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 x t, 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 в 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 n° to 1, being n' (w+w.) Hence n' (w+w) — } n3 w = n3 w-n' (n-1) w, the load which the bridge itself would bear at the middle point.

=

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

extent.

- 140000 114000 =

=

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

force divided by the mass or weight of the body. So, if a body of 2 lbs. weight be acted upon by a motive force of 40, the accelerating force is 20; but, if the same force of 40 act upon another body of 4 lbs. weight, the accelerating force is then only 10; that is, it is only half the former, and will produce only half the velocity.

SECTION I.-Uniform Motions.

1. The space described by a body moving uniformly, is represented by the product of the velocity into the time: and in comparing two, we say

2. In regard to momenta, m varies, as bv, or

M: m :: BV : b v.

Example. Two bodies, one of 10, the other of 5 pounds, are acted upon by the same momentum, or receive the same quantity of motion 30. They move uniformly, the first for 8 seconds, the second for 6: required the spaces described by both.

Here
3
=
= v, and 30
Then T V = 3 x 8 =

=6=v

24 s; and t v = 6 x 6 36 =s. Thus the spaces are 24 and 36 respectively,

SECTION II.-Motion uniformly accelerated.

1. Motion uniformly accelerated, is that of a material point or body subjected to the continual action of a constant force.

2. In this motion the velocity acquired at the end of any time whatever is equal to the product of the accelerating force into the time; and the space described is equal to the product of half the accelerating force into the square of the time.

3. The spaces described in successive seconds of time are as the odd numbers, 1, 3, 5, 7, 9, &c.

4. Gravity is a constant force, whose effect upon a body falling freely in a vertical line is represented by g; and the motion of such body is uniformly accelerated.