This consideration only takes place in the horizontal situa tion of the body ; in the vertical, the fibres of the base all break at once ; so that the absolute weight of the body must exceed the united resistance of all its fibres : a greater weight is therefore required here than in the horizontal situation ; that is, a greater weight is required to overcome their united resistance than to overcome their several resistances one after another. The differenoe between the two situations arises hence, that in the horizontal there is an immoveable point, or line, as a centre of motion, which is not in the vertical.
Varignon has improved on the system of M. Mariotte, and shown, that to Galileo's system it adds the consideration of the centre of percussion. The comparison of the centres of gravity with the centres of percussion afford a fine view, and set the whole doctrine in a most agreeable light.
In each system, the base, by which the body breaks, moves on the axis of equilibrium, which is an immoveable line in the same base ; but in the second, the fibres of this base are continually stretching more and more, and that in the same ratio as they recede farther and fitrther from the axis of equi librium ; and of consequence, are still exerting a greater and greater part of their whole force.
These unequal extensions, like all other forces, must hate some common centre where they all meet, and, with regard to which, they make equal efforts on each side; and, as they are precisely in the same proportion as the velocities which the several points of a rod moved circularly would have to each other, the centre of extension of the base, by which the body breaks or tends to break, must be the same with the centre of percussion. Galileo's hypothesis, according to which the fibres are supposed to stretch equally, and break all at once, corre ponds to the case of a rod moving parallel to itself, where the centre of' extension or percussion does not appear, as being confounded with the centre of gravity.
The base of fraction being a surface, whose particular nature determines its centre of percussion, it is necessary that this should be first known, to find on what point of the ver tical axis of that base it is placed, and how lia• it is from the axis of equilibrium. Indeed, we know in the general, that it always acts with so much the more advantage as it is farther from it ; because it acts by a longer arm of a lever ; and, of consequence, it is the unequal consistency of the fibres in M. Mariotte's hypothesis which produces the centre of percussion ; but this unequal resistance is greater or less, according as the centre of percussion is placed more or less high on the vertical axis of the base, in the different surfaces of the base of the fracture.
To express this unequal resistance, accompanied with all the variation it is capable of, regard must be had to the ratio between the distance of the centre of percussion from the axis of equilibrium, and the length of the vertical axis of the base; in which ratio, the first term, or the numerator, is always less than the second, or the denominator ; so that the ratio is always a fraction less than unity ; and the unequal resist ance of the fibres in M. Mariotte's hypothesis is so much the greater, or, which amounts to the same, approaches so much nearer to the equal resistance in Galileo's•hypothesis, as the two terms of the ratio are nearer to an equality.
Hence it follows, that the resistance of bodies in M. Mari otte's system is to that in Galileo's, as the least of the terms in the ratio is to the greatest. Hence, also, the resistance being less than what Galileo imagined, the relative weight must also be less ; so that the proportion already mentioned, between the absolute and relative weight, cannot subsist in the new system, without an augmentation of the relative weight, or a diminution of the absolute weight ; which dimi nution is had by multiplying the weight by the ratio, which is always less than unity. This done, we find that the abso lute weight, multiplied by the ratio, is to the relative weight as the distance of the centre of gravity of the body from the axis of equilibrium, is to the distance of the centre of gravity of the base of the fracture from the same axis; which is precisely the same thing with the general formula given by M. Varignon for the system of M. Mariotte. In effect, after conceiving the relative weight of a body, and its resistance equal to its absolute weight, as two contrary powers applied to the two arms of a lever, in the hypothesis of Galileo, there needs nothing to convert it into that of M. Mariotte, but to imagine that the resistance, or the absolute weight, is become less, everything else remaining the same. One of the most curious, and perhaps the most useful questions in this research, is to find what figure a body must have, that its resistance may be equal in all its parts, whether it be loaded with an additional weight, or as only sustaining its own weight.