Proven by experiment.

It is possible that equal weights hung on unequal arms remain equal in power; and this will happen when the junction of these balance arms is angular at its pole.

It is proved. Let the balance g a c have an angular junction at its pole a; and let the greater arm be a g and the lesser a c, and the length of the greater arm is double the lesser arm. Now I will hang at the centre of the lesser arm the weight n, and at the end of the greater arm the weight m; and each weight is found equal and equally distant from the central line d h; as is shown by the definition of the circle that the lines or spaces p g and p o are equal. And the gravity of the arms is also proved: although they are double one to the other, they perform the office of equal gravity, because of the position where the centres of their gravities are found, that is e and b: e is the centre of gravity of the greater arm, and b of the lesser arm; for their distances are double one to the other from the central line d h, as is plainly seen that a b is double the e f. And thus we have concluded what was promised.

It is proved that the centres of gravity of the arms are, by distance, in double proportion from the central line, by the pyramid which, being cut in the middle by the centre at e f, that cut is half of the base g p: which base is equal to a b.

Rule of the angular balance.

The angular balance is that whose junction of its arms is angular; in which angle its pole is established.

Always the distances that the opposite ends of the angular balance have from the central line of its pole will be in the same proportion as that of the length of its arms between themselves, that is, if one of the arms is 2 and the other is 4, the end of the greater arm will be twice as near to the central line as the lesser arm was.

The central line is that which is imagined to extend with its straightness between the centres of the accidental gravity of the suspended bodies and to end at the centre of the world; which passes through all those threads that suspend the heavy bodies, passing also through the centre of that body and through the middle of the thread that suspends it, and ends with the end of the straightness of the said thread.

a b e f is the support of the heavy body d; and e shall be the centre of the world. The central line is said to be that which arises at the upper beginning of the straightness of the support, and passes through the middle of the thickness of that support, and passes through the centre of the body sustained by it.

Take the lesser arm and as much of the greater arm adjoining that lesser one, and at the centre of the remainder you will give a counterweight at the end of the lesser arm.

Let the lesser arm be p r. Take from the greater arm m p which is equal to the lesser arm, and to the weight of the remainder you will give a counterweight at the end of the lesser arm, that is at r. This counterweight will have such proportion to the weight a m as the spaces enclosed between the centre of gravity of a m, that is n p with p r, which is a sesquialteral proportion, that is 3 and 2. And so hang 3 at r against the 2 of n. And the opposite weights will remain equal in power.

Battista Alberti says, in a work of his titled Ex ludis rerum mathematicarum, that when the balance a b c has the arms b a and b c in double proportion, then the weights hung at its ends that dispose it in such a way are in the same proportion as the arms themselves; but it is reversed, that is, the greater weight on the lesser arm. To which thing experience and reason show him to be a false proposition; because where he puts the opposite weights 2 against 4 on the balance that itself weighs 6 pounds, it needs to be 7 against 2; and thus the balance will remain still with equal resistance of the arms. And here that author erred, for not making mention of the weight of the beam of the balance, which is unequal in weight.

As many times as the lesser arm of such a balance goes into the greater, so many times the greater of the weights that dispose it in such a way contains within itself the lesser, according to Battista Alberti.