SUPPL

SUPPL. VOL. I. Part I.

The ring or circumference consists of pieces of oak 12 inches broad and 6 thick.

The stretcher LL is 12 inches square.

The straining piece GH is also 12 by 12.

The lower struts 10 by 8.

The king post 12 by 12.

The upper struts 10 by 6.

The bridles 20 by 8.

These dimensions are French, which is about $\backslash frac\{1\}\{3\}$th larger than ours, and the superficial dimensions (by which the section and the absolute strength is measured) is almost $\backslash frac\{1\}\{3\}$th larger than ours. The cubic foot, by which the stones are measured, exceeds ours nearly $\backslash frac\{1\}\{3\}$th. The pound is deficient about $\backslash frac\{1\}\{3\}$th. 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 160 pounds per foot. Mr Pitot, by a computation (in which he has committed a mistake), says, that only $\backslash frac\{1\}\{3\}$ths of this weight is carried by the frame. We believe, however, that this is nearer the truth than Mr Couplet's assumption of $\backslash frac\{2\}\{3\}$ths already mentioned.

Mr Pitot farther 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 707520 pounds, which he reduces to $\backslash frac\{1\}\{3\}$ths, or 555908 pounds.

The absolute force of each of the lower struts is 576000 (at 7200 per inch), and that of the curves 518400. Mr 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 struts can safely bear by the rule delivered in the article CARPENTRY. We therefore proceed as follows:

Measure off by a scale of equal parts $as,\; at$, each 576000, and add $t$ to 518400. Complete the parallelogram $avxs$, and draw the vertical $xc$, meeting the horizontal line $aC$ in $c$. Make $cb$ 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 $av$ is the united force of the curve AP and the strut IE, and $as$ is the strength of IG. These two are equivalent to $ax$. $xb$ 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 = 2850000 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 5 inches by 4. Even this would have carried twice the weight, and

B b

would

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 432000, and that of the curve is 518400; therefore, having drawn $Mv$ parallel to the strut $ON$, make $Mv\; =\; 432000$, and $Mv\; =\; 432000\; +\; 518400$. Complete the parallelogram $Mvrv$. 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 $=\; 1160000$. 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. Mr 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 of 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 $F$. 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 bridge. 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) is constructed on the same principle precisely for an elliptical arch ($\backslash Delta$). The calculation of its strength is nearly the same also; only the two upper struts of a side being parallel, the parallelogram $Mvrv$ (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 $Mv$ of fig. 1.—$N.\; B.$ The calculation proceeds on the supposition that the short straining piece $MM$ makes but one firm body with the king post. Mr Pitot employed this piece (we pre-

fume) 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 Mr 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 $CZ\; =\; \backslash frac\{1\}\{2\}\; CY$. Describe the How to
semicircle $ZAEY$, and make $ZS\; =\; ZAE$. $S$ is the centre constructed
of the side arches, each of 60 degrees. The centre $T$ of such an
the arch, which unites these two, is at the angle of an
equilateral triangle $STS$.

This construction of Mr Pitot's makes a handsome oval, and very near an ellipse, 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, 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 chord $PD$, and if a line $PRr$ 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 ellipse? The most likely method for this is to find the very point $P$ where the ellipse cuts the circle $ADbe$. The easiest way for the artist is to describe an arch of a circle $a\; m$, having $AB$ for its radius, and the remote focus $f$ for its centre. Then set one foot of the compasses on any point $P$, and try whether the distance $PF$ from the nearest focus $F$ is exactly equal to its distance $Pm$ 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 $DPr\; =\; PDC$, 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 $Dd$ parallel to $AB$, cutting the circle in $d$. Draw $ed$, cutting $AC$ in $N$. Draw $CG$ parallel to $Ae$, and make the

the angle $CGi\; =\; ADa$. Bisect $CN$ in $O$, and join $Oi$. Make $OM,\; OM\text{'}\; =\; Oi$, and draw $MP,\; MP\text{'}$ perpendicular to $AB$. These ordinates will cut the circle $ADbe$ in the points $P$ and $P\text{'}$, where it is cut by the ellipse. We leave the demonstration as a geometrical exercise for the dilettante.

We said, that this centering of Mr 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 $agbb$ 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 $gib$ of any use, if the king post above it be strapped to the tie-beam and straining fill. Perhaps the inventor considered the king post as a pillar, and wished to secure the tie-beam against its cross strain. This centering, however, must be allowed to be very well composed; and we expect that the well-informed reader will join us in preferring it to Mr Pitot's, both for simplicity of principle, for scientific propriety, and 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 centering 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 archstones will close on the haunches, and will almost relieve the lower centering, 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 Mr Perronet, a very celebrated French architect.

Mr 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 case, 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 equili-

brium). 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. Mr Perronet does not assume the invention to himself; but says, that it was invented and practised by Mr Maniard 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 centering employed for the Centering bridge of Cravant. The arches are elliptical, of 60 feet span and 20 feet rise. The archstones are four feet thick, and weigh 176 pounds per foot. The truss-beams were from 15 to 18 feet long, and their section was 9 inches by 8. Each half of the king posts was about 7 feet long, and its section 9 inches by 8. The whole was of oak. The five trusses were 5$\backslash frac\{1\}\{2\}$ feet asunder. The whole weight of the arch was 135000 lbs. which we may call 600 tons (it is 5.8). This is about 112 tons for each truss. We must allow near 90 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$\backslash frac\{1\}\{2\}$ 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 437 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 58$\backslash frac\{6\}\{10\}$ 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 centering. 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

Center. 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 re-filled.

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 centering 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 row 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 archstones on each side up to the posts $HC,\; bc$; the angles $E$ and $e$ are pressed down, and the beams $EOF,\; eoF$ push up the point $F$. This cannot rise without bending the beams $EOF,\; eoF$; because $O$ and $o$ are held down by the double king posts, which grasp the beams between them. There is there-

fore 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. Mr 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 Mr 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. Mr Perronet was obliged to load the crown of the centering 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 45 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 an arch of 90 feet span and 28 feet rise. The trusses were 7 feet apart, and the arch was $4\backslash frac\{1\}\{2\}$ thick; so that the unreduced load on each frame was very nearly 235 tons. The scantling of the struts was 15 by 12 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.

Mr Perronet has shewn 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 shew, 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 Mr Perronet in St Maxence, the centering of the bridge of St Maxence, exhibited in fig. 5. n^o 2. than in that of Nogent, fig. 5. n^o 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 $cd$, $cd$, when the arch is nearly completed. This is an excellent centering, 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 centering of the bridge of Neully,

Neuilly, near Paris, also by Perronet. The arch has 120 feet span, and 30 feet rise, and is 5 feet thick. The frames are 6 feet apart, and each carries an absolute (that is, not reduced to $\backslash frac\{1\}\{4\}$ or to $\backslash frac\{1\}\{2\}$) load of 350 tons. The strut beams are 17 by 14 inches in scantling. The king posts are of 15 by 9 each half; and the horizontal bridles, which bind the different frames together in five places, are also 15 by 9 each half. There are eight other horizontal binders of 9 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 26 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. Mr Perronet had doubts 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 13 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 20 courses were laid on each side, and about 16 tons laid on the crown of each frame, it sunk about an inch. When 46 courses were laid, and the crown loaded with 50 tons, it sunk about half an inch more. It continued sinking as the work advanced; and when the keystone was set it had sunk 13$\backslash frac\{1\}\{2\}$ inches. 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 frame sagged.

Fig. 7. represents the centering of the bridge of Orleans. The arch has 100 feet span, and rises 30, and the arch-stones are 6 feet long. It is the construction of Mr 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 polygon 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 instructive much light on this point, and is very instructive. Mr Hupeau died before any of the arches were carried farther than a very few of the first courses. Mr 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. Mr 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 centering 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 e g b employed only for giving a slight support to this great angle, so as not to allow it to exceed 180°. But Mr Perronet found, that the joint e, at the foot of the post E c, 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

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. Mr Perronet says that he put these checks 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 6 feet thick, and weighing 170 pounds per foot, we learn the load. The beams were 16 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 trusses 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 advantage 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 checks or fishes), it is better to use ties than struts, because ties never bend.

We do not approve of Mr 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 Mr 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 centering of the bridge at Blackfriars, London. The span of the arch is 100 feet, and its height from the spring is about 43. 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 con-

struction. The leading maxim, in the present example, seems to be, that every part of the arch shall be supported by a simple truss of 1000 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 12 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 key-stones, 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 Mlyne'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 centering after the keystones are all set, and on the gradual transference of the pressure from the boards of the centering to the joints of the archstones.

While all the archstones lie on the centering, the lower courses are also leaning pretty strongly on each other. But the mortar is hardly compressed in the joints; and 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 centering is gradually withdrawn, all the archstones 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

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 centering, the joints close, the archstones begin to butt on each other, and to force aside the lateral courses. This abutment gradually increasing, the pressure on the haunches of the centering 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 centering quits the arch, gradually, from the bottom to the top, by its own retiring from it, but the arch also quits the centering by changing its shape. If the centering 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 equilibrio when the centering is removed. It is therefore necessary to form the centering 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 centering is warped by the load laid on it, and continually increasing on each side. The first pressure on the centering 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 centering is very flexible; but this occurred to Mr 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 centering. Were it removed in an instant, all would probably come down; for the archstones are not yet abutting on each other, and the joints in the middle are open below. Mr Perronet's method appears to us to be very judicious. He began to detach the centering at the very bottom, on each side equally, where the pressure on the centering is very slight. He cut away the blocks which were immediately under each archstone. 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 centering. This be-

ing no longer supported, sunk inward, till it was stopped by the abutment which it found on the archstones 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 centering. 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 centering, their outer joints closing in the meanwhile. The haunches were now pressing pretty hard on the archstones nearer the crown. He then proceeded more slowly, destroying the blocks and bridgings of these upper archstones. 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 fofset, 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 centering and the archstones. 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 the three uppermost courses from the centering: for the whole elasticity of the centering 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 13 inches. It sunk six inches more when the blocks and bridgings were taken out, and 14 when the little standards were destroyed, and 14 more next day; so that the whole sinking of the pendulous arch was 94 inches, besides what it had sunk by the bending and compression of the centering.

The crown of the centering 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 60 feet of it formed an arch of a circle whose radius was 244 feet. Hence Mr 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 every thing proceeded in the same way. It amounted in all to 204 inches, of which 12 inches were owing

Center. owing to the compression and bending of the centering.

²⁶ The foregoing observations exemplified.
In fig. 5. n^o 1. may be observed an indication of this procedure of the masonry. There may be noticed a horizontal line a c, and a diagonal a b. 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 b' c' shews the shape which it took by the sinking of the centering. 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 shews the form which the dotted line took by the striking of the centering. 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 shews 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 Mr 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 Mr Perronet's method of removing the centering, 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 centering, to a great distance from the crown, was so little, that the archstones 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 archstones 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 centerings 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 centering 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 Mynne, 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, no not even by greater compression.

²⁷ Here we beg leave to mention our notions of the connection that is formed by mortar composed of lime and 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 Maelstricht in 1770, while they were cutting up a masonry of a part of the fortifications more than 300 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 limewater, 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, and 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?

²⁸ We think therefore that it is of great consequence to keep the joints in their first state if possible; and that the strength (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 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

— particular archstones 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. Mr 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 centering, it may be open at the outer joint—Remove the centering—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 shews 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. Mr 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 every thing 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 Neully 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 mad 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