of gold or of gossamer, will form itself into precisely the same curve, and must consequently, be governed by the same laws of tension and gravity, according to the density of the substance employed in its composition.

Since every small element of the Catenarian chain is of the same weight, it is evident that the tension at each inferior point, must be less than that at the point immediately above it, so that, in strictness, the tension excited in the chain by means of its own weight, varies regularly from the points of suspension where it is greatest, to the lowest point or vertex of the curve where it is the least. It was the attentive consideration of this circumstance, that led Mr. Dredge to the discovery of the tapering chain, and a minute examination of the laws of nature, in forming the bones and ribs of animals, and the branches and leaves of plants, suggested the idea of the diagonal rods.

It would be utterly impossible in the construction of a bridge, to taper the chain in the same regular order that the diminution of tension indicates; it is therefore, the practice of the inventor, to make the chain of sufficient strength at the bases or points of support, by putting the requisite number of bars in the first links adjacent to the towers, and to diminish the number by one in each successive link, until it terminates in a single bar at the middle of the bridge. In this way he avoids the difficulty of the gradual taper, and the principle is notwithstanding, as closely imitated as the most delicate case of practice can ever require.

In consequence of a letter from the Right Honourable Lord Western to Lord Viscount Melbourne, then at the head of Her Majesty's Government, the Commissioners of Woods and Forests have been induced to notice the subject, and under their directions, five bridges on the principles of the tapering chain, have been erected in the Regent's Park, two of them 75 feet span each, over the canal at the extremities of the Zoological Gardens, and three in the interior of the park, across the ornamental waters, the largest of which has a span of 150 feet. These bridges are highly convenient for opening communications between different parts of the public promenades, and with a little attention to the

workmanship, would have furnished splendid examples of the justness of the principle which Mr. Dredge has had the good fortune to introduce to public notice.

In the course of last winter, (1842,) a bridge of large dimensions for general traffic, was thrown across the river Leven in Dumbartonshire North Briton; this bridge is 200 feet in span and 20 feet wide, with 92 feet of reverse curvature, or 46 feet on each side of the river, making in all 292 feet of suspended roadway.

The chains were begun about the beginning of January, and it was opened to the public on the 22nd of the following month, having occupied a period of seven weeks, and for the most part of very stormy and inclement weather.

The specimens now executed on this principle, are quite sufficient to prove its superiority over the old system of Catenarian practice; and we hope to hear no more of those objections to its general adoption that have been thrown out by those who, to all appearance, have suffered their minds to become stultified by sheer envy. The Continental Engineers are more liberal than those of our own country, and the inventor is daily receiving testimonials from men of the highest eminence, that cannot fail to prove extremely gratifying to his feelings.

The grand secret of the invention lies, in transmitting the strain through the road-way, to the points of suspension, by means of the oblique diagonal bars,-and this is the very thing that has elicited the strongest opposition. The tapering of the chain is admitted to be only what the nature of the curve suggests, but the obliquity of the suspending-rods is still affirmed, by men of no mean pretensions, to be totally subversive of every mechanical law, although the soundness of the principle has been attested by the immutable evidences of mathematical demonstration.

The design of what follows, is to give the general reader some idea of the mathematical principles upon which the merit of the invention rests; and to this end, it has been deemed expedient to introduce the subject with the simplest cases of tension, as in this way, the mind will be gradually prepared to receive the im

portant truth, which terminates in the formation of the tapering chain.

PROPOSITION 1.-The direct tension of any string, cord, or chain, in whatsoever manner it may be produced :

Is measured by that weight or force which will stretch it just as much, being applied at one extremity of the cord or chain, and sustained by it, while the cord itself is sustained at rest in a vertical position by the other extremity.

This is obviously the simplest case in which the tension of a cord can be exhibited, and the import of the proposition will be readily gathered from the following illustration :

Let B, fig. 1, represent a block of wood, stone, or any other immoveable object, so posited as to admit a cord or chain to hang freely in a direction perpendicular to the horizon. Into this block is fixed a hook or staple, at н, and from this hook a cord c is suspended, having a weight w attached to the lower extremity.

Fig. 1.

H

Now, since the cord c, is suffered to hang freely from the hook at H, and by its tenacity, to support the weight w, acting in the direction of gravity, it must be stretched by a force, the intensity of which is equivalent to the weight applied to it; therefore, by whatever name, or in whatever manner, the straining force is expressed, it is said to be the measure of the tension excited in the cord c.

Thus, if the magnitude or value of the weight w, be expressed in tons, the tension of the cord c, is said to be equivalent to as many tons as are contained in the weight that stretches it, the weight being always supposed to act in the direction of gravity. PROPOSITION 2.-If a string, cord, or chain, be no where attached to any thing but at its extremities :—

The tension produced by any force acting upon it, will be the same at every point in its length, disregarding the effect produced by its own gravity or weight.

The truth of what is here enunciated, is manifest from the

drawing, fig. 1; for the action of the weight w, on the cord c, is precisely the same as the re-action of the block B, in an opposite direction; and, consequently, since the action of the weight w, is transmitted to the block B, through the medium of the cord C, it follows, that every part of the communicating cord must sustain a tension equal to the weight w.

These two propositions involve the fundamental principles of tension; and the variations that are shown to exist in different systems, can only be obtained in consequence of a difference in position, or in the direction of the force by which the tension is produced.

PROPOSITION 3.-If a cord or chain be fastened by its extremities to two immoveable objects, and stretched by a weight applied at any point in it :

The tensions on the two portions of the cord, between the weight and the points of suspension, are proportional to the secants of the angles which their directions make with the plane of the horizon.

Let A P B, fig. 2, be a cord, fastened by its extremities to the immoveable objects a, and B, and let w, be a weight, suspended at the point P, through the point P, at which the weight is applied; draw the straight line m n, perpendicular to P R, and consequently parallel to the

Fig. 2.

horizon; then, by the proposition, the tension in a P, is to the tension in B P, as the secant of the angle A P m, is to the secant of the angle B Pn, these being respectively the angles which the directions of the cords A P, and в P, make with the horizon m n.

In the straight line PR, which marks the direction of the straining force, let P R, be assumed as the linear representative of the weight w; produce a P, and в P, the opposite portions of the sustaining cord, and upon PR, as a diagonal, describe the parallelogram P C RD; then is P D, the tension on the cord a P, and PC, or its equal DR, the tension on B P. Therefore, the weight w, and the tensions on A P, and в P, the opposite portions of the sustaining cord, are to each other, respectively, as P R, P D,

and RD, the sides of the triangle PD R, which, by the writers on mechanics, is denominated the triangle of forces.

Let PD, and PC, the linear equivalents of the tensions in a p, and B P, be resolved into the two forces P E, D E, and P F, DF, of which P E, and PF, are perpendicular, and D E, CF, parallel to the horizon; then, since PCR D, is a parallelogram, the forces D E, and c F, are equal to each other, but lying in opposite directions with respect to the vertical line P R, they destroy each other's effects, and thereby annihilate the horizontal force.

Since D E, is parallel to the straight line m n, the angles P DE, and RDE, are respectively equal to A P m, and в Pn; therefore, if DE, be made the radius of the circle Er s, the tensions PD, and R D, become the secants of the angles P D E, and R D E, to that radius; so that, according to the proposition, the tensions in the opposite portions of the sustaining cord, are to one another, as the secants of the angles which their directions make with a straight line, drawn parallel to the plane of the horizon.

PROPOSITION 4.-If a cord or chain be fastened by its extremities to two immoveable objects, and stretched by two weights, applied at different points in it:

The tensions in the extreme portions of the cord, are precisely the same as they would be, if a single weight equivalent to the sum of both, were applied at the point where the extended directions of the cords would intersect each other.

Let the cord A P P1 B, fig, 3, be fastened by its extremities to the immoveable objects at A, and B, and let the weights w, and w1, be suspended from the points P, and p1. Produce A P, and B P1, the extreme portions of the cord, until they meet each other in the point a; then, if the

Π

Fig. 3.

weight w, equal to the sum of the two weights w, and w1, be suspended from the point e, the tensions in AP, and в Pl, will be precisely the same as when the weights are applied separately at the points P, and p1. This is a very beautiful property, and furnishes the ground-work of the theory of tension. Since the