upon it; also let G' D be a perpendicular passing through a', and F E drawn from F parallel to it. Then

2. PROP. If the arch fF F'f' tend to fall vertically in one piece, removing the sections ƒ F, f' F'; if a be the weight of the semi-arch F K k, and P that of the pier up to the joint f F, the equilibrium will be determined by these two equations: viz.

where ƒ is the measure of the friction, or the tangent of the angle of repose of the material, and the first equation is that of the equilibrium of the horizontal thrusts, while the latter indi. cates the equilibrium of rotation about the exterior angle a of the pier.

3. PROP. If each of the two semi-arches r k, k, r', tend to turn about the vertex k of the arch, removing the points F, F', the equilibrium of horizontal translation, and of rotation, will be respectively determined by the following equations: viz.

4. Hence it will be easy to examine the stability of any cemented arch, upon the hypothesis of these two propositions. Assume different points, such as F in the arch, for which let the numerical values of the equations (1) and (2), or (3) and (4), be computed. To ensure stability, the first members of the respective equations must exceed the second; those parts will be weakest, where the excess is least.

If the figure be drawn on a smooth drawing pasteboard, upon a good sized scale, the places of the centres of gravity may be found experimentally, as well as the relative weights of the semi-arch and piers, and the measures of the several lines from the scale employed in the construction.

If the dimensions of the arch were given, and the thickness of the pier required, the same equations would serve; and different thicknesses of the pier might be assumed until the first members of the equations come out largest.

The same rules are applicable to domes, simply taking the ungulas instead of the profiles.

Models.

From an experiment made to ascertain the firmness of the model of a machine, or of an edifice, certain precautions are necessary before we can infer the firmness of the structure itself.

The classes of forces must be distinguished; as whether they tend to draw asunder the parts, to break them transversely, or crush them by compression. To the first class belongs the stretching suffered by key-stones, or bonds of vaults, &c. to the second, the load which tends to bend or break horizontal or inclined beams; to the third the weight which presses vertically upon walls and columns.

PROB. 1. If the side of a model be to the corresponding side of the structure, as 1 to n, the stress which tends to draw asunder, or to break transversely the parts, increases from the smaller to the greater scale, as 1 to n3; while the resistance of those ruptures increases only as 1 to no.

The structure, therefore, will have so much less firmness than the model as n is greater.

If w be the greatest weight which one of the beams of the model can bear, and w the weight or stress which it actually sustains, then the limit of n will be n =

W

--

พ

PROB. 2. The side of the model being to the corresponding side of the structure as 1 to n, the stress which tends to crush the parts by compression, increases from the smaller to the greater scale, as 1 to n3, while the resistance increases only in the ratio of 1 to n.

Hence, if w were the greatest load which a modular wall or column could carry, and w the weight with which it is actually loaded; then the greatest limit of increased dimensions would be found from the expression n =

W

--

w

If, retaining the length or height n h, and the breadth n b, we wished to give to the solid such a thickness xt, as that it should not break in consequence of its increased dimensions, we should have x = n2 ✔

W

In the case of a pilaster with a square base, or of a cylin

drical column, if the dimension of the model were d, and of the largest pillar, which should not crash with its own weight when n times as high, x d, we should have

These theorems will often find their application in the profession of an architect or an engineer, whether civil or military.

3. Suppose, for an example, it were required to ascertain the strength of Mr. Smart's "Patent Mathematical Chain-bridge," from experiments made with a model. In this ingenious construction, the truss-work is carried across from pier to pier, so that the road-way from A to B, and thence entirely across, shall be in a horizontal plane, and all the base bars, diagonal bars, hanging bars, and connecting bolts, shall retain their own respective magnitudes throughout the structure. The annexed representation of half the bridge so exhibits the construction as to supersede the necessity of a minute verbal description.

Now, let represent the horizontal length of the model, (say 12 feet,) from interior to exterior of the two piers, w its weight (say 30 pounds), w the weight it will just sustain at its middle point B before it breaks (say 350 lbs.) Let n the length of a bridge actually constructed of the same material as the model, and all its dimensions similar: then, its weight will be n3 w, and its resisting power to that of the model, as n3 to 1, being = n2 (w + 1⁄2 w.) Hence n' (w + } w) — § n3 w = no − ǹ n2 (n − 1) w, the load which the bridge itself would bear at the middle point.

W

Suppose n = 20, or the bridge 240 feet long, and entirely similar to the model; then we shall have (400 × 350) — 200 (20 1) 30 140000 114000 26000 lbs. = 11 tons 124 cwt., the load it would just sustain in the middle point of its

extent.

=

=

Note. This bridge is, in fact, a suspension bridge, and would require brace or tie-chains at each pier. A considerable improvement upon its construction, by Colonel S. H. Long, of the American Engineers, is described in the Mechanic's Magazine, vol. xiii. or No. 368.

CHAPTER X.

DYNAMICS.

1. THE mass of a body is the quantity of matter of which it is composed.

The knowledge of the mass of a body is given to us by that property of matter which we call inertia; and which being greater or less as the mass is greater or less, we regard as an index of the mass itself.

2. Density is a word by which we indicate the comparative closeness or otherwise of the particles of bodies. Those bodies which have the greatest number of particles, or the greatest quantity of matter, in a given magnitude, we call most dense; those which have the least quantity of matter, least dense; Density and weight are regarded as correlatives; so that the heaviest bodies of a given size, are the most dense, the lightest bodies, the least dense.

Thus lead is more dense than freestone; freestone more dense than oak; oak more dense than cork.

3. When bodies are impelled by certain forces, they receive certain velocities, and move over certain spaces, in certain times. So that body, force, velocity, space, time, are the subjects of investigation in Dynamics; and in mathematical theorems, they are usually represented by the initial letters, b, f, v, s, t: or, if two or more bodies, &c. are compared, two or more corresponding letters в, b, b', v, v, v', &c. are employed in the formulæ. Gravity, which is a separate force incessantly acting, is represented by g; and momentum, or quantity of motion, by m, this being the effect produced by a body in motion.

Force is distinguished into motive and accelerative, or retardive.

4. Motive force, otherwise called momentum, or force of percussion, is the absolute force of a body in motion, &c.; and is expressed by the product of the weight or mass of matter in the body multiplied by the velocity with which it moves. But 5. Accelerative force, or retardive force, is that which respects the velocity of the motion only, accelerating or retarding it; and it is denoted by the quotient of the motive