If two cylinders of the same matter, having their bases and lengths in the same proportion, be suspended horizontally; it is evident, that the greater has more weight than the lesser, both on account of its length, and of its base. But it has less resistance on account of its length, considered as a longer arm of a lever, and has only more resistance on account of its base ; therefore it exceeds the lesser in its bulk and weight more than in resistance, and, consequently, it must break more easily.
hence we see why, upon making models and machines in small, people are apt to be mistaken as to the resistance and strength of certain horizontal pieces, when they come to execute their designs in large, by observing the same propor tion as in the small. Galileo's doctrine of resistance, there fore, is no idle speculation, but becomes applicable in archi tecture, and other arts.
The weight required to break a body placed horizontally, being always less than that required to break it in a vertical situation ; and this weight being greater or less, according to the ratio of the two arms of the lever, the whole theory is always reducible to this : viz, to find what part of the absolute weight the relative weight is to be, supposing the figure of the body known ; which indeed is necessary, because it is the figure that determines the two centres of gravity, or the two arms of the lever. For if the body, e. gr. were a cone, its centre of gravity would not be in the middle of its axis, as in the cylinder ; and, if it were a semi-para bolical solid, neither would its centre of gravity be in the middle of its length or axis, nor the centre of gravity of its base in the middle of the axis of its base. But still, where soever these centres fall in the several figures, the two arms of the lever are estimated accordingly.
It may be here observed, that if the base, by which the body is fastened into the wall, be not circular, but, e. gr. parabolical, and the vertex of the parabola be at the top, the motion of the fracture will not be on an immovable point, but on a whole immovable line ; which may be called the axis of equilibrium; and it is with regard to this, that the distances of the centres of gravity are to be determined.
Now, a body horizontally suspended, being supposed such as that the smallest addition of weight would break it, there is an equilibrium between its positive and relative weight ; and, of consequence, those two opposite powers are to each other reciprocally as the arms of the lever to which they are applied. On the other hand, the resistance of a body is always equal to the greatest weight which it will sustain in a vertical situation without breaking, i. e. is equal to its absolute weight. Therefore, substituting the absolute weight for the resistance, it appears that the absolute weight of a body, suspended horizontally, is to its relative weight as the distance of the centre of gravity from the axis of equilibrium is to the distance of the centre of gravity of its base from the same axis.
The discovery of this important truth, at least anequiva lent to it, and to which this is reducible, we owe to Galileo. From this fundamental proposition are easily deduced several consequences ; as, for instance, that if the distance of the centre of gravity of the base from the axis of equilibrium be half the distance of the centre of gravity of the body, the relative weight will only be half the absolute weight ; and that a cylinder of copper, horizontally suspended, whose length is double the diameter, will break, provided it weigh halt' what a cylinder of the same base, 4S01 fathoms long, weighs.
On this theory of resistance, which we owe to Galileo, M. Mariotte made a very ingenious remark, which gave birth to a new system. Galileo supposes, that where the body breaks, all the fibres break at once ; so that the body always resists with its whole absolute force, or with the whole force that all its fibres have in the place where it is to be broken. But M. Mariotte, finding that all bodies, even glass itself, bend before they break, shows that fibres are to be considered as so many little bent springs, which never exert their whole force till stretched to a certain point, and never break till entirely unbent. IIence, those nearest the axis of equili brium, which is an immovable line, are stretched less than those thrther off; and, of consequence, employ a less part of their force.