Find the center of gravity of the whole system, D, and join C to D.Now since the weights A and B are at rest, the line CD will be perpendicular to the horizon.Therefore when the balance is at ECF, the line CD will be at CG, and, since this is not perpendicular to the horizon, the balance ECF will return to ACB.The same will happen if the center C is placed above the balance as at H.
![](https://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler?fn=/permanent/archimedes/monte_mecha_084_en_1969/figures/084.01.024.1.jpg&dw=200&dh=200)
Now if the curve or angle ACB shall be beneath the line AB, we may show in the same way that the balance ECF, whether its center is at C or H, must move downward on the side of F.
And if the angle ACB is above the line AB at the center of the balance H, and if the line CH sustains the balance, when the balance is moved to EKF, it will return to ACB.
![](https://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler?fn=/permanent/archimedes/monte_mecha_084_en_1969/figures/084.01.024.2.jpg&dw=200&dh=200)
But if the center of the balance is D, one may move the balance in any way, and wherever it is placed, it will rest.
![](https://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler?fn=/permanent/archimedes/monte_mecha_084_en_1969/figures/084.01.024.3.jpg&dw=200&dh=200)
Then if the point H shall be beneath the line AB, the balance EKF will move downward on the side of F.
![](https://digilib.mpiwg-berlin.mpg.de/digitallibrary/servlet/Scaler?fn=/permanent/archimedes/monte_mecha_084_en_1969/figures/084.01.024.4.jpg&dw=200&dh=200)
And for similar reasons if the angle ACB shall be beneath the line AB, and the center of the balance is at H, and the balance is sustained by the line CH, then, if the balance is moved from this position, it will move downward on the side of the lower weight.And if the center of the balance is at D, it will rest wherever it is left.And if the center is at K and the balance is moved from that position, it will return in any event to the same place.All which things are manifest from what we said at the beginning.Similarly if the center of the balance is placed in one of the arms of the balance, either within or without, or any other way, we shall find the same things.