Alteration is often an intermediate step between a temporary change and a complete fracture. There are many substances, which, after bending to a cer• tain extent, are no longer capable of resuming their original form : and in such cases it generally hap pens that the alteration may be increased without limit, until complete fracture takes place, by the continued operation of the same force which has be gun it, or by a force a little greater. Those sub stances which are the most capable of this change, are called ductile, and the most remarkable are gold, and a spider's web. When a substance has under• gone an alteration by means of its ductility, its stiff ness, in resisting small changes on either side, re mains little or not at all altered. Thus if' the stiff ness of a. spider's web, in resisting torsion, were suf ficient at the commencement of an experiment, to cause it to recover itself, after being twisted in an angle of ten degrees, it would return ten degrees, and not more, after having been twisted round a thou sand times. The ductility of all substances, capa ble of being annealed, is greatly modified by the effects of heat hard steel, for example, is incompar ably less subject to alteration than soft, although in some cases more liable to fracture ; so that the de gree of hardness requires to be proportioned to the uses for which each instrument is intended : al though it was proved by Coulomb, and has since been confirmed by other observers, that the primitive stiff ness of steel, in resisting small flexures, is neither in creased nor diminished by any variation in its temper.
The strength of a body is measured by the force required completely to overcome the corpuscular powers concerned in the aggregation of its particles and it is jointly proportional to the primitive stiff: ness, and to the toughness of the substance ; that is, to the degree in which it is capable of change of form without. permanent alteration. •It becomes however of importance in some cases, to consider the measure of another kind of strength, which has sometimes been called resilience, or the power of resisting a body in motion, and which is proportional to the strength and the toughness. conjointly, that is, to the stiffness and the square of the toughness. Thus if we double the length of a given beam, we reduce its absolute strength to one half; and its stiffness to one eighth ; but since the toughness, or the space, through which it will continue to resist, is quadrupled, the resilience will be doubled, and it would require a double weight to fall from the same height, or the same weight to fall from a double height, in order to overcome its whole resistance.
If we wish to determine the resilience of a body from an experiment on its strength, we must mea sure ..the distance through which it recedes or is bent, previously to its fracture ; and it may be shown that a weight, which is capable of breaking it by pressure, would also break it by impulse if it moved with the velocity acquired by falling from a height equal to half the deflection. Thus if a beam or bar were broken by a weight of 100 pounds, after being bent 6 inches, without' alteration, it would also be broken by a weight of 100 pounds falling from a height of s inches, or moving in a horizontal direc tion with a velocity of 4 feet in a aecond, or by a weight of 1 pound falling from a height of es; - This substitution of velocity for quantity of matter has however one limit, beyond which the velocity must over the resistance, without regard to the quantity of matter, and this limit is derived from the time required for the successive propagation of the pressure through the •different parts of the substance, in order that they may participate in the resistance. Thus if a weight fell on the end of a bar or column with a velo city of 100 feet in a second, and the substance could only be compressed of its length, without being crushed, it is obvious that the pressure must be propagated through the substance, with a veloci ty of 20,000 feet in a second, in order that it might resist the stroke ; and, in general, a substance will be crushed or penetrated by any velocity exceeding that which is acquired by a body falling from a height, which is to half that of the modulus of elastici ty of the substance, as the square of the greatest pos.. sible change of length is to the whole length. From the consideration of the effect of rigidity in lessening the resilience of bodies, we may understand how a diamond, which is capable of resisting an enormous pressure, may be crushed with a blow of a small hammer, moving with a moderate velocity. It is, remarkable that, for the same substance in different forms, the resilience is in most cases simply propor tional to the bulk or weight, while almost every other kind of resistance is capable of infinite varia tion by change of form only. ' The elaborate investigations of Mr Lagrange, re specting the strength and the strongest forms of columns, appear to have beep conducted upon prin ciples• not altogether unexceptidnable ; but it is much easier to confute the results than to follow the steps of the computations. One great error is the supposition that columns are to be considered as elastic beams, bent by a longitudinal force ; while, in reality, a stone column is never slender enough to be bent by a force which it - can bear without being crushed : and even for such columns as are capable of being bent by a longitudinal force, Mr Lagrange's determinations are in several instances inadmissible ; he asserts, for example, that a cylin der is the strongest of all possible forms, and that a cone is stronger than any conoid of the same bulk ; but it appears to be demonstrable in a very simple manner, and upon incontestable principles, that a conoidal form may be determined, which shall be stronger than either a cone or a cylinder of the same bulk.