If the weights of two hard bodies of equal consistency are inversely proportional to their velocities, the forces of the percussions carried out by these bodies at a perpendicular and median incidence on the same hard body are equal.

Let two bodies A and B equally hard and consistent (Table 23.​1, Fig. 73) fall at a perpendicular and median incidence on the plane surface FK of a subjacent body, the body A with an impetus D, the body B at a velocity C. The ratio of the impulses C/D is equal to the inverse ratio of the body masses or weights B/A. I claim that the energies or the moments of the percussions carried out on the plane surface FK are equal. The ratio of the compressive moment of the body A to the compressive moment of the body B is the product of the ratio of the body mass of A to the body mass or weight of B and the ratio of the impulses D/C. The bodies indeed compress not only by their masses or weight but also by the impetus at which they move. By hypothesis, the ratio of the weights A/B is equal to the inverse ratio of the impulses C/D and these two inverse proportions result in a ratio of equality. Consequently, the compressive moments of the bodies A and B are equal. But the forces of the percussions carried out at a perpendicular and median incidence are measured by the compressive moments or forces of the colliding bodies. Consequently, the body A at a velocity D carries out a percussion on the plane surface FK equally strong as that carried out by the body B at a velocity C.

Instead of the subjacent plane surface FK let us suppose two equal nails F and K which must be driven in into the body FD yielding equally. Or rather the points F and K which sustain the blows are the extremities of the equal arms of a balance supported at G. The bodies A and B are suspended and pull the extremities of the scales R and S about the mid-point X which is the fulcrum. If they fall at the same instant, two percussions of equal strength obviously occur at this instant. They drive in the nails F and K equally. They also impel and pull the extremities of the scales equally. Therefore, the arms do not tip but remain horizontal since, as was shown, the percussive forces and the percussions themselves were absolutely equal and equally strong. This occurs at this particular instant of time at which the two percussions are carried out although, later, when the velocities are extinct, the heavier weight of A tips the scales. This, however, has nothing to do with the percussion since a percussion is an action resulting from the weight and the impetus. From what was described above it is more than evident that this action must occur at one instant of time in hard and inelastic bodies.

This proposition is confirmed by the demonstration thus presented. If one chose to investigate it experimentally, the following artifice may be used. Two pendulums AB and CD of equal lengths are taken but the iron globe B is bigger than D (Table 23.​1, Fig. 74). Both pendulums are suspended from the same horizontal line AC. In the vertical plane ACG a balance with equal arms EF rotating horizontally about the fulcrum G is added. Then the pendulums B and D are unequally moved from the vertical plane ACG so that the ratio of the height of the globe D from its lowest position F to the height of the globe B from its lowest position E is equal to the inverse ratio of the squares of the weights B and D. The fallen distances are proportional to the squares of the velocities acquired at the end of these falls. Therefore, the ratio of the impetus acquired by the weight D at the end of its fall over the arc DF to the impetus acquired by the weight B at the end of its fall over the arc BE is equal to the inverse ratio of the bodies B/D. Thus, the moments of the percussions carried out at the same instant on the extremities of the scales EF are equal. Therefore, the balance with equal arms does not tip. These equal pendulums are appropriate because they describe the arc BE and the arc DF in the same time. If they are released from the extremities B and D at the same instant, they will also carry out percussions at E and F at the same instant. The truth of the above demonstration thus can be confirmed at will.

I do not think it necessary to remind again that percussions of equal moments do not persist for some length of time. They are verified only at the split instant when they occur. After the blow of each of both bodies the action of their impulses stops. Only unequal compressions resulting from the force of gravity remain. It must also not be forgotten and it must be repeated again and again that the proposition is verified only if the striking bodies and those which sustain a perpendicular and median percussion are absolutely hard and inelastic. If they can be squeezed and compressed in any way, in the time following the first instant of the collision, subsequent continuous small percussions while the bodies constrict and condense can result in discrepancies in energy and in duration.

After this preamble I arrive at the main proposition.

The force and energy of percussion is greater than any finite force which compresses by virtue of gravity only without any displacement.

A huge heavy body A is supposed to be suspended from the extremity R of a balance RS with equal arms and the fulcrum of which is X (Table 23.​1, Fig. 75). The body A is supported by the hardness of the subjacent plane FK. Obviously the scales are not subjected to any force from the body A and thus remain immobile as if they were pulled by nothing. Two equal and opposite forces, that of the compressing weight of the body A and that of the resistance and firmness of the subjected plane FK cancel each other, or rather the action of the compression of the body A is prevented. Therefore, the scales are tipped by nothing or by no force. A small body B is then supposed to be hung from the other extremity of the scales S but immobile and without downwards movement since as much is withdrawn from the weight of A as is the gravity of B. The scales RS do not tip towards S because there remains an excess of gravity of A over B. This excess is supported by the underlying plane. What has been achieved in such instance is a diminution of the compression of the plane K by the body A. The plane FK thus exerts less resistance than previously. Then the very small weight B at the extremity S of the scales is not inactive but is provided with some downwards impetus. For the percussion to be efficient, it must be shown that a force of compression and percussion at a perpendicular and median incidence of the body B on the plane FK exceeds the energy of the simple gravity of any big body A compressing the subjacent plane without any displacement. The heavy body A is supposed to travel downwards the distance RZ so that the ratio of the impetus D finally acquired in this fall to the impetus C acquired in the fall of B through SL is equal to the inverse ratio of the weights B/A. Consequently, the bodies A and B carry out on the given plane FK percussions equally strong and of equal moments. Obviously the compressive force of the gravity alone is increased by the movement and impetus, and acquires huge forces in so far as its movement is more accelerated. Thus, the compressive force is composed of the force of gravity and the motive virtue of the movement or impetus. Therefore, the compressive force of the body A moving at a velocity D is greater than the compressive force of the same body A deprived of any velocity and immobile. Thus, the energy of the percussion which the body A moving at a velocity D carries out on the plane FK is greater than that with which the same body A immobile and deprived of impetus compresses the same plane. But the energy of the percussion carried out on the same plane by the body B moving at a velocity C is equal to the force of percussion carried out by the body A at a velocity D. Consequently, the energy of the percussion carried out on the plane FK by the body B moving at a velocity C is greater than the force with which the body A immobile and deprived of any movement compresses the same plane since in the process of the instantaneous percussion the scales necessarily tip and their extremity R is raised. Therefore, the body A is raised and leaves its contact with the subjacent plane. Thus, the force and the moment of the percussion of B exceed the force of the huge gravity alone of A immobile. Q.E.D.