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forming what is called a dolphin at eacli end of the pier,flie piers and abutments should be made in all cases suffi-ciently strong to resist the thrust of the arch. In cases ofsmall foot-bridges, where even the distance between thesupports should be as much as 20 or 30 feet, longitudinalscarfed girders may be laid upon the caps of the piles. UnderSuch circumstances, as we have seen, there is nothing but theheight, or perpendicular pressure, to be provided for ; andtlie same may be said of timber-bridges of greater width, fortoads, and even for railways, provided the distance betweenthe piers does not greatly exceed 10 or 15 feet; beyond thatopening, however, bridges are usually sustained by struts ortension rods, or the roadway-timbers are trussed so as to; i exert an oblique pressure upon the supports; indeed, in all' instances of the kind, where the bays are formed upon the; principle of compression, or tension, the piers must be so\ formed as to counteract the tendency constantly exerted toJ force them out of their perpendicular position ; this must be! done either by making them of sufficient weight and strengthj so as to overcome, by their inertia, any force that may be: j exerted against them, or else to counterbalance the efforts of' i °ne bay, or arch, acting in one direction, by a similar andj similarly-acting arch, or timber-frame, exerting an equal and\! l*ke force in a contrary direction. The former of thesej! Methods is used in the abutments of a bridge, while the latteri j 18 invariably adopted with respect to piers.

! J The roadway of timber-bridges is usually a flooring ofboards laid upon the joists ; for, in cases where sand andstones are employed, it is found that their weight, together"nth the humidity they engender, causes the timbers of suchbridges speedily to decay. This, however, is far from beinga general rule ; and many splendid erections of this descrip-tion are rapidly being destroyed, owing to a want of attentionto this important particular. Some have proposed to covertbe surface of the roadway with lead, iron, copper, &c., buttbe increased expense will be a great obstacle to their fre-quent introduction : we would recommend the wood pave-ment to be laid as a covering for the roadway of all timber-bridges. The parapet or handrail of these bridges is frequentlyof wood, or it may be of cast and wrought iron ; now, how-ler, that it has been shown how important an addition totbe strength of a bridge the sides of a beam are, and that ita cts usefully in the direction of its depth, if it has onlyEfficient breadth to prevent its yielding laterally, we would8u ggest that in future it should be made available to sustaintbe bridge, in addition to its present purposes of ornamenta nd protection.

Draw Bridges of timber, made after the manner of a"°at, to draw up or let down, as occasions serve, before theSate of a town or castle, may be made in several differentm a ys, but the most common are made with plyers twice theength of the gate, and a foot in diameter. The inner space* s traversed with a cross, which serves for a counterpoise ;a "d the chains which hang from the extremities of the plyers0 lift up or let down the bridge, are of iron or brass. Suchbfidges are to be seen at Stirling and Edinburgh castles, atle Tower of London , at the castle of Vincennes, and many°ffier regularly-fortified places. In navigable rivers, it is8 °ttietime8 necessary to form the middle arch of bridges withmovable platforms, to be raised occasionally, in order toet the masts and rigging of ships pass the bridge. Therej, re two piers which support the drawbridge, one of the plat-,° r ms of which is raised, and the other let down, having a8a m for its plyer. To this drawbridge are suspended twom° v able braces, which, resting on the supports, press againsta bracket and thereby strengthen it.bt is one of the first principles of statics, that if two forces

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are represented, both in amount and direction, by the sidesof a parallelogram, then their resultant will be representedby the diagonal of such parallelogram, likewise in directionand intensity. Now, if we reduce the polygonal framing, ofwhich many wooden bridges are formed, to their simplestform, that of two beams abutting against one another, andresting upon two solid supports at their lower ends, we shallfind that, inasmuch as the forces will act in the direction ofthe beams, their weight or gravity tending to keep the endsof the beams against each other by preventing them fromrising, will be represented by a vertical line passing throughtheir points of contact, which will be the diagonal of theparallelogram formed by the beams and lines drawn parallelto them at such a point in each as would limit the length ofthe line to what would fitly represent the force acting inthe direction of the beam, and it is evident that the beamsbeing in equilibrio, their horizontal thrusts must be equaland contrary, and consequently will be neutralized by theirjoint action.

The fact is, in a polygonal framing of any kind, there arethree forces exerted at every joint; one in the direction ofeach beam, and one, a vertical force, acting by gravity, whichbeing the diagonal of the parallelogram constructed upon thebeams, is the resultant of the other two, and which, by actingas a counterpoise to the others, keeps the system in equilibrio.

To construct a polygon in any particular instance, we mustknow the direction and amount of each of the forces, andtheir points of application ; when these are given, we mayreadily construct the successive sides of a polygonal bridge.It is a well-known fact that gravity always acts vertically ;we may, therefore, take a line perpendicular to the horizon,and mark upon it a number of divisions, each one represent-ing the effect of gravity at every successive joint ; having,then, found a point at such a distance from it as will repre-sent the horizontal strains which, for a system of forces inequilibrio, are always equal, and connecting that point withthe several divisions marked on the vertical line, we shallhave a number of right lines which, both in direction andintensity, will respectively represent the force acting at eachjoint or angle of the framework. Then, by drawing linesparallel to these, placing them one after the other, we shallconstruct a polygonal frame which will be in equilibrio, bear-ing in mind that the compressions of any two sides of thepolygon are reciprocally as the sines of the angles which theyform with the vertical lines, or directly as the secants oftheir inclination to the horizon.

In order to find the resultant of two forces acting in agiven angle, we must add to the sum of the squares of theforces twice their products multiplied into the co-sine of theangle, and extract the square root of the sum, and this willgive the required resultant. Again ; to ascertain the anglebetween the resultant and one of the composants, we mustdivide the product of the sine of the given angle and thelesser force by the product of the co-sine of that angle, andthe same force to which the greater force must be added.

It has been maintained, and apparently with a great dealof truth, and several eminent men who have studied the sub-ject deeply have coincided in the opinion, that where an archis constructed of several key-stones or voussoirs, and remainsin equilibrio, its various parts being so adjusted as to cause itto remain without change of position, that a catenary or funi-cular curve will be found to be wholly included somewherebetween its intrados and its extrados. Now, as a woodenbridge, formed on the principle of compression, either by apolygonal frame or with a timber arch, should be made toconform as much to the principle of the arch as possible, soas to make it as strong as can be, with a uniform pressure at