If several elastic and resilient rings of equal strength are compressed by the same weight, on one side one ring alone and on the other side several rings set horizontally on a plane surface, the ratio of the compression which the ring alone undergoes to that which each of the others sustaining the weight together undergoes is equal to the inverse ratio of all the rings acting together to that of the one ring alone.

Any number of rings of equal strength, for example of iron, AB, CD, EF, GH are supported by the same firm plane surface VX (Table 23.​1, Fig. 81). Firstly, a weight R compresses the first three rings AB, CD, EF set horizontally side by side and parallel. Secondly, the same weight compresses the single ring GH. I claim that the ratio of the compression which the ring GH undergoes to the compression which any of the associated rings AB or CD or EF undergoes is equal to the ratio of the number of rings resisting together the weight R to the single ring GH. The rings AB, CD and EF are supposed to be equal, at equal distances and equally strong. All of them are compressed by the opposite force R and thus resist it equally. Since the whole force of R is exerted, not on one alone of these rings, but on all of them, the weight of R must be considered as divided into as many parts as there are rings supporting it and each of these parts is supported by one ring. Therefore, if there are three rings, any of them AB carries a third of the weight R. It is thus compressed, not by the whole weight R, but by a third of it. On the other hand, the single ring GH is compressed by all the weight R superimposed on it. Thus, the compression which the ring GH undergoes is three times the compression on AB just as the number of rings which resist together is three times the single ring GH. The same is true for any other number of rings. Q.E.D.

The conditions are the same. If the compressing weight is struck by an external body with equal energy, the rings are further compressed in inverse ratio to their number. Moreover, the resilience of the single ring is as many times the resilience of any of the associated rings as there are associated rings.

The conditions are the same. The supported body R is struck by the same body Z with the same impetus (Table 23.​1, Fig. 81). Obviously, the percussion Z is equivalent to an added weight. The two percussions ZZ are supposed to be equally strong. They compress the subjacent rings as if two other equal weights ZZ were superimposed on the equal weights RR. Therefore, the new compression occurring in the single ring GH is equal to as many times the compression occurring in any of the associated rings AB as there are rings acting together. Since all rings are supposed to be of equal strength, the energies by which they can resile after compression are equal and the more the ring GH is compressed beyond any of the associated rings AB, the more strongly and the higher it resiles. Since, after the percussion, the impulses ZZ are extinguished, the rings can resile according to the measure of their compression and bending in repelling the residual weights RR. Therefore, the ring GH must react more and propel further the weight R than any of the associated rings AB. Its reaction must be exactly as many times that of each of the associated rings as there are associated rings.

But these rings are compressed and resile very differently when superimposed on each other and compressed by the same weight.

If several elastic and resilient rings of equal strength superimposed on each other vertically are compressed by the same heavy body, any of them undergoes the same compression as that which the weight would exert if it compressed one ring alone.

Let several iron rings, for example AB, CD, EF, GH of the same strength be supported by an absolutely hard and rigid plane surface VX (Table 23.​1, Fig. 82). The rings are supposed to be weightless. The heavy bodies R, R equal to each other are set on the first ones superimposed on each other vertically and on the last single one. Unquestionably all the rings are bent and compressed. I claim that any of them is compressed as is the single ring GH, by the same incumbent body R. Not only the ring AB is compressed and squeezed by the incumbent weight R but also all the next ones. The lowest point B of the upper ring AB is supported by the apex C of the next ring CD which resists it with as much strength as the stable plane surface VX does since the point C of the ring CD by its immobility and firmness plays the role of a stable and firm plane surface. Therefore, the compressive force of the weight R acts equally against the resistance of the ring AB as if it was directly supported by the plane surface VX. For the same reason, the next ring CD is supported by the apex E of the following ring EF also immobile and the upper one is compressed by the weight R. Consequently, it is compressed and supported like the ring AB. Finally, the ring EF is supported by the stable ground VX and is compressed at E by the same weight R. Consequently, it bends equally (since the upper rings are supposed to be weightless). Thus, it is obvious that any of them is bent as if the others were absent and thus is bent as is the ring GH.

However, it must be noticed that, although several rings are equally bent by the same weight as if one only of them was compressed by this weight, nevertheless the lowering of the weight R is greater than the lowering which would occur in the instance of the superimposition on one single ring. This lowering would be as many times that of one single ring as there are rings.

The conditions are the same. If the compressing weight is later struck by an external body, all the rings are equally compressed and, moreover, any of them by resiling repels upwards the compressing body with a force equal to that with which one single ring bent by the same percussion would repel it.

The conditions are the same. A percussion occurs at ZZ with the same energy (Table 23.​1, Fig. 82). Clearly, all the rings are further compressed and bent by the force of this percussion as if it were by a new weight. The new deflections are equal to each other and to the one which would occur in the single ring GH compressed and struck by the same forces (this is shown by the same argument which was used above). All the rings are supposed to be the same and of equal strength, and thus equally resilient. Thus, after extinction of the energy of percussion, any of them by resiling with an equal force will repel the compressing body R upwards. The dilatation or the resilience of the ring AB occurs against two forces both compressing it, namely against the obstacle of the apex C of the lower ring CD and against the weight R. Point C of the subjacent ring, while repelling point B of the upper ring upwards, plays the role of the stable ground VX. Thus, two opposite forces C and B impel each other. Since they are equal, none of them prevails. Therefore, the ring AB while dilating only repels the compressing body R which is raised as it is by the single ring GH until, the difference between the force of the resiling body and the force of the weight R being consumed, the forces are balanced. For the same reason, the next ring CD, while attempting to dilate, propels point B of the upper ring upwards and the apex E of the subjacent ring downwards. The ring is squeezed from above and from below by two opposite and equal forces E and B. Point E replaces the subjacent immobile plane surface VX and, therefore, propels upwards point B which cannot be propelled without all the ring AB being expelled together with the superimposed weight R. Thus, the ring CD which was equally compressed and, therefore, resiles with the same energy as the ring AB, will raise the weight R by as much, before balance of the forces is achieved. The distance by which the weight R is raised will be measured by the two dilatations, i.e. that of AB and that of CD. The same is true for the dilatation and resilience of the lower ring EF. Obviously thus any of the rings overcomes the resistance R with an equal energy and repels it not less than the single ring GH does.

The action of superimposed rings thus appears to be different from the action of the same rings arranged horizontally and supporting the same weight. In the former instance the compressing weight R is supported and repelled back equally strongly by one single ring as it is by countless rings compressing each other. Although there are several rings which act together against the same resistance of the body R, their effort is neither decreased nor subdivided. They act with the same effort as if one single ring was acting. This of course could be considered as paradoxical if we had not been convinced by the above demonstration.

Here it must also be noticed that a great number of rings vertically superimposed do not carry out a stronger repulsion of the weight R but only propel it the higher and the more quickly the more there are rings.

Any huge weight, since it is like an aggregate of countless compressible and resilient rings, will indeed be compressed by an occurring percussion and will then spontaneously dilate and be raised by the strength and consistency of the minute rings.

The aggregate formed by the rings AB, CD and EF together with the superimposed weight R is conceived as if it were a column or a rock or any other solid body undergoing compression and resilience (Table 23.​1, Fig. 82). This mass is struck at Z by an external body so that all the superimposed rings are compressed and bent. After extinction of the impetus of percussion, the minute ring EF obviously must dilate and resile with as much energy as necessary to repel the whole incumbent mass RD. But it seems absurd and incredible that the force, strength and consistency by which the minute ring EF can resile by dilating be able to propel the very huge mass RD since exactly as much hardness and repulsive force are required as the magnitude of the energy of gravity of the whole compressing mass. This, although seeming incredible, cannot be denied unless we intend to deny experience and demonstration. Indeed, any huge hard mass when struck undergoes vibration and agitation. Vibration of any body cannot be conceived without its very frequent compression and dilation either transverse or perpendicular. Then to and fro movements must necessarily be admitted, i.e. depressions and elevations of the huge weight. Since these are opposite movements, they must necessarily result from opposite impulses. Since no other extrinsic upwards impeller can be imagined, this force and energy by which elastic bodies resile must necessarily be resorted to despite the exiguity of the propelling machine. Most obviously the force and energy of resilience are not different in air which otherwise seems to have no consistency. Air accumulated in the very consistent and hard metal water-mains of aqueducts can indeed dilate with enough force to disrupt and break these metal pipes. If one attempted to disrupt such a metal water-main by using a wedge driven in by an incumbent weight, some thousands of pounds would not be enough. Consequently, the energy which air generates there by dilating exceeds the force of a weight of as many thousands of pounds. This action of course seems to be surprising since, although all the substance of air is made of resilient coils or small machines, the action of resilience by which it disrupts a copper pipe must not be measured by all the enclosed and squeezed mass of air. Such a strong and powerful disruption is indeed carried out by the strength, hardness and rigidity of the very last small machines composing the air, while resiling. They impel directly the internal surface of the pipe. The other particles of air indeed do not increase the force and energy of resilience but its velocity alone, as was shown. All the machines superimposed on each other exert the same force of resilience as that exerted by the thinnest layer of particles of air which is thinner than any paper. This copper pipe the resistance of which exceeds so many thousands of pounds is overcome and disrupted by the strength of this layer of resilient air thinner than any paper. Anybody would of course find that wonderful and incredible.

We experience the same phenomenon in the pneumatic catapult recently invented. This catapult ejects an iron ball with such force that it drills a board of fair thickness. If one attempts to perforate the same board with the same stone by superimposing a weight, some thousands of pounds may not be enough. Consequently, in the catapult compressed air while resiling dilates with so much force that this is equivalent to the force of the mentioned immense weight and would unquestionably impel and raise the same incumbent weight. It must also be noticed that not all the mass of air enclosed there corresponds to the force of the operation, but only the thinnest layer of air which is in direct contact with and propels the weight. This would be impelled by the very thin envelope of air as well as by all the mass enclosed there, as I said. No doubt, in a stronger catapult, air could be compressed much more by a greater compressive force. Therefore, the force which would dilate in resiling can be considered as being much greater.