construction that has yet been thought of. One advantage it possesses that is very precious; any piece that fails may be taken out, and replaced by another, without disturbing the rest, and without the smallest risk. On the whole, we think it a very valuable addition to British carpentry. The method here practised, both for joining the parts of one beam and for framing the different beams together, suggests the most firm and light constructions for dome-roofs that can be conceived; incomparably superior to any that have yet been erected. The whole may be framed, without a nail or a spike, into one net-like shell, that cannot even be pulled in pieces.

When the width of the river exceeds what is thought practicable by a single truss, we must then combine, either by simple addition or by composition, different trusses together. We compose a bridge by simple addition when we make a frame of carpentry of an unchangeable and proper shape, to serve as one of the arch-stones of a bridge of masonry. This may easily be comprehended by looking at fig. 22. Each of the frames A, B, C, D, must be considered as a separate body, and all are supported by their mutual abutment. The nature of the thing is not changed, although we suppose that the rails of the frame B, instead of being mortised into an upright b', unconnected with the frame C, is mortised into the upright e' of that frame; the direction and intensity of the mutual pressures of the two frames are the same in both cases; accordingly, this is a very common form of small wooden bridges. It is usual, indeed, to put diagonal battens into each; but we believe that this is more frequently done to please the eye than to produce an unalterable shape of each frame.

To an unskilful carpenter this bridge does not seem essentially different from the centring of M. Hupeau for the bridge of Orleans; and indeed, in many cases, it requires reflection, and sometimes very minute reflection, to distinguish between a construction which is only an addition of frame to frame till the width be covered, from a construction where one frame works on the adjoining one transversely, pushing it in one part, and drawing it in another. The ready way for an unlettered artist to form a How to disjust notion of this point, is to examine whether he may distinguish saw through the connecting piece b' from one end to the bridges other, and make them two separate frames. Whenever formed according to this cannot be done without that part opening, it is a construction by composition. Some of the beams are on the ferent mestretch; and iron straps, extending along both pieces, are thods. necessary for securing the joint. The bridge is no longer a piece of masonry, but a performance of pure carpentry, depending on principles peculiar to that art. Equilibration is necessary in the first construction; but, in the second, any inequality of loading is made ineffectual for hurting the edifice, by means of the stretch that is made to operate on some other piece. We are of opinion that this most simple employment of the distinguishing principle of carpentry, by which the beams are made to act as ties, will give the most perfect construction of a wide bridge. One polygon alone should contain the whole of the abutments, and one other polygon should consist entirely of ties; and the beams which form the radii, connecting the angles of the two polygons, complete the whole. By confining the attention to these two simple objects, the abutments of the outer polygon, and the joints of the inner one, may be formed in the most simple and efficient manner, without any collateral connections and dependencies, which divide the attention, increase the complication, and commonly produce unexpected and hurtful strains. It is for this reason that we have so frequently recommended the centring of the bridge of Orleans. Its office will be completely performed by a truss of the form of fig. 23, where the polygon ABCDEF, consisting of two layers of beams, if one is not sufficient, contains the whole abutments, and the other A b c d e F is nothing but an iron rod. In this construction, the obtuseness of the angles of the lower polygon is rather an advantage. The braces G c, G d, which are wanted for trussing the middle of the outer beams, will effectually secure the angles of the exterior polygon against all risk of change. The reader must perceive that we have now terminated in the best general form, when some moderate declivity is not an insuperable objection. When this is the case, we recommend the general plan of the centring of the bridge of Orleans. We would make the bridge (we speak of a great bridge) consist of four trusses, two to serve as the outsides of the bridge, and two inner trusses, separating the carriage-way from the footpaths. The road should follow the course of the lower polygon, and the main truss should form the rails. It might look strange, but we are here speaking of strength; and evident, but not unwieldy strength, once it becomes familiar, is the surest source of beauty in all works of this kind.

(B. B. B.)

CENTRESIMA USURA means a rate of interest which in a hundred months became equal to the principal, that is, where the money is laid out at one per cent. per month, answering to what in our style would be called twelve per cent. The Romans reckoned their interest not by the year, but by the month.