« PreviousContinue »
that its resistance be not overcome by the effort of the corresponding power; and, as a superabundant resistance would be a lost and use less force, it follows that the most advantageous form which can be given to a solid so disposed is that which makes it resist, equally, at each point of its length, to the effort of the weight with which it is charged. In forming bodies of equal resistance, we form them of the least volume; and thus we may act conformably to the views of nature, whose operations seem all to be carried into effect with the greatest œconomy of means.
The distinction of the productions of nature from those of art appears in this ;-in the former, the cause and effect essentially agree; the one cannot undergo any modification without the other experiencing some change; or, to speak more precisely, a new effect always results from a new cause:-in the productions of human industry, on the contrary, there is no necessary proportion between the effect and cause: if, for example, a determinate weight is to be raised, it is indifferent whether we use the thread which has precisely the adequate force, or the cable which has a superabundant force; while, if the same weight had rested naturally suspended, it would have done so by means of fibres peculiarly appropriated in their or ganization to this object, and whose disposition would have presented the most advantageous form. Perfection resides in a single point, at which nature arrives without effort; while man is obliged, in repeated trials, to pass over an immense space which separates him from it.
By the meditation of these operations, we recognize the boundless sagacity which directs them; and often we lift up a corner of the veil that conceals from us the eternal laws to which nature is subject. Galileo was thus conducted "as by the band" (to use his own expression) to discover the uniform acceleration of heavy bodies; and Maupertuis was thus led to discover the laws of the refraction of light and of the shock of bodies; for the principle of the least action established by this geometrician is, in fact, only the summary enunciation of the ideas conceived before him by Galileo. Euler, having meditated on these ideas, has generalised the principle of Maupertuis, by his application of it to the motion of projectiles; and lastly, La Grange, regarding this principle not so much as a metaphysical truth, as a simple and natural result of the laws of mechanics, has given to it in these latter times a new degree of extension.
The theory of the solids of equal resistance is in truth only an application of the same principle; since, among an infinite number of bodies of a determinate resistance, the solids of equal resistance are those which contain the least quantity of matter. To reduce to this form, then, as much as circumstances permit, all bodies used in the construction of machy intended to augment our forces, is to approach to the perfection which so distinctly characterises all the works of nature: in a word, it is to act conformably to her views.
In the mechanism of the animal economy, has not nature, then, forgotten so advantageous a form? The feathers of birds offer to us a sensible example. We must consider them as solids inserted at their extremities, and employed to beat the air during the flight of the bird: since the air re-acts on them, the case is the same as if they were sub
Ject to the action of a certain pressure distributed along all the points of their length, and tending to produce a fracture. That they may be as light as possible, it is necessary that they resist equally the effort acting on them; and in fact it is observed that the axis of all the feathers is a solid, of which the bases of fracture decrease, according to a certain law, from its origin to its extremity; to which there is always correspondent a base of fracture equal (c), a characteristic property of the solids of equal resistance.'
The theory invented by Galileo was not advanced by any of his immediate disciples. The first experiments on the resistance of solids were made by Wurtzius, a Swede; and Blondel, a French architect, was the second author on the subject. In 1669, Marchetti wrote on the resistance of solids; and in 1712, Père Grandi, the adversary of Marchetti, and the critical examiner of his writings, published a treatise which has been inserted in the complete edition of the works of Galileo. In 1680, Mariott made several experiments; and, perceiving that their results did not accord with those given by the theory of Galileo, he proposed to substitute a new hypothesis concerning the nature of the resistance of the fibres of bodies.
Mariott was followed by the great Leibnitz, a man destined to enlarge, illustrate, and inform every subject which he con templated. Observing that a state of inflexion preceded the rupture of bodies, he supposed bodies to be composed of extensive fibres which, at the instant of their rupture, resist proportionally to their extension; hence, by a simple calculation, he obtained results conformable to the observations of Mariott.
On the subject of the resistance of solids, Bernouilli also wrote; and he proved that the hypothesis of Leibnitz was not more generally admissible than that of Galileo. In the Memoirs of the Academy of Paris for 1705, he gave a paper on elastic curves, but did not apply his demonstrations to the resistance of solids; this application was however made by Euler, in his treatise de curvis elasticis; and he gave the method of determining the absolute elasticity or moment of elasticity, by virtue of which, solids resist their inflexion with more or less energy, This elasticity cannot be deduced in finite terms from the equa tion of the elastic curve which is not integrable: but, since the moment of elasticity is independent of the curvature of bodies, there is no reason why the curvature should not be deemed so small, that the element of the curve may coincide with that of the abscissa; in which case, the expression for the radius of curvature becoming more simple, an equation is obtained that is easily integrable; and thence the value sought, which is a function of the co-ordinates of the curve, and of the charge producing the inflexion,
In 1769, La Grange found the same expression that Euler had used for the resistance of an elastic spring, pressed parallel to its length, when engaged in inquiring whether the practice of all architects, from the time of Vitruvius, (viz. of swelling the column near the third of its height,) was founded on the circumstance of this form contributing to its strength. The result of La Grange's inquiries was, that, if the swell of pillars increases the elegance of their form, it adds nothing to their strength; and that, neglecting their own weight in the expression for the charge that compresses them, the cylindrical figure is that which essentially suits them.
A curious and important memoir, by Coulomb, treats of the resistance of those solids which are not composed of flexible fibres, but of particles adherent to each other; as stones, minerals, &c.
- The last labours of mathematicians on this subject were those of the illustrious Euler; who gave three memoirs in the Petersburg Acts, to determine the height of a prismatic or cylindrical column, such as it is at the moment of its bending beneath its own weight.
Mariott, Belidor, Mussembrock, and Buffon, have made experiments on the resistance of solids. The experiments of the latter are most valuable for number and accuracy; and, had he known the theory of elastic curves, he would probably have determined the absolute elasticity: but the co-operation of calculation and experiment is rare.
The first section of the present work relates to the resistance of solids, and is purely theoretical.-The general expressions for the absolute and relative resistances of solids are calculated, and considered relatively to the hypothesis of Galileo and Leibnitz. It is judiciously observed that experience alone can indicate the modifications to be made in these formulas, since no body is either perfectly hard or perfectly elastic. An illustration of the two hypotheses is exhibited by means of an indefinite-number of levers; and the expressions are calculated accordingly, and shewn to agree with those which arise from a fluxionary process. Galileo's theory is applied to several examples: it is manifested that, of an hollow cylinder, where the diameters of the two circles remain the same, the resistance is greatest when the interior circle touches the exterior in the lowest part, Leibnitz's theory is likewise applied to several examples. The resistance of solids is next considered in those cases in which the power of producing a fracture acts at their extremities; and it is demonstrated that the weight, capable of producing the rupture of a prism firmly inserted at its parts of support, is double that which is necessary to break the same
body, freely sustained on the same parts: a result which accords with the experiments of Mariott.
M. GIRARD then proceeds to shew that the formulas of the resistances, on the theories of Galileo and Leibnitz, agree in this, that the resistance of the rectangular bases of fracture are as the squares of the heights multiplied by their length. At the end of the section designed to determine the relation between weights which, compressing solids of given dimensions parallel to their length, are able to make them bend, we have a digres sion on elastic curves, extracted from Euler's treatise. The relation above mentioned was deduced by this geometrician, as a consequence from the theorems appertaining to the theory of elastic curves. M. GIRARD extracts only what is indispen sably necessary to lead him to the proposition which he had in view, and which is thus announced; that the dimensions of the bases, and the elasticity of a series of columns, remaining the same, the weights which they can support before their flexure will be in the inverse ratio of the squares of their respective lengths.'
The rigorous equation of the elastic curve is of this form: (my2 + n) y' which admits no integration :
if, however, the solid be considered in its first degrees of inflexion, the element of the curve (z) is nearly equal to the element of the abscissa (x); in which case, the differential equation may be integrated, and the absolute elasticity determined.
In discussing the absolute negative resistance of solids charged solely with their own weight, M. GIRARD notices what neglect of circumstances led Euler to the paradox, that a heavy solid, prismatic, or cylindrical, standing vertically on a fixed plane, cannot bend beneath its own weight, to whatever height it is raised *.
The Second Section considers the solids of equal resistance, and is purely theoretical: their general equation is zy2=F (x), z being the breadth of the base of any rectangular fracture, y its height, x the distance of the base of the lever from the extremity of the arm to which the weight is applied, and F(x) denoting a function of x. An application is made of this equation to cases in which the weight is constant; or in which the weight or charge is some function of the arm of the lever to
The absolute negative resistance of solids, in contradistinction to the absolute positive, is that of the fibres of a body against the action of a power tending to compress them, and in a direction parallel to their length.
which it is applied. Solids of equal resistance are considered relatively to their weight. Solids of equal absolute negative resistance are also discussed, and an application is made to the form of pillars. The result of the investigation concerning the form of pillars we have already stated: the method of M. GIRARD is, first, to determine the form of pillars having only their own weight to sustain, and then that of those having their own and an added weight. The pillars, concerning the form of which the inquiry is made, are those which resist equally in all their bases of rupture; their form is conoidal, engendered by the revolution of a logarithmic curve about its axis: but, by reason of the excessive magnitude of the subtangent of the curve relatively to the height, the generating arc nearly coincides with a line parallel to the axis, and the conoid nearly with the cylinder.
The Third Section contains experiments on the resistance of solids. The first and principal object is to determine the value of Ek', which represents the absolute elasticity: this value cannot
be deduced from the equation y⋅=−P(~2+cx+ƒ)x• 'Br−P(+a+f)
taken rigorously: but, by supposing the solid to be in the first degrees of inflexion, the equation may be integrated, and the value Ek deduced.
This section contains likewise the description of the apparatus by means of which the experiments were made. It is difficult to form a complete and precise idea of a complicated machine, even with plates and a verbal description: but it would be vain to expect that verbal description alone could convey an adequate notion of the machinery, its parts, and the whole; we must therefore refer our readers, for satisfaction on this point, to M. GIRARD's book.
The description of the machinery is followed by a detail of experiments, and an explanation of the tables constructed according to the new French system of weights and mea
Section 4th and last relates to the circumstances which attend the inflexion of bodies supposed to be perfectly elastic. What had been determined in the preceding sections concerned merely the equilibrium between the resistance and charge: but the equili brium does not take place instantaneously; the inflexion of solids has always a certain duration; and, reasoning by analogy, its motion ought to be subject to some law of continuity. To obtain the time which must pass before the equilibrium is established, M. GIRARD